? ; ■I 1 > 11 ■ reference collection book KC Kansas city public library kansas city, missouri iP^ ^U' Ns^ Sii/ From the collection of the n R m o Jrre linger V * library p San Francisco, California 2008 f • « fl • • • • • « • ■ • HE BEL Li:iii8L^^S:'^T-,,E\M p.', j'i'— ' , meat journal l^^r/ A IN ^OTED TO THE SCIENTIFIC ^^r>^ AND ENGINEERING »ECTS OF ELECTRICAL COMMUNICATION U M E XXXV JANUARY 1956 tf k k--- • ' t. N U M B E R-lv DiflPused Emitter and Base Silicon Transistors J ^' ^ '^ ^ ^^^° M. TANENBAUM AND D. E. THOMAS 1 A High-Frequency Diffused Base Germanium Transistor c. a. lee 23 Waveguide Investigations with Millimicrosecond Pulses a. c. beck 35 Experiments on the Regeneration of Binary Microwave Pulses o. B. delange 67 Crossbar Tandem as a Long Distance Switching System a. O. ADAM 91 Growing Waves Due to Transverse Velocities J. R. pierce and l. r. walker 109 Coupled Helices j. s. cook, r. kompfner and c. f. quatb 127 Statistical Techniques for Reducing the Experiment Time in Re- liability Studies MILTON sobel 179 A Class of Binary Signaling Alphabets david slepian 203 Bell System Technical Papers Not Published in This Journal 235 Recent Bell System Monographs 242 Contributors to This Issue 244 COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY ; , * -^ -^ f - -.r » ' J " -' • THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD F. E. K A P P E L, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Executive Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. MCMILLAN, Chairman H. R. HUNTLEY A. J. BUSCH F. R. LACK A. C. DICKIESON J. R. PIERCE R. L. DIETZOLD H. V. SCHMIDT K. E. GOULD C. E. SCHOOLEY E. L GREEN G. N. THAYER EDITORIAL STAFF J. D. TEBO, Editor M. E. s T R I E B Y, Managing Editor R. L. SHEPHERD, Production Editor THE" BELL SYSTEM TECHNICAL JOURNAL is pubUshed six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV JANUARY 1956 number 1 Copyright 1956, American Telephone and Telegraph Company Diffused Emitter and Base Silicon Transistors* By M. TANENBAUM and D. E. THOMAS (Manuscript received October 21, 1955) Silicon n-p-n transistors have been made in which the base and emitter regions were produced by diffusing impurities from the vapor phase. Tran- sistors with base layers 3.8 X 10~ -cm thick have been made. The diffusion techniques and the processes for making electrical contact to the structures are described. The electrical characteristics of a transistor with a maximum alpha of 0.97 and an alpha-cutoff of 120 mc/sec are presented. The manner in which some of the electrical parameters are determined by the distribution of the doping impurities is discussed. Design data for the diffused emitter, dif- fused base structure is calcidated and compared with the rneasured char- acteristics. INTRODUCTION The necessity of thin base layers for high-frequency operation of tran- sistors has long been apparent. One of the most appealing techniques for controlling the distribution of impurities in a semiconductor is the dif- fusion of the impurity into the solid semiconductor. The diffusion co- efficients of Group III acceptors and Group V donors into germanium and silicon are sufficiently low at judiciously selected temperatures so * A portion of the material of this paper was presented at the Semiconductor Device Conference of the Institute of Radio Engineers, Philadelphia, Pa., June, 1955. 2 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 that it is possible to envision transistors with base layer thicknesses of a micron and frequency response of several thousand megacycles per second. A major deterent to the application of diffusion to silicon transistor fabrication in the past was the drastic decrease in lifetime which generally occurs when silicon is heated to the high temperatures required for dif- fusion. There was also insufficient knowledge of the diffusion parameters to permit the preparation of structures with controlled layer thicknesses and desired dopings. Recently the investigations of C. S. Fuller and co- workers have produced detailed information concerning the diffusion of Group III and Group V elements in silicon. This information has made possible the controlled fabrication of transistors with base layers suffi- ciently thin that high alphas are obtained even though the lifetime has been reduced to a fraction of a microsecond. In a cooperative program with Fuller, diffusion structures were produced which have permitted the fabrication of transistors whose electrical behavior closely approxi- mates the behavior anticipated from the design. This paper describes these techniques which have resulted in high alpha silicon transistors with alpha-cutoff of over 100 mc/sec. 1.0 FABRICATION OF THE TRANSISTORS Fuller's work has shown that in silicon the diffusion coefficient of a Group III acceptor is usually 10 to 100 times larger than that of the Group V donor in the same row in the periodic table at the same tem- peratures. These experiments were performed in evacuated silica tubes using the Group III and Group V elements as the source of diffusant. Under these conditions a particular steady state surface concentration of the diffusant is produced and the depth of diffusion is sensitive to this concentration as well as to the diffusion coefficient. The experiments show that the effective steady state surface concentration of the donor impurities produced under these conditions is ten to one hundred times greater than that of the acceptor impurities. Thus, by the simultaneous diffusion of selected donor and acceptor impurities into n-type silicon an n-p-n structure will result. The first n-la,yer forms because the surface concentration of the donor is greater than that of the acceptor. The p-laycr is protluced because the acceptor diffuses faster than the donor and gets ahead of it. The final n-region is simply the original background doping of the n-type silicon sample. It has been possible to produce n-p-n structures by the simultaneous diffusion of several combinations of donors and acceptors. Often, however, the diffusion coefficients and surface concentrations of the donors and acceptors are such that opti- 1 C. S. Fuller, private communication. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 3 mum layer thicknesses (see Sections 3 and 4) are not produced by simul- taneous diffusion. In this case, one of the impurities is started ahead of the other in a prior diffusion, and then the other impurity is diffused in a second operation. With the proper choice of diffusion temperatures and times it has been possible to make n-p-n structures with base layer thicknesses of 2 X 10~* cm. The uniformity of the layers in a given specimen is better than ten per cent of the layer thickness. Fig. 1 illustrates the uniformity of the layers. This figure is an enlarged photograph of a view perpendicular to the surface of the specimen. A bevel which makes an angle of five degrees with the original surface has been polished on the specimen. This angle magnifies the layer thickness by 11.5. The layer is defined by an etchant which preferentially stains p-type silicon^ and the width of the layer is measured with a calibrated microscope. After diffusion the entire surface of the silicon wafer is covered with the diffused n- and p-type layers, see Fig. 2(a). Electrical contact must now be made to the three regions of the device. The base contact can be made by polishing a bevel on the specimen to expose and magnify the base layer and then alloying a lead to this region by the same tech- f.^ *f^'- *; '>i i * /i n-TfPE DIFFUSED LAV^ER fo-t^^E*OiFFUSED LAYER i»# OF^GIt^L n-TYPE CRYSTAl. I 1 EQUIVALENT TO 2 X lO"'* CM LAYER THICKNESS Fig- 1 — Angle section of a double diffused silicon wafer. The p-type center ayer is approximately 2 X 10-< cm thick. 4 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 niques employed in the fabrication of grown junction transistors. Fig. 2(b). However, a much simpler technique has been evolved. If the sur- face concentration of the donor diffusant is maintained below a certain critical value, it is possible to alloy an aluminum wire directly through the diffused n-type layer and thus make effective contact to the base layer, Fig. 2(c). Since the resistivity of the original silicon wafer is one to five ohm-cm, the aluminum will be rectifying to this region. It has been experimentally shown that if the surface concentration of the donor diffusant is less than the critical value mentioned above, the aluminum will also be rectifying to the diffused n-type region and the contact becomes merely an extension of the base layer. The n-layers produced by diffusing from elemental antimony are below the critical concentration and the direct aluminum alloying technique is feasible. -n + TYPE DIFFUSED LAYER -p-TYPE DIFFUSED LAYER n + n+ -ALUMINUM WIRE p + ALUMINUM DOPED REGROWTH LAYER n-TYPE (b) ,^- ALUMINUM WIRE P + ALUMINUM DOPED , REGROWTH LAYER ^M'nY ^-i-r n-TYPE (c) Fig. 2 — ■ Schematic illustralioii of (a) double diffused n-p-n wafer, (b) angle section method of making base contact, and (c) direct alloying method of making base contact. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS AU-Sb PLATED POINT VAPORIZED Al LINE 0.005 CM WIDE t MM Fig. 3 — Mounted double diffused transistor. Contact to the emitter layer is achieved by alloying a film of gold containing a small amount of antimony. Since this alloy will produce an n-type regrowth layer, it is only necessary to insure that the gold- antimony film does not alloy through the p-type base layer, thus shorting the emitter to the collector. This is controlled by limiting the amount of gold-antimony alloy which is available by using a thin evaporated film or by electroplating a thin film of gold-antimony alloy on an inert metal point and alloying this structure to the emitter layer. Ohmic, contact to the collector is produced by alloying the silicon wafer to an inert metal tab plated with a gold-antimony alloy. 6 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The transistors whose characteristics are reported in this paper were prepared from 3 ohm-cm n-type siHcon using antimony and ahmiinum as the diffusants. The base contact was produced by evaporating alumi- num through a mask so that a hne approximately 0.005 X 0.015 cm in o lateral dimensions and 100,000 A thick was formed on the surface. This aluminum line was alloyed through the emitter layer in a subsequent operation. The wafer was then alloyed onto the plated kovar tab. A small area approximately 0.015 cm in diameter was masked around the line and the wafer was etched to remove the unwanted layers. The unit was then mounted in a header. Electrical contact to the collector was made by soldering to the kovar tab. Contact to the base was made with a tungsten point pressure contact to the alloyed aluminum. Contact to the emitter was made by bringing a gold-antimony plated tungsten point into pressure contact with the emitter layer. The gold-antimony plate was then alloyed by passing a controlled electrical pulse between the plated point and the transistor collector lead. Fig. 3 is a photograph of a mounted unit. 2.0 ELECTRICAL CHARACTERISTICS The frequency cutoffs of experimental double diffused silicon tran- sistors fabricated as described above are an order of magnitude higher than the known cutoff frequencies of earlier silicon transistors. This is shown in Fig. 4 which gives the measured common base and common emitter current gains for one of these units as a function of frequency. The common base short-circuit current gain is seen to have a cutoff fre- quency of about 120 mc/sec. The common emitter short-circuit current gain is shown on the same figure. The low-freciuency current gain is better than thirty decibels and the cutoff frequency which is indicated by the freciuency at which the gain is 3 db below its low-frequency value is 3 mc/sec. This is an exceptionally large common emitter band- width for a thirty db common emitter current gain and is of the same order of magnitude as that obtained with the highest frequency ger- manium transistors (e.q., p-n-i-p or tetrode) which had been made prior to the diffused base germanium transistor. ^ Tlio iiicroasp in (•oiiiinon haso current gain ahovc unity (indicated by current gain in decibels being positive) in the vicinity of 50 mc/sec is caused by a reactance gain error in the common base measurement. This error is caused by a combination of the emitter to ground parasitic capacitance and the i)ositive reactance com- ponent of the transistor input impedance resulting from phase shift in the ali)ha current gain. ' C. A. Lee, A High-Frequency Diffused Base Germanium Transistor, see page 23. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS z < o I- z LJ a. cr D O 40 30 20 (0 -\0 -20 -30 Ie = 3 MA Vc = 10 VOLTS COMMON^ EMITTER N 'OCCB — ^ ^^ OCq = 0.9716 ['=^"=106MC l-Ofg \ facb = i20MC \ COMMON BASE \ \ \ 0.1 0.2 0.5 1.0 2 6 10 20 50 100 200 FREQUENCY IN MEGACYCLES PER SECOND 500 1000 Fig. 4 — ■ Short-circuit current gain of a double diffused silicon n-p-n transistor as a function of frequency in the common emitter and common base connections. Fig. 5 shows a high-freciueiicy lumped constant equivalent circuit for the double diffused silicon transistor whose current gain cutoff char- acteristic is shown in Fig. 4. External parasitic capacitances have been omitted from the circuit. The configuration is the conventional one for junction transistors with two exceptions. A series resistance rj has been added in the emitter circuit to account for contact resistance resulting from the fact that the present emitter point contacts are not perfectly ohmic. A second resistance r/ has been added in the collector circuit to account for the ohmic resistance of the n-type silicon between the col- lector terminal and the effective collector junction. This resistance exists in all junction transistors but in larger area low frequency junction transistors its effect on alpha-cutoff is sufficiently small so that it has been ignored in equivalent circuits of these devices. The collector RC Ce = TmmF Pq -]AU) Cc = 0.52//^F r ' _ ,50 co Tg = 150; a J^C( •Le '%=QOCO COMMON BASE CURRENT GAIN CUT-OFF FREQUENCY ■ 120 MC Ic = 3 MA Vc = 10 VOLTS Fig. 5 ~ High-frequency lumped constant equivalent circuit for a double diffused silicon n-p-n transistor. 8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 cutoff caused by the collector capacitance and the combined collector body resistance and base resistance is an order of magnitude higher than the measured alpha cutoff frequency and therefore is not too serious in impairing the very high-frecjuency performance of the transistor. This is due to the low capacitance of the collector junction which is seen to be approximately 0.5 mmf at 10 volts collector voltage. The base resistance of this transistor is less than 100 ohms which is quite low and compares very favorably with the best low frequency transistors reported previously. The low-frequency characteristics of the double diffused silicon tran- sistor are very similar to those of other junction transistors. This is il- lustrated in Fig. 6 where the static collector characteristics of one of these transistors are given. At zero emitter current the collector current is too small to be seen on the scale of this figure. The collector current 45 40 35 30 25 20 15 10 -5 le=0 2 4 6 8 10 12 ] J 14/ ^ J^ ^ y^ ^ 2 4 6 8 10 12 14 CURRENT, If, IN MILUAMPERES Fig. 6 — Collector characteristics of a double diffused silicon n-p-n tran- sistor. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 9 0.98 0.94 0.90 0.86 a 0.82 0.78 0.74 0.70 T=150°C, ^ ^ ^ ^ ^ 7 <^ y ^ ^ ^ \ / 9/ y 24, 5M 65-W /> 7 /24.5 t35^y\ 7 15ol / / 1 1 1 _L. 1 1 1 ,1 0.1 0.2 0.4 0.6 1 2 4 6 8 10 20 CURRENT, Ig, IN MILLIAMPERES Fig. 7 — Alpha as a function of emitter current and temperature for a double diffused silicon n-p-n transistor. under this condition does not truly saturate but collector junction re- sistance is very high. Collector junction resistances of 50 megohms at reverse biases of 50 volts are common. The continuous power dissipation permissible with these units is also shown in Fig. 6. The figure shows dissipation of 200 milliwatts and the units have been operated at 400 milliwatts without damage. As illus- trated in Fig. 3 no special provision has been made for power dissipation and it would appear from the performance obtained to date that powers of a few watts could be handled by these iniits with relatively minor provisions for heat dissipation. However, it can also be seen from Fig. 6 that at low collector voltages alpha decreases rapidly as the emitter current is increased. The transistor is, therefore, non-linear in this range of emitter currents and collector voltages. In many applications, this non-linearity may limit the operating range of the device to values below those which would be permissible from the point of view of con- tinuous power dissipation. Fig. 7 gives the magnitude of alpha as a function of emitter current for a fixed collector voltage of 10 volts and a number of ambient tem- peratures. These curves are presented to illustrate the stability of the parameters of the double diffused silicon transistor at increased ambient temperatures. Over the range from 1 to 15 milliamperes emitter current and 25°C to 150°C ambient temperature, alpha is seen to change only 10 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 by approximately 2 per cent. This amounts to only 150 parts per million change in alpha per degree centigrade change in ambient temperature. The decrease in alpha at low emitter currents shown in Fig. 7 has been observed in every double diffused silicon transistor which has been made to date. Although this effect is not completely understood at present it could be caused by recombination centers in the base layer that can be saturated at high injection levels. Such saturation would result in an increase in effective lifetime and a corresponding increase in alpha. The large increase in alpha with temperature at low emitter currents is con- sistent with this proposal. It has also been observed that shining a strong light on the transistor will produce an appreciable increase in alpha at low emitter currents but has little effect at high emitter currents. A strong light would also be expected to saturate recombination centers which are active at low emitter currents and this behavior is also con- sistent with the above proposal. 3.0 DISCUSSION OF THE TRANSISTOR STRUCTURE Although the low frequency electrical characteristics of the double diffused silicon transistor which are presented in Section 2 are quite similar to those usually obtained in junction transistors, the structure of the double diffused transistor is sufficiently different from that of the grown junction or alloy transistor that a discussion of some design principles is warranted. This section is devoted to a general discussion of the factors which determine the electrical characteristics of the tran- sistors. In Section 4 the general ideas of Section 3 are applied in a more specialized fashion to the double diffused structure and a detailed cal- culation of electrical parameters is presented. One essential difference between the double diffused transistor and grown junction or alloy transistors arises from the manner in which the impurities are distributed in the three active regions. In the ideal case of a double-doped grown junction transistor or an alloy transistor the concentration of impurities in a given region is essentially uniform and the transition from one conductivity type to another at the emitter and collector junctions is abrupt giving rise to step junctions. On the other hand in the double diffused structure the distribution of impurities is more closely described by the error function complement and the emitter and collector junctions are graded. Tlu\se differences can have an appre- ciable influence on the electrical beha\'ior of the transistors. Fig. 8(a) shows the probable distribution of donor impurities, No , and acceptor impurities, A''^ , in a double diffused n-p-n. Fig. 8(b) is a DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 11 DONORS ACCEPTORS DISTANCE (a) DISTANCE *• (b) Fig. 8 — Diagrammatic representation of (a) donor and acceptor distributions and (b) uncompensated impuritj- distribution in a double diffused n-p-n tran- sistor. plot of Nd — Na which would result from the distribution in Fig. 8(a). Kromer has shown that a nonuniform distribution of impurities in a semiconductor will produce electric fields which can influence the flow of electrons and holes. For example, in the base region the fields between the emitter junction, Xe , and the minimum in the Nd — Na curve, x', will retard the flow of electrons toward the collector while the fields between this minimum and the collector jvmction, Xc , will accelerate the flow of electrons toward the collector. These base laj^er fields will affect the transit time of minority carriers across the base and thus contribute * H. Kromer, On Diffusion and Drift Transistor Theory I, II, III, Archiv. der Electr. Ubertragung, 8, pp. 223-228, pp. 363-369, pp. 499-504, 1954. 12 THE BELL SYSTEM TECHNICAL JOUENAL, JANUARY 1956 to the fre(iuency response of the transistor. In addition the base re- sistance will be dependent on the distribution of both diffusants. These three factors are discussed in detail below. Moll and Ross have determined that the minority current, /,„ , that will flow into the base region of a transistor if the base is doped in a non- uniform manner is given by f N(x) dx where rii is the carrier concentration in intrinsic material, q is the elec- tronic charge, V is the applied voltage, Dm is the diffusion coefficient of the minority carriers, and the integral represents the total number of uncompensated impurities in the base. The primary assumptions in this derivation are (1) planar junctions, (2) no recombination in the base region, and (3) a boundary condition at the collector junction that the minority carrier density at this point equals zero. It is also assumed that the minority carrier concentration in the base region just adjacent to the emitter junction is equal to the equilibrium minority carrier density at this point multiplied by the Boltzman factor exp (qV/kT). It is of special interest to note that Im depends only on the total number of uncom- pensated impurities in the base and not on the manner in which they are distributed. In the double diffused transistor, it has been convenient from the point of ease of fabrication to make the emitter layer approximately the same thickness as the base layer. It has been observed that heating sili- con to high temperatures degrades the lifetime of n- and p-type silicon in a similar manner. Both base and emitter layers have experienced the same heat treatment and to a first approximation it can be assumed that the lifetime in the two regions will be essentially the same. Thus as- sumptions (1) and (2) should also apply to current flow from base to emitter. If we assume that the surface recombination \'elocity at the free surface of the emitter is infinite, then this imposes a boundary condition at this side of the emitter which under conditions of forward bias on the emitter is equivalent to assumption (3). Thus an equation of the form of (3.1) should also give the minority current flow from base to emitter. Since the emitter efficiency, y, is given by ^ J. Tj. Moll and I. M. Ross, The J)opendencc of Transistor Paramotors on tlie Distribution of Base Layer liesistivity, Proc. I.R.E. in press. 8 G. Bemski, private comnmnication. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 13 /m (emitter to base) -y = . . . /^(emitter to base) + /„j(base to emitter) proper substitution of (3.1) will give the emitter efficiency of the double diffused n-p-n transistor, 1 7 = J-'n Z).^''^-^"^ dx p .6 (3.2) \ (No - iVj dx In (3.2), Dp is the diffusion coefficient of holes in the emitter, /)„ is the diffusion coefficient of electrons in the base and the ratio of integrals is the ratio of total uncompensated doping in the base to that in the emitter. A calculation of transit time is more difficult. Kromer has studied the case of an aiding field which reduces transit time of minority carriers across the base region and thus increases frequency response. In the double diffused transistor the situation is more complex. Near the emitter side of the base region the field is retarding (Region R, see Fig. 8) and becomes aiding (Region A) only after the base region doping reaches a maximum. The case of retarding fields has been studied by Lee and by MoU.^ At present, the case for a base region containing both types of fields has not been solved. However, at the present state of knowledge some speculations about transit time can be made. The two factors of primary importance are the magnitude of the built-in fields and the distance over which they extend. In the double diffused transistor, the widths of regions R and A are determined by the surface concentrations and diffusion coefficients of the diffusants. It Can be shown by numerical computation that if region R constitutes no more than 30-40 per cent of the entire base layer width, then the overall effect of the built-in fields will be to aid the transport of minority car- riers and to produce a reduction in transit time. In addition the absolute magnitude of region R is important. If the point x' should occur within an effective Debye length from the emitter junction, i.e., if x' is located in the space charge region associated with the emitter junction, then the retarding fields can be neglected. The base resistance can also be calculated from surface concentrations and diffusion coefficients of the impurities. This is done by considering the base layer as a conducting sheet and determining the sheet con- ' J. L. Moll, private communication. 14 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 ductivity from the total number of uncompensated impurities per square centimeter of sheet and the approjiriate moliility weighted to account for impurity scattering. 4.0 CALCULATION OF DESIGN PARAMETERS To calculate the parameters which determine emitter efficiency, transit time, and base resistance it is assumed that the distribution of uncom- pensated impurities is given by N(x) = Nicrfc f - N-2erJc^ + Nz (4.1) where A^i and A^2 are the surface concentrations of the emitter and base impurity diffusants respectively, Li and L^ are their respective diffusion lengths, and Nz is the original doping of the semiconductor into which the impurities are diffused. The impurity diffusion lengths are defined as Li = 2 V/M and L2 = 2 ^Ddo (4.2) where the D's are the respective diffusion coefficients and the f's are the diffusion times. Equation (4.1) can be reduced to r(^) = Ti erfc I - Ta erfc X^ + 1 (4.3) where For cases of interest here, r(^) will be zero at two points, a and 13, and will have one minimum at ^'. In the transistor structure the emitter junction occurs at ^ = ^v and the collector junction occurs at ^ = (3. Thus the base width is determined by 13 — a. The extent of aiding and retarding fields in the base is determined by ^'. The integral of (4.3) from 0 to a, I\ , and from o to ^, I2 , are the integrals of interest in (3.2) and thus determine emitter efficiency. In addition I2 is the integral from which base resistance can be calculated. The calculations which follow apply only for values of ri/r2 and To greater than ten. Some of the simplifying assumptions which are made will not apply at lower values of these parameters where the distribution of both diffusants as well as the background doping affect the structure in all three regions of the device. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 15 4.1 Base Width From Fig. 8 and (4.3) it can be seen that for r2 ^ 10, a is essentially independent of r2 and is primarily a function of T1/T2 and X. Fig. 9 is a plot of a versus ri/r2 with X as the parameter. The particular plot is for r2 = 10 . Although as stated a is essentially independent of r2 , at lower values of r2, a may not exist for the larger values of X, i.e., the p-layer does not form. In the same manner, it can be seen that ^ is essentially independent of T]/T2 and is a function only of r2 and X. Fig. 10 is a plot of /3 versus F^ with X as a parameter. This plot is for Ti/Fo = 10 and at larger Fi/Fo , /3 may not exist at large X. 10" \0' 10 r2=)o'' /// // / ^ ::i ll r / / m 0/ / ' > /os/ 1 i 1 /// 'o.e/ / f 0.7/ / /// / / <.e I w. W / / / 1.0 1.4 1.8 2.2 2.6 a 3.0 3.4 3.8 Fig. 9 — Emitter layer thickness (in reduced units) as a function of the ratio of the surface concentrations of the diffusing impurities (ri/r2) and the ratio of their diffusion lengths (X). 16 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The base width W = ^ — a can be obtained from Figs. 9 and 10. a, 13 and iv can be converted to centimeters by nuiltiplying by the appropriate value of Li . 4.2 Emitter Efficiency With the hmits a and /3 determined above, the integrals h and 1 2 can be calculated. Examination of the integrals shows that h is closely pro- portional to ri/r2 and also to r2 . On the other hand I2 is closely propor- tional to r2 and essentially independent of ri/r2 . Thus, the ratio of /2//1 which determines 7 depends primarily on ri/r2 . Fig. 11 is a plot of the constant /2//1 contours in the ri/T2 — X plane for lo/h ii^ the range from — 1.0 to —0.01. The graph is for r2 = 10 . Since from (3.2) 7 = 1 1 _ ^h Dnh (4.4) for an n-p-n transistor, and assuming Dp/Dn = /^ for silicon, then to' (0- 10' 10 1' 1 \= ..J\ 0.6- 0.5- ::ffl M \u |6 In 1 1° 1 \\\ ( 0.2 0.1 '/// /// 0.01/ ill 7 / / /// / / / 10 20 50 100 200 500 1000 Fig. 10 — (Collector junction dopth (in rodurod units) as a function of the sur- face concuMit.ration (in reduced units) of llie dilfusaiit wliicli determines the con- ductivity type of the l)ase layer (I'.') and liie ratio of tlie dilTusioii lengths (X) of the tAvo diffusing inii)urifies. DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 10" 17 10 H Ta 10 10 r2 = io'* 2 w \v V 2 ? 1 ^ 1 \\ \ t ^. \ \I2/I 1 2 V ,\ N-0 VO.05 02 -i.o\ -0.3S^ 32X^0 '\ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 11 — ^Dependence of emitter efficiency upon diffusant surface concentra- tions and diffusion lengths. The lines of constant /2//1 are essentially lines of constant emitter efficiency. The ordinate is the ratio of surface concentrations of the two diffusants and the abscissa is the ratio of their diffusion lengths. /2//1 = — 1.0 corresponds to a 7 of 0.75 and /2//1 = —0.01 corresponds to a 7 of 0.997. 4.. 3 Base Resistance It was indicated above that I2 depends principally on r2 and X. Fig. 12 is a plot of the constant I2 contours in the r2 — X plane for I2 in the range from —10^ to —10. The graph is for Ti/To = 10. The base layer sheet conductivity, cjb , can be calculated from these data as Qb = —qtihTjiNz (4.5) where q, L\ and A^3 are as defined above and /I is a mobility properly weighted to account for impurity scattering in the non-uniformly doped base region. The units of gb are mhos per square. 18 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 10- 1 2= -10,00^ / / 7/ 1 / -5000/ r / / // / / / -1000/ // // / / 1 2 1 / /-5oa / / / / / 1 ^/^^ / / / /I ^/ / / 1 1 10 / / // v. /-ioy V 11 / / / /, // /-/ , (I 5 // /, -^ /J / V/ / 2 ^ ^ ^ f^ u 10 102 r / / ^ / / ^ 5 — 1 0 ^/ V r 10 / / 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Fig. 12 — Dependence of base layer sheet condiictivitj^ on diffusant surface concentrations and diffusion lengths. The lines of constant Ii are essentiallj' lines of constant base sheet conductivity. The ordinate is the surface concentration (in reduced units) of the diffusant which determines the conductivity type of the base layer and the abscissa is the ratio of the diffusion lengths of the two difi'using impurities. 4.4 Transit Time With a knowledge of where the minimum value, ^', of (4.3) occurs, it is possible to calculate over what fraction of the base width the fields are retarding. The interesting quantity here is 13 - a ^ is a function of ri/r2 and X and varies only very slowly with ri/r2 . a is also a function of ri/r2 and X and varies only slowly with ri/r2 . The most rapidly changing part of bJi is l^ which depends primarily on r2 as noted above. Fig. 13 is a plot of the constant LR contours in the r2 — X plane for values of A/2 in the range 0.1 to 0.3. This graph is DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 19 lor data with ri/r2 = 10. As ri/r2 increases at constant r2 and X, AR decreases slightly. At ri/r2 = 10\ the average change in AR is a decrease of about 25 per cent for constant r2 and X when AR ^ 0.3. The error is larger for values of AR greater than 0.3. It was noted above that when AR becomes greater than 0.3, the retarding fields become dominant. Therefore, this region is of slight interest in the design of a high frequency transistor. 4.5 A Sample Design By superimposing Figs. 11, 12 and 13 the ranges of r2 , ri/r2 and X which are consistent with desired values of y, gt and AR can be deter- 0.7 Fig. 1.3 — Dependence of the built-in field distribution on concentrations and diffusion lengths. The lines of constant aR indicate the fraction of the base layer thickness over which built-in fields are retarding. The ordinate is the surface concentration (in reduced units) of the diffusant which determines the conductiv- ity type of the base layer and the abscissa is the ratio of the diffusion lengths of the two diffusing impurities. 20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 mined by the area enclosed by the specified contour lines. It is also possible to compare the measured parameters of a specific device and observe how closely they agree with what is predicted from the estimated concentrations and diffusion coefficients. This is done below for the transistor described in Sections 1 and 2. The comparison is complicated by the fact that the exact values of the surface concentrations and diffusion coefficients are not known {Precisely enough at present to permit an accurate evaluation of the design theory. However, the following values of concentrations and diffusion coefficients are thought to be realistic for this transistor. iVi = 5 X 10^' /)i = 3 X 10"'' /i = 5.7 X lO' iV2 = 4 X 10'' Di = 2.5 X 10"" t^= 1.2 X lO' Nz = 10'' From these values it is seen that Ti/ra = 12.5; r, = 400; X = 0.6 From Fig. 9, a = 1.9 and from Fig. 10, /3 = 3.6 and therefore w = 1.7. Measurement of the emitter and base layer dimensions showed that these layers were approximately the same thickness which was 3.8 X 10" cm. Thus the ifieasured ratio of emitter width to base width of unity is in good agreement with the ^'alue of 1.1 predicted from the assumed con- centrations and diffusion coefficients. From Fig. 11, lo/h ~ —0.01. If this value is substituted into (4.4), 7 = 0.997. This compares with a measured maximum alpha of 0.972. From Fig. 12, lo = —15. Assuming an average hole mobility of 350 cm' /volt. sec. and evaluating Li from the measured emitter thickness and the calculated a, (4.5) gives a value of gb = 1.7 X 10^ mhos per square. The geometry of the emitter and base contacts as shown in Fig. 3 makes it difficult to calculate the effective base resistance from the sheet conductivity even at very small emitter currents. In addition at the very high inje{;tion levels at which these transistors are operated the calculation of effective base resistance becomes very difficult. However, from the geometr}^ it would be expected that the effective base re- sistance would l)c no greater than 0.1 of the sheet resistivity or 600 ohms. This is about seven times larger than the measured \'alue of 80 ohms reported in Section 2. From Fig. b3, A/^ is approximately 0.20. Thus there should be an over- all aiding elfect of the built-in fields. In addition the impurity gradient at the emitter junction is believed to be approximately lO'Vcm and the DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 21 space charge associated with this gradient will extend approximately 2 X 10 ■' cm into the base region. The base thickness over which re- tarding fields extend is AR times the base width or 7.6 X 10~^ cm. Thus the first quarter of region R will be space charge and can be neglected. The frequency cutoff from pure diffusion transit is given by 2A3D ,. , where W is the measured base layer thickness. Assuming D — 25 cmVsec for electrons in the base region, ,/'„ = (w mc/sec. Since the measured cutoff was 120 mc/sec, the predicted aiding effect of the built-in field is evidently present. These computations illustrate how the measured electrical parameters can be used to check the values of the surface concentrations and dif- fusion coefficients. Conversely knowledge of the concentrations and diffusion coefficients aid in the design of devices which will have pre- scribed electrical parameters. The agreement in the case of the transistor described above is not perfect and indicates errors in the proposed values of the concentrations and diffusion coefficients. However, it is sufficiently close to be encouraging and indicate the value of the calculations. The discussion of design has been limited to a very few of the important parameters. Junction capacitances, emitter and collector resistances are among the other important characteristics which have been omitted here. Presumably all of these quantities can be calculated if the detailed structure of the device is known and the structure should be susceptible to the type of analysis used above. Another fact, which has been ignored, is that these transistors were operated at high injection levels and a low level analysis of electrical parameters was used. All of these other factors must be considered for a detailed understanding of the device. The object of this last section has been to indicate one path which the more detailed analysis might take. 5.0 CONCLUSIONS By means of multiple diffusion, it has been possible to produce silicon transistors with alpha-cutoff above 100 mc/sec. Refinements of the described technicjues offer the possibility of even higher frequency per- formance. These transistors show the other advantages expected from silicon such as low saturation currents and satisfactory operation at high temperatures. The structure of the double diffused transistor is susceptible to design 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 analysis in a fashion similar to that which has been applied to other junc- tion transistors. The non-uniform distribution of impurities produces significant electrical effects which can be controlled to enhance appre- cial)ly the high-frequency behavior of the devices. The extreme control inherent in the use of diffusion to distribute im- purities in a semiconductor structure suggests that this technique will become one of the most valuable in the fabrication of semiconductor devices. ACKNOWLEDGEMENT The authors are indebted to several people who contributed to the work described in this paper. In particular, the double diffused silicon from which the transistors were prepared was supplied by C. S. Fuller and J. A. Ditzenberger. The data on diffusion coefficients and concen- trations were also obtained by them. P. W. Foy and G. Kaminsky assisted in the fabrication and mounting of the transistors and J. M. Klein aided in the electrical characterization. The computations of the various solutions of the diffusion equation, (4.3), were performed by Francis Maier. In addition many valuable discussions with C. A. Lee, G. Weinreich, J. L. Moll, and G. C. Dacey helped formu- late many of the ideas presented herein. A High-Frequency Diffused Base Gernianiuni Transistor By CHARLES A. LEE (Manuscript received November 15, 1955) Techniques of impurity diffusion and alloying have been developed which make possible the construction of p-n-p junction transistors utilizing a diffused surface layer as a base region. An important Jeature is the high degree of dimensional control obtainable. Diffusion has the advantages of being able to produce uniform large area junctions which may be utilized in high power devices, and very thin surface layers which may be utilized in high-frequency devices. Transistors have been made in germanium which typically have alphas of 0.98 and alpha-cutoff frequencies of 500 mcls. The fabrication, electrical characterization, and design considerations of these transistors are dis- cussed. INTRODUCTION Recent work ■ concerning diffusion of impurities into germanium and silicon prompted the suggestion that the dimensional control in- herent in these processes be utilized to make high-frecjuency transistors. One of the critical dimensions of junction transistors, which in many cases seriously restricts their upper freciuency limit of operation, is the thickness of the base region. A considerable advance in transistor proper- ties can be accomplished if it is possible to reduce this dimension one or two orders of magnitude. The diffusion constants of ordinary donors and acceptors in germanium are such that, with'n realizable tempera- tures and times, the depth of diffused surface layers may be as small as 10" cm. Already in the present works layers slightly less than 1 micron (10~ cm) thick have been made and utilized in transistors. Moreover, the times and temperatures required to produce 1 micron surface laj^ers permit good control of the depth of penetration and the concentration of the diffusant in the surface layer with techniciues described below. If one considers making a transistor whose base region consists of such 23 24 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 a diffused surface layer, several problems become immediately apparent : (1) Control of body resistivity and lifetime during the diffusion heat- ing cycle. (2) Control of the surface concentration of the diffusant. (3) INIaking an emitter on the surface of a thin diffused layer and controlling the depth of penetration. (4) Making an ohmic base contact to the diffused surface layer. One approach to the solution of these problems in germanium which has enabled us to make transistors with alpha-cutoff frequencies in excess of 500 mc/sec is described in the main body of the paper. An important characteristic feature of the diffusion technique is that it produces an impurity gradient in the base region of the transistor. This impurity gradiant produces a "built-in" electric field in such a direction as to aid the transport of minority carriers from emitter to collector. Such a drift field may considerably enhance the frequency response of a transistor for given physical dimensions. The capabilities of these new techniques are only partially realized by their application to the making of high frequency transistors, and even in this field their potential has not been completely explored. For example, with these techniques applied to making a p-n-i-p structure the possibility of constructing transistor amplifiers with usable gain at frequencies in excess of 1,000 mc/sec now seems feasible. DESCRIPTION OF TRANSISTOR FABRICATION AND PHYSICAL CHARACTERIS- TICS As starting material for a p-n-p structure, p-type germanium of 0.8 ohm-cm resistivity was used. From the single crystal ingot rectangular bars were cut and then lapped and polished to the approximate dimen- sions: 200 X 60 X 15 mils. After a slight etch, the bars were washed in deionized water and placed in a vacuum oven for the diffusion of an n-type impurity into the surface. The vacuum oven consisted of a small molybdenum capsule heated by radiation from a tungsten coil and sur- rounded by suitable radiation shields made also of molybdenum. The capsule could be baked out at about 1,900°C in order that impurities detrimental to the electrical characteristics of the germaniinn be evapo- rated to sufficiently low levels. As a source of n-type impurity to be placed with the p-type bars in the molybdenum oven, arsenic doped germanium was used. The rela- tively high vapor pressure of the arsenic was reduced to a desirable range (about lO"* nun of Ilg) by diluting it in germanium. The use of ger- manium eliminated any additional problems of contamination by the A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 25 dilutant, and provided a convenient means of determining the degree of dilution by a measurement of the conductivity. The arsenic concentra- tions used in the source crystal were typically of the order of 10 '-10^^/cc. These concentrations were rather high compared to the concentrations desired in the diffused surface layers since compensation had to be made for losses of arsenic due to the imperfect fit of the cover on the capsule and due to some chemical reaction and adsorption which occurred on the internal surfaces of the capsule. The layers obtained after diffusion were then evaluated for sheet con- ductivity and thickness. To measure the sheet conductivity a four-point probe method^ was used. An island of the surface layer was formed by masking and etching to reveal the junction between the surface layer and the p-type body. The island was then biased in the reverse direction with respect to the body thus effectively isolating it electrically during the measurement of its sheet conductivity. The thickness of the surface layer was obtained by first lapping at a small angle to the original surface (3^-2°~l°) and locating the junction on the beveled surface with a thermal probe; then multiplying the tangent of the angle between the two sur- faces by the distance from the edge of the bevel to the junction gives the desired thickness. Another particularly convenient method of measuring the thickness' is to place a half silvered mirror parallel to the original sur- face and count fringes, of the sodium D-Yme for example, from the edge of the bevel to the junction. Typically the transistors described here were prepared from diffused layers with a sheet conductivity of about 200 ohms/square, and a layer thickness of (1.5 ± 0.3) X 10~ cm. When the surface layer had been evaluated, the emitter and base con- tacts were made using techniques of vacuum evaporation and alloying. o For the emitter, a film of aluminum approximately 1,000 A thick was evaporated onto the surface through a mask which defined an emitter area of 1 X 2 mils. The bar with the evaporated aluminum was then placed on a strip heater in a hydrogen atmosphere and momentarily brought up to a temperature sufficient to alloy the alimiinum. The emitter having been thus formed, the bar was again placed in the masking jig and a film of gold-antimony alloy from 3,000 to 4,000 A thick was evaporated onto the surface. This film was identical in area to the emitter, and was placed parallel to and 0.5 to 1 mil away from the emitter. The bar was again placed on the heater strip and heated to the gold-germanium eutectic temperature, thus forming the ohmic base contact. The masking jig was constructed to permit the simultaneous evaporation of eight pairs of contacts on each bar. Thus, using a 3-mil diamond saw, a bar could be cut into eight units. 20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Each unit, with an alloyed emitter and base contact, was then soldered to a platinum tab with indium, a sufficient quantity of indium being- used to alloy through the n-type surface layer on the back of the unit. One of the last steps was to mask the emitter and base contacts with a 6- to 8-mil diameter dot of wax and form a small area collector junction by etching the unit attached to the platinum tab, in CP4. After washing in solvents to remove the wax, the unit was mounted in a header designed to allow electrolytically pointed wire contacts to be made to the base and emitter areas of the transistor. These spring contacts were made of 1-mil phosphor bronze wire. ELECTRICAL CHARACTERIZATION Of the parameters that characterize the performance of a transistor, one of the most important is the short circuit current gain (alpha) ver- sus frequency. The measured variation of a and q:/(1 — a) (short-circuit current gain in the grounded emitter circuit) as a function of frequency for a typical unit is shown in Fig. 1 . For comparison the same parameters for an exceptionally good unit are shown in Fig. 2. In order that the alpha-cutoff frequency be a measure of the transit time of minority carriers through the active regions of the transistor, any resistance-capacity cutoffs, of the emitter and collector circuits, must lie considerably higher than the measured /„ . In the emitter circuit, an external contact resistance to the aluminum emitter of the order of 10 U1 _J LU eg o lij Q •4U ( 30 20 >-( — , 4.3 MC UNIT 0-3 p- n-p Ge Ie = 2 MA Vc =-10 VOLTS ao= 0.982 ' 1 s S. 1-a 6 DB OCT/> PER ^' VE ■> ^s 1 0 0 -10 l«l w > \ 46 3 M( 1 ; ^ * 0.1 0.2 0.4 0,6 1 2 4 6 8 10 20 40 60 100 200 400 1000 FREQUENCY IN MEGACYCLES PER SECOND Fig. 1 — The grounded emitter and grounded base response versus frequency for a typical unit. A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 27 40 30 10 _l LU 5 20 o LJJ Q 10 o- ^ « 3.4 M C UNIT M-2 p-r Ie = 2MA 1-p Ge N -— oc 1 \-oc Vc=-10 VOLTS OCo- 0.980 6Db\ PER A OCTAVE ^N ^'s oc i-C v^ 540 MC ^\ \ -10 0.1 0.2 0.4 0.6 1 2 4 6 e 10 20 40 60 100 200 400 1000 FREQUENCY IN MEGACYCLES PER SECOND Fig. 2 — The grounded emitter and grounded base response versus frequency for an exceptionally good unit. to 20 ohms and a junction transition capacity of 1 fx^xid were measured. The displacement current which flows through this transition capacity reduces the emitter efficiency and must be kept small relative to the injected hole current. With 1 milliampere of ciu"rent flowing through the emitter junction, and conseciuently an emitter resistance of 26 ohms, I the emitter cutoff for this transistor was above 6,000 mc/sec. One can now see that the emitter area must be small and the current density high to attain a high emitter cutoff freciuency. The fact that a low base resistance requires a high level of doping in the base region, and thus a high emitter transition capacity, restricts one to small areas and high current densities. In the collector circuit capacities of 0.5 to 0.8 ^l^xid at a collector volt- age of — 10 volts were measured. There was a spreading resistance in the collector body of about 100 ohms which was the result of the small emitter area. The base resistance was approximately 100 ohms. If the phase shift and attenuation due to the transport of minority carriers through the base region w^ere small at the collector cutoff frequency, the (effective base resistance would be decreased by the factor (1 —a). The collector cutoff frequency is then given by where Cc = collector transition capacity and Re = collector body spreading resistance. 28 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 However, in the transistors described here the base region produces the major contribution to the observed alpha-cutoff frequency and it is more appropriate to use the expression 2irCcin + Re) where n = base resistance. This cutoff frequency could be raised by in- creasing the collector voltage, but the allowable power dissipation in the mounting determines an upper limit for this voltage. It should b noted that an increase in the doping of the collector material would raise the cutoff since the spreading resistance is inversely proportional to Na , while the junction capacity for constant collector voltage is only pro- portional to Na . The low-frequency alpha of the transistor ranged from 0.95 to 0.99 with some exceptional units as high as 0.998. The factors to be con- sidered here are the emitter efficiency y and the transport factor (3. The transport factor is dependent upon the lifetime in the base region, the recombination velocity at the surface immediately surrounding the emitter, and the geometry. The geometrical factor of the ratio of the emitter dimensions to the base layer thickness is > 10, indicating that solutions for a planar geometry may be assumed.^ If a lifetime in the base region of 1 microsecond and a surface recombination velocity of 2,000 cm/sec is assumed a perturbation calculation gives iS = 0.995 The high value of ^ obtained with what is estimated to be a low base region lifetime and a high surface recombination velocity indicates that the observed low frecjuency alpha is most probably limited by the emitter injection efficiency. As for the emitter injection efficiency, within the accuracy to which the impurity concentrations in the emitter re- growth layer and the base region are known, together with the thick- nesses of these two regions, the calculated efficiency is consistent with the experimentally observed values. Considerations of Transit Time An examination of what agreement (^xists between the alpha-cutoff frequency and the physical measurements of the base region involves the me(;hanism of transport of minority carriers through the active regions of the transistor. The "active regions" include the space charge A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 29 region of the collector junction. The transit time through this region is no longer a negligible factor. A short calculation will show that with — 10 volts on the collector junction, the space charger layer is about 4 X 10"^ cm thick and that the frequency cutoff associated with trans- port through this region is approximately 3,000 mc/sec. The remaining problem is the transport of minority carriers through the base region. Depending upon the boundary conditions existing at the surface of the germanium during the diffusion process, considerable gradients of the impurity density in the surface layer are possible. How- ever, the problem of what boundary conditions existed during the diffu- sion process employed in the fabrication of these transistors w^ill not be discussed here because of the many uncertainties involved. Some quali- tative idea is necessary though of how electric fields arising from impurity gradients may affect the frequency behavior of a transistor in the limit of low injection. If one assumes a constant electric field as would result from an ex- ponential impurity gradient in the base region of a transistor, then the continuity eciuation may be solved for the distribution of minority carriers.* From the hole distribution one can obtain an expression for the transport factor j3 and it has the form /3 = e" r? sinh Z -{- Z cosh Z where 1, Ne IqE ^"2^^iV; = 2^^' z ^ [i^ + ,r' IV' Ne = donor density in base region at emitter junction Nc = donor density in base region at collector junction E = electric field strength Dp = diffusion constant for holes w = width of the base layer 30 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 A plot of this function for various values of rj is shown in Fig. 3. For ?? = 0, the above expression reduces to the well known case of a uniformly doped base region. The important feature to be noted in Fig. 3 is that relatively small gradients of the impurity distribution in the base layer can produce a considerable enhancement of the frequency response. It is instructive to calculate what the alpha-cutoff f recjuency would be for a base region with a uniform distribution of impurity. The effective thickness of the base layer may be estimated by decreasing the measured thickness of the surface layer by the penetration of the space charge region of the collector and the depth of the alloyed emitter structure. Using a value for the diffusion constant of holes in the base region appro- priate to a donor density of about 10 Vcc, 300 mc/s ^fa^ 800 mc/s This result implies that the frecjuency enhancement due to "built-in" fields is at most a factor of two. In addition it was observed that the alpha-cutoff frequency was a function of the emitter current as shown in Fig. 4. This variation indicates that at least intermediate injection *~ :;^;~->i^ k.^ ^ Nv ^ N "\ V \ \ \ V ^ \ \ >v, 0.2 A \ \ K \ \ \ i \ K \ \ \ 0.08 - ^ \— ^ A — \ — v\- 0.06 0.04 - ^^ \ ^, ^ \ ^ 1\ 4i r 0.02 \ \ \ V \ \ V 0.01 1 1 1 \ \ 1 1 > 1 1 1 1 _L 0.1 0.2 0.4 0.6 0.8 1 6 8 10 20 40 60 80 100 w2 <^-U} -g- , (RADIANS) Fig. .3 — The variation of | i3 | ver.sii.s frequency for various values of a uniform drift field in the base region. A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 31 in _i LU m o LU a z b n =7^" '^^-^^ S— 1' i f \ ^ ' ' ; Q ■ ■_;;;; -t Fv Rl k -5 UNIT 0-3 p-n-p Ge o Ie = 2 MA A Ie=0.8MA D Ig=0.4MA \ k^ ^ \ \ \ 10 Vc = -K ) VOLTS 1 1 \ 1 1 1 10 20 30 40 50 60 80 100 200 300 400 FREQUENCY IN MEGACYCLES PER SECOND 600 800 1000 Fig. 4 current. The variation of the alpha-cutoff frequency as a function of emitter levels exist in the range of emitter current shown in Fig. 4. The conclu- sion to be drawn then is that electric fields produced by impurity gradients in the base region are not the dominant factor in the transport of minority carriers in these transistors. The emitter current for a low level of injection could not be deter- mined by measuring /„ versus /« because the high input impedance at very low levels was shorted by the input capacity of the header and socket. Thus at very small emitter currents the measured cutoff fre- quency was due to an emitter cutoff and was roughly proportional to the emitter current. At /e ^ 1 ma this effect is small, but here at least intermediate levels of injection already exist. A further attempt to measure the effect of any "built-in" fields by turning the transistor around and measuring the inverse alpha proved fruitless for two reasons. The unfavorable geometrical factor of a large collector area an a small emitter area as well as a poor injection effi- ciency gave an alpha of only a = 0.1 Secondly, the injection efficiency turns out in this case to be proportional to oT^^'^ giving a cutoff freciuency of less than 1 mc/sec. The sciuare-root dependence of the injection efficiency on freciuency may be readily seen. The electron current injected into the collector body may be expressed as Je = qDnN 1 -)- iu^Te 1/2 where q = electronic charge 32 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Dn ^ diffusion constant of electrons Vi = voltage across collector junction Tic = density of electrons on the p-type side of the collector junction Te = lifetime of electrons in collector body Le = diffusion length of electrons in the collector body Since the inverse cutoff frequency is well below that associated with the base region, we may regard the injected hole current as independent of the frequency in this region. The injection efficiency is low so that 7 ;^ ^ « 1 J e Thus at a frequency where then cor, »1 I -1/2 An interesting feature of these transistors was the very high current densities at which the emitter could be operated without appreciable loss of injection efficiency. Fig. 5 shows the transmission of a 50 millimicro- second pulse up to currents of 18 milliamperes which corresponds to a current density of 1800 amperes/cm". The injection efficiency should remain high as long as the electron density at the emitter edge of the base region remains small compared to the acceptor density in the emitter regrowth layer. When high injection levels are reached the in- jected hole density at the emitter greatly exceeds the donor density in th(> base region. In order to preserve charge neutrality then p ^ n where p = hole density n = electron density As the inject(Hl hole density is raised still further the electron density will eventually become comparable to the acceptor density in the emitter regrowth layer. Tlie density of acceptors in the emitter regrowth A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 33 30 46 60 75 90 TIME IN MILLIMICROSECONDS >" 0 9 "^ V 4 '^ \^ / 18 V / -15 15 30 45 60 75 90 TIME IN MILLIMICROSECONDS 105 120 136 Fig. 5 — Transmission of a 50 millimicrosecond pulse at emitter currents up to 18 ma by a typical unit. (Courtesy of F. K. Bowers). region is of the order of and this is to be compared with injected hole density at the base region iside of the emitter junction. The relation between the injected hole density and the current density may be approximated by^ J. w where pi = hole density at emitter side of base region w = width of base region jA short calculation indicates that the emitter efficiency should remain 'high at a current density of an order of magnitude higher than 1,800 |amp/cm'. The measurements were not carried to higher current densities jbecause the voltage drop across the spreading resistance in the collector was producing saturation of the collector junction. CONCLUSIONS Impurity diffusion is an extremely powerful tool for the fabrication of high frequency transistors. Moreover, of the 50-odd transistors which 34 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 were made in the laboratory, the characteristics were remarkably uni- form considering the ^•ariations usually encountered at such a stage of development. It appears that diffusion process is sufficiently controllable that the thickness of the base region can be reduced to half that of the units described here. Therefore, with no change in the other design parameters, outside of perhaps a different mounting, units with a 1000 mc/s cutoff frequency should be possible. ACKNOWLEDGMENT The author wishes to acknowledge the help of P. W. Foy and W. Wieg- mann who aided in the construction of the transistors, D. E. Thomas who designed the electrical equipment needed to characterize these units, and J. Klein who helped with the electrical measurements. The numerical evaluation of alpha for drift fields was done by Lillian Lee whose as- sistance is gratefully acknowledged. REFERENCES 1. C. S. Fuller, Phys. Rev., 86, pp. 136-137, 1952. 2. J. Saby and W. C. Dunlap, Jr., Phys. Rev., 90, p. 630, 1953. 3. W. Shocklej', private communication. 4. H. Kromer, Archiv. der Elek. tlbertragung, 8, No. 5, pp. 223-228, 1954. 5. R. A. Logan and M. Schwartz, Phys. Rev., 96, p. 46, 1954 6. L. B. Valdes, Proc. I.R.E., 42, pp. 420-427, 1954. 7. W. L. Bond and F. M. Smits, to be published. 8. E. S. Rittner, Pnys. Rev., 94, p. 1161, 1954. 9. W. M. Webster, Proc. I.R.E., 42, p. 914, 1954. 10. J. M. Early, B.S.T.J., 33, pp. 517-533, 1954. Waveguide Investigations with Millimicrosecond Pulses By A. C. BECK (Manuscript received October 11, 1955) Pulse techniques have been used for many waveguide testing 'puryoses. The importance of increased resolution hy means of short pulses has led to the development of equipment to generate, receive and display pidses about 5 or 6 millimicroseconds lo7ig. The equipment is briefly described and its resolution and measuring range are discussed. Domi7ia7it mode waveguide and antenna tests are described, and illustrated. Applications to midtimode waveguides are then considered. Mode separation, delay distortion and its equalization, and mode conversion are discussed, and examples are given. The resolution obtained with this equipment provides information that is difficult to get by any other means, and its use has proved to be very helpfid in ivaveguide investigations. CONTENTS 1 . Introduction 35 2. Pulse Generation 36 3. Receiver and Indicator 41 4. Resolution and Measuring Range 42 5. Dominant Mode Waveguide Tests 43 6. Testing Antenna Installations 45 7. Separation of Modes on a Time Basis 48 8. Delay Distortion 52 9. Delay Distortion Ecjualization 54 10. Measuring Mode Conversion from Isolated Sources 57 11. Measuring Distril)uted Mode Conversion in 1 ong Waveguides 61 12. Concluding Remarks 65 1. INTRODUCTION Pulse testing techniques have been employed to advantage in wave- guide investigations in numerous ways. The importance of better resolu- tion through the use of short pulses has always been apparent and, from the first, eciuipment was employed which used as short a pulse as pos- sible. Radar-type apparatus using magnetrons and a pulse width of about one-tenth microsecond has seen considerable use in waveguide research, and many of the results have been published.' • - 35 36 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 To improve the resolution, work was initiated some time ago by S. E. Miller to obtain measuring equipment which would operate with much shorter pulses. As a result, pulses about 5 or 6 millimicroseconds long became available at a frequency of 9,000 mc. In a pulse of this length there are less than 100 cycles of radio frequency energy, and the signal occupies less than ten feet of path length in the transmission medium. The RF bandwidth required is about 500 mc. In order to obtain such bandwidths, traveling wave tubes were developed by J. R. Pierce and members of the Electronics Research Department of the Laboratories. The completed amplifiers were designed by W. W. Mumford. N. J. Pierce, R. W. Dawson and J. W. Bell assisted in the design and construc- tion phases, and G. D. Mandeville has been closely associated in all of this work. 2. PULSE GENERATION These millimicrosecond pulses have been produced by two different types of generators. In the first equipment, a regenerative pulse gener- ator of the type suggested by C. C. Cutler of the Laboratories was used.^ This was a very useful device, although somewhat complicated and hard to keep in adjustment. A brief description of it will permit comparisons with a simpler generator which was developed a little later. A block diagram of the regenerative pulse generator is shown in Fig. 1. The fundamental part of the system is the feedback loop drawn with heavy lines in the lower central part of the figure. This includes a travel- ing wave amplifier, a waveguide delay line about sixty feet long, a crystal expander, a band-pass filter, and an attenuator. This combination forms an oscillator which produces very short pulses of microwave energy. Between pulses, the expander makes the feedback loop loss too high for oscillation. Each time the pulse circulates around the loop it tends to shorten, due to the greater amplification of its narrower upper part caused by the expander action, until it uses the entire available band width. A 500-mc gaussian band-pass filter is used in the feedback loop,^ of this generator to determine the final bandwidth. An automatic gain control operates with the expander to limit the pulse amplitude, thus preventing amplifier compression from reducing the available expansion. To get enough separation between outgoing pulses for reflected pulse measurements with waveguides, the repetition rate would need to be too low for a practical delay fine length in the loop. Therefore a r2.8-mc fundamental rate was chosen, and a gated traveling wave {\\\)v ampfifier was used to reduce it to a 100-kc rate at the output. This amplifier is kept in a cutoff condition for 127 pulses, and then a gate pulse restores I i t WAVEGUIDE TESTING WITH MILLIMIf'ROSECOND PULSES 37 it to the normal amplifying condition for fifty millimicroseconds, during which time the 128th pulse is passed on to the output of the generator as shown on Fig. 1. The synchronizing system is also shown on Fig. 1. A 100-kc quartz crystal controlled oscillator with three cathode follower outputs is the basis of the system. One output goes through a seven stage multiplier to get a 12.8-mc signal, which is used to control a pulser for synchroniz- ing the circulating loop. Another output controls the gate pulser for the output traveling wave amplifier. Accurate timing of the gate pulse is obtained by adding the 12.8-mc pulses through a buffer amplifier to the gate pulser. The third output synchronizes the indicator oscilloscope sweep to give a steady pattern on the screen. Although this equipment was fairly satisfactory and served for many OSCILLATOR AND CATHODE FOLLOWERS 100 KC I 1 MULTIPLIER 100 KC TO 12.8 MC SYNC PULSER 0.02 A SEC 12.8 MC 500 MC BANDPASS FILTER GATE PULSER 0.05 USEC 100 KC A BUFFER AMPLIFIER "1 CRYSTAL EXPANDER U AGC I WAVEGUIDE DELAY LINE TW TUBE ■Y^ MILLI/iSEC/ 9000 MC/' PULSES 12.8 MC RATE MlLLIyUSEC 9000 MC PULSES 100 KC RATE GATED TW TUBE SYNC SIGNAL TO INDICATOR SCOPE Fig. 1 — Block diagram of the regenerative pulse generator. 38 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 testing purposes, it was rather complex and there were some problems in its construction and use. It was difficult to obtain suitable microwave crystals to match the waveguide at low levels in the expander. Tliis would make it even more difficult to build this type of pulse generator for higher frequency ranges. Stability also proved to be a problem. The frequency multiplier had to be very well constructed to avoid phase shift due to drifting. The gate pulser also required care in design and construction in order to get a stable and flat output pulse. It was some- what troublesome to keep the gain adjusted for proper operation, and the gate pulse time adjustment required some attention. The pulse frequency could not be changed. For these reasons, and in order to get a smaller, lighter and less complicated pulse generator, work was carried out to produce pulses of about the same length by a simpler method. If the gated output amplifier of Fig. 1 were to have a CW instead of a pulsed input, a pulse of microwave energy would nevertheless appear at the output because of the presence of the gating pulse. This gating pulse is applied to the beam forming electrode of the tube to obtain the gating action. If the beam forming electrode could be pulsed from cutoff to its normal operating potential for a very short time, very short pulses of output energy could be obtained from a continuous input signal. How- ever, it is difficult to obtain millimicrosecond video gating pulses of suf- ficient amplitude for this purpose at a 100-kc repetition rate. A traveling-wave tube amplifies normally only when the helix is within a small voltage range around its rated dc operating value. For voltages either above or below this range, the tube is cut off. When the helix voltage is raised through this range into the cutoff region beyond it, and then brought back again, two pulses are obtained, one during a small part of the rise time and the other during a small part of the return time. If the rise and fall times are steep, very short pulses can be obtained. Fig. 2 shows the pulse envelopes photographed from the indicator scope screen when this is done. For the top trace, the helix was biased 300 volts negatively from its normal operating potential, then pulsed to its correct operating range for about 80 millimicroseconds, during which time normal amplification of the CW input signal was ob- tained. The effect of further increasing the helix video pulse amplitude in the positive direction is shown by the succeeding lower traces. The envelope dips in the middle, then two separated pulses remain — one during a part of the rise time and one during a part of the fall time of helix voltage. The pulses shown on the bottom trace have shortened to about six millimicroseconds in length. The helix pulse had a positive amplitude of about 500 volts for this trace. 1 WAVEGUIDE TESTIXG WITH MILUMICROSErOXD PULSES 39 Since only one of these pulses can be used to get the desired repetition rate, it is necessary to eliminate the other pulse. This is done in a simi- lar manner to that used for gating out the undesired pulses in the re- generative pulse generator. However, it is not necessary to use another amplifier, as was required there, since the same tube can be used for this purpose, as well as for producing the microwave pulses. Its beam forming electrode is biased negatively about 250 volts with respect to the cathode, and then is pulsed to the normal operating potential for about 50 millimicroseconds during the time of the first short pulse ob- tained by gating the helix. Thus, the beam forming electrode potential has been returned to the cutoff value during the second helix pulse, which is therefore eliminated. Il A block diagram of the resulting double-gated pulse generator is shown in Fig. 3. Comparison with Fig. 1 shows that it is simpler than the regenerative pulse generator, and it has also proved more satisfactory in operation. It can be used at any frequency where a sig- nal source and a traveling-wave amplifier are available, and the pulse Fig. 2 — Envelopes of microwave pulses at the output of a traveling wave am- lifier with continuous wave input and helix gating. The gating voltage is higher or the lower traces. 40 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 frequency can be set anywhere within the bandwidth of the travehng- wave ampUfier by tuning the klystron oscillator. The pulse center frequency is shifted from that of the klystron os- cillator frequency by this helix gating process. An over-simphfied but helpful explanation of this effect can be obtained by considering that the microwave signal voltage on the helix causes a bunching of the elec- tron stream. This^ bunching has the same periodicity as the microwave signal voltage when the dc helix potential is held constant. However, since the helix voltage is continuously increased in the positive direction during the time of the first pulse, the average velocity of the last bunches of electrons becomes higher than that of the earlier bunches in the pulse, because the later electrons come along at the time of higher positive helix voltage. This tends to shorten the total length of the series of bunches, resulting in a shorter w^avelength at the output end of the helix and therefore a higher output microwave frequency. On the second pulse, obtained when the helix voltage returns toward zero, the process is reversed, the bunching is stretched out, and the frequency is de- creased. This second pulse is, however, gated out in this arrangement by the beam-forming electrode pulsing voltage. The result for this particular tube and pulse length is an effective output frequency ap- proximately 150 mc higher than the oscillator frequency, but this figure is not constant over the range of pulse frequencies available within the amplifier bandwidth. OSCILLATOR AND CATHODE FOLLOWERS 100 KC KLYSTRON OSCILLATOR 9000 MC BEAM FORMING ELECTRODE PULSER HELIX PULSER ^ PULSED TW TUBE MILLI/aSEC 9000 MC PULSES SYNC SIGNAL TO INDICATOR SCOPE Fig. 3 — Block diugram of the double-gated traveling wave tube millimicro- second pulse generator. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 41 3. RECEIVER AND INDICATOR The receiving equipment is shown in Fig. 4. It uses two traveUng- wave amplifiers in cascade. A wide band detector and a video amplifier then follow, and the signal envelope is displayed by connecting it to the vertical deflecting plates of a 5 XP type oscilloscope tube. The video amplifier now consists of two Hewlett Packard wide band dis- tributed amplifiers, having a baseband width of about 175 mc. The second one of these has been modified to give a higher output voltage. The sweep circuits for this oscilloscope have been built especially for this use, and produce a sweep speed in the order of 6 feet per micro- second. An intensity pulser is used to eliminate the return trace. These parts of the system are controlled by a synchronizing output from the pulse generator 100-kc oscillator. A precision phase shifter is used at the receiver for the same purpose that a range unit is employed in radar systems. This has a dial, calibrated in millimicroseconds, which moves the position of a pulse appearing on the scope and makes accurate measurement of pulse delay time possible. Fig. 4 also shows the appearance of the pulses obtained with this equipment. The pulse on the left-hand side of this trace came from the PULSE SIGNAL 9000 MC SYNC SIGNAL 100 KC TW TUBES VIDEO AMPLIFIER INTENSITY PULSER 0.05/USEC 100 KC PRECISION PHASE SHIFTER SWEEP GENERATOR DOUBLE-GATED PULSE REGENERATIVE PULSE Fig. 4 — Block diagram of millimicrosecond pulse receiver and indicator. The idicator trace photograph shows pulses from each type of generator. 42 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 newer double-gated pulse generator, while the pulse on the right was produced by the regenerative pulse generator. It can be seen that they appear to have about the same pulse width and shape. This is partly due to the fact that the video amplifier bandwidth is not c^uite adequate to show the actual shape, since in both cases the pulses are slightly shorter than can be correctly reproduced through this amplifier. The ripples on the base line following the pulses are also due to the video amplifier characteristics when used with such short pulses. 4. RESOLUTION AND MEASURING RANGE Fig. 5 shows a piece of equipment which was placed between the pulse generator and the receiver to show the resolution which can be obtained. This waveguide hybrid junction has its branch marked 1 connected to the pulse generator and branch 3 connected to the receiver. If the two side branches marked 2 and 4 were terminated, substantially no energy would be transmitted from the pulser straight through to the receiver. However, a short circuit placed on either side branch will send energy through the system to the receiver. Two short circuits were so placed that the one on branch 4 was 4 feet farther away from the hybiid junc- tion than the one on branch 2. The pulse appearing first is produced l)y a signal traveling from the pulse generator to the short circuit on branch 2 and then through to the receiver, as shown by the path drawn with short dashes. A second pulse is produced by the signal which travels BRANCH 2 SHORT CIRCUIT BRANCH FROM PULSER TO RECEIVER FIRST PULSE PATH SECOND PULSE PATH SHORT CIRCUIT DOUBLE-GATED PULSES REGENERATIVE PULSES Fig. 5 — W;iv(!guicle hyhriil ciicuil- uscxl to demonstrate resululion of milli- microsecond pulses. Trace photographs of pulses from each type of generator ;iie shown. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 43 TO RECEIVER \ TE° IN 3"DIAM copper GUIDE (ISO FT LONG) Fig. 6 — Waveguide arrangement and oscilloscope trace photos showing pres- ence and location of defective joint. The dominant mode (TEn) was used with its polarization changed 90 degrees for the two trace photos. from the pulse generator through branch 4 to the short circuit and then to the receiver as shown by the long dashed line. This pulse has traveled 8 feet farther in the waveguide than the first pulse. This would be equiva- lent to seeing separate radar echoes from two targets about 4 feet apart. Resolution tests made in this way \vith the pulses from the regenerative pulse generator, and from the double-gated pulse generator, are shown on Fig. 5. With our video amplifier and viewing equipment, there is no appreciable difference in the resolution obtained using either type of pulse generator. The measuring range is determined by the power output of the gated amplifier at saturation and by the noise figure of the first tube in the receiver. In this equipment the saturation level is about 1 watt, and the noise figure of the first receiver tube is rather poor. As a result, received pulses about 70 db below the outgoing pulse can be observed, which is I enough range for many measurement purposes. 5. DOMINANT MODE WAVEGUIDE TESTS Fig. 6 shows the use of this equipment to test 3'^ round waveguides such as those installed between radio repeater equipment and an an- tenna. This particular 150-foot line had very good soldered joints and was thought to be electrically very smooth. The signal is sent in through a transducer to produce the dominant TEn mode. The receiver is con- nected through a directional coupler on the sending end to look for any 44 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 7 — Defective joint caused by imperfect soldering which gave the reflec- tion shown on Fig. 6. reflections from imperfections in the line. The overloaded signal at the left of the oscilloscope trace is produced by leakage directly through the directional coupler. The overloaded signal on the other end of this trace is produced by the reflection from the short circuit piston at the far end of the waveguide. The signal between these two, which is about 45 db down from the input signal, is produced by an imperfect joint in the waveguide. The signal polarization was oriented so that a maxi- mum reflection was obtained in the case of the lower trace. In the other trace, the polarization was changed by 90°. It is seen that this particular joint produces a stronger reflection for one polarization than for the other. By use of the precision phase shifter in the receiver the exact location of this defect was found and the particular joint that was at fault was sawed out. Fig. 7 shows this joint after the pipe had been cut in half through the middle. The guide is quite smooth on the inside in spite of the discoloration of some solder that is shown here, but on the left-hand side of the illustration the open crack is seen where the solder did not run in properly. This causes the reflected pulse that shows on the trace. The fact that this crack is less than a semi-circumference in length causes the echo to be stronger for one polarization than for the other. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 45 Fig. 8 shows the same test for a 3" diameter ahiminum waveguide 250 feet long. This line was mounted horizontally in the test building with compression couplings used at the joints. The line expanded on warm days hut the friction of the mounting supports was so great that it pulled open at some of the joints when the temperature returned to normal. These open joints produced reflected pulses from 40 to 50 db down, which are shown here. They come at intervals equal to the length of one section of pipe, about 12 feet. Some of these show polarization effects where the crack was more open on one side than on the other, but others are almost independent of polarization. These two photo- graphs of the trace were taken with the polarization changed 90°. Fig. 9 shows the same test for a 3" diameter galvanized iron wave- guide. This line had shown fairly high loss using CW for measure- ments. The existence of a great many echoes from random distances indicates a rough interior finish in the waveguide. Fig. 10 shows the kind of inperfections in the zinc coating used for galvanizing which caused these reflections. 6. TESTING ANTENNA INSTALLATIONS The use of this equipment in testing waveguide and antenna installa- tions for microwave radio repeater systems is shown in Fig. 11. This particular work was done in cooperation wdth A. B. Crawford's antenna research group at Holmdel, who designed the antenna system. A direc- tional coupler was used to observe energy reflections from the system under test. In this installation a 3" diameter round guide carrying the TEu mode was used to feed the antenna. Two different waveguide TE,, IN 3"D1AM aluminum GUIDE (250 FT LONG) Fig. 8 — Reflections from several defective joints in a dominant (TEn) mode waveguide. The two trace photos are for polarizations differing by 90 degrees. 46 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 TO RECEIVER ^n ^— a^IS^rv^ i— ^ TE ■^-^Bi 21 I 10 TE,° IN 3" DIAM GALVANIZED IRON GUIDE (250 FT LONG) Fig. 9 ■ — Multiple reflections from a dominant (TEn) mode waveguide with a rough inside surface. The two trace photos are for polarizations differing by 90 I degrees. joints are shown here. In addition, a study was being made of the re- turn loss of the transition piece at the throat of the antenna which • connected the 3" waveguide to the square section of the horn. The I waveguide sections are about 10 feet long. The overloaded pulse at the left on the traces is the leakage through the directional coupler. The Fig. 10 — Rough inside surface of a galvanized iron waveguide which produced the reflections shown on Fig. 9. I WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 47 other echoes are associated with the parts of the system from which they came by the dashed Hues and arrows on the figure. A clamped joint in the line gave the reflection shown next following the initial overloaded pulse. A well made threaded coupling in which the ends of the pipe butted squarel,y is seen to have a very much lower reflection, scarcely observable on this trace. Since there is ahvays reflection from the mouth and upper reflector parts of this kind of antenna, it is not possible to measure a throat transition piece alone by conventional CW methods, as the total reflected power from the system is measured. Here, use of the resolution of this short pulse equipment completely separated the reflection of the transition piece from all other reflections and made a measurement of its performance possible. In this particular case, the reflection from the transition is more than 50 db down from the incident signal which represents very good design. As can be seen, OPEN APERTURE FIBERGLASS COVER OVER APERTURE REFLECTION APPEARS -^TO COME FROM 16 FT N FRONT OF HORN MOUTH DIRECTIONAL TRANSDUCER CLAMPED THREADED ROUND-TO COUPLER JOINT COUPLING SQUARE TRANSITION Fig. 11 — Waveguide and antenna arrangement with trace photos showing re- flections from joints, transition section, and cover. 48 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 the reflection from the parabohc reflector and mouth is also finite low, and this characterizes a good antenna installation. The extra reflected pulse on the right of the lower trace on Fig. 11 appeared when a fiberglas weatherproof cover was installed over the open mouth of the horn. This cover by itself would normally produce a troublesome reflection. However, in this antenna, it is a continuation of one of the side walls of the horn. Consequently, outgoing signals strike it at an oblique angle. Reflected energy from it is not focused by the parabolic section back at the waveguide, so the overall reflected power in the waveguide was found to be rather low. However, measuring it with this equipment, we found that an extra reflection appeared to come from a point 16 feet out in front of the mouth of the horn when the cover was in place. This is accounted for by the fact that energy re- flected obliquely from this cover bounces back and forth inside the horn before getting back into the waveguide, thus traveling the extra distance that makes the measurement seem to show that it comes from 16 feet out in front. 7. SEPARATION OF MODES ON A TIME BASIS If a pulse of energy is introduced into a moderate length of round waveguide to excite a number of modes which travel with different group velocities, and then observed farther along the line, or reflected from a piston at the end and observed at the beginning, separate pulses will be seen corresponding to each mode that is sent. This is illustrated ! t r t t TE„ TMo,TE2, TM„ TE3, (TEoi) ^NOT EXCITED TO RECEIVER =^ ^-^ =^^ t ;ft t TMj, TE4I TE,2 TM02 TM3, AND TE5, TOO WEAK TO SHOW TE, •^^ " PROBE 3 DIAM ROUND GUIDE COUPLING (WILL SUPPORT 12 MODES) Fig. 12 — Arrangement for showing mode separation on a time basis in a multi- mode waveguide. The pulses in the trace ])]io(o have all traveled to the iiisloii and back. The earlier outgoing pulse due to direelional coupler unbalance is not shown. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 49 in Fig. 12. In this arrangement energy was sent into the round line from a probe inserted in the side of the guide. This couples to all of the 12 modes which can be supported, with the exception of the TEoi circular electric mode. The sending end of the round guide was terminated. A directional coupler is connected to the sending probe so that the return from the piston at the far end can be observed on the receiver. Because of the different time that each mode takes to travel one round trip in this waveguide, which was 258 feet long, separate pulses are seen for each mode. The pulses in this figure have been marked to show which mode is being received. The time of each pulse referred to the outgoing pulse was measured and found to check very well with the calculated time. The formula for the time of transit in the waveguide for any mode is: T = 0.98322V'1 - VnJ [where T = time in millimicroseconds L = length of pulse travel in feet Vnm ^^ A /Ac X = operating wavelength in air Ac = cutoff wavelength of guide for the mode involved. [ Table I — Calculated and Measured Value of Time for One Round Trip Time in Millimicroseconds Mode Designation Calculated Measured 1 TEn 545 545 2 TMoi 561 561 3 TE,i 587 587 4 TMn 634 634 5 TEoi 634 . 6 TE31 665 665 7 TM21 795 793 8 TE4: 835 838 9 TE12 838 10 TM„2 890 890 11 TMn 1461 — 12 TE51 1519 — 50 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The calculated and measured \'alue of time for one round trip is given in Table I. In this experiment the operating wavelength was 3.35 centimeters This was obtained by measurements based on group velocit}' in a num- ber of guides as well as information about the pulse generator com- ponents. It represents an effective wa\'elength giving correct time of travel. The pulse occupies such a wide bandwidth that a measurement of its wavelength is difficult by the usual means. The dashes in the measured column indicate that the mode was not excited by the probe or was too weak to measure. These modes do not appear on the oscilloscope trace photograph. The relative pulse heights can be calculated from a knowledge of the probe coupling factors and the line loss. The probe coupling factors as given by M. Aronoff in unpublished work are expressed by the following For TE„„, modes: P = 2.390 r—^ i For TM„^ modes: TV- L a -. j\. nm ^ "flu X X P = 1.195€„ — - where P = ratio of probe coupling power in mode nm to that in mode TEn n = first index of mode being calculated Knm = Bessel function zero value for mode being calculated = Td/\c X = wavelength in air X(, = wavelength in the guide for the mode involved ' Xc = cutoff wavelength of guide for the mode involved €„ = 1 for w = 0 €„ = 2 for n ?^ 0 , d = waveguide diameter Formulas for guide loss as given by S. A. Schelkunoff on page 390 of his book Elect romagnelir Waves for this case where the resistivity of the aluminum guide is 4.14 X 10~^ ohms per cm cube are: WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 51 For TE„,„ modes: a = 3.805 ! — 2 2 + V.an ) (1 " Vnm) \l\n,n — n / For TM„,„ modes: a = 3.805(1 - VnJy''' where: a = attemiation of this aluminum guide in db n — first index of mode being calculated Knm — Bessel function zero value for mode being calculated = TrtZ/Xc Vnm = A/Ac X = operating wavelength in air Xc = cutoff wavelength of guide for the mode involved d = waveguide diameter Table II gives the calculated probe coupling factor, line loss, and rela- tive pulse height for each mode. In the calculation of the latter, wave elUpticity and loss due to mode conversion were neglected, but the heat loss given by the preceding formulas has been increased 20 per cent for all modes, to take account of surface roughness. Relative pulse heights were obtained by subtracting the relative line loss from twice the rela- tive probe coupling factor. The relative line loss is the number in the itable minus 2.33 db, the loss for the TEn mode. The actual pulse heights on the photo of the trace on Fig. 12 are in fair agreement with these calculated values. Differences are probably due to polarization rotation in the guide (wave ellipticity) and conver- sion to other modes, effects which were neglected in the calculations, and which are different for different modes. Calculated pulse heights with this guide length, except for modes near cutoff, vary less than the probe coupling factors, because line loss is high when tight probe coupling exists. This is to be expected, since both are the result of high fields near the guide walls. The table of round trip travel time shows that the TE41 and TE12 modes are separated by only three millimicroseconds after the round trip in this waveguide. They would not be resolved as separate pulses by this e(iuipment. However, the table of calculated pulse heights shows that the TE41 pulse should be about 22 db higher than the TE12 pulse. 52 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table II — Calculated Probe Coupling Factor, Line Loss and Pulse Height for Each Mode Mode Mode Relative Probe 1.2 X Theoretical Calculated Relatix e Number Designation Coupling Factor, db Line Loss, db Pulse Heights, db 1 TEu 0 2.33 0 2 TMoi +0.32 4.88 -1.91 3 TE2, +2.86 4.85 +3.20 4 TMu +2.80 5.51 +2.42 5 TEo, — 00 1.73 — 00 6 TE31 +4.82 8.21 +3.76 7 TM2, + 1.82 6.92 -0.95 8 TE41 +6.80 13.86 +2.07 9 TE12 -8.73 4.70 -19.83 10 TM02 -1.68 7.74 -8.77 11 TMsi -0.82 12.71 -12.02 12 TE51 + 10.14 32.09 -9.48 Since the TE12 pulse is so weak, it would not show on the trace even if it were resolved on a time basis. Coupling to the TM02 mode is rather weak, and the gain was increased somewhat at its position on the trace to show its time location. 8. DELAY distortion Another effect of the wide bandwidth of the pulses used with this equipment can be observed in Fig. 12. The pulses that have traveled for a longer time in the guide are in the modes closer to cutoff, and are on the right-hand side of the oscilloscope trace. They are broadened and distorted compared with the ones on the left-hand side. This effect is due to delay distortion in the guide. This can be explained by refer- ence to Fig. 13. On this figure the ratio of group velocity to the velocity in an unbounded medium is shown plotted as a function of frequency for each of the modes that can be propagated. The bandwidth of the transmitted pulse is indicated by the vertical shaded area. It will he noticed that the spacing of the pulses on the oscilloscope trace on Fig. 12 from left to right in time corresponds to the spacing of the group velocity curves in the bandwidth of the pulse from top to bottom. De- lay distortion on these curves is shown by the slope of the line across the pulse bandwidth. If the line were horizontal, showing the same group velocity at all points in the band, there would be no delay distortion. The greater the difference in group A-elocity at the two edges of the band, the greater the delay distortion. The curves of Fig. 13 indicate WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 53 I that there should be increasing amounts of delay distortion reading ifrom top to bottom for the pulse bandwidth used in these experiments. ;The effect of this delay distortion is to cause a broadening of the pulse. Examination of the pulse pattern of Fig. 12 shows that the later pulses corresponding in mode designation to the lower curves of Fig. 13 do in- deed show a broadening due to the increased delay distortion. One method of reducing the effect of delay distortion is to use frequency division multiplex so that each signal uses a smaller bandwidth. Another way, suggested by D. H. Ring, is to invert the band in a section of the waveguide between one pair of repeaters compared with that between an adjacent pair of repeaters so that the slope is, in effect, placed in the opposite direction, and delay distortion tends to cancel out, to a first order at least. The (luantitative magnitude of delay distortion has been expressed by S. Darlington in terms of the modulating base-band frequency needed to generate two side frequencies which suffer a relative phase error of 180° in traversing the line. This would cause cancellation of a single frequency AM signal, and severe distortion using any of the 1.0 PULSE BANDWIDTH — >. <— ^^ ^— UJ ^0.9 Q. o ^ 0.5 >- o 3 0.4 m > ^0.3 o (r o ^0.2 o io., 0 ■^^H;;^ . — /^ o^^ ^ ^ ^ ^^ / / y \a X y / ^ ^ / / / 6 ^ / 7 / f // < f/> \ ^-'^'fA y^. / / 1 / 1 'L f4 // 1 / / 1 1 /A '/ // // L ^i / 1 7 \\ 1 3 4 FREQUENCY 5 6 7 8 9 10 IN KILOMEGACYCLES PER SECOND 12 1 Fig. 1.3 — Theoretical group velocity vs. frequency curves for the 3" diameter ivaveguide used for the tests shown on Fig. 12. The vertical shaded area gives the bandwidth for the millimicrosecond pulses employed in that arrangement. 54 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 ordinary modulation methods. Darlington gives this formula: ^) ^^^^ iLLi/ Vnm where : jB = base bandwidth for 180° out of phase sidebands / = operating frequency (in same units as jB) X = wavelength in air L = waveguide length (in same units as X) Vnm = X/Xe Xc = cutoff wavelength for the mode involved With this equipment, the base bandwidth of the pulse is about 175 mc, and when/5 from the formula above is about equal to or less than this, pulse distortion should be observed. The following Table III gives fB calculated from this formula for the arrangement shown on Fig. 12. It is interesting to note that pulses in the TMu and TE31 modes, for which jB is less than the 175-mc pulse bandwidth, are broadened, but not badly distorted. For the higher modes, where jB is much less than 175 mc, broadening and severe distortion are evident. Another example is given in the next section. 9. DELAY DISTORTION EQUALIZATION If the distance which a pulse travels in a waveguide is increased, its delay distortion also increases. Since the group velocity at one edge of the band is different than at the other edge of the band, the amount by which the two edges get out of phase with each other increases with the total length of travel, causing increased distortion and pulse broaden- ing. The Darlington formula in the previous section shows that jB varies inversely as the square root of the length of travel. This efTect is shown on Fig. 14. In this arrangement the transmitter was connected to the end of a 3" diameter round waveguide 107 feet long through a small hole in the end plate. A mode filter was used so that only the TEoi mode would be transmitted in this Avaveguide. Through another small hole in the end plate polarized 90° from the first one, and rotated 90° around tlu^ plate, a directional coupler was connected as shown. The direct through guide of this directional coupler could be short cir- cuited with a waveguide shorting switch. Energy reflected from this fl WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 55 Table III - — Calculatee > Values of fB foe the Arrangement Shown in Fig. 1 2 Mode Number Mode Designation /B Megacycles Remarks 1 TEn 324.0 2 TMoi 237.7 3 TEn 174.9 4 TMu 124.1 5 TEoi 124.1 Not excited 6 TE31 105.2 7 TMoi 65.9 8 TE41 59.1 9 TEi, 58.6 Veiy weakly excited 10 TMoo 51.8 11 TM3: 21.3 Not observed 12 TE51 20.0 Not observed NUMBER OF R( 3UND TRIPS TAPERED DELAY DISTORTION EQUALIZER WAVEGUIDE SHORTING SWITCH 1/ 'M ^ te; >T0 RECEIVER NOT EQUALIZED (SWITCH CLOSED) EQUALIZED (SWITCH OPEN) TEqiIN 3 DIAM ROUND GUIDE (107 FT LONG) Fig. 14 — The left-hand series of pulses shows the build up of delay distortion with increasing number of round trips in a long waveguide. The right-hand series shows the im]irovement obtained with the tapered delay distortion equalizer shown at the right. 56 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 switch was then taken through the directional coupler to the receiver as shown by the output arrow. The series of pulses at the left-hand photograph of the oscilloscope traces was taken with this waveguide i shorting switch closed. The top pulse shows the direct leakage across the inside of the end plate before it has traveled through the 3" round guide. The next pulse is marked one round trip, having gone therefore 214 feet in the TEoi mode in the round waveguide. The successive pulses have traveled more round trips as shown by the number in the center between the two photographs. The effect of increased delay distortion broadening and distorting the pulse can be seen as the numbers increase. The values of fB from the Darlington formula in the previous section for these lengths are given in Table IV. It will be noticed that pulse broadening, and eventually severe dis- tortion, occurs as fB decreases much below the 175-mc pulse band- width. The effect is gradual, and not too bad a pulse shape is seen until fB is about half the pulse bandwidth, although broadening is very evident earlier. When the waveguide short-circuiting switch was opened so that the tapered delay distortion equalizer was used to reflect the energy, in- stead of the switch, the series of pulses at the right was observed on the indicator. It will be noted that there is much less distortion of these, pulses, particularly toward the bottom of the series. The ones at the top, have less distortion than would be expected, probably because of fre-, quency modulation of the injected pulse. The equalizer consists of a long gradually tapered section of waveguide which has its size reduced to a point beyond cutoff for the frequencies involved. Reflection takes place at the point of cutoff in this tapered guide. For the high frequency part of the pulse bandwidth, this point is farther away from the short- ing switch than for the low frequency part of the bandwidth. Conse- quently, the high frequency part of the pulse travels farther in one round trip into this tapered section and back than the low frequency part of Table IV — Values of fB from the Darlington Formula FOR the Arrangement Show^n in Fig. 14 li Round Trip Number JB Megacycles Round Trip Number fB Megacycles 1 2 3 4 5 185.8 131.4 107.3 92.9 83.1 6 7 8 9 10 75.8 70.2 65.7 61.9 58.7 j 1 1 WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 57 he pulse. This increased time of travel compensates for the shorter ime of travel of the high frequency edge of the band in the 3" round .vaveguide, so equalization takes place. Since this waveguide close to cutoff introduces considerable delay distortion by itself, the taper effect nust be made larger in order to secure the equalization. This can be ilone by making the taper sufficiently gradual. This type of equalizer ntroduces a rather high loss in the system. For this reason it might le used to predistort the signal at an early level in a repeater system, ilqualization by this method was suggested by J. R. Pierce. .0. MEASURING MODE CONVERSION FROM ISOLATED SOURCES I One of the important uses of this equipment has been for the meas- irement of mode conversion. W. D. Warters has cooperated in develop- ng techniques and carrying out such measurements. One of the prob- ems in the design of mode filters used for suppressing all modes except ;he circular electric ones in round multimode guides is mode conversion. Since these mode filters have circular symmetry, conversion can take alace only to circular electric modes of order higher than the TEoi mode. This conversion is, however, a troublesome one, since these higher Drder circular modes cannot be suppressed by the usual type of filter. An arrangement for measuring mode conversion at such mode filters rom the TEoi to the TE02 mode is being used with the short pulse equip- :nent. This employs a 400-foot long section of the b" diameter line. Be- ause the coupled- line transducer available had too high a loss to TE02 , a 3ombined TEoi — TE02 transducer was assembled. It uses one-half of :he round waveguide to couple to each mode. Fig. 15 shows this device. The use of this transducer and line is illustrated in Fig. 16. Pulses in :he TEoi mode are sent into the waveguide by the upper section of the transducer as shown. Some of the TEoi energy goes directly across to ohe TE02 transducer and appears as the outgoing pulse with a level down about 32 db. This is useful as a time reference in the system and s shown as the outgoing pulse in the photo of the oscilloscope trace ibove. The main energy in the TEoi mode propagates down the line as hown by dashed line 2, which is the path of this wave. Most of ohis energy goes all the way to the reflecting piston at the far end and ohen returns to the TE02 transducer where it gives a pulse which is narked TEoi round trip on the trace photograph above. Two thirds of ;he way from the sending end to the piston, the mode filter being meas- ired is inserted in the line. When the TEoi mode energy comes to this node filter, a small amount of it is converted to the TE02 mode. This 58 THE BELL SYSTEM TECHNICAL JOURNAL Fig. 15 — A special experimental transducer for injecting the TEoi mode and' receiving the converted TE02 mode in a 5" diameter waveguide. continues to the piston by path 4 (with dashed Hnes and crosses) and then returns and is received by the TE02 part of the transducer. This appears on the trace photo as the TE02 first conversion. When the main TEoi energy reflected by the piston comes back to the mode filter, conversion again takes place to TE02 • This is shown by path 3 hav- ing dashed lines and circles. This returns to the TE02 part of the trans- ducer and appears on the trace photo as the TE02 second conversion. In addition, a small amount of energy in the TE02 mode is generated by the TEoi upper part of the transducer. It is shown by path 5, having' OUTGOING PULSE TEoi ROUND TRIP TE02 SECOND CONVERSION TE02 FIRST CONVERSION TE02 ROUND TRIP ' MODE FILTER Fig. 16 — Trace photos and waveguide paths traveled when measuring TEoi, to TE02 mode conversion at a mode filter with the transducer shown on Fig. 15 All WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 59 jihort dashes. This goes down through the waveguide to the far end Ijiston and back, and is received by the TE02 transducer and shown as [he pulse marked TE02 round trip. The pulse marked TEoi round trip las a time separation from the outgoing pulse which is determined by ,he group velocity of TEoi waves going one round trip in the guide. The |rEo2 round trip pulse appears at a time corresponding to the group /elocity of the TE02 mode going one round trip in the guide. Spacing the node filter two-thirds of the way down produces the two conversion :)ulses equally spaced between these two as shown in Fig. 16. The first ponversion pulse appears at a time which is the sum of the time taken or the TEoi to go down to the filter and the TE02 to go from the filter uo the piston and back to the receiver. Because of the slower velocity bf the TE02 , this appears at the time shown, since it was in the TE02 node for a longer time than it was in the TEoi mode. The second con- [/ersion, which happened when TEoi came back to the mode filter, comes jiarlier in time than the first conversion, since the path for this signal ivas in the TEoi mode longer than it was in the TE02 mode. This arrange- ,nent gives very good time separation, and makes possible a measure- Inent of the amount of mode conversion taking place in the mode filters, viode conversion from TEoi to TE02 as low as 50 to 55 db down, can be neasured with this equipment. Randomly spaced single discontinuities in long waveguides can be ocated by this technique if they are separated far enough to give in- lividually resolved short pulses in the converted mode. Fig. 17 shows CONVERSION FIRST CONVERSION AT FAR END SECOND CONVERSION AT NEAR END SQUEEZED AT NEAR END SQUEEZED SECTION SECTION SQUEEZED SECTION TO RECEIVER TEJo — *- TEq, TE2, -• »- TE,o NEAR END 250 FT OF FAR END TRANSDUCER COUPLED LINE SQUEEZED 3"DIAM ROUND SQUEEZED TRANSDUCER SECTION GUIDE SECTION Fig. 17 — Arrangement used to explain the measurement and location of mode onversion from isolated sources. A deliberately squeezed section was placed t each end of the long waveguide, producing the pulses shown in the trace photo. 60 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 an arrangement having oval sections deliberately placed in the wave- ' guide in order to explain the method. Pure TEoi excitation is vised, and the converted TE21 mode observed with a coupled line transducer giv- ; ing an output for that mode alone. ; Let us consider first what would happen with the far-end squeezed; section alone, omitting the near-end squeezed section from considera- • tion. The injected TEoi mode signal would then travel down the 250 , feet of 3" diameter round waveguide to the far end with substantially, no mode conversion at the level being measured. At this point it goes through the squeezed section. Conversion now takes place from the TEou, mode to the TE21 mode. Both these modes after reflection from the piston travel back up the waveguide to the sending end. The group velocity of the TE21 mode is higher than the group velocity of the TEoi mode, so energy in these two modes separates, and if a coupling system were used to receive energy in both modes, two pulses would appear, with at time separation between them. In this case, since the receiver is con- nected to the line through the coupled line transducer which is responsive only to the TE21 mode, only one pulse is seen, that due to this mode alone. This is the center pulse in the trace photograph at the top of Fig. 17. If only one mode conversion point at the far end of the guide exists, only this one pulse is seen at the receiver. It would be spaced a distance away from the injected outgoing pulse that corresponds m:^ time to one trip of the TEoi mode down to the far end and one trip of || the TE21 mode from the far end back to the receiver. Now let us consider what would happen if the near-end squeezed sec- tion alone were present. When the TEqi wave passes the oval section! just beyond the coupled line transducer, conversion takes place, andi the energy travels down the line in both the TEoi and the TE21 modes,:; at a higher group velocity in the TE21 mode. These two signals are re- flected by the piston at the far end and return to the sending end. The TE21 signal comes through the coupled line transducer and appears as the pulse at the left of the photo shown on Fig. 17. Now the TEoi energy has lagged behind the TE21 energy, and when it gets back to the near- end squeezed section, a second mode conversion takes place, and TE21 mode energy is produced which comes through the coupled line trans-: ducer and appears at the receiver at the time of the right hand pulse. The spacing between these two pulses is equal to the difference in round trip times between the two modes. In general, for a single conversion source occurring at any point in the line, two pulses will appear on the scope. The spacing between these pulses corresponds to the difference in group velocity between the modes. WAVEGUIDE TESTING AVITH MILLIMICROSECOND PULSES 61 {from the point of the discontiimity down to the piston at the far end, land then back to the discontinuity. If the discontinuity is at the far lend, this time difference becomes zero, and a single pulse is seen. By i [making a measurement of the pulse spacing, the location of a single i icon version point can be determined. [ In the arrangement illustrated in Fig. 17, two isolated sources of j conversion existed. They were spaced far enough apart so that they \ were resolved by this equipment, and all three pulses were observed. The two outside pulses were due to the first conversion point. The center pulse was caused by the other squeeze, which was right at the reflecting |:)iston. If this conversion point had been located back some distance rom the piston, it would have produced two conversion pulses whose 'spacing could be used to determine the location of the conversion point. I The coupled-line transducers are calibrated for coupling loss by send- ng the pulse through a directional coupler into the branch normally ised for the output to the receiver. This gives a return loss from the lirectional coupler equal to twice the transducer loss plus the round rip line loss. 1. MEASURING DISTRIBUTED MODE CONVERSION IN LONG WAVEGUIDES ; Measurements of mode conversion from TEoi to a number of other nodes have been made with 5" diameter guides using this equipment, rhe arrangement of Fig. 18 was set up for this purpose. This is the same IS Fig. 17, except that a long taper was used at the input end of the 5" waveguide, and a movable piston installed at the remote end. One of the converted modes studied with this apparatus arrange- uent was the TMu mode, which is produced by bends in the guide, rhis mode has the same velocity in the waveguide as the TEoi mode. Therefore energy components converted at different points in the line tay in phase with the injected TEoi mode from which they are converted, rhere is never any time separation between these modes, and a single TO RECEIVER ■ I ■■'■ '■■■ ^ ^^i Si TEro— *TE^, COUPLED LINE -^^p^P, 5„ 0,^^^ MOVABLE TRANSDUCER TRANSDUCER HOLMDEL LINE PISTON FOR THE MODE BEING MEASURED Fig. 18 — Arrangement used for measuring mode conversion in the 5" diameter aveguides at Holmdel. 62 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 narrow pulse like the transmitted one is all that appears on the indicator oscilloscope. It is not possible from this to get any information about the location or extent of the conversion points in the line. Moving the far end piston does not change the relative phases of the modes, so no changes are seen in indicator pattern or pulse level as the piston is moved. For the Holmdel waveguides, which are about 500 feet long, the total round trip T]\In mode converted level varies from 32 to 36 db below the input TEoi mode level over a frequency range from 8,800 to 9,600 mc per second. All the other modes have velocities that are different than that of the TEoi mode. ^Vhen mode conversion takes place at many closely spaced points along the waveguide, the pulses from the various sources overlap, and phasing effects take place. In general, a filled-in pulse much longer than the injected one is observed. The maximum possible, but not necessary, pulse length is equal to the difference in time re- quired for the TEoi mode and the converted mode to travel the total waveguide length being observed. The phasing effects within the broad- ened pulse change its height and shape as a function of frequency and line length. Measurements of mode conversion from TEoi to TE31 in these wave- guides illustrate distributed sources and piston phasing effects. The TE3, mode has a group velocity 1.4 per cent slower than the TEoi mode. For a full round trip in the 500-foot lines, assuming conversion at the imput end, this causes a time separation of about two and one half pulse widths between these two modes. The received pulse is about two and a half times as long as the injected pulse, indicating rather closely spaced sources over the whole line length. For one far-end piston posi- tion, the received pattern is shown as the upper trace in Fig. 19. As the piston is moved, the center depressed part of the trace gradually ImK. 10 — Hocoivcd pulsr patterns willi llic .irraiijicnuMit of Fig. IS used for studying conversion to tlie Tlvn mode. WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 63 rises until the pattern shown in the lower trace is seen. As the piston is moved farther in the same direction the trace gradually changes to have the appearance of the upper photo again. Moving the far-end piston changes the phase of energy on the return trip, and thus it can be made to add to, or nearly cancel out, conversion components that originated ahead of the piston. When the time separation becomes great enough to prevent overlapping in the pulse ^^^dth, phasing effects cannot take place, therefore, the beginning and end of the spread-out received pulse are not affected by moving the piston. Energy converted at the sending end of the guide travels the full round trip to the piston and back in the slower TE31 mode, and thus appears at the latest time, which is at the right-hand end of the received pulse. Conversion at the piston end returns at the center of the pulse, and conversion on the return trip comes at earlier times, at the left-hand part of the pulse. The TEoi mode has less loss in the guide than the TE31 mode. Since the energy in the earlier part of the received pulse spent a greater part of the trip in the lower loss TEoi mode before conversion, the output is higher here, and slopes off toward the right, where the later returning energy has gone for a longer distance in the higher loss mode. The pulse height at the maximum shows the converted energy from that part of the line to be between 30 and 35 db below the incident TEoi energy level over the measured band\\ddth. Measurements of mode conversion from TEoi to TE21 in these wave- guides show these same effects, and also a phasing effect as a function of frequency. The TE21 mode has a group velocity 2.4 per cent faster than the TEoi mode. For a full round trip in the guides, this is a time separation of about four pulse mdths between the modes. At one fre- quency and one far-end piston position, the TE21 response shown as the top trace of Fig. 20 was obtained. Moving the far-end piston gradually changed this to the second trace from the top, and further piston mo- tion changed it back again. This is the same kind of piston phasing effect observed in the TE31 mode conversion studies. The irregular top of this broadened pulse indicates fewer conversion points than for the TE31 mode, or phasing effects along the guide length. Since the TE21 mode has a higher group velocity' than the TEoi mode, energy converted at the beginning of the guide returns at the earlier or left-hand part of the pulse, and conversions on the return trip, having traveled longer in the slower TEoi mode, are on the right-hand side of the pulse. This is just the reverse of the situation for the TE31 mode. Since the loss in the TE21 mode is higher than in the TEoi mode, the right side of this broad- ened pulse is higher than the left side, as the energy in the left side has 64 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 gone further in the higher loss TE21 mode. Conversions from the piston end of the guide return in the center of the pulse, and only in this re- gion do piston phasing effects appear. As the frequency is changed the ' pattern changes, until it reaches the extreme shape shown in the next- to-the-bottom trace, with this narrower pulse coming at a time corre- sponding to the center of the broadened pulse at the top. Further fre- quency change in the same direction returns the shape to that of the top traces. At the frequency giving the received pulse shown on the next-to-the-bottom trace, moving the far-end piston causes a gradual change to the shape shown on the lowest trace. This makes it appear as if the mode conversion were coming almost entirely from the part of the guide near the piston end at this frequency. The upper traces appear to show that more energy is converted at the transducer end of the waveguide at that frequency. It would seem that at certain frequencies some phase cancellation is taking place between conversion points spaced closely enough to overlap within the pulse width . At frequencies between the ones giving traces like this, the appearance is more like that shown for the TE31 mode on Fig. 19 except for the slope across the top of the pulse being reversed. The highest part of this TEoi pulse is Fiff. 20 — Received pulse patterns witli the urrangemeiit of Fig. 18 used for studying conversion to the TE21 mode. WAVEGUIDE TESTING WITH MILLIMICKOSECOND PULSES 65 24 to 27 db below the injected TEoi pulse level for the 5" diameter Holmdel waveguides. 12. CONCLUDING REMARKS The high resolution obtainable with this millimicrosecond pulse equipment provides information difficult to obtain by any other means. These examples of its use in waveguide investigations indicate the possibilities of the method in research, design and testing procedures. It is being used for many other similar purposes in addition to the illus- tratio)is given here, and no doubt many more uses will be found for such short pulses in the future. REFERENCES 1. S. E. Miller and A. C. Beck, Low-loss Waveguide Transmission, Proc. I.R.E., 41, pp. 348-358, March, 1953. 2. S. E. Miller, Waveguide As a Communication Medium, B. S. T. J., 33, pp. 1209- 1265, Nov., 1954. 3. C. C. Cutler, The Regenerative Pulse Generator, Proc. I.R.E., 43, pp. 140- 148, Feb., 1955. 4. S. E. Miller, Coupled WaveTheory and Waveguide Applications, B. S. T. J., 33, pp. 661-719, May, 1954. Experiments on the Regeneration of Binary Microwave Pulses By O. E. DeLANGE (Manuscript received September 7, 1955) A sifnple device has been produced for regenerating binary pulses directly at microwave frequencies. To determine the capabilities of such devices one of them was included in a circidating test loop in which pidse groups were passed through the device a large number of titnes. Residts indicate that even in the presence of serious noise and bandwidth limitations pidses can be regenerated many times and still shotv no noticeable deterioration. Pic- tures of circulated pidses are included which illustrate performance of the regenerator. INTRODUCTION The chief advantage of a transmission system employing Ijinary pulses resides in the possibility of regenerating such pulses at intervals along the route of transmission to prevent the accumulation of distortion due to noise, bandwidth limitations and other effects. This makes it possible to take the total allowable deterioration of signal in each section of a long relay system rather than having to make each link sufficiently good to prevent total accumulated distortion from becoming excessive. This has been pointed out by a number of writers. i-- W. M. GoodalP has shown the feasibility of transmitting television signals in binary form. Such transmission reciuires a considerable amount of bandwidth; a seven digit system, for example, would require trans- mission of seventy million pulses per second. This need for wide bands makes the microwave range an attractive one in which to work. S. E. Miller* has pointed out that a binary system employing regeneration might prove to be especially advantageous in waveguide transmission. 1 B. M. Oliver, J. R. Pierce and, C. E. Shannon, The Pliilosophv of PCM, Proc. I. R.E., Nov., 1948. '^ L. A. Meacham and Iv Peterson, An Experimental Multichannel Pulse Code Modulation System of Toll Quality, B. S. T. J., Jan. 1948. ' W. M. Goodall, Television l)y Pulse Code Modulation, B. S. T. J., Jan., 1951. * S. E. Miller, Waveguide as a Communication Medium, B. S. T. J., Nov., 1954. 67 68 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 INPUT FILTER AUTOMATIC GAIN CONTROL REGENERATOR DETECTOR TIMING WAVE GENERATOR FILTER OUTPUT Fig. 1 — A typical regenerative repeater shown in block form. That the Bell System is interested in the long-distance transmission of television and other broad-band signals is evident from the number of miles of such broad-band circuits, both coaxial cable and microwave radio, ^ now in service. These circuits provide high-grade transmission because each repeater was designed to have a very fiat frequency charac- teristic and linear phase over a considerable bandwidth. Furthermore, these characteristics are very carefully maintained. For a binary pulse system employing regeneration the requirements on flatness of band and linearity of phase can be relaxed to a considerable degree. The compo- nents for such a system should, therefore, be simpler and less expensive to build and maintain. Reduced maintenance costs might well prove to be the chief virtue of the binary system. Since the chief advantage of a binary system lies in the possibility of regeneration it is obvious that a very important part of such a system is the regenerative repeater employed. Fig. 1 shows in block form a typical broad-band, microwave repeater. Here the input, which might come from either a radio antenna or from a waveguide, is first passed through a proper microwave filter then amplified, probably by a traveling-wave amplifier. The amplified pulses of energy are regenerated, filtered, am- plified and sent on to the next repeater. The experiment to be described here deals primarily with the block labeled "Regenerator" on Fig. 1. In these first experiments one of our main objectives was to keep the repeater as simple as possible. This suggests regeneration of pulses directly at microwave frequency, which for this experiment was chosen to be 4 kmc. It was suggested by J. R. Pierce and W. D. Lewis, both of Bell Telephone Laboratories, that further simplification might be made possible by accepting only partial instead of complete regeneration. This suggestion was adopted. For the case of complete regeneration each incoming pulse inaugurates a new pulse, perfect in shape and correctly timed to be sent on to the 'A. A. Roetken, K. D. Smith and R. W. Friis, The TD-2 System, B. S. T. J., Oct., 1951, Part II. REGENERATION OF BINARY MICROWAVE PULSES 69 next repeater. Thus noise and other disturbing effects are completely eliminated and the output of each repeater is identical to the original signal which entered the system. For the case of partial regeneration incoming pulses are retimed and reshaped only as well as is possible with simple equipment. Obviously the difference between complete and partial . regeneration is one of degree. One object of the experiment was to determine how well such a partial regenerator would function and what price must be paid for employing partial instead of complete regeneration. The regenerator developed consists simply of a waveguide hybrid junction with a silicon crystal diode in each side arm. It appears to meet the requirement of simplicity in that it combines the functions of amplitude slicing and pulse retiming in one unit. A detailed description of this unit will be given later. Al- though the purpose of this experiment was to determine what could be accomplished in a very simple repeater we must keep in mind that superior performance would be obtained from a regenerator which ap- proached more nearly the ideal. For some applications the better re- generator might result in a more economical system even though the regenerator itself might be more complicated and more expensive to produce. METHOD OF TESTING The regeneration of pulses consists of two functions. The first function is that of removing amplitude distortions, the second is that of restoring each pulse to its proper time. The retiming problem divides into two [parts the first of which is the actual retiming process and the second ! that of obtaining the proper timing pulses with which to perform this lifunction. In a practical commercial system timing information at a [repeater would probably be derived from the incoming signal pulses. There are a number of problems involved in this recovery of timing pulses. These are being studied at the present time but were avoided in the experiment described here by deriving such information from the local synchronizing gear. Since the device we are dealing with only partially regenerates pulses it is not enough to study the performance of a single unit — we should •like to have a large number operating in tandem so that we can observe 'what happens to pulses as they pass through one after another of these Tegenerators. To avoid the necessity of building a large number of units the pulse circulating technique of simulating a chain of repeaters was j employed. Fig. 2 shows this circulating loop in block form. 70 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 HYBRID JUNCTION ' NO. 3 CW OSCILLATOR (4 KMC) TRAVELING WAVE AMPLIFIER (NOISE GENERATOR) Fig. 2 — The circulating loop. To provide RF test pulses for this loop the output of a 4 kmc, cw oscillator is gated by baseband pulse groups in a microwave gate or modulator. The resultant microwa\-e pulses are fed into the loop (heavy line) through hybrid junction No. 1. They are then amplified by a trav- eling-wave amplifier the output of which is coupled to the pulse regen- erator through another hybrid junction (No. 2). The purpose of this hybrid is to provide a position for monitoring the input to the regen- erator. A monitoring position at the output of the regenerator is pro- vided by a third hybrid, the main output of which feeds a considerable length of waveguide which provides the necessary loop delay. At the far end of the waveguide another hybrid (No. 4) makes it possible to feed noise, which is derived from a traveling-wave amplifier, into the loop. The combined output after passing through a band pass filter is ampli- REGENEKATION OF BINARY MICROWAVE PULSES 71 fied by another traveling-wave amplifier and fed back into the loop in- put thus completing the circuit. The synchronizing equipment starts out with an oscillator going at approximately 78 kc. A pulse generator is locked in step with this os- cillator. The output of the pulser is a negative 3 microsecond pulse as shown in Fig. 3A. After being amplified to a level of about 75 volts this pulse is applied to the helix of the first traveling-wave tube to re- I duce the gain of this tube during the 3-microsecond interval. Out of each 12.8/xsec interval pulses are allowed to circulate for O.S/xsec but are blocked I for the remaining 3Msec thus allowing the loop to return to the quiescent i condition once during each period as shown on Figs. 3A and 3C. The S^sec pulse also synchronizes a short-pulse generator. This unit delivers pulses which are about 25 millimicroseconds long at the base and spaced by 12.8/isec, i.e., Avith a repetition frequency of 78 kc. See Fig. 3B. In order to simulate a PCM system it was decided to circulate pulse CIRCULATING INTERVAL 9.8/ZS QUENCHING INTERVAL -3//S-*| (A) GATING CYCLE (B) SHORT SYNCHRONIZING PULSES --24 GROUPS OF PULSES (C) CIRCULATING PULSE GROUPS GROUP GROUP GROUP 1 2 3 lOOMyUS ^ k ^^-o.4;uS-^^ I (D) PULSE GROUPS (EXPANDED) ■ ' |300M/US| I I I (E) TIMING WAVE (40MC) EXPANDED 0 TIME Fig. 3 — Timing events in the circulating loop. 72 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 groups rather than individual pulses through the system. These were derived from the pulse group generator which is capable of delivering any number up to 5 pulses for each short input pulse. These pulses are about 15 milli-microseconds long at the base and spaced 25 milli-micro- seconds apart. The amplitude of each of these pulses can be adjusted independently to any value from zero to full amplitude making it pos- sible to set up any combination of the five pulses. These are the pulses which are used to gate, or modulate, the output of the 4-kmc oscillator. The total delay around the waveguide loop including TW tubes, etc.,' was 0.4)usec or 400 milli-microseconds. This was sufficient to allow time between pulse groups and yet short enough that groups could circulate 24 times in the available 9.8jLtsec interval. This can be seen from Figs. 3C and 3D. The latter figure shows an expanded view of circulating pulse groups. The pulses in Group 1 are inserted into the loop at the beginning of each gating cycle, the remaining groups result from circu- lation around the loop. When all five pulses are present in the pulse groups the pulse repeti- tion frequency is 40 mc. (Pulse interval 25 milli-microseconds). For this condition timing pulses should be supplied to the regenerator at the rate of 40 million per second. These pulses are supplied continuously and not in groups as is the case with the circulating pulses. See Fig. BE. In order to maintain time coincidence between the circulating pulses and the tim- ing pulses the delay around the loop must be adjusted to be an exact multiple of the pulse spacing. In this experiment the loop delay is equal to 16-pulse intervals. Since timing pulses are obtained by harmonic generation from the quenching frequency as will be discussed later this frequency must be an exact submultiple of pulse repetition frequency. In this experiment the ratio is 512 to 1. Although the above discussion is based on a five-pulse group and 40-mc repetition frequency it turned out that for most of the experi- ments described here it was preferable to drop out every other pulse, leaving three to a group and resulting in a 20-mc repetition frequency. The one exception to this is the limited-band-width experiment which will be described later. - For all of the experiments described here timing pulses were derived from the 78-kc quenching frequency by harmonic generation. A pulse with a width of 25 milli-microseconds and with a 78-kc repetition fre- quency as shown in Fig. 3B supplied the input to the timing wave gen- erator. This generator consists of several stages of limiting amplifiers all tuned to 20 mc, followed by a locked-in 20-mc oscillator. The output of the amplifier consists of a train of 20-mc sine waves with constant ampli- til REGENERATION OF BINARY MICROWAVE PULSES 73 tude for most of the 12.8Msec period but falling off somewhat at the end of the period. This-train locks in the oscillator which oscillates at a con- stant amplitude over the whole period and at a frequency of 20 mc. Timing pulses obtained from the cathode circuit of the oscillator tube pro^'ided the timing waves for most of the experiments. For the experi- ment where a 40-mc timing wave was required it was obtained from the, 20 mc train by means of a frequency doubler. For this case it is necessary for the output of the timing wave generator to remain constant in ampli- tude and fixed in phase for the 512-pulse interval between synchronizing pulses. In spite of the stringent requirements placed upon the timing equip- ment it functioned well and maintained synchronism over adequately long periods of time without adjustment. PERFORMANCE OF REGENERATOR Performance of the regenerator under various conditions is recorded on the accompanying illustrations of recovered pulse envelopes. The first experiment was to determine the effects of disturbances which arise at only one point in a system. Such effects were simulated by adding disturbances along with the group of pulses as they were fed into the circulating loop from the modulator. This is equivalent to having them occur at only the first repeater of the chain. Some of the first experiments also involved the use of extraneous pulses to represent noise or distortion since these pulses could be syn- chronized and thus studied more readily than could random effects. In , Fig. 4A the first pulse at the left represents a desired digit pulse with ' its amplitude increased by a burst of noise, the second pulse represents ' a clean digit pulse, and the third pulse a burst of noise. This group is at 1 the input to the regenerator. Fig. 4B shows the same group of pulses ' after traversing the regenerator once. The pulses are seen to be shortened due to the gating, or retiming, action. There is also seen to be some ampli- tude correction, i.e. the two desired pulses are of more nearly the same j amplitude and the undesired pulse has been reduced in relative ampli- tude. After a few trips through the regenerator the pulse group was rendered practically perfect and remained so for the rest of the twenty- four trips around the loop. Fig. 4C shows the group after 24 trips. In 'another experiment pulses were circulated for 100 trips without deteri- oration. Nothing was found to indicate that regeneration could not be repeated indefinitely. Figs. 5 A and 5B represent the same conditions as those of 4 A and 4B 74 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 4 — Effect of regeneration on disturbances which occur at only one re- peater. A — Input to regenerator, original signal. B — Output of regenerator, first trip. C — Output of regenerator, 24th trip. Fig. 5 — l']ffect of regeneration on disturbances which occur at only one re- peater. A — Input to regenerator, first four groups. B — Output of regenerator, first four groups. C — Output of regenerator, increased input level. REGENERATION OF BINARY MICROWAVE PULSES 75 Fig. 6 — Effect of regeneration on disturbances which occur at only one re- peater. A — Input to regenerator, original signal. B — ^ Output of regenerator, first trip. C • — Oi^tput of regenerator, 24th trip. except that the oscilloscope sweep has been contracted in order to show the progressive effects produced by repeated passage of the signal through the regenerator. Fig. 5B shows that after the pulses have passed through the regenerator only twice all visible effects of the disturbances have been removed. Fig. 5C shows the effect of simply increasing the RF pulse input to the regenerator by approximately 4 db. The small "noise" pulse which in the previous case was quickly dropped out because of being below the slicing level has now come up above the slicing level and so builds up to full amplitude after only a few trips through the regenerator. Note that in the cases shown in Figs. 4 and 5 discrimination against unwanted pulses has been purely on an amplitude basis since the gate has been unblocked to pulses with amplitudes above the slicing level whenever one of these distiu'bing pulses was present. For Fig. 6A conditions are the same as for Fig. 4A except that an ad- ditional pulse has been added to simulate intersymbol noise or inter- ference. Fig. 6B indicates that after only one trip through the regenerator the effect of the added pulse is very small. After a few trips the effect is completely eliminated leaving a practically perfect group which con- tinues on for 24 trips as shown by Fig. 6C. For the intersymbol pulse, discrimination is on a time basis since this interference occurs at a time 76 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 7 — Effect of regenerating in amplitude without retiming. A — Outputof regenerator, no timing, firt trip. B — Output of regenerator, no timing, 10th trip. Output of regenerator, no timing, 23rd trip. when no gating pulse is present and hence finds the gate blocked regard- less of amplitude. To show the need for retiming the pictures shown on Figs. 7 and 8 were taken. These were taken with the amplitude slicer in operation but with the pulses not being retimed. Figs. 7A, 7B and 7C, respectively, show the output of the slicer for the first, tenth and twenty-third trips. After ten trips, there is noticeable time jitter caused by residual noise in the system; after 23 trips this jitter has become severe though pulses are still recognizable. It should be pointed out that for this experiment no noise was purposely added to the system and hence the signal-to- noise ratio was much better than that which would probably be encoun- tered in an operating system. For such a system we would expect time jitter effects to build up much more rapidly. For Fig. 8 conditions are the same as for Fig. 7 except that the pulse spacing is decreased by the addition of an extra pulse at the input. Now, after ten trips, time jitter is bad and after 23 trips the pulse group has become little more than a smear. This increased distortion is probably due to the fact that less jitter is now required to cause overlap of pulses. There may also be some effects due to change of duty cycle. For Fig. 9 there was neither slicing nor retiming of pulses. Here, pulse groups deteriorate very rapidly to nothing more than blobs of energy. Note that there is an increase of i REGENERATION OF BINARY MICROWAVE PULSES 77 Fig. 8 — ■ Effect of regenerating in amplitude without retiming. A — Output of regenerator, no timing, first trip. B — Output of regenerator, no timing, 10th trip. C — Output of regenerator, no timing, 23rd trip. Fig. 9 — Pulses circulating through the loop without regeneration. A — Origi- nal input. B — 4th trip without regeneration. C — 20th to 24th trip without re- generation. '8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 iWWWMMMWWIWWMilJflM^ II . rlilllT- i \m....: iniTiinr- IH. Fig. 10 — The regeneration of band-limited pulses. A — Input to regenerator, first two groups. B — Output of regenerator, first two groups. C — Output of regenerator, 24th trip. amplitude with each trip around the loop indicating that loop gain was slightly greater than unity. Without the sheer it is difficult to set the gain to exactly unity and the amplitude tends to either increase or de- : crease depending upon whether the gain is greater or less than unity. Results indicated by the pictures of Fig 9 are possibly not typical of a properly functioning system but do show what happened in this par- . ticular sj^stem when regeneration was dispensed with. Another important function of regeneration is that of overcoming . band-limiting effects. Figs. 10 and 11 show what can be accomplished. . For this experiment the pulse groups inserted into the loop were as shown i| at the left in Fig. lOA. These pulses were 15 milli-microseconds wide at the base and spaced by 25 milli-microseconds which corresponds to a j repetition frequency of 40 mc. After passing through a band-pass filter these pulses were distorted to the extent shown at the right in Fig. lOA. From the characteristic of the filter, as shown on Fig. 12, it is seen that the bandwidth employed is not very different from the theoretical min- imum required for double sideband transmission. This minimum char- acteristic is shown by the dashed lines on Fig. 12. Fig. lOB shows that at the output of the regenerator the effects of band limiting have been removed. This is borne out by Fig. IOC which shows that after 24 trips the code group was still practically perfect. It should l)e pointed out that the pulses traversed the filter once for each trip around the loop, REGENERATION OF BINARY MICROWAVE PULSES 79 Fig. 11 — The regeneration of band-limited pulses. A — Input to regenerator, first two groups. B — Output of regenerator, first two groups. C — Output of re- generator, 24th trip. that is for each trip the input to the regenerator was as shown at the right of Fig. lOA and the output as shown by Fig. lOB. It is important to note that Fig. 12 represents the frequency characteristic of a single hnk of the simulated system. The pictures of Fig. 11 show the same experi- ment but this time with a different code group. Any code group which we could set up with our five digit pulses was transmitted equally well. In order to determine the breaking point of the experimental system, broad-band noise obtained from a traveling-wave amplifier was added into the system as shown on Fig. 2. The breaking point of the system is the noise level which is just sufficient to start producing errors at the output of the system.* The noise is seen to be band-limited in exactly the same way as the signal. With the system adjusted to operate properly the level of added noise was increased to the point where errors became barely discernible after 24 trips around the loop. Noise level was now reduced slightly (no errors discernible) and the ratio of rms signal to rms noise measured. Fig. 13A shows the input to the regenerator for the 23rd and 24th trips with this amount of noise added. Note that the noise has * The type of noise employed has a Gaussian amplitude distribution and there- fore there was actually no definite breaking point — the rate at which errors Oc- curred increased continuously as noise amplitude was increased. The breaking point was taken as the noise level at which errors became barely discernible on the viewing oscilloscope. More accurate measurements made in other experiments indicate that this is a fairly satisfactory criterion. 80 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 28 24 IT) aJ2o 03 O 16 to If) g 12 a. UJ 5 8 ll 1 — 1 \ i A \ ^ 1 1 1 1 1 / < \ 1 1 1 / / \ 1 / < V h* --20 M 1 --20MC *] 1/ v. 1 / 1 ■~ TX^ ^ ■rrD*^ /J 3950 3960 3970 3980 3990 4000 4010 4020 4030 4040 FREQUENCY IN MEGACYCLES PER SECOND Fig. 12 — Characteristics of the band-pass microwave filter. m % I JYYYYYYTin Fig. 13. — The regeneration of pulses in the presence of broad-hand, random noise added at each repeater. A — Ini)ut to regenerator, 23rd and 24th trijis, broad-band noise added. B — Ini)ut to regenerator, 23rd and 24th trips, no added noise. C — 20-mc timing wave. \ KEGENERATION OF BINARY MICROWAVE PULSES 81 Fig. 14 — The regeneration of pulses in the presence of interference occurring at each repeater. A — Original signal with added moduhited carrier interference. B — Input to regenerator, 24th trip, niochilatod carrier interference. C — Output of regenerator, 24th trip, modulated carrier interference. produced a considerable broadening of the oscilloscope trace. Fig. 13B shows the same pulse groups with no added noise. These photographs are included to give some idea as to how bad the noise was at the l;)reaking point of the system. Of course maximum noise peaks occur rather infre- quently and do not show on the photograph. At the output of the re- generator effects due to noise were barely discernible. This output looked so much like that shown at Fig. 14C that no separate photograph is shown for it. Figs. 14A, 14B and 14C show the effects of a different type of inter- ference upon the system. This disturbance was produced by adding into the system a carrier of exactly the same frequency as the signal carrier (4 kmc) but modulated by a 14-mc wave, a frequency in the same order as the pulse rate. Here again the level of the interference was adjusted to be just below the l)reaking point of the system. A comparison between Figs. 14B and 14C gives convincing evidence that the regenerator has substantially restored the waveform. For the case of the interfering signal a ratio of signal to interference of 10 db on a peak-to-peak basis was measured when the interference was just below the breaking point of the system. This, of course, is 4 db above the theoretical value for a perfect regenerator. For the case of 82 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 broad-band random noise an rms signal to noise ratio of 20 dl) was meas- ured.* This compares Avith a ratio of 18 db as measured by Messrs. Meacham and Peterson for a system employing complete regeneration and a single repeater, f Recently, A. F. Dietrich repeated the circulating loop experiment at a radio frequency of 11 kmc. His determinations of required signal-to- noise ratios are substantially the same as those reported here. From the various experiments we conclude that for a long chain of properly func- tioning regenerative repeaters of i-he type discussed here practically perfect transmission is obtained as long as the signal-to-noise ratio at the input to each repeater is 20 db or better on an rms basis. In an operat- ing system it might be desirable to increase this ratio to 23 db to take care of deficiencies in automatic gain controls, power changes, etc. From the experiments we also conclude that the price we pay for using partial instead of complete regeneration is about 3 to 4 db increase in the required signal-to-noise ratio. In a radio system which provides a fading margin this penalty would be less since the probability that two or more adjacent links will reach maximum fades simultaneously is very ' small. Under these conditions only one repeater at a time would be near the breaking point and the system would behave much as though the repeater provided complete regeneration. TIMING Although we have considered the problem of retiming of signal pulses up to now we have not discussed the problem of obtaining the necessary ' timing pulses to perform this function, but have simpl}^ assumed that a source of such pulses was available. As w^as mentioned earlier timing I pulses would probably be derived from the signal pulses in a practical »^ system. These pulses would be fed into some narrow band amplifier tuned to pulse repetition frequency. The output of this circuit could be made to be a sine wave at repetition frequency if gaps between the input pulses were not too great. Timing pulses could be derived from this sine wave. This timing equipment could be similar to that used in these ex- periments and described earlier. Further study of the problems of ob- taining timing information is being made. * For Gaussian noise it is not possible to specif.y a theoretical value of minimum S/N ratio without specifying the tolerable percentage of errors. For the number of errors detectable on the oscilloscope it seems rasonable to assume a 12 db peak factor for the noise. The peak factor for the signal is 3 db. The 6 db peak S/N which would be required for an ideal regenerator then becomes 15 db on an rms basis. t L. A. Meacham and E. Peterson, B. S. T. J., p. 43, Jan., 1948. " KEGENERATION OF BINARY MICROWAVE PULSES 83 ' GATING PULSE INPUT OUTPUT Fig. 15A — Low-frequency equivalent of the partial regenerator. DESCRIPTION OF REGENERATOR This device regenerates pulses by performing on them the operations of ''slicing" and retiming. An ideal slicer is a device with an input-output characteristics such as shown by the dashed lines of Fig. 15C. It is seen that for all input levels below the so-called slicing level transmission through the device is zero but that for all amplitudes greater than this value the output level is finite and constant. Thus, all input voltages which are less than the slic- ing level have no effect upon the output whereas all input voltages greater than the slicing level produce the same amplitude of output. Normally conditions are adjusted so that the slicing level is at one-half INPUT LEVEL Fig. 15B — Characteristics of the separate branches with ditterential bias. 84 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 INPUT LEVEL Fig. 15C — Resultant output with differential bias. BRANCH 2 BRANCH 1 RESULTANT INPUT LEVEL Fig. 15D — Characteristics of the separate branches and resultant output with equal biases. of peak pulse amplitude — then at the output of the slicer there will be no effect whatsoever from disturbances unless these disturbances exceed half of the pulse amplitude. It is this slicing action which removes the amplitude effects of noise. Time jitter effects are removed by retiming, i.e., the device is made to have high loss regardless of input level except at those times when a gating pulse is present. Fig. 15A shows schematically a low-frequency equivalent of the re- generator used in these experiments. Here an input line divides into two identical branches isolated from each other and each with a diode shunted across it. The outputs of the two branches are recombined through neces- sary isolators to form a single output. The phase of one branch is re- versed before recombination, so that the final output is the difference between the two individual outputs. Fig. 15B shows the input-output characteristics of the two branches when the diodes are biased back to be non-conducting by means of bias voltages Vi and V2 respectively. For low levels the input-output char- acteristic of both branches will be linear and have a 45° slope. As soon REGENEKATION OF BINARY MICROWAVE PULSES 85 as the input voltage in a branch reaches a vakie equal to that of the back bias the diode will start to conduct, thus absorbing power and decrease the slope of the characteristic. The output of Branch 1 starts to flatten off when the input reaches the value Vi , while the output of Branch 2 does not flatten until the input reaches the value V2 . The combined output, which is equal to the differences of the two branch outputs, is then that shown by the solid line of Fig. 15C and is seen to have a transi- tion region between a low output and a high output level. If the two branches are accurately balanced and if the signal voltage is large com- pared to the differential bias V2 — Vi the transition becomes sharp and the device is a good slicer. If the two diodes are equally biased as shown on Fig. 15D the outputs of the two branches should be nearly equal regardless of input and the total output, which is the difference between the two branch outputs, will always be small. Fig. 16 shows a microwave equivalent of the circuit of Fig. 15A. In the microwave structure lengths of wave-guide replace the wire lines and branching, recombining and isolation are accomplished by means of hybrid junctions. The hybrid shown here is of the type known as the lA junction. Fig. 17 shows another equivalent microwave structure employing only one hybrid. This is the type used in the experiments described here. The [output consists of the combined energies reflected from the two side jarms of the junction. With the junction connected as shown phase rela- Itionships are such that the output is the difference between the reflec- GATING PULSE ^(— r-V\^^^ RF INPUT ARM PROBE TERMINATION I ARM 4 I— vw-^ Fig. 16 — Microwave regenerator. 86 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 tions from the two side arms so that when conditions in the two arms are identical there is no output. The crystal diodes coupled to the side arms are equivalent to those shunted across the two lines of Fig. 15A. Fig. 18, which is a plot of the measured input-output characteristic of the regenerator used in the loop test, shows how the device acts as a combined sheer and retimer. Curve A, ol)tained with equal biases on the two diodes, is the characteristic with no gating pulse applied i.e. the diodes are normally biased in this manner. It is seen that this condition produces the maximum of loss through the device. By shifting one diode bias so as to produce a differential of 0.5 volt the characteristic changes to that of Curve B. This differential bias can be supplied by the timing pulse in such a way that this pulse shifts the characteristic from that shown at A to that shown at B thus decreasing the loss through the de- vice by some 12 to 15 db during the time the pulse is present. In this way the regenerator is made to act as a gate — though not an ideal one. We see from curve B that with the differential bias the device has the characteristic of a slicer — though again not ideal. For lower levels of input there is a region over which the input-output characteristic is square law with a one db change of input producing a two db change of output. This region is followed by another in which limiting is fairly pronounced. At the 8-db input level, which is the point at which limiting sets in, the loss through the regenerator was measured to be approxi- mately 12 db. The characteristic shown was found to be reproducible both in these experiments at 4 kmc and in those bj'- A. F. Dietrich at 11 kmc. For a perfect slicer only an infinitesimal change of input level is re- GATING PULSE ■AAV-i_ ARM 2 RF OUTPUT Fig, 17 — Microwave regenerator employing a single hybrid junction. REGENERATION OF BINARY MICROWAVE PULSES 87 ID m o LU a D 3 o -10 -12 -14 -16 -18 -20 -22 -24 V, = 0.5 V2 = 0 <-- 12 DB LOSS i>— m UJ (0 Hi' Hi' X oo 2-mmiiiii o (J X a UJ UJQ i£2 U.UJ °5 QO uicr -lU. _j - MADISON 2- 1234 STEP-BY-STEP PRIMARY CENTER Fig. 8 — Prefixing. 104 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 f/l 10 with Madison and Milwaukee, Wisconsin, in area 414 and Belle Plaine Crossbar Tandem in Chicago, Illinois, in area 312. An economical trunking plan may provide for direct circuits from Chicago to each place. If only three-digit translation were provided in the Chicago switching equipment, the route to both places would be selected as a result of the translation of the 414 area code alone and, therefore, calls to central offices reached through Madison, would need to be routed via Milwaukee. This involves not only the extra trunk mileage, ])ut also the use of an extra switching point. With six-digit translation, both the area code and the central office code are analyzed, making it possible to select the direct route to either city. Six-digit translation in crossbar tandem will involve primarily the use of a foreign area translator and a marker. The translator will have a capacity for translation of five foreign areas and for 60 routes to each area. Since the translator holding time is very short, one translator is sufficient to handle all of the calls requiring six-digit translation, but two are always provided for hazard and maintenance reasons. On a call requiring six-digit translation the first three digits are CASE 1 ^ DIGITS RECEIVED t 2 3 -IMPULSING 4 5 REGISTERS - 6 7 8 9 10 \ X 0 X A B X X X X X . . ; i. OUTPULSING CONTROL ;Si : Sd S J Sa - S3 So . S / - So b» * oiu CASE 2 DIGITS RECEIVED OUTPULSING CONTROL 0 DIGITS CODE CONVERTED AA AB AC ;: PS1 X' PS2 ;:PS3 ;:S4 ):S6 ~:S6 ;;S7 ::S8 10 59 ;:S10 CASE 3 DIGITS RECEIVED DIGITS PREFIXED AB AC B' C OUTPULSING CONTROL i PS2 : : P B PS3 •:SI ':S2 ::S3 :;S4 : : S5 :'S6 ■;S7 Fig. 9 — Method used for outpulsing digits. CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 105 translated in the marker and the second three digits in a foreign area translator which is associated with the marker. Fig. 11 shows, in simpli- fied form, how this translation is accomplished. The first three digits, corresponding to the area code, are received by a relay code tree in the marker which translates it into one of a thousand code points. This code point is cross-connected to the particular relay of the five area relays A(3-A4 which has been assigned to the called area. A foreign area translator is now connected to the marker and a corre- sponding area relay is operated in it. The translator also receives the called office code from the sender via the marker and by means of a relay code tree similar to that in the marker translates the office code to one of a thousand code points. This code point plus the area relay is sufficient to determine the actual route to be used. As shown on the sketch, wires from each of the code points are threaded through trans- formers, two for each area. When the marker is ready to receive the route information, a surge of current is sent through one of these threaded wires which produces a voltage in the output winding to ionize the T- and U- tubes. Only the tubes associated with the area involved in the translation pass current to operate one each of the eight T- and U- relays. This information is passed to the marker and registered on corresponding tens and units relays. These operate a route relay which WISCONSIN MICH. J ILLINOIS CHICAGO = ' f BELLE \ 1 AREA IplaINeJ \312 I ^- — 1 I N D. ROUTE WITHOUT 6 DIGIT TRANSLATION ROUTE WITH 6 DIGIT TRANSLATION Fig. 10 — Six-digit translation. 106 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Fig. 11 — Method used for foreign area translation. CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 107 provides all the information necessary for routing the call to the central office involved. CUSTOMER DIRECT DISTANCE DIALING Crossbar tandem will provide arrangements permitting customers in step-by-step offices to dial their own calls anywhere in the country. Centralized automatic message accounting previously mentioned will be used for charging purposes. While the basic plan for direct distance dialing provides for the dialing of either seven or ten digits, it will be necessary for the customer in step-by-step areas to prefix a three-digit directing code, such as 112, to the called number. This directing code is required to direct the call through the step-by-step switches to the crossbar tandem office so that the seven or ten digit number can be registered in the crossbar tandem office. When a customer in a step-by-step office originates a call to a distant customer whose national number is 915-CH3-1234, he first dials the directing code 112 and then the ten-digit number. The dialing of 112 causes the selectors in the step-by-step office to select an outgoing trunk to the crossbar tandem office. The incoming trunk in the crossbar tandem office has quick access to a three-digit register. The register must be connected during the interval between the last digit of the directing code and the first digit of the national number to insure registration of this number. This arrangement is used to permit the customer to dial all digits without delay and avoids the use of a second dial tone. If this arrangement were not used, the customer would be required to wait after dialing the 112 until the trunk in the tandem crossbar office could gain access to a sender through the sender link circuit which would then signal the customer to resume dialing by returning dial tone. After recording the 915 area code digits in the case assumed, the CH3-1234 portion of the number is registered directly in the tandem sender which has been connected to the trunk while the customer was dialing 915. When the sender is attached to the trunk, it signals the three-digit register to transfer the 915 area code digits to it via a con- nector circuit. Thus when dialing is complete, the entire number 915- CH3-1234 is registered in the sender. Crossbar tandem is being arranged to serve customers of panel and No. 1 crossbar offices for direct distance dialing. At the present time, ten digit direct distance dialing is not available to these customers because the digit storing equipments in these offices are limited to eight digits. Developments now under way, will provide arrangements for expanding the digit capacity in the local offices so that ultirnately 108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 calls from custoniers in panel and No. 1 crossbar offices may be routed through crossbar tandem cr other equivalent offices to telephones anywhere in the country. CONCLUSION The new features developed for crossbar tandem will adapt it to switching all types of traffic at many important switching centers of the nationwide toll network. Of the 225 important toll switching centers now contemplated, it is expected that about 80 of these will be ecjuipped with crossbar tandem. REFERENCES 1. Collis, R. E., Crossbar Tandem System, A.I.E.E. Trans., 69, pp. 997-1004, 1950. 2. King, G. v.. Centralized Automatic Message Accounting, B.S.T.J., 33, pp. 1331-1342, 1952. 3. Nunn, W. H., Nationwide Numbering Plan, B.S.T.J., 31, pp. 851-859, 1952. 4. Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, B. S.T.J. , 31, pp. 832-850, 1952. 5. Shipley, F. F., Automatic Toll Switching Systems, B.S.T.J., 31, pp. 860-882, 1952. 6. Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- ments in Alternate Routing Trunk Networks, B.S.T.J., 33, pp. 277-302, 1954. 7. Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Laboratories Record, 32, pp. 51-57, Feb. 1954. Growing Waves Due to Transverse Velocities By J. R. PIERCE and L. R. WALKER (Manuscript received March 30, 1955) This paper treats propagation of slow waves in two-dimensional neu- tralized electron floiv in which all electrons have the same velocity in the direction of propagation hut in which there are streams of two or more veloci- ties normal to the direction of propagation. In a finite beam in which ' electrons are reflected elastically at the boundaries and in which equal dc currents are carried by electrons with transverse velocities -\-Ui and — Wi , there is an antisi/mmetrical growing ivave if Up ~ {rUi/Wf and a symmetrical growing wave if y- i{Tu,/wy Here cop is plasma frequency for the total charge density and W is beam width. INTKODUCTION i It is well-known that there can be growing waves in electron flow when the flow is composed of several streams of electrons having different velocities in the direction of propagation of the waves. ' While Birdsall considers the case of growing waves in electron flow consisting of streams which cross one another, the growing waves which he finds apparently occur when two streams have different components of velocity in the direction of propagation. This paper shows that there can be growing waves in electron flow consisting of two or more streams with the same component of velocity in the direction of wave propagation but with different components of velocity transverse to the direction of propagation. Such growing Avaves can exist when the electric field varies in strength across the flow. Such waves could result in the amplification of noise fluctuations in electron ' flow. They could also be used to amplify signals. 109 110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Actual electron flow as it occurs in practical tubes can exhibit trans- verse velocities. For instance, in Brillouin flow, ' • if we consider electron motion in a coordinate system rotating with the Larmor frequency we see that electrons with transverse velocities are free to cross the beam repeatedly, being reflected at the boundaries of the beam. The trans- verse \-elocities may be completely disorganized thermal velocities, or they may be larger and better-organized velocities due to aberrations at the edges of the cathode or at lenses or apertures. Two-dimensional Brillouin flow allows similar transverse motions. It would be difficult to treat the case of Brillouin or Brillouin-like flow with transverse velocities. Here, simpler cases with transverse velocities will be considered. The first case treated is that of infinite ion-neutra- lized two-dimensional flow with transverse velocities. The second case treated is that of two-dimensional flow in a beam of finite width in which the electrons are elastically reflected at the boundaries of the beam. Growing waves are found in both cases, and the rate of growth may be large. In the case of the finite beam both an antisymmetric mode and a symmetric mode are possible. Here, it appears, the current density required for a growing wave in the symmetric mode is about ^^ times as great as the current density required for a growing wa^•e in the anti- symmetric mode. Hence, as the current is increased, the first growing waves to arise might be antisymmetric modes, which could couple to a symmetrical resonator or helix only through a lack of symmetry or through high-level effects. 1 . Infinite two-dimensional flow Consider a two-dimensional problem in which the potential varies sinusoidally in the y direction, as exp{—j^z) in the z direction and as exp (jut) with time. Let there be two electron streams, each of a negative charge po and each moving with the velocity ?/o in the z direction, but with velocities Wi and —ih respectively in the y direction. Let us denote ac quantities pertaining to the first stream by subscripts 1 and ac quan- tities pertaining to the second stream by subscripts 2. The ac charge density will be denoted by p, the ac velocity in the y direction by y, and the ac velocity in the z direction by i. We will use linearized or small-signal equations of motion.^ We will denote differentiation with respect to ?/ by the operator D. The equation of continuity gives jupi = -D(piUi + po?yi) + j|8(piWo + pnii) (1.1)1 jcopo = -D{-p-iHi -\- pi)lj':d + il3(P2''o + Poi2) (1.2) t; GROWING WAVES DUE TO TRANSVERSE VELOCITIES 111 Let US define dx = i(co - ^u,) + u,D (1.3) do = ./(w - i8wo) - uj) (1.4) We can then rewrite (1.1) and (1.2) as f/iPi = Poi-Diji + j(3zi) (1.5) dopi = Pi^{ — Dy2 + .7/3i2) (1.0) We will assume that we are dealing ^^•ith slow waves and can use a po- tential V to describe the field. We can thus write the linearized equations of motion in the form r/iii = -j-^F (1.7) m d2h = -j-^V (1.8) m drlji = - DV (1.9) m d,y, = 1 DV (1.10) w From (1.5) to (1.10) we obtain ^m = ~ PoiD' - ^')V (1.11) m d'p2= --poiD'- ^')V (1.12) m Now, Poisson's equation is {D' - ^')V = _^L±£! (1.13) From (1.11) to (1.13) we obtain {D' - /3^)y = - Kco/ (^1 + ^^ (D' - /3^)7 (1.14) 9 ^ — Z— po 2 m Wp = e Here Wp is the plasma frequency for the charge of both beams. (1.15) 112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Either or else (2)' - /3')7 = 0 — C0„" (c/l" + ^2") ^ 2 di^ d.} We will consider this second case. W(< should note from (1.3) and (1.4) that d{ = u^-D^ - (co - /5(/„)" + 2yD(co - |8?/.o)«i ^2^ = ? y Ug where Ue is the extreme value of lu; these correspond to intersections 3 and 3' in Fig. 2. The other waves, two per value of Un , may be unat- tenuated or a pair of increasing and decreasing waves, depending on the values of the parameters. If CO pn -yhir? > 1 there will be at least one pair of increasing and decreasing waves. It is not clear what will happen for a Maxwellian distribution of veloci- ties. However, we must remember that various aberrations might give a very different, strongly peaked velocity distribution. Let us consider the amount of gain in the case of one pair of transverse velocities, ±i/i . The equation is now 2 2 7 Ui C0„2 [ 1 + CO — |3wo )•] [ ■ - (^OI (1.27) Let /5 = ^+i^ Wo Wo (1.28) 114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 1 .u 0.9 0.8 \ \ 0.7 0.6 \ \ \, \ ^ 0.5 0.4 0.3 \^ \ >s. \ V 0.2 \ > \ 0.1 \ \ 0 \ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 v2 m Fig. 2 This relation defines e. Equation (1.27) becomes 2 2 0}J 1 - e^ (1 + e^)^ ^'-''^ In Fig. 2, e is plotted versus the parameter y^Ui/oip^. We see that as the parameter falls below unity, e increases, at first rapidly, and then more slowly, reaching a value of ±1 as the parameter goes to zero (as cop' goes to infinity, for instance). It will be shown in Section 2 of this paper that these results for infinite flow are in some degree an approximation to the results for flow in narrow beams. It is therefore of interest to see what results they yield if applied to a beam of finite width. If the beam has a length L, the voltage gain is The gain G in db is G = 8.7 '^ € db Wo (1.30) (1.31) GROWING WAVES DUE TO TRANSVERSE VELOCITIES 115 Let the width of the beam be W. We let Thus, for n = 1, there is a half -cycle variation across the beam. From (1.31) and (1.32) G = 27.s(^^^\ne db (1.33) Now L/uo is the time it takes the electrons to go from one end of the beam to the other, while W/ui is the time it takes the electrons to cross the beam. If the electrons cross the beam A'' times iV = ^4 (1-34) Thus, G = 27.SNnedb (1.35) While for a given value of e the gain is higher if we make the phase vary many times across the beam, i.e., if we make n large, we should note that to get any gain at all we must have 2 . //iTTUlV 0)r> > (1.36) W If we increase oop , which is proportional to current density, so that cop passes through this value, the gain will rise sharply just after cOp" passes through this value and will rise less rapidly thereafter. .?. A Two-Dimensional Beam of Finite Width. Let us assume a beam of finite width in the ^/-direction ; the boundaries lying a,t y = ±^o • It will be assumed also that electrons incident upon these boundaries are elastically reflected, so that electrons of the incident stream (1 or 2) are converted into those of the other stream (2 or 1). The condition of elastic reflection implies that yi = -h (2.1) Zi = 22 Sit y = ±2/0 (2.2) and, in addition, that Pi = p2 at y = ±?/o . (2.3) since there is no change in the number of electrons at the boundary. 116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The equations of motion and of continuity (1.7-1.12) may be satisfied by introducing a single quantity, ^, such that V = dx dzV (2.4) ii = -J - /3 d, ^2^ (2.5) m zi = —j — di di\p (2.6) m yi=-d, d^Dyp (2.7) m 112=- di d^Di^ (2.8) m Pi m poiD' - ^') dirl^ (2.9) P2 = -- Po(i)' - n di'rl^ (2.10) m Then, if we introduce the symbol, 12, for co — jSuo yi + y^ = 2j-d,d2D^yp (2.11) ' m h- Z2 = 2j - di diUiD^ (2.12) m PI - P2 = 2j- po{D' - l3')uiQDi^ (2.13) m It is clear that if Drjy = D^xl^ = 0 y = ±yo (2.14) the conditions for elastic reflection will be satisfied. The equation satis- fied by rf/ may now be found from Poisson's equation, (1-13), and is {D' - /3^) dx' di^P = '-^{D'- fi'){d,' + di)^l. we or {D' - ^')[{u,'D' + ny + coJiu.'D' - n')] = 0 (2.15) which is of the sixth degree in D. So far four boundary conditions have, been imposed. The remaining necessary pair arise from matching the GROWING WAVES DUE TO TRANSVERSE VELOCITIES 117 internal fields to the external ones. For y > ijo V = Voe-'^'-e~^" (2.16) and Similarlv ^ + i37 = 0 at 2/ = 2/0 dy dV — - ^V = 0 at y = -7/0 (2.17) dy The most familiar procedure now would be to look for solutions of (2,15) of the form, e''^. This would give the sextic for c (c' - /3')[(WiV + nY + a;/(niV - n')] = 0 (2.18) with the roots c = ±|8, ±ci , ±C2 , let us s^y. We could then express \p as a linear combination of these six solutions and adjust the coefficients to satisfy the six boundary equations. In this way a characteristic equa- tion for l3 would be obtained. From the S3anmetry of the problem this has the general form F(l3, Ci) = F(i3, C2), where Ci and Co are found from ; (2.18). The discussion of the problem in these terms is rather laborious and, if we are concerned mainly with examining qualitatively the onset of increasing waves, another approach serves better. From the symmetry of the equations and of the boundary conditions we see that there are solutions for \p (and consequently for V and p) which are even in y and again some which are odd in y. Consider first the even solutions. We will assume that there is an even function, ^i(y), periodic in y with period 2yo , which coincides with \l/(y) in the open interval, —yo< ■^ ^^ >C ^ "\ ^ '^ ^ 3 4 5 6 7 8 9 Fig. 5 J 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G left. Clearly in the regions marked + which lie to the left of every curve given by (2.30), the sum is positive and we cannot have roots. Let us examine the sum in the region to the right of the n = 0 curve and to the left of all others. On the line, A;^ = J4» the sum is positive, since the first term is zero. On any other line, k' = constant, the sum goes from + °° at the n = 1 curve monotonically to — oo at the n = 0 curve, so that somewhere it must pass through 0. This enables us to draw the zero- sum contours qualitatively in this region and they are indicated in Fig. 3. We are now in a position to follow the variation in the sum as k varies at fixed 5 . It is readily seen that for 5 < 0.25, because —wz is negative in the region under consideration, there will be four real roots, tw^o for positive, two for negative k. For 5' slightly greater than 0.25, the sum has Fig. 6A GROWING WAVES DUE TO TRANSVERSE VELOCITIES 123 a deep minimum for k = 0, so that there are still four real roots unless z is very large. For z fixed, as 5^ increases, the depth of the minimum de- creases and there will finally occur a 5" for which the minimum is so shal- low that two of the real roots disappear. Call z(0) the value of ziork = 0, write the sum as 2(5^ k^) and suppose that 2(5o^ 0) = —irziO), then for small k we have S(5^ e) = -«(0) + (6^ - 8o') §, + k'§,= -«(0) -"^ k do^ dk^ Ua as dB dk^ ^ = ^± / ".^(^-^0^) + '^ a/ dk' y The roots become complex when aA-2 S.2 J 2 (Ul/Uo) 0 = do — 52 as d8^ dB Since Ui/uq may be considered small (say 10 per cent) it is sufficient to look for the values of 5o^. When k = 0 we have -TZ = 2X) 2z z (n + y,y z^ + 52 irz" z'-\-in-\- y^r (n -1- y^y- - s' ' H ^ + i ^ 0 \in + 3^)2 - 52 ^ (n + 1^)2 + zy (5 tan -Kb -\- z tanh irz) z" + 52 Fig. 4 shows the solution of this equation for various 2(0) or oiyo/iruo . Clearly the threshold 5 is rather insensitive to variations in uyo/ir^io . Equation (2.28) may be examined by a similar method, but here some complications arise. Fig. 5 shows the infinity curves for n = 0, 1, 2, 3; the n = 0 term being of the form k^/k^ — 8^. The lowest critical region in 5^ is the neighborhood of the point fc^ = 6^ = ]^i, which is the intersec- tion of the n = 0 and n = 1 lines. To obtain an idea of the behavior of 124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G the left hand side (l.h.s.) of (2.28) in this area we first see how the point k^ = f = 1^ can be approached so that the l.h.s. remains finite. If we put k^ = H + £ and a' = ^ + ce and expand the first two dominant terms of (2.28), then adjust c to keep the result finite as f -^ 0 we find = 1 3^' - 5 ^ ~ 4 32^ + 1 c varies from — % to \i as z goes from 0 to c» , changing sign at 2^ = %. Every curve for which the l.h.s. is constant makes quadratic contact with the Jine 5" — V3 = c(/v" — ]i) at Jc' = 5' = I/3. If we remember that the l.h.s. is positive for A;' = 0, 0 < 5" < 1 and for A;^ = 1, 0 < 5^ < 1, 1 2 lik 3 w-oX k^ / 1 y( I 3 SHADED AREAS // NEGATIVE yV X /' / // / /I / / / X \ n = i^v 0 3 3 Fig. 6B GROWING WAVES DUE TO TRANSVERSE VELOCITIES 125 since there are no negative terms in the sum for these ranges and again that the l.h.s. must change sign between the n = 0 and n — I Unes for any k^ in the range 0 < k^ < 1 (since it varies from T oo to ±0°), this information may be combined with that about the immediate vicinity of 5 = k = V^ to enable us to draw a Hue on which the l.h.s. is zero. This is indicated in Figs. 6A and 6B for small z and large z respec- tively. It will be seen that the zero curve and, in fact, all curves on which the l.h.s. is equal to a negative constant are required to have a vertical tangent at some point. This point may be above or below /c^ = ^ (de- pending upon the sign of c or the size of z) but always at a 3^ > ^. For 5 < H there are no regions where roots can arise as we can readily see by considering how the l.h.s. varies with k"^ at fixed 5^ For a fixed d^ > }/s we have, then, either for k^ > ]4 or k^ < V^, according to the size of z, a negative minimum which becomes indefinitely deep as 5^ -^ ^. Thus, since the negative terms on the right-hand side are not sensitive to small changes in 5^, we must expect to find, for a fixed value of the l.h.s., two real solutions of (2.28) for some values of 5^ and no real solutions for some larger value of 5 , since the negative minimum of the l.h.s. may be made as shallow as we like by increasing 6". By continuity then we expect to find pairs of complex roots in this region. Rather oddly these roots, which will exist certainly for 5' sufficiently close to V^ + 0, will disappear if 5^ is sufficiently increased. REFERENCES 1. L. S. Nergaard, Analysis of a Simple Model of a Two-Beam Growing-Wave Tube, RCA Review, 9, pp. 585-601, Dec, 1948. 2. J. R. Pierce and W. B. Hebenstreit, A New Type of High-Frequency Amplifier, B. S. T. J., 28, pp. 23-51, Jan., 1949. 3. A. V. Haeff, The Electron-Wave Tube — A Novel Method of Generation and Amplification of Microwave Energy, Proc. I.R.E., 37, pp. 4-10, Jan., 1949. 4. G. G. Macfarlg,ne and H. G. Hay, Wave Propagation in a Slipping Stream of Electrons, Proc. Physical Society Sec. B, 63, pp. 409-427, June, 1950. 5. P. Gurnard and H. Huber, Etude E.xp^rimentale de L'Interaction par Ondes de Chargd^d'Espace au Sein d'Un Faisceau Electronique se Deplagant dans Des Champs Electrique et Magn^tique Croisfe, Annales de Radio^lectricite, 7, pp. 252-278, Oct., 1952. 6. C. K. Birdsall, Double Stream Amplification Due to Interaction Between Two Oblique Electron Streams, Technical Report No. 24, Electronics Research Laboratory, Stanford University. 7. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Mag- netic Fields, Phys. Rev., 67, pp. 260-266, 1945. 8. J. R. Pierce, Theory and Design of Electron Beams, 2nd Ed., Chapter 9, Van Nostrand, 1954. 9. J. R. Pierce, Traveling-Wave Tubes, Van Nostrand, 1950. Coupled Helices By J. S. COOK, R. KOMPFNER and C. F. QUATE (Received September 21, 1955) An analysis of coupled helices is presented, using the transmission line approach and also the field approach, with the objective of providing the tube designer and the microwave circuit engineer with a basis for approxi- mate calcidations. Devices based on the presence of only one mode of propa- gation are briefly described; and methods for establishing such a mode are given. Devices depending on the simultaneous presence of both modes, that is, depending on the beat wave phenomenon, are described; some experi- mental results are cited in support of the view that a novel and useful class of coupling elements has been discovered. CONTENTS 1. Introduction 129 2. Theory of Coupled Helices 132 2.1 Introduction 132 2.2 Transmission Line Equations 133 2.3 Solution for Synchronous Helices 135 2.4 Non-Synchronous Helix Solutions 137 2.5 A Look at the Fields 139 2.6 A Simple Estimate of b and x 141 2.7 Strength of Coupling versus Frequency 142 2.8 Field Solutions 144 . 2.9 Bifilar Helix 146 2.10 Effect of Dielectric Material between Helices 148 2.11 The Conditions for Maximum Power Transfer 151 2.12 Mode Impedance 152 3. Applications of Coupled Helices 154 3.1 Excitation of Pure Modes 156 3.1.1 Direct Excitation 156 3.1.2 Tapered Coupler 157 3.1.3 Stepped Coupler 158 3.2 Low Noise Transverse Field Amplifier 159 3.3 Dispersive Traveling Wave Tube 159 3.4 Devices Using Both Modes 161 3.4.1 Coupled Helix Transducer 161 3.4.2 Coupled-Helix Attenuator 165 4. Conclusion 167 Appendix I Solution of Field Equations 168 II Finding r I73 III Complete Power Transfer 175 127 128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 GLOSSARY OF SYMBOLS a Mean radius of inner helix h Mean radius of outer helix h Capacitive coupling coefficient Bio, 20 shunt susceptance of inner and outer helices, respectively Bi, 2 Shunt susceptance plus mutual susceptance of inner and outer helices, respectively, Bm + Bm , Boo + B^ Bm Mutual susceptance of two coupled helices c Velocity of light in free space d Radial separation between helices, h-a D Directivity of helix coupler E Electric field intensity F Maximum fraction of power transferable from one coupled helix to the other F(ya) Impedance parameter 7i, 2 RF current in inner and outer helix, respectively K Impedance in terms of longitudinal electric field on helix axis and axial power flow L ]\Iinimum axial distance required for maximum energy transfer from one coupled helix to the other, X6/2 Axial power flow along helix circuit Radial coordinate Radius where longitudinal component of electric field is zero for transverse mode (about midway between a and b) Return loss Radial separation betw^een helix and adjacent conducting shield Time RF potential of inner and outer helices, respectively • Inductive coupling coefficient Series reactance of inner and outer helices, respectively Series reactance plus mutual reactance of inner and outer helices, respectively, Xio + Xm , X20 + Xm Mutual reactance of two coupled helices Axial coordinate Impedance of inner and outer helix, respectively Attenuation constant of inner and outer helices, respectively General circuit phase constant; or mean circuit phase constant. Free space phase constant Axial phase constant of inner and outer helices in absence of coupling, V^ioXio , VBioXio p r f R s t F1.2 X Xva, 20 Xl, 2 Xm Z Zil, 2 Oil, 2 ^0 ^10. 20 COUPLED HELICES 129 181 , 2 May be considered as axial phase constant of inner and outer helices, respectively (Sft Beat phase constant jSc Coupling phase constant, (identical with ^b when /3i = JS2) I3ce Coupling phase constant when there is dielectric material be- tween the helices /3d Difference phase constant, [ /3i — /32 [ (8f Axial phase constant of single helix in presence of dielectric ^t, ( Axial phase constant of transverse and longitudinal modes, re- spectively 7 Radial phase constant jt, ( Radial phase constant of transverse and longitudinal modes, respectively r Axial propagation constant Tt. ( Axial propagation constant for transverse and longitudinal coupled-helix modes, respectively e Dielectric constant e' Relative dielectric constant, e/eq En Dielectric constant of free space X General circuit wavelength; or mean circuit wavelength, \/XiX2 Xo Free space wavelength Xi, 2 Axial wavelength on inner and outer helix, respectively X6 Beat wavelength Xc Coupling wavelength (identical with Xb when (5i = /So) yj/ Helix pitch angle i/'i, 2 Pitch angle of inner and outer helix, respectively CO Angular frequency 1. INTRODUCTION Since their first appearance, traveling-wave tubes have changed only very little. In particular, if we divide the tube, somewhat arbitrarily, into circuit and beam, the most widely used circuit is still the helix, and the most widely used transition from the circuits outside the tube to the circuit inside is from waveguide to a short stub or antenna which, in turn, is attached to the helix, either directly or through a few turns of increased pitch. Feedback of signal energy along the helix is prevented by means of loss, either distributed along the whole helix or localized somewhere near the middle. The helix is most often supported along its whole length by glass or ceramic rods, which also serve to carry a con- ducting coating ("aquadag"), acting as the localized loss. We therefore find the following circuit elements within the tube en- velope, fixed and inaccessible once and for all after it has been sealed off: 130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 1 . The helix itself, determining the beam voltage for optimum beam- circuit interaction ; 2. The helix ends and matching stubs, etc., all of which have to be positioned very precisely with relation to the waveguide circuits in order to obtain a reproducible match ; 3. The loss, in the form of "aquadag" on the support rods, which greatly influences the tube performance by its position and distril)ution. In spite of the enormous bandwidth over which the traveling-wave tube is potentially capable of operating — a feature new in the field of microwave amplifier tubes — it turns out that the positioning of the tube in the external circuits and the necessary matching adjustments are rather critical; moreover the overall bandwidths achieved are far short of the obtainable maximum. Another fact, experimentally observed and well-founded in theory, rounds off the situation: The electro-magnetic field surrounding a helix, i.e., the slow wave, under normal conditions, does not radiate, and is confined to the close vicinity of the helix, falling off in intensity nearly exponentially with distance from the helix. A typical traveling-wave tube, in which the helix is supported by ceramic rods, and the whole enclosed by the glass envelope, is thus practically inaccessible as far as RF fields are concerned, with the exception of the ends of the helix, where provision is made for matching to the outside circuits. Placing objects such as conductors, dielectrics or distributed loss close to the tube is, in general, observed to have no effect whatsoever. In the course of an experimental investigation into the propagation of space charge waves in electron beams it was desired to couple into a long helix at any point chosen along its length. Because of the feebleness of the RF fields outside the helix surrounded by the conventional sup- ports and the envelope, this seemed a rather difficult task. Nevertheless, if accomplished, such a coupling would have other and even more im- portant applications; and a good deal of thought was given to the problem. Coupled concentric helices were found to provide the solution to the problem of coupling into and out of a helix at any particular point, and to a number of other problems too. Concentric coupled helices have been considered by J. R. Pierce, who has ti'cated the problem mainly with transverse fields in mind. Such fields were thought to be useful in low-noise traveling-wave tube devices. Pierce's analysis treats the helices as transmission lines coupled uniformly over their length by means of nuitual distributed capacitance and inductance. Pierce also recognized that it is necessary to wind the COUPLED HELICES l,']! two helices in opposite directions in order to obtain well defined trans- verse and axial wave modes which are well separated in respect to their velocities of propagation. Pierce did not then give an estimate of the velocity separation which might be attainable with practical helices, nor did anybody (as far as we are aware) then know how strong a coupling one might obtain with such heUces. It was, therefore, a considerable (and gratifying) surprise^' ^ to find that concentric helices of practically realizable dimensions and separa- tions are, indeed, very strongly coupled when, and these are the im- portant points, (a) They have very nearly equal velocities of propagation when un- coupled, and when (b) They are wound in opposite senses. It was found that virtually complete power transfer from outer to inner helix (or vice versa) could be effected over a distance of the order of one helix wavelength (normally between i^fo and 3^^o of a free-space wavelength. It was also found that it was possible to make a transition from a co- axial transmission line to a short (outer) helix and thence through the glass surrounding an inner helix, which was fairly good over quite a con- siderable bandwidth. Such a transition also acted as a directional coupler, RF power coming from the coaxial line being transferred to the inner helix predominantly in one direction. Thus, one of the shortcomings of the "conventional" helix traveling- wave tube, namely the necessary built-in accuracy of the matching parameters, was overcome by means of the new type of coupler that might evolve around coupled helix-to-helix systems. Other constructional and functional possibilities appeared as the work progressed, such as coupled-helix attenuators, various tj^pes of broadband couplers, and schemes for exciting pure transverse (slow) or longitudinal (fast) waves on coupled helices. One central fact emerged from all these considerations: by placing part of the circuit outside the tube envelope with complete independence from the helix terminations inside the tube, coupled helices give back to the circuit designer a freedom comparable only with that obtained at much lower frequencies. For example, it now appears entirely possible to make one type of traveling wave tube to cover a variety of frequency bands, each band requiring merely different couplers or outside helices, the tube itself remaining unchanged. Moreover, one tube may now be made to fulfill a number of different 132 THE BELL SYSTEM TECHNICAI- JOURNAL, JANUARY 1956 functions; this is made possible by the freedom with which couplers and attenuators can be placed at any chosen point along the tube. Considerable work in this field has been done elsewhere. Reference will be made to it wherever possible. However, only that work with which the authors have been intimately connected will be fully reported here. In particular, the effect of the electron beam on the wave propaga- tion phenomena will not be considered. 2. THEORY OF COUPLED HELICES 2.1 Introduction In the past, considerable success has been attained in the under- standing of traveling wave tube behavior by means of the so-called "transmission-line" approach to the theory. In particular, J. R, Pierce used it in his initial analysis and was thus able to present the solution of the so-called traveling-wave tube equations in the form of 4 waves, one of which is an exponentially growing forward traveling wave basic to the operation of the tube as an amplifier. This transmission-line approach considers the helix — or any slow- wave circuit for that matter — as a transmission line with distributed capacitance and inductance with which an electron beam interacts. As the first approximation, the beam is assumed to be moving in an RF field of uniform intensity across the beam. In this way very simple expressions for the coupling parameter and gain, etc., are obtained, which give one a good appreciation of the physically relevant quantities. A number of factors, such as the effect of space charge, the non-uniform distribution of the electric field, the variation of circuit impedance with frequency, etc., can, in principle, be calculated and their effects can be superimposed, so to speak, on the relatively simple expressions deriving from the simple transmission line theory. This has, in fact, been done and is, from the design engineer's point of view, quite satisfactory. However, phj^sicists are bound to be unhappy over this state of affairs. In the beginning was Maxwell, and therefore the proper point to start from is Maxwell. So-called "Field" theories of traveling-w^ave tubes, based on Maxwell's equation, solved with the appropriate boundary conditions, have been worked out and their main importance is that they largely confirm the results obtained by the inexact transmission line theory. It is, however, in the nature of things that field theories cannot give answers in terms of COUPLED HELICES 133 simple closed expressions of any generality. The best that can be done is in the form of curves, with step-wise increases of particular param- eters. These can be of considerable value in particular cases, and when exactness is essential. In this paper we shall proceed by giving the "transmission-line" type theory first, together with the elaborations that are necessary to arrive at an estimate of the strength of coupling possible with coaxial helices. The "field" type theory will be used whenever the other theory fails, or is inadequate. Considerable physical insight can be gotten with the use of the transmission-line theory; nevertheless recourse to field theory is necessary in a number of cases, as will be seen. It will be noted that in all the calculations to be presented the presence of an electron beam is left out of account. This is done for two reasons: Its inclusion would enormously complicate the theory, and, as will eventually be shown, it would modify our conclusions only very slightly. Moreover, in practically all cases which we shall consider, the helices are so tightly coupled that the velocities of the two normal modes of propaga- tion are very different, as will be shown. Thus, only when the beam velocity is very near to either one or the other wave velocity, will growing-wave interaction take place between the beam and the helices. In this case conventional traveling wave tube theory may be used. A theory of coupled helices in the presence of an electron beam has been presented by Wade and Rynn,^ who treated the case of weakly coupled helices and arrived at conclusions not at variance with our views. 2.2 Transmission Line Equations Following Pierce we describe two lossless helices by their distributed series reactances Xio and A'20 and their distributed shunt susceptances Bio and ^20 . Thus their phase constants are /3io = V^ioA'io Let these helices be coupled by means of a mutual distributed reac- tance Xm and a mutual susceptance B^ , both of which are, in a way which will be described later, functions of the geometry. Let waves in the coupled system be described by the factor jut — Tj; e e \v here the F's are the propagation constants to be found. 134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 The transmission line equations may be written: r/i - jB,V, + jB„y2 = 0 rFi - iXi/i + jXJo = 0 r/o - JB0V2 + jB„yi = 0 TV2 - jXJa + jXJ, - 0 where B, - 5io + 5« Bo = B20 + Bm X2 = X20 -f" Xm 1 1 and 1 2 are eliminated from the (2.2.1) and we find F2 ^ + (r- + XiBi + x^Bj Fi F2 (2.2.1) X\Bm + B%Xm + (r- + X2S2 + x^Bj XlBm + 5lX„ (2.2.2) (2.2.3) These two equations are then multipUed together and an expression for r of the 4th degree is obtained : r' + (XiBi + X2B2 + 2Z,„Bjr' + (X1Z2 - Xj){B,B2 - Bj) = 0 We now define a number of dimensionless quantities: (2.2.4) B, BiB. Xm = h' = (eapacitive coupling coefficient)' = X = (inductive coupling coefficient) XiXo B\Xi = ^1, B2X2 = (82' X1B1X2B2 = 13^ = (mean phase constant) With these substitutions we obtain the general equation for T~ T' = 13' 2 \(3-r ^ I3{' ^ y 4v^2'^^/3i^ _ (2.2.5) + 26.r - (1 - .r-)(l - U') COUPLED HELICES 135 (2.2.6) If we make the same substitutions in (2.2.2) we find Fi T ZiL /3(/3i?> + /3o:r) . where the Z's are the impedances of the heUces, i.e., Z,. = VXJB, 2.3 Solution for Synchronous Helices Let us consider the particular case where (Si = (S-z = |S. From (2.2.5) we obtain r' = -I3\l + xb db (x + b)] (2.3.1) Each of the above values of T" characterizes a normal mode of propaga- tion involving both helices. The two square roots of each T" represent waves going in the positive and negative directions. We shall consider only the positive roots of T , denoted Tt and Tt , which represent the forward traveling waves. Ttj = i/3Vl + xb ± {x + b) (2.3.2) If a: > 0 and 6 > 0 I r, I > |/3i, I r,| < 1^1 Thus Vt represents a normal mode of propagation which is slower than the propagation velocity of either helix alone and can be called the "slow" wave. Similarly T( represents a "fast" wave. We shall find that, in fact, X and b are numerically equal in most cases of interest to us; we therefore write the expressions for the propagation constants r. = M^ + H(-^ + b)] (2.3.3) r. = Ml - Viix + b)] If we substitute (2.3.3) into (2.2.6) for the case where /3i = (82 = /3 and assume, for simplicity, that the helix self-impedances are equal, we find that for r = Tt Y% _ for r = T; F2 -— = -f 1 Yx ^ 136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Thus, the slow wave is characterized by equal voltages of unlike sign on the two helices, and the fast wave by equal voltages of like sign. It fol- lows that the electric field in the annular region between two such coupled concentric helices will be transverse for the slow wave and longitudinal for the fast. For this reason the slow and fast modes are often referred to as the transverse and longitudinal modes, respectively, as indi- cated by our subscripts. It should be noted here that we arbitrarily chose h and x positive. A different choice of signs cannot alter the fact that the transverse mode is the slower and the longitudinal mode is the faster of the two. Apart from the interest in the separate existence of the fast and slow waves as such, another object of interest is the phenomenon of the simul- taneous existence of both waves and the interference, or spatial beating, between them. Let V2 denote the voltage on the outer hehx; and let Vi , the voltage on the inner halix, be zero at z = 0. Then we have, omitting the common factor e'" , (2.3.4) Since at 2 = 0, Fi = 0, Vn = — V(^ . For the case we have considered we have found Fa = — V^ and Vn = V^ . We can write (2.3.4) as Fi = I {e~'^' - e-^n V, = ^ {e''^' + e-'n (2.3.5) F2 can be written = Ye-"'''''^''^''' cos [-jj^(r, - Vi)z\ In the case when x = 6, and /Si = /32 = /8 F2 = Ye"'^' cos Wiix + h)^z\ (2.3.6) Correspondingly, it can be shown that the voltage on the inner helix is y, = j\Tfr^^' sin Wiix + h)^z\ (2.3.7) The last tAvo equations exhibit clearly what we have called the spatial beat phenomonou, a wave-like transfer of power from one helix to thc^ COUPLED HELICES . 137 other and back. We started, arbitrarily, with all the voltage on the outer helix at 2 = 0, and none on the inner; after a distance, z', which makes the argument of the cosine x/2, there is no voltage on the outer helix and all is on the inner. To conform with published material let us define what we shall call the "coupling phase-constant" as ^, = ^{h + x) (2.3.8) From (2.3.3) we find that for (Si = ^2 = |S, and x = h, Tt - Ti = jl3c 2.4 Non-Synchronous Helix Solutions Let us now go back to the more general case where the propagation velocities of the (uncoupled) helices are not equal. Eciuation (2.2.5) can be written: Further, let us define (2.4.1) r- = -^- [1 + (1/2)A + xb ± V(l + xb)A + (1/4)A2 + (6 + xy] where L /3 _ In the case where x = h, (2.4.1) has an exact root. r,, , = j^ [Vl + A/4 ± 1/2 Va + (a; + by] (2.4.2) We shall be interested in the difference between Tt and Tt, Tt-Tf = j^ Va + (x + by- (2.4.3) Now we substitute for A and find Tt- Tc = j V(^i - ^2y + ^M& + 4' (2.4.4) Let us define the "beat phase-constant" as: Pb = V(/3i - /32)2 + nb + xy so that r, - r, = jA (2.4.5) (3a = \ i5i - iSo 138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 and call this the "difference phase-constant," i.e., the hase constant cor- responding to two uncoupled waves of the same frequency but differing phase velocities. We can thus state the relation between these phase constants : ^b' = &I + ^c (2.4.6) This relation is identical (except for notation) with expression (33) in S. E. Miller's paper. ^ In this paper Miller also gives expressions for the voltage amplitudes in two coupled transmission systems in the case of unequal phase velocities. It turns out that in such a case the power trans- fer from one system to the other is necessarily incomplete. This is of particular interest to us, in connection with a number of practical schemes. In our notation it is relatively simple, and we can state it by saying that the maximum fraction of power transferred is (2.4.7) or, in more detail, iS/ + iSc- (^1 - iS2)- + ^Kh + xY This relationship can be shown to be a good approximation from (2.2.6), (2.3.4), (2.4.2), on the assumption that h ^ x and Zx 'PH Z2 , and the further assumption that the system is lossless; that is, I 72 I ^ + I Fi I ^ = constant (2.4.8) We note that the phase velocity difference gives rise to two phenomena : It reduces the coupling w^avelength and it reduces the amount of power that can be transferred from one helix to the other. Something should be said about the case where the two helix imped- ances are not equal, since this, indeed, is usually the case with coupled concentric helices. Equation (2.4.8) becomes: I F2 1 _^ \Vx\_ ^ (3Qj^g^^j^^ (2.4.9) Z2 Z\ Using this relation it is found from (2.3.4) that F2 , /Zi FiT z, (1 ± Vl - /^) (2.4.10) When Ihis is combined with (2.2.6) it is found that the impedances droj) out with the voltages, and that "F" is a function of the |S's only. In other COUPLED HELICES 139 words, complete power transfer occurs when ,81 = /So regardless of the relative impedances of the helices. The reader will remember that (3io and (820 , not jSi and ^o , were defined as the phase constants of the helices in the absence of each other. If the assumption that h ^ x is maintained, it will be found that all of the de- rived relationships hold true when (Sno is substituted for /3„ . In other words, throughout the paper, /3i and /So may be treated as the phase con- stants of the inner and outer helices, respectively. In particular it should be noted that if these ciuantities are to be measured experimentally each helix must be kept in the same environment as if the helices were coupled ; onl}^ the other helix may be removed. That is, if there is dielectric in the annular region between the coupled helices, /Si and ^2 must each be measured in the presence of that dielectric. Miller also has treated the case of lossy coupled transmission systems. The expressions are lengthy and complicated and we believe that no substantial error is made in simply applying his conclusions to our case. If the attenuation constants ai and ao of the two transmission systems (helices) are equal, no change is required in our expressions; when they are unequal the total available power (in both helices) is most effectively reduced when ^4^'^l (2.4.11) Pc This fact may be made use of in designing coupled helix attenuators. 2.5 A Look at the Fields It may be advantageous to consider sketches of typical field distribu- tions in coupled helices, as in Fig. 2.1, before we go on to derive a quanti- tative estimate of the coupling factors actually obtainable in practice. Fig. 2.1(a) shows, diagrammatically, electric field lines when the coupled helices are excited in the fast or "longitudinal" mode. To set up this mode only, one has to supply voltages of like sign and equal ampli- tudes to both helices. For this reason, this mode is also sometimes called the "(+-f) mode." Fig. 2.1(b) shows the electric field lines when the helices are excited in the slow or "transverse" mode. This is the kind of field required in the transverse interaction type of traveling wave tube. In order to excite this mode it is necessary to supply voltages of equal amplitude and opposite signs to the helices and for this reason it is sometimes called the "(-| — ) mode." One way of exciting this mode consists in connecting one 140 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 helix to one of the two conductors of a balanced transmission line ("Lecher"-line) and the other hehx to the other. Fig. 2.1(c) shows the electric field configuration when fast and slow modes are both present and equally strongly excited. We can imagine the two helices being excited by a voltage source connected to the outer (a) FAST WAVE (longitudinal) (b) SLOW WAVE (transverse) (C) fast and slow waves combined SHOWING SPATIAL "BEAT" PHENOMENON Fig. 2.1 — Typical electric field distributions in coupled coaxial helices when thej^ are excited in: (a) the in-phase or lonfritudinal mode, (b) the out-of-phase or transverse mode, and (c) both modes equally. COUPLED HELICES 141 helix only at the far left side of the sketch. One, perfectly legitimate, view of the situation is that the RF power, initially all on the outer helix, leaks into the inner helix because of the coupling between them, and then leaks back to the outer helix, and so forth. Apart from noting the appearance of the stationary spatial beat (or interference) phenomenon these additional facts are of interest: 1) It is a simple matter to excite such a beat- wave, for instance, by connecting a lead to either one or the other of the helices, and 2) It should be possible to discontinue either one of the helices, at points where there is no current (voltage) on it, without causing reflec- tions. 2.6 A Simple Estimate of h and x How strong a coupling can one expect from concentric helices in prac- tice? Quantitatively, this is expressed by the values of the coupling fac- tors X and h, which we shall now proceed to estimate. A first crude estimate is based on the fact that slow-wave fields are known to fall off in intensity somewhat as c where (3 is the phase con- stant of the wave and r the distance from the surface guiding the slow wave. Thus a unit charge placed, say, on the inner helix, will induce a charge of opposite sign and of magnitude -Pib-a) on the outer helix. Here h = mean radius of the outer helix and a = mean radius of the inner. We note that the shunt mutual admittance coupling factor is negative, irrespective of the directions in which the helices are wound. Because of the similarity of the magnetic and electric field distributions a current flowing on the inner helix will induce a simi- larly attenuated current, of amplitude on the outer helix. The direction of the induced current will depend on whether the helices are woimd in the same sense or not, and it turns out (as one can verify by reference to the low-freciuency case of coaxial coupled coils) that the series mutual impedance coupling factor is nega- tive when the helices are oppositely wound. In order to obtain the greatest possible coupling between concentric helices, both coupling factors should have the same sign. This then re- fiuires that the helices should be wound in opposite directions, as has been pointed out by Pierce. When the distance between the two helices goes to zero, that is to say, 142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 .if they lie in the same surface, it is clear that both coupling factors h and x will go to unity. As pointed out earlier in Section 2.3, the choice of sign for h is arbi- trary. However, once a sign for h has been chosen, the sign of x is neces- sarily the opposite when the helices are wound in the same direction, and vice versa. We shall choose, therefore, the sign of the latter depending on whether the helices are wound in the same direction or not. In the case of unequal velocities, (5, the propagation constant, would be given by 1^ = VM~2 (2.6.2) 2.7 Strength of Coupling versus Frequency The exponential variation of coupling factors with respect to frequency (since /3 = co/y) has an important consequence. Consider the expression for the coupling phase constant /3. = I3{b + x) (2.3.8) or l/3e| = 2/3^"^^'""^ (2.7.1) The coupling wavelength, which is defined as Ac is, therefore, 27r (2.7.2) or Xc- -e X, = ;^ g(2./x)u.-«) (2.7.3) where X is the (slowed-down) RF wavelength on either helix. It is con- venient to multiply both sides of (2.7.1) with a, the inner helix radius, in order to obtain a dimensionless relation between /3c and /3: ^,a = 2/3ac~^''"''°^"" (2.7.4) This relalion is j)l()Ued on Fig. 2.2 for several values of b/a. COUPLED HELICES 143 3.00 2.75 2.50 2.25 2.00 /3ca 1.75 1.50 1.25 i.OO 0.75 0.50 0.25 ^^ — - / / / / / / / / / l-y / / / / / J / / / /(/Jc3)max / / / / / 1 / / / / / ^ / b = 1.5 / / / / f ^ , V / ^^ '" \ 1 -\ "^^^ — 1 75 L 2.0 ■\ / "^ ■-^ 3.0 — - 0.5 1.5 2.0 2.5 /3a 3.0 3.5 4.0 4.5 5.0 Fig. 2.2 — Coupling pliase-constant plotted as a function of the single helix phase-constant for synchronous helices for several values of b/a. These curves are based on simple estimates made in Section 2.7. There are two opposing tendencies determining the actual physical length of a coupling beat-wavelength: 1) It tends to grow with the RF wavelength, being proportional to it in the first instance; 2) Because of the tighter coupling possible as the RF wavelength increases in relation to the heli.x-to-helix distance, the coupling beat- wavelength tends to shrink. Therefore, there is a region where these tendencies cancel each other, and where one would expect to find little change of the coupling beat- wavelength for a considerable change of RF freciuency. In other words, the "bandwidth" over which the beat-wavelength stays nearly constant can be large. This is a situation naturally very desirable and favorable for any device in which we rely on power transfer from one helix to the other by 144 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 means of a length of overlap between them an integral number of half beat-wavelengths long. Ob^'iously, one will design the helices in such a way as to take advantage of this situation. Optimiun conditions are easily obtained by dijfferentiating ^c with respect to (3 and setting d^c/d^ equal to zero. This gives for the optimum conditions ^opt — 1 b — a (2.7.5) or Pc opt 2e h — a = 2e ')8opt (2.7.6) Equation (2.7.5), then, determines the ratio of the helix radii if it is re- quired that deviations from a chosen operating frequency shall have least effect. 2.8 Field Solutions In treating the problem of coaxial coupled helices from the transmis- sion line point of view one important fact has not been considered, namely, the dispersive character of the phase constants of the separate helices, /3i and fS-i . By dispersion we mean change of phase velocity with frequency. If the dispersion of the inner and outer helices were the same it would be of little consequence. It is well known, however, that the dispersion of a helical transmission line is a function of the ratio of helix radius to wavelength, and thus becomes a parameter to be considered. When the theory of wave propagation on a helix was solved by means of Maxwell's equations subject to the boundary condition of a helically conducting cylindrical sheath, the phenomenon of dispersion first made its appearance. It is clear, therefore, that a more complete theory of /i 'V^ 'TV Fig. 2.3 — ShoMtli liolix arrangement on which the field equations are based. COUPLED HELICES 145 coupled helices will require similar treatment, namely, Maxwell's equa- tions solved now with the boundary conditions of two cylindrical heli- cally conducting sheaths. As shown on Fig. 2.3, the inner helix is specified by its radius a and the angle 1^1 made by the direction of conductivity with a plane perpendicular to the axis; and the outer helix by its radius h (not to be confused with the mutual coupling coefficient 5) and its corresponding pitch angle i/'-j . We note here that oppositely wound helices require opposite signs for the angles \f/i and i/'o ; and, further, that helices with equal phase velocities will ha\'e pitch angles of about the same absolute magnitude. The method of solving Maxwell's equations subject to the above men- tioned boundary conditions is given in Appendix I. We restrict our- selves here to giving some of the results in graphical form. The most universally used parameter in traveling-wave tube design is a combination of parameters: /3oa cot \pi where (So = 27r/Xo , Xo being the free-space wavelength, a the radius of the inner helix, and xpi the pitch angle of the inner helix. The inner helix is chosen here in preference to the outer helix because, in practice, it will be part of a traveling-wave tube, that is to say, inside the tube envelope. Thus, it is not only less accessible and changeable, but determines the important aspects of a traveling-wave tube, such as gain, power output, and efficiency. The theory gives solutions in terms of radial propagation constants which we shall denote jt and yt (bj^ analogy with the transverse and longitudinal modes of the transmission line theory). These propagation constants are related to the axial propagation constants ^t and j3( by Of course, in transmission line theory there is no such thing as a radial propagation constant. The propagation constant derived there and de- noted r corresponds here to the axial propagation constant j^. By analogy with (2.4.5) the beat phase constant should be written How^ever, in practice ^0 is usually much smaller than j3 and Ave can there- fore write with little error iSfc = 7e — li for the beat phase constant. For practical purposes it is convenient to 146 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 J.OU _^^ 1 3i.Z0 COT ^2 _ „„„ ;:^ ^ COTV'i -0.90.. \ !^ -0.82,^ ^ :J^ 2.80 >^ . "^ "a"'" A s ^ N. \. \^ i & \ \ ^ ■^ 0.82 w ^ = -0.98 COT^, ^ 0.90 ^V ^ // / \ \ v J, / \ \ \, t \ f \ f 0.5 1.0 1.5 2.0 2.5 /3oaCOTi^, 3.0 3.5 4.0 4.5 Fig. 2.4.2 — Beat phase-constant plotted as a function of /3oa cot ^i-i . These curves result from the solution of the field equations given in the appendix. For hia = 1.5. line of Curve B in Fig. 2.5. Again the coupling phase constant j3c is given by the difference of the individual phase constants: ^cO- — /3oa cot \f/ — ya (2.9.1) which is plotted in Fig. 2.6. Now note that when /So <3C 7 this equation is accurate, for it represents a solution of the field equations for the helix. From the simple unsophisticated transmission line point of view no coupling between the two helices would, of course, have been expected, since the two helices are identical in every way and their mutual capacity and inductance should then be equal and opposite. Experiments confirm the essential correctness of (2.9.1). In one experi- ment, which was performed to measure the coupling wavelength for the l)ifilar helices, we used helices with a cot 1/' = 3.49 and a radius of 0.036 cm which gave a value, at 3,000 mc, of ^oa cot i^ = 0.51 . In these experi- ments the coupling length, L, defined by (/3oa cot xp — 7a) — = TT a was measured to be 15.7o as compared to a value of 13.5a from Fig. 2.6. At 4,000 mc the measured coupling length was 14.6a as compared to 148 THE BELL vSYSTEM TECHNICAL JOURNAL, JANUARY 1956 1.20 b a 1.76 ^ ^^ ^ ^ X, / y ^ ^ S. X 1.00 / V \ ^. -\ / / \ P> •^.82 0.80 r — \ ^ <5. COT^ COT^ N ^ = -0.9 1 k^ "^^^ 0.90 (0 0.60 a >s^ X <0 '^ 0.40 ^ . 0.20 0 < D 0 5 1 0 1 5 2 .0 2 .5 ^1 3.0 3.5 4.0 4 Fig. 2.4.3 — Beat phase-constant plotted as a function of ^^a cot -^x . These curves result from the solution of the field equations given in the appendix. For hi a = 1.75. 12.6a computed from Fig. 2.6, thus confirming the theoretical prediction rather well. The slight increase in coupling length is attributable to the dielectric loading of the helices which were supported in quartz tubing. The dielectric tends to decrease the dispersion and hence reduce /3,. . This is discussed further in the next section. 2.10 Effect of Dielectric Material hetween Helices In many cases which are of interest in practice there is dielectric ma- terial between the helices. In particular when coupled helices are used with traveling-wave tubes, the tube envelope, which may be of glass, quartz, or ceramic, all but fills the space between the two helices. It is therefore of interest to know whether such dielectric makes any difference to the estimates at which we arrived earlier. We should not be surprised to find the coupling strengthened by the presence of the di- electric, because it is known that dielectrics tend to rob RF fields from the surrounding space, leading to an increase in the energy flow through the dielectric. On the other hand, tlio dielectric tends to bind the fields closer to the conducting medium. To find a qualitative answer to this question we have calculated the relative coupling phase constants for two sheath helices of infinite radius separated by a distance "d" for 1) COUPLED HELICES 149 1.00 b -a-^.u ^ ^^ ^ ^ j^ ^ COT Tp2 ^ ^ C 0.60 )^ 1 0.40 m y ^ COT }^, ^ 1 > -^ S^ ^ . , — -0. 90 =- -- V ^, i ^ ^0^ 98 0.20 1 0 1 ( 3 0 5 1 0 1 5 2 0 >oac 2.5 3 .0 3 .5 4 0 4. Fig. 2.4.4 — Beat phase-constant plotted as a function of /3oa cot ^i . These curves result from the solution of the field equations given in the appendix. For b/a = 2.0. the case with dielectric between them having a relative dielectric con- stant e' = 4, and 2) the case of no dielectric. The pitch angles of the two helices were \p and —xp, respectively; i.e., the helices were assumed to be synchronous, and wound in the opposite sense. ■ Fig. 2.7 shows a plot of the ratio of /3,,.//3, to ^d^ versus /3o (f//2) cotiA, 1.00 0.80 to n «5. 0.60 II m i 0.40 0.20 b a-o.u ^y ^ >< ^ y COT ^2 ^ \>^ ^-- COT 5^, -^ r /, ^ ^ ==^ "^^ N^ ^^ y^ ^ ^ ^ f/ ^ ^ N. ^ -c ).90 -^ r \ -^ _ -o.s ?8 , ^ 0.5 1.0 f.5 2.0 2.5 /JoacoT;^, 3.0 3.5 4.0 4.5 Fig. 2.4.5 — ■ Beat phase-constant plotted as a function of (3o« cot ^\ . These curves result from the solution of the field equations given in the appendix. For Va = 3.0. 150 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 2.4 0.5 ).0 1.5 2.0 2. 5 3.0 3.5 /3oaCOT^ 4.0 4.5 Fig. 2.5 — Propagation constants for a bifilar helix plotted as a function of /3oa cot i/-! . The curves illustrate, (A) the dispersive character of the in-phase mode and, (B) the non-dispersive character of the out-of -phase mode. where ^^ is the coupling phase-constant in the presence of dielectric, /3j is the phase-constant of each helix alone in the presence of the same dielectric, ^c is the coupling phase-constant with no dielectric, and (3 is the phase constant of each helix in free space. In many cases of interest /3o(d/2) cot lA is greater than 1.2. Then 3£ + 1" _2£' + 2_ g—(v'2« '+2-2)^0 (dl2) cot \l/ (2.10.1) Appearing in the same figure is a similar plot for the case when there is a conducting shield inside the inner helix and outside the outer, and separated a distance, "s," from the helices. Note that c? = 6 — a. It appears from these calculations that the effect of the presence of dielectric between the helices depends largely on the parameter /So (d/2) cot \{/. For values of this parameter larger than 0.3 the coupling wave- length tends to increase in terms of circuit wavelength. For values smaller than 0.3 the opposite tends to happen. Note that the curve representing (2.10.1) is a fair approximation down to /3o(c?/2) cot i/' = 0.6 to the curve representing the exact solution of the field equations. J. W. Sullivan, in unpublished work, has drawn similar conclusions. COUPLED HELICES 151 2.11 The Conditions for Maximum Power Transfer The transmission line theory has led us to expect that the most efficient power transfer will take place if the phase velocities on the two helices, prior to coupling, are the same. Again, this would be true were it not for the dispersion of the helices. To evaluate this effect we have used the field equation to determine the parameter of the coupled helices which gives maximum power transfer. To do this we searched for combinations of parameters which give an equal current flow in the helix sheath for either the longitudinal mode or the transverse mode. This was suggested by L. Stark, who reasoned that if the currents were equal for the indi- vidual modes the beat phenomenon would give points of zero RF current on the helix. The values of cot T/'2/cot 4/i which are required to produce this condi- tion are plotted in Fig. 2.8 for various values of b/a. Also there are shown values of cot ^2/cot \{/i required to give equal axial velocities for the helices before they are coupled. It can be seen that the uncoupled velocity of the inner helix must be slightly slower than that of the outer. A word of caution is* necessary for these curves have been plotted without considering the effects of dielectric loading, and this can have a rather marked effect on the parameters which we have been discussing. The significant point brought out by this calculation is that the optimum u.^o r N 0.24 0.20 / \ \ / N \ \ \ \ \ \ \ \ \ L \ i \ \ \ \ \ \ \ \ \ V \ \ ,25\ yp \ \ \ XT' \ \ \ \ \ ^ \ ' \ \ \ \ \ \ \ ' 1 \ \ \ \ \ \ \ \ k \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ N, \ \ \ \ s > \ \ \ \, V, ^.0 \\ \ N s^5 \ \ 1 \ \ 1=1.2^ ^^^ ^ ^^ ^ '""'^-^ ^^ "^^ >., '- ^^ ^ ■ ^^ ^==^ 3. APPLICATION OF COUPLED HELICES When we come to describe devices which make use of coupled helices we find that they fall, quite naturally, into two separate classes. One COUPLED HELICES 155 class contains those devices which depend on the presence of only one of the two normal modes of propagation. The other class of devices depends on the simultaneous presence, in roughly equal amounts, of both normal modes of propagation, and is, in general, characterized by the words "spatial beating." Since spatial beating implies energy surging to and fro between inner and outer helix, there is no special problem in exciting both modes simultaneously. Power fed exclusively to one or the other /bo a COT jfi, Fiji;. 2.10 — The relation l)et\veen the impedance in terms of the transverse field between conpled helices excited in the out-of -phase mode, and the impedance in terms of the longitudinal field on the axis shown as a function of /3oa cot tpi . 156 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 helix will inevitably excite both modes equally. When it is desired to excite one mode exclusively a more difficult problem has to be solved. Therefore, in section 3.1 we shall first discuss methods of exciting one mode only before going on to discuss in sections 3.2 and 3.3 devices using one mode only. In section 3.4 we shall discuss devices depending on the simultaneous presence of both modes. 3.1 Excitation of Pure Modes 3.1.1 Direct Excitation In order to set up one or the other normal mode on coupled helices, voltages with specific phase and amplitudes (or corresponding currents) E|(f) E|(o) 10^ 5 10^ 10^ 10' 10 10" COT ip? ■ — = -0.90 COT i^, 1 / / 1 ' l-.o/ 1 L L 1 / J l\.2b / J / J / ^ '^ 3 A /ho a COT 1fi^ Fig. 2.11 — -The relation Ijetween the impedance in terms of the longitudinal field between couj)led helices excited in the in-phase mode, and the impedance in terms of the longitudinal field on the axis shown as a function of /3offl cot \pi . COUPLED HELICES 157 have to be supplied to each helix at the input end. A natural way of doing this might be by means of a two-conductor balanced transmission line (Lecher-line), one conductor being connected to the inner helix, the other to the outer helix. Such an arrangement would cause something like the transverse (-| — ) mode to be set up on the helices. If the two con- ductors and the balanced line can be shielded from each other starting some distance from the helices then it is, in principle, possible to intro- duce arbitrary amounts of extra delay into one of the conductors. A delay of one half period would then cause the longitudinal ( + + ) mode to be set up in the helices. Clearly such a coupling scheme would not be broad-band since a frequency-independent delay of one half period is not realizable. Other objections to both of these schemes are: Balanced lines are not generally used at microwave frequencies; it is difficult to bring leads through the envelope of a TWT without causing reflection of RF energy and without unduly encumbering the mechanical design of the tube plus circuits; both schemes are necessarily inexact because helices having different radii will, in general, require different voltages at either input in order to be excited in a pure mode. Thus the practicability, and success, of any general scheme based on the existence of a pure transverse or a pure longitudinal mode on coupled helices will depend to a large extent on whether elegant coupling means are available. Such means are indeed in existence as will be shown in the next sections. 3.1.2 Tapered Coupler A less direct but more elegant means of coupling an external circuit to either normal mode of a double helix arrangement is by the use of the so-called "tapered" coupler.^' ^' ^^ By appropriately tapering the relative propagation velocities of the inner and outer helices, outside the inter- action region, one can excite either normal mode by coupling to one helix only. The principle of this coupler is based on the fact that any two coupled transmission lines support two, and only two, normal modes, regardless of their relative phase velocities. These normal modes are characterized by unequal wave amplitudes on the two lines if the phase velocities are not equal. Indeed the greater the phase velocity difference and /or the smaller the coupling coefficient between the lines, the more their wave amplitudes diverge. Furthermore, the wave amplitude on the line with the slower phase velocity is greater for the out-of-phase or trans- verse normal mode, and the wave amplitude on the faster line is greater 158 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G for the longitudinal normal mode. As the ratio of phase constant to coupling constant approaches infinity, the ratio of the wave amplitudes on the two lines does also. Finally, if the phase velocities of, or coupling between, two coupled helices are changed gradually along their length the normal modes existing on the pair roughly maintain their identity evan though they change their character. Thus, by properly tapering the phase velocities and coupling strength of any two coupled helices one can cause the two normal modes to become two separate waves, one existing on each helix. For instance, if one desires to extract a signal propagating in the in- phase, or longitudinal, normal mode from two concentric helices of equal phase velocity, one might gradually increase the pitch of the outer helix and decrease that of the inner, and at the same time increase the diameter of the outer helix to decrease the coupling, until the longitudinal mode exists as a wave on the outer helix only. At such a point the outer helix may be connected to a coaxial line and the signal brought out. This kind of coupler has the advantage of being frequency insensitive ; and, perhaps, operable over bandwidths upwards of two octaves. It has the disadvantage of being electrically, and sometimes physically, quite long. 3.1.3 Stepped Coupler There is yet a third way to excite only one normal mode on a double helix. This scheme consists of a short length at each end of the outer helix, for instance, which has a pitch slightly different from the rest. This has been called a "stepped" coupler. The principle of the stepped coupler is this: If two coupled transmis- sion lines have unlike phase velocities then a wave initiated in one line can never be completely transferred to the other, as has been shown in Section 2.4. The greater the velocity difference the less will be the maxi- mum transfer. One can choose a velocity difference such that the maxi- mum power transfer is just one half the initial power. It is a characteristic of incomplete power transfer that at the point where the maximum trans- fer occurs the waves on the two lines are exactly either in-phase or out-of- phase, depending on which helix was initially excited. Thus, the condi- tions for a normal mode on two equal-velocity helices can be produced at the maximum transfer point of two unlike velocity helices by initiating a wave on only one of them. If at that point the helix pitches are changed to give equal phase velocities in both helices, with equal current or volt- age amplitude on both helices, either one or the other of the two normal modes will be propagated on the two helices from there on. Although the COUPLED HELICES 159 pitch and length of such a stepped coupler are rather critical, the re- quirements are indicated in the equations in Section 2.4. The useful bandwidth of the stepped coupler is not as great as that of the tapered variety, but may be as much as an octave. It has however the advantage of being very much shorter and simpler than the tapered coupler. 3.2 Low-Noise Transverse-Field Amplifier r One application of coupled helices which has been suggested from the very beginning is for a transverse field amplifier with low noise factor. In such an amplifier the EF structure is required to produce a field which is purely transverse at the position of the beam. For the transverse mode there is always such a cylindrical surface where the longitudinal field is zero and this can be obtained from the field equation of Appendix II. In Fig. 3.1 we have plotted the value of the radius f at which the longi- tudinal field is zero for various parameters. The significant feature of this plot is that the radius which specifies zero longitudinal field is not constant with frequency. At frequencies away from the design frequency the electron beam will be in a position where interaction with longitudinal components might become important and thus shotnoise power will be introduced into the circuit. Thus the bandwidth of the amplifier over which it has a good noise factor would tend to be limited. However, this effect can be reduced by using the smallest practicable value of b/a. Section 2.12 indicates that the impedance of the transverse mode is very high, and thus this structure should be well suited for transverse field amplifiers. 3.3 Dispersive Traveling-Wave Tube Large bandwidth is not always essential in microwave amplifiers. In particular, the enormous bandwidth over which the traveling-wave tube is potentially capable of amplifying has so far found little application, while relatively narrow bandwidths (although quite wide by previous standards) are of immediate interest. Such a relatively narrow band, if it is an inherent electronic property of the tube, makes matching the tube to the external circuits easier. It may permit, for instance, the use of non-reciprocal attenuation by means of ferrites in the ferromagnetic resonance region. It obviates filters designed to deliberately reduce the band in certain applications. Last, but not least, it offers the possibility of trading bandwidth for gain and efficiency. A very simple method of making a traveling-wave tube narrow-band 160 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 0.5 1.W 1.8 ^ ^ 1.7 COT \p. _^ ^ ^ <^ T^ = -0. COT ^, 82^ ^ ^ ^ ^ l=- 1.6 ^ ^ ^ -0.90 ^ ^ ^ ^ * 1.5 — - -0.9 8 ' ^-' ' ^ ^ 1.4 -^ COT ^i'2 ^-^ = -0.82 COT UJ^ , -0.9j , ^- 1.3 . — "ZH ' 1 -0.98 1.2 _ "71 | = ,.25 — ^= CO' r 1//. — T-" H COT 5^, = -0.82 - -0.90 ■ ' / ^ 1 / / i.n -0 .98 ~ 1.0 1.5 2.0 2.5 3.0 /3o a COT j^. 3.5 4.0 4.5 5.0 Fig. 3.1 — The radius r at which the longitudinal field is zero for transversely excited coupled coaxial helices. is by using a dispersive circuit, (i.e. one in which the phase velocity varies significantly with frequency). Thus, we obtain an amplifier that can be limed by varying the beam voltage; being dispersive we should also expect a low group velocity and therefore higher circuit impedance. Calculations of the phase velocities of the normal modes of coupled concentric helices presented in the appendix show that the fast, longitu- dinal or (+ + ) mode is highly dispersive. Given the geometry of two such coupled helices and the relevant data on an electron beam, namely current, voltage and beam radius, it is possible to arrive at an estimate of the dependence of gain on frecjuency. Experiments with such a tube showed a Ijandwidth 3.8 times larger than the simple estimate would show. This we ascribe to the presence COUPLED HELICES 161 of the dielectric between the helices in the actual tube, and to the neglect of power propagated in the form of spatial harmonics. Nevertheless, the tube operated satisfactorily with distributed non- reciprocal ferrite attenuation along the whole helix and gave, at the center frequency of 4,500 mc/s more than 40 db stable gain. The gain fell to zero at 3,950 mc/s at one end of the band and at 4,980 mc/s at the other. The forward loss was 12 db. The backward loss was of the order of 50 db at the maximum gain frequency. 3.4 Devices Using Both Modes In this section we shall discuss applications of the coupled-helix princi- ple which depend for their function on the simultaneous presence of both the transverse and the longitudinal modes. When present in substantially equal magnitude a spatial beat-phenomenon takes place, that is, RF power transfers back and forth between inner and outer helix. Thus, there are points, periodic with distance along each helix, where there is substantially no current or voltage; at these points a helix can be terminated, cut-off, or connected to external circuits without detriment. The main object, then, of all devices discussed in this section is power transfer from one helix to the other; and, as will be seen, this can be ac- complished in a remarkably efficient, elegant, and broad-band manner. 3.4.1 Coupled-Helix Transducer It is, by now, a well known fact that a good match can be obtained between a coaxial line and a helix of proportions such as used in TWT's. A wire helix in free space has an effective impedance of the order of 100 ohms. A conducting shield near the helix, however, tends to reduce the helix impedance, and a value of 70 or even 50 ohms is easily attained. Pro\'ided that the transition region between the coaxial line and the helix does not present too abrupt a change in geometry or impedance, relatively good transitions, operable over bandwidths of several octaves, can l)e made, and are used in practice to feed into and out of tubes em- ploying helices such as TWT's and backward-wave oscillators. One particularly awkward point remains, namely, the necessity to lead the coaxial line through the tube envelope. This is a complication in manufacture and reciuires careful positioning and dimensioning of the helix and other tube parts. Coupled helices offer an opportunity to overcome this difficulty in the form of the so-called coupled-helix transducer, a sketch of which is shown in Fig. 3.2. As has been shown in Section 2.3, with helices having 162 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 the same velocity an overlap of one half of a beat wavelength will result in a 100 per cent power transfer from one helix to the other. A signal in- troduced into the outer helix at point A by means of the coaxial line will be all on the inner helix at point B, nothing remaining on the outer helix. At that point the outer helix can be discontinued, or cut off; since there is no power there, the seemingly violent discontinuity represented by the 'open" end of the helix will cause no reflection of power. In practice, un- fortunately, there are always imperfections to consider, and there will often be some power left at the end of the coupler helix. Thus, it is de- sirable to terminate the outer helix at this point non-reflectively, as, for instance, by a resistive element of the right value, or by connecting to it another matched coaxial line which in turn is then non-reflectively ter- minated. It will be seen, therefore, that the coupled-helix transducer can, in principle, be made into an efficient device for coupling RF energy from a coaxial line to a helix contained in a dielectric envelope such as a glass tube. The inner helix will be energized predominantly in one direction, namely, the one away from the input connection. Conversely, energy traveling initially in the inner helix will be transferred to the outer, and made available as output in the respective coaxial line. Such a coupled- helix transducer can be moved along the tube, if required. As long as the outer helix completely overlaps the inner, operation as described above should be assured. By this means a new flexibility in design, operation and adjustment of traveling-wave tubes is obtained which could not be achieved by any other known form of traveling-wave tube transducer. Naturally, the applications of the coupled-helix transducer are not restricted to TWT's only, nor to 100 per cent power transfer. To obtain Fig. 3.2 — A simple coupled helix transducer. COUPLED HELICES 1G3 power transfer of proportions other than 100 per cent two possibilities are open: either one can reduce the length of the synchronous coupling helix appropriately, or one can deliberately make the helices non-syn- chronous. In the latter case, a considerable measure of broad-banding can be obtained by making the length of overlap again equal to one half of a beat-wavelength, while the fraction of power transferred is deter- mined by the difference of the helix velocities according to 2.4.7. An application of the principle of the coupled-helix transducer to a variable delay line has been described by L. Stark in an unpublished memo- randum. Turning again to the complete power transfer case, we may ask: How broad is such a coupler? In Section 2.7 we have discussed how the radial falling-off of the RF energy near a helix can be used to broad-band coupled-helix devices which depend on relative constancy of beat-wavelength as frequency is varied. On the assumption that there exists a perfect broad-band match between a coaxial line and a helix, one can calculate the performance of a coupled-helix transducer of the type shown in Fig. 3.2. Let us define a center frequency co, at which the outer helix is exactly one half beat-wavelength, \b , long. If oj is the frequency of minimum beat wavelength then at frequencies coi and co2 , larger and smaller, respectively, than co, the outer helix will be a fraction 5 shorter than }i\b , (Section 2.7). Let a voltage amplitude, Vo , exist at the point where the outer helix is joined to the coaxial line. Then the magnitude of the voltage at the other end of the outer helix will be | F2 • sin (x5/2) | which means that the power has not been completely transferred to the inner helix. Let us assume complete reflection at this end of the outer helix. Then all but a fraction of the reflected power will be transferred to the inner helix in a reverse direction. Thus, we have a first estimate for the "directivity" defined as the ratio of forward to backward power (in db) introduced into the inner helix: D = 10 log sin" (3.4.1.1) We have assumed a perfect match between coaxial line and outer helix; thus the power reflected back into the coaxial line is proportional to sin^(x5/2). Thus the reflectivity defined as the ratio of reflected to incident power is given in db by i^ = 10 log sin' ^ (3.4.1.2) 164 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 For the sake of definiteness, let us choose actual figures: let /3a = 2.0. and hi a = 1.5. And let us, arbitrarily, demand that R always be less than -20 db. This gives sin (7r5/2) < 0.316 and 7r5/2 < 18.42° or 0.294 radians, 8 < 0.205. With the optimum value of (Sea = 1.47, this gives the mini- mum permissible value of I3ca of 1.47/(1 + 0.205) = 1.22. From the graph on Fig. 2.2 this corresponds to values of jSa of 1.00 and 3.50. Therefore, the reflected power is down 20 db over a frequency range of aj2/aji = 3,5 to one. Over the same range, the directivity is better than 10 to one. Suppose a directivity of better than 20 db were required. This requires sin (7r5/2) = 0.10, 8 = 0.0638 and is obtained over a fre- quency range of approximately two to one. Over the same range, the reflected power would be down by 40 db. In the above example the full bandwidth possibilities have not been used since the coupler has been assumed to have optimum length when jSctt is maximum. If the coupler is made longer so that when I3ca is maxi- mum it is electrically short of optimum to the extent permissible by the quality requirements, then the minimum allowable (S^a becomes even smaller. Thus, for h/a =1.5 and directivity 20 db or greater the rea- lizable bandwidth is nearly three to one. When the coupling helix is non-reflectively terminated at both ends, either by means of two coaxial lines or a coaxial line at one end and a resistive element at the other, the directivity is, ideally, infinite, irrespec- tive of frequency; and, similarly, there will be no reflections. The power transfer to the inner helix is simply proportional to cos (t8/2). Thus, under the conditions chosen for the example given above, the coupled- helix transducer can approach the ideal transducer over a considerable range of frequencies. So far, we have inspected the performance and bandwith of the coupled-helix transducer from the most optimistic theoretical point of view. Although a more realistic approach does not change the essence of our conclusions, it does modify them. For instance, we have neglected dispersion on the helices. Dispersion tends to reduce the maximum at- tainable bandwidth as can be seen if Fig. 2.4.2 rather than Fig. 2.2 is used in the example cited above. The dielectric that exists in the annular region between coupled concentric helices in most practical couplers may also affect the bandwidth. In practice, the performance^ of coupled-hc^lix transducers has been short of the ideal. In the first place, the match from a coaxial line to a helix is not perfect. Secondly, a not inappreciable fraction of the RF power on a real wire helix is propagated in the form of spatial harmonic COUPLED HELICES 165 28 26 24 22 20 18 )6 in _i LU m (4 u 12 10 r\ \ \ \ \ ' * / ' * / 1 t / [\ n [ 1 I 1 1 \j ^ \ Wf \ 1 \ \ I / I / \ / \1^ U~ / / / \ .' 1 A \J \- / \ \ A / Vi \ \ \ 1 1 / 1 p OUPLER DIRECTIVITY ETURN LOSS \ \ 1 J \ A V I / l 1.5 2.5 3 4 FREQUENCY IN KILOMEGACYCLES Fig. 3.3 — • The return loss and directivity of an experimental 100 per cent coupled-helix transducer. wave components which have variations with angle around the helix- axis, and coupling between such components on two helices wound in opposite directions must be small. Finally, there are the inevitable me- chanical inaccuracies and misalignments. Fig. 3.3 shows the results of measurements on a coupled-helix trans- ducer with no termination at the far end. 3.4.2 Coupled-Helix Attenuator In most TWT's the need arises for a region of heavy attenuation somewhere between input and output; this serves to isolate input and output, and prevents oscillations due to feedback along the circuit. Be- cause of the large bandwidth over which most TWT's are inherently capable of amplifying, substantial attenuation, say at least 60 db, is 166 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 required over a bandwidth of maybe 2 octaves, or even more. Further- more, such attenuation should present a very good match to a wave on the heHx, particularly to a wave traveling backwards from the output of the tube since such a wave will be amplified by the output section of the tube. Another requirement is that the attenuator should be physically as short as possible so as not to increase the length of the tube unneces- sarily. Finally, such attenuation might, with advantage, be made movable during the operation of the tube in order to obtain optimum performance, perhaps in respect of power output, or linearity, or some other aspect. Coupled-helix attenuators promise to perform these functions satis- factorily. A length of outer helix (synchronous with the inner helix) one half of a beat wavelength long, terminated at either end non-reflectively, forms a very simple, short, and elegant solution of the coupled-helix attenuator problem. A notable weakness of this form of attenuator is its relatively narrow bandwidth. Proceeding, as before, on the assumption that the attenuator is a fraction 8 larger or smaller than half a beat wavelength at frequencies coi and W2 on either side of the center frequency co, we find that the fraction of power transferred from the inner helix to the attenu- ator is then given by (1 — sin" (ir8/2)). The attenuation is thus simply A = sin^ (I) For helices of the same proportions as used before in Section 3.4.1, we find that this will give an attenuation of at least 20 db over a frequency band of two to one. At the center frequency, coo , the attenuation is in- finite; — in theory. Thus to get higher attenuation, it would be necessary to arrange for a sufficient number of such attenuators in tandem along the TWT. More- over, by properly staggering their lengths within certain ranges a wdder attenuation band may be achieved. The success of such a scheme largely depends on the ability to terminate the helix ends non-reflectively. Con- siderable work has been done in this direction, but complete success is not yet in sight. Another basically different scheme for a coupled-helix attenuator rests on the use of distributed attenuation along the coupling helix. The diffi- culty with any such scheme lies in the fact that unequal attenuation in the two coupled helices reduces the coupling between them and the moi'c they differ in respect to attenuation, the less the coupling. Naturally, one COUPLED HELICES 167 would wish to have as Httle attenuation as practicable associated with the inner helix (inside the TWT). This requires the attenuating element to be associated with the outer helix. Miller has shown that the maxi- mum total power reduction in coupled transmission systems is obtained when ai — 0:2 where ai and 012 are the attenuation constants in the respective systems, and ^b the beat phase constant. If the inner helix is assumed to be loss- less, the attenuation constant of the outer helix has to be effectively equal to the beat wave phase constant. It turns out that 60 db of attenuation requires about 3 beat wavelengths (in practice 10 to 20 helix wave- lengths). The total length of a typical TWT is only 3 or 4 times that, and it will be seen, therefore, that this scheme may not be practical as the only means of providing loss. Experiments carried out Avith outer helices of various resistivities and thicknesses by K. M. Poole (then at the Clarendon Laboratory, Oxford, England) tend to confirm this conclusion. P. D. Lacy" has described a coupled helix attenuator which uses a multifilar helix of resistance material together with a resistive sheath between the helices. Experiments were performed at Bell Telephone Laboratories with a TWT using a resistive sheath (graphite on paper) placed between the outer helix and the quartz tube enclosing the inner helix. The attenua- tions were found to be somewhat less than estimated theoretically. The attenuator helix was movable in the axial direction and it w^as instructive to observe the influence of attenuator position on the power output from the tube, particularly at the highest attainable power level. As one might expect, as the power level is raised, the attenuator has to be moA-ed nearer to the input end of the tube in order to obtain maximum gain and power output. In the limit, the attenuator helix has to be placed right close to the input end, a position which does not coincide with that for maximum low-level signal gain. Thus, the potential usefulness of the feature of mobility of coupled-helix elements has been demonstrated. 4. CONCLUSION In this paper we have made an attempt to develop and collect together a considerable body of information, partly in the form of equations, partl}^ in the form of graphs, which should be of some help to workers in the field of microwave tubes and devices. Because of the crudity of the assumptions, precise agreement between theory and experiment has not 168 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 been att-aiiu>(l iiur can it l)c expected. Nevertheless, the kind of physical phenomena occurring with coupled helices are, at least, qualitatively described here and should permit one to develop and construct various types of (lexices with fair chance of success. ACKNOWLEDGEMENTS As a final note the authors wish to express their appreciation for the patient work of Mrs. C. A. Lambert in computing the curves, and to G. E. Korb for taking the experimental data. Appendix i i. solution of field equations In this section there is presented the field equations for a transmission system consisting of two helices aligned with a common axis. The propa- gation properties and impedance of such a transmission system are dis- cussed for various ratios of the outer helix radius to the inner helix radius. This system is capable of propagating two modes and as previously pointed out one mode is characterized by a longitudinal field midway between the two helices and the other is characterized by a transverse field midway between the tw^o helices. The model which is to be treated and shown in Fig. 2.3 consists of an inner helix of radius a and pitch angle \pi which is coaxial with the outer helix of radius 6 and pitch angle \j/2 . The sheath helix model will be treated, wherein it is assumed that helices consist of infinitely thin sheaths which allow for ciuTent flow- only in the direction of the pitch angle \p. The components of the field in the region inside the inner helix, be- tween the two helices and outside the outer helix can be written as follows — inside the inner helix H,, = BrIoM (1) E., = B^hM (2) H,, = j - BMyr) (3) 7 Hr, = ^^ BMyr) (4) 7 E,, = -j "^ BMyr) (5) 7 Er, = -^ BJ,(rr) (()) 7 COUPLED HELICES 169 and between the two helices H,, = BMrr) + BJuirr) ' (7) E., = BJoiyr) + B^oiyr) (8) H,, = ^~ [B,h(yr) - B^^(yr)] (9) 7 Hr, = -^ [53/1(7/0 - BJuiyr)] (10) 7 E,, = - J ^ [B^hiyr) - BJuiyr)] (11) 7 Er, = -^ [BMyr) - BJv,{yr)\ (12) 7 and outside the outer hehx H.^ = B,Ko(yr) (13) E,, = 58/vo(7r) (14) ^.s = -J- BsK,{yr) (15) 7 Hr, = ^^ 5,Ki(7r) (16) 7 ^,, = i — BJuiyr) (17) 7 ^r« = ^^ 58Ki(7r) (18) 7 With the sheath helix model of current flow only in the direction of wires we can specify the usual boundary conditions that at the inner and outer helix radius the tangential electric field must be continuous and per- pendicular to the wires, whereas the tangential component of magnetic field parallel to the current flow must be continuous. These can be written as E, sin t/' + E^ cos ^ = 0 (19) ' E, , E^ and (H, sin \f/ -f H^ cos \p) be equal on either side of the helix. By applying these conditions to the two helices the following equations are obtained for the various coefficients. 170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 First, we will define a more simple set of parameters. We will denote Io(ya) by /oi and h{yh) by /02 , etc. Further let us use the notation introduced by Humphrey, Kite and James" in his treatment of coaxial helices. Poi ^ laiKoi P02 = ToiKa2 Rq = I01K02 Pn = InKn P12 = InKu Ri = /iii^i2 and define a common factor (C.F.) by the equation r(/3oa cot hY p p (/3oa cot ^pif cot i/'z „ r, \_ (yay {jay cot t^i + Ro' — PoiP (20) .,] (21) With all of this we can now write for the coefficients of equations 1 through 18: y ju j8oa cot \pi 1 02 U iQoa cot 1^1 7oi/vi2 RiSoa cot i^i) y M ""to C.F. L ^4 _ _ • / £_ /3oa cot 1^1 /pi/ii r( B^~ -^ T M 7^ C.F. L" (7a)^ 5 5 (7a)'^ (/3oa cot 1^2)^ cot 1A2 p cot ;^i J P12 — jPo2 ■] B5 B, Bt Ro C.F. Ro — ((Soa cot xl/iY cot 1/' (7a^) (/3oa cot 1^2) cot l/' ;«'] (7a)^ 12 — -P02 B7 _ • . /£ i3oa cot lAi 1 /oi r 5; ~ "^ y M 7a C.F. K12 L Bs _ (|8oa cot i/'i)" cot 1/^2 /pi "" B2 {yay coT^i C.F.Po P02R1 — P02R1 - cot l/'2 cot i/'i cot l// cot \l/ 2R0 - P12R0 (22) (23) (24) (25) (26) (27) (28) The last equation necessary for the solution of our field problem is the transcendental equation for the propagation constant, 7, which can be COUPLED HELICES 171 written Ro — (i8o a cot \J/iY cot ^2 „ (yaY cot 4/1 [ = P02 - (jSo a cot \p2) D ? Vi -^ 12 Poi - (/3oa cot ^0" (yay _ (29) 11 The solutions of this equation are plotted in Fig. 4.1. There it is seen that there are two values of 7, one, yt , denoting the slow mode with transverse fields between helices and the other, yt , denoting the fast mode with longitudinal fields midway between the two helices. 5.0 4.S 4.0 3.5 3.0 ra 2.6 2.0 1.5 1.0 0.5 4 = 1.25 // / COT 5^2 COT ^1 0.82 0.90 0.98 ^ #■ /t // A na / f A y f / r / I / if < A / -<^^ ^ •y L -- •=**^ 0.5 1.0 1.6 2.0 2.5 3.0 /3o a COT yj 3.5 4.0 4.5 5.0 Fig. 4.1.1 —-The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a function of |3oa cot ^i-i for several values of hja = 1.25. 172 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 These equations can now be used to compute the power flow as defined by P = }4 Re j E XH' which can be written in the form dA (30) r^;^(o)T L ^'p J fo © ^^-' ''' (31) where [F{ya, yb)] = (( W + (i8oa cot i/' {yar ^ /n^) (In' - /oi/2i)(C.F.)- - A'02' + 240 (C.F.)' (i8oa cot 1/^1)' (t«)'^ /or/n- r (80a cot ;^iY ' /Vl2" i^O - ya ((Soft cot i^i)' cot \p2 (ya)'' cot i/'i Rx - ) (/02/22 — /12') 4" (/ii — /01/21) , /p (/3oa cot 1A1)- cot \i/2 p Wp (^0^ ^'0^' "^2)'' p (ya)'^ cot i/'i (7a)^ ( - ) i'lInKu + /02/V22 + /22X02) — (2/iiKii + /01K21 + /21/voi) ot ^2)'^ p T (32) 2 , (^ofl cot l/'i) J ■> •'01 i- 7 r;; ^11 (l3oa cot - I (K02K22 — K12 ) — (K01K21 — Kn) .a, + (/3oa cot i^i)" A^ ■ 2 , (/Soa cot i/'2)" J 2 J. 2 (7a)'^ cot 1/^2 p J. I 02itl — -— r- i 12A0 cot 1^1 [/Vo2A'22 — /V12"] In (32) we find the power in the transverse mode by using values of COUPLED HELICES 173 5.0 0.5 2.0 2.5 3.0 /3o a COT y/ 5.0 Fig. 4.1.2 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a function of ^ofl cot \}/i when h/a — 1.50. yt obtained from (29) and similarly the power in the longittidinal mode is found by using values of yi . II. FINDING r When coaxial helices are used in a transverse field amplifier, only the transverse field mode is of interest and it is important that the helix parameters be adjusted such that there is no longitudinal field at some radius, f, where the cylindrical electron beam will be located. This condi- tion can be expressed by equating Ez to zero at r = f and from (8) BMyr) + B^,{yf) = 0 (33) 174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 which can be written with (25) and (26) as (jSott cot ipiY cot \f/2 K(i2 Ri [ 02 ilO (7a)- cot \{/i = /oi Ri loM (/3oa cot \l/2)- ■I 02 — 7 rr, rn (34) Koiyf) This equation together with (29) enables one to evaluate f/a versus j8oa cot \l/i for various ratios of b/a and cot i^2/cot xpi . The results of these calculations are shown in Fig. 3.1. 5.0 4.5 4.0 3.5 3.0 7a 2.5 2,0 0.5 Fig. 4.1. .3 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a finiclion of 0oa cot \{/i when b/a = 1.75. i COUPLED HELICES 175 5.0 7a 2.0 2.5 3.0 /Oo <3 COT ^, 3.5 4.0 4.5 5.0 Fig. 4.1.4 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coaxial sheath helices given as a function of /3oa cot yp\ when 6/a = 2.0. III. COMPLETE POWER TRANSFER For coupled heli.x applications we require the coupled helix parame- ters to be adjusted so that RF power fed into one helix alone will set up the transverse and longitudinal modes equal in amplitude. For this condition the power from the outer helix will transfer completely to the inner helix. The total current density can be written as the sum of the current in the longitudinal mode and the transverse mode. Thus for the inner helix we have -i&li J a = Jate-'''' + Jate .-J^<2 (35) 17G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 7?, 2.5 Fig. 4.1.5 — The radial propagation constants associated with the transverse and longitudinal modes on coupled coa.xial sheath helices given as a function of /3oa cot i/-! when hi a = 3.0. and for the outer helix For complete power transfer we ask that •J hi — J hi when Jo is zero at the input {z = 0) or Jbt _ Jbt J at J at \ (36) \ (37) COUPLED HELICES 177 Now J at is equal to the discontinuity in the tangential component of magnetic field which can be written at r = a J at = {H,z cos ^i — //^5 sin \pi) — (H,i cos i/'i - H^o sin \f/i) \^'hich can be written as Ja( = - (H,i - H,3)a((cot i/'i + tau xj/i) slu \Pi (38) and similarily at r = h Jb( = — (H^7 — H,s)b({cot \p2 + tan 4^2) sin i/'2 (39) Equations (38) and (39) can be combined with (37) to give as the condi- tion for complete power transfer At = -At (40) where ^ = V (yay / ni) (T J^ _i- r V \( T? (/3oa cot cos ^ilJZ^ , (48) and J, = J.ce-'''^'^'^''"' cos M/3i^ (49) where we have defined iSfcO = {yta — jta) (50) This value of /S^ is plotted versus /3oa cot i/'i in Fig. 2.4. 178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 BIBLIOGRAPHY 1. J. R. Pierce, Traveling Wave Tubes, p. 44, Van Nostrand, 1950. 2. R. Kompfner, Experiments on Coupled Helices, A. E. R. E. Report No. G/M98, Sept., 1951. 3. R. Kompfner, Coupled Helices, paper presented at I. R. E. Electron Tube Conference, 1953, Stanford, Cal. 4. G. Wade and N. Rynn, Coupled Helices for Use in Traveling-Wave Tubes, I.R.E. Trans, on Electron Devices, Vol. ED-2, p. 15, July, 1955. 5. S. E. Miller, Coupled Wave Theory and Waveguide Applications, B.S.T.J., 33, pp. 677-693, 1954. 6. M. Chodorow and E. L. Chu, The Propagation Properties of Cross-Wound Twin Helices Suitable for Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., Stanford Univ., June, 1953. 7. G. M. Branch, A New Slow Wave Structure for Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., Stanford Univ., June, 1953. G. M. Branch, E.xperimental Observation of the Properties of Double Helix Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., Univ. of Maine, June, 1954. 8. J. S. Cook, Tapered Velocity Couplers, B.S.T.J. 34, p. 807, 1955. 9. A. G. Fox, Wave Coupling by Warped Normal Modes, B.S.T.J., 34, p. 823, 1955. 10. W. H. Louisell, Analysis of the Single Tapered Mode Coupler, B.S.T.J., 34, p. 853. 11. B. L. Humphrey's, L. V. Kite, E. G. James, The Phase Velocity of Waves in a Double Helix, Report No. 9507, Research Lab. of G.E.C., England, Sept., 1948. 12. L. Stark, A Helical-Line Phase Shifter for Ultra-High Frequencies, Technical Report No. 59, Lincoln Laboratory, M.LT., Feb., 1954. 13. P. D. Lacy, Helix Coupled Traveling-Wave Tube, Electronics, 27, No. 11, Nov.. 1954. Statistical Techniques for Reducing the Experiment Time in Reliability Studies By MILTON SOBEL (Manuscript received September 19, 1955) Given two or more processes, the units from which fail in accordance with an exponential or delayed exponential law, the problem is to select the partic- ular process with the smallest failure rate. It is assumed that there is a com- mon guarantee period of zero or positive duration during which no failures occur. This guarantee period may be known or unknown. It is desired to accomplish the above goal in as short a time as possible without invalidating certain predetermined probability specifications. Three statistical techniques are considered for reducing the average experiment time needed to reach a decision. 1 . One technique is to increase the initial number of units put on test. This technique will substantially shorten the average experiment time. Its effect on the probability of a correct selection is generally negligible and in some cases there is no effect. 2. Another technique is to replace each failure immediately by a new unit from the same process. This replacement technique adds to the book- keeping of the test, but if any of the population variances is large (say in comparison with the guarantee period) then this technique will result in a substantial saving in the average experiment time. 3. A third technique is to use an appropriate sequential procedure. In many problems the sequential procedure results in a smaller average experi- ment time than the best non-sequential procedure regardless of the true failure rates. The amount of saving depends principally on the ^'distance'" between the smallest and second smallest failure rates. For the special case of two processes, tables are given to show the proba- bility of a correct selection and the average experiment time for each of three types of procedures. Numerical estimates of the relative efficiency of the procedures are given by computing the ratio of the average experiment time for two procedures of different type with the same initial sample size and satisfying the same probability specification. 179 180 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 INTRODUCTION This paper is concerned with a study of the advantages and disad- vantages of three statistical techniques for reducing the average dura- tion of hfe tests. These techniques are: 1. Increasing the initial number of units on test. 2. Using a replacement technique. 3. Using a sequential procedure. To show the advantages of each of these techniques, we shall consider the problem of deciding which of two processes has the smaller failure rate. Three different types of procedures for making this decision will be considered. They are: Ri , A nonsequential, nonreplacement type of procedure E,2 , A nonsequential, replacement type of procedure Rs , A sequential, replacement type of procedure Within each type wq will consider different values of n, the initial number of units on test for each process. The effect of replacement is shown by comparing the average experiment time for procedures of type 1 and 2 with the same value of n and comparable probabilities of a correct selection. The effect of using a sequential rule is shown by com- paring the average experiment time for procedures of type 2 and 3 with the same value of n and comparable probabilities of a correct selection. ASSUMPTIONS 1. It is assumed that failure is clearly defined and that failures are recognized without any chance of error. 2. The lifetime of individual units from either population is assumed to follow an exponential density of the form f{x; e,g) =\ e-^^-")/" iov x -^ g f(x; e,g) = 0 iorx 0 represents the unknown parameter which distinguishes the two different processes. Let Ox ^ do denote the ordered values of the unknown parameter 6 for the two processes; then the ordered failure rates are given by Xi = 1/(01 + {/) ^ Xo = 1/(02 -f g) (2) 3. It is not known which process has the parameter di and which has the parameter dt . REDUCING TIME IN RELIABILITY STUDIES 181 4. The parameter g is assumed to be the same for both processes. It may be known or unknown. 5. The initial number n of units put on test is the same for both pro- cesses. 6. All units have independent lifetimes, i.e., the test environment is not such that the failure of one unit results in the failure of other units on test. 7. Replacements used in the test are assumed to come from the same population as the units they replace. If the replacement units have to sit on a shelf before being used then it is assumed that the replacements are not affected by shelf-aging. CONCLUSIONS 1. Increasing the initial sample size n has at most a negligible effect on the probability of a correct selection. It has a substantial effect on the average experiment time for all three types of procedures. If the value of n is doubled, then the average time is reduced to a value less than or equal to half of its original value. 2. The technique of replacement always reduces the average experi- ment time. This reduction is substantial when ^ = 0 or when the popu- lation variance of either process is large compared to the value of g. This decrease in average experiment time must always be weighed against the disadvantage of an increase in bookkeeping and the necessity of having the replacement units available for use. 3. The sequential procedure enables the experimenter to make rational decisions as the evidence builds up without waiting for a predetermined number of failures. It has a shorter average experiment time than non- sequential procedures satisfying the same specification. This reduction brought about by the sequential procedure increases as the ratio a of the two failure rates increases. In addition the sequential procedure always terminates with a decision that is clfearly convincing on the basis of the observed results, i.e., the a posteriori probability of a correct selection is always large at the termination of the experiment. SPECIFICATION OF THE TEST Each of the three types of procedures is set up so as to satisfy the same specification described below. Let a denote the true value of the ratio 61/62 which by definition must be greater than, or equal to, one. It turns out that in each type of procedure the probability of a correct selection depends on 6i and 62 only through their ratio a. 182 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1950 1. The experimenter is asked to specify the smallest value of a (say it is a* > I) that is worth detecting. Then the interval (1, a*) represents a zone of indifference such that if the true ratio a lies therein then we would still like to make a correct selection, but the loss due to a wrong selection in this case is negligible. 2. The experimenter is also asked to specify the minimum value P* > \'2 that he desires for the probability of a correct selection whenever a ^ a*. In each type of procedure the rules are set up so that the proba- bility of a correct selection for a = a* is as close to P* as possible without being less than P*. The two constants a* > 1 and \2 < P* < 1 are the only quantities specified by the experimenter. Together they make up the specification of the test procedure. EFFICIENCY If two procedures of different type have the same value of n and satisfy the same specification then we shall regard them as comparable and their relative efficiency will be measured by the ratio of their average experiment times. This ratio is a function of the true a but we shall consider it only for selected values of a, namely, a = 1, a = a* and a = CO . PROCEDURES OF TYPE Ri — • NONSEQUENTIAL, NONREPLACEMENT "The same number n of units are put on test for each of the two pro- cesses. Experimentation is continued until either one of the two samples produces a predetermined number r (r ^ n) of failures. Experimenta- tion is then stopped and the process with fewer than r failures is chosen to be the better one." Table I — Probability of a Correct Selection — Procedure Type Ri (a = 2, any g '^ 0, to be used to obtain r for a* = 2) n r = 1 r = 2 r = 3 r = i 1 0.667 . — . 2 0.667 0.733 — — 3 0.667 0.738 0.774 — 4 0.667 0.739 0.784 0.802 10 0.667 0.741 0.78!) 0.825 20 0.667 0.741 0 . 790 0.826 00 0.667 0.741 0.790 0.827 Note: The value for ?• = 0 is obviously 0.500 for any n. REDUCING TIME IN RELIABILITY STUDIES 183 We shall assume that the number n of units put on test is determined by non -statistical considerations such as the availability of units, the availability of sockets, etc. Then the only unspecified number in the above procedure is the integer r. This can be determined from a table of probabilities of a correct selection to satisfy any given specification (a*, P*). If, for example, a* = 2 then we can enter Table I. If n is given to be 4 and we wish to meet the specification a* = 2, P* = 0.800 then we would enter Table I with n — 4 and select r = 4, it being the smallest value for which P ^ P*. The table above shows that for the given specification we would also have selected r = 4 for any value of n. In fact, we note that the proba- bility of a correct selection depends only slightly on n. The given value of n and the selected value of r then determine a particular procedure of type Ri , say, Ri(n, r). The average experiment time for each of several procedures R\{n, r) is given in Table II for the three critical values of the true ratio a, namely, a = \, a = a* and a = oo . Each of the entries has to be multi- plied by 6-1 , the smaller of the two d values, and added to the common guarantee period g. For n = oo the entry should be zero (-\-g) but it was found convenient to put in place of zero the leading term in the asymptotic expansion of the expectation in powers of I/71. Hence the entry for n = 00 can be used for any large n, say, n ^ 25 when r ^ 4. We note in Table II the undesirable feature that for each procedure the average experiment time increases with a for fixed 62 . For the se- quential procedure we shall see later that the average experiment time is greater at a = a* than at either a = 1 or a = 00 . This is intuitively more desirable since it means that the procedure spends more time when the choice is more difficult to make and less time when we are indifferent or when the choice is easy to make. PROCEDURES OF TYPE R2 — NONSEQUENTIAL, REPLACEMENT "Such procedures are carried out exactly as for procedures oiRi except that failures are immediately replaced by new units from the same population." To determine the appropriate value of r for the specification a* = 2, P* = 0.800 when g = 0 we use the last row of Table I, i.e., the row marked n = ^ , and select r = 4. The probability of a correct selection for procedures of type Ro is exactly the same for all values of n and de- pends only on r. Furthermore, it agrees wdth the probability for pro- cedures of type Ri with n = co so that it is not necessary to prepare a separate table. PL, II H PM H K P o 0 .^ Pi d 1 '^ I «3 'a 2 S CC Cl t-- o 00 r^ r-i o O '^ (N o ^ Ci o o »o CO o CO 00 ^ (M CD .-I o deo Oi r^ r^ CD ^ ^ lO o ■* CO i-i o rH 00(N CO CO CD CO o CO 00 CO »o o 00 o CO ^ o 1— I .— I o oco (M ^ t^ C5 (M t^ ■* O CO "tH lO O C^ T— I CO T-H O O O — I O CD 1— I O O O O '-H t^ t^ CO iM -^ O t— I 1— t CD CO CD lO a; kO CO r-H o (M O O O O O T-H O O CO o o o o O O CO lO O lO o O lO CO (M T-H O O ^ o o o o o ^ t- CO (M t^ t^ CO t^ CD CO (M CD CD CO CD CD CO C^ >— I O O CD d> d> d CD d> d> d> s II a o o r^ vo o lO o O >0 CD (M lO (M O lO (M »-< 1-^ O O lO ooooooo --H iM CO "* O O 184 REDUCING TIME IN RELIABILITY STUDIES 185 Table III — Value of r Required to Meet the Specification (a*, P*) FOR Procedures of Type R2 (g = 0) a* p* 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 0 1.45 0 1.50 2.00 2.50 0 3.00 0.50 0 0 0 0 0 0 0 0 0 0 0.55 14 4 2 2 1 1 1 1 1 1 1 1 1 0.60 55 15 7 5 3 3 2 2 2 1 1 1 1 0.65 126 33 16 10 7 5 4 3 3 3 1 1 1 0.70 232 61 29 17 12 9 7 6 5 4 2 1 1 0.75 383 101 47 28 19 14 11 9 7 6 3 2 1 0.80 596 157 73 43 29 21 17 13 11 9 4 2 2 0.85 903 238 111 65 44 32 25 20 16 14 5 3 3 0.90 1381 363 169 100 67 49 37 30 25 21 8 5 4 0.95 2274 597 278 164 110 80 61 49 40 34 12 7 5 0.99 4549 1193 556 327 219 160 122 98 80 68 24 14 10 It i.s also unnecessary to prepare a separate table for the average ex- periment time for procedures of type R2 since for g = 0 the exact values can be obtained by substituting the appropriate value of n in the ex- pressions appearing in Table II in the row marked n = oo . For example, for /( = 2, /• = 1 and a = 1 the exact value for ^ = 0 is 0.500 62/2 = 0.250 62 , and for n = 3, r = 4, a = 00 the exact value for g = 0 is 4.000 62/3 = 1.333 62 . It should be noted that for procedures of type R2 we need not restrict our attention to the cases r ^ n but can also con- sider r > //. Table III shows the value of r recjuired to meet the specilication (a*, F*) with a procedure of type R2 for various selected values of a* and P*. procedures of type R3 — sequential, replacement Let D{t) denote the absolute difference between the number of fail- ures produced by the two processes at any time t. The sequential pro- cedure is as follows: "Stop the test as soon as the inequality Dit) ^ In [P*/{1 - P*)] In a (3) is satisfied. Then select the population with the smaller number of fail- ures as the better one." To get the best results we will choose (a*, P*) so that the right hand member of the inequality (3) is an integer. Otherwise we would be operat- ing with a higher value of P* (or a smaller value of a*) than was specified. 186 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table IV — Average Experiment Time and Probability of a Correct Selection — Procedure Type R3 (a* = 2, P* = 0.800, ^ = 0) (Multiply each average time entry by d^) n a = 1 a = 2 a = 00 1 2.000 2.400 2.000 2 1.000 1.200 1.000 3 0.667 0.800 0.667 4 0.500 0.600 0.500 10 0.200 0.240 0.200 20 0.100 0.120 0.100 oc 2.000/w 2.400/n 2.000/n Probability 0.500 0.800 1.000 For example, we might choose a* = 2 and P* = 0.800. For procedures of type R3 the probability of a correct selection is again completely in- dependent of n; here it depends only on the true value of the ratio a. The average experiment time depends strongly on n and only to a limited extent on the true value of the ratio a. Table IV gives these quantities for a = 1, a = 2, and a = 00 for the particular specification a* = 2, p* = 0.800 and for the particular value ^ = 0. efficiency We are now in a position to compare the efficiency of two different types of procedures using the same value of n. The efficiency of Ri rela- tive to R2 is the reciprocal of the ratio of their average experiment time. This is given in Table V for a* = 2, P* = 0.800, r = 4 and n = 4, 10, 20 and 00 . By Table I the value P* = 0.800 is not attained for n < 4. In comparing the sequential and the nonsequential procedures it was found that the slight excesses in the last column of Table I over 0.800 Table V — Efficiency of Type Ri Relative to Type R2 {a* = 2, P* = 0.800, r = 4:,g = 0) { n a = 1 a = 2 a = 00 4 10 20 00 0.501 0.837 0.925 1.000 0.495 0.836 0.917 1.000 0.480 0.835 0.922 1.000 I REDUCING TIME IN RELIABILITY STUDIES 187 Table VI — Efficiency of («* = 2, P* Adjusted Ri Relative To R^ = 0.800, ^ = 0) n a = 1 a = 2 a = 00 4 10 20 00 0.615 0.754 0.818 0.873 0.575 0.708 0.768 0.822 0.419 0.528 0.573 0.612 had an effect on the efficiency. To make the procedures more comparable the values for r = 3 and r = 4 in Table I were averaged with values p and 1 — p computed so as to give a probability of exactly 0.800 at a = a*. The corresponding values for the average experiment time were then averaged with the same values p and 1 — p. The nonsequential pro- cedures so altered will be called "adjusted procedures." The efficiency of the adjusted Ri relative to Rz is given in Table VI. In Table VI the last row gives the efficiency of the adjusted procedure 7^2 relative to Rz . Thus we can separate out the advantage due to the replacement feature and the advantage due to the sequential fea- ture. Table VII gives these results in terms of percentage reduction of average experiment time. We note that the reduction due to the replacement feature alone is greatest for small n and essentially constant with a while the reduction Table VII — Per Cent Reduction in Average Experiment Time DUE TO Statistical Techniques (a* = 2,P* = 0.800, ^ = 0) a K Reduction due to Replacement Feature Alone Reduction due to Sequential Feature Alone Reduction due to both Replacement and Sequential Features 1 4 10 20 00 29.5 13.7 6.3 0.0 12.7 12.7 12.7 12.7 38.5 24.6 18.2 12.7 2 4 10 20 00 30.1 13.9 6.6 0.0 17.8 17.8 17.8 17.8 42.5 29.2 23.2 17.8 cc 4 10 20 00 31.5 13.6 6.3 0.0 38.8 38.8 38.8 38.8 58.1 47.2 42.7 38.8 188 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 due to the sequential feature alone is greatest for large a and is inde- pendent of n. Hence if the initial sample size per process n is large we can disregard the replacement techniciue. On the other hand the true value of a is not known and hence the advantage of sequential experi- mentation should not be disregarded. The formulas used to compute the accompanying tables are given in Addendum 2. ACKNOWLEDGEMENT The author wishes to thank Miss Marilyn J. Huyett for considerable help in computing the tables in this paper. Thanks are also due to J. W. Tukey and other staff members for constructive criticism and numerical errors they have pointed out. Addendum 1 In this addendum we shall consider the more general problem of select- ing the best of k exponential populations treated on a higher mathemati- cal level. For k = 2 this reduces to the problem discussed above. DEFINITIONS AND ASSUMPTIONS There are given k populations H, (^ = 1, 2, • • • , k) such that the life- times of units taken from any of these populations are independent chance variables with the exponential density (1) with a common (known or unknown) location parameter g ^ 0. The distributions for the k popu- lations are identical except for the unknown scale parameter 6 > 0 which may be different for the k different populations. We shall consider three different cases with regard to g. Case 1 : The parameter g has the value zero (g = 0). Case 2: The parameter g has a positive, known value (g > 0). Case 3: The parameter g is unknown (g ^ 0). Let the ordered values of the k scale parameters be denoted by di^ e.-^ ■■■ ^ dk (4) where equal values may be regarded as ordered in any arbitrary manner. At any time / each population has a certain number of failures associated with it. Let the ordered values of these integers be denoted by ri = ri{t) so that I ri g r2 ^ • • • ^ r-fc (5) ^ i REDUCING TIME IN RELIABILITY STUDIES 189 For each unit the life beyond its guarantee period will be referred to as its Poisson life. Let Li{t) denote the total amount of Poisson life observed up to time t in the population with Vi failures (z = 1, 2, • • • , fc). If two or more of the r^ are equal, say Vi = rj+i = • • • = r^+y , then we shall assign r, and L; to the population with the largest Poisson life, ri+i and L^+i to the population with the next largest, • • • , ri+_, and Lj+,- to the population with the smallest Poisson life. If there are two or more equal pairs (ri , Li) then these should be ordered by a random device giving equal probability to each ordering. Then the subscripts in (5) as well as those in (4) are in one-to-one correspondence with the k given populations. It should be noted that Li(t) ^ 0 for all i and any time t ^ 0. The complete set of quantities Li{t) {i = 1, 2, • • • , k) need not be ordered. Let a = 61/62 so that, since the 6i are ordered, a ^ 1. We shall further assume that : 1 . The initial number n of units put on test is the same and the start- ing time is the same for each of the k populations. 2. Each replacement is assumed to be a new unit from the same popu- lation as the failure that it replaces. 3. Failures are assumed to be clearly recognizable without any chance of error. SPECIFICATIONS FOR CASE 1 : gf = 0 Before experimentation starts the experimenter is asked to specify two constants a* and P* such that a* > 1 and l'^ < P* < 1. The procedure Ri = Rsin), which is defined in terms of the specified a* and P*, has the property that it will correctly select the population with the largest scale parameter with probability at least P* whenever a ^ a*. The initial number n of units put on test may either be fixed by nonstatistical con- siderations or may be determined by placing some restriction on the average experiment time function. Rule Rs : "Continue experimentation with replacement until the inequality k ^ ^*-(^.-a) ^ (1 _ p*)/p* (6) i=2 is satisfied. Then stop and select the population with the smallest num- ber of failures as the one having the largest scale parameter." 190 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Remarks 1. Since P* > Y2 then (1 — P*)/P* < 1 and hence no two popula- tions can have the same vahie ri at stopping time. 2. For A: = 2 the inequality (6) reduces to the inequalitj^ (3). 3. The procedure 7^3 terminates onl}^ at a failure time, never between failures, since the left member of (G) depends on t only through the quantities 7-i{t). 4. After experimentation is completed one can make, at the lOOP per cent confidence level, the confidence statement ds ^ di S a* 9, (or di/a"" ^ ds S e,) (7) where 6s is the scale parameter of the selected population. Numerical Illustrations »l/4 Suppose the preassigned constants are P* = 0.95 and a* = 19' 2.088 so that (1 - P*)/P* = ^9- Then for A; = 2 the procedure is to stop when r-i — ri ^ 4. For A; = 3 it is easy to check that the procedure reduces to the simple form: "Stop when ?'2 — ri ^ 5". For A; > 3 either calculations can be carried out as experimentation progresses or a table of stopping values can be constructed before experimentation starts. For A: = 4 and A; = 5 see Table VIII. In the above form the proposed rule is to stop Avhen, for at least one Table VIII — Sequential Rule for P* = 0.95, a* = 19 A: = 4 fc = 5 1/4 r2 — ri rs — ri n — ri 5 5 9 5 6 6 6 6 6 ri — ri ra — ri n — ci Ti — n 5 5 9 10 5 5 10 10 5 6 6 8 5 6 7 7 5 7 7 7 6 6 6 6 * Starred rows can be omitted without affecting the test since every integer in these rows is at least as great as the corresponding integer in the previous row. They are shown here to ilhistrate a systematic method which insures that all the necessary rows are included. REDUCING TIME IN RELIABILITY STUDIES 191 row (say row j) in the table, the observed row vector (r^ — Vi , Ts — Ti , ■ ■ ■ , Vk — z'l) is such that each comyonent is at least as large as the corresponding component of row j. Properties of Rs for k = 2 and g = 0 For A- = 2 and ^ = 0 the procedure Rs is an example of a Sequential Probability Ratio test as defined by A. Wald in his book.^ The Average Sample Number (ASN) function and the Operating Characteristics (OC) function for Rs can be obtained from the general formulae given by Wald. Both of these functions depend on di and 0-2 only through their ratio a. In our problem there is no excess over the boundary and hence Wald's approximation formulas are exact. When our problem is put into the Wald framework, the symmetry of our problem implies equal proba- bilities of type 1 and type 2 errors. The OC function takes on comple- mentary values for any point a = 61/62 and its reciprocal 62/61 . We shall therefore compute it only for a ^ 1 and denote it by P{a). For a > 1 the quantity P(a) denotes the probability of a correct selection for the true ratio a. The equation determining Wald's h function is 1 + a 1 + a for which the non-zero solution in h is easily computed to be h{a) = }^ (9) In a Hence we obtain from Wald's formula (3:43) in Reference 5 s a Pia) = -^^ (10) where s is the smallest integer greater than or equal to S = In [PV(1 - P*)]/ln a* (11) In particular, for a = 1"^, a* and 00 we have Pi^^) = 1/2, ^(«*) ^ P*, P(^) = 1 (12) ^\'e have written P(l"^) above for lim P{x) as x -^ 1 from the right. The procedure becomes more efficient if we choose P and a* so that *S' is an integer. Then s ^ S and P(a*) = P*. Letting F denote the total number of observed failures required to 192 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 terminate the experiment we obtain for the ASN function and, in particular, for a = 1, oo E(F; 1) = s- and E{F; oo) = s (14) It is interesting to note that for s = 1 we obtain E{F; a) = 1 for all a ^ 1 (15) and that this result is exact since for s = 1 the right-hand member S \ of (3) is at most one and hence the procedure terminates with certainty ' immediately after the first failure. ' As a result of the exponential assumption, the assumption of replace- ; ment and the assumption that ^ = 0 it follows that the intervals between \ failures are independently and identically distributed. For a single popu- ' lation the time interval between failures is an exponential chance vari- ; able. Hence, for two populations, the time interval is the minimum of j two exponentials which is again exponential. Letting r denote the i (chance) duration of a typical interval and letting T denote the (chance) j total time needed to terminate the procedure, Ave have E{T; a, 62) = E{F; a)E(r; a, d^) = E{F; a) (^^^ (f^) (16) I Hence Ave obtain from (13) and (14) E{T; a, 02) = - -^ ^^^ for a > 1 (17) n a — 1 a* + 1 E{T; 1, d,) = ^ and E{T; <^, 0,) = ^ (18> For the numerical illustration treated above Avith k = 2 we have na) = ^-^ (19) : P(l+) = ^; P(2.088) = 0.95; P(oo) = 1 (20) EiF-a) = 4^^4^ = 4^--+ Vy + '^ (21) a— la*-f-l a*-t-l E{F; 1) = 16.0; /iXF; 2.088) = 10.2; E{F; 00) = 4 (22), REDUCING TIME IN RELIABILITY STUDIES 193 E(T; 1, ^2) = — ; E{T; 2.088, 6^ = — ; n n (23) n For /.• > 2 the proposed procedure is an application of a general se- quential rule for selecting the best of A- populations which is treated in [1]. Proof that the probability specification is met and bounds on the probability of a correct decision can be found there. CASE 2: COMMON KNOWN ^ > 0 In order to obtain the properties of the sec^uential procedure R:>. for this case it will be convenient to consider other sequential procedures. Let (S = 1/6-2 — 1/^1 so that, since the di are ordered, jS ^ 0. Let us assume that the experimenter can specify three constants a*, /3* and P* such that a* > 1, /3* > 0 and ^ 2 < -P* < 1 ai^d a procedure is de- sired which will select the population with the largest scale parameter with probability at least P* whenever we have both a ^ a* and i3 ^ /3* The following procedure meets this specification. Rule Rs': "Continue experimentation with replacement until the inec^uality fi «*-(^i-'-i>e-^*(^i-^i)^ (l_p*)/p* (24) 1=2 is satisfied. Then stop and select the population with the smallest nimiber of failures as the one having the largest scale parameter. If, at stopping time, two or more populations have the same value ri then select that particular one of these with the largest Poisson life Li ." Remarks 1 . For k = 2 the inequality reduces to (r, - n) In a* + (Li - L2) 13* ^ In [P*/a - P*)] (25) If <7 = 0 then Li = Li for all t and the procedure R/ reduces to R3 . 2. The procedure R/ may terminate not only at failures but also be- tween failures. 194 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 3. The same inequality (24) can also be used if experimentation is carried on without replacement, one advantage of the latter being that there is less bookkeeping involved. In this case there is a possibility that the units will all fail before the inequality is satisfied so that the procedure is not yet completely defined for this case. One possibility in such a situation is to continue experimentation with new units from each population until the inequality is satisfied. Such a procedure will terminate in a finite time with probability one, i.e., Prob{ T > To} -^0 as To — > 00, and the probability specification will be satisfied. 4. A procedure R3 (ni , n-z , ■ • • , rik , ti , t2 , • • • , tk) using the same inequality (24) but based on dilTerent initial sample sizes and/or on different starting times for the initial samples also satisfies the above probability specification. In the case of different starting times it is required that the experimenter wait at least g units of time after the last initial sample is put on test before reaching any decision. 0. One disadvantage of R3 is that there is some (however remote) possibility of terminating while ri = r2 . This can be avoided by adding the condition r^ > n to (24) but, of course, the average experiment time is increased. Another way of avoiding this is to use the procedure R3 which depends only on the number of failures; the effect of using R3 when g > 0 will be considered below. 6. The terms of the sum in (24) represent likelihood ratios. If at any time each term is less than unity then we shall regard the decision to select the population with n failures and Li units of Poisson life as opti- mal. Since (1 — P*)/P* < 1 then each term must be less than unity at termination. Properties of Procedure Rz for k = 2 p The OC and ASN functions for Rs will be approximated by comparing R3' with another procedure R/ defined below. We shall assume that P* is close to unity and that g is small enough (compared to d^) so that the probability of obtaining two failures within g imits of time is small enough to be negligible. Then we can write approximately at termination Li^nT - r,g {i = 1, 2, • • • , A:) (26) and Li - Li ^ (r, - r,)g (i = 2, 3, • • • , A:) (27) Substituting this in (24) and letting 5* = a* c^*" (28) suggests a new rule, say R/' , which we now define. REDUCING TIME IN RELIABILITY STUDIES 195 h'ule R/ "Continue experimentation with replacement until the inequality k X 6*-(^i-'-i) ^ (1 - P*)/P* (29) is satisfied. Then stop and select the population with n failures as the one with the largest scale parameter." For rule Rz" the experimenter need only specify P* and the smallest value 5* of the single parameter 8 = ^' e''''"''-''"''' = ae'^ (30) 62 that he desires to detect with probability at least P*. We shall approximate the OC and ASX function of R/' for k = 2 by computing them under the assumption that (27) holds at termina- tion. The results will be considered as an approximation for the OC and ASN functions respectively of R/ for /,■ = 2. The similarity of (29) and (6) immediately suggests that we might replace a* by 5* and a by 5 in the formulae for (6). To use the resulting expressions for R^ we would compute 5* as a function of a* and /3* by (28) and 5 as a function of a and /3 by (30). The similarity of (29) and (6) shows that Z„ (defined in Reference 5, page 170) under (27) with gr > 0 is the same function of 5* and 5 as it is of a* and a when g = 0. To complete the justification of the above result it is sufficient to show that the individual increment ^ of Z„ is the same function of 5* and 8 under (27) with ^ > 0 as it is of a* and a when ^ = 0. To keep the increments independent it is necessary to as- sociate each failure with the Poisson life that follows rather than with the Poisson life that precedes the failure. Neglecting the probability that any two failures occur ^^•ithin g units of time we have two values for z, namely ^ -(.nt-g)/ei -ntl$2 z = log^^^ = -log 5 (31) and, interchanging 61 and ^2 , gives z — log 5. Moreover 196 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 r r - e-(— «)/«v"^^^^ dx dy Jg Jg 6-1 Prob \z = -logSj ^2 -0[92(n-l)+9l"l/9lfl2 _i_ ^1 -H9in+Bi(n-l)]l9ie2,) - e + - e /o9\ 1 + 5 Thus the OC and ASN functions under (27) with g > 0 bear the same relation to 5* and 5 as they do to a* and a when ^ = 0. Hence, letting w denote the smallest integer greater than or equal to ^ In [P*/(l - P*)] ^ \n[P*/{l-P*)] In 8* gl3* + In a* ^' ' we can write (omitting P* in the rule description) | 7^15; /?/ («*, /5*){ ^ P{5; /^.^"(S*)! ^ ^-^^^ (34) 1 {So) ^ \8 - l/\5"' +1/ w~ for 5=1 W'e can approximate the average time between failures by I and the average experiment time by « E{T; /?/(«*, ^*)} ^ E{F; R,'(a*, 0*)\ [^.^ f^'^ _^ ^'^, (37) n{Oi -T 02 -f- zg) Since 5 ^ 1 then 5"(1 + 5") is an increasing function of w and by (33) it is a non-increashig function of 5*. By (28) 5* ^ a* and hence, if we disregard the approximation (34), P{8; AV(«*)1 - ^!^{py^/_p.^y..n^* ^ P{S;R/m} (38) Clearly the rules Ri{a*, P*) and R/ {a*, P*) are equivalent so that for g > 0 we haA-e P{8;R-s{a*)} ^ P{8;R/ia*)] (39) REDUCING TIME IN RELIABILITY STUDIES 197 and hence, in particular, letting 8 = 8* in (38) we have P{8*;R,(a*)} ^ P{8*;R,"(8*)] ^ P* (40) since the right member of (34) reduces to P* when W is an integer and 5 = 5*. The error in the approximations above can be disregarded when g is small compared to 02 . Thus we have shown that for small values of g/d2 the probability specification based on (a*, ^*, P*) is satisfied in the sense of (40) if we use the procedure Rsia*, P*), i.e., if we proceed as if It would be desirable to show that w^e can proceed as if g = 0 for all values of g and P*. It can be shown that for swfficiently large n the rule Ri{a*, P*) meets it specification for all g. One effect of increasing n is to decrease the average time E{t) between failures and to approach the corresponding problem without replaceme^it since g/E{T) becomes large. Hence we need only show that Ri{a*, P*) meets its specification for the corresponding problem without replacement. If we disregard the information furnished by Poisson life and rely solely on the counting of failures then the problem reduces to testing in a single binomial whether 6 = di for population IIi and 6 = do for population 112 or vice versa. Let- ting p denote the probability that the next failure arises from 111 then we have formally tia'-V = -. — ; — versus Hi-.p = 1 + a ^ 1 + a For preassigned constants a* > I and P* (V2 < P* < 1) the appropri- ate sequential likelihood test to meet the specification: "Probability of a Correct Selection ^ P* whenever a ^ a*" (41) then turns out to be precisely the procedure Rsia*, P*). Hence we may proceed as if gr = 0 when n is sufficiently large. The specifications of the problem may be given in a different form. Suppose 01* > 02* are specified and it is desired to haxe a probability of a correct selection of at least P* whenever ^1 ^ 0i* > 02* ^ 02 . Then we can form the following sequential likelihood procedure R3* which is more efficient than Rsia*, P*). Rule /?3*.- "Continue experimentation without replacement until a time t is reached at which the inequality 198 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 is satisfied. Then stop and select the population with ri failures as the population with d = di". It can be easily shown that the greatest lower bound of the bracketed quantity in (42) is 0i*/^2*. Hence for di*/d2* = a* and P* > i 2 the time required by Rz*{6i*, 62*, P*) ivill always be less than the time required by R,(a*,P*). Another type of problem is one in which we are given that 6 = di* for one population and d = 62* for the A; — 1 others where 6]* > 62* are specified. The problem is to select the population with 6 = di*. Then (42) can again be used. In this case the parameter space is discrete with k points only one of which is correct. If Rule R3* is used then the probability of selecting the correct point is at least P*. Equilibrium Approach When Failures Are Replaced 9 Consider first the case in which all items on test are from the same exponential population with parameters (6, g). Let Tnj denote the length of the time interval between the j^^ and the j + 1^* failures, (j = 0, 1, • • • ), where n is the number of items on test and the 0*'' failure de- notes the starting time. As time increases to infinity the expected number of failures per unit time clearly approaches n/(0 + g) which is called the equilibrium failure rate. The inverse of this is the expected time between failures at equilibrium, say E{Tn^). The question as to how the quanti- ties E{Tnj) approach E(Tn^) is of considerable interest in its own right. The following results hold for any fixed integer 71 ^ 1 unless explicitly stated otherwise. It is easy to see that ^^(^i) ^ E{TnJ ^ E(T„o) (43) since the exact values are respectively e /, e-^-^'^/^X ^ g+d ^ , d < ^ 9+ - (44) n — 1 \ n / n n In fact, since all units are new at starting time and since at the time of the first failure all units (except the replacement) have passed their guarantee period with probability one then ^(^i) ^ E(Tnj) S E{Tn,) (j ^ 0) (45) If we compare the case g > 0 with the special case g = 0 we obtain E{2\j) ^ - (y= 1,2, •••) (46) n REDUCING TIME IN RELIABILITY STUDIES 199 and if we compare it with the non-replacement case {g/Q is large) we obtain ^(n,) ^ -^. (i = 1, 2, • . • , n - 1). (47) These comparisons show that the difference in (46) is small when g/0 is small and for j < n the difference in (47) is small when g/d is large. It is possible to compute E{Tnj) exactly for g ^ 0 but the computa- tion is extremely tedious for j ^ 2. The results for j = 1 and 0 are given in (44). Fori = 2 E(Tn2) = n (n + 2)(/i - 1) -(n-2)gie 1 - ' ' ': -e n + Vl^iI g-(«-i)p/^ ri-2_ -un-i),ie I {n>2) n — \ v?{n — 1) and 2{n-l)glB (48) E{T,.^ = ^ - ^ [1 - ^e-'" + e-'"'\ (49) For the case of two populations with a common guarantee period g we can write similar inequalities. We shall use different symbols a, h for the initial sample size from the populations with scale parameters Oi , O2 respectively even though our principal interest is in the case a = b = n say. Let Ta,b.j denote the interval between the j^^ and j -f P* fail- ures in this case and let X, = l/di (i = 1,2). We then have for all values of a and b [aXi + b\o]-' ^ E(TaXj) ^ E(Taxo) = g + [aXi + b\,]-' (j = 0,1,2, ■■■, ^) (50) J?(T ^ (gl + g){e2 + g) .riN a{92 -h 9) + b{di + g) The result for E(Ta,b.i) corresponding to that in (43) does not hold if the ratio di/62 is too large; in particular it can be shown that -0[(a-l)Xi+6X2l-l E{T.,b..) = ^ "^^ ^' ^ aXi + 6X2/ \(a — l)Xi 4- 6X2 _ Xie aXi + 6X2 + / ^X2 Y 1 \r x^e-''^'^^''-''''-'- (52) ,aXi -\- bX2/\aXi + (& — 1)^2 L 0X1 + ^^2 is larger than E{Ta,h.J for a = 6 = 1 when ^/^i = 0.01 and g/di = 0.10 200 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 SO that QilQi = 10. The expression (52) reduces to that in (44) if we set di = 02 = 6 and replace a and h by n/2 in the resulting expression. Corresponding exact expressions for E(Ta.b,j) for j > 1 are extremely tedious to derive and unwieldy although the integrations involved are elementary. If we let g —^ oo then we obtain expressions for the non- replacement case which are relatively simple. They are best expressed as a recursion formula. E(.Ta,bj) = — , ,. ETa-\,b,}-l + m^ ^"— ^^ = '^ (53) EiT.,b.d = "^^ ^ aXi + 6X2 (a — l)Xi + 6X2 I 0X2 1 ( h > ^^ "^ aXi + 6X2 aXi + (6 - 1)X2 ' = (54) E(Tafij) ^ g + di/a fori ^ a and j = 0 (55) E{Ta,oJ = dr/(a -j) for 1 ^ i ^ a - 1 (56) Results similar to (55) and (56) hold for the case a = 0. The above results for gr = 00 provide useful approximations for E{Ta,b,j) when g is large. Upper bounds are given by M E{Ta,bj) ^ [aXi + (6 - i)X2r (i = 1, 2, • • • , h) (57) E(Ta.bj+b) ^ [(a - j)Xr' (i = 1, 2, • . • , a - 1). (58) Duration of the Experiment For the sequential rule R^' with k = 2 we can now write down approxi- mations as well as upper and lower bounds to the expected duration E{T) of the experiment. From (50) I g + ..5^;^.\ s E(T) = E /?(r.,,) c-l n(Xi -f X2) ^ '''^ ' ~ § '^^^ "'"'^^ (59) + \FA¥; 5) - c]i!;(T„,„,.) where c is the largest integer less than or equal to E{F\ 5). The right ex- pression of (59) can be approximated by (53) and (54) if g is large. If c < 2n then the upper bounds are given by (57) and (58). A simpler j REDUCING TIME IN RELIABILITY STUDIES 201 upper bound, which holds for all \'aliies of c is given by E{T) ^ E{F- b)E{Tn,n..) = E{F; 8) (g + ^^ (60) CASE 3: COMMON UNKNOWN LOCATION PARAMETER ^ ^ 0 In this case the more conservative procedure is to proceed under the assumption that 0 we need only add g to this result. This result was used to compute E(T) in table lA f or a = 1 and a = 2. For a = oo the expression simplifies to E{T) = e^rC: ± erl ^~^^'^\ (66) which can be shoAvn to be equivalent to E{T) = e,f: ^— (67) REFERENCES 1. Bechhofer, R. E., Kiefer, J. and Sobel, M., On a Type of Sequential Multiple Decision Procedures for Certain Ranking and Identification Problems with k Populations. To be published. 2. Birnbaum, A., Statistical methods for Poisson processes and exponential populations, J. Am. Stat. Assoc, 49, pp. 254-266, 1954. 3. Birnbaum, A., Some procedures for comparing Poisson processes or popula- tions, Biometrika, 40, pp. 447-49, 1953. 4. Girshick, M. A., Contributions to the theory of sequential analj'sis I, Annals Math. Stat., 17, pp. 123-43, 1946. 5. Wald, A., Sequential Analysis, John Wiley and Sons, New York, 1947. I A Class of Binary Signaling Alphabets By DAVID SLEPIAN (Manuscript received September 27, 1955) A class of binary signaling alphabets called "group alphabets" is de- scribed. The alphabets are generalizations of Hamming^ s error correcting codes and possess the following special features: {1) all letters are treated alike in transmission; {2) the encoding is simple to instrument; (3) maxi- mum likelihood detection is relatively simple to instrument; and (4) in certain practical cases there exist no better alphabets. A compilation is given of group alphabets of length equal to or less than 10 binary digits. INTRODUCTION This paper is concerned with a class of signahng alphabets, called "group alphabets," for use on the symmetric binary channel. The class in question is sufficiently broad to include the error correcting codes of Hamming,^ the Reed-Muller codes," and all "systematic codes''.^ On the other hand, because they constitute a rather small subclass of the class of all binary alphabets, group alphabets possess many important special features of practical interest. In particular, (1) all letters of the alphabets are treated alike under transmission; (2) the encoding scheme is particularly simple to instru- ment; (3) the decoder — a maximum likelihood detector — is the best I possible theoretically and is relatively easy to instrument; and (4) in certain cases of practical interest the alphabets are the best possible theoretically. It has very recently been proved by Peter Elias^ that there exist group alphabets which signal at a rate arbitarily close to the capacity, C, of the symmetric binary channel with an arbitrarily small probability of error. Elias' demonstration is an existence proof in that it does not show explicitly how to construct a group alphabet signaling at a rate greater than C — e with a probability of error less than 5 for arbitrary positive 5 and e. Unfortunately, in this respect and in many others, our understanding of group alphabets is still fragmentary. In Part I, group alphabets are defined along with some related con- 203 204 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 cepts necessary for their understanding. The main results obtained up to the present time are stated without proof. Examples of these concepts are given and a compilation of the best group alphabets of small size is presented and explained. This section is intended for the casual reader. In Part II, proofs of the statements of Part I are given along with such theory as is needed for these proofs. The reader is assumed to be familiar with the paper of Hamming, the basic papers of Shannon* and the most elementary notions of the theory of finite groups. Part I — Group Alphabets and Their Properties 1.1 INTRODUCTION We shall be concerned in all that follows with communication over the symmetric binary channel shown on Fig. 1. The channel can accept either of the two symbols 0 or 1 . A transmitted 0 is received as a 0 with probability q and is received as a 1 w'ith probability p — 1 — g : a trans- mitted 1 is received as a 1 with probability q and is received as a 0 with probability p. We assume 0 ^ p ^ ^^. The "noise" on the channel operates independently on each symbol presented for transmission. The capacity of this channel is C = 1 + P log2P + q log29 bits/symbol (1) By a K-leUer, n-place binary signaling alphabet we shall mean a collec- tion of K distinct sequences of n binary digits. An individual sequence of the collection will be referred to as a letter of the alphabet. The integer K is called the size of the alphabet. A letter is transmitted over the channel by presenting in order to the channel input the sequence of n zeros and ones that comprise the letter. A detection scheme or detector for INPUT X OUTPUT Fig. 1 — The symmetric binary channel. A CLASS OF BINARY SIGNALING ALPHABETS 205 a given /v-letter, n-place alphabet is a procedure for producing a sequence of letters of the alphabet from the channel output. Throughout this paper we shall assume that signaling is accomplished with a given /i-letter, n-place alphabet by choosing the letters of the alphabet for transmission independently with equal probability l/K. Shannon^ has shown that for sufficiently large n, there exist K-letter, n-place alphabets and detection schemes that signal over the symmetric binary chaimel at a rate R > C — e for arbitrary £ > 0 and such that the probability of error in the letters of the detector output is less than any 5 > 0. Here C is given by (1) and is shown as a function of p in Fig. 2. No algorithm is known (other than exhaustvie procedures) for the construction of A'-letter, /i-place alphabets satisfying the above inequalities for arbitrary positive 8 and e except in the trivial cases C — 0 and C = 1. 1.2 THE GROUP -S„ There are a totality of 2" different w-place binary sequences. It is fre- quently convenient to consider these sequences as the vertices of a cube of unit edge in a Euclidean space of n-dimensions. For example the 5- place sequence 0, 1, 0, 0, 1 is associated with the point in 5-space whose o.e 0.6 0.4 0.2 Fig. 2 — The capacity of the symmetric binary channel. C = 1 + p log2 p + {I - p) log2 (1 - p) 206 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 coordinates are (0, 1, 0, 0, 1). For convenience of notation we shall gen- erally omit commas in writing a sequence. The above 5-place sequence will be written, for example, 01001. We define the product of two n-ylace hinarij sequences, aicii • • • a„ and ^1^2 • ■ • bn as the n-place binary sequence fli + hi , a-i ■]- h-i , ■ ■ • , ttn + hn Here the a's and 6's are zero or one and the + sign means addition modulo 2. (That is 0 + 0=1 + 1 = 0, 0+1 = 1+0=1) For example, (01101) (00111) = 01010. With this rule of multiplication the 2" w-place binary sequences form an Abelian group of order 2". The elements of the group, denoted by Ti , T'2 , • • • , Tin, say, are the n-place binary sequences ; the identity element I is the sequence 000 • • • 0 and IT, = Til = T. ■ T,Tj = TjTr, TiiTjT,) = iTiTj)Tk ; the product of any number of elements is again an element; every ele- ment is its own reciprocal, Ti = Tf^, TI = /. We denote this group by Bn . All subgroups of Bn are of order 2 where k is an integer from the set 0, 1, 2, • • • , n. There are exactly N{n, k) = (2" - 2") (2" - 2') (2" - 2') • • • (2" - 2'-') (2^ - 2»)(2'^ - 20(2* - 22) = N(n, n — k) {2" - 2'-') (2) distinct subgroups of Bn of order 2 . Some values of N(n, k) are given in Table I. Table I — Some Values of A^(n, k), the Number of Subgroups OF Bn OF Order 2''. N(n, k) = N{n, n — k) n\k 0 1 2 3 4 5 2 3 1 3 7 7 1 4 15 35 15 1 5 31 155 155 31 1 6 63 651 1395 651 63 7 127 2667 IISU 11811 2667 8 255 10795 97155 200787 97155 9 511 43435 788035 3309747 3309747 10 1023 174251 6347715 53743987 109221651 000 000 000 000 000 000 000 100 100 100 010 010 001 no 010 001 oil 001 101 no on 110 101 111 on 111 111 101 A CLASS OF BINARY SIGNALING ALPHABETS 207 1.3 GROUP ALPHABETS An ?i-place group alphabet is a 7v-letter, n-place binary signaling alpha- bet whose letters form a subgroup of Bn . Of necessity the size of an n-place group alphabet is /v = 2 where k is an integer satisfying 0 ^ k ^ n. By an (n, k)-alphahet we shall mean an n-place group alphabet of size 2^. Example: the N{3, 2) = 7 distinct (3, 2)-alphabets are given by the seven columns (i) (ii) (iii) (iv) (v) (vi) (vii) (3) 1.4 STANDARD ARRAYS Let the letters of a specific (n, /i:)-alphabet be Ai = / = 00 • • • 0, Ao , As , • ■ ■ , A^ , where ju = 2 . The group Bn can be developed accord- ing to this subgroup and its cosets: /, A2, A3, ■■• ,A^ S2 , S2A2 , S2A3 , • • • , S2A^ Sz , S3A2 , S3A3 , • • • , SsA^ Bn = ; (4) Sr f SyA2 , SpAz , • ' • , SfAfi In this array every element of Bn appears once and only once. The col- lection of elements in any row of this array is called a coset of the (n, k)- alphabet. Here *S2 is any element of B„ not in the first row of the array, S3 is any element of Bn not in the first two rows of the array, etc. The elements S2 , S3 , • • • , Sy appearing under I in such an array will be called the coset leaders. If a coset leader is replaced by any element in the coset, the same coset will result. That is to say the two collections of elements Si , ^1^2 , SiSz ; ■ • ■ , SiA^ and SiA,, , (SiAu)A2 , (SiAMs ,■■■ {SiAk)A, are the same. 208 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G We define the weight Wi = w{Ti) of an element, Ti , of Bn to be the number of ones in the n-place binary sequence T,- . Henceforth, unless otherwise stated, we agree in dealing with an ar- ray such as (4) to adopt the following convention: the leader of each coset shall be taken to be an . . element of minimal weight in that coset. Such a table will be called a standard array. Example: Bi can be developed according to the (4, 2)-alphabet 0000, 1100, 0011, nil as follows (6) 0000 1100 0011 nil 1010 Olio 1001 0101 1110 0010 1101 0001 1000 0100 1011 0111 )W"ever, ^^-e should write. for exan 0000 1100 0011 nil 1010 0110 1001 0101 0010 1110 0001 1101 1000 0100 1011 0111 (7) The coset leader of the second coset of (6) can be taken as any element of that row since all are of weight 2. The leader of the third coset, how- ever, should be either 0010 or 0001 since these are of weight one. The leader of the fourth coset should be either 1000 or 0100. 1.5 THE DETECTION SCHEME Consider now communicating with an (n, fc) -alphabet over the sym- metric binary channel. When any letter, say A,, of the alphabet is transmitted, the received sequence can be of any element of B„ . We agree to use the following detector: if the received element of Bn lies in column i of the array (4), the detector prints the letter Ai ,i = 1,2, • • • , ju. The array (4) is to (8) be constructed according to the convention (5). The following propositions and theorems can be proved concerning signaling with an (n, /c)-alphabet and the detection scheme given by (8). 1.6 BEST DETECTOR AND SYMMETRIC SIGNALING Define the probability /,• = ((Ti) of an element Ti of Bn to be A = ^wi^n-uf ^yYiere p and q are as in (1) and Wi is the weight of Ti . Let A CLASS OF BINARY SIGNALING ALPHABETS 209 Qi , i = 1 , 2, • • • , jLi be the sum of the probabilities of the elements in the iih. column of the standard array (4). Proposition 1. The probability that any transmitted letter of the (n, A;) -alphabet be produced correctly by the detector is Qi . Proposition 2. The equivocation^ per symbol is 1 ** Hy{x) = — S Qi log2 Qi n i=i Theorem 1 . The detector (8) is a maximum likelihood detector. That is, for the given alphabet no other detection scheme has a greater average probability that a transmitted letter be produced correctly by the de- tector. Let us return to the geometrical picture of w-place binary sequences as vertices of a unit cube in n-space. The choice of a i^-letter, n-place alphabet corresponds to designating K particular vertices as letters. Since the binary sequence corresponding to any vertex can be produced by the channel output, any detector must consist of a set of rules that associates various vertices of the cube with the vertices designated as letters of the alphabet. We assume that every vertex is associated with some letter. The vertices of the cube are divided then into disjoint sets, Wi , Wi , • • • , Wk where Wi is the set of vertices associated with tth letter of the signaling alphabet. A maximum likelihood detector is char- acterized by the fact that every vertex in Wi is as close to or closer to the iih. letter than to any other letter, i = 1,2, • • • , K. For group alpha- bets and the detector (8), this means that no element in the iih. column of array (4) is closer to any other A than it is to ^i , z = 1, 2, • • • , ;u. Theorem 2. Associated with each {n, /(;)-alphabet considered as a point configuration in Euclidean n-space, there is a group of n X n orthogonal matrices which is transitive on the letters of the alphabet and which leaves the unit cube invariant. The maximum likelihood sets 1^1 , W2 , • • • Wn are all geometrically similar. Stated in loose terms, this theorem asserts that in an (n, A;)-alphabet every letter is treated the same. Every two letters have the same number of nearest neighbors associated with them, the same number of next nearest neighbors, etc. The disposition of points in any two W regions is the same. 1.7 GROUP ALPHABETS AND PARITY CHECKS Theorem 3. Every group alphabet is a systematic^ code: every syste- matic code is a group alphabet. 210 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 We prefer to use the word "alphabet" in place of "code" since the latter has many meanings. In a systematic alphabet, the places in any letter can be divided into two classes : the information places — A; in number for an (n, /c)-alphabet — and the check positions. All letters have the same information places and the same check places. If there are k information places, these may be occupied by any of the 2 /v-place binary sequences. The entries in the n — k check positions are fixed linear (mod 2) combinations of the entries in the information positions. The rules by which the entries in the check places are determined are called parity checks. Examples: for the (4, 2)-alphabet of (6), namely 0000, 1100, 0011, nil, positions 2 and 3 can be regarded as the informa- tion positions. If a letter of the alphabet is the sequence aia^a^ai , then ai = a2 , tti = az are the parity checks determining the check places 1 and 4. For the (5, 3)-alphabet 00000, 10001, 01011, 00111, 11010, 10110, 01100, 11101 places 1, 2, and 3 (numbered from the left) can be taken as the information places. If a general letter of the alphabet is aiazazaiai , then a4 = a2 -j- as , Ob = ai -j- a2 -|- ^3 . Two group alphabets are called equivalent if one can be obtained from the other by a permutation of places. Example: the 7 distinct (3, 2)- alphabets given in (3) separate into three equivalence classes. Alpha- bets (i), (ii), and (iv) are equivalent; alphabets (iii), (v), (vi), are equiva- lent; (vii) is in a class by itself. Proposition S. Equivalent (n, fc) -alphabets have the same probability Qi of correct transmission for each letter. Proposition 4- Every (n, /c) -alphabet is equivalent to an (n, k)- alphabet whose first k places are information places and whose last n — k places are determined by parity checks over the first k places. Henceforth we shall be concerned only with (n. A;) -alphabets w^hose first k places are information places. The parity check rules can then be written k ai = S Tij-ay , t = /b -j- 1, • • • , n (9) where the sums are of course mod 2. Here, as before, a typical letter of the alphabet is the sequence aia^ • ■ - ttn . The jn are k(n — k) quantities, zero or one, that serve to define the particular (n, A;)-alphabet in question. 1.8 MAXIMUM LIKELIHOOD DETECTION BY PARITY CHECKS For any element, J\ of Bn we can form the sum given on the right of (9). This sum maj^ or may not agree with the symbol in the ?'th place of A CLASS OF BINARY SIGNALING ALPHABETS 211 T. If it does, we say T satisfies the tth-place parity check; otherwise T fails the zth-place parity check. When a set of parity check rules (9) is giN'cii, we can associate an (n — /i^-place binary sequence, R{T), with each element T of 5„. We examine each check place of T in order starting with the (k -\- 1 )-st place of T. We write a zero if a place of T satisfies the parity check; we write a one if a place fails the parity check. The re- sultant sequence of zeros and ones, written from left to right is R(T). We call R(T) the parity check sequence of T. Example: with the parity rules 04 = 02 -j- 03 , 05 = Oi -j- 02 -j- c^s used to define the (5, 3)-alphabet in the examples of Theorem 3, we find i?(11000) = 10 since the sum of the entries in the second and third places of 11001 is not the entry of the fourth place and since the sum of Oi = 1, 02 = 1, and 03 = 0 is 0 = 05 . Theorem 4- Let I, A2 , • • • ^^^ be an {n, /c)-alphabet. Let R{T) be the parity check sequence of an element T of B„ formed in accordance with the parity check rules of the (n, /c) -alphabet. Then R(Ti) = R(T2) if and only if Ti and T2 lie in the same row of array (4). The coset leaders can be ordered so that R{Si) is the binary symbol for the integer i — 1. As an example of Theorem 4 consider the (4, 2)-alphabet shown with its cosets below 0000 1011 0101 1110 0100 nil 0001 1010 0010 1001 0111 1100 1000 0011 1101 0110 The parity check rules for this alphabet are 03 = oi , 04 = Oi -j- ^2 • Every element of the second row of this array satisfies the parity check in the third place and fails the parity check in the 4th place. The parity check sequence for the second row is 01. The parity check for the third row is 10, and for the fourth row 11. Since every letter of the alphabet satisfies the parity checks, the parity check sequence for the first row is 00. We therefore make the following association between parity check sequences and coset leaders 00 -^ 0000 = Si 01 -^ 0100 = S2 10 -^ 0010 = S, 11 -^ 1000 = ^4 1.9 INSTRUMENTING A GROUP ALPHABET Proposition 4 attests to the ease of the encoding operation involved 212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 with the use of an (n, fc) -alphabet. If the original message is presented as a long sequence of zeros and ones, the sequence is broken into blocks of length k places. Each block is used as the first k places of a letter of the signaling alphabet. The last n-k places of the letter are determined by fixed parity checks over the first k places. Theorem 4 demonstrates the relative ease of instrumenting the maxi- mum hkelihood detector (8) for use with an (n. A:) -alphabet. When an element T of Bn is received at the channel output, it is subjected to the n-k parity checks of the alphabet being used. This results in a parity check sequence R{T). R(T) serves to identify a unique coset leader, say Si . The product SiT is then formed and produced as the detector out- put. The probability that this be the correct letter of the alphabet is Qi . 1.10 BEST GROUP ALPHABETS Two important questions regarding (n, fc)-alphabets naturally arise. What is the maximum value of Qi possible for a given n and k and which of the N(n, k) different subgroups give rise to this maximum Qi? The answers to these questions for general n and k are not known. For many special values of n and k the answers are known. They are presented in Tables II, III and IV, which are explained below. The probability Qi that a transmitted letter be produced correctly by the detector is the sum, Qi = ^i f{Si) of the probabilities of the coset leaders. This sum can be rewritten as Qi = 2Zi=o «« P^Q^~^ where a, is the number of coset leaders of weight i. One has, of course, ^a, = v = / y) \ T? ' 2^"'' for an (n, /(;)-alphabet. Also «> ^ ( . ) = -7-7 — '■ — n- ! since this is the \t / tlin — t) number of elements of Bn of weight i. The (Xi have a special physical significance. Due to the noise on the channel, a transmitted letter, A, , of an (n, /c)-alphabet will in general be received at the channel output as some element T of Bn different from Ai .li T differs from Ai in s places, i.e., if w{AiT) = s, we say that an s-tuple error has occurred. For a given (n, fc)-alphabet, ai is the number of i-tuple errors which can be corrected by the alphabet in question, i = 0, 1,2, ■ • • , n. Table II gives the a{ corresponding to the largest possible value of Qi for a given k and ?i for k = 2,3, •••w— l,n = 4--- ,10 along with a few other scattered values of n and k. For reference the binomial coeffi- cients ( . ) are also listed. For example, we find from Table II that the best group alphabet with 2 =16 letters that uses n = 10 places has a A CLASS OF BINARY SIGNALING ALPHABETS 213 1 A Q C 'J ** Q probability of correct transmission Qi = q + lOg p + 39g p" + l-Ag'p . The alphabet corrects all 10 possible single errors. It corrects 39 of the possible f .^ j = 45 double errors (second column of Table II) and in addition corrects 14 of the 120 possible triple errors. By adding an addi- tional place to the alphabet one obtains with the best (11, 4)-alphabet an alphabet with 16 letters that corrects all 11 possible single errors and all 55 possible double errors as well as 61 triple errors. Such an alphabet might be useful in a computer representing decimal numbers in binary form. For each set of a's listed in Table II, there is in Table III a set of parity check rules which determines an {n, A)-alphabet having the given a's. The notation used in Table III is best explained by an example. A (10, 4)-alphabet which realizes the a's discussed in the preceding para- graph can be obtained as follows. Places 1, 2, 3, 4 carrj- the information. Place 5 is determined to make the mod 2 sum of the entries in places 3, 4, and 5 ecjual to zero. Place 6 is determined by a similar parity check on places 1, 2, 3, and 6; place 7 by a check on places 1, 2, 4, and 7, etc. It is a surprising fact that for all cases investigated thus far an {n, k)- alphabet best for a given value of p is uniformly best for all values of p, 0 ^ p ^ 1 2. It is of course conjectured that this is true for all n and /,-. It is a further (perhaps) surprising fact that the best {n, fc) -alphabets are not necessarily those with greatest nearest neighbor distance be- tween letters when the alphabets are regarded as point configurations on the n-cube. For example, in the best (7, 3)-alphabet as listed in Table III, each letter has two nearest neighbors distant 3 edges away. On the other hand, in the (7, 3)-alphabet given by the parity check rules 413, 512, 623, 7123 each letter has its nearest neighbors 4 edges away. This latter alphabet does not have as large a value of Qi , however, as does the (7, 3)-alphabet listed on Table III. The cases /.; = 0, 1, /? — 1, n have not been listed in Tables II and III. The cases k = 0 and k = n are completely trivial. For k = 1, all n > 1 the best alphabet is obtained using the parity rule a> = 03= • • • = a„ = oi . If n = '2j, If n = 2j + 1, Qi = i: (^') pY-\ For k = n — 1, /; > 1. the maximum Qi is Qi = g"~ and a parity rule for an alphabet realizing this Qi is o„ = oi . If the a's of an (/<, A)-alphabet are of the form a, = ( . j , i = 0, 1, 214 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table II — Probability of No Error with Best Alphabets, Qi = 2Z «»P*2"~' (?) k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 * = 10 i 0 ai (li a a, ai Oi fli ai a; n = 4 1 1 1 4 3 0 1 1 1 n = 5 1 2 0 5 10 1 5 2 1 3 1 1 71 = 6 1 2 6 15 6 9 6 1 3 0 1 1 1 1 1 n = 7 1 2 3 7 21 25 7 18 6 7 8 7 3 0 1 1 1 1 1 1 n = 8 1 2 3 8 28 56 8 28 27 8 20 3 8 7 7 3 0 1 1 1 1 1 1 1 1 9 9 9 9 9 7 3 n = 9 2 3 4 36 84 126 36 64 18 33 21 22 6 0 1 1 1 1 1 1 1 1 1 10 10 10 10 10 10 7 3 n = 10 2 3 4 45 120 210 45 110 90 45 64 8 39 14 21 5 0 1 1 1 1 1 1 1 1 1 11 11 11 11 11 11 7 3 n = 11 2 3 4 5 55 165 330 462 55 165 226 54 55 126 63 55 61 20 4 0 1 1 1 1 1 1 1 1 12 12 12 12 12 7 3 n = 12 2 3 4 5 66 220 495 792 66 220 425 300 66 200 233 19 3 A CLASS OF BINARY SIGNALIHG ALPHABETS 215 2, • • • , j, «j+i = f some integer, aj+o = ay+s = • • • = «„ = 0, then there does not exist a 2 -letter, w-place alphabet of any sort better than the given (n, A)-alphabet. It will be observed that many of the a's of Table II are of this form. It can be shown that Proposition 5 ii n -\- I „ /"t"! q 1^2"^* — 1 there exists no 2'''-letter, n-place alphabet better than the best (n, /c) -alphabet. When the inequality of proposition 5 holds the a's are either «o = 1, ""'' - 1, all other « = 0; or ao = 1, «i = (Vj , «2 = 2"~' - 1 - , all other a = 0; or the trivial ao = 1 all other a = 0 which holds uhen k = n. The region of the n — k plane for which it is known that (n, A-)-alphabets cannot be excelled by any other is shown in Table IV. 1.11 A DETAILED EXAMPLE As an example of the use of {n, A") -alphabets consider the not un- realistic case of a channel with -p = 0.001, i.e., on the average one binary digit per thousand is received incorrectly. Suppose we wish to transmit messages using 32 different letters. If we encode the letters into the 32 5-place binary sequences and transmit these sequences without further encoding, the probability that a received letter be in error is 1 — (1 _ pf = 0.00449. If the best (10, 5)-alphabet as shown in Tables II and III is used, the probability that a letter be wrong is 1 — Qi = 1 - r/" - lOgV - 21gy - 24/)' - 72p' + • • • = 0.000024. Thus by reducing the signaling rate by ^^, a more than one hundredfold re- duction in probability of error is accomplished. A (10, 5)-alphabet to achieve these results is given in Table III. Let a typical letter of the alphabet be the 10-place sequence of binary digits aia2 ■ • • agttio . The symbols aia^Ozaia^ carry the information and can be any of 32 different arrangements of zeros and ones. The remaining places are determined by 06 = ai -j- a-i -j- a4 -j- ^5 a? = tti -j- oo -f a4 -j- as as = ai -j- a2 + a.3 + Os ag = Oi + 02 4- Qi -j- 0,4 Oio = Oi + a-i -j- 03 4- 04 4- «5 To design the detector for this alphabet, it is first necessary to deter- mine the coset leaders for a standard array (4) formed for this alphabet. •Jl t-l a pa < M Ph < cc o H O H ti; O H I— I -< Ph P3 t^ 00 -f ^ cc CC C^) !M O t^ X lO a; t^ oc 00 C2 ^ ^ CC CC (N C^l t- GC lO ic lO -r -f -^ CC CT CC C^ CM C^I ;C 1^ X c: ^ cc -+ -f -^ cc -f -^ cc cc -r -^ cc re cc (M OCD t^OO C5 C^l ex re C^l CM C^l re-rocot^ ce-^iocot^oc C^l C^l ' >o re f lO CO re T lO CO t^ oC' iCi CO oc 210 1—1 1—1 ^CM 1-H 1-H cO'f -* CM CM CO 1-1 1-l T— 1 1-H 1-H Ot-h Oi-HCM 1—1 1—1 I-H 1-H 1-H 00 ^cot-oo ^^iCiO 134 0 124 1 123 12351 0 123 1 124 2134 Ol ^ 01 .—1 1—1 "^ 1-H 1-H 1-H r^ t- t- t^ coco CO CO lO •^iO-* ^^10^-^ CO "^^00 CO "''^ CO CO CO -* 't< jvj ^'*(N^ '^'* CM CM CM CO(M^ «^^^ COCM^^^ ^-^o -^-^O-H ^'~' 0--HCM GOO-J T-1 00 01 1-1 1—1 00 02 1-H 1-H 1-H CO CO iC ^-t OCOCO^^ iOiO>Oj^ '^'^'^coco -^-^^0^ '^'^^cacM CO!M C^ ^ CO — 1 iCi 'I* lOiOiO'*^ j^ •r-^ c^cc ^ CO CM iM O) ^ CO t^ 00 02 ^ •* ^ CO ^co CO^ ^ -* CM '^'cOCM 'f CM (M CO CO CO ^ "* coco j^ 1— H CO T-H 1— 1 ^H C^l 0 1— 1 CM r-( CM .-< ^ 1— 1 10 CO t^ 00 cr. 10 CO t^ 00 0 rH 1-H CO coco CO CO CM (N (M CO CM CM CI CO CO r— 1 coco CM T—l T~i o*o*c— 1 CO ■* 10 CO t> 00 Oi i-H r— ( CO-*lOCOt^00C2l-Hr-l^ 0 I-l CM T— 1 1— ( 1— t II II II e e e 217 218 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Table IV — Region of the n-k Plane for Which it is Known THAT [n, fc)-ALPHABETS CaNNOT Be EXCELLED k 30 29 28 • • • 27 .... 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 \ 0 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 n This can be done by a \'ariety of special methods which considerably reduce the obvious labor of making such an array. A set of best »S's along with their parity check symbols is given in Table V. A maximum likelihood detector for the (10, 5)-alphabet in question forms from each received sequence 6162 • • • &10 the parity check symbol C1C2C3C4C5 where Ci = h 4- ^h 4- ^3 + Ih + ^5 C2 = 67 -(- 6i -]- h-i + hi \- Ih Cs = &8 + ^^1 + h -j- Ih + ^5 Ci = bg + hi 4- h-i -i- h-.i -(- hi C5 = />in + hi + />, + h, 4- hi 4- 65 According to Table V, if CiC-jCiAf'b contains less than three ones, the de- tector should brint hih^kihih^ . The detector should piint (/m 4- 1)^2^3^4'':. if the parity check sequence C1C2C3C4C5 is either 11111 oi- 11110; the dv- A CLASS OF BINARY SIGNALING ALPHABETS 219 Table V — Coset Leaders and Parity Check Sequences FOR (10, 5) -Alphabet ClCiCsCiCb ^ s CIC2C3C4C6 5 00000 0000000000 11100 0000100001 10000 0000010000 11010 0001000001 01000 0000001000 11001 0001000010 00100 0000000100 10110 0010000001 00010 0000000010 10101 0010000010 00001 0000000001 10011 OOIOOOOIOO 1 1000 0000011000 OHIO 0100000001 10100 0000010100 01101 0100000010 10010 0000010010 01011 0100000100 10001 0000010001 00111 0100001000 01100 0000001100 11110 1000000001 01010 0000001010 11101 OOOOIOOOOO 01001 0000001001 11011 OOOIOOOOOO 00110 0000000110 10111 0010000000 00101 0000000101 01111 0100000000 00011 0000000011 mil 1000000000 tector should print 61(62 -j- l)b3lhh^ if the parity check sequence is 01111, 00111, 01011, 01101, or OHIO; the detector should print hMb-i + 1)6465 if the parity check sequence is 10111, 10011, 10101, or 10110; the de- tector should print 616263(64 -j- 1)65 if the parity check sequence is 11011, 11001, 11010; and finally the detector should print 61626364(65 -j- 1) if the parity check sequence is 11101 or 11100. Simpler rules of operation for the detector may possibly be obtained by choice of a different set of S's in Table V. These quantities in general are not unique. Also there may exist non-equivalent alphabets with simpler detector rules that achieve the same probability of error as the alphabet in question. I'vrt II — Additional Theory and Proofs of Theorems of Part I ' 2.1 the abstract group Cn It will be helpful here to say a few more words about Br, , the group of n-place binary sequences under the operation of addition mod 2. This j group is simply isomorphic with the abstract group Cn generated by n \ commuting elements of order two, say ai, a-2 , ■ ■ ■ , a„ . Here a,:ay = .4,) = E Pr{Y -> A, \ X -^ AdPr{X -^ A^ = ^ E QU,A,) = 4, since E Q^A.A^ = E QUi) = 1. This last follows from the group property of the alphabet. Therefore /i(lO = -- E P>iy -^ A,) log Pr{Y -^ A,) = - bits/symbol. n n It follows then from (10) that h{X I Y) = h(Y I X) The computation of h(Y \ X) follows readily from its definition h{Y I X) = E Prix -^ AdhiY \ X -^ Ai) i = -E Prix -> AdPriY -^ Aj \ X -> Ai) log PHY -^Aj I X-^Ai) = -^,1211 PriN ->AiScAj) log E PriN -> AiS„,Aj) I = -^,ZQiAiAj)'}ogQiAiAj) Zi ij = - EQU,)logQ(A,) i Each letter is n binary places. Proposition 2, then follows. 2.3 DISTANCE AND THE PROOF OF THEOREM 1 Let A and B be two elements of Bn ■ We define the distance, diA, B), between A and B to be the weight of their product, d{A, B) = w(AB) (11) The distance between .4 and B is the number of places in which A and B difTer and is jnsl the "Hamming distance." ^ In terms of the n-cube, diA, B) is Ihe minimum mmiber of edges that must be traversed to go A CLASS OF BINARY SIGNALING ALPHABETS 223 from vertex ^4 to vertex B. The distance so defined is a monotone fnne- tion of the Euchdean distance between vertices. It follows from (11) that if C is any element of B„ then d{A,B) = cJ(A(\BC) (12) This fact shows the detection scheme (8) to be a maximum likelihood detector. By definition of a standard array, one has d(Si , I) ^ d(S,Aj , I) for all i and j The coset leaders were chosen to make this true. From (12), d(S, , I) = d(SiA,„S,- , / .4„.^S,) = d(SiA,n , A,„) d(SAj , /) - diS^AjSiAm , I SiAJ = diAjA,n , SiAr.) = d{SiAm , A() where Af = AjA^ . Substituting these expressions in the inecjuality above yields d(SiAm , A„,) ^ d(SiAm , At) for all i, m, I This equation says that an arbitrary element in the array (4) is at least as close to the element at the top of its column as it is to any other letter of the alphabet. This is the maximum likelihood property. 2.4 PROOF OF THEOREM 2 Again consider an (n, /c) -alphabet as a set of vertices of the unit n-cube. Consider also n mutually perpendicular hyperplanes through the cen- troid of the cube parallel to the coordinate planes. We call these planes "symmetr}^ planes of the cube" and suppose the planes numbered in accordance with the corresponding parallel coordinate planes. The reflection of the vertex with coordinates (ai , a^ , • • • , a^ , • • • , a,j) in symmetry plane i yields the vertex of the cube whose coordinates are (ai , oo , ■ • • , a, -j- 1, • • • , 0,0 . More generally, reflecting a given vertex successively in symmetry planes i, j, k, ■ • ■ yields a new vertex whose coordinates differ from the original vertex precisely in places i, j, k ■ ■ ■ . Successive reflections in hyperplanes constitute a transfor- mation that leaves distances between points unaltered and is therefore a "rotation." The rotation obtained by reflecting successively in sym- metry planes ?', j, k, etc. can be represented by an ?i-place symbol having a one in places ?', j, k, etc. and a zero elsewhere. We now regard a given {n, /j)-alphabet as generated by operating on the vertex (0, 0, • • ■ , 0) of the cube with a certain collection of 2 ro- 224 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 tation operators. The symbols for these operators are identical with the sequences of zeros and ones that form the coordinates of the 2 points. It is readily seen that these rotation operators form a group which is transitive on the letters of the alphabet and which leave the unit cube invariant. Theorem 2 then follows. Theorem 2 also follows readily from consideration of the array (4). For example, the maximum likelihood region associated with / is the set of points I, So , S3 , • • • , Sy . The maximum likelihood region asso- ciated with A; is the set of points Ai , AiS^ , AiSs , ■ • ■ , AiSy . The rotation (successive reflections in symmetry planes of the cube) whose symbol is the same as the coordinate sequence of Ai sends the maximum likelihood region of / into the maximum likelihood region oi Ai , i = 1, 2, • • • , M. 2.5 PROOF OF THEOREM 3 That every systematic alphabet is a group alphabet follows trivially from the fact that the sum mod 2 of two letters satisfying parity checks is again a letter satisfying the parity checks. The totality of letters satis- fying given parity checks thus constitutes a finite group. To prove that every group alphabet is a systematic code, consider the letters of a given (w, /c) -alphabet listed in a column. One obtains in this way a matrix with 2 rows and n columns whose entries are zeros and ones. Because the rows are distinct and form a group isomorphic to Ck , there are k linearly independent rows (mod 2) and no set of more than h independent rows. The rank of the matrix is therefore h. The matrix therefore possesses k linearly independent (mod 2) columns and the remaining n — k columns are linear combinations of these A;. Main- taining only these k linearly independent columns, we obtain a matrix of k columns and 2*' rows with rank k. This matrix must, therefore, have k linearly independent rows. The rows, however, form a group under mod 2 addition and hence, since k are linearly independent, all 2" rows must be distinct. The matrix contains only zeros and ones as entries; it has 2 distinct rows of k entries each. The matrix must be a listing of the num- bers from 0 to 2^^ — 1 in binary notation. The other n — k columns of the original matrix considered are linear combinations of the columns of this matrix. This completes the proof of Theorem 3 and Proposition 4. 2.6 PROOF OF THEOREM 4 To prove Theorem 4 we first note that the parity check sequence of the product of two elements of Bn is the mod 2 sum of their separate A CLASS OF BINARY SIGNALING ALPHABETS 225 parity check sequences. It follows then that all elements in a given coset have the same parity check sequence. For, let the coset be Si , SiA2 , SiAz , ■ ■ • SiA^ . Since the elements I, A^ , A3, • • • , A^ all have parity check sequence 00 • • • 0, all elements of the coset have parity check R(Si). In the array (4) there are 2" cosets. We observe that there are 2"~* elements of Bn that have zeros in their first k places. These elements have parity check symbols identical with the last n — k places of their symbols. These elements therefore give rise to 2"~ different parity check symbols. The elements must be distributed one per coset. This proves Theorem 4. 2.7 PROOF OF PROPOSITION 5 If n ^ 2"-' - we can explicity exhibit group alphabets having the property mentioned in the paragraph preceding Proposition 5. The notation of the demon- stration is cumbersome, but the idea is relatively simple. We shall use the notation of paragraph 2.1 for elements of Bn , i.e., an element of Bn will be given by a list of integers that specify what places of the sequence for the element contain ones. It will be convenient furthermore to designate the first k places of a sequence by the integers 1, 2, 3, • • • , k and the remaining n — k places by the "integers" 1', 2', 3', • • • , r, where ( = n — k. For example, if n = 8, /c = 5, we have 10111010^ 13452' 10000100^ 11' 00000101 ^ 1'3' Consider the group generated by the elements 1', 2', 3', • • • , (' , i.e. the 2' elements /, 1', 2', ■■■,(', 1'2', 1'3', • • • , 1'2'3' ■■■('. Suppose these elements listed according to decreasing weight (say in decreasing order when regarded as numbers in the decimal system) and numbered consecutively. Let Bt be the zth element in the list. Example: if ( ^ 3, Ih = 1'2'3', B2 = 2'3', B, = 1'3', B, = 1'2', B, - 3', B, = 2', B, - 1'. Consider now the (n, /^-alphabet whose generators are ISi , 2B, ,W,, ■■• , kBk We assert that if 22G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 >r% n—k .. — 2 - this alphabet is as good as any other alphabet of 2 letters and n places. In the first place, we observe that every letter of this (n, A-)-alphabet (except /) has unprimed numbers in its symbols. It follows that each of the 2' letters /, 1', 2', • ■ • , (', V2', ■■■ , V2' ■■■ (' occurs in a different coset of the given (n, A-)-alphabet. For, if two of these letters appeared in the same coset, their product (which contains only primed numbers) would have to be a letter of the (n, k) alphabet. This is impossible since every letter of the (/i, A) alphabet has unprimed numbers in its symbol. Since there are precisely 2 cosets we can designate a coset by the single element of the list Bi , Bi , ■ • ■ , B-ii = I which appears in the coset. We next observe that the condition 71 ^ 2 — guarantees that J5a+i is of weight 3 or less. For, the given condition is equivalent to '-■-©-o-o-e We treat several cases depending on the weight of Bu+i . If Bk+\ is of weight 3, we note that for i = 1,2, • • • , A-, the coset con- taining Bi also contains an element of weight one, namely the element i obtained as the product of Bi with the letter iBi of the given (n, A;)- alphabet. Of the remaining (2 — A') 5's, one is of weight zero, C are of weight one, f j are of weight 2 and the remaining are of weight 3. We have, then an = 1, ai = f + A- = n. Now every B of weight 4 occurs in' the list of generators \Bi , 2B-2 , • • • , kBk . It follows that on multi- plying this list of generators by any B of weight 3, at least one element of weight two will result. (E.g., (l'2'3')(il'2'3'40 = j4') Thus every coset with a B of weight 2 or 3 contains an element of weight 2 and a2 = 2 — ao — cn] . The argument in case Bk+i is of weight two or one is similar. 2.8 MODULAR REPRESENTATIONS OF C„ In order to explain one of the methods used to obtain the best (//, A)- alphabets listed in Tal)les II and III, it is necessary to digress here lo present additional theory. I A CLASS OF BINARY SINGALING ALPHABETS 227 It has been remarked that every (n, /v)-alphabet is isomorphic with Ck . Let us suppose the elements of Ci, hsted in a column starting with / and proceeding in order /, 1, 2, 3, • • • , /.', 12, 13, ■••,(/.•— 1)/,-, 123, , 123 • • • k. The elements of a given (n, A-)-alphabet can be paired off with these abstract elements so as to preserve group multipli- cation. This can be done in many different ways. The result is a matrix with elements zero and one with 7i columns and 2 rows, these latter being labelled by the symbols /, 1,2, • • • etc. What can be said about the columns of this matrix? How many different columns are possible when all (n, A)-alphabets and all methods of establishing isomorphism with Ck are considered? In a given column, once the entries in rows 1,2, • • • , /,• are known, the entire column is determined by the group property. There are therefore only 2 possible different columns for such a matrix. A table showing these 2 possible columns of zeros and ones will be called a modular repre- senfafion table for Ck ■ An example of such a table is shown for /,• = 4 in Table VI. It is clear that the colunuis of a modular representation table can also be labelled by the elements of Ck , and that group multiplication of these column labels is isomorphic with mod 2 addition of the columns. The table is a symmetric matrix. The element with row label A and column label B is one if the symbols A and B have an odd number of different numerals in common and is zero otherwise. Every (n, /c)-alphabet can be made from a modular representation table by choosing w columns of the table (with possible repetitions) at least k of which form an independent set. Table VI — Modular Representation Table for Group C4 I 12 3 4 12 13 14 23 24 34 123 124 134 234 1234 I 1 2 3 4 12 13 14 23 24 34 123 124 134 234 1234 0 0 0 0 0 0 0 0 0 0 0 0 0 0 n 0 1 0 0 0 1 1 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 0 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 0 1 1 0 0 1 1 1 () 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 1 1 1 0 0 0 0 1 u 228 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 We henceforth exclude consideration of the column / of a modular representation table. Its inckision in an (n, /v)-alphabet is clearly a waste of 1 binary digit. It is easy to show that every column of a modular representation table for Ch contains exactly 2 " ones. Since an (n, /v)-alphabet is made from n such columns the alphabet contains a total of n2 '~ ones and we have Proposition 6. The weights of an (n, /c)-alphabet form a partition of n2''~^ into 2* — 1 non-zero parts, each part being an integer from the set 1,2, ■■■ ,n. The identity element always has weight zero, of course. It is readily established that the product of two elements of even weight is again an element of even weight as is the product of two ele- ments of odd weight. The product of an element of even weight with an element of odd weight yields an element of odd weight. The elements of even weight of an (n, A;) -alphabet form a subgroup and the preceding argument shows that this subgroup must be of order 2*" or 2*""^ If the group of even elements is of order 2''~\ then the collec- tion of even elements is a possible (n, k — l)-alphabet. This (n, k — 1) alphabet may, however, contain the column / of the modular represen- tation table of Ck-i ■ We therefore have Proposition 7. The partition of Proposition 6 must be either into 2^ — 1 even parts or else into 2 " odd parts and 2^—1 even parts. In the latter case, the even parts form a partition of a2 "" where a is some integer of the set k — I, k, ■ • • , n and each of the parts is an in- teger from the set 1, 2, • • • , n. 2.9 THE CHARACTERS OF Ck Let us replace the elements of Bn (each of which is a sequence of zeros and ones) by sequences of 4-1 's and — I's by means of the following substitution The multiplicative properties of elements of Bn can be preserved iti this new notation if we define the product of two 4-1,-1 symbols to be the symbol whose tth component is the ordinary product of the ?'th compo- nents of the two factors. For example, 1011 and 01 10 become respectively -11 -1 -1 and 1 -1 -11. We have (-11 -1 -1)(1 -1 -11) = (-1 -11 -1) 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1 A CLASS OF BINARY SIGNALING ALPHABETS 229 corresponding to the fact that (1011) (0110) = (1101) If the +1,-1 symbols are regarded as shorthand for diagonal matrices, so that for example -11 -1 -1 then group multiplication corresponds to matrix multiplication. (While much of what follows here can be established in an elementary way for the simple group at hand, it is convenient to fall back upon the established general theory of group representations for several proposi- tions. The substitution (13) converts a modular representation table (col- umn / included) into a square array of +l's and — I's. Each column (or row) of this array is clearly an irreducible representation of Ck ■ Since Ck is Abelian it has precisely 2 irreducible representations each of degree one. These are furnished by the converted modular table. This table also furnishes then the characters of the irreducible representations of Ck and we refer to it henceforth as a character table. Let x"(^) be the entry of the character table in the row labelled A and column labelled a. The orthogonality relationship for characters gives E x'{A)/{A) = 2'8., ACCk Z x%A)x"(B) = 2'b ) = ^5, whereas a-z = 39, so the given alphabet does not correct all possible double errors. In the standard array for the alphabet, 39 coset leaders are of weight 2. Of these 39 cosets, 33 have only one element of weight 2; the remaining 6 cosets each contain two elements of weight 2. This is due to the two elements of weight 4 in the given group, namely 1289 and 3467. A portion of the standard array that demonstrates these points is 1289 3467 12 89 • 18 29 • 19 28 . 34 67 36 47 37 46 ] • In order to have a smaller probability of error than the exhibited alphabet, it is necessary that a (10, 4)-alphabet have an a^ > 39. We proceed to show that this is impossible by consideration of the weights of the letters of possible (10, 4)-alphabets. We first show that every (10, 4)-alphabet must have at least one ele- ment (other than the identity, /) of weight less than 5. By Propositions • ') and 7, Section 2.8, the weights must form a partition of 10-8 = 80 into 1 5 positive parts. If the weights are all even, at least two must be less than 6 since 14-6 = 84 > 80. If eight of the weights are odd, we see from 8-5 + 7-() = 82 > 80 that at least one weight must be less than 5. 232 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 An alphabet with one or more elements of weight 1 must have an «2 ^ 36, for there are nine elements of weight 2 which cannot possibly be coset leaders. To see this, suppose (without loss of generality) that the alphabet contains the letter 1. The elements 12, 13, 14, • • • 1 10 can- not possibly be coset leaders since the product of any one of them with the letter 1 yields an element of weight 1 . An alphabet with one or more elements of weight 2 must have an ai S 37. Suppose for example, the alphabet contained the letter 12. Then 13 and 23 must be in the same coset, 14 and 24 must be in the same coset, ■ • • , 1 10 and 2 10 must be in the same coset. There are at least eight elements of weight two which are not coset leaders. Each element of weight 3 in the alphabet prevents three elements of weight 2 from being coset leaders. For example, if the alphabet contains 123, then 12, 13, and 23 cannot be coset leaders. We say that the three elements of weight 2 are "blocked" by the letter of weight 3. Suppose an alphabet contains at least three letters of weight three. There are several cases: (A) if three letters have no numerals in common, e.g., 123, 456, 789, then nine distinct elements of weight 2 are blocked and a-2 S 36; (B) if no two of the letters have more than a single numeral in common, e.g., 123, 345, 789, then again nine elements of weight 2 are blocked and a-2 ^ 36; and (C) if two of the letters of weight 3 have two numerals in common, e.g., 123, 234, then their product is a letter of weight 2 and l)y the preceding paragraph ao ^ 37. If an alphabet contains exactly two elements of weight 3 and no elements of weight 2, the elements of weight 3 block six elements of weight 2 and 0:2 ^ 39. The preceding argument shows that to be better than the exhibited alphabet a (10, 4)-alphabet with letters of weight 3 must have just one such letter. A similar argument (omitted here) shows that to be better than the exhibited alphabet, a (10, 4)-alphabet cannot contain more than one element of weight 4. Furthermore, it is easily seen that an alphabet containing one element of weight 3 and one element of weight 4 must have an ao ^ 39. The only new contenders for best (10, 4)-alphabet are, therefore, alphabets with a single letter other than / of weight less than 5, and this letter must have weight 3 or 4. Application of Propositions 6 and 7 show that the only possible weights for alphabets of this sort are: 35 6 and 5 46' where 5' means seven letters of weight 5, etc. We next show that there do not exist (10, 4)-alphabets having these weights. Consider first the suggested alphabet with weights 35 6'. As explained in Section 2.9, from such an alphabet we can construct a matrix repre- sentation of ('4 having the character x(/) = 10, one matrix of trace 4, A CLASS OF BINARY SIGNALING ALPHABETS 233 seven of trace 0 and seven of trace —2. The latter seven matrices cor- respond to elements of even weight and together with / must represent a subgroup of order 8. We associate them with the subgroup generated by the elements 2, 3, and 4. We have therefore x(/) = 10, x(2) = x(3) = x(4) = x(23) = x(24) = x(34) = x(234) = -2. Examination of the symmetries involved shows that it doesn't matter how the remaining Xi ai"e associated with the remaining group elements. We take, for example x(l) = 4, x(12) = x(13) = x(14) = x(123) = x(124) = x(134) = x(1234) = 0. Now form the sum shown in equation (14) with /3 = 1234 (i.e., with the character x^" obtained from column 1234 of the Table VI by means of substitution (13). There results c?i234 = V-i which is impossible. There- fore there does not exist a (10, 4) -alphabet with weights 35 6 . The weights 5 46 correspond to a representation of d with character x(/) = 10, 0^, 2, ( — 2)^ We take the subgroup of elements of even weight to be generated by 2, 3, and 4. Except for the identity, it is clearly im- material to w^hich of these elements we assign the character 2. We make the following assignment: x(/) = 10, x(2) = 2, x(3) = x(4) = x(23) = x(24) = x(34) = x(234) = -2, x(l) = x(12) = x(13) = x(14) = x(123) = x(124) = x(134) = x(1234) = 0. The use of equation (14) shows that ^2 = \'2 which is impossible. It follows that of the 53,743,987 (10, 4)-alphabets, none is better than the one listed on Table III. Not all the entries of Table III were established in the manner just demonstrated for the (10, 4)-alphabet. In many cases the search for a l)est alphabet was narrowed down to a few alphabets by simple argu- ments. The standard arrays for the alphabets were constructed and the best alphabet chosen. For large n the labor in making such a table can be considerable and the operations involved are highly liable to error when performed by hand. I am deeply indebted to V. M. Wolontis who programmed the IBM CPC computer to determine the a's of a given alphabet and who pa- tiently ran off many such alphabets in course of the construction of Tables II and III. I am also indebted to Mrs. D. R. Fursdon who eval- uated many of the smaller alphabets by hand. 234 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 REFERENCES 1. R. W. Hamming, B.S.T.J., 29, i)p. 147-160, 1950. 2. I. S. Reed, Transactions of tlie Piofossional (iroup on Information Tlieorv, ^ PGIT-4, PI). 3S-49, 1954. 3. See section 7 of R . W. Hamniinji's paper, loc. cit. 4. I.R.E. Convention Record, I'art 4, pp. 37-45, 1955 National Convention, March, 1955. 5. C. E. Shannon, B.S.T.J., 27, pp. 379-423 and pp. 623-656, 1948. 6. Birkhoff and MacLane, A Snrvey of Modern Algebra, Macmillan Co., New York, 1941 . Van der Waerden, Alodern Algebra, Ungar Co., New York, 1953. Miller, Bliclifeldt, and Dickson, Finite Groups, Stechert, New York, 1938. 7. This theorem has been previously noted in the literature by Kiyasu-Zen'iti, Research and Development Data No. 4, Ele. Comm. Lai)., Nippon Tele. Corp. Tokyo, Aug., 1953. 8. F. D. Murnaghan, Theory of Group Representations, Johns Hopkins Press, Baltimore, 1938. E. Wigner, Gruppentheorie, Edwards Brothers, Ann Arbor, Michigan, 1944. I Bell System Technical Papers Not Published in This Journal Allen, L. J., see Fewer, D. R. Alllson, H. W., see Moore, G. E. Baker, W. 0., see Winslow, F. H. Barstow, J. M.^ Color TV How it Works, I.R.E. Student Quarterly, 2, pp. 11-16, Sept., 1955. Basseches, H.^ and ^McLean, D. A. Gassing of Liquid Dielectrics Under Electrical Stress, Ind. c^- Engg. Chem., 47, pp. 1782-1794, Sept., 1955. Beck, A. C} Measurement Techniques for Multimode Waveguides, Proc. I.R.E., MRI, 4, pp. 325-6, Oct. 1, 1955. Becker, J. A.^ The Life History of Adsorbed Atoms, Ions, and Molecules, N. Y. Acad. Sci. Ann., 58, pp. 723-740, Sept. 15, 1955. Hlackwell, J. H., see Fewer, D. R. BooRSE, H. A., see Smith, B. HozoRTii, R. M.,' Getlin, B. B.,' Galt, J. K.,' Merritt, F. R.,' and- ^'ager, W. a.' Frequency Dependence of Magnetocrystalline Anisotropy, Letter to the Editor, Phys. Rev., 99, p. 1898, Sept. 15, 1955. 1. Bell Telephone Laboratories, Inc. 235 236 THE BELL SYSTEM TECHXICAL JOURNAL, JANUARY 1956 BozoRTH. R. M.\ TiLDEX, E. F..' and Williams, A. j/ Anisotropy and Magnetostriction of Some Ferrites, Phys. Rev., 99, pp. 17S8-1798, Sept. 15, 1955. Bridgers, H. E.,^ and Kolb, E. D.^ Rate-Grown Germanium Crystals for High-Frequency Transistors, Letter to the Editor, J. Appl. Phys., 26, pp. 1188-1189, Sept., 1955. j BULLIXGTOX, K.^ Characteristics of Beyond-the-Horizon Radio Transmission, Pioc. I.R.E., 43, pp. 1175-1180, Oct., 1955. BULLIXGTOX, K.^ IXKSTER, W. J.,^ and DVRKEE, A. L.^ Results of Propagation Tests at 505 Mc and 4,090 Mc on Beyond- Horizon Paths, Proc. I.R.E., 43, pp. 1306-1316, Oct., 1955. Calbick, C. J.' Surface Studies with the Electron Microscope, X. Y. Acad. Sci. Ann., 58, pp. 873-892, Sept. 15, 1955. Cass, R. S., see Fewer, D. R. DuRKEE, A. L., see Bullington, K. Fewer, D. R..' Blackwell. J. H..' Allex. L. J..^ and Cass, R. S." Audio-Frequency Circuit Model of the 1-Dimensional Schroedinger Equation and Its Sources of Error, Canadian J. of Pins., 33, pp. 483- 491, Aug., 1955. Francois, E. E., see Law, J. T. Davis, J. L., see Suhl, H. Galt, J. K., see Bozorth, R. "SI., and Yager, W. A. Garn, p. D.,' and Hallixe, Mrs. E. W.' Polarographic Determination of Phthalic and Anhydride Alkyd Res- ins, Anal Cliem., 27, pp. 15()3-15G5, Oct., 1955. 1. Bell Telephone Laboratories, Inc. 4. University of Western Ontario, London, Canada 5. Bell Telephone Company of Canada, Montreal TECHNICAL PAPERS 237 Getlin, B. B., see Bozorth, R. M. GlANOLA, V. F} Application of the Wiedemann Effect to the Magnetostrictive Coupling of Crossed Coils, J. Appl. Phys., 26, pp. 1152-1157, Sept., 1955. Goss, A. J., see Hassion, F. X. Green, E. I.^ The Story of 0, American Scientist, 43: pp. 584-594, Oct., 1955. Halline, Mrs. E. W., see Garn, P. D. Harrower, G. A.^ Measurement of Electron Energies by Deflection in a Uniform Electric Field, Rev. Sci. Instr., 26, pp. 850-854, Sept., 1955. Hassion, F. X.,^ Goss, A. .1.,^ and Trumbore, F. A.^ The Germanium-Silicon Phase Diagram, J. Phys. Chem., 59, p. 1118, Oct., 1955. Hassion, F. X.,^ Thurmond, C. D.,^ and Trumbore, F. A.^ On the Melting Point of Germanium, J. Phys. Chem., 59, p. 1076, Oct., 1955. Hines, I\I. E.,' Hoffman, G. W.,' and Saloom, J. A.^ Positive-Ion Drainage in Magnetically Focused Electron Beams, J. Appl. Phys., 26, pp. 1157-1162, Sept., 1955. Hoffman, G. W., see Hines, M. E. Inkster, W. J., see Bullington, K. Kelly, M. J.' Training Programs of Industry for Graduate Engineers, Elec. Engg., 74, pp. 866-869, Oct., 1955. KoLB, E. D., see Bridgers, H. E. 1. Bell Telephone Laboratories, Inc. 1 238 THE BELL SYSTEM TECHXICAL JOURXAL, JANUARY 1 9 of) Law, J. T./ and Francois, E. E.' Adsorption of Gasses and Vapors on Germanium, X. Y. Acad. Sci. Ann., 58, pp. 925-936, Sept. 15, 1955. LovELL, Miss L. C, see Pfann, W. G. Matreyek, W., see Winslow, F. H. McLean, D. A., see Basseches, H. Merritt, F. R., see Bozorth, R. M., and Yager, W. A. Meyer, F. T.' An Improved Detached-Contact Type of Schematic Circuit Drawing, A.LE.E. Commun. ct Electronics, 20, pp. 505-513, Sept., 1955. Miller, B. T.' Telephone Merchandising, Telephony, 149, pp. 116-117, Oct. 22, 1955. Miller, S. L.^ Avalanche Breakdown in Germanium, Phys. Rev., 99, pp. 1234-1241, Aug. 15, 1955. Moore, G. E.,^ and Allison, H. W.^ Adsorption of Strontium and of Barium on Tungsten, J. Chem. Phys., 23, pp. 1609-1621, Sept., 1955. Neisser, W. R.,^ Liquid Nitrogen Coal Traps, Rev. Sci. Instr., 26, p. 305, Mar., 1955. Ostergren, C. N." Some Observations on Liberahzed Tax Depreciation, Telephony, 149, pp. 16-23-37, Oct. 1, 1955. Ostergren, G. N. Depreciation and the New Law, Telephony, 149, pp. 96-100-104-108, ; Oct. 22, 1955. I Rape, N. R., see Winslow, F. H. 1. Bell Telephone Laboratories, Inc. 2. American Telephone and Telegraph Co. \\ technical papers 239 Pedekskn, L. Aluminum Die Castings for Carrier Telephone Systems, A.I.E.E. Commun. & Electronics, 20, pp. 434-439, Sept., 1955. Peters, H.^ Hard Rubber, Tnd. and Engg. Chem., Part II, pp. 2220-2222, Sept. 20, 1955. Pfann, w. c;.' Temperature-Gradient Zone-Melting, J. Metals, 7, p. 961, Sept., 1955. Pfann, W. G.,' and Lovell, Miss L. C.^ Dislocation Densities in Intersecting Lineage Boundaries in Ger- manium, Letter to the Editor, Acta. Met., 3, pp. 512-513, Sept., 1955. Pierce, J. P.' Orbital Radio Relays, Jet Propulsion, 25, pp. 153-157, Apr., 1955. Poole, K. M.' Emission from Hollow Cathodes, J. Appl. Phys., 26, pp. 1176-1179, Sept., 1955. Saloom, J. A., see Hines, M. E. Slighter, W. P.^ Proton Magnetic Resonance in Polyamides, J. Appl. Phys., 26, pp., 1099-1103, Sept., 1955. Smith, B./ and Boorse, H. A. Helium II Film Transport. II. The Role of Surface Finish, Phys. Rev. 99, pp. 346-357, July 15, 1955. Smith, B.,^ and Boorse, H. A. Helium II Film Transport. IV. The Role of Temperature, Phys. Rev., 99, pp. 367-370, July lo, 1955. SuHL, H.,^ Van Uitert, L. G.,^ and Davis, J. L.^ Ferromagnetic Resonance in Magnesium-Manganese Aluminum Fer- rite Between 160 and 1900 Mc, Letter to the Editor, J. Appl. Phys., 26, pp. 1181-1182, Sept., 1955. 1. Bell Telephone Laboratories, Inc. 6. Columbia University, New York City 240 THE EELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 Thurmond, C. D., see Hassion, F. X. TiDD, W. H/ I Demonstration of Bandwidth Capabilities of Beyond -Horizon Tropo- spheric Radio Propagation, Proc. I.R.E., 43, pp. 1297-1299, Oct., 1955. Tien, P. K.,' and Walker, L. R.' Large Signal Theory of Traveling -Wave Amplifiers, Proc. I.R.E., 43, p. 1007, Aug., 1955. TiLDEN, E. F., see Bozorth, R. M. Trumbore, F. a., see Hassion, F. X. IThlir, a., Jr.^ Micromachining with Virtual Electrodes, Rev. Sci., Instr., 26, pp. 965-968, Oct., 1955. Ulrich, W., see Yokelson, B. J. Van Uitert, L. G., see Siihl, H. Walker, L. R., see Tien, P. K. Weibel, E. S.' Vowel Synthesis by Means of Resonant Circuits, J. Acous. Soc, 27, pp. 858-865, Sept., 1955. Williams, A. J., see Bozorth, R. M. WiNSLow, F. H.,' Baker, W. O.,^ and Yager, W. A.^ Odd Electrons in Polymer Molecules, Am. Chem. Soc, 77, pp. 4751- 4756, Sept. 20, 1955. WiNSLow, F. II.,' Baker, W. O.,' Rape, N. R.' and Matreyek, W.' Formation and Properties of Polymer Carbon, J. Polymer Science, 16, p. 101, Apr., 1955. Yager, W. A., sec Bozorth, R. M. 1. Bell Tc;l(;i)li()ne liaboratorics, Inc. TECHNICAL PAPERS 241 Yagkr, W. a./ Galt, J. K/ and Merritt, F. R.' Ferromagnetic Resonance in Two-Nickel-Iron Ferrites, Phys. Rev., 99, pp. 1203-1209, Aug. 15, 1955. YoKELSON, B. J.,^ and Ulrich, W.^ Engineering Multistage Diode Logic Circuits, A.I.E.E. Commun. & Electronics, 20, pp. -466-475, Sept., 1955. 1. Bell Telephone Laboratories, Inc. Recent Monographs of Bell System Technical Papers Not Published in This Journal* Arnold, W. O., and Hoefle, R. R. A System Plan for Air Traffic Control, ]\Ionograph 2483. Beck, A. C. Measurement Techniques for Multimode Waveguides, ]\Ioiiograph 2421. Becker, J. A., and Brandes, R. G. Adsorption of Oxygen on Tungsten as Revealed in Field Emission Microscope, Alonogiaph 24U3. Boyle, W. S., see Germer, L. H. Brandes, R. G., see Becker, J. A. Brattain, W. H., see Garrett, C. G. B. Garrett, C. G. B., and Brattain, W. H. Physical Theory of Semiconductor Surfaces, Monograph 2453. Gerner, L. H., Boyle, W. S., and Kisliuk, P. Discharges at Electrical Contacts — II, Monograph 2499. Hoefle, R. R., see Arnold, W. 0. KisLiuK, P., see Germer, L. H. Linvill, J. G. Nonsaturating Pulse Circuits Using Two Junction Transistors, Mono- graph 2-17."). I * Copies of these monographs may 1)0 ()l)l;tin(Ml on request to the Pul)licat ion Department, Hell Telephone Laboratories, Iiie., 463 West Street, New York 14, N. Y. The numbers of the monographs should be given in all requests. 242 MONOGRAPHS 243 Mason, W. P. Relaxations in the Attenuation of Single Crystal Lead, Monograph 2454. Mkykr, F. T. An Improved Detached-Contact-Type of Schematic Circuit Drawing, Monograph 2456. VoGEL, F. L., Jr. Dislocations in Low-Angle Boundaries in Germanium, Monograph 2455. Walker, T.. R. Generalizations of Brillouin Flow, Monograph 2432. Warner, A. W. Frequency Aging of High -Frequency Plated Crystal Units, Monograph 2474. Weibel, E. S. On Webster's Horn Equation, Monograph 2450. Contributors to This Issue A. C. Beck, E.E., Rensselaer Polytechnic Institute, 1927; Instructor, Rensselaer Polytechnic Institute, 1927-1928; Bell Telephone Labora- tories, 1928 -. With the Radio Research Department he was engaged in the development and design of short-wave and microwave antennas. During World War II he was chiefly concerned with radar antennas and associated waveguide structures and components. For several years after the war he worked on development of microwave radio repeater systems. Later he worked on microwave transmission developments for broadband communication. Recently he has concentrated on further developments in the field of broadband communication using circular waveguides and associated test equipment. J. S. Cook, B.E.E., and M.S., Ohio State University, 1952; Bell Telephone Laboratories, 1952 -. Mr. Cook is a member of the Research in High-Frequency and Electronics Department at Murray Hill and has been engaged principally in research on the traveling- wave tube. Mr. Cook is a member of the Institute of Radio Engineers and belongs to the Professional Group on Electron Devices. 0. E. DeLange, B.S. University of Utah, 1930; M.A. Columbia Uni- versity, 1937; Bell Telephone Laboratories, 1930 — . His early work was principally on the development of high-frequency transmitters and re- ceivers. Later he worked on frequency modulation and during World War II was concerned with the development of radar. Since that time he has been involved in research using broadband systems including microwa^'e and baseband. Mr. DeLange is a member of the Institute of Radio Engineers. R. KoMPFNER, Engineering Degree, Technische Hochschule, Vienna, 1933; Ph.D., Oxford, 1951; Bell Telephone Laboratories, 1951 -. Be- tween 1941-1950 he did work for the British Admiralty at Birmingham University and Oxford University in the Royal Naval Scientific Service. He invented the traveling-wave tube and for this achievement Dr. Kompfner i-eceived the 1955 Duddcll Medal, bestowed by the Physical Society of England. In the Laboratoi'ies' Research in High Frequency 244 CONTRIBUTORS TO THIS ISSUE 245 and Electronics Department, he has continued his research on vacuum tubes, particularly those used in the microwave region. He is a Fellow of the Institute of Radio Engineers and of the Physical Society in London. Charles A. Lee, B.E.E., Rensselaer Polytechnic Institute, 1943; Ph.D., Columbia University, 1953; Bell Telephone Laboratories, 1953-. When Mr. Lee joined the Laboratories he became engaged in research concerning solid state devices. In particular he has been developing techniques to extend the frequency of operation of transistors into the microwave range, including work on the diffused base transistor. During World War II, as a member of the United States Signal Corps, he was concerned with the determination and detection of enemy counter- measures in connection with the use of proximity fuses by the Allies. He is a member of the American Physical Society and the American Institute of Physics. He is also a member of Sigma Xi, Tau Beta Pi and Eta Kappa Nu. John R. Pierce, B.S., M.S. and Ph.D., California Institute of Tech- nology 1933, 1934 and 1936; Bell Telephone Laboratories, 1936-. Ap- pointed Director of Research — Electrical Communications in August, 1955. Dr. Pierce has specialized in Development of Electron Tubes and Microwave Research since joining the Laboratories. During World War li II he concentrated on the development of electronic devices for the [I Armed Forces. Since the war he has done research leading to the develop- ;j ment of the beam traveling- wave tube for which he was awarded the h 1947 Morris Liebmann Memorial Prize of the Institute of Radio Engi- [li neers. Dr. Pierce is author of two books: Theory and Design of Electron Ij Beams, published in second edition last year, and Traveling Wave Tubes il (1950). He was voted the ''Outstanding Young Electrical Engineer of [| 1942" by Eta Kappa Nu. Fellow of the American Physical Society and J the I.R.E. Member of the National Academy of Sciences, the A.I.E.E., I Tau Beta Pi, Sigma Xi, Eta Kappa Nu, the British Interplanetary So- il ciety, and the Newcomen Society of North America. C. F. QuATE, B.S., University of Utah 1944; Ph.D., Stanford Uni- i versity 1950; Bell Laboratories 1950-. Dr. Quate has been engaged in rj research on electron dynamics — the study of vacuum tubes in the ;| microwave frequency range. He is a member of I.R.E. I David Slepian, University of Michigan, 1941-1943; M.A. and Ph.D., li Harvard LTniversity, 1946-1949; Bell Telephone Laboratories, 1950-. Dr. 24G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1950 Slepian has been engaged in mathematical research in communication theory, switching theory and theory of noise. Parker Fellow in physics. Harvard University 1949-50. Member of I.R.E,, American Mathemati- cal Society, the American Association for the Advancement of Science and Sigma Xi. Milton Sobel, B.S., City College of New York, 1940; M.A., 1946 and Ph.D., 1951, Columbia University; U. S. Census Bureau, Statistician, 1940-41; U. S. Army War College, Statistician, 1942-44; Cohunbia Uni- versity, Department of Mathematics, Assistant, 1946-48 and Research Associate 1948-50; Wayne University, Assistant Professor of Mathe- matics, 1950-52; Columbia University, Department of Mathematical Statistics, Visiting Lecturer, 1952; Cornell University, fundamental re- search in mathematical statistics, 1952-54; Bell Telephone Laboratories, 1954-. Dr. Sobel is engaged in fundamental research on life testing reliability problems with special application to transistors and is a con- sultant on many Laboratories projects. Member of Institute of Mathe- matical Statistics, American Statistical Association and Sigma Xi. Morris Tanenbaum, A.B., Johns Hopkins University, 1949; M.A., Princeton University, 1950; Ph.D. Princeton University, 1952; Bell Telephone Laboratories, 1952-, Dr. Tanenbaum has been concerned with the chemistry and semiconducting properties of intermetallic com- pounds. At present he is exploring the semiconducting properties of silicon and the feasibility of silicon semiconductor devices. Dr. Tanen- baum is a member of the American Chemical Society and American Physical Society. He is also a member of Phi Lambda LTpsilon, Phi Beta Kappa and Sigma Xi. Donald E. Thomas, B.S. in E.E., Pennsylvania State College, 1929; M.A., Columbia University, 1932; Bell Telephone Laboratories, 1929- 1942, 1946-. His first assignment at the Laboratories was in submarine cable development. Just prior to World War II he became engaged in the development of sea and airborne radar and continued in this work I until he left for military duty in 1942. During World War II he was made ' a member of the Joint and Combined Chiefs of Staff Committees on Radio C-ountermeasures. Later he was a civilian memlior of the Depart-' ment of Defense's Research and Development Board Panel on Electronic Countermeasures. Upon rejoining the Laboratories in 1946, Mr. Thomas was active in the development and installation of the first deep sea re- peatered submarine telephone cable, hctwcen Key West and Havana,' COXTIUBUTOKS TO THIS ISSUE 247 which went into service in 1950. Later he was engaged in the develop- ment of transistor devices and circuits for special applications. At the present time he is working on the evaluation and feasibility studies of new types of semiconductors devices. He is a senior member of the I.R.E. and a member of Tau Beta Pi and Phi Kappa Phi. Laurence R. Walker, B.Sc. and Ph.D., McGill University, 1935 and 1939; LTniversity of California 1939-41; Radiation Laboratory, Massachusetts Institute of Technology, 1941-45; Bell Telephone La- boratories, 1945-. Dr. Walker has been primarily engaged in the develop- ment of microwave oscillators and amplifiers. At present he is a member of a physical research group concerned with the applied physics of solids. Fellow of the American Physical Society. IHE BELL SYSTEM Jechnical journal VOTED TO THE SC I E N T I FIC^^^ AND ENGINEERING PECTS OF ELECTRICAL COMMUNICATION LUME XXXV MARCH 1956 NUMBER 2 An Experimental Remote Controlled Line Concentrator \.f^ y A^E. JOEL, JR. 249 Transistor Circuits for Analog and Digital Systems F. H. BLECHER 295 Electrolytic Shaping of Germanium and Silicon a. uhlir, jr. 333 A Large Signal Theory of Traveling-Wave Amplifiers p. k. tibn 349 A Detailed Analysis of Beam Formation with Electron Guns of the Pierce Type w. e. danielson, j. l. rosenfeld and j. a. saloom 375 Theories for Toll Traffic Engineering in the U.S.A. r. i, Wilkinson 421 Crosstalk on Open -Wire Lines W, C, BABCOCK, ESTHER RENTROP AND C. S. THAELER 515 Bell System Technical Papers Not Published in This Journal 519 Recent Bell System Monographs 527 Contributors to This Issue 531 COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD F. K. K A P P E L, President Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Executive Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. MCMILLAN, Chairman A. J. BUSCH H. R. HUNTLEY A. C. DICKIBSON F. R. LACK R. L. DIETZOLD J. R. PIERCE K. E. GOULD H. V. SCHMIDT E. I. GREEN C. ESCHOOLEY R. K. HON AM AN G. N. THAYER ED ITORI AL STAFF J. D. TEBO, Editor M. E. s T R I E B Y, Managing Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Qeo F. Craig, President; S. Whitney Landon, Secretary; John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV MARCH 1956 number 2 Copyright 1958, American Telephone and Telegraph Company An Experimental Remote Controlled Line Concentrator By. A. E. JOEL, JR. (Manuscript received June 30, 1955) Concentration, which is the process of connecting a number of telephone lines to a smaller number of switching paths, has always been a funda?nental function in switching systems. By performing this function remotely from the central office, a new balance between outside plant and switching costs may be obtained which shows promise of providing service more economi- cally in some situations. The broad concept of remote line concentrators is not new. However, its solution with the new devices and techniques now available has made the possibilities of decentralization of the means for switching telephone con- nections very promising. Three models of an experimental equipment have been designed and con- structed for service. The models have included equipment to enable the evalua- tion of new procedures required by the introduction of remote line concentra- tors into the telephone plant. The paper discusses the philosophy, devices, and techniques. CONTENTS 1 . Introduction 250 2. Objectives 251 3. New Devices Emploj^ed 252 4. New Techniques Emploved 254 5. Switching Plan ". 257 249 250 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 6. Basic Circuits 261 a. Diode Gates 261 b. Transistor Bistable Circuit 262 c. Transistor Pulse Amplifier 263 d. Transistor Ring Counter 264 e. Crosspoint Operating Circuit 266 f . Crosspoint Relay Circuit 267 g. Pulse Signalling Circuit 268 h. Power Supply 269 7. Concentrator Operation 270 a.Line Scanning 270 b. Line Selection 272 c. Crosspoint Operation and Check 273 8. Central Office Circuits 274 a. Scanner Pulse Generator 279 b. Originating Call Detection and Line Number Registration 280 c. Line Selection 282 d. Trunk Selection and Identification 284 9. Field Trials 286 10. Miscellaneous Features of Trial Equipment 287 a. Traffic Recorder, b. Line Condition Tester 288 c. Simulator, d. Service Observing 290 e. Service Denial, f . Pulse Display Circuit 291 1. INTRODUCTION The equipment which provides for the switching of telephone connec- tions has ahvays been located in what have been commonly called "cen- tral offices". These offices provide a means for the accumulation of all switching equipment required to handle the telephone needs of a com- munity or a section of the community. The telephone building in which one or more central offices are located is sometimes referred to as the "wire center" because, like the spokes of a wheel, the wires which serve local telephones radiate in all directions to the telephones of the community. A new development, made possible largely by the application of de- vices and techniques new to the telephone switching field, has recently been tried out in the telephone plant and promises to change much of . the present conception of "central" offices and "wire" centers. It is known as a "line concentrator" and provides a means for reducing the amount of outside plant cables, poles, etc., serving a telephone central office by dispersing the switching equipment in the outside plant. It is not a new concept to reduce outside plant by bringing the switching equipment closer to the telephone customer but the technical difficulties of maintaining complex switching equipment and the cost of controlling" such equipment at a distance have in the past been formidable obstacles to the development of line concentrators. With the invention of low power, small-sized, long-life devices such as transistors, gas tubes, and sealed relays, and their application to line concentrators, and with the development of new local switching systems with greater flcxibilit}', it has been possible to make the progress described herein. REMOTE CONTROLLED LINE CONCENTRATOR 251 2. OBJECTIVES Within the telephone offices the first switching equipment through which dial lines originate calls concentrates the traffic to the remaining equipment which is engineered to handle the peak busy hour load with the appropriate grade of service.^ This concentration stage is different for different switching systems. In the step-by-step system^ it is the line ' finder, and in the crossbar systems it is the primary line switch.^ Pro- 1 posals for the application of remote line concentrators in the step-by- i step system date back over 50 years/ Continuing studies over the years have not indicated that any appreciable savings could be realized when such equipment is used within the local area served by a switching center. When telephone customers move from one location to another within a local service area, it is desirable to retain the same telephone numbers. The step-by-step switching system in general is a unilateral arrangement where each line has two appearances in the switching equipment, one for originating call concentration (the line finder) and one for selection of the line on terminating calls (the connector) . The connector fixes the line number and telephone numbers cannot be readily reassigned when moving these switching stages to out-of-office locations. Common-control systems^ have been designed with flexibility so that the line number assignments on the switching equipment are independ- ent of the telephone numbers. Furthermore, the first switching stage in the office is bilateral, handling both originating and terminating calls through the same facilities. The most recent common-control switching system in use in the Bell System, the No. 5 crossbar,^ has the further advantage of universal control circuitry for handling originating and terminating calls through the line switches. For these reasons, the No. 5 crossbar system was chosen for the first attempt to employ new tech- niques of achieving an economical remote line concentrator. A number of assumptions were made in setting the design require- ments. Some of these are influenced by the characteristics of the No. 5 crossbar system. These assumptions are as follows: 1. No change in customer station apparatus. Standard dial telephones to be used with present impedance levels, transmission characteristics, dial pulsing, party identification, superimposed ac-dc ringing,^ and sig- naling and talking ranges. 2. Individual and two-party (full or semi-selective ringing) stations to be served but not coin or PBX lines. 3. Low cost could best be obtained by minimizing the per line equipment in the central office. AMA^ charging facilities could be used but to avoid per station equipment in the central office no message reg- ister operation would be provided. 252 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 4. Each concentrator would serve up to 50 lines with the central office control circuits common to a number of concentrators. (Experimental equipment described herein was designed for 60 lines to provide addi- tional facilities for field trial purposes.) No extensive change would be made in central office equipment not associated with the line switches nor should concentrator design decrease call carrying capacities of exist- ing central office equipment. 5. To provide data to evaluate service performance, automatic traffic recording facilities to be integrated with the design. 6. Remote equipment designed for pole or wall mounting as an addi- tion to existing outside plant. Therefore, terminal distribution facilities would not be provided in the same cabinet. 7. Power to be supplied from the central office to insure continuity of telephone service in the event of a local power failure. 8. Concentrators to operate over existing types of exchange area fa- cilities without change and with no decrease in station to central office service range. 9. Maintenance effort to be facilitated by plug-in unit design using the most reliable devices obtainable. 3. NEW DEVICES EMPLOYED »! I Numerous products of research and development were available for this new approach. Only those chosen will be described. For the switching or "crosspoint" element itself, the sealed reed switch was chosen, primarily because of its imperviousness to dirt.* A short coil magnet with magnetic shield for increasing sensitivity of the reed switches were used to form a relay per crosspoint (see Fig. 1). A number of switching applications^ '^^ for crosspoint control using small gas diodes have been proposed by E. Bruce of our Switching Re- search Department. They are particularly advantageous when used in an "end marking" arrangement with reed relay crosspoints. Also, these diodes have long life and are low in cost. One gas diode is employed for operating each crosspoint (see Fig. 6). Its breakdown voltage is 125v ± lOv, A different tube is used in the concentrator for detecting marking potentials when termination occurs. Its breakdown potential is lOOv ± lOv. One of these tubes is used on each connection. Signaling between the remote concentrator and the central office con- trol circuits is performed on a sequential basis with pulses indicative of the various line conditions being transmitted at a 500 cycle rate. This frequency encounters relatively low attenuation on existing exchange area wire facilities and j^et is high enough to transmit and receive in- formation at a rate which will not decrease call carrjdng capacitj^ of the REMOTE CONTROLLED LINE CONCENTRATOR 253 Fig. 1 — Reed switch relay. central office equipment. To accomplish this signaling and to process the information economically transistors appear most promising. Germanium alloy junction transistors were chosen because of their ; improved characteristics, reliability, low power requirements, and mar- gins, particularly when used to operate with relays.^^ Both N-P-N and P-N-P transistors are used. High temperature characteristics are par- ticularly important because of the ambient conditions which obtain on pole mounted equipment. As the trials of this equipment have progressed, 254 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table I— Transistor Characteristics Code No. Type and Filling Alpha Max. Ico at 28V and 65°C Emitter Zener Voltage at 20^=1 M1868 M1887 p-n-p Oxygen n-p-n Vacuum 0.9-1.0 0.5- .75 150 Ma 100 Ma >735 >735 considerable progress has been made in improving transistors of thi.s type. Table I summarizes the characteristics of these transistors. For directing and analyzing the pulses, the control employs semicon- ductor diode gate circuits." The semiconductor diodes used in these circuits are of the silicon alloy junction type,^^ Except for a few diode.s operating in the gas tube circuits most diodes have a breakdown voltage requirement of 27v, a minimum forward current of 15 ma at 2v and a maximum reverse current at 22v of 2 X 10^^ amp. 4. new techniques employed The concentrator represents the first field application in Bell System telephone switching systems which departs from current practices and techniques. These include: Fig. 2 — Transistor packages, (a) Diode unit, (b) Transistor counter, (c) Transistor amplifiers and bi-stable circuits, (d) Five trunk unit. REMOTE CONTROLLED LINE CONCENTRATOR 255 1. High speed pulsing (500 pulses per second) of information between switching units. 2. The use of plug-in packages employing printed wiring and encap- sulation. (Fig. 2 shows a representative group of these units.) 3. Line scanning for supervision with a passive line circuit. In present systems each line is equipped with a relay circuit for detecting call orig- inations (service requests) and another relay (or switch magnet) for indicating the busy or idle condition of the line, as shown in Fig. 3(a). The line concentrator utilizes a circuit consisting of resistors and semi- conductor diodes in pulse gates to provide these same indications. This circuit is shown in Fig. 3(b). Its operation is described later. The pulses for each line appear at a different time with respect to one another. These pulses are said to represent "time slots." Thus a different line is examined each .002 second for a total cycle time (for 60 lines) of .120 second. This process is known as "line scanning" and the portion of the circuit which produces these pulses is known as the scanner. Each of the circuits perform the same functions, viz., to indicate to the central office equipment when the customer originates a call and for terminating calls to indicate if the line is busy. 4. The lines are divided for control and identification purposes into twelve groups of five lines each. Each group of five lines has a different pattern of access to the trunks which connect to the central office. The ten trunks to the central office are divided into two groups as shown in Fig. 4. One trunk group, called the random access group, is arranged in a random multiple fashion, so that each of these trunks is available to approximately one-half of the lines. The other group, consisting of two trunks, is available to all lines and is therefore called the full access group. The control circuitry is arranged to first select a trunk of the random access group which is idle and available to the particular line to which a connection is to be made. If all of the trunks of this random ac- cess group are busy to a line to which a connection is desired, an attempt is then made to select a trunk of the full access group. The preference order for selecting cross-points in the random access group is different for each line group, as shown in the table on Fig. 4. By this means, each trunk serves a number of lines on a different priority basis. Random ac- cess is used to reduce by 40 per cent the number of individual reed relay crosspoints which would otherwise be needed to maintain the quality of service desired, as indicated by a theory presented some years ago.^^ 5. Built-in magnetic tape means for recording usage data and making call delay measurements. The gathering of this data is greatly facilitated by the line scanning technique. 256 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CROSSBAR CROSSPOINT OR SWITCH CONTACTS -^ TO LINE -^ TO OTHER CENTRAL OFFICE EQUIPMENT 9 9 r ^ LR ■^f- CO c HI "H 1_ (a) ■:l LINE BUSY SERVICE REQUEST I + 5V CROSSPOINT ■^ TO LINE -^ TO CENTRAL OFFICE ^4- -X -16V -16 VOLTS -NORMAL (RECEIVER ON HOOK) -3 VOLTS -AWAITING SERVICE (RECEIVER OFF HOOK) -16 VOLTS -CROSSPOINT CLOSED (RECEIVER OFF HOOK) S\. -¥^ -16 V ^ LINE BUSY + 15 VOLT TIME SLOT PULSE FROM SCANNER GATE SERVICE REQUEST Fig. 3 — (a) Relay line circuit, (b) Passive line circuit. REMOTE CONTROLLED LINE CONCENTRATOR 257 5. SWITCHING PLAN The plan for serving lines directly terminating in a No. 5 Crossbar office is shown in Fig. 5(a). Each line has access through a primary line switch to 10 line links. The line links couple the primary and secondary switches together so that each line has access to all of the 100 junctors to the trunk link switching stage. Each primary line switch group accommodates from 19 to 59 lines (one line terminal being reserved for no-test calls). A line link frame contains 10 groups of primary line switches.^* . The remote concentrator plan merely extends these line links as trunks to the remote location. However, an extra crossbar switching stage is introduced in the central office to connect the links to the secondary line switches with the concentrator trunks as shown in Fig. 5(b). Since each line does not have full access to the trunks, the path chosen by the marker to complete calls through the trunk link frame may then be independent of the selection of a concentrator trunk with access to the line. This arrangement minimizes call blocking, simplifies the selection of a matched path by the marker, and the additional crossbar switch hold magnet serves also as a supervisory relay to initiate the transmission of disconnect signals over the trunk. In addition to the 10 concentrator trunks used for talking paths, 2 additional cable pairs are provided from each concentrator to the central office for signaling and power supply purposes. The use of these two pairs of control conductors is described in detail in Section 6g. The concentrator acts as a slave unit under complete control of the central office. The line busy and service request signals originate at the LINE 60 LINES I o.-»-o 0 5 9 7 '^ / ^ / ■v / p, ■^ i' \. ^ V •y \ / s f \ ^ \ ,• \ '^ ^ 1 > f < > 1 2 3 5 6 8 9 10 11 Fig 4. — Concentrator trunk to line crosspoint pattern and preference order CONCENTRATOR TRUNKS 9 9 9 9 9 9 9 9 9 9 9 9 8 8 8 8 8 8 8 8 8 8 8 8 6 0 5 4 7 5 3 1 4 7 2 1 7 3 1 5 2 0 6 4 6 5 0 3 1 7 2 3 6 2 4 0 0 6 3 5 0 4 6 2 3 7 1 6 2 4 1 7 1 5 6 8 VERTICAL GROUPS OF FIVE LINES EACH " ORDER OF PREFERENCE GAS TUBE REED -RELAY CROSS POINTS 10 11 258 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 49 LINES TEN GROUPS OF LINES 49 LINES I CENTRAL OFFICE LINE LINK FRAME LINE SW 1 CON- NECTOR 1 1 1 — -- 1 1 CON- NECTOR I TO MARKER TRUNK LINK FRAME Fig. 5(a) — No. 5 crossbar system subscriber lines connected to line link frame. 60 LINES 60 LINES 60 0 TRUNK LINK FRAME TO MARKER CONCENTRATOR LINE LINK FRAME Fig. 5(b) — No. 5 crossbar system subscriber lines connected to remote line concentrators. REMOTE CONTROLLED LINE CONCENTRATOR 259 Fig. 6 — Line unit construction. concentrator only in response to a pulse in the associated time slot or when a crosspoint operates (a line busy pulse is generated under this condition as a crosspoint closure check). The control circuit in the central office is designed to serve 10 remote line concentrators connected to a single line link frame. In this way the marker deals with a concen- trator line link frame as it would with a regular line link frame and the marker modifications are minimized. The traffic loading of the concentrator is accomplished by fixing the 260 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Fig. 7(a) — Line unit. number of trunks at 10 and equipping or reassigning lines as needed to obtain the trunk loading for the desired grade of service. The six cross- points, the passive line circuit and scanner gates individual to each line are packaged in one plug-in unit to facilitate administration. The cross- points are placed on a printed wiring board together with a comb of plug contacts as shown in Fig. 6. The entire unit is then dipped in rubber and encapsulated in epoxy resin, as shown in Fig. 7(a). This portion of the unit is extremely reliable and therefore it may be considered as expendable, should a rare case of trouble occur. The passive line circuit and scanner gate circuit elements are mounted on a smaller second printed wiring plate (known as the "line scanner" plate, see Fig. 7(b) which fits into a recess in the top of the encapsulated line unit. Cir- Fig. 7(b) — Scanner plate of the line unit shown in Fig. 7 (a). REMOTE CONTROLLED LINE CONCENTRATOR 261 cuit connection between printed wiring plates is through pins which ap- pear in the recess and to which the smaller plate is soldered. 6. BASIC CIRCUITS a. Diode Gates All high speed signaling is on a pulse basis. Each pulse is positive and approximately 15 volts in amplitude. There is one basic type of diode gate circuit used in this equipment. By using the two resistors, one con- denser and one silicon alloy junction diode in the gate configuration shown in Fig. 8, the equivalents of opened or closed contacts in relay circuits are obtained. These configurations are known respectively as enabling and inhibiting gates and are shown with their relay equivalents ill Figs. 8(a) and 8(b). In the enabling gate the diode is normally back biased by more than the pulse voltage. Therefore pulses are not transmitted. To enable or INPUT ENABLING GATE CIRCUIT CI OUTPUT (a) ENABLING GATE SYMBOL INPUT OUTPUT CONTROL EQUIVALENT RELAY CIRCUIT OUTPUT INPUT f CONTROL CHli^ INPUT INHIBITING GATE CIRCUIT Cl OUTPUT INHIBITING GATE SYMBOL INPUT OUTPUT CONTROL EQUIVALENT RELAY CIRCUIT OUTPUT DhUHl Fig. 8 — Gates and relay equivalents. 262 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 open the gate the back bias is reduced to a small reverse voltage which is more than overcome by the signal pulse amplitude of the pulse. The pulse thus forward biases the diode and is transmitted to the output. The inhibiting gate has its diode normally in the conducting state so that a pulse is readily transmitted from input to output. When the bias is changed the diode is heavily back biased so that the pulse amplitude is insufficient to overcome this bias. The elements of 12 gates are mounted on a single printed wiring board w4th plug-in terminals and a metal enclosure as shown in Fig. 2(a). All elements are mounted in one side of the board so that the opposite side may be solder dipped. After soldering the entire unit (except the plug) is dipped in a silicone varnish for moisture protection. b. Transistor Bistable Circuit Transistors are inherently well adapted to switching circuits using but two states, on (saturated) or off.^^ In these circuits with a current gain greater than unity a negative resistance collector characteristic can be obtained which will enable the transistor to remain locked in its conduct- ing state (high collector current flowing) until turned off (no collector current) by an unlocking pulse. At the time the concentrator develop- ment started only point contact transistors were available in quantity. Point contact transistors have inherently high current gains (>1) but the collector current flowing when in the normal or unlocked condition (Ico) was so great that at high ambient temperatures a relay once op- erated in the collector circuit would not release. Junction transistors are capable of a much greater ratio of on to off current in the collector circuit. Furthermore their characteristics are amenable to theoretical design consideration.^^ However, the alpha of a simple junction transitor is less than unity. To utilize them as one would | a point contact transitor in a negative resistance switching circuit, a combination of n-p-n and p-n-p junction transistors may be employed, i see Fig. 9(b). Two transistors combined in this manner constitute a ' "hooked junction conjugate pairs." This form of bi-stable circuit was j used because it requires fewer components and uses less power than an Eccles-Jordan bistable circuit arrangement. It has the disadvantage of a single output but this was not found to be a shortcoming in the design of circuits employing pulse gates of the type described. In what follows the electrodes of the transistor will be considered as their equivalents shown in Fig. 9(b). The basic bi-stable circuit employed is shown in Fig. 10. The set REMOTE CONTROLLED LINE CONCENTRATOR 263 EMITTER COLLECTOR EMITTER n-p-n COLLECTOR BASE fa) POINT CONTACT TRANSISTOR Ic BASE (b) CONJUGATE PAIR ALLOY JUNCTION TRANSISTORS C _ 0C> 1 Fig. 9 — Point contact versus hooked conjugate pair. pulse is fed into the emitter (of the pair) causing the emitter diode to conduct. The base potential is increased thus increasing the current flowing in the collector circuit. When the input pulse is turned off the base is left at about —2 volts thus maintaining the emitter diode con- ( lucting and continuing the increased current flow in the collector circuit. The diode in the collector circuit prevents the collector from going positive and thereby limits the current in the collector circuit. To reset, a positive pulse is fed into the base through a pulse gate. The driving of tlie base positive returns the transistor pair to the off condition. c. Transistor Pulse Amplifier This circuit (Fig. 11) is formed by making a bi-stable self resetting circuit. It is used to produce a pulse of fixed duration in response to a TRANSISTORS p-n-p SET RESET I-5V -I6V F/F Fig. 10 — Transistor bi-stable circuit. 264 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 pulse of variable width (within limits) on the input. Normally the emitter is held slightly negative with respect to the base. The potential difference determines the sensitivity of the amplifier. When a positive input pulse is received, the emitter diode conducts causing an increase in collector current. The change in bias of the diode in the emitter circuit permits it to conduct and charge the condenser. With the removal of the input pulse the discharge of the condenser holds the transistor pair on. The time constant of the circuit determines the on time. When the emitter potential falls below the base potential, the transistor pair is turned off. The amplifiers and bi-stable circuits or flip-flops, >as they are called more frequently, are mounted together in plug-in packages. Each pack- age contains 8 basic circuits divided 7-1, 6-2, or 2-6, between amplifiers and fhp-flops. Fig. 2(c) shows one of these packages. They are smaller than the gate or line unit packages, having only 28 terminals instead of 42. The transistors for the field trial model w^ere plugged into small hear- ing aid sockets mounted on the printed wiring boards. For a production model it w^ould be expected that the transistors w^ould be soldered in. d. Transistor Ring Counter By combining bi-stable transistor and diode pulse gate circuits to- gether in the manner shown in Fig. 12 a ring counter may be made, with INPUT p-n-p ^w ^vW-" I + 5V OUTPUT -16 V INPUT OUTPUT Fig. 11 — Transistor pulse amplifier. REMOTE CONTROLLED LINE CONCENTRATOR 265 COUNT INPUT lie STAGE NUMBER 3 NOTE: LEADS A-0 TO A-4 ARE OUTPUT LEADS OF RESPECTIVE STAGES 1 I I \ r s 's 's 's 's Fig. 12 — Ring counter schematic. a bi-stable circuit per stage. The enabling gate for a stage is controlled by the preceding stage allowing it to be set by an input advance pulse. The output signal from a stage is fed back to the preceding stage to turn it off. An additional diode is connected to the base of each stage for re- setting when returning the counter to a fixed reference stage. A basic package of 5 ring counter stages is made up in the same frame- work and with the same size plug as the flip-flop and amplifier packages, see Fig. 2(b). A four stage ring counter is also used and is the same package with the components for one stage omitted. The input and out- put terminals of all stages are available on the plug terminals so that the stages may be connected in any combination and form rings of more than 5 stages. The reset lead is connected to all but the one stage which is considered the first or normal stage. Other transistor circuits such as binary counters and square wave generators are used in small quantity in the central office equipment. They will not be described. 266 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CONCENTRATOR LINE BUSY CENTRAL OFFICE TO ALL CROSSPOINTS / SERVED BY TRUNK + 130 V VG VF I L..1.. / TO ALL CROSSPOINTS FOR SAME LINE SELECTION FROM " CENTRAL OFFICE i-65V I + 100V Fig. 13 — Crosspoint operating circuit. e. Crosspoint Operating Circuit The crosspoint consists of a reed relay with 4 reed switches and a gas diode (Fig. 1). The selection of a crosspoint is accomplished by marking with a negative potential ( — 65 volts) all crosspoints associated with a line, and marking with a positive potential ( + 100 volts) all crosspoints associated with a trunk (Fig. 13). The line is marked through a relay circuit set by signals sent over the control pair from the central office. The trunk is marked b}^ a simplex circuit connected through the break contacts of the hold magnet of the crossbar switch associated with the trunk in the central office. Only one crosspoint at a time is exposed to 165 volts which is necessary and sufficient to break down the gas diode to its conducting state. The reed relay operates in series with the gas diode. A contact on the relay shunts out the gas diode. When the marking- potentials are removed the relay remains energized in a local 30-voll circuit at the concentrator. The holding current is approximately 2.5 ma. This circuit is designed so that ringing signals in the presence or ab- sence of lino marks will not falsely fire a crosspoint diode. Furthonnoi'o, REMOTE CONTROLLED LINE CONCENTRATOR 267 a line or trunk mark alone should not be able to fire a crosspoint diode on a busy line or trunk. When the crosspoint operates, a gate which has been inhibiting pulses is forward biased by the —65 volt signal through the crosspoint relay winding. The pulse which initiates the mark operations at the concentra- tor then passes through the gate to return a line busy signal to the central office over this control pairs which is interpreted as a crosspoint closure check signal. f. Crosspoint Release Circuit The hold magnet of the central office crossbar switch operates, remov- ing the +100- volt operate mark signal after the crosspoint check signal is received. A slow release relay per trunk is operated directly by the hold magnet. When the central office connection in the No. 5 crossbar system releases, the hold magnet is released. As shown in Fig. 14, with the hold magnet released and the slow release relay still operated, a — 130- volt signal is applied in a simplex circuit to the trunk to break down a gas tube provided in the trunk circuit at the concentrator. This tube in CONCENTRATOR CENTRAL OFFICE TO ALL CROSSPOINTS SERVED BY SAME TRUNK 130V I Fig. 14 — Crosspoint release circuit. 268 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 breaking down shunts the local holding circuit of the crosspoint causing it to release. The — 130-volt disconnect signal is applied during the release time of the slow release relay which is long enough to insure the release of the crosspoint relay at the concentrator. The release circuit is individual to the trunk and independent of the signal sent over the control pairs. g. Pulse Signalling Circuits To control the concentrator four distinct pulse signals are transmitted from the central office. Two of these at times must be transmitted simultaneously, bvit these and the other two are transmitted mutually exclusively. In addition, service request and line busy signals are trans- mitted from the concentrator to the central office. The two way trans- mission of information is accomplished on each pair by sending signals in each direction at different times and inhibiting the receipt of signals when others are being transmitted. To transmit four signals over two such pairs, both positive and nega- CONTROL PAIR NO. 1 VF M LB D SR -16V VG CONTROL PAIR NO. 2 16 V M CONCENTRATOR AMPLIFIERS I CENTRAL OFFICE AMPLIFIERS PER CONCENTRATOR Fig. 15. — Signal transmission circuit. REMOTE CONTROLLED LINE CONCENTRATOR 269 tive pulses are employed. Diodes are placed in the legs of a center tapped transformer, as shown in Fig. 15, to select the polarity of the trans- mitted pulses. At the receiving end the desired polarity is detected by taking the signal as a positive pulse from a properly poled winding of a transformer. The amplifier, as described in Section 6c responds only to positive pulses. If pulses of the same polarity are transmitted in the other direction over the same pair, as for control pair No. 1, the outputs of the receiving amplifier for the same polarity pulse are inhibited whenever a pulse is transmitted. As shown in Fig. 15, the service request and line busy signals are transmitted from the concentrator to the central office over one pair of conductors as positive and negative pulses respectivel3^ The trans- mission of these pulses gates the outputs of two of the receiving ampli- fiers at the concentrator to permit the receipt of the polarized signals from the central office. This prevents the pulses from being used at the sending end. A similar gating arrangement is used with respect to the signals when sent over this control pair from the central office. The pulses designated VG or RS never occur when a pulse designated SR or LB is sent in the opposite direction. The transmission of the VF pulse over control pair No. 2 is processed by the concentrator circuit and becomes the SR or LB pulses. Li section 7 the purpose of these pulses is described. The signaling range objective is 1,200 ohms over regular exchange area cable including loaded facilities from sfation to central office. h. Power Supply Alternating current is supplied to the concentrator from a continuous service bus in the central office. The power supply path is a phantom circuit on the two control pairs as shown in Fig. 16. The power trans- former has four secondary windings used for deriving from bridge rectifiers four basic dc voltages. These voltages and their uses are as fofiows: —16 volts (regulated) for transistor collector circuits and gate biases, -|-5 volts (regulated) for transistor base biases, -|-30 volts (regu- , lated) for crosspoints holding circuits and — 65 volts for the marking and operating of the line crosspoints. For this latter function a reference to the central office applied -flOO volt trunk mark is necessary. The refer- ence ground for the concentrator is derived from ground applied to a simplex circuit on the power supply phantom circuit. Series transistors and shunt silicon diodes with fixed reference breakdown voltages are I used to regulate dc voltages. 270 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Total power consumption of the concentrator is between 5 and 8 watts depending upon the number of connections being held. 7. CONCENTRATOR OPERATION a. Line Scanning The sixty lines are divided into 12 groups of 5 lines each. These group- ings are designated VG and VF respectively corresponding to the vertical group and file designations used in the No. 5 crossbar system. Each concentrator corresponds to a horizontal group in that system. To scan the lines two transistor ring counters, one of 12 stages and one of 5 stages, are employed as shown in Fig. 17. These counters are driven from pulses supplied from the central office control circuits and only one stage in each is on at any one time. The steps and combinations of these counters correspond to the group and file designation of a par- ticular line. Each 0.002 second the five stage counter (VF) takes a step and between the fifth and sixth pulse the r2-stage counter (VG) is stepped. Thus the 5-stage counter receives 60 pulses or re-cycles 12 times in 120 milliseconds while the 12-stage counter cycles but once. Each line is provided with a scanner gate. The collector output of each each stage of the VG counter biases this gate to enable pulses which are generated by the collector circuit of the 5-stage counter to pass on -65V + 30V + SV -I6V 115 V AC MOTOR GENERATOR TO COMMERCIAL AC REGULATORS Fig. 16 — Power supply transmission circuit. REMOTE CONTROLLED LINE CONCENTRATOR 271 to the gate of the passive line circuit, Fig. 3(b). If the line is idle the pulses are inhibited. If the receiver is off-hook requesting service (no (•rosspoint closed) then the gate is enabled, the pulse passes to the service request amplifier and back to the central office in the same time slot as the pulse which stepped the VF counter. If the line has a receiver off-hook and is connected to a trunk the pulse passes through a contact of the crosspoint relay to the line busy amplifier and then to the central office in the same time slot. At the end of each complete cycle a reset pulse is sent from the central office. This pulse instead of the VG pulse places the 12-stage counter in its first position. It also repulses the 5 stage VF counter to its fifth stage so that the next VF pulse will turn on its first stage to start the next j cycle. The reset pulse insures that, in event of a lost pulse or defect in a counter stage, the concentrator will attempt to give continuous ser- \'ice without dependence on maintaining synchronism with the central I office scanner pulse generator. Fig. 18(a) shows the normal sequence of I line scanning pulses. , When a service request pulse is generated, the central office circuits t] 04 r VF 5- STAGE COUNTER 03 TO 10 INTERMEDIATE GATES EACH V 02 I 01 00 TO 5 GATES EACH I 1 23456 789 10 I I I I I I I I I I I I I I I I I I I I VG 12-STAGE COUNTER 59\ 58 57 56 55 GATE PER LINE ■ FEEDS PASSIVE LINE CIRCUITS / VG RESET VF FROM CENTRAL OFFICE Fig. 17 — Diode matrix for scanning lines. 272 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 common to 10 concentrators interrupt the further transmission of the vertical group pulse so that the line scanning is confined to the 5 lines in the vertical group in which the call originated. In this way the cen- tral office will receive a service request pulse at least every 0.010 sec as a check that the call has not been abandoned while awaiting service. Fig. 18(b) shows the detection of a call origination and the several short scan cycles for abandoned call detection. b. Line Selection When the central office is ready to establish a connection at the con- centrator a reset pulse is sent to return the counters to normal. In gen- eral, the vertical group and vertical file pulses are sent simultaneously to reduce holding time of the central office equipment and to minimize marker delays caused by this operation. For this reason the VG and VF pulses are each transmitted over different control pairs from the central office. The same polarity is used. On originating calls it is desirable to make one last check that the call has not been abandoned, while on terminating calls it is necessary L* 120MS >| ■M k-2MS I PULSE ' 012340123401234 0123401234 VF - VG LB RS J__l I I I I I I I I I I I I I I 1 I I I I 1 I L 1^ \1 ^ (a) REGULAR LINE SCANNING VF 123401 23401 23401 2340123401 _l_l 1 I I I I I I I I I I I I I I I I I L_l I I I I ,5 ,6 VG 1 1 — LB- RS- SR- M- (b) CALL ORIGINATION SERVICE REQUEST FROM LINE 6/3 12 3 0 1 I I I I I VG- LB RS — 1>- SR h 1° 1' 1^ |3 1^ 1^ 1^ jLi. M - H^-l^-- "7 RESULTS FROM CONC CONTROL RECEIVED 'OPERATE ""CROSSPOINT 'NORMAL SCANNING CKT AT CENTRAL OFFICE ONLY IF CROSSPOINT CLOSURE IS RESUMED RECEIVING FROM MKR VG , LINE 6/3 HAS INDICATION VF, HG INFORMATION BECOME BUSY (C) LINE SELECTION FOR LINE 6/3 Fig. 18 — Pulse sequences, (a) Regular, (b) Call origination, (c) Line selection. REMOTE CONTROLLED LINE CONCENTRATOR 273 to determine if the line is busy or idle. These conditions are determined in the same manner as described for line scanning since a service re- quest condition would still prevail on the line if the call was not aban- doned. If the line was busy, a line busy condition would be detected. However to detect these conditions a VF pulse must be the last pulse transmitted since the stepping of the VF counter generates the pulse which is transmitted through an enabled line selection and passive line circuit gates. Fig. 18(c) shows a typical line selection where the num- ber of VF pulses is equal to or less than the number of VG pulses. In all other cases there is no conflict and the sending of the last VF pulse need not be delayed. On terminating calls, the line busy indication is returned to the central office within 0.002 sec after the selection is com- plete. During selections the central office circuits are gated to ignore any extraneous service request or line busy pulses produced as a result of steps of the VF counter prior to its last step. c. Crosspoint Operation and Check Associated with each concentrator transistor counter stage is a reed relay. These relays are connected to the transistor collector circuits through diodes of the counter stages when relay M operates. The con- tacts of these reed relays are arranged in a selection circuit as shown in Fig. 19 and apply the —65 volt mark potential to the crosspoint relays of the selected line. After a selection is made as described above a "mark" pulse is sent from the central office. This pulse is transmitted as a pulse of a different polarity over the same control pair as the VF pulses. The received pulse after amplification actuates a transistor bistable circuit w^hich has the M reed relay permanently connected in its collector circuit. The bi-stable circuit holds the M relay operated during the crosspoint opera- tion to maintain one VF and one VG relay operated, thereby applying — 65 volts to mark and operate one of the 6 crosspoint relays of the selected line as described in section 6e, and shown on Fig. 13. The operation and locking of the crosspoint relay with the marking potentials still applied enables a pulse gate associated with the holding circuit of the crosspoint relays in each trunk circuit. The mark pulses are sent out continuously. This does not affect the bi-stable transistor circuit once it has triggered but the mark pulse is transmitted through the enabled crosspoint closure check gate shown in Fig. 20 and back to the central office as a line busy signal. With the receipt of the crosspoint closure check signal the sending 274 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of the mark pulses is stopped and a reset pulse is sent to the concentra- tor to return the mark bi-stable circuit, counters and all operated selec- tor relays to normal. The concentrator remains in this condition until it is resynchronized with the regular line scanning cycle. A complete functional schematic of the concentrator integrating the circuits described above is shown in Fig. 21. Fig. 22(a) and (b) show an experimental concentrator built for field tests. 8. CENTRAL OFFICE CIRCUITS The central office circuits for controling one or more concentrators are composed of wire spring relays as well as transistors, diode and reed VG RS VF M -20V -20 V o- VF-5 STAGE COUNTER r -65V n 04 03 02 00 6 RELAY I W-, wo w<- ,^, -o p PACKAGE J '-|„p_„p_^_p-_-p-^Ui TO CONTACTS OF 4 INTERMEDIATE RELAYS 6 RELAY PACKAGE TO CONTACTS OF 4 . INTERMEDIATE RELAYS n L 59 n 58 i^ 34 33 32 31 30 , 29 28 27 26 25 I 5^ 57 56 55 TO 4 INTERMEDIATE RELAYS ' 0 -/ *■ I I I 2 b^'" / 7_ 8_ 9 10 cr LU h- z o o < I- OJ I (J > I TO 4 INTERMEDIATE il RELAYS P^ig. 19 — Line selection and marking. I REMOTE CONTROLLED LINE CONCENTRATOR 275 relay packages similar to those used in the concentrator. The reed relays are energized by transistor bi-stable circuits in the same manner as described in Section 7c. The reed relay contacts in turn operate wire spring relays or send the dc signals directly to the regular No. 5 crossbar marker and line link marker connector circuits. Fig. 23 shows a block diagram of the central office circuits. A small amount of circuitry is provided for each concentrator. It consists of the following: 1. The trunk connecting crossbar switch and associated slow relays for disconnect control. 2. The concentrator control triuik circuits and associated pulse ampli- fiers. 3. An originating call detector to identify which concentrator among the ten served by the frame is calling. 4. A multicontact relay to connect the circuits individual to each concentrator with the common control circuits associated with the line link frame and markers. The circuits associated with more than one concentrator are blocked out in the lower portion of Fig. 23. Much of this circuitry is similar to the relay circuits now provided on regular line link frames in the No. 5 crossbar system.^ Only those portions of these blocks which employ the new techniques will be covered in more detail. These portions consist of the following: 1. The scanner pulse generator. 2. The originating line number register. T TO ALL TRUNK LINES + 30V A/vV U j^Wv- -65V T I I I I I CONCENTRATOR TRUNK I I I I I i Fig. 20 — Crosspoint closure check. aoidzio nvbiNBo oi iinoaio ONnvNois via 276 Fig. 22(a) — Complete line concentrator unit. r 5 -STAGE COUNTER 12 -STAGE COUNTER -fO TRUNK CiftCUITS AMPLIFIERS RECTIFIERS Fig. 22(b) — Identification of units within the line concentrator. 277 a. Q-tU O z o o 1 2 o< z o CO UJ O <:^ o: u tu z D. o (J UJ£t Q. 10 I ^-; C-: 0 n W liJO IOq Q I I I J o o LJJ o u a. o ^ cc I- z UJ o z o u Z 111 Di- ce I- l-< >^av/^l Asng i3S3a 9A dA VAVVV tr zy= Di- CEZ I-LIJ Nl 9H lAIS "11 Z CEUJ I 1 UJ CO S3 (O u 90 ui- UJ z -lO ujo (/I Hj UJ(- ZU -iUJ 9A HH dA d: p uJuJt; z -id: < DUJ en "J ; < UJ CC O O UJ zmCE ^UjS cc 2 cc On IdA cc o I- O UJ z z O O < CC U. z _J UJ ^^ 1- O u o _l UJUJ ^tr oruj ^ Fig. 3 — Single loop feedback amplifier. voltage gain of an operational amplifier. Fig. 4 shows a generalized op- erational amplifier with N inputs. With this configuration, IN j=i L TTT he r, I Zj (S) where Ej , j = 1,2, • ■ • , N, are the N input voltages referred to the ground node. Zj,j— 1,2, • ■ ■ , N, are the A^ input impedances ZiN is the input impedance of the amplifier measured at the summing node with the feedback loop opened. Eo //i sc UT 'IN la = A Eovr = Zk /sc ~ / (3 Rl Zovt (5>) (10) TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 301 where Zovr is the output impedance of the amplifier measured with the feedback loop opened. The expression for the output voltage is obtained by combining (7), (8), (9), and (10). E, OUT N y = zL ^i 7" ;=1 ^i A^ + 3 = 1 ^1 _ (iir where A^ = A 1 - 'IN \Ri + /OUT 1 _^ ^ + ^^^ Rl Zovr IA/3 is equal to the current returned to the summing node when a unit Ei Z, MN 1/3 Zk I IN Zn Zls J7 1/5 Zk Equt NORTON EQUIVALENT CIRCUIT Fig. 4 — Generalized operational amplifier. icurrent is placed into the base of the first transistor stage (/in = 1). If I A^ 1 is much greater than ] Zj^'/Zr \ and 1 + L 'IN then N Eqvt — ~ 2^ J^j nT (12) y=i The accuracy of the operational amplifier depends on the magnitude of AjS and the precision of the components used in the input and feedback networks as can be seen from (11). There is negligible interaction between the input voltages because the input impedance at the summing node is equal to Zin' divided by (1 — A^)? This impedance is usually negligibly tsmall compared to the impedances used in the input circuit. * In general, E,- and Eout are the Laplace transforms of the input and output fvoltages, respectively. 302 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 2.2. Methods Used to Shape the Loop Current Transmission An essential consideration in the design of a feedback amplifier is the provision of adequate margins against instability. In order to accomplish this objective, it is necessary to choose a criterion of stability. In Ap- pendix I it is shown that it is convenient and valid to base the stability of single loop transistor feedback amplifiers on the loop current trans- mission. In order to calculate the loop current transmission of the dc amplifier, the feedback loop is opened at a convenient point in the cir- cuit, usually at the base of one of the transistors, and a unit current is injected into the base (refer to Fig. 24). The other side of the opened loop is connected to ground through a resistance (r^ -j- r^) and voltage Veli • In many instances, the voltage re/4 can be neglected. If | Zj? | and 3=1 Zj I are much greater than | Z IN then A/3 is very nearly equal to the loop ciu'rent transmission. For absolute stability^ the amplitude of the loop current transmission must be less than unity before the phase shift (from the low frequency value) exceeds 180°. Consequently, this charac- teristic must be controlled or properly shaped over a wide frequency 10 _J LU O LU Q < o \- z LU a. o 40 U), a;,' Wa ^{\-\-S)u)^ \ ao ^" ■\ \, \J i "^ \ 30 1- ao+cT t- — . ao \ 7~' \ \ ^ AM PL ITUC E \ \ 20 10 i-ao + 7_ AMPLITUDE' (WITH LOCAL FEEDBACK) \ \ \ \ S < PHASE (WITH LOCAL FEEDBACK) phase\ \ y \ > \ \ ao 0 ^'^ f' \+S 1 -270° X — . s \ \ N d «/c -10 20 30 40 "^ N \ ^ ^ \ ^ \ \ •180 -200 -220 •240 •260 uj _l 2 < -280 •300 LU < I I Q- -320 ■340 ,02 2 5 ,q3 2 ^ ,o4 •=; S ,q5 t S ,q6 5 ■/^4 2 5 ,„s 2 5 ,„6 2 5 ^q7 2 5 jq8 FREQUENCY IN CYCLES PER SECOND Fig. 5 — Current transmission of a common emitter stage. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 303 band. In addition, it is desirable that the feedback fall off at a rate equal to or less than 9 db per octave in order to insure that the dc aniplifier has a satisfactory transient response. Three methods of shaping are described in this paper; local feedback shaping, interstage network shaping, and (3 circuit shaping. Local feed- back shaping will be described first. The analysis starts by considering the current transmission of a common emitter stage, ecjuivalent circuit shown in Fig. 2(b). If the stage operates into a load resistance Rl , then to a good approximation the current transmission is given by where Gr = r" = ^ ~ ^° +/ (13)^ ^^ 1 + ^ + '^ wi a)aCOc(l — ao -\- 8) RL+Te 8 = COl = (1 - ao + 8) 1 + 5,1 -^ ^ alpha-cutoff frequency Ztt 1 Uc (7?x, + re)Cc It is apparent from expression (13) that if (1 — Oo + 8) is less than 0.1, then the current gain of the common emitter stage falls off at a rate of 6 db per octave with a corner frequency at wi .f A second 6 db per octave cutoff with a corner frequency at [co^ + (1 + 5)aJc] is introduced by the p" term in the denominator of (13). A typical transmission characteristic is shown in Fig. .5. The current gain of the common emitter stage is unity at a frequency equal to ao 1 +5 I 1 * Expressions (13) and (14) are poor approximations at frequencies above ' coo/27r. ' t Strictly speaking the corner frequency is equal to 01/2 tt. However, for sim- plicity, corner frequencies will be expressed as radian frequencies. 304 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Since the phase crossover of A|S* is usually placed below this frequency, the principal effect of the second cutoff is to introduce excess phase. This excess phase can be minimized by operating the stage into the smallest load resistance possible, thus maximizing Wc . j An undesirable property of the common emitter transmission charac- teristic is that the corner frequency coi occurs at a relatively low fre- quency. However, the corner frequency can be increased by using local feedback as shown in Fig. 6(a). Shunt feedback is used in order to pro- vide a low input impedance for the preceding stage to operate into. The amplitude and phase of the current transmission is controlled prin- cipally by the impedances Z\ and Z2 . If | A& \ is much greater than one, and if /3 ;^ ^1/^2 , then from (7) the current transmission of the stage is approximately equal to — Z2/Z1 . Because of the relatively small size of A^ for a single stage, this approximation is only valid for a very limited range of values of Zi and Z2 . If Zi and Zi are represented as resistances R\ and Ri , then the current transmission of the circuit is given to a good approximation by tto h. _ R2 1 — gp + 7 ^^ = /i= ~{R2 + n)r_^p_^ v' where 7 = coi = COc = Co/ COaCOcCl — Oo + 7), R\ + Te _,Rl + Te R2 + ^6 I (14) (/?2 + rb)rc i22 + n (1 +ao + ro + 7) 1 + 7 1 {R, -f re)Cc i By comparing (14) with (13), it is evident that the negative feedback has reduced the low-frequency current gain from ao/(l — ao) (5 may usually be neglected) to ( R2 \ I «0 \ _ , ^2 R2 + rj \1 - ao + 7/ ^1 + re (if 7 > 1 - ao) .-•! * The phase crossover of A/3 is equal to the frequency at which the phase shift of A/3 from its low-frequency value is 180°. I TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 305 The half power frequency, however, has been increased from 1— Oo , 1— Oo + 7 t:^ 1 + 7 , 1 as shown by the dashed curves in Fig. 5.* The bandwidth of the common emitter stage can be increased without reducing the current gain at dc and low-frequencies by representing Zi by a resistance Ri , and Z2 by a resistance R2 in series with a condenser C2 . If I/R2C2 is much smaller than co/, then the current transmission of the stage is given by (14) multiplied by the factor P 1 + C04 P (15) where 602 Wi H^^i 1 - cro + Ri + re C2(/?2 + r6)(l - ao + 7) The current transmission for this case is plotted in Fig. 6(b). The con- denser d introduces a rising 6 db per octave asymptote with a corner frequency at wi . At dc the current gain is equal to ao 1 — ao + 5 A second method of shaping the loop current transmission char- acteristic of a feedback amplifier is by means of interstage networks. These networks are usually used for reducing the loop current gain at relatively low frequencies while introducing negligible phase lag near the gainf and phase crossover frequencies. Interstage networks should be designed to take advantage of the variable transistor input impedance. The input impedance of a transistor in the common emitter connection * In Figs. 5 and 6(b), the factor R^/iRi + n) is assumed equal to unity. This is ' a good approximation since in practice R2 is equal to several thousand ohms while rt is equal to about 100 ohms. t The gain crossover frequency is equal to the frequency at which the magni- tude of Al3 is unity. 306 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 is given by the expression 'INP UT = ?'6 + ^e(l — Gi) (16)1 where Gj is the current transmission given by (13). If Gi at dc is much' greater than 1, then the input impedance and the current transmission of the common emitter stage fall off at about the same rate and with approximately the same corner frequency (wi). The input impedance finally reaches a limiting value equal to r^ + Vb . A particularly useful interstage network is shown in Fig. 7(a). This network is analyzed in Appendix II and Fig. 7(b) shoAvs a plot of the 60 50 40 30 20 Z < z UJ 10 tr 0 cr D U -10 -20 -30 (a: EQUIVALENT CIRCUIT \ \ (b) \ AMPLITUDE an \ (WITHOUT LOCAL \ FEEDBACK) 1-ao+d" ^ - ^" \ ■*•> 1 ^ s .AMPLITUDE ^4 ^>CiL ^'' ,^_ •— ^ ■ ■~^ r"**^ cvz r^ i / "^v ^ ' 1 >^ / V \ \ \ X V ^ ao i-ao+ 7 - / A / ^ \ s. \ k \ \ \ / / PH/ >s> / \ \ \ \ V \ \ PHASE N \ (WITHOUT LOCAL \ s FEEDBACK) s w k. - ^-. ■"••^^, 120 140 -160 10 UJ m cr -180 liJ Q Z -200 ^ z < -220 , 10 2 5p2 5,2 5.2 5,2 5 -!•= in^ m^ in5 10'= lO-^* 10^ 10= FREQUENCY IN CYCLES PER SECOND lO'' -240 ' 260 - -280 10' Fig. 6 — Negative feedback applied to a common emitter stage. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 307 resulting current transmission. The amplitude of the transmission falls off at a rate of 6 db per octave with the corner frequency C05 determined by C'3 and the low frequency value of the transistor input impedance. The inductance L3 introduces a 12 db per octave rising asymptote with a corner frequency at C03 = WLsCs . The corner frequencies C03 and C05 are selected in order to obtain a desirable loop current transmission characteristic (specific transmission characteristics are presented in Sec- tions 3.0 and 4.0). The half power frequency of the current transmission of the transistor, wi , does not. appear directly in the transmission char- acteristic of the circuit because of the variation in the transistor input impedance with frequency. The overall (3 circuit of the feedback amplifier can also be used for i-ao+(J I ^ s LU Q z I - X X V 1 1 / / / 1 1 1 1 1 _ \ \ \ \ \ s,. / cu,(rb+ ' Te \ \-do+l W \ .PHASE ^ Tb+le-l-Ra-K^iLa ^^ / \. *^.., — ^** X — - N - -135 10 LU UJ isog z < -225 1}^ < I a. -270 102 " "^ \0^ " = 10^ -^ = 105 FREQUENCY IN CYCLES PER SECOND Fig. 7 — Interstage shaping network. lO'' 308 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 shaping the loop current transmission. If the feedback impedance Zk (Fig. 4) consists of a resistance Rk and condenser Ck in parallel, then the loop current transmission is modified by the factor 1 + CO; 1 + ^ COS (17) where C07 = C08 = RkCk (Rl_±_Rk) RlRkC K Since Zk affects the external voltage gain of the operational amplifier, (11), the corner frequency C07 must be located outside of the useful fre- quency band. Usually it is placed near the gain crossover frequency in order to improve the phase margin and the transient response of the amplifier. In Sections 3.0 and 4.0, the above shaping techniques are used in the design of specific operational amplifiers. 3.0. THE SUMMING AMPLIFIER 3.1. Circuit Arrangement The schematic diagram of a dc summing amplifier is shown in Fig. 8. From the discussion in Section 2.0 it is apparent that each common emitter stage will contribute more than 90 degrees of high-frequency phase lag. Consequently, while the magnitude of the low-frequency : feedback increases with the number of stages, this is at the expense of , the bandwidth over which the negative feedback can be maintained. It is possible to develop 80 db of negative feedback at dc with three common emitter stages. This corresponds to a dc accuracy of one part in 10,000. In addition, the feedback can be maintained over a broad enough band in order to permit full accuracy to be attained in about 100 microseconds. Thus it is evident that the choice of three stages repre- sents a satisfactory compromise between accuracy and bandwidth ob- jectives. The output stage of the amplifier is designed for a maximum power dissipation of 75 milliwats and maximum voltage swing of ±25 volts I TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 309 when operating into an external load resistance equal to or greater than 50,000 ohms. A p-n-p transistor is used in the second stage and n-p-n transistors are used in the first and third stages. This circuit arrangement makes it possible to connect the collector of one transistor directly to the base of the following transistor without introducing appreciable interstage loss. ''Shot" noise" and dc drift are minimized by operating the first stage at the relatively low collector current of 0.25 milliamperes. The 110,000-ohm resistor provides the collector current for the first stage, and the 4,700-ohm resistor provides 3.8 milliamperes of collector current for the second stage. The series 6,800-ohm resistor between the xcond and third stages, reduces the collector to emitter potential of the second stage to about 4.5 volts. The loop current transmission is shaped by use of local feedback ap- plied to the second stage, by an interstage network connected between the second and third stages, and by the overall (3 circuit. The 200-ohm resistor in the collector circuit of the second stage is, with reference to Fig. 6(a), Zi . The impedance of the interstage network can be neglected since it is small compared to 200 ohms at all frequencies for which the local feedback is effective. The interstage network is connected between the second and third stages in order to minimize the output noise voltage. ^^'ith this circuit arrangement, practically all of the output noise voltage iE 250 K IN + 33V 5MUf Hf- 20on n-p-n 250 K 2.4 K 200 n 0.01/U.F p-n-p ■llOK 100 K POT. MANUAL ZERO SET I + 33V I + 4.5V OUT 5>UH -45V -27V +33V Fig. 8 — ■ DC summing amplifier. 310 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 120 100 UJ U 80 LU o z " 60 < < IS LU a: tr D U Q. o o _) 40 20 ■20 -40 ,-' ../>' --T .-' — 364 LOCAL _ FEEDBACK"^- ^-' .-1 41,000 -^ ^ d 6 630 \, . 12 N \ ?,000 --.., s^-> ^-. 'S '-.. \ \ V \ \ V 2ND •^^ STAGE ^ s \ -.. "-. ^-. 1ST & 3RD STAGES N \ 0.5/ZF ^■-S;:-.-, ^ \, \ \ > \ 10' 10-^ 10- 10' 10' FRFOUENCY IN CYCLES PER SECOND Fig. 9 — Gain-frequency asymptotes for summing amplifier. is generated in the first transistor stage. If the transistor in the first stage has a noise figure less than 10 db at 1,000 cycles per second, then the RMS output noise voltage is less than 0.5 millivolts. Fig. 9 shows a plot of the gain-frequency asymptotes for the sum- ming amplifier determined from (13), (14), (15), (17), and (A6) under the assumption that the alphas and alpha-cutoff frequencies of the tran- sistors are 0.985 and 3 mc, respectively. The corner frequencies intro-' duced by the 0.5 microfarad condenser in the interstage network, thel local feedback circuit, and the cutoff of the first and third stages are so located that the current transmission falls off at an initial rate of about' 9 db per octave. This slope is joined to the final asymptote of the loop transmission by means of a step-type of transition.^ The transition is provided by 3 rising asymptotes due to the interstage shaping network, and the overall /S circuit. An especially large phase margin is used in order to insure a good transient performance. Fig. 10 shows the amplitude and phase of the loop current trans- mission. When the amplitude of the transmission is 0 db, the phase angle is -292°, and when the phase angle is —360°, the amplitude is 27.5 db TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 311 100 LU m u LU Q <^ z < H Z UJ a. a. D o Q. O " o _l 80 60 40 20 20 -40 — ■~-^ ^•"v > \ ^ AM PLITL DE \ 1 \ \ \ s > \. ._ s >^ ^r •—'' y^ "N phase 'PHASE 'nCROSSOVER s \ \ V / GAIN^-N^ CROSSOVER N -27.5 DB 95 = -360° sv ■160 -200 to LU -240 ^ O LU Q -280 7 •320 -360 -400 ■440 10= FREQUENCY IN CYCLES PER SECOND 10^ 10' Fig. 10 — Loop current transmission of the summing amplifier. below 0 db. The amplifier has a 68° phase margin and 27.5 db gain margin. In order to insure sufficient feedback at dc and adequate margins against instability, the transistors used in the amplifier should have alphas in the range 0.98 to 0.99 and alpha-cutoff frequencies equal to or greater than 2.5 mc. 3.2. Automatic Zero Set of the dc Summing Amplifier The application of germanium junction transistors to dc amplifiers does not eliminate the problem of drift normally encountered in vacuum tube circuits. In fact, drift is more severe due principally to the varia- tion of the transistor parameters alpha and saturation current with temperature variation. Even though the amplifier has 80 db of negative feedback at dc, this feedback does not eliminate the drift introduced by [the first transistor stage. Because of the large amount of dc feedback, the collector current of the first stage is maintained relatively constant. The collector current of the transistor is related to the base current by the equation Ic = /c + a I — a 1 — a (18) [The saturation current, Ico , of a germanium junction transistor doubles (approximately for every 11°C increase in temperature. The factor a/(l — a) increases by as much as 6 db for a 25°C increase in tempera- 312 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ture. Consequently, the base current of the first stage, Ih , and the output voltage of the amplifier must change with temperature in order to main- ' tain Ic constant. The drift due to the temperature variation in a can be reduced by operating the first stage at a low value of collector current. With a germanium junction transistor in the first stage operating at a collector current of 0.25 milliamperes, the output voltage of the amplifier drifts about ±1.5 volts over a temperature range of 0°C to 50°C. It is possible to reduce the dc drift by using temperature sensitive elements in the amplifier. • In general, temperature compensation of a transistor dc amplifier requires careful selection of transistors and critical adjust- ment of the dc biases. However, even with the best adjustments, tem- perature compensation cannot reduce the drift in the amplifier to within typical limits such as ±5 millivolts throughout a temperature range of i 0 to 50°C. In order to obtain the desired accuracy it is necessary to use an automatic zero set (AZS) circuit. t Fig. 11 shows a dc summing amplifier and a circuit arrangement fori reducing any dc drift that may appear at the output of the amplifier. The output voltage is equal to the negative of the sum of the input volt- ages, where each input voltage is multiplied by the ratio of the feedback resistor to its input resistor. In addition, an undesirable dc drift voltage ^ is also present in the ovitput voltage. The total output voltage is ^o.t = -i:^y|^ + Adrift (1!))^ In order to isolate the drift voltage, the A^ input voltages and the output voltage are applied to a resistance summing network composed of re- sistors Ro , Ri , R2 , • • • , Rn ■ The voltage across Rs is equal to Es=^ Adrift (20) if R,«Ro,R/; j = 1,2, ■■' ,N and RoRj = RkR,'; j = 1,2, ■■■ ,N The voltage E, is amplified in a relatively drift-free narrow band dc amplifier and is returned as a drift correcting voltage to the input of the dc summing amplifier. If the gain of the AZS circuit is large, the drift voltage at the output of the summing amplifier can be made very small. Fig. 12 shows the circuit diagram of a summing amplifier which uses a mechanical chopper in the AZS circuit.^^ The AZS circuit consists of a TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 313 resistance summing network, a 400-cycle synchronous chopper, and a tuned 400-cycle amplifier. Any drift in the summing amplifier will pro- duce a dc voltage Es at the output of the summing network. The chopper converts the dc voltage into a 400 cycles per second waveform. The fundamental frequency in the waveform is amplified by a factor of about 400,000 by the tuned amplifier. The synchronous chopper rectifies the sinusoidal output voltage and preserves the original dc polarity of Eg . The rectified voltage is filtered and fed back to the summing amplifier as an additional input current. The loop voltage gain of the AZS circuit at dc is about 54 db. Any dc or low-frequency drift in the summing amplifier is reduced by a factor of about 500 by the AZS circuit. The drift throughout a temperature range of 0 to 50°C is reduced to ±3 millivolts. Since the drift in the summing amplifier changes at a relatively slow rate, the loop voltage gain of the AZS circuit can be cutoff at a relatively low frequency. In this particular case the loop voltage gain is zero db at about 10 cycles per second. 4.0. THE INTEGRATOR 4.1. Basic Design Considerations The design principles previously discussed are illustrated in this sec- tion by the design of a transistor integrator for application in a voltage VvV -OUT Fig. 11 — DC summing amplifier with automatic zero set. 314 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 315 encoder. The integrator is required to generate a 15-volt ramp which is linear and has a constant slope to within one part in 8,000. This ramp is to have a slope of 5 millivolts per microsecond for an interval of 3,000 microseconds. The first step in the design is to determine the bandwidth over which the negative feedback must be maintained in order to realize the desired output voltage linearity. The relationship between the output and input voltage of the integrator can be obtained from expression (11) by sub- stituting (1/pc) for Zk and R for Zj (refer to Fig. 1). £l-C'outJ — pRC A/3 + Zr^'pC 1 - AjS + -nN_ R (21) where ce[£'ouT] and JSiii'iN] are the Laplace transforms of the output and input voltages, respectively. In order to generate the voltage ramp, a step voltage of amplitude E is applied to the input of the integrator. The term Zy^ jR is negligible compared to unity at all frequencies. Therefore, £L-£'outJ — E \ A& + EZ IN 1 '^-RC Ll - A&\ pR \\ - A^_ It will be assumed that A/3 is given by the expression -K (22) A^ = V )0 + ^T (i + -M(i + ^ (23) Expression (23) implies that A/3 falls off at a rate of 6 db per octave at low frequencies and 12 db per octave at high frequencies. The output \ voltage of the integrator, as a function of time, is readily evaluated by substituting (23) into (22) and taking the inverse Laplace transform of the results. A good approximation for the output voltage is ^OUT — E RC + 2K ^-[(2w2+«l)(/2] ^;„ -x/W sm Vk> OJo ■iC02M ER (24)^ IN R [1 _ e-(-i'W _!_ g-[(2<-2+.i)t/2i ^Qg ^Tkc.,!] The linear voltage ramp is expressed by the term — (Et/RC) . The additional terms introduce nonlinearities. The voltage ramp has a slope of 5 millivolts per microsecond for E = —21 volts, R = 42,000 ohms, * In evaluating jE'out it was assumed that Zm' was equal to a fixed resistance Rin' , the low frequency input resistance to the first common emitter stage. A complete analysis indicates that this assumption makes the design conservative. 31G THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 and C = 0.1 microfarads. For these circuit values, and K = 10,000 (corresponding to 80 db of feedback) the nonhnear terms are less than 1/8,000 of the linear term (evaluated when / = 4 X 10"^ seconds) if /i ^ 30 cycles per second, J2 ^ 800 cycles per second, and if the first 1000 microseconds of the voltage ramp are not used. Consequently, 80 db of negative feedback must be maintained over a band extending from 30 to 800 cycles per second in order to realize the desired output voltage linearity. 4.2. Detailed Circuit Arrangement Fig. 13 shows the circuit diagram of the integrator. The method of biasing is the same as is used in the summing amplifier. The 200,000-ohm resistor provides approximately 0.5 milliamperes of collector current for the first stage. The 40,000-ohm resistor provides approximately 0.9 milliamperes of collector current for the second stage. The output stage is designed for a maximum power dissipation of 120 milliwatts and for an output voltage swing between —5 and +24 volts when operating into a load resistance equal to or greater than 40,000 ohms. J+'08V • + 108V 42 K D2 44- C 0.01>(/F o.l/iF 2.4K 270 K I + I08V 1MEG 200n \ — vw 2>U.F 200 K rVWA/^An j 100 K [ POT. I I OUT -10.5V + 108V + 4.5V •45V -10.5V Fig. 13 — Integrator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 317 1 !!3 { LU I 5 u ai a 1 z < 140 120 N 100 .^^ \ AMPLITUDE v. 80 ^""^ \ \ \ \ N \, 60 \ \ > \ \ \ s 40 20 \ '^— -- ,'' S "-s S, PHASE ^ \ \ ■\ PHASE . CROSSOVER 0 -20 -40 GAIN-" CROSSOVER \ \ — ?n HR 95=- 360° ■80 -120 160 ■200 ■240 •280 UJ _J z < LU lO -320 < I Q. -360 -400 ■440 10 2 S .- 2 5 .^3 2 5 ,^^ 2 ^ 105 2 ^ ,0« ' ' 10^ lO'^ w FREQUENCY IN CYCLES PER SECOND Fig. 14 — - Loop current transmission of the integrator. The negative feedback in the integrator has been shaped by means of local feedback and interstage networks as described in Section 2.2. The loop current transmission has been calculated from (13), (14), (15), and (A6) and is plotted in Fig. 14. The transmission is determined under the assumption that the alphas of the transistors are 0.985 and the alpha- cutoff frequencies are three megacycles. Since the feedback above 800 cycles per second falls off at a rate of 9 db per octave, the analysis in Section 4.1 using (23), is conservative. The integrator has a 44° phase margin and a 20 db gain margin. In order to insure sufficient feedback between 30 and 800 cycles per second and adequate margins against instability, the transistors used in the integrator should have alphas in the range 0.98 to 0.99 and alpha-cutoff frequencies equal to or greater than 2.5 megacycles. The silicon diodes Di and D2 are rec^uired in order to prevent the integrator from overloading. For output voltages between —4.0 and 21 volts the diodes are reverse biased and represent very high resistances, of the order of 10,000 megohms. If the output voltage does not lie in this range, then one of the diodes is forward biased and has a low resistance, of the order of 100 ohms. The integrator is then effectively a dc amplifier with a voltage gain of approximately 0.1. The silicon diodes affect the 318 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 linearity of the voltage ramp slightly due to their finite reverse resistances and variable shunt capacities. If the diodes have reverse resistances greater than 1000 megohms, and if the maximum shunt capacity of each diode is less than 10 micromicrofarads (capacity with minimum reverse voltage), then the diodes introduce negligible error. As stated earlier, the integrator generates a voltage ramp in response to a voltage step. This step is applied through a transistor switch which is actuated by a square wave generator capable of driving the transistor well into current saturation. Such a switch is required because the equivalent generator impedance of the applied step voltage must be very small. A suitable circuit arrangement is shown in Fig. 15. For the par- ticular application under discussion the switch *S is closed for 5,000 microseconds. During this time, the voltage E = —217 appears at the input of the integrator. At the end of this time interval, the transistor switch is opened and a reverse current is applied to the feedback con- denser C, returning the output voltage to —4.0 volts in about 2500 micro- seconds. An alternate way of specifying a low impedance switch is to say that the voltage across it be close to zero. For the transistor switch, con- nected as shown in Fig. 15, this means that its collector voltage be within FIRST STAGE OF DC AMPLIFIER 10.5V 50 K 150 K ' — WV-HVW RESIDUAL VOLTAGE BALANCE (TO AZS) Fig. 15 — Input circuit arrangement of the integrator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 319 one millivolt of ground potential during the time the transistor is in saturation. Xow, it has been shown that when a junction transistor in the common emitter connection is driven into current saturation, the minimum voltage between collector and emitter is theoretically equal to — in - (25) q oci where k is the Boltzmann constant, T is the absolute temperature, q is the charge of an electron ((kT/q) = 26 millivolts at room temperature), and ai is the inverse alpha of the transistor, i.e., the alpha with the emitter and collector interchanged. There is an additional voltage drop across the transistor due to the bulk resistance of the collector and emitter regions (including the ohmic contacts). A symmetrical alloy junction transistor with an alpha close to unity is an excellent switch because both the collector to emitter voltage and the collector and emit- ter resistances are very small. At the present time, a reasonable value for the residual voltage* be- tween the collector and emitter is 5 to 10 millivolts. This voltage can be eliminated by returning the emitter of the transistor switch to a small negative potential. This method of balancing is practical because the voltage between the collector and emitter of the transistor does not change by more than 1.0 millivolt over a temperature range of 0°C to 50°C. 4.3. Automatic Zero Set of the Integrator A serious problem associated with the transistor integrator is drift. The drift is introduced by two sources; variations in the base current of the first transistor stage and variations in the base to emitter potential of the first stage wdth temperature. In order to reduce the drift, the input resistor R and the feedback condenser C must be dissociated from the base current and base to emitter potential of the first transistor stage. This is accomplished by placing a blocking condenser Cb between point T and the base of the first transistor as shown in Fig. 15. An automatic zero set circuit is required to maintain the voltage at point T equal to zero volts. This AZS circuit uses a magnetic modulator known as a "magnettor."^^ A block diagram of the AZS circuit is shown in Fig. 16. The dc drift current at the input of the amplifier is applied to the magnettor. The carrier current required by the magnettor is supplied by a local transistor * The inverse alphas of the transistors used in this application were greater than 0.95. 320 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 oscillator. The useful output of the magnettor is the second harmonic of the carrier frequency. The amplitude of the second harmonic signal is proportional to the magnitude of the dc input current and the phase of the second harmonic signal is determined by the polarity of the dc input current. The output voltage of the magnettor is applied to an active filter which is tuned to the second harmonic frequency. The signal is then amplified in a tuned amplifier and applied to a diode gating circuit. Depending on the polarity of the dc input current, the gating circuit passes either the positive or negative half cycle of the second harmonic signal. In order to accomplish this, a square wave at a repetition rate equal to that of the second harmonic signal is derived from the carrier oscillator and actuates the gating circuit. A circuit diagram of the AZS circuit is shown in Figs. 17(a) and 17(b). The various sections of the circuit are identified with the blocks shown in Fig. 16. The active filter is adjusted for a Q of about 300, and the gain of the active filter and tuned amplifier is approximately 1000. The AZS circuit provides ±1.0 volt of dc output voltage for ±0.05 microamperes of dc input current. The maximum sensitivity of the circuit is limited to ±0.005 microamperes because of residual second harmonic generation in the magnettor with zero input current. When the transistor integrator is used together with the magnettor AZS circuit, the slope of the voltage ramp is maintained constant to within one part in 8,000 over a temperature range of 20°C to 40°C. 5.0. The Voltage Comparator The voltage comparator is one of the most important circuits used in analog to digital converters. The comparator indicates the exact time that an input waveform passes through a predetermined reference level. It has been common practice to use a vacuum tube blocking oscillator as a voltage comparator. ^^ Due to variations in the contact potential, heater voltage, and transconductance of the vacuum tube, the maximum DC INPUT AC MAGNETTOR ACTIVE FILTER \ GATING CIRCUIT ^ A ■~ OSCILLATOR GATING PULSE DC OUTPUT Fig. 16 — Block diagram of AZS circuit. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 321 accuracy of the circuit is limited to about ±100 millivolts. By taking advantage of the properties of semiconductor devices, the transistor blocking oscillator comparator can be designed to have an accuracy of ±5 millivolts throughout a temperature range of 20°C to 40°C. 5.1. General Descri'ption of the Voltage Comparator Fig. 18 shows a simplified circuit diagram of the voltage comparator. Except for the silicon junction diode D\ , this circuit is essentially a transistor blocking oscillator. For the purpose of analysis, assume that the reference voltage Vee is set equal to zero. When the input voltage V, is large and negative, the silicon diode Di is an open circuit and the jiuic- tion transistor has a collector current determined by Rb and Ebb [Expres- sion (18)]. The base of the transistor resides at approximately —0.2 volts. As the input voltage Vi approaches zero, the reverse bias across the diode Di decreases. At a critical value of Vi (a small positive poten- tial), the dynamic resistance of the diode is small enough to permit the circuit to become unstable. The positive feedback provided by trans- 1 former Ti forces the transistor to turn off rapidly, generating a sharp I output pulse across the secondary of transformer T-z . When Vi is large and positive, the diode Di is a low impedance and the transistor is main- tained cutoff. In order to prevent the comparator from generating more than one output pulse during the time that the circuit is unstable, the natural period of the circuit as a blocking oscillator must be properly chosen. Depending on this period, the input voltage waveform must have a certain minimum slope when passing through the reference level in order to prevent the circuit from misfiring. I The comparator has a high input impedance except during the switch- 1 ing interval.* When Vi is negative with respect to the reference level, the \ input impedance is equal to the impedance of the reverse biased silicon i diode. When Vi is positive with respect to the reference level, the input I impedance is equal to the impedance of the reverse biased emitter and ! collector junctions in parallel. This impedance is large if an alloy ; junction transistor is used. During the switching interval the input im- ■ pedance is equal to the impedance of a forward biased silicon diode in series with the input impedance of a common emitter stage (approxi- mately 1,000 ohms). This loading effect is not too serious since for the circuit described, the switching interval is less than 0.5 microseconds. The voltage comparator shown in Fig. 18 operates accurately on voltage waveforms with positive slopes. The voltage comparator will operate accurately on waveforms with negative slopes if the diode and * The switching interval is the time required for the transistor to turn off. 322 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 note: all capacitors and inductors IN tuned circuits have a tolerance of ±0.1% Fig. 17(a) — AZS circuit. battery potentials are reversed and if an n-p-n junction transistor is used. 5.2. Factors Determining the Accuracy of the Voltage Comparator Fig. 19 shows the ac equivalent circuit of the voltage comparator. In the equivalent circuit Ri is the dynamic resistance of the diode Di , Rg is the source resistance of the input voltage, and R2 is the impedance of TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 323 the load R^ as it appears at the primary of the transformer T2 . Ri is a function of the dc voltage across the diode Z)i . At a prescribed value of Ri , the comparator circuit becomes unstable and switches. The relation- ship between this critical value of Ri and the transistor and circuit parameters is obtained by evaluating the characteristic equation for the circuit and by determining the relationship which the coefficients of the equation must satisfy in order to have a root of the equation lie in the right hand half of the complex frequency plane. To a good approxima- tion, the critical value of Ri is given by the expression R, -\-R„ + n = Mao RiCc -\- (26) N'^Rr where M is the mutual inductance of transformer Ti and R2 — ly h^l Since the transistor parameters which appear in expression (26) have only a small variation with temperature, the critical value of Ri is independent of temperature (to a first approximation). It will now be shown that the comparator can be designed for an ac- curacy of ±5 millivolts throughout a temperature range of 20°C to 40°C. In order to establish this accuracy it will be assumed that the critical value of 7^1 is equal to 30,000 ohms. This assumption is based on the 30/iF TO LC FILTER IN MAGNETTOR NPUT CIRCUIT 4/iF +33V I+33V Fig. 17(b), 900-cycle carrier oscillator. 324 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 data displayed in Fig. 20 which gives the volt-ampere characteristics of a silicon diode measured at 20°C and 40°C. Throughout this temperature range, the diode voltage corresponding to the critical resistance of 30,000 ohms changes by about 30 millivolts. Fortunately, part of this voltage variation with temperature is compensated for by the variation in voltage Vb-e between the base and emitter of the junction transistor. From Fig. 18, V, = Vo - Vb-e + Ve (27) For perfect compensation (Vi independent of temperature), Vb-e should have the same temperature variation as the diode voltage Vd . Experi- REFERENCE I LEVEL -I ADJUSTMENT i+ Fig. 18 — Simplified circuit diagram of voltage comparator. Fig. 19 — Equivalent circuit of voltage comparator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 325 0.7 _) O > 0.6 R, = 30,000 OHMS 20°C > u:o.5 < I- _l o > o 0.3 2 3 4 5 6 DIODE CURRENT, Ip, IN MICROAMPERES Fig. 20 — Volt-ampere characteristic of a silicon junction diode. mentally it is found that Yh-e for germanium junction transistors varies by about 20 millivolts throughout the temperature range of 20°C to 40°C. Consequently, the variation in Yi at which the circuit switches is ±5 millivolts. It is apparent from Fig. 20 that the accuracy of the comparator in- creases slightly for critical values of R\ greater than 30,000 ohms, but decreases for smaller values. For example, the accuracy of the comparator is ±10 millivolts for a critical value of U\ equal to 5,000 ohms. In gen- eral, the critical value of R\ should be chosen between 5,000 and 100,000 ohms. 5.3. A Practical Yoltage Comparator Fig. 21 shows the complete circuit diagram of a voltage comparator. The circuit is designed to generate a sharp output pulse* when the input voltage waveform passes through the reference level (set by Yee) with a positive slope. The pulse is generated by the transistor switching from the "on" state to the "off" state. To a first approximation the amplitude of the output pulse is proportional to the transistor collector current during the "on" state. When the input voltage waveform passes through the reference level with a negative slope an undesirable negative pulse is generated. This pulse is eliminated by the point contact diode D2 . The voltage comparator is an unstable circuit and has the properties * For the circuit values shown in Fig. 21, the output pulse has a peak amplitude of about 6 volts, a rise time of 0.5 microseconds, and a pulse width of about 2.0 microseconds. 32G THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of a free running blocking oscillator after the input voltage Vi passes through the reference level. After a period of time the transistor will return to the "on" state unless the voltage Vi is sufficiently large at this time to prevent switching. In order to minimize the required slope of the hiput waveform the time interval between the instant Vi passes through the reference level and the instant the transistor would naturally switch to the "on" state must be maximized. This time intei-val can be con- trolled by connecting a diode D3 across the secondary winding of trans- former Ti . When the transistor turns off, the current which was flowing through the secondary of transformer Ti(Ic) continues to flow through the diode D3 so that L2 and D3 form an inductive discharge circuit. The point contact diode D3 has a forward dynamic resistance of less than 10 ohms and a forward voltage drop of 0.3 volt. If the small forward re- sistance of the diode is neglected, the time required for the current in the circuit to fall to zero is T = 0.3 (28) During the inductive transient, 0.3 volt is induced into the primary of transformer Ti (since N = 1) maintaining the transistor cutoff. The duration of the inductive transient can be made as long as desired by increasing L2 . However, there is the practical limitation that increasing L2 also increases the leakage inductance of transformer Ti , and in turn, I I -4.5V 5.1K 250A :iD2 >3K A-l- OUTPUT PULSE V- INPUT WAVEFORM PULSE AMPLITUDE^, ADJUSTMENT^ •^ 2.5 MEG POT. I- jr ee' Ij, = 4 MILS L, = L2= 5 MILLIHENRIES L', = L2= 5 MILLIHENRIES COEFFICIENT OF COUPLING = 0.99 REFERENCE g^ LEVEL ''adjustment MA 1 I I -46V I I -t-1.5V 100 OHM POT. I I -1.5V Fig. 21 — Voltage comparator. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 327 increases the switching time. The circuit shown in Figure 21 does not misfire when used with voltage waveforms having slopes as small as 25 millivolts per microsecond, at the reference level. 6.0. A TRANSISTOR VOLTAGE ENCODER 6.1. Circuit Arrangement The transistor circuits previously described can be assembled into a voltage encoder for translating analog voltages into equivalent time intervals. This encoder is especially useful for converting analog informa- , tion (in the form of a dc potential) into the digital code for processing in a digital system. Fig. 22 shows a simplified block diagram of the encoder. The voltage I'amp generated by the integrator is applied to amplitude selector number one and to one input of a summing amplifier. The amplitude selector is a dc amplifier which amplifies the voltage ramp in the vicinity of zero volts. Voltage comparator number one, which follows the amplitude selector, generates a sharp output pulse at the exact instant of time that the voltage ramp passes through zero volts. The analog input voltage, which has a value between 0 and —15 volts,* is applied to the second input of the summing amplifier. The output voltage of the summing amplifier is zero whenever the ramp INTEGRATOR N0.1 N0.1 3000^65 SUMMING AMPLIFIER AMPLITUDE SELECTORS VOLTAGE COMPARATORS ANALOG INPUT VOLTAGE 0-^-16V N0.2 N0.2 Fig. 22 — • Simplified block diagram of voltage encoder. * If the analog input voltage does not lie in this range, then the voltage gain of the summing amplifier must be set so that the analog voltage at the output of the summing amplifier lies in the voltage range between 0 and +15 volts. 328 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 voltage is equal to the negative of the input analog voltage. At this instant of time the second voltage comparator generates a sharp output pulse. The time interval between the two output pulses is proportional to the analog input voltage if the voltage ramp is linear and has a con- stant slope at all times. 6.2. The Amplitude Selector i The amplitude selector increases the slope of the input voltage wave- form (in the vicinity of zero volts) sufficiently for proper operation of the voltage comparator. The amplitude selector consists of a limiter and a dc feedback amplifier as shown in Fig. 23. The two oppositely poled silicon diodes Di and D2 , limit the input voltage of the dc amplifier to about ±0.65 volts. The dc amplifier has a voltage gain of thirty, and so the maximum output voltage of the amplitude selector is limited to about ±19.5 volts. The net voltage gain between the input and output of the amplitude selector is ten. The principal requirement placed on the dc amplifier is that the input current and the output voltage be zero when the input voltage is zero. This is accomplished by placing a blocking condenser Cb between point T and the base of the first transistor stage, and by using an AZS circuit to maintain point T at zero volts. The dc and AZS amplifiers are identical in configuration to the amplifiers shown in Fig. 12. The dc amplifier is 50 K -VvV 50 K :|N D 1:: SILICON DIODES Dp 1.5 MEG Cb 250 /ZF 500 K I OUT I V^^ »— AAA^ 50 K 1.5 MEG -1 Fig. 23 — Block diagram of the amplitude selector. TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 329 designed to have about 15.6 db less feedback than that shown in Fig. 10 since this amount is adequate for the present purpose. The bandwidth of the dc ampHfier is only of secondary importance because the phase shifts introduced by the two amplitude selectors in the voltage encoder tend to compensate each other. 6.3. Experimental Results The accuracy of the voltage encoder is determined by applying a precisely measured voltage to the input of the summing amplifier and by measuring the time interval between the two output pulses. The maxi- mum error due to nonlinearities in the summing amplifier and the voltage ramp is less than ±0.5 microseconds for a maximum encoding time of 3,000 microseconds. An additional error is introduced by the noise voltage generated in the first transistor stage of the summing amplifier. The ! RMS noise voltage at the output of the summing amplifier is less than 0.5 millivolts. This noise voltage produces an RMS jitter of 0.25 micro- I seconds in the position of the second voltage comparator output pulse. ; The over-all accuracy of the voltage encoder is one part in 4,000 through- ' out a temperature range of 20°C to 40°C. 1 I i ACKNOWLEDGEMENTS ! I The author wishes to express his appreciation to T. R. Finch for the ^ advice and encouragement received in the course of this work. D. W. ! Grant and W. B. Harris designed and constructed the magnettor used ' in the AZS circuit of the integrator. I Appendix I I RELATIONSHIP BETWEEN RETURN DIFFERENCE AND LOOP CURRENT i TRANSMISSION } In order to place the stability analysis of the transistor feedback ampli- fier on a sound basis, it is desirable to use the concept of return differ- ence. It will be shown that a measurable quantity, called the loop current transmission, can be related to the return difference of aZc with reference Ve .*• t In Fig. 24, N represents the complete transistor network exclusive of the transistor under consideration. The feedback loop is broken at the input to the transistor by connecting all of the feedback paths to * In this appendix it is assumed that the transistor under consideration is in the common emitter connection. The discussion can be readily extended to the other transistor connections. t This fact was pointed out by F. H. Tendick, Jr. 330 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Te+rb ^6^4 ( 'V -aZcLb N COMPLETE AMPLIFIER EXCLUSIVE OF THE TRANSISTOR IN QUESTION Fig. 24 — Measurement of loop current transmission. ground through a resistance (/•<; + n) and a voltage r J4 • Using the nomenclature given in Reference 8, the input of the complete circuit is designated as the first mesh and the output of the complete circuit is designated as the second mesh. The input and output meshes of the transistor under consideration are designated 3 and 4, respectively. The loop current transmission is equal to I3', the total returned current when a unit input current is applied to the base of the transistor. The return difference for reference Ve is equal to the algebraic differ- ence* between the unit input current and the returned current h'. 1 3 is evaluated by multiplying the open circuit voltage in mesh 4 (produced by the unit base current) by the backward transmission from mesh 4 to mesh 3 with zero forward transmission through the transistor under consideration. The open circuit voltage in mesh 4 is equal to (re — aZc). The backward transmission is determined with the element aZc , in the fourth row, third column of the circuit determinant, set equal to Ve . Hence, the return difference is expressed as A43 Fr' = 1 + {aZc - re) (Al)t Fr' = A''* + {aZc - r.)A 43 ir', (A2) Fr'.= A^'' = 1+ Tr' (A3) The relative return ratio Tr', is equal to the negative of the loop current transmission and can be measured as shown in Fig. 24. The voltage reh takes into account the fact that the junction transistor is not perfectly * The positive direction for the returned current is chosen so that if the original circuit is restored, the returned current flows in the same direction as the input current. t A''« is the network determinant with the element aZc in the fourth row, third column of the circuit determinant set equal to r, . TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 331 unilateral. Fortunately, in many applications, this voltage can be neg- lected even at the gain and phase crossover frequencies. In the case of single loop feedback amplifiers. A""* will not have any zeros in the right hand half of the complex frequency plane. A study of the stability of the amplifier can then be based on F^-, or T^-, . Appendix II INTERSTAGE NETWORK SHAPING This appendix presents the analysis of the circuit shown in Fig. 7(a). The input impedance of the common emitter connected junction tran- sistor is given by the expression ^iNPUT = n-\- re(l - Gl) (A4) where Gi is the current transmission of the common emitter stage, ex- pression (13). The current transmission A of the complete circuit is equal to A = ^ = ^ I\ Zz -\- ^ IN PUT G, (A5) where Z3 = i?3 + V^ + (l/p<^3). Combining (13), (A4), and (A5) yields ao A = 1 — ao + 5 1 + C03 + V \ W5/ I, Wl (A6) + p^ W5 , CsOO^in + Te -\- R3) _C0iC03- + PCO5 where WaWc(l — tto -}- 6) J ' CO3^C0aC0c(l — ^Q "j- 6) j ^ ^ Rl + Te COl = Wc = CO3 OJs = (1 - ao + 5) 1 + 6 _^ 1 1 (R^ + r,)Co 1 1 ~. . ^« C (1 - ao + 5)J^ 332 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Expression (A6) is valid if l/ws ^ 1/coi + RzCz . The denominator of the expression indicates a falUng 6 db per octave asymptote with a corner, frequency at ws . The second factor in the denominator can be approxi- mated bj^ a falHng 6 db per octave asymptote with a corner frequency at COl 1 ^^ n + (1 - ao + 5) ] n -\- Te -^ Rz -\- W1L3 pkis additional phase and amplitude contributions at higher f recjuencies due to the y and p terms. If COzCzRz then the circuit has a rising 12 db per octave asymptote with a corner frequency at C03 . Fig. 7(b) shows the amplitude and phase of the current transmission. REFERENCES 1. Felker, J. H., Regenerative Amplifier for Digital Computer Applications, Proc. I.R.E., pp. 1584-1596, Nov., 1952. 2. Korn, G. A., and Korn, T. M., Electronic Analog Computers, McGraw-Hil Book Company, pp. 9-19. 3. Wallace, R. L. and Pietenpol, W. J., Some Circuit Properties and Applications of n-p-n Transistors, B. S.T.J. , 30, pp. 530-563, July, 1951. 4. Shockley, W., Sparks, M. and Teal, G. K., The p-n Junction Transistor, Physical Review, 83, pp. 151-162, July, 1951. 5. Pritchard, R. L., Frequenc}' Variation of Current-Amplification for Junction Transistors, Proc. I.R.E., pp. 1476-1481, Nov., 1952. 6. Early, J. M., Design Theory of Junction Transistors, B.S.T.J., 32, pp. 1271- 1312, Nov., 1953. 7. Sziklai, G. C, Symmetrical Properties of Transistors and Their Applications, Proc. I.R.E., pp. 717-724, June, 1953. 8. Bode, H. W., Network Analysis and Feedback Amplifier Design, Van Nos- trand Co., Inc., Chapter IV. 9. Bode, H. W., Op. Cit., pp. 66-69. 10. Bode, H. W., Op. Cit., pp. 162-164. 11. Bargellini, P. M. and Herscher, M. B., Investigation of Noise in Audio Fre- quency Amplifiers Using Junction Transistors, Proc. I.R.E., pp. 217-226,' Feb., 1955. 12. Bode, H. W., Op. Cit., pp. 464-468, and pp. 471-473. 13. Keonjian, E., Temperature Compensated DC Transistor Amplifier, Proc: I.R.E., pp. 661-671, April, 1954. 14. Kretzmer, E. R., An Amplitude Stabilized Transistor Oscillator, Proc. I.R.E.,« pp. 391-401, Feb., 1954. i 15. Goldberg, E. A., Stabilization of Wide-Band Direct-Current Amplifiers for Zero and Gain, R.C.A. Review, June, 1950. 16. Ebers, J. J. and Moll, J. L., Large Signal Behavior of Junction Transistors. Proc. I.R.E., pp. 1761-1772, Dec, 1954. 17. Manlej', J. M., Some General Properties of Magnetic Amplifiers, Proc. I.R.K. March, 1951. 18. M.I.T., Waveforms, Volume 19 of the Radiation Laboratories Series. McGraw Hill Book Company, pp. 342-344. Electrolytic Shaping of Germanium , and Silicon ^ By A. UHLIR, JR. i (Manuscript received November 9, 1955) Properties of electrolyte-semiconductor barriers are described, with em- phasis on germanium. The use of these barriers in localizing electrolytic ! etching is discussed. Other localization techniques are mentioned. Electro- lytes for etching germanium and silicon are given. I INTRODUCTION I I Mechanical shaping techniques, such as abrasive cutting, leave the surface of a semiconductor in a damaged condition which adversely affects the electrical properties of p-n junctions in or near the damaged j material. Such damaged material may be removed by electrolytic etch- ing. Alternatively, all of the shaping may be done electrolytically, so that no damaged material is produced. Electrolytic shaping is particu- [ larly well suited to making devices with small dimensions. I A discussion of electrolytic etching can conveniently be divided into [■ two topics — the choice of electrolyte and the method of localizing the ji etching action to produce a desired shape. It is usually possible to find 1 an electrolyte in which the rate at which material is removed is accurately proportional to the current. For semiconductors, just as for metals, the I choice of electrolyte is a specific problem for each material ; satisfactory j electrolytes for germanium and silicon will be described. The principles of localization are the same, whatever the electrolyte used. Electrolytic etching takes place where current flows from the semiconductor to the electrolyte. Current flow may be concentrated at I certain areas of the semiconductor-electrolyte interface by controlling the flow of current in the electrolyte or in the semiconductor. LOCALIZATION IN ELECTROLYTE Localization techniques involving the electrolytic current are appli- cable to both metals and semiconductors. In some of these techniques, 333 334 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 the localization is so effective that the barrier effects found with n-type semiconductors can be ignored; if not, the barrier can be overcome by light or heat, as will be described below. If part of the work is coated with an insulating varnish, electrolytic etching will take place only on the uncoated surfaces. This technique, often called "masking," has the limitation that the etching undercuts the masking if any considerable amount of material is removed. The i same limitation applies to photoengraving, in which the insulating coat- ing is formed by the action of light. The cathode of the electrolytic cell may be limited in size and placed close to the work (which is the anode). Then the etching rate will be greatest at parts of the work that are nearest the cathode. Various shapes can be produced by moving the cathode with respect to the I work, or by using a shaped cathode. For example, a cathode in the form | of a wire has been used to slice germanium. Instead of a true metallic cathode, a "virtual cathode" may be used to localize electrolysis.^ In this technique, the anode and true cathode are separated from each other by a nonconducting partition, except for a small opening in the partition. As far as localization of current to the anode is concerned, the small opening acts like a cathode of equal size and so is called a virtual cathode. The nonconducting partition may include a glass tube drawn down to a tip as small as one micron diameter but nevertheless open to the flow of electrolytic current. With such a tip as a virtual cathode, micromachining can be conducted on a scale comparable to the wavelength of visible light. A general advantage of the virtual cathode technique is that the cathode reaction (usually hydrogen evolution) does not interfere with the localizing action nor with observation of the process. :| In the jet-etching technique, a jet of electrolyte impinges on the work.^'* The free streamlines that bound the flowing electrolyte are governed primarily by momentum and energy considerations. In turn, the shape of the electrolyte stream determines the localization of etch- ing. A stream of electrolyte guided by wires has been used to etch semi- conductor devices.^ Surface tension has an important influence on the free streamlines in this case, PROPERTIES OF ELECTROLYTE-SEMICONDUCTOR BARRIERS The most distinctive feature of electrolytic etching of semiconductors is the occurrence of rectifying barriers. Barrier effects for germanium will be described; those for silicon are qualitatively similar. The voltage-current curves for anodic n-type and p-type germanium ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 335 [in 10 per cent KOH are shown in Fig. 1. Tlie concentration of KOH [is not critical and other electrolytes give similar results. The voltage 'drop for the p-type specimen is small. For anodic n-type germanium, ! however, the barrier is in the reverse or blocking direction as evidenced by a large voltage drop. The fact that n-type germanium differs from p-type germanium only by very small amounts of impurities suggests that the barrier is a semiconductor phenomenon and not an electro- i chemical one. This is confirmed by the light sensitivity of the n-type 1 voltage-current characteristic. Fig. 2 is a schematic diagram of the ! arrangement for obtaining voltage-current curves. A mercury-mercuric loxide-10 per cent KOH reference electrode was used at first, but a gold (wire was found equally satisfactory. At zero current, a voltage Vo exists j between the germanium and the reference electrode ; this voltage is not [included in Fig. 1. I The saturation current Is , measured for the n-type barrier at a \moderate reverse voltage (see Fig. 1), is plotted as a function of tempera- Iture in Fig. 3. The saturation current increases about 9 per cent per [degree, just as for a germanium p-n junction, which indicates that the I 40 35 30 ^25 Lil O 20 15 10 1 12 OHM-CM n-TYPE / DAR\<. 1 / / 1 1 1 1 1 / 1 1 1 1 1 WITH ; LIGHT ^' 1 1 I 1 1 1 n i 1 1 / / P- FYPE 10 20 30 40 50 60 CURRENT FLOW IN MILLIAMPERES PER CM^ Fig. 1 — Anodic voltage-current characteristics of germanium. 336 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1956 current is proportional to the equilibrium density of minority carriers (holes). The same conclusion may be drawn from Fig. 4, which shows that the saturation current is higher, the higher the resistivity of the n-type germanium. But the breakdown voltages are variable and usu- ally much lower than one would expect for planar p-n junctions made, for example, by alloying indium into the same n-type germanium. Breakdown in bulk junctions is attributed to an avalanche multipli- cation of carriers in high fields.^ The same mechanism may be responsible for breakdown of the germanium-electrolyte barrier; low and variable breakdown voltages may be caused by the pits described below. The electrolyte-germanium barrier exhibits a kind of current multi- plication that differs from high-field multiplication in two respects: it occurs at much lower reverse voltages and does not vary much with voltage.^ This effect can be demonstrated very simply by comparison with a metal-germanium barrier, on the assumption that the latter has a current multiplication factor of unity. This assumption is supported by experiments which indicate that current flows almost entirely by hole flow, for good metal-germanium barriers. The experimental arrangement is indicated in Fig. 5(a) and (b). The voltage-current curves for an electrolyte barrier and a plated barrier on the same slice of germanium are shown in Fig. 5(c).* The curves for the REFERENCE ELECTRODE CATHODE LIGHT Fig. 2 — Arrangement for obtaining voltage current characteristics. * In Fig. 5 the dark current for the phited barrier is much hirger than can be exphained on the basis of hole current; it is even higher than the dark current for the electrolyte barrier, which should be at least 1.4 times the hole current. This excess dark current is believed to be leakage at the edges of the plated area and probably does not affect the intrinsic current multiplication of the plated barrier as a whole. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 337 10 2 o a. 01 a. to Ui oc LU Q. 5 < _) m cc tr 3 U z o cc 3 (0 •> I 10" / { - / - 1 / 7 / / / - /% - / / / / n/ ^ y i<:i 0 10 20 30 40 50 60 TEMPERATURE IN DEGREES CENTIGRADE Fig. 3 — Temperature variation of the saturation current of a barrier between 5.5 ohm-cm n-type germanium and 10 per cent KOH solution. illuminated condition were obtained by shining light on a dry face of a slice while the barriers were on the other face. The difference between the light and dark currents is larger for the electrolyte-germanium bar- rier than for the metal-germanium barrier, by a factor of about 1.4. The transport of holes through the slice is probably not very different for the two barriers. Therefore, a current multiplication of 1.4 is indi- cated for the electrolyte barrier. About the same value was found for temperatures from 15°C to 60°C, KOH concentrations from 0.01 per cent to 10 per cent, n-type resistivities of 0.2 ohm-cm to 6 ohm-cm, light currents of 0.1 to 1.0 ma/cm^, and for O.IN indium sulfate. Evidently the flow of holes to the electrolyte barrier is accompanied by a proportionate return flow of electrons, which constitutes an addi- tional electric current. Possible mechanisms for the creation of the electrons will be discussed in a forthcoming article. 338 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 7 > 4 LU o > I 0,5 1.0 1 CURRENT F 5 2.0 2.5 3.0 3.5 4.0 LOW IN MILLIAMPERES PER CM^ 4.5 Fig. 4. — Anodic voltage -current curves for various resistivities of germanium. SCRATCHES AND PITTING The voltage- current curve of an electrolyte-germanium barrier is very sensitive to scratches. The curves given in the illustrations were : obtained on material previously etched smooth in CP-4, a chemical I etch.* '' If, instead, one starts with a lapped piece of n-type germanium, the electrolyte-germanium barrier is essentially "ohmic;" that is, the voltage drop is small and proportional to the current. A considerable reverse voltage can be attained if lapped n-type germanium is electrolytically etched long enough to remove most of the damaged germanium. How- ever, a pitted surface results and the breakdown voltage achieved is not as high as for a smooth chemically-etched surface. The depth of damage introduced by typical abrasive sawing and lapping was investigated by noting the voltage-current curve of the Br2 Five parts HNO3 , 3 parts 48 per cent HF, 3 parts glacial acetic acid, ^0 P^-^t ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 339 electrolyte-germanium barrier after various amounts of material had been removed by chemical etching. After 20 to 50 microns had been re- moved, further chemical etching produced no change in the barrier characteristic. This amount of material had to be removed even if the lapping was followed by polishing to a mirror finish. The voltage-current curve of the electrolyte-germanium barrier will reveal localized damage. On the other hand, the photomagnetoelectric (PME) measurement of I -< — REFERENCE ELECTRODE CATHODE- -- -^ ■< ■y GLASS TUBING CEMENTED TO Ge E LECTROLYT z i N-Ge ■^ 1 1 1 1 (a) ELECTROPLATED INDIUM METAL TO N-Ge CONTACT ELECTROLYTE TO N-Ge BARRIER (c) 0 2 4 6 CURRENT, I, IN MILLIAMPERES PER CM 2 Fig. 5 — Determination of the current multiplication of the barrier between 6 ohm-cm n-type germanium and an electrolyte. 340 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Fig. 6 — Electrolytic etch pits on two sides of 0.02-inch slice of n-type germa- nium. Half of the slice was in contact with the electrolyte. surface recombination velocity gives an evaluation of the average con- dition of the surface. A variation of the PME method has been used to study the depth of abrasion damage; the damage revealed by this method extends only to a depth comparable to the abrasive size. A scratch is sufficient to start a pit that increases in size without limit if anodic etching is prolonged. However, a scratch is not necessary. Pits are formed even when one starts with a smooth surface produced by chemical etching. A drop in the breakdown voltage of the barrier is noticed when one or more pits form. The breakdown voltage can be restored by masking the pits with polystyrene cement. Evidence that the spontaneous pits are caused by some features of the crystal, itself, was obtained from an experiment on single-crystal n-type germanium made by an early version of the zone-leveling process. A slice of this material was electrolytically etched on both sides, after preliminary chemical etching. Photographs of the two sides of the slice are shown in Fig. 6. Only half of the slice was immersed in the electro- lyte. The electrolytic etch pits are concentrated in certain regions of the slice — the same general regions on both sides of the slice. It is interesting that radioautographs and resistivity measurements indicate high donor concentrations in these regions. Improvements, including more intensive stirring, were made in the zone-leveling process, and the electrolytic etch pit distribution and the donor radioautographs have been much more uniform for subsequent material. Several pits on a (100) face are shown in Fig. 7. The pits grow most rapidly in (100) directions and give the spiked effect seen in the illustra- tion. Toiler prolonged etching, the spikes and their branches form a com- plex network of caverns beneath the surface of the germanium. High-field carrier generation may be responsible for pitting. A locally ELECTROLYTIC SHAPING OF GERMAXIUM AND SILICON 341 Fig. 7 — Electrolytic etch pits on n-type germanium. high donor concentration would favor breakdown, as would any con- cavity of the germanium surface (which would cause a higher field for a given voltage) . Very high fields must occur at the points of spikes such jas those shown in Fig. 7. The continued growth of the spikes is thus favored by their geometry. Microscopic etch pits arising from chemical etching have been corre- ;lated with the edge dislocations of small-angle grain boundaries. A I specimen of n-type germanium with chemical etch pits was photomicro- graphed and then etched electrolytically. The etch pits produced elec- trolytically could not be correlated with the chemical etch pits, most of which were still visible and essentially unchanged in appearance. Also, no correlation could be found between either kind of etch pit and the locations at which copper crystallites formed upon immersion in a copper sulfate solution. Microscopic electrolytic etch pits at dislocations j in p-type germanium have been reported in a recent paper that also I mentions the deep pits produced on n-type germanium.^* y Electrolytic etch pits are observed on n-type and high-resistivity silicon. These etch pits are more nearly round than those produced in germanium. In spite of the pitting phenomenon, electrolytic etching is success- 342 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 I fully used in the fabrication of devices involving n-type semiconductors. Pitting can be reduced relative to "normal" uniform etching by any agency that increases the concentration of holes in the semiconductor. Thus, elevated temperatures, flooding with light, and injection of holes by an emitter all favor smooth etching. SHAPING BY MEANS OF INJECTED CARRIERS I Hole-electron pairs are produced when light is absorbed by semi- conductors. Light of short wavelength is absorbed in a short distance, while long wavelength light causes generation at considerable depths. The holes created by the light move by diffusion and drift and increase the current flow through an anodic electrolyte-germanium barrier at whatever point they happen to encounter the barrier. In general, more holes will diffuse to a barrier, the nearer the barrier is to the point at which the holes are created. For n-type semiconductors, the current due to the light can be orders of magnitude greater than the dark cur- rent, so that the shape resulting from etching is almost entirely deter- mined by the light. As shown in Fig. 3, the dark current can be made very small by lowering the temperature. An example of the shaping that can be done with light is shown in Fig. 8. A spot of light impinges on one side of a wafer of n-type germanium or silicon. The semiconductor is made anodic with respect to an etching electrolyte. Accurately concentric dimples are produced on both sides of the wafer. Two mechanisms operate to transmit the effect to the oppo- site side. One is that some of the light may penetrate deeply before generating a hole-electron pair. The other is that a fraction of the car- riers generated near the first surface will diffuse to the opposite side. By varying the spectral content of the light and the depth within the \ \ -n-TYPE SEMICONDUCTOR LIGHT I I Fig. 8 — Double dimpling with light. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 343 wafer at which the light is focused, one can produce dimples with a vari- ,'ety of shapes and relative sizes. I It is obvious that the double-dimpled wafer of Fig. 8 is desirable for {the production of p-n-p alloy transistors. For such use, one of the most [important dimensions is the thickness remaining between the bottoms of the two dimples. As has been mentioned in connection with the jet- I etching process, a convenient way of monitoring this thickness to de- Itermine the endpoint of etching is to note the transmission of light of [suitable wavelength.^ There is, however, a control method that is itself [automatic. It is based on the fact that at a reverse-biased p-n junction [Or electrolyte-semiconductor barrier there is a space-charge region that is practically free of carriers. When the specimen thickness is reduced so that space-charge regions extend clear through it, current ceases to flow and etching stops in the thin regions, as long as thermally or op- tically generated carriers can be neglected. However, more pitting is to be expected in this method than when etching is conducted in the pres- ence of an excess of injected carriers. A p-n junction is a means of injecting holes into n-type semiconduc- tors and is the basis of another method of dimpling, shown in Fig. 9. The p-n junction can be made by an alloying process such as bonding an acceptor-doped gold wire to germanium. The ohmic contact can be made by bonding a donor-doped gold wire and permits the injection of a greater excess of holes than would be possible if the current through the p-n junction were exactly equal to the etching current. Dimpling without the ohmic contact has been reported.^ 14 OHMIC CONTACT p-n JUNCTION Fig. 9 — Dimpling with carriers injected by a p-n junction. 344 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CONTROL BY OHMIC CONDUCTION The carrier-injection shaping techniques work very well for n-typei material. It is also possible to inject a significant number of holes intos rather high resistivity p-type material. But what can be done about: p-type material in general, short of developing cathodic etches? ] The ohmic resistivity of p-type material can be used as shown in Fig.!^ 10. More etching currect flows through surfaces near the small contact than through more remote surfaces. A substantial dimpling effect is observed when the semiconductor resistivity is equal to the electrolyte resistivity, but improved dimpling is obtained on higher resistivity semiconductor. This result is just what one might expect. But the math- ematical solution for ohmic flow from a point source some distance from a planar boundary between semi-infinite materials of different conduc- tivities shows that the current density distribution does not depend on the conductivities. An important factor omitted in the mathematical solution is the small but significant barrier voltage, consisting largely of electrochemical polarization in the electrolyte. The barrier voltage is; approximately proportional to the logarithm of the current density; while the ohmic voltage drops are proportional to current density. Thus,- high current favors localization. ELECTROLYTES FOR ETCHING GERMANIUM AND SILICON » The electrolyte usually has two functions in the electrolytic etching of an oxidizable substance. First, it must conduct the current necessary for the oxidation. Second, it must somehow effect removal of the oxida- tion product from the surface of the material being etched. The usefulness of an electrolytic etch depends upon one or both of: ANY CONTACT, PREFERABLY OHMIC ^//yyyy//y/y/y////////y////y/////yyyyyyyyyyy7^ Fig. 10 — Dimpling by ohmic conduction. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 345 the following situations — the electrolytic process accomplishes a reac- tion that cannot be achieved as conveniently in any other way or it permits greater control to be exercised over the reaction. Accordingly, chemical attack by the chosen electrolyte must be slight relative to the electrochemical etching. A smooth surface is probably desirable in the neighborhood of a p-n junction, to avoid field concentrations and lowering of breakdown voltage. Therefore, a tentative requirement for an electrolyte is the production of a smooth, shiny surface on the p-type semiconductor. Such \ an electrolyte will give a shiny but possibly pitted surface on n-type j specimens of the same semiconductor. The effective valence of a material being electrolytically etched is ; defined as the number of electrons that traverse the circuit divided by the number of atoms of material removed. (The amount of material ! removed was determined by weighing in the experiments to be described.) If the effective valence turns out to be less than the valence one might predict from the chemistry of stable compounds, the etching is sometimes said to be "more than 100 per cent efficient." Since the anode reactions in electrolytic etching may involve unstable intermediate compounds and competing reactions, one need not be surprised at low or fractional effective valences. Germanium can be etched in many aqueous electrolytes. A valence of almost exactly 4 is found. That is, 4 electrons flow through the circuit for each atom of germanium removed. For accurate valence measure- ments, it is advisable to exclude oxygen by using a nitrogen atmosphere. Potassium hydroxide, indium sulfate, and sodium chloride solutions are among those that have been used. Sulfuric acid solutions are prone to ) yield an orange-red deposit which may be a suboxide of germanium/* I Similar orange deposits are infrequently encountered with potassium I hydroxide. Hydrochloric acid solutions are satisfactoiy electrolytes. The reaction I product is removed in an unusual manner when the electrolyte is about 2N hydrochloric acid. Small droplets of a clear liquid fall from the etched regions. These droplets may be germanium tetrachloride, which is denser than the electrolyte. They turn brown after a few seconds, perhaps be- cause of hydrolysis of the tetrachloride. Etching of germanium in sixteen different aqueous electroplating electrolytes has been mentioned. Germanium can also be etched in the partly organic electrolytes described below for silicon. One would expect that silicon could be etched by making it the anode in a cell with an aqueous hydrofluoric acid electrolyte. The seemingly 346 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 | ) likely oxidation product, silicon dioxide, should react with the hydro-! fluoric acid to give silicon tetrafluoride, which could escape as a gas. In fact, a gas is formed at the anode and the silicon loses weight. But the gas is hydrogen and an effective valence of 2.0 ± 0.2 (individual deter- minations ranged from 1.3 to 2.7) was found instead of the value 4 that i might have been expected. The quantity of hydrogen evolved is con- sistent with the formal reaction Si —> Si"*"'" + me (electrochemical oxidation) Si+™ + (4-to)H+ -^ Si+' + Vz (4-m)H2 (chemical oxidation) where m is about two. The experiments were done in 24 per cent to 48 per cent aqueous solutions of HF at current densities up to 0.5 amp/cm^. The suggestion that the electrochemical oxidation precedes the chemi- cal oxidation is supported by the appearance and behavior of the etched surfaces. Instead of being shiny, the surfaces have a matte black, brown, or red deposit. At 40 X magnification, the deposit appears to consist of flakes of a; resinous material, tentatively supposed to be a silicon suboxide. A re- markable reaction can be demonstrated if the silicon is rinsed briefly in water and alcohol after the electrolytic etch, dried, and stored in air for as long as a year. Upon reimmersing this silicon in water, one can observe the liberation of gas bubbles at its surface. This gas is presumed to be hydrogen. To initiate the reaction it is sometimes necessary to dip the specimen first in alcohol, as water may otherwise not wet it. The speci- mens also liberate hydrogen from alcohol and even from toluene. Thus, chemical oxidation can follow electrolytic oxidation. But chemical oxidation does not proceed at a significant rate before thei current is turned on. Smooth, shiny electrolytic etching of p-type silicon has been obtained; with mixtures of hydrofluoric acid and organic hydroxyl compounds,; such as alcohols, glycols, and glycerine. These mixtures may be an- hydrous or may contain as much as 90 per cent water. The organic additives tend to minimize the chemical oxidation of the silicon. They; also permit etching at temperatures below the freezing point of aqueous solutions. They lower the conductivity of the electrolyte. For a given electrolyte composition, there is a threshold current density, usually between 0.01 and 0.1 amps/cm , for smooth etching.; Lower current densities give black or red surfaces with the same hy- drogen-liberating capabilities as those obtained in aqueous hydrofluoric acid. ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 347 In general, smooth etching of siHcon seems to result when the effective valence is nearly 4 and there is little anodic evolution of gas. The elec- I trical properties of the smooth surface appear to be equivalent to those ! of smooth silicon surfaces produced by chemical etching in mixtures of i nitric and hydrofluoric acids. On the other hand, the reactive surface [produced at a valence of about 2, with anodic hydrogen evolution, is I capable of practically shorting-out a silicon p-n junction. The electrical j properties of this surface tend to change upon standing in air. ACKNOWLEDGEMENTS Most of the experiments mentioned in this paper were carried out by my wife, Ingeborg. An exception is the double-dimpling of germanium by light, which was done by T. C. Hall. The dimpling procedures of Figs. 9 and 10 are based on suggestions by J. M. Early. The effect of light upon electrolytic etching was called to my attention by 0. Loosme. W. G. Pfann provided the germanium crystals grown with different degrees of stirring. REFERENCES 1. J. F. Barry, I.R.E.-A.I.E.E. Semiconductor Device Research Conference, Philadelphia, June, 1955. 2. A. Uhlir, Jr., Rev. Sci. Inst., 26, pp. 965-968, 1955. 3. W. E. Bailey, U. S. Patent No. 1,416, 929, May 23, 1922. 4. Bradley, et al. Proc. I.R.E., 24, pp. 1702-1720, 1953. 5. M. V. Sullivan and J. H. Eigler, to be published. 6. S. L. Miller, Phys. Rev. 99, p. 1234, 1955. 7. W. H. Brattain and C. G. B. Garrett, B.S.T.J., 34, pp. 129-176, 1955. 8. E. H. Borneman, R. F. Schwarz, and J. J. Stickler, J. Appl. Phvs., 26, pp. 1021-1029, 1955. 9. D. R. Turner, to be submitted to the Journal of the Electrochemical Society. 10. R. D. Heidenreich, U. S. Patent No. 2,619,414, Nov. 25, 1952. 11. T. S. Moss, L. Pincherle, A. M. Woodward, Proc. Phys. Soc. London, 66B, p. 743, 1953. 12. T. M. Buck and F. S. McKim, Cincinnati Meeting of the Electrochemical Society, Mav, 1955. 13. F. L. Vogel, W. G. Pfann, H. E. Corey, and E. E. Thomas, Phys. Rev., 90, p. 489, 1953. 14. S. G. Ellis, Phys. Rev., 100, pp. 1140-1141, 1955. 15. Electronics, 27, No. 5, p. 194, May, 1954. 16. F. Jirsa, Z. f. Anorg. u. AUgemeine Chem., Bd. 268, p. 84, 1952. \ A Large Signal Theory of Traveling Wave Amplifiers Including the Effects of Space Charge and Finite Coupling Between the Beam and the Circuit By PING KING TIEN Manuscript received October 11, 1955) The non-linear behavior of the traveling-wave amplifier is calculated in this paper by numericalhj integrating the motion of the electrons in the presence of the circuit and the space charge fields. The calculation extends the earlier work by Nordsieck and the srnall-C theory by Tien, Walker and Wolontis, to include the space charge repulsion between the electrons and the effect of a finite coupling between the circuit and the electron beam. It however differs from Poulter's and Rowers works in the methods of calcu- lating the space charge and the effect of the backward wave. The numerical work was done using 701 -type I.B.M. equipment. Re- sults of calcidation covering QC from 0.1 to 0.4, b from 0.46 to 2.56 and k from 1.25 to 2.50, indicate that the saturation efficiency varies between 23 per cent and 37 per cent for C equal to 0.1 and between 33 per cent and Jf.0 per cent for C equal to 0.15. The voltage and the phase of the circuit wave, the velocity spread of the electrons and the fundamental component of the charge-density modidation are either tabulated or presented in curves. A method of calculating the backward wave is provided and its effect fully discussed. 1. INTRODUCTION Theoretical evaluation of the maximum efficiency attainable in a traveling-wave amplifier requires an understanding of the non-linear behavior of the device at various working conditions. The problem has been approached in many ways. Pierce/ and later Hess,^ and Birdsalf and Caldwell investigated the efficiency or the output power, using cer- tain specific assumptions about the highly bunched electron beam. They either assume a beam in the form of short pulses of electrons, or, specify 349 350 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 an optimum ratio of the fundamental component of convection current to the average or d-c current. The method, although an abstract one, generally gives the right order of the magnitude. When the usual wave concept fails for a beam in which overtaking of the electrons arises, we may either overlook effects from overtaking, or, using the Boltzman's transport equation search for solutions in series form. This attack has been pursued by Parzen and Kiel, although their work is far from com- plete. The most satisfying approach to date is Nordsieck's analysis.' Nordsieck followed a typical set of "electrons" and calculated their velocities and positions by numerically integrating a set of equations of motion. Poulter has extended Nordsieck equations to include space charge, finite C and circuit loss, although he has not perfectly taken into account the space charge and the backward wave. Recently Tien, Walker, and Wolontis have published a small C theory in which "elec- trons" are considered in the form of uniformly charged discs and the space charge field is calculated by computing the force exerted on one disc by the others. Results extended to finite C, have been reported by Rowe,^*^ and also by Tien and Walker.^^ Rowe, using a space charge expression similar to Poulter's, computed the space charge field based on the electron distribution in time instead of the distribution in space. This may lead to appreciable error in his space charge term, although its influence on the final results cannot be easily evaluated. In the present analysis, we shall adopt the model described by Tien, Walker and Wolontis, but wish to add to it the effect of a finite beam to circuit coupling. A space charge expression is derived taking into account the fact that the a-c velocities of the electrons are no longer small com- pared with the average velocity. Equations are rewritten to retain terms involving C. As the backward wave becomes appreciable when C in- creases, a method of calculating the backward wave is provided and the effect of the backward wave is studied. Finally, results of the calculation covering useful ranges of design and operating parameters are presented and analyzed. 2. ASSUMPTIONS To recapitulate, the major assumptions which we have made are: 1. The problem is considered to be one dimensional, in the sense that the transverse motions of the electrons are prohibited, and the current, velocity, and fields, are functions only of the distance along the tube and of the time. 2. Only the fundamental component of the current excites waves on the circuit. A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 351 3. The space charge field is computed from a model in which the helix is replaced by a conducting cylinder, and electrons are uniformly charged discs. The discs are infinitely thin, concentric with the helix and have a radius equal to the beam radius. 4. The circuit is lossfree. These are just the assumptions of the Tien-Walker-Wolontis model. In addition, we shall assume a small signal applied at the input end of a long tube, where the beam entered unmodulated. What we are looking for are therefore the characteristics of the tube beyond the point at which the device begins to act non-linearly. Let us imagine a flow of electron discs. The motions of the discs are computed from the circuit and the space charge fields by the familiar Newton's force equation. The elec- trons, in turn, excite waves on the circuit according to the circuit equa- tion derived either from Brillouin's model^ or from Pierce's equivalent circuit. The force equation, the circuit equation, and the equation of conservation of charge in kinematics, are the three basic equations from which the theory is derived. 3. FORWARD AND BACKWARD WAVES In the traveling-wave amplifier, the beam excites forward and back- ward waves on the circuit. (We mean by "forward" wave, the wave which propagates in the direction of the electron flow, and by "back- ward" wave, the wave which propagates in the opposite direction.) Because of phase cancellation, the energy associated with the backward wave is small, but increases with the beam to circuit coupling. It is there- fore important to compute it accurately. In the first place, the waves on the circuit must satisfy the circuit equation dH^(z,t) 2d'V{z,t) „ d'p^iz, t) ,v Here, V is the total voltage of the waves. Vo and Zo are respectively the phase velocity and the impedance of the cold circuit, z is the distance along the tube and t, the time, p^ is the fundamental component of the linear charge density. V and p„ are functions of z and /. The complete solution of (1) is in the form Viz) = Cre'^'' + (726 "^"^ + e —-y— J e " po,{^) dz ^2) + e " —^ j e p^{z) dz 352 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 where the common factor e^"' is omitted. To = j{co/vo), j = \/— 1 and w is the angular frequency. Ci and C2 are arbitrary constants which will be determined by the boundary conditions at the both ends of the beam. The first two terms are the solutions of the homogeneous equation (or the complementary functions) and are just the cold circuit waves. The third and the fourth terms are functions of electron charge density and are the particular solution of the equation. Let us consider a long traveling-wave tube in which the beam starts from z = 0 and ends at 2; = D. The motion of electrons observed at any particular position is periodic in time, though it varies from point to point along the beam. To simplify the picture, we may divide the beam along the tube into small sections and consider each of them as a current element uniform in z and periodic in time. Each section of beam, or each current element excites on the circuit a pair of waves equal in ampli- tudes, one propagating toward the right (i.e., forward) and the other, toward the left. One may in fact imagine that these are trains of waves supported by the periodic motion of the electrons in that section of the beam. Obviously, a superposition of these waves excited by the whole beam gives the actual electromagnetic field distribution on the circuit. One may thus compute the forward traveling wave at z by summing all the waves at z which come from the left. Stated more specifically, the forward traveling energy at z is contributed by the waves excited by the current elements at the left of the point z. Similarly the backward travel- ing energy, (or the backward wave) at z is contributed by the waves excited by the current elements at the right of the point z. It follows obviously from this picture that there is no forward wave at 2 = 0 (except one corresponding to the input signal), and no backward wave at 2 = D. (This implies that the output circuit is matched.) With these boundary conditions, (1) is reduced to z) = Finput e " + e ° — -— / e " po,{z) Z Jo dz + /-^J e-%.(.) (3) dz Equations (2) and (3) have been obtained by Poulter.^ The first term of (3) is the wave induced by the input signal. It propagates as though the ; beam were not present. The second term is the voltage at z contributed by the charges between 2 = 0 and 2 = 2. It is just the voltage of the forward wave described earlier. Similarly the third term which is the voltage at 2 contributed by the charges between z = z and 2 = D is the voltage of the backward wave at the point 2. Denote F and B respec- A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 353 tively the voltages of the forward and the backward waves, we have F{z) = Fi„put e-'"^ + e-^»^ ^« r e'^' p^z) dz (4) Z Jo Biz) = e^- ^° £ e-^-p„(e) dz (5) It can be shown by direct substitution that F and B satisfy respectively the differential equations dz Vo dt 2 (9^ (6) dB(z, t) 1 a5(2, 0 Zo ap„(2, 0 (92 1^0 di 2 dt (7) We put (4) and (5) in the form of (6) and (7) simply because the differ- ential equations are easier to manipulate than the integral equations. In fact, we should start the analysis from (6) and (7) if it were not for a physical picture useful to the understanding of the problem. Equations (6) and (7) have the advantage of not being restricted by the boundary conditions at 2; = 0 and D, which we have just imposed to derive (4) and (5). Actually, we can derive (6) and (7) directly from the Brillouin model in the following manner. Suppose Y, I and Zo are respectively the voltage, current and the characteristic impedance of a transmission line system in the usual sense. (V + /Zo) must then be the forward wave and {V — IZo) must be the backward wave. If we substituted F and B in these forms into (1) of the Brillouin' s paper,^^ we should obtain exactly (6) and (7). It is obvious that the first and third terms of (2) are respectively the complementary function and the particular solution of (6), and similarly the second and the fourth terms of (2) are respectively the comple- mentary function and the particular solution of (7). From now on, we shall overlook the complementary functions which are far from syn- chronism with the beam and are only useful in matching the boundary conditions. It is the particular solutions which act directly on the elec- tron motion. With these in mind, it is convenient to put F and B in the form Fiz, t) = -j~ [aiiij) cos

)—)] d(f> sgn (iy, 360 A LARGE SIGNAL THEORY OF TRAVELING- WAVE AMPLIFIERS 361 equipment. The problem was programmed by Miss D. C. Legaus. The cases computed are listed in Table I in which m and m2 are respectively Pierce's .xi and iji , and A,(d — iny) and tj at saturation will be discussed later. All the cases were computed with A^ = 0.2 using a model based on 24 electron discs per electronic wavelength. To estimate the error involved in the numerical work, Case (10) has been repeated for 48 elec- trons and Cases (10) and (19) for Ay = 0.1. The results obtained by using different numbers of electrons are almost identical and those ob- tained by varying the inter\'al A// indicate a difference in A (y) less than 1 per cent for Case (10) and about 6 per cent for Case (19). As error generally increases with QC and C the cases listed in this paper are limited to QC = 0.4 and C = 0.15. For larger QC or C, a model of more electrons or a smaller interval of integration, or both should be used. 7. POWER OUTPUT AND EFFICIENCY Define A(ij) = HVa,(yy + aM' -0(y)=i^n-'^-^ + by ^^^^ aiiy) We have then F{z,t) = ^A{y) cos ^ -^t- e{y) Uo (22) The power carried by the forward wave is therefore 2CA'hVo (23) (f) = \Z/o/ average and the efficiency is Eff. = ?£^^ = 2CA' or ^ = 2CA' (24) In Table I, the values of A(y), 6{y) and y at the saturation level are listed for every case computed. We mean by the saturation level, the distance along the tube or the value of y at which the voltage of the forward wave or the forward traveling power reaches its first peak. The Eff./C at the saturation level is plotted in Fig. 1 versus QC, for k = 2.5, h for maximum small-signal gain and C = small, 0.05, 0.1, 0.15 and 2. It is also plotted versus h in Fig. 2 for QC = 0.2, k = 2.5 and C = small, 0.1 and0.15, and in Fig. 3 for QC = 0.2, C = 0.1 and k = 1.25 and 2.50. In Fig. 2 the dotted curves indicate the values of h at Avhich 1 362 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195G 4.5 0.5 Fig. 1 — The saturation eff./C versus QC, for k = 2.5, h for maximum small- signal gain and C = small, 0.1, 0.15 and 0.2. ixx = Ml (max), 0.94 jui(max), 0.67 iui(max) and 0.3 /ii(niax), respectively. It is seen that Eff./C decreases as C increases particularly when h is large. It is almost constant between k = 1.25 and 2.50 and decreases slowly for large values of C when QC increases. The (Eff./C) at saturation is also plotted versus QC in Fig. 4(a) for small C, and in Fig. 4(b) for C = 0.1. It should be noted that for C = 0.1 the values of Eff./C fall inside a very narrow region say between 2.5 to 3.5, whereas for small C they vary widely. 8, VELOCITY SPREAD In a traveling-wave amplifier, when electrons are decelerated by the circuit field, they contribute power to the circuit, and when electrons are accelerated, they gain kinetic energy at the expense of the circuit power. It is therefore of interest to plot the actual velocities of the fastest and the slowest electrons at the saturation level and find how they vary with the parameters QC, C, b and k. This is done in Fig. 5. These veloci- ties are also plotted versus y for Case 10 in Fig. 6, in which, the A(y) curve is added for reference. 9. THE BACKWARD WAVE AND THE FUNDAMENTAL COMPONENT OF THE ELECTRON CHARGE DENSITY Our calculation of efficiency has been based on the power carried by the forward wave only. One may, however, ask about the actual power A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 363 6.0 5.5 5.0 4.5 4.0 3.5 EFFI. C 3.0 (SAT.) 2.5 2.0 1 .5 1.0 0.5 1 QC = 0.2 1 1 A- — K=2.5 \ SMALI " *^r^ \ 1 Sa y^ 1 1 \ _/^ I Ji A 1 /^ \ 1 \ / \ t \ / \ \ ^ / \ \ X f \ ( \ \ \ \ ^ ' \ \ C = 0.1 '\ \ \ \ , lyj \ \ \ JT"^ \ \ V C=0.15 \ \ \ \ \ \ >"1 = 1 AX) /"1=C K 1 1 ).94/Z.(MAX) .at^ / 1 \^ /t/i = 0.67//i(MAX) \ //, = 0.3//i(MAX) 0.5 1.0 1.5 b 2.0 2.5 3.0 Fig. 2 — The saturation eff./C versus fe, for k = 2.5, QC = 0.2, and C = small, 0.1 and 0.15. The dotted curves indicate the values of h for m = \, 0.94, 0.67, and 0.3 of ;ui(max) respectively. output in the presence of the backward wave. For simphcity, we shall use the approximate solution (12b) which can be written in the form B{z, t) ^ Real Component of ZqIq c 4C 2(1 + hC) dax(y)Y ^ (da,{y)\- j^^-v,.-,y+j^\ (12d) with tan ^ = dij (laiiyT dy , dy dchiyY dy , As mentioned earlier that the complete solution of (6) is obtained by adding to (12b) a complementary function such that -yu 1+ r Qz ZqIq + c 4C 2(1 + bC) dy:) ^\dy ) ' -hy+ji (25) 364 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 EFFI. c (SAT.) 3 QC = o.2 C = 0.1 J<_=K25. 3- 2.50 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 b Fig. 3 — The saturation eff./C versus b, for QC = 0.2 C = 0.1 and k = 1.25 and 2.50. If the output circuit is matched by cold measurements, the backward wave must be zero at the output end, z = D. This determines Ci , that is, „ ZqIq c ^1 = ~^rPT or Cie jut+Toz 4C 2(1 + bC) ZqIq C //dai(t/)Y I /da2{y)Y ro(2+bc)D+ji dai{y)V /da2{y)y 4C 2(1 + 6C) y \ dy )z=o \ dy Jz^d (26) The backward wave therefore consists of two components. One compo- o 7 (a) C = SMALL ^- Ml = 0.67 /U,(MAX) D 5 ^^;;:^ ^^ EFFI. C 4 /U, = 0.94//i(MAX) ^^^ 1 (SAT.) ^ 2 ■"Zr^AtlC^AX) " 0 (b) C = o.i = 0.94//, (MAX) 1 Xj fea,^_^-VZi = 0.67 /Z, (MAX) >U, = /i|(MAX)- 1 —===3 ^^^ 0.1 0.2 QC 0.3 0.4 0 0.1 0.2 QC 0.3 0.4 Fig. 4 — The saturation eff./C versus QC for h corresponding jui = 1, 0.94 and 0.67 of Mi(max), (a) for C = small, (b) for C = 0.1. A LARGE SIGNAL THEORY OF TRAVELING- WAVE AMPLIFIERS 365 nent is coupled to the beam and has an amplitude equal to Zolo C IC 2(1 + bC) VX^'Y + K^y / \dy) which generally grows with the forward wave. It thus has a much larger amplitude at the output end than at the input end. The other component is a wave of constant amplitude, which travels in the direction opposite to the electron flow with a phase velocity equal to that of the cold cir- cuit. At the output end, 2 = Z), both components have the same ampli- tude but are opposite in sign. One thus realizes that there exists a re- flected wave of noticeable amplitude, in the form of (26), even though the output circuit is properly matched by cold measurements. Under j such circumstances, the voltage at the output end is the voltage of the forward wave and the power output is the power carried by the forward wave only. This is computed in (23). Since (26) is a cold circuit wave it may be eliminated by properly ad- c[-w], ■C[w], 5.0 4.5 4.0 3.5 5 3.0 2 o 9- 2.5 1.5 1.0 0.5 (a) ; / y / / ( r' ,.--- ( L"1 .-'■ (b) j^ V / / / y 1 Qw '"--^ ^-"^ (c) J i / / ^ / / ,''^ 1 / ( f r 1 1 1 1 < f 0.1 0.2 0.3 0.4 0.5 1.0 QC 1.5 2.0 2.5 0 b 0.05 0.10 0.15 0.20 Fig. 5 — Cw(y, r\ / ^ \ / \ CASE 10 QC = 0.2 C = 0.1 b = 0.875 k = 2.5 MAXC(-W) / ,-' ''s \ \ /A(y) \ \ , 1 1 / /\ S // y / X- ./ ,^ "^AXCW i / / / y r ■7 / / A y ^-' ^ ^ :z=^ — ** — ^ 1.4 1.2 1.0 0.8 ID < 0.6 0.4 0.2 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 y Fig. 6 — Cw{y, (pa) of the fast and the slowest electrons versus y for Case (10). A{y) is also plotted in dotted lines for reference. justing the impedance of the output circuit. This may be necessary in practice for the purpose of avoiding possible regenerative oscillation. In doing so, the voltage at 2 = D is the sum of the voltage of the forward wave and that of the particular solution of the backward wave. In every case, the output power is always equal to the square of the net voltage actually at the output end divided by the impedance of the output cir- cuit. We find from (14), (15) and (16) that the fundamental component of electron charge density may be written as f s. \ h ( . dai{y) . da2(y)\ = Real component of 1/0 dai{y) dy , + doM dy (26) jo)—Toz—by+Ji ) where —Io/uq is the dc electron charge density, po . If (26) is compared with (12d) or (12c), it might seem surprising that the particular solution of the backward wave is just equal to the funda- A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 367 1.6 t.5 1.4 1.3 1.2 1.1 1.0 Pq 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.2 1.1 1.0 0.9 Pq 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 CASES 2, 10,19 (a) k = 2.5 - C = 0.1 b-»MAX n^ /' \ r \ J ' ^ r \ r\ \ / f V 1] s — QC=o.i/ / f 0.2 // r 0.4 // 1 V // \ 7 \ <^ L / \ A \ ^ ^ CASES 9, 0,14 (c) QC=0.2 - k=2.5 b-»MAX//| rv -\ 1 u \ r C=o,s/// \ / rf-o 10 \ ///o.05 i II k A f / ^ 8 0 4 y CASES 1C ,11,12 iH r V r\ (b) QC = o.2 C = o.i k = 2.5 r k\ (A \ // \ / y ^ c \ / \f A \ \ /^, = >U,MAx/^ ' / A / \ // / \ / f \ \ A / I / J /09« / 11 y / \ \ . // / / 11 / \J 17 // Ai.^i ^^ "w 1 /' y ^^ .^^ -^ >^ '^1 = 0.3X/,MAX 1 1 7 8 y 10 11 12 13 14 15 Fig. 7(a) — p^/po versus ?/, (a) using QC as the parameter, for A; = 2.5, C = 0.1, and 6 for maximum small-signal gain (Cases 2, 10, and 19) ; (b) using h as the param- eter, for k = 2.50, C = 0.1 and QC = 0.2 (Cases 10, 11, 12 and 13); and (c) using C as the parameter, for k = 2.50, QC = 0.2 and h for maximum small-signal gain (Cases 9, 10 and 14). 368 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 mental component of the electron charge density of the beam multiplied by a constant / Zq/o C 2uo 2wo\ h) (27) V 4C 2(1 + hC) The ratio of the electron charge density to the average charge density, P«(2) Po 2319^21 5 17/^,1 9 ^ +e Fig. 8(a) — y versus 1 1 \ \ 1 j i r 1 ll i i i5!r 1 ti23l 11 1 ,3 _ -1 9 in L. J 3 15 17 ig/sii 1 i i 23 -10 -9 -8 -7 -4 -3 0 1 10 Fig. 8(c) — y versus

^ \ — p \ \ \ \ v\ I :: ''^: N\ "t ^ :\ \ \ -^^ K\l 1 H. '■.-■; 1 1 1 Ui y V t 1 J. 1 ^W ^\ / / )) r i t l-^i \V d \ r // 1 ( 1 i : 1 \\ ; 1 / i w \\ , :' ■,J; W I 1 1 1 — p- rl - 1 \ / ] \ -— - 1 1 1 1 \\\ 1 % 1 1 1 1 1 1 1 t n , 9 1 1 1 15; f ,'21 123 / !l V' 3 15 17 jl9 21 23 -1 0 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 0 1 : 3 3 - i 5 6 7 3 9 y-by Fig. 8(e) — y versus

(A-5) dy where p* is the conjugate of p. After substituting (A-1) to (A-5) into the working equations (15) to (18) and carrying out considerable algebraic work, we obtain exactly Pierce's equation. 2 (1 + jC/i)(l + bC) innn \ ah \^ r\ r\ (j - >iCfi -h j}/ibC)(ti + jb) provided that + CO —k\((>(.y ,)—)] 0 (A-7) • di^ sgn (^(?/, .i?o + «/)) - 9?(^, ') and Vsdr, r) be the three potential solutions where: (1) Vaif, r) is the solution for the case of no space charge with Ai and cathode at zero potential and A 2 at potential C, (2) Vb{r^ r) is the solution for the case of no space charge with A2 * Verbal disclosure. BEAM FORMATION WITH ELECTRON GUNS 381 and cathode at zero potential and Ai at potential B, and (3) Vsc(f, r) is the soUition when space charge is present but when Ax , A^ , and cathode are all grounded. If the configuration of charge which contributes to Vs<-(f, r) is that corresponding to ideal Pierce type flow, then we can use the principle of superposition to give the Langmuir potential, VL(r, r): VUr, r) = Vcif, r) + V,{f, r) + V..{f, r) (2) Furthermore, the potential configuration for the case where ^i and A2 are at potentical C can be written V =V.-\-^V, + F(.c)' (3) where the functional notation has been dropped and F(sc)' is the po icntial due to the new space charge when Ai and A2 are grounded. We are now ready to use the fact that F(sc)' may be well approximated 1)3' Fsc which is easily obtained from (2). This substitution may be justified by noting that the space charge distribution in a gun using a \'oltage C for Ai does not differ significanth^ from the corresponding dis- tribution when Ai is at voltage B except in the region near and beyond A-i where the charge density is small anyway (because of the high electron velocities there). Substituting Fsc as given by (2) for F(sc)' in (3) then gives V Vi 1 B, V, (4) We have thus obtained an expression, (4), for the potential at an arbi- ANODE A2 v=c ANODE A, V = B CATHODE Fig. 1(a) — ■ Electrode configuration for anode lens evaluation in Section 2>A. 382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ■i trary point in our gun in terms of the well known solution for space charge limited flow between two concentric spheres, Vl , and a potential distribution, Vb , which does not depend on space charge and can there- fore be obtained in the electrolytic tank. Once the potential distribution is found, electron trajectories may be calculated, and an equivalent lens sj^stem found. Equation (4) is used in this way in Part C as one basis for estimating a correction to the Davisson equation. (It will be noted that i (4) predicts a small but finite negative field at the cathode. This is be- cause the space charge density associated with Fsc is slightly greater near the cathode than that associated with F(sc)' , and it is this latter space charge which will make the field zero at the cathode under real space charge limited operation. Equation (4), as applied in Part C of this section, is used to give the voltage as a function of position at all points except near the cathode where the voltage curves are extended smoothly to make the field at the cathode vanish.) B. Use of a False Cathode in Treating the Anode Lens Problem Before evaluating the lens effect by use of (4), it will be useful to de- velop another approach which is a little simpler. The evaluation of the lens effect predicted by both methods will then be pursued in Part C where the separate results are compared. In Part A we noted that no serious error is made in neglecting the dif- ference between the two space charge configurations considered there because these differences were mainly in the very low space charge region near and beyond A2 . It similarly follows that we can, with only 1 a small decrease in accuracy, ignore the space charge in the region near and beyond A2 so long as we properly account for the effect of the high space charge regions closer to the cathode. To place the foregoing obser- vations on a more quantitative basis, we may graph the Langmuir po- tential (for space charge limited flow between concentric spheres) versus the distance from cathode toward anode, and then superpose a plot of the potential from LaPlace's equation (concentric spheres; no space charge) which will have the same value and slope at the anode. The La- Place curve will depart significantly from the Langmuir in the region of the cathode, but will adequately represent it farther out." Our experi- ence has shown that the representation is "adequate" until the difference between the two potentials exceeds about 2 per cent of the anode voltage. Then, since space charge is not important in the region near the anode for the case of a gridded Pierce gun, corresponding to space charge limited flow between concentric spheres, it can be expected to be similarly unimportant for cases where the grid is replaced by an aperture. Let us I BEAM FORMATION WITH ELECTRON GUNS 383 therefore consider a case where electrons are emitted perpendicularly and with finite velocity from what would be an appropriate spherical equipotential between cathode and anode in a Pierce type gun. So long as (a) there is good agreement between the LaPlace and Langmuir curves at this artificial cathode and (b) the distance from this artificial cathode to the anode hole is somewhat greater than the hole diameter, we will liiid that the divergent effect of the anode hole will be very nearly the same in this concocted space charge free case as in the actual case where space charge is present. (The quantitative support for this last state- ment comes largely from the agreement between calculations based on this method and calculations by method A.) The electrode configura- tion is shown in Fig. 1(b), and the potential distribution in this space charge free anode region can now be easily obtained in the electrolytic j tank. This potential distribution will be used in the next section to pro- ^•ide a second basis for estimating a correction to the Davisson equation. C. Calculation of Anode Lens Strength by the Two Methods The Davisson equation, (1), may be derived by assuming that none of the electric field lines which originate on charges in the cathode-anode region leave the beam before reaching the ideal anode plane where the voltage is F, and that all of these field lines leave the beam symmetrically and radially in the immediate neighborhood of the anode. Electrons I are thus considered to travel in a straight line from cathode to anode, and then to receive a sudden radial impulse as they cross radially diverg- ing electric field lines at the anode plane. A discontinuous change in CATHODE ANODE A2 V = C ANODE A, v = c (b) ^ FALSE CATHODE Fig. 1(b) — The introduction of a false cathode at the appropriate potential lUows the effect of space charge on the potential near the anode hole to be satis- :ictorily approximated as discussed in Section 3i?. 384 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 slope is therefore produced as is common to all thin lens approximations. The diverging effect of electric field lines which originate on charges which have passed the anode plane is then normally accounted for by the universal beam spread curve/" In our attempt to evaluate the lens effect more accurately, we will still depend upon using the universal beam spread curve in the region following the lens and on treating the ; equivalent anode lens as thin. Consequently our improved accuracy must come from a mathematical treatment which allows the electric field lines originating in the cathode-anode region to leave the beam grad- ually, rather than a treatment where all of these flux lines leave the beam , at the anode plane. In practice the measured perveances, P(= I/V^'^), of active guns of the type considered here have averaged within 1 or 2 per cent of those predicted for corresponding gridded Pierce guns. There- fore the total space charge between cathode and anode is much the same with and without the use of a grid, even though the charge dis- tribution is not the same in the two cases. The total flux which must leave our beam is therefore the same as that which will leave the cor- , responding idealized beam and we may write yp = I EndA = TT/VFidea/ (5) w^here En is the electric field normal to the edge of the beam, ra = rdfa/fc) is the beam radius at the anode lens, and Videai is the magnitude of the field at the corresponding gridded Pierce gun anode. To find the appropriate thin lens focal length we will now find the total integrated transverse impulse which would be given to an elec- tron which follows a straight-line path on both sides of the lens (see Fig. 2), and we will equate this impulse to wAw where An is the transverse velocity given to the electron as it passes through the equivalent thin lens. In this connection we will restrict our attention to paraxial elec- trons and evaluate the transverse electric fields from (4) and from the tank plot outlined in Section B, respectively. The total transverse im- pulse experienced by an electron can be written f Fn dt = e [ —dl (()) J Path J Path U where u is the velocity along the path and Fn is the force normal to the path. We will usually find that the correction to (1) is less than about 20 per cent. It will therefore be worthwhile to put (6) in a form which in effect allows us to calculate deviaiions from Fu as given by (1) instead BEAM FORMATION WITH ELECTRON GUNS 385 1 of deriving a completely new expression for F. In accomplishing this piir- f pose, it will be helpful to define a dimensionless function of radius, 6, by - = 1 + 5, r and a dimensionless function of voltage, f, by (7a) (7b) where Ta is the radius at the anode lens when the lens is considered thin, and T^'x is a constant voltage to be specified later. (Note that the quan- tities 5 and f are not necessarily small compared to 1.) Using u = \/2r]V, and substituting for -y/V from (7b) we obtain f En dl 4 r , = 7~7tW / ^"^1 + r + 5 + rs) ^z (8) where use has also been made of (7a) in the form 1 = r(l + d)/ra . Now, as outlined above, we equate this impulse to 771 An, and we obtain ^» = WW. (/ ''■'' '' + / ''"'■'^ + ' + ^'' 'i (9) CATHODE Fig. 2 — The heavy line represents an electron's path when the effect of the .•mode hole may be represented by a thin lens, and when space charge forces are iihsent in the region following the anode aperture. For paraxial electrons, the (negative) focal length is related to the indicated angles by (y = 0 + Ta/F). 386 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 CENTER OF ~~ CURVATURE OF CATHODE SURFACE Fig. 3 — The gun parameters used in Section SC for comparing two methods of evaluating the effect of the anode lens. The first integral can be obtained from (5) ; hence, if we are able to choose Vx so that the second integral vanishes, we may write: Au = raV'2riVx The reciprocal of the thin lens focal length is therefore i _ ^ _ ^' F ~ ~raUf ^ ~^VWf (10) where w/ and F/ are the final velocity and voltage of the electron after it leaves the lens region. The real task, then, is to use the potential distribution in the gun as obtained by the methods of Part A or Part B above to find the value of V X which causes the last integral in (9) to vanish : To compare the two focal lengths obtained by the methods of Part A and B respectively, a specific tank design of the type indicated in Fig. 1 was carried out. The relevant gun parameters are indicated in Fig. 3. Approximate voltages on and near the beam axis were obtained as indicated in Parts A and B, above, with the exception that in the superposition method, A, special techniques were used to subtract the effect of the space charge lying in the post-anode region (because the effect of this space charge is accounted for separately as a divergent force in the drift region*). From these data, * See Section 4B. BEAM FOKMATION WITH ELECTRON GUNS 387 800 805 810 815 820 825 830 835 840 845 850 855 860 Fig. 4 — Curves for finding the value of Fx to be used in equation (10) for the set of gun parameters of Fig. 3. l)oth the direction and magnitude of the total electric field near the beam axis were (with much labor) determined. Once these data had been obtained, a trial value was selected for Vx , and the corresponding local length was calculated by (10). This enabled the electron's path through the associated thin lens to be specified so that, at this point in the procedure, both r and V were known functions of ^, and the quan- tities 8 and f were then obtained as functions of € from (7). Finally the second integral in (9) was evaluated for the particular Vx chosen, and then the process was repeated for other values of Vx . Fig. 4 shows curves whose ordinates are proportional to this second integral and whose abscissae are trial values for Vx . As noted above, the appropriate value for Vx is that value which makes the ordinate vanish, so that we obtain T'c = 813 and 839 for methods A and B, respectively. The percentage difference in the focal lengths obtained by the two methods is thus only 1 .6 per cent, and the reasonableness of making calculations as outlined in Part B is thus put on a more quantitative basis. Even calculations based on the method of Part B are tedious, and we naturally look for simpler methods of estimating the lens effect. In this fonnection we have found that Vx is usually well approximated by the \alue of the potential at the point of intersection between the beam axis and the ideal anode sphere. The specific values of the potential at this point as obtained by the methods of Parts A and B were 814 and 827, respectively. It will be noted that these values agree remarkably well with the values obtained above. Furthermore, very little extra effort is required to obtain the potential at this intersection in the false cathode case: I 388 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Electrolytic tank measurements are normally made in the cathode- anode region to give the potential variation along the outside edge of the electron beam (for comparison with the Langmuir potential) ; hence, by tracing out a suitable equipotential line, the shape of the false cathode can easily be obtained. With the false cathode in place and at the proper potential, the approximate value for Vx is then obtained by a direct tank measurement of the potential at an axial point whose distance from the true cathode center is (fc — fa) as outlined above. Although finite elec- tron emission velocities typically do not much influence the trajectory of an electron at the anode, they do nevertheless significantly alter the beam in the region beyond. It is in this affected region where experi- mental data can be conveniently taken. We must, therefore, postpone a comparison of lens theory with experiment until the effect of thermal velocities has been treated. At that time theoretical predictions com- bining the effects of both thermal velocities and the anode lens can be made and compared with experiment. Such a comparison is made in Section 6. 4. TREATMENT OF BEAM SPREADING, INCLUDING THE EFFECT OF THERMAL ELECTRONS Jn Section 2 the desirability of having an approach to the thermal spreading of a beam which would be applicable under a wide variety of conditions was stressed. In particular, there was a need to extend ther- mal velocity calculations to include the effects of thermal velocities even when electrons with high average transverse velocities perturb the beam size by as much as 100 or 200 per cent. Furthermore, a realistic mathe- matical description which would allow electrons to cross the axis seemed essential. The method described below is intended adequately to answer these requirements. A. The Gun Region The Hines-Cutler method of including the effect of thermal velocities on beam size and shape leads one to conclude that, for usual anode voltages and gun perveance, the beam density profile in the plane of the anode hole is not appreciably altered by thermal velocities of emis- sion. (This statement will be verified and put on a more quantitative basis below.) Under these conditions, the beam at the anode is ade- quately described by the Hines-Cutler treatment. We will therefore find it convenient to adopt their notation where possible, and it will be worthwhile to review their approach to the thermal problem. BEAM FORMATION WITH ELECTRON GUNS 389 It is assumed that electrons are emitted from the cathode of a therm- ionic gun with a IMaxwelhan distribution of transverse velocities ZTTfC 1 where Jc is the cathode current density in the z direction, T is the cath- jode temperature, and v^: and Vy are transverse velocities. The number iof electrons emitted per second with radially directed voltages between V and V + dV is then -(.Ve/kT) (S) ^J. = /.e— -^^^(^^j (12) Now in the accelerating region of an ideal Pierce gun (and more generally I in any beam exhibiting laminar flow and having constant current density ()\'er its cross section) the electric field component perpendicular to the axis of symmetry must vary linearly with radius. Conseciuently Hines and Cutler measure radial position in the electron beam as a fraction, ^, of the outer beam radius (re) at the same longitudinal position, r = fire (13) The laminar flow assumption for constant current densities and small beam angles implies a radius of curvature for laminar electrons which so varies linearly with radius at any given cross section so that a Substituting for r from (13), (14) becomes rfV , /2 dre\ dfj. d^^VcTt)dt=^ ^^^^ where Ve and dr /dt can be easily obtained from the ideal Langmuir solution. Since the eciuation is linear in /x, we are assured that the radial position of a non-ideal electron that is emitted with finite transverse velocity from the cathode center (where ^ = 0) will, at any axial point, be proportional to dii/dt at the cathode. Let us now define a quantity "o-" such that n = a/re is the solution to (15) with the boundary conditions /Xr = 0 and _ 1 where the subscript c denotes evaluation at the cathode surface, k is 390 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Boltzman's constant, T is the cathode temperature in degrees Kelvin, and m is mass of the electron. For the case ixc = 0, but with arbitrary initial transverse velocity, we will then have /^\ ^^nl_ /kf ^^^'^ Tc y m Plence we can express a in terms of the thermal electron's radial po- sition (r), and its initial transverse velocity, Vc , y m _ y . . - . /kT dt } f The quantity a can now be related to the radial spread of thermal electrons (emitted from a given point on the cathode) with respect to an electron with no initial velocity: By (11) we see that the number of electrons leaving the cathode with dji/dt = Vc/ve is proportional to Vc exp —Vcm/2kT. Suppose many experiments were conducted where all electrons except one at the cathode center had zero emission velocity, and suppose the number of times the initial transverse velocity of the single thermal electron were chosen as Vc , is proportional to Vc exp — Vcm/2kT. Then the probability, P{r), that the thermal electron would have a radial position between r and r -\- dr when it arrived at the transverse plane of interest would be proportional to Vc exp —Vc^(m/2kT). Here Vc is the proper transverse velocity to cause arrival at radius r, and by (17) we have a y m so that the probability becomes Pir) = J.e-^^'''-'^ d (^Q (18) We therefore identify cr with the standard deviation in a normal or Gaussian distribution of points in two dimensions. At the real cathod(\ thermal electrons are simultaneously being emitted from the cathode surface with a range of transverse velocities. However, if a as definml above is small in comparison with r,. , the forces experienced by a ther- mal electron when other thermal electrons are present will be very nearly BEAM FORMATION WITH ELECTRON GUNS 391 2.0 1.8 1.6 > 1.4 t 1.2 \%y 1.0 0.8 0.6 0.4 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 Fig. 5 — Curves useful in finding the transverse displacement of electron tra- i jectories at the anode of Pierce-type guns. i tlie same as the forces involved in the equations above. Thus if o- <3C J'e , (18) may be taken as the distribution, in a transverse plane, of those electrons which were simultaneously emitted at the cathode center. I Furthermore, the nature of the Pierce gun region is such that electrons emitted from any other point on the cathode will be similarly distributed \\ ith respect to the path of an electron emitted from this other point w ith zero transverse velocity (so long as they stay within the confines , of the ideal beam). Hines and Cutler have integrated (15) with n^ = 0 ' and {dn/dt)c = 1 to give g/ {fc\/kT/'2eV^ at the anode as a function of ; /", /fo . This relationship is included here in graphical form as Fig. 5. , For a large class of magnetically shielded Pierce-type electron guns, including all that are now used in our traveling wave tubes, Ve/a at the anode is indeed found to be greater than 5 (in most cases, greater than 10) so that evaluation of a at the anode of such guns can be made with considerable accuracy by the methods outlined above. One source of error lies in the assumption that electrons which are emitted from a point at the cathode edge become normally distributed about the cor- responding non-thermal (no transverse velocity of emission) electron's path, and with the same standard deviation as calculated for electrons from the cathode center. In the gun region where Ve/a tends to be large this difference between representative a- values for the peripheral and central parts of the beam is unimportant, but it must be re-examined in tlie drift region following the anode. 392 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 We have already investigated the region of the anode hole in some detail in Section 3 and have found it worth while to modify the ideal Davisson expression for focal length of an equivalent anode lens. In particular, let us define a quantity F by F = focal length = Fd/T (19) where Fd is the Davisson focal length. Thus T represents a corrective factor to be applied to Fd to give a more accurate value for the focal length. In so far as any thin lens is capable of describing the effects of diverging fields in the anode region, we may then use the appropriate optical formulas to transfer our knowledge of the electron trajectories (calculated in the anode region as outlined above) to the start of the drift region. In particular, -f (20) where {dr/dz)i and {dr/dz)^ are the slopes of the path just before and just after the lens, and r is the distance from the axis to the point where the ideal path crosses the lens plane. B. The Drift Region Although Te/a- was found to be large at the anode plane for most guns of interest, this ratio often shrinks to 1 or less at an axial distance of only a few beam diameters from the lens. Therefore, the assumption that electron trajectories may be found by using the space charge forces which would exist in the absence of thermal velocities of emission (i.e., forces consistant with the universal beam spread curve) may lead to very appreciable error. For example, if ecjual normal (Gaussian) distributions of points about a central point are superposed so that the central points are equally dense throughout a circle of radius Te , and if the standard de- viation for each of the normal distributions is cr = r^ , the relative density of points in the center of the circle is only about 39 per cent of what it would be Avith a < (re/5). In order to minimize errors of this type we have modified the Hines- Cutler treatment of the drift space in two ways: (1) The forces influenc- ing the trajectories of the non- thermal electrons are calculated from a progressive estimation of the actual space charge configuration as modi- fied by the presence of thermal electrons. (2) Some account is taken of the fact that, as the space charge density in the beam becomes less uni- form as a function of radius, the spread of electrons near the center of the beam increases more rapidly than does the corresponding spread BEAM FORMATION WITH ELECTRON GUNS 393 farther out. Since item (1) is influenced by item (2), the specific as- sumptions involved in the latter case will be treated first. When current density is uniform across the beam and its cross section changes slowly with distance, considerations of the type outlined above for the gun region show that those thermal electrons which remain within the beam will continue to have a Gaussian distribution with re- spect to a non-thermal electron emitted from the same cathode point. When current density is not uniform over the cross section, we would still like to preserve the mathematical simplicity of obtaining the current density as a function of beam radius merely by superposing Gaussian distributions which can be associated with each non-thermal electron. To lessen the error involved in this simplified approach, we will arrive at a value for the standard deviation, a (which specifies the Gaussian distribution), in a rather special way. In particular, a at any axial po- sition, z, will be taken as the radial coordinate of an electron emitted from the center of the cathode with a transverse velocity of emission given by, ve = y- — (21) m It is clear from (17) that for such an electron, r = o- in the gun region. From (18), the fraction of the electrons from a common point on the cathode which will have r ^ a in the gun region is 2 fraction = [ e'^'-'"-''^ d ^= I - e'"' = 0.393 (22) If re denotes the radial position of the outermost non-thermal electron and if 0- > /■,, , the "a--electron" will be moving in a region where the space charge density is significantly lower than at the axis. We could, of course, have followed the path of an electron with initial velocity equal to say 0.1 or 10 times that given in (21) and called the correspond- nig radius O.lcr or lOo-. The reason for preferring (21) is that about 0.4 or nearly half of the thermal electrons emitted from a common cathode point will have wandered a distance less than a from the path of a non- thermal electron emitted from the same cathode point, while other thermal electrons will ha\'e wandered farther from this path; conse- quently, the current density in the region of the o--electron is expected to be a reasonable average on which beam spreading due to thermal \elocities may be based. With this understanding of how a is to be cal- culated, we can proceed to the calculation of non-thermal electron trajectories as suggested in item (1). 394 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 The non-thermal paths remain essentially laminar, and with r^ de- noting the radial coordinate of the outermost non-thermal electron, we will make little error in assuming that the current density of non-ther- mal electrons is constant for r < Ve . Consequently, if equal numbers of thermal electrons are assumed to be normally distributed about the cor- responding non-thermal paths, the longitudinal current density as a function of radius can be found in a straightforward way by using (18). The result is J ^ ^_(..,,..) n" R ^-(«^/2.^)^^ frR\ ^ /R\ ^23) Jd Jo a \a^/ \(t/ where /o is the zero order modified Bessel function and the total current is Id = TTVe Jd ' Equation (23) was integrated to give a plot of Jr/Jo versus r/a, with re/a as a parameter and is given as Fig. 6 in Reference 6. It is reproduced here as Fig. 6. Since the only forces acting on elec- trons in the drift region are due to space charge, we may write the equa- tion of motion as where Er is the radial electrical field acting on an electron with radial coordinate r. Since the beam is long and narrow, all electric lines of force may be considered to leave the beam radially so that Er is simpl}^ ob- tained from Gauss' law. Equation (24) therefore becomes -— = --^— / 2irp dr = -— ! — / ■ Iirr dr dt^ zireor Jo Zireor Jo \/2t]V a. (25) 2irenr Jo 27reor From (23) we note that the fraction of the total current within any radius depends only on fe/o- and j'/ct: :il dr ^ / J0')2irr ar / xo ,/o r = - = H-) f '- r.J(r)2.rdr ^''''° (2«) ' Jo ■•r I a C '^dV^]^Fr-j- \(X a t \ BEAM FORMATION WITH ELECTRON GUNS 395 Fig. 6 — Curves showing the current density variation with radius in a beam I which has been dispersed by thermal velocities. Here r« is the nominal beam radius, I r is the radius variable, and ^ Fr = 0.995/ ^ ^ /^ ^ ^ ^ /^ / rz ^ / >> / ^ z:^ ^ :^ / y. ^ /y 'A %: ^ ;^ ^ Xy '^. ^^ ^^ ^^ w i^ /^ oao^ ^ 1^ ^- oo^ = ^^ 10 re/0- Fig. 7 — Curves showing the fraction, Fr , of the total beam current to be found within any given radius in a beam dispersed by thermal velocities as in Fig. 6. consequently the continuous solution for r^ and r„ (= a) as one moves axially along the drifting beam involves the simultaneous solution of two equations : (fve d~a d^ KFr./re KFJa (28) BEAM FORMATION WITH ELECTRON GUNS 397 0.36 0.32 0.28 0.24 0.16 0.12 0.08 0.04 0 \ \ 1 \ \ \ \ V V. --- — ■ 8 10 12 14 16 I Fig. 8 — A curve showing the effect of a quantity related to the space charge • force (in the drift region) on a thermal electron with standard deviation a. (See 'equation 28.) which are related by the mutual dependence of Fr^ and Fa on re/a. F„ and Frjve are plotted in Figs. 8 and 9. We may summarize the treatment of the drift region, then, as follows: 1 (a) The input values of r^ and rgJ at the entrance to the anode lens jare obtained from the Pierce gun parameters r^ and 6, while the value of a and aJ at the lens entrance can be obtained as mentioned above by integrating (15) from the cathode, where Mc = 0 and (dfx/dt)c = 1, to the anode plane. (The minus subscripts on r' and a' indicate that these slopes are being evaluated on the gun side of the lens; a plus sub- script will be used to indicate evaluation on the drift region side of the lens.) The values of Ve and a on leaving the lens will of course be their entrance values in the drift region, and the effect of the lens on r/ and a' is simply found in terms of the anode lens correction factor T by use of (20). The value of a at the anode can be obtained from (17) if n is known there. In this regard, (15) can be integrated once to give = 1_/M dt " " r\dt)c{r,/r,y (29) 398 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 LL 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 . — — ■-^ \ X ^ \ I / \ \ J \ / / A 7 \ / \ / \ \ \ \ \ \ \ \ f.,/(reA) \ \ \ s ^, > "-. '"-.. ^-*^^ ■•—.^ 1/ 1 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.20 0.18 1? 0.1 6 0.14 0.12 0.10 0.08 0.06 0.04 0.02 6 7 8 9 10 11 12 13 14 Fig. 9 — Showing quantities related to the effect of the space charge force in the drift region on the outermost non-thermal electron. (See equation 28.) i We can now substitute for transit time in terms of distance and Lang- muir's well known potential function/^ —a. The value of this parameter, for the case of spherical cathode-anode geometry in which we are in- terested, depends only on the ratio fe/f which is equal to Vc/rg . (Because of their frerjuent use in gun design, certain functions of —a are included here as Table I.) In terms of —a, then, the potential in the gun region BEAM FORMATION WITH ELECTRON GUNS 399 Fable I Table of Functions of —a Often Used in Electron Gun Design fc/f (-«)2 (- a)V3 (- a)2/3 difc/r) 1.0 0.0000 0.0000 0.0000 0.0000 1.025 0.0006 0.0074 1.05 0.0024 0.0179 0.134 1.075 0.0052 0.0306 0.173 1.10 0.0096 0.0452 0.212 1.392 0.590 1.15 0.0213 0.0768 0.277 1.20 0.0372 0.1114 0.334 1.767 0.716 1.25 0.0571 0.1483 0.385 1.30 0.0809 0.1870 0.432 2.031 0.790 1.35 0.1084 0.2273 0.476 1.40 0.1396 0.2691 0.519 2.243 0.874 1.45 0.1740 0.3117 0.558 1.50 0.2118 0.3553 0.596 2.423 0.886 1.60 0.2968 0.4450 0.667 2.583 0.915 1.70 0.394 0.5374 0.733 2.725 0.939 1.80 0.502 0.6316 0.795 2.855 0.954 1.90 0.621 0.7279 0.853 2.975 0.970 2.00 0.750 0.8255 0.908 3.087 0.982 2.10 0.888 0.9239 0.961 3.192 0.993 2.20 1.036 1.024 1.012 3.292 1.003 2.30 1.193 1.125 1.061 3.388 1.012 2.40 1.358 1.226 1.107 3.481 1.020 2.50 1.531 1.328 1.152 3.570 1.028 2.60 1.712 1.431 1.196 3.655 1.034 2.70 1.901 1.535 1.239 3.738 1.039 2.80 2.098 1.639 1.280 3.817 1.044 2.90 2.302 1.743 1.320 3.894 1.048 3.00 2.512 1.848 1.359 3.968 1.052 3.1 2.729 1.953 1.397 4.040 1.056 3.2 2.954 2.059 1.435 4.111 1.059 3.3 3.185 2.164 1.471 4.180 1.062 3.4 3.421 2.270 1.507 4.247 1.064 3.5 3.664 2.376 1.541 4.315 1.066 3.6 3.913 2.483 1.576 4.377 1.068 3.7 4.168 2.590 1.609 4.441 1.070 3.8 4.429 2.697 1.642 4.501 1.072 3.9 4.696 2.804 1.674 4.563 1.074 4.0 4.968 2.912 1.706 4.621 1.076 400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 may be written df df i-aaY'^ dt 'V2nV a/2;^ {-a) 2/3 (30) (31) (32) so that upon substitution from (29) and (31), (17) becomes Fig. 5, which has been referred to above, shows O-a . /2eVa 'fcV 'kf- as a function of {fc/fa) as obtained from (32), and allows o-„ to be de- termined easily. Using (20), the value of re+' is given by / Tea , F -,.= -^^_,. = ,/_g-l) (33) where dg is the half-angle of the cathode (and hence the initial angle which the path of a non-thermal edge electron makes with the axis). We may write for 1/Fd 1 V fe /d(-aY"\ Fo 4F 4(-aa)^/VV\rf(fc/r-) 7a (34) In Fig. 10 we plot —falFr, as a function of fjfa for easy evaluation of re+' in (33). Taking the first derivative of (32) with respect to ^, we ob- tain an expression for aJ. Using this in conjunction with (20) and (34) we find 0-+ = Y (r<^i + C2) I (35) where cira d{fc/f) /3 and ^-i/f. ft -(-''"/ (-a)2/3_ ! Ci and C2 are plotted as functions of fc/fa in Fig. 11. (b) After choosing a specific value for r and evaluating K = rj/c/ . BEAM FORMATION WITH ELECTRON GUNS 401 Q LL lU I.O 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 \ \ V \ \ ~~~- ■--- 1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 40 rcAa Fig. 10 — Curve used in finding ?•«+', the direction of a nonthermal edge elec- tron as it enters the drift region. (See equation 33.) (27r€o(277 Fa) ''), (28) is integrated numerically using the BTL analog com- puter to obtain a and r^ as functions of axial distance along the beam, (c) Knowing a and Ve , other beam parameters such as current dis- tribution and the radius of the circle which would encompass a given percentage of the total current can be found from Figs. 6 and 7. X tvi U 20 15 10 5 0 -5 -10 -15 -20 -25 -30 POLYNOMIAL REPRESENTATION (ACCURATE WITHIN 2°/o) -OR c, & C2 ,'''' C, = 4.13 fc/ra + 2.67 C2 = 0.635(r^/faf-13.56 rc/fa + 19-33 , ,-' .' ' .-''' .'-' ,'-' \ ^^ -' ^^-' < ^v ■^ X ,.-' ^** ,^-' ''H '^ "^ ^ ^ ^^ \> ^ 20 18 16 14 12 rO O 10 X (J 8 2 0 10 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 tc/fa Fig. 11 — Curves used in evaluating o-+', the slope of the trajectory of a thermal electron with standard deviation a as it enters the drift region. (See equation 35.) 402 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 5. NUMERICAL DATA FOR ELECTRON GUN AND BEAM DESIGN A. Choice of Variables Except for a scaling parameter, the electrical characteristics of an ideal Pierce electron gun are completely determined when three param- eters are specified, e.g., fc/fa , perveance, and Va/T. Also, for the simp- lest case r is equal to 1 so that (since K depends only on gun perveance) in this case no additional parameter is needed. This implies that nor- malized values of ?-/, a, a', and K at the drift side of the anode lens are not independent. If, however, the value of F at the anode lens is taken as an additional variable, four parameters plus simple scaling are re- quired before complete predictions of beam characteristics can be made. In assembling analog computer data which would adequately cover values of fc/fa , perveance, and Va/T which are likely to be of interest to us in designing future guns, we chose to present the major part of our data with T fixed at 1.1. This has seemed to be a rather typical value for r, and by choosing a specific value we decrease the total number of significant variables from 4 to 3. (The effect of variations in T on the minimum radius which contains 95 per cent of the beam is, however, included in Fig. 16 for particular values of Va/T and perveance.) Al- though the boundary conditions for our mathematical description of the beam in a drift space are simplest when expressed in terms of Vg , r/, a and ct', we have attempted to make the results more usable by express- ing all derived parameters in terms of fc/fa , s/Va/T, and the perveance, P. B. Tabular Data The rather extensive data obtained from the analog computer for the r = 1.1 case and for practical ranges in perveance, Ve/T, and fc/fa are summarized in Tables IIA to E where the parameters r^ and a which specify the beam cross section are given as functions of axial distance from the anode plane. Some feeling for the decrease in accuracy to be expected as the distance from the anode plane increases can be obtained by reference to Section 6B where experiment and theory are compared over a range of this axial distance parameter. C. Graphical Data, Including Design Charts and Beam Profdes In typical cases, the designer of Pierce electron guns is much more concerned with the beam radius at the axial position where it is smallest (and in the axial position of this minimum) than he is in the general BEAM FORMATION WITH ELECTRON GUNS 403 jspreadiug of the beam with distance. This is true because, in microwave beam tubes, the beam from a magnetically shielded Pierce gun normally enters a strong axial magnetic field near a point where the radius is a minimum, so that magnetic focusing forces largely determine the beam's subsequent behavior. The analog computer data has therefore been re- processed to stress the dependence of the beam's minimum diameter and the corresponding axial position of the minimum on the basic design iparameters fdfa , perveance, and s/Va/T. As a first step in this direc- tion, the radius, rgs , of a circle which includes 95 per cent of the beam : I current is obtained as a function of axial position along the beam. Such idata are shown graphically in Fig. 12. Finally, the curves of Fig. 12 are . lused in conjunction with the tabular data to obtain the "Design Curves" of Fig. 13 where all of the pertinent information relating to the beam at its minimum diameter is presented. \D. Example of Gun Design Using Design Charts Assume that we desire an electron gun with the following properties : anode voltage Va = 1,080 volts, cathode current Ip = 7.1 ma, and mini- mum beam diameter 2(r95)min = 0.015 inches. Let us further assume a cathode temperature T = 1080° Kelvin, an available cathode emission density of 190 ma per square cm, and an anode lens correction factor of r = 1.1. From these data we find -x/Va/T = 1.0, perveance P = 0.2 X 10"^ amps/(volts)^''" and (r95)min/''c = 0.174. Reference to the de- sign chart, Fig. 13, now gives us the proper value for fc/fa : using the upper set of curves in the column for y/Va/T =1.0 we note the point of intersection between the horizontal line for {rgr^^i^/rc = 0.174 and the perveance line P = 0.2, and read the value of fc/fa (= 2.8) as the corresponding abscissa. The convergence angle of the gun, de , is now simply determined fi'om the equation^^ de = cos-^ {\ - t|^ X 10^) (37) {Qe is found to be 13.7° in this example) and the potential distribution in the region of the cathode can be obtained from (30). When this point has been reached, the gun design is complete except for the shapes of the beam forming electrode and the anode, which are determined with the aid of an electrolytic tank in the usual way. The radius of the anode hole which will give a specified transmission can be found by obtaining (re/a)a through the use of Fig. 5, and then choosing the anode radius from Fig. 7. In practical cases where (rf/a)a > 3.0, ^' O O z o I-H e o w o O CO a Q O 12; < O > m IS < Q o P a, o O o < o m PQ "5 d II > 1 I-H 1 II ■! ' o -1" bl>? ^H (M lO O C35 O M C^ t^ C^ GO GOt^cOCDiOcOCOr^GO'Cl^COCO (M C^l C^l C^) 'M C^J !M C^l C0^'*-rtoco GOiO^OOiOC^rt-tl^OO-liO (M U3 ,-H CO lO lO (M CO CD CO t^ COOC0t^O'f--H--HCO'-HG0'*'Q0 ^Hi — ^r-HOOOOOO' — ^' — ^1 — 1 1 II M 1 COCOC^i-h^OOOO^Ht-hiMOI J 1 1 1 1 1 «| a O(N^CD00O(M-*<£>Q0OC<) T— 1 1-H T— t T— 1 T-H C^ C^l 0(M^COGOOrt(N^cDO'*00 t-Ht-Ht-Hi-H^05(MIM q b|=? O IC O to iO lO cr. (M C-j t^ C^ C-j -f-fOiOcOt^QOOi— iiOOCS OCOI-^O-IOOt^ OOCSl COUO I- t^ l-^ CCi O) C~. ^ C^ C^ f 1^ O CD CO !MCM(MC^llMO-1COCOCOCOCO-*-flO O OOOOOOOOOOO OOO OOOOOOOOOOO >,"|b lOOiOOOC^OOiOCOOt^-HtO coocooooioco.-H^-rooc2 OiO(M t^COiCOC^tXCOl-- t^T— lCDT-HCDT-HCD^!MT-H-fCDT-HTt< ^,-,^^000 0 0000 1 1 1 1 COCOC^(M--HrtOOOOOO^^ 1 1 1 1 1 "k= OiM'Tt? t^iOOOOt— iiOOO>000 C^-t^CDGOOiM-t'l^OiMCOO ■*CO^(MCOGO^'fiM'*COO'-H COt^GOClO'-HCO-flOt^C^IOOCD (M(M(MC^C0C0C0COC0C0-*-rt-»O OOOOOOOOOOOO OOOOOOOOOOOOO -"lb OGOlO-*rtlcOClC^OOlO(M O^'^IMOOC'OiOCOC^^ coJCM(M q bk" CO 00 1-H CO O rH C5 rH -H04^k0t^0 coo lOiCiOiCiOO cOt^ dddodo dd COOiO(NOOOOOOCi(NcOCO(M lOt^GOO^COiCt^OiOT-HGO 0 iCCi 005 O5C0 t^c^ CO Tfi lO lO CO CO o o oo o OOt^OOOC^OO-*iMOoot^iOTt0 ooooooooooo 00 iC O (M O O lO lO lO o OlMOCOrCOCOiMiMGOCO 1 1 1 1 1 1 1 rt ^ ^ .— 1 C^5 C^l (M 00>OO00^^(MOiOC;»O-f dooooooooooo lO lO O (M O lO IC 1 1 1 1 1 1 1 T-lrHi-l.-H,-l(N(M(MCO lOQOCOOCiOC^lC^COiO'— 10? OJ I-Hi— I.— Ir-Hr-Hi— It— I,— IC^CO-^IOCO dooooooooooo o O COOOCOOCOCiOOt-^COt^ Tt< CO IM -* O «3 CO O 1^ IM lO t^ 00 r^^ccoi— i^^oooO' — ^1 — ^1 — ^^H I I I I I O^OClMCOTt^iOcOO'^COiMCO 1— ii— ii— irHi— itNoac^coco INO— <000(MO(NO'*COC^ CO-^iOCOOOO^COCiCOiOiO ,-ii-irtT-Hrt^c^c^c^co-r'O OOOOOOOOOOOO t^i^asrHt-c^cioooiOGOrt lO t^ rt 00 CO t^ O-l C-.' CO O --H CO I>»C^O00t^-*OO (N-*»000OC0«5O'O^00CC>^ 1-1.— iT-^,— (C^l(M(MCOCOTt<'*>OCO dodoooooooooo >— I IC "—I 00-*C^(MiCO-tiOt^iCC004^ t^CCHCThlCOC^C^CSIrtrt^rtr-l O-rt<0CCOOOOcC>»0 00<»COlO4OTt<^':fCOCOCO O-*i00(MCDO' fa o o ;z <1 1/3 PL, ^^ |x 1—1 go g II CL, o O o < 63 o fa > fa IS Ph fa o w. pq bi « ;ib ;ib b| e «h ^1 b wl b d > I^OOiOOOOt^t^COCAiOC^O ot^cocoio»oiocc!r^cx:ic:o COCOCOCOCOCOCOCOCOCOCO-* OOOOOOOOOOOO (MC0i0iO'*i0c000«5OC0l^ iOOOO«5(MrtiOco^C:C^iO Csj^T-HOOOOi-ifMiMCOCO I I I I I I I 0!M-1">OOt^CC'OiM-f»OCO OC^(M00(MC5iO>O-t»C»0(M^Ht^M<0OClC05C'COOCOCOCO dd dooooooooo O^COOOCOGCC^CO00iMC^C2t-^^HOO-^ t^GOOC^iOOO'— iiOO-^0»C COCO-t''^'^-^iOkOCOOt~-t~- oooooooooooo oioc^icor^t^ocococj-*!^ 0(MCC-rOt-~iOiMOOC^CO (MC^rtrti-iOOOOOOO O(NTt^cC>00O(M'titr>00O(M .— ( T— I ^H r-l ^H C^ C^ COO^';D(M-*'CiiOiOO COC2(M>CO-. COt^C^t^CO coco-^-^-^fiOioasot-- oooooooood lOI^-^tOlN'— iCOt^-^CO t^COOt^iOCO-— iC500I^ CaiMC^'— ii— ii— (1— lOOO OlM^O00O(M-*O00 b|^ OOi^OOiOcOOOCO iO00COt^ <~) <6 O CM II E~i a > o ''U" OOiCOOOiO(NOOiOiC?ib ooooooot^LOiooaiooiic >— iC^tMCOOIM'^OOCOOCM-^ Ot^-^fM^O^COCO-^-^iOiC 1 1 1 1 1 1 1 1 looooq— icooc-ieocc*^ 1 1 1 1 1 1 1 «k« . — ^ 1 — ^ 1— * 1 — 1 ^^ OlM'^'OCOt^OOOiM^cOCOO 1 — ^ ^H 1-H ^H 1 — ^ C^ bk« 00(M^OCO^'#C^CDI^(MOGOt^ CTsOiOOO^Cq^OOCOCiiOOcO OOO^^'-^rt'— irt(N»o o ^ c^) CO t-^ ^ CD ^ r^ CO c--' ^1— iC^(MC^C. OOOOOOOOOOOOOO ^1 b t^C2CO'fflt^iOC»-tiO'*co(M'-i^ododd III 'O^COOliM'— ii— i^OOOOO 1 1 1 >'!>? O? (M>OOC500t^OOO-fCO»0(MO'CiOiO»C>CiO coO'-Hcot^o^oo.-icocMot^co --lOCSOOt^t^cOCO'OiC'*^^ T— 1 1—1 "k"* O.-i(NTt O II a > o CO* II > o *>l^ ^■^iCOO^OiOiOO ^cococo^cc>o»o^ T— ti— It— irHi — ^^HC^^(^^co ooooooooo iMOOOO-HOiMlM'OOO t^coor^ooooc^cico OOOOOi-H^C^C-lCO oooooooooo ^Ib Mill lO lO i:iG0O(M OfN-^M^iOCDOOOlM OCO'^-'J^iCiOOOOlM^ 1-H 1-H r-H re H^ 00 QO 00 GCi CI (M I^ Ol t-iOO00^'*(M00 ■*'*'rri'#0«COO ^^^^(MOICO-^ d d d d d d d d rt^i-H^iCOOOOiOt^iM t^t^t^r^ooo-*c^ooi05 oooooi-Hi-Hc^icoco^ oooooooo OOOOOOOOOOO ^Ib •Ot^-^iOOOOOOS iccc-— iiooor^eo 1 1 1 1-Ht^OlOiOCOOO''— 1 t-^'rt^(Md'-iOiO iMcq'^.-Hc^oooicaiOi-H -^Ol^C^O^i-HIMiMfOCO 1 1 1 1 1 1 «l^ O(M^5O00O(M't< 1-H i-H 1— ( OIN^iOCOt^OOOiM'^CO 1-H 1-H 1-H 1-H H2 COQOiCOO^OOiOOO l^f^OOO'tiCsiO'^OO oooiooooor-Hococoo CO-*'^COt^OCOCOCi^ C0(MC0OiCt^02»0OOO CO^iOt^OiCOOt^t^t^OO .-H,— li-H,— lrtC^C^COTf0 oooooooooo OOOOOOOOOOO OOOOOOOOOOOOO ^1 b 03C^O--tt^-*0iO00f- 1000»0 lOiOOOlMO iOiCCO»OrtO^^OC OC' c<) ^ CO c; ^ -^ r- o -*< t^ J CO CO CO -Tf COCO^COO-^OCD-^CO-^COOO COt^OOO^C^COiOt^O^COiO OOOO^i-Hi-Hi-Hi-Hr-HC^OllM OOOOOOOOOOO OOOOOOOOOOOOO ^1 b O lO ^ t^ lO CO rt Ci -*COCO'-Hoooo.-H»oocDco'-Hocr- oor^cDioio-^-^-^cococo !OCO(M00500000I--t^l:^OcO 1-H 1 — 1 ^S 1-H "12 OC^-^COOOOOlTfi OtM-^COOOOlM-'t'COOOO f-H 1— t 1-H I-H r— 1 (M OIM^COOOO q l-H 11 s > o II E-. ! =3 > q bk« <£>00iMCD'-hOC^JOOO oo:c3c;'0'-Hcococ2!N CO r^ lO X' ^ CO to CO 00 o O O CI Cj Ol lO C CO t^ c^ ^OOO^^'-H(M(NC0 (MOXOcOOI^tOiOtC 1 lOtO-f-fl^iMCD^CO^ O O OOO T-H l-H (M (NCO ; o oooooooooo OOOOOOOOOO OOOOOOOOOO i g .rib Id O »0 lO t^ X X t^Xt^tOCi-^COCl^cO to tO-^-^COcDi-Ht^i-HCOiO o 1 1 1 1 1 1 C3COCOO^CO-+-fiOiO 1 1 1 1 1 1 C-. COt^^C^^-HiOiOlO ^^ 1 1 1 1 II «k" Oi-HIMCO-^iOCOt^XO O l-H (M CO -* to CD t^ X OJ o O i_J CO bk« OiC(MCOO-HiOOOO O CT. CI 0-. -H CO CO O -^ 00 C^4^rt^C^lC^(MCOC0CO OOOOOOOOOO OXcDCi-t'iO >0(MX o o c; o o l-H >o o TO c-j 1-hOOO' — ^1 — ^i^^hC^cO C5XO(M)Xi- OOOOOOr-Hrt(MiMr| OOOOOOOOOO ooooooooooq .?! b icr^t'-c^ioootDio t^COC;iMCO^t^(McO lO O (M OiOXi-HXCOCiXt^i-H O to to IOC" cocMXcoco-r^tocooi o^a.coT-HO(Neocoeoe4CSlt^cOCDt^^ -— I'-HOlC^liMCO-t'iOI^ C-jCi^iC»0-+i^O OO^H^HfMcOiOI^ X O CO(M --t< CO '^ to to X 1^ CO Ol c^ OOOO^COtot^ ' <6 cS <=) Analog Computer Data foe Perveance = 0.4 X 10~* ooooooooo OOOOOOOO ;?1 b OiOOiOOOiOiO^O >— iC^iOt^'— iCOl— i-:fCO iOCC'—iOOOi— '^-^i— 1 1 1 1 1 COCOC^iOcOOCO-* OCOCOOOi-Hi-Hi-H 1 1 1 1 o o to t^ to t^ O to X O O i O (N to l-H O O O rt ^^ 1 1 II "k' OC^'*iOCOCOOiM-r l-H l-H l-H OiM-to iC to OOi-HCOXt^O-*OX o^^rtrtC^^ior^x t^ o to O X to to "TfiOtOl^OCOCOOt^tO'' ooooo^c^-t^toix: OOOOOOOOOO ooooooooooe .-"lb (MOOOfflt^OOfNiO (^S«5COC^'*Or-(COTt< o II u 1 K bk= CO O CO ^ (M (M CO t^ (M CO 05 GO "O iM ^ C^ C^ CO CO -t< »o <~> <6i ^ Q Q <6> i^i XXC^OXOCOCOCOO XOCOtOXC^iOOCOX Or-Hrtrtr^C^CaC o T-H II 0 > o II > q bk« ^ OQ C' C-l CO -+ o o iO-t'^~vuril-j—i<:0'—i COCNO'-H»CO0 0 t^ I^ t^ t^ t^ 00 ^ t^ (>) 00 OOOOOOt-i,-i(M(M OOCOiO>OO^COiO»C COCOCOCOlOXi-*-— (00 OOOOOO^C^Ol OOOOOOOOO oooooooooo OOOOOOOOO .!lb O O lO C^^OC-.O-*'— il^iOO t^ t^ CD 1-1 05 i-H CD 00 lo Tt< r^ 1-1 im' CO lO lo ic ^^ 11,11 OCO(NOO(NCO'*>0 1 II 1 1 COl^^OOr-ico^iOCO 1 1 1 1 1 «I2 OT-Hi-KNC^eOTtHiCiCD »0 (N lOOOOO O.-irtiM(N(Ne0TjOCiCO(Mi— i^M OOOOOi-HC^COrfiO OOOOOOOOO OOOOOOOOOO OOOOOOOOOO K-lb Qccoco-r>oocccor- ^ lO lO 00 r^o-^ocot^'i^^ioco lOt^O-^t^i-iCOOi— iC-lCO COTfi^OOiMIMCOCO Mill COOOCOi— IOt— KNCOCOCO 1 1 M 1 1 t^CO^-IMOfNC^COCOCOCO ^^ 1 1 1 1 1 1 «l^ Or-l(^^(^^|^^TflOcD^^ O'-i(M(MC0C0'*i0c0t^ 0'-i(MC0 O O iO00 lOiCOO lOCDOOiMO r-((McOT}H t^(MT-HOOT-Hr-H(M OT-i(Ncoeo-*»OcDl:^ooc:i bk» itSCO CD OOCl C0»Ct^0:>'*O'X>00— 1 1-Hl— IrHi— lC^C«5-*CDO 1--CO COOOOO CDt^O(Nt^^t^CO OOt— irti— iCOiCOO ■<:t■ t^iCTtHCO(M(Mi— ii— (1— 1 »OOCD»C^COCO(M(M--H.-H «k" O(Me0TfU5cD00O(N 0(N-*iOCOl^OOOlM-*iCO I— ( j-H 1— t I— 1 II e bk= lO lO kC iC o o (M CO 1-1 t- -t^ cq .-1 .-1 0:i C^ fO Tt< o ooooo (N coo O lOiOCl CO 00 i-H '^^ t^ rt lO O O T-H --1 ^ C^ (M d c> d d d d d ^HC^cDi— lOlM'+'O CO^iCt^O^COCO OOOOOi— ii— ii— < OOOOOOOO K^b O lO 1— 1 O Oi O i-H O ^ "5 -^ CO im' CO J \ l\\ \ \^ \ y ^ ^V A \ \ \ \ \ . 'S ^ 1 ii f 10/ / 1 "v V ^// r ^0 y # \s \\ \ \; \ \ V \ > A. N \ ^ N M^ vl '^i 0 i ^ ); i ^ ^ t? N ^ II 01 lO oil fVi, ' m ^ (\l >? t>J o CO 6 d 5_0l X SOO=d o o (\J 00 '- 0 _0l X 10 = d 410 \ \ \ x^ \ N ^^ \ ""''^ '^ \ V/ ^ ^ U ^ / V \, -^ \ \ \ l1 Vv "^^^^\" fe o> = ^«^^-"~. '^^ k ^ ^ \ \ \ ^\ t^ .-^ \- °^ ^ ^ \ / ^ \ W ^ V \v. "0^ '■^ 3 k^ V ^ ^ 0-^ Ij^ M^ Jj w ^ ^ \J ^ ^r^^ ^ \ vs; ^. \ "A o <\J CO ^ o o o o (\J OO d d CO &-; Q t- CO > T) fl 'J c3 N I5~ 1^ o (\J ST ^^ — ' a) a 0) > Lh «1) OJ a a • p— 1 r^ CO o -t^ rt Si -il ^ !> ^ o %-. OJ o ^3 a o o CO .2 O -C CO CO > 3 o fci) • r-t fa 411 q > > \ \ 1 \ \ ^ \ . ^-- t^ ^ J^ IJ?£!\ I ^ tsl > V \ \ \ \ \"^ ^ L ^^ \ ^ ^V. K ^ C> ■- > ^ •i \& ^ ^N K\ ■^ ^ d II 1— ^ ^ V s. > ^ ^ 1^ \ y ~A^ K ^ > o N CO 9_01X 90:=c) 412 d |5- o d > C3 O • i-t 03 c3 > o OS c a faC o u 0) o o o3 3 CO > C o o3 o3 O -a EC a> > O O .— I bb BEAM FORMATION WITH ELECTRON GUNS 413 we find less than 1 per cent anode interception if anode hole radius = 0.93 r^a + 2o-a (38) Additional information about the axial position of (r95)min and the cur- rent density distribution in the corresponding transverse plane is con- tained in Fig. 13. The second set of curves in the \/Va/T = 1 column gives Zm\n/Tc — 2.42 for this example, so that we would predict Zmin = distance from anode to (r95)inin = 0.104'' The remaining 3''^ and 4*^^ sets of curves in the ■\/Va/T = 1 column allow us to find o- and re/a- at ^min . In particular we obtain a = 0.0029" and I'e/o = 0.8, and use Fig. 6 to give the current density distribution at 2min .* Section VI contains experimental data which indicate a some- what larger value for 2m in than that obtained here. However the pa- rameter of greatest importance, (r95)niin , is predicted with embarrassing precision. For those cases in which additional information is required about the beam shape at axial points other than ZnVin , the curves of Fig. 12 or the data of Table II may be used. 6. COMPARISON OF THEORY WITH EXPERIMENT In order to check the general suitability of the foregoing theory and the usefulness of the design charts obtained, several scaled-up versions of Pierce type electron guns, including the gun described in Section 5D, were assembled and placed in the double-aperture beam analyzer de- scribed in Reference 7. A. Measurement of Current Densities in the Beam Measurements of the current density distributions in several trans- verse planes near Smin were easily obtained with the aid of the beam analyzer. The resulting curve of relative current density versus radius at the experimental 2min is given in Fig. 14 for the gun of Section 52). (This curve is further discussed in Part C below.) For this case, as well as for all others, special precautions were taken to see that the gun was functioning properly : In addition to careful measurement of the size and position of all gun parts, these included the determination that the dis- tribution of transverse velocities at the center of the beam was smooth * When j'c/o- < 0.5, the current density distribution depends almost entirely on a, and, in only a minor way, on the ratio Te/a- so that in such cases this ratio need not be accurately known. q / / / / / / / A y V / i / ^ / y y / /a ^ ^ ^ 4 <^^ 1 1 1 / '/ // 1 / YfA/. h (^ Y // ^*^.. 00 :-5> ^^~5 ~^^^^r if'i^ — CO ID m ^ do o o o d d d d p n It] it- in IS- J d II u. o inlnj O > n / / V/ 1 2 < / / // // 1 / / A / / 1 D X 1 "^ 1 O / \ ^ 4 / ^ '^ ^ ^ /■ -f 1 1 1 3 CO ID in '3- (O (\J - d O O O o o / A / / // A y oy 1 / / fo Y / \ / ^^-^ ~~, 0j/S6(N,^j) - <0 o ^ i It'-' 2 m 11-. ' 4 « Tj- fO (M o- 414 / 1, V/ ' / / / // A / o 1 d /d ( \ V u 1 V \ \^ ^^ \ ll;.:: o CM d d d NIIAI iV CO o d o d o 1 J CM ^^ J f 00 6 'J- d o \ /§ ' If) (D O — d O CO ^ d /d is«p o CO (D lU •^ lii^ ry o NIW IV in ^ fO N I IN IV i?/^J fvi 415 416 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 12 11 10' z UJ o 5 UJ > ^ \PREDICTED ^■o-.. -IN \ measured"' ^ V 4 \ ^. N V \ k \ "V. \ \ } ^ ^.. P" ^-Gr.^ ==^ 01 23456789 RADIUS IN MILS Fig. 14 — Current density distribution in a transverse plane located where the 95 per cent radius is a minimum. The predicted and measured curves are normal- ized to contain the same total current. (The corresponding prediction from the universal beam spread curve would show a step function with a constant relative current density of 64.2 for r < 1.2 mils and zero beyond.) The gun parameters are given in Section 5D. and generally Gaussian in form, thereby indicating uniform cathode emission and proper boundary conditions at the edge of the beam near the cathode. The ejffect of positive ions on the beam shape was in every I case reduced to negligible proportions, either by using special pulse techniques, or by applying a small voltage gradient along the axis of the beam. B. Comparison of the Experiinentally Measured Spreading of a Beam with that Predicted Theoretically From the experimentally obtained plots of current density versus radius at several axial positions along the beam, we have obtained at each position (by integrating to find the total current within any radius) a value for the radius, rgs , of that circle which encompasses 95 per cent of the beam. For brevity, we call the resulting plots of rgs versus axial distance, "beam profiles". The experimental profile for the giui de- scribed in Section 5D is shown as curve A in Fig. 15(a). Curve B shows the profile as predicted by the methods of this paper and obtained from Fig. 12. Curve C is the corresponding profile which one obtains by the Hines-Cutler method, and Curve D represents Tq^ as obtained from the BEAM FORMATION WITH ELECTRON GUNS 417 CO 20 18 16 14 12 t- 8 2 0 I 50 45 40 35 if) 30 Z 25 l? 20 15 10 (a) GUN PARAMETERS: fc/fa=2.8 s / ^ \, (C)j / 1 / e = i3.7° VVa/T-i.o \ ^> k / / / [B]/ r rc = 0.043" (A) EXPERIMENT (B) METHODS OF THIS PAPER (C) HINES-CUTLER METHOD (D) UNIVERSAL BEAM SPREAD CURVE \ ^^ V / / / / / \ N s. <; >^ ^ '4 / ^ <. \ \, "^ ^>3e \ \, y /(D) \ \ y y "~~- ^^ ^ 40 80 120 160 200 240 Z, DISTANCE FROM IDEAL ANODE IN MILS 280 320 (b) i /(C) y GUN PARAMETERS: f c/fa = 2.5 1 1 1 / / e = 8° 1.0 / 1 * y ^B) ^/V, /T- \ x^ V a/ rc = 0.150" / / / f y /^ V ^ V ^ ^***^^ •■ • • } \ X X "^ , -» -^ <^ — ■^ (A) \ ^^ .^ y ^-- ^^ ^ (D) 100 200 300 400 500 600 Z, DISTANCE FROM IDEAL ANODE IN MILS 700 800 Fig. 15 — Beam profiles (using an anode lens correction of r = 1.1 and the gun parameters indicated) as obtained (A) from experiment, (B) bj^ the methods of this paper, (C) Hines-Cutler method, (D) by use of the universal beam spread curve. universal l^eam spread curve'" (i.e., under the assumption of laminar flow and gradual variations of beam radius with distance) . Note that in each case a value of 1.1 has been used for the correction factor, r, repre- senting the excess divergence of the anode lens. The agreement in (/'95)min as obtaiucd from Curves A and B is remarkably good, but the axial position of (r95)min in Curve A definitely lies beyond the correspond- 418 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ing inininumi position in Curve B. Fortunately, in the gun design stage, one is usually more concerned with the value of (r95)min than with its exact axial location. The principal need for knowing the axial location of the minimum is to enable the axial magnetic field to build up suddenly in this neighborhood. However, since this field is normally adjusted ex- perimentally to produce best focusing, an approximate knowledge of 2m in is usually adequate. In Fig. 15b we show a similar set of experimental and theoretical beam profiles for another gun. The relative profiles are much the same as in Fig 15a, and all of several other guns measured yield experimental points similarly situated with respect to curves of Type B. C. Comparison of Experimental and Theoretical Current Density Dis- tributions where the Minimum Beam Diameter is Reached In Fig. 14 we have plotted the current density distribution we would have predicted in a transverse plane at ^min for the example introduced in Section 5Z). Here the experimental and theoretical curves are nor- malized to include the same total currents in their respective beams. The noticeable difference in predicted and measured current densities at the center of the beam does not appreciably alter the properties such a beam would have on entering a magnetic field because so little total current is actually represented by this central peak. D. Variation of Beam Profile with T All of the design charts have been based on a value of T = 1.1, which is typical of the values obtained by the methods of Section 3. When appreciably different values of F are appropriate, we can get some feel- ing for the errors involved, in using curves based on T = 1.1, by refer- ence to Fig. 16. Here we show beam profiles as obtained by the methods of this paper for three values of F. The calculations are again based on the gun of Section 5D, and a value of just over 1.1 for F gives the ex- perimentally obtained value for (r95)min . 7. SOME ADDITIONAL REMARKS ON GUN DESIGN In previous sections we have not differentiated between the voltage on the accelerating anode of the gun and the final beam voltage. It is important, howovei', that the separate functions of these two voltages be kept clearly in mind: The accelerating anode determines the total current drawn and largely controls the shaping of the beam; the final beam voltage is, on the other hand, chosen to give maximum interaction between the electron beam and the electromagnetic waves traveling along the slow wave circuit. As a consequence of this separation of func- , BEAM FORMATION WITH ELECTRON GUNS 419 0.006 0.02 0.18 0.20 0.22 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Z, DISTANCE FROM IDEAL ANODE IN INCHES Fig. 16 — Beam profiles as obtained by the methods of this paper for the gun parameters given in Section bD. Curves are shown for three values of the anode lens correction, viz. T = 1.0, 1.1, and 1.2. tions, it is fouiicl that some beams which are difficult or impossible to obtain with a single Pierce-gun acceleration to final beam voltage may be obtained more easily by using a lower voltage on the gun anode. The acceleration to final beam voltage is then accomplished after the beam has entered a region of axial magnetic field. Suppose, for example, that one wishes to produce a 2-ma, 4-kv beam with (rgs/rc) = 0.25. If the cathode temperature is 1000°K, and the gun anode is placed at a final beam voltage of 4 kv, we have \^Va/T = 2 and P = 0.008. From the top set of curves under \^Va/T = 2 in Fig. 13, we find (by using a fairly crude extrapolation from the curves shown) that a ratio of fc/fa'^ 3.5 is required to produce such a beam. The value of {ve/o-) at Zmin IS therefore less than about 0.2 so that there is little x'mblance of laminar flow here. On the other hand we might choose r, = 250 volts so that a/fT^ = 0.5 and P = 0.51. From Fig. 13* we than obtain fc/fa = 2.6 and (re/o-)min = 0.8 for the same ratio of '■'joAc(= 0.25). While the flow could still hardly be called laminar, it is (•(jnsiderably more ordered than in the preceding case. Here we have in- cluded no correction for the (convergent) lens effect associated with the post-anode acceleration to the final beam voltage, F = 4 kv. Calculations of the Hines-Cutler type will always predict, for a given set of gun parameters and a specified anode lens correction, a minimum beam size which is larger than that predicted by the methods of this ])aper. Nevertheless, in many cases the difference between the minimum sizes predicted by the two theories is negligible so long as the same anode lens correction is used. The extent to which the two theories agree ob- 420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 viously depends on the magnitude of Velo. When rel(T as calculated by the Hines-Cutler method (with a lens correction added) remains greater than about 2 throughout the range of interest, the difference between the corresponding values obtained for rgs will be only a few per cent. For these cases where rja does not get too small, the principal advan- tages of this paper are in the inclusion of a correction to the anode lens formula and in the comparative ease with which design parameters may be obtained. In other cases r^la may become less than 1, and the theory presented in this paper has extended the basic Hines-Cutler approach so that one may make realistic predictions even under these less ideal conditions where the departure from a laminar-type flow is quite severe. ACKNOWLEDGMENT We wish to thank members of the Mathematical Department at B.T.L., particularly H. T. O'Neil and Mrs. L. R. Lee, for their help in programming the problem on the analog computer and in obtaining the large amount of computer data involved. In addition, we wish to thank J. C. Irwin for his help in the electrolytic tank work and both Mr. Irwin and W. A. L. Warne for their work on the beam analyzer. REFERENCES 1. Pierce, J. R., Rectilinear Flow in Beams, J. App. Phys., 11, pp. 548-554, Aug., 1940. 2. Samuel, A. L., Some Notes on the Design of Electron Guns, Proc. I.R.E., 33, pp. 233-241, April, 1945. 3. Field, L. M., High Current Electron Guns, Rev. Mod. Phys., 18, pp. 353-361, July, 1946. 4. Davisson, C. J., and Calbick, C. J., Electron Lenses, Phys. Rev., 42, p. 580, Nov., 1932. 5. Helm, R., Spangenburg, K., and Field, L. M., Cathode-Design Procedure for Electron Beam Tubes, Elec. Coram., 24, pp. 101-107, March, 1947. 6. Cutler, C. C, and Hines, M. E., Thermal Velocity Effects in Electron Guns, Proc. I.R.E., 43, pp. 307-314, March, 1955. 7. Cutler, C. C, and Saloom, J. A., Pin-hole Camera Investigation of Electron Beams, Proc. I.R.E., 43, pp. 299-306, March, 1955. 8. Hines, M. E., Manuscript in preparation. 9. Private communication. 10. See for example, Zworykin, V. K., et al.. Electron Optics and the Electron Microscope, Chapter 13, Wiley and Sons, 1945, or Klemperer, O., Electron Optics, Chapter 4, Cambridge Univ. Press, 1953. 11. Brown, K. L., and Siisskind, C., The Effect of the Anode Aperature on Po- tential Distribution in a "Pierce" Electron Gun, Proc. I.R.E., 42, p. 598, March, 1954. 12. See, for example, Pierce, J. R., Theory and Design of Electron Beams, p. 147, Van Nostrand Co., 1949. 13. See Reference 6, p. 5. 14. Langmuir, I. L., and Blodgett, K., Currents Limited by Space Charge Be- tween Concentric Spheres, Phys. Rev., 24, p. 53, July, 1924. 15. See Reference 12, p. 177. 16. See Reference 12, Chap. X. Theories for Toll Traffic Engineering in the U.S.A.* By ROGER I. WILKINSON (Manuscript received June 2, 1955) Present toll trunk traffic engineering practices in the United States are reviewed, and various congestion formulas compared with data obtained on long distance traffic. Customer habits upon meeting busy channels are noted and a theory developed describing the probable result of permitting subscribers to have direct dialing access to high delay toll trunk groups. Continent-wide automatic alternate routing plans are described briefly, in which near no-delay service will permit direct customer dialing. The presence of non-random overflow traffic from high usage groups co7nplicates the estimation of correct quantities of alternate paths. Present methods of solving graded multiple problems are reviewed and found unadaptable to the variety of trunking arrangements occurring in the toll plan. Evidence is given that the principal fluctuation characteristics of overflow- type of non-random traffic are described by their mean and variance. An approximate probability distribution of simultaneous calls for this kind of non-random traffic is developed, and found to agree satisfactorily with theo- retical overflow distributions and those seen in traffic simidations. A method is devised using ^^ equivalent random''^ traffic, which has good loss predictive ability under the "lost calls cleared" assumption, for a diverse field of alternate route trunking arrangements. Loss comparisons are made with traffic simulation residts and with observations in exchanges. Working curves are presented by which midti-alternate route trunking systems can be laid out to meet economic and grade of service criteria. Exam- ples of their application are given. Table of Contents 1 . Introduction 422 2. Present Toll Traffic Engineering Practice 423 * Presented at the First International Congress on the Application of the Theory of Probability in Telephone Engineering and Administration, Copen- hagen, June 21, 1955. 421 422 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 3. Customers Dialing on Groups with Considerable Delay 431 3.1. Comparison of Some Formulas for Estimating Customers' NC Service on Congested Groups 434 4. Service Requirements for Direct Distance Dialing by Customers 436 5. Economics of Toll Alternate Routing 437 6. New Problems in the Engineering and Administration of Intertoll Groups Resulting from Alternate Routing 441 7. Load-Service Relationships in Alternate Route Systems 442 7.1. The "Peaked" Character of Overflow Traffic 443 7.2. Approximate Description of the Character of Overflow Traffic 446 7.2.1. A Probability Distribution for Overflow Traffic 452 7.2.2. A Probability Distribution for Combined Overflow Traffic Loads 457 7.3. Equivalent Random Theory for Prediction of Amount of Traffic Over- flowing a Single Stage Alternate Route, and Its Character, with Lost Calls Cleared 461 7.3.L Throwdown Comparisons with Equivalent Random Theory on Simple Alternate Routing Arrangements with Lost Calls Cleared 468 7.3.2. Comparison of Equivalent Random Theory with Field Results on Simple Alternate Routing Arrangements 470 7.4. Prediction of Traffic Passing Through a Multi-Stage Alternate Route Network 475 7.4.1. Correlation of Loss with Peakedness of Components of Non- Random Offered Traffic 481 7.5. Expected Loss on First Routed Traffic Offered to Final Route 482 7.6. Load on Each Trunk, Particularly the Last Trunk, in a Non-Slipped Alternate Route 486 8. Practical Methods for Alternate Route Engineering 487 8.1. Determination of Final Group Size with First Routed Traffic Offered Directly to Final Group 490 8.2. Provision of Trunks Individual to First Routed Traffic to Equalize Service 491 8.3. Area in Which Significant Savings in Final Route Trunks are Real- ized by Allowing for the Preferred Service Given a First Routed Traffic Parcel 494 8.4. Character of Traffic Carried on Non-Final Routes 495 8.5. Solution of a Typical Toll Multi-Alternate Route Trunking Arrange- ment : Bloomsburg, Pa 500 9. Conclusion 505 Acknowledgements 506 References 506 Abridged Bibliography of Articles on Toll Alternate Routing 507 Appendix I: Derivation of Moments of Overflow Traffic 507 Appendix II: Character of Overflow when Non-Random Traffic is Offered to a group of Trunks 511 1. INTRODUCTION It has long been the stated aim of the Bell System to make it easily and economically possible for any telephone customer in the United States to reach any other telephone in the world. The principal effort in this direction by the American Telephone and Telegraph Company and its associated operating companies is, of course, confined to inter- connecting the telephones in the United States, and to providing com- munication channels between North America and the other countries of the world. Since the United States is some 1500 miles from north to fSOuth and 3000 miles from east to west, to realize even the aim of fast THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 423 and economical service between customers is a problem of great magni- tude; it has engaged our planning engineers for many years. There are now 52 million telephones in the United States, over 80 per cent of which are equipped with dials. Until quite recently most telephone users were limited in their direct dialing to the local or immediately sur- rounding areas and long distance operators were obliged to build up a circuit with the aid of a "through" operator at each switching point. Both speed and economy dictated the automatic build-up of long toll circuits without the intervention of more than the originating toll oper- ator. The development of the No. 4-type toll crossbar switching system with its ability to accept, translate, and pass on the necessary digits (or lujuivalent information) to the distant office made this method of oper- ation possible and feasible. It was introduced during World War II, and now by means of it and allied equipment, 55 per cent of all long distance calls (over 25 miles) are completed by the originating operator. As more elaborate switching and charge-recording arrangements were developed, particularly in metropolitan areas, the distances which cus- tomers themselves might dial measurably increased. This expansion of the local dialing area was found to be both economical and pleasing to the users. It was then not too great an effort to visualize customers dialing to all other telephones in the United States and neighboring countries, and perhaps ultimately across the sea. The physical accomplishment of nationwide direct distance dialing which is now gradually being introduced has involved, as may well be imagined, an immense amount of advance study and fundamental plan- ning. Adequate transmission and signalling with up to eight intertoll trunks in tandem, a nationwide uniform numbering plan simple enough to be used accurately and easily by the ordinary telephone caller, pro- ^ ision for automatic recording of who called whom and how long he talked, with subsequent automatic message accounting, are a few of man}^ problems which have required solution. How they are being met is a romantic story beyond the scope of the present paper. The references given in the bibliography at the end contain much of the history as well as the plans for the future. • 2. PRESENT TOLL TRAFFIC ENGINEERING PRACTICE There are today approximately 116,000 intertoll trunks (over 25 miles in length) in the Bell System, apportioned among some 13,000 trunk groups. A small segment of the 2,600 toll centers which they interconnect is shown in Fig. 1. Most of these intertoll groups are presently traffic engineered to operate according to one of several so-called T-schedules: T-8, T-15, T-30, T-60, or T-120. The number following T (T for Toll) is 424 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 KEY O TOLL CENTERS INTERTOLL TRUNK GROUPS Fig. 1 — Principal intertoll trunk groups in Minnesota and Wisconsin. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 425 4 5 6 7 8 9 10 NUMBER OF TRUNKS 30 40 50 Fig. 2 — Permitted intertoU trunk occupancy for a 6.5-minute usage time per message. the expected, or average, delay in seconds for calls to obtain an idle trunk in that group during the average Busy Season Busy Hour. In 1954 the system "average trunk speed" was approximately 30 seconds, re- sulting from operating the majority of the groups at a busy-hour trunk- ling efficiency of 75 to 85 per cent in the busy season. The T-engineering tables show permissible call minutes of use for a 426 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 wide range of group sizes, and several selections of message holding times. They were constructed following summarization of many obser- vations of load and resultant average delays on ringdown (non-dial) intertoll trunks.^ Fig. 2 shows the permissible occupancy (efficiency) of various trunk group sizes for 6.5 minutes of use per message, for a va- riety of T-schedules. It is perhaps of somfe interest that the best fitting curves relating average delay and load were found to be the well-known Pollaczek-Crommelin delay curves for constant holding time — this in spite of the fact that the circuit holding times were far indeed from having a constant value. A second, and probably not uncorrected, observation was that the per cent "No-Circuit" (NC) reported on the operators' tickets showed consistently lower values than were measured on group-busy timing de- vices. Although not thoroughly documented, this disparity has generally been attributed to the reluctance of an operator to admit immediately the presence of an NC condition. She exhibits a certain tolerance (very difficult to measure) before actually recording a delay which would recjuire her to adopt a prescribed procedure for the subsequent handling of the call.* There are then two measures of the No-Circuit condition which are of some interest, the "NC encountered" by operators, and the "NC existing" as measured by timing devices. It has long been observed that the distribution of numbers n of simul- taneous calls found on T-engineered ringdown intertoll groups is in re- markable agreement with the individual probability terms of the Erlang "lost calls" formula, f n — a ' a e fin) = ^-^^ (1) e E- n=o n! where c = number of paths in the group, a' = an enhanced average load submitted such that a'[l — Ei^c(a')] = L, the actual load carried, and Ei^cid') = fie) = Erlang loss probability (commonly called Er- lang B in America). An example of the agreement of observations with (1) is shown in Fig. 3, where the results of switch counts made some years ago on many ringdown circuit groups of size 3 are summarized. A wide range of "sub- * Upon finding No-Circuit, an operator is instructed to try again in 30 seconds and GO seconds (before giving an NC report to the customer), followed by addi- tional attempts 5 minutes and 10 minutes later if necessary. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 427 0.10 0.2 0.5 1.0 2 AVERAGE "submitted" LOAD IN ERLANIGS Fig. 3 — Distributions of simultaneous calls on three-trunk toll groups at .\lbany and Buffalo. I nit ted" loads a' to produce the observed carried loads is required. On Fig. 4 are shown the corresponding comparisons of theory and obser- vations for the proportions of time all paths are busy ("NC Existing") for 2-, 4-, 5-, 7-, and 9-circuit groups. Good agreement has also been ob- served for circuit groups up to 20 trunks. This has been found to be a stable relationship, in spite of the considerable variation in the actual practices in ringdown operation on the resubmission of delayed calls. Since the estimation of traffic loads and the subsequent administration of ringdown toll trunks has been performed principally by means of Group Busy Timers (which cumulate the duration of NC time), the Erlang relationship just described has been of great importance. 428 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 With the recent rapid increase in operator dialed intertoll groups, it might be expected that the above discrepancy between " % NC encoun- tered" and "% NC existing" would disappear — for an operator now initiates each call unaware of the momentary state of the load on any particular intertoll group. By the use of peg count meters (which count calls offered) and overflow call counters, this change has in fact been observed to occiu'. ]\Ioreo^'er, since the initial re-trial intervals are com- monly fairly short (30 seconds) subsequent attempts tend to find some of the previous congestion still existing, so that the ratio of overflow to peg count readings now exceeds slightly the "% NC existing." This situation is illustrated in Fig. 5, which shows data taken on an operator- 1.0 AVERAGE SUBMITTED LOAD Fig. 4 — Observed proportions of time all trunks were busy on Albany and Buffalo groups of 2, 4, 5, 7, and 9 trunks, THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 429 u z o z I- UJ HI 5 ul _i _i < o U- o z o (- cc o a. o tr a. 0.001 12 14 L = LOAD CARRIED IN ERLANGS 18 Fig. 5 — Comparison of NC data on a 16-trunk T-engineered toll group with various load versus NC theories. dialed T-engineered group of 16 trunks between Newark, N. J., and Akron, Ohio. Curve A shows the empirically determined "NC encoun- tered" relationship described above for ringdown operation; Curve B gives the corresponding theoretical "NC existing" values. Lines C and D give the operator-dialing results, for morning and afternoon busy hours. The observed points are now seen generally to be significantly above Curve B.* At the same time as this change in the "NC encountered" was occur- ring, due to the introduction of operator toll dialing, there seems to have l)een little disturbance to the traditional relationship between load * The observed point at 11 erlangs which is clearly far out of agreement with the remainder of the data was produced by a combination of high-trend hours and an hour in which an operator apparently made many re-t^rials in rapid suc- cession. 430 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 u. 10 z o o 5 o m rvj ti _r < (- z LU z o z o i tr UJ I/) § o «--- LIMIT OF OBSERVED DATA i [ oiT / / / / / / / / / / / y / / /' ^•^ /^ ^ ^ ««- ^ ^ Tt^^ ^•^^ ^ s:;^ 8 If) o 0> o (0 (O o If) o in o o in tvj o in SBIONII^ 1- a3AO SidlAjaiiV dO iN3D«3d THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 431 carried and " % NC existing." C. J. Truitt of the A.T. & T. Co. studied i a number of operator-dialed T-engineered groups at Newark, New Jersey, in 1954 with a traffic usage recorder (TUR) and group-busy timers, and found the relationship of equation (1) still good. (This analysis has not been published.) A study by Dr. L. Kosten has provided an estimate of the probability that when an NC condition has been found, it will also appear at a time T later." When this modification is made, the expected load-versus-NC relationship is shown by Curve E on Fig. 5. (The re-trial time here was taken as the operators' nominal 30 seconds; with 150-second circuit-use time the return is 0.2 holding time.) The observed NC's are seen to lie slightly above the E-curve. This could be explained either on the basis that Kosten's analysis is a lower limit, or that the operators did not strictly observe the 30-second return schedule, or, more probably, a combination of both. 3. CUSTOMERS DIALING ON GROUPS WITH CONSIDERABLE DELAY It is not to be expected that customers could generally be persuaded to wait a designated constant or minimum re-trial time on their calls which meet the NC condition. Little actual experience has been accumulated on customers dialing long distance calls on high-delay circuits. However, it is plausible that they would follow the re-trial time distributions of customers making local calls, who encounter paths-busy or line-busy signals (between which they apparently do not usually distinguish). Some information on re-trial times was assembled in 1944 by C. Clos by observing the action of customers who received the busy signal on 1,100 local calls in the City of New York. As seen in Fig. 6, the return times, after meeting "busy," exhibit a marked tendency toward the exponential distribution, after allowance for a minimum interval required for re- dialing. An exponential distribution with average of 250 seconds has been I fitted by eye on Fig. 6, to the earlier ■ — and more critical — customer re- turn times. This may seem an unexpectedly long wait in the light of indi- vidual experience; however it is probably a fair estimate, especially since, following the collection of the above data, it has become common practice for American operating companies in their instructional lit- erature to advise customers receiving the busy signal to "hang up, wait a few minutes, and try again." The mathematical representation of the situation assuming exponen- I tial return times is easily formulated. Let there be .r actual trunks, and 432 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 imagine y waiting positions, whore y is so large that few calls are re- jected.* Assume that the offered load is a erlangs, and that the calls have exponential conversation holding times of unit average duration. Finally \ let the average return time for calls which have advanced to the waiting > positions, be 1/s times that of the unit conversation time. The statistical j equilibrium equation can then be written for the probability j\m, n) (j that m calls are in progress on the x trunks and n calls are waiting on the y storage positions: ■ /(w, n) = aj{m — 1, n) dt + s(w + l)/(m — 1, n + 1) dt ''■) + (m + \)J{m + 1, n) dt + a/(.r, n - 1) dH^ (2) + [1 - (a*** + sn**) dt - m dt]f(m, n) ^ where 0 ^ m ^ .-r, 0 ^ w ^ //, and the special limiting situations are recognized by: ■* Include term only when m — x **■ Omit sn when m = x *** Omit a when m = x and n = y Equation (2) reduces to (a*** + snifif + m)f{m; n) = af{m — 1, n) 1 + s(n + l)/(m - 1, w + 1) (3) + (m + l)/(w + 1, n) + af(x, n - !)•, Solution of (3) is most easily effected for moderate values of x and y by first setting f(x, ?/) = 1 .000000 and solving for all other /(/?? , ?? ) in X y terms of /(o:, ?/). Normahzing through zl 11f(m, n) = 1.0, then gives m=0 n=0 the entire f(m, n) array. The proportion of time "NC exists," will, of course be Z Six, n) (4) n=0 and the load carried is L = Xl X wi/(m, n) (5) The proportion of call attempts meeting NC, including all re-trials * The quant itjr y can also be chosen so that some calls are rejected, thus roughly describing those calls abandoned after the first attempt. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A, 433 will be W{x, a, s) = Expected overflow calls per unit time Expected calls offered per unit time Z (a + sn)jXx, n) - , ./ ^ ^^^ sn -\- af{x, y) n=0 X y S 2 (« + sn)f(m, n) a -{- sn m=0 71=0 X y in which n = ^ 2^ nf(7n, n). And when y is chosen so large that/(.r, y) 7H = 0 71=0 is negligible, as we shall use it here, L = a W(x, a, s) = sn a -\- sn (5') (6') 1^ 0.5 < "^O 0.4 ilZ Oo ZZ 0.3 Ol- pllJ o5 0.2 Q. o ? 0.1 6 TRUNKS / // APOISSON ' ^1 P(C,L) 5=0.6 2 4 6 8 L=LOAD CARRIED IN ERLANGS APOISSON P(C,L) fly >^- f I6j _, 8 10 12 14 L = LOAD CARRIED IN ERLANGS Fig. 7 — ■ Comparison of trunking formulas. 434 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 I This formula provides a means for estimating the grade of service which customers might he expected to receive if asked to dial their calls over moderate-delay or high-delay trunk groups. For a circuit use length of 150 seconds, and an average return time of 250 seconds (as on Fig. 6), both exponential, the load-versus-proportion-NC curves for 6 and IG trunks are given as curves (3) on Fig. 7. For example with an offered (= carried) load of a = 4.15 erlangs on 6 trunks we should expect to find 27.5 per cent of the total attempts resulting in failure. For comparison with a fixed return time of NC-calls, the IF-formula curves for exponential returns of 30 seconds (s = 5) and 250 seconds (s = 0.6) averages are shown on Fig. 5. The first is far too severe an assumption for operator performance, giving NC's nearly double those actually observed (and those given by theory for a 30-second constant return time). The 250-second average return, however, lies only slightly above the 30-second constant return curve and is in good agreement with the data. Although not logically an adequate formula for interpreting Peg Count and Overflow registrations on T-engineered groups under operator dialing conditions, the IF-formula apparently could be used for this purpose with suitable s-values determined empirically. 3.1. Comparison of Some Formulas for Estimating Customers' NC Service on Congested Groups , 1 As has been previously observed, a large proportion of customers who receive a busy signal, return within a few minutes (on Fig. 6, 75 per cent of the customers returned within 10 minutes). It is well known too, that under adverse service conditions subscriber attempts (to reach a par- ticular distant office for example) tend to produce an inflated estimate of the true offered load. A count of calls carried (or a direct measurement of load carried) will commonly be a closer estimate of the offered load than a count of attempts. An exception may occur when a large propor- tion of attempts is lost, indicating an offered load possibly in excess even of the number of paths provided. Under the latter condition it is diffi- cult to estimate the true offered load by any method, since not all the attempts can be expected to return repeatedly until served; instead, a significant number will be abandoned somewhere through the trials. In most other circumstances, however, the carried load will prove a reason- ably good estimate of the true offered load in systems not provided with alternate paths. This is a matter of especial interest for both toll and local operation in America since principal future reliance for load measurement is ex- THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 435 pected to be placed on automatically processed TUR data, and as the TUR is a switch counting device the results will be in terms of load carried. Moreover, the quantity now obtained in many local exchanges is load carried.* Visual switch counting of line finders and selectors off- normal is widely practiced in step-by-step and panel offices; a variety of electromechanical switch counting devices is also to be found in crossbar offices. It is common to take load-carried figures as equal to load-offered when using conventional trunking tables to ascertain the proper pro- vision of trunks or switches. Fig. 7 compares the NC predictions made by a number of the available load-loss formulas when load carried is used as the entry variable. The lowest curves (1) on Fig. 7 are from the Erlang lost calls formula El (or B) with load carried L used as the offered load a. At low losses, say 0.01 or less, either L or a = L/[l — Ei(a)] can be used indiscrimi- nately as the entry in the Ei formula. If however considerably larger losses are encountered and calls are not in reality "cleared" upon meet- ing NC, it will no longer be satisfactory to substitute L for a. In this circumstance it is common to calculate a fictitious load a' to submit to the c paths such that the load carried, a'[I — Ei^dd')], equals the desired L. (This was the process used in Section 2 to obtain " % NC existing.") The curves (2) on Fig. 7 show this relation ; physically it corresponds to an initially offered load of L erlangs (or L call arrivals per average hold- ing time), whose overflow calls return again and again until successful but without disturbing the randomness of the input. Thus if the loss from this enhanced random traffic is E, then the total trials seen per holding time will be L(l + ^ + ^' -f • • •) = L/(l - E) = a', the ap- parent arrival rate of new calls, but actually of new calls plus return attempts. The random resubmission of calls may provide a reasonable descrip- tion of operation under certain circumstances, presumably when re-trials are not excessive. Kosten^ has discussed the dangers here and provided upper and lowxr limit formulas and curves for estimating the proportions of NC's to be expected when re-trials are made at any specified fixed leturn time. His lower bounds (lower bound because the change in con- gestion character caused by the returning calls is ignored) are shown by open dots on Fig. 7 for return times of 1.67 holding times. They lie above curves (2) (although only very slightly because of the relatively long return time) since they allo\\- for the fact that a call shortly returning * In fact, it is difficult to see how any estimate of offered load, other than carried load, can be obtained with useful reliability. 436 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 after meeting a busy signal will have a higher probability of again find- ing all paths busy, than would a randomly originated call. The curves (3) show the TF-formula previously developed in this sec- tion, which contemplates exponential return times on all NC attempts. The average return time here is also taken as 1 .67 holding times. These curves lie higher than Kosten's values for two reasons. First, the altered congestion due to return calls is allowed for; and second, with exponential returns nearly two-thirds of the return times are shorter than the aver- age, and of these, the shortest ones will have a relatively high probability of failure upon re-trying. If the customers were to return with exponen- tial times after waiting an average of only 0.2 holding time (e.g., 30 seconds wait for 150-second calls) the TT^-curves would rise markedly to the positions shown by (4). Curves (5) and (6) give the proportions of time that all paths are busy (equation 4) under the T'F-formula assumptions corresponding to NC curves (3) and (4) respectively; their upward displacement from the random return curves (2) reflects the disturbance to the group congestion produced by the non-random return of the delayed calls. (The limiting position for these curves is, of course, given by Erlang's E2 (or C) delay formula.) As would be expected, curve (6) is above (5) since the former contemplates exponential returns with average of 0.2 holding time, as against 1.67 for curve (5). Neither the (5)-curves nor the open dots of constant 30-second return times show a marked increase over curves (2). This appears to explain why the relationship of load carried versus "NC existing" (as charted in Figs. 3 and 4) was found so insensitive to vari- able operating procedures in handling subsequent attempts in toll ring- down operation, and again, why it did not appreciably change under operator dialing. Finally, through the two fields of curves on Fig. 7 is indicated the Poisson summation P{c, L) with load carried L used as the entering variable. The fact that these values approach closely the (2) and (3) sets of curves over a considerable range of NC's should reassure those who have been concerned that the Poisson engineering tables were not useful for losses larger than a few per cent.* 4. SERVICE REQUIREMENTS FOR DIRECT DISTANCE DIALING BY CUSTOMERS As shown by the TF-curves (3) on Fig. 7, the attempt failures by cus- tomers resulting from their tendency to re-try shortly following an NC * Reference may be made also to a throwdown by C. Clos (Ref. 3) using the return times of Fig. 6; his "% NC" results agreed closely with tlie Poisson pre- dictions. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 437 would be expected to exceed slightly the values for completely random re-trials. These particular curves are based on a re-trial interval of 1.67 times the average circuit-use time. Such moderation on the part of the customer is probably attainable through instructional literature and other means if the customer believes the "NC" or "busy" to be caused by the called party's actually using his telephone (the usual case in local practice). It would be considerably more difficult, however, to dissuade the customer from re-trying at a more rapid rate if the circuit NC's should generally approach or exceed actual called-party busies, a con- dition of which he would sooner or later become aware. His attempts might then be more nearly described by the (4) curves on Fig. 7 cor- responding to an average exponential return of only 0.2 holding time — or e\en higher. Such a result would not only displease the user, but also result in the requirement of increased switching control equipment to handle many more wasted attempts. If subscribers are to be given satisfactory direct dialing access to the iiitertoll trunk network, it appears then that the probability of finding XC even in the busy hours must be kept to a low figure. The following engineering objective has tentatively been selected: The calls offered to the ^'final" group of trunks in an alternate route system should receive no more than 3 per cent NC(P.03) during the network busy season busy hour. (If there are no alternate routes, the direct group is the "final" route.) Since in the nationwide plan there will be a final route between each of some 2,600 toll centers and its next higher center, and the majority of calls offered to high usage trunks will be carried without trying their final route (or routes), the over-all point-to-point service, while not easy to estimate, will apparently be quite satisfactory for cus- tomer dialing. 5. ECONOMICS OF TOLL ALTERNATE ROUTING In a general study of the economics of a nationwide toll switching plan, made some years ago by engineers of the American Telephone and Tele- graph Company, it was concluded that a toll line plant sufficient to give ihe then average level of service (about T-40) with ordinary single-route procedures could, if operated on a multi-alternate route basis, give the desired P.03 service on final routes with little, if any, increase in toll line investment.* On the other hand to attain a similar P.03 grade of service by liberalizing a typical intertoll group of 10 trunks working presently * This, of course, does not reflect the added costs of the No. 4 switching equip- I nient. 438 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 at a T-40 grade of service and an occupancy of 0.81 would recjuire an increase of 43 per cent (to 14.3 trunks), with a corresponding decrease in occupancy to 0.57. The possible savings in toll lines with alternate routing are therefore considerable in a system which must pro\'ide a service level satisfactory for customer dialing. In order to take fullest advantage of the economies of alternate rout- ing, present plans call for five classes of toll offices. There will be a large number of so-called End Offices, a smaller number of Toll Centers, and progressively fewer Primary Centers (about 150), Sectional Centers (about 40) and Regional Centers (9), one of which will be the National Center, to be used as the "home" switching point of the other eight Regional Centers.* Primary and higher centers will be arranged to per- form automatic alternate routing and are called Control Switching Points (CSP's). Each class of office will "home" on a higher class of office (not necessarily the next higher one) ; the toll paths between them are called "final routes." As described in Section 4, these final routes will be provided to give low delays, so that between each principal toll point and ever}' other one there will be available a succession of approximatelj' P.03 engineered trunk groups. Thus if the more direct and heavily loaded interconnecting paths commonly provided are busj- there will still be a good chance of making immediate connection over final routes. Fig. 8 illustrates the manner in which automatic alternate routing will operate in comparison with present-day operator routing. On a call from Syracuse, X. Y., to Miami, Florida, (a distance of some 1,250 miles), under present-day operation, the Syracuse operator signals Albany, and requests a trunk to Miami. With T-schedule operation the Syracuse- Miami traffic might be expected to encounter as much as 25 per cent NC during the busy hour, and approximately 4 per cent NC for the whole day, producing perhaps a two-minute over-all speed of serA-ice in the busy season. With the proposed automatic alternate routing plan, all points on the chart will have automatic switching systems. f The customer (or the operator until customer dialing arrangements are completed) will dial a ten-digit code (three-digit area code 305 for Florida plus the listed Miami seven-digit telephone number) into the Jiiachine at Syracuse. The various routes which then might conceivably be tried automatically * Sec the hihlio^rajjliy ( i);irticulMily Pilliod and Truitt) for details of tlie general trunkinji plan. t The notation uscmI on the diagram of Fig. 8 is: Opon firclo — Primary Center (Syracuse, Miamij; Triangle — Sectional Center (All)an\-, Jacksonville); Sqviare — Regional Center (White Plains, Atlanta, St. Louis; St. Louis is also the Na- tional Center). THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 439 PRESENT OPERATOR ROUTING '^^ AUTOMATIC ALTERNATE ROUTING white Plains N. Y.) Miami Miami Fig. 8 — Present and proposed methods of handling a call from Syracuse, N. Y., to Miami, Florida. are shown on the diagram numbered in the order of trial; in this par- ticular layout shown, a maximum of eleven circuit groups could be tested for an idle path if each high usage group should be found NC. Dotted lines show the high usage roiites, which if found busy will overflow to the final groups represented by solid lines. The switching ecjuipment at each point upon finding an idle circuit passes on the required digits to the next machine. While the routing possibilities shown are factual, only in rare instances would a call be completed over the final route via St. Louis. Even in the busy season busy hour just a small portion of the calls would be expected to be switched as many as three times. And only a fraction of one per cent of all calls in the busy hour should encounter NC. As a result the service will be fast. When calls are handled by a toll operator, the cus- 440 THE BELL SYSTEM TECHNICAL JOURNAL; MARCH 1956 tomer will not ordinarily need to hang up when NC is obtained. When he himself dials, a second trial after a short wait following NC should have a high probability of success. Not many situations will be as complex as shown in Fig. 8; commonly several of the links between centers will be missing, the particular ones retained having been chosen from suitable economic studies. A large number of switching arrangements Avill be no more involved than the illustrative one shown in Fig. 9(a), centering on the Toll Center of Bloomsburg, Pennsylvania. The dashed lines indicate high usage groups from Bloomsburg to surrounding toll centers; since Bloomsburg "homes" on Scranton this is a final route as denoted by the solid line. As an exam- ple of the operation, consider a call at Bloomsburg destined for Williams- port. Upon finding all direct trunks busy, a second trial is made via Harrisburg; and should no paths in the Harrisburg group be available, a third and final trial is made through the Scranton group. In considering the traffic flow of a network such as illustrated at Bloomsburg it is convenient to employ the conventional form of a two- stage graded multiple having "legs" of varying sizes and traffic loads individual to each, as shown in Fig. 9(b). Here only the circuits im- mediately outgoing from the toll center are shown; the parcels of traffic (a) GEOGRAPHICAL LAYOUT WILLIAMSPORT I SCRANTON BLOOMSBURG HARRISBURG PA. (b) GRADED MULTIPLE SCHEMATIC FRACKVILLE HAZLETON WILKES- BARRE PHILADELPHIA FINAL GROUP TO SCRANTON H.U. GROUP TO HARRISBURG .1 M t I NO. TRUNKS IN H.U. GROUPS I [T] [jF] [^ [A] [T] [28 1 rsl m LOAD TO AND FROM ^^^ .^. ^^ ^ DISTANT OFFICE (CCS) "^^^ '^' ^^ ^'^^ ^^' '^0 '^3 836 228 154 DISTANT OFFICE SCRN HBG PTVL SHKN SNBY WMPT FKVL HZN WKSB PHLA Fig. 9 liiirg, Pa. Aulonialic ;ilU'riiaie routing for direct distance dialing at Blooms- THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 441 calculated for each further connecting route will be recorded as part of the offered load for consideration when the next higher switching center is engineered. It is implicitly assumed that a call which has selected one of the alternate route paths will be successful in finding the necessary paths available from the distant switching point onward. This is not quite true but is believed generally to be close enough for engineering piu'poses, and permits ignoring the return attempt problem. 6. NEW PROBLEMS IN THE ENGINEERING AND ADMINISTRATION OF INTER- TOLL GROUPS RESULTING FROM ALTERNATE ROUTING With the greatly increased teamwork among groups of intertoll trunks which supply overflow calls to an alternate route, an unexpected increase or flurry in the offered load to any one can adversely affect the service to all. The high efficiency of the alternate route networks also reduces their overload carrying ability. Conversely, the influence of an underprovision of paths in the final alternate route may be felt by many groups which overflow to it. With non-alternate route arrangements only the single groups having these flurries would be affected. Administratively, an alternate route trunk layout may well prove easier to monitor day by day than a large number of separate and in- dependent intertoll groups, since a close check on the service given on the final routes only may be sufficient to insure that all customers are being served satisfactorily. When rearrangements are indicated, how- SIMPLE PROGRESSIVE GRADED MULTIPLE GRADED MULTIPLE (a) (b) t t t t t t tt t t tl ILLUSTRATIVE INTERLOCAL AND INTERTOLL ALTERNATE ROUTE TRUNKING ARRANGEMENT; (c) (d) t t t t t = ,-"" ^ tttl It ttl 1 t Fig. 10 — Graded multi])los .•nid altornaic route trunking nrrangeinoiits. I 442 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 ever, the determination of the proper place to take action, and the de sirable extent, may sometimes be difficult to determine. Suitable traffic measuring devices must be provided with these latter problems in mind For engineering purposes, it will be highly desirable: (1) To be able to estimate the load-service relationships with any specified loads offered to a particular intertoll alternate routing network; and (2) To know the day-to-day busy hour variations in the various groups' offered loads during the busy season, so that the general grade of service given to customers can be estimated. The balance of this paper will review the studies which have been made in the Bell System toward a practicable method for predicting the grade of service given in an alternate route network under any given loads. Analyses of the day-to-day load variations and their effects on customer dialing service are currently being made, and will be reported upon later. ?; 7. LOAD-SERVICE RELATIONSHIPS IN ALTERNATE ROUTE SYSTEMS In their simplest form, alternate route systems appear as symmetrical graded multiples, as shown in Fig. 10(a) and 10(b). Patterns such as these have long been used in local automatic systems to partially over- come the trunking efficiency limitations imposed by limited access switches. The traffic capacity of these arrangements has been the sub- ject of much study by theory and "throwdowns" (simulated traffic studies) both in the United States and abroad. Field trials have sub- stantiated the essential accuracy of the trunking tables which have resulted. In toll alternate route systems as contemplated in America, however, there will seldom be the symmetry of pattern found in local graded multiples, nor does maximum switch size generally produce serious limitation on the access. The ''legs" or first-choice trunk groups will vary widely in size; likewise the number of such groups overflowing calls jointly to an alternate route may cover a considerable range. In all cases a given group, whether or not a link of an alternate route, will have one or more parcels of traffic for which it is the first-choice route. [See the right-hand parcel of offered traffic on Fig. 10(c).] Often this first routed traffic will Ijc the bulk of the load offered to the group, which also serves as an alternate I'oute for other traffic. The simplest of the approximate formulas developed for solving the local graded multiple problems are hopelessly unwieldy when applied to such arrangements as shown in Fig. 10(d). Likewise it is impracticable i THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 443 to solve more than a few of the infinite variety of arrangements by means of "throwdowns." However, for both engineering (planning for future trunk provisions) I and administration (current operating) of trunks in these multi-alternate routing systems, a rapid, simple, but reasonably accurate method is (required. The basis for the method which has been evolved for Bell System use will be described in the following pages. 7.1. The "Peaked" Character of Overflow Traffic The difficulty in predicting the load-service relationship in alternate route systems has lain in the non-random character of the traffic over- flowing a first set of paths to which calls may have been randomly offered. This non-randomness is a well appreciated phenomenon among traffic engineers. If adecjuate trunks are provided for accommodating the momentary traffic peaks, the time-call level diagram may appear as in Fig. 11(a), (average level of 9.5 erlangs). If however a more limited j number of trunks, say a: = 12, is provided, the peaks of Fig. 11(a) will be Ichpped, and the overflow calls will either be "lost" or they may be j handled on a subsequent set of paths y. The momentary loads seen on 2/ then appear as in Fig. 11(b). It will readily be seen that a given average i load on the y trunks will have quite different fluctuation characteristics i than if it had been found on the x trunks. There will be more occurrences of large numbers of calls, and also longer intervals when few or no calls are present. This gives rise to the expression that overflow traffic is "peaked." Peaked traffic requires more paths than does random traffic to operate at a specified grade of delayed or lost calls service. And the increase in paths required will depend upon the degree of peakedness of the traffic involved. A measure of peakedness of overflow traffic is then required which can be easily determined from a knowledge of the load offered and the number of trunks in the group immediately available. In 1923, G. W. Kendrick, then with the American Telephone and I Telegraph Company, undertook to solve the graded multiple problem ■through an application of Erlang's statistical eciuilibrium method. His i principal contribution (in an unpublished memorandum) was to set up I the equations for describing the existence of calls on a full access group \oi X -{- y paths, arranged so that arriving calls always seek service first iu the .T-group, and then in the ^/-group when the x are all busy. Let f{m, n) be the probability that at a random instant m calls exist j on the x paths and n calls on the y paths, when an average Poisson load 444 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of a erlangs is submitted to the x -\- y paths. The general state equation for all possible call arrangements, is (a* + m + n)f{m, n) = (w + l)/(m + 1, n) + (n + l)/(m, w + 1) + ajim — 1, n) + aj{x, n — 1)% (7) in which the term marked {%) is to be included only when m = x, and * indicates that the a in this term is to be omitted when in -\- n = x -{- y. m and n may take values only in the intervals, 0 -^ m ^ x;Q -^ n -^ y. As written, the equation represents the "lost calls cleared" situation. (a) RANDOM TRAFFIC 10 00 AM < I if) Q. a. 2 to 10 00 A M 10 30 TIME OF DAY (b) PEAKED OVERFLOW TRAFFIC PI -^ 10 30 TIME OF DAY Fig. 11 — Production of peakedness in overflow traffic. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 445 By choosing x -]- y large compared with the submitted load a a "lost calls held" situation or infinite-overflow-trunks result can be approached as closely as desired. Kendrick suggested solving the series of simultaneous equations (7) by determinants, and also by a method of continued fractions. However little of this numerical work was actually undertaken until several years later. Early in 1935 Miss E. V. Wyckoff of Bell Telephone Laboratories be- came interested in the solution of the (x -\- 1)(^/ + 1) lost calls cleared simultaneous equations leading to all terms in the /(m, n) distribution. She devised an order of substituting one equation in the next which pro- vided an entirely practical and relatively rapid means for the numerical solution of almost any set of these equations. By this method a con- siderable number of /(m, n) distributions on x, y type multiples with varying load levels were calculated. From the complete m, n matrix of probabilities, one easily obtains the distribution 9m{n) of overflow calls when exactly m are present on the lower group of x trunks; or by summing on m, the d{n) distribution with- out regard to m, is realized. A number of other procedures for obtaining the/(m, n) values have been proposed. All involve lengthy computations, very tedious for solution by desk calculating machines, and most do not have the ready checks of the WyckofT-method available at regular points through the calculations. In 1937 Kosten^ gave the following expression for /(m, n) : /(», n) = (- l)V.fe) i (i) M^- "f^'l., (8) i=0 (Pi^l{x)ipi(x) where (po{x) = x^—a a e xl ; and for i > 0, ;=o \ J / (.^• - J)i These equations, too, are laborious to calculate if the load and num- 1 K^rs of trunks are not small. It would, of course, be possible to program a modern automatic computer to do this work with considerable rapidity. The corresponding application of the statistical equilibrium equations to the graded multiple problem was visualized by Kendrick who, how- ever, went only so far as to write out the equation for the three-trunk 446 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 case consisting of two subgroups of one trunk each and one common overflow trunk. Instead of solving the enormously elaborate system of equations de- scribing all the calls which could simultaneously be present in a large multiple, several ingenious methods of convoluting the X 6(n) = Z/(w, n) overflow distributions from the individual legs of a graded multiple have been devised. For example, for the multiple of Fig. 10(a), the probability of loss Pi as seen by a call entering subgroup number i, is approximately, Pi = 2 £ e.Ar)-rl^{z -r) +J: d.Ar) (9) r=0 z=y T—y in which \l/{z — r) is the probability of exactly z — r overflow calls being present, or wanting to be present, on the alternate route from all the subgroups except the zth, and with no regard for the numbers of calls present in these subgroups. The ^x,i(^) = jiixi , r) term, of course, con- templates all paths in the particular originating call's subgroup being occupied, forcing the new call arriving in subgroup i to advance to the alternate route. This corresponds to the method of solving graded mul- tiples developed by E. C. Molina^ but has the advantage of overcoming the artificial "no holes in the multiple" assumption which he made. Similar calculating procedures have been suggested by Kosten.* These computational methods doubtless yield useful estimates of the resulting service, and for the limited numbers of multiple arrangements which might occur in within-office switching trains (particularly ones of a sym- metrical variety) such procedures might be practicable. But it would be far too laborious to obtain the individual overflow distributions Q{n), and then convolute them for the large variety of loads and multiple arrangements expected to be met in toll alternate routing. 7.2. Approximate Description of the Character of Overflow Traffic It was natural that various approximate procedures should be tried in the attempt to obtain solutions to the general loss formula sufficiently accurate for engineering and study purposes. The most ol^vious of these is to calculate the lower moments or semi-invariants of the loads over- flowing th(; sul)groups, and from them construct approximate fitting * Kosten gives the above approximation (9), which he calls Wb^, Jis an upper limit to the blocking. He also gives a lower limit , Wr, in which z = // throughout (References 4, 5). THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 447 I distributions for 6{n) mid dx(;n). Since each such overflow is independent I of the others, they may be combined additively (or convokited), to ob- [tain the corresponchng total distribution of calls appearing before the , I alternate route (or common group) . It may further be possible to obtain I [an approximate fitting distribution to the sum-distribution of the over- flow calls. The ordinary moments about the 0 point of the subgroup overflow distribution, when m of the x paths are busy, are found by V ta'im) = 2 njim, n) (10) When an infinite number of |/-paths is assumed, the resulting expres- sions for the mean and variance are found to be:* Number of x-paths busy unspecified :'\ Mean = a = a-Ei,^{a) (11) Variance = v = a[l — a -{- a{x -\- I -\- a — a)'^] (12) All x-paths occupiedi Mean = a^^ = a[x - a + 1 -\- aEiMf^ (13) Variance = v^ = ax[l — ax + 2a(x + 2 + a^ — a)~^] (14) Equations (11) and (12) have been calculated for considerable ranges 1 of offered load a and paths x. Figs. 12 and 13 are graphs of these results. i For example when a load of 4 erlangs is submitted to 5 paths, the aver- I age overflow load is seen to be a = 0.80 erlang, the same value, of I course, as determined through a direct application of the Erlang Ei formula. During the time that all x paths are busy, however, the over- flow load wdll tend to exceed this general level as indicated by the value of ax = 1.41 erlangs calculated from (13). Similarly the variance of the overflow load will tend to increase when the x-paths are fully occupied, * The derivation of these equations is given in Appendix I. t The skewness factor may also be of interest : ilz l^i: 3/2 ^" + "-"^"' +a^ (15) x+1 +a- a \x + 2\{x-a)'^^-2{x-a) + x + 2 + {x^-2-a)a + 3(1 -a) I + a(l - a)(l - 2a) o K:i' \ . . . t i > . wm Mm ^ ' \'' '^ 'mV \ I ■ . \ m \ ^. \ \ q o 6 |r ly\v\\ \ . . \ • ^ \ \ \ ■ ■■ \, r- '\ iD o '^ 0) '^ * \ « , \ \ \ \ o \ F?^ \ \ X, a v^ X V^ 'S ■f 'x^ ^^ ro "^ ■ ■■■^ ■^ ."-^.^ "■\ ^^N "^ z < _] a: LU q: LU a ■ < i - o : < ) ir ■ > ■; < . lO o o ■■,,. II > < (M 1 t\J T— 1 X o bJQ S ^x. X ■ v>r.m,^Mt«.f,.i.,sxrrfri o o o o ~ p 6 6 6 d d S5NVla3 Nl 'SHlVd X ONISSVd QVOl 30Va3AV = » 449 SHlVd X ONISSVd avOI dO 30NVIHVA = A 450 o in 6 ■ ^^ 1:1 i:, !• > o o X o ro o \ X \ \ \ o \- \ ' \ \ s V ■■V. V x N \ \ \ \ X \ \ . \ ^ V • X \ CM ro in o z < _l a. oi o n z Q lU cc ico li- ifvj o i a < o _i UJ (M ^ CK > < o ■n \ \ ', ' II o t-- vo n r^ SHiVd X ONISSVd avO"l dO 30NVIbVA=A 451 q r~; m ^ n - 6 6 6 6 6 o bO C O O O > o o a •I— I 1— t bi) 452 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 as shown by ?; = 1.30, and Vx = 1.95. In all cases the variances v and Vx will exceed the variance of corresponding Poisson traffic (which would have variances of a and ax respectively). 7.2.1. A Prohahility Distribution for Overflow Traffic It would be of interest to be able, given the first several descriptive parameters of any traffic load (such as the mean and variance and skew- ness factors of the overflow from a group of trunks), to construct an approximate probability distribution d{n) which would closely describe the true momentary distribution of simultaneous calls. Any proposed fitting distribution for the overflow from random traffic offered to x trunks, can, of course, be compared with . ^ X determined from (7) or (8). Suitable fitting curves should give probabilities for all possitive in- tegral values of the variable (including zero) , and have sufficient unspeci- fied constants to accommodate the parameters selected for describing the distribution. Moreover, the higher moments of a fitting distribution should not diverge too radically from those of the true distribution ; that is, the "natural shapes" of fitting and true distributions should be simi- lar. Particularly desirable would be a fitting distribution form derived with some attention to the physical circumstances causing the ebb and flow of calls in an overflow situation. The following argument and der- ivation undertake to achieve these desiderata.* A Poisson distribution of offered traffic is produced by a random arrival of calls. The assumption is made or implied that the probability of a new arrival in the next instant of time is quite independent of the number currently present in the system. When this randomness (and correspond- ing independence) are disturbed the resulting distribution will no longer be Poisson. The first important deviation from the Poisson would be expected to appear in a change from variance = mean, to variance ^ * A two-parameter function which has the ability to fit quite well a wide variety of true overflow distributions, has the form t(n) = Kin + l)''e-^(''+i) in which K is the normalizing constant. The distribution is displaced one unit from the usual discrete generalized exponential form, so that ^(0) 9^ 0. The ex- pression, however, has little rationale for being selected a priori as a suitable fitting function. I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 453 mean. Corresponding changes in the higher moments would also be expected. WTiat would be the physical description of a cause system with a vari- ance smaller or larger than the Poisson? If the variance is smaller, there must be forces at work which retard the call arrival rate as the number of calls recently offered exceeds a normal, or average, figure, and which increase the arrival rate when the number recently arrived falls below the normal level. Conversely, the variance will exceed the Poisson's .should the tendencies of the forces be reversed.* This last is, in fact, a rough description of the incidence rates for calls overflowing a group of trunks. Since holding times are attached to and extend from the call arrival instants, calls are enabled to project their influence into the future; that is, the presence of a considerable number of calls in a system at any in- stant reflects their having arrived in recent earlier time, and now can be used to modify the current rate of call arrival. Let the probability of a call originating in a short interval of time dt be Po.n = [a + (n — a)co(n)] dt where n = number of calls present in the system at time t, a = base or average arrival rate of calls per unit time, and w(n) = an arbitrary function which regulates the modification in call origination rate as the number of calls rises above or falls below a. Correspondingly, let the probability that one of n calls will end in the short interval of time dt be which will be satisfied in the case of exponential call holding times, with mean unity. Following the usual Erlang procedure, the general statistical equilibrium equation is (16) Jin) = /(n)[(l - Po.n){l - Pe,n)\ + /(« " l)Po,n-l(l " Pe.n-l) -Vj{n+ 1)(1 - Po.„+i)P,,„+i which gives (Po,„ + P.,„)/(n) = Po,«-i/(n - 1) + Pe,n+xKn + 1) i ignoring terms of order higher than the first in dt. * The same thinking lias been used by Vaiilot^ for decreasing the call arrival I rate according to the number momentaril}^ present; and by Lundquist^ for both increasing and decreasing the arrival rate. 454 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 (17) Or, [a + (n — a)w(?i) + ??.]/(n) = [a + (n -a- l)co(n - l)]/(/^ - 1) + (n + l)/(7i + 1) The choice of aj(n) will determine the solution of (17). Most simply, co(n) = k, making the variation from the average call arrival rate directly- proportional to the deviation in numbers of calls present from their average number. In this case, the solution for an unlimited trunk group becomes, with a' = a{l — k), a (a + k) -■■ [a + {n - 1)A;] fin) = n! ^^ , , a' (a' + k) , a' (a' + k)(a' + 2k) , 1 + « H ^t; H ^ TT, + (18) 2! ' 3! which may also be written after setting a" = a'/k = a(l — k)/k, as a'ia' + 1) • • • [a" + (n - 1)]A;" fin) = n! (19); (1 - k)- The generating function (g.f.) of (19) is Z/(n)r = (1 - kT)-"" n=0 (1 - k)--" which is recognized as that for the negative binomial, as distinguished from the g.f., P (i + ? tX (1/g)^ for the positive binomial. The first four descriptive parameters of /(w) are: Order Moment about Mean Descriptive Parameter 1 Ml = 0 Mean = n = a (20) 2 M2 = variance, v = a/(l — k) Std Devn, v \ _ \ • \ •\ \ n) \ \ ^ \ » \\ \ \ • \ ^ V \ 0.01 - \ V5 • v\ \s:=io \\ \\ \ . ^ V \> V • \ • \ ^^ \ n^ 0.001 _J M \ i 1 \ l> \V 1 1 0 t 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n = NUMBER OF SIMULTANEOUS CALLS Fig. 14 — Probability distributions of overflow traffic with 5 erlangs offered to 1, 2, 5, and 10 trunks, fitted by negative binomial. I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 457 the agreement, of course, is poor since the non-randomness of the over- flow here is marked, having an average of 1.88 and a variance of 3.84. Comparison of Negative Binomial with Overflow Distributions Observed hi/ llirowdoivns and on Actual Trunk Groups Fig. 18 shows a comparison of the negative binomial with the over- How distributions from four direct groups as seen in throwdown studies, 'ilie agreement over the range of group sizes from one to fifteen trunks is seen to be excellent. The assumption of randomness (Poisson) as shown by the dot values is clearly unsatisfactory for overflows beyond more than two or three trunks. A number of switch counts made on the final group of an operating toll alternate routing system at Newark, New Jersey, during periods when few calls were lost, have also shown good agreement with the neg- ative binomial distribution. 7.2.2. A Probability Distribution for Combined Overflow Traffic Loads It has been shown in Section 7.2.1 that, at least for load ranges of wide interest, the negative binomial with but two parameters, chosen to agree Fx(§n) 0.01 0.001 TION OMIAL BUTION 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n= NUMBER OF SIMULTANEOUS CALLS Fig. 15 — Probability distributions of overflow traffic with 5 erlangs offered to 1, 2, 5, and 10 trunivs, when all trunks are busy; fitted by negative binomial. 458 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 with mean and variance, gives a satisfactory jfit to the distribution of traffic overflowing a group of trunks. It is now possible, of course, to convohite the various overflows from any number of groups of varying sizes, to obtain a combined overflow distribution. This procedure, how- ever, would be very clumsy and laborious since at each switching point in the toll alternate route system an entirely difl"erent layout of loads and high usage groups would require solution; it would be unfeasible for practical working. We return again to the method of moments. Since the overflows of the several high usage groups will, in general, be independent of one another, the iih semi-invariants Xi of the individual overflows can be combined to give the corresponding semi-invariants A, of their total, Ai — iXi + 2X1 + (27) Or, in terms of the overflow means and variances, the corresponding parameters of the combined loads are Average = A' = ai -{- az + ■ ■ ■ (28) Variance = V = vi + V2 + • • - (29) TRUE DISTRIBUTION NEGATIVE BINOMIAL FITTING DISTRIBUTION 0.001 2 3 4 5 6 7 8 9 10 II 12 13 14 15 n = NUMBER OF SIMULTANEOUS CALLS Fig. 16 — Probability distributions of overflow traffic: 3 erlangs offered to 2 trunks, and 9.6 erlangs offered to 10 trunks. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 459 With the mean and variance of the combined overflows now deter- mined, the negative binomial can again be employed to give an approxi- mate description of the distribution of the simultaneous calls (p{z) offered to the common, or alternate, group. The acceptability of this procedure can be tested in various ways. One way is to examine whether the convolution of several negative binomials (representing overflows from individual groups) is sufficiently well fitted by another negative binomial with appropriate mean and variance, as found above. It can easily be shown that the convolution of several negative bi- nomials all with the same over-dispersion (variance-to-mean ratio) but not necessarily the same mean, is again a negative binomial. Shown in Table I are the distribution components and their parameters of two examples in which the over-dispersion parameters are not identical. The third and fourth semi-invariants of the fitted and fitting distributions, are seen to diverge considerably, as do the Pearsonian skewness and kurtosis factors. The test of acceptability for traffic fluctuation description comes in comparing the fitted and fitting distributions which are shown on Fig. 19. Here it is seen that, despite what might appear alarming dis- 0(n) 0.01 O.OOI TRUE DISTRIBUTION NEGATIVE BINOMIAL FITTING DISTRIBUTION • RANDOM TRAFFIC, 8=1.9 a = 9.6 = 3.84 I 2 3 4 5 6 7 8 9 10 II 12 n = NUMBER OF SIMULTANEOUS CALLS Fig. 17 — Probability density distributions of overflow traffic from 10 trunks, fitted by negative binomial. 460 THE BELL SYSTEM TECHNICAL JOUENAL, MARCH 1956 parities in the higher semi-invariants, the agreement for practical traffic purposes is very good indeed. Numerous throwdown checks confirm that the negative binomial em- ploying the calculated sum-overflow mean and variance has a wide range over which the fit is quite satisfactory for traffic description purposes. Fig. 20 shows three such trunking arrangements selected from a con- siderable number which have been studied by the simulation method. Approximate!}^ 5,000, 3,500, and 580 calls were run through in the three examples, respective!}' . Tlie overflow parameters obtained !)y experiment are seen to agree reasonably well with the theoretical ones from (28) and (29) when the numbers of calls processed is considered. On Fig. 21 are sliown, for the first arrangement of Fig. 20, distributions of simultaneous offered calls in each subgroup of trunks compared with the corresponding Poisson; the agreement is satisfactory as was to be expected. The sum distribution of the overflows from the eight subgroups is given at the foot of the figure. The superposed Poisson, of course, is a poor fit; the negative binomial, on the other hand, appears quite accept- able as a fitting curve. 1.0 0.8 0.6 P 2n 1 TRUNK- a = \.22 3 TRUNKS- a = 2.24 0.4 - 0.2 ■ 1.0 0.8 - 0.6 0.4 0.2 234501 234 n=NUMBER OF SIMULTANEOUS CALLS THEORY OBSD V\ ( ) ( ) AVG 0.67 0.63 VAR 0.77 0.60 i • RANDOM TRAFFIC \, a = 0.67 THEORY OBSD c- ) ( — 1 u AVG 0.55 0.51 VAR 0.77 0.63 \\ • RANDOM TRAFFIC a= D.55 v^^ P^n 1.0 15 TRUNKS- a \ THEORY = 11.46 OBSD '-O .\ ( H ( ) 0.8 *\ AVG 0.81 0.80 '-'•® 'A VAR 1.88 1.42 0.6 "\\, • RANDOM TRAFFIC °-^ \l a=o.8i 0.4 0.4 0.2 0.2 0 • ^'^v,.^^^^ _ , n 9 TRUNKS- a = 6.21 THEORY OBSD ( -) ( ) AVG 0.52 0.46 VAR 1.00 1.48 . RANDOM TRAFFIC a = 0.52 4 68 10 024 68 n=NUMBER OF SIMULTANEOUS CALLS 10 12 Fifj;. 18 — Ovorflow (li.-^ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 n = NUMBER OF SIMULTANEOUS CALLS OFFERED TO THE ALTERNATE ROUTE 17 Fig. 21 — Comparison of theoretical and throwdown dis(ril)utions of simul- taneous calls offered to direct groups and to tlieir first alternate route (OST No. 1). THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 465 read from Fig. 25, will be an integer. This causes no trouble and S should be carried along fractionally to the extent of the accuracy of result de- sired. Reading *S' to one-tenth of a trunk will usually be found sufficient for traffic engineering purposes. Example 1: Suppose a simple graded multiple has three trunks in each of two subgroups, which overflow to C common trunks, where C = 1, P^n OST NO. 6 THEORY OBSD AVG 5.02 5.06 VAR 9.95 7.90 • RANDOM TRAFFIC, a = 5.0 -OBSD -NEGATIVE BINOMIAL 2 4 6 8 10 12 14 16 18 n = NUMBER OF SIMULTANEOUS CALLS P?n --OBSD OST N0.14 THEORY OBSD ( ) ( ) AVG 2.83 2.87 VAR 3.35 3,34 RANDOM TRAFFIC, a = 2.8 -NEGATIVE BINOMIAL 2 4 6 8 10 12 14 16 18 n = NUMBER OF SIMULTANEOUS CALLS Fig. 22 — Combined overflow loads off'ered to alternate-route OST trunks from lirect interoffice trunks; negative binomial theory vs throwdown observations. t«3. V, ta2,y 2y^2 f a,,i Fig. 23 — A full access group divided at several points to examine the traffic character at each point. 466 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 2 or 3. A load of a erlangs is submitted to each subgroup, a having the values 1, 2, 3, 4 or 5. What grade of service will be given? Solution: The load overflowing each subgroup, when a = 1 for example, has the characteristics a = 0.0625 and y = 0.0790. Then A' = 2a = 0.125 and V — 2v = 0.158. Reading on Fig. 26 gives the Ecjuivalent Random values oi A = 1.04 erlangs, S = 2.55 trunks. Reading on Fig. 12.1 with C + *S = 3.55 when C = 1, and A = 1.04, we find a' = 0.0350 and oi' liflx + a-^ = 0.0175. We construct Table II in which loss values pre- dicted by the Equivalent Random (ER) Theory are given in columns (3), (5) and (7). For comparison, the corresponding exact values given by Neovius* are sho\vn in columns (2), (4) and (6). (Less exact loss s (OR X) (a) ta,v (b) fa'.v ta,v (c) fa'.V |A fa, f; la, faafaa 134*35* J Fig. 24 — Various high usage trunk group arrangements producing the same total overflow a, v. figures were given previously by Conny Palm^°. The agreement is seen to be excellent for engineering needs for all values in the table. Example 2: Suppose in Fig. 24(b) the random offered loads and paths are as given in Table III; we desire the proportion of overflow and the overflow load characteristics from an alternate route of 5 trunks. Solution: The individual overflows ai , vi ; a^ , v-i ; and as , Vz are read from Figs. 12 and 13 and recorded in columns (4) and (5) of the table. The a and v columns are totalled to obtain the sum-overflow average A' and variance V . The Equivalent Random load A which, if submitted to S trunks would produce overflow A', V , is found from Fig. 26. Finally, with A submitted io S -\- C trunks the characteristics a' and y', of the load overflowing the C trunks are found. The numerical values obtained * Artificial Traffic Trials Using Digital Computers, a paper presented by G. Neovius at the First International Congress on the Application of the Theory of Probability on Telephone Engineering and Administration, Copenhagen, June, 1955. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 467 Table II^ — Calculation of Loss in a Simple Graded Multiple g = 2, Xi = X2 = S, ai = a2 = a = 1 to 5, C = 1 to 3 T nafl Submitted to each Proportion of Each Subgroup Load which Overflows = a'/(.ai + ai) Subgroup in Erlangs a C = 1 C = 2 C = 3 True ER True ER True ER (1) 1 2 3 5 5 (2) 0.01737 0.11548 0.24566 0.35935 0.44920 (3) 0.0175 0.115 0.246 0.363 0.445 (4) 0.00396 0.05630 0.16399 0.27705 0.37336 (5) 0.0045 0.057 0.163 0.279 0.370 (6) 0.00077 0.02438 0.10212 0.20535 0.30308 (7) 0.00088 0.024 0.103 0.210 0.305 for this example are shown in the lower section of Table III. As before, of course, the "lost" calls are assumed cleared, and do not reappear in the system. Example 3: A load of 18 erlangs is offered through four groups of 10-point selector switches to twenty- two trunks which have been desig- nated as "high usage" paths in an alternate route plan. Which of the trunk arrangements shown in Fig. 27 is to be preferred, and to what extent? Solution: By successive applications of the Equivalent Random method the overflow percentages for each of the three trunk arrange- ments are determined. The results are shown in column 2 of Table IV. The difference in percentage overflow between the three trunk plans is small; however, plan 2 is slightly superior followed by plans 3 and 1 in Table III — Calculation of Overflows from a Simple Alternate Route Trunk Arrangement Subgroup Number Offered Load in Erlangs a Number of Trunks X Overflow Loads a V 1 2 3 3.5 5.7 6.0 15.2 3 6 9 1.41 1.39 0.45 3.25 1.98 2.40 0.85 5.23 Description of load offered to alternate route: A' = 3.25, V = 5.23. ]'"quivalent straight multiple: S = 5.8 trunks, A = 8.00 erlangs (from Fig. 26). Overflow from C = 5 alternate route trunks (enter Figs. 12 and 13 with A = 8.0 and S + C = 10.8: a' = 0.72, v' = 1.48. Proportion of load to commons which overflows = 0.72/3.25 = 0.22. Proportion of offered load which overflows = 0.72/15.2 = 0.0475. 468 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 PROPORTION OVERFLOWING N0.1 E.R THEORY NEOVIUS THROWDOWNS f — ^- • • • • —m- • . • . A = 18 <^ l^ — •- • • • » [ 1 BESK PUNCHED 1 CARDS ■ ■ » -*- 0.123 0.118 0.114 NO. 2 A = 18 < fr: : : 1 1 1 l~: : : n I ■ 1 -^0.113 0.110 0.110 N0.3 f"*' ■ 1 1 n 1 1 -»-0.118 0.113 0.111 l::::imii ' Fig. 27 — Comparison of losses on three graded arrangements of 22 trunks. that order. The results of extensive simulations made by Neovius on the three trunk plans are available for comparison.* The values so obtained are seen to be very close to the ER theoretical ones ; moreover the same order of preference among the three plans is indicated and with closely similar loss differentials between them. 7.3.1. Throwdown Comparisons with Equivalent Random Theonj on Simple Alternate Routing Arrangements with Lost Calls Cleared Results of manuallj' run throwdowns on a considerable number of non-symmetrical single-stage alternate route arrangements are available. Some of these were shown in Fig. 20; they represent part of a projected multi-alternate route layout (to be described later) for outgoing calls from the local No. 1 crossbar Murray Hill-6 office in New York to all other offices in the metropolitan area. The paths hunted over initially are called direct trunks; they overflow calls to Office Selector Tandem (GST) groups, numbered from 1 to 17, which are located in widely dispersed central office buildings in the Greater New York area. * Loc. cit. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 469 Table IV — Loss Comparison of Graded Arrangements Estimates of Percentage of Load Overflowing Plan Number ER Theory Neovius Throwdowns BESK Computer (262144 calls) Punched Cards (10,000 calls) (1) 1 2 3 (2) 12.3 11.3 11.8 (3) 11.81 10.98 11.25 (4) 11.4 11.0 11.1 Table V — Comparison of Theory and Throwdowns for the Parameters of Loads Overflowing the Common Trunks in Single-Stage Graded Multiples OST (Alternate) Route Group No. of Groups of Total No. of Trunks Total Load Offered to Direct Trunks Total Overflow Load from OST Group No. of trunks Direct Trunks in Direct Groups Erlangs Approximate No. of Calls Theory Throwdown no. (in 2.7 hours) a' v' a' v' 1 6 8 91 68.91 4950 2.00 5.50 2.36 6.52 2 3 3 45 37.49 2690 2.10 5.60 2.05 6.36 3 6 6 80 60.62 4355 1.50 4.00 1.30 5.67 4 3 6 52 38.49 2765 2.30 5.20 2.08 6.43 5 3 3 17 12.51 900 0.45 0.83 0.49 1.02 6 4 7 64 48.62 3490 2.50 5.90 2.36 4.88 7 8 12 78 57.42 4125 2.20 5.60 1.71 4.08 8 6 9 16 12.96 930 0.82 1.63 0.81 1.11 9 1 2 22 16.96 1220 1.30 2.60 1.02 1.73 10 5 6 10 9.52 685 0.78 1.40 1.05 2.07 11 8 13 16 16.43 1180 1.90 3.80 2.77 7.29 12 8 9 2 6.88 495 0.70 1.30 0.81 1.83 13 5 15 33 21.42 1540 1.75 3.30 1.16 2.01 14 2 7 11 8.05 580 1.46 2.20 1.63 2.14 15 9 15 8 11.97 860 1.60 3.25 1.55 4.12 16 11 22 34 27.46 1970 1.75 4.00 1.34 2.26 17 3 7 4 5.81 420 1.53 2.31 1.43 1.80 26.64 58.42 25.92 61.32 In Table V are given certain descriptive data for the 17 OST trunk arrangements showing numbers of legs of direct trunks, total direct trunks, the offered erlangs and calls, and the mean and variance of the alternate routes' overfiovvs, as obtained by the ER theory and by throwdowns.* The throwdown a' and v' values of the OST overflow * Additional details of this simulation study are given in Section 7.4. 470 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 i.a ^O 0.2 EQUIVALENT RANDOM THEORY ERLANG THEORY- 1 2 3 4 5 6 7 8 9 10 It 12 13 14 15 16 17 ALTERNATE ROUTE (OST) NUMBER Fig. 28 — Comparison of theoretical and throwdown overflows from a number of first alternate routes. were obtained by 36-second switch counts of those calls from each OST group which had come to rest on subsequent alternate routes. On Fig. 28 is shown a summary of the observed and calculated pro- portions of "lost" to "offered" traffic at each OST alternate route group. As may be seen from the figure and the last four columns of Table V, the general agreement is quite good ; the individual group variations are probably no more than to be expected in a simulation of this magnitude. An assumption of randomness (which has sometimes been argued as returning when several overflows are combined) for the load offered to the OST's gives the Erlang Ei loss curve on Fig. 28. This, as was to be expected, rather consistently understates the loss. Since "switch-counts" were made on the calls overflowing each OST, the distributions of these overflows may be compared with those esti- mated by the Negative Binomial theory having the mean and variance predicted abo\'e for the overflow. Fig. 29 shows the individual and cumu- lative probability distributions of the overflow simultaneous calls from the first two OST alternate routes. As will be seen, the agreement is quite good even though this is traffic which has been twice "non-ran- domized." Comparison of the observed and calculated overflow means and variances in Table V indicates that similar agreement between observed and theoretical fitting distributions for most of the other OST's would be found. 7.3.2. Comparison of Equivalent Random Theory with Field Results on Simple Alternate Routing Arrangements _ Data were made available to the author from certain measurements made in 1941 by his colleague C. Clos on the automatic alternate routing trunk arrangement in operation in the Murray Hill-2 central office in New York. Mr. Clos observed for one busy hour the load carried on THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 471 several of its OST alternate rovite groups (similar to those shown in Table V for the Murray Hill-6 office, but not identical) by means of an electromechanical switch-counter having a six-second cycle. During each hour's observation, numbers of calls offered and overflowing were also recorded. Although the loads offered to the corresponding direct trunks which ()^'erflowed to the OST group under observation were not simultaneously measured, such measiu'ements had been made previously for several hours so that the relative contribution from each direct group was closely known. In this way the loads offered to each direct group which produced the total arriving before each OST group could be estimated with considerable assurance. From these direct group loads the character (mean and variance) of the traffic offered to and overflowing the OST's was predicted. The observed proportion of offered traffic which over- flowed is shown on Fig. 30 along with the Equivalent Random theory prediction. The general agreement is again seen to be fairly good al- though with some tendency for the ER theory to predict higher than observed losses in the lower loss ranges; perhaps the disparity on in- (n) 0.5 0.4 0.3 0.2 0.1 0 OST N0.1 THEORY OBSD AVG VAR 2.00 2.36 5.50 6.52 RANDOM TRAFFIC ^--NEGATIVE BINOMIAL -THROWDOWN OST NO. 2 THEORY OBSD AVG VAR 2.10 5.60 2.05 6.36 >RAND0M TRAFFIC THROWDOWN -NEGATIVE BINOMIAL 10 15 0 5 n = NUMBER OF SIMULTANEOUS CALLS 15 p^n -NEGATIVE BINOMIAL -THROWDOWN -NEGATIVE BINOMIAL THROWDOWN 10 15 0 5 n = NUMBER OF SIMULTANEOUS CALLS 15 Fig. 29 — Distributions of loads overflowing from first alternate (OST) groups; negative binomial theory versus throwdown observations. 472 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195G dividual OST groups is within the limits one might expect for data based on single-hour observations and for which the magnitudes of the direct group offered loads required some estimation. The assumption of random traffic offered to the OST gives, as anticipated, loss predictions (Erlang £"1) consistently below those observed. More recently extensive field tests have been conducted on a working toll automatic alternate route system at Newark, New Jersey. High usage groups to seven distant large cities o\'erflowed calls to the New- ark-Pittsburgh alternate (final) route. Data describing the high usage groups and typical system busy hoiu- loads are given in Table ^T. (The loads, of course, varied considerably from day to day.) The size of the Pittsburgh route varied over the six weeks of the 1955 tests from 64 to 71 trunks. Altogether the system comprised some 255 intertoll trunks. Observations were made at the Newark end of the groups by means of a Traffic Usage Recorder — making switch counts every 100 seconds — and by peg count and o^'erflow registers. Register readings were photo- graphically recorded by half-hourly, or more frequent, intervals. To ^- r^ t: 0 0 ro 0 CI (^ m O) ro Q Q n LU UJ UJ LU LU CD LU (D 03 NO. TRUNKS 13 12 8 7 3 8 3 4 3 OFFERED JavG 7.55 7.19 5.22 3.81 2.06 7.79 2.36 4.09 2.4 LOAD |vAR 13.58 15.66 6.59 7.30 2.51 18.54 2.77 4.59 5.90 Fig. 30 — -Observed tandem ovciflow.s in nlicriKilc llill-2 (New York) 1940-1941. loulc study at Murray THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 473 Table VI — High Usage Groups and Typical System Busy Hour Loads High Usage Group, Newark to: Length of Direct Route (Air Miles) Nominal Size of Group (Number of Trunks) Typical Offered Load (erlangs) Baltimore 170 560 395 1375 470 1100 1170 18 42 27 33 37 26 5 19 Cincinnati Cleveland Dallas Detroit Kansas City New Orleans 43 26 34 36 23 4 compare theory with the observed overflow from the final route, esti- mates of the offered load A' and its ^-ariance V are required. In the present case, the total load offered to the final route in each hour was estimated as A' = Average of Offered Load Peg Count of Calls Offered to Pittsburgh Group (Peg Count of Offered Calls) — (Peg Count of Overflow Calls) X Average Load Carried by Pittsburgh Group The variance V of the total load offered to the final route was estimated for each hour as V' = Variance of Offered Load 7 7 = A' — 2 «i + 2 Vi i=l where «» and Vi are, respectively, the average and variance of the load overflowing from the tth high usage group. (The expression. A' — 7 ^ «i , is an estimate of the average — and, therefore of the variance 1=1 — of the first-routed traffic offered directly to the final route. Thus the total variance, V, is taken as the sum of the direct and overflow com- ponents.) Using A', V and the actual number, C, of final route trunks in service, the proportion of offered calls expected to overfloAv was calcu- lated for the traffic and trunk conditions seen for 25 system busy hours from February 17 to April 1, 1955 on the Pittsburgh route. The results are displayed on Fig. 31, where certain traffic data on each hour are given in the lower part of the figure. The hours are ordered — for con- venience in plotting and viewing — by ascending proportions of calls overflowing the group; observed results are shown by the double line 474 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 0.001 3 5 7 9 11 13 15 17 19 21 23 25 NO. P'BGH TRKS 71 70 65 71 65 71 65 69 64 64 70 65 64 71 68 65 64 65 64 70 65 65 65 65 65 HOURS BY AMT. OF OBS'D loss EST'd LOAD fAVG. 50 54 55 56 55 63 55 58 54 54 68 60 63 74 76 74 76 83 91 102 109105 101 115124 OFFERED War. 82 95 85 89 98 101 84 98 97 89110 10588125 121140114 141 175182 170 176 179 199197 Fig. 31 — Final route (Newark-Pittsburgh) overflows in 1955 toll alternate^ route study. I THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 475 curve. The superposed single line is the corresponding estimate by EE, theory of the hour-to-hour call losses. As may be seen, theory and ob- servation are in good agreement both point by point and on the average over the range of losses from 0.01 to 0.50. The dashed line shows the prediction of final route loss for each hour on the assumption that the offered traffic A' was random. Such an assumption gives consistently low estimates of the existing true loss. As of interest, a series of heavy dots is included on Fig. 31. These are the result of calculating the Poisson Summation, P{C,L), where L is the average load carried on, rather than offered to, the C trunks. It is inter- esting that just as in earlier studies in this paper on straight groups of intertoll trunks (for example as seen on Fig. 7), the Poisson Summation with load carried taken as the load offered parameter, gives loss values surprisingly close to those observed. Also, as before, this summation has a tendency to give too-great losses at light loadings of a group and too- small losses at the heavier loadings. ; 7.4 Prediction of Traffic Passing Through a Midti-Stage Alternate Route Network I In the contemplated American automatic toll switching plan, wide I advantage is expected to be taken of the efficiency gains available in i multi-alternate routing. Thus any procedure for traffic analysis and prediction needs to be adaptable for the . more complex multi-stage arrangements as well as the simpler single-stage ones so far examined. Extension of the Equivalent Random theory to successive overflows is easily done since the characterizing parameters, average and variance, of the load overflowing a group of paths are ahvays available. Since few cases of more than single-stage automatic alternate routing are yet in operation in the American toll plant, it is not readily possible to check an extension of the theoiy with actual field data. Moreover col- lecting and analyzing observations on a large operating multi-alternate route system would be a comparatively formidable experiment. However, in New York city's local interoffice trunking there is a very considerable development of multi-alternate routing made possible by the flexibility of the marker arrangements in the No. 1 crossbar switching system. None of these overflow arrangements has been observed as a whole, simultaneously and in detail. The Murray Hill-2 data in OST groups reviewed in Section 7.3.2 were among the partial studies which have been made. In connection with studies made just prior to World War II on these 476 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table VII — Sum of Direct Group Overflow Loads, Offered to OST's Average. Variance Theory 86.06 129.5 Observed 87.12 127.4 local multi -alternate route systems, a throwdown was made in 1941 on a proposed trunk plan for the Murray Hill-6 office. The arrangement of : trunks is shown on Fig. 32. Three successive alternate routes, Office Selector Tandems (OST), Crossbar Tandem (XBT), and Suburban: Tandem (ST), are available to the large majority of the 123 direct trunk groups leading outward to 169 distant offices. (The remaining 46 parcels of traffic did not have direct trunks to distant offices but, as indicated on the diagram, offered their loads directly to a tandem group.) A total of 726 trunks is involved, carrying 475 erlangs of traffic. A throwdown of 34,001 offered calls corresponding to 2.7 hours of traffic was run. Calls had approximate exponential holding times, averag- ing 135 seconds. Records were kept of numbers of calls and the load from the traffic parcels offered to each direct group, as they were carried or passed beyond the groups of paths to which they had access. Loads car- ried by each trunk in the system were also observed by means of a 36- second "switch-count." (The results on the 17 OST groups reported in Section 7.3.1 were part of this study.) Comparisons of observation and theory which are of interest include the combined loads to and overflowing the 17 OST's. Observed versus calculated parameters (starting with theory from the original direct group submitted loads) are given in Table VII. The agreement is seen to be very good. The corresponding comparison of total load from all the OST's is given in Table VIII. Again the agreement is highly satisfactory. Not all of the overflow from the OST's was offered to the 22 crossbar tandem trunks; for economic reasons certain parcels by-passed XBT andf were sent directly to Suburban Tandem.* This posed the problem of breaking off certain portions of the overflow from the OST's, to be added"' again to the overflow from XBT. An estimate was needed of the contri bution made by each parcel of direct group traffic to any OST's over flow. These were taken as proportional to the loads offered the OST by each direct group (this assumes that each parcel suffers the same over- * In the toll alternate route system by -passing of this sort will not occur. Tt U'lnntuunu u L \'^\\\\\\\'' \va p^^^^^ \\nii\^ S m T^ m: Tr± 7 J '''.': ; , ±±± ^PP tf+Ff- 4^ :« tn ^ '/V//'/-'//./'/''///;^: fl NO. 16 NO. 17 SPECIAL TANDEM tt inttttiiittttMitiftt ttinit tit t tttttttttt ttttt (V OJ ^O ^^»*^u-^^OsO r- u-^Ty fv^ O OJ fVOJ O^ rNvO TO-* -t OJ C^ V\vO (V-*fV C^OiJ^<*\<*\fH TO^OC^OJU^r^ ^ -^ i/n f^ OtO to r^*rfc"~'-*r^r^rsi CT^^OJi-tiH OO -* r^ i-H ^ O rH ^ rH.-H .-H rH rH rHrH O O O OO O O O rH (-1 ^ O O O O r^rH O O OOOOOOOOOO OOOOO 27.46 5.81 5.44 0.31 5.99 1.96 tOvO -JvO rH fV f'^0^_JO>JD O^ r^rH (Vr- ifNvO »A(NJryNO -*fV (NiTO Of^Of^O^ -* ^ ^ -* (N* ONf^ "^tO C^ t*^C^ <^ *'^ C^CM^^TO >-i t£) E-« cr: tn CO ,_j ti3 < a. i*; M MJM*s<:w mmooo crossbar office. FINAL ,0 TANDEM TRUNKS 5 INTERMEDIATE TANDEM TRUNKS 10 FIRST ALTERNATE 5 ■ nUTE(OST) TRUNKS SUBURBAN TANDEM N0.1 N0.2 N0.3 N0.4 N0.5 N0.6 NO. 7 N0.8 N0.9 NO.IO N0.11 N0.12 N0.I3 N0.14 DIRECT INTEROFFICE TRUNKS 15 r. 10rzE 5:E: 1 --- m ill ]M \M\ Vw timii ttimlmtt tiitifti! ti flmi tttitttttiiit tttitttft tmmmntti tiiittt tmiitittittit tmtmitmitmtni imttt tii i ititnttit ftitt Y I k.ni.L' IMnu ■-! fv IV iH r>j >0 J t^ i-t^.H p^tov>r^s>r^ O (v rt o (N to mr^pj O cvj ^ -tto ''^-4 O rsi tT>C'tJ't~-t^C^NO&''">-i -* (m lAi-i r\ O O- r~ f^ 'O [ERLAN6S) f|JOOO£>r-t^O 'OcvtJ. rHC'O'OO-Ov i^-0~0,0'0 J ■/■OO Ot^fv-^Cr'ovrH O<0 t^i-^f^r^rufOcjpHu^ pJc-j^^^^^OcJ (> 68.91 37.49 60.62 38.49 12.51 48.62 57.42 DESTINATION *Cr^-*t^r>^,^ r.^>r OFFICES S£S253S£ gS? ~-»f«Ot- Jr" mff-tO toO^iT ^rHrHO rH p- ^ -i f-i rt -J r4 -^ ,H ^d O -4 rH r4 O O O O O O «/^ (V cJ .-* O O O -- r+^ f"* O OO O (VrHodcHO ^ddOrHOOOOOOOOOO 12.96 16.96 9.52 16.43 6.88 21.42 8.05 11.97 27.46 5.81 5.44 0.31 5.99 1. T Q Ita. tfc< C3 O Z OW < f) Fig. 32. — Multi-alternate route trunking arrangcinenl at Murray Hill — 6 (New York) local No. 1 crossbar office. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 477 flow probability). The variance of this overflow portion by-passing XBT was estimated by assigning to it the same variance-to-average ratio as was found for the total load overflowing the OST. Subtracting the means and variances so estimated for all items by-passing XBT, left an approxi- mate load for XBT from each OST. Combining these corrected overflows gave mean and variance values for offered load to XBT, Observed values Table VIII - -Sum of Loads Overflowing OST's Theory Observed Avftraere 26.64 58.42 25.92 Variance 61.32 Table IX — Load Offered to Crossbar Tandem I Average. Variance Theory 25.18 47.67 Observed 25.51 56.10 0.10 r -RANDOM TRAFFIC -THROWDOWN ,--NEGATIVE BINOMIAL to 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS P^n THEORY OBSD 1.0 I — - — -^ .^^^ ( ) ( ) ^X AVG 25.18 25.51 0.8 ^V VAR 47.67 56.10 0.6 VS. 0.4 0.2 0 , , 10 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS Fig. 33 — Distribution of load offered to crossbar tandem trunks; negative bi- nomial theory versus throwdown observations. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 477 flow probability). The variance of this overflow portion by-passing XBT was estimated by assigning to it the same variance-to-average ratio as was found for the total load overflowing the OST. Subtracting the means ' and variances so estimated for all items by-passing XBT, left an approxi- mate load for XBT from each OST. Combining these corrected overflows gave mean and variance values for offered load to XBT, Observed values Table VIII - - Sum OF Loa Ds Overflowing OST's Theory Observed Average Variance 26.64 58.42 25.92 61.32 Table IX . — Load Offered to Crossbar Tandem Theory Observed Average 25.18 47.67 25.51 Variance 56.10 0.10 r ^-.--RANDOM TRAFFIC -THROWDOWN --NEGATIVE BINOMIAL 10 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS THEORY OBSD ( ) ( ) AVG 25.18 25.51 VAR 47.67 56.10 Pin 10 20 30 40 50 n = NUMBER OF SIMULTANEOUS CALLS Fig. 33 — Distribution of load offered to crossbar tandem trunks; negative bi- nomial theory versus throwdown observations. 478 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table X — Load Overflowing Crossbar Tandem Average. Variance , Observed 6.47 33.48 and those calculated (in the above manner) are given in Table IX. Fig. 33 shows the distribution of XBT offered loads, observed and calcu- lated. The agreement is very satisfactory. The random traffic (Poisson) distribution, is of course, considerably too narrow. In a manner exactly similar to previous cases, the Ecjuivalent Random load method was applied to the XBT group to obtain estimated param- eters of the traffic overflowing. Comparison of observation and theory at this point is given in Table X. Fig. 34 shows the corresponding observed and calculated distributions 0.15 0.10 f(n) 0.05 )RANDOM TRAFFIC THEORY OBSD AVG 6.55 6.47 VAR 23.80 33.48 ^'NEGATIVE BINOMIAL 0 5 10 15 20 25 30 35 n=NUMBER OF SIMULTANEOUS CALLS P^n _^^RANDOM TRAFFIC --NEGATIVE BINOMIAL THROWDOWN 0 5 10 15 20 25 30 35 n = NUMBER OF SIMULTANEOUS CALLS Fig. 34 — Distribution of calls from crossbar tandem trunks; negative binomial theory versus throwdown observations. ! THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 479 of siniiiltaneoiis calls. The agreement again is reasonably good, in spite of the considerable disparity in variances. The overflow from XBT and the load which by-passed it, as well as some other miscellaneous parcels of traffic, were now combined for final offer to the Suburban Tandem group of 17 trunks. The comparison of parameters here is again available in Table XI. On Fig. 35 are shown the observed and calculated distributions of simultaneous calls for the load offered to the ST trunks. The agreement is once again seen to be very satisfactory. We now estimate the loss from the ST trunks for comparison with the actual 'proportion of calls which failed to find an idle path, and finally Table XI — Load Offered to Suburban Tandem Average. . Variance . Theory 15.38 42.06 Observed 14.52 48.53 THEORY OBSD f(n) P^n 10 20 30 40 n = NUMBER OF SIMULTANEOUS CALLS I.O ^ ^ \ 0.8 " ^ , --NEGATIVE BINOMIAL 0.6 V ^-THROWDOWN \ \ 0.4 0.2 x^^^ 0 1 " -r-^ 10 20 30 40 n^NUMBER OF SIMULTANEOUS CALLS 50 Fig. 35 — Distribution of load offered to suburban tandem trunks; negative linomial theory versus throwdown observations. 480 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Table XII - — Grade of Service on ST Group Theory Obser- vation Observation Load submitted (erlangs) Load overflowing (er- langs) Proportion load over- flowing 15.38 3.20 0.209 14.52 2.63 0.181 Number of calls sub- mitted 1057 Number of calls over- flowing 200 Proportion of calls over- flowing 0.189 Table XIII — Grade of Service on the System Total load submitted Total load overflowing Proportion of load not served Theory Observed 475 erlangs 3.20 erlangs 0.00674 34,001 calls 200 calls 0.00588 compare the proportions of all traffic offered the system which failed to find a trunk immediately. See Tables XII and XIII. After these several and varied combinations of offered and overflowed loads to a system of one direct and three alternate routes it is seen that 'i the final prediction of amount of load finally lost beyond the ST trunks is gratifyingly close to that actually observed in the throwdown. The prediction of the system grade of service is, of course, correspondingly good. It is interesting in this connection to examine also the proportions I overflowing the ST group when summarized by parcels contributed from the several OST groups. The individual losses are shown on Fig. 36; they appear well in line with the variation one would expect from group to group with the moderate numbers of calls which progressed this far through the multiple. 0.4 0.3 octr o^ 0.2 So a ^0.1 ,-THEORY =0.21 ._>. • • --AVG OBSD = 0.19 12 4 6 8 10 12 14 16 18 20 OST GROUP NUMBER Fig. 36 — Overflow calls on third alternate (ST) route. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 481 7.4.1 Correlation of Loss with Peakedness of Components of Non-Ran- dom Offered Traffic Common sense suggests that if several non-random parcels of traffic are combined, and their joint proportion of overflow from a trunk group is P, the parcels which contain the more peaked traffic should experience overflow proportions larger than P, and the smoother traffic an overflow proportion smaller than P. It is by no means clear however, a priori, the extent to which this would occur. One might conjecture that if any one parcel's contribution to the total combined load is small, its loss would be caused principally by the aggregate of calls from the other parcels, and consequently its own loss would be at about the general average loss P, and hence not very much determined by its own peakedness. The Murray Hill-6 throwdowai results may be examined in this respect. The mean and variance of each OST-parcel of traffic, for example, arriving at the final ST route was recorded, together with, as noted before, its own proportion of overflow from the ST trunks. The variance/mean over- dispersion ratio, used as a measure of peakedness, is plotted for each parcel of traffic against its proportion of loss on Fig. 37. There is an un- doubted, but only moderate, increase in proportion of overflow with increased peakedness in the offered loads. It is quite possible, however, that by recognizing the differences be- tween the service given various parcels of traffic, significant savings in final route trunks can be effected for certain combinations of loads and trunking arrangements. Of particular interest is the service given to a parcel of random traffic offered directly to the final route when compared 04 o oc_l 0.3 UJUJ >u °% °ia2|- zo o< OO0.1 o a. • • • 0.5 1.0 1.5 2.0 2.5 3.0 3.5 V/a OF EACH OST PARCEL REACHING ST TRUNKS Fig. 37 — Effect of peakedness on overflow of a parcel of traffic reaching an ilternate route. 482 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 with that received by non-random parcels overflowing to it from high usage groups. 7.5 Expected Loss on First Routed Traffic Offered to Final Route The congestion experienced by the first-routed traffic offered to the final group in a complex alternate route arrangement [such as the right hand parcels in Figs. 10(c) and (d)] \vill be the same as encountered in a series of random tests of the final route by an independent observer, that is, it will be the proportion of time that all of the final trunks are busy. As noted before, the distribution of simultaneous calls n (and hence the congestion) on the C final trunks produced by some specific arrange- ment of offered load and high usage trunks can be closely simulated by that due to a single Equivalent Random load offered to a straight group of aS -f C trunks. Then the proportion of time that the C trunks are busy in such an equivalent system provides an estimate of the corres- ponding time in the real system ; and this proportion should be approxi- mately the desired grade of service given the first routed traffic. Brockmeyer has given an expression (his equation 36) for the pro- portion of time, Rx , in a simple S -\- C system with random offer A, and "lost calls cleared," that all C trunks are busy, independent of the condition of the *S-trunks: R, = f{S,C,A) = Ii,x,s+cKA) — — where m=o \ m / (S — m However, (rdS) is usually calculated more readily step-by-step using the formula _^ ^^ ;-.^ V '*^^>v •^ 'N^ Y^x. X \ V vv \ \ \ \ \ \ \ \ A = 30 C = 15 s \s\^ \ .' ^ \\v \ n, -^ ^ f- \ >■ \ \ \ \ "S^ \ V \ \ \ \ \ [a \ ""') \ \ \ ^ \X' ^"' o\ \ \ \ \ \ \ A?T iO \ \ \ \ \ R, \'C-. \ \ I A = 20^ rc = io^^ V \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ \ \ \ \ \ \ \ \ ' \^ \ \ \ \ \ A \ \ V A = 10,^ C = io \ \ \ \ \ 10 15 20 25 S = *equivalent"number of paths 30 35 Fig. 38 — Comparison of Ri and R2 losses under various load and trunk con- ditions. Table XIV— The R2/R1 Ratios for A = 2C A C Ri/Ri when R2 = 0.05 10 20 30 5 10 15 10.6 3.25 2.44 THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 485 Table XV — Comparison of E.R. Theory and Throwdowns on Disparity of Loss Between High Usage Overflow and Random Offer to a Final Group (8 trunks in each high usage group; 9 final trunks serving 2.0 erlangs high usage overflow and 2.0 erlangs first routed traffic.) Number of Groups of 8 High Usage Trunks ER Theory {A' = 4.0) Tange V A 5 R2=a7A' i?i Rh.u, ~ 2R-L- Ri Rh.u.—Rl= 2{R2 - Ry) Throwdown Rh.u. - Ri (1) (2) (3) (4) (5) (6) (7) (8) (9) 1 5.77 7.51 4.17 0.0375 0.0251 0.0499 0.0248 0.0180 2 5.80 7.50 4.25 0.0383 0.0255 0.0511 0.0256 0.0247 3 5.74 7.44 4.08 0.0369 0.0248 0.0490 0.0242 0.0286 4 5.68 7.30 3.91 0.0362 0.0247 0.0477 0.0230 0.0276 5 5.64 7.20 3.80 0.0355 0.0242 0.0468 0.0226 0.0245 6 5.58 7.06 3.64 0.0350 0.0240 0.0460 0.0220 0.0221 7 5.55 7.00 3.56 0.0345 0.0238 0.0452 0.0204 0.0202 8 5.51 6.91 3.45 0.0335 0.0236 0.0434 0.0198 0.0188 9 5.47 6.81 3.34 0.0325 0.0231 0.0419 0.0188 0.0177 10 5.45 6.76 3.29 0.0312 0.0225 0.0399 0.0174 0.0166 Limited data are available showing the disparity of Ri and Ro in actual operation in a range of load and trunk values well beyond those for which Ri values have been calculated. Special peg count and over- flow registers were installed for a time on the final route during the 1955 Newark alternate route tests. These gave separate readings for the calls from high usage groups, and for the first routed Newark to Pittsburgh calls. Comparative losses for 17 hours of operation over a wide range of loadings are shown on Fig. 39. The numbers at each pair of points give the per cent of final route offered traffic which was first routed (random). In general, approximately equal amounts of the two types of traffic were offered. In 6 of the hours almost identical loss ratios were observed, in 7 hours the overflow-from-high-usage calls showed higher losses, and in 4 hours lower losses, than the corresponding first routed calls. The non-random calls clearly enjoyed practically as good service as the random calls. This result is not in disagreement with what one might expect from theory. To compare directly with the Newark-Pittsburgh case we should need curves on Fig. 38 expanded to correspond to A', V values of (50, 85) to (120, 200). Examining the mid-range case of C = 65, A' = 70, V = 120, we find A = 123, >S = 54. Here A is approximately 2C; extrapolat- ing the A = 2C curves of Fig. 38 to these much higher values of A and C suggests that R2/R1 w^ould be but little different from unity. 486 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 It is clear from the above theory, throwdowns, and actual observa- tion that there are certain areas where the service differences given first routed and high usage trunk overflow parcels of traffic are significant. In Section 8, where practical engineering methods are discussed, curves are presented which permit recognition of this fact in the determination of final trunk requirements. 7.6 Load on Each Trunk, Particularly the Last Trunk, in a Non-Slipped Alternate Route In the engineering of alternate route systems it is necessary to deter- mine the point at which to limit a high usage group of trunks and send the overflow traffic via an alternate route. This is an economic problem whose solution requires an estimate of the load which will be carried on 1.0 0.5 z o il' 0.05 a. UJ > o z o I- cc o a. O a. a- O.OiO 0.005 0.00)0 6 64 56 8' 57 61( OL 65^ 69, 40 56 o 50 ,58 41 58 69 8 64 52 s 38 6 66 49 8 52 O FIRST ROUTED TRAFFIC (NUMBERS INDICATE PER CENT OF TOTAL WHICH IS FIRST ROUTED) • OVERFLOW TRAFFIC FROM 7 HIGH USAGE GROUPS 60 70 80 90 100 110 120 A'= ESTIMATED OFFERED LOAD TO PITTSBURGH IN ERLANGS (INCLUDING RETRIALS) Fig. 39 — Comparison of losses on final route (Newark to Pittsburgh) for high usage overflow and first routed traffic. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 487 the last trunk of a straight high usage group of any specified size, carry- ing either first or higher choice traffic or a mixture thereof.* The Equivalent Random theory readily supplies estimates of the loads carried by any trunk in an alternate routing network. After having found the Equivalent Random load A offered to *S + C trunks which corresponds to the given parameters of the traffic offered to the C trunks, it is a simple matter to calculate the expected load i on any one of the C trunks if they are not slipped or reversed. The load on the ith trunk in a simple straight multiple (or the S + jth. in a divided multiple of *S lower and C upper trunks), is A- = Is+j = A[E^,s+j-M) - Ex,s+j{A)] (33) where Ei,n(A) is the Erlang loss formula. A moderate range of values of ■Ci versus load A is given on Figure 40. f Using this method, selected comparisons of theoretical versus observed loads carried on particular trunks at various points in the Murray- Hill-6 throwdown are shown in Fig. 41 ; these include the loads on each of the trunks of the first two OST groups of Fig. 32, and on the second and third alternate routes, crossbar and suburban tandem, respectively. The agreement is seen to be fairly good, although at the tail end of the latter two groups the observed values drop aw^ay somewhat from the theoretical ones. There seems no explanation for this beyond the possi- bility that the throwdown load samples here are becoming small and might by chance have deviated this far from the true values (or the arbitrary breakdown of OST overflows into parcels offered to and by- passing XBT may well have introduced errors of sufficient amount to account for this disparity). As is well known, (33) gives good estimates of the loads carried by each trunk in a high usage group to which random (Poisson) traffic is offered; this relationship has long been used for the purpose in Bell System trunk engineering. 8. PRACTICAL METHODS FOR ALTERNATE ROUTE ENGINEERING To reduce to practical use the theory so far presented for analysis of alternate route systems, working curves are needed incorporating the * The proper selection point will be where the circuit annual charge per erlang of traffic carried on the last trunk, is just equal to the annual charge per erlang of traffic carried by the longer (usually) alternate route enlarged to handle the overflow traffic. t A comprehensive table of /< is given by A. Jensen as Table IV in his book "Moe's Principle," Copenhagen, 1950; coverage is for / ^ 0.001 erlang, z = 1(1)140; A = 0.1(0.1)10, 10(1)50, 50(4)100. Note that n + 1, in Jensen's notation, equals i here. 488 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 6 <0 6 ID 6 6 6 soNvibB Ni viNnai Hi-n 3hi no agiaavD avon = '-Tf THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 489 pertinent load-loss relationships. The methods so far discussed, and proposed for use, will be briefly reviewed. The dimensioning of each high usage group of trunks is expected to be performed in the manner currently in use, as described in Section 7.6. The critical figure in this method is the load carried on the last high usage trunk, and is chosen so as to yield an economic division of the offered load between high usage and alternate route trunks. Fig. 40 is one form of load-on-each-trunk presentation suitable for choosing eco- nomic high usage group size once the permitted load on the last trunk is established. The character (average a and variance v) of the traffic overflowing each high usage group is easily found from Figs. 12 and 13 (or equivalent - OST GROUP NO. 2 1.U OST GROUP NO.l 0.5 - ^ 0 4 5 6 TRUNK NUMBER CROSSBAR TANDEM GROUP Z 0 O 2 4 6 8 10 12 14 16 18 20 22 TRUNK NUMBER SUBURBAN TANDEM GROUP 12 4 6 8 10 12 14 16 TRUNK NUMBER Fig. 41 — Comparison of load carried by each alternate route trunk; theory versus throwdowns. 490 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 tables). The respective sums of the overflow a's and v^s, give A' and V by (28) and (29); they provide the necessary statistical description of traffic offered to the alternate route. According to the Equivalent Random method for estimating the alter- nate route trunks required to provide a specified grade of service to the overflow traffic A', one next determines a random load A which when submitted to S trunks will yield an overflow with the same character {A', V) as that derived from the complex system's high usage groups. An alternate route of C trunks beyond these S trunks is then imagined. The erlang overflow a', with random offer A, to S + C trunks is found from standard i^i-formula tables or curves (such as Fig. 12). The ratio R2 = a! I A' is a first estimate of the grade of service given to each parcel of traffic offered to the alternate route. As discussed in Sec- tion 7.5, this service estimate, under certain conditions of load and trunk arrangement, may be significantly pessimistic when applied to a first routed parcel of traffic offered directly to the alternate route. An improved estimate of the overflow probability for such first routed traffic was found to be R\ as given by (30). 8,1 Determination of Final Group Size with First Routed Traffic Offered Directly to the Final Group When first routed traffic is offered directly to the final group, its service Ri will nearly always be poorer than the overall service given to those other traffic parcels enjoying high usage groups. The first routed traffic's service will then be controlling in determining the final group size. Since Ri is a function of *S, C and A in the Equivalent Random solution (30), and there is a one-to-one correspondence of pairs of A and S values with A' and V values, engineering charts can be constructed at selected service levels Ri which shoAv the final route trunks C required, for any given values of A' and V. Figs. 42 to 45 show this relation at service levels of Ri = 0.01, 0.03, 0.05 and 0.10, respectively.* * On Fig. 42 (and also Figs. 46-49) the low numbered curves assume, atjfirst sight, surprising shapes, indicating that a load with given average and variance would require fewer trunks if the average were increased. This arises from the sensitivity of the tails of the distribution of offered calls, to the V'/A' peaked- ness ratio which, of course, decreases with increases in A'. For example, with C = 4 trunks and fixed V = 0.52, the loss rapidly decreases with increasing A': A' V'/A' A S a' a' /A' 0.28 0.33 0.40 0.52 1.86 1.58 1.30 1.00 6.1 3.0 1.42 0.52 10. 5.0 2.03 0 0.0155 0.0081 0.0036 0.0008 0.055 0.025 0.009 0.002 THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 491 These four Ri levels would appear to cover the most used engineering range. For example, if the traffic offered to the final route (including the first routed traffic) has parameters A' = 12 and V ^ 20, reading on Fig. 43 indicates that to give P = 0.03 "lost calls cleared" service to the first routed traffic, C = 19 final route trunks should be provided. (For random traffic (F' = A' = 12), 17.8 trunks would be required.) Other charts, of course, might be constructed from which Ri could be read for specific values of A', V and C. They would become voluminous, however, if a wide range of all three variables were required. 8.2 Provision of Trunks Individual to First Routed Traffic to Equalize Service If the difference between the service Ri given the first routed parcel of traffic and the service given all of the other parcels, is material, it may be desirable to take measures to diminish these inequities. This may readily be accomplished by setting aside a number of the otherwise full access final route trunks, for exclusive and first choice use of the first routed traffic. High usage groups are now provided for all parcels of traffic. The alternate route then services their combined overflow. The overall grade of service given the ith. parcel of offered traffic in a single stage alter- nate route system will then be approximately '* Pi = Ei,Xi{ai)R2 = EiXiia.)^, (34) Thus the service will tend to be uniform among the offered parcels when all send substantially identical proportions of their offered loads to the alternate route. And the natural provision of "individual" trunks for the exclusive use of the first routed traffic would be such that the same pro- portion should overflow as occurs in the associated high \isage groups. This procedure cannot be followed literally since high usage group size is fixed b}^ economic considerations rather than any predetermined overflow value. The resultant overflow proportions will commonly vary over a considerable range. In this circumstance it would appear reason- able to estimate the objective overflow proportion to be used in estab- lishing the individual group for the first routed traffic, as some weighted average h of the overflow proportions of the several high usage groups. Thus with weights g and overflow proportions h, h = ^'^' + ^'^' + • ' • (35) ^1 + ^2+ • • • * Although not exact, this equation can probably be accepted for most engi- neering purposes where high usage trunks are provided for each parcel of traffic. \ \ •* \ / \ O-l SO 0 o N ■-^ / \ . V— k -^ / \ a — ' ^^ GO eg . r— ^ "< \ b \ ;;;;::; "^ . — — r"— b' q to o 6 si - ■ -— [ _^ — — ' \_^^ 1 rj \ >. s; imp- 0^ S \ O/V \ ■*> ^ ^ \ 2\/ K en "t o (\J 7 < 1 (M Ct (\J UJ Z o (\j 1- -) C) 00 tr 1 < ID z or Al Lll 11 11 o O Q < () _l on UJ O < (r LU > < II ^ < tn It t\J o to (C ^f r\j O n ro n ri (M (\J OJ OJ i\J ainoH nvNid Oi aaaaddo avoi dO 30NviavA=,A - \ y\ 0 0.5 1.0 \ \ \ . \ \ \ "V A \ X ^___ ,^ ^ \ ^ ^ ><: -^ ^ \ A. ^ OJ s^ --- "^ ^ -\ c 3 d O s OJ ^ sS I^ -^ h. OJ >c .—- - ^ -;:: --^ b^ '""^o ^^.; -N. ^0*-, ^ ^v,^ s ^ — ^ \ ^ . .'-^ >X — VL ^ '< Xj ^ D o o cr < z fe '■ DO < o _1 UJ o < q: <0 UJ > < II n c ^\^ i nv ' r~- ' ^ 1 f\i' "* ;. =^-^ — k J ■^A, 1 2" ■^^ O/v Oi ^ t^ 1 (OV^ \ L._ . ' ^f\10JO<0(£i^OJO rrionrvjruryrvirvj — — — ^ — ainoa ivNid oi asaaddo avoi dO 30NvidVA=,A \ \ q - in d o - \ \ \. ^ \ i r ' ^ ^ \ ' . N ^ ^ ! ^1 >^ . . \ \ ^ ( J ^^'^s,. — - , X U1 O "1 o o rvj 2 N ' — ' ^s 2 <>1 S^ — \ ^% >==:: — ■ ^ i 04., ^ —^ OA/ V; a^Vfc. ^s ^ <. \ ^ k 1 1 1 1 1 ^ 0 J c > a t C 3i 5 r t 0 J r VNId U C 'J 0 0± Q: 5 a J )HBdd( D u D OVO 4 3 •< n do 93 i 1 30NV U c avA = -A D U 5 ■< X ^ J o Tf bO O CO a 3 -1^ (\J q: LU t-i U- , , o LL o < C to o e^-i _l o 0} UJ r-. o o • < •t;o 10 LU .2-^ > >o < o ■f II 1 i^ rvj "5 o O bD « (O o Cli E (O <\j -T3 0) ^ If) o 4J /. o < » OJ _l ■4-3 ft tc LU !-. 7 «3 o OJ on LU > 1- -5 00 g O -u _| to < ^ (D ^i C Li- ? P -fj ■o II y II Ph c^ ^ Ct-H '^ O . n> u CJ • fH t^H > tH 0) < ^ ^ *\ »n o lo c ^ ^ , ^ ^ ^ \ '^'^. x] ^^ ^ X ^vj y / \ ^ ^ 0 \ ^ ^*nJ ^ to k a> (v «> (M •f (d (\J o 7 < f\l _J f\l ir ?■ O fVJ ILI t- -) en C) ir 1 < c ^ ^ \ V -;;; ^-^ ^^ ^ '"Vo ^^^. H,° ">r ^ /-^ ^ X. ^ 7 ^^^^. '0 r / / U/v ^ l^ V > V i^ ^^ k OO OJ lO C\J <* ?• < (\J _1 a UJ 7 o (\I UJ t- "3 OO <1 a. \ (0 < z u. 2 P n Ul (\1 cr UJ u. u. CJ O n < o -J OO UJ O < fr (O UJ > 4. K> ■* fVJ o CO u s-i, ^ ^ ^" - u. u. O O Q '-^'^q ^'^. i^K "^ ^ , N _J Oa, ^ s< y It ^ ^ >s II r^ ^ is •c' 4^ ^ ^ r r r VI c r 3 r 0 VJ r 0 ■< y r i a D il s ^ f r y c 3 a 3 U a ^ f N o ainoa nvNid ojl oaaaddo avoi do 30NviyvA=,A ■^ rvi o CO - O 0) o 05 l-H CO ^ lO • CCI O --I CO CO "O CO lO I-H >* ,X ^ lO ^ (M o I-H CO m o -4-* o dcO I-H I-H d T-H J OO "3 -tj pq o a l> II £3 03 % 3 o ^ O. 02 O OOOcO -* t^ M •na iO(M OOOT I-H (M CO Iz; 5g di-n'dd d d o o 2^ c4 t^ s I-H T-H II H c<> 05 P - ^ (M ic ic o CO t^ o II o o rrj coco o t^ dec t^i-H 05 T-H > H §§• P> be O li ti H ►J o E kC *< ^ CO bO ^ co' r-H « II W tn J^ O OCDCO t^ Oi Eh •n.ss cot^co o ^ T-H ^ CM O bO bO (N Oi^ t-- (MC^COOO '^ lO ^— ' H-t L- C ,,-^ l>-0^'* rt< "^i tJ !U (0 M l-H _ CO O r-f O T-H CO CO 00^ >o • rH n fS (M '^ OCO CO (M CO o to XI :3 •♦J < Eh CJ •4-* a ■*-» to 5 - ■4J 1- o (U c ft > ° s s bfi »-. 3 -Q u "H.S •C o 1 ^ o o 11^ o3 P^WJ^Ph THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 503 I Y" = 25.60, we obtain the trunk requirements: ! Rx Method 23.8 trunks i?2 Method 24.8 trunks Thus the more precise method of sokition here yields a reduction of 1 .0 in 25 trunks, a saving of 4 per cent, as had been predicted. The above calculation is on a Lost Calls Cleared basis. Since the over- flow direct traffic calls will return to this group to obtain service, to as- sure their receiving no more than 3 per cent 'NC, the provision of the final route would theoretically need to be slightly more liberal. An esti- mate of the allowance required here may be made by adding the ex- pected erlangs loss A for the direct traffic (most of the final route over- flow calls which come from high usage routes will be carried by their respective groups on the next retrial) to both the A" and Y" values previously obtained, and recalculating the trunks required from that point onward. (In fact this could have been included in the initial com- putation.) Thus: A = 0.03 X 10.14 = 0.30 erlang A'" = 16.27 + 0.30 = 16.57 erlangs V" = 25.60 + 0.30 = 25.90 erlangs Again consulting Figs. 43 and 47 gives the corresponding final trunk values Ri Method 24.1 trunks R2 Method 25.1 trunks Of the above four figures for the number of trunks in the Scranton route, the i?i-Method with retrials, i.e., 24.1 trunks, would appear to give the best estimate of the required trunks to give 0.03 service to the poorest service parcel. Solution (h) : With High Usage Group Provided for First Routed Traffic Following the procedure outlined in Section 8.2, we obtain an average of the proportions overflowing to the final route for all offered load par- cels. The individual parcel overflow proportion estimates are shown in the last column of Table XVII; their unweighted average is 0.112. With a first routed offer to Scranton of 10.14 erlangs, a provision of 12 high usage trunks will result in an overflow of a = 1.26 erlangs, or a propor- tion, of 0.125 which is the value most closely attainable to the objective 0.112. With 12 trunks the overflow variance is found to be 2.80. 504 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Replacing 10.14 in columns 7 and 8 of Table XVII with 1.26 and 2.80, respectively, gives new estimates characterizing the offer to the final route. A" = 7.39 and V" — 18.26. We now proceed to insure that the poorest service parcel obtains 0.03 service. This occurs on the Phila- delphia and Harrisburg groups, which overflow to the final group ap- proximately 0.224 of their original offered loads. The final group must then, according to equation (34) be engineered for R2 = 0.03/0.224 = 0.134 service. This value lies above the highest R2 engineering chart (Fig. 49, R2 = 0.10), so an ER calculation is indicated. The Equivalent Random average is 28.6 erlangs, and S = 23.5 trunks. We determine the total trunks S -\- R which, with 28.6 erlangs offered, will overflow 0.134(7.39) = 0.99 erlang. From Fig. 12.2, 35.6 trunks are required. Then the final route provision should be C = 35.6 — 23.5 = 12.1 trunks; and a total of 12 + 12.1 or 24.1 Scranton trunks is indicated. Simplified Alternative Solution: In Section 8.2 a simplified approxi- mate procedure was described using a modified probability P' for the average overall service for all parcels of traffic, instead of P for the poor- est service parcel. Suppose P' = 0.01 is chosen as being acceptable. Then P' 0 01 «' = T = am = oo^" Interpolating between the R2 = 0.05 and 0.10 curves (Figs. 48 and 49) gives with A" = 7.39 and F" = 18.26, C = 13.4, the number of final trunks required. Again the same result could have been obtained by making the suitable ER computation. It may be noted that if P' had been chosen as 0.015 (one-half of P), R2 would have become 0.134, exactly the same value found in the poorest-service-parcel method. The final trunk provision, of course, would have again l)een 12.1 trunks. Disscussion In the first solution above, 24.1 full access final trunks from Blooms- burg to Scranton were refiuired. The service on the first routed traffic was 0.03; however, the service enjoyed by the offered traffic as a whole was markedly better than 0.03. The corresponding ER calculation shows (.4 = 28.3, .S -\- C = 12.3 + 24.1) a total overflow of a" = 0.72 erlangs, or an overall service of 0.72/91.21 = 0.008. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 505 In the second solution, 12 high usage and 12.1 common final, or a total of 24.1, trunks were again required, to give 0.03 service to the poorest service parcels of offered load. The overall service here, however, was 0.99/91.21 .= 0.011. Thus, with the same number of paths provided, in the second solution (high usage arrangement) the overall call loss was 40 pes cent larger than in the first solution,* However, it may well be desirable to accept such an average service penalty since by providing high usage trunks for the first routed traffic, the latter's service cannot be degraded nearly so readily should heavy overloads occur momentarily in the other parcels of traffic. 9. CONCLUSION As direct distance dialing increases, it will be necessary to provide intertoll paths so that substantially no-delay service is given at all times. To do this economically, automatic multi-alternate routing will replace the present single route operation. Traffic engineering of these compli- cated trunking arrangements will be more difficult than with simple intertoll groups. One of the new problems is to describe adequately the non-random character of overflow traffic. In the present paper this is proposed to be done by employing both mean and variance values to describe each par- cel of traffic, instead of only the mean as used heretofore. Numerous comparisons are made with simulation results which indicate that ade- quate predictive reliability is obtained by this method for most traffic engineering and administrative purposes. Working curves are provided by which trunking arrangements of considerable complexity can readily j be solved. A second problem requiring further review is the day-to-day variation i among the primary loads and their effect on the alternate route system's I grade of service. A thorough study of these variations will permit a re- I evaluation of the service criteria which have tentatively been adopted. j A closely allied problem is that of providing the necessary kind and [ amounts of traffic measuring devices at suitable points in the toll alter- ! nate route systems. Requisite to the solution of both of these problems ! is an understanding of traffic flow character in a complex overflow-type I * The actual loss difference may be slightly greater than estimated here since i in the first solution (complete access final trunks), an allowance was included for i j return attempts to the final route by first routed calls meeting an 0.03 loss, while 1 in the second solution (high usage group for first routed traffic) no return at- i| tempts to the final route were considered. These would presumably be small since I I only 1 per cent of all calls would overflow and most of these upon retrial would be ij handled on their respective high usage groups. 506 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of trunking plan, and a method for estimating quantitatively the essential fluctuation parameters at each point in such a system. The present paper has undertaken to shed some light on the former, and to provide an approximate j^et sufficiently accurate method by which the latter can be accomplished. It may be expected then that these studies, as they are developed, will provide the basis for assuring an adequate direct dis- tance dialing service at all times with a minimum investment in intertoll trunk facilities. ACKNOWLEDGEMENTS The author wishes to acknowledge the technical and mathematical as- sistance of his associates, Mrs. Sallie P. Mead, P. J. Burke, W. J. Hall, and W. S. Hayward, in the preparation of this paper. Dr. Hall provided the material on the convolution of negative binomials leading to Fig. 19. Mr. Hayward extended Kosten's curve E on Fig. 5 to higher losses by a calculating method involving the progressive squaring of a probability matrix. The author's thanks are also due J. Riordan who has summarized | some of the earlier mathematical work of H. Nyquist and E. C. INIolina, as well as his own, in the study of overflow load characteristics; this appears as Appendix I. The extensive calculations and chart constructions are principally the work of Miss C. A. Lennon. REFERENCES 1. Rappleye, S. C, A Study of the Delays Encountered bj'^ Toll Operators in Ob- taining an Idle Trunk, B. S.T.J. , 25, p. 539, Oct., 1946. 2. Kosten, L., Over de Invloed van Herhaalde Oproepen in de Theorie der Blok- keringskausen, De Ingenieur, 59, j). 1'j123, Nov. 21, 1947. 3. Clos, C, An Aspect of the Dialing Behavior of Subscribers and Its Effect on the Trunk Plant, B. S.T.J. , 27, p. 424, July, 1948. 4. Kosten, L., Uber Sperrungswahrscheinlichkeiten bei Staffelschaltungen, E.N.T., 14, p. 5, Jan., 1937. 5. Kosten, L., Over Blokkeerings-en Wachti)rol>lemen, Thesis, Delft, 1942. 6. Molina, E. C, Appendix to: Interconnection of Telephone Systems — Graded Multiples (R. I. Wilkinson), B.S.T.J., 10, p. 531, Oct., 1931. 7. Vaulot, A. E., Application du Calcul des Probabilites a I'Exploitation Tele- phonique. Revue Gen. de I'Electricite, 16, p. 411, Sept. 13, 1924. 8. Lundcpiist, K., General Theorv for Telephone Traffic, Ericsson Technics, 9, p. Ill, 1953. 9. Berkeley, G. S., Traffic and Trunking Principles in Automatic Telei)hony, 2nd revised edition, 1949, Ernest Benn, Ltd., London, Chapter V. 10. Pahu, C., Calcul I']xact de la Perte dans les Groupes de Circuits Echelonn^s, lOricsson Technics, 3, ]). 41, 1936. 11. Brockmever, 1']., The Simph> Overflow Problem in the Theory of Telephone Traffic! Teleteknik, 5, ji. 361, December, 1954. THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 507 ABRIDGED BIBLIOGRAPHY OF ARTICLES ON TOLL ALTERNATE ROUTING Clark, A. B., and Osborne, H. S., Automatic Switching for Nationwide Telephone Service, A.I.E.E., Trans., 71, Part I, p. 245, 1952. (Also B.S.T.J., 31, p. 823, Sept., 1952.) Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, A.I.E.E. Trans., 71, Part I, p. 248, 1952. (Also B.S.T.J., 31, p. 832, Sept., 1952.) Nunn, W. H., Nationwide Numbering Plan, A.I.E.E. Trans., 71, Part I, p. 257, 1952. (Also B.S.T.J., 31, p. 851, Sept., 1952.) Clark, A. B., The Development of Telephony in the United States, A.I.E.E. Trans., 71, Part I, p. 348, 1952. Shiplev, F. F., Automatic Toll Switching Systems, A.I.E.E. Trans., 71, Part I, p. '261, 1952. (Also B.S.T.J., 31, p. 860, Sept., 1952.) Myers, O., The 4A Crossbar Toll System for Nationwide Dialing, Bell Lab. Record, 31, p. 369, Oct., 1953. Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Lab. Record, 32, p. 51, Feb., 1954. Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- ments in Alternate Routing Trunk Networks, B.S.T.J., 33, p. 277, March, 1954. Molnar, I., Some Recent Advances in the Economy of Routing Calls in Nation- wide Dialing, A.E. Tech. Jl., 4, p. 1, Dec, 1954. Jacobitti, E., Automatic Alternate Routing in the 4A Crossbar System, Bell Lab. Record, 33, p. 141, April, 1955. Appendix I* DERIVATION OF MOMENTS OF OVERFLOW TRAFFIC This appendix gives a derivation of certain factorial moments of the c(iuilibrium probabilities of congestion in a di^dded full-access multiple used as a basis for the calculations in the text. These moments were de- rived independently in unpublished memoranda (1941) by E. C. Molina (the first four) and by H. Nyquist; curiously, the method of derivation here, which uses factorial moment generating functions, employs auxili- ary relations from both Molina and Nyquist. Although these factorial moments may be obtained at a glance from the probability expressions given by Kosten in 1937, if it is remembered that pw = |:(-i)'-'(';)^, (1.1) where p{x) is a discrete probability and M (k) is the A;th factorial moment of its distribution, Kosten does not so identify the moments and it may 1)0 interesting to have a direct derivation. Starting from the equilibrium formulas of the text for f(;ni, n), the l)robability of m trunks busy in the specific group of x trunks, and n in Prepared by J. Riordan. 508 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 the (unlimited) common group, namely {a -{- m -\- n)f(m, n) — (w + l)f(m + 1, n) — (n + l)/(m, n + 1) — af(in — 1, n) = 0 (1-2) « (a -{- X -{- n)j{x, n) — af{x, n — 1) \ - (n -\- l)f(x, n + 1) - af(x - 1, n) = 0 and /(m, n) = 0, m < 0 or n < 0 or m > x, factorial moment generating function recurrences may be found and solved. With m fixed, factorial moments of n are defined by M(fc)(m) = E {n)kf{m, n) (1.3) n=0 or alternatively by the factorial moment exponential generating function M{m, 0 = Z MUm)t'/k\ = £ (1 + 07K n) (1.4) ] fc=0 n=0 I In (1.3), {n)k = n{n — 1) • • • (n — /c + 1) is the usual notation for a \ falling factorial. Using (1.4) in equations (1.2), and for brevity D = d/dt, it is found that a^ m ^- tD)M{in, t) - (m + l)M{m + 1, t) - aM(m - l,t) = 0 (1.5) (x - at -\- tD)M{x, t) - aM{x - \,t) = 0 which correspond (by equating powers of t) to the factorial moment re- currences {a-\- m^ k)M^kM) - (m + l)Ma)(w + 1) - ailf (fc)(m - 1) = 0 (1.6) (x + k)M(k)(x) - akM^k-i)ix) - aMik)(x - 1) = 0 Notice that the first of (1.6) is a recurrence in m, which suggests (fol- lowing Molina) introducing a new generating function defined by Gdu) = T.M^k){m)u'^ (1.7) THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 509 Using this in (1.5), it is found that (a -h k - au + (u - l)~\ GM = 0 (1.8) Hence 1 dGM^^^J^ ^j_g^ Gk(u) du I — u and, by easy integrations, Gk{u) = ce"" (1 - ur\ (1.10) with c an arbitrary constant, which is clearly identical with Gk(0) = M(.)(0). Expansion of the right-hand side of (1.10) shows that il/a,(m) = Ma)(0) Z "^ •^. , "" ■„ = Ma,(0)a-.(m), (1.11) j=o \ J / {m - j)l if o LL o LLI < LJJ II 0.04f/ AQ-- 0.02 TRUN 10 .1 0:3 TRUNKS .1 0.3 1.0 3 a,= AVERAGE IN 10 .1 0.3 TRUN 10 .1 0.3 1.0 3 OF OFFERED TRAFFIC ERLANGS Fig. 51 — Mean and variance of overHovv load when non-random traffic is offered to a group of trunks. 512 THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 513 ing the character of non-random traffic. An approximate solution of the problem is offered based on this method. Suppose a random traffic a is offered to a straight multiple which is divided into a lower Xi portion and an upper X2 portion, as follows: T «2 , V2 X2. ] OCl,Vi u From Nyquist's and Molina's work we know the mean and variance of the two overflows to be: ai = a-Ei^xiia) = a a"» •ril Vi = ai\ 1 — ai -\ ■ — - L Xi — a + ai + IJ a2 = a-Ei,xi+x2(0') V2 = aol I — a2 -\ j j : — r L xi + a;2 - a + 0:2 4- IJ Since ai and vi completely determine a and Xi , and these in turn, with X2 , determine 02 and Vo , we may express 02 and V2 in terms of only ai , Vi , and X2 . The overflow characteristics (0:2 and V2), are then given for a non-random load (ai and Vi) offered to x trunks as was desired. Fig. 51 of this Appendix has been constructed by the Equivalent Ran- dom method. The charts show the expected values of 0:2 and I'o when ai , Vi (or vi/ai), and X2 , are given. The range of ai is only 0 to 5 er- langs, and v/a is given only from the Poisson unity relation to a peaked- ness value of 2.5. Extended and more definitive curves or tables could readily, of course, be constructed. The use of the curves can perhaps best be illustrated by the solution of a familiar example. Example: A load of 4.5 erlangs is submitted to 10 trunks; on the "lost calls cleared" basis; what is the average load passing to overflow? Solution: Compute the load characteristics from the first trunk when 4.5 erlangs of random traffic are submitted to it. These values are found to be a\ = 3.G8, vi = 4.15. Now using ai and vi (or vi/ai = 4.15/3.68 = 1.13) as the offered load to the second trunk, read on the chart the param- eters of the overflow from the second trunk, and so on. The successive overflow values are given in Table XVIII. 514 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 The proportion of load overflowing the group is then 0.0472/4.50 = 0.0105, which agrees, of course, with the Erlang £^i,io(4.5) value. The successive overflow values are shown on the chart by the row of dots along the a2 and V2 1-trunk curves. Instead of considering successive single-trunk overflows as in the ex- ample above, other numbers of trunks may be chosen and their over- flows determined. For example suppose the 10 trunks are subdivided into 2 + 3 + 2-1-3 trunks. The loads overflowing these groups are given in Table XIX. Again the overflow is 0.0472 erlang, or a proportion lost of 0.0105, which is, as it should be, the same as found in the previous example. The values read in this example are indicated by the row of dots marked 1, 3, 6, 8 on the 2-trunk and 3-trunk curves. The above procedure and curves should be of use in obtaining an esti- mate of the character of the overflow traffic when a non-random load is offered to a group of paths. I Table XVIII — Successive Non-Random Overflows Characteristics of Load Offered to Trunk No. i (same as overflow from previous trunk) Trunk Number i Average Variance Ratio of variance to average 1 4.50 4.50 1.00 (Random) 2 3.68 4.15 1.13 3 2.92 3.68 1.26 4 2.22 3.11 1.40 5 1.61 2.46 1.53 6 1.09 1.80 1.64 7 0.694 1.19 1.72 8 0.406 0.709 1.75 9 0.217 0.377 1.74 10 0.106 0.180 1.70 Overflow 0.0472 0.077 1.64 Table XIX — Sucessive Non-Random Overflows Trunlc Number No. Trunks in Next Bundle Offered Load Cliaracteristics (same as overflow from previous trunk) i Average Variance Ratio of variance to average 1 3 6 8 Overflow 2 3 2 3 4.50 2.92 1.09 0.406 0.0472 4.50 3.68 1.80 0.709 0.077 1.00 (Random) 1.26 1.64 1.75 1.64 Crosstalk on Open-Wire Lines By W. C. BABCOCK, ESTHER RENTROP, and C. S. THAELER (Manuscript received September 29, 1955) Crosstalk on open-wire lines results from cross-induction between the circuits due to the electric and magnetic fields surrounding the wires. The limitation of crosstalk couplings to tolerable magnitudes is achieved by systematically turning over or transposing the conductors that comprise the circuits. The fundamental theory underlying the engineer- ing of such transposition arrangements was presented by A. G. Chapman in a paper entitled Open-Wire Crosstalk published in the Bell System Technical Journal in January and April, 1934. There is now available a Monograph (No. 2520) supplementing Mr. Chapman's paper which reflects a considerable amount of experience re- sulting from the application of these techniques and provides a basis for the engineering of open-wire plant. The scope of the material is indi- cated by the following: TRANSPOSITION PATTERNS This describes the basic transposition types which define the number and locations of transpositions applied to the individual open-wire circuits. TYPES OF CROSSTALK COUPLING Crosstalk occurs both within incremental segments of line and be- tween such segments. Furthermore, the coupling may result from cross- induction directly from a disturbing to a disturbed circuit or indirectly by way of an intervening tertiary circuit. On the disturbed circuit the crosstalk is propagated both toward the source of the original signal and toward the distant terminal. A knowledge of the relative importance of the various types of coupling is valuable in establishing certain time- saving approximations which facilitate the analysis of the total cross- talk picture. 515 516 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 TYPE UNBALANCE CROSSTALK Crosstalk is measured in terms of a current ratio between the disturb- ing and disturbed circuits at the point of observation. Crosstalk between open-wire circuits is also generally computed in terms of a current ratio (cu) but it is also convenient to refer to it in terms of a coupling loss (db). The coupling in crosstalk units (cu) is the product of three terms: a coefficient dependent on wire configuration; a type unbalance depend- ent on transposition patterns; and frequency. The coefficient represents the coupling between relatively untransposed circuits of a specified length (1 mile) at a specific frequency (1 kc). The type unbalance is a measure of the inability to completely cancel out crosstalk by intro- ducing transpositions because of interaction effects between the two halves of the exposure and because of propagation effects, primarily phase shift. Type unbalance is expressed in terms of a residual unbalance in miles and the frequency is expressed in kilocycles. The coefficients applicable to lines built in accordance with certain standardized specifications are available in tabular form. When it is desired to obtain coefficients for other types of line, it is possible to compute approximate values which may be modified by correction factors to indicate the relationship between the computed values and measurements on carefully constructed lines. Expressions for near-end type unbalance for certain simple types of exposures are developed and the formulas for all types of exposures are given. In addition, the values for near-end type unbalance are tabulated at 30° line angle intervals for lines where the propagation angle is iu 2,880° or less. The principal component of far-end crosstalk between well transposed circuits results from compound couplings involving tertiary circuits. Again the expressions are developed for some of the exposures involving a few transpositions and the procedure for obtaining the formulas for any type of exposure is shown. Formulas are included for the types of exposures encountered in normal practice and the numerical values of far-end type unbalance are given at 30° intervals for line angles up to 2,880°. SUMMATION OF CROSSTALK The procedures referred to thus far evaluate the crosstalk occurring within a limited length of line known as a transposition section. In practice, however, a line is transposed as a series of sections. It is neces- sary, therefore, to determine how the crosstalk arising within the several CROSSTALK ON OPEN- WIRE LINES 517 sections and that arising from interactions between the sections tend to combine. In a series of like transposition sections there is a tendency for the crosstalk to increase systematically, sometimes reaching in- tolerable magnitudes. This tendency can be controlled to a degree by introducing transpositions at the junctions between the sections, thus cancelling out some of the major components of the crosstalk. Complete cancellation is impossible because of interaction and propagation effects. ABSORPTION Since very significant couplings exist by way of tertiary circuits, it is possible for crosstalk to reappear on the disturbing circuit and thus strengthen or attenuate the original signal. This gives rise to the ap- pearance of high attenuation known as absorption peaks in the line loss characteristic at certain critical frequencies. The evaluation of such pair-to-self coupling requires the use of coefficients which differ from those between different pairs and these are given for standard configura- tions. STRUCTURAL IRREGULARITIES It is impracticable to maintain absolute uniformity in the spacing between wires and in the spacing of transpositions. Thus there are un- avoidable variations in the couplings between pairs from one transposi- tion interval to the next. This in turn reduces the effectiveness of the measures to control the systematic or type unbalance crosstalk and produces what is known as irregularity crosstalk. Since the occurrence of structural irregularities tends to follow a random distribution, it is possible to evaluate it statistically and procedures for doing so are in- cluded. In addition to this direct effect of structural irregularities, there is a component of crosstalk resulting from the combination of systematic and random unbalances. A method is developed for estimating the magnitude of this important component of crosstalk. EXAMPLES In order to demonstrate how the procedures and data are used in solving practical problems, there is included the development of a transposition system to satisfy certain assumed conditions. This is carried through to the selection of transposition types for one transposi- tion section and the selection of suitable junction transpositions. Additional examples of transposition engineering are given in the form 518 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 of several transposition systems which have been widely used in the Bell System. These include: Exposed Line — for voice frequency service. CI — for voice frequency and carrier service up to 30 kc. J5 — for voice frequency and carrier operation up to 143 kc. 01 ■ — for voice frequency and compandored carrier operation up to 156 kc. RIC — suitable for exchange lines with a limited number of carrier assignments. Altogether, the theory, explanatory material, formulas and compre- hensive data included in the Monograph make it possible to estimate open-wire crosstalk couplings and provide the necessary background for the development of new transposition systems. P I Bell System Technical Papers Not Published in This Journal Alsberg, D. A.^ 6-KMC Sweep Oscillator, I.R.E. Trans., PGI-4, pp. 32-39, Oct., 1955. Anderson, J. R.,i Brady, G. W.,^ Merz, W. J.,^ and Remeika, J. P.^ Effects of Ambient Atmosphere on the Stability of Barium Titanate, J. Appl. Phys., Letter to the Editor, 26, pp. 1387-1388, Nov., 1955. Anderson, 0. L.,^ and Andreatch, P.^ stress Relaxation in Gold Wire, J. Appl. Phys., 26, pp. 1518-1519, Dec, 1955. Anderson, P. W.,^ and Hasegawa, H.^ Considerations on Double Exchange, Phys. Rev., 100, pp. 675-681, Oct. 15, 1955. Anderson, P. W.^ Electromagnetic Theory of Cyclotron Resonance in Metals, Phys. Rev., Letter to the Editor, 100, pp. 749-750, Oct. 15, 1955. Andreatch, P., see Anderson, 0. L. Augustine, C. F., see Slocum, A. Barstow, J. M.* The ABC's of Color Television, Proc. I.R.E., 43, pp. 1574-1579, Nov., 1955. Bartlett, C. A.2 Closed-Circuit Television in the Bell System, Elec. Engg., 75, pp. 34-37, Jan., 1956. 1. Bell Telephone Laboratories, Inc. 2. American Telephone and Telegraph Company. 5. University of Tokyo, Japan. 519 520 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Becker, J. A.^ Adsorption on Metal Surfaces and Its Bearing on Catalysis, Advances in Catalysis, 1955, Nov., 1955. Bommel, H. E.i Ultrasonic Attenuation in Superconducting and Normal-Conducting Tin at Low Temperatures, Phys. Rev., Letter to the Editor, 100, pp. 758-759, Oct. 15, 1955. Bemski, G.^ Lifetime of Electrons in p-Type Silicon, Phys. Rev., 100, pp. 523-524, Oct. 15, 1955. Bennett, W. R.^ Steady State Transmission Through Networks Containing Periodi- cally Operated Switches, Trans. I.R.E., PGC.T., 2, pp. 17-21, Mar., 1955. Bommel, H. E.,i Mason, W. P.,* and Warner, A. W., Jr.' Experimental Evidence for Dislocation in Crystalline Quartz, Phys. Rev., Letter to the Editor, 99, pp. 1895-1896, Sept. 15, 1955. Bradley, W. W., see Compton, K. G. Brattain, W. H., see Buck, T. M., and Pearson, G. L. Brady, G. W., see Anderson, J. R. Brown, W. L.' Surface Potential and Surface Charge Distribution from Semicon- ductor Field Effect Measurements, Phys. Rev., 100, pp. 590-591, Oct. 15, 1955. Buck, T. M.,' and Brattain, W. H.' Investigations of Surface Recombination Velocities on Germanium by the Photoelectric Magnetic Method, J. Electrochem. Soc, 102, pp. 636-640, Nov., 1955. Cetlin, B. B., see Gait, J. K. Charnes, a., see Jacobson, M. J. 1. Bell Telephone Laboratories, Inc. I TECHNICAL PAPERS 521 CoMPTON, K. G.,^ Mendizza, a./ and Bradley, W. W.' Atmospheric Galvanic Couple Corrosion, Corrosion, 11, pp. 35-44, Sept., 1955. CoRENzwiT, E., see Matthias, B. T. Dail, H. W., Jr., see Gait, J. K. Dillon, J. F., Jr.,^ Geschwind, S.,^ and Jaccarino, V.^ Ferromagnetic Resonance in Single Crystals of Manganese Ferrite, Phys. Rev., Letter to the Editor, 100, pp. 750-752, Oct. 15, 1955. Dodge, H. F.^ Chain Sampling Inspection Plan, Ind. Quality Control, 11, pp. 10-13, Jan., 1955. Dodge, H. F.^ Skip-lot Sampling Plan, Ind. Quality Control, 11, pp. 3-5, Feb., 1955. Fagen, R. E.,^ and Riordan, J.^ Queueing Systems for Single and Multiple Operation, J. S. Ind. Appl. Math., 3, pp. 73-79, June, 1955. Fine, M. E.^ Erratum: Elastic Constants of Germanium Between 1.7° and 80°K J. Appl. Phys., Letter to the Editor, 26, p. 1389, Nov., 1955. 1 Flaschen, S. S.^ A Barium Titanate Synthesis from Titanium Esters, J. Am. Chem. Soc, 77, p. 6194, Dec, 1955. Fletcher, R. C.,^ Yager, W. A.,* and Merritt, F. R.^ Observation of Quantum Effects in Cyclotron Resonance, Phys. Rev., Letter to the Editor, 100, pp. 747-748, Oct. 15, 1955. Franke, H. C.i Noise Measurement on Telephone Circuits, Tele-Tech., 14, pp. 85-97, Mar., 1955. 1. Bell Telephone Laboratories, Inc. 522 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Galt, J. K.,1 Yager, W. A./ Merritt, F. R./ Cetlin, B. B.,» and Bail, H. W., .Tr.^ Cyclotron Resonance in Metals: Bismuth, Phys. Rev., Letter to the Editor, 100, pp. 748-749, Oct. 15, 1955. Geller, S.,^ and Thurmond, C. D.' On the Question of a Crystalline SiO, Am. Chem. Soc. J., 77, pp. 5285-5287, Oct. 20, 1955. Geschwind, S., see Dillon, J. F. Harker, K. J.^ Periodic Focusing of Beams from Partially Shielded Cathodes, I.R.E. Trans., ED-2, pp. 13-19, Oct., 1955. Hasegawa, H., see Anderson, P. W. Haynes, J. R.,^ and Hornbeck, J. A.^ Trapping of Minority Carriers in Silicon II: n-type Silicon, Phys. Rev., 100, pp. 606-615, Oct. 15, 1955. Hornbeck, J. A., see Haynes, J. R. Israel, J. 0.,^ Mechline, E. B.,^ and Merrill, F. F.^ A Portable Frequency Standard for Navigation, I.R.E. Trans., PGI-4, pp. 116-127, Oct., 1955. Jaccarino, v., see Dillon, J. F. Jacobson, M. J.,' Charnes, A., and Saibel, E.^ The Complete Journal Bearing With Circumferential Oil Inlet, Trans. A.S.M.E., 77, pp. 1179-1183, Nov., 1955. James, D. B., see Neilson, G. C. KoHN, W.,^ and Scheciiter, D.^ Theory of Acceptor Levels in Germanium, Phys. Rev., Letter to the Editor, 99, pp. 1903-1904, Sept. 15, 1955. 1. Bell Telephone Laboratories, Inc. 4. Carnegie Institute. TECHNICAL PAPERS 523 Law, J. T.,1 and Meigs, P. S.^ The Effect of Water Vapor on Grown Germanium and Silicon n-p Junction Units, J. Appl. Phys., 26, pp. 1265-1273, Oct., 1955. Leavis, H. W.i Search for the Hall Effect in a Superconductor: II — Theory, Phys. Rev., 100, pp. 641-645, Oct. 15, 1955. LiNViLL, J. G.,^ and Mattson, R. H.^ Junction Transistor Blocking Oscillators, Proc. I.R.E., 43, pp. 1632- 1639, Nov., 1955. Logan, R. A.^ Precipitation of Copper in Germanium, Phys. Rev., 100, pp. 615-617, Oct. 15, 1955. Logan, R. A.,^ and Schwartz, M.^ Restoration of Resistivity and Lifetime in Heat Treated Germanium, J. Appl. Phys., 26, pp. 1287-1289, Nov., 1955. McCall, D. W., see Shulman, R. G. Mason, W. P., see Bommel, H, E. Matthias, B. T.,^ and Corenzwit, E.^ Superconductivity of Zirconium Alloys, Phys. Rev., 100, pp. 626-627, Oct. 15, 1955. Mattson, R. H., see Linvill, J. G. Mays, J. M., see Shulman, R. G. Mechline, E. B., see Israel, J. 0. Meigs, P. S., see Law, J. T. Mendizza, a., see Compton, K. G. Merrill, F. F., see Israel, J. 0. ' Merritt, F. R., see Fletcher, R. C., and Gait, J. K. Merz, W. J., see Anderson, J. R. 1. Bell Telephone Laboratories, Inc. 524 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Moll, J. L.^ Junction Transistor Electronics, Proc. I.R.E., 43, pp. 1807-1818, Dec, 1955. J MuMFORD, W. W.,^ and Schafersman^, R. L.^ ^ Data on Temperature Dependence of X-Band Fluorescent Lamp Noise Sources, I.R.E. Trans., PGI-4, pp. 40-46, Oct., 1955. Neilson, G. C.,^ and James, D. B.^ Time of Flight Spectrometer for Fast Neutrons, Rev. Sci. Instr., 26, pp. 1018-1023, Nov., 1955. Nesbitt, E. A.,^ and Williams, H. J.^ New Facts Concerning the Permanent Magnet Alloy, Alnico 5, Conf . on Magnetism and Magnetic Materials, T-78, pp. 205-209, Oct., 1955. Nesbitt, E. A.,^ and Williams, H. J.^ Shape and Crystal Anisotropy of Alnico 5, J. Appl. Phys., 26, pp. 1217-1221, Oct., 1955. OWNES, C. D.i Stability of Molybdenum Permalloy Powder Cores, Conf. on Mag- J netism and Magnetic Materials, T-78, pp. 334-339, Oct., 1955. Pearson, G. L.,^ and Brattain, W. H.^ History of Semiconductor Research, Proc. I.R.E., 43, pp. 1794-1806, Dec, 1955. Pederson, L.^ Aluminum Die Castings in Carrier Telephone Systems, Modern Metals, 11, pp. 65, 68, 70, Sept., 1955. Prince, M. B.^ High-Freauency Silicon Aluminum Alloy Junction Diode, Trans. I.R.E., ED-2, pp. 8-9, Oct., 1955. Remeika, J. P., see Anderson, J. R. RiORDAN, J., see Fagen, R. E. 1. Bell Telephone Laboratories, Inc. 6. University of British Columbia, Vancouver, Canada. TECHNICAL PAPERS 525 Saibel, E., see Jacobson, M. J. ScHAFERSMAN, R. L., See Mumford, W. W. Schechter, D., see Kohn, W. Schelkunoff, S. A.^ On Representation of Electromagnetic Fields in Cavities in Terms of Natural Modes of Oscillation, J. Appl. Phys., 26, pp. 1231-1234, Oct., 1955. Schwartz, M., see Logan, R. A. Shulman, R. G.,1 Mays, J. M.,i and McCall, D. W.^ Nuclear Magnetic Resonance in Semiconductors: I — ^ Exchange Broadening in InSb and GaSb, Phys, Rev., 100, pp. 692-699, Oct. 15, 1955. Slocum, A.,^ and Augustine, C. F.^ 6-KMC Phase Measurement System For Traveling Wave Tube, Trans. I.R.E., PGI-4, pp. 145-149, Oct., 1955. Thurmond, C. D., see Geller, S. Uhlir, a., Jr.^ Micromachining with Virtual Electrodes, Rev. Sci. Instr., 26, pp. 965-968, Oct., 1955. Ulrich, W., see Yokelson, B, J, Van Uitert, L. G.^ DC Resistivity in the Nickel and Nickel Zinc Ferrite System, J. Chem. Phys., 23, pp. 1883-1887, Oct., 1955. Van Uitert, L. G.^ Low Magnetic Saturation Ferrites for Microwave Applications, J. Appl. Phys., 26, pp. 1289-1290, Nov., 1955. Wannier, G. H.^ Possibility of a Zener Effect, Phys. Rev., Letter to the Editor, 100, p. 1227, Nov., 15, 1955. 1. Bell Telephone Laboratories, Inc. 526 the bell system technical journal, march 1956 Wannier, G. H.^ Threshold Law for Multiple Ionization, Phys. Rev., 100, pp. 1180, Nov. 15, 1955. Warner, A. W., Jr., see Bommel, H. E. Williams, H. J., see Nesbitt, E. A. i Yager, W. A., see Fletcher, R. C, and Gait, J. K. YoKELSON, B. J.,^ and Ulrich, W.^ Engineering Multistage Diode Logic Circuits, Elec. Engg., 74, p. 1079, Dec, 1955. 1. Bell Telephone Laboratories, Inc. Recent Monographs of Bell System Technical I Papers Not Published in This Journal* Allison, H. W., see Moore, G. E. Baker, W. 0., see Winslow, F. H. Basseches, H., and McLean, D. A. Gassing of Liquid Dielectrics Under Electrical Stress, Monograph 2448. BozoRTH, R. M., TiLDEN, E. F., and Willlams, A. J. Anistropy and Magnetostriction of Some Ferrites, Monograph 2513. Bradley, W. W., see Compton, K. G. CoMPTON, K. G., Mendizza, a., and Bradley, W. W. Atmospheric Galvanic Couple Corrosion, Monograph 2470. Davis, J. L., see Suhl, H. Fagen, R. E., and Riordan, John Queueing Systems for Single and Multiple Operation, Monograph 2506. Fine, M. E. Elastic Constants of Germanium Between 1.7° and 80°K, Monograph 2479. FoRSTER, J. H., see Miller, L. E. Galt, J. K., see Yager, W. A. II' Geballe, T. H., see Morin, F. J. * Copies of these monographs may l)e obtained on request to the Publication Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14, N. Y. The numbers of the monographs should be given in all requests. 527 528 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 GlANOLA, U. F. Use of Wiedemann Effect for Magnetostrictive Coupling of Crossed Coils, Monograph 2492. Green, E. I. The Story of Q, Monograph 2491. GuLDNER, W. G., see Wooten, L. A. Harrower, G. a. Measurement of Electron Energies by Deflection in a Uniform Electric Field, Monograph 2495. Haus, H. a., and Robinson, F. N. H. The Minimum Noise Figure of Microwave Beam Amplifiers, Mono- graph 2468. Hines, M. E., Hoffman, G. W., and Saloom, J. A. Positive-ion Drainage in Magnetically Focused Electron Beams, Monograph 2481. Hoffman, G. W., see Hines, M. E. Kelly, M. J. Training Programs of Industry for Graduate Engineers, Monograph 2512. Law, J. T., and Meigs, P. S. Water Vapor on Grown Germanium and Silicon n-p Junction Units, Monograph 2500. McAfee, K. B., Jr. Attachment Coefficient and Mobility of Negative Ions by a Pulse Techniaue, Monograph 2471. McLean, D. A., see Basseches, H. Meigs, P. S., see Law, J. T. Mendizza, a., see Compton, K. G. Merritt, F. R., see Yager, W. A. MONOGRAPHS 529 Miller, L. E., and Forster, J. H. Accelerated Power Aging with Lithium-Doped Point Contact Transis- tors, Monograph 2482. Miller, S. L. Avalanche Breakdown in Germanium, Monograph 2477. Moore, CI. E., see Wooten, L. A. Moore, G. E., and Allison, H. W. Adsorption of Strontium and of Barium on Tungsten, Monograph 2498. MoRiN, F. J., and Geballe, T. H. Electrical Conductivity and Seebeck Effect in Nio.so Fe2.2o04 , Mono- graph 2514. Morrison, J., see Wooten, L. A. Nesbitt, E. a., and Williams, H. J. Shape and Crystal Anisotropy of Alnico 5, Monograph 2502. Olmstead, p. S. Quality Control and Operations Research, Monograph 2530. Pearson, G. L., see Read, W. T., Jr. Pfann, W. G. Temperature Gradient Zone Melting, Monograph 2451. Poole, K. M. Emission from Hollow Cathodes, Monograph 2480. Read, W. T., Jr., and Pearson, G. L. ^ The Electrical Effects of Dislocations in Germanium, Monograph ! 2511. RiORDAN, John, see Fagen, R. E. Robinson, F. N. H., see Haus, H. A. Saloom, J. A., see Hines, M. E. 530 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 SCHELKUNOFF, S. A. Electromagnetic Fields in Cavities in Terms of Natural Modes of Oscillation, INlonograph 2505. Sears, R. W. A Regenerative Binary Storage Tube, jNIonograph 2527. '< Slighter, W. P. Proton Magnetic Resonance in Polyamides, Monograph 2490. SuHL, H., Van Uitert, L. G., and Davis, J. L. Ferromagnetic Resonance in Magnesium-Manganese Aluminum Ferrite Between 160 and 1900 mc, Monograph 2472. Tilden, E. F., see Bozorth, R. M. Treuting, R. G. Some Aspects of Slip in Germanium, Monograph 2485. Uhlir, A., Jr. Micromachining with Virtual Electrodes, Monograph 2515. Van Uitert, L. G., see Suhl, H. Walker, L. R. Power Flow in Electron Beams, Monograph 2469. Williams, A. J., see Bozorth, R. M. Williams, H. J., see Nesbitt, E. A. WiNSLOW, F. H., Baker, W. O., Yager, W. A. Odd Electrons in Polymer Molecules, Monograph 2486. WooTEN, L. A., Moore, G. E., Guldner, W. G., and Morrison, J. Excess Barium in Oxide-Coated Cathodes, Monograph 2497. Yager, W. A., see Winslow, F. H. Yager, W. A., Galt, J. K., and Merritt, F. R. Ferromagnetic Resonance in Two Nickel-Iron Ferrites, Monograph 2478. Contributors to This Issue Armand 0. Adam,* New York Telephone Company, 1917-1920; West- ern Electric Company, 1920-24; Bell Telephone Laboratories; 1925-. Mr. Adam tested local dial switching systems before turning to design j on the No. 1 and toll crossbar systems. From 1942 to 1945 he was as- sociated with the Bell Laboratories School For War Training. Since then he has been concerned with the design and development of the marker for the No. 5 crossbar system. Currently he is supervising a group I doing common control circuit development work for the crossbar tandem I switching system. i Wallace C. Babcock, A.B., Harvard University, 1919; S.B., Harvard University, 1922. U.S. Army, 1917-1919. American Telephone and Tele- i graph Company, 1922-1934; Bell Telephone Laboratories, 1934-. Mr. Babcock was engaged in crosstalk studies until World War IL Afterward , he was concerned with radio countermeasure problems for the N.D.R.C. ' Since then he has been working on antenna development for mobile radio and point-to-point radio telephone systems and military projects. I Member of I.R.E. and Harvard Engineering Society, , Franklin H. Blecher, B.E.E., 1949, M.E.E., 1950 and D.E.E., , 1955, Brooklyn Polytechnic Institute; Polytechnic Research and De- ' velopment Company, June, 1950 to July, 1952; Bell Telephone Labora- I tories 1952-. Dr. Blecher has been engaged in transistor network de- I velopment. His principal interest has been the application of junction [ transistors to feedback amplifiers used in analog and digital computers. He is a member of Tau Beta Pi, Eta Kappa Nu and Sigma Xi and is an associate member of the I.R.E. W. E. Danielson, B.S., 1949, M.S., 1950, Ph.D, 1952, California Institute of Technology; Bell Laboratories 1952-. Dr. Danielson has been j engaged in microwave noise studies with application to traveling-wave [ tubes and he has been in charge of development of traveling-wave tubes * Inadvertently, Mr. Adam's biography was omitted from the January issue of the Journal in which his article, "Crossbar Tandem as a Long Distance Switch- ing Equipment," appeared. 531 532 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 for use at 11,000 megacycles since June of 1954. He is the author of articles published by the Journal of Applied Physics, Proceedings of the I.R.E., and the B.S.T.J., and he is a Member of the American Physical Society, Tau Beta Pi, and Sigma Xi. Amos E. Joel, Jr., B.S., Massachusetts Institute of Technology, 1940; M.S., 1942; Bell Telephone Laboratories, 1940-. IMr. Joel's first assignment was in relay engineering. He then worked in the crossbar test laboratory and later conducted fundamental development studies. During World War II, he made studies of communications projects and from 1944 to 1945 designed circuits for a relay computer. Later he prepared text and taught a course in switching design. The next two years were spent designing AM A computer circuits, and since 1949 Mr. Joel has been engaged in making fundamental engineering studies and directing exploratory development of electronic switching systems. He was appointed Switching Systems Development Engineer in 1954. Member of A.I.E.E., I.R.E., Association for Computing Machinery, and Sigma Xi. Esther M. Rentrop, B.S., 1926, Louisiana State Normal College. Miss Rentrop joined the transmission group of the Development and Research Department of the American Telephone and Telegraph Com- pany in 1928, and transferred to Bell Laboratories in 1934. In both com- panies she has been concerned principally wdth control of crosstalk, both in field studies and transposition design work. During World War II, she assisted in problems of the Wire Section, Eatontown Signal Corps Laboratory at Fort Monmouth, and later she worked on other military projects at the Laboratories for the duration of the war. Miss Rentrop is presently a member of the noise and crosstalk studies group of the Out- side Plant Engineering Department and is engaged in studies of inter- ference prevention. Jack L. Rosenfeld is a student in electrical engineering at the Mas- sachusetts Institute of Technology. He will receive the S.M. and S.B. degrees in 1957. He has been with Bell Telephone Laboratories on co- operative assignments in microwave tube development and electronic central office during 1954 and 1955. He is a student member of the I.R.E. and a member of Tau Beta Pi and Eta Kappa Nu. Joseph A. Saloom, Jr., B.S., 1948, M.S., 1949, and Ph.D., 1951, all in Electrical Engineering, University of Illinois. He joined Bell Labora- tories in 1951. Mr. Saloom worked on electron tube development at' CONTRIBUTORS TO THIS ISSUE 533 Murray Hill until 1955 with particular emphasis on electron beam studies. He is now at the Allentown, Pa., laboratory where he is en- gaged in the development of microwave oscillators. Member of the Institute of Radio Engineers, Sigma Xi, Eta Kappa Nu, Pi Mu Epsilon. Charles S. Thaeler, Moravian College, 1923-25, Lehigh University 1925-28, E.E., 1928. During the summer of 1927 he was employed by the Bell Telephone Company of Pennsylvania, returning there after gradua- tion, where he was concerned with transmission engineering and the Toll Fundamental Plan. In 1943 he was on loan to the Operating and Engineering Department of the A.T.&T. Co., working on toll transmis- sion studies. From 1944 to the present he has been with the Operating and Engineering Department and is currently engaged in toll circuit noise and crosstalk problems on open wire and cable systems. Mr. Thaeler is an Associate Member of A.I.E.E., and member of Phi Beta I Kappa, Tau Beta Pi, and Eta Kappa Nu. Ping King Tien, B. S., National Central University, China, 1942; M.S., 1948, Ph.D., 1951, Stanford University; Stanford Microwave ]>aboratory, 1949-50; Stanford Electronics Research Laboratory, 1950- 52; Bell Telephone Laboratories, 1952-. Since joining the Laboratories, ' Dr. Tien has been concerned with microwave tube research, particularly t raveling- wave tubes. In the course of this research he has engaged in studies of space charge wave amplifiers, helix propagation, electron beam focusing, and noise. He is a member of Sigma Xi. Arthur Uhlir, Jr., B.S., M.S. in Ch.E., Illinois Institute of Tech- jnology, 1945, 1948; S.M. and Ph.D. in Physics, University of Chicago, ' 1950, 1952. Dr. Uhlir has been engaged in many phases of transistor development since joining the Laboratories in 1951, including electro- I chemical techniques and semiconductor device theory. Since 1952 he has participated in the Laboratories' Communications Development I'laining Program, giving instruction in semiconductors. Member of American Physical Society, Sigma Xi, Gamma Alpha, and the Institute ' of Radio Engineers. Roger I. Wilkinson, B.S. in E.E., 1924, Prof. E.E., 1950, Iowa State College; Northwestern Bell Telephone Company, 1920-21; American Telephone and Telegraph Company, 1924-34; Bell Telephone Labora- tories, 1934-. As a member of the Development and Research Depart- iinent of the A.T.&T. Co., Mr. Wilkinson specialized in the applica- tions of the mathematical theory of probability to telephone problems. 534 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 Since transferring to Bell Telephone Laboratories in 1934, he has con- tinued in the same field of activity and is at present Traffic Studies Engineer responsible for probability studies and traffic research. For two years during World War II, in a civilian capacity, he engaged in opera- tions analysis studies for the Far East Air Forces in the South Pacific, for which he received the Medal for Merit. He has also served as a con- sultant to the Air Force, the Navy and the Air Navigation Delevopment Board. Mr. Wilkinson is a member of A.I.E.E., American Society for Engineering Education, American Statistical Association, Institute of Mathematical Statistics, Operations Research Society of America, Amer- ican Society for Quality Control, Eta Kappa Nu, Tau Beta Pi, Phi Kappa Phi and Pi ]\Iu Epislon. I I i 1 p 1 cr FIG. 25 EQUIVALENT RANDOM LOAD A AND TRUNKS S, FROM NON-RANDOM LOAD A',V' 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 A=AVERAGE RANDOM LOAD IN ERLANGS » Copyright 1955 by Bel] Telephone Laboratories, Incorporated Fig. 25 - Equivalent random load A and number of trunks S, from non-random load A', V - random loads 0 to 50 erlangs FIG. 26 EQUIVALENT RANDOM LOAD A AND TRUNKS S, FROM NON-RANDOM LOAD A'V 3 4 5 6 7 A = AVERAGE RANDOM LOAD IN ERLANGS 10 Copyright 1955 by Bell Telephone Laboratories, Incorporated Fig 26 - Equivalent random load A and number of trunks S. from non-nuidom load A'. V - random loads 0 to 10 erlangs [HE BELL SYSTEM Jechnical journal fIvOTED TO THE SC I E N T I FIC^W^ AND ENGINEERING [PECTS OF ELECTRICAL COMMUNICATION KANSAS C"^ MO' t*M— IM I III I (ILUME XXXV MAY 1956 NUMBERS Chemical Interactions Among Defects in Germanium and Silicon H. REISS, C. S. FULLER AND F. J. MORIN 535 Single Crystals of Exceptional Perfection and Uniformity by Zone Leveling D. c, bennett and b, sawyer 637 Diffused p-n Junction Silicon Rectifiers M. b. prince 661 The Forward Characteristic of the PIN Diode d. a. kleinman 685 A Laboratory Model Magnetic Drum Translator for Toll Switch- ing Offices F. J. buhrendorf, h. a. henning and o. j. murphy 707 Tables of Phase of a Semi-Infinite Unit Attenuation Slope D. E. THOMAS 747 Bell System Technical Papers Not Published in This Journal 751 Recent Bell System Monographs 759 Contributors to This Issue 762 COPYRIGHT 195< AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD F. R. KAPPEL, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNBELY, ExecutivB Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. MCMILLAN, Chairman A. J. BUSCH A. C. DICKIESON B. L. DIETZOLD K. E. GOULD E. I. GREEN R. E. HONAMAN H. R. HUNTLEY F. R. LACK J. R. PIERCE H. V. SCHMIDT G. N. THAYER EDITORIAL STAFF J. D. TEBO, Editor M . E. 8TRIEBY, Managing Editor R. L. SHEPHERD, Prodvction Editor THE BELL SYSTEM TECHNICAL JOURNAL is pubUshed six timea a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Se<»etary; John J. Scan- Ion, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A» THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV MAY 1956 number 3 Copyright 1956, American Telephone and Telegraph Company Chemical Interactions Among Defects in Germanium and Silicon By HOWARD REISS, C. S. FULLER, and F. J. MORIN Interactio7is among dejects in germanium and silicon have been investi- gated. The solid solutions involved hear a strong resemblance to aqueous solutions insofar as they represent media for chemical reactions. Such phenomena as acid-base neutralization, complex ion formation, andion pair- ing, all take place. These phenomena, besides being of interest in themselves, are uscfid in studying the properties of the semiconductors in which they occur. The following article is a blend of theory ami experime7it, and de- scribes developments in this field during the past few years. CONTENTS I . Introduction 536 IL Electrons and Holes as Chemical Entities 537 in. Application of the Mass Action Principle 546 IV. Further Applications of the Mass Action Principle 550 V. Complex Ion Formation 557 VI. Ion Pairing 565 VII. Theories of Ion Pairing 567 VIII. Phenomena Associated with Ion Pairing in Semiconductors 575 IX. Pairing Calculations 578 X. Theory of Relaxation 582 XI. Investigation of Ion Pairing by Diffusion 591 535 536 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 XII. Investigation of Ion Pairing by Its Effect on Carrier Mobility 601 XIII. Relaxation Studies 607 XIV. The Effect of Ion Pairing on Energy Levels 610 XV. Research Possibilities 611 Acknowledgements 613 Appendix A — The Effect of Ion Pairing on Solubility 613 Appendix B — Concentration Dependence of Diffusivity in the Pres- ence of Ion Pairing 617 Appendix C — Solution of Boundary Value Problem for Relaxation. . 619 Appendix D —Minimization of the Diffusion Potential 623 Appendix E — Calculation of Diffusivities from Conductances of Diffusion Layers 626 Glossary of Symbols 630 References 634 I. INTRODUCTION The effort of Wagner' and his school to bring defects in solids into the domain of chemical reactants has provided a framework within which • various abstruse statistical phenomena can be viewed in terms of the intuitive principle of mass action.^ Most of the work to date in this field ' has been performed on oxide and sulfide semiconductors or on ionic com- '[ pounds such as silver chloride. In these materials the control of defects ■ (impurities are to be regarded as defects) is not all that might be desired, i and so with a few exceptions, experiments have been either semiquanti- . tative or even qualitative. i With the emergence of widespread interest in semi-conductors, cul- : minating in the perfection of the transistor, quantities of extremely pure , single crystal germanium and silicon have become available. In addition the physical properties, and even the quantum mechanical theory of the behavior of these substances have been widely investigated, so that a great deal of information concerning them exists. Coupled with the fact that defects in them, especially impurities, are particularly susceptible to control, these circumstances render germanium and silicon ideal sub- stances in which to test many of the concepts associated with defect I interactions. This view was adopted at Bell Telephone Laboratories a few years ago when experimental work was first undertaken. Not only has it been possible to demonstrate quantitatively the validity of the mass action principle applied to defects, but new kinds of interactions have been discovered and studied. Furthermore new techniques of measurement have been developed which we feel open the way for broader investiga- tion of a still largely unexplored field. In fact solids (particularly semiconductors like germanium and silicon) CHEMICAL INTERACTIONS AMONG DEFECTS IN Gg AND Si 537 appear in every respect to provide a medium for chemical reactivity similar to liquids, particularly water. Such pehnomena as acid-base reac- tions, complex ion formation, and electrolyte phenomena such as Debye Hiickel effects, ion pairing, etc., all seem to take place. Besides the experiments theoretical work has been done in an attempt to define the limits of validity of the mass action principle, to furnish more refined electrolyte theories, and most importantly, to provide firm theoretical bases for entirely new phenomena such as ion pair relaxation processes. The consequence is that the field of diamond lattice^ semiconductors which has previously engaged the special interests of physicists threatens to become important to chemists. Semiconductor crystals are of interest, not only because of the specific chemical processes occurring in these substances, but also because they serve as proving grounds for certain ideas current among chemists, such as electrolyte theory. On the other hand renewed interest is induced on the part of physicists because chem- ical effects like ion pairing engender new physical effects. The purpose of this paper is to present the field of defect interaction as it now stands, in a manner intelligible to both physicists and chem- ists. However, this is not a review paper. Most of the experimental re- sults, and particularly the theories which are fully derived in the text or the appendices are entirely new. Some allusion will be made to published work, particularly to descriptions of the results of some previous theories, in order to round out the development. The governing theme of the article lies in the analogy between semiconductors and aqueous solutions. This analogy is useful not so j much for what it explains, but for the experiments which it suggests. : More than once it has stimulated us to new investigations. 1 In our work we have made extensive use of lithium as an impurity. This is so because lithium can be employed with special ease to demon- strate most of the concepts we have in mind. This specialization should not obscure the fact that other impurities although not well suited to the performance of accurate measurements, will exhibit much of the same behavior. II. ELECTRONS AND HOLES AS CHEMICAL ENTITIES Since electrons and holes'* are obvious occupants of semiconductors I like germanium and silicon, and are intimately associated with the pres- [ence of donor and acceptor impurities,^ it is fitting to inciuire into the f roles they may play in chemical interactions between donors and ac- 538 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 ceptors. This question has been discussed in two papers,^- ® and only its principle aspects will be considered. To gain perspective it is convenient to consider a system representing the prototype of most systems to be discussed here. Consider a single crystal of silicon containing substitutional boron atoms. Boron, a group III element, is an acceptor, and being substitutional cannot readily dif- fuse^ at temperatures much below the melting point of silicon. If this crystal is immersed in a solution containing lithium, e.g., a solution of lithium in molten tin, lithium will diffuse into it and behave as a donor. Evidence suggests that lithium dissolves interstitially in silicon, thereby accounting for the fact that it possesses a high diffusivity^ at a tempera- ture where boron is immobile, for example, below 300°C. When the lithium is uniformly distributed throughout the silicon its solubility in relation to the external phase can be determined. Throughout this process boron remains fixed in the lattice. If both lithium and boron were inert impurities the solubility of the former would not be expected to depend on the presence or absence of the latter, for the level of solubility is low enough to render (under ordinary circumstances) the solid solution ideal.* On the other hand the impurities exhibit donor and acceptor behaviors respectively, and some unusual effects might exist. We shall first speculate on the simplest possi- bility in this direction, with the assistance of the set of equilibrium reac- tions diagrammed below.* , Li{Sn) «=± Li{Si) t± Li+ + e~ + B{Si) :f±B- + e+ (2.1) Ti eV At the left lithium in tin is shown as Li(Sn). It is in reversible equilib- rium with Li(Si), un-ionized lithium dissolved in silicon. The latter, in turn, ionizes to yield a positive Li'^ ion and a conduction electron, e~. Boron, confined to the silicon lattice as B(Si) ionizes as an acceptor to give B" and a positive hole, e"*". The conduction electron, e~, may fall into a valence band hole, e"*", to form a recombined hole-electron pair, e"^e~. This process and its reverse are indicated by the vertical equilibrium at the right. All of the reactions in (2.1), occuring within the silicon crystal are describable in terms of tansitions between states in the energy band dia- A glossary of symbols is given at the end of this article. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 539 gram of silicon, exhibited in Fig. 1. The conduction band, the valence band, and the forbidden gap are shown. Lithium and boron both intro- duce localized energy states in the range of forbidden energies. The state for lithium lies just below the bottom of the conduction band while that for boron lies just over the top of the valence band. The separations in energy between most donors or acceptors and their nearest bands are of the order of hundredths of an electron volt while the breadths of the for- bidden gaps in germanium or silicon are of the order of one electron volt. Process 1 in Fig. 1 involving a transition between the donor level and conduction band corresponds to the ionization of lithium in (2.1). Proc- ess 2 is the ionization of boron while process 3 represents hole-electron recombination and generation. The various energies of transition are the heats of reaction of the chemical-like changes in (2.1). Proceeding in the chemists fashion one might argue as follows concern- ing (2.1). If e'^e' is a stable compound, as it is at fairly low temperatures, then its formation should exliaust the solution of electrons, forcing the set of lithium equilibria to the right. In this way the presence of boron, supplying holes toward the formation of e'^e", increases the solubility of lithium. In fact if e"*" is regarded as the solid state analogue of the hydro- gen ion in aqueous solution, and e~ as the counterpart of the hydroxyl ion, then the donor, lithium, may be considered a base while boron, may be considered an acid. Furthermore e'*"e~ must correspond to water. Thus the scheme in (2.1) is analogous to a neutralization reaction in which the weakly ionized substance is e'*"e~. If the immobile boron atoms were replaced by immobile donors, e.g., I phosphorus atoms, a reduction, rather than an increase, in the solubility IT BORON LEVELS (ACCEPTORS) x : ;w>/.-v v.^;i::-.:-:VX;^;;;v valence band v. DISTANCE Fig. 1 — Energy band diagram showing the chemical equilibria of (2.1). 540 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 of lithium might be expected on the basis of an oversupply of electrons (i.e., by the common ion effect^"). In that case we would have a base displacing another base from solution. The intimate comparison between this kind of solution and an aqueous solution is worth emphasizing not so much for what it adds to one's understanding of the situation but rather for the further effects it sug- gests along the lines of analogy. These additional phenomena have been looked for and found, and Mill be discussed later in this article. The scheme shown in (2.1) should be applicable, in principle, to other donors and acceptors and to germanium and other semiconductors as well as silicon. Furthermore the external phase may be any one of a suit- al)le variety, and need not even be liquid. Other systems, however, are not as convenient, especially in regard to the ease of equilibration of an impurity over the parts of an heterogeneous system. The lengths to which one can go in comparing electrolytes and semiconductors are discussed in a recent paper." In order to quantify the scheme of (2.1) it seems natural to invoke the law of mass action. Treatments in which holes and electrons are in- volved in mass action expressions are not new, although systems forming such perfect analogies to aqueous solutions do not seem to have been discussed in the past. For example, in connection with the oxidation of copper Wagner " writes 4Cu -f O2 ^ 2CU2O -f 40" + 4e+ (2.2) in which D ~ is a negatively charged cation vacancy in the CU2O lattice, and e"^ is a hole. Wagner proceeds to invoke the law of mass action in order to compute the oxygen pressure dependence in this system. In another example Baumbach and Wagner^^ and others have investi- gated oxygen pressure over non-stoichiometric zinc oxide. They consider the possible reactions 2ZnO ;=± 2Zn + O2 t\ u 2Z?i+ i^ 2Zn++ -f 2e" (2.3) + 2e- and apply the law of mass action. In (2.3) the various states of Zii are presumably interstitial. Kroger and Vink have recently considered the problem in oxides and sulfides in a rathcM- general way. However in none of the oxide-sulfidc systems has it been possible to achieve really quantitative results. In CHEMICAL INTEKACTIONS AMONG DEFECTS IN Ge AND Si 541 contrast silicon and germanium offer possibilities of an entirely new order. The advent of the transistor has not only provided large supplies of pure single crystal material, but it has also made available a store of funda- mental information concerning the physical properties of these sub- stances. For example, data exists on their energy band diagrams includ- ing impuritj^ states — also on resistivity — impurity density curves, diffusivities of impurities, etc. Furthermore, the amount of ionizable impurities can be controlled within narrow limits, and can be changed at will and measured accurately. Consequently it is reasonable to assume that experiments on germanium and silicon will be more successful than similar investigations using other materials. A t this point it is in order to examine whether or not the treatment of electrons and holes as normal chemical entities satisfying the law of mass action is altogether simple and straightforward. This problem has been investigated by Reiss who found the treatment permissible only as long as the statistics satisfied by holes and electrons remain classical. The validity of this contention can be seen in a very simple manner. Consider a system like that in (2.1). Let the total concentration of donor (ionized and un-ionized) be No , the concentration of ionized donor be D"*", the concentration of conduction electrons be n, and that of valence band holes be p. Let A''^ and A~ denote the concentrations of total ac- ceptor and acceptor ions respectively. Finally, let a be the thermody- namic activity'^ of the donor (lithium in (2.1)) in the external phase. Then, corresponding to the heterogeneous equilibrium in which lith- ium distributes itself between the two phases we can write ^» - ^" = K, (2.4) a in which Ko depends on temperature, but not on composition. This as- sumes the semiconductor to be dilute enough in donor so that the ac- tivity of un-ionized donor can be replaced by its concentration. No — D^. For the ionization of the donor we can write the mass action relation, Z)+ n and for the acceptor. Nd - D+ A~p = Kd (2.5) = Ka (2.6) iVx - A- while for the electron-hole recombination equilibrium np = Ki (2.7) 542 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 In (2.5), (2.6), and (2.7) all the i^'s are independent of composition. To these equations is added the charge neutrality condition, D+ + p = A~ + 7i (2.8) Equations (2.4) through (2.8) are enough to determine No in its de- pendence on Na , «, and the various K's. Together they represent the mass action approach. To demonstrate their validity it is necessary to appeal to statistical considerations. Thus Nd — D^, the concentration of un-ionized donor is really the density of electrons in the donor level of the energy diagram for the semi- conductor. According to Fermi statistics this density is given by No- D+ = No/{l + M exp \{Eu - F)/kT]} . (2.9) in which Ed is the energy of the donor level, F is the Fermi level, k, the Boltzmann constant, and T, the temperature. Furthermore, accord- ing to Fermi statistics, n, the total density of electrons in the conduction band is n = E ^y {1 + exp [{Ei - F)/kT]} (2.10) where Qi is the density of levels of energy, Ei , in the conduction band, and the sum extends over all states in that band. Similar expressions are available for the occupation of the acceptor level and the valence band. F is usually determined by summing over all expressions like (2.9) and (2.10) and equating the result to the total number of electrons in the system. This operation corresponds exactly to applying the conserva- tion condition, (2.8). It is obvious from the manner of its determina- tion that F depends upon No — D^y n, etc. If we now form the expression on the left of (2.5) by substituting for each factor in it from (2.9) and (2.10), it is obvious that the result de- pends in a very complicated fashion upon F, and so cannot be the con- stant, Kd , independent of composition, since in the last paragraph F was shown to depend on composition. On the other hand if attention is confined to the limit in which classical statistics apply^ the unities in the denominators of (2.9) and (2.10) can be disregarded in comparison to the exponentials, and those equations become 1 No - /)+ = 2Noe''"\-'''"'' (2.11); and n = e I ^"'' Z 9ie~"'"" (2.12) I CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 543 respectively. Moreover, from (2.11) i)+ ^ Nn[l - 2e"'^e-^'"'^] = Nu (2.13) where the second term in brackets is ignored for the same reason as unity in the denominators of (2.9) and (2.10). Substituting (2.11) through (2.13) into (2.5) yields D^n _ ?^-- (2.14) in which the right side is truly independent of composition, since F has cancelled out of the expression. Similar arguments hold for (2.6) and (2.7). Therefore in the classical limit the law of mass action is valid, at least insofar as internal equilibria are concerned. We have next to examine the validity of (2.4) which is really the law of mass action applied to the heterogeneous equilibrium between phases. Substitution of (2.11) into (2.4) leads to the prediction a = ^"^^ {e"''}No = K{e"''}Nu (2.15) in the classical case, if (2.4) is valid. In order to confirm (2.15) it is neces- sary to evaluate the chemical potentials of the donor in the external phase and in the semiconductor, and equate the two. The resulting ex- pression should be equivalent to (2.15). Since a is the activity of the donor in the external phase its chemical potential in that phase is, by definition, M = fl'iT, p) + kT in a (2.16) where /i°, the chemical potential in the standard state, may depend on temperature and pressure, but not on composition. To compute the chem- ical potential in the semiconductor statistical methods must once more be invoked. Thus, according to (2.13), donor atoms are nearly totally ionized in the classical case, so that the addition of a donor atom to the semiconductor amounts to addition of two separate particles, the donor ion and the electron. The chemical potential of the added atom is there- fore the sum of the potentials of the ion and the electron separately. Since the ions are supposedly present in low concentration the latter can serve as an activity, ^^ and in analogy to (2.16) we obtain for the ionic chemical potential MD+ = hd+\T, p) -f kT (n Z)+ (2.17) 544 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Furthermore, it is well established'" that the Fermi level plays the role of chemical potential, ju* , for the electron ile = F (2.18) Thus the chemical potential for the donor atom is y^D^ + M. = MB+' + kTfnD^ + F (2.19) = Mz)+° + kT inNn + F = /x/>+" + kT In [e''"'^]Nn where (2.13) has been used to replace D'^ by Nd . We note that the ac-| tivity of the donor atom must be {e^^'^\Nn (2.20)1 with e^""^ playing the role of an activity coefficient." Equating ixd given by (2.19) to n in (2.16) results in the equation a = exp[(M.>+° - ix')/kT]{e'^"]Nu (2.21)| which can be made identical to (2.15) by identifying exp[(Mz.+° - n')/kT] with K of that expression. Thus in the classical case the law of mass] action is applicable to the heterogeneous equilibrium. When classical statistics no longer apply it is still possible to evaluatei Nd — D'^, using the full expression (2.9). Therefore the solubility Nd J of the donor can still be determined if (2.4) remains valid. To decidef this question it is necessary to evaluate hd , the chemical potential of j the donor in the semiconductor under non-classical conditions. Thisl problem is not as simple as those treated above, but it can be solved,™ and the detailed arguments can be found in Reference 5. Here we shall be content with quoting the results. However, before doing this the non- classical counterpart of (2.15) will be written by combining (2.9) with! (2.4). The result is a = [K,/{1 + yi exp[(^„ - F)/kT\\]ND (2.22), and if (2.4) is valid (2.22) should be derivable by equating n to the| proper value of (Xd . Since in the non-classical case a finite portion of the donor states are'' occupied by electrons, the introduction of an additional average donoi atom is no longer equivalent to adding two independent particles whose chemical potentials can be summed. In the statistical derivation of ni it is therefore necessary to evaluate the total free energy of the semi- CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 545 j conductor phase, and to differentiate this with respect to No , keeping I temperature and pressure fixed.* The result is ' ^^ = juz,+° + kT in Nd (2 23) j + F - kT In [1 + 2 Qx\^[- {Ed- F)/kT]] I in which it has been assumed that the concentration of impurity is j sufficiently low so that the solution would be ideal if the impurity could not ionize. In the classical case the exponential in the logarithm is small t compared to unity and (2.23) becomes identical with (2.19), as it should. 1 In the totally degenerate case the exponential dominates the unity and we have ^^ = {^^+0 -{- Ed - kTin2] + kT (uNd I (2.24) = fil -{- kTin Nd ' which is the chemical potential of an un-ionized component of a dilute j * An interesting by-product of this derivation (discussed in Reference 5) is the I fact that the Fermi level, F, is hardly ever the Gibbs free energy per electron for the electron assembly, although it is always the electronic chemical potential, in 1 the sense that it measures the direction of flow of electrons. This arises because I the Gibbs free energy is not alwa3-s a homogeneous function^^ of the first degree in ; the mole numbers (electron numbers). Thus if the number of electrons in the as- sembly is N, the Gibbs free energy, G, is given by G =^ NF + kT Z 1 T.N In ■- hi where the sum is over all energy levels, j, referred to an invariant standard level. ' V is the volume of the system, w/ is the total number of states at thejth level, and , hj is the number of unoccupied states (holes) at the yth level. For F to be the free ! energy per electron the term involving the sum must vanish so that But this can only happen when N CO; = KjV where K^ is independent of V. This requirement is formally met in the case of the free electron gas where the electrons have been treated as independent particles in a box so that CO,- = [8mo"' TT E dE/2h^V where mo is the electron mass, and h, Plank's constant. Since this is the case most frequently dealt with in thermodynamic problems it has been customary to think of F as the free energy per electron, although even here the truth of the contention depends on the assumption of particle in the box behavior. At the other extreme, it is obvious that co, for a level corresponding to the deep closed shell states of the atoms forming a solid cannot depend at all on the ex- ternal volume since they are essentially localized. In computing the free energy of the semiconductor phase it is necessary to understand carefully subtleties of this nature. 546 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 solution, as it should be for the degenerate case in which ionization is suppressed. Equating iid in (2.23) to n in (2.16) yields _ jH exp [(>■„/ - M° + Bo)/kT]\ . " - \ 1 + J/, exp [(£ - FVATJ / ^" ('-^S' which is identical Avith (2.22) if A'o is taken to be }i exp[(Mz>+° - M° + Eo)/kT] (2.26) Thus one arrives at the conclusion that the law of mass action remains valid for the heterogeneous equilibrium even when it fails for the homo- geneous internal equilibria. This is a fairly important result since it implies that solubilities can give information on the behavior of the Fermi level and hence on the distribution of electronic energy levels, even under conditions of de- generacy. The chemical potential specified by (2.23) is of course important in itself, for treating any equilibrium (external or internal) in which the donor may participate. One last remark is in order. This concerns the treatment of heterogene- ous equilibria involving some external phase, and the surface^^ rather than the body of a semiconductor. In such treatments it has been customary to compute the chemical potential of an ionizable adsorbed atom by summing the ion chemical potential and the Fermi level, as in (2.19). This is no more possible if the statistics of the surface states are non- classical, then it is possible when considering non-classical situations involving the body of the crystal. Care must therefore be exercised also in the treatment of surface equilibria. The above discussion has shown that there are extensive ranges of conditions under which holes and electrons obey the law of mass action, and behave like chemical entities. In the next section some of the con- sequences of this fact will be developed. III. APPLICATION OF THE MASS ACTION PRINCIPLE Equations (2.4) through (2.8) will now be used to determine how, in the classical case, the solubility. No , of lithium in (2.1) depends upon Na the concentration of boron in silicon. In the experiments to be de- scribed, the systems are classical, and the donors and acceptors there- fore so thoroughly ionized that No can be replaced by D and Na by A~. Insertion of (2.4) into (2.5) yields D+n = aKoKo = K* (3.1) CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 547 since a is maintained constant. Furthermore (2.7) can be written as np = /vi = ni (3.2) where Wj is obviously the concentration of holes or electrons under the condition that the two are equal. It is called the intrinsic concentration of holes or electrons. The values of rii in germanium and silicon have been determined by Morin.^^' ^® Fig. 2 gives plots of the logarithms of n,- in germanium and silicon versus the reciprocals of temperature. These re- sults are necessary for subsequent calculations. Since A''^ and A" are assumed equal, we may dispense with (2.6). The one remaining equation is then (2.8) which we adopt unchanged. These three relations, (3.1), (3.2), and (2.8) are sufficient to determine D^ or Nd as a function of A" or Na • The only undetermined parameter in the set is K* and this can be evaluated by measuring the solubility, D"^, in the absence of acceptor, i.e., under the condition that A~ is zero. The symbol Do^ is used to designate this value of D'^. In Reference 6 it is shown that Z)/ = K*/(K* + n^y or K* = (Doy/2 + {{Doy/4: + ni'iDoy}'" (3.3) Eliminating K* by the use of this relation it is further shown in Ref- erence 6 that A- ' 1 + VI + (2n,/i)o+)^ ,^^, V/2 (^-4) D+ = + _1 + Vl + {2ni/Do+y_ +\2\ + (Do") which is the required relation between donor solubility and acceptor concentration. Examination of (3.4) reveals several simple features, the more import- ant of which we list below: (1) When A~ (the acceptor doping) is sufficiently large so that {Do^Y in the second term can be ignored relative to the term in A~, (3.4) reduces to that of a straight line with slope Knowledge of this slope is equivalent to knowledge of Do . (2) Wlicre the straight lino portion of the D^ \'ersus A~ curve is in- 548 THE BELL SYSTEM TECHNICAL JOURNAL, AL\Y 1956 volved, the temperature dependence of the solubiHty, D'^, enters only through the ratio, ni/Do^ . If this ratio is very small, then D^ ^ A~ (3.6) and the solubility is independent of temperature. In this condition Z)"^ may approximate A~ by being either slightly less or slightly greater than the latter. Details are given in Reference 6. (3) Whereas D^ at small values of doping may be an increasing func- tion of temperature, it may, depending on the system, be a decreasing function of temperature at high dopings. Thus doping may change the sign of the temperature coefficient of solubility. Because of this, doping sometimes may prevent precipitation of a donor when a semiconductor is cooled, since the latter becomes an undersaturated rather than a supersaturated solution of impurity. Details are given in Reference 6. (4) It is also shown in Reference 6 that for the acceptor to have any effect on the solubility of the donor the concentration of A~ should satisfy the following criterion A- > (Do"*" or m) (3.7) Do or Hi being used depending on which is greater. Obviously at high 10 '9 10 10' 18 5 '0'^ O lO'S 10' 10 13 10 12 \ \ \ \ \ GERMA NIUM ^ V \ 5IL jcon\ \ \ \ i 0.001 0.002 0.003 0-004 i/TEMPERATURE in degrees KELVIN Fig. 2 — Temperature dependences of intrinsic carrier concentrations in ger-, manium and silicon. "ft CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 549 temperatures when rii achieves a very large value it may not be possible to have A~ exceed n, , and no effect due to the acceptor will be observable. This is simply a mathematical reflection of the fact that the hypothetical compound e'^e~ in (2.1) is highly dissociated at high temperatures so that the holes contributed by the acceptor cannot cause the exhaustion of electrons in the solution. In Reference 6 the system described in (2.1) was investigated for the purpose of testing (3.4). The concentrations, Z)"*" and A~, of lithium and boron respectively were determined by measuring the electrical resis- tivities of the crystal specimens before and after immersion in molten tin contaning lithium. Some typical results of these experiments are shown in Fig. 3 which contains three Z)"*" versus A~ isotherms for the temperatures 249°, 310°, and 404°C. For the case shown the tin phase contained 0.18 per cent lithium by weight. The points in the figure represent experimental findings, while the drawn curves are based on theory. The agreement between theory and Ij experiment is very good, in fact the overall accuracy appears to be bet- ter than 1 per cent. These isotherms are only a few of a large group ob- tained at different temperatures and with differently proportioned ex- ternal phases. The accuracy in all of these is of the same order. I Various of the features of (3.4) listed above are apparent in the curves of Fig. 3. For example at large values of ^~ the curves are straight lines, thus validating (3.5). Also, the inversion of the temperature coefficient of solubility with doping is apparent for the curves cross one another, md whereas, at low dopings (low A~) the solubility is an increasing func- on of temperature, at high dopings it decreases with increasing tempera- ture. Finally we note that D'^ remains more or less independent of A~ until A~ exceeds n,- , confirming (3.7). Values of n,- appear in the Figure. The possible increases in solubility above Do^ are really quite large. For example in Fig. 3 the largest increase is of the order of a factor of 10^ However in some experiments increases of 10 have been observed. These effects truly represent profound interactions between impurities which are present in highly attenuated form. Thus the number of atoms per cubic centimeter in crystal silicon is of the order of 5 X 10 cm" . Interactions at doping levels as low as 10^* cm~^, as appear in Fig. 3, therefore take place at atom fraction levels of about 2 X 10 . In Fig. 4 we show a curve of lithium solubility at room temperature in gallium-doped germanium. The curve is wholly experimental; no attempt has been made to apply theory. The symbols D and A~ are once more used for the donor and acceptor. In this case the curve again exhibits some of the general features required by (3.4). The measure- / 550 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 10 18 17 10 r<5 o a. io'® UJ Q. + 10 ,15 to'' 10 ,13 POINTS -EXPERIMENTAL THEORY LUTE BATH "C nL=6.16Xl0'4 Dl_ D 249 J i A 404 "C nL = 2.06X 10'6 J "^ ^ ^ ^ — ff ^ / _ — D y^ 10" 10' 15 10" 10 ,17 10^ ,18 A" PER CM3 Fig. 3 — Isotherms showing the solubility of lithium Z)+, in silicon as a func- tion of boron doping A~, for an external phase of tin containing 0.18 per cent lithium. ments were made by saturating gallium-doped germanium crystals with lithium by alloying lithium to the germanium surface at a high tempera- ture, and letting it diffuse in. Following this the crystals were cooled and lithium was allowed to precipitate to equilibrium. In this case the external solution is the precipitate and is of unknown composition. If the straight line portion of the curve is used to determine D^/A~ appearing in (3.5), the value of Do"*" associated with the precipitate as an external phase can be computed by using the value of n, obtained from Fig. 2 for 25°C. The latter is 3 X lO'' cm"', and the measured D^/A" is 0.85. Application of (3.5) then leads to a value of Dq'^ of 6.6 X 10^' + cm at 25°C. Since the highest value of D measured in Fig. 4 is 5.5 X 10 cm , the solubility increase here shows a factor of 10 . Interaction is already apparent at values of A~ as low as 10^ cm~*, and since there are 4.4 X 10 cm~ atoms per cubic centimeter in pure germanium this represents interaction at levels of atom fraction as low as 2 X 10~ . IV. FURTHER APPLICATIONS OF THE MASS ACTION PRINCIPLE In the last section the possibility was mentioned of inverting the sign of the temperature coefficient of solubility, and so preventing impurity CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 551 10 19 10" 10" 5 10'- to'' o / y V / / / Y / A _^ 10 13 10'' 10'S 10 16 10 17 10 t& to' |19 GALLIUM CONCENTRATION IN CM" Fig. 4 — Room temperature isotherm showing the solubility of lithium in germanium as a function of gallium doping, the external phase being an alloy of lithium and germanium. The curve merely shows locus of experimental points. precipitation which might normally occur upon cooling a crystal speci- men. An experiment demonstrating this effect is described in Reference 6. Two specimens of germanium, one without added acceptor, and the other containing gallium at an estimated concentration of 1.3 X 10 cm" , were saturated with lithium. Table I compares the changes in lithium content observed in these samples with the passage of time. After 25 days no apparent precipitation had occurred in the gallium doped speci- men, while precipitation was almost complete in the other. This result suggests a practical scheme for measuring the concentra- tion of lithium along the solidus curve of the lithium-germanium phase diagram, i.e., the solubility of lithium in solid germanium when the ex- ternal phase is also composed of germanium and lithiimi, and probably represents the liquidus phase. This measurement, though desirable, has not been performed before because lithium, diffused into germanium at an elevated temperature, precipitates when the specimen is cooled. 552 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Table I Ga Cone, (cm-s) Li Cone, after saturation (cm-3) Li Cone, after 4 days at room Temp. (cm"3) Li Cone, after 25 days at room Temp, (cm"') 0 1.3 X lO's 1.4 X 10i« 8.0 X 1018 9.0 X 1015 8.0 X 1018 1.1 X 1016 8.0 X 1018 Resistivities then measure only the dissolved lithium although the true solubility at the temperature of saturation includes the precipitated material. However, we have seen that germanium suitably doped with gallium will not lose lithium by precipitation. Therefore the experiment might be performed in doped germanium. The only difficulty with this sugges- tion lies in the fact that doping chayiges the solubility. This objection can be overcome through use of (3.4). In terms of that equation D'^ would be measured in the presence of gallium whereas Do"^, the solubility in undoped germanium, is required. But according to (3.4) if Z) , n, , and A~ (gallium concentration) are known Do"*" can be computed. In fact solving (3.4) for Do yields ^+ D^(D-^ - A-) + Do"- = / D^iD"- - A-) + (Dyn,' V' rii + D^{D^ - A~) + / Z)^(D^ - A-) -\ 2 (4.1) i +\2 2 + aryn^ The plan is therefore self-evident. Samples of germanium of known ! suitable gallium contents A~ are to be saturated with lithium at various \ temperatures. If a judicious choice of gallium content is made the lith- ium will not precipitate when the specimen is cooled. Therefore the value * of D^ characteristic of the saturation temperature can be determined ' through resistivity measurements performed at room temperature. Taking nj from Fig. 2 it then becomes possible to calculate Do using (4.1). I The crystal specimens employed were cut in the form of small rec-l tangular wafers of dimensions, approximately 1 cm X 0.4 cm X 0.1 cm. " On the surfaces of these, small filings of lithium were distributed densely enough so that their average separation was less than the half thickness of the specimen's smallest dimension. The filings Avere alloyed to the germanium specimen by heating in dry helium for 30 seconds at 530°C. ■ Then the crystals w^ere permitted to saturate with lithium by diffusion from the alloy at some chosen lower temperature. After the period of saturation which ranged from one half hour to as long as 1G8 days, de- CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 553 Table II T°C. po ohm cm A- (cm-3) p ohm (cm) Z>+cm-3 I»o+ (cm-') 25 6.6 X 10" 100 0.0523 2.2 X 10'' 0.0735 .9 X 10i« 2.5 X 10'^ 200 0.44 1.3 X 10i« 0.90 7.8 X 1015 4.6 X lOK* 250 0.1494 4.7 X 1016 0 652 3.9 X 10>6 2.6 X 1016 300 0.042 2.9 X 10'' O.IOS 2.15 X 10" 7.3 X 1016 500 0.00614 4.5 X lO's 0.0340 4.13 X lO's 1 7 X 1018 608 0.00577 5.0 X IQis 0.049 4.78 X 10i« 2.8 X 1018 650 0.00584 4.3 X W 0.0178 3.75 X lO's 2.4 X lO's pending on the temperature, the specimen surface was lapped smooth with carborundum paper. Resistivities were then measured by means of a two point probe. Table II collects the data showing T, the temperature of saturation in degrees centigrade, po the resistivity before saturation, .4" the gallium concentration computed from po, p the resistivity after saturation, and D^ the lithium concentration computed from p. The final column shows Do"^ computed using (4.1) and Fig. 2. In Table II the 25°C value of Dq^ has been taken as the value com- puted in section III in connection with Fig. 4. It might be thought (in view of a later section in this paper) that the 25° and 100°C values of Do are not as reliable as the others because at the low temperatures involved the solubility of lithium may be influenced by ion pairing as well as electron-hole equilibria. However, Appendix A shows that the possible error is small. In Fig. 5 Dq^ is plotted against temperature using these data. The plot is the curve labeled GaT = 0, and the open circles were obtained by in- serting the measured D^ values (crosses) into (4.1). We notice that the curve has a maximum in the neighborhood of 600°C. The occurrence of a maximum, is a necessity if Dq^ is to pass to zero, as it must at the melting point of germanium. It is also worth noticing that Do"*" near room temperature lies in the range of order 10^^ cm~^, but that its meas- urement has been effected at concentrations as high as 10^^ cm~^ This ! illustrates another application of the electron-hole equilibrium, namely in the determination of solubilities. \18 and 10 cm This 3 With Do in our possession it is interesting to return to (3.4) and to calculate D^ as a function of temperature for various levels of A has been done for values of A" equal to 10^^ 10^ ^ 10^^, The curves so obtained appear in Fig. 5, labeled Ga" = 10'", 10'°, 10 1 10 cm" , respectively. Their most striking common feature is the mini- I mum which appears below 200°C. This minimum introduces a new prob- o<^0<><>^6^^^\:SX\\«^ Fig. 7 — Distribution of lithium after an extended period of diffusion at a temperature lower than the alloying temperature — showing leakage out of the crystal in the one case (no-skin) and conservation in the other. 558 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Fig. 8 — Photograph of experimental situation described schematically in Fig. 7. to the possibility of interactions between the donor and acceptor ions themselves. For example, in (2.1) direct interaction of Li^ and B" above 600°C may be possible, especially in view of the mobility of Li^ . Such a reaction was indicated in the work of Reiss, Fuller, and Pietruszkie- 34 wicz. Fig. 9 is of assistance in understanding the nature of these observa- tions. In it are shown plots of the solubility of lithium in silicon. In this case the situation is similar to that involved in the germanium curves of Fig. 5 because the external phase is composed of silicon and lithium and is probably of the liquidus composition. It is formed by simply alloying lithium to the silicon surface. In Fig. 9, Curve A, illustrates how solubility depends on temperature when the silicon is undoped. Curve B, unlike A, is not an experimental plot, i.e., it is not supposed to represent the locus of the points through which it seems to pass. In- stead it has been calculated from the theory expounded below. The points themselves are experimental and represent solubility measurements on silicon doped with boron to the level 1.9 X 10 cm" I CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 559 Curve A possesses a maximum (just as the Dq^ curve of Fig. 5) in the neighborhood of 650°C. A marked disparity is apparent between solu- bihties in undoped and doped sihcon, the sohibiHty in the latter bemg greater. Below 500°C this disparity is easily understood. It stems from the electron-hole equilibrium considered previously. However the high solubility in doped silicon at high temperatures is not explicable on this basis since the crystal becomes intrinsic, and e'^e~ is mostly dissociated. To account for this phenomenon Reiss, Fuller, and Pietruszkiewicz invoked the idea of interaction between Li'^ and B". They presented the following argument. At low temperatures lithium ions occupy the interstices of the silicon • 1 EXPERIMENTAL /\ /•••^ p 9 B 7 / \\ ^ v\ 1 / \\ 7 L r \ \ 11 \\ t 6 11 h V \ ^ / / I \ b// \ / /A • 3 / / \ / 1 \ / i 1 ° \ y • ^ / o O c ) 1 1 1 1 1 \ \ \ \ \ • • 1/ • \ 9 J \ • i \ \ 8 7 / / I \ / \ / \ / / ^ / r \ t 1 \ •\ 2 200 400 600 800 1000 TEMPERATURE IN DEGREES CENTIGRADE 1200 Fig. 0 — Plots showing the soluliilitj' of lithium in silicon us a function of tem- perature. The external phase is an alloy of lithium and silicon. Curve A is for un- doped silicon. The locus of the points in B is for silicon doped with about 1.9 X 10^* cm~^ boron. 560 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 lattice as in Fig. 10. In an interstitial position lithium can approach an oppositely charged boron, but the interaction will be, at the most, coulombic so that an ion pair will form (see later sections). A covalent bond is unable to appear not only because there are no electrons avail- able for it, but also because the lithium ion cannot move to a position where it can satisfy the tetrahedral symmetry inherent in sp^ hybridiza- tion. Calculations (of the sort appearing in the later sections of this paper) show that at high temperatures, at the ion densities involved, ion pairs of the kind depicted in Fig. 10 are completely dissociated. Suppose, however, that as temperature is raised vacancies dissolve in the silicon lattice, and that one such vacancy occupies a position near Fig. 10 — Schematic diagram of a silicon lattice showing a lithium ion in an interstitial position near a substitutional boron ion, as it occurs in an ion pair. a boron ion, as in Fig. 11, a slight modification of Fig. 10 in which the dots represent electrons (dangling bonds). Unpaired electrons such as these might capture an electron from the valence band of silicon so that the vacancy acquires a negative charge and behaves like an acceptor. It is reasonable to suppose that the positive lithium ion will move into this negative vacancy, in the tetrahedral position, and form a covalent bond as in Fig. 11. The lithium-boron complex so formed retains a nega- tive charge and is thus a complex ion. If the specimen were extrinsic at these high temperatures, there would still appear to be as many net acceptors as before the addition of lithium.* If the LiB~ compound is stable enough (a question to which we shall * It is possible that rapid cooling may quench some of these LiB acceptors into the crystal at room temperature. If this is so it should be possible to investigate the associated energy level by Hall measurements in the interval of time before ' the complexes anneal out. Similar phenomena might be observed in germanium. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 561 return below) to hold the lithium atom, the solubility of lithium will be determined principally by the density of boron atoms. At low tempera- tures, \-a(*an('ies are reabsorbed and the lithium atoms return to their interstitial positions, at quenched-in densities corresponding to the tem- peratures of equilibration. However, boron acceptors now appear to be compensated since interstitial lithium behaves as a donor. This renders it feasible to measure the concentration of lithium by the determination of resistivity. The overall reaction may be written in the form w^ + B^ + n + t" - = LiB~ (5.1) in which D represents a vacancy. This equilibrium can be grafted onto (2.1) so that the latter becomes (ignoring un- ionized lithium and boron) Li (external) - P) = fi (9.4) where Na and No are, respectively, the total densities of acceptors and donors. This equation has the following solution for P/Nd , the fraction of donors paired. P_ No 1 = o 1 + 1 , Na' nNo Nd, /i 1 + 1 N, mo + ¥j-k ^'-'^ Inspection of (9.5) reveals that for given A^^ and 0, P/Nd is a decreasing function of increasing No . Very often, P/Nd is measured in an experiment, and from this it is desired to calculate a, the distance of closest approach. For such pur- poses the form (9.5) is not very convenient. In fact an entirely different procedure is to be preferred. Suppose P/Nd is denoted by 6, and 6 is substituted into (9.4), into which (9.2) has been inserted. We obtain logio Q{(x) = logio d {Na - eND)(i - e) ] (9.6) A knowledge of 6 thus suffices to determine logic Q(a), from which, in turn, a can be determined by interpolation in Table III. Then (9.3) can be used for the evaluation of a. * This is a situation which cannot arise in liquids, since there, charge balance -.must be maintained by the ions themselves. It can occur when the ions are of Idifferent charge, but then things are complicated by the formation of triplets, etc., in addition to pairs. 580 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Table IV r°K Q (cm') r°K n (cm3) too 2.2 X 102 400 2.3 X 10-" 150 6.45 X 10-7 500 1.54 X 10-18 200 3.42 X 10-1' 600 3.0 X 10-19 225 1.28 X 10-12 700 1.03 X 10-19 250 8.79 X 10-" 800 4.7 X 10-2" 300 1.61 X 10-16 Experiments which will be described later indicate that in germanium, gallium and lithium can approach as close as 1.7 X 10 cm, Usmg this value of a, and k = 16, g = 4.77 X 10~^ statcoulombs, the values of Q appearing in Table IV were computed from (9.2) With these values, P/Nd , the fraction of donors paired can be com- puted from (9.5) as a function of temperature and N a for the simplest case, i.e., the one for which A''^ = No . Fig. 15 contains plots showing these dependences. It must be remembered that all other things remain- ing the same P/Nd will be greater than the values shown in Fig. 15 when Nd < Na ' A rather important integral to which reference shall be made later is x^ exp {q/KkTx) • 1 dx (9.7) The integral appearing in (7.14) is a special case of (9.7) with ri = a, and Ti = h. I(r2 , n) has been evaluated over a considerable range. To facili- tate matters the transformation X = (q^'/KkT) X (9.8) has been employed. In this notation n and ro transform to pi and p2 , and I{r, , n) = {g'/KkTY r X' exp (1/X) dX = {q/KkTYHp^ , pi) (9.9) •'pi Figs. 16 and 17 contain plots of i{p2 , 0.05) out to p2 = 5. The choice of pi equal to 0.05 was rather unfortunate since for k = 16, and T = 300°K it corresponds to pi = 2.5 X 10~^ cm. Since acceptors like gallium possess values in respect to lithium as low as 1.7 X 10~ cm i(p2 , 0.05) is not much use in these cases. The choice 0.05 was made before the ex- perimental data on gallium was available. Below we shall describe a method for extending t(p2 , pi) to cases where n is less than 2.5 X 10 cm. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 581 Ga IN Ge a = i.7 X to'^CM q=4.77 xio"'° e.s.u. 1 0 100°K 0.9 150°H / ^ ^ / y^ / / / / / / / / 200°K. / / / / / // 0.8 / / / / f / / / / 1 i 1 ' / Q ^ 0.7 1 1 f 1 / / 1 // / / 250 / / / / // |o.a o < q: 0.5 LL 0.4 0.3 0.2 O.t 1 / / / / / / / / / / 300/ 1 / // / / 1 / / / r 400/ 500/ 7, ,-6 00 / / 1 / / / // 700°K / / / / /, // 1 / / / / J 7 / / y f / / J // 0 y y •^ \y ^ / ■^ y i/ 10 10 10" 10 12 10'^ 10' 10 15 10 16 10 17 10 18 N IN CM"^ 10'9 10' Fig. 15 — Fraction of ions paired, assuming equal densities of positive and negative ions, calculated as a function of temperature and concentration from equation (9.5). The situation illustrated might apply to gallium and lithium in germanium in view of the choice of a and /c. Fig. 16 covers the range from pi = 0.05 to 0.08 and involves a logarithmic scale because of the sharp variation of i in this range. (This points up the sensitivity of the degree of pairing to the magnitude of a.) Fig. 17 extends the curve to pi = 5. When pi exceeds 5, i{pi. , 0.05) can be obtamed from the formula i{pi , 0.05) = 3865 + ^' + ^' (9.10) In order to determine i{pi , pi) when pi ^ 0.05, the following formula may be used. i(p2 , Pi) = i(p2 , 0.05) - 2(pi , 0.05) (9.11) 582 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Finally for cases in which pi < 0.05, Table III can be used. Thus ^(P2 , pi) = Qil/pi) - Q(20) + t(p2 , 0.05) (9.12) where 1/pi , and 20 are a values in Table III. X. THEORY OF RELAXATION In Section VIII attention was drawn to the fact that ion pairing in semiconductors can be made to occur slowly enough so that its kinetics can be followed. It is possible to characterize these kinetics by a relaxa- tion time r, which we shall endeavor to calculate in the present section. 4000 2000 1000 800 600 500 400 300 In o d <5. 100 80 60 50 40 30 20 10 ^ ^ 0.050 0.055 0.060 0.065 Pa 0.070 0.075 0.080 Fig. 16 — Plot, for small values of P2 of i(p2 , 0.05) from (9.9). CHEMICAL INTEKACTIONS AMONG DEFECTS IN Ge AND Si 583 3920 3900 ^^3880 O c£^3860 3840 3820 3800 / y ^ y 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Fig. 17 — Plot, for larger values of p-i , of i(p2 , 0.05) from (9.9). Suppose a system is first maintained at a temperature high enough to prevent pairing, and then, at an instant designated as zero time, is suddenly chilled to a temperature at which pairing takes place. One thereby has a system which would normally contain pairs but which finds itself with donors and acceptors which are uniformly and randomly distributed. Since the donors are assumed mobile, a process ensues whereby they drift toward acceptors until an equilibrium is established in which each acceptor develops an atmosphere of donors with density c(r), given by (7.7). This final state in which the atmosphere is fully developed is the paired state characteristic of the lower temperature. The relaxation time to be defined must measure the interval required for the near completion of the above process. In order to acquire physical feeling for the phenomenon, we begin with some simple considerations. In particular a system will be dealt with containing equal numbers of positive and negative ions. This restriction can be lifted later. Now, to a first approximation the pairing phenomenon may be re- garded as a trapping process in which mobile, positive donor atoms are captured by the negative acceptors. Thus, suppose each acceptor is imag- ined to possess a sphere of influence of radius R, beyond which its force field may be considered negligible, and inside which a positive ion is to be regarded as captured. This picture immediately emphasizes certain sub- tleties which require discussion before further progress can be made. In the crudest sense one might reason that the probability of an en- counter between a positive ion and a negative trap would depend on the 584 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 product of the densities of both. These densities must be equal because when a positive ion is trapped the resulting ion pair is neutral so that a trap is eliminated simultaneously. If these equal densities are designated by n, we arrive at the second order rate law - f = '^^^ (^°-'> where /ca is a suitable constant, and t is time. This law would be perfectly valid if the mean free path of a mobile positive ion were large compared to the distance between ions and the probability of sticking on a first encounter were small. The trapping cross-section rather than the movement prior to trapping would de- termine the trapping rate. In this case the rate would certainly depend on the concentrations of both the traps and the ions being trapped. On the other hand, in our case, not only is the mean free path of a positive ion much smaller than the distance between ions, but the sticking probability is high. A given ion must diffuse or make many ran- dom jumps before encountering a trap and upon doing so is immediately captured. Therefore, the rate of reaction is diffusion controlled. Because of the random jump process a given mobile ion is most likely to be captured by its nearest neighbor during the first half of relaxation, and relative to the degree of advancement of the trapping process, the density of traps may be considered constant. This leads to first order kinetics rather than second,* i.e., to - ^ = /cin (10.2) at where n is the density of untrapped ions. By definition ki is the fraction of ions captured in unit time, i.e., the probability that one ion will be captured per unit time. Its reciprocal must be the average lifetime of an ion. This lifetime r = I (10.3) ki shall be defined as the relaxation time for ion pairing. A rough calculation of T can be made quickly. Thus, suppose that the initial concentrations of donors and acceptors are equally A^. About each fixed acceptor can be described a sphere of volume, 1/A^. On the average this sphere should be occupied by one donor which according to what has been said above, will eventually be captured by the acceptor at the center. In the mind, all * The phenomenon stems from the fact that first and second order processes are almost indistinguishable during the first half of the reaction, but also from the fact that the diffusion control prevents the process from being a ti-ue second or- der one, although its departure from second order may be small. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 585 the spheres can be superposed so that an assembly of donors A'' in num- ber is contained in the volume 1/A^, at the density A'^ . The problem of relaxation is then the problem of diffusion of these donors to the sink of radius R, at the center of the volume. The bounding shell of the sphere may be considered impermeable, thus enforcing the condition that each donor shall be trapped by its nearest neighbor. Since the diffusion prob- lem has spherical symmetry the radius, r, originating at the center of the sink at the origin may be chosen as the position coordinate. At r = R, the density, p, of diffusant may be considered zero. The radius, L, of the volume, 1/A^, is so large compared to R, that in the initial stages of diffusion L may be regarded as infinite. In spherical diffusion to a sink from an infinite field, a true steady state is possible, and this steady state is quickly arrived at when the radius, R, of the sink is small. Under this condition concentration is described by p = A -- (10.4) r where A and B are constants. Furthermore at early times n is still N, the initial concentration at r = L ^ oo , so that p(oo) = AT' (10.5) In addition we know that p(R) = 0 (10.6) These boundary conditions suffice to determine A and B in (10.4), and yield P = N' 1-^ r (10.7) Now the rate of capture (—(dn/dt) in (10.2)) is obviously measured by the flux of ions into the spherical shell of area, 4tR', which marks the boundary of the sink. This flux is given according to Fick's law by 4.^R'D, (^-^) = - ^ (10.8) where Do is the diffusivity of the donor. Substituting (10.7) into (10.8) yields r2 r- - C?n 4:tN'RDo = - 4^ (10.9) dt During the initial stages of trapping the right side of (10.2) may be 586 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 written as ^lA'', i.e., hN^-'^ (10.10) Equating the left sides of (10.9) and (10.10) gives A;i = 4:7rNRDo or ^ = r = A A^r. (10-11) It now remains to choose a value for the capture radius, R. A reason- able guess may be made as follows: Around each acceptor there is a coulomb potential well of depth V = -(IIkt . (10.12) Since the average thermal energy is kT, it seems reasonable to regard an ion as trapped when it falls to a depth kT in this well. Thus, inserting kT on the left of (10.12) and R for r on the right leads to R = qlKkT (10.13) and upon substitution in (10.11) we obtain KkT (10.14) 4xgWZ)o This result, obtained by crude reasoning, is actually quite close to the more rigorous value derived below. Furthermore, the above derivation is useful in providing insight into the physical meaning of the relaxation time. The chief difficulty with the preceding lies in the arbitrary choice of 72, and is a direct consequence of the long range nature of coulomb forces. Another difficulty arises because the distribution of donors about ac- ceptors is eventually specified by (7.7) so that at r = i^ = q/KkT Be Since this slope has a negative value the trap exhibits some aspects of a source rather than a sink which could only produce a positive concen- tration gradient. This last objection will not be serious when h is very small since, then the final value of c{r) beyond r = q/KkT = R will be effectively zero, as would be required for a perfect sink. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 587 The last point raises still another question: What happens when the sink is not perfect, i.e. where the equilibrium state does not involve complete pairing? All these difficulties can be removed by a more sophisticated treatment of the diffusion problem. Thus, retain the sphere of volume, 1/A^, en- closing A^ donors at the density N^. However, the equations of motion of these donors are altered to account for the fact that besides diffusing they drift in the field of the acceptor at the origin. Thus the flux density of donors will be given by /*(r,o = -z).|^^+l^ (10.16) where R has been substituted for q/KkT. Equation (10.16) is obtained by adding to the diffusion component, — Lfo — dr of the flux density, the drift component, Mog where hq is the mobility of a donor ion and —q/nr' is the field due the acceptor at the origin. The Einstein relation^" Mo - qD,/hT (10.17) has also been used to replace mo with Do . The spherical shell bounding the volume, 1/iV, of radius L = {^y (10.18) is regarded as impermeable, so we obtain the boundary condition J*{L, t) = 0. (10.19) Furthermore an arbitrary inner boundary, r = i?, is no longer defined but use is made of the real boundary, r = a, i.e., the distance of closest approach, at which is applied the condition J*ia, 0-0 (10.20) As before, the initial condition may be expressed as p = N- t = 0 a < r < L (10.21) 588 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 51 The continuity equation, in spherical coordinates takes the form r^ dr dt Substitution of (10.16) into (10.22) gives, finally, L I h 1^ + 7^ A = 1 % (10.23) r^dr\ dr '^ j Do dt Equations (10.23), (10.21), (10.20) and (10.19) form a set defining a boundary value problem, the solution of which is p(7\ t), from which, in turn, J*(r, t) can be computed. It then remains to compute (dn/dt) in (10.2) from J*. The former is not simply AttR^J* (as in (10.8)) because now J* is not defined unambiguously, being a function of r. J*{R, t) might be employed but then the method is no less arbitrary than the simple one described above. Fortunately, nature eliminates the dilemma. It is a peculiarity of spherical diffusion, when the sink radius is much smaller than the radius of the diffusion field, that after a brief transient period, 47rr'J*(r), except near the boundaries of the field, becomes practically independent of r, and depends only on t. This feature is elaborated in Appendix C. Since in our case the radius of the field is of order, L, and the effective radius of the sink is of order, R, and L » R, it may be expected that this phe- nomenon will be observed. In fact its existence has been assumed previ- ously in the derivation of (10.4). Under such conditions it does not matter how the radius of the sink is defined so long as 4:irR^ is multiplied by J*{R) and not the value of J* at some other location. The boundary value problem, (10.23), (10.21), (10.20), (10.19) is solved in Appendix C, and it is shown there that the value of 47rr"J*(?') obtained after the transient has passed is closely approximated by 4xrV*(r) = -^C^° e-"' (10.24) with where M ^ .kTiN-M) ,^) 47r5W2/)o = l/47r [ r exp [q/KkTr] dr (10.26) CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 589 The close connection between M defined by (10.26) and h defined by (7.17) is apparent. Thus in (7.17) when r = L, exp[-47rrW/3] is e~\ and for larger values of r this exponential quickly forces the convergence of the integral. Therefore the values of h and M will be almost equal. This is not surprising since they are meant to be the same thing, i.e., the average concentration, c(oo), of donors at infinite distance in the equilibrium atmosphere of an acceptor. Both quantities are computed so as to conserve charge in this atmosphere. At large values of N, M proves to be much smaller than A^ so that (10.25) reduces to (10.14), validating the crude treatment, for r in (10.24) is obviously the relaxation time. This is easily seen by writing -^ = -4,rrV*(r) = ^^« .""^ (10.27) at kkT from which one derives by integration n = M + (AT - M)e~"'' (10.28) According to (10.28) at ^ = 0, n = A", the correct initial density for unpaired ions. At ^ = 00,72 = M, also the correct density, i.e., the density at large values of ?-, when equilibrium is achieved. Obviously r plays the role of the relaxation time, since by differentiation of (10.28) din - M) _{n- M) ^^^^9) dt T which is to be compared with (10.2) and (10.3). Values oiM can be computed using formulas (9.10), (9.11), and (9.12) and Figs. 16 and 17 since the integral in (10.26) is one of the i integrals I'lg. 18 shows some values of M, computed in this way for the tempera- tures 206°, 225°, 250°, and 300°K, for a semiconductor where the value of a = 2.5 X 10~^ cm, k = 16, and q = 4.77 X 10"^" statcoulombs. The plots are of M versus N. Note that the values of M are generally much less than A", the disparity increasing with lower temperatures and larger A. It is also possible to calculate t for the above system in its dependence upon A" and T. To do this the value of Z)o must be known as a function of temperature. Fuller and Severiens have measured the diffusivities of lithium in germanium and silicon down to about 500°K. These data plot logarithmically against \/T as excellent straight lines. In Fig. 19, we show an extrapolation of the line for lithium in germanium down to the neighborhood of 200°K. From this figure it is possible to read values of 590 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 10" 300° K a = 2.5X10"8 CM >f=16 ^ ^ 4^16 /^ / X' ,^ / 250 1 5 LJ J /-lis /v U ^Ql5 / ^ z 225 2 ^ J ,^14 20 6° K 1015 10' 10 16 10" N IN CM"3 10'' 10 19 Fig. 18 — Dependence of constant M defined by (10.26) on temperature and concentration, for particular values of a and k. Do for germanium to which the system of Fig. 18 refers, since k has been chosen at 16. Using Figs. 18 and 19, Fig. 20 was computed. It shows t plotted in seconds versus A^ for the same temperatures appearing in Fig. 18. These curves show that at values of N as low as 10 cm" relaxation times are short enough to be observable down to 200°K, being at the most some 50 hours in extent. The value of N makes a big difference.' For example at 200°K the relaxation time is only 4 minutes with A'' = 10^^ cm~^ Presumably, at 10 cm~ , relaxation could be observed down to much lower temperatures. It is interesting to note that insofar as M hardly appears in r, the latter is independent of the distance of closest approach, a. Since a is to some extent empirical this is a fortunate circumstance, and the measure- ment of T may provide an accurate means of determining, A^, Do , k, or q, whichever parameter is regarded as unknown. Furthermore k as a macroscopic parameter has real meaning in r since the forces involved may be regarded as being applied over the many lattice parameters separating the di-ifting donor from its acceptor. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 591 This section will be closed by indicating how the restriction to systems containing equal numbers of donors and acceptors might be lifted. Thus, suppose Na exceeds No • Then there will be Na — Nd mobile holes main- taining charge neutrality. To a first approximation these will screen the N^ — Nd uncompensated acceptor ions so that the No donors will see effectively only A''!, acceptors. Thus in first approximation r can be com- puted for this system by replacing N in the preceding formulas by Nd • Of course it is possible that there will be a further effect. Thus the mobile holes will probably shield some of the compensated acceptors as well. This Avill lead to a further (probably small) reduction in t, over and above that obtained by replacing N by Nd - We shall not go into this in the present paper, because in most of the experiments performed Nd was near Na - In the few^ exceptions the crude correction, suggested above, can be used. XI. INVESTIGATION OF ION PAIRING BY DIFFUSION Most of the theoretical tools required for the study of ion pairing have now been provided, and attention will be turned to experiments which 10' - 7 iO 10-8 Q 0,0-9 JJ UJ CL 10- 5 ,n-ii 10" o 10' Q ■12 10" 10" 10-'5 o.oot TEMPERATURE IN DEGREES KELVIN 600 500 400 350 300 250 200 0.002 0.003 0.004 0.005 t/TEMPERATURE IN DEGREES KELVIN Fig. 19 — Diffusivitj' of lithium in germanium extrapolated from the data of Fuller and Severiens. 592 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 have been performed in this field. A fairly large group of these exist, and it remains to describe them in detail. We shall begin with the study of the diffusion of lithium in p-type germanium. At the outset a matter having to do with the diffusion 'potential de- mands attention. This is the potential which arises, for example, in p-type material, because the mobility of a hole is so much greater than the mobility of a lithium ion. In consequence, holes diffuse into regions containing high concentrations of lithium more rapidly than lithium ions can diffuse out to maintain space charge neutrality. As a result such re- o 2 O o ai in 105 \ \ \ s. 3 = 2.5X10-8 CM /C=16 10^ \ \ \ N \ > \ V, \ s. 103 \ s \ 206°K ^ \ \ \ \ <'' 1 \ 102 N \ \ \, \ \ s \ 250°K \ 10 \ ■\ N \ ^ \,, ^ 00°K n-' \ \ 10'5 N IN CM-3 10' Fig. 20 — Relaxation time as a function of temperature and concentration com- puted from equation (10.25) using the data of Figs. 18 and 19. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 593 gions develop positive potentials and a field exists tending to expel lithium. This causes the lithium to drift as well as diffuse so that Fick's laW^ is no longer valid. The most that can be done toward the elimination of diffusion poten- tials is to minimize them so that no local space charge exists. At equilib- rium, this corresponds to the condition^^ Nd - Na = 2ni smh(qV/kT) (11.1) where V is the local electrostatic potential. It is always permissible to assume that fast moving electrons and holes are in equilibrium relative to diffusing ions. If a material which is p-type everywhere is being con- sidered, (11.1) can be simplified to Na - Nj, = Ui exp [-qV/kT] (11.2) In Appendix D it is proved that (11.2) will be valid everywhere within a region where N a is constant and greater than Nd , provided that No does not fluctuate through ranges of the order Na in a distance less than (11.3) Under most conditions of experiment I will be of the order of 10~ cm. Unfortunately many of the experiments described in this section (par- ticularly those performed at 25°C.) involve diffusion layers as thin as 10"^ cm. As a result space charge will exist and the diffusion potential will not always be minimized. Even if it is minimized so that (11.2) is satisfied the residual field will still aid diffusion and lead to higher ap- parent diffusivities. Therefore the effect cannot be ignored even when minimization has been achieved. In the absence of space charge the drift component of flux density due to the field is easily computed. It will be given by -M ^ No (11.4) dx According to (11.2) _dV _ kT dNp dx ~ q{NA - Nd) dx so that (11.4) becomes nkT / Nd \ ONd ^ _fiokT / _ _P \ / Nd \ dNp ^'a - No) dx q \ Nd) \Na - Nd) dx Nd \ dNp - Nd) dx (11.5) (ii.rO 594 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 where (7.15) and the Emstein relation have been used, and Do is the diffusivity in the absence of pairing. P/Nd in (11.6) can be evaluated using (9.5) so that the coefficient pre- ceding (dNo/dx) contains No as the only variable. In Appendix B it is shown that ion pairing itself leads to severe de- partures from Fick's law.*^ In fact the diffusion flux density in the pres- ence of pairing is given by -2 l^" - ^- + 5) l/i(^° - iV. - i) + '-f° (11.7) dx Here again the diffusivity is specified by the factors preceding (dNo/dx) and, though variable, depends only on No , the local concentration of diffusant. Adding the two coefficients appearing in (11.6) and (11.7) the value of the diffusivity, D, in the presence of both pairing and diffusion potential is obtained. Thus D-^\l + (11.8) It is obvious from (11.8) that even in the absence of space charge D is an extremely complicated function of Nd , and will be much more com- plex if space charge needs to be considered. When Nd « A^.i (11.8) re- duces to Comparison with equation (B15) shows that when (11.8) is true (i.e., in the absence of space charge) the diffusion potential may be ignored for Nd <3C Na • Comparison of (B14) with (B15) shows how much D can vary with Nd when ion pairing occurs. The proper study of diffusion in the presence of ion pairing should be augmented by a mathematical analysis, accounting for the concentra- tion dependent diffusivity. Since this dependence is complicated the resulting boundary value problem must be solved numerically, and this CHEMICAL INTEKACTIONS AMONG DEFECTS IN Ge AND Si 595 represents a formidable task. Although work along these lines is being done we shall content ourselves, in this article, with a less quantitative approach. The following plan has been followed. A rectangular wafer of semiconductor uniformly doped with ac- ceptor to the level, Na , is uniformly saturated with lithium to a level, Nd , slightly less than Na • Thus, the resulting specimen is well compen- sated but not converted to n-type. Lithium is then allowed to diffuse out of the specimen, and because of the thinness of the wafer, this process may be regarded as plane-parallel diffusion normal to its large surfaces. Low resistivity p-type layers therefore develop near the sur- faces. If the thin ends of the wafer are put in contact with a source of current, current will flow parallel to its axis, so that the equipotential surfaces will be planes normal to this axis. The flow of current will be one dimensional because the inhomogeneity in lithium distribution oc- curs in the direction normal to its flow (see Fig. 21). If two probe points are placed at a fixed distance apart on the broad surface of the wafer (see Fig. 21), then the conductance measured be- tween them is a reflection of the total number of carriers in the low resistivity layers, i.e., a measure of the total amount of lithium which has diffused out. A more detailed connection between this conductance and diffusivity is derived in Appendix E. For the moment, however, attention will be confined to the description of the general plan of ex- periment. According to the formulas derived in the early parts of this section, and also to (B14) and (B15), the diffusivity is something like Do/2 in the CURRENT CURRENT, (I) Fig. 21 — Diagram illustrating measurement of dependence of diffusivity on ion pairing (see Section XI). 596 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 bulk of the wafer where Nd almost equals Na , but is as low as Do/(l + ^Na) near the surface where Nd « A''^ . If Q,Na is very much larger than unity as it will be under conditions where appreciable pairing oc- curs, the diffusivity will, therefore, be much smaller near the surface than at the high end of the diffusion cui-ve, deeper within the specimen. The surface will then offer resistance to diffusion, and it may be expected that the measured value of the diffusivity Avill correspond more closely to the slow process near the surface rather than to the faster process occurring deeper in the semiconductor. Of course this cannot be entirely true because the resistance at the surface coupled with the lack of re- sistance inside the wafer will tend to steepen the concentration gradient near the surface. This wdll give the impression of a diffusivity somewhat higher than the one corresponding to the surface. If the current flowing in the wafer under the conditions of measure- ment is I, and the potential measured between the points is V, then the conductance between the points is S = I/V. (11.10) In Appendix E it is shown (under the assumption that D is constant) that S/S. = 1 + ?:?«|v^ (1^) V^ (lUl) where So is the conductance after the specimen is saturated with lithium, but before any lithium has diffused out, and S^ is the con- ductance before lithium has been added. Na is the uniform concentration of acceptor, and Nd° is the initial uniform concentration of lithium, while d is the thickness of the wafer. ?? is a correction factor which arises be- cause the mobility of holes varies from point to point in the wafer, as the density of lithium varies. There are two extreme types of variation. The first takes place in a specimen in which, at room temperature (where the conductance measurement is made) ion pairing is complete. Then the local density of impurity scatterers will be A''^ — Nd ■ At the other extreme no ion pairing occurs, and the density of scatterers is Na + Nd. The nature of t> depends on how much pairing is involved. In Fig. 22 d^ has been evaluated in its dependence on Nd° for the extreme cases men- tioned. Furthermore it has been assumed then that Nd is given by a Fick's law solution of the diffusion problem, and that diffusion begins in a nearly compensated specimen. The first thing to notice is that ?? is not very different from unity in CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 597 1.7 1.6 .5 - 1.4 1.3 1.2 1. 1 1.0 0.9 0.8 0.7 10 16 y-00 -r r I Fr t Ai 1 yUI,IMCIl^S;v.ii^>3^.'>, t Jo //(n)/ erf c i di ^0 y / ^ y ^ y ^ ^^AIRIN G -"^ NON PAIklNfc. ^ ^ 10 17 5 6 10 18 Nd in cm' Fig. 22 — Plots of correction factor ??, required to compensate for the depend- ence of hole mobility on the density of scattering centers along a diffusion curve. I? is plotted against the initial density of donor and is shown for the two extreme cases of pairing and no pairing. either extreme, and therefore closer to unity in some intermediate situ- ation. In any event the correct value of ?? can be read from Fig. 22 if the experiments involve either extreme at the measurement temperature. This has, in fact, been approximately the case in our experiments, in which pairing is almost complete at the temperature where conduc- tances have been measured. According to (11.11) a plot of S/2o against 's/l should be a straight line of slope 2.256^V^/2«>A^D°\ S = d \:eoNa (11.12) f Measurement of S therefore affords a measure oi D. Of course the ap- ! parent D obtained in this manner can never represent anything beyond ' some average quantity having the general significance of a diffusi\-ity. This follows from the previous discussion concerning the non-constancy 598 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 of D. The only exception to this statement occurs in connection with high temperature experiments (above 200°C.) where both pairing and the diffusion potential are of little consequence. The mere fact that 2/2o plots as a straight line against -s/t is not evidence for the constancy of D. In Appendix E it is shown that a straight line will result, even when ion pairing is important, provided that the diffusion potential is based on the no-space- charge condition, i.e. provided that D varies only through its dependence on Nd • On the other hand, the last statement implies that the existence of a straight line relationship is evidence that the diffusion potential has at least been minimized. The most careful experiments were performed in germanium doped to various levels with gallium, indium, and zinc as acceptors. The ger- manium specimens were cut in the form of rectangular wafers of ap- proximate dimensions (1.25 cm X 0.40 cm X 0.15 cm). Fresh lithium filings, were evenly and densely spread on one surface of the wafer, and alloyed to the germanium by heating for 30 seconds at 530°C in an at- mosphere of dry flowing helium. Then the other surface was subjected to similar treatment. After this the specimen was sealed in an evacuated pyrex tube and heated at a predetermined temperature for a predetermined period of time. The temperature was chosen, according to Fig. 5, so that the saturated specimen would still be p-type and just barely short of being fully compensated. Also attention was paid to the problem of avoiding precipitation on cooling. The time of saturation was determined from an extrapolation of the known lithium diffusion data, in germanium, of Fuller and Severiens^^ which is plotted in Figure 19 for the range ex- tending from about 0° to 300°C. After saturation the sealed tube was dropped into water and cooled, t It was opened and the wafer ground on both sides, first with No. 600 ' Aloxite paper, and then with M 303)^ American Optical corundum abrasive paper. The final thicknesses of the specimens ranged from 0.025 ' to 0.075 cm, the thinnest samples being used for the runs at the lowest temperature. If the specimen is quite thin and highly compensated it is possible in principle to measure very small diffusivities (as low as 10~ cm /sec) i within a period of several hours. This is so because the low resistivity layer formed near the surface, although thin, will carry a finite share of the current in thin compensated specimens. On the other hand, additional o difficulties arise. Diffusion layers as small as 100 A may be involved. If the surface is microscopically rough, diffusion will not be plane-parallel; CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 599 and the measured diffusivity will appear larger than the real diffusivity. This condition can be partially corrected by etching the surface chemi- cally until it is fairly smooth. When dealing with such thin layers, the no-space-charge assumption becomes invalid and the diffusion potential ought really to be considered. Considering all the difficulties, i.e., concentration dependence of diffusion coefficient, possible existence of space charge, and roughness of surface, it is apparent that only qualitative effects are to be looked for in the dif- fusivities which have been measured. The most that can be predicted is that for specimens containing a given amount of acceptor, the measured D (some average quantity) should be less than Do , the disparity increasing with decreasing tem- perature. At high temperatures D should converge on Do . Furthermore, at a given temperature D should decrease with an increase in concen- tration of acceptor. These tendencies are in line with the idea that reduc- tion of temperature or increase of doping leads to an increase in pairing. Runs Avere carried out on specimens etched with Superoxol^^ at the temperatures 25°, 100°, and 200°C. In the 25°C run the wafer was allowed to remain in the measuring apparatus under the two probe points in air, and S was measured from time to time. At 100°C the specimen was immersed in glycerine containing a few drops of HCl, the temperature of the bath being controlled. Periodic removal from the bath facilitated the measurement of 2. At 200°C glycerine was again used as a sink for lithium, the sample being removed periodically for measurement. Fig. 23 illustrates some typical plots of 2/So versus \/t. They are all satisfactorily straight. Fig. 24 shows a plot of log Do against \/T, extrapolated from the data of Fuller and Severiens.^^ In this illustration, Aalues of log D (obtained from the above measurements by determining the slopes S and employing (11.12)) are also plotted at the temperatures of diffusion. For ■& the case of complete pairing was assumed. The first thing to note is that the points for log D all lie below log Do except at 200°C and satisfy the qualitative requirement outlined above.* Moreover they drop further below log Do as the temperature is reduced, ^\■hile at 200°C they have almost converged on log Do . The results for zinc are particularly interesting. Zinc is supposed to have a double negative charge in germanium. Hence we would expect very intense pairing to occur. This is indicated in the difi'usion data where the sample containing zinc at the rather low level, A^^ = 2.7 * The long range nature of the interaction forces becomes evident when one considers that the diffusivities are being altered by impurity (acceptor) concen- trations of the order of 1 part per million. I 600 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 UJ D O o o tr o 4.5 4.0 3.5 3.0 2.5 MM 2.0 1.5 1.0 / / / / / / / / 200° c/ / / / / / / A / f / / / / /zb" c / / / 7 / / // / / 1.14 1.12 1.10 1.08 1.06 > a. D O o LL WW 1.04 1.02 1.00 10 15 20 25 30 35 1(/t IN SECONDS 40 45 50 55 Fig. 23 — Curves illustrating the observed linear dependence of S/2o on the \/~t. X 10^^ cm~', shows a large reduction in diffusivity even at temperatures as high as 200°C. The difficulties discussed in this section serve to emphasize the im- portance of a direct transport experiment in which lithium atoms nni- jormlij distributed throughout germanium or silicon, uniformly doped with acceptor, are caused to migrate by an electric field, and their mobilities measured. Because of the uniform dispersion of solutes the mobility will be constant everywhere. Furthermore no diffusion poten- tial will be involved, and also the refined formula (7.25) can be applied. There are, however, many difficulties associated with the performance of this type of measurement. In closing it may be mentioned that a few much less careful experi- ments of the kind described here have been performed in boron-doped silicon. The results indicate ion pairing in a qualitative way but more definite experiments are needed. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 601 XII. INVESTIGATION OF ION PAIRING BY ITS EFFECT ON CARRIER MOBILITY II In Section VIII attention was called to the fact that ion pairing should influence the mobility of holes, because each pair formed, reduces the number of charged impurities by two. Thus, a specimen previously doped with acceptor, might, if sufficient lithium is added, exhibit an increase in hole mobility, even though the addition of lithium implies the addition of more impurities. This effect has been observed in connection with the Hall mobility of holes in germanium. Two specimens of germanium were cut from adjacent positions in a single crystal doped with gallium to the level 3 X 10^^ cm~l One of these iwas saturated with lithium through application of the same procedure TEMPERATURE IN °C 200 100 25 r 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.S 3.0 3.2 Y X 10^ ^3.4 Fig. 24 — Plot of diffusivity of lithium in undoped germanium as a function of temperature — also showing points for apparent diffusivities of lithium in variously doped specimens. 602 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 employed in section V. Hall mobilities of the two specimens were meas- ured^^ down to below 10°K. Cooling was carried out slowly to permit as much relaxation into the paired state as possible (see Section X). Li Fig. 25 plots of the Hall mobilities versus temperature of both specimens are presented. Curve A is for the sample containing 2.8 X lO" cm~* lithium. It therefore contained about 5.8 X 10^^ cm~^ total impurities as compared to the control sample whose curve is shown as B in Figin-e 25 and which contained only 3 X 10 cm" impurities. The lithium doped bridge exhibits by far the higher Hall mobility for holes (except at very low temperatures where poorly understood phe- nomena occur). In fact at 40°K the sample containing lithium shows a hole mobility 16 times greater than that of the control at the correspond- ing temperature. Rough analysis of the relative mobilities at T = 100°K indicate '^2 X 10 cm scattermg centers in the control sample and 5 X 10 cm" scattering centers in the sample containing pairs. This experiment has been repeated with other specimens doped to different levels with gallium and even with other acceptors, and leaves no doubt that a mechanism which is most reasonably assumed to be pairing, is removing charged impurities from the crystal. The phenomenon we have just described suggests an excellent method for testing the ion pairing formula derived in Sections VII and XI, for it 10' Q Z o u o > DC liJ a cvj 5 10 u CD O 5 < X 10' - i r '^'^-^ ■v.^ - 1 ^^ ^^ - 1 \. X s_ / ^ -^ _^ ^ N) r / "*" "^ / / : 40 80 120 160 200 240 280 TEMPERATURE IN DEGREES KELVIN 320 Fig. 25 — Plot of Hall mobility as a function of temperature for germanium containing 3 X 10^' cm"' gallium. Curve A is for a sample containing 2.8 X 10''' cm~^ lithium. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 603 enables us to determine at what temperature, at given values of Na and Nd , P/Nd is exactly 0.5. Thus consider the fact that, all other things being equal, the control bridge and the one containing added lithium will exhibit equal Hall mobilities at a given temperature when the con- centrations of charged impurities are identical in both of them. Now the concentration of such impurities in the control is simply Na • The con- centration in the bridge containing lithium is Na + No- 2P (12.1) The quantity 2P is removed from Na + Nd , because each time a pair forms two charged scatterers are eliminated. The condition that the ; .scattering densities in both bridges be equal is then simply I Na = NA-\-Nn-2P or ■^ = 0.5 (12.2) N D i Therefore if plots of Hall mobilities versus temperature such as those I appearing in Figure 25 are continued until they cross, the temperature I of crossing marks the point at which P/Nd is 0.5. In Fig. 26 typical crossings of this kind are shown. They are for two I ••17 ! different gallium doped germanium crystals, one containing 3 X 10 cm~^ gallium and the other 9 X 10^^ cm~^. The curves for the controls and lithium saturated samples in each case are shown as plots of the logarithm of Hall mobility against logarithm of absolute temperature. : The lines plotted in this manner are straight. The lithium content of 1he bridge containing 9 X 10^^ cm~^ gallium was 6.1 X 10 cm~ while that in the bridge with 3 X 10^^ cm"^ gallium was 2.8 X 10^^ cm~l All , of these concentrations were obtained from Hall coefficient measurements < m the controls and the lithium doped specimens. As the temperature is increased the mobilities of the samples with lithium are reduced and approach the mobilities of the controls. This happens because pairs dissociate and more charged impurities appear. 1 inally when P/Nd is exactly 0.5 the curves cross. In Fig. 27 we notice that mobility measurements were not performed right up to the cross point, but that the straight lines have been extrapolated. This procedure was adopted of necessity, because of the high diffusivity of lithium. Thus, Inference to Fig. 5 shows that the solubility in doped germanium de- i (leases to a minimum as the temperature is raised from room tempera- t ure, and there is danger of precipitation. For this reason the measure- 604 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 a z o o Ud I/) I o > a. Ill a. 2 o ffl o 5 Ga CONTROL \ = 9X10'^CM-A k 2600 ^^ i a, CALCULATED \ ^ CROSS POINT A =1.73X10-8CM 2000 \ \L ' \ \ 1500 \ V \ \ \ 1000 CONTROL ^ Ga = 3x 10"CM" \ \ \ \ A 900 H w \ \ \ 800 700 600 \ \ V V A \ a, = 1.71xlO-8CM--A. 500 1 w 100 150 ?00 250 300 400 TEMPERATURE IN DEGREES KELVIN 500 600 Fig. 26 — Illustration of cross over phenomenon for germanium samples con- taining gallium. Sample 314 contains 9 X 10^* cm"^ gallium and sample 302 con- tains 3 X 10'^ cm"^ Samples 316 and 301 are the corresponding samples to which lithium has been added. ments were not carried to high temperatures.* In addition the value of the Hall coefficient was carefully checked at each temperature to see if it had changed. Since the reciprocal of the Hall coefficient measures the carrier density any reduction in its value would have implied loss of compensation, or precipitation of lithium. Over the measured points no appreciable variation of Hall coefficient was noted. Fortunately, the pairing relaxation time is quite small (less than a second) at the high temperatures involved so that it wasn't necessaiy to hold the samples at these temperatures for long periods in order to achieve pairing equilibrium. The times involved were too short for the occurrence of phase equilibrium characterized by precipitation. The above discussion points up some of the care that must be taken to obtain reliable measurements. Another factor which enters the pic- ture is the possible existence of a precipitate in the lithium doped bridge. * In boron-doped germanium the cross-over was actually observed — no extra- polation having been necessary, because the temperature of intersection was suffi- ciently low. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 605 During the course of our experiments it was discovered that precipitates have a profound effect on carrier mobility, reducing it so severely, that the mobility of the lithium doped bridge may never even rise above that of the control. Great care must be exercised in the preparation of suitable bridges to avoid the presence of precipitated lithium. Thus it may be necessary to saturate the bridge at a very low temperature (see Section IV, Figure 5) so that it is somewhat undersaturated at room tempera- ture. This means that diffusion periods of weeks may be involved. In Fig. 26 the sample with Na = 9 X lO'' cm~^ and No = 6.1 X lO'* cm~^ has P/Nd = 0.5 at 348°K, while the sample with AT^ = 3 x lO" cm"* and No = 2.8 X 10^^ cm~* is half-paired at 440°K. This is to be expected, the more heavily doped specimen remaining paired up to higher temperatures. Using (9.6) and (9.3) it is possible to calculate a, the distance of closest approach of a gallium and lithium ion, from each of the measured cross points. Thus in (9.6) we set 6 = 0.5, and Na , No and T to correspond to each of the cases described. Having logio Q((x), a can be determined by in- terpolation in Table III and a then determined from (9.3). Of course k is taken to be 16. Carrying through this procedure in connection with Fig. 26 leads to the satisfying result that a = 1.71 X 10~^ cm for the heavily doped sample and 1.73 X 10"* cm for the lightly doped one. The values of Q, appearing in Table IV based on a = 1.7 X 10" cm there- fore correspond to gallium. Not only is this result satisfying because the two a's agree so well even though the samples involved were so different in constitution, but also because it is expected on the basis of the addition of known particle radii. Thus according to Pauling^^ the tetrahedral covalent radius of gallium is 1.26 X 10~* cm while the ionic radius of lithium is 0.6 X 10" cm. Since gallium is presumably substitutional in a tetrahedral lattice we use its tetrahedral covalent radius, and since lithium is probably in- terstitial we use the ionic radius. The sum of the two is 1.86 X 10 cm which compares very favorably with the values of a quoted above. This result constitutes good evidence that lithium is interstitial, for if it were somehow substitutional we might expect a to be something like a germanium-germanium bond length which is 2.46 X 10" cm. Such a value of a would lead to profoundly different crossing temperatures (of the order of 100° lower) so that it is not very likely. One further point needs mention. This is the fact that as the two ions approach very closely, the concept of the uniform macroscopic dielectric constant, k, loses its meaning. In fact, the binding energy should be in- creased (as though K were reduced). Crude estimates of the magnitude 606 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 of this effect based on a dielectric cavity model show it to be of the order of some 10 or 15 percent of the energy computed on the assumption of the dielectric continuum, the increased binding energy showing up as a reduced value of a. This may account for the fact that the observed a, at 1.7 X 10" cm is less than the theoretical value, 1.86 X 10" cm. The above example shows the ion pairing phenomenon in action as a structural tool, useful in investigating isolated impurities. In particular the demonstration that lithium is interstitial is interesting. The values of a have much more meaning as independent parameters in solids than they have in liquids, where a given ion may be surrounded by a sheath of solvating solvent molecules. Under the latter conditions the value of (i can only be determined through application of the ion pairing theor}- itself. Of course, certain unusual situations arise in solids also, and values of a (determined from ion pairing) are valuable indications of structural peculiarities. Similar experiments have been performed on specimens doped with indium and boron. The results of all our investigations on the cross-over phenomenon are tabulated in Table V. In the table the first column lists the acceptor involved, and the second and third the appropriate concentrations of impurities. The fourth column contains the cross-over temperature, while the fifth, the measured value of a determined from it. The last column lists the values of a to be expected on the basis of the addition of tetrahedral covalent radii to the ionic radius of lithium — all of which appear in Pauling. The reliability of the measurements are in the order gallium, alumi- num, boron, and indium. The principal reason for this is that the indium crystal was not grown specially for this work and was somewhat non- uniform. Of the two values obtained for a we tend to place more confi- Table V Acceptor Acceptor cone. Lithium cone. Cross-over Temp. Measured a Pauling a (cm-3) (cm-3) (°C.) (cm) (.cm; B 7.0 X 1016 5.9 X 10i« 338 2.05 X 10-8 1.48 X 10-8 B 7.0 X 10i« 5.54 X 10" 320 2.27 X 10-8 1.48 X 10-8 B 7.0 X 10'« 5.85 X 10" 330 2.16 X 10-8 1.48 X 10-8 Al 9.5 X 10" 9.0 X 10" 350 1.68 X 10-8 1.86 X 10-8 Ga 3.0 X 101^ 2.8 X 10" 440 1.71 X 10-8 1.86 X 10-8 Ga 9.0 X 10" 6.1 X 10" 348 1.73 X 10-8 1.86 X 10-8 In 8.3 X 10'7 1.9 X 10" 476 1.61 X 10-8 2.04 X 10-8 In 3.3 X 10" 2.68 X 10" 426 1.83 X 10-8 2.04 X 10-8 CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 607 dence in 1.83 X 10~^ than in 1.61 X 10~^ cm. More work is necessary, however, before a real decision can be made. A feature of Table V is the fact that gallium, aluminum, and indium exhibit orthodox behavior, i.e., the measured a's are in both cases slightly less than those expected on the basis of the addition of radii. The in- ternal consistency of the theory gains support from the fact that gal- lium and aluminum behave similarly as the Pauling a's tabulated in Table V predict. In fact if 1.83 X 10~ cm is taken as the more reliable indium value the three cases fail to match the Pauling radii by about the same amount, a result which implies that the disparity is due to the same cause, i.e., failure of the dielectric continuum concept. Another feature of Table V is the fact that boron is out of line to the extent that the measured a exceeds the Pauling a by 50 per cent. A pos- sible explanation is the following. The tetrahedral radii of boron and o o germanium are poorly matched (0.88 A and 1.26 A, respectively). The strain in the boron-germanium bond may appear as a distortion of the germanium atom in such a way as to increase the effective size of the boron ion. This strain was mentioned before in Section V where it was invoked to explain the stability of LiB~ complex in silicon. XIII, RELAXATION STUDIES The relaxation time discussed in Section X has been studied experi- mentally. The following procedure was used. A specimen was warmed to 350°K where a considerable amount of pair dissociation occurred, and then cooled quickly by plunging into liquid nitrogen. It was then rapidly transferred to a constant temperature bath, held at a temperature where pair formation took place at a reasonable rate, and the change in sample conductivity (as pairing took place) was measured as a function of time. The principle upon which this measurement is based is the following. At a given temperature the occurrence of pairing does not change the carrier concentration, only the carrier mobility. As a result the measure- ment of conductivity is effectively a measurement of relative mobility. During relaxation the densities of charged impurities are changed, at the most, by amounts of the order of 50 per cent. Over this range, the mobil- ity may be considered a linear function of scatterer density. The depend- ence of conductivity on time, as pairing takes place, must be of the form c ^ a„-^ e-"' (13.1) where cr^ is the conductivity when ^ = co , and r is the relaxation time de- fined in section X while $ is some unknown constant, depending among 608 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 other things on the initial state of the system. Equation (13.1) is based on the assumption that the number of charged scatterers decays as a first order process, and that cr is a linear function of this number, relative to the exponential dependence on time. The first order character of pairing is fortunate for it renders the measurement of r independent of a knowledge of $, i.e. independent of the initial state of the system. This is not only fortunate from the point of view of calculation but from experiment, since it is almost impossible to prepare a specimen in a well defined initial state. The unimportance of is best seen by plotting the logarithm of (T„ — (T against time. According to (13.1) this plot is specified by log (o-« - cr) = log * + - T (13.1) Thus the reciprocal of its slope measures r, and $ is not involved. Fig. 27 illustrates the data for a typical run plotted in this manner. The sam- ple is one containing about 9 X lO'^ cm~^ gallium and the experiment was performed at 195°K (dry ice temperature). Notice that the curve is absolutely straight out to 3500 minutes, demonstrating beyond a doubt that the process is first order. The relaxation time computed from its slope is 1.51 X 10 seconds as against a value calculated by the methods 1 I I o l.Or-: 0.6 0.4 0.2 0.1 1 "V,^^ \ TEm Go T = T = P = 195° = lO'^CM" 1.66 X 10^ 1.51 X 10^ 3 SEC (mE/C SEC (CAL vSURED) culated) \ ^\ '.!• 500 1000 1500 2000 TIME IN MINUTES 2500 3000 3500 Fig. 27 — Plot of log (o-„ — ) N (cm-«) Nd (cm-») Nd* (cm-») P (cm-») 10i« 10" 0.99 X 10" 0.99 X 1017 3.2 X 101* 1.6 X 10" 1.26 X 10'* 0.88 X 10" 3 X 101* . 1.6 X 10" CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 617 lively, 10^^ and 10^^ cm~^ gallium ions available for pairing, the pairing process did \'ery little to increase the solubility. If the constant 9, is exceedingly large as is probably the case for a multiply charged acceptor, it is possible that ion paring will have a meas- urable effect on solubility. Appendix B concentration dependence of diffusivity in the presence of ion PAIRING In Section VIII it was mentioned that the diffusivity of a mobile donor like lithium is concentration dependent when the donor participates in a pairing equilibrium with an immobile acceptor. In this appendix we propose to investigate the nature of the dependence. Consider a semiconductor, uniformly doped to the level, Na , with acceptor. Let the local density of mobile donor be Noix), x being the position coordinate. If P(x) is the local pair concentration, then the local density of free diffusible ions is {No — P). The flux of these diffusing ions then depends upon the gradient (assuming Fick's law^^) of {N d — P). Thus, if Do is the diffusivity of free donor, i.e. the diffusivity in the absence of pairing, then the flux density is /=-D.£(^[^) (Bl) dX If we apply (9.4) to the present case we can write O = ^ ^ - {No - P) + .V. {Na - P){Nu - P) [{Na - No) + (A^;, - P)]{No -P) ^ ^^ ifrom which it is possible to solve for {No — P). Thus Substitution of (B3) into (Bl) yields Do ^=-2 \[Nn-NA + 1 + 0/ /I dNi dx (B4) llf ion pairing was not thought of, the flux density would have been writ- ten in terms of the gradient of the total concentration. No ■ f= -D^-p^ (B5) dx 618 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 where D is the diffusivity. Comparison of (B5) with (B4) leads to the relation D = Do 1 + K^^- N^ + i/l(^«- (B6) so that D depends on the local concentration, No , of diffusant. It is interesting to explore the limiting forms of D when No « Na and when Nd = Na . In the latter case (B6) reduces to 1 + ^ -f y 4fi2 ^ 12 _ (B7) while (B3) becomes N, ^20 y 402 ^ j2 (B8) Substituting the left side of (B8) for the denominator involving the radical in (B7) leads to - = T° 1 + 2(Na - P)0 + IJ But according to (B2), when Na = Nd , P (Na - P)0 = N, (B9) (BIO) so that (B9) becomes « = l" 1 + 2P N. + 1 (B12) Now in case the degree of pairing is high (which is, of course, the case we are interested in) P will be almost equal to Na so that 2P Na- P (B13) will be a very large number. If this is so the second term in brackets in (B12) can be set equal to zero and we have D, D = (B14) CHEMICAL INTERACTIONS AMONG DEFECTS IN Gg AND Si 619 In the other extreme with No « Na (B6) becomes -f 1 + «-^^ 4/laWJ Do 1 + ^Na (B15) Since Q Na can exceed unity by a large amount it is evident that the re- lation in (B15) predicts a large reduction in diffusivity towards the front end of a diffusion curve where Nd « A^^ , and (B14) a smaller re- duction in Do where Np may be close to Na . That part of the medium near the front of the diffusion curve acts therefore like a region of high resistance, confining the diffusant to the back end where the resistance is low. Appendix C solution of boundary value problem for relaxation In Section X equations (10.23), (10.21), (10.20), and (10.19) defined a boundary value problem which we reproduce here, except that (10.20) and (10.19) have been written more completely with the aid of (10.16). Thus r^ dr \ dr . Do dt dp , R n T r- + ^P=0, r = L, dr r^ r = a p = N\ t == 0, a ' (C8) I and if we assign the subscript 77 to the G going with r? the most general solution of (CI) and (C2) will be P = Z A,Gr,(r)e-"'''°' (C9) where the A,, are arbitrary constants so determined that (C3) is satisfied. Equation (C9) shows that in reality there exists, for this problem, a spectrum of relaxation times, I/t^'Do . After a brief transient period many of the higher order terms will decay away and eventually only the first two terms will have to be considered. Finally when equilibrium is at- tained only the first term Avill survive. The last statement implies that 77 = 0, is an allowable eigenvalue, i.e., that the first term is independent of time. That this is so can be proved by solving (C5) for 17 = 0, and substituting the result in (C7). Thus Go(r) = exp (f^^ (CIO) and this does satisfy (C7). p can then be approximated after the transient by p = Ao exp (j^ + ^1 Gi(r)e-''i-^'" (Cll) from which it is obvious that the relaxation time dealt with in section X is T = -hr (C12) In principle it should be possible to evaluate Gi by the straightforward solution of (C5) and determination of the second eigenvalue through substitution of this solution in (C7). In fact this represents a rather un- pleasant task since G is a confluent hypergeometric function. Therefore we shall follow an alternative route based on the assumption that by the time (Cll) applies the flux 4xr"J*(r), where J* is given by (10. IG), is almost independent of r. The reader is referred to some related papers ' CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 621 for the justification of this view. Briefly it is permissible, after a short transient period, in spherical diffusion, whenever the dimensions of the diffusion field are large compared to the dimension of the sink. This re- sults from the fact that in spherical diffusion from an infinite field a real steady state is reached after a brief transient period. In contrast, in plane-parallel diffusion to a sink from an infinite field, a steady state is never reached. Substituting (Cll) into (10.16) then yields J* == -AAie-'"'''"' ("^ + ^ G^ (C13) \dr r^ / Multiplying J* by 47rr" and demanding that the product be independent of r, leads to the relation r'^+RG^ = 8 (C14) dr where 8 is constant. The solution of (C14) is G, = exp g) + I (C15) This is a sufficient approximation for Gi . 1 The constants r]i , Ao , Ai , and 5 must now be determined. To accom- 'plish this we note that (C2) which specifies that the boundaries at r = a ,and r = L, are impermeable is equivalent to the condition that ions be 'conserved with the interval (a, L), or that 47r ( r'p dr = N (C16) Ja \ \fter infinite time p is specified by the first term of (Cll) and when this is inserted into (CI 6) the result is Ao = NM (C17) |,vhere M is defined by (10.26). ': Substitution of (C17) and (CIS) into (Cll) gives p = NM exp (R/r) + (ai exp (R/r) + ^ j e""^'"'"' (C18) Now (C3) applied to (C18) demands NM + Ai = 0 (C19) ^ ^ N' (C20) R 622 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Of course this presumes that the approximation contained in (C18) is valid down to very small values of time. This assumption is well founded as the transient does vanish after a rather short time. Inserting (C19) and (C20) in (C18) then gives us p = NM exp {R/r) + N[N - M exp iR/r)]e-'"''">' (C21) in which only 771 remains to be determined. Substitution of (C21) into (C16), recalling the definitions of M and L, shows that it already satisfies (C16) for any time, t. Thus (C16) cannot be used for determining 771 . On the other hand we note from (C21) that as soon as r becomes of order, R, p becomes almost independent of r, being given p = N{N + (N - M)e-'"'''''} (C22) Since L is of the order lOR or greater, this means that throughout most of the volume, l/N (in fact throughout 0.999 1/A^) p is independent of r. Effectively, the entire volume 1/iV has been drained of ions, i.e., they have been trapped. The total ion content at time t, may then be taken as the product of p, given by (C22), with 1/iV, that is, N + (N - M)e-'""'''' (C23) : The time rate of change of this content must be given by the flux Airr J*. ^[Ar+ (iV - M)e-''^'^°'] , , (C24)l = -mDo(N - M)e-'"^°' = 47rrV*(r, t) = -AwRN^D^e-''"'''' in which (C21) has been substituted into (10.16) to pass from the third to the fourth expression. Comparing the second and fourth term of (C24) reveals or 1 KkTjN - M) "" ~ 771'Do ~ Wn'Do the value quoted in (10.25). (C26) chemical interactions among defects in ge and si 623 Appendix D minimization of the diffusion potential In Section V the statement was made that equation (11.2) was a valid approximation everywhere within a p type region, provided that No did not fluctuate through ranges of order A^^ in shorter distances than = 4/^ (Dl) This statement will now be proved. The electrostatic potential is determined by the space charge equation 31 dx^ where we assume that the material is everywhere p-type so that the elec- tron density, n, does not enter the right side of (D2). Furthermore, the mobility of holes is so much greater than that of donor ions that the for- mer may be considered to always be at equilibrium with respect to the distribution of the latter. Boltzmann's law^^ may then be applied to p. The result is p = Na exp [-qV/kT] (D3) where the potential is taken to be zero when p = A^^ . Choose an arbitrary point, Xo , where the potential is Vo and investi- gate (D2) in its neighborhood. We wish to determine the conditions under which the right side of (D2) may be approximated by zero, i.e., the "no- space-charge condition," in this neighborhood. The limits of the neigh- borhood will be defined such that \V - Vol = \n\ ^ kT/2q (D4) so that, in it, the exponential in (D3) can be linearized p = Na exp [- gVo/kT] (l - ||) (D5) jThen (D2) becomes i~ = ^ Ina [1 - exp (- qVo/kT)] - Noix) + w ^^p ^~ ^^°/^^^ u\ (D6) 624 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 The no space charge condition in the defined region is therefore ^ ^ Arexp(,yoAT)\ ^, ^ ,kT\ exp(-gFoAr)-l \ q^A / \q/ exp {- qVo/kT) To simplify notation define expi-qVo/kT] = 70 (D8) Next expand both No and u in Fourier series i 00 Nd = ^ As sin sx + Bs cos sx (D9) s=0 00 u = 2Z «3 sin s-x + jSa cos sa: (DIO) »=o Substitution of (D9) and (DIO) into (D6) and equating coefficients of like terms leads to the set of relations /3o = 4^ [^^(-^0 - 1) + ^J (1^11) K \1 + (s2/V47r-7o)/ Now the wavelength of the sth component in (D9) is X. = 27r/s (D14)' If N'd contains no important components of wavelength shorter than Vto (D15) the Bk for such components may be set equal to zero. But then the only- terms which appear in (D12) and (D13) are terms where the denomina- tors which (with the aid of (D14)) may be written as may be set equal to k. Thus we have in place of (D12) and (D13) a, = ^As= 4^ As (D17) CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 625 ^. = 5-i 5, = -P- B, (D18) KIT qJS AlO The requirement that No contain no Fourier terms of wavelength shorter than (D15) is obviously the condition that No never pass from its maximum to its minimum value in a distance shorter than D(15). Since we are assuming that Nd may at places be of order Na , and at others, of order zero, this amounts to the condition that No does not fluctuate over ranges comparable with A^^ in distances shorter than (D15). The use of (Dll), (D17), and (D18) in (DIO) yields kT u qNaJo NAiyo — I) + '^ (As sin sx + B^ cos sx) «=o J , (D19) ^ kT (70 - 1) ,kT_ND q (to) qyo Na which by reference to the definition (D8) for 70 proves to be identical with (D7), the no-space- charge condition. Equation (D19) is only true when No does not fluctuate through ranges of order, Na , in distances smaller than //\/7o . This distance de- pends on 7o and thus on the point where V = Vo , whose neighborhood is being explored. Thus, we may say that there will be no space charge at all points whose Vo is such as to fix 70 at a value such that 70 > .-2- (D20) Amin where Xmin is the minimum wavelength which needs to be considered in the Fourier expansion of Nd . In terms of the definition of 70 this means Vo <—in^ ■ (D21) q (?■ Thus, at all points where Fo is less than the right side of (D21) the no space charge approximation will hold. (D21) shows, that in the limit when Xmin goes toward zero, i.e. when the infinite series must be used for N n , the right side of (D21) will approach — 00 and Fo will satisfy (D21) hardly anywhere. Thus space charge will exist almost eveiy where. I In most diffusion problems the extremes of potential will occur in re- Igions where there is no space charge. Thus in one extreme N d may equal jO.9 N A and in the other it may equal zero. If there is no space charge in (these extremes we may write for them NA-Nn = V = Na exp i-qV/kT) (D22) in which (D3) has been used. Setting N d equal to zero in one extreme 626 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 yields F = 0. In the other extreme No = 0.9 A'' a so that we get l-T 7 = ^ ^n 10 (D23) Q This therefore is the largest value which Vo may assume in our case. Inserting the expression in D21 in place of Vo we end with the relation 10 < ^ (D24) Thus provided that in the distribution being considered Xmin > 3.5^ (D25) there will be no space charge anywhere. At high temperatures 0.1 Na may be less than rii . Under these condi- tions (D24) should be replaced by 12a. ^ Amrn ^j^26) rii P and in the limit that rii becomes very large it is obvious that (D26) will always be satisfied. The rule to be enunciated for the cases we shall be interested in is the one given in section XI, i.e. that no space charge will exist provided that X min is no less than order, /. Appendix E calculation of diffusivities from conductances of diffusion LAYERS In this appendix equation (11.12) will be derived. In the first place we note that the dependence of Nd on position x, and time t, will be of the form Nc{x/\/t) at any stage of the diffusion process. This results from a theorem due to Boltzmann^^ that when the dependence of D upon X and t is of the form D(Nd), i.e., the dependence is through Nd , and a semi-infinite region extending from x = 0 to a: = oo is being considered, then, in the case of plane parallel diffusion, the only variable in the prob- lem will be x/\/}. Although the wafers considered in Section XI are of finite thickness d, the stages of diffusion investigated are such that the two regions of loss near the surfaces have not contacted each other. As a result the system behaves like two semi-infinite regions backed against one another, and the preceding arguments hold. The conductance 2, defined in section XI will be proportional to the integral of the product of the local carrier 1 CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 627 density by the local mobility. Thus S = CO / ix(x, t)[NA - ND{x,t)]dx (El) where co is a proportionality constant and n(x, t) is the local mobility. An upper limit of d/2 rather than d is used because of symmetry. The local mobility will vary because No , and therefore the local density of charged impurity scatterers, varies. Let No be the initial uniform den- sity (before any diffusion out) of donors, and write (El) as pan S = CO n(x, t)[NA - No + No° - Nn(x, t)] dx ''0 = CO / IX{X, t)[NA. - Nd] dx + CO / li{x, t)[ND Jo Jo (E2) - NdCxjOj^x The second integral on the right of (E2) is given the upper limit co , because in the experiments we wish to perform No — No becomes zero long before x reaches d/2. Now in the first integral on the right of (E2) we may set fjL(x, t) equal to the constant value no , which it assumes in the bulk of the wafer, be- cause the breadth of the depletion layer near the surface (in which (i(x, t) departs from juo) is small compared to d/2. The same thing can- not be done in the second integral since the integrand vanishes beyond the depletion layer and the total contribution comes from that layer. We thus obtain 2 = com''(N^ - Nz,°) d/2 + C0 X "VV?, .^"° - ^° ivi)] "" (E3) In the integral in (E3) both /x and No are represented as functions of x/-\/i, the latter because of what has been said above, and the former, because it is a function of the latter. Defining V = x/2\/Dt (E4) in which D is constant, and substituting in (E3) gives finally 2 = co/Xo(N^ - N/)f//2 + 2o^\/Di f m('')[Nz,° - Nx,(^)]rf^ (E5) Jo 628 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Since the definite integral is a constant (E5) shows that S is a Hnear function of s/t^ a fact mentioned in section XL In order to make use of the measured dependence of 2 on -sfi to determine diffusivities, the functions y.{y) and A^d(v) must be specified. For the latter we shall assume the Fick's law solution " Ar„ = AT^" erf v (E6) going with constant Z), and "N d = 0 as a boundary condition at a; = 0 at the surface. (In section XI the limitations of this assumption in the presence of ion pairing and diffusion potential are discussed.) The v dependence of \x is more complicated. In general, we shall be concerned with electrical measurements in two extreme cases. In the first case ion pairing, under the condition of measurement, is everywhere com- plete so that the local density of scatterers will be given by l^A - NM (E7) In the other case ion pairing will be entirely absent, so that the local scatterer density, will be specified by N^ + NoM (E8) In all experiments A^^ will be only slightly greater than Nd so that it may be replaced by this quantity. Doing this, and substituting (E6) into (E8) and (E9) gives No' erfc V = N{v) (E9) for the scattering density in the ion pairing case, and Nn'a + erf v) = N(v) (ElO) for the no pairing case. Since almost all our experiments have been in germanium we now specialize our attention to that substance. However, the procedure in- voked below can be applied to silicon as well. The dependence of hole mobility, n, on scattering density, A^, for ger- manium at room temperature is shown in Fig. 30 taken from Prince's data.^^ The integral in (E5) assumes the form Nz," [ fxCNiv)) eric vdv. (Ell) Jo Choosing N{v) as either (E9) or (ElO) and using Fig. 30 together with a tabic of error functions makes the numerical evaluation of (Ell) possible. Since N(v) given by (E9) or (ElO) depends on No, so will the integral. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 629 Q Z y <;uuu in > (600 Lll ^^ ^^ N s. 5 (200 z \ \, \ in lU d 800 I u. \ V. \ V O > \ t 400 _) CD O 5 h- 0 X o 10' 10'^ (0'^ (0'° (0' CONCENTRATION OF IONIZED IMPURITIES IN CM"^ (0" Fig. 30 — Plot of hole-drift mobility in germanium as a function of ionized impurity concentration after Prince. The numerical evaluation has been performed for a range of Nd^ in both the pairing and non-pairing cases. In this manner it has been pos- sible to evaluate the "correction factor" t^ defined by the following equa- tion / niv) erfc V dv = t?M«> / erfc v Jq Jo = t?Moc(0.563) dv (E12) where /i^ is the mobility in the presence of A^^ scatterers. Fig. 22 contains plots (for germanium) of i}(ND°) versus A^d" for both the pairing and non- pairing cases. It is seen that t> is never much different from unity. Equation (E5) can now be written as 2 = a;Mo(A^^ - ND°)d/2 + COM [i .l2St}N ^W D]\/t (E13) Defining 2o = coMoCN^ - ^z>°)d/2 ^00 — X (E14) (E15) it is obvious that So is the conductance before any donor has diffused 630 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 out and S^ after all the donor has been diffused out. With these defini- tions (E16) becomes ./..., ?-^ (^^-N^,) V. (-) = Calling the slope of this curve S leads to the result I or using (E14) and (E15) ) D = ( _^^iAZ£_ ) (E19) This is equivalent to equation (11.12). Glossary of Symbols a distance of closest approach of two ions of opposite sign A constant in expression for p in section on relaxation theory,' A~ concentration of ionized acceptors ^0 Ar, going with t? = 0 Ai Ar, going with rji Aj, constant preceding the Tjth eigenfunction in solution of the! relaxation problem A, coefficient of sin sx in Fourier expression for No h q^/2KkT, position of minimum in g(r) B constant in expression for p in section on relaxation theory B~ boron ion \ B(Si) un-ionized boron in silicon ; Bs coefficient of cos sx in Fourier expression for No c(r) concentration of positive ions in atmosphere of a negative ion C concentration of LiB~ d thickness of wafer in diffusion experiment D diffusivity of donor ion in the most general sense Do diffusivity of donor ion m the absence of pairing D"*" concentration of ionized donors Do"*" value of D^ in the absence of acceptor D*^ concentration of mobile donor ions where V = 0 e~ conduction band electron valence band hole .+ CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 631 , e^e" recombined hole-electron pair E energy level in electron gas Ed ionization energy of a donor I Ea ionization energy of an acceptor Ei energy level in conduction band j E{r) chance that volume 47rr /3 will not contain an ion I / flux density t F Fermi level — also constant in equation (7.21) ! Qi density of states of energy Ei in conduction band ' g{r) nearest neighbor distribution function at equilibrium I G Gibbs free energy of electron assembly GaT gallium ion in germanium (?„ space dependent part of relaxation eigenfunction ; Go G, for 77 = 0 Gi Gr, for r? = 171 h Plank's constant — also used for normalizing constant in c(r) hj number of holes in the jth energy level H net local density of fixed donors i{p2 , pi) £^/(r2 , n) I field current in diffusion measurement I(r2 , ri) integral for ion pairing calculations taken between ri and rz J(r) current in the atmosphere of a nearest neighbor J* flux density of ions being trapped k Boltzmann's constant fci first order rate constant in relaxation theory ^2 second order rate constant in relaxation theory Ko distribution coefficient of donor between semiconductor and external phase Ki electron-hole recombination equilibrium constant Ka ionization constant of acceptor Kd ionization constant of donor Kj constant relating wy to volume, V K* product oi Kd , Ko, and a I screening length for diffusion potential L Debye length — also used for radius of volume, 1/A^ Li^ lithium ion Li(Sn) lithium in molten tin Li{Si) un-ionized lithium in silicon LiSi lithium-silicon complex LiB un-ionized LiB~ 632 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 IAB~ lithium-boron complex ion in semiconductor [Li^BT] lithium-boron ion pair [Li^GaT] lithium-gallium ion pair mo normal mass of electron mp effective mass of a hole M normalizing constant in relaxation theory n concentration of conduction electrons — also used for density of untrapped ions in relaxation rii intrinsic concentration of electrons N A total acceptor concentration Nd total donor concentration Nd total solubility of donor in undoped semiconductor — also used for initial density of donors in diffusion experiments N ion concentration in an electrolyte solution — also used for initial value of n in relaxation — also used for concentration of im- mobile donors in Appendix A Nd* solubility of donor in absence of ion pairing in Appendix A p concentration of holes P concentration of ion pairs q charge on an ion Q{a) tabulated integral for computing 9, r distance between positive and negative ions in a pair ri a particular value of r rt a particular value of r R capture radius of an ion in relaxation S slope of 2/So versus -\/f curve S^ time dependent part of relaxation eigenfunction belonging to eigenvalue t] t time T temperature u V -Vo V electrostatic potential — also used for volume — also used for potential difference between probe points — also used for] potential energy of a positive in neighborhood of negative ion Vo electrostatic potential where x = Xq X variable of integration - — same as r also rectangular position coordinate Xo special value of a:. 1 CHEMICAL INTEEACTIONS AMONG DEFECTS IN Ge AND Si 633 zja — also used for thermodynamic activity of donor in external phase coefficient of sin sx in Fourier expression for u constant in exponential in LiB~ equilibrium constant constant in exponential in expression for vacancy concentra- tion /3s for s = 0 coefficient of cos 8X in fourier expression for u pre-exponential factor in LiB~ equilibrium constant pre-exponential factor in expression for vacancy concentra- tion exp[-gFoAT] non-equilibrium nearest neighbor distribution function constant appearing in Appendix C eigenvalue in relaxation problem second eigenvalue in set of q fraction of donor paired correction factor for variable carrier mobility dielectric constant xje 2ir/s, wavelength of sth component of fourier series wavelength of component of fourier series for No , having minimum wavelength chemical potential of donor in an external phase — also used for mobility of donor ion — also used for local carrier mo- bility chemical potential of donor in external phase in standard state chemical potential of donor ion chemical potential of donor ion in the standard state chemical potential of an electron chemical potential of donor atom in semiconductor chemical potential of donor atom in standard state mobility of donor atom at infinite dilution — also used for carrier mobility in diffusion experiments before diffusion carrier mobility in diffusion experiments after all diffusant has diffused out x/2VDt e/r. LiBT equilibrium constant resistivity of gallium-doped germanium after saturation with 634 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 lithium — also used for local charge density in Poisson's equation — also used for density of diffusing positive ions in relaxation Po resistivity of gallium-doped germanium before saturation with lithium . J Pi n/e I a conductivity during relaxation | a^ conductivity in relaxed state S conductance between probe points So conductance before diffusion begins in diffusion experiments; S„ conductance after diffusion is over in diffusion experiments T relaxation time $ constant in relaxation formula for conductivity "^ local electrostatic potential in ionic atmosphere w proportionality constant connecting conductance between probe points with integral over carrier concentration CO; number of states in jih. level of electronic energy diagram 0 ion pairing equilibrium constant D vacant lattice site in covalent crystal n~ negatively charged cation vacancy REFERENCES 1. Wagner, C, Z. Phys. Chem., B21, p. 25, 1933, B32, p. 447, 1936. 2. Taylor, H. S., and Taylor, H. A., Elementary Physical Chemistry, p. 343, Van Nostrand, 1937. 3. Shockley, W., Electrons and Holes in Semiconductors, p. 6, Van Nostrand, 1950. 4. Shockley, W., Electrons and Holes in Semiconductors, Van Nostrand, 1950. 5. Reiss, H., J. Chem. Phys., 21, p. 1209, 1953. 6. Reiss, H., and Fuller, C. S., J. Metals, 12, p. 276, 1956. 7. Fuller, C. S., and Ditzenberger, J. A., J. App. Phys., 25, p. 1439, 1954. 8. Fuller, C. S., and Ditzenberger, J. A., Phys. Rev., 91, p. 193, 1953. 9. MacDougall, F. H., Thermodynamics and Chemistry, p. 143, Wiley, 1939. 10. Miller, F. W., Elementary Theory of Qualitative Analysis, p. 102, Century. Company, New York, 1929. 11. Fuller. C. S., Record of Chemical Progress, 17, No. 2, 1956. 12. Wagner, C, and Grunewald, K., Z. Phys. Chem., B40, p. 455, 1938. 13. von Baumbach, H. H., and Wagner, C, Z. Phys. Chem., 22B, p. 199, 1933. 14. Kroger, F. A., and Vink, H. J., Physica, 20, p. 950, 1954. 15. MacDougall, F. H., Thermodynamics and Chemistry, p. 258, Wiley, 1939. 16. Shockley, W., Electrons and Holes in Semiconductors, p. 231, Van Nostrand, 1950. 17. Mayer, J. E., and Maver, M. G., Statistical Mechanics, p. 120, Wiley, 1940. 18. MacDougall, F. H., thermodynamics and Chemistry, p. 137, Wiley, 1939. 19. Lewis, G. N., and Randall, M. C, Thermodynamics, p. 258, McGraw Hill, 1923. CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 635 20. Mayer, J. E., and Mayer, M. G., Statistical Mechanics, p. 121, Wiley, 1940. 21. MacDougall F. H., Thermodynamics and Chemistry, p. 261, Wiley, 1939. 22. MacDougall, F. H., Thermodynamics and Chemistry, p. 25, Wiley, 1939. 23. Engell, H. J., and HouiTe. K., Z. Electrochem., 56, p. 366, 1952, 57, p. 762, 1953. 24. Shockley, W., Electrons and Holes in Semiconductors, p. 15, Van Nostrand, 1950. 25. Morin, F. J., and Malta, J. P., Phys. Rev., 94, p. 1525, 1954. 26. Morin, F. J., and Malta, J. P., Phys. Rev., 96, p. 28, 1954. •^7. Shockley, W., Electrons and Holes in Semiconductors, p. 86, Van Nostrand, 1950. 28. Shockley, W., Electrons and Holes in Semiconductors, p. 88, Van Nostrand, 1950. 29. Fowler, R. H., Statistical Mechanics, p. 48, Cambridge, 1929. 30 Slater. J. C, and Frank, N. H., Introduction to Theoretical Physics, p. 212, McGraw Hill, 1933. 31. Shockley, W., B.S.T.J., 28, p. 435, 1949. 32. Fuller, C. S., and Ditzenberger, J. A., J. App. Phys., May, 1956. 33. Shulman, R. G., and McMahon, M. E., J. App. Phys., 24, p. 1267, 1953. 34. Reiss, H., Fuller, C. S., and Pietrusczkiewcz, A. J., J. Chem. Phys. (in press). 35. Eyring, H., Walter, J., and Kimball, G. E., Quantum Chemistry, p. 231, Wiley, 1946. 36. Pauling, L., The Nature of the Chemical Bond, p. 179. Cornell, 1942. 37. Debye, P., and Huckel, E., Phys. Z., 24, p. 195, 1923. 38. Kirkwood, J. G., J. Chem. Phys., 2, p. 767, 1934. 39. Briggs, H. B., Phys. Rev., 77, p. 287, 1950. 40. Briggs, H. B., Phys. Rev., 77, p. 287, 1950. 41. Wyman, Phys. Rev., 35, p. 623, 1930. 42. Bjerrum, N., Kgle. Danske Vidensk. Selskab., 7, No. 9, 1926. 43. Fuoss, R. M., Trans. Faraday Soc, 30, p. 967, 19.34. 44. Reiss, H., J. Chem. Phys. (in press). 45. Reiss, H., J. Chem. Phys. (in press). '46. Shockley, W., and Read, W. T., Jr., Phys. Rev., 87, p. 835, 1952, Haynes, J. R., and Hornbeck, J. A., Phys. Rev., 90, p. 152, 1953, 97, p. 311, 1955. 47. Harned and Owen, The Physical Chemistry of Electrolytes, p. 123, A. C. S. Monograph, 1950. 48. Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, p. 209, Oxford, 1948. i49. Glasstone, S., Textbook of Physical Chemistry, p. 1231, Van Nostrand, 1940. '50. Shockley, W., Electrons and Holes in Semiconductors, p. 300, Van Nostrand, 1950. 51. Slater, J. C, and Frank, N. H., Introduction to Theoretical Physics, p. 186, I McGraw Hill, 1933. ;52. Fuller, C. S., and Severiens, J. C, Phys. Rev., 95, p. 21, 1954. l53. Shockley, W., B.S.T.J., 28, p. 435, 1949. ")4. Shockley, W., Electrons and Holes in Semiconductors, p. 258, Van Nostrand, 1950. ■ |55. Theuerer, H. C, U. S. Pat. No. 2542727. '56. Margeneau, H., and Murphy, G. M., The Mathematics of Physics and Chem- istry, p. 213, Van Nostrand, 1943. 157. Margeneau, H., and Murphy, G. M., The Mathematics of Physics and Chem- i istry, p. 72, Van Nostrand, 1943. 58. Reiss;" H., and LaMer, V. K., J. Chem. Phys., 18, p. 1. 1950. ,59. Reiss, H., J. Chem. Phys., 19, p. 482, 1951. [60. Carslaw, H. S., and Jaeger. J. C. Conduction of Heat in Solids, p. 40, Oxford, 1948. tU. Boltzmann, L., Ann. Phys., 53, p. 959, 1894. 62. Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, p. 41, Oxford, 1948. • 3. Prince, M. B., Phys. Rev., 92, p. 681, 1953, 93. p. 1204, 1954. lU. Tvler,W. W., and Woodbury, H. H.,Bull. Am. Phys. Soc, 30, No. 7, p. 32, 1955. 'i.'). Debye, P. P., Conwell, E. M., Phys. Rev., 93, p. 693, 1954. 636 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 66. Shockley, W., Electrons and Holes in Semiconductors, Chapter 8, Van Nos- trand, 1950. 67. Shockley, W., Electrons and Holes in Semiconductors, Chapter 16, Van Nos- trand, 1950. 68. Geballe, T. H., and Morin, F. J., Phys. Rev., 95, p. 1085, 1954. 69. Conant,J. B., The Chemistry of OrganicCompounds,p. 196,Macmillan, 1939. 70. Valenta, M., and Ramasastry, C, Symposium on Semiconductors, Meeting I.M.D., and A.I.M.E., Feb. 20, 1956. 71. Longini, R. E., and Green, R., Phys. Rev. (in press). I Single Crystals of Exceptional Perfection and Uniformity by Zone Leveling By D. C. BENNETT and B. SAWYER (Manuscript received January 23, 1956) The zone-leveling process has been developed into a simple and effective tool, capable of growing large single crystals having high lattice perfection and containing an essentially uniform distribution of one or more desired impurities. Experimental work with germanium is discussed, and the possi- bility of broad application of the principles involved is indicated. IXTRODUCTION The first publication describing the concept of zone melting appeared about four j^ears ago.^ As there defined, the term zone melting designates a class of solidification techniques, all of which involve the movement of one or more liquid zones through an elongated charge of meltable ma- terial. This simple concept has opened a whole new field of possibilities for utilizing the principles of melting and solidification. The first zone melting technique to gain widespread usage was one for zone refining germanium by the passage of a number of liquid zones in succession through a germanium charge. This process may be quite prop- erly compared to distillation, the essential difference being that the change in phase is from solid to liquid and back, instead of from liquid to vapor and back. The zone refining technique has been eminently suc- cessful in the purification of germanium. Harmful impurity concentra- tions are of the order of one part in 10^". This is mainly because all the impurities whose segregation behavior in freezing germanium has been measured have segregation coefficients (see equation 1) differing from 1 by an order of magnitude or more.^ During the zone refining ; operation, these impurities collect in the liquid zones and are swept with them to the ends of the charge, which may be later removed. 1 Pfann, W. G., Trans. A.I.M.E., 194, p. 747, 1952. ^ Burton, J. A., Impurity centers in Germanium and Silicon, Physica, 20, p. 845, 1954. 637 638 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 This paper deals with a second zone melting process, zone leveling,^' ^ which has gained usage somewhat more slowly than zone refining, but which has proved to be a highly effective tool for distributing desired impurities uniformly throughout a charge. For this process, only one liquid zone is used and its composition is adjusted to produce the desired impurity concentration in the material which is solidified from the liq- uid zone. Appropriate precautions are taken to insure the production of single crystals, if the material is desired in this form. Since the invention of zone leveling, the process has been developed into a precision tool and as such it has become a preferred practical method for growing germanium single crystals of uniform donor or ac- ceptor content. It is the purpose of this paper to discuss the technical development of this process, which has had two chief objectives: (1) the attainment of the greatest possible uniformity of donor and/or acceptor impurity distribution in the crystal ; and (2) the attainment of a germa- nium crystal lattice with a minimum of imperfections of all kinds. The presentation will cover the principles involved, the means developed and results achieved toward these objectives in that order. The first applications of the principles of zone melting have been in the field of semiconductor materials processing, chiefly because there are tio other known refining techniques capable of meeting the extremely stringent purity requirements necessary for material to be used in semi- conductor devices. Nevertheless, it is clear that these relatively simple and very effective zone melting techniques are beginning to find a wide variety of useful applications throughout the general fields of metallurgy and chemical engineering. BASIC PRINCIPLES The basic concept, theory and experimental confirmation of zone level- ing have been well covered in previous publications.'- ' Accordingly, the intention here is only to repeat the salient points of the theory with a special emphasis on the assumptions involved since it will be necessary to refer to them. Fig. 1 is a schematic drawing of a zone leveling operation showing a liquid zone of constant volume containing a solute whose concentration is Cl . As the zone moves a distance Ax an increment of germanium is melted at the right end, and another is frozen at the left end. The concentration of solute in the newly frozen Ax of solid solution is Cs • The distribution coefficient k is now conveniently defined as the ratio » Pfann, W. G., and Olsen, K. M., Physical Review, 89, p. 322, 1953. SINGLE CRYSTAL BY ZONE LEVELING of these solute concentrations: k = 639 (1) When A- < 1 , the freezing interface may be regarded as a filter permit- ting only a fraction A: of the solute concentration in the liquid to pass into the growing solid and rejecting the rest to remain in the liquid. If the unmelted charge of solvent is pure — ■ that is, if no solute passes into the zone at the melting interface it is readily seen that the liquid zone will be : gradually depleted of its solute impurity content during passage through the charge. An expression for the solute concentration in the solid, Cs , deposited there by the passage of one zone, for the case of "starting charge into ' pure solvent" has been derivec^ based on the following assumptions: (1) The liquid volume is constant (both cross section of charge and zone length I are constant). (2) k is constant. ! (3) Mixing in the liquid is complete (i.e. concentration in the liquid is uniform). (4) Diffusion in the solid is negligible. I The expression is i Cs = kCo e-'"'" (2) where Clq is the initial concentration of impurities in the liquid, I is the zone length, and x is the distance moved by the solidifying interface. A set of Cs versus x/l curves is shown in Fig. 2 for various k's. From this ' figure it is readily seen that when k is small the decay of Cs is slow (i.e., I the depletion of Cl is slight). Largely because of this consideration, most of the practical work re- I ported in this paper has utilized solutes in germanium having low segre- MOVING HEATER ■ *■ cs^ '///////////, / / liquid zone "and impurity y SEED r3-_^L^F SOLID Ge CHARGE ^W%M. AX-* «--£—> Fig. 1 — Schematic of zone leveling operation. 640 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 gation coefficients (usually antimony, whose k = 0.003 as donor, and indium whose k = 0.001 as acceptor). However, the principles of zone leveling are broad and capable of application to any solvent-solute sys- tem within the range of solubilities of its solid and liquid phases. The general method of attack' is first to find that composition of the liquid zone which will deposit the desired solid solution. Secondly, if one or more of the segregation coefficients involved is not small, the liquid zone must be maintained at its proper composition by admixing to the solid charge the same solutes that the zone will deposit in its product. Thus the solutes that are removed from the liquid zone at the freezing end will be replenished at the melting end. The above mathematical treatment leads one to expect an essentially uniform solute distribution throughout a zone leveled crystal for the case under discussion in which k is small and the zone moves through a charge of pure solvent as indicated in Fig. 2. Irregular variations of Cs along the length or over the cross-section of the ingot are not predicted. The CO = 1.0 m 0.8 !< 0.6 _J UJ °' 0.4 z in O z" o ^ cr (- z UJ o z o u 0.2 o.to 0.08 0.06 0.04 0.02 0.01 - Cs = kC, . p-Wi I Clq=i except for k = o.oi 1 ^l ^ sK -r \ 1 \ ^ \. .^k=o .01, Cl 0='° . -" T- ^ NT - \ N Sr-* _ai - 1 \ \ v ...,___^ - ' 1 \ 0. ^\ \, k=5.o\ \ \ \ ZONE -LENGTHS SOLIDIFIED, X/£ 10 Fig. 2 — Solute concentration curves predicted for zone leveling with a start- ing charge of solute into pure solvent. SINGLE CRYSTAL BY ZONE LEVELING 641 treatment is not concerned with lattice imperfections in the ingot such as dislocations, lineage, or grain boundaries. The predictions the theory does make have been well verified by experiment insofar as it has been possible to meet the assumptions enumerated above. However, as with most assumptions, their validity is sensitive to the experimental condi- tions, particularly in the cases of the first three. Much of the develop- ment effort, especially that toward improving resistivity^ uniformity, has been directed toward controlling the process so that these assump- tions will be as nearly valid as possible. Early experiments in zone leveling yielded crystals good enough to meet device reciuirements of that time. However, as semiconductor de- vices were designed to meet tighter design requirements, the demands on the germanium material grew" more critical. Under these circum- stances, it became necessary to examine the requirements on the product of the process and what precautions would be necessary to insure that its operation was under sufficient control. Accordingly, we shall chscuss first the requirements on semiconductor material and then those critical as- pects of the leveling operation which must be controlled to insure quality of the final product. liEQUIREMENTS ON GERMANIUM FOR SEMICONDUCTOR USES The basic electrical bulk property of a germanium crystal is its con- ductivity or the reciprocal of that quantity, its resistivity. For a great majority of semiconductor uses, an extrinsic conductivity* is required in addition to the 3^o ohm"~^ cm~' intrinsic conductivity that results at room temperature from thermal excitation of electron-hole pairs in pure •iermanium. An extrinsic conductivity may be either n-type or p-type. Both of these may be produced by trace impurities distributed through- out the crystal, the n-type by donor impurities and the p-type by accep- tor impurities. At room temperature donors give rise to conduction elec- trons and the acceptors to conduction holes which are free to move within the germanium crystal. If both donors and acceptors are present in the same crystal, the resulting electrons and holes recombine, leaving essentially the extrinsic conductivity contributed by the excess of one over the other, that is by | No — A''^ i . The fundamental requirement is, then, to control the net donor and 1lie acceptor balance, | No — A^4 I , tea predetermined value throughout the crystal. For most applications, the conductivity is to be increased by one or two orders of magnitude above the 27°C intrinsic value. An idea of the donor or acceptor concentrations involved may be acquired * Shockley, W., Electrons and Holes in Semiconductors. 642 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 from noting that a conductivity of H ohm"^ cm""^ (that is, a conductivity increased by one order of magnitude) corresponds to a No — Na concen- tration of 7 parts per biUion. The next most commonly measured bulk property of germanium is the lifetime of minority carriers,^ i.e., the time constant for decay by recombination of a surplus population of minority carriers artificially introduced into the crystal. Minority carriers are holes in n-type ger- manium or electrons in p-type germanium. This time constant may be regarded reasonably as a figure of merit for the crystal, being an indica- tion of its freedom both from certain chemical impurities and from crys- tal faults, since these act as catalysts to the electron-hole recombination reaction. Normally, the highest possible lifetime is desired. Thus it be- comes important to take extreme precautions during handling and proc- essing of the germanium to avoid contamination, particularly by such known recombination center elements as nickel and copper and it is also important to avoid crystal lattice faults such as dislocations, line- age, and grain boundaries. Another observable c^uantity has recently been gaining acceptance as a more definite indication of mechanical crystal perfection than the mi- nority carrier lifetime measurement. This is the etch pit density count, €, (see Fig. 3) which is observed microscopically on an oriented (111) surface of a Ge crystal that has been etched three minutes in an agitated CP-4 etch (20 parts by volume concentrated HNO3 , 12 parts concentrated HF, 12 parts concentrated acetic acid, and 3^^ part Br2). There is strong evidence that the etch pits are formed at the intersections of dislocations with the surface of the crystal. While an etch pit count probably indicates only certain edge dislocations which intersect the sur- face of the crystal, it is at least a relative indication of the total dis- location density, and thus appears to be a highly useful index of crystal lattice perfection. In the last year, evidence of a strong correlation has been observed between certain electrical properties of alloy junctions, especially the l)reakdown voltage, and the etch pit density of the material on which the alloy junction is made. Accordingly, material to be used for alloy junction transistors is now selected on the basis of its maximum etch pit count and its freedom from lineage, twin, and grain boundaries. The usual device test requirements on n- or p-type Ge material vary 5 Valdes, L. B., Proc. I.R.E., 40, p. 1420, 1952. « Vogel, F. L., Read W. T., and Lovell, L. C, Phys. Rev., 94, ]). 1791, 1954. ' Vogel, F. L., Pfann, W. G., Corey, H. E., Thomas, E. E., Physical Review, 90, p. 489, 1953. * Zuk, P., and Westberg, R. W., private communication. SINGLE CRYSTAL BY ZONE LEVELING 643 Fig. 3. — Microphotograph of Typical Etch Pits on (111) Plane. from device to device, but may be summarized as follows: (1) Composition — The donor-acceptor balance No — Na must be accurately controlled so that the resistivity, p, of the crystal is uniform and falls within acceptable tolerance limits. (2) Macro Perfection — The crystal shall contain no grain boundaries, lineage, or twinning. (3) Micro Perfection — The etch pit density, e, must be lower than a certain empirically determined maximum. (4) Lifetime of Minority Carriers ■ — r, must usually be above a certain minimum, although in many cases this minimum may be as low as a few microseconds. Assuming macro perfection a consideration of these requirements leads directly to the two general objectives mentioned in the intro- duction of this paper: composition uniformity and control, and crystal lattice perfection. A third objective, high chemical purity, might also be inferred from the lifetime requirement, but the results obtained by zone refining raw material and by fairly standard laboratory techniques of cleaning and baking of furnace parts at high temperature have been .-satisfactory. Hence this objective has required little development effort. We proceed to a discussion of critical aspects of zone leveling in the light of the two major development objectives. COMPOSITION UNIFORMITY AND CONTROL The experimental development work described in this paper has been •oncerned with the distribution of two trace impurities, indium and anti- G44 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 mony, in a pure element, germanium. The traces are generally desired in concentrations varying from 1 to 100 parts per billion, (p = 35 to 0.35 cocm). These amounts are too small to be detected by chemical or spectrographic means, but are readily detectable by electrical resistiv- ity measurements. Although this application of zone leveling is very specific, it should be possible, as we have already suggested, to apply the experimental results to be described to more general systems. The sub- ject of uniformity is conveniently discussed in two sections: (a) longi- tudinal resistivity uniformity, and (b) cross-sectional uniformity. (a) Longitudinal Composition Uniformity It has already been shown, by (2), that if the k is small, the variation in Cs over four or five zone lengths should be slight. This should be true either if a charge of pure germanium is used, or if a charge containing the same impurity present in the liquid zone is used, provided that the charge concentration of this impurity is of the same order of magnitude as that sought in the product. Where the solute has a small k, the leveling action of the zone is strong and the large C'l, that is required is relatively unaffected by variations of the order of Cs ■ The primary cause of observed variations in the longitudinal resistiv- ity is fluctuation of the volume of the liquid zone. If this volume increases for any reason, the solute dissolved in it will be diluted. On the other hand, if the volume decreases, which can occur only when some of the liquid freezes and if k is small, most of the zone's solute will be concen- trated in the smaller volume. Thus for small A;'s the concentration of solute in the liquid zone, Cl , varies inversely with the zone's volume. If Cl is to be constant, the volume must be constant, i.e. assumption (1) must be valid. Unfortunately, the zone volume is directly affected by many variables, namely temperature fluctuation and drift, fluctuation in growth rate, variation in the cross-section of the unmelted charge, variation in the inert gas flow, and even cracks in the unmelted charge. For optimum control of longitudinal resistivity uniformity, it is, therefore, necessary to control all of these variables. The remainder of this section will consider their control. Toward minimizing the effect of temperature variation on the zone volume, it is important to consider both the means of overall temperature control and the design of the temperature field which melts the liquid zone. It is clear that variation of the temperature field as a whole will directly affect the length of the liquid zone. Accordingly, it will be im- portant to use a precision temperature controller in order to maintain a SINGLE CRYSTAL BY ZONE LEVELING 645 constant zone length. The controller used here is a servo system that cycles the power on and off about ten times a second, adjusting the on fraction of the cycle according to the demands of a control thermo- couple. The sensitivity of the controller is ±0.2°C at 940°C. With a liquid zone about 4 centimeters long and a temperature gradient of about 10°C per centimeter at the solidification interface, this degree of control should introduce longitudinal resistivity variations no greater than ±0.3 per cent. When other requirements permit, it is possible to design a temperature contour to minimize the effects of control fluctuations. When the tem- perature gradients at the ends of the liquid zone are small, a slight change in the general temperature of the system will cause a relatively large change in the position of the solid-liquid interface. On the other hand, when the gradient is steep, the shift in position of the interface will be small. It is with this consideration in mind that a temperature gradient of about 130°C/cm is provided at the melting end of the liquid zone (Fig. 4). A steep gradient has the added advantage that it provides a large heat flux which is capable of supplying or removing the heat of solidification even at relatively fast leveling rates. Thus, a steep temper- ature gradient serves effectively to localize a solid-liquid interface. Other considerations, soon to be discussed, dictate that a small temperature ti;radient (about 10°C/cm) must be used at the freezing end of the zone. Accordingly, high precision of temperature control is required to properly stabilize the position of this solid-liquid interface. Variation in the cross-section of the liquid zone may be controlled by using a boat with uniform cross-section, and by using as charge material which has been cast into a mold of controlled cross-section. Less precise control is obtained by using ingots from the zone refining process which were produced in a boat matched to the zone leveler boat. Even when care is used to maintain a uniform height of the zone refined ingot, the control is less precise than in a casting. A constant and uniform growth rate is important toward obtaining uniform longitudinal resistivity because segregation coefficients vary with growth rate.^" This is especially true in the case of the/c forantimony. Under steady state conditions, the growth rate is the rate at which the boat is pulled through the heater. A stiff pulling mechanism is required in order that the slow motion be steady. In the apparatus described here, a syncronous motor, operating through a gear reduction to drive a lead ■ screw, has served to pull the boat smoothly over polished quartz rods. "Pfann, W. G., J. Metals, 5, p. 1441, 1953. " Burton, J. A., Kolb, E. D., Slichter, W. P., Struthers, J. D., J. Chem. Phys., 21, p. 1991, Nov., 1953. 646 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 a; a > o o > o bC SINGLE CRYSTAL BY ZONE LEVELING 647 The true growth rate may be affected by factors that cause variations from steady state growth such as temperature and gas flow fluctuations. The need to control these variables has already been mentioned because of their effect on zone volume; their effect on growth rate is thus a second reason for their control. Cracks or similar discontinuities in the unmelted charge act as barriers to heat flow. Thus they cause a local rise in temperature and lengthening of the liquid zone as the crack approaches the zone, until it is closed by melting. The resulting transient increase in liquid volume (and in p of the product) may be of the order of 10 per cent. (b) Cross-Sectional Com'position Uniformity Difficulty may be expected in controlling the cross-sectional uniform- ity of the zone leveled ingot chiefly when the third assumption is invalid, i.e., when Cl throughout the liquid is non-uniform. As shown in the next paragraph, the true Cl must always rise locally near the solidifying inter- face due to the solute diffusion which is necessary when k < \. However, it is possible to improve the validity of assumption 3 both by slowing Ihe groAvth rate and by stirring the liquid zone. One can form an estimate of a theoretically reasonable growth rate in terms of the rate of diffusion of impurities in liquid germanium. It should be noted that movement of a liquid zone containing a solute whose segregation coefficient is small implies a general movement by diffusion of essentially all the solute atoms away from the solidifying interface at a speed ecjual to the rate of motion of the zone. Even slow zone motion corresponds to a high diffusion flux of the solute through the Uquid. As a consequence, the solute concentration must rise in front of the advancing solidification interface to a concentration Cl' (see Fig. 5) until a concentration gradient is reached sufficient to provide a diffusion flux equal to the growth rate. Fick's Law of diffusion is useful here to calculate the extent of the rise in C/,/ at the growth interface, assuming the liquid to be at rest. The ratio of the maximum concentration to the bulk concentration may be taken from Fig. 5. If the maximum is to l)c no greater than 10 per cent above the mean, a maximum growth rate of 2 X 10~^ mils per second or 7 X 10"^ inches/hour would be r(3(juired. Clearly, this rate is far too slow to provide an economical means of growing single crystals. For a practical process, it will be neces- sary to use non-equilibrium conditions at growth rates that must result HI appreciable concentration differences within the liquid zone. Of course, the slower the growth rate the smaller will be the diffusion gradient and the higher will be the expected cross-sectional uniformity. 648 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 SOLID Cs=k(x)CL=kCo)CL fV n LIQUID ^*''*>*..,,_^^^^^ Cl (AVERAGE)-^ tQUILIBRIUM DISTRIBUTION COEFFICIENT ~ Cu (AVERAGE) *• MOTION OF INTERFACE 1 FOR GROWTH RATE = X Cs= k(0)CL(AVERAGE) ..EQUILIBRIUM DISTANCE, X k(x) = Cs (7L(ave) (3) In practice, however, the situation is complicated by the existence of convection currents in the liquid zone. It is true that these currents tend to stir the liciuid zone and thereby to minimize the concentration gradient within it. However, the currents are not uniform over the growing inter- face and they carry liquid of varying concentrations past the interface, causing fluctuations in Cs • Since these convection currents cannot be eliminated, one turns to the alternative of using forced stirring of the liquid zone. Such a forced stirring is readily available when RF induc- tion heating is used by allowing the RF field to couple directly with the Fig. 5 — Solute concentration in solid and liquid at equilibrium and at finite growth rates. If the liquid were static, that is, without any currents, it should be I; possible to obtain a uniform, controlled solute concentration in the solid even at appreciable growth rates, merely by adjusting the average con- centration in the liquid to arrange that the Cl obtained at the growing interface will be the desired one. Instead of working with the equilibrium distribution coefficient ko , one works with an effective distribution co- efficient k(x) for the given growth rate, x: SINGLE CRYSTAL BY ZONE LEVELING 649 liquid zone." The resulting stirring currents are shown schematically in Fig. 6. It is seen that the liquid is mo\'ed from the center of the zone along its axis toward both ends. There it passes radially outward across the interface and returns along the outside of the zone to its center. These stirring currents are faster than convection currents and tend to minimize the rise of Cl at the solidification interface and to improve the uniformity of Cl and of crystal growth conditions in general over the freezing interface. CRYSTAL LATTICE PERFECTION A single edge dislocation in germanium may be regarded as a line of free valence bonds. The dislocation line is believed to have about -i X lO" potential acceptor centers per centimeter, producing a space charge in the neighboring germanium and strongly modifying its semiconductor properties. A lineage boundary (a term found useful to designate a low angle grain boundary) is a set of regularly spaced dislocations, and may I be regarded as a surface of p-type material. Since the basic electrical properties of a semiconductor, resistivity (and also minority carrier life- time) are drastically out of control at dislocations and arrays of disloca- I tions, it is easy to understand why these lattice imperfections are un- ' desirable in crystals to be used for most semiconductor purposes. The attainment of high perfection in germanium lattices may conven- iently be discussed in two parts: first, the growth of a single crystal of i high perfection and, second, the preservation of the crystal's perfection (luring its cooling to room temperature. ; The problem of growing a single crystal in the zone leveler is basically one of arranging conditions so that the liquid germanium solidifies only Fig. 6 — Stirring currents in liquid induced by RF induction heater. " Brockmeir, K., Aluminium, 28, p. 391, 1952. 12 Read, W. T., Phil. Mag. 45, p. 775, 1954. 650 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 i 1 z o SLOW GROWTH UJ< 11 Cj. Oiu (/)U z o o T GRADIENT NO. 2 REGION OF \ ^ < CONSTITUTIONAL > \^ SUPER COOLING ^^^"^ LIQUIDUS DISTANCE, X DISTANCE, X Fig. 7 — Schematic solute concentration and temperature curves in liquid, . near freezing interface, illustrating constitutional supercooling. The left edge of each diagram represents the solid-liquid interface. on the single crystal germanium seed. In order to achieve this situation, it is essential that no stable nuclei form. Thus, not only must the tem- perature of the liquid zone be above its freezing point everywhere except at the interface, but the liquid must also be free of foreign bodies that can ct as nuclei. Furthermore, temperature fluctuations are to be avoided. The requirement that the liquid temperature be above its freezing point necessitates a slow growth rate because of what has been termed "constitutional supercooling."^^ This phenomenon can best be described with the aid of Fig. 7. The freezing point of a liquid is depressed by in- creasing concentration of solutes having /c's less than unity. Because of the rise in Cl near the solidifying interface, the freezing point is more depressed in this region than that in the bulk of the liquid zone as shown in Fig. 7. It has also been shown^^ for crystals growing in one dimension that the temperature gradient in the liquid decreases for increasing growth rates. The temperature gradients for two growth rates are plotted on Fig. 7. It can be seen that where the growth rate is slow and the temperatin-e " Chalmers, B., J. Metals, 6, No. 5, Section 1, May, 1954. " Burton, J. A., and Slichter, W. P., private communication. SINGLE CRYSTAL BY ZONE LEVELING 651 gradient is steep, the temperature of the liquid is above its liquidus (freezing point curve) throughout the Hqiiid, and no stable nuclei can form. However, increasing the growth rate decreases the temperature gradient, while it depresses the liquidus. If the temperature gradient is reduced to that indicated for fast grow^th, a region of constitutional supercooling will exist in front of the solidifying interface where nuclei can form and grow. The freezing of such a crystallite onto the growing crystal marks the end of single crystal growth. A foreign body may also initiate polycrystalline growth. A natural site for nucleation by foreign bodies is the wall of the boat, close to the growth interface. Here the liquid germanium is in contact with foreign matter at temperatures approaching its freezing point. It was found by D. Dorsi that germanium single crystals could be grown satisfactorily in a smoked quartz boat, at growth rates up to 2 mils per second. However, uniform- ity considerations mentioned previously make it desirable to zone level at much slower rates. It is believed that scattered dislocations may be produced in a single crystal germanium lattice by three chief mechanisms. They may be prop- agated from a seed into the new lattice as it grows; they may result from various possible growth faults; but probably the most important mechanism in this work is plastic deformation of the solid crystal. The lirst cause may be minimized by selecting the most nearly perfect seeds available, the second by using slow growth rates, and the third by mini- mizing stresses in the crystal. The first hint that plastic deformation in the crystal might be an im- portant source of dislocations came from the study of crystals pulled from the melt by the Teal-Little technique. Frequently when sections of crystals grown in the [111] direction were etched in CPi the pits were arrayed in a star pattern, Fig. 8(a), in which the pits appeared on lines — not randomly distributed. This coherent pattern suggested strongly that the lines were caused by dislocations in slip planes which had been ac- tive in plastic deformation of the crystal. The slip system of germanium has been determined to be the <110> directions on {111} planes.^^ If the periphery of the crystal is assumed to be in tension, it is possible to calculate the relative shear stress pattern in each slip system of the 3 {111 { planes which intersect the (111) section plane. The results of these calculations are summarized in Fig. 8(b) which shows a polar plot of the largest resolved shear stresses for these planes and also their traces in the section plane. The agreement with the observed star pattern is striking. 15 Treuting, R. G. Journal of Metals, 7, p. 1027, Sept., 1955. 652 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Fig. 8 (a)- from melt). Star Pattern on (111) plane (etched cross-section of crystal pulled The peripheral tension assumed in the above paragraph may be seen to be quahtatively reasonable upon consideration of the heat flow pat- tern of the crystal during growth. Heat must enter the crystal by conduction through its hottest surface, the gro^\^ng interface, which is a 940°C isotherm. It must leave through all the other surfaces by radiation and conduction. Therefore, these surfaces must be cooler than their adjacent interiors, and cross-sections of the crystal must have cooler peripheries than cores because of the heat escaping from the peripheral surfaces. Due to thermal contraction the cooler periphery must be in tension and the core in compression. In zone leveled crystals the distribution of etch pits on a (111) section was not dense or symmetric enough to display a star pattern. However, it was reasoned that since thermal contraction stresses appeared to play a major role in the production of dislocations in pulled crystals through plastic deformation in the available slip systems, the same mechanism might be playing a significant role in zone leveled crystals. .' "-.The only stresses in a zone leveled ingot other than those due to the weight of the crystal itself must be those due to non-uniformities in SINGLE CRYSTAL BY ZONE LEVELING 653 thermal contraction. Consider a small increment of the length of a newly formed zone leveled crystal as heat flows through it from its hotter to its colder ends while the crystal moves slowly through the apparatus. Heat flows in by conduction from the higher temperature germanium adjacent to it. Heat leaves not only by conduction out the other end, but also by conduction and radiation from the ingot surface. Because of this latter heat loss, there is a radial component as well as a longitudinal component to the temperature gradient. The cooler surface contracts resulting, as above, in peripheral tension and internal compression. Clearly if the radial component of heat flow could be eliminated, there would be no peripheral contraction. Accordingly, the most desirable temperature dis- tribution is one whose radial heat flow is zero, i.e., a case of purely axial or one dimensional heat flow, which implies a uniform temperature gradient along the axis of the ingot. In practice, it is difficult to obtain a uniform axial temperature gradient except for the special case of a very small one. This may be obtained fairly easily by the use of an ap- / / / ffr / / / ^— TRACE OF (ill) -TRACE OF (Tll) r / / / 7 / / ' ^^^N — 7 ' / / / * ///'// \ , _, N / / / /trace of (iii) '', / / / [oTi] resolved s STRESS (u stress (Tm)- STRESS (lTl) - -" IT)- / / \\\\//// " ^' \\v/// \\v/ \v/ Fig. 8(b) — Resolved shear stress and slip-plane traces on (111) Plane. u. I- <: H C, < 654 SINGLE CRYSTAL BY ZONE LEVELING 655 propriate heater. The heater designed for this purpose is called an after- heater and is shown in Figs. 4 and 9. The after-heater reduces the heat loss by radiation and radial conduc- tion from the crystal maintaining the entire crystal at a temperature only slightly below its melting point throughout its growth. After zone leveling has been completed, the entire ingot is cooled slowly and uniformly. Of course, a finite temperature gradient must exist at the liquid-solid interface. The gradient at the interface of the leveler shown in Figs. 4 and 9 is about 10°C per centimeter and the maximum gradient, about yi inch into the solid, is 30°C per centimeter. The gradient de- creases slowly to nearly zero within the after-heater, as can be seen in the measured temperature curve of Fig. 4. A ZONE LEVELING APPARATUS AND TECHNIQUE FOR GERMANUIM The apparatus required for zone leveling is basically simple. A single crystal seed, the desired impurities, and a germanium charge, are held in a suitable container in an inert atmosphere. Provision is supplied for either moving a heater along the charge or the charge container through a heater. The heater may be either an electric resistance type or a radio frequency induction type. The resistance heater offers the advantage of economy while the induction heating offers the advantage of direct in- ductive stirring of the melted zone by the RF field, which, as mentioned previously, is helpful in attaining uniformity of impurity distribution, and is therefore to be preferred for critical work. Schematic drawings of an RF powered zone leveler following in general the original design by K. M. Olsen are shown in Fig. 9 in two useful configurations. The outer clear quartz tube serves to support the inner members of the apparatus and also to contain the inert atmosphere for which nitrogen, hydrogen, helium, or argon, can serve. For this appara- tus, a quartz boat is used to contain the germanium, since it permits inductive stirring of the liquid germanium by the RF field. The auxiliary fore and after heaters, which are made of graphite, have special purposes discussed in the two preceding sections. A typical boat used in this ap- paratus is about 16" long, is smoked on the inside, and is made of thin- walled clear quartz of V I.D. and of semi-circular cross-section. A normal charge of zone refined Ge and seed is about 12 inches long and weighs about 500 gm. A photograph of the assembled apparatus appears in Fig. 10. For the best results in crystal perfection and resistivity uniformity, the apparatus is run with the full length after-heater and at a slow pull rate, 0.09 mils per second (approximately 1" in three hours). For some- what less critical demands a pull rate 10 times faster is used, with a short- ened after-heater or none at all. If it is desired to reproduce a resistivity obtained in the zone leveler, it is very convenient to reuse the solidified zone containing the impurity addition that yielded the desired resistivity. This solid zone, if undam- aged (when cut from the finished ingot), will contain all of the sohite that was not deposited during the ingot run. When it is remelted next to a seed the solute will redissolve into the liquid to yield very nearly SINGLE CRYSTAL BY ZONE LEVELING 657 Fig. 11. — Photograph of zone leveled single crystal ingot. the same Cl , provided that the zone vokime is accurately reproduced. In this way it is readily possible to resume leveling as before and hence virtually to reproduce a desired resistivity. For the small k solutes, In and Sb , discussed in this paper the loss of d in one leveling run is so small as to be insignificant compared to other sources of error in this quantity. ' PILOT PRODUCTION RESULTS The capabilities of the zone leveling equipment and techniques just I described may be evaluated with reasonably good accuracy on the basis 1 of the measurement results obtained from more than 300 single crystal 1 ingots so produced. Over 200 of these crystals were grown in the after- heater at the "slow" growth rate of 0.09 mils per second. The rest were I grown with a short after-heater or none at all at a growth rate about ten times greater. The ingots to be measured (see Fig. 11) were usually 4-6 inches long after removing seeds and solidified zones (i.e., 2-3 zone lengths), and were cut into 1 inch lengths. The p, r, and e measurements were taken ^ on the flat ends of these segments. The results of the observations will ' be summarized and discussed in terms of the four device test require- ments described earlier. (1) Compositional Uniformity The resistivity measurements were taken with a calibrated 4-point probe technique at five locations on each ingot cross-section (center, top, bottom and each side). The spacing between adjacent points of the probe was 50 mils. Accordingly, these measurements would be insensitive to p fluctuations in the material of this order or smaller. However, an investi- gation by potential probing techniques, of Ge filaments cut from zone leveled ingots indicates that p fluctuations in zone leveled material are '6 L. B. Valdes, Proc. I.R.E., 42, p. 420, 1954. " Erhart, D. L., private communication. G58 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Table I — Average Resistivity Variations (A) Along length axis. Grand Length Average ± 10%. Growth Rate Mils per Second 0.9 0.8 0.09 n-Type ±% 9.9 7.6 9.0 No. of Ingots 27 12 108 p-Type ±% 10.9 17.4 9.3 No. of Ingots 33 16 137 Average ± /o 10.4 13.2 9.2 (B) Over Cross-Section Growth Rate n-Type p-Type Average Mils per Second ±% No. of Ingots ±% No. of Ingots ±% 0.9 0.8 0.09 9.5 8.3 4.3 22 12 93 8.5 6.9 2.3 30 14 122 8.9 7.5 3.2 generally coarse — • changing over distances 2 to 5 times larger in dimen- sion than the 50 mil dimension in question. Thus the p data summarized here should give a reasonably valid representation of the true p variations in the ingots measured. Table I summarizes the resistivity variations recorded as percentages of the mean resistivity of each ingot. These variations are separated into those observed (a) along the length axis and (b) over the cross-section, for the different growth conditions and resistivity types. It is readily seen that the average variation along the length, about ±10 per cent, is larger than the average cross-sectional variation. The variations are not systematic along the length of the ingot and are chiefly due to fluctuation in the length of the liquid zone. An appreciable part of this variation is due to the effect, mentioned earlier, of discon- tinuities in the unmelted charges between 1 inch lengths of crystals that were being releveled. A smaller length variation of p, about ±7 per cent, was observed in those ingots grown from continuous charges. Part B of the table shows that the variation of p over the cross-section is sensitive to the growth rate in the range covered. For slow growth, it is small, and one would reasonably expect that if further improvement in p variation were required, it should first be sought by improving the control of the zone length. (2) Macro Perfection Macro perfection of the pilot production product is extremely high. There were essentially no cases of polycrystallinity, or twinning, except SINGLE CRYSTAL BY ZONE LEVELING 659 300 200 100 600 400 200 100 5 80 Z 60 tu 40 I- 400 LU LL _l 200 < liJ 100 < 80 60 40 20 10 FAST p (a) NO AFTER-HEATER FAST n 1 SLOW n + p in Q z o o LU 10 o cc u (b) 5" AFTER -HEATER ^ — ^ ^^ .-'-"'' ^— — "^ ':^ ^" "slow p " ^^ t / .^--^ ^^Low n (C) 12" AFTER-h HEATER ^^ — — "^ -•''' ^t ,-"" ^^^00""''''''''^ ^ x' ^.^ SLOW Pj -' > ^^ y'^ ^/^Low n y ■v ''V / 2 3 4 5 DISTANCE FROM SEED IN INCHES Fig. 12 — Average minority carrier lifetime plotted against distance from seed for 2-8 ohm cm crystals grown with 12", 5" and no after-heaters. for clearly attributable causes such as power or equipment failure. There were few cases of lineage in the short after-heater and virtually none in the full after-heater, while lineage is not uncommon in ingots grown with no after-heater. (3) Micro Perfectio7i Table II summarizes the etch pit density, e, measurement results. In general, it can be seen that with the after-heater one can expect etch pit counts of the order of 1,500 pits per cm- which is lower than results without an after-heater by about an order of magnitude (and lower than Table II — Average Etch Pit Densities, e Growth Rate Mils per Second « Ave a No. of Ingots (12" after-heater) (5" after-heater) No after-heater 0.09 0.09 0.9 0.9 1560 3800 7000 11000 770 1600 1900 6600 39 3 3 6 G60 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 e's of pulled Ge crystals by about two orders of magnitude) . The lowest average count that has been observed is 40 pits per cm^ This crystal was found to have the smallest X-Ray rocking-curve widths observed in germanium at Bell Telephone Laboratories — very nearly the the- oretically ideal mdths. The perfection indicated is exceptional — com- parable to that of selected quartz crystals. (4) Lifetime of Minority Carriers T data are summarized in Fig. 12 in which are plotted averages of the r measurements on the ingot sections against distance from the seed. One sees a systematic rise in t along the length axis of an ingot grown slowly in the after-heater. This is interpreted to indicate that the ingot is being slowly contaminated with chemical recombination centers during its long wait inside the after-heater at high temperatures. If improvement were needed in lifetime, it should be sought first by increasing the chemi- cal cleanliness precautions, which were nonetheless strict in this work. SUMMARY A zone leveler has been developed to provide growth conditions suit- able for the production of quality germanium single crystals. The crys- tals are nearly uniform and have exceptionally high lattice perfection, jri Similar levelers are in use in production. '' The apparatus developed has been used to supply germanium single crystals for experiments and for the pilot production of a variety of point contact, alloy, and diffusion transistors. The machine operating at slow growth rate with an after-heater can produce one 6-inch 250-gm crystal per day. For less critical demands, it can produce several longer crystals per day. Evaluation of the product indicates that resistivity variation on a cross-section of the ingot can be ±3 per cent and that along the length axis it can be controlled to ±7 per cent if a continuous charge is used. Furthermore, the crystals contain no grain boundaries or lineage and the scattered etch pit densities average about 1,500 per cm-. Thus, the zone leveling process has proved to be simple, efficient, and capable of more than meeting the present specifications for quality germanium single crystals. ACKNOWLEDGMENTS i The authors arc indebted for the help and cooperation of many people, especially that of L. P. Adda and D. L. Erhart who guided the evaluation of zone leveled material summarized above, and that of F. W. Bergwall through whose patient effort and suggestions the machine worked. Diffused p-n Junction Silicon Rectifiers By M. B. PRINCE (Manuscript received December 12, 1955) Diffused p-n junction silicon rectifiers incoryorating the feature of con- ductivity modulation are being developed. These rectifiers are made by the liiffusion of impurities into thin wafers of high-resistivity silicon. Three \ development models with attractive electrical characteristics are described irhich have current ratings from 0 to 100 amperes with inverse peak voltages qreater than 200 volts. These devices are attractive from an engineering stand- point since their behavior is predictable, one process permits the fabrication of an entire class of rectifiers, and large enough elements can be processed so that power dissipation is limited only by the packaging and mounting ■of the unit. l.n INTRODUCTION 1.1 The earliest solid state power rectifier, the copper oxide rectifier, was introduced in the 1920's. It found some applications where effi- ciency, space, and weight requirements were not important. In 1940 the selenium rectifier was introduced commercially and overcame to a great extent the limitations of the copper oxide rectifier. As a result, the selenium rectifier has found wide usage. In early 1952 a large area licrmanium^ junction diode was announced which showed further im- ' provements in efficiency, size, and weight. In addition it shows promise of greater reliability and life as compared to the earlier devices. How- ever, all of these devices have one drawback in that they cannot operate 111 ambient temperatures greater than about 100°C. Also in 1952, the silicon alloy^ junction diode was announced and was shown to be capable of operating at temperatures over 200°C. However it was a small area device and could not handle the large power that the other devices could rectify. During the past three years development has been carried on by several laboratories in improving the size and power capabilities of these alloy diodes. In early 1954 the gaseous diffu- ' Hall, R. N., Proc. I.R.E., 40, p. 1512, 1952. 2 Pearson, G. L., and Sawyer, B., Proc. I.R.E., 40, p. 1348, 1952. 661 662 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 (a) FORWARD REVERSE a: a: D U (b) /rs y^^ Vb v„ VOLTAGE Fig. 1 — (a). Ideal rectifier, (b). Semiconductor rectifier. sion technique^ for producing large area junctions in silicon was an- nounced. This technique lends itself very readily to controlling the position of junctions in silicon. An early rectifier^ made by this tech-- nique was one half cm^ in area and conducted 8 amperes at one volt in i the forward direction and about 2 milliamperes at 80 volts in the re-- verse direction. The series resistance of this device was approximately' 0.07 ohms. 1.2 In order to understand quantitatively the problems associated! with power rectifier development, consider Fig. 1(a) which shows what I an engineer would like in the way of an ideal rectifier. It will pass a large amount of current in the forward direction without any voltage. 3 Pearson, G. L., and Fuller, C. S., Proc. I.Il.E., 42, No. 4., 1954. I DIFFUSED p-> JUNCTION SILICON RECTIFIERS 663 drop aiul will pass no current for any applied voltage in the reverse direction. At present no device with this characteristic exists. A typical semiconductor rectifier has a characteristic of the type shown in Fig. 1(b). In these devices there is a forward voltage, Vo , that must be de- veloped before appreciable current will flow and a series resistance, Rs, thru which the current will flow. In the reverse biased direction there is a current that will flow due to body and surface leakage and that usually increases with reverse voltage. At some given reverse volt- age, Vb, the device will break down and conduct appreciable currents. To have an efficient rectifier, Vo and Rs should be as small as possible and Vb should be as large as can be made; also, the reverse leakage cur- rents should be kept to a minimum. According to semiconductor theory. To depends mainly upon the energy gap of the semiconductor, in- creasing with increasing energy gap. Rs consists of two parts; body re- sistance of the semiconductor and resistance due to the contacts to the semiconductor. The higher the resistivity of the semiconductor, the higher is the body resistance part of Rs ■ The leakage currents in the reverse direction depend to some extent on the energy gap of the semi- conductor, being smaller with larger energy gap; and Vb depends most strongly on the resistivity of the semiconductor, being larger for higher resistivity material. Another factor that is important in the choice of the semiconductor is the ability of devices fabricated from the semi- conductor to operate at high temperatures; high temperature operation of devices improves with larger energy gap semiconductors. Thus there are two compromises to be made in choosing the material (energy gap) and resistivity of the semiconductor. 1.3 This paper reports on a special class of rectifiers in which im- proved performance has been obtained. These devices are made by using the diffusion technique with silicon. The diffusion process permits both accurate geometric control and low resistance ohmic contacts, which in turn makes it possible to reduce Rs to very small values inde- pendent of the resistivity of the initial silicon. Therefore, high resis- tivity material can be used to obtain high Vb ■ An explanation of this result is given in Section 3. Silicon permits small reverse currents and high temperature operation. Its only drawback is that Fo ^^ 0.6 volts. Rectifiers made of silicon with the diffusion technique are able to pass j hundreds of amperes per square centimeter continuously in the forward ( direction in areas up to 0.4 square centimeter. One type of device whose i area is 0.06 cm- readily conducts ten amperes with less than one volt forward drop. The forward current voltage characteristic of this family of rectifiers follows an almost exponential characteristic indicating that 664 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 | Rs is extremely small (<0.05 ohms). Although the measured reverse currents are greater than those predicted by theory for temperatures up to 100°C, the reverse losses are low and do not affect the efficiency appreciably. ,1 1.4 The diodes made by the diffusion of sihcon are very attractive' from an engineering standpoint for several reasons. First of all, their ; behavior is predictable from the theory of semiconductor devices, as are junction transistors. This makes it possible to design rectifiers of given electrical, thermal, and mechanical characteristics. Secondly, : rectifier elements of many sizes are available from the same diffused < wafers making it possible to use the same diffusion process, material, and equipment for a range of devices. Thirdly, large enough elements can be processed so that the power dissipation in the unit is limited only by the thermal impedance of mount and package. 2.0 DIFFUSION PROCESS t 2.1 It will be shown in 3.2 that the forward characteristic of these devices is practically independent of the type (n or p) and resistivity of the starting material. The reverse breakdown voltage of a silicon p-n junction depends primarily on the resistivity of the lightly doped region. With these two considerations in mind; that is, to fabricate rectifiers having the desirable excellent forward characteristic and at the same time high reverse breakdown voltage, high resistivity siUcon is used as ; the starting material for the diffused barrier silicon rectifiers. Single crystal material has been found to give a better reverse characteristic than multicrystalline material. Also, it has been found that p-type ma- terial has yielded units with a better reverse characteristic than n-type material. Therefore, in the remainder of this paper, we will limit dis-';' cussion to rectifiers made from high resistivity, single crystalline, p-type /"i silicon. We will designate this material as ir type silicon. 2.2 In addition to the fine control one has in the diffusion process (see 2.4), the process lends itself admirably to the semiconductor recti-' fier field in as much as the distribution of impurities in this process re- ; suits in a gradual transition from a degenerate semiconductor at the' surface of the material to a non-degenerate semiconductor a short dis- tance below the surface. This condition permits low resistance ohmic metallic; contacts to be made to the surfaces of the diffused silicon. In order to create a p-n junction in the x silicon, it is necessary to diffuse donor imjiurities into one side of the slice. Although several donor type imi)urities have been diffused into siUcon, all the devices discussed I DIFFUSED p-n JUNCTION SILICON RECTIFIERS 665 in this paper were fabricated by using phosphorus as the donor impurity. In order to make the extremely low resistance contact to the tt side of the junction that is desirable in rectifiers, acceptor nnpurities are dif- fused into the opposite side of the x silicon slice. Boron was selected from the several possible acceptor type impurities to use for the fabri- cation of these devices. A configuration of the diffused slice is shown in Figure 2. 2.3 It will be shown in Section 3 that there are limits to the thick- nesses of the three regions, N-{-, x, P+, due to the nature of the opera- tion of these rectifiers With present techniques, it is necessary to keep ^LOW-RESISTANCE CONTACTSn / . . _ -\ ACTIVE p-n JUNCTION Fig. 2 — Diffused silicon rectifier configuration. the thickness of the t region to the order of two or three mils (thou- sandths of^an inch). 2,4 In the diffusion process of introducing impurities in silicon for the purpose of creating junctions or ohmic contacts, the diffusant is deposited on the silicon and serves as an infinite source. The resulting concentration of the diffusant is given by 9 rx/\/iDt , c = Cc 1 - 4- / ^ dy V TT ''0 (1) = Co erf c y where C = concentration at distance x below surface Co = concentration at surface D = diffusion constant for impurity at temperature of dif- fusion 66G THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 t = total time of diffusion X y = /jyr- = variable of integration A plot of C/Co = erfc y versus y is given in Fig. 3. Co is the surface solu- bility density and depends upon the tempers are of the diffusion proc- ess/ At some depth, Xj , the concentration C equals the original im- purity concentration where the silicon will change conductivity type resulting in a junction. In order to obtain desirable depths of the dif- fused layers, A^+ and P+, it is necessary to diffuse at temperatures in the range of 1000°G to 1300°C for periods of hours. With such periods it is obvious that the diffusion process lends itself to easy control and reproducibiUty. .3.0 CONDUCTIVITY MODULATION 3.1 It is well known that the series resistance of a power rectifier is the most important electrical parameter to control and should be made as small as possible for several reasons. The series resistance consists essentially of two parts; the body resistance of the semiconductor and the contact resistance to the semiconductor. In the early stages of recti- fier development both parts of the series resistance contributed about equally to the total series resistance. However, methods were soon found to reduce the contact resistance. It then became apparent that in order to reduce the body resistance, the geometry would have to be changed and the resistivity chosen carefullJ^ By going to larger, thinner wafers it was possible to reduce this body resistance. However, the cost of pure silicon made it important that conductivity modulation (described below) be incorporated in these devices as a method for reducing the body resistance. Our initial attempts were successful due to the fact that higher lifetime of minority carriers could be maintained in the ex- tremely thin wafers that were used as compared to the lifetime remain- ^ ing after the diffusion process in thicker wafers. 3.2 A complete mathematical description of the I-V characteristic for the conductivity modulated rectifier is practically impossible due to the fact that the equations are transcendental. However, it is easy to understand the operation of the device physically. When the device is biased in the forward direction, electrons from the heavily doped N-\- region are injected into the high resistivity ir region. If the lifetime for these electrons in the tt region is long enough, the electrons will diffuse across the w region and reach the P-f region * Fuller, C. S., and Ditzenberger, J. A., J. Appl. Phys., 25, p. 143!), li)54. DIFFUSED jy-n JUNCTION SILICON RECTIFIERS 667 II 10 10 10" 10 10" 10 2 3 4 5 6 7 0.4 O.f 1.6 2.0 y 2.4 3.2 3.6 4.0 Fig. 3. — Error function complement. with little recombination. To maintain electrical neutrality, holes are jinjected into the x region from the P+ region. These extra mobile car- riers (both eleictrons and holes) reduce the effective resistance of the tt jlayer and thus decrease the voltage drop across this layer. The higher (he current density, the higher is the injected mobile carrier densities Mid therefore, the lower is the effective resistance. It is for this reason iliat the process is termed conductivity modulation. This effect tends 'to make the voltage drop across the tv region almost independent of the current, resistivity, and semiconductor type. When the junction is biased in the reverse direction, a normal re- verse characteristic with an avalanche breakdown is expected and ob- served. 3.3 The forward characteristic of a typical \uiit is plotted semi- logarithmically in Fig. 4. The best fit to the low current data can be G68 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 expressed as / = 7oe«^'^'=^ where I = current thru unit 7o = constant q = charge of electron V = voltage across unit k = Boltzmann's constant T = absolute temperature and 1< .V < 2. (2) 10 10" to a UJ Q. < UJ <£. o 10 -2 10" 10 10 10 y / / /^ / / / / / / f / r y / / / qv/i.29kT I = Ioe / / / / v -6 0.2 0.3 0.4 0.5 0.6 VOLTS 0.7 0.8 0.9 Fig. 4 — Forward characteristic of silicon power rectifier. DIFFUSED p-n JUNCTION SILICON EECTIFIERS CG9 I O < UJ u z < HI tr 100 50 20 10 1.0 0.5 0.2 O.t 0 0.05 0.02 0.01 1 k \ N \, T k S." \ s. N \ V °\ \, S k N \ \ 0.001 0.01 0.1 1.0 CURRENT, Idci'N amperes 10 Fig. 5 — Small signal resistance versus dc forward current. The departure of the high current data from the exponential charac- teristic is due to the contact resistance. Another interesting measure- ment of the forward characteristic is given in Fig. 5 where the small signal ac resistance is plotted as a function of the forward dc current for a typical rectifier element. The departure from the simple rectifier theory^ where iV = 1 is not surprising inasmuch as p-n junctions made by various methods and of different materials almost always have A^ > 1. Several calculations have been carried out using different assumptions and all indicate that the forward characteristic is independent of the type and resistivity of the middle region as long as the diffusion length for minority carriers is the order of or larger than the thickness of the region. 3.4 In order to go to higher reverse breakdown voltages (>500 volts) it is necessary to use still higher resistivity starting material. It might be expected that intrinsic silicon will be used for the highest reverse breakdown voltages when it becomes available. How^ever, in this case " Shockley, W., B.S.T.J., 28, p. 435, 1949. II 070 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1950 thick wafers are necessary since the reverse biased junction space charge region extends rapidly with voltage for almost intrinsic material, and high lifetime is necessary in order to get the conductivity modulation effect in these thick w^afers. Therefore at present it is necessary to com- promise the highest reverse breakdown voltages with the lowest for- ward voltage drops, in a similar manner to that discussed in Section 1. However this is now done at a different order of magnitude of voltage and current density. | 4.0 FABRICATION OF MODELS 4.1 It has been pointed out in Section 1.2 that a low series resistance, Rs , is desirable and that it is composed of two parts; the body resistance and the contact resistance. In Section 3 a method for reducing the body resistance was described. The contact resistance can also be made very ! low. It has been found to be very difficult to solder low temperature solders (M.P. up to 325°C) to silicon with any of the standard commer- cial fluxes. However, it is quite easy to plate various metals to a surface of silicon from an electroplating bath or by an electro-less process^ to ! which leads can readily be soldered. Some metals used for plating con- tacts are rhodium, gold, copper, and nickel. This type of contact yields a low contact resistance. Another techniciue that has shown some prom- ise for making the necessary extremely low resistance contact is the hydride fluxing method.'^ 1 4.2 A wafer which may be about one inch in diameter is ready to be i diced after it is prepared for a soldering operation. Up to this point all ; the material may undergo the same processing. Now it is necessary to ' decide how the prepared material is to be used; whether low current i ('^l amp) devices or medium or high current ('^10-50 amps) devices;. are desired. The common treatment of all material for the entire class i of rectifiers is one reason these devices are highly attractive from a . manufacturing point of view. | The dicing process may be one of several techniques; mechanical! cutting with a saw, breaking along preferred directions, etching alonii given paths with chemical or electrical means after suitable maskiiiti methods, etc. In the case of mechanical damage to the exposed junc- tions, the dice should be etched to remove the damaged material. The dice are cleaned by rinses in suitable solvents and are then ready for « Brenner, A., and Riddell, Grace E. J., Proc. American Electroplaters' Society, 33, p. 16, 1946,34,1). 156, 1947. ' Sullivan, M. V., Hydrides as Alloying Agents on Silicon, Semiconductor Symposium of the Electrochemical Society, May 2-5, 1955. I DIFFUSED p-n JUNCTION SILICON RECTIFIERS 671 assemlily into the mechanical package designed for a given current rating. 4.3 The dice may be tested electrically before assembly by using pressure contacts to either side. Pressure contacts have been considered for packaging the units; however, this type of contact was dropped from development due to mechanical chemical, and electrical instabilities. 4.4 The drawbacks of the pressure contact make it important to find a solder contact that does not have the same objections. The solder used should have a melting point above 300°C, be soft to allow for different coefficients of expansion of the silicon and the copper connections, wet the plated metal, and finally, be chemically inactive even at the high temperature operation of the device. These recjuirements are met with many solders in a package that is hermetically sealed. This combina- tion of a solder and a hermetically sealed package has been adopted for the intermediate development of the diffused silicon power rectifiers. 5.0 ELECTRICAL PERFORMANCE CHARACTERISTICS 5.1 Before describing the electrical properties of these diodes, let us consider some of the physical properties of a few members of the class. r~=^^T:) SMALL 0-1 AMPERES MEDIUM 1-10 AMPERES LARGE 10-100 AMPERES Fig. 6 — Development silicon rectifiers. 672 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 10 - 10 -^ 10 ' 10 CURRENT IN AMPERES 1 10' Fig. 7 — I-V characteristic of medium size rectifier. Fig. 6 shows a picture of three sizes of units that will be discussed in this section together with the range of currents that these units can con- duct. The actual current rating will depend upon the ability of the de- vice to dispose of the heat dissipated in the unit. A description of how the rating is reached is given in Section 6. The smallest device has a silicon die that is 0.030'' by 0.030" in area 10- 10' 10 to I- o > 10 10 SMALL REVERSE ^- _^^ . LARGE _«' ^•"f-^ y"^"^ / • / / f // / / / / / / / / f 1 FORV VARD *^ '-^^S^ ~ '^^ lO"" 10' 10 ° to " 10 ^ 10 10 '^ CURRENT IN AMPERES 10 10 Fig. 8 — I-V characteristics of devehjpment rectifiers. DIFFUSED y-n JUNCTION SILICON RECTIFIERS 673 i I and all the units have dice about 0.005" thick. The medium size device has a wafer 0.100" by 0.100" in area. The largest device has a element 0.250" by 0.250" in area. It is obvious that a range of die size could have been chosen for any of these rectifiers. However, electrical and thermal considerations have dictated minimum sizes and economic considera- tions have suggested maximum sizes. The actual sizes are intermediate in value and appear to be satisfactory for the given ratings. 5.2 Of fundamental importance to users of these rectifiers are the for- ward and reverse current — voltage characteristics. These characteris- tics of the medium size iniit are shown in Fig. 7 for two temperatures, 25°C and 125°C, using logarithmic scales. It can be seen that in the forward direction at room temperature, 25°C, more than 20 amperes are conducted with a one volt drop in the rectifier. At the higher tem- perature more current will be conducted for a given voltage drop. In the reverse direction, this particular unit can withstand inverse voltages as high as 300 volts before conducting appreciable currents (>1 ma) even at 125°C. A comparison of the current-voltage characteristics for the three different size units is shown in Fig. 8 where again the informa- tion is plotted on logarithmic scales. This information was obtained at 25°C. One can observe that the reverse leakage current varies directly as the area of the device and the forward voltage drop varies inversely as the area. These relations are to be expected; however, the reverse characteristics indicate that surface effects are probably effecting the exact shape of the curves. The changes in the forward characteristics can be attributed to the contacts and the internal leads of the packages. The breakdown voltage can be adjusted in any size device by the proper choice of starting material and therefore no significance should be placed on the different breakdown voltages in Fig. 8. SILICON GERMANIUM SELENIUM Fig. 9 — Semiconductor rectifiers of different materials. 674 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 CURRENT IN AMPERES Fig. 10 — Rectifier characteristics at 25°C. It is quite interesting to compare these units with germanium and selenium rectifiers that are commercially available. To make the com- parison as realistic as one can, we have chosen to compare the smallest silicon vuiit with a commercially available germanium unit and a six element selenium rectifier stack rated at 100 milliamperes. The com- parative size of these units can be seen in Fig. 9. Curves of the forward and reverse characteristics at 25°C are given in Fig. 10. Similar curves taken at 80°C are given in Fig. 11 and at 125°C in Fig. 12. It can be seen that the forward characteristic is best for the germanium device at all temperatures and that the reverse currents are least for the silicon rectifier. The selenium rectifier is a poor third in the forward direction. However, if one has to operate the device at 125°C, only the silicon de- vice will be satisfactory in both the forward and reverse directions. 5.3 Capacitance measurements of all the silicon units have been made at different reverse voltages and temperatures. The temperature depend- ence is negligible. However, as expected in semiconductor rectifiers, the capacitance varies inversely with the voltage according to the rela- tion VC^ = constant where 2 < N < 3. Measurements are given in Fig. 13 for a group of medium size units. The other units made from the same resistivity mat(n'ial have capacitances that vary dii'cctly as their areas. DIFFUSED p-n JUNCTION SILICON RECTIFIERS 675 5.4 The reverse breakdown voltage, Vb , of these devices is controlled by the choice of resistivity of the starting material and the depth of diffusion of the junction. By keeping the resistivity of the initial p-type silicon above 20 ohm-cm., it is possible to keep Vb above 200 volts. Units have been made with Vb greater than 1,000 volts. The deeper diffusion causes the junction to be more "graded"^ and therefore re- quire a greater voltage for the breakdown characteristic. This is in line with the capacitance measurements where the exponent indicates that the junction is neither a purely abrupt junction which would result in an exponent of two nor a constant gradient junction which would result in an exponent of three. 5.5 Another interesting measurement, which is related to the life- time of minority carriers in the high-resistivity region and the frequency response, is the recovery time of these devices. During a forward bias on a p-n junction, excess minoritj^ carriers are injected into either region. When the applied voltage polarity is reversed, these excess minority carriers flow out of these regions, giving rise initially to a large reverse current until the excess carriers are removed. The magnitude and time variation of this current will depend to some extent upon the level of the forward current but mostly upon the circuit resistance. If one ad- justs the circuit resistance such that the maximum initial current in CURRENT IN AMPERES Fig. 11 — Rectifier characteristics at 80°C. 676 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 CURRENT IN AMPERES Fig. 12 — Rectifier characteristics at 125°C. the reverse direction is equal to the forward current before reversing i the polarity of the junction, then the reverse current will have a con- i stant magnitude, limited ])y the circuit resistance, for a time known as •' the recovery time before it decays to a small steady-state value. Fig. 14 . shows graphically this effect. The recovery time in diffused junctions |i is found to be in the range of less than 0.1 microsecond to more than 4 Q 200 < < O tr 100 U i 80 O a 60 o 2 40 UJ U z < 20 < a. < 10 u 1 6 8 10 20 40 60 VOLTS 100 200 400 600 1000. Fig. 13 — Capacitance versus reverse voltage in medium size rectifer. DIFFUSED p-n JUNCTION SILICON RECTIFIERS 677 z LU CE q: D O FORWARD REVERSE y^ RECOVERY •*■-- TIME *■ -If TIME, t — *- Fig. 14 — Recovery effect in silicon rectifiers. microseconds. It can be shown that the longer recovery times are associ- ated with higher Hfetimes of minority carriers. More interesting, how- ever, is the fact that these devices will have their excellent rectification characteristics to frequencies near the reciprocal of the recovery time. Measurements have been made of the rectification ability of typical small and medium size units by using the circuit shown in Fig. 15. The results of normalized rectified current versus frequency are given in [Fig. 16 and it is seen that these units could be used to rectify power up to 1 kc/sec without any appreciable loss of efficiency. \ 5.6 It is interesting to note that many of the electrical measurements ,inade with the diffused barrier silicon rectifiers are self-consistent and jean be related to simple concepts of semiconductor theory. As an exam- !ple, experimental measurements indicating variations of recovery time I of units are related to variations in minority carrier lifetime which in turn are related to experimental variations in the forward characteristic OSCILLATOR AAA- 1000 n Fig. 15 — Rectification measuring circuit. G78 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 >■ u 2 Z o I- < u UJ > < 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 < » — & ^ a I i « i > ▲ • MEDIUM ▲ SMALL < i 1 Ua ^ ^ • A ▲ 1 • ▲ 1 ▲ 10 I 10' to'' 10' 10- 10° 10' FREQUENCY IN CYCLES PER SECOND Fig. 16 — Relative rectification efficiency versus frequency. of these same devices. Such relationships among the measurable param- eters of these devices make it possible to design and control the elec- trical characteristics of the units and therefore make them extremely attractive from an engineering point of view. 6.0 MECHANICAL AND THERMAL DESIGN 6.1 In order to have a device that is usable for more than experimen- tal purposes, it is necessary that it be packaged in a mechanically stable; structure and that the heat generated in the combined unit should not' lead to a condition ^vhere the device no longer has its desirable charac- teristics. In earlier sections of this paper several mechanical require- ments of a satisfactory package have been suggested. These may be repeated at this point. First, pressure contacts are not satisfactory; sec- ond, oxidizing ambients are to be avoided; third, approximately one watt per ampere of forward current is generated and must be disposed ; and fourth, the package must be electrically satisfactory. The first rc- (juirement is met by using soldered contacts. Since these rectifiers are, usable at temperatures over 200°C, a solder was chosen that has a melt- ing point over 300°C. The second recjuirement necessitated the use of a hermetic seal structure. If the seal is truly hermetic, no gases can DIFFUSED p-7l JUNCTION SILICON RECTIFIERS 679 enter or leave the package and thus no changes of the device due to the enclosed gas should occur as long as the gas does not react with the sili- con, solder or package. However, no seal is absolutely vacuum tight and thus care should be used in choosing a package design so that mini- mum effects should occur to the electrical properties during the use of the device. The third requirement of the disposal of the internally de- veloped heat suggested the use of copper due to its high thermal conduc- tivity. However, a small package alone is capable of dissipating only a small amount of heat without reaching a temperature that is too high for the device. This necessitates the use of cooling fins in conjunction with the device to make use of its electrical properties. This thermal requirement demands a package to which thermal fins can be attached. This is met by having the package contain a bolt terminal to which thermal fins can be attached or by which the unit can be mounted to a chassis for cooling. The fourth requirement consists of two parts; the package must have two leads that are electrically separated from one another and the leads must be sufficiently heavy to conduct the maxi- mum currents. The first of these requirements is met by using glass-to- metal seals in the package and the second is met by using copper leads of sufficiently heavy cross-section. The resulting packages for the units discussed in this paper are shown in Fig. 6. It should be remembered that the packages are only intermediate development packages and that further work will probably alter these both in size and in shape. How- ever, all the requirements mentioned will be applicable to any package. 6.2 The units pictured in Fig. 6 have a range of dc current ratings associated with them. The lower rating of each device corresponds to the maximum rating of the next smaller device. Of course, the larger units could be used for smaller current applications; however, such use M'ould be like using a freight car to haul a pound of coal. The maximiuu rating of each de^'ice has been arbitrarily chosen for it to operate with a reasonable sized cooling fin at an ambient of 125°C and no forced air or water cooling. It is known that the ratings could be increased by either method of forced cooling. It has been found that a copper con- vection cooling fin is able to dissipate 8 milliwatts per square inch per degree centigrade. This cooling rate is obtained from the difference be- tween the average temperature of the fin and the ambient temperature over the effective exposed area of the fin. For example, a copper fin S}4, inches scjuare when mounted so that both surfaces are effective for cooling will })e able to dissipate ten watts and at the same time prevent the temiK'rature of the fin from exceeding 50°C above the ambient tem- perature. Another thermal drop is found between the junction and the 680 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 base of the package. This temperature difference depends mostly on the material of the base and its geometry. In the devices presented this drop is not more than 15°C at the maximum rated current. Thus the largest drop in temperature occurs between the cooling fin and the ambient which means that the design of the cooling fin is the controlling factor in the operating junction temperature of the rectifier. 6.3 It is possible to use the devices without an attached cooling fin. In this case, the maximum current is limited essentially by the size of the package. The small rectifier package is designed for 3^ watt dissipa- tion and therefore the maximum current that should be rectified is about 500 milliamperes. The medium size unit will comfortably rectify 1 am- pere without any additional cooling and the large rectifier unit will conduct 3 amperes under the same conditions. 7.0 RELIABILITY AND LIFE MEASUREMENTS 7.1 One of the desired properties of any device is that it should op- erate satisfactorily at its rating for a long period of time. The above general statement contains many implications which should be made specific for the devices under consideration in this paper. By stating that these devices should operate satisfactorily we mean that they should not age during operation; that is, the forward and reverse char- acteristics at any temperature should not change with time. The state- ment implies that a rating has been established for the units. Further- more, a "long period of time" has to be defined. There are applications where a few hours is considered a long time as in some military appli- cations. However, in most Bell System applications, a long period of time may be 20 years or approximately 200,000 hours. Clearly, in the short time since these rectifiers have been developed, it is impossible to make a fair statement as to their reliability and their life expectancy. , However, it is possible to present some results of some early experi- | ments and describe where and how the units have lived and died. It is ] this information that we will present in this section. It is a common ex- I perience that during the early development of any new component, i there are many units that do not satisfy all the requirements of the de- | sired end product. These units will generally deteriorate very rapidly | on life testing due to some electrical or mechanical instability. The units used for life testing have been screened to remove the above men- tioned unstable devices. 7.2 The life tests consist of four types; shelf tests at room tempera- ture and at 150°C, forward characteristic tests, reverse characteristic I DIFFUSED p-n JUNCTION SILICON RECTIFIERS 681 tests, and load tests. The last tests are really the important tests; how- ever, these require the dissipation of large quantities of power in the load to test only a few devices. Therefore only a few units were tested in this condition and the majority tested under other conditions. The several units under load test have been operating for six months with no noticeable change in their characteristics. These devices are the small and medium size development units. The large rectifiers would require about 10 kilowatts of dissipation each in a load to give them a fair load test. The shelf tests at room temperature and at a temperature of i50°C have been running for six months and have indicated that most of the units remain practically constant. There have been some units that improve on standing but there is no method of predicting which ones will improve. Some units get worse on standing; however, most of these can be predicted from the initial tests since these units usually have a noisy reverse characteristic near the reverse breakdown voltage. The units that change differ only in their reverse characteristic; the forward characteristic changes are not detectable indicating that the contacts are stable. The changes in the reverse characteristic are probably due to the trapping of ions and vapors on the surface of the devices during the packaging operation. Another source of these variations is due to the non-hermeticity of the glass-to-metal seals allowing gases to diffuse into the package where they may cause changes in the reverse charac- teristic. These leaks have been found in many early units and new as- semblies are being tried at present. f The forward characteristic life test was considered a good test since the device is subject to practically all the internal power dissipation without reciuiring the relatively high load dissipation. It is tests of this nature that allow one to rate the various size devices. The medium size rectifiers that ran at 15 amperes in this test failed after three months of testing; whereas no units running at 5 and 10 amperes have failed during the six months since the tests have started although their re- verse characteristics have changed slightly. It should be noted that most of the change of reverse characteristic occurred during the first test period of two weeks. These changes are probably due to the causes mentioned in the above paragraph. Reverse characteristic tests have been running for several months on a group of 10 small rectifiers which we feel have a better gas tight seal than the other development units. The voltage has been adjusted on these units such that they are pulsed into the breakdown region with a 682 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 maximum current of one millianipere. None of these units show any ap- preciable change. 7.3 All of those tests in the past sub-section had to do with continu- ous dc or ac power being supplied to the units under test. However, in actual operation the units may be subject to voltage pulses due to power line pulses, accidental shorts, etc. In order for the rectifier to be useful, it should be able to take an overload for a period of time suffi- ciently long to allow a protective device to operate. Pulse tests have been performed on the medium size rectifier. These devices are able to with- stand over 300 amperes for times of the order of 50 microseconds. How- ever, the fastest circuit breakers operate in about 20 milliseconds and for this period, these units can stand onl}^ approximately 50 amperes before failing. Since these units have such a low forward resistance at the operating currents (Fig. 7), any small increase in voltage across the diode will change the current through the device to a very large cjuan- tity. Therefore series protective resistances may be necessary where the possibility of short-circuiting the device is high. Such operation would reduce the efficiency of the unit and is to be avoided if possible. Another type of protection may be afforded through the use of a high impedance, high current inductor. This type of protection is quite bulky and heavy and suitable only for stationary apparatus. Another common possibility of burnout of the devices occvu's when using a capacitance input in conjunction with the rectifier. When the circuit is turned on, large currents will flow to charge up the capacitors and consequently burn out the rectifiers. One possible protection from such operation is the use of a series resistance in conjunction with a time delay relay. The| series resistance will limit the initial capacitor charging current and the time delay relay will short out the resistance after the capacitors have reached near their maximum charge. 7.4 Dissection of burned out units have indicated that the failure takes place through small spots on the device. This can be explained by the fact that some small areas of the device have slightly better forward characteristics. These areas will tend to conduct most of the forward current. Therefore most of the power will be dissipated there and these areas will become even more conducting leading to a channeling of the forward current through these spots with the consequent burnout. The best way to avoid such mishaps would be to make a more uniform de- vice. Experiments are in process along this line. Another less satisfactory - method would be the control of contact resistance such that the current would be limited in any particular area by the contact resistance. Simi- lar ideas must be considered when paralleling these diffused junctioiii DIFFUSED p-n JUNCTION SILICON RECTIFIERS 683 silicon rectifiers. It is possible to use these devices in parallel if oni' ad- justs the lead resistances such that no one unit will be allowed to con- duct much more than its share of the current. 7.5 As a conclusion to this section, it should be noted that these rec- tifiers are expected to have a long life when operated within their rat- ings. They are able to operate for short periods of time (seconds) at five times their rated currents. Since the rectifiers have an extremely small series resistance, they should be protected against accidental surges and turning on to a capacitance input filter. 8.0 SUMMARY 8.1 The development rectifiers described in the article are silicon diffused p-n junction rectifiers. These devices together with associated cooling fins can be used to rectify a complete range of currents from 0 to 50 amperes in a single phase, half wave rectifier circuit. They can be used in more complex rectification circuits to yield even more dc cur- rent. Also, they are able to withstand at least 200 volts peak in the in- verse direction and operate satisfactorily at temperatures as high as 200°C. Furthermore, one process of diffusion and plating is sufficient for all the devices of the class. This makes it possible for one diffusion and plating line to feed material for all the rectifiers in a manufacturing operation. 8.2 The rectifiers discussed behave according to the theory of semi- conductor devices which makes it possible to design them for given electrical, thermal, and mechanical characteristics. One failure to meet ideal theory of a p-n junction is with the forward characteristic. 8.3 The diffused silicon type of rectifier has been compared with germanium and selenium units and has better reverse characteristics at all temperatures. In the forward direction, the germanium units have a smaller voltage drop for any given current than the silicon rectifiers but the silicon devices are capable of operating at much higher tem- peratures, thereby permitting higher overall current densities than the germanium devices. 8.4 The diffused silicon rectifiers are capable of use in any rectifier application where dc currents up to the order of 100 amperes are re- fjuired and where inverse peak voltages up to 200 volts are encountered. Another imoortant use for these devices will be in the magnetic ampli- fier application where the low reverse currents of silicon will enable large amplification factors to be realized. Since the forward character- istics of these devices are so uniform, they can be used in voltage ref- erence circuits that require voltages near 0.6 volts and in circuits uti- 684 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 lizing the exponential character of the forward characteristic. However, as is to be expected from devices with the characteristics described in this paper, the most immediate apphcation will be found in power sup- plies. ACKNOWLEDGMENTS It is obvious that the work reported in this paper is not the result of one man's labor. Much of the stimulus and many of the ideas are those of K. D. Smith. Other members of the Semiconductor Device Department who have contributed considerably to the development of these devices are R. L. Johnston, R. Ruhson, and R. C. Swenson. D. A. Kleinman, J. L. Moll and I. M. Ross have been most helpful in dis- cussing the theoretical aspects of these devices. The author wishes to thank H. R. Moore for his suggestions on protecting the silicon rectifiers against large overloads. The Forward Characteristic of the PIN Diode By D. A. KLEINMAN (Manuscript received January 18, 1956) A theory is given for the forward current-voltage characteristic of the PIN diffused junction silicon diode. The theory predicts that the device should obey a simple PN diode characteristic until the current density approaches 200 amp /cm?. At higher currents an additional potential drop occurs across the middle region proportional to the square root of the current. A moderate ' amount of recomhiriation in the middle region has little effect on the charac- teristic. It is shown that the middle region cannot lead to anomalous char- acteristics at low currents. t INTRODUCTION In some diode applications it is desirable to have a very low ohmic re- sistance as well as a high reverse breakdown voltage. A device meeting these requirements, in which the resistance is low because of heavily doped P"^ and A^"^ contacts and the breakdown \'oltage is high because of a lightly doped layer between the contacts, has been described by M. B. Prince. The device is shown schematically in Figure la and con- sists of three regions, the P^ contact, the middle P layer, and the A'"'' contact. The device is called a PIN diode because the density P of un- compensated acceptors in the middle region is much less than P"*" or iV"*" and in normal forward operation much less than the injected carrier density.^ We shall let the edge of the P^P junction in the middle region be oj = 0, and the edge of the PN^ junction in the middle region be x = w. Thus the region 0 ^ .r ^ w is space charge neutral and bounded at each end by space charge regions whose width is of the order of the Debye length 1 Prince, M. R., Diffused p-?i .Junction Silicon Rectifiers, B.S.T.,J., page 661 of this issue. ^ A device witli similar geometry has been discussed by R. N. Hall, Proc. I.R.r:., 40, p. 1512, 1952. 685 G8G THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 (K/^eP) 1/2 l.o X 10" cm. (1) where A' is tlie dielectric constant, c is the electronic charge, and )3 is the constant ^ = e/kT = n„/D,, = fxp/Di (2) which at room temperature is 38.7 \'olt~\ We shall denote points in the P and A'' contacts on the edges of the space charge regions by oo and WW respectively. Thus Uoo is the electron density in the P^ contact at the junction, and Ho is the electron density at the same junction in the middle region. Similarly p„.,„ is the hole density at the junction in the A^ contact and pu, is the hole density at the junction in the middle region. We shall denote equilibrium carrier densities in the three regions by np+ , 7ip , Pp , Pn+ . Typical values for the parameters characterizing N" (a) w X- > -V, oo/ 1 WW V= 1=0 (b) WW Vj f X f Vw+Vp / > — — ~-____________^^ / V-Vi 00/ 1 " ^ v„ ! (Cj w V\^. 1 — SchcuKitic represent al ion ol' (lie IMX (li(Ki(! with tli(> 1'+ and N+ con- tacts regarded as extending to infinity. (1)) .shows the; tdect rostatic potential in equilibrium and (c) show.s the potential when a forward current Hows. THE FORWARD CHARACTERISTIC OF THE FIX DIODE 687 the device are W'^2 X 10"^ cm P -- 10'' cm"' (3) Ar+ p+ ^ 10^' cm"' L„ , Lj, ^ 10"^ cm where L„ , Lp are minority carrier diffusion lengths in the contacts. The present treatment makes three distinct approximations. The first is to neglect the voltage drop in the contacts. The highest currents ordi- I narily used are of the order of 500 amp/cm" which should produce an ohmic drop in the contacts of about 1 volt/cm. Since the entire diode has a length of about 0.01 cm we are neglecting only about 0.01 ^'olts in this I approximation. The second approximation is to regard the Debye length as small compared to w and the diffusion lengths L„ , Lp . li L„ , Lp are as small as the typical values given in (3) the error made in this approximation is not completely negligible. Nevertheless, we use the approximation be- cause it enables us to regard the device as three relatively large neutral regions and two relatively narrow space charge regions. The behavior of the device can then be determined by solving for the diffusion and drift of carriers in the neutral regions subject to boundary conditions con- necting the carrier densities across the space charge layers. The third approximation is to neglect any increase in majority carrier density in the contacts due to injection of minority carriers. This approxi- mation is valid until the current density approaches 5 X 10 amp/cm", which is well above anticipated operating currents. It is conceivable that in some junctions all the current may flow through small active spots at which the current density is ^'ery high, perhaps exceeding the above figure. In such cases the current flow is two or three dimensional and the present analysis would not apply. It is also necessary to assume some law for carrier recombination. We shall assume that recombination in the contacts is linear in the injected minority carrier density din ■ n — np+ rKJ ax T Modification of the theory to suit other recombination laws is simple in principle, although considerable analytical complications might be en- countered. It seems most likely that in silicon FN junctions the re- combination actuallv is nonlinear. It can be shown that if the rccombi- 688 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 nation follows some power v of the injected density the forward characteristic of a simple PN junction is of the form exp W^^iv + 1)F] (6) Thus nonlinear recombination can account for the observation that in silicon diodes the slope of V versus log / is usually much less than /3. Our purpose here is not to study this interesting effect, but to study those effects which are due to the presence of the middle region. Therefore, we assume linear recombination for the sake of simplicity. In the last sec- tion we give a brief consideration of what to expect in the case of non- linear recombination in the contacts. Recombination in the middle region will also be assumed to be linear in the injected carrier density, but this assumption is not critical, since it turns out that a moderate amount of recombination in the middle region does not change the quali- tative behavior of the device. BASIC EQUATIONS Fig. 1(b) shows the electrostatic potential V{x) for the equilibrium case 7 = 0. The potential is constant except in the space charge layers. If w^e call the potential of the middle region zero, the P^ and N^ contacts are at the potentials — Vi and Vi respectively, where /3Fi - In (P^/pp) (7) ^F2 = {n (N^/np) Figure Ic shows the potential when a forward current I flows and a forward bias F is produced across the device. We shall define the poten- tial so that the A^"^ contact remains at V2 , which puts the P"^ contact at potential F — Fi . The potential at a point x is then given by V{x) = V2- r E{x) dx (8) "WW where E{x) is the electric field assumed zero in the contact regions x > WW and x < 00. The applied bias F consists of three terms F = Vo + Vp + F,„ (9) ' This potential distribution has been discussed b^y A. Herlet and E. Sp(MiI L^e-'^'-'^' -{- ^ I U„ Jo X (14) Dn JOO I Shockley, W., B.S.T.J., 28, p. 435, 1949. 690 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Since \/Lp « 1 we can write for the junction at x = id piw) = e-''- U^y-^ - (P- - y^^"") j e'^'dx w /?u>to6 p "WW (15)1 I floo = Tioiup /np)e Vo = V. e^'» Wu, ^ np e'""- where 0(X/Lp) means a term of order X/Lp . Thus we see that if we may: neglect X/Lp and X/L„ we have the following simple boundary conditions at the junctions (16). It is clear that in order to divide the device into three neutral regions we must also be able to neglect \/w. Finally, we have the condition of space charge neutrality p - n = P (17) It can be shown that the term K~ dE/dx is of order (A/L)" or (X/w) | and therefore negligible in our approximation. Therefore (17) is the Poisson equation for the middle region in our approximation. When we > use (17) we are not saying that E{x) is constant but only that K~ dE/dx is negligible compared to p(x) and 7i(x). The basic eciuations then are (10), (11), (13), (16), (17). Large Injection, No Recomhinalion In this section we consider current densities of the order of magnitude of those that flow in normal operation of the diode as a power rectifier. These currents inject large densities of electrons and holes into the middle region greatly increasing its conductivity. The result is that the \'oltage drop Vp is small even though the normal resistivity of the middle region is high. For this reason the device has been called a conductivit}' modulated rectifier. Also in this section we shall neglect recombination in the middle region, which makes In{x) and Ip{x) constant and greatly simplifies the analysis. The effect of recombination is to remove carrieis and increase the drop across the middle region. Therefore, it is desirable to keep recombination in the middle region as low as possible. THE FORWARD CHARACTERISTIC OF THE PIN DIODE 691 0Vo n.y = ppe Equations (13) can be written /. -f- 6/, dn In — bl-o dx 2Dn where b = Dn/Dp . Combining (19) and (21) gives the equations Ho = niHn/Insf"" Tin, = 7li(Ip/IpsY'' where nf = rippp is a constant, and also /3Fo = Vz (n -^ ^ Pp ^ns Up Ips (18) Under conditions of large injection we can say n» P, p» P rioo » np Pww » Pn so that (11) becomes In = Insirioo/np'^) ^p — J^ psKPww/pN ) and (17) becomes n{x) = p{x) 0 ^ X ^ w (20) Equation (16) becomes 7ioo = Uo^np /np)e (19) (21) (22) (23) (24) I'^rom the first equation (22) we have SY, = k+J^ f A^ (25) 2Dn .'o nix) 692 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Upon invoking the second equation of (22) we get fiVp = i" "^ f ^^ /n — (26) In — Olp Ho and n^=^no+ ^" ~ ^^^ w;. (27) We see that Vp is always positive in sign whatever the sign of /„ — hip . We now define a parameter (28) 7 = rio/ny, and a device constant li = J-ns/J-ps Then from (23) and (10) h/Ip = i^T' " 1 + i^T^ 1 _/ 1 + Ry^ (29) (30) Combining (23), (27) and (30) gives the equation for 7 as a function of total current 7=1- In — bip W 2Dn Un _ n (7/7 j^ - 1 ^^^^ y /o VI + &(7/7»)^ where 7co' = &/i2 (32) and /o is a unit of (particle) current density characteristic of the device Z. = i^^ = 4 m ^ (33) A typical value for e /o in a silicon diode is e /o '-^ 200 amp/cm" (3-1'^ based on (3). i THE FORWARD CHARACTERISTIC OF THE PIN DIODE 693 From (26) the potential drop in the middle region can be written (35) 0Vp = y 2-^n7 '00 From (24) and (30) KVo + 7.) = tn -\-(n- T + ^n h + Hy/yJ' J-ps (36) Thus the total applied bias y as a function of total current density / is given by /5F = (iij - V^ ^^ ^ + ^^ 1 , J / v> + ^^^ T- (^^) /o 7^ - Too 1 + 0(7/7 J- Ips where y{I) is the (positive) solution of (31). Thus far we have referred the problem of the V — I characteristic to the problem of calculating 7(7) from (31). We see that in the limits of high and low current 7 approaches the limits 7 -> 1 / « /o (38) 7 — ^ 7oo i » io and in general lies between these limits. A good approximate solution is readily obtained by replacing (31) with the cjuadratic equation 7=1- 4(7/700)' - 1] z = {i/ur (1 + hr" which has the solution V7co' + 4(1 + z)zyJ - 7. 7 = 2z (40) A plot of this solution is shoAvn in Fig. 2 as a function of z for 7oo = x^"^, 7oo = 2. Since 7(7) is bounded by unity and 700 , which usually will be of order unity, we can reject some of the dependence of V upon 7 and re- tain only its essential dependence upon 7. This appears in the first and second terms of (37). By means of (31) this second term can be written /n7 (7/7.)^ + 1 ^^' [7 - 1 \/l + 6(7/7jd T h 7 (41) Retaining only the essential dependence on 7 we write this equation i87p = C(7/7„)^^' (42) 694 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 2.0 0.5 lo Vi+ b Fig. 2 — The function 7(2) given by equation (40) for two choices of 7^, . 45 40 35 /3V 30 25 20 Sl ^^ 0.01 0.02 0.04 0.06 0.1 0.2 0.4 0.6 0.8 1 I/Io 4 6 8 10 20 40 60 100 Fig. 3 — The voltage-current cliaracteristic of the PIN diode accortling to equation (44). The dashed line represents an ideal PN diode and c/q '^ 200 amp/ THE FORWARD CHARACTERISTIC OF THE PIN DIODE 095 where C is a constant representing the slowly varying coefficient of ' 7„) " in (41). We choose C snch that (42) becomes exact at high cur- I _nt density when ^Vp is large C = -^^ S- (43) 7^ - 1 V& + 1 When we regard the third and fom-th tei-nis of (37) together as a constant jSFc Ave obtain the simplified voltage-current characteristic /3F = fn ^ + C j/^ + 0Vo (44) 0 In this approximation it is unnecessary to evaluate 7(7) from (31). Fig. 3 shows plots of jSV versus I/Io calculated from (44). For plotting the curves the \'alue (' =1.1 was used. To choose a value for 13V c we put 7=1, which gives 1 + fc(7/7oo)- 1 + i^ so that ^Vc -^ HIo/(Ins + Ips)] (4G) which has the value 27 in silicon according to the values in (3). The dot- ted line is the asymptote approached by the curve at low current densities 0V -^ (n ■[ I « h (47) This is the characteristic of a simple PA'' junction Avhen ;Ve retain now to the cjuestion of when the large injection conditions (18) are satisfied. Let us suppose 7 is much less than 7o so that 7 '^ 1, 7„/7;, ^ R. It follows from (30) and (23) that rio ^ n,, ^ ni[I/(Ins + Ips)f'~ (48) Now let us set /?„ » P Avhich gives a condition on the current density I » (P/mY (L,s ^ Ips). (49) Setting /;„„ » Hp'^, Pu-w » Vn'^ gives 7 » 7,„ + Ip, . (50) Usually P » 7ii so that (49) includes (50). When numbers are put in 696 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 from (3) we get the condition for large injection el » 0.07 amp/cm^ in Si (51) Since tliis current in (51) is much less than do , we may quite properly speak of large injection n^ P and small currents / « !„ at the same time. Let us denote by ICM = iP/Uif (Ins + Ips) (52) the current density at which conductivity modulation starts to be im- portant. Then we may distinguish three ranges of current: (a) very small current / < I cm for which large injection analysis does not apply; (b) low current I cm < I < h for which large injection analysis applies, but the voltage drop Vp in the middle region is negligible; (c) large current I > lo for which Vp is sizable. The treatment of this section has covered ranges (b) and (c) . Range (c) (as treated here) does not extend to infinity but only up to current densities of the order L 8 X 10* amp/cm'^ p so that the diffusion currents in the contacts may be treated as a small injection. Small Injection, No Recomhinaiion In this section, we shall cover ranges (a) and (b) in current density. We must go back to the basic equations, but we shall make use of two facts that have come out of the large injection analysis: (a) jSFp is negli- gible when / « /o ; (b) 7 = no/n^ ^ 1 which means n(x) and p(x) are essentially constant in the middle region 0 ^ x ^ w when I <^ lo . When we set no = n^o, Po = Pw (53) equations (16) give us noo = npV^^"^""^ (54) Pww — Pn ^ Then (11) gives I = I„^ I^= (/„, + /,,) [/^'^o+^-Li] (55) Now Vo + Vw is the total applied bias when Vp can be neglected ; there- THE FOKWAKD CHARACTERISTIC OF THE PIN DIODE 697 fore we obtain the characteristic PV = ^n ( \_ + l) (56) which is vaHd until 7 approaches lo . Of course we would not have ob- tained this ideal characteristic of a simple PN junction had w^e taken recombination into account; our result depends upon the constancy of n(x) and pix) in the middle region. For the case of no recombination in the middle region (56) and (44) cover ranges (a), (b) and (c). Instead of (44) the more exact expression (37) could be used requiring the evalu- ation of y{I) from (31). It seems that the extra refinement is of no help in understanding the device and unnecessary in treating experimental data. Therefore, we shall adopt (44) and the approximations leading to it as a model for treating the more complicated recombination case. That is, we shall seek a generalization of (44) which takes recombination into account in a sufficiently good approximation. Large Injection with Recombination We are interested in determining the effect of recombination in the middle region upon the operating characteristics of the device. Therefore we go immediately to the large injection case n = p. Equation (16) be- come ny, = npe^^"" p^w = n„,(pivVpp)e^^"' (57) no = ppe^^'^ noo = no(np'^/np)e^^° t which gives /3(Fo + F„) = (nin^nolnl) (58) We shall assume that recombination is linear in the injected carrier density to simplify the calculation. It will be possible, later to approxi- mate bimolecular recombination by using an appropriate value for the lifetime r corresponding to the injected carrier density. Therefore we write O'i-n ui p n (rj(S\ dx dx T Eliminating In{x) by use of (13) gives the equation for n(x) (60) dn n dx^ L % 698 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 where L is the effective diffusion length in the middle region L = [2Dn r/ib + 1)]^'=^ (61) The solution of (60) may be written no sinh (w — z) + w„, sinh z n{z) sinh CO (62) where z = x/L is the position variable and w = w/L is the length of the middle region in units of L. Fig. 4 shows several of these solutions for the In equation (60) and the solution (62) we have neglected the equi- librium carrier densities Up , pp . The criterion for the validity of this approximation is sinh Hco « (rio/P), inJP) (63) X/w Fig. 4 — Tlie carrier (l(Misi(y accordiiii;- (o ('(|ua1i(>n (iS'l) for the case 7?u and several values of co. = n.u THE FORWARD CHARACTERISTIC OF THE PIN DIODE 699 arrived at by considering the minima in the sokitions for co » 1. This is really a criterion for conductivity modulation, so we shall assume hence- forth that it is satisfied. We now modifv (13) by setting n = p and eliminating E{x) by use of (22) , . . 67 + 2Dnn{x) ^-^'^ = — mh — J , . I — 2Dnn (aO where n'(x) = dn/dx. Inserting these currents into (22) gives E{x) and integrating gives the potential drop Vp in the middle region (6 + 1)D„ Jo n 6+1 no This is the generalization of (26) for linear recombination. The direct evaluation of (58) and (65) in terms of the total current / leads to a very complicated expression for the applied voltage. It will be jshown in the next section that this result reduces in its simplest approxi- mate form retaining only the essential dependence on w to the formula I Jn.9 + ips V io(<^) iwhich is identical with (44) except that the characteristic current density lis a function of oj 7(co) = /o^(w) (67) g{o:>) = cosh - tan ^ f sinh — 1 _^' + - - 6 ^ 48 (Fig. 5 shows a plot of ^(co). These results show that if co < 1 as we might lexpect in a good diode recombination has no significant effect on the jforward voltage-current characteristic in the conductivity modulation range of operation. 700 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 195G 3 l.U ^^ ^ 0.8 \ N. > \ s 0.6 \ s. N \ 0.4 \ N^ N \ 0.2 0 0.5 1.0 1.5 2.0 2.5 3.0 Fig. 5 — The function (j(co) of equation (67). Analysis We denote r = no rii From (11) and (67) /p(co) = Ips^ , ln(0) = Ins^ (68) (69) By means of (62) and (64) Ave eliminate /„ and Ip and obtain the equa- tions (b + l)Ipsf = / - Ir{^ cosh ic - t) (b + l)RIpsf = hi + Iri^ - r cosh co) where Ir is a (particle) current density 2Dnni Ir = L sinh CO (70) (71) In principle we could solve (70) for ^ and f as functions of I with R and oj as parameters; this would determine ^V through (58) and (65) and complete the problem. First we shall rewrite these equations in terms of 7 as in the analysis of the second section. THE FORWARD CHARACTERISTIC OF THE PIN DIODE 701 If we eliminate / from equations (24) we get J . Ir h cosh O) + 1 nr 2 olps + J r+l " ''^ + 7 which can be solved for ^ Ir cosh CO + 6 1 b + 1 (72) y _ Ir cosh CO + ^ To — 7 ^ ~ 7;:. 6 + 1 /?7- - ^ where pi' 70 = 6 cosh cj + 1 cosh CO + 5 Substituting (73) into (70) gives the equation satisfied by y Ry' cosh CO + 1\ , ^ / [(7/700)' - 1]' 7 r> o , — T ) (7 - 70) = 7 Ry"^ + cosh CO / " ' "' /oo ^7^ + cosh co ■where loo is a characteristic (particle) current density (73) (74) (75) J^ 00 — -' o CO sinh CO cosh u -\- h h + 1 (76) Now the solution of (75) has two branches which as / -^ 0 approach N'alues given by a) b) 7 -^ 70 Ry" cosh CO + 1 Ry- + cosh CO (77) As I increases the first branch remains positive and approaches 700 as / — > 00 . The second branch becomes negative and approaches —700 . Therefore, we choose that branch which satisfies 7(0) = 70 = b cosh CO + 1 6+ 1 y(cc) =y^= {h/Rf" 7 > 0 (78) ( )n this Ijranch 7 always lies between 70 and y^ , and 7 never approaches the quantity in (77b). Therefore we replace Ry by b (as if 7 = 700) in II 702 THE HELL SV8TEM TECHNICAL JOUKXAL, MAY 195G the first factor on th(> h ft of (7")), and obtain the siniplei- form 7 - 7(1 /oo y/Ry- + cosh CO (79) which is the generalization of (31). The drop ^Vp in the middle region given by (()o) can be written ^^^ = r^ ^^ ^ + J-^Tx yj;^ V/^y + cosh 0, FM (80) AN'here Fc^ii) comes from / dx/ n and is defined p ( \ _ f Mill Jo 7 siiih fw(l — u)] + sinh [ww] In 7 snm [(jo(] 1 + Q 1 + Q (n 1 + e"Q 1 - e^Q (81) or 2 \/l — 27 cosh CO + 7- tan~' e"Q — tan~^ Q \/27 cosh CO — 1 — 7- The first form applies when 7 > r", or 7 < c~", and the second applies when e"" < 7 < <»", and Q is the (luantity Q = 1 — ye' ■\/\ 1 — 27 cosh CO + 7^ It can readily b(> shown that when co -^ 0 In 7 /'^o(7) = Thus when co = 0 (80) reduces to 7 - 1 (82) (83) /3T> Ch 7 /> — 1 r, (7 - 1) + 7-16 + 2 (n 7 (7/700)' - 1 6+ 1 -^ \//?7- + 1 l/i (84) .7-1 Ry- -{- I which is identical with (il). It is also clear that (7!J) reduces to (31) as the recombination goes to zc ro. f'inally we write from (58) ^(Fo + V„.) = tn yt = (n /- + (n ^ ., / . (85) 1 p, Ry- + cosh CO THE FORWARD CHARACTERISTIC OF THE PIN DIODE 703 2.6 2.0 K 1.5 3 LL 1.0 0.5 s. ^ \ v\ \\ s\^^ \ V^ '^ V X ^ N X h ^ "^ 'v >^ ^<^ ^ ^*-^ ^ ^ ^ ^^ 1 1 1 1 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 4 5 6 8 10 Fig. 6 — The function F„{y) of equation (81) for several values of co. which reduces to (3G) when co = 0. Thus the whole theory reduces cor- rectly in the case w = 0. The function F„(t) is plotted in Fig. 6 for several values of w including 05 = 0. The expansion of F^{y) to order co" is FM = -^ - ifM 7—1 4 (t + 1) - 27 /(t) = (n 7 (86) (7 - 1)-^ 1 = 1 - 27 fn - + • • • 7 i Our next step is to eliminate from (80) and (85) unimportant depen- i dencies on I which would be difficult or impossible to detect experi- I mentally. If in (85) we let 7 = 1, cosh co = 1 we get ^(Fo + F,„) = tn (87) In (80) we drop the first term (as if 7 = 1) and in the second term we 704 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 put By^ = 6 (as if 7 = 7„) and FM = PM, 2 1 ^i ^^ /3Fp = ^-pp 4/ J- V6 + cosh CO FM (88) In this way we retain the correct form of dependence on w, but throw out the dependence on / that comes from 7(/). It can be shown from| (81) that tan ^ f sinh - sinh ~ (89) 2 4 CO CO = 1 - — + — - + 12 ^ 180 ^ Thus we define the characteristic (particle) current density of the device /o(co) = {b + l)7oo (b + cosh co)F„(l)2 (90) and (88) can be written n2 CO _F^(1) sinh co_ = /o ja ^ ,_^ bf X. C rr -a ^- o T1 ^ o rt 1 < fcX UJ () +^ z < ^ 1- 1/1 o Q a III hi 2 a! 1^ ^ MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 715 may also be represented as a function of time. This time-rate-of-change of flux within the coils of the head generates a voltage which is of the order of 50 millivolts peak-to-peak in the case of the translator. This volt- age, after amplification, appears as shown in line C of Fig. 3. The trace shown is that which appears at the "linear output" monitor jack of a translator reading amplifier, and includes a phase inversion, character- istic of a three stage amplifier. Such a curve is readily recognized as being quite similar in shape to the first derivative of the normal error-function and hence we may infer that the magnetic condition of the drum surface, at least as interpreted by the head, may be portrayed by a bell-shaped curve, previously mentioned, similar to the error-function itself. The residual magnetic irregularity pictured in cell 4 resulting from writing a "0" over a "i" will induce a voltage in the winding of the head having a different amplitude and wave shape from that occasioned by reading a "i." It is sketched out approximately to scale in Fig. 3 and is seen to be a smaller twinned- version of the "i" signal. Its amplitude or- dinarily lies in the range of }4o to 3^ of that of the " T' signal, and for about the middle third of the cell its instantaneous polarity is opposite to that which a "i" signal would have. These facts suggest at least two means of discriminating between the voltage signals obtained for the two code values: (a) on the basis of amplitude difference, and (b) on the basis of instantaneous polarity difference determined or sampled within a particular epoch in each cell. The method adopted for the translator is that of simple amplitude threshold. The threshold value indicated by the dotted line in Fig. 3, is set so that the strongest of the residual signal outputs never exceeds it \\ hile, at the same time, the greatest possible proportion of the positive- going lobe of a " 1" signal is allowed tp produce an output. The threshold output stage of the amplifier is also arranged for limiting and this has the effect of blunting the peaks of the applied signals. The over-all result of these actions is shown by the shape of the signals in line D of Fig. 3. Cell packing may be of major economic importance in a large installa- tion. The general effect of making recordhigs closer and closer together is that the presence or absence of one of the recordings in a series has an increasing influence on the size and shape of the signals reproduced from its neighbors on either side. In the translator, the cells are spaced "JO niilli-inches center-to-center along the track and the influence of action in one cell on the amplitude of reproduction from neighboring cells is never more than about 10 per cent. The trace of line C, Fig. 3, is drawn for this cell spacing and shows a slight inflection at the transition between the output voltage occasioned by reading cell 3, and the voltage obtained 716 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 from the "^" which was originally written in cell 4. In many applica tions a much larger "influence factor" may be tolerable, but this usually] requires greater elaboration of the signal detecting devices. The cell size is also influenced by physical constants such as design of the head, prop- erties of the medium, and dimensional clearances. A discussion of suchj factors is outside the scope of this paper but it is not unreasonable to i hope for an improvement of two-to-one in packing factor in future de-» signs. Reading Synchronization The magnetic drum used for the translator provides 80 tracks. About sixteen microseconds is required for each cell in a track to pass under its head. Information occupying the same slot on the drum (so-called be- J cause of its obvious relationship to the term "time-slot" commonly used in the digital computer field) is presented at the various heads essenti- ally, but not exactly, simultaneously. Departure from exact simultaneity is occasioned by small variations in the shapes and amplitudes of the output waves shown typically as line C in Fig. 3, and by small time-vari- ations occurring in the writing process, as applied to the ^^arious tracks. To achieve exact simultaneity, as required for certain subsequent op- erations of the translator circuitry, narrow "Read Synchronizing" pulses are produced by the synchronizing circuit previously mentioned. These pulses are located, within the time boundaries of the cells, so that they fall approximately at the center of the broad output pulses from the reading amplifiers and thus permit the latter to be sampled. This rela- tionship is indicated in lines D'and E of Fig. 3. Similar pulses, slightly displaced in time, are used to control the writing operations, and are des- ignated "Write Synchronizing" pulses. The necessity for the time-shift is apparent from an examination of lines A and E of Fig. 3. This condensed explanation of the technology of magnetic drum digi- tal data storage devices, particularly as applied to the translator drum, should serve as sufficient background for the description of the translator wherein the drum is but one part of a large ensemble of apparatus. THE JOH WHICH THE CAKD TRANSLATOR NOW DOES It will be advantageous to examine very briefly the card translator and its functions in the No. 4A toll switching system so that the analogous operation of the magnetic di'um equivalent may be more readil,y ex- plained. A more detailed description is given in Reference 4. 'l'li(> (I'Munnds of nation wid(> toll dialing rcniuire a \'ery extonsi\-c vvp- MAGNETIC DRUM TRANSLATOR FOR TOLL SAVITCHING OFFICES 717 ertoire of translations between destination codes and routing instruc- tions, and it must be possible to change the routing instructions with ease. The card translator fulfills these requirements. Each individual translation item is contained on a metallic card; the output code of rout- ing instructions is in the form of selectively enlarged perforations in the perforated field of the card, arranged so as to be read by photoelectric means, and the input code, which identifies the card for purposes of selec- tion, appears in the form of tabs projecting downward from the bottom edge. Each card is capable of holding a total of 154 bits of information, input and output, and somewhat over 1,000 cards are stacked in a bin in each card translator mechanism. It is possible to classify the elements of any translator into three broad categories: the memory unit, the translation selecting unit, and the trans- lation delivery unit. In the card translator the memory unit is, of course, I the group of cards; the translation selecting unit consists of code bars, ' electro-mechanically actuated, for displacing a selected card sufficiently ' so that it may be "read." It also contains a network of relays which per- form the function of checking the authenticity of the input codes applied I to the code bars. The translation delivery unit consists, in the main, of a number of output channels, each originating with a light beam for prob- I iiig one of the code elements (a bit of output information) on the card. [ Each output channel contains a photo-transistor, a transistor amplifier, a cold cathode gas tube circuit which has been designated a "channel output detector" and a register relay. The register relays perform work ' functions and therefore are located separately from the translator; some are in the decoders, others in the markers. ! In the 4A office, the card translator is one of several items of common I control ecjuipment which cooperate to establish the talking connections. ( )ther items are the sender, the decoder, and the marker. The sender re- I ceives and registers and subsequently transmits the decimal digits of the ! called designation; the decoder receives the code digits (from 3 to 0 in ' number) from the sender and submits them to the translator for con- ; \ersion into information needed for the proper routing of the call; and 1 the marker selects an outgoing trunk and establishes a transmission path by operating the crossbar switches. Since this common control equipment is associated with any one call for only the short interval necessary to j establish the talking-circuit connection, its speed of operation is a matter of considerable importance. It is ob\^ious that the decoder is the intermediary between the trans- I lator and the remainder of the office. Each decoder, of which there are a maximum of 18 in a large office, has exclusively associated with itself a 718 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 card translator mechanism ; each of these mechanisms contains an identi- cal repertory of translations. Each decoder also has available, through connectors, a common pool of translators containing a large quantity of less-often used information. In order to better understand the duties that a magnetic drum translator must be expected to perform it will now be convenient to follow, in a highly abbreviated manner, a typical opera- tion of the decoder and its associated card translator. The first translation on an incoming call is performed using the first three decimal digits accumulated by a sender. As soon as three digits are available the sender connects to a decoder which immediately signals its individual translator to perform certain mechanical chores in preparation for selecting a card. There are several sequencing signals between the de- coder and translator during the complete cycle of a translation (several of these signals must be synthesized by the drum translator); acting on one of these signals from the translator, the decoder passes the input code from the sender, adding certain supplemental information of its own. The three decimal digits of the input code are in checkable combina- tions of two leads energized in each of three groups of five leads connected to the translator. The supplementary information supplied by the de- coder is in a similar checkable combination on six leads. None of the re- maining leads in the total of 38 is energized, since the translation being described involves only three code digits. In the translator, the input code actuates the card selecting mechanism and also operates relays whose contacts are wired with a checking net- work which confirms that the input code, and the responsive operation of the code bars, is an authentic combination. This is done by establish- ing a path to operate a "code bar check" relay, cbk. (This relay retains the same identity in the magnetic drum translator.) Acting upon the authenticity check, the card translator proceeds to select a card, and signals the decoder to begin timing for a possible non- appearance. When the card is in a position to be read, the decoder is sig- naled on two "index" channels, ind. The decoder now "reads" the card by applying 130 volt battery to the coils of its register relays; the re- quired relays operate through the ionized cold-cathode gas tubes in the translator, and lock up, extinguishing the gas tubes. '{'he first card dropped may provide information sufficient for complet- ing the connection; in this circumstance the decoder will then call in a marker. The first card, however, may specify that more digits are re- quired and the decoder will so instruct the sender. The sender, unless it already has the necessary digits, is then dismissed by the decoder which also instructs the translator to restore itself to normal. MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 719 Six-digit translations are obtained in a manner similar to that des- cribed above except that the checking network on the relays is switched to check for six rather than three digits. In some instances the decoder must refer to one of the translators in the common pool of "foreign area translators" in order to obtain the reciuired information. Frequently, sev- eral different cards must be dropped successively before a route is finally established for the outgoing call. With the above description as a background, we may proceed to discuss the magnetic drum translator. THE ANALOGOUS FUNCTIONS OF THE MAGNETIC DRUM TRANSLATOR The magnetic drum translator is essentially a device which performs a translation by making a selection from a recurrent pattern of electrical pulses generated by a magnetic drum unit. A schematic diagram of the magnetic drum translator, as arranged for direct substitution for a card translator, is shown in Fig. 4. In this diagram, the system is divided into three principal functional components: (a) the drum memory assembly which produces (from the outputs of 80 reading amplifiers and a timing unit) a repetitive pattern of electrical pulses representing all the transla- tions on the drum, both input codes and corresponding output codes; (b) the translation selecting unit which reads that portion of the pulse pat- tern representing input codes and acts to identify the unique code group which matches the incoming information from the decoder ; (c) the trans- lation delivery unit which, under control of the translation selecting unit, gates-out the particular pulses of the corresponding output code from the continuous stream of microsecond pulses, and converts them into signals capable of operating the register relays in the decoder. To maintain direct interchangeability, two items of apparatus were adopted virtually without change from the card translator. These are the (ODE CHECK RELAYS which accept and check input information, and the CHANNEL OUTPUT DETECTORS comprising cold-cathodc gas tubes and as- sociated transformers. This allows input and output terminal facilities to the decoder to be the same for both translators. It should be noted that the magnetic drum memory assembly differs significantly in one functional respect from the binful of cards in the card translator. When a selected card is being read by the photo-electric cells in the output channels, no other cards are available. In the drum trans- lator, all translations are continuously available and if a number of trans- lation selecting and translation delivery circuits are employed, all may obtain translations from a common drum memory assembly at the same time without interference. This feature could not be demonstrated in the 720 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1950 test set-iip as planned, but il would have been incorporated in any test which includcnl more than one decoder in an office. In such an arrange- ment, the various units illustrated in Fig. 2, except the drum memory assembly-, would be furnished to each decoder. One drum memory assem- l)ly (;iii(l lui (Miici-gency standby) would supply Ihc pattei'n of electrical MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 721 pulses to all translation selecting and translation delivery circuits in mul- tiple. The object of such an arrangement, naturally, is to employ the mag- netic drum system in the most economical manner. A further extension along the same lines would involve relay switching of the pulse circuits TRANSLATION SELECTING UNIT "n MATCH UNIT NO.t MATCH UNIT — NO. 8 I — T — 1 ' I f f I t ♦ T T L AND -GATE AND -GATE AND-GATE 'A" PULSE GENERATOR -MATCH PULSE CBKM DISABLE ^~j»j BIAS SLOT- SPANNING MEMORY 3_J AND-GATE "B" PULSE GENERATOR CODE CHECK RELAYS H 0 K> X T I I I INPUT CODE CHECKING NETWORK I I H X T X CBK 1 I IND B ^ i t X si U. DC oS tr block diagram. 722 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 to give access to the emergency drum memory, or to a "foreign area" memory where such extra memory capacity is necessary. Let us now return to the discussion of Fig. 4 and consider the assign- ment of the translation information to the drum surface where it is stored. Recall that the drum surface is effectively divided into a grid by the co- ordinates of tracks, each passing under an individual write-read magnetic head, and "slots," each defined by the appearance of a timing pulse in a rhythmic train synchronized from the drum itself, and that the "cells," at the coordinate intersections, each accommodate one bit of code infor- mation. Since each card in the card translator accommodates 38 bits of input | code and 116 bits of output, about 160 cells, divided in the ratio of one cell for input to every three cells for output, must be assigned to each i translation item. One simple and direct assignment would be to place the entire translation item in a single slot composed of 160 cells. With i this layout the slot containing the desired translation would be identi- fied by reading, or "matching" the input code, and during this same in- 1 terval the output information in the same slot would be gated-out to the I translation delivery circuits. A 1 ,000-translation drum would then be' long and narrow, and far too many reading amplifiers would be required. ' Another evident arrangement would be to assign the entire input code: to the first of each group of four slots proceeding under the heads, with the output code following in the next three slots. Such an allocation would require only 40 reading amplifiers but the drum necessary for the desired capacity, with the cell-spacing chosen, would have been larger' in diameter than the mechanical designers cared to undertake in their first trial. A logical choice, therefore, was to place each translation item in a pair of adjacent slots, and this was done, although it was later recog- nized that other, more sophisticated, arrangements might offer eertainj advantages. In Fig. 4, the apparent location of one translation item is sketched inj relation to the drum surface. This sketch is not drawn to scale, since thef slot width is actually only 0.020 inch, and the track width is comparable. It is also geographically inaccurate; actually the cells of any one slot arei positioned in four quadrants on the drum, the associated heads being! positioned in four stacks for mechanical reasons. However, all of the cells in a time slot pass under all of the heads at the same instant and the| presentation of Fig. 4 was adopted for the sake of clarity. Note, then, that the input code and one-third of the output code arel recorded in the first or a slot of a slot-pair passing undcn- the reading^ heads, and that the remaining two-thirds of the output code occupies MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 723 the B slot which immediately follows. The parallel (simultaneous) pres- entation of the entire input code to the translation selecting unit permits that unit to indicate, by a pulse, that the translation item is the one de- sired and to gate-out the output code in the same slot while it is still passing under the heads. Having thus identified the first slot of a trans- lation item, it is a simple matter to pro\'ide the facilit}^ for gating-out the remaining information recorded in the next succeeding slot. It will be seen, from the circuit arrangement shown, that the transla- tion selecting unit also receives a portion of the output code recorded in the second slot of each pair. It is therefore necessary to distinguish be- tween the A and b slots of a pair. This is most conveniently done by the Timing Unit, which is provided with two outputs, the pulses defining the slots appearing alternately at these outputs. One output lead is cho- sen to define all the a slots and it is routed to the translation selecting unit to provide a portion of the pulse-pattern required for complete and proper identification of an input code. The action of the magnetic drum translator in making a translation may now be traced by following the block diagram of Fig. 4. The decoder, of course, gives the same preliminary signals as for the card translator, but these are ignored by the drum translator, because it is continuously presenting all 1024 translations at the rate of 30,000 per second and need not take any preparatory steps, provided its relays have returned to normal after the last translation. The normal state of the relays is checked by means of a circuit through their contacts; if this circuit is complete, the decoder receives the signal to apply the input code as soon as it seizes the translator. A more elaborate checking arrangement could } have made this signal conditional upon other tests, such as a "standard 1 translation," to determine that the electronic circuitry (in bulk) was functioning properly, but it was not considered worthwhile to do so in I the system described here. 1 The decoder, then, furnishes the input code of the desired translation 1 item, causing certain of the relays labeled code check relays in Fig. 4 ! to operate. Contacts on these relays are interwired to provide the same checking network as in the card translator, and a check on the authen- ticity of the input code will be evidenced by operation of the relay labeled CBK. This event is signaled to the decoder so that it may start its "no- I card" timer action. When cbk closes, it also operates a chatter-free mer- cury-contact relay, cbkm, in the translation selecting unit, permitting that unit to produce an output at the appropriate time. Each code-check relay which operates applies a positive voltage to one of the input ter- minals of a "match" unit in the translation selecting unit. For each of 724 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 these input terminal there is a complementary terminal to which are applied negative-going pulses from one of the drum memory reading am- plifiers. As will be explained later, advantage is taken of this comple- mentary arrangement to obtain a signal indicating a match between either, (1) an operated code relay and a pulse from the reading amplifier, or (2) a nonoperated relay and no pulse from the reading amplifier. All of these signals, from 40 sections of the match units, are combined in a cascade of "and" gates; when all indicate a match, the translation se- lecting unit delivers an output "match" pulse. Since this match pulse is not strong enough to enable 40 gates in the output channels, it is passed to a "pulse generator" (a regenerative pulse repeater) which produces, virtually coincident in time, a powerful "a" gate-opening pulse. Note that both the "a" and the similar "b" pulse generators are enabled to operate only when the input code is authentic, as evidenced by the operated code check relay cbkm. In an unrestricted magnetic drum translator design this identifying pulse would cause immediate registry of part of the desired information. Here, however, is evidenced one of the penalties for having a direct one- for-one substitution for a card translator. The decoder and card transla- tor function in a definite sequence; one of the steps in this sequence is initiated by the ind signal from the translator which informs the decoder that the selected card is properly "indexed" so that it may be "read." Therefore, in the case of the drum translator, to preserve this sequence, the selected translation is permitted to pass unheeded, except that the IND signal is synthesized from the identifying b gate-opening pulse. This operation closes one relay, indb, through a special output channel (top- most one in Fig. 4) provided for the purpose. The decoder, thus notified that the desired translation is available, applies battery to its register relays, and the output channels are completely enabled for a subsequent registry of the desired information. The output information is usually registered during its next passage, one drum-revolution after initial identification of the item. The action of identifying the translation is again as described above, and there remains only to follow the operation in the output channels. E\'en before the translation selecting unit has initiated the identifying gate-opening pulse, reading amplifiers which are required to deliver an output code have each commenced delivery of a pulse to their corresponding gate terminals in the and gate and pulse stretcher units. (See Fig. 4). When these pulse signals have reached a stable maximum, the gate-opening pulse (a or b depending on the slot which is being read at the moment) is free to pass through the gates and to trigger the pulse stretchers. The , MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 725 latter devices, each containing a single transistor in a monostable circuit arrangement, deliver 12-volt pulses lasting about a millisecond. The pulse stretchers from which an output code is not required are not trig- gered, owing to the absence of pulses from the corresponding reading amplifiers. The remainder of the output channel, as previously stated, is borrowed directly from the card translator, and the action is similar. In the output detector, a transformer steps-up the 12-volt pulse signal to a voltage more than sufficient to establish a discharge in the control gap of a cold-cathode gas tube. Since the decoder has applied voltage through a relay coil to the main gap, the discharge transfers, and the resultant current flow operates the relay. The operated relay, which may be in the decoder, registers the code and locks to ground through an auxiliary contact. This ' action also extinguishes the gas tube, thereby extending its life. : Except for relay operation, all of the activity described here for two drum revolutions repeats itself for every subsequent drum revolution I for as long as the code check relay cbkm remains operated. However, ! once the code is registered, no further use is made of the pulses in the output channels. When the decoder has made use of the translation, it transmits a sig- I lull which is used in the code-check relay system to indicate when all re- ;lays are properly restored. In the card translator this signal is also used to restore the selected card, but in the drum translator this operation, of course, is not required. . idministration Equipment I To utilize the magnetic drum translator as described above, it is obvi- ous that some means for writing-in the translations is as necessary to t he drum as a card punch is to the card translator. Although a selective S writing, or "Administration Unit" was required, a highly efficient design \\ as not essential to the experiment. Consequently there was constructed a separate, portable aggregation of essential basic electronic circuits, ; arranged for manual control, but designed with a view to possible ex- 'leusion to fully automatic operation. This equipment will be described ill a later section. I QIJIPMENT AND CIRCUIT DESIGN DETAILS OF THE TRANSLATOR (nticrdl Description 'i'lie entire translator is mounted on an 11-foot by 32-inch bay and has licen made to conform to telephone central office practices as far as pos- 726 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Fici. 5 — FvowtM- easing containing ])artial complement of reading amplifiers, liming unit, tilani(Mit transformers and hlowers. Koceptacle at right end of (>acli amidificr mounting strip allows Administration unit to connect directl}' to mag- netic heads associated with those amplifiers. sible; except for the presence of the drum unit at the base of the i-ack, its appearance is not unhke that of other racks found in central offices. Mounted directly above the drum unit is a casing of conventional de- sign (shown open in Fig. 5) which houses the reading amplifiers, timing unit, filament transformers, and a self-contained forced-air ventilating MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 727 Fig. 6 — Upper casing containing translation selecting unit, and partial com- plement of pulse stretchers and channel detectors. system. A second casing, (Fig. 6), located directly above the first, houses the translation selecting unit, pulse stretchers, and channel output detec- tors. The various plug-in components used in these sections are shown in Fig. 7. At the top of the rack are located the code-check input relays, fuses and terminal blocks. 728 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 195G r^ 1 -=5. ■^^.J /^ Fig. 7 — Plug-in units. Left to right, reading amplifier, match unit varistor cluster, individual varistor, match and-gate, transistor, and pulse stretcher. In wiring the rack, use of individually-shielded conductors was held to a minimum. The cable between the drum unit and the reading ampli- fiers was composed of standard switchboard wire, shielded as a unit by removable sheet-metal enclosures, thus greatly reducing the bulk as com- pared to the usual bundle of coaxial cables. The remainder of the wiring, which carries relatively high-level signals from unit to unit within the frame was also in the form of cables of switch- board wire; this type of wiring was tried as an experiment for micro- second pulse work, and was found to be successful in this instance. Under normal conditions the entire translator, with the exception of the tube filaments and drum drive motor, operates from the standard plant batteries of +130 and —48 volts. Commercial 60-cycle power is normally used for filaments and motor; the motor is duplex and is de- signed to transfer automatically to the 48-volt plant battery in case of' power failure, and the same provision would have to be made for the filaments in the event of a telephone plant installation. i Magnetic Drum Unit \ The magnetic drum unit is located at the bottom of the rack, as shown i in Fig. 1; a close-up view with one of the covers removed is shown in Fig. 8. A mounting casting supports the machine directly on the floor, straddling the lower member of the rack so that no load is imposed on the rack structure. The drum rotates about a vertical axis and is housed in two cast-iron end-bells spaced by a cast-iron shell. The end-bells carryi^ the bearings for the drum, and serve to mount the motor, while the shell- 1 casting rigidly locates the magnetic heads, each very close to the drum surface. This design requires a minimum of floor space, insures accurate bearing alignment, provides a convenient location for the magnetic heads, and permits the use of tightly-fitting gasketed covers to exclude i MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 729 Fig. 8 — Magnetic diiun unit pnrllN- uncovered to show magnetic iieads and wiring terminals. dirt iiiid foreign material from the magnetic drum surface and the bear- ings. The Ke-hp motor dri\'es the drum through a spring-diaphragm coupling. The drum is comprised of a stress-reheved iron casting of high dimen- -^ional stabiHty, a press-fitted steel shaft, and a ^^ie" thick brass outer , sliell which carries the magnetic recording medium. Since both drum and 730 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 housing are of similar materials, and have almost identical temperature- expansion coefficients, it is expected that pole-tip-to-drum clearance will remain unchanged under normal conditions of service. The drum, which is 12.8" in diameter, 10" long, and weighs 150 pounds, is dynamically balanced and runs without sensible vibration. Commercial super-precision angular-contact ball bearings, two at each end, are used to mount the drum in its housing. The lower bearings are arranged to share the thrust load imposed by the weight of the drum, and the upper bearings are mounted opposing each other, and are pre- loaded one against the other. The upper bearings serve only as radial constraints, the outer races being free to move axially. This type of con- struction results in a finished unit having a total runout of only a few ten-thousandths of an inch without the necessity of machining the drum on its own bearings. For the experimental installation, the bearings were grease-packed at assembly and can be expected to function satisfactorily during any reasonable test period. If, however, such a drum unit were made a permanent part of the telephone plant, other provisions have been considered which wovdd insure adequate lubrication over a much more extended period. The magnetic coating used on the drum is an electro-deposited alloy of cobalt and nickel (90 per cent Co-10 per cent Ni) approximately 0.0003" thick. This coating was selected because of its hardness, strength, uniformity, and desirable magnetic characteristics. The thickness of the coating is such as to result in a satisfactory cell-size without undue sacri- fice in output. The purpose of the brass sleeve mentioned previously is to form a nonmagnetic surface between the magnetic coating and the cast-iron core since, if the coating were applied directly to a ferro-mag- netic material, its effectiveness would be greatly reduced by the shunting effect of the base material. The brass sleeve also serves to facilitate plat- ing the drum, since brass, unlike cast-iron, is amenable to the electro- plating process. Read-Write Heads One of the read-write heads is shown in Fig. 9. The magnetic structured consists of three rectangular bars of laminated material, arranged in theij form of a triangle (as schematically represented in Fig. 2). Two legs of j this triangle carry single-layer coils which are series-connected. These; two legs also serve as pole-tips, being pointed at the end and separated ( by an air gap. The third leg sorx-es to complete the magnetic circuit and, f in assembly, is butted tightly against the other members by means of a-, Icafspring. MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 731 Fig. 9 — Magnetic head and mounting bracket showing means of adjustment. I The magnetic structure is assembled on a nickel-silver plate to which have been soldered two copper shoes which serve to locate the pole pieces and shield the pole-tips, thereby focusing the recording flux to some de- gree. After adjustment of the pole-tips, the assembly is clamped in a I sandwich by means of a second, smaller nickel-silver plate. As is evident i from the illustration, this magnetic assembly is in turn assembled to a mounting bracket which contains facilities for precisely adjusting the clearance between pole-tips and drum surface. The pole- tips of the head are 0.050" wide and the tracks are on 0.10'' centers, leaving a nominal value of 0.050" between tracks to allow for misalignment of heads and for flux-spreading. Heads which are physi- i cally adjacent in each of the four corner stacks are mounted on 0.40" centers, but the stacks are offset with respect to one another, thereby interlacing the tracks on the drum. The read-write heads have been designed expressly for use in liigh- speed digital recording. Very thin laminations are used and this, coupled with carefully prescribed manufacturing techniques, results in a head having a satisfactory frequency response for the very short pulses em- ployed. When used as a transducer to convert electrical pulses to mag- 732 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 | i netic flux, it is capable of responding faithfully to frequencies approach- ing ten megacycles per second. j The Timing Wheels and Associated Heads | The synchronizing pulses derived from the drum originate from aj ol2-tooth soft-steel gear mounted at the top end of the drum. In com-^'i bination with a polarized reproducing head, the gear generates a timing i signal which proA'ides means for permanently locating the various cells ' used to store information on the drum surface. The polarized head differs from those used on the drum proper, being of a form which is conven- tional in tone-generators where, as in this instance, a sinusoidal output is desired. | A second gear is mounted at the bottom of the drum, carrying a single \ tooth of the same proportions as the teeth on the upper gear. In combina- ! tion with a polarized reproducing head, otherwise quite similar to those used on the drum proper, this single tooth provides a signal once per rev- olution of the drum which (as will be shown later) is necessary for the operation of the administration unit. The Reading Am-plifier One of the 80 plug-in reading amplifiers is pictured at the far left in Fig. 7. It employs two twin-triode vacuum tubes, and consists of a three- :^ stage ac-coupled linear broad-band feedback amplifier, followed by aj threshold output stage. As shown in the circuit schematic of Fig. 10, the two halves of vi and] the left-hand half of V2 constitute the linear broad-band amplifier. A suitable choice of coupling elements insures that the amplification ^^ill| diminish, with decreasing frequency, at a controlled rate for frequenciesj below a few hundred cycles per second. It is unnecessary to provide am- plification at low frequencies, since the signals to be handled have noJ low-frequency components, and it is undesirable to do so from the stand- point of hum pickup. There is about 20db of feedback in the important part of the frequency range and the amplifier is thus substantially sta- bilized against variations of gain due to change in operating voltages and aging of tubes. The over-all operating voltage gain of the linear stages, with feedback, is about 56 db; the 3 db points are approximately 300 c/sec and 700 kc/sec. The grid of the fourth stage of the reading amplifier is coupled to the output of the linear amplifier and is biased to about twice the plate-cur- rent cut-off value. The output signal from the plate of this stage, occa- MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 733 sioned by reading a "1", will be a negative-going pulse of approximately 40-volt amplitude from a standing potential equal to the plate supply, + 130 volts. As a precaution against false signals, an externally-mounted plate-feed resistor is provided to establish at the output a condition cor- responding to that of no signal present when the amplifier is removed from its receptacle. Timing Unit The timing unit accepts an approximately sinusoidal timing-wave sig- nal from the upper timing head, and converts this signal into two pulse- trains, each having 1,024 narrow pulses per drum revolution, designated as A sync and b sync, alternating in time and available on separate out- puts for controlling all the rest of the circuit action of the translator. A block-schematic indicating how the pulse trains are produced is shown in Fig. 1 1 . The general procedure for converting from a sine-wave to a synchro- nous train of short pulses, two per cycle of input, may be traced through the upper channel of the drawing. The signal, as represented by voltage trace 1, is amplified and clipped until a steep-sided square wave is ob- tained; this wave, trace 2, is applied to a push-pull phase inverter from which a pair of oppositely-phased outputs is obtained. Each of the two outputs is then differentiated by means of an r-c network, and the nega- + 130V ^pvw INPUT FROM HEAD OUTPUT A A A -' ^S"^ Fig. 10 — Reading am])lirior circuit. 734 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 s o3 1 a MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 735 tive-goiug spikes, traces 3 and 4, are combined in a negative-going or gate of crystal diodes. These spikes, trace 5, are used to trigger a cathode-coupled single-shot multivibrator, designed to give a rectangular pulse of about one micro- second duration. The multivibrator drives a pair of identical out- put stages: one furnishes the recjuired a sync pulses to other equipment in the translator bay, and the other delivers its output to a coaxial con- nector so that, when required, the pulses may be furnished to the admin- istration unit. I The B sync pulse-train is produced in the lower channel shown in Fig. 11. After some linear amplification, a part of the original input sine-wave is applied to a vacuum tube integrator circuit. The constants of the inte- grator are such that it provides very nearly a quarter-period of phase shift even if the drum varies from its nominal speed. The output of the integrator is then treated in the same manner as that described for the direct input, with the result that the required b sync pulses are produced. The timing unit also contains a third channel which accepts the once- per-revolution signal from the special head adjacent to the single-tooth wheel. The output of this channel provides the fiducial signal, on a low- impedance basis, for administrative operations. The Translation Selecting Unit This unit, which appears as the bottom panel in the photograph. Fig. 6, performs a number of successive steps in making its selection. These are: (1) recognition of a match between input information from a decoder .seeking a translation, and the unique corresponding information from .the drum, selected from the flow of continuously-presented information; '(2) production of a gate-opening pulse whose leading edge is substanti- ally coincident in time with the leading edge of the particular a sync 1 pulse corresponding to the entry for which the match occurred; (3) acti- \ation of a slot-spanning pulse circuit to bridge the time interval until jthe next-following b slot; (4) production, at a separate output, of another igate-opening pulse whose leading edge is substantially coincident in time with the leading edge of the identified b sync pulse. These actions will now be considered individually. (1) Recognition of Match Responsibility for this function is divided among a group of eight match-units operating with their associated differential amplifiers. Each match-unit is capable of comparing the inputs from five code-relays with the potentially-matching outputs of five reading amplifiers. A circuit schematic of one of the units, with its associated differential 736 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 amplifier and some of the connected apparatus, is shown in Fig. 12. The j uppermost channel on this diagram is typical of all five channels. Re- sistors Ri to R5 are proportioned so that the potential at point c assumes a value of +115 volts for either of the two acceptable conditions of match: (1) code-relay unoperated and reading amplifier not drawing plate current, or (2) code relay operated and reading amplifier drawing a pulse of plate current. Whenever either of the two possible conditions of mismatch exists, the potential at point c assumes a value about 15 volts higher or lower, depending on the nature of the mismatch. Resistor r6 is introduced for protective purposes only. Varistor vri limits the nega- +130V! CODE-CHECK RELAYS Fig. 12 — Match unit and differential amplifier circuit. MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 737 tive voltage excursion at point b, during a pulse, so that it never goes below +105 volts. This establishes the uniform pulse amplitude among Ithe forty match channels which is necessary for proper functioning of the unit. To detect and recognize the voltage conditions at the five junction points, two varistor gates and a differential amplifier are employed. One gate, comprising six varistors including vr6 and VRii, will transmit the type of mismatch signal which is more positive than +115 volts. This signal is dc-coupled to the left-hand grid of the differential amplifier as illustrated in Fig. 12. The type of mismatch signal which is less positive than +115 volts is blocked by this gate but is transmitted through the other gate to the right-hand grid. The threshold for this discriminating action is established by application of a fixed nominal potential of +115 volts to varistors VRii and vri2. At match, the output of each of the two gates presents a potential of + 115 volts to the differential amplifier. The differential amplifier is bi- ased (by inequality of r7 and rs) so that for this condition the right- hand triode is conducting, and the output potential is lower than the plate supply voltage. Positive-going mismatch signals on the left-hand [grid, or negative-going signals on the right-hand grid are then equally jeffective in cutting oft' the right-hand triode, causing the output voltage to rise to plate supply potential signifying a mismatch. The outputs from the differential amplifiers of the eight match units are combined with the a sync pulses in a system of and gates, as illus- trated in Fig. 4. A match-pulse output from this system thus signifies that conditions for match have been uniquely determined for 40 pairs [of items. Thus the match unit, in total, is capable of distinguishing be- jtween all binary combinations of 40 bits or approximately 10'^ items al- I though when a self-checking code is employed, as in the translator appli- cation, many of these combinations are inadmissible. (2) The A Gate-Opening Pulse Occurrence of the match-pulse, as just described, indicates that the 40 items constituting one-half the contents of one of the a slots match the incoming input code; it is then desired to spill out from the other half of this same a slot the information which is also appearing at ampli- Ifier outputs at that instant. This is done by means of gates opened by the action of a gate-opening pulse, triggered by the match pulse. The a gate-opening pulse is only a few microseconds in duration and normally is produced only once per revolution of the drum; a quiescent [blocking-oscillator was chosen as the type of circuit best suited for this [purpose. Whenever the code-check relays are operated in an authentic 738 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 code combination, relay cbkm is operated, ^emo^•ing a disabling bias from the driver stage of the blocking oscillator. When in this condition, each occurrence of the match pulse will trigger the blocking oscillator, thereby producing the a gate-opening pulse once per drum revolution. (3) Slot-Spanning Pidser Whene^'er an a gate-opening pulse has acted to permit read-out of i information from half of the proper a slot, it is also desired to read out ; all the information from the next-following b slot. The first step toward : doing this is to cause the a gate-opening pulse to trigger a single-shot , multixibrator whose characteristic period is long enough to just bridge i the time until the next slot appears. The output of this pulser is combined I with the B sync pulses in an and gate so that the selected b pulse, cor- i responding to the wanted b slot, can be used to trigger another gate- . opening blocking-oscillator just as the match pulse was used to trigger t the A gate-opening blocking-oscillator. j (4) The B Gate-Opening Pulse | The outputs of all the reading amplifiers must be gated for the b slot. | Hence the b gate-opening pulse must operate twice as many gates as the . A gate-opening pulse and must be correspondingly more powerful. This ■ requirement is met by using the same circuit design with parallel output tubes. Pulse Stretchers and Channel Detectors Fig. 13 presents a simplified schematic of one of the translator output channels, together with certain of the relays in the decoder. Package-wise, the pulse stretchers combine two functions: that of an and gate with two inputs and a threshold feature, and that of a single-shot multivibrator for amplifying and lengthening the short input pulse from the gate. A single point-contact transistor provides the necessary gain for the monostable action. The inputs to the and gate come from sources which supply nega- tive-going pulses from a standing potential of +130 volts. When one or the other, but not both, of these sources supphes a pulse, a larger portion i of the current being supplied to resistor ri must be drawn from the non- active source; this extra demand causes a small \oltage drop which be- comes evident at the gate output. The resultant weak false signal is pre- vented from affecting the transistor pulser by the action of threshold diode VRi which is normally back-biased a few volts by the potential di- \-ider r2, r3. Small negative-going signals from the gate will not over- come the bias and will therefore be greatly attenuated; normal gate-out- put pulses, occasioned by coincidence of pulses at both inputs will, MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 739 however, overcome the bias and will be transmitted to the transistor monostable circuit. When triggered at the base, the transistor delivers a pulse of about one millisecond duration to the load represented by the input transformer and the channel detector gas tube and thus provides the drive required to initiate ionization in the control gap of the gas tube. When brought into action, the transistor serves as a switch to connect capacitor c to collector supply resistor r6. The voltage change, occasioned by the re- sultant flow of current in r6, is communicated to the transformer primary through a blocking capacitor and a current limiting resistor. As capacitor c charges, the voltage at the transistor emitter will approach the collector supply potential at an approximately exponential rate. When the di- minishing flow of emitter current can no longer maintain the transistor in its low-impedance mode, it reverts to its pre-triggered condition, and the timing capacitor c is then discharged, primarily through forward- conducting varistor vr2 and resistors r5 and r4. Owing to the necessity of using early-production samples of the type of point-contact transistor chosen for this application, the associated circuitry for biasing the emitter into the normal non-conducting state is somewhat more elaborate than that which might have sufficed with later samples whose characteristics were more closely controlled. The principal components of the channel detector are a step-up trans- PULSE STRETCHER CHANNEL DETECTOR ASSOCIATED EQUIPMENT IN DECODER OR MARKER INPUT Fig. 13 — Pulse stretcher and channel detector circuit, i Fig. 14 — Administration unit. Three eo-ax leads entering under shelf bringi] A, B and F pidses from translator. Cable leading to plug with bail-handle resting; on shelf serves to connect writing amplifier output to magnetic heads in translatori| Bottom cable connects to 60-cycle source which supplies all power, 740 MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 741 former designed for the audio frequency range, and a cold-cathode gas tube. The starter-anode of the gas tube has a dc bias of about +24 volts with respect to its cathode to reduce the value of pulse voltage required to ionize it. When +130 volt battery is applied via the winding of the channel rela.y to the main anode of the gas tube, ionization established in the starter gap by the pulse stretcher signal will transfer to the main gap and cause the relay to operate. Closure of one of the relay make- contacts serves to divert the winding current from the gas tube directly to ground, thereby extinguishing the tube and prolonging its life. Other contacts, not shown, make the registered information available. Co7npo7ients A full complement of the electronic apparatus described in the last few sections utilizes plug-in components in the following quantities: Twin-triode electron tubes 186 Cold-cathode gas tubes 121 Germanium varistors 552 Point-contact transistors 120 Only one type of each of these components is used in the translator; this uniformity greatly simplifies the maintenance problem and imposed little if any handicap on the circuit designs. ADMINISTRATION EQUIPMENT Whenever it is desired to add, or to change, a translation item on the drum, the auxiliary administration unit pictured in Fig. 14 is connected to the translator by three shielded cables, shown leaving the rack just under the shelf, and a ten-conductor cable, shown with its plug resting ' on the shelf. The shielded cables convey the a and b sync pulses and the 1 once-per-drum-revolution fiducial f pulse to the administrator. The ten- conductor cable, with plug, is used to establish paths extending directly i to magnetic heads on the drum. During the recording of any one com- ! plete translation item on the drum, this plug is successively shifted to each of nine multi-connector jacks located in the amplifier compartment I )f the translator. '■ The manual controls are located just above the shelf. At the right are the two keys for ordering a writing operation, one for the a slot and an- other for the B slot of the chosen pair. If either key is lifted, it will order Ihe entry of a magnetic mark (write " 1"). If depressed, the key will order ! the removal of a mark (write "0"). It is obvious that the translation is 742 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 I- -o ^ ounce tr i-q: _iOO LU>.l/)Z 5 o > , UJ UJ_j5 Q.UJ u Q < UJ s a o to a <5 10 bb HOivnsNvai do iind onii^ii ox MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 743 inserted piecemeal by working in each track successively. The manual switching operation of connecting a single pair of writing amplifiers to each of eighty magnetic heads, in turn, is accomplished partly by setting the nine-position switch shown at the center of the panel, and partly by sliifting the plug of the ten-conductor cable. At the left are two signal lights which serve as alarms to warn the operator of possible incorrect functioning of the equipment. The operation of the administration unit can best be traced with the aid of the schematic block-diagram of Fig. 15. A ten-stage binary counter is supplied with b sync pulses from the translator; the 1,024 possible states of the counter are traversed in the course of exactly one revolu- tion of the translator drum. The f pulse from the translator will, mid- way between two b pulses, set all counter stages to zero, once per revolu- tion. After the first such reset, however, if the counter is working properly, it will always have returned to the zero condition just before the occurrence of the f pulse, by having counted 1,024 b pulses; under these conditions the f pulse, though still initiating reset action, does not change the state of the counter. The basis for the alarm signals mentioned above is a circuit arranged to detect if a change of state is occasioned by the F pulse. Associated with the counter is a coincidence circuit with a keyboard on which may be set up any "address" between 0 and 1,023. When the count of B pulses ecjuals the address set up on the keyboard, the coinci- dence circuit delivers a pulse which persists until the next b pulse alters the count; this coincidence pulse spans the time of occurrence of an a pulse, and is used in the read sync selector to gate-out a "selected" a pulse uniquely assigned to the address set up on the keyboard. A slot- spanning pulser, triggered bj^ the selcted a pulse, gates-out the associ- ated "selected" b pulse. These selected pulses, which occur once per revolution of the drum, are passed through gates under control of bistable electron-tube pairs which can be set by the manual writing keys and are re-set by the writing action itself. This insures that the desired action takes place only once per key operation, instead of repeating, once per drum-revolution, as long as the keys are held operated. The manually-gated unique selected a or selected B sync pulse is then slightly delayed in time to become a selected write- sync pulse. It is passed on through further gates under direct control of the writing keys, and is emplo3''ed as an input to a writing amplifier. A pair of writing amplifiers is provided, one to write " 1" and the other to write "0"; the circuits are identical quiescent blocking-oscillators shar- ing a common output transformer, and one or the other is triggered into 744 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 195G action by the write-sync pulses. The output transformer supplies the writing current pulses, under control of the selector switch, to the chosen magnetic head. Arrangements are provided for synchronizing an oscil- loscope to display the writing current pulses or the voltage outputs from the head at the selected address, as required. When a new translation item is to be entered, or an existing one al- tered, the address corresponding to the desired slot-pair is determined from a card-index, or ledger, listing all items on the drum. The address keyboard is then set to the assigned number, thereby singling-out the desired slot pair so that the writing operation can proceed as described above. During this procedure, the monitoring oscilloscope may be used for verifying the new entry, two cells at a time. Over-all verification is accomplished by exercising the translator through facilities already avail- able in the toll switching office. There is nothing about this procedure which precludes the use of automatic facilities for performing the admin- istration. There is also no fundamental need to take the translator out of routine service during the administration operation, since each writing operation disables the equipment for only a few microseconds and would rarely delay a translation by as much as one drum revolution. CONCLUSION After short preliminary tests, the equipment described and pictured was installed in the switching systems laboratory at Bell Laboratories. A rapid-transfer arrangement permitted direct interchangeability with a card translator in a skeletonized model of a toll switching office. A testing program was then begun entailing continuous 24-hours-per- day operation of the magnetic drum translator for approximately one year. After an initial shakedown period during which wiring faults and other minor troubles were recognized and cleared, many millions of trans- lations were handled with only a small proportion of failures. The accu- mulated data on failure rate and cause was significant, being one of the primary objectives of the experiment. An analysis of the data indicated the desirability of certain simple design changes in the existing circuitiy and established a basis for the selection of future designs. If, ill the future, consideration is given to the design of ciniipineiil of this type for some specific application, new electronic developments must also be taken into account. Many more types of transistors are now available than when the present design was undertaken, and some of the newer types have capabilities which make them obvious candidates for many of the jobs now done in the translator with electron tubes. Such a substitution would not only increase reliability and decrease power con- MAGNETIC DRUM TRANSLATOR FOR TOLL SWITCHING OFFICES 745 sumption, but since transistors are essentially current-operated devices t hey would seem to be particularly suitable for working with microsecond [)ulses in the environment of existing relay-equipped offices where the majority of interference transients are capacitively-propagated voltage- tlisturbances. Evaluation of the magnetic drum reveals it to be a safe and vevy roli- ;il)le means of storing several hundred thousand bits of information. Dur- ing the course of these tests, the drum functioned perfectly, and the trans- lations that were recorded at the beginning of the test were retained until near the end, when they were deliberately altered. During this interval of nearly continuous operation there was no detectable deterioration, or iliange in the signals obtained from the drum. The results obtained from the tests of this particular drum translator indicate that the associated circuitry, working with microsecond pulses, ran be designed to measure up to the exacting standards demanded for i telephone office apparatus, whether the application be that of a magnetic (hum translator or some other type of equipment. i;eferences 1. W. D. Lewis, Electronic Computers and Telephone Switching, Proc. I.R.E., 41, pp. 1242-1244; Oct., 1953. '2. W. A. Malthaner and H. E. Vaughan, An Automatic Telephone System Em- ploying Magnetic Drum Memory, Proc. I.R.E., 41, pp. 1341-1347; Oct., 1953. '■\. .J. H. McGuigan, Combined Reading and Writing on a Magnetic Drum, Proc. I.R.E., 41, pp. 1438-1444; Oct., 1953. 4. L. N. Hampton and J. B. Newsom, The Card Translator for Nationwide Dial- ing, B. S. T. J., 32, pp. 1037-1098; Sept., 1953. I Tables of Phase of a Semi-Infinite Unit Attenuation Slope By D. E. THOMAS (Manuscript received February 24, 1956) Five and seven place tables of the integral B(x,) = ' log 1 +a: 1 — X dx X which gives the 'phase associated with a semi-infinite unit slope of attenua- tion, are now available in monograph form. The usefulness of this integral and its tabulation are discussed. H. W. Bode' has shown that on the imaginary axis, the vahies of the imaginary part of certain functions of a complex variable may be ob- tained from the corresponding values of the real part, and vice versa. This theorem was immediately recognized as a powerful tool in the com- munications and network fields. The most generally useful function which was given by Bode for use in applying this theorem to the solution of communications problems, is the phase associated with a semi-infinite unit slope of attenuation. This is given by the integral 1 r':=Xc 5(.T.) = - log 1 -\-x (J/Jy / 1 \ X 1 - X where: 5(:i-c) is the phase in radians at frequency /c , x = ^ ,x, = ^^ < 1.0 Jo Jo and fo = the frequency at which the semi-infinite unit slope begins The usefulness of Integral (1) is illustrated by some of the communica- tion problems which stimulated its accurate tabulation. iT 1 Bode, H. W., Network Aiuily.sis and Feedback Amplifier Design, D. Van Nos- trand Co., Inc., New York, 1945, Chap. XIV. 2 Ibid: Chap. XV, pp. 342-343. 747 748 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 When the development program on deep sea repeatered .submarine telephone cable systems was reactivated at the close of World War II, one of the first problems to present itself was the detei-mination of the delay distortion of a transatlantic repeatered cable system. The only | means then known of obtaining an answer to this problem was by com- puting the minimum phase of the system from its predictable attenua- tion characteristic, using Bode's straight line approximation method,* and then determining the delay distortion from the non-linear portion of this minimum phase. However, the non-linear phase is such a small part of the total phase, that a five figure accuracy tabulation of Integral (1) was needed for a satisfactory determination of the non-linearity. The necessary table was therefore compiled. A mmierical computation was used to evaluate the integral because of the simplicity of its integrand. The minimum phase of the projected transatlantic repeatered telephone cables was then computed using this table and the anticipated delay dis- tortion was determined from the non-linear portion of this minimum phase. About this time the delay ecjualization of coaxial cable systems for television transmission became a pressing problem. Bode's techniciue proved to be the simplest means for determining the delay to be equal- i ized and so the existing phase table was immediately put to use in the ji coaxial cable delay ecjualization program. The increasing use of the tables led to a decision to publish them in in The Bell System Technical Journal.^ In order to make the tables more generally useful, the published paper included a tabulation of the phase in radians as well as in degrees. The radian tables can, for example, I be used to determine the reactance characteristic associated with a given resistance characteristic of a minimum reactance impedance function. Because of the demand for higher accuracy which occasionally arose after the publication of the five place tables, it was decided to undertake the computation of seven-place tables. These tables were also computed lunnerically using intervals selected to give at least ±1 accuracy in the final figure. The complete tables require forty-nine pages for tabulation. Since it is probable that only a fraction of the Journal readers would need these tables, it did not seem desirable to publish the actual tables iti the Journal. They are therefore being put)lished in original monograph form as Bell System Monograph 2550 entitled "Tal)les of Phase of a Semi-Infinite Unit Attenuation Slope." The phase is tabulated in the ■' Ibid : Chap. XV. ■* Tliomas, D. K., Tables of I'liase Associated willi a Semi Inliiiile I'nil Slope of Attenuation, B. S.T.J. , 26, pp. 870-899, Oct., 1947. ^ This Monograph will be available about June 15, 1956. TABLES OF PHASE 749 monograph both in degrees and radians for values of/ greater than /o as well as for/ less than/o . The tabular intervals are 0(0.001) 0.600 (0.0005) 0.9000 (0.0001) 0.9940 (0.00005) 0.99800 (0.00001) 1.00000. These intcr\'als were selected to permit linear interpolation for intermedi- ate values of the phase to an accuracy of the same order as the accuracy of the tabulated values, i.e., ±1 in the last place. The original Journal article discussed the construction of the tables and the errors involved in the numerical evaluation of Integral (1), described and illustrated the use of the tables, and gave five-place tabulations of the integral. I'his entire article is therefore included in Monograph 2550 for complete- ness along with the newer seven-place tables. B. A. Kingsbury^ has pointed out that the Integral (1) which is tabu- lated in the phase tables in question is useful in other than the communi- cations and network fields. A bibliography covering other possible fields of interest is given in an article by Murakami and Corrnigton. ACKNOWLEDGMENT The author is indebted to R. W. Hamming of the Mathematical Ke- search Department w'ho supervised the computation of the seven place tables, to Miss R. A. Weiss who planned, programmed, ran, and checked the IBM computations of the tables and to Miss J. D. Goeltz who com- puted the ten-figure accuracy check points required for the construction of the tables. He also wishes to acknowledge the support and encourage- ment given to the project by R. L. Dietzold and P. H. Richardson, and the continued interest and helpful comments of B. A. Kingsbury. ^ Kingsbury, B. A., private communication. 'Murakami, T., and Corrington, M. S., Relation Between Amplitude and Phase in Electrical Networks, R.C.A. Review, 9, pp. 602-631, Dec, 1948. Bell System Technical Papers Not Published in This Journal Anderson, P. W./ and Suhl, H/ Instability in the Motion of Ferromagnets at High Microwave Power Levels, Phys. Rev., Letter to the Editor, 100, pp. 1788-1789, Dec. 15, 1955. Andrus, J., see Bond, W. L. I I Beaciiell, H. C, see Veloric, H. S. i I Beck, A. C.,^ and Mandeville, G. D.^ I Microwave Traveling Wave Tube Millimicrosecond Pulse Generators, I I.R.E. Trans., MTT-3, pp. 48-51, Dec, 1955. I I j Benedict, T. S.^ Single-Crystal Automatic Diffractometer — Part II, Acta Cryst., 8, pp. 747-752, Dec. 10, 1955. Bennett, W. R.^ Application of the Fourier Integral in Circuit Theory and Circuit Problems, I.R.E. Trans., CT-2, 3, pp. 237-243, Sept., 1955. Biondi, F. J.' Corrosion-Proofing Electronic Parts Against Ozone, Ceramic Age, 66, p. 39, Oct., 1955. Bond, W. L.^ ! Single-Crystal Automatic Diffractometer — ^Part I, Acta Cryst., 8, pp. 741-746, Dec. 10, 1955. ' Bell Telephone Laboratories, Inc. 751 752 the bell system technical journal, may 1956 Bond, W. L.,' and Andrus, J/ Photographs of the Stress Field Around Edge Dislocations, Phys. Rev., Letter to the Editor, 101, p. 1211, Feb. 1, 1950. Boyle, W. S., See Gernier, L. H. Boyle, W. S.,^ and Haworth, F. E/ Glow-to-Arc Transitions, Phys. Rev., 101, pp. 935-938, Feb. 1, 1950. Bozorth, R. M.^ The Physics of Magnetic Materials, Elec. Engg., 75, pp. 134-140, Feb. 1956. Bridgers, H. E.^ A Modern Semiconductor — Single Crystal-Germanium, Chem. and Engg. News, 34, p. 220, Jan., 1956. BuRRUS, C. A.,^ and Gordy, W.^ Millimeter and Submillimeter Wave Spectroscopy, Phys. Rev., 101, pp. 599-603, Jan. 15, 1956. Chynoweth, a. G. I Dynamic Method for Measuring the Pyroelectric Effect with Special Reference to Barium Titanate, J. Appl. Phys., 27, ])i). 78 84, Jan., 1956. Cutler, C. C.^ Spurious Modulation of Electron Beams, Proc. I.R.E., 44, pp. 61-64, Jan., 1956. Davis, H. M., see Wernick, J. H. Duncan, R. A.,^ and Stone, J. A., Jr.' a Survey of the Application of Ferrites to Inductor Design, Proc. I.R.E., 44, pp. 4-13, Jan., 1956. ' Bell Telephone Laboratories, Inc. ^ Duke University. 01 TECHNICAL PAPERS 753 Fehek, G./ Fletcher, R. C./ and Gere, E. A/ Exchange Effects in Spin Resonance of Impurity Atoms in Silicon, Phys. Rev., Letter to the Editor, 100, pp. 1784-1785, Dec. 15, 1955. Feldmann, W. L., see Pearson, G. L. Fkaver, D. R.' Design Principles for Junction Transistor Audio Power Amplifiers, l.R.E. Tran.s., AU-3, pp. 183-201, Nov.-Dec, 1955. Flaschen, S. S.,^ and Van Uitert, L. G.' New Low Contact Resistance Electrode, J. Appl. Phys., Letter to the Editor, 27, p. 190, Feb., 195(5. Fletcher, R. C., see Feher, G. Fry, T. C.^ Mathematics as a Profession Today in Industry, Am. ]\Iath. Monthly, 63, pp. 71-80, Feb., 1956. Fuller, C. S., see Reiss, H. Geballe, T. H., see Hrotowski, H. J. Gere, E. A., see Feher, G. Germer, L. H.,^ and Boyle, W. S.^ Short Arcs, Nature, Letter to the Editor, 176, p. 1019, Nov. 26, 1955. Germer, L. H.,^ and Boyle, W. S.^ Two Distinct Types of Short Arcs, J. Appl. Phys., 27, pp. 32-39, Jan., 1956. GlANOLA, U. F. Photovoltaic Noise in Silicon Broad Area p-n Junctions, .1. Appl. Phys., 27, pp. 51 53, Jan., 1950. GoRDY, W., see Burrus, C. A. 1 Bell Telephone Laboratories, Inc. 754 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Hagelbarger, D. W., see Pfann, W. G.; Shannon, C. E.; and Wil- liams, H. J. Hagstrum, H. D/ Electron Ejection from Metals by Positive Ions, Appl. Sci. Res. B5, Nos. 1-4, pp. 16-17, 1955. Haworth, F. E., see Boyle, W. S. Herring, C.,^ and Vogt, E.^ Transport and Deformation Potential Theory for Many-Valley Semi- conductors with Anisotropic Scattering, Phys. Rev., 101, pp. 944- 961, Feb. 1, 1956. Herrmann, D. B., see Williams, J. C. Holden, a. N.,^ Merz, W. J.,^ Remeika, J. P.,' and Matthias, B. T.^ Properties of Guanidine Aluminum Sulfate Hexahydrate and Some of its Isomorphs, Phys. Rev., 101, pp. 962-967, Feb. 1, 1956. Horotowski, H. J.,^ Morin, F. J.,^ Geballe, T. H.,^ and Wheatley, G. H.' Hall Effect and Conductivity of InSb, Phys. Rev., 100, pp. 1672-1677, Dec. 15, 1955. Ingram, S. B. The Graduate Engineer His Training and Utilization in Industry Elec. Engg., 75, pp. 167 170, Feb., 1956. Kaplan, E. L.^ Transformation of Stationary Random Sequences, Mathematicai Scandinavica, 3, FASCl, pp. 127-149, June, 1955. Lewis, H. W.^ ; Superconductivity and Electronic Specific Heat, Phys. Rev., 101, pp.^ 939-940, Feb. 1, 1956. Mandeville, G. D., see Beck, A. C. 1 Bell Telephone Laboratories, Inc. TECHNICAL PAPERS 755 Matthias, B. T., see Holden, A. N. Merz, W. J., see Holden, A. N. Miller, L. E. Negative Resistance Regions in the Collector Characteristics of the Point-Contact Transistor, Proc. I.R.E., 44, pp. 65-72, Jan., 195G. Moll, J. L./ and Ross, I. M.' The Dependence of Transistor Parameters on the Distribution of Base Layer Resistivity, rioc. I.R.E., 44, pp. 72-78, Jan., 1950. Montgomery, H. C, See Pearson, G. L. iMoRiN, F. J., see Hrotowski, H. J. MuMFORD, W. W.,^ and Schaferman, R. L/ Data on the Temperature Dependence of X-Band Fluorescent Lamp Noise Sources, I.R.E. Trans., MTT-3, pp. 12-16, Dec, 1955. Xesbitt, E. a., see Williams, H. J. ( )lmstead, p. S.^ QC Concepts Useful in OR, lud. Qual. Cent., 12, pp. 11, 14-17, Oct., 1955. < )WENS, C. D.' Stability Characteristics of Molybdenum Permalloy Powder Cores, Elec. Engg., 74, pp. 252-256, Feb., 1956. Pearson, G. L.,^ Montgomery, H. C.,^ and Feldmann, W. L.^ Noise in Silicon p-n Junction Photocells, J. Appl. Phys., 27, pp. 91-92, Jan., 1956. Pfann, W. G.,^ and Hagelbarger, D. W.^ Electromagnetic Suspension of a Molten Zone, J. Appl. Phys., 27, pp. 12-17, Jan., 1956. ' Bell Telephone Laboratories, Inc. 756 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 ^ QUINLAN, A. L.^ * Roll- Welding Precious Metals for Telephone Contacts, Elec. Engg., 75, pp. 154-157, Feb., 1956. Reiss, 11.,^ and Fuller, C. S.^ The Influence of Holes and Electrons on the Solubility of Lithium in Boron-Doped Silicon, .J. of Metals, 12, p. 276, Feb., 1956. Remeika, J. P., see Holden, A. N. Ross, I. M., see Moll, J. L. ScHAFERMAN, R. L., See Mumford, W. W. SCHAWLOW, A. L.^ I Structure of the Intermediate State in Superconductors, Phys. Rev., 101, pp. 573-580, Jan. 15, 1956. ScHAWLow, A. L.,^ and Townes, C. H.* Effect on X-Ray Fine Structure of Deviations from a Coulomb Field near the Nucleus, Phys. Rev., 100, pp. 1273-1280, Dec. 1, 1955. Shannon, C. E.,^ and Hagelbarger, D. W.^ Concavity of Resistance Functions, J. Appl. Phys., 27, pp. 42-43, j Jan, 1956. SiMKiNS, Q. W.,^ and Wogelsong, J. H.^ Transistor Amplifiers for Use in a Digital Computer, Proc. I.R.E., 44, pp. 43-54, Jan., 195(). Snoke, L. R.^ Specific Studies on the Soil-Block Procedure for Bioassay of Wood Preservatives, Appl. Mierubiology, 4, pp. 21-31, Jan., 1956. i SOUTHWORTH, G. C.^ Early History of Radio Astronomy, Sei. Mo., 82, pp. 55-66, Feb., 1956. |j ^ Bell Telephone Laboratories, Inc. ■'' Western Electric Company. •* Columbia University. h TECHNICAL PAPERS 757 Stone, H. A., see Duncan, R. A. SuHL, H., see Anderson, P. W. Thomas, E. E/ Tin Whisker Studies Observation of some Hollow Whiskers and Some Sharply Irregular External Forms, Letter to the Editor, Acta Met., 4, p. 94, Jan., 1956. TowNES, C. H., see Schawlow, A. L. TOWNSEND, M. A.^ A Hollow Cathode Glow Discharge with Negative Resistance, Appl. Sci. Research, Sec. B, 5, pp. 75-78, 1955. \'aldes, L. B.^ Frequency Response of Bipolar Transistors with Drift Fields, Proc. I.R.E., 44, pp. 178-184, Feb., 1956. \'an Uitert, L. G., see Flaschen, S. S. \'eloric, H. S.,^ and Beachell, H. C. Absorption Isotherms, Isobars and Isoteres of Diborane on Palladium on Charcoal and Boron Nitride, J. Phys. Chem., 60, p. 102, Jan., 1956. \'oGELSONG, J. H., see Simkins, Q. W. \ ogt, E., see Herring, C. W'eibel, E. S.' Strains and the Energy in Thin Elastic Shells of Arbitrary Shape for Arbitrary Deformation, Zeitchrift f. Mathematik and Physik, 6, pp. 153-189, May 25, 1955. W'krnick, J. H.,^ and Davis, H. M.'' Preparation and Inspection of High-Purity Copper Single Crystals, J. Appl. Pliys., 27, pp. 144-153, Feb., 1956. ' Bell Telephone Laboratories, Inc. ^ University of Delaware. ^ Penn State University. 758 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 Wheatley, G. H., see Hrotowski, H. J. Williams, H. J.,' Heidenreich, R. D.,^ and Nesbitt, E. A.^ Mechanism by which Cobalt Ferrite Heat Treats in a Magnetic Field,!: J. Appl. Phys., 27, pp. 85-89, Jan., 1956. Williams, J. C.,^ and Herrmann, D. B. Surface Resistivity of Non-Porous Ceramic and Organic Insulating' Materials at High Humidity with Observations of Associated Silver Migration, I.R.E. Trans., PGRQC-6, pp. 11-20, Feb., 1956. Wood, Mrs. E. A.' A Heated Sample-Holder for X-Ray Diffractometer Work, Rev. Sci.j Instr., 27, p. 60, Jan., 1956. ^ Bell Telephone Laboratories, Inc. decent Monographs of Bell System Technical Papers Not Published in This Journal* Anderson, P. W., and Hasegawa, H. Considerations on Double Exchange, Monograph 2532. Baker, W. O., see Winslow, F. H. Barstow, J. M. The ABC's of Color Television, Monograph 2529. Bemski, G. Lifetime of Electrons in p-type Silicon, Monograph 2534. Bennett, W. R. Application of the Fourier Integral in Circuit Theory, Monograph 2533. Brattain, W. H., see Pearson, G. L. Brown, W. L. Surface Potential and Surface Charge Distribution from Semicon- ductor Field Effect Measurements, Monograph 2501. Bullington, K. Characteristics of Beyond-the-Horizon Radio Transmission, Mono- graph 2494. Bullington, K., Inkster, W. J., and Durkee, A. L. Propagation Tests at 505 mc and 4,090 mc on Beyond-Horizon Paths, Monograph 2503. * Copies of these monographs may be obtained on request to the Publication Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14. N. Y. The numbers of the monographs should be given in all requests. 759 760 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 DuRKEE, A. L., see Biillington, K. Freynik, H. S., see Gohn, G. R. (Jkllkh, S., and Thurmond, (■. D. j On the Question of the Existence of a Crystalline SiO, Monograph 2530. Gohn, G. R., Guerard, J. P., and Freynik, H. S. The Mechnical Properties of Wrought Phosphor Bronze Alloys, Monograph 2531. I Guerard, J. P., see Gohn, G. R. ^' Hasegawa, H., see Anderson, P. W. Haynes, J. R., see Hornbeck, J. A. Hornbeck, J. A., and Haynes, J. R. - Trapping of Minority Carriers in Silicon, Monograph 2368. Inkster, W. j., see Bullington, K. Lewis, H. W. Search for the Hall Effect in a Superconductor. II. Theory, Mono- graph 2523. LiNviLL, J. G., and Mattson, R. H. Junction Transistor Blocking Oscillators, Monograph 2487. Logan, R. A. Precipitation of Copper in Germanium, Monograph 2524. Logan, R. A., and Schwartz, M. Restoration of Resistivity and Lifetime in Heat-Treated Germanium Monograph 2525. Mattson, R. H., see Linvill, J. G. MONOGRAPHS 761 Mays, J. M., see Shulman, R. G. McCall, D. W., see Shulman, R. (i. Moll, J. L. Junction Transistor Electronics, Monograph 2537. Pearson, G. L., and Brattain, W. H. History of Semiconductor Research, Monograph 2538. Sandsmark, p. I. ElHpticity on Dominant-Mode Axial Ratio in Nominally Circiilar Waveguides, Monograph 2539. Schwartz, M., see Logan, R. A. Shulman, R. G., Mays, J. M., and McCall, D. W. Nuclear Magnetic Resonance in Semiconductors. I, Monograph 2528. Thurmond, C. D., see Geller, S. Van Uitert, L. G. Low Magnetic Saturation Ferrites for Microwave Applications, Mono- graph 2504. \ax Uitert, L. G. Dc Resistivity in the Nickel and Nickel Zinc Ferrite System, Mono- graph 2540. :Weible, E. S. Vowel Synthesis by Means of Resonant Circuits, Monograph 2541. WiNSLow, F. IL, Baker, W. 0., and Yager, W. A. Odd Electrons in Polymer Molecules, Monograph 2486, Vager, W. a., see Winslow, F. H, Contributors to This Issue Donald C. Bennett, B.S. 1949 and M.S. 1951, Rensselaer Poly- technic Institute; Battelle Memorial Institute, 1951-1952; Bell Tele- phone Laboratories, 1952-. Mr. Bennett has been engaged in the de- velopment of processes for producing single crystals suitable for use in transistors. He is a member of the American Institute of Mining and Metallurgical Engineers. F. G. BuHRENDORF, B.S.M.E. and M.E., Cooper Union Inst. Tech. 1925. Bell Telephone Laboratories 1925-. Mr. Buhrendorf's early Labo- ratories work included the design of switchboard apparatus and sound recording and reproducing equipment ; among the latter were the Mirro- phone and the stereophonic equipment demonstrated at the New York World's Fair. During World War II he was concerned with the design of mechanical components of a number of radar systems, particularly antenna drives and range units. After the war he resumed his work on high-quality sound reproduction and more recently has devoted his efforts to the design of magnetic drum units for digital data storage and special machinery for the purification and production of single-crystal semiconductors. He is a New York State Professional Engineer. Calvin S. Fuller, B.S. 1926 and Ph.D. 1929, University of Chicago. Bell Telephone Laboratories, 1930-. His early work was on organic in- sulating material, after which he made studies of plastics and synthetic rubber including investigations of the molecular structure of polymers and the development of plastics and rubbers. Since 1948 Dr. Fuller has concentrated on semiconductor research and the development of semi- conductor devices. His work led to a techniciue of diffusing impurities -. into the surface of a silicon wafer, a preparation basic to the Bell Solar (j Battery and other silicon devices. He is a member of the A.C.S., an associate member of the A.P.S. and a member of the A.A.A.S. H. A. Henning, B.S. in ElcctrocluMuical Engineering, Pennsylvania State College 1926; Columbia University 1930-33. Bell Telephone 762 CONTRIBUTORS TO THIS ISSUE 763 Laboratories, 1926-. Mr. Henning's early Laboratories work was con- nected with the development of high-quality sound recording and re- producing equipment and techniques. During this interval he developed the 9A disc phonograph reproducer. Other pre-war experience included development of telephone voice recorders, noise reduction studies of the dynamics of teletype equipment, and design of coin collector slug rejec- tors and coin disposal relays. During World War II he was concerned with improvements to the sound power telephone, and later with develop- ment of specialized magnetic sound recording- reproducing systems. After the war he resumed his work on high quality sound recording equipment and supervised the design of the 2A lateral disc feedback recorder. More recently he has been concerned with the principles and design of magnetic drum digital data storage and apparatus. He is cur- rently engaged in investigating the application of square hysteresis loop magnetic cores to digital computer systems. David, A. Kleinman, S.B. in Chemical Engineering, 1946, S.M. in Mathematics, 1947, Massachusetts Institute of Technology; Ph.D. in physics, Brown LTniversity, 1952. Dr. Kleinman joined Bell Telephone Laboratories at Murray Hill in Jul}^ 1953. Since then he has studied theory of transistor devices and has been engaged in research in the band ■theory of solids in the Solid State Electronics Research Department. I He is a member of the American Physical Society. F. J. MoRiN, B.S. and M.S., University of New Hampshire, 1939 and 1940; University of Wisconsin, 1940-1941; Bell Telephone Laboratories, 1041-. During World War II, Mr. Morin was involved in research on j elemental and oxide semiconductors and the development of thermistor materials. Since that time he has worked on fundamental investigations into the mechanism of conduction in silicon, germanium and oxide semi- -conductors. Mr. Morin is a member of the American Chemical Society and the American Physical Society. i 0. J. Murphy, B.S. in Electrical Engineering, University of Texas, 1927; Columbia University, 1928-31. Bell Telephone Laboratories, 1927-. Mr. Murphy's early Laboratories projects included studies of \'oice-operated switching devices, effects of transmission delay on two- Way telephone conversation, and voice-frequency signaling systems. 1 )uring World War II he was concerned with design and development of ihe M-9 electrical gun director and related projects. After the war he resumed his research work on signaling systems and more recently has 764 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 concentrated on the design of magnetic drum digital data storage ap- paratus and circuits. He is a member of the A.I.E.E., a senior member of the T.R.E., and is a licensed professional engineer. M. B. Prince, A.B., Temple University, 1947; Ph.D., Massachusetts Institute of Technology, 1951; Bell Telephone Laboratories, 1951-1956; National Semiconductor Products, 1956-. Between 1949-51 he was a research assistant at the Research Laboratory of Electronics at M.LT, where he was concerned with cryogenic research. At Bell Telephone Laboratories, Dr. Prince was concerned with the physical properties of semiconductors and semiconductor devices and was associated with the development of silicon devices, including the Bell Solar Battery and the silicon power rectifier. Dr. Prince is a member of the LR.E., the Ameri- can Physical Society, and Sigma Xi. Howard Reiss, B.A., New York University, 1943; Ph.D., Columbia University, 1949; Instructor and Assistant Professor in Chemistry, Boston University, 1949-51; Head of the Fundamental Research Sec- tion, Celanese Corporation, 1951-52; Bell Telephone Laboratories, 1952-. Dr. Reiss is engaged in the theoretical chemistry of defects in semiconductors. He is a member of the American Chemical Society, the American Physical Society, Sigma Xi and Phi Lamda Upsilon. Baldwin Sawyer, B.E., Yale University, 1943; D.Sc, Carnegie Insti- tute of Technology, 1952; Manhattan Project, University of Chicago, 1943-1946; Instructor and Research Associate in Physics, Carnegie In- stitute of Technology, 1948-1951; Bell Telephone Laboratories, 1951- Dr. Sawyer's first work at the Laboratories was on the development of semiconductor devices, especially the silicon alloy junction diode. Since 1953 he has been in charge of a group at Allentown concerned with the growth, measurement and characterization of germanium and silicon crystals for use in semiconductor devices. He is a member of the Ameri- can Physical Society, the American Institute of Mining and Metallurgi- cal Engineers, Tau Beta Pi, Sigma Xi, and an associate of the LR.E. Donald E. Thomas, B.S. in E.E., Pennsylvania State University, 1929; M.A., Columbia LTniversity, 1932; Bell Telephone Laboratories, 1929-. Mr. Thomas specialized in the development of repeatei'ed sub- marine cable systems until 1940 when he became engaged in the de\'elop- ment of sea and airborne radar. In 1942 he entered military service where he was active in electronic countermeasures research and development. CONTRIBUTORS TO THIS ISSUE 765 Following the war he took part in the development and installation of ' the first deep-sea repeatered submarine telephone cable system between i Key West and Havana. During this period he also served as a civilian ! member of the Department of Defense's Research and Development Board Panel on Electronic Countermeasures. At present Mr. Thomas is engaged in characterization and feasibility evaluation of research models of semiconductor devices. He is a senior member of the I.R.E. and a member of Tau Beta Pi and Phi Kappa Phi. rHE BELL SYSTEM Jechnical journa^ mOTEH TO THE SC I E N T I FlC^^r^ AND ENGINEERING JPECTS OF ELECTRICAL C OM M U N IC AT Io4j, EJ ^(JQ .J C-UME XXXV JULY 1956 NVMi*R4 The Effect of Surface Treatments on Point-Contact Transistors J. H. FORSTER AND L. E. MILLER 767 The Design of Tetrode Transistor Amplifiers J. G. LINVILL AND L. G. SCHIMPF 813 The Nature of Power Saturation in Traveling Wave Tubes C. C. CUTLER 841 The Field Displacement Isolator s. weisbaum and h. seidel 877 Transmission Loss Due to Resonance of Loosely-Coupled Modes in a Multi-Mode System a. p. king and e. a. marcatili 899 Measurement of Atmospheric Attentuation at Millimeter Wave- lengths A. B. CRAWFORD AND D. C, HOGG 907 A New Interpretation of Information Rate j. l, kelly, jr. 917 Automatic Testing of Transmission and Operational Functions of Intertoll Trunks H. H. FELDER, a. j. PASCARELLA AND H. F. SHOFFSTALL 927 Intertoll Trunk Net Loss Maintenance Under Operator Distance and Direct Distance Dialing H. H. FELDER AND E, N. LITTLE 955 Bell System Technical Papers Not Published in This Journal 973 Recent Bell System Monographs 979 Contributors to This Issue V 985 COPYRIGHT 1954 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD F. R. KAPPEL, President, Western Electric Company M. J. KELLY, President, Bell Telephone Laboratories E. J. McNEELY, Executive Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. McMillan, Chairman A. J. BUSCH A. C. DICKIESON R. L. DIETZOLD K. B. GOULD E. I. GREEN R. K. HONAMAN H. R. HUNTLEY F. R. LACK J. R. PIERCE H. V. SCHMIDT G. N. THAYER EDITORIAL STAFF J. D. TEBO, Editor M.E. STRiEBY, Managing Editor R. L. SHEPHERD, Production Editor t THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; John J Scan- Ion, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed , in U. S. A. § THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV JULY 1956 number 4 Copyright 1956, American Telephone and Telegraph Company The Effect of Surface Treatments on Point-Contact Transistor Characteristics By J. H. FORSTER and L. E. MILLER (Manuscript received January 23, 1956) A description is given of the electrical properties of formed point con- facts on germanium. A useful technique for observation of the equipotentials surrounding such contacts is described. The contrasting properties of donor- free and donor-doped, contacts, used as diodes or transistor collectors are emphasized. It is shown that unformed point contacts {which have electrical properties largely determined by a surface barrier layer) , may exhibit analogous dif- ferences. Such changes are produced by chemical treatments calcidated to influence properties of a soluble germanium oxide film on the surface. The above information is applied to a study of transistor forming as it is done in present point-contact transistor processing. It is shown that high yields from the forming process can be expected on oxidized surfaces, and (hat chemical ivashes which remove soluble germanium oxide drastically lower forming yields. These and, other effects are evaluated as sources of variability in forming yield. Table of Contents I [l . Introduction 768 2. Pro]ierties of Formed Point Contacts 770 I 2.1 Effects of Electrical Forming on Point Contacts 770 1 2.2 Donor-Free and Donor-Doped Contacts 774 767 768 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G 2.2.1 Potential Probes 774 2.2.2 Use of the Copper Plating Technique 776 2.3 Under-Formed and Over-Formed Contacts 781 3. Properties of Unformed Point Contacts 783 3.1 Physical Properties of Metal -Semiconductor Contacts 783 3.2 Experimental Procedures 785 3.3 Experimental Results 786 3.3.1 Unformed Transistors on Superoxol-Etched Surfaces 786 ' 3.3.2 Unformed Transistors on CP4-Etched Surfaces 789 3.3.3 Diode Characteristics on Electro-Etched Surfaces 789 3.3.4 Output Characteristic Anomalies 789 3.3.5 Floating Potential Measurements 790 '■ 3.3.6 Contamination of Collector Points and Surfaces 792 3.4 Discussion of Experimental Results 794 3.4.1 Effects of the Chemical Ti-eatment on the Superoxol-Etched Surfaces 794 3.4.2 CP4-Etched Surfaces 795 : 4. Relation of Germanium Surface Properties to Transistor Forming 796 4.1 Pilot Production Problems 796 4.2 Experimental Results 797 4.2.1 Pilot Process Forming Yields 797 4.2.2 Relation of Unformed Diode Characteristics to Transistor "Formability " 801 4.2.3 Controlled Ambient Experiments 804 4.2.4 A Statistical Survey Experiment on Transistor Forming 805 4.2.5 Effect of Contamination Before Etching 806 ' 4.3 Conclusions 807 < 5. General Concluding Remarks 808 5.1 Point-Contact Transistors with High Current Gain 809 5.2 Current Multiplication in Unformed Transistors 809 ' 5.3 Surface Properties and Transistor Forming 810 , 1. INTRODUCTION j The point-contact transistor, on the basis of several years use in the I field in Bell System applications, has proved itself to be rugged and de- ■ pendable. For certain military applications, a lasting demand exists for high-speed point-contact transistors. The adaptation of cartridge type units to a hermetically sealed structure has been completed, with further benefits to reliability. To date, the point-contact transistor is one of the few transistors to successfully pass all military specifications for shock, vibration, and high acceleration. Thus, although there are at present limitations to the electrical characteristics that can be built into a point-contact transistor which make it unsuitable for use in some switching circuits, there are many applications in which this type of transistor can give consistent and reliable performance. In fact, applica- tions exist w^herein the specific requirements are uniquely satisfied by the point-contact transistor. However, the basic operational principles of this kind of device arc not as well understood as would be desirable for facilitating develop- mental studies for manufacture. Although considerable effort has been I POINT-CONTACT TRANSISTOR SURFACE EFFECTS 769 expended towards the analysis and understanding of the physical mecha- nisms of the point-contact transistor since its announcement in 1948, a complete design theory for these transistors is not available. This lack probably I'esults partially from a more general interest in the readily designable junction transistor types, and partially from the relative complexity of the device itself. Actual!}^ the physical mechanisms which account for the operation of this device have their counterparts in at least three basically unique devices: the point diode, the junction tran- sistor, and the filamentary transistor. Thus, although the empirical knowledge of point-contact transistor design and operation is large enough to allow a reasonable degree of designability, and manufacture of these transistors in large quantities is possible, there are, from time to time, manufacturing problems which are often difficult to solve without sound theoretical understanding of the physical mechanisms which make the device work. This article is concerned with describing the results of a general study of the physical properties of a few specific kinds of point contacts. The kinds of contact studied have been those of specific interest to those concerned with manufacture and processing of point-contact transis- tors. This investigation was conducted in parallel with the final develop- ment for manufacture of the hermetically sealed point-contact transis- tor. The study of these properties has led to practical solutions of several problems encountered during manufacture of point-contact transistors, and has provided experimental data which is of interest in consideration of the basic physical mechanisms involved in the operation of the point- contact transistor. The work to be described, primarily experimental in nature, follows in Sections 2, 3 and 4. In section 2, the properties of formed, or electri- cally pulsed point contacts, and their relation to the source of output characteristic anomalies often responsible for lowering forming yields in point-contact transistor production is discussed. The properties of point contacts which have received no electrical forming in the conven- tional sense are considered in section 3. The electrical properties of these contacts, used as diodes or transistor collectors, are shown to be de- I pendent on chemical history of the etched germanium surface. Thus I "chemical forming" of point contacts is possible. Section 4 deals with application of these results to forming problems which arise during manufacture of point-contact transistors. The important relation be- tween the chemical history of the surface and the forming on that sur- face is considered. i 770 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 2. PROPERTIES OF FORMED POINT CONTACTS 2.1 Effects of Electrical Forming on Point Contacts The simplest form of point-contact transistor collector is a metal to semiconductor contact which has not been subjected to excessive power dissipation either in short high energy pulses, or in the form of more prolonged aging at lower power levels. Such contacts will be referred to as vuiformed contacts, and their properties will be discussed in detail in Section 3. Unformed point-contact transistors sometimes exhibit power gain, but in general they are not suitable for use as active devices be- cause the gain, although it may be highly variable from unit to unit, is usually low. The electrical characteristics of such contacts depend on a metal-semiconductor contact at the semiconductor surface, and control of these properties requires exacting control of surface preparation, sur- rounding ambient, and mechanical stability of the point. In early experiments, Brattain used electrical forming to improve both the power gain and stability of the transistor. For present purposes, the process of electrical forming will be defined as the passage of a short pulse of reverse current through a point contact which produces perma- nent changes in the electrical properties of the contact. This is usually accomplished by charging a condenser to several hundred volts, and] subsequently discharging it through a resistor in series with the transis- tor collector. Bardeen and Pfann," investigating electrical forming of phosphor bronze points on etched germanium surfaces, indicate, as a possible explanation of their data, that the forming pulse changes the height of the potential barrier at the germanium surface. This would, in absence of large surface conductivity, increase the reverse current through the point and increase the efficiency of hole collection by the point.^ Thus, the formed point may, according to theory, act as a col- lector with a current multiplication (a) greater than unity. Thermal and potential probing of an ?i-germanium surface under a formed phos- phor bronze point indicates, according to Valdes, that an appreciable volume of germanium is converted to p-type conduction. Thus, the reverse current through a formed point probably depends on the char- acteristics of a p-n junction a small distance from the point, rather than on a potential barrier at the germanium surface. A characteristic of the point-contact transistor is that the current gain can be substantially greater than unity. The current gain, a, i^^ usually defined as the current multiplication at constant voltage, that is: dl a = die i (1)1 POINT-CONTACT TRANSISTOR SURFACE EFFECTS 771 where /c and le are the collector and emitter currents. The a can be con- sidered as the product of three terms, that is: a = aSy (2) where 7 and /i represent the injection efficiency and transport factor respectively for minority carriers. The term a^- is the "intrinsic" current multiplication of the collector itself. As mentioned above, there are theoretical reasons to account for an ai as large as (1 -f h), where h is the ratio tin/y^p of the mobilities of electrons and holes, and thus the term ai may be roughly as large as 3.1. The average current gain, a, taken over a large interval of emitter current, is seldom found to be greater than this value, and is usually about 2.5. However, the small signal a at low emitter current usually is found to be considerably larger than 3.1. Several mechanisms have been proposed to account for this excess current gain at low emitter bias in formed transistors. The most generally known of these are the p-n hook hypothesis and the trapping model. ' The experiments to be described in this section will be concerned pri- marily with the characteristics of formed points as transistor collectors, and thus with the transport factor /S. The subject of the origin of the intrinsic «»■ will be discussed further in a later section. The experiment of Valdes indicates that the properties of a formed point contact depend on the physical properties of a small region of ger- manium near the point, produced by impurity diffusion from the point or imperfections introduced during the formmg pulse. A highly idealized representation of the physical situation is shown in Fig. 1 . This is a radial model of a formed point contact on a semi-infinite block of n-germanium (respectively p) , with a hemispherical p-layer (radius c:^ ro) . The electron and hole concentrations in the formed layer near the junction are desig- nated as Up and p. If a reverse bias Vc is applied to the point, a potential difference F(ri) — F(r2) = Vj results from the resistance of the junction at ro . For r ^ ro , at distances well outside ?'o , the potential V{r) and the magnitude of the field E(r) are given by where / is the total current through the point. For I Fo - V(n) I « 1 F^ I, V{r2) ^V,-Vj, 772 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 and the junction resistance limits the magnitude of the drift field that can be set up near the point. For example, if the lifetime t„ of electrons in the p-layer^ is substantially lower than Tp , that of holes in the ger- manium bulk, the reverse current density across the junction can be increased by an increase in Up , and junction resistance lowered. Pfann reports a substantial mcrease in the reverse current of formed point contacts with donor concentration of the point wire. The increase in rip will depend on the distribution of donors in the p-layer after the forming pulse. A high donor concentration near the collector point may -V, (r) \/ Vc V^ ^(H) f ^^ X) -V2) r 1 \ \ \ \ s r, 1 r^ ~~ 1 (b) Fig. 1 — Formed point contact under reverse bias — schematic representation. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 773 form an 7i-type inversion layer under the point (p-n hook) which, when the point is under reverse bias, acts as an electron emitter. Such a situa- tion might arise as a result of diffusion of impurities from the collector point at the high temperature reached during the forming pulse. An acceptor element, such as copper, with a high diffusion coefficient might penetrate substantially farther into the germanium than donor elements such as phosphorous or antimony^^ with lower diffusion con- stants. Thus, the donor concentration near the point might be substan- tially higher than the acceptor concentration if the solubility of the acceptor element is low. On the other hand, an appreciable number of donor atoms may pene- trate the germanium as far as do the acceptors. Thus, the equilibrium value of Wp may be increased simply by decreasing the effective concentra- tion of acceptors in the p-layer. Such a case might arise when a collector point such as copper is doped with a suitable amount of a donor element with a large diffusion coefficient and limited solubility. The observation of regions of melted germanium^ under heavily formed points gives evidence for a somewhat different interpretation of the forming process. It has been suggested that forming is essentially a remelt process. For example, forming of a phosphor-bronze point may produce a copper-germanium eutectic, allowing the introduction of a sizeable phosphorus concentration in the remelt region which is main- tained after freezing. Thus the depth of penetration of the donor ele- ment depends upon the size of the remelt region, and the penetration of the acceptor element depends upon its solid state diffusion coefficient. This mechanism can lead either to the formation of a p-n hook, or at least to a Iyer of p-germanium with a high equilibrium electron concen- tration. Whatever the reason for the decrease in resistance of the collector barrier, if it is sufficient, the magnitude of E(r) for r > r^ can be increased by forming to sufficient value to ensure efficient collection of holes and a transport factor /3 close to unity. It would then be expected that for a formed donor-free point, such as the beryllium-copper alloy points often used as unformed emitters, the formed p-region would have a high acceptor concentration, n^ would be small, and under reverse bias, the magnitude of V j would be large, with I /co I , I V{r^ I , and average a small, [solid curve. Fig. 1(b). On the other hand, a formed phosphor bronze point of the kind conven- tionally used to make transistor collectors, should exhibit under reverse bias, a lesser magnitude of V j , with | /eo | , | Vir^) \ , and a as much as an order of magnitude larger, (dashed line in Fig. 1(b)]. 774 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 2.2 Donor-Free and Donor-Doped Contacts The qualitative picture of the conventional formed contact given above has been substantially supported by the work of Valdes, who ob- served a large increase in floating potential near the reverse biased col- lector after the forming pulse and a substantial p-region in the bulk of the germanium after forming. Experiments have been directed to a comparison of the properties as diodes and collectors, between two kinds of points. Phosphor bronze points of the type used as transistor collectors, and beryllium copper points, normally used as emitters, were investigated. Thus a direct com- parison can be made between donor-doped and donor-free points which have been given similar forming pulses. The forming pulses were of the capacitor discharge type, with voltage and RC values similar to those used in conventional transistor forming. The points used were of the cantilever variety, and the n-germanium was zone-leveled material in the 3 to 4 ohm-cm range. Two points were supported in a double-ended micro-manipulator which allowed freedom of movement in 3 dimensions for each point. 2.2.1 Potential Probes Conventionally, point-contact transistors are made on a superoxol- etched wafer. This etch leaves a rough surface which is unsuitable for accurate potential probing. Some measurements were made of the float- ing potentials on this kind of surface, but accurate results were difficult | to obtain. In a later section it is shown that the kind of etch used in surface preparation can have profound effects on the degree of forming obtained. However, it is shown that forming characteristics of an "aged" CP4-etched surface are quite similar to the superoxol surface. Thus this kind of surface was used, since its topographical uniformity allows very reproducible results in the measurement of floating potentials. Fig. 2 is a comparison of the floating potentials for the two kinds of transistor points examined. The log-log plot shows the magnitude of the floating potential, Vp , near the reverse biased collector as a function of r, the distance of the probe from the collector measured between centers of the two points. The bars represent the uncertainty in measurement of the linear distance. Three curves are shown. The lowest Curve I repre- sents the potential near a Be-Cu point formed with a conventional form- ing pulse. Curve II is a plot of the potential near a similarly formed phos- phor bronze point, while Curve III represents data obtained using such a point more heavily formed. In all cases the magnitude of the floating potential decreases inversely as the distance from the point, and is given POINT-CONTACT TRANSISTOR SURFACE EFFECTS 775 2.0 1.0 0.8 0.6 0.5 0.4 0.3 0.2 0.1 0.08 1- 0.06 n 0.05 > z 0.04 > 1 0.03 0.02 0.01 0.008 0.006 0.005 0.004 0.003 0.002 0.001 - \ - III,Ic(0. -'0) =-1.5M \ \ 1 \ i V - \ K^^ V - k: 1 \ HJc^Oi "10^ =-1.0 MA ^ >1 S, \ ^^ s. H H ^J ■1 ^^ S H \ - ^ - I,lc(0, -10) = -0.08MaN 1 — I -^, H^ \ \ ■% \ \ 1 1 ^ 1 ■0^ O.l 0.2 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 8 10 r IN MILS 20 Fig. 2 — Comparison of floating potentials near formed points. by pI/2Trr where p is Avell A\ithiii the range of the measured resistivity (3-4 ohm-cm). Thus the effect of adding the donor to the point wire is to increase the reverse current and increase the floating potential near the point by an order of magnitude. One would therefore expect an accompanying 776 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 5.0 4.5 4.0 3.5 3.0 < J 2-5 2.0 1.5 1.0 0.5 0 [ Vc =-10 VOLTS \ \ \ V N II,HF -UNFORMED ' - J k I.HjOs-UNFORMED Y \ ? \ ^'— — — « 1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 CURRENT, le, IN MILLIAMPERES 5.0 Fig. 3 — Comparison of alpha-emitter-current characteristics of formed points. ,' I increase in the drift field near the point and a corresponding increase in a. Fig. 3 indicates that such is the case. The small signal a is plotted as a function of emitter current in Curves I and II. The point spacing in this case is 2.5 mils. It is interesting to note that the peak at low emitter currents is present in both cases, in spite of the fact that presence of a p-n hook is not likely when the Be-Cu point is formed. It is thus apparent that the forming the Be-Cu point produces a struc- ture which more closely resembles a p-n junction. The effect of adding the donor is to reduce the resistance of the junction. Further contrast between these two kinds of contacts is demonstrated by comparing for- ward currents through the contacts and their capacities. In Table I, a summary of all the contrasting properties is given. All values quoted are representative values. 2.2.2 Use of the Copper Plating Technique During the investigation of these contact properties, an interesting way of illustrating their physical properties was developed. This technique, borrowed from junction transistor technology, can be used to identify visually the boundary between the formed region and the bulk genua- f^ nium in a metallographic section of a point-contact transistor. It further appears that modifications of the technique will enable determination of k: POINT-CONTACT TRANSISTOR SURFACE EFFECTS 777 Table 1 Contact Formed Be Cu Formed Phosphor Bronze /CO —10) ma -0.01 ma -1.0 ma 2.8 ma 0.25 0.1 3.0 MMf - 1.0 ma T„(6 —5) ma -14.0 ma 7e(0, +0.5) ma Peak value of a 0.8 4.5 a (5.0, -10) Capacity (Fc = -5F) 1.7 < 0.1 jUMf the equipotentials surrounding a collector or emitter point under bias, and visualization of current flow patterns in point contact transistors under bias operating conditions. Use of this technique in identification of formed transistor properties is quite simple. A transistor container (including only the completed header, wafer, and point-contact structure) is filled with araldite plastic, which is allowed to harden. The collector point is then electrically formed. The plastic is necessary to ensure that the collector point does not subse- quently move from the formed area. The can itself is then embedded in a plastic block, which is lapped down to expose a cross section of the unit. Fig. 4(a) and (b). Both the collector point and the base electrode are well masked. Fig. 5. A droplet of CuS04 solution of fairly low concentration is placed on the germanium, so that it is in physical contact only with with the germanium and the masking plastic. In order to identify the formed region, a reverse bias of 20 volts or so is applied between the collector point and the base contact for a time usually of 0.1 second or less. Actually, best results have been obtained by applying the reverse PLASTIC -=r- PLASTIC BLOCK (a) Fig. 4 — Preparation of a transistor for copper plating. n 8 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 bias in the form of a condenser discharge pulse. Care must be taken to avoid changes in contact characteristics resulting from the plating pulse. The deposit of copper does not appear instantly after pulse ap- phcation, but may require several seconds before becoming visible. At the instant the deposit becomes visible, the plating solution is washed ofi-. Fig. G(a) and 6(b) show the results of the plating operation on a formed collector point and a formed emitter point. Both pulses were similar to, though somewhat "heavier" than those usually used to form transistors. These units were plated under the conditions illustrated in Fig. 5(a). The floating potential in the vicinity of the reversed bias point can be measured as a function of the distance, r, from its center, using an aux- iliary tungsten point. Qualitatively this potential is shown as a function of the distance, r, in Fig. 5(b). In this case most of the drop in magnitude of the potential appears within a radius, r, less than 0.002 inches, pro- vided surface conductivity is small. The conductivity of the plating solu- tion is kept small to ensure that the potential distribution in the ger- manium is not altered by presence of the solution. Under these conditions, it is assumed that, although copper ions in solution are at- tracted towards the highly negative regions of the germanium, the main current flow is through the germanium, except for regions of high poten- tial gradients. In these regions some of the current will be carried by ions in the solution, by -passing the region. If the formed region bound- ary is a sharp p-n junction, one would expect a plating pattern as ob- served in Fig. 6(b) and 6(d), as is observed with the donor-free emitter point. For the more complicated structure produced by forming the COLLECTOR POINT MASKING FORMED REGION BOUNDARY CU SO4 SOLUTION MASKING n-Ge 17" ■'-BASE CONTACT -V(r) (b) Fig. 5 — Experimental conditions for copixT ])lating. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 779 Vr = -20 VOLTS Vp = -20 VOLTS 0.25% CUSO4 (PULSE TIME =: 10//S) 025"y<, CUSO4 (PULSE TIME = lO/ZS) Mi LS Vq = -20 VOLTS (PULSE TIME = 10/uS 1 Vg = -20 VOLTS [PULSE TIME = 10/US) Fig. 6 — CopiJer plated formed layers in point-contact transistors. collector, the pattern obtained is more difficult to interpret, Fig. 6(a) and (c). However, in both cases the disturbed areas are roughlj^ compa- rable in shape and size. Differences in the forward characteristics of the collector and emitter points may also be graphically observed by means of the plating tech- nique. In Figs. 7(a) and 7(b) are sketches of patterns obtained by applj'- ing forward bias to contacts for plating. In this case a more concentrated solution is used, and the plating time is longer. In Fig. 7(a) is shown the pattern obtained when an unformed collector point is biased for- 780 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 ward during the plating pulse. The copper deposits to within the order of a diffusion length from the emitter point. Fig. 7(b) shows the pattern obtained by plating the region near a forward biased formed collector. Here again the copper has deposited over practicallj^ all of the base wafer surface, except for a much smaller hemispherical region near the collector point. By adjustment of the plating time and solution concentration, the almost radial field in the bulk germanium under a reverse-biased col- lector point can be detected. Under similar conditions, an emitter point biased to the same voltage shows a plating pattern similar to that of Fig. 6(b), with little evidence of the radial field. This would be expected from the potential plots shown in Fig. 2. These techniques serve merely to illustrate graphically the differences in the two types of contact. Although both points when formed give rise to a formed region in the bulk germanium of similar size and shape, the diode characteristics of the junction under the donor-doped point are degraded. The plating technique may also be adjusted to allow sensitivity to the current flow pattern in a transistor with both points biased to operating values. The example shown in Fig. 8 demonstrates visually the bulk nature of the current flow in the point contact transistor. Here the cop- per plates out on the negative regions of the crystal and is noticeably absent from the regions of high hole density under the emitter point. In the region to the left of the collector indicated by the arrow, the plating is partially obscured by masking. The size of the copper-free region under the emitter point may be i-educed to substantially zero for the same /, by increasing the bias applied to the collector. (a) t^^^, COLLECTOR POINT (BEFORE FORMING) "^"-S COLLECTOR POINT {AFTER FORMING) Fig. 7 — The effect of forming and current flow in point-contact collectors. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 781 2.3 Under-Formed and Over-Formed Contacts One of the problems encountered in the large-scale manufacture of point-contact transistors is the variation in the forming yield. Thus, forming to a specified criterion of transistor performance does not always result in a uniform product. Although considerable care may be taken to ensure uniformity of all bulk properties and forming technique, a large variation may be encountered in the output characteristics of the tran- sistors. In Section 4, a prime factor in determining the efficiency of form- ing is shown to be the chemical history of the germanium surface. Un- controllable variations in surface conditions may therefore often account for much of the variations in results of a specific forming technique. Such variations often manifest themselves merely as differences in degree, but may show up as differences in kind, takmg the form of anoma- lous output characteristics. These have been classified by L. E. Miller^^ into three qualitatively different phenomena. The first of these, referred COLLECTOR PLATING INHIBITED IN THIS AREA BY MASKING COPPER PLATED AREA VERY HEAVY PLATE UNDER COLLECTOR EMITTER - UNPLATED AREA UNDER EMITTER GERMANIUM TO BASE OHMIC CONTACT 0 5 SCALE IN MILS UNIT OPERATED AT LOW le; PLATED 0.25'Vo CUSO4, 20 SECONDS le = 0.5 MA, Vc = -20V, Oi - 0.1 Fig. 8 — Flow geometry for a low alpha point-contact transistor. 782 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 to as the type (1) anomaly, is of interest here since it represents a col- lector contact whose physical properties are between the extremes listed in Section 2.2. Miller has shown that the source of this kind of outpvit characteristic can be identified as the formed area under the collector point. Essentially this anomaly consists of an abrupt rise in the current gain as the collector voltage Vc is increased at constant emitter current. Be- yond the critical value of Vc , the characteristic of the unit resembles that of a well formed transistor. One is led to consider that such a con- tact is under-formed, in the sense that at low Vc , collection of holes is inadequate. Further support is lent to such a definition by the data of Miller, which shows a definite increase in the occurrence of anomalous units with a decrease in the Ico of the contact. Such an increase occurs regardless of whether the Ico decrease is obtained by decreasing the donor concentration of the point wire, or by increasing the time constant of the forming pulse. In Table II are compared collector capacity and Ico measurements made in units with and without output characteristic anomalies. The capacity of these anomalous collectors also appears to range between the two extremes listed in Table I. Thus there is evidence that these collectors are intermediate between the extremes cited in Table I in the sense that at low reverse biases the drift field is low, and the properties of the formed barrier resemble those of a formed donor- free point. The results of detailed investigation of the properties of such anoma- lous characteristics now being conducted will be published at a later date. The present experimental results indicate that the instability oc- curs when the extra current to the collector. Ale , reaches a critical value. In this respect, increasing the transport factor (3, by increasing Vc , or increasing the emitter current are equivalent. At a roughly critical Ale , the transition between a low a and a higher value of a occurs. After the transition, the unit behaves like a conventional point contact transistor, with a current multiplication on the order of (1 + &) at higher values of le . Thus the origin of this kind of anomaly may lie in the lowering of the formed barrier by the space charge of the holes, a mechanism sug- gested by Bardeen. Table II Idle = 0, Fc = -10 volts) Typical Transistor Typical Anomalous Transistor . ] .0 Ilia 0.2 ma Cede = 0, ^0 = -10 volts) (I. 1 nix( 0.5 fi/jif POINT-CONTACT TRANSISTOR SURFACE EFFECTS 783 The other anomalous collector characteristics considered by Miller have their origin in the relation between the transport factor and the properties of the emitter at various operating conditions. In view of the relations existing between the occurrence of these anomalies and the Ico of the collector contact, there is some justification for classification of these contacts as "over-formed." 3. PROPERTIES OF UNFORMED POINT CONTACTS 3.1 Physical Properties oj Metal- Semiconductor Contacts The classical ideas on the nature of the rectifying metal-semiconductor contact have undergone substantial revision since the consideration by Bardeen of the importance of surface states and the work on the point contact transistory by Bardeen and Brattain. According to Bardeen's model, the nature of the space charge layer at such a contact is to be considered largely independent of the metal used for contact, and is pri- maril}^ dependent on the charge residing in localized states at the ger- manium surface. Thus the rectifying properties of the metal semiconduc- tor contact in air are expected to be largely independent of the work function of the contact metal. The question of the exact nature of the surface charges is not yet read- ily answerable. Charges may arise which consist of electrons and holes residing in surface states of the type proposed by Tamm.^' On the other hand, other surface charges may arise as a result of adsorbed impurity ions, or from adsorbed atoms or molecules having electrical dipole mo- ments. Brattain and Bardeen^ have shown that the space charge layer is dependent on the surrounding ambient and have indicated that charge may reside on the outer surface of a film (presumably an oxide laj^er) at the germanium surface as well as in surface states of the type men- tioned above, which are presumably those responsible for surface re- combination processes. Thus, it is the surface charge on the semiconductor, rather than the nature of the metal, which primarily determines the nature of the po- tential barrier which exists at a metal semiconductor junction. A schematic electron energy diagram for the contact between a metal and an 7i-type semiconductor is shown in Fig. 9. The potential barrier ^0 , and the nature of the space charge layer in the semiconductor are determined by the surface charge system and the bulk properties of the semiconductor. In turn, the surface charge system is dependent upon such factors as the ambient at the germanium surface and the chemical history of the surface. 784 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 -METAL n-TYPE SEMICONDUCTOR- Fig. 9 — Electron energy diagram for a metal -semiconductor contact. The experiments of Brown^ indicate that the presence of charge on the surface of p-germanium can alter the space charge in the crystal near its surface and, in some cases, produces an inversion layer of n- germanium at the surface. Garrett and Brattain^° have shown that a change of ambient from sparked oxygen to dry oxygen to wet oxygen can increase Ico and floating potential on n-p-n junction transistors, and the process is reversible. Their interpretation is that sparked oxygen builds up a film, presumably germanium oxide. Oxygen atoms on the surface, negatively charged, can give rise to a p-type inversion layei on n-germanium. Moisture apparently counteracts this negative charge, and humid oxygen can cause an n-type inversion layer on p-germanium, which can be removed with a dry oxygen ambient. Thus, the electrical resistance of an unformed metal-germanium con- tact on an etched germanium surface can be expected to be extremely sensitive to any chemical treatment which tends to affect the constitu- tion of the oxide layer present on the surface, regardless of the metal used for contact in air. Bardeen and Brattain,^ in early transistor ex- periments, have shown that such is the case. They have used transistor collector points on germanium surfaces which, after etching, were sub- jected to an oxidation treatment (heating in air). In this section are described experiments which seem to indicate that the reverse resistance of unformed diodes on etched n-germanium sur- faces can be decreased by chemical surface treatment, and the magnitude of the floating potential near such contacts is increased to sufficient ex- tent that the point can serve as a multiplying collector. Average a for POINT-CONTACT TRANSISTOR SURFACE EFFECTS 785 these points approaches values found in electrically formed collectors. Subsequent parts of this section will be concerned with description of the experiments involved and comparison of the electrical characteristics of these points with those of conventionally formed points. The effections of electrical forming on donor-doped and donor-free point contacts have been described in earlier sections. It has been stressed that the addition of the donor element to the point results in a contact with degraded diode characteristics, but which serves as an excellent collector. The possibility of an analogous situation in an unformed point collec- tor exists, with the electrical forming of the donor-doped point being replaced by a suitable chemical treatment of the surface. The experi- ments described below indicate that such is the case. 3.2 Experimental Procedures The germanium used in these experiments was zone-leveled material. The n-germanium was in the 3 to 4 12-cm range. Originally, experiments were run using slices, about 0.025 in thickness, soldered on flat brass blocks, with the brass well masked with polystyrene. Germanium dice, already mounted on standard base-header assemblies used in a hermetic- seal transistor process pilot line, were also used. The ground surface of a slice was given a three-minute chemical etch (CP4 or superoxol), washed in pure water (conductivity <0.1 micromho), and blown dry in a nitrogen stream. This surface could then be exposed for several minutes to 24 per cent HF, hot zinc chloride-ammonium chlo- ride solder flux, or other chemical treatments as the experiment might require. These solutions were applied to the slice or die in the form of large droplets, so the solution did not come in contact even with the masking. Later, in order to make doubly sure that contamination from the base or base contact was not involved, all experiments were repeated using a two-inch length of a zone leveled bar with a base contact soldered on one end, and the other end, freshly ground between treatments, used as the surface under examination. The etching was done by lowering one end of the bar about one-half inch into the etch, leaving the contact end a good distance from the etch. The etched surface could subsequently be exposed to any desired chemical treatment. After the chemical treat- ment, the sample surface was again washed in low conductivity water for several minutes and blown dry with nitrogen. The sample, after chemical treatment, was placed on a double ended manipulator base, used to control the position and pressure of two canti- 786 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 « 'J lever points on the treated surface. The electrical characteristics of a beryllium copper point, operating as transistor collector on the treated surface, could then be investigated. An auxiliary etched tungsten point doubled as a potential probe and as an emitter. A switching arrangement allowed oscilloscope presentation of the h-Vc collector family and the alpha-emitter current sweep, measurement of the emitter floating poten- tial on a high impedance VTVM, and determination of other transistor parameters for any desired position of the emitter point. Phosphor bronze collector points were not used since it was found that, on certain chemically etched surfaces the mere application of a negative bias of 15-40 volts for a few seconds sometimes is sufficient to cause elec- trical forming of the point in the sense that Ico and average a are in- creased by an appreciable amount. The beryllium copper points were carefully cleaned to prevent con- tamination by donor elements. Their cleanHness was then tested by other methods described in Section 3.3.6. With this arrangement, most of the electrical properties of a given manipulator unit could be inspected during the time the unit ''survived." These electrical measurements were made in room air (R. H. between 20 and 30 per cent), although provision was made for directing a con- tinuous stream of dry nitrogen at the points and surrounding surface. 3.3 Experimental Results 3.3.1 Unformed Transistors on Superoxol Etched* Surfaces A striking difference was observed in the electrical characteristics of unformed collector points on the \'arious n-germanium surfaces ex- amined. In particular, surprisingly large values of 7c.(0, —10) and /c(6, —5), (the latter taken as a measure of average a), were encountered on the superoxol etched surface subsequently "soaked" for about 10 minutes with 24 per cent HF. At these locations the unformed transis- tor action was quite similar to that observed with a conventional phos- phor bronze point formed on a freshly etched surface. These large values were found only in specific locations on the treated surface, there being a random fluctuation of 7,(0 —10) and /c(6, —5) with location of the points on the surface. However, no such large values of these parameters were found (together) on surfaces freshly etched in superoxol. The a as a function of emitter current for the unformed points (2.5 mil spacing) on a superoxol etched surface, before (Curve I) and after (Curve II) HF treatment is shown in Fig. 10. Comparison with One part 30 por cent H2O2 , one part 48 per cent HF and four parts water. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 787 4.5 4.0 3.5 3.0 < 2.5 I Q. < 2.0 1.5 1.0 0.5 K Vr =-10 VOLTS \ \ k \ > II,Ph Br -FORMED M ' — [ ~ <^^ ' I, Be CU- FORMED > 1 g 1 < 9 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 CURRENT, Ig, IN MILLIAMPERES 4.5 5.0 Fig. 10 collectors. Comparison of alpha-emitter-current characteristics for unformed Curve II, Fig. 3, indicates that the «(/£), obtained after the HF treat- ment, is comparable to that of a phosphor bronze collector formed con- ventionally on the same etched surface before treatment. (It turns out that conventional electrical forming on the etched surface after the HF treatment is more difficult, and in cases as referred to above, where the a is not initially high, requires an excessive number of pulses to bring the a to a normal value.) In Table III are listed the maximum and minimum values of some transistor parameters found on the same superoxol-etched surface be- fore and after the HF treatment (point spacing about 2 mils). It is seen that the effect of the subsequent HF treatment after the superoxol etch is at least in some locations on the treated surface to in- crease the /c(0, —10) and the average a, in some cases to values ap- proaching those encountered in conventionally formed point-contact transistors. There is also a lowering of the forward current of the un- formed collector point after the HF treatment. It is not to be implied from this table that the Ico is always found to be low on fresh superoxol- etched surfaces. Actually high values of 7c(0, —10) have been occa- sionally found on surfaces freshly etched in superoxol. However, these collectors seldom have high values of average a, and it is suspected that here the higher reverse current is associated with excessive surface conductivity. Treatment of such a surface with HF always serves to increase the average a, and decrease the forward emitter current, with 788 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Table III Parameter /c(0, -10) ma. . /c(6, —5) ma. . , 7c (0, +0.5) ma. Peak value of a a (5.0, -10).... After 3 Min. Supero.xol Etch Max. Value Observed -0.16 -7.0 2.8 3.0 0.50 Min. Value Observed -0.06 -0.95 2.0 0.15 0.09 After Subsequent 10 Min. "Soak" in 24% HF Max. Value Observed - 0.98 -13.5 2.3 6.0 2.0 Min. Value Observed -0.20 -7.0 1.3 2.0 0.5 no significant changes in the extreme values of IdO, —10) encountered initially. Some of the unformed units have collector families quite simi- lar to those of an electrically formed point-contact transistor. However, the resemblance ends when stability of operation is considered. Wlien the unformed units are operated in room ambient, hysteresis loops are occasionally observed, either in the Ic-Vc output characteristic sweep, or the a-emitter current sweep. This hysteresis can be eliminated by di- recting a stream of dry nitrogen across the germanium surface in the , vicinity of the points. It is not known whether the hj^steresis is thermal or electrolytic in nature. The operation of these unformed units, even ; in the absence of hysteresis, is extremely erratic and unstable. Operating ^> a unit at a high power level will cause loss of a and Ico , and mechanical shock delivered to the collector point while the unit operates under bias may cause loss or gain of a and Ico • In cases where Ico (and a) are low when the collector point is initially set down on the treated surface, an increase in Ico and a may be brought about by mechanical motion of the point, (such as "tapping" the manipulator base, or dragging the point across the surface). In other cases the high a and Ico are found immedi- ately after the point is set doAvn on the freshly treated surface, without any such procedure. None of these effects is observed to an appreciable degree on a freshly etched surface without further treatment. The effect of zinc chloride-ammonium chloride solder flux on fresh superoxol-etched surfaces was also investigated. In this case, after the etch, the surface was immersed in almost boiling solder flux for about ten minutes. The effect of this surface treatment on the performance of the unformed transistors was entirely similar to the results quoted in connection with the HF treatment. The treatment increased the reverse collector current and average a, and decreased the forward collector current, on the average. Magnitudes of /c(0, —5) as high as 14 ma were observed on surfaces treated in this way. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 789 3.3.2 Unformed Transistors on CPi-Etched Surfaces With reference to unformed point contact properties, the CP4-etched surface is not at all similar to the superoxol etched surface. If two beryl- lium-copper points are put down on a ground surface freshly etched; in CP4 , and operated as a transistor, high values of 7c(0, — 10) and /<.(6, —5) are often encountered. However, after an hour or so in room air, both these parameters decrease and after an overnight exposure to room air, the properties of the surface with regard to the transistor ac- tion resemble those of a surface freshly etched in superoxol. At this point, a treatment in 24 per cent HF will return /c(0, —10) and 7c(6, —5) to their originally high values. These effects are summarized in Table IV. 3.3.3 Diode Characteristics on Electro-Etched Surfaces It has been found that the rectification properties of unformed point diodes may also be changed conveniently by changing the conditions during an electrolytic etch in KOH solution. These results are summa- rized in Table V which represents typical variation in reverse current, Ir , with surface variation attainable by adjusting the current density and etching time. In each case the measurements represent data taken on germanium cut from adjacent sections of the same ingot and given the surface treatment noted in the table. In general the electro-etched and chemically etched results agree; that is, any treatment which ap- pears most likely to leave an oxide film (such as the use of a high current density during electro-etching) will yield a diode with improved rectifica- tion characteristics. I 3.3.4 Output Characteristic Anomalies In the process of examining these chemically treated surfaces, some i of the superoxol-etched n-germanium surfaces were given additional Table IV Value after 3 Min. CP4 Etch Value after 16 Hrs. in Room Air Value after 10 Min. in 24% HF Max. Value Observed Min. Value Observed Max. Value Observed Min. Value Observed Max. Min. /e (0,-10) ma.. /c(6, —5) ma. . Peak value of a (5.0, -10)''^' -1.7 -13.3 4.5 1.8 -0.30 -11.0 2.5 1.0 -0.10 -7.0 2.0 1.0 -0.04 -2.0 0.75 0.25 -1.0 -17.5 9.0 2.0 -0.06 -8.0 3.0 0.75 790 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G Table V Etch Treatment in 0.1% KOH /, (-10 volts) 10 ma for 30 sec 5 ma for 30 sec 2.5 ma for 30 sec 5 ma for 1 min -0.16 ma -0.37 -0.55 -0.04 2.5 ma for 1 min 1 . 75 ma for 1 min -0.18 -0.74 treatments in H2O2 (superoxol strength). In general, no great differences were observed in the unformed alpha and /c(0, — 10) after the treatment. However, in isolated cases, unformed units made on etched p-gerraanium treated in this way exhibit output characteristic anomalies of the type characterized by Miller as type (1). It was later found that the same surface treatment can produce a similar result on etched 11-germanium surfaces, again only in isolated locations on the surface. An output char- acteristic of this form is shown in Fig. 11. This unformed unit was made on a superoxol-etched n-germanium surface with a subsequent three- minute soak in H2O2 . This characteristic was extremely sensitive to variation in point pressure. Miller has also referred to output anomalies of types (2) and (3), which are usually associated with close point spacing in conventional point- contact transistors. Such types of anomaly have been observed in un- formed units (with high average alpha) made on HF treated surfaces. 3.3.5 Floating Potential Measurements In all cases where the /c(0, —10) and average alpha on etched sur- faces are increased by the HF or solder flux treatment, these increases are accompanied by an increase in the magnitude of the floating poten- tial near the reverse-biased collector. In Fig. 12 the magnitude of the floating potential Vp of a sharp tungsten probe near the reverse-biased collector is shown as a function of r, the distance of the probe from the collector (r is approximately the distance between the center of the two point contacts). The surface used in this experiment was prepared by chemical polish for three minutes in CP4 and subsequent storing in room air for sixteen hours. This provided a smooth surface which resembled, at least with regard to electrical characteristics, a freshly etched super- oxol surface. Curve I represents the potential-distance plot for an unformed BeCu point on the aged superoxol-etched surface. Curve II represents a similar plot for an unformed BeCu point taken after the surface was given a ten-minute soak in 24 per cent HF. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 791 The measured resistivity of the germanium used in this experiment was 3.3 to 3.6 fl-cm. It can be seen from Curves I and II that increase in the magnitude of the floating potential near the unformed point on the etched surface after the HF treatment is, to a rough extent, propor- tional to the increase in /c(0, — 10) produced by the treatment. Values of 2irVpr/I taken from lines of slope ( — 1) drawn for best fit through points on the individual curves give reasonable agreement with the measured resistivity. For curve I, 27r F^r// = 3.3 ohm-cm, and for Curve II, 2TrVpr/I = 3.5 ohm-cm. By comparing Curves I and II of Fig. 2 with Curves I and II of Fig. 12, it can be seen that the effect of treating the surface under the un- formed point with HF is analogous to adding donor to the formed point -10 CURRENT, Ic.lN MILLIAMPERES -7 -6 -5 -4 -3 -2 Fig. 11 ■ — Type (1) collector anomaly observed in unformed unit (n-t,\])e germanium) . I 792 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 0.20 0.10 0.08 0.06 0.05 0.04 0.03 O 0.02 > t 0.010 0.008 0.006 0.005 0.004 0.003 0.002 0.001 ^N - 1 — c \, - \ S^ H, Ic(0,-10) = -0.9M A ^ 1 \ -OH \ ^ fO' N '^^ - \ - i-K>^ k I, Ic(0,-10) = -0.07MA^ i-C k N K V >■ 1 1 1 1 I 0.1 0.2 0.3 0.4 0.6 0.8 1.0 r 2 IN MILS 4 5 6 8 10 20 Fig. 12 — Comparison of floating potentials for unformed point-contact col- lectors. on the etched surface. It seems reasonable to ascribe the increased nega- tive floating potential after the HF treatment to an increase in current density through the surface under the point, rather than to any increase in surface conductivity. It is worth noting that on superoxol-etched sur- faces, the negative floating potential near an unformed collector point can often be increased by an order of magnitude by blowing a stream of dry nitrogen near the point. This effect may possibly be a result of excess surface conductivity, but in these cases is not accompanied by any appreciable changes in IdO, — 10) or average alpha. 3.3.6 Contamination of Collector Points and Surfaces Past experience Avith use of point-contacts as transistor collectors indi- cates that experiments may often be confused or confounded by unsus- POINT-CONTACT TRANSISTOR SURFACE EFFECTS 793 pected contamination of the points used. For this reason, particular attention was given to chemical processing of the beryllium copper points used in the preceding experiments. These points were chemically cleaned to remove oxides and unwanted contaminants, and carefully washed before use. Several lots were processed at different times, and all experi- ments repeated on the different lots, with no contradictory results. It is particularly important that the point be free from donor elements, since it has been observed that that phosphor bronze points or "poi- soned" beryllium copper points washed with a lithium chloride solution often exhibit on superoxol etched surfaces a kind of "forming" after the application of reverse bias. The symptoms of this are a sudden increase in I CO which take place as the reverse bias is increased above 15-20 volts. The alpha emitter current sweep shows evidence of excessive noise in such a case, and it is not until the collector is given a conventional form- ing pulse that this excessive noise is ehminated, and the unit becomes stable in operation. A donorless point can be reasonably identified by the fact that elec- trical pulsing, heavy or light, will not increase the initially low average alpha on a superoxol-etched surface to values much above 1 .0, although J CO may be increased or decreased depending on the type of condenser discharge used. The beryllium copper points used were tested on super- oxol-etched surface to make sure they showed no tendency to form electrically. If high values of alpha can be found when these points are used as un- formed collectors on the surfaces treated in HF or solder flux, the ques- tion arises whether such values may be attributable to presence of a donor element left on the surface in some mysterious way by the chemi- cal treatment. If such is the case, the donor might, at high enough re- \ erse bias, be responsible for an increased alpha in a manner similar to that observed in connection with the forming in under bias of phosphor bronze collectors on etched surfaces. Two precautions were taken in this connection. No reverse bias greater than 10 volts was ever applied in- tentionally to these collectors during experiments (with exception of the iunit in Figure 11), and secondly, forming characteristics of both phos- jphor bronze points and the beryllium copper points on this type of sur- face were investigated. It was found that on a superoxol surface treated with HF or the solder iflux, a phosphor bronze point would form to a high average a, but this invariably required more forming pulses than on a superoxol etched [surface. "One-shot" forming is common for a superoxol etched surface, 'whereas after the HF or solder-flux treatment, forming to high average 794 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Table VI /c(0, -10). a(5.0, -10) Noisv Point Beryllium Copper Collector Phosphor Bronze Collector Occasion Before Forming -0.75 1.5 Yes After Four Forming Pulses -0.50 0 Yes Before Forming -0.80 1.5 Yes After Four Forming Pulses -2.5 2.0 No alpha invariably requires at least three and sometimes many more-, ''shots", although it can be done. This type of formed unit does not exhibit excessive noise in the a-I^ sweeping gear. However, pulsing of the beryllium copper points on the latter kind of surface, in similar fash- ion, invariably results in loss of alpha and never eliminates the exces- sive noise. Initially, the pulsing decreases the Ico magnitude, but con- tinued pulsing will eventually cause large increases in this case. These results provide circumstantial evidence, at least, that the treated sur- face and the point are operationally free of any donor element and that the transistor collector barrier involved is at the germanium surface. For example, in Table VI are given some typical data obtained during pulsing of points on a superoxol-etched surface after treatment with near boiling zinc chloride-ammonium chloride solder-flux. A tungsten emitter was used.* 3.4 Discussion of Experimental Results .3.4.1 Effects of the Chemical Treatment on the Superoxol-Etched Surfaces It might be presumed that an inversion layer and a relatively high surface conductivity is responsible for the increase in negative floating potential and reverse current observed on the superoxol-etched n-ger- manium surface after the HF treatment. On the other hand, if it be assumed that at the etched surface, in room air, an inversion layer ex- ists which does not introduce excessive surface conductivity, one can say that the effect of the HF treatment is merely to raise the surface potential, (i.e., to reduce the barrier height for electrons). This might * Alpha values are usually lower in any given situation when the conventional chisel-type beryllium copper emitter point is replaced by an etched tungsten point. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 795 account for the increase in reverse current density* and a proportional increase in the magnitude of the floating potential near the point. In this case the geometry of current flow across the contact should remain relatively unchanged as indicated by the floating potential measure- ments. In this way the effect of the HF treatment is somewhat analogous to the addition of a small donor concentration near the surface to coun- teract the inversion layer. Since soluble oxide layers' have been identified on etched germanium surfaces, it is not unlikely that HF (known to dissolve germanium oxide)" might act to reduce the effective thickness of an oxide layer. Such a hypothesis is in agreement with the results of other experimenters/^ who have attributed a surface inversioji layer under the point of an n-germanium rectifier to the presence of germa- nium oxide. They have presumed the oxide is essential to the formation of a good point contact rectifier. The fact that, for a given ambient, the surface potential is determined by the oxide layer thickness has been postulated b}- Ivingston.''* 3.4.2 CPi-Etched Surfaces Sullivan,"" in connection with an experimental investigation of hu- midity stability of electrolytically-etched and chemically-etched p-n grown junction diodes, shows that CP4 chemically-etched surfaces be- come more stable \\ith respect to humidity variation after humidity exposure and cycling at room temperature. Referring to the fact that electron diffraction studies fail to reveal a crystalline oxide film on CP4 chemically-polished surfaces and to the results of Law,"*" which indicate that oxide films may be formed slowly at room temperature on exposure 1 0 water vapors, he attributes the changes of stability on the CP4 polished surface to the building up of an oxide film. If such a change can take place on the CP4 chemically-polished surface on exposure to humid room air, then the results of Section 3.3 can be understood under the assump- tion that the action of the HF treatment is to remove the oxide film. After the chemical polish, values of /c(0, —10) and average alpha for the unformed units are high, as might be expected if the polishing opera- tion leaves the germanium surface with no appreciable oxide film. As the oxide film builds up on continued exposure to room air, both of these parameters are reduced. The subsecjuent application of HF tends to lestore these parameters to their original ^'alues by removal of some of this oxide film. Thus, the results of this section are in accord with the * Evidence for an increase in surface recombination velocity on HF treated surfaces is given in Section 4.2.3. 796 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 hypothesis discussed in the previous section to account for the effect of HF on the unformed transistors. Such evidence, however, is at best only indirect evidence for the build- up of an oxide layer on prolonged exposure to room air. In experiments Avith grown p-n junction diodes, the authors have found great variations in the length of time required for the electrical properties of the diodes to recover after short wash periods in low conductivity water. Thus the slow changes mentioned above may at this point result from simply a longer time required for the surface to "dry out" after the washing treat- ment. However, a substantial difference in the physical properties of the oxide layer left by the two etches concerned is still implied. In this con- nection it is also worth noting that hysteresis effects appear primarily in unformed units made on HF treated surfaces. The results of these experiments have important implications in the technology of point contact transistors. The results of an application of these results to transistor forming procedures are given in the following section. 4. RELATION OF GERMANIUM SURFACE PROPERTIES TO TRANSISTOR FORMING 4.1 Pilot Production Problems The pilot production and early manufacturing stages of cartridge- type point-contact transistors has generally been characterized by peri- ods during which the forming yields have been very high and similar periods of very low yield. Often these alternate periods occurred during the use of germanium taken from the same rod-grown or zone-leveled crystal. Considerable effort has been expended in attempting to corre- late these variations in yield to variations, from crystal to crystal, or in different portions of the same crystal, or such bulk properties as re- sistivity or minority carrier lifetime. Although these properties of ger- manium do have some effect on device parameters such as average alpha, reverse emitter current, and Ico , there has not been any positive indica- tion that variations in yield are attributable to the amount of variation of bulk properties normally found in the germanium which meets the specifications of the particular device concerned. This problem was compounded during the early stages of the develop- ment of the process for hermetically sealing the point-contact transistor. It was found that although reasonable yields were obtained in the car- tridge process, equivalent transistors in the hermetically sealed structure were made only with greatly reduced yield. Further, although micro- POINT-CONTACT TRANSISTOR SURFACE EFFECTS 797 manipvilator units could be made with no difficulty, the same material fabricated into a completed structure showed completely different char- acteristics. In the course of investigation of this problem, it was ofund that the nature of the germanium surface treatment and specifically treatments calculated to produce or react with germanium oxide can profoundly affect the "formability" of the germanium surface as well as a number of other transistor parameters in the fabricated units. It is the purpose of this section to emphasize the importance of con- sidering the surface properties of germanium in attempting to solve such specific problems of development encountered in devices of this type. In particular, the striking variability of transistor forming on etched germanium surfaces subjected to varying chemical treatments and am- bients will be described, as well as the effects of such pre-forming treat- ments on the parameters of the finished units. The experiments discussed in the previous section indicate how changes in the double layer at the germanium surface can influence the characteristics of an unformed point diode. In turn, the experiments below indicate how the character- istics of the unformed diode are related to the device properties of the transistor collector produced by forming the diode. 4.2 Experimental Results 4.2.1 Pilot Process Forming Yields The forming yield of a point-contact transistor is determined by the \'alues of the acceptance criteria and the allowable limits for each of these. Often, different criteria as well as different forming techniques are used j for different transistors, so that direct comparison of results is quite I complex. There are, however, certain common requirements placed on all point-contact transistors: (a) The unit is formed so that the average alpha is roughly two or more. The collector current at a relatively high emitter current and low ! collector voltage is usually an approximate measure of this value, /c(6, —5) for example. (b) The collector current with no emitter current flowing should be as low as is commensurate with the first objective. The other transistor parameters are either directly or indirectly re- lated to these. The number of pulses required to achieve the minimum forming objective, therefore, is one direct measure of the formability of a particular transistor; the average alpha obtained after pulsing is an- other. However, one must consider both average alpha and Ico , since \\ hile forming to a given average alpha, the Ico may increase prohibi- 798 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 19 ^ 18 O z tr 1 7 o u. \^ 16 < 1 5 5 z m I "t. 1 3 T 12 tr U 1 1- 10 A, micromanipulator/ \ n ^ /\ N /^ i ) -\ \ l\ r N r \/ \ 1 Y A U V header/ k ^ k A U ^^ Ly ^^ -\ /; U1 M 1 V V' V / KJ ' yX UNITS ? / N J W 4 6 8 10 12 14 16 GROUP FROM SAME ZONE LEVELED BAR 18 20 Fig. 13 — A pilot production process control chart. tively. In later sections the authors have adopted the ratio Ic(Q, —5) /c(0, —20) as a measure of the success of the forming. The 2N21 transistor is a hermetic seal version of a point -contact medium-speed switching transistor. During the early stages of the de- velopment of this device, it became evident that although similar ger- manium and point wire are used for l^oth structures, the electrical parameters by which the devices are characterized belong to different universes. However, if the geometery of the 2N21 unit is duplicated in a manipulator transistor, the resulting device parameters do resemble those of the earlier unsealed unit. It is therefore likely that an unknown variable in the 2N21 process is responsible for the different universes mentioned above. The effect of such a variable is shown in Fig. 13, which shows a chart of a continuous process control. Each point represents the average of foin- different units sampled at the particular point in the process denoted in the legend. The micromanipulator data represents measurements taken on wafers which have been processed up to but not including point wire attachment. The curve denoted ''header" repre- sents data taken immediately after the point-wire attachment. This is one additional process step beyond th(^ point at which the manipulator data was found. It is evident from this curve that a severe degradation POINT-CONTACT TRANSISTOR SURFACE EFFECTS 799 Table VII Treatment None ZnCl2-NH4Cl Flux Flux and heat .... Ave. No. of Pulses to Form 2 3 7 Average /c(6, -5) -13.8 ma -13.5 -10.4 Average /c(0, -20) -1.7 ma -1.8 -6.0 Fig. of Merit /c(6, -5)/ IciO, -20) 8.1 7.5 1.7 in the attainable average alpha has oeciirred even though the forming objective was the same. Finally, the curve denoted "unit" represents data on the first four completed units out of the same group from which the manipulator and header samples were taken, A slight decrease in average alpha is ol^serA^d at this point. However, previous experience has indicated that this is an expected effect caused by the addition of the impregnant. This chart suggests that the point soldering operation in the process is causing a significant degradation in the formability of transistors passing through this step.* This process step consists of placing the germanium wafer, which has already been etched and mounted on the header, in a point alignment tool. The point spacing and force is adjusted and the points are then soldered to the header point-wire support. In the early stages of this process a corrosive zinc chloride-ammonium chloride solder flux was necessary to obtain efficient soldering. The effect of this solder flux on the formability of micromanipulator transistors made on such surfaces is shown in Table VII. These units were formed to the acceptance cri- terion of Vc(S, —5.5) ^ 2.0 volts. Each figure represents the average of ten imits treated in the same way. The value of the use of a figure of merit such as suggested earlier is illustrated in this table. Since the average alpha (denoted here by /c(6, —5) is related to the forming objective, one might presumably keep forming until the average alpha was the same as for an easily formed transistor. In this case Ico tends to increase. Under these conditions, if one examined only average alpha, the data might easily be misleading. From an examination of the figures of merit in Table VII one concludes that the corrosive flux plus a heating cycle tends to degrade the ger- manium surface to such an extent that transistors are formed only with great difficulty. The finiction of a flux during the soldering process is to remove any * Curves of this nature have also been obtained by N. P. Burcham in in- vestigation of soldering flux effects in hermetically sealed point contact transis- tor processes. 800 TEH BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Table VIII Treatment No. of Pulses to Form Average /c(6, -5) Average /c(0, -20) Figure of Merit /c(6. -5)/ IciO. -20) 3 min. in normal superoxol etch 1 min. in 48% HF 2 4 1 — 15.5 ma -10.2 -17.7 -0.69 ma -3.2 -1.9 22.4 3.2 1 min. in 30% H2O2 9.3 oxides which are present so that a good solder joint may be made. Since the oxide on chemically-etched germanium is likely of the soluble form, one might assume that the results of Table VII imply that the action of the flux and heat tends to dissolve or remove this layer. Also implied by the data is that the presence of such an oxide layer is essential to efficient forming. The experiments summarized in Table VIII further substantiate this hypothesis. These data represent manipulator transistors made on the same germanium wafers which had been treated in succession to a normal superoxol etch, a treatment in 48 per cent hydrofluoric acid, and a treat- ment in hydrogen peroxide, superoxol strength. Since the soluble form of germanium dioxide is known to react with hydrofluoric acid, it is presumed that the action of the HF is to partially or wholly remove any oxide left by the etch. The H2O2 tends to restore the original surface conditions left by the etch. Each figure represents the average of five transistors formed to the 2N21 acceptance criterion, (Vc(S, —5.5) ^ 2.0 volts). In this case the hydrogen peroxide treated units have an extremely high average alpha, but the Ico is also higher than for normally etched units. In terms of the device properties, a unit with a more or less typical average alpha with a low Ico is more desirable than the one Avith an extremely high average alpha but accompanying high Ico • It has not been determined whether the Ico would be lower for the superoxol treated units if it had been possible to form to the same average alpha as the normally etched units. This is an important piece of device design in- formation which is currently under investigation. It is clear from these experiments that the nature of the germanium surface, and most probably the nature of the germanium oxide layer on it, to a large extent, determines the properties of the transistor formed on this surface. Direct application of this knowledge to the fabrication process of the hermetically sealed point contact transistor has been carried out by N. P. Burcham. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 801 4.2.2 Relation of Unformed Diode Characteristics to Transistor "Forma- bility" From the results of the previous sections, it appears that superoxol- etched germanium surfaces treated with reagents in which germanium dioxide is soluble provide point contact diode characteristics unsuited to electrical pulse forming. Part of this difficulty, manifested in the in- ability to reach a specified value of average a without a prohibitive in- crease in I CO , probably results from a lower injection efficiency, 7, for the emitter on such a surface. This seems reasonable in view of the lower forward and higher reverse currents indicated in Table III produced by an HF soak. In Section 4.2.3 evidence will be shown that surface recom- bination is greater on n-type germanium surfaces treated with HF. This effect can also lead to difficulty in forming to high a without increase in Ico , since, for the same drift field, one would expect more minority carriers to die at the surface during their transit to the collector. On the other hand, there is evidence for believing that the nature of the forming process itself may be quite different on an HF treated sur- face. Fig. 14(a) shows the time dependence of the collector voltage dur- ing a typical condenser discharge forming pulse. The envelope of the voltage pulse follows roughly an exponential de- cay of a condenser-resistor series combination. However, inspection shows that during the discharge time, the resistance of the combination 400 300 _l o > 200 z > 100 V (a) v Ih r\ y'«^ t, V y r ^ r- — I— ( tri ..LU Z liJ uj Q- q: < D o z (b) n / — s A^ ^U l> ~^ S- 50 100 150 200 250 300 350 TIME IN MICROSECONDS 400 450 500 Fig. 14 — Collector current and voltage versus time for a condenser discharge forming pulse. 802 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 if) o > 2 > 300 200 100 V \ \ \ V ^- 0 . 0 50 100 150 200 250 300 350 400 450 500 TIME IN MICROSECONDS Fig. 15 — Forming voltage pulse for HF treated surface. undergoes a succession of breakdown and recovery intervals. In Fig. l-l(b) is the accompanying plot of current against time. Comparison of these two plots shows that following the application of the voltage, the resistance of the contact decreases until a rather sudden more rapid de- crease in resistance occurs, taking place at time h . In view^ of this time scale, the first decrease can be attributed to a heating of the contact, a form of thermal breakdoAvn at the metal-semiconductor surface.^^ Any reason invoked to account for the second more rapid decrease in resistance must account for the short time (a few /xs) in which this change occurs. In any event, shortly after the second "breakdown," a quenching results, with the collector resistance returning to a \'alue nearer to its original value. This sequence of events is roughly repeated until the condenser is discharged. The properties of the contact at nominal reverse voltage and currents are usually changed as soon as one such condenser discharge pulse has occurred, and often one such pulse is sufficient to reach the forming ob- jective. A typical forming pulse obtained under similar conditions to those for Fig. 14 is shown in Fig. 15, with the exception that the surface has been treated in HF for a few minutes. On this case it is apparent that the second, rapid breakdown is entirely absent. The well-defined form- ing pulse of Fig. 14 is usually obtained on surfaces with good pre-forming diode characteristics, and results in production of a usable transistor. From results of the previous sections it is well established that etched surfaces treated with reagents in which germanium dioxide is soluble provide point contact diode characteristics unsuited to electrical pulse forming. It is often assumed, on the basis of the results of Valdes,^ that forming effects result from the diffusion of impuiities from the point into the semiconductor during the forming pulse. Since the high temperature required for such diffusion results from the power dissipated at the metal POINT-CONTACT TRANSISTOR SURFACE EFFECTS 803 to semiconductor contact, more efficient forming probably results on surfaces which display very low initial saturation currents. On surfaces which produce a poor initial rectifying diode, the local energy of the forming pulse may be dissipated too far out into the bulk of the semi- conductor. This situation would result in inefficient forming. Since the low-voltage diode characteristics and the forming are proba- l)ly related, one should be able to predict the "foi-mability" of any par- ticular surface. Fig. 16 shows that this can be done qualitatively. In the (!,raph each point represents the average of at least five units formed on electro-etched surfaces to the forming objective, Fc(3, —5.5) ^ 2.0 ^'olts. Fig. 16(a) represents the reverse emitter current before forming plotted on a log scale versus the percentage of units taking more than five pulses to form. The reverse emitter current rather than the reverse col- lector current is a desirable preforming parameter to use since this pre- 100 80 Z 60 UJ o cr LU 40 D. 20 (a) PERCENTAGE OF UNITS TAKING MORE THAN 5 PULSES TO FORM TO Vc(3,-5.5)<2 y y /. ^•" X • _ .^ — • t 1 dV, (b) = IGURE OF MERIT FOR THE SAME ZA • FORMED TRANSISTORS I 20 "o 16 V • \ k^ 6_ 8 • ^ s N ^ • « * ""^•■> * 4 ■■■ • 0 1 1 1 1 0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.6 0.8 1.0 CURRENT, Igl'^OjO) IN MILLIAMPERES (BEFORE FORMING) Fig. IG — Relation of forming to pre-forming characteristics: electro-etched surfaces. 804 THE BELL SYSTEM TECHNICAL JOUKNAL, JULY 1956 } eludes any premature forming which could occur. This curve shows that a low reverse emitter current (high back impedance) is associated with easy forming and that a high reverse emitter current is associated with hard forming. Fig. 16(b) represents data on the same group of units with /c(6, — 5)//c(0, —20) plotted versus the reverse emitter current on a log abscissa. It is significant to note that the figure of merit is consist- ently high for units with low reverse emitter current and low for units with high reverse emitter currents. It was possible to achieve this wide range in reverse currents on the same material by adjusting the current density in the manner summarized by Table V. In each case a high cur- rent density results in the low reverse currents. Some other oxidizing agents may be used interchangeably with the materials just discussed. A dilute nitric acid solution produces a surface on which excellent diode properties are observed and good forming re- sults on these surfaces. It has also been found that a treatment in potas- sium cyanide results in a surface which appears to be well oxidized. There are, however, some indications that certain chemical treatments tend, more than others, to passivate the germanium surface to any sub- sequent treatment. Although it has been shown that variations in the surface oxide layer markedly affect the transistor made on that particular surface, varia- tions in forming yield such as illustrated by the manipulator line in Fig. 13 are still unaccounted for. The etching procedure in the fabrication of the point contact transistor has always been one of the most carefully controlled steps. It therefore becomes necessary to examine the process for some subtle interaction between the germanium surface and the ambient to which the surface is subjected during processing. 4.2.3 Controlled Ambient Experiments The experiment summarized by Fig. 17 represents a "dry box" ex- periment designed to investigate the effect of ambient on the forming yield. Ten germanium wafers were mounted on hermetic seal headers, they were electro-etched, and then five treated for one minute in HF. The wafers were rinsed in deionized water, dried for three minutes in a stream of nitrogen, and placed in a nitrogen dry box where the relative humidity was maintained at less than 1 per cent. One micromanipulator transistor was formed on each wafer immediately and then at subsequent intervals of one day, always in widely different locations on the wafer. These manipulations Mere carried out inside the dry box using rubber gloves so that at no time was the RH greater than 1 per cent. After two days the box was opened to room air and the experiment continued. POINT-CONTACT TRANSISTOR SURFACE EFFECTS 805 ETCHED FOR 1 MINUTE IN IN 0.1% KOH AT 5 MA. SAME ETCH FOLLOWED BY 1 MINUTE IN 48% HF < 5 z oi cr ^ DRY Ns" ROOM AIR ''V ■~9— (a) Ie(+0.5, 0) BEFORE FORMING 16 if) _l D 12 O 8 d: LJJ CD 5 4 D z D 2 5 z UJ 5 ^ r' -9" ""V (b) NUMBER OF FORMING PULSES REQUIRED TO FORM TO Vc(3,-5.5)<2 V (c) ^ FORMING ^ "^ y^ ■— - ■"^ \ C '~~«. ~^o •o- >-— -- > 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 DAYS AFTER ETCHING 5.0 5.5 6.0 Fig. 17 — Effect of storage ambient on transistor characteristics — electro- etched surfaces. Each point on Fig. 17 represents the average of five units on five differ- ent wafers. The difference in the electrical properties of the two surfaces in air already noted in previous sections is observed. In addition an increase in surface recombination is indicated on the HF treated surface by a decrease in the turn-off-time measurement (TOT).* Finally, any influence of ambient on the electrical properties of the two surfaces used is ap- parently small. 4.2.4 A Statistical Survey Experiment on Transistor Forming The experiment described here was designed to check some of the effects noted in earlier sections as well as to investigate possible interac- tions between the germanium surface and various ambients experienced during the processing of point contact units. The experimental design * TOT is a nonparametric measurement indicative of the switching speed when used in a specific circuit. 806 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1950 Table IX — Experimental Design of Randomized Block Experiment Surface Treatments A B C D E F Ambient Shelf Conditions Electro 1 min. H2O2 after Normal Etch 1 min. H2O2 1 min. HF after Normal Etch 1 min. HF Normal Etch 5 ma. after after Superoxol Etch for 1 min. in 0.1% KOH Normal Etch and Shelf in Ambient Normal Shelf in Ambient Formed immediately X X X X X X after treatment Formed after shelf in X X X X X X room ambient Formed after shelf X X X X X X over drierite Formed after shelf in X X X X X X dry No Formed after shelf at X X X X X X 76.5% RH Note: Shelf represents storage for 24 hours. used is a 5 X 6 randomized block experiment with multiple subgroups.^* Table IX shows the general plan of the experiment. The six columns represent different etch treatments, and the five rows represent some pos- sible variations in storage conditions. Each subgroup represents five transistors, and the experiment represents a total of 150 transistors made on germanium from the same zone-leveled slice, given 30 different treat- ments. Although nine measurements were made for each transistor, the figure of merit appeared to be most significantly dependent on the treatments. As expected from the results already quoted, the major variability was found in units formed on surfaces freshly treated with HF, with considerable improvement in formability during storage. However, the looked for influence of storage ambients does not appear when the column F has been removed from consideration. One concludes that the variation between treatments is small, and the effect of ambient is even less than the effect of the treatments. Thus when surface treatment does not vary to extremes, the effect of storage ambient is relatively minor. Thus variations found in such experiments as exemplified on the manipu- lator line in Fig. 13 must be attributed to a still unknown factor. 4.2.5 Effect of Contamination Before Etching Since etching removes the damaged surface and is usually done with highly corrosive materials, it seems unlikely that any contamination POINT-CONTACT TRANSISTOR SURFACE EFFECTS 807 before etching could affect the efficiency of etch. There have, however, ])een some indications that this does occur. Certain chemical treatments appear to passivate the surface to any subsequent treatment, for ex- ample, the results in Sections 4.2.3 and 4.2.4. The electro-etched sur- face followed by an HF treatment does not change rapidly with time in room air, while the superoxol-etched .'^urface followed by an HF treat- ment changes quite rapidly. Surfaces which have been etched in CP4 and subseciuently treated in HF appear to be as stable as electro-etched surfaces. Subsec^uent treatments in superoxol do not appear to result in significant changes in the surface characteristics. Experiments on un- ctched germanium wafers indicate that none of the components of CP4 alone will prevent normal etching, but if an unetched Avafer is treated with a combination of 50 per cent nitric acid plus 48 per cent HF for a few moments, the surface will be stabilized as to retard the formation of the normal pyramidal etch pattern when the eurface is etched in super- oxol etch. Taken together these ol)servations may imply that certain types of oxide surfaces are more stable than others and perhaps may even ])e passivated to subsequent environmental conditions. With this background of information it becomes more believable that chemical treatments before etching could affect the surface of the ger- manium resulting from the subseciuent etching. It is not unreasonable to believe that any variation in surface potential resulting from pre-etch treatment might influence the reaction between the etchant and the germanium. An experiment was performed using gold-bonded bases to isolate the contribution of the solder flux normally used in the base- wafer attachment. Twenty wafers from the same slice were divided into four subgroups of five. The groups were treated in such a way that any effects of HF or solder flux soaking before superoxol etching could ])e detected. The results of this experiment do indicate that presence of flux before etching significantly affects the collector currents and turn-off time of transistors made on such surfaces. Although there was no apparent dif- ference in forming yield between sub-groups, it is felt that this variation would show up as a difference in forming yield in a process where the forming efficiency is decreased somewhat by the impregnant. 4. .3 Conclusions Treatment of an etched surface with germanium dioxide solvents such as HF or KOH degrades the surface to such an extent that transistor forming efficiency is decreased. A similar effect is produced by corrosive flux and heat. Thus, pre-forming measurements may be used to predict 808 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 the formability of a particular germanium surface. It is shown that poor diode characteristics are usually associated with poor forming yields. One convenient way of controlling the diode characteristics to ensure successful forming is to etch electrolytically. High current density results in the most desirable surface characteristics. Electro-etched germanium which has been subsequently treated in hydrofluoric acid shows little tendency to oxidize either in room air or dry nitrogen ambient, while superoxol-etched germanium, given the same HF treatment, changes quite rapidly in room air presumably due to oxidation of surface. Sulli- van^^ has also observed differences in the stability of electro-etched and chemically-treated surfaces. Different surfaces can be prepared chemically which show more than the amount of variation normally found in pilot and manufacturing process lines. However, extreme variations in storage ambients have relatively little significant effects on any of these surfaces. It is therefore concluded that although certain chemical treatments may affect forming, the variations in process yields are not attributable to interaction between the germanium surface and storage ambients. The results of Sections 4.2.2 and 4.2.3 suggest the possibility of passi- vation of the germanium surface. An electro-etched surface followed by an HF treatment exhibits a higher degree of stability to ambient than does a superoxol-etched surface treated in the same way. Treat- ment of a lapped germanium surface with two components of CP4 (HF -f HNO3) will inhibit subsequent etching in superoxol. The possibility that contamination before etching may affect the char- acteristics of the germanium surface after etching is considered. Experi- ments show that contamination of the germanium with corrosive zinc chloride-ammonium chloride flux before etching significantly affects the rectification properties of the germanium surface obtained after etching. The surface recombination velocity (in so far as it is determinative of the turn-off time of the transistor) is also significantly affected. However, on the basis of the results quoted here, it is not possible to conclude that such contamination can account for an appreciable amount of the unassignable variability in forming yields experienced in pilot and manu- facturing process lines involving soldered base-wafer connections. 5. GENERAL CONCLUDING REMARKS The experiments which have been described have implications which are important in both design and processing of point-contact transistors. These are summarized below: POINT-CONTACT TEANSISTOR SURFACE EFFECTS 809 5.1 Point-Contact Transistors with High Current Gain In most switching applications the combination of high current gain and low reverse current is desirable. The measurements of current gain, taken together with the potential probe measurements in Section 2.2.1, indicate that, for the structures used here, the reverse collector current at operating voltage must be large enough to set up a substantial drift field before efficient collection of holes can occur. If this condition is not met, either the unit has low gain at all values of emitter current (un- formed), or develops a bistability of the kind described in Section 2.3 (partially formed). For a given structure, the drift field can be increased by increasing resistivity of the germanium at the expense of increased base resistance. Here thermal stability of the contact also provides a limit. A more likely expedient, in the case of germanium, is to decrease the area of the formed collector junction by using sharper points and modified forming technique. The limits here are produced by reliability requirements for mechanical stability of the point structure. 5.2 Current Multiplication in Unformed Transistors Many experiments have reported on junction transistors with high current gains which are attributable to the p-n hook mechanism. The high values of current gain observed with conventionally formed point contact transistors have been attributed to various mechanisms, among , which is the hypothesis of a p-n hook structure, primarily in the bulk of the germanium, introduced by the pulsing of the donor-doped point. In particular, at small emitter currents small signal a-values in conven- tionally formed collectors may reach values as high as ten, and values of a as large as 100 are encountered in formed collectors exhibiting anoma- lous output characteristics. However, the average a over a 6-ma emitter current range is usually near the value of 3.1 which would be expected from the mobility ratio of holes and electrons with the Type-A transis- tor geometry. The increase in reverse current of a formed collector by t addition of donor to the point wire may result from the production of a hook structure. However, information is needed concerning the impor- tance of the hook structure in accounting for the high values of a en- countered at low emitter currents, or in connection with collector char- acteristic anomalies in conventionally formed point-contact transistors. The unformed transistors discussed in this article differ from electri- cally formed units in that the collector barrier is the one at the metal- semiconductor surface. It has been found that certain chemical treat- ments can produce a collector barrier which allows an increased reverse 810 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 current flow and a substantial drift field near the emitter. Some of these units show an a vahie at all emitter currents quite comparable in magni- tude to that of conventionally formed collectors, and surface treatment alone can also introduce in these unformed collector characteristics anomalies similar to those found in some formed units. It is difficult to visualize a p-n hook structure arising at the germanium surface as a re- sult of the chemical treatments discussed. If such a possibility is pre- cluded, the p-n hook mechanism does not seem necessary to the attain- ing of high a values at low emitter currents, or an a emitter current dependence of the kind normally observed in anomaly-free units. To account for values of a obtained with unformed collectors at low emitter currents, other mechanisms, such as the suggestion of Shockley, involv- ing hole trapping in the germanium under the collector point *"' ^ or the suggestion of Van Roosbroeck, involving conductivity modulation, might in this case be more suitable. Further, unformed transistors made by appropriate chemical treat- ments can duplicate qualitatively the electrical characteristics of con- ventionally formed units, including alpha-emitter cvuTent dependence and output characteristic anomalies of types (1), (2) and (3). These phenomena can thus occur under circumstances where a well-defined hook structure is improbable. 5.3 Surface Properties and Transistor Fornmig It has been found that a major factor in determining the forming yield of point-contact transistors is the chemical history of the surface. Thus in processing of point-contact transistors, major attention should be paid to ensuring chemical control of the base wafer surface if the forming yield is to be kept high. On the other hand, considerable variation may apparently be tolerated in storage ambients. Of course it has not been shown that such variations in storage conditions do not have an efl'ect on subsequent reliability of the product. Processes which permit expos- ure of surfaces to solder fumes either before or after etching are to be regarded with suspicion. Monitoring of the reverse emitter diode char- acteristics should prove useful as a means of securing proper control of the pre-forming surface. ACKNOWLEDGEMENT The authors wish to acknowledge the help of M. S. Jones, who carried out many of the experiments mentioned here, and N. Carthage who did the electroetching work. The continued support and encouragement of N. J. Herbert has been greatly appreciated. 1- POINT-CONTACT TRANSISTOR SURFACE EFFECTS 811 KEFERENCES 1. J. Bardeen and W. H. Brattain, Physical Principles Involved in Transistor action, Phvs. Rev. 75, p. 1213, April 15, 1949. 2. J. Bardeen and W. G. Pfann, Effects of Electrical Forming on the Rectifying Barriers of n- and p-Germanium Transistors, Phys. Rev. 77, p. 401-402, Feb. 1, 1950. 3. W. Shockley, Electrons and Holes in Semiconductors, D. VanNostrand Com- pany, New York, N. Y., p. 110. 4. Reference 3, p. 111. 5. L. B. Valdes, Transistor Forming Effects in n-Type Germanium, Proc. I.R.E. 40, p. 446, April, 1952. 6. W. Shocklev, Iheoiies of High Values of Alpha for Collector Contacts on Geimanium, Phys. Rev. 78, p. 294-295, May 1, 1950. 7. W. R. Sittner, Current Midti] licalion in the Type A Transistor, Proc. I.R.E. , 40, pp. 448-454, April, 1952. 8. Valdes (Reference 5j reports large concentrations of copper present in the p-germanium under heavily formed phosphor-bronze points. 9. W. G. Pfann, Significance of Composition of Contact Point in Rectifjing Junctions on Germanium, Phys. Rev. 81, p. 882, March 1, 1951. 10. C. S. Fuller and J. D. Struthers, Copper as an Acceptor Element in Germa- nium, Phys. Rev. 87, p. 526, Aug. 1, 1952. 11. C. S. Fuller, Diffusion of Acceptor and Donor Elements into Germanium, Phys. Rev. 86, p. 136, April 1, 1952. 12. Reference 5, p. 448. 13. Personal communication, H. E. Corey, Jr. 14. L. E. Miller, Negative Re.sistance Regions in the Collector Characteristics of Point Contact Transistors, Proc. I.R.E., 40, p. 65-72, Jan. 1, 1956. 15. Reference 1, p. 1225. 16. John Bardeen, Surface States and Rectification at a Metal Semiconductor Contact, Phys. Rev., 71, p. 717-727, May, 15, 1947. 17. I. Tamm, iiber eine Mogliche Art der Elektronenbindung an Kristallober- flitchen, Physik, Zeits, Sowjetunion, 1, 1932, p. 733. 18. W. H. Brattain and J. Bardeen, Surface Properties of Germanium, B. S. T. J. 32, pp. 1-41, Jan., 1953. 19. W. L. Brown, n-T}'pe Surface Conductivity on p-Tvpe Germanium. Phvs. Rev. 91, pp. 518-527, Aug. 1, 1953. 20. W. H. Brattain and C. G. B. Garrett, private communication. 21. R. D. Heidenreich, private communication. 22. O. H. Johnson, Germanium and its Inorganic Compounds, Chem. Rev. 51, pp. 431-469, 1952. 23. M. Kikurchi and T. Onishi, A Thermo-Electrical Study of the Electrical Forming of Germanium Rectifiers, J. App. Phys., 24, pp. 162-166, Feb., 1953. 24. R. H. Kingston, Water-Vapor Induced n-Type Surface Conductivity on p- Type Germanium, Phys. Rev., 98, 1766-1775, June 15, 1955. 25. M. V. Sullivan, personal communication. 26. J. T. Law, A Mechanism for Water Induced Excess Reverse Current on Grown Germanium n-p Junctions, Proc. I. R. E., 42, pp. 1367-1370, Sept., 1954. 27. E. Billig, Effect of Minority Carriers on the Breakdown of Point Contact ^ Rectifiers, Phys. Rev. 87, p. 1060, Sept. 15, 1952. 28. G. W. Snedcor, Statistical Methods, The Iowa State College Press, Ames, Iowa, 1946. 29. W. VanRoosbroeck, Design of Transistors with Large Current Amplification, J. App. Phys., 23, p. 1411, Dec, 1952. The Design of Tetrode Transistor Amplifiers By J. G. LINVILL and L. G. SCHIMPF (Manuscript received March 7, 1956) The design of tetrode transistor amplifiers encounters problems of the type that occurs with other transistor uses. Desired frequency characteristics, limitations of parasitic elements, and other practical considerations impose constraints on the range of terminations that can he employed. With many transistors, one can terminate a transistor so that it will oscillate without external feedback; this oscillation or other exceedingly sensitive terminations must be avoided. The two-port parameters of the transistor in any orientation in which it is to be used constitute the fixed or given information which is the starting point of the amplifier design. Using this starting point, methods are de- veloped by which one can select, on simple bases, the kinds of terminations that will be suitable. To facilitate the design of amplifiers, a set of charts has been developed from which one can read power gain and input impedance as functions of the load termination. Illustrative tetrode amplifiers are described. These include a common base 20-mc video amplifier, a common-emitter 10-mc video amplifier, an IF amplifier centered at SO mc, and an IF amplifier centered at 70 mc. Pre- dicted and measured gains are compared. INTRODUCTION Junction tetrode transistors^ of the type currently produced for re- search purposes at Bell Telephone Laboratories are suitable for high- frequency applications. They are being studied for use in video ampli- fiers, as IF amplifiers where the center frequency is below 100 mc, for oscillators up to 1,000 mc and for very fast pulse circuits. Their application in amplifiers brings up design considerations similar to those encountered for other transistors but with differences resulting 1 R. L. Wallace, L. G. Schimpf and E. Dickten, A Junction Transistor Tetrode for High-Frequency Use, Proc. I.R.E., 40, pp. 1,395-1,400, Nov. 1952. 813 814 THE BELL SYSTEM TECHNICAL JOUENAL, JULY 1956 from different parameter values and variation. The analysis presented in this paper regarding amplifier design was motivated by the study of t(>trodes, but the results are ecjuall}^ applicable for other types. The design of an amplifier begins with a characterization of the transistor which is suital)le for the study of its performance as an am- plifier. From this characterization, or functional representation, one CORRESPONDING QUANTITIES I n n E h„ z„ yii 911 h,2 Z12 yi2 912 ha, Z2, y2, 92, h22 Z22 y22 922 I, I, E, E, E, E, I, I, E2 I2 E2 I2 I2 E2 I2 E2 Es Es Is Is Zs Zs Ys Ys Yl Zl Yl Zl Zl Zl yl Yl Yo Zo Yo Zo RELATIONSHIPS BELOW ARE BETWEEN QUANtrjl IN COLUMN I. CORRESPONDING RELATIONSHIPS] WRITTEN DIRECTLY FOR CORRESPONDING QUA^^ IN ANY OTHER COLUMN. (1) E,= I,h„ + E2h,2 (2) l2= I,h2,+E2h22 h,2h2, (3) Zl= h„ - (4) Yo (5) 1,= Yl + 1^22 h,2h2i ^zz Zs + h„ h2,EsYL (h„ + Zs)(h22 + YL)-h,2h2, Fig. 1 — Two-port parameters with summary of relationships. THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 815 determines the potentialities of amplifiers employing the transistor and designs a suitable amplifier circuit. This step in^'olves answering two (luestions: What performance, maximum power gain for instance, is it possible to obtain? What source and loatl impedances should the tran- sistor be associated with? Two-Port Parameters of Transistors For circuit applications, the two-port parameters are the most con- venient for characterization of the transistor. These parameters implicitly but completely characterize the device from the performance standpoint. Four sets of two-port parameters are illustrated in Fig. 1. Any set can be calculated from any other set, and the choice of the set to employ is determined only by convenience in the use of available measuring eciuip- ment and the preference of the designer. The relationships between parameters, input and output impedances, voltage and current ratios are summarized on Fig. 1. The same expressions given there for /i's can be used for any parameter set so long as one uses the corresponding quantities applicable to the desired parameter set. Though the transistor can be operated as an amplifier with the base, emitter or collector common between the input and output terminal pairs, the two-port parameters for any of the connections can be used to calculate the parameters for any other connection. For determination of the two-port parameters of tetrode transistors, R. L. Wallace suggested the use of two-terminal impedance measure- ments with subsec^uent calculation of the two-port parameters of in- terest from these. The impedances indicated in Fig. 2 have proved simple to measure at typical operating points with conventional high- : frecjuency bridges. These impedances have been measured at a set of frequencies extending to 30 mc. Because of the number of transistors measured it has been economical to program a digital computer to cal- culate two-port parameters and other ciuantities of interest from the measured two-terminal impedances. THE RELATIONSHIPS OF TRANSISTOR PARAMETERS TO AMPLIFIER PER- FORMANCE Any of the sets of two-port parameters implicitly characterize all of I the linear properties of the transistor for the range of frequencies for which the parameters ha\'e been measured. As mentioned before, it is necessary to translate the parameters into answers to the following questions. How much amplification can the transistor give at a particular 816 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 frequency? What impedance should it be supplied from? What impedance should it feed? What gain will be obtained using a pair of impedances different from the optimimi ones? The answering of these and related (juestions amounts to establishing a convenient means of translating the parameter values into the ciuantities of interest applying to the amplifier. Such a convenient translating means for solving these problems is described in this section. Earlier explicit solutions to special cases of the problem are well known. Wallace and PietenpoP have given simple expressions in terms of the transistor parameters for matching input and output impedances and the maximum available gain when the transistor has purely real parame- ters. An implicit solution for optimum source and load impedances for maximum gain in the complex case has been known for a long time. It is simply that the transistor be terminated at the input and output by conjugate matching impedances. The implicit nature of this solution arises from the fact that the input impedance is a function of the load impedance, and the output impedance is a function of the source impe- dance for transistors wdth internal feedback. The solution for optimum source and load impedance from this approach amounts to the solution of simultaneous quadratic equations with complex unknoA\nis and be- comes involved. c ■) / le ■ > r -X "h J) h- v^ V( Fig. 2 — Two terminal impedance measurements for determination of two-port parameters. ^ R. L. Wallace and W. J. Pietenpol, Some Circuit Properties of n-p-n Transis- tors, Proc. I.R.E., 39, pp. 753-67, July, 1951. THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 817 From the approach to the problem taken in this paper, one solves first for the maximum power gain and subsequently determines the optimum terminations. It turns out that the solutions leads to explicit relation- ships for optimum performance and terminations and also leads to charts from which power gains and input impedance can be read for any terminations. In all expressions to be developed, the h parameters are used. Pre- cisely the same expressions can be obtained for z's, y's, or ^'s provided that one uses the corresponding quantities in the table of Fig. 1. The maximum power gain is a quantity of primary interest in tran- sistors since the transistor ordinarily has a resistive component in its driving-point impedance. Thus voltage or current amplification is con- strained by the limited power gain attainable. In some cases, however, because of the inherent feedback internal to the device, instability can result simply from proper passive terminations without application of any additional feedback. Such cases are distinct because of this property. Transistors exhibiting this possibility are said to be potentially unstable at the frequency in question. A quantity of interest presented here and derived later is a particular power gain defined for /i-parameters as power out _ Poo _ | h 21 power in Pm 4:hnrh22r — 2ReQinh2i) (1) where hnr and h^ir mean the real part of hn and of /122 • ReQinhi^ means the real part of the product of /ii2 and hn . Unless the amplifier is po- tentially unstable, the quantity Poo/Pio is M'ithin 3 db of the maximum available gain for the transistor. The matter of potential instability of the transistor is of great interest. Certainly the transistor is potentially unstable if Poo/-Pio is negative. Otherwise potential instability is indicated by greater than unity values of the criticalness factor C = 2^ PiO h2i (2) If the transistor is not potentially unstable the maximum available gain is Ko(Poo/Pio) where K, = ^(1 - ^[^'^ (3) For O^C^ 1,1 ^ Ko ^ 2. A plot of Kg as a function of C is shown in Fig. 3. The function is seen to be exceedingly flat near Kg = 1 for 818 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 C between zero and 0.6. Thus the value Pm/Pio in the majority of cases ^^•here the transistor is not potentially unstable is a close approximation to the maximum available gain. The optimum source and load impedances can be expressed in terms of the transistor parameters and other quantities given in terms of them by the following relationships where the transistor is not potentially unstable. G 1 irg i — hnhn) = e je Zs opt = Zin = hn hnh 21 CKgG^ Zh'>'>r Yl opt = —ho-z + 2h 22r 1 CKgG 2 (4)^ (5) (6) Though explicit relationships for ideal terminations and for the maxi- mum power gain which one can achieve with a transistor are of interest, such terminations limit the band width of the amplifiers. Therefore, it is important to have convenient means for evaluating power gain and input impedance for other than ideal terminations in order to realize a desired bandwidth. A chart which facilitates computation of these quan- tities is now developed from an analysis which leads to the other results quoted above. Kf 2.0 1.5 / / 1.0 0.5 0 0.2 0.4 0.6 c 0.6 1.0 1.2 Fig. 3 — K^ plotiod ms a f unci ion of C. ^ If —hxih^i \s c + jd, then d = tau~'(r//c) ; G = c' and —hvihn is the conjugate of —hiohii . THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 819 Power Flow in a Two-Port Device A convenient point of departure in the analysis of power amplification iu a transistor or other linear two-port device is the arrangement shown ill Fig. 4. The two-port is supplied by a unit current at the frequency of interest and at reference phase at the input terminal pair. The output of the two-port is connected to a voltage source of the same frequency. The input-current and output-voltage time functions are ti = Re\/2€'"' = ReV2li£'"' (7) iiid = ReV2(a + jb)e^"' = ReV2(L -f jM) (-AA \2h22r/ jolt (8) = Re\^E2e jat In (8), L and M are introduced for simplicity in some later relation- ships. The whole analysis is essentially a study of power flow in the circuit shown in Fig. 4 as L and M of (8) are varied. All possible terminations and excitations can be simulated simply by varying L and M. Under some conditions the voltage source will absorb power; under others it w ill supply power to the two-port. Ordinarily the current source supplies power to the two-port, but for appropriate ranges of L and M if the two- l)ort is potentially unstable, the transistor may supply power both to the current source and the voltage source. The problem of evaluating maxi- mum power gain is simply finding the values of L and M corresponding lo the greatest ratio of power out to power in. The load impedance to which this situation corresponds is E-il — l-i . The input impedance for t his condition is simply £"1//! , and the optimum source impedance is the complex conjugate of the latter quantity. Ii=i+jo I £2= a+jb = Fig. 4 — A two-port device .supplied bj^ a current source and feeding into a voltage source. 820 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Fig. 5 — Sketch of power output as a function of L and M. PROJECTION IN L-M PLANE OF GRADIENT LINE IS G OR Arg-h,2h2, ^21^12 SLOPE OF PLANE ALONG G IS 2h 22r Fig. 6 — Sketch of power input as a function of L and M. THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 821 The output power can be readily evaluated in terms of L and M. h = /l/i21 + ^2/i22 (9) /2 = (1 + mn + (L + jM)h,, tM (10) I Powerout = Po = fl«(-^2/!) (11) ^ ^^ [-(L -jM)h, ^^ _ , ^ ^j,^ ^^^ (12) L 2/i22r 4/i22r J '21 r /t2 , tit2\ ^21 |2 On the basis of (13) the power output plotted as a function of L and M is a paraboloid as shown in Fig. 5, having the pertinent dimensions indicated there. Only within the circle centered at L = 1, ikf = 0 and passing through the origin does one obtain positive power output. The apex of the paraboloid corresponds to P, = P,„ = IM (14) 4/i22r The input power can similarly be evaluated in terms of L and M. El = hhn + E^hn (15) = (1 + jO)hn +{L+ jM) t^ hn (16) Power in = Pi = Re[E,h] (17) { — h2^)hn Pi = Re hn + (L + jM) 2h 22r (18) 7 T Ti (^12^2l) , Tirr \h\2h2V f-,rs\ = hur - LRe — T — + MIm -— — (19) where /m[(/ii2/i2i)/2/i22r] means the imaginary part of the expression in parenthesis. On the basis of Eq. 19 the input power plotted as a function of L and M is simply an inclined plane having the properties indicated on Figure 6. Since Figures 5 and 6 turn out to be such simple geometrical figures the problem of finding the point of maximum ratio of Po to P, is very simple and other interpretations are easy to make. First, a negative value of Pio{Pi at 1, 0) certainly indicates potential instability for both input and output terminations receive power from the two-port. Even if the plane of P.- intersects the L-M plane within the unit circle centered at 822 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G 1, 0, then the two-port is potentially unstable since on one side of the intersection both input and output terminations receive power from the two-port. The change in Pi from the minimum value found on the unit circle centered at 1, 0 to Pio divided by Pjo is the criticalness factor, C. A^alues of C greater than unity indicate potential instability. The power input at 1, 0 is -PiO 2hnrh22r — Re{hnh2i) 2h 11r Using (14) and (20), one obtains 00 /i21 Pio 4/iiir/i22r — 2Re{hi2h2i) hnfh 12«21 C 2/l2 '2hiirfl22r — Ke{hi2'l21J = 2 00 PiO hu h2i (20) (21) (22) 2h 22r Now if the plane of power input, Fig. G, is parallel to the L-M plane and above it, certainly the point of maximum power gain is the apex of the paraboloid, 1, 0 in Fig. 5. If the plane is incliued but alwaj's above the unit circle centered at 1, 0 certainly the point of maximum power gain is downward along the gradient line which lies above the point 1, 0. This must be so since for any contour of equal power out (a circle of fixed elevation around the paraboloid) the minimum power input (or greatest gain) lies along the line of steepest descent from 1, 0 in Fig. 6. Thus the problem of evaluation of the maximum available gain reduces to the simple problem of finding the abscissa of Fig. 7 where the ratio of ordinates of the parabola and straight line is a maximum. The parabola Pl or Po PROJECTION IN L-M PLANE OF GRADIENT LINE OF PLANE THROUGH 1,0 Fig. 7 — Section of paraboloid and inclined plane of Figs. 5 and 6. THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 823 and straight line are sections of the paral)oloid and plane through the gradient line of the plane over 1, 0. A straightforward analysis indicates that the point in the L-M plane \\ here the maximum of Po/Pi occurs is at L + jAI = 1 - CKgG (23) ^\liere these quantities are defined as in (2), (3), and (4). The power gain cit this optimum point is Kg times that obtained at 1, 0. One finds that 1 he maximum gain is only two times Pm/Pio even if C approaches unity A\ hich corresponds to the marginal case of potential instability. The analysis just described leads to the maximum values of power I gain and to the best terminating impedances. For many design problems 1 liese answers are a guide but one may prefer to use other than optimum ' \alues for other compelling reasons. For such a case charts from which I one can get the pertinent quantities are very helpful. P^^IGUE lN^DEGRee3 G2+jB2= YL+h22 ^22 — i'22n +jh22l Fig. 8 ■ — Gain and impedance chart. 824 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 -1.0 Pl=Plo(i + cx) C = 2 Pqo LO h„ hs, ■0.8 -0.6 '21 P'lo 4h„rh22r-2Reh,2h2i -0.4 x=o ■0.2 LO 0.2 0.4 0.6 0.8 ANGLE OF"G"iN L-M PLANE IS: ARG -^^z^z^ Fig. 9(a) — Input power as a function of X. Development of Transmission and Impedance Charts The same point of departure employed in the evakiation of optimum cases leads to a convenient set of charts. Equation 12 shoAvs that a setj of concentric circles centered at 1, 0 are loci in the L-M plane of constant power output for a unit current source at the input. It is convenient to plot these as is done on Figure 8, showing Pq as a fraction of Poo • h 21 Po Poo l-(L-lf- M' (24) 4/i 22r Since Yl , the load admittance, is —I2/E2 , using (10) one obtains -h = Yl = -A22 + 2h 22r E2 - ' L+ jM Now it is clear that if one defines G2 and B2 by 2/l?2r ^2=^2+ JB2 = Yl + h22 = L + jM (25) (26) THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 825 loci of constant real and imaginary parts of Y2 become the mutually orthogonal circles shown in Fig. 8. Thus the value of L + jM is deter- mined by the load admittance and two-port parameters. Contours representing constant input power, with equal increments of power between successive contours, are always parallel equally-spaced lines in the L-M plane. However, as may be seen from (19) and Fig. 6 different cases have different directions for the line normal to the con- tours, (the gradient line) and also different power increments for a given spacing of equal-power-input contours. It is convenient to define a new \ariable A^ which is the component along the gradient line of the vector starting at L = 1, M = 0 and going to L, M. Thus Pi = P.o(l + CX) (27) Equation 27 suggests Fig. 9(a) which shows loci of constant power input plotted as a function of X. If Fig. 9(a) is shown on a transparent ma- 90 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 80 ^ 70 ^ K ^ ' z„-h„= \^ 4( \ 30 ^ . V,20 \ V-10 - n 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 h,ah 2"21 2h 22r ^(L.jM) ^n22r R,+jx, 1.8 Fig. 9(b) — Input impedance as a function of L and M. 826 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 terial, its center (at X = 0 along the gradient line) can be superposed with the point L = 1, ilf = 0 of Fig. 8. With the gradient line of Fig. 9(a) oriented at the argument of —hnh^i , or S in the L-M plane, one can easily determine graphically the power gain at any point in the L-M plane compared to the power gain at 1, 0. With Fig. 9(a) superposed on Fig. 8 as just described the viewer gets a bird's-eye-impression of tlic paraboloid of power output and the inclined plane of power input simul- taneously. With such a bird's-eye view, it is easy to assess possibilities for power gain with all possible angles of load termination. The evaluation in input impedance is done through use of (16) from which is obtained ^ = Z,. = hn + (L + jM) t^^ or Zin = hn + (L +jM)(e-n hnh 21 Zhf22r (28) (29) For evaluating the second component of (29), it is convenient to have a second transparent overlay, Fig. 9(b), consisting of a rectangular grid to the same scale as the L-M plane. Fig. 8, with coordinates marked as (Zin — hn) Ri and hiihii 2/l22r hiohn 2/l22r This overlay is placed over the L-M plane with the ^1 hioh 12fl'21 2ihj22r axis making the angle d with respect to the L axis. Thus on the rec- tangular overlay for any point in the L-M plane, one reads Zin — /ill hiih ■21 2/?.,. r.\i;TI('UL.\U DESIGNS OF TETKODE TRANSISTOR AMPLIFIERS IMie charts and optimum rc^lationships developed in the preceding section are convenient starting points in the design of amplifiers. They THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 827 do not ordinarily constitute a finished solution, however, since practical constraints frequently modify the design used. Moreover, all of the relationships are expressed on a single frequency basis, and many times the amplifier must operate over a range of frequencies broad enough that parameters change significantly over the range. Four amplifier designs are described in this section: a single stage, common-base, 20-mc ^ddeo amplifier; a common-emitter, 10-mc video amplifier; an IF amplifier at 30 mc and a 60 to 80-mc IF amplifier. Parameter measurements made with bridges support the first three designs. Parameter values and associated constants of a typical tetrode transistor are given in Table I. The quantities shown there reveal some interesting facts about the typical tetrode transistor represented. First, in the common-base connection the tetrode is potentially unstable at 30 mc but not at the lower fre(|uencies. The common-emitter amplifier is potentially unstable at 1 and 3 mc. Second, the power gains of common- emitter and common-base stages are about the same at 30 mc, the com- mon-emitter connection giving more gain at low frequencies. The matter of potential instability requires further consideration from a practical point of view. Potential instability at a frequency neither implies that a stable amplifier cannot be built at that particular fre- quency, nor does it imply that one can obtain an unlimited amount of stable amplification at that frequency. It does mean that by simul- taneously tuning output and input one can adjust for oscillation. The region of potential instability corresponds to a region in which the input resistance may be negative for appropriate loads. Instability is avoided in the physical amplifier if one supplies the amplifier from a sufficiently high impedance that the input loop impedance always has a positive real part. To operate the amplifier Avith such a load that it presents a negative resistance to the source is attended by the difficulty that the amplification is more sensitive to changes in the source impedance than it is when the input resistance is positive. Hence the possible higher gain with internal positive feedback goes along with a greater sensitivity to changing termination impedance. ! A Common-Base 20-Mc Video Amplifier- The data presented in Table 1 gives a ciuite comprehensive picture of possibilities for amplifier designs. To it must be added a practical fact. It is difficult to connect the load impedance without adding about 2 jujuf of capacitance. This means that any termination considered must include about this amount of capacitance. By a theorem regarding 1 828 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 CQ «0 ' CO fe^fe h 7 y-i O • rH ^-vO-'-v .^t^ . ^i-H ^O 0(N •,— ^ o Oi-H , ■* Tt< .-H rO ^•'^l , •<>> I 00 + Ti-H • I 1 ' + 00 -* t^o • OiO coo O OOOOO -lOCO •-IIM 1 COO -Tt^ IC CO I-H •■"*!•* • O lO •0-— 1 ^-'Tt<.-i i-H ,-1 -^O ^-"* O "-H CO o CO CO 1 CO • o I-H CO 1 O f I-H ^ • ^o o •'-^ " T-H ^~N y—t to o . _o H o • ^^•-"x O .l-H^-v O ^l-H -00 ,_i (M-'^Tf.-H o C^ ' -r-i ■'^J , ■^•<-s « u iC »Oo • OOO CIO o ■ 00 Tfi T-H r^ 00 i-f H Oi-H 1 (M t^ • »0 f^ (M,-H -COIN -iM '-I UJN^ 1 ^— COO CO l> (M>^C^-~^,-iO>-i <-< o eo CO 1 A 5^ o"3 1 O 1 Eh •-'So th (b !z; • O I— 1 T-H < cop^-v _C5 _^-^ H -^ • "^"5 O •cOt>. S ^ O Or-< , (N O o i-.v_-io .-( 1 (N Eh <1 CO o o 1 •= CO «0 2co<|r> t t CC • 22'-" rH rH CO ^^o . • • < ^.dcs" ^co 00 o • <^ . o . Q VH OO , 00 ^-jT^cooi C^ c~> • ■«-» ;g ■'^s . ■* . <; •-^+co + 1 +'"^+ cc "Ti-i -t^ C5 1 1 >o o 1 coo ^ • O .Ot^o o O • . CO T-H I— 1 I— 1 tf OS T-i 1 CO '-I • O IQ CO CO C^ 00 CO -I 00 w Tt<^— • 1 ^^COO(N 1-H CO ^ 'I— I^I-Hl— 1 1 CO PS g 1 «» CO "» < Sioi, i=. ^ K ■ ?^'-* 1-H 1—1 ,-^o . • • ^ CD CO 00 ^ \q • '-I . d OO , lO CO (m' CO CD 1— I-C--V • -c^ ^ . CO , 1— 1 •-^+co + 1 +'^+ %. W i_ C5 i-i-H • CO 00 1 1 CI o 1 -^ OOCO •O -Ot-o o o • -r^ooi-H 05 hJ CI r-( 1 CO ■— 1 • >— < CO . ^^.-H ^— -l-H O 1 CO <1 t-i 0) += o (D HJ ;§ CO c9 *§ pq w o 0 •» s S ~^ „. - ^< C3 o ,^ '-' -e (M 3 rf C< -1 « S «> r«; (M S r ^ " cS'CLt:-, 1 o -< •< •*? -< *^ "O ' g "s; -sj -s; rjg Wh ) 54 58 62 66 70 74 78 82 FREQUENCY IN MEGACYCLES PER SECOND 86 Fig. 20 — Gain of a 3-stage band pass amplifier working between 75-ohm im- pedances. Each stage uses the circuit shown on Fig. 19. (0 cc o \- Z 3 < cr I- u. ^2 a. UJ CD Z 1 4 — 5 6 7 8 9 10 n GAIN IN DECIBELS 12 Fig. 21 — Variation of gain for a grouj) of transistors used in the circuit of Fig. 19. THE DESIGN OF TETRODE TRANSISTOR AMPLIFIERS 839 6 - tr ^ o (- to o q:3 ID ^2 p5!^ p: : J jwp 8 9 10 11 12 13 14 13 16 NOISE FIGURE IN DECIBELS Fig. 22 — Noise figure for a group of transistors used in the circuit of Fig. 19. If) A 1^3 or o 1- " 10 ;:;-:::o; z ^ — p*r<*-| v:::-:;:; < :":■:::::■: CL (- U- C> CC^ ~ i!i :-:-x:;-: Hi ■■■y-yA ::::■:;:":: m 2 D Z 1 " 0 1 1 5 6 7 8 NOISE FIGURE IN DECIBELS 10 Fig. 23 — Noise figure for a group of transistors used in a 10-mc bandpass amplifier. 840 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 between various transistors, 18 tetrodes were measured in the first stage of the amplifier. If the measured gain of each transistor is rounded off to the nearest db and the number of transistors having this gain plotted as the abscissa, the results shown on Fig. 21 are obtained. Of the 18 transistors measured, 11 have a gain of 8 db or greater. Similar data has been obtained on the noise figure of the same 18 transistors, the results being shown on Fig. 22. In general, the transistors having the highest gain also have the lowest noise figure. The noise figure depends to some extent on the source impedance but a 75-ohm source results in a noise figure which is within a few tenths of a db of the minimum. The value of the noise figure does not vary a great deal as the collector voltage and emitter current are changed except that if the collector voltage is lowered below 6 or 8 volts the gain decreases and in general the noise figure in- creases. Noise Figure at 10 Mc. Although not described here, bandpass amplifiers centered at 10 mc with a 200-kc pass band have been constructed using tetrode transistors. A gain of slightly over 20 db per stage can be realized at this frequency. The noise figure of transistors tried in this circuit is shown on Fig. 23, the data being shown in the same manner as described above. At 10 mc the noise figures are lower than at 70 mc. The remarks made above con- cerning variation of noise figure with operating conditions also apply to this case. ACKNOWLEDGMENTS We are happy to acknowledge the advice and encouragement given us by R. L. Wallace, Jr., and others in the Laboratories. We also wish to express our thanks to E. Dickten who fabricated the transistors used to obtain the experimental data presented. W. F. Wolfertz made the transistor parameter measurements used in the computations. R. H. Bosworth and C. E. Scheideler were responsible for construction of the circuits and some of the gain measurements. We also wish to thank W. R. Bennett for his aid in preparing the manuscript. The Nature of Power Saturation in Traveling Wave Tubes By C. C. CUTLER (Manuscript received February 2, 1956) The non-linear operating characteristics of a traveling wave tube have been studied using a tube scaled to low frequency and large size. Measure- ments of electron beam velocity and current as a function of RF phase and amplitude show the mechanism of power saturation. The most important conclusions are: I. There is an optimum set of parameters (QC = 0.2 and yro = 0.6) giving the greatest efficiency. II. There is a best value of the gain parameter "C" which leads to a best efficiency of about 38 per cent. III. A picture of the actual spent beam modidation is now available which shows the factors contributing to traveling wave tube power saturation. INTRODUCTION The highest possible efficiency of the travehng wave tube has been estimated from many different points of view. In his first paper on the subject^ J. R. Pierce showed that according to small signal theory, when the dc beam current reaches 100 per cent modulation an efficiencj^ of , = § (1) is indicated,* and thus the actual efficiency might be limited to some- thing like this value. Upon later consideration" he concluded that the ac convection current could be twice the dc current and that one might expect an efficiency of r) = 2C (2) He also considered the effects of space charge, and concluded on the * Symbols are consistent with Reference 2 and are listed at the end of this paper. 841 842 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 same basis that under high space charge and elevated voltage conditions, efficiencies might be as high as 77 = 8C (3) J. C. Slater^ on the other hand considered the motion of electrons in a traveling wave and concluded that the maximum possible reduction in beam velocity would also indicate a limiting efficiency of 2C. Taking a more realistic account of the electron velocity, Pierce showed that these considerations lead to a value of V = -^yiC (4) which, since t/i ranges between —}'2 and —2, leads to the same range of values as the other predictions. None of these papers purport to give a physical picture of the over- loading phenomenon, but only specify clear limitations to the linear theory. L. Brillouin on the other hand found a stable solution for the flow of electrons bunched in the troughs of a traveling wave. This he supposed to represent the limiting high level condition of traveling wave tube operation. His results give an efficiency of V = 2hC (5) In the first numerical computations of the actual electron motion in a traveling wave tube in the nonlinear region of operation, Nordsieck pre- dicted efficiencies ranging between 2.5 and 7 times C and showed that there would be a considerable reduction in efficiency for large diameter beams, due to the non-uniformity of circuit field across the beam diame- ter. He also gave some indication of the electron dynamics involved. Improving on this line of attack, Poulter calculated some cases includ- ing the effect of space charge and large values of C. Tien, Walker and Wolontis carried computations still further for small values of C by including the effect of small beam radii upon the space charge terms, and showed that space charge and finite (small) beam radii result in much smaller efficiencies than were previously predicted. J. E. Rowe^ got similar results and gave more information on the effects of finite values of C. Computations for large values of C by Tien showed that a serious departure from the small C conditions takes place above values of C = 0.1 if space charge is small (i.e., below QC = 0.1) and above C = 0.05 for larger values of space charge. They indicated that a maximum value of efficiency as high as 40 per cent should be possible using C = 0.15, QC = 0.1 and elevated beam voltages. These five papers give some insight into the electron dynamics of power NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 843 saturation, but still involve questionable approximations which make it desirable to compare predictions with the actual situation. Theoretical considerations of the effects of attenuation upon efhciency have not led to conclusions coming even close to the observed results. Measured characteristics^^' ^^ show that the effect of attenuation is very large, but that attenuation may be appropriately distributed to attain stability and isolation between input and output of the tube without de- grading the output power. There are also several papers in the French and German periodicals which deal with the question of traveling wave tube efficienc3^ Some of these are listed in References 12 through 20. This paper describes measurements of efficiency and of beam modula- tion made on a traveling wave tube scaled to large size,* and low fre- quencies. The construction of the tube, shown in Fig. 1, and the measure- ment of its parameters were much more accurate than is usual in the design of such tubes. The results are believed to be generally applicable to tubes having similar values of the normalized parameters. OUTPUT TERMINATION INPUT TERMINATION INTERMEDIATE TAP VACUUM HEADER / ^ VELOCITY ANALYZER SAMPLE OF HELIX SUPPORTS SECTION OF FOCUSING SOLENOID Fig. 1 — The scale model traveling wave tube. The tube is 10 feet long with a c-opper helix supported by notched glass tubing from an aluminum cylinder over- wound with a focusing solenoid. It is continuously pumped and readilj^ demount- able. See Appendix. 844 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Two kinds of measurements are described. First, the efficiency and power output are determined for various conditions of operation, and second the spent beam ac velocity and current are measured. The prin- cipal results are shown in Figs. 2 to 4 which give the obtainable effi- ciencies, and in Figs. 7 to 10 which show some of the factors which con- tribute to power saturation. These figures are discussed in detail later. The most significant phenomenon is the early formation of an out-of- phase bunch of electrons which have been violentl}^ thrown back from the initial bunch, absorbing energy from the circuit wave, and inhibiting its growth. The final velocity of most of the electrons is near to that of the circuit wave which would lead to a value of limiting efficiency t] = —2yiC (6) if the wave velocity maintained its small signal value. Actually the wave slows down, under the most favorable conditions giving rise to a some- what higher efficiency. For other conditions, space charge, excess elec- tron velocity, or nonuniformity of the circuit field enter in various ways to prevent the desired grouping of electrons and result in lower effi- ciencies. The observed efficiencies are a rather complicated function of QC, yvo and C. To compare with efficiencies obtained from practical tubes one must account for circuit attenuation and be sure that some uncontrolled factor such as helix non-uniformity and secondary emission is not seri- ously affecting the tubes' performance. Measured efficiencies of several carefully designed tubes have been assembled and are compared with the results of this paper in Table I. The results of these measurements compare fa^'orably with the com- putations of Tien, Walker and Wolontis , and of Tien . There are, how- ever some important differences which are discussed in a later section. TRAVELING WAVE TUBE EFFICIENCY MEASUREMENTS Reasoning from low level theory, efficiency should be a function of the gain parameter, "C," the space charge parameter "QC," the circuit, attenuation, and (for large beam sizes), the relative beam radius "yro ." It was soon found that efficiency is a much more complicated function of y)\i than expected. The iiiilial ()l)jecti\-e was to detoiniine the effect of QC, C, and yr^ separately on efficiency, but it A\'as necessary to gi^'e a much more general coverage of these parameters, not assuming an>' of them to be small. Most of the measurements ha\^e been made with small \alues of loss NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 845 Table I Laboratory Freq. mc. QC yrr, c V meas- ured V (from Fig. 3) 7) (From Fig. 3 with allowance for circuit attenuationio McDowell* 4,000 0.27 0.62 0.078 19.5 26 21.6 6,000 0.29 0.8 0.058 13.2 16.2 12.5 Brangaccio and Cutlerf Danielson and Watson* 4,000 11,000 0.61 0.35 0.87 1.2 0.041 0.05 11 6.6 6 7 6 4.8 R. R. Warnecke^e, n, is 870 0.32 0.3 .125 27 33 33 W. Kleen and W. Frizes 4,000 0.5 0.43 0.05 7.8 11.5 5.7 W. KleenJ L. Bruck§ 4,000 3,500 0.2 0.19 0.94 0.6 0.1 0.065 20 15 26 23 22 18.5 Hughes Aircraft Co. 3,240 0.19 0.94 0.12 39 31 29 9,000 0.15 1.3 0.11 25 15.5 12.7 * At Bell Telephone Laboratories. t Reference 10 (a slight beam misalignment could account for most of this difference). t Siemens & Halske, Munich, Germany. § Telefunken, Ulm, Germany. and of the gain parameter, where efficiency is proportional to C, as ex- pected from small-signal small-C predictions. This reduces the problem to a determination of -q/C versus QC and 7ro . Many measurements of this kind have been made, and the data are summarized in Figs. 2 and 3, with efficiency shown as a function of QC and yro . In Fig. 2 we have the efficiency when the beam voltage is that which gives maximum low-level gain. Fig. 3 shows the efficiency ob- tained when the beam potential is raised to optimize the power output, and contours of constant efficiency have been sketched in. There is significantly higher efficiency than before in the region of maximum effi- ciency, but not much more elsewhere. Fig. 4 shows how efficiency varies with C for a small value of QC, a representative value of 7ro , and with beam voltage increased to maxi- mize the output. This indicates a maximum of about 38 per cent at C = 0.14. Some of the computed results of Tien, Walker and Wolontis, and of Tien are also indicated in the figures. Their results generally indicate somewhat greater efficiencies than were observed, but in the most sig- i nificant region the comparison is not too bad as will be seen in a later section. The measurements are for conditions having negligible circuit loss near the tube output. There are no new data on the effect of loss, but earlier results'** have been verified by measurements at Stanford Uni- versity and are still believed to be a satisfactory guide in tube design. 846 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 3.0 0.5 Fig. 2 — Values of efficiency/C as a function of QC andyro at the voltage giving maximum gain per unit length. The shaded contours and triangular points are from the computations^ of Tien, Walker and Wolontis. The circled points are from the measurements and the line contours are estimated lines of constant efficiency. The most significant difference is for large beam radii, where the RF field varies over the beam radius in a way not accounted for in the computations. SPENT BEAM CHARACTERISTICS The scale model traveling wave tube was followed by a velocity an- alyzer as sketched in Fig. 5 and described in the Appendix. A sample of the beam at the output end of the helix is passed through a sweep cir- cuit to separate electrons according to phase, and crossed electric and NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 847 magnetic fields to sort them according to velocity. The resulting beam draws a pattern on a fluorescent screen as shown in Fig. 6 from which charge density and velocity can be measured as a function of signal phase. The velocity coordinate is determined by photographing the ellipse with several different beam potentials, as in Fig. 6(a), and the phase coordinate is measured along the ellipse. From pictures like this a complete determination of electron behavior is obtained from the linear region up to and above the saturation level. The results of such a run are plotted in Fig. 7. The upper lefthand 2.0 Fig. 3 — Values of efficiency/C as u function of QC and yro at elevated beam voltage. Raising the beam voltage has little effect at large QC and small yro , and less than expected anywhere. Again the triangular points are from Tien, Walker and Wolontis,^ and the line contours are estimated from the measured data. 848 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 A \ \. 40% EFFICIENCY \ \ s. n a S-& 1 s. ^ o U u a.ZM—a on" ^ ---..'^ ^ D D c ^^).^ *^.. --- o o "^■. □ TAKEN WITH 7ro = 0.78 QC=0,1 o TAKEN WITH 7ro=0.41 QC=0.06 A FROM TIEN REFERENCE 9 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 i.a Fig. 4 — Efficiency/C for large values of C and with elevated beam voltage. Efficiency seriously departs from proportionality to C at C = 0.14, where a maxi- mum efficiency of about 38 per cent is measured. MAGNETIC SHIELD COIL ELECTROSTATIC ELECTRON LENS (3) FLOURESCENT SCREEN, NOTCHED GLASS RODS (3) DEFLECTING PLATES DEFLECTING COILS DEFLECTING PLATES Fig. 5 — The velocity analyzer. A sample of the spent electron beam is ac- celerated to a high potential, swept transversely with a synchronous voltage, sorted with crossed electric and magnetic fields, and focused onto a fluorescent screen. pattern, Fig. 7(a), is representative of the low level (linear) conditions (22 db below the drive for saturation output) . The dashed curve repre- sents the voltage on the circuit, inverted so that electrons can be vis- ualized as rolling down hill on the curve. The phase of this voltage rela- tive to the electron ac velocity is computed from small signal theory, but NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 849 everything else in Fig. 7, including subsequent variations of phase, are measured. The solid line patterns represent the ac velocity, and the shaded area, the charge density corresponding to that velocity. Thus in each pattern we have a complete story of (fundamental) circuit voltage, electron velocity and current density as a function of phase, for a par- ticular signal input level. The velocity and current modulations at small signal levels check calculated values well, and it is not difficult to visu- alize the dynamics giving this pattern. Consider first the situation in the tube at small signal amplitudes. At the input an unmodulated electron beam enters the field of an elec- tromagnetic wave moving with approximately the same velocity as the electrons. The electrons are accelerated or decelerated depending upon their phase relative to the wave, and soon are modulated in velocity. The velocity modulation causes a bunching of the electrons near the potential maxima (i.e., the valleys in the inverted potential wave shown) and these bunches in turn induce a new electromagnetic wave com- ponent onto the circuit roughly in ciuadrature following the initial wave. I'he addition of this component gives a net field somewhat retarded from the initial wave and larger in amplitude. Continuation of this process Fig. 6 — Velocity analyzer patterns. The beam sample is made to traverse an ellipse at }i the signal frequency. Current density modulation appears as intensity variation, and velocity variation as vertical deflection from the ellipse. 2 1 0 - I -2 2 1 0 -1 -2 -22 DB (a) — 5^!stt __ -_^w^^ p^^~^ ' 1 ^-LLfcggq 1 ! 1 i 1 ij=rr— T3aij^gj4j^^_l_j^ 1 1 -,1^^ 1 -17 DB (b) 1 ^iiTTTTT SCCl-r-^ — -=,=e^^ K ±3=-- , -^ ^^^^^^^^ < D U tr < _] UJ cc Q Z < u o z < I o o o z o ct > UJ a. 1 0 - 1 -2 2 , 1 •0 - I -2 -3 2 I 0 - 1 -2 -3 1 ^ -12 DB (d) '" ~ ~~ ~^3^ ^' "■^. ^ -- ■ --^ ^ ^^ ^ 4H- -^e^7 ~ W^'-M iiiiirrr :^^^ ^LU- ■ , ^^"^'"""'^'^""^ -10DB (e) ,---^' ~--^>^ A '^ \ tt J 1^ fffi ~-_ toil-- fflfflfe*^^ Lcn Jt^^ 'III 1 1 M 1 L-W^-^ ^ -8DB (D / ^ '^ "X^'^ ^ I ^ ^ _^^^ --. i^ :^^^ Jrii. -^ .rrrr-^,^ ^ i^ tti^ *€-^fe^^ 1 <:^ J- ' 1 ^ — 1 -4 DB (h) ' -«. 0 1 ^ y / ^, \ ^ / _ _^flrr -- ^ L^ ^-^ 2 1"^^ ^ ^ 3 ! ^^^ ^ ^ 240 180 120 60 0 RELATIVE PHASE IN 60 120 DEGREES 180 240 Fig. 7 — Curves of current and velocity as a function of phase for various input levels. The velocity becomes multivalued at a very low level, a tail forming a nucleus for a second electron bunch which eventually caused saturation in the output. For this run C = 0.1 Q,C = 0.06, t^o = 0.4 and h = 0.26. 850 - 1 -2 -3 X' J ^ -2 DB ^ _ ^ / (0 • • r \ \, 1 ^g-^ K .- X ^vt^ xii^^^Li-UJ f^lO- -3 ^ ^--L--^^ -1 DB X^ -, ^' .< ^ (^) < 7 fei-'. 1-^ ^ ■'■■■ 11 .^ / 1 1^ T < o > o > Q Z < UJ z < I u u o -J dJ > z o a. \- o UJ . _^ 1 3 DB J >':^-';^ ^ (m) > ^-' (¥ ^^ -^ ^'-^ -<• 4; yv~ - ^ "J^LK ^v- ^ I \ ^"^ > I- < UJ 2 1 0 - 1 -2 -3 xS^ 6 DB^ ^ ^-^"^ ~ /^ ^■- U' (n) \ ^dr ^^ * y ^f ^ K 7 y r 3 2 1 0 - 1 -2 -3 240 9 DB (0) L _ -^V^ -^^ ~^^^-- r ~~~~" x' ^^ /\ /^" ^ 7^ ^ X y ^5^ ffe. ^ • x^ y 160 120 60 0 60 120 RELATIVE PHASE IN DEGREES 180 240 851 852 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 may be seen to give a resultant increasing wave traveling somewhat slower than the initial wave, and thus slower than the electron velocity. Returning to Fig. 7 we see that electrons in the decelerating field [from +30 to +210° in Fig. 7(a)] have been slowed down, and because of their initial velocity being faster than the wave velocity, have moved forward in the wave giving a region of minimum velocity somewhat in advance of the point of maximum retarding field (greatest negative slope in the wave potential). Also, bunching due to acceleration and deceleration of electrons has produced a maximum of electron current density which, because of the initial excess electron velocity, is somewhat to the right of the potential maximum (downward). As the level is increased the modulation increases and at 17 db below saturation drive, Fig. 7(b), some nonlinearity is evident. The velocity and current are no longer sinusoidal, but show the beginnings of a cusp in the velocity curve and a definite non-sinusoidal bunching of the electrons in the retarding field region (between +30 and 210°). In the next pattern, Fig. 7(c), at 14 db below saturation a definite cusp has formed with a very sharp concentration of electrons extending sig- nificantly below the velocities of the other electrons. We already have a wide range of velocities in the vicinity of the cusp, and at this level the single valued velocity picture of the traveling wave tube breaks down. Although it cannot be distinctly resolved, the study of many such pic- tures leaves little doubt that the cusp and its later development is really a folding of the velocity line. The next pattern at 12 db below saturation drive. Fig. 7(d), shows a greater development of the spur and a somewhat greater consolidation of current in the main bunch between +60° and +180°. It is interesting that the velocity in this region has not changed significantly. In order for this to be true the space charge field must just compensate for the circuit field. In the vicinity of the 60° point the space charge field ob- viously must reverse, accounting for the very sharp deceleration evident in the very rapid development of the low velocity spur. The decelerating field must be far from that of the wave, inasmuch as the electrons just behind the cusp are much more sharply decelerated than those preced- ing the cusp. We conclude that there are very sharply defined space charge fields much stronger than the helix field. At this relatively low drive, the velocity spread has already achieved its maximum peak value. The succeeding three patterns show a continuing growth of the spur, a continued bleeding of electrons from the higher velocity regions, and a consolidation of the main bunch just in advance of the spur. Presum- ably the increased concentration of space charge in the bimch has kept NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 853 pace with the increasing hehx field, so that the net decelerating field still balances to nearly zero. At 4 db below the saturation drive, Fig. 7(h), the spur has moved well into the accelerating region, and has been speeded up. The main bunch of electrons is still to the right of the spur, and has been consolidated into a 60° interval. The few electrons in ad- vance of this region evidently no longer find the space charge field suffi- cient to balance the circuit field, and are being decelerated into a second low velocity loop. The next three patterns show a continued growth of this second low velocity loop, further consolidation of the 'main bunch', and^the rapid formation of a second bunch in the accelerating field at the end of the spur. It is interesting that at saturation drive, Fig. 7(k) the two bunches are very nearly equal, and in equal and opposite circuit fields, nearly 180° apart. The reason for the saturation is that while the main bunch is still giving up energy to the wave, the new one is absorbing energy at an equal rate. The fundamental component of electron current is evi- dently small, and is in quadrature with the circuit field. The current density in the dashed regions is less than 1 per cent of that in the bunches, and probably more than 95 per cent of the electrons are in the two bunches. Two new effects are observable at this level. The second elec- tron bunch has begun to come apart, presumably because of strong lo- calized space charge forces. These forces are also evident in the kink in the velocity pattern drawn by the fast electrons at the same phase as the second bunch. Since the majority of the current is in the two bunches at a reduced velocity of ^^ = -1.1 2FoC one would expect an output efficiency of ^ = 2.2C The actual measured efficiency RF power output DC power input was 2.0 C. Under the conditions described, (6) would give 1.4 C. At still higher drive levels the pattern continues to develop, electrons from the first bunch falling back into the second, which in turn continues to divide, one part accelerating ahead into a new spur, and the other I 854 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 1 i slowing down and falling further back in phase. At 9 db above satura- > tion, Fig. 7(o), the pattern is quite complex, and at still higher levels it is utterly indescribable. ■ It is interesting that the ^'elocity gives a line pattern, even though a ' multi^'alued one. It is reasonable to suppose that the development of the spur is really a folding of the \'elocity line so that the spur is really a double line. Thus, at the 9 db level, and at 0° phase, for instance, there must be electrons originating from five different parts of the initial dis- ; tribution. In an attempt to verify this the resolution of the velocity an-il alyzer was adjusted so that a difference in velocity of 2 per cent of the- overall spread could be observed, but there was no positive indication of r- more than one velocity associated with any line shown. : There has been a long-standing debate as to whether or not electrons j are trapped in the circuit field, or continue to override the w^ave at large amplitudes. The observations indicate that with low values of space charge and near synchronous voltage the electrons are effectively trapped: in the wave until well above saturation amplitude. In other circum- stances this is not the case, as we shall see. SPACE CHARGE EFFECTS The data of Fig. 7 were taken with a very small value of the space charge parameter QC, so small in fact as to be almost negligible as far as low level operation is concerned. Yet the space charge forces evidently played a very strong role in the development of the velocity and currenti' patterns. It is doubtful that space charge would ever be negligible in thisii respect, because if the space charge parameter were smaller, the bunch-i ing would be more complete, the electron density in the bunch would be greater limited only by the balance of space charge field and circuit field in the bunch. The effect of decreaising QC further therefore is a greater localization of the space charge forces, rather than a reduction of their magnitude, at least until the bunch becomes short compared to the beam radius. Increasing the value of the space charge parameter has quite the op-' posite effect. In Fig. 8 are shown three velocity-cm-rent distributions ati the saturation level, for different A-alues of QC. It can be seen that a re-' suit of increased space charge is a greater spread of velocities, and a wider phase distribution of current. With the introduction of space charge, the velocity difference between the electrons and the circuit wave at low levels is increased. Consequently electrons spend a longer time in the decelerating field before beingj thrown back in the low velocity spur, and thus lose more energy. Thel i NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 855 -^ r- ' ""' ' ^ NO (a)QC = 0.064 / > \ Vo ^-' \^ Hi y L -^ w_ 1 /-^ 1 \ r /; I- »W J* y y y o^niij\[\ o o ^ y y y^ ^^ *^ o v^^ UJ z < I o >- u o _l z o tr H U _] UJ LLl > _l LU cr o ^'"^ ^^ 1 (b) QC=0.22 y \ Vo ^x^ - -^ u— o— -Vvw -^ 0\^ ^ ° 1 c ^ ,--;; >. .-«-' V o ° o 3 ^ ^^ ^^--^^^ ^^^^Wf*^*^ - I -2 -3 -4 240 o o o (c) o QC = 0.48 o o o ° o o " \° o o o o /^ o o \° ^0 ° oc' ^^ \ o o °° ^ m Hlx o \ o o""^ "vw^X 0 -^^^ \ ° ° It, fe-ffig "^^ ^ t^^^ n < o % ^ o o o ^^ ^^ ° 180 120 60 0 60 120 RELATIVE PHASE IN DEGREES 180 240 Fig. 8 — A comparison sliowing the effect of the space charge parameter QG 1 on the velocity and current at overload. The points represent the disc electrons of { the computations^ of Tien, Walker and Wolontis. For this run 7ro = 0.4 and h is I chosen for maximum X\ . I greater reduction of velocity results in a faster and farther retarding of I the current in the spur before the retarded electrons recover velocity in i the accelerating region. Also the larger space charge forces prevent as I tight bunching of the electrons anywhere, so that at overload they are •spread over a much wider phase interval (about 360° for QC = 0.5). ! Space charge also prevents electrons from the forward part of the bunch j from being trapped so that more electrons escape ahead of the decelerat- ' ing field and more current is found in the upper half of the velocity j curve. This very likely is the reason that efficiency decreases when QC \ is increased above about 0.3. 856 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 EFFECT OF BEAM SIZE In small signal operation, decreasing the beam radius below that which assures a constant circuit field throughout the beam has no effect except that accounted for by its effect on QC. Fig. 9 shows that for large signals, however, it has a pronounced effect. When the beam is made smaller (with QC maintained by changing frequency and beam current), the slowed up tail is formed at a much lower signal level (not shown), by a very few electrons which begin to collect in the accelerating region before the beam is strongly modulated. As the level is increased, the current is redistributed, more going into the tail without much alteration in the shape of the velocity pattern, and with no strong bunching at any part of the curve. This result is exaggerated in Fig. 9(c) by measuring with a o o > (J) z < I o >- H; O O _I LU > z o tr o UJ _l UJ > UJ (a) 7ro = o.64 ^ — ■ - ^, ^^ ^ \ % \ > 1^ V' I ^ /<^M^ j|'"vw~ ' ^^ ^^ ^ttin ^% ^^ 240 180 120 60 0 60 120 RELATIVE PHASE IN DEGREES 1 1 (b) •5fro=0.22 / *"*\, y \ Vo ^ --" ^ 3 V, ^ ^*^^ ^ ^ "-'-^ Ui w ^**^ '^Jjy^ ^ \ \ I ^^^ ,riin4+ft: (C) 7 '0 = 0.06 ^ ^icnxct; ■0!^ ^3^ Vo [???*?s^rt ^ '>'■ V ^IL rtjff# ■ip> p> 4^ ^\. , — L L. W^-^i' ^ ^•^^m-rf 180 240 Fig. 9 — Curves of current and velocity as influenced by yr^ . Space charge becomes a very potent factor near overload, especially when the beam is small. For this run QC = 0.34 and 6 = 1.0. NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 857 ridiculously small beam. By comparison with curves taken for larger beams, the tail is diminutive, electrons are much more uniformly dis- tributed over all velocities and phases, and a peculiar splitting of veloci- ties in the main bunch is found. The latter indicates that electrons entering from the higher velocity region move forward in the bunch, and the rest gradually retard. The smaller reduction in velocities, and the spread of electrons into the higher velocity regions is consistent with the lower efficiency measured (Fig. 2). To explain the observed difference in high level performance of tubes with different size beams we must consider the character of the ac longi- tudinal space charge field. The coulomb field from an elemental length of an electron beam is inversely proportional to the square of the dis- tance from the element E = Const 7-^, (7) provided (z — Zi) » ro and {z — Zi) « a. For (z — zi) not small compared to a, (i.e., circuit radius not awfully large) the field would drop even faster with (z — zi) due to the shielding effect of the circuit. On the other hand, very near to the beam element {z — zi ! o z < I o > u o _l z § I- u LU _l UJ UJ > < UJ \ \ \ (a) b = o x" ^ ' \ Vq.v, ^r-K -^w /r^:':?- Ih -■ ^^m. -iii > . ^% ^^^ -1 -2 -3 2 1 0 -1 -2 -3 -4 r"'' ,-'''' \ \ (b) b = 0.77 ^ .'' \ \ Vo /^^ y m/ M^ N" V, vw ^ A "\ /t'' y:a ^ V ^ ...***^ ^^X (c) b = ).56 ^^^ \ Vo 1 .___ . ._ (|^m^^e*^=*** Vw (171 N V, i k. / 240 180 120 60 0 60 120 RELATIVE PHASE IN DEGREES 180 240 Fig. 10 — The influence of beam velocity on ac velocity and current. When the velocity is raised too high, the electrons are not effectively trapped by the wave, and override into the accelerating field. With large QC and/or small 7ro the elec- trons override in any case, and little is gained by increasing h. For this case QG = 0.13 andTfo = 0.21. ^main effect being to push more electrons forward into the accelerating j region. KLECTRIC FIELD IN THE BEAM ' Besides telling a clear story of the non-linear dynamics of the traveling , wave tube, the foregoing curves contain a lot of information about aver- jage current and velocity distributions. From the current or velocity curves we can in turn deduce the distribution of longitudinal electric field in the beam. Figs. 11(a) and (b) show the instantaneous current as a function of phase, taken from the curves of Figs. 8(a) and (b). The infinite differential in the velocity curve necessarily gives a pole in the charge density (at about 88°). The total charge in the vicinity of the 860 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 z UJ (T a. O UJ > u EC (a) A h ^lo+L / n V > ^ ^ ^ — '■ / \ o > y o i-\ o u u -J I > LJ < I- erg UJ -2 -6 (b) p\ ^ ,/^ \ \ ^ CIRCUIT ' FIELD // r \ y )^'-- V TOTAL FIELD \\ « 1 V -180 120 -60 0 60 RELATIVE PHASE IN DEGREES 120 180 Fig. 11 — AC current and electric field in the beam. The upper curve comes directly from Fig. 8(a). The lower curve is deduced by an approximate method from the velocity curve of Fig. 8(a). The double value below 90° is partly due to inconcistency between the two parts of the velocity curve, and partly due to the nature of the approximation. pole, and the range of the space charge force (dependent upon QC and 7ro) determines its effect upon the electron dynamics. Most of the current is incorporated in the two bunches nearly 180° apart, as we have seen, each bunch having a current density many times the average. We might obtain the space charge fields from the current density, but this would require a rather definite knowledge of the characteristic space charge field versus distance as influenced by beam diameter. It would also be pushing the accuracy of charge density measurement, which is crude at best. A better way is to compute the electron accelera- tion from the velocity curves. This may be done by taking two velocity patterns at slightly different signal levels, and tracing electrons from one to the next, using the measured velocity to determine the relative phase shift of any electron. In the appendix it is shown that a close approximation to this is E^ = 2/3CYo [ (Fo - FJ + A7' 2FoC (10) NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 861 where the parameters are all obtained from a single velocity curve, and is field strength in volts meter at phase $ is the value of the ordinate of the velocity characteristic of of interest (Figures 7 to 10) and is the ^'alue of the ordinate corresponding to the wave velocity. (To be precise, the wave velocity at the associ- ated output level, but to a reasonable approximation, that of the wave velocity at low levels. (This value is indicated by Vw in the velocity curves.) i The total electric field has been computed for the case of Figs. 8(a) and (b) and is given in Figs. 11(b) and 12(b) together with the circuit field .calculated for the associated power level and plotted with an arbitrarily chosen phase. In each case it is seen that the space charge field is com- parable in magnitude to the circuit field, is far from sinusoidal, and z UJ cr q: D O UJ > 111 cc 1 (a) 1 . T 4- i io + i* ^_^ A Ao J ^"^ -^■v;^-- ^ ■ — v jL _^^ u u UJ -J I UJ H o > UJ I- tc < H CE 9 CIRCUIT FIELD r\ (b) y^ ;- \l\ / ^ K V / / /] \ \ / y '^y \V V,. ^^ TOTAL FIELD u V 1 \\ -J ■180 -120 -60 0 60 RELATIVE PHASE IN DEGREES 120 180 Fig. 12 — AC current and electric field in the beam deduced from Fig. 8(b). The greater space charge results in a less defined bunch, and smoother space charge field than in Fig. 11. 862 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 10 >° o en < CO o -10 OUTPUT POWER -.^ ' X / Si RELATIVE HARMONIC CURRENT ,2 ND rK \ ,ST CALCULATED 1ST x-^ srd'^-^ V6 \ X -30 -20 - 10 0 10 RELATIVE INPUT LEVEL IN DB FROM SATURATION DRIVE Fig. 13 —^Curves of output level, fourier component amplitudes of beam cur-ij rent, and peak velocity as a function of input level for low space charge. These-: curves were deduced from Fig. 8 (a). j 0.6 Fig. 14 — Maximum velocity reduction as a function of space charge (from Fig. J 8). The velocity reduction is about 3.5 iji . i NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 863 agrees qualitatively with what would be expected from the associated curve of beam current. To determine the curves of Figs. 11 and 12 is rather stretching the accuracy of the measurements as can be seen by the large discrepancy in the field calculated from the two parts of the velocity curve which of course should be identical. The figures do give an interesting qualitative picture of traveling wave tube behavior however, and are included here for that reason. OVERALL VELOCITY SPREAD Of more practical importance is the overall velocity spread in the spent beam. It is often desirable to reduce the power dissipation in a traveling wave tube by operating the collector at a potential below that of the electron beam, and it is interesting to see how far one might go. Fig. 13 shows how the velocity reduction of the slowest electron, together with the output level and fourier current components of beam current vary with input level. For small amplitudes, the low level theory ac- curately predicts the velocity, but near overload, as we have seen, the minimum \'elocity drops sharply to a value several times lower than that projected from small signal theory. The maximum velocity spread dependence upon the space charge parameter QC is shown in Fig. 14. Similar data for values of the other parameters may be obtained from the velocity diagrams. From the foregoing data, one can deduce the amount of reduction of collector potential that should be theoretically possible wdthout turning back any electrons. An idealized unipotential anode could collect all the current at a potential AF (in the foregoing figures) above the cathode, decreasing the dissipated power by a factor of AF/Fo below the dc beam power. STOPPING POTENTIAL MEASUREMENTS Information on spent beam velocity has also been obtained by a stop- ping-potential measurement at the collector of a more conventional 4,000-mc traveling wave tube.* Two fine mesh grids were closely spaced to a flat collecting plate, and collector current was measured as a func- tion of the potential of second grid. The first grid was very dense, to prevent reflected electrons from returning into the helix. One curve taken with this arrangement is shown in Fig. 15 and for comparison we have * Similar measurements have been reported by Atsumi Kondo, Improvement of the Efficiency of the Traveling Wave Tube, at the I.R.E. Annual Conference on Electron Tube Research, Stanford University, June 18, 1953. 864 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 plotted the distribution predicted from Fig. 9(b). The RF losses in the 4,000-mc tube were not neghgible, and probably account for slightly smaller power output and greater proportion of higher velocity electrons. COMPARISON WITH COMPUTED CURVES Non-linear calculations of traveling wave tube behavior have been made by Tien, Walker and Wolontis' and by Tien^ covering the same region of parameter values as is reported here. In Figs. 2, 3, 4 and 9 are shown some of their data on our coordinates. The similarity of the results over much of the range is rather reassuring. It is interesting that in order to make the computations it was necessary to assume two space charge factors, just as was found experimentally. There are, however, some significant differences: 1. In general, the computed values give a higher ^alue of efliciency than is measured, by about 25 per cent. Thus, the computations indicate rwm(m^ ^ Hill zero signal characteristic: IDEAL ^>4 MEASURED ^j I I l( -400 0 400 800 STOPPING GRID VOLTAGE I 1200 ■ F^^' A ZZ. '^^^\^^ current versus stopping potential. The oscilloscope curve IS tor a 4,0n0-mc tube, and the other that predicted from the scale model meas- urements. By integrating current as a function of velocit v for Figs. 7-10 stopping potential distributions can be deduced for other conditions NATUEE OF POWER SATURATION IN TRAVELING WAVE TUBES 865 3.5 3.0 2.5 2.0 c 1.5 (.0 0.5 • ^ • \ \ ,j .. ^, \ \ S V 0.2 0.4 0.6 0.8 1.0 1.2 7ro 1.4 1.6 1.8 2.0 2.2 Fig. 16 — Efficiency versus 7ro for small QC. The dashed curve is proportional to the amount of beam current in the circuit field strength having at least 85 per cent of the intensity at the edge of the beam. This illustrates the fact that for large beams only the edge of the beam is effective. that with the reasonable vahies of QC = .25 and 7ro = 0.8 {kr = 2.5), the efficiency would be about 3.8C, whereas the measured value is 3.1C. 2. The largest discrepancy in the measured and computed value of r]/C is for large values of yro (small kr), where the computations show a steady increase in efficiency instead of a sharp decrease. This arises be- cause the computational model assumed the electric field to be uniform across the beam, whereas in the actual tube it varies as loiyr), and for large values of 77-0 the field is weak near the beam axis. This effect is shown in Fig. 16 where rj/C is plotted versus yro for small values of QC, on the same scale with a curve proportional to the square of the fraction of the beam within a cylindrical shell such that 1 - Io(yn) hiyro) = 0.85 (11) where ri is the inside radius, and ro the outside beam radius (i.e., the fraction of the beam in a field greater than 85 per cent of that at the beam edge). No serious studies of velocity were made for large beams, but on cur- sory examination it was evident that the beam modulation varied con- siderably over the cross section when the beam was very large, and scarcely at all when it was smaller than around yro < 0.8. 866 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 3. The observed effect of small beam radius upon efficiency is not as pronounced as was found in the computations. The reason is not kno^^^l but may be due to modulation of the beam diameter at large signal levels. This effect would be neghgible with the larger Tro's, due to the focusing fields being relatively much larger. 4. The computations, and also those of Nordsieck, Poulter and Rowe^ indicate a much higher efficiency than has been observed at elevated beam voltages and small C and QC. The reason for this may be that the limited number of "electrons" used in the computational models fail to adequately account for the very sharp space charge cusp that forms under low QC conditions, or that interpolation between their points should not be linear, as assumed in making the comparison. On the other hand it would be difficult to be sure that nonuniformities in electron emission were not influencing the measurements in the case of the large beams by giving a larger QC than calculated. 5. The increase in efficiency to be had by elevation of beam voltage is much smaller than is indicated by the computations. This may be a real difference, or it may be that at elevated voltages, the measurements are beginning to feel the influence of overloading in the attenuator. The margin of safety on attenuator overloading is not as great as one would like at the higher frequencies. 6. The velocity curves, Fig. 8, compare the computed and measured data on three runs. For small QC, Fig. 8(a), the agreement is remarkably good considering the fact that in the computation only 24 "electrons" were used to describe a rather complicated function. The effect of the lumping of space charge in the artificial 'disc' electrons causes a scatter- of points which is different from that in an actual tube as is especially apparent in Figure 8c. In spite of this the computational results indicate a velocity spread and current distribution not greatly different from that observed. CONCLUSIONS The large scale model traveling wave tube is a means for the deter- mination of non-hnear behavior, and has been valuable in determining relationships and limitations important to efficient operation of such tubes. It has shown that there is a broad optimum in tube parameters around C = 0.14 QC = 0.2 and 7ro = 0.5 for which values it is possible to obtain efficiencies well above 30 per cent. The measured ac beam velocity and current near overload show that it is unlikely that signifi- cant increase in efficiency can be obtained by any simple expedients such NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 867 as operations on the helix pitch alone, or the use of an auxiliary output circuit. The results being in normalized form, are believed to be generally applicable to conventional traveling wave tube design. With determina- tion of an equivalence yi beams, they should even be a useful guide in the design of tubes using hollow beams or other configurations. The work described could not have been done without the able assist- ance of G. J. Stiles and L. J. Heilos and the helpful council of many of my colleagues at Bell Telephone Laboratories. Appendix scale model tube design There were a larger number of factors to be accounted for in the de- sign of this tube. Its proportions should be such as to make it repre- sentative of the usual design of traveling wave tube. Its size should be such as to make it easy to define the electron beam boundary, and to dissect the beam. The size should also be such that the electron beam velocity analysis could be done before the beam character would be changed either by space charge, or its velocity spread. The voltage should he low so that further acceleration in the velocity analyzer would not lead to an inconveniently high voltage. Finally, the availability of suit- able measuring gear over a 3-1 frequency range, and the size of the laboratory must be considered. All of these factors led to low frequency operation, limited principally by the laboratory size and the mechanics of construction. A moderate perveance of around 0.2 X 10~ was taken, with a 7a of 1.2 and 7ro of 0.8 in a representative helix with small impedance reduc- tion due to dielectric and space harmonic loading. This is representative of practical tube design in the microwave range and is centered on the parameter values of most general interest. At a frequency of 100 mc and a beam potential of 400 volts this resulted in a helix 10 feet long and l}^ inches in diameter, with an electron beam 1 inch in diameter. The choice of frequency was finally determined by the availability of meas- uring equipment, and the voltage was selected to give a convenient size for dissection of the electron beam. By changing frequency, beam current, and beam diameter it was ! possible to cover a reasonable range of yro , and QC, and to make some observations into the region of large C operation. In all of the measurements described, a very strong uniform magnetic field was used to confine the beam, and therefore scaling of the magnetic 868 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 focusing field need not be considered. The electron beam was produced in a gridded gun and is thus near to the ideal confined flow, which is the only focusing arrangement which is known to determine a reasonably- uniform boundary to the beam. The beam size and straightness was checked using a fluorescent screen at the collector end. \ NORMALIZING FACTORS The measurements described are expressed relative to the linear theory, in Pierce's' notation, which are generally used in the design of traveling wave tubes. Thus, instead of being presented in the terms of measurement or simply normalized to efficiency, perveance, impedance, etc., they are expressed in terms of C, QC, yro , etc., with normalized fields, currents and velocities. In this way the results become adjuncts to the linear theory and are more easily applied to tube design. Electron velocity is plotted on the same scale as the relative velocity parameters b and yi used in low level theory, (i.e., normalized to AV/2VoC). Effi- ciency is normalized as r]/C, which for C less than 0.1 is relatively inde- pendent of C. Field strength in the linear region is proportional to v't {ri being efficiency measured at the appropriate signal level). Solving the equation for C , >3 E lo ^ ~WP2Vo ^^^^ gives us V ^ (13) C /3C2F, which we use as the normalizing parameter for electric field. Circuit po- tential is the integral of circuit field over a quarter period, giving a normalized parameter V/VoC . For convenience in the use of common coordinates, circuit potential was plotted as —V/2VoC^ in Figure 7. The other curves are plotted as values relative to dc quantities or to saturation level. Strictly speaking, the results hold only for tubes having the same pro- portions as the model. Practically, however, as long as the helix imped- ance and radius {ka or ya) are not different by orders of magnitude from the values used, and as long as the perveance is low (below 2 X 10~^ for NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 869 instance), the results are believed to be significant for tubes having the indicated values of 7ro and QC. Il HELIX IMPEDANCE It is important to the measurements to have an accurate evaluation [of the helix impedance. Several methods of measurement have been discussed in the literature.^^ ' ^^ That described by R. Kompfner was I selected, wherein the circuit impedance is correlated with the beam current and voltage which gives a null in the output signal. When the beam voltage and current are adjusted to give zero transmission for a lossless section of helix (neglecting space charge) CN = 0.314 and 5F/Fo = 1/A^. Using the measured length of the helix, and measuring the voltage and current giving the null in signal transmission, we can compute C, and thus the impedance and velocity (synchronous voltage) of the hehx. The impedance was calculated by P. K. Tien,^' and the results are compared in Fig. 17. The measured impedance at the high frequency end was much too low until space charge in the beam was accounted for in interpreting the measurements. Fortunately, in the absence of attenu- 1000 800 600 400 ^ 200 UJ u7 100 o z < Q Hi Q. 5 80 60 40 20 10 - ^ - N - \( - CALCULATED - \ \ \ \ - \ - \ - MEASURED POINTS- -^ V - N I 40 80 FREQUENCY 120 160 200 MEGACYCLES PER SECOND 240 Fig. 17 — Helix impedance as a function of frequency. The impedance was calculated taking into account dielectric loading and wire size. It was measured using the Kompfner dip method, taking account of space charge. 870 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 ation, the conditions for start of oscillation in a backward wave oscillator are the same as for the output null in a traveling wave tube. Space charge was first accounted for using the results of H. Heffner'^' " giving an excellent check between predicted and computed helix impedance. Later C. F. Quate showed that the same measurement could be used to de- termine the space charge parameter QC as well as the helix impedance. Since thermal velocity effects and the uncertainty of some of the assump- tions used in evaluating the small signal effects of space charge cast some doubt on the proper evaluation of this term, further measurements were made on this factor, and a satisfactory correlation between the ob- served value of QC and that computed from the Fletcher^'^ curves was obtained. TOTAL ACCELERATING FIELDS From the velocity characteristics shown in Figs. 7 through 10, we can deduce the electron accelerations, and thus the electric fields at any point. While the curves are actually diagrams of velocity as a function of phase, they closely correspond to the velocity- time or distance distri- bution of the electrons in the traveling wave tube. Knowing these charac- teristics we can deduce the motion of any element of charge, and thus the force under which it moves. It is observed that over most of the curve the shape of the velocity pattern does not change nearly so rapidly as the redistribution of electrons within the pattern. Thus, we can approxi- mate the situation at any amplitude by assuming the velocity pattern to be constant, and that electrons move within the pattern according to simple particle dynamics. This is a good approximation except where the acceleration is high (i.e., vertical crossings of the wave velocity line). Consider then an element of the velocity pattern at phase $i and velocity (wo + Aw). In an interval dt this element will move a distance (wo + Aw) dt (14) and will change velocity by du = E -dt (15) m At the same time the wave will have moved a distance v dt, resulting in a relative change in phase between wave and current element of d^ = ^(uo - V -\- Aw) dt (16) In terms of equivalent differences the term in brackets can be written («. - . + A„) = a/7T7. c (^° - ;: + "0 NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 871 from (16) and (17) we can write: du du d^ It d^ dt d ( AV d

' ^ ^ y ■::; A, / ^ / / / / / / / / / ^ f / ^, / / 0.1 0.2 0.3 0.4 0.5 0.6 QC 0.7 0.8 0.9 1.0 1.1 Fig. 18 — Increasing wave propagation factors used in interpreting the meas- urements. These are the maximum value of x\ and the corresponding value of 6 and y\ for given values of QC. ^ periment because a simple control of sensitivity was important in order I to study velocity differences ranging from 1 per cent up to as much as 1 100 per cent of the dc beam velocity. The velocity analyzer is sketched in Fig. 5. It consists of an aperture which transmits only a few microamperes of the electron stream; a mag- netic pole piece (not shown) terminating the focusing field; a pair of horizontal deflection plates; an electrostatic lens system; pole pieces and j deflection plate to provide a region with crossed electric and magnetic 'fields; and finally a drift tube, a post deflection acceleration electrode ,aiid fluorescent screen. The whole assembly is raised 1,000 volts above the helix potential and the 0.001 " aperture is very close to the end of the helix, so that the electrons are very quickly accelerated to a high voltage. V>Y this means, the region of debunching outside of the helix field is kept t)clow 1.4 radians transit angle and the velocity spread within the ana- lyzer is reduced by a factor of four. Space charge within the analyzer is <'iitirely negligible because of the small current transmitted. In order to discriminate in phase before the electrons are scrambled 87*1 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 due to their spread in velocity, the horizontal sweeping plates are mounted just as close to the aperture as is deemed practical. The ob- served velocity spreads in the beam were such as to give less than 0.2 radians error in phase under the worst conditions. The horizontal deflecting plates were driven synchronously with a sub-harmonic of the RF input to the helix, and the resulting deflection served to separate electrons according to phase in the final display. Placing the focusing lens after the deflection plates results in a con- siderable reduction in deflection sensitivity. However, undesirable mag- nification of the pinhole aperture dictated that the lens could not be close to it, and it was important to initiate the deflection as early as possible. The lens consists of three discs, the center one being biased to about 800 volts above the mean voltage of the rest of the system. Immediately after the lens there are two iron pole pieces and two insu- lated electric deflection plates which extend parallel to the beam for IJ^ inches. The pole pieces provide a dc magnetic field up to about 20 gausses induced by small coils outside of the envelope, and the electric deflection plates are biased with up to a corresponding 50 volts dc polarized to oppose the magnetic deflection of the beam. The electric and magnetic fields are adjusted so that the normal unmodulated electron beam tra- verses the region with no deflection and strikes the center of the fluores- cent screen. In the crossed field region 1= W^2^Fo. (26) Electrons having greater or lesser velocity are deflected parallel to the electric field, and give a corresponding deflection from the center of the fluorescent screen. To get a display in which the various elements are not hopelessly en- tangled, it was necessary to sweep the trace in an initial ellipse at a subharmonic rate. The sweep voltage was applied to the horizontal de- : flection plates, with just a little applied to the vertical plates through a , phase shifter. The relative phase of any part of the trace was measured k from the ellipse, and the velocity sensitivity was calibrated by observing | the ellipse deflection as a function of the dc beam potential, as shown f in Fig. 6(a). There is a small error due to the sensitivity of deflection to ( velocity, and due to distortion of the ellipse by fringing fields. ( In order to measure velocity and current density in the displayed pat- ;; tern, the fluorescent screen was photographed, and the negati^'e pro- h jected in a microcomparator. It was assumed that with the small ciu'rents P used, the light intensity was proportional to current, and the film i linearity was calibrated by making exposures of several different dura- j tions. The trace density was measured with a densitometer, sweeping NATURE OF POWER SATURATION IN TRAVELING WAVE TUBES 875 over the trace width to account for variations in focus for different parts of the pattern. Admittedly, the process is not very accurate, but it does give a rough measure of current density and helps considerably in in- terpreting the observed velocity patterns. NOMENCLATURE a Circuit radius 6 Parameter relating electron velocity to that of the cold circuit wave Uq — Vi/uqC = AF/2FoC B Magnetic field (8 the axial phase constant co/^i C The gain parameter = (E^/2l3^P) (/o/4Fo) 7 Radial phase constant = ^ = co/^i 8i Complex propagation constant for the increasing wave I E Electric field A'* Electric field at phase $ c/m Charge to mass ratio of the electron i h Beam current in amperes /„( ) Modified Bessel function ' /,> Tien's constant k, = 2/7^0 l.ri Circuit circumference measured in (air) wavelengths X Number of wavelengths 7] Maximum efficiency f]' Efficiency at an intermediate power level ^ P RF power obtainable from the circuit j, QC Space charge parameter I q Charge per unit length in the electron beam /■ Radial distance from the axis j Vq Beam radius t Time variable Electron velocity DC beam velocity V AC velocity of the electron beam t\ Wave velocity Fo DC beam voltage 7',„ Voltage corresponding to the wave velocity AF ^^oltage difference corresponding to the difference in velocity of an electron and the dc beam velocity bV Difference between synchronous voltage and that giving the Kompfner dip $ Relative phase z Distance measured along the beam "0 876 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 REFERENCES 1. Pierce, J. R., Theory of the Beam Type Traveling Wave Tube, Proc. I.R.E., 35, pp. 111-123, Feb., 1947. 2. Pierce, J. R., Traveling Wave Tubes, D. VanNostrand Co., Chapter XII. 3. Slater, J. C, Microwave Electronics, D. VanNostrand Co., 1950, pp. 298. 4. Brillouin, L., The Traveling Wave Tube (Discussion of Waves of Large Ampli- tudes), J. Appl. Phys., 20, p. 1197, Dec, 1949. 5. Nordsieck, A., Theory of the Large Signal Behavior of Traveling Wave Ampli- fier, Proc. I.R.E., 41, pp. 630-647, May, 1953. 6. Poulter, H. C, Large Signal Theory of the Traveling Wave Tube, Tech. Report No. 73 Electronics Research Laboratory, Stanford University, Stanford, California, Jan., 1954. 7. Tien, P. K., Walker, L. R., and Wolontis, V. M., A Large Signal Theory of Traveling Wave Amplifiers, Proc. I.R.E., 43, pp. 260-277, Mar. 1955. 8. Rowe, J. E., A Large Signal Analysis of the Traveling Wave Amplifier, Tech- nical Report No. 19, Electron Tube Laboratory, Universit.y of Michigan. 9. Tien, P. K., A Large Signal Theory of Traveling Wave Amplifiers Including! the Effects of Space Charge and'Finite C, B.S.T.J., 34, Mar., 1956. 10. Brangaccio, D. J., and Cutler, C. C, Factors Affecting Traveling Wave Tube Power Capacity, Trans. I.R.E. Professional Group of Electron Devices, PGED 3, June, 1953. 11. Crumly, C. B., Quarterly Status Progress Report No. 26, Electronics Re- search Laboratory, Stanford Universitj', Stanford, California, pp. 10-12. 12. Doehler, O., et Kleen, W., Phenomenes non Lin^aires dans les Tubes a Propa- gation D'onde" Annales de Radioelectricit^ (Paris), 3, pp. 124-143, 1948. 13. Doehler, O., et Kleen, W., Surle Rendement du Tube a Propagation D'onde," Annales de Radio^lectricite, Tome IV No. 17 Juillet, 1949 pp. 216-221. 14. Berterotidre, R., et Convert, G., Sur Certains Effets de la Charge D'espace dans les Tubes a Propagation D'onde, Annales de Radio^lectricit^, Tome V, No. 21, Juillet, 1950. 15. Klein, W., und Friz, W., Beitrag zum Verhalten von Wanderfeldrahren bei Hohen Engangspegeln, F.T.Z., pp. 349-357, July, 1954. 16. Warnecke, R. R., L'^volution des Principes des Tubes Electroniques Modernes pourMicro-ondes, Convegno di Elellronica e Televisione, Milano, p. 12-17, Aprile, 1954. 17. Warnecke, R. R., Sur Quelques R^sultats R^cemment Obtenus dans le Do- maine des Tubes Electroniques pour Hyperfrequences, Annales de Radio- ^lectricite. Tome IX, No. 36, Avril, 1954. 18. Warnecke, R., Guenard, P., and Doehler, O., Phenomenes fondamentaux dans les Tubes k onde Progressive, Onde Electrique, France, 34, No. 325, p 323-338, 1954. 19. Briick, L., und Lauer, R., Die Telefunken Wanderfeldrohre TL6, Die Tele funken-Rohre Heft 32, pp. 1-21, Februar, 1955. 20. Briick, L., Vergleich der Verschiedenen Formeln fiir den Wirkungsgrad einer Wanderfeldrohre, Die Telefunken-Rohre Heft 32, pp. 23-37, Februar, 1955 21. Cutler, C. C, Experimental Determination of Helical Wave Properties, Proc I.R.E. , 36, pp. 230-233, Feb., 1948. 22. Kompfner, R., On the Operation of the Traveling Wave Tube at Low Level Journal British I.R.E., 10, p. 283, Aug.-Sept., 1950. 23. Tien, P. K., Traveling-Wave Tube Helix Impedance, Proc. I.R.E., 41, pp 1617-1623, Nov., 1953. 24. Heffner, H., Analysis of the Backward-Wave Traveling-Wave Tube, Proc I.R.E., 42, pp. 930-937, June, 1954. 25. Johnson, H. R., Kompfner Dip Conditions, Proc. I.R.E., 43, p. 874, July, 1955 26. Quate, C. F., Power Series Solution and Measurement of Effective QC in Traveling-Wave Tubes, Oral presentation at Conference on Electron Tube Research, University of Maine, June, 1954. 27. Fletcher, R. C, Helix Parameters in Traveling Wave Tube Theory, Proc. I.R.E., 38, pp. 413-417, Apr., 1950. 28. Birdsall, C. K., and Brewer, G. R., Traveling Wave Tube Characteristics for Finite Values of C, Trans. I.R.E., PGED-1, pp. 1-11, Aug., 1954. 29. Pierce, J. R., Traveling Wave Oscilloscope, Electronics, 22, Nov., 1949. I The Field Displacement Isolator By S. WEISBAUM and H. SEIDEL (Manuscript received February 7, 1956) A nonreciprocal ferrite device (field displacement isolator) has been con- structed with reverse to forward loss ratios of about 150 in the region from 5,925 to 6,425 mc/sec. The forward loss is of the order of 0.2 dh while the reverse loss is 30 dh. These results are obtained by using a single ferrite element, spaced from the sidewall of the guide. The low forward loss suggests the existence of an electric field nidi at the location of a resistance strip on one face of the ferrite. We discuss the various conditions, derived theoretically, under which the electric field null may be obtained and utilized. Further- more, a method of scaling is demonstrated which permits ready design to other frequencies. I. INTRODUCTION The need for passive nonreciprocal structures has long been recog- nized.^ In the microwave field, Hogan's gyrator' paved the way for an increasingly important class of such devices. The isolator, in particular, has emerged as one of the more useful ferrite components. It performs the function, as its name implies, of isolating the generator from spurious mismatch effects of the load. Unlike lossy pads, which consume generator power, the isolater provides a unidirectionally low loss transmission path. A. G. Fox, S. E. IMiller and M. T. Weiss'' have pointed out that non- reciprocal ferrite devices may exploit any of the following waveguide effects : 1. Faraday rotation 2. Gyromagnetic resonance 3. Field displacement 4. Nonreciprocal phase shift In the present paper we shall discuss an isolator, based upon the field displacement effect, which was developed to meet the following require- ments for a proposed microwave relay system (5,925-6,425 mc/sec): 1 . Forward loss 0.2 db 877 878 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 2. Reverse loss 20 db 3. Return loss 30 db The field displacement isolator employs an ordinary rectangular | waveguide and requires no specialized adaptation to the rest of the guide system. It is relatively compact and does not require excessive magnetic fields. In contrast to the field displacement structure of Ref- erence 3, in which a symmetrically disposed pair of ferrite slabs is used, the present unit (see Fig. 1) contains only a single slab. Other differences of a more substantial nature may be noted — in the present case the slab is displaced from the guide wall, it occupies a partial height of the waveguide, and it employs a novel disposition of the absorption material on one face. These features result in a broadband device. In the analysis presented in this paper the isolator field characteristics for a full height slab are determined by exact solution of Maxwell's equations, as opposed to the "point-field" perturbation approximation used in Reference 3. An exact solution of the partial height geometry of the experimental device would be exceedingly difficult to obtain. How- ever, such a solution did not appear to be essential for this investigation since good correspondence has been obtained between the experimental results and the idealized full height slab calculations. The following performance of the isolator was obtained from 5,925- 6,425 mc/sec: 1. Forward loss --^ 0.2 db PERMANENT MAGNET V/y/////y/^y>///////y///y///////////y'////y^A ._FERRITE ^ RESISTANCE I ""COATING L- '////////////////////////////////////////y'yy//A h = 0.550 IN. ^■ = 0. 180 IN. b = 0.074 IN. L= 1.590 IN. 3 = 0.795 IN. T I I I I I S I Fig. 1 — Field displacement isolator. THE FIELD DISPLACEMENT ISOLATOR 879 2. Reverse loss ^ 30 db 3. Return loss ^^ 30 db The extremely low forward loss strongly suggested the existence of an electric field null in the plane of the resistance material. Consequently, a theoretical investigation of the null condition was made and a set of criteria estal)lished for the existence and utilization of the null. (E. H. Turner^ independently developed the same null conditions.) An exten- sion of the analysis leads directly to a set of scaling laws which permits the ready design of isolators of comparable performance at other fre- quency bands. i II. DESCRIPTION OF OPERATION I In Section IIA we will show how the "point-field" approach^ is used to predict the ciualitative behavior of the structure and in Section IIB we will apply a more rigorous analysis to the determination of the op- timum design parameters. .1. Qualitative Prior to introducing the actual isolator configuration, we shall re- ^'iew some elementary properties both of the ferrite medium and of an unloaded rectangular waveguide. It is in terms of these properties that we can understand, in a qualitative sense, the interaction of an rf wave with a ferrite in such a waveguide. Since the behavior of a ferrite medium in the presence of a static magnetic field and a small rf field has been discussed in the literature^ the following resume is not intended to be detailed. It is presented, however, to maintain continuity. If a static magnetic field is applied to a ferrite medium the unpaired electron spins, on the average, will line up with the field. If now an rf magnetic field, transverse to the dc field, excites the spin system these 1 electrons will precess, in a preferential sense, about the static field. The precession gives rise to components of transverse permeability at right { angles to the rf magnetic field, leading to a tensor characterization of the medium. This tensor has been given by Polder^ and may be diag- onalized in terms of circularly polarized wave components. Correspond- , ing to the appropriate sense of polarization we use the designation -|- ' and — . When the polarization is in the same sense as the natural pre- cessional motion of the spin system, gyromagnetic resonance occurs for an appropriate value of the static magnetic field. The scalar permea- bilities /i_ and M+ are shown in Fig. 2 as functions of the internal static magnetic field as would be observed at an arbitrary frequency. 880 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 o u~ 1 r 1 t i ^^X^+ / Hres Hi Fig. 2 — Permeability versus magnetic field. Clearly, in employing a ferrite medium, we intend to use the basic dif- ference between the scalar permeabilities ix^ and /i_ . To this end we may exploit the fact that the magnetic field configuration at any given point in a rectangular waveguide is, in general, elliptically polarized. Travel- ing loops of magnetic intensity appear in Fig. 3 for the fundamental (TEio) mode. At point P an observer sees a counterclockwise elliptically polarized magnetic intensity if the wave is traveling in the (+y) direc- tion.* The propagating wave may be decomposed into two oppositely rotating circularly polarized waves of different amplitudes: + O For propagation in the ( — y) direction the rj polarization is reversed: + O Let us now consider the actual experimental configuration shown in Fig. 4 (the partial height geometry was chosen on an experimental basis, in that it gave VSWR considerably less than that for a full height ferrite slab). The precession of the spin magnetic moments is counterclockwise * It is evident that a point converse to P exists symmetrically to the right of center. This is utilized in a double slab isolator which has been investigated by fi S. Weisbaum and H. Boyet, I.R.E., 44, p. 554, April, 1956. ' THE FIELD DISPLACEMENT ISOLATOR 881 looking along the direction of the dc magnetic field shown in Fig. 4. Since the major component of circular polarization for (+?/) propagation is also counterclockwise the permeability will be less than unity for this direction of propagation. This occurs provided we are using small static fields, as is readily verified from Fig. 2. The permeability will be greater than unity for ( — y) propagation. Physically, this is equivalent to energy being crowded out of the ferrite for ( + ^) propagation and to energy being crowded in, in the reverse direction. The electric field will thus be distorted as shown in a qualitative way in Fig. 5. The vertical dimen- sion in this figure serves both to identify the guide configuration and to provide an ordinate for the electric field intensity. The fields as shown in Fig. 5 merely represent a (iualitative picture of the distributions in the guide and are not intended to be exact. There is no question, however, that the electric fields at the ferrite face are dif- Fig. 3 — Magnetic field configuration — Dominant TEjo mode. Hoc FERRITE ELEMENT ^,w/^////j/y/////Ay/yyy^//^^^////^^^^^^^^^^^y/-'/w////y'^. RESISTANCE STRIP ;^/////,v.vy/yy/-^^^/'y/yy^/V////'-^y^yy^vAvyyyy^^/yy/////-^////^^ -4<-d-A , 1_ Fig. 4 — Experimental configuration. 882 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G FERRITE-K- Fig. 5 — Electric field distortion. ferent in magnitude corresponding to the two directions of propagation. Hence, if resistance material is placed at the interior face of the ferrite (see Fig. 1) we may expect to absorb more energy in one direction of propagation. B. Analysis of Electric Field Null:-Full Height Ferrite The description we have given in Section IIA is based on a perturba- tion approach and does not take into account the higher order interac- tion effects of the ferrite and the propagating wave. In this section we consider an analysis of the idealized case, namely that of a full height ferrite slab, and impose the condition of an electric field null at the face of the ferrite for the forward direction of propagation. While this too does not represent the true experimental situation, we believe it to be a better approximation than the "point-field" perturbation viewpoint. The fields of the various regions shown in Fig. 6 are described as follows : E^ (1) sm a]X E,^'^ = Ae-'"'''' -^ 5e'"^"' where x = x ^,"^ = V sin aix" where x" = x - L a (II -1) where aj = transverse wave number in the j*'^ region a = transverse dimension from narrow wall to ferrite face L = broad waveguide dimension X = variable dimension along broad face z = height variable A, B, V = constants Setting up the wave equation, there results THE FIELD DISPLACEMENT ISOLATOR 883 2 7^2 a2 = K tr / 2 — iMr kr') + ai" (II -2) where nr and /iv are the relative diagonal and off-diagonal terms of the Polder tensor, respectively, K is the free space wave number and £r is the relative dielectric constant. Mr = 1 + 4:7rMsyo}o kr = ± 4:irMsyo} 7 = 2.8 X 10^ cycles/sec/oersted 4iTrMs = saturation magnetization in gauss Ho = static magnetization in oersteds COo = yHo 27r X K = The following transcendental equation results from satisfying the boundary conditions on E and H:^ (II -3) tan aia\jjLT(X2 + kr^ tan a28] + (/Mr — kr) ai tan a2B tan ai6 + = 0 Oil (|3^ — K'HrSr) tan aia tan q:25 + ai(ju,Q;2 — Av/3 tan aaS) where /3 is the propagation constant. The minimum nontrivial value of an causing a null to appear at the ferrite face is ai = t/g. Placing this value in (II • — • 3) produces the fol- lowing transcendental equation for the null: TT / 2 a (jur — kr) tan a28 UrOCi — kr^ tan a28 -j- tan aih =0 (II -4) y///////////////^//////////////////////" "// ••• •; ' >v2v; ^////y/////yy/////y//////////w/vy//'/////,v/>'>///>/^////y/'^ =. a .4..4.-b-J Fig. 6 — Full height geometry. 884 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 where h = L-a-8. Equation (II — 4) demonstrates that the null condi- tion is nonreciprocal since, in general, the solutions differ for Av positive and kr negative. The quantity Av has the same sign as the direction of the dc magnetic field; reversing the sign of Av is equivalent to reversing \ the direction of propagation. A numerical analysis of etiuation (II • — 4) has led to the conclusion that the null condition is most broadband when | /x^ | < | Av |.* We use the criterion \ Hr\ = [ A\ | to determine a critical magnetic field: Hc = --^tM8 (11 — 5) 7 Clearly we require co/7 > AttIMs for physically realizable solutions. The saturation magnetization (47rM's) is subject to the following: 1 . A choice of too large a 4^ttMs might create a mode problem and in addition will not satisfy the limit on AtvMs implied in (II — 5). 2. 4xil/s must be sufficiently large so that the field needed to make I jUr I < I Av I not be excessive. 3. y\/H(H + 4:tM) (this being the slab resonance frequency for small slab thickness^) must be sufficiently far from the operating fre- quency to avoid loss due to resonance absorption. In addition, this con- dition improves the frequency insensitivity of the null. Further analytic considerations are presented in Section IV. III. EXPERIMENTAL DESIGN CONSIDERATIONS Aside from the partial height nature of the slab, there are two other basic factors in the experimental situation which are not present in the analysis of Section IIB (see also Section IV). First, the ferrite has both finite dielectric and magnetic loss. Second, higher order modes may be present. These deviations from the simplified analysis are by no means trivial and it would not be surprising if one found a considerable modifi- cation of the analytic results. As it turns out, there are broad areas of general agreement between the theoretical and experimental results and in no case examined here does one find a basic inconsistency. In considering the various parameters which must be adjusted to optimize the broadband performance of the isolator we will point out, where possible, how the theoretical results are modified by the factors men- tioned above. The parameters of interest are: * This is partially evident from eciuation II — 4. The quantity fj.r \ must be less than | A;, | if the angle (aib) is to be small and in the first quadrant. Second quadrant solutions cause the guide cross section to be excessively large, with attendant higher mode complication. I THE FIELD DISPLACEMENT ISOLATOR 885 A. The saturation magnetization {4:TrMs) and the applied magnetic field (Hnc). B. The ferrite height. C. The thickness (5) of the ferrite and its distance (b) from the nearest side wall. D. The placement of the resistance material and its resistivity (p). E. The length of the ferrite (^). A. 4:TrMs and Hoc Theoretically, minimum forward loss occurs with a true null at the face of the loss film and has been given in the condition \ fXr\ < \ kr\. Although this inequality is required in the full height slab analysis, lexperiment (Fig. 7) indicates the low loss region to be so broad as to extend well into the low field, or | /Ur | > \K\ region. There is inherent loss in the ferrite so that a more accurate statement of the bandwidth of operation is that in which the losses in the film are of equal order to the ferrite losses at the band edges. Even discounting ferrite losses, it will be shown in Section IV that we have a good analytic basis for the observed broadness of the low loss region. In general, there- fore, we need not be as restrictive as the null analysis of Section JIB would imply. It is not surprising then that optimum operation actually occurs in the region | /x^ | > \ kr\. There are several reasons why this may be so: 1. Shift of operation occurs due to the partial height nature of the ferrite slab. 2. Reverse loss has a peak in the low field region, requiring a compro- mise of low forward loss and high reverse loss for best isolation ratios (see Fig. 8). 3. Optimum compromise between low ferrite loss and low film loss must be made. The internal magnetic field, determining | ju^ | and | Av |, differs from the applied field by the demagnetization of the ferrite slab. Although not ellipsoidal, it may nonetheless be considered to have an average demag- netization which has been computed, for this case, to be 460 oersteds. A further complication in knowledge of the internal field is the proximity effect of the pole pieces. This latter correction was obtained experi- mentally and, all in all, it was determined that the internal field for optimum operation was of the order of 300 oersteds. For the given ferrite and the I'ange of frequency of operation, this internal field corres- ponds to the condition that \ ij.,- \ > | Av |, as stated above. Taking all effects into account, it was found that optimum permanent magnet design occurred for an air gap field of 660 oersteds. 886 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 _i LLI CO o UJ Q tf) O _l Q a. < cr O u. ferrite: R) 4;rMs=t700 GAUSS 6r=0.160 IN. h = 0.550 IN. 1 = 5.000 IN. 1 1 1 AT 5925 MC PER SECy f / / AT 6425 MC PER SEC A / / ^ ^*w 660 OERSTEDS _^ , y^ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 MAGNETIZING CURRENT IN AMPERES 3.2 3.6 Fig. 7 — Forward loss versus magnetizing current. Using the experimental values 4:tMs — 1,700 gauss and internal mag- netic field = 300 oersteds, the frequency at which ferromagnetic reso- nance occurs was estimated to be about 2200 mc/sec. This value is suf-ii ficiently far from our operating range (5,925-6,425 mc/sec) that we| would expect a negligible loss contribution due to resonance absorption.}, This is confirmed by the low forward loss actually observed. B. Ferrite Height We have already pointed out that when the ferrite height is reduced from full height a more reasonable VSWR is obtained. This is due to the fact that we have relieved the stringent boundary requirements at the 60 ^ 50 LU O UJ 40 if! 30 (/) o _J UJ 20 CO CC UJ > yj 10 (^ k. 660 OERSTEDS ^— -< AT 6425 MC PER SEC 7 Fl N \, / l\ 1 > ^_ __ N k AT 5925 MC PER SEcS; ^.. V 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 MAGNETIZING CURRENT IN AMPERES Fig. 8 — Reverse loss versus magnetizing current. THE FIELD DISPLACEMENT ISOLATOR 887 top and bottom faces of the ferrite and approach, in a sense, a less critical rod type geometry. A ferrite height of 0.550" gave a VSWR -^ 1.05 (iver the band. With full height slabs (0.795"), A^SWR values as high as 10:1 have been observed for typical geometries. I j C. 8 and h Experimentally, we have examined various ferrite thicknesses at dif- ferent distances from the sidewall until optimum broadband performance . jwas obtained. Table I shows the ferrite distance from the wall which • I gave the best experimental results (highest broadband ratios, low for- ,ward loss, high reverse loss) for each thickness 8 of one of the BTL I materials. It is interesting to note that the empirical ctuantity 8 -\- 6/2 — I Table I 5 (mils) b (mils) s + 1 (mils) t (mils) S + ~2t (mils) 201 11 206.5 3 200.5 189 35 206.5 3 200.5 186 42 207.0 3 201.0 176 65 208.5 3 202.5 189 42 210.0 6 198.0 2t, where t is the thickness of the resistive coating, is very nearly con- stant (within a few mils) for the stated range of 8 and for this type of [design.* i In Section IV a theoretical calculation using the null condition at i 6175 mc/sec for a full height ferrite gives 5 = 180 mils b = 38.7 mils so that 8 -f b/2 = 199.3 mils. In the theoretical case t is assumed to be very small. It will be noted that the theoretical result for 8 + b/2 (with small t) agrees quite well with the experimental 5 -f 6/2 — 2t. The ques- tion of the possible phj^sical significance of this quantity is being investigated. D. Placement of Resistance Material and Choice of Resistivity The propagating mode with a full height ferrite slab is of a TEo variety, the zero subscript indicating that no variation occurs with re- * In one design of the isolator we used a General Ceramics magnesium manga- nese ferrite with 5 = 0.180", b = 0.074" and t = 0.009" so that 8 + b/2 - 2t = 199 mils, in good agreement with Table I. 888 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Fig. 9 — Distribution of jsmall tangential electric fields at interior ferrite faceJ Fig. 10 — Resistance configuration. spect to height. A field null in this construction therefore extends across the entire face of the full height ferrite and all of this face is then "active" in the construction of an isolator. This field situation no longer accurately; applies to the partial height slab. The departure of the ferrite from the top wall creates large fringing fields extending from the ferrite edges, and large electric fields may exist tangential to the ferrite face close to these edges. We would therefore expect the null condition to persist only in a small region about the vertical center of the ferrite face. We may, how- ever, also expect longitudinally fringing modes (TM-like) to be scattered! at the input edge of the ferrite slab so that a longitudinal field maximum will exist at the central region of the ferrite. However, this is a higher: mode, so that this maximum decays rapidly past the leading edge. Considering all the effects, the distribution of small tangential electric fields at the ferrite face may be expected to appear as shown in Fig. 9. Experimentally, we have utilized this low loss region and have avoided the decay region of the higher TM-like modes by using the resistance configuration shown in Fig. 10. The resistivity is uniform and about 75 POLYSTYRENE- COPPER PLATE RESISTANCE STRIP FERRITE V/M///////^^^^J/^^^.';^/^J^^//^/////^^^^?^??j//?^/^9r»'/ m Fig. 11 — Elimination of longitudinal components. THE FIELD DISPLACEMENT ISOLATOR 889 30 25 20 15 10 FREQ H = 6425 MC = 1150 OERSTEDS 477Ms= 1700 GAUSS d= 0.180 IN. h= 0.550 IN. 1=5.000 IN. / A k / s \, / V • ^^ ■-— < > y / C / 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 LENGTH OF LOSS FILM IN INCHES 4.5 5.0 5.5 Fig. 12 — Attenuation versus length of resistance strip. 'ohms/square. Variations of about d=30 ohms/square about this value result in little deterioration in performance. Some further discussion of the perturbed dominant mode is of interest. jWe may think of the height reduction as primarily a dielectric discon- itinuity where we have effectively added a negative electric dipole den- Isity to a full height slab. Since this addition is smaller for the forward case (where there was initially a small electric field) than for the reverse case, we may expect the longitudinal components to be smaller for the forward propagating mode. The other type of longitudinal electric field, 50 -I 40 O UJ O , 30 If) if) O -■ 20 LU If) a. LU I '0 a.^__ REVERSE LOSS FORWARD LOSS .J^ 0.5 0.4 0.3 If) If) o 0.2 0.1 Q a: < cc o 5900 6000 6100 6200 6300 6400 FREQUENCY IN MEGACYCLES PER SECOND Fig. 13 — Loss versus frequency. 6500 890 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Fig. 14 — Isolator model. which occurs due to the scattering of the TM-Uke lono;itudinal modes, decays rapidly and is not of consequence in an experiment now to be described. This experiment was designed to demonstrate the nonre- ciprocal nature of the longitudinal electric fields associated with the dis- torted dominant mode. It also shows that the existence of these com- ponents is significant as a loss mechanism for the reverse direction of propagation in the isolator. The geometry employed is shown in Fig. 11. The copper plate was inserted to minimize longitudinal electric field components, and we may therefore expect to obtain less reverse loss than in the condition of its absence. The result of this experiment was that the reverse loss decreased from about 25 db* without the plate to 18 db | with the plate. The forward loss was unaffected. E. Determination of Length Given a dominant mode distribution in a waveguide, attenuation will be a linear function of length, once this mode has been established. Con- sequently, one would expect that doubling the loss film length would double the isolator reverse loss. The isolator does not exhibit this be- havior, however, as is illustrated in Fig. 12. This occurrence might be explained by the appearance of still another longitudinal mode, peculiar in form to gyromagnetic media alone, which propagates simultaneously with the transverse electric mode, and is essentially uncoupled to the loss material. The maximum reverse loss * This experiment was conducted with a different ferrite than that employed ini' the eventual design. THE FIELD DISPLACEMENT ISOLATOR 891 thus obtainable is limited by the scattering into this mode. The charac- ter of these singular modes will be discussed in a subsequent paper. Results The performance of the isolator as a function of frequency is shown in Fig. 13. Fig. 14 shows a completed model of the isolator. IV. FURTHER ANALYSIS While an exact characteristic equation is obtainable for the overall geometry of the full height isolator, including the lossy film, the ex- pressions which result are sufficiently complex to be all but impossible to handle. However, if the resistance film is chosen to have small conduc- tivity we may utilize a simple perturbation approach in which the field at the ferrite face is assumed to be unaffected by the presence of the loss film. A quantity rj may then be defuied* so that . = LM (IV- 1) For small conductance values ri is proportional to attenuation to first order in either direction of propagation, Er , in equation (IV ^ — ■ 1), is the electric field adjacent to the film and P is the power flowing across the guide cross section. The loss in the ferrite material is not taken into account in this approximation, but it would naturally have a deteriorat- ing effect on the isolator characteristics. The ratio of the values of 77 corresponding to backward and forward direction of propagation defines the isolation ratio, given in db/db, for the limit of very small conductivity. Fig. 15 shows a calculated curve of the forward value of 17 and Fig. 16 shows the backward case. The isolation ratio shown in Fig. 17 dem- onstrates surprisingly large bandwidth for values of the order of 200 db/db. Fig. 18 portrays propagation characteritics for both forward and backward power flows and provides the interesting observation, in conjunction with Fig. 16, that peak reverse loss occurs in the neighbor- hood of X = \g . Fig. 19 is a plot of ai , the transverse wave number, over the fre- quency range. The flatness of the forward wave number means that the position of null moves very little with frequency across the band. Hence the lossless transmission in the forward direction is broadband. Since the forward and backward wave numbers have such radically different See Appendix 892 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 0.6 0.5 0.4 ^f 0.3 0.2 0.1 I 5800 5900 6000 6100 6200 6300 6400 FREQUENCY IN MEGACYCLES PER SECOND 6500 Fig. 15 — Relative attenuation — forward direction rates of variation, a simple adjustment of parameters may be made to cause the forward null and maximum reverse attentuation to appear at the same frequency, resulting in an optimum performance. The occurrence of the reverse maximum loss in the region of X = Xj may roughly be explained as follows. As the transverse air wave number decreases, the admittance of the guide, defined on a power flow basis, increases. The electric field magnitude distribution must therefore gener- ally decrease in such a fashion as to cause the overall power flow to re- 600 500 400 300 200 100 0 /^ ^ s. / N ) f / / / / f ^ X 5600 5900 6000 6100 6200 6300 6400 FREQUENCY IN MEGACYCLES PER SECOND 6500 Fig. 16 — Relative attenuation — backward direction THE FIELD DISPLACEMENT ISOLATOR 893 lU 5 ^ V 2 ^ ^ < o ^ < cr 2 g 10^ 1- o J t- \. X. / ^v 1 ^ / / <0 2 5 / / / 2 10 v / 1 5900 6000 6t00 6200 6300 6400 FREQUENCY IN MEGACYCLES PER SECOND 6500 Fig. 17 — Ideal isolation characteristics. main constant.. On the other hand as the transverse air wave number decreases through real vahies, the electric field adjacent to the ferrite becomes relatively large. At X = X^ the distribution is linear with rela- ]\ tively large dissipation at the ferrite face. As the transverse air wave number increases through imaginary values the distribution becomes exponential such that the field adjacent to the ferrite is always the maxi- mum for the air region and the growth of the field at the face of the ferrite would not seem to be so great as formerly. One would therefore expect a maximum reverse loss somewhere in the region X = Xy . The abo^^e considerations plus the transcendental equation for the null show consistency with the experimental design values which were: 8 = 0.180" L = 1.59 47rilf s = 1 ,700 gauss Using Hue = 000 oersteds in the calculation we obtain the spacing from the guide wall h = 0.0387". The fact that we used 600 oersteds for the full height slab calculation as opposed to the internal field of 300 oersteds found experimentally for the partial height slab should not be a source of confusion. It has been indicated earlier that the peak reverse loss shifts 894 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 1.4 t.2 1.0 0.8 0.6 0.4 0.2 ^^ ^ BACKWARD PROPAGATION,^-^ ^ ^ - m 'forward PROPAGATION ~""~" ... 5900 6000 6100 6200 6300 6400 FREQUENCY IN MEGACYCLES PER SECOND 6500 Fig. 18 — Ferrite isolator characteristics. 5800 5900 6000 6100 6200 6300 FREQUENCY IN MEGACYCLES PER SECOND 64001 Fig. 19 — Transverse characteristics of a ferrite isolator THE FIELD DISPLACEMENT ISOLATOR 895 with ferrite height reduction. It is not inconsistent therefore to choose 600 oersteds for the full height analysis in contrast to the value deter- mined from the experiment. V. SCALING Once the optimum set of parameters has been decided upon for a given frequency range (e.g., 5,925-6,425 mc/sec, 5 = 0.180", b = 0.074", ( = 5", h = 0.550", 4wMs = 1,700 gauss, Hoc = 660 oersteds) it is a simple matter to scale these parameters to other frecpency ranges. From Maxwell's equations: Curl H = icceE + gE Curl^ = -iwT-H where T is the permeability tensor, and g is the conductivity in mhos/ meter. The first of Maxwell's equations suggest that frequency scaling may be accomplished by permitting both the curl and the conductance to grow linearly with respect to frequency. The curl, which is a spatial derivati^'e operator, may be made to increase appropriately by shrinking all dimensions by a 1/co factor, which will keep the field configuration the same in the new scale. Having imposed this condition on the first equation we must satisfy the second of Maxwell's equations by causing 7" to remain unchanged with frequency. T is a tensor given as follows for a cartesian coordinate system: (Hr ikr 0\ -ikr Mr 0 (V— 1) 0 0 1/ for a magnetizing field in the z direction. The components may be ex- panded in the following fashion: Mr = 4 X ^ ^ (V— 2) h = CO m - ^ 896 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 where AtMs is the saturation magnetization in gauss and y is the mag- netomechanical ratio. The Polder tensor evidently remains unchanged if Ms and H are both scaled directly with frequency. Since the field distributions are assumed unchanged relative to the scale shift, normal and tangential E and H field components continue to satisfy the appropriate boundary equalities at interfaces. Then, in- voking the uniqueness theorem, the guide characteristics are only as presumed and the model has been properly scaled as a function of fre- quency. The scaling equations are: di =2 C02 d2 gi = 032 92 Ms, = CO2 Ms, Ho)i = COl C02 {Ho)2 (V-3) where d is any linear dimension. CONCLUSION An isolator with low forward loss and high reverse loss can be con- structed by a proper choice of parameters. Once a suitable design has been reached the scaling technique can be used to reach a suitable design for other frequencies. As yet, a theoretical analysis of this problem has been carried out only for a full height ferrite. ACKNOWLEDGMENT We would like to thank F. J. Sansalone for his assistance in developing the field displacement isolator. We would also like to thank Miss M. J. Brannen for her competent handling of the numerical computations. APPENDIX It is desirable to establish an isolator figure of merit. A simple quan- tity characterizing the isolator action is the normalized rate of power THE FIELD DISPLACEMENT ISOLATOR 897 loss in the resistive strip, for an idealized ferrite, in the low conductive limit of such a strip. Let 12 Er V = where 77 is the appropriate quantity, Er is the field at the resistance, and P is the total power flow across the guide cross-section. This figure of merit is related to the rate of change of the attenuation constant (A) with respect to strip conductance in the following manner: p = 0.0434377/1 (db)(ohms)/cm where h is the fractional height of the loss strip, and g is the reciprocal of the surface resistivity in ohms/square. The total power flow may be divided into integrations of the Poynting vector over the three regions of the guide cross-section. The following results are obtained normalized to Er = sin aia: Region 1: 0 ^ a; ^ a p (1) _ Region 2: a ^ x ^ a -\- 8 (2) /3 / /Xr3 ^ 2co/xo / sin 2aia\ V 2^7 / , 2 , , 2n , sin 2a28 fXridi^ — di) — -^ a2{2did^ + 1 — cos 2a25 y.r{2dd2) + ^ id,' - d,') Region Z: a -\- b ■^ x S L p (3) _ ^y — /3 2cJ)Uo h - sin 2 a]b\ {d\ cos aih -\- di sin a-^ where and 2q;i d\ = sin aia sin aj) do = yird-l [{nr — kr)oci cos aitt -f- /cr/3 siu aia] 898 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 BIBLIOGRAPHY 1. Tellegen, B. D. H., Philips Res. Rep., 3, 1948. 2. Hogan, C. L., B.S.T.J. 31, 1952. 3. Fox, A. G., Miller, S. E., and Weiss, M. T., B.S.T.J. 34, p. 5., Jan. 1955. 4. Turner, E. H,, URSI Michigan Symposium on Electromagnetic Theory, June, 1955. 5. Polder, D., Phil. Mag., 40, 1949. 6. Lax, B., Button, K. J., Roth, L. M., Tech Memo No. 49, M.I.T. Lincoln Lab-' oratory, Nov. 2, 1953. 7. Kittel, C, Phys. Rev., 73, 1948. Transmission Loss Due to Resonance of Loosely-Coupled Modes in a Multi-Mode System By A. P. KING and E. A. MARCATILI (Manuscript received Januarj' 17, 1956) In a multi-mode transmission system the presence of spurious modes which reso7iafe in a closed environment can produce an appreciable loss to the principal mode. The theory for the evaluation and control of this effect under certain conditions has been derived and checked experimentally in the particularly interesting case of a TEoi transmission system, where mode conversion to TE02 , TE03 • • • is produced by tapered junctions between two sizes of waveguide. INTRODUCTION In a transmission system, the presence of a region which supports one or more spurious modes can introduce a large change in the trans- mission loss of the principal mode when the region becomes resonant for one of the spurious modes. This phenomenon can occur even when the mode conversion is low and the waveguide increases in cross section smoothly to a region which supports more than one mode. In general, the conditions required to resonate the various spurious modes are not fulfilled simultaneously and, in consequence, interaction takes place between the principal mode and only one of the spurious modes for each resonating frequency. Under these conditions the resonating environ- ment can be visualized as made of only two coupled transmission lines, one carrying the desirable mode and the other the spurious one. This simplification makes it possible to calculate the transmission loss as a function of (1) the coefficient of conversion between the two modes and (2) the attenuation of the modes in the resonating en\-ironment. The theory has shown good agreement with the measurement of transmission loss of the TEoi mode in a pipe wherein a portion was tapered to a larger diameter which can support the TE02 mode. 899 900 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 TRANSMISSION LOSS OF A WAVEGUIDE WITH A SPURIOUS MODE RESONATING REGION Let US consider a single-mode waveguide connected to another of different cross-section that admits two modes. Since these two modes are orthogonal, the junctions may be considered as made of three single- mode lines connected together, provided we define the elements of the scattering matrix properly. The three modes, or lines in which they travel, are indicated by the subscripts 0, 1, and 2, as shown in Fig. 1. If ao , tti , Qi and 6o , &i , ?>2 are the complex amplitudes of the electric field of the incident and reflected waves respectively, then "6o' ao 6i = [S] ai h. _«2_ where Too Foi ro2 [S] = Toi Tu ri2 (1) _ro2 Flo r22_ is the scattering matrix.^ This specific type of change of cross section may be treated as a three-port junction. Now, if a length / of a two mode waveguide is terminated sym- metrically at both ends mth a single mode waveguide (Fig. 2), each joint is described by the same matrix (1), and the connecting two mode wave guide has the following scattering matrix: (10 0 e-'"' 0 0 ^-m 0 0 0 0 0 0 e 0 0 ^-ie. 0 in which je, = y,(. = («i +JiSl)f je2 = 72^ = (a2 + jS-^t, 7i and 72 are the propagation constants of modes 1 and 2. • N. Marcuvitz, Waveguide Handbook, 10, M.I.T., Rad. Lab. Series, McGraw- Hill, New York, 1951, pp. 107-8. TRANSMISSION LOSS DUE TO RESONANCE OF CONVERTED MODES 901 Matrices 1 and 1' describe the system completely and from them, the transmission coefficient results, ao — 1 oie rlze -jlBiA* 1 + ri2 1 22 1 Too r* 1 02 r22 o ri2 (2) [1 - (r?2 - TnT,,)e-'''^^'''Y - (rue"^'' + r226-^''^)^ where A = 1 + £02^ .roi/ -y(92-«i) A * is the complex conjugate of A Toi* is the complex conjugate of Toi ro2* is the complex conjugate of ro2 Furthermore, let us make the following simplifying assumptions I |ro2 /3=2 |3=0 I^U, -,/l. 00 = 0 (3) «1 (4) if w = n = 0, 1, 2 (5) if m ^ n Equation (3) indicates that if in Fig. 1, lines 1 and 2 were matched, line 0 would also be matched looking toward the junction. Equation (4) states that almost all the transmission is made from 0 to 1 , or that there is small mode conversion to the spurious mode 2. Equation (5) assumes that the transition is nondissipative. The first two conditions are ful- filled when the transition is made smoothly. The last is probably the most stringent one, especially if the transition is a long tapered wave- guide section, but it is always possible to imagine the transition as lossless and attribute its dissipation to the waveguides. 0- — - — bo ^ — aa —2 Fig. 1 — Schematic of a three-port junction. 902 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G From (2), (3), (4) and (5) V = I r,o |2 2|ri.2|2(l - cos ^) 1 + T22 J«l12 1 - [rooe"' - + Tne"'''] (6) Avliere i 22 1 22 6 (f = 61 — 02 — t 1 - 2|ri2|2(l - cosha^) (9) l + B'\ r22 Ye 2 — 2a2^ where E = 1 + ri2 •l 22 -at C 1 - ri2 i 22 -a( a = ai — a2 For the most important practical case, that is, when the maximum value attainable by cosh aC is of the order of 1, and knowing from (3), (4) and (5) that 12 ro2 r (1 - I ro2 p) r22 1' ^ 1 - 2 I ro2 904 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 7^201ogio(l + 2lro2re""0 (10) 1 - 2 I ro2 Kl + cosh ap \ 1 _ e-2"2^ + 2 I ro2 I'd + e-"')e-'"''l From this expression we deduce (a), / is strongly reduced when a^C » ro2. ^ (b), Attenuation in line 1 is not an important factor until ail and I ck:i — a2 1 ^ ^re of the order of 1 . In other words, for low attenuation in both lines, aot assumes a major importance in the determination of I because it influences the conditions of resonance. That the effect of ail is small is shown in Fig. 3 (dotted line for the particular case ai = 0:2/4). In order to handle the general problem, (10) has been plotted in Fig. 3. We can enter with any two and obtain the third following quantities: Zi , relative insertion loss in db; 10 logio e~^"^ , attenuation in db of the spurious mode in the resonating environment; and 20 logio ro2 conversion level at the junction, in db, of power in the spurious mode relative to that in the first line. APPLICATION OF THESE RESULTS TO A TEqi TRANSMITTING SYSTEM The results of the preceding section have been checked experimentally by measuring the relative insertion loss of different lengths of %" di- ameter round waveguide tapered at both ends to round waveguides of J4.6" diameter. This waveguide is shown in Fig. 4 with a schematic diagram of the measuring set. In the round transmission line A-B, section A will propagate only TEoi . Section B, which has been expanded by means of the conical taper Ti , can support TE02 and TE03 in addition to the principal TEoi mode. This section is a closed region to the spurious modes ( TE02 , TE03) whose length can be adjusted to resonate each one of these modes. A sliding piston provides a means for varying the length, /, of section B. &^ T T X fl e- ./^ 1 2 r~y TE 01 B TEo, TE02 TE03 1 RECEIVER Fig. 4 — Circuit used to measure TEoi insertion loss due to resonance of the TE02 and TE03 modes. TRANSMISSION LOSS DUE TO RESONANCE OF CONVERTED MODES 905 •lOr U° -12 -14 -16 -16 -20 -22 -24 -26 -28 O -30 O ^ -32 -34 -36 -38 -40 -42 -44 ■46 -48 -50 \\ 1 ; PtEo2 / PtEo3 ( \ \ hTEo,-|. t d \ \^ s. *- L \ \ k \ \ i \ \ \ \ \ \ ^ s \ \ k \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ s \ \^°' \ TEo\ \ \^ D = 2" TEo) \ \ V "7" 7" \T6 \ \ V \ 7 k \ \ 1 \ \ \ \ k \TEo3 ie \ \ V" \ \ \ V \ \ \ 7"\ 8 \ \ \ \ k d=*-^ V \ \ \ \ \ \ V \ \ \ \^ 8 \ \ \ \ \ \ \ \ \ \ 1 1 \ \ \ 1 1 0.1 0.2 0.3 0.4 0.6 0.8 1.0 2 3 TAPER LENGTH, L, IN METERS 5 6 8 10 Fig. 5 — Mode conversion of TE02 and TE03 relative to TEoi generated by a conical taper. The relative levels of TE02 and TE03 conversions, which have been calculated from unpublished work of S. P. Morgan are shown plotted in Fig. 5 for the waveguide sizes employed in the millimeter wavelength band. The conversions, 20 logio ro2 , are plotted in terms of the TE02 and TE03 powers relative to the TEoi mode power and are expressed in db as a function of the taper length L, in meters. Fig. 6 shows the theoretical and experimental values obtained for TEoi relative insertion loss. Since the minimum length of pipe tested is 906 THE BELL SYSTEM TECHNICAL JOUENAL, JULY 1956 -102 8 UJ u UJ O to to o -10 z o z > < _l cr -t TEo2 ATTENUATION RELATIVE INSERTION - / l(M) (DB) 0.37 -0.024 1.135 -0.074 2.39 -0.157 3.73 -0.245 LOSS (DB) -6.0 -3.2 - 1.7 - 1.3 - EXPERIMENTAL . DATA ^ \ 2 3 - - 'N, ^ ^ V ro2=-27DB - N - X - 1 N N - ^ N \ 2 \ \, EORETICAL VALUES lASURED VALUES \ 4 o ME - \ - s - , 1 1 _L 1 1 1 _L 1 1 1 1 \ I 1 -10" 6 8-, -10 2 -10 ■1 ATTENUATION OF SPURIOUS MODE IN DECIBELS Fig. 6 Theoretical and measured relative insertion loss in the TEoi trans- mission system of Fig. 4. several times the length of the tapers, the losses in the transitions are fairly small compared to the losses in the multimode guide and this justifies assumption (5). The resonance due to the other modes is too small to be appreciable. This is understandable since, according to (10), the value of the mode conversion for the TE03 (Fig. 5) and the attenua- tion for the shortest length of pipe tested, the calculated relative inser- tion loss is less than —0.1 db. CONCLUSIONS The resonance of spurious modes in a closed environment can produce a large insertion loss of the transmitting mode. In a fairly narrow band device it is possible to avoid this problem by selecting a proper wave- guide size for the closed environment. In a broad-band system the losses can be minimized by providing a high attenuation and a low mode con- version for the spurious mode. For example, it may be noted, by refer- ring to Fig. 3, that mode conversion as high as —20 db with a spurious mode loss of — 8 db results in only an —0.1 db insertion loss for the transmitting mode. Measurement of Atmospheric Attenuation at Millimeter Wavelengths By A. B. CRAWFORD and D. C. HOGG (Manuscript received September 20, 1955) A frequency -modulation radar technique especially suited to measure- ment of atmospheric attenuation at millimeter wavelengths is described. This two-way transmission method employs a single klystron, a single an- tenna and a set of spaced corner reflectors whose relative reflecting properties are known. Since the method does not depend on measurements of absolute antenna gains a7id power levels, absorption data can he obtained more readily and with greater accuracy than by the usual one-way transmission methods. Application of the method is demonstrated by measurements in the 6 -mm to 6-mm wave band. The residts have made it possible to assign an accurate value for the line-breadth constant of oxygen at atmospheric pressure; the constant appropriate to the measurements lies between 600 and 800 MCS per atmosphere. INTRODUCTION It is well known that certain bands in the microwave region are at- tenuated considerably due to absorption by water vapour and oxygen in the atmosphere. A theory of absorption for both gases was given by Van Vleck.^ Numerous measurements have been made on the gases when confined to waveguides or cavities- and several when unconfined in the free atmosphere.^ Nevertheless, there is some uncertainty regarding the line-breadth constants which should be used in calculating water vapour and oxygen absorption. In particular, at atmospheric pressure there is doubt as to the amount of absorption on the skirts of the bands where the absorption is small. The present work was undertaken to test a new method of measurement and to improve the accuracy of experimental data measured in the free atmosphere. The method of measurement is one of comparison of reflections from 907 908 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 spaced corner reflectors whose relative reflecting properties are known. The free-space attenuation is readily calculated and any measured at- tenuation in excess of this represents absorption by the atmospheric gases. A description of the method and the apparatus is followed by a dis- cussion of data taken in the wavelength range 5.1 to 6.1 mm (which in- cludes the long wavelength skirt of the oxygen absorption band centered at 5 mm). These data, when compared with the theory,^ indicate that the line-broadening constant of oxygen at atmospheric pressure is of the order of 600 mc. Some rain and fog attenuation measurements at a wavelength of 6.0 mm are included. METHOD The experimental setup is shown in Fig. 1. It consists of a high-gain antenna for both transmitting and receiving and a pair of spaced corner reflectors. Corner reflectors can be built to have good mechanical and electrical stability, and their reflecting properties are relatively insensi- tive to slight misalignments. The reflectors are mounted well above the ground to ensure free-space propagation conditions. At the outset, the relative reflecting properties of the corner reflectors are measured by placing them side by side at a convenient distance (cfi for example) from the antenna. By alternately covering one and the other with absorbent non-reflecting material and measuring the reflected sig- nals, the relative effective areas are determined. The reflectors are then separated as shown and consecutive measurements are made of the sig- nals returned from each reflector. From these measurements, knowing the distances di and d^ and the calibration of the reflectors, one determines the attenuation over the path d2-di in excess of the free-space attenua- tion.* This excess, in the absence of condensed water in the air, repre- sents absorption by the atmosphere. The power received from the reflector at distance di is, A' A,'' Pi = Pt -tt^- Q(K dd X di where A and Ai are the effective areas of the antenna and corner-reflector respec- tively, and Pr is the transmitted power; Q(\, d\) is a loss factor which accounts for atmospheric absorption. A similar relation holds for the power received from the reflector at distance ^2 . The ratio of the received powers is then, '2 [aJ \dj f^= (t^) It) QlKid^"- d.)\ ATMOSPHERIC ATTENUATION AT MILLIMETER WAVELENGTHS 909 The accuracy of the measurements Avill be affected, of course, by spuri- ous refiections in the neighborhood of the corner-refiectors. The sites for the experiment were chosen to minimize such refiections and checks were made by observing the decrease in the return signals when the corner- refiectors were covered by absorbent material. In all cases, the back- ground reflections were at least 30 db below the signal from the corner- reflector. The method of measuring the reflected signals is illustrated in Fig. 2. The transmitted signal is frequency modulated in a saw tooth manner with a small total frequency excursion, F. The signal reflected from the near corner-reflector is delayed \ni\\ respect to the transmitted signal by a time, n , equal to twice the distance to the reflector divided by the velocity of light. During a portion, Ti — rx , of the sa^^i:ooth cycle, there is a constant frequency difference, /, between the transmitted and re- ceived signals, {f/F = ti/Ti). Power at this frequency is produced by mixing the initial source signal with the delayed received signal and am- plifying the difference frequency in a narrow-band amplifier centered at frequency /. The output of this amplifier is, therefore, a pulse at fre- quency /, of length Ti — n and repetition rate 1/Ti . To measure the signal returned from the far corner-reflector it is neces- sary merely to increase the period of the sawtooth modulation propor- tionate to the increase in distance. The frequency excursion, F, re- mains the same; hence the average power output of the transmitter is unchanged. As may be seen in Fig. 2, the freciuency difi"erence, /, between the transmitted and received signals is unchanged; thus the same am- plifier and output meter can be used for the two cases. Another advan- tage in changing only the sawtooth repetition rate is that the delay is the same fraction of a period in both cases; therefore the duty cycle is unchanged and the intermediate frequency pulses can be detected by either an average or a peak measuring device. Since the beat frequency, /, is not affected by slow changes in the fre- ANTENNA EFFECTIVE AREA CORNER REFLECTOR R1 EFFECTIVE AREA At SOURCE \>- -d,- HI _i_ CORNER REFLECTOR R2 EFFECTIVE AREA Ag Fig. 1 — Siting arrangement for the atmospheric absorption measurements. 910 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 qiiency of the transmitter, the bandwidth of the intermediate frequenc}^ amplifier need be only wide enough to take care of non-linearity in the sawtooth modulation. A signal-to-noiso advantage is obtained by the use of the narrow-band amplifier. Table I gives the distances, heights and effective areas of the reflectors as well as the sa^^iiooth repetition rates that were used in the experiment. The frequency excursion of the sawtooth modulation was 5.8 mc. It will be noted that three reflectors were used; this was done to pro- vide a long path (comparison of reflections from Rl and 7?3) for wave- lengths at which the absorption was relatively low, and a short path (comparison of Rl and R2) for wavelengths at which the absorption was high. The small reflector, Rl, was one foot on a side; the large reflectors, 7?2 and i?3, were about 5.6 feet on a side. Fig. 3 is a set of side-by-side measurements sho^^ing the reflecting properties of the large reflectors relative to the small one for the wavelengths at which they were used. J APPARATUS A schematic diagram of the wa^•eguide and electronic apparatus is shown in Fig. 4; Fig. 5 is a photograph of the waveguide eciuipment so mounted that it moves as a unit with the horn antenna. The antenna is adjusted in azimuth and elevation by means of the milling ^-ise at the bottom of the photograph. The box at the left contains the transmitting tube, a low voltage reflex klystron* which has an average power output of about 12 milliwatts over its 5.1- to 6.1-mm tuning range. About 2 mil- liwatts of the klystron output is fed through a 6-db directional coupler to a balanced converter that contains two wafer-type millimeter rectifier units, t The remainder of the power proceeds into a 3-db coupler which TRANSMITTED DELAYED SIGNAL RETURN SIGNAL 1 1 ' 1 ^/ -1 / / X >\ / A ^ y^ -.^ 1 \ // 1 1 / / 1 »x / 1 y ^ 1 // 1 F ^X- y y/ \ A/ // /' ' y^ ^-'f ^ y' / u /v f 1/ /v ; ^^"^ 0^ 1^' ^--T,— - T\W TIME *■ < -T2 — - — > <72> % NEAR REFLECTOR FAR REFLECTOR Fig. 2 — Transmitted and reflected frequency-modulated signals. * This klystron was developed by E. D. Reed, Electron Tube Development Department, Murray Hill Laboratory. t These millimeter-wave rectifiers were developed by W. M. Sharpless, Radio Research Department, at the Holmdel Laboratory. ATMOSPHERIC ATTENUATION AT MILLIMETER WAVELENGTHS Table I 911 Reflector Distance Height Effective Area (Average) Sawtooth Rep. Rate Intermediate Frequency-f Rl R2 R3 km di = 0.59 do = 1.36 ds = 2.87 m 6.7 21.5 75 «»2 0.05 0.67 0.79 kc 33 14.4 6.8 kc 750 750 750 has the antenna on one arm and an impedance composed of an adjustable attenuator and shorting phniger on another arm. This impedance is ad- justed to balance out reflections from the antenna so that a negligible amount of the power flowing toward the antenna enters the converter which is on the remaining arm of the coupler. The delayed energy that re-enters the antenna after reflection from a corner reflector passes through the 3-db coupler to the converter. The intermediate frequency amplifier shown in Fig. 4 operates with a bandwidth of 300 kc centered at/=750 kc. The output of the amplifier is fed to a sciuare law detector and meter for accurate measurement and to an oscilloscope for checking operation of the equipment. Oscillograms of the pulses obtained from the three corner reflectors are shown in Fig. 6; these are all on the same time scale. The gap between the pulses is the delay, r, shown schematically in Fig. 2. 12.6 12.4 IXJ o LU Q z < LU < 12.2 12.0 11.8 u LU < CC 1.5 11.4 1.2 11.0 \ A? A, "-— ^, — ^ THEORETICAL A, A, < <^ y \ 1 1 Y A3 \ -r^ MEASURED^ A, \, s J. b K /- /^- MEASURED r A, \ V 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 WAVELENGTH, \, IN MILLIMETERS Fig. 3 — Calibration of corner-reflectors R2 and R3 using Rl as a standard. 912 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 SAWTOOTH GENERATOR SYNC OSCILLOSCOPE ^ r DETECTOR AMPLIFIER f = 750 KC PRECISION ATTENUATOR BALANCED TO UNBALANCED TRANSFORMER HYBRID . JUNCTION [> \ -0 BALANCED CONVERTER X REED KLYSTRON 6DB COUPLER RECORDER METER ADJUSTABLE SHORT X 3 DB COUPLER ANTENNA Fig. 4 — Schematic diagram of frequency-modulation radar. Fig. 5 — Waveguide apparatus and antenna. ' ATMOSPHERIC ATTENUATION AT MILLIMETER WAVELENGTHS 913 ^^- W^ -^W S : li Wi W^- WmWai 'i^ Rl R3 Fig. 6 — 750-kc pulses corresponding to the data in Table I. Fig. 7 shows the conical horn-lens antenna supported by two bearings to allow adjustment of azimuth and elevation angles. The aperture of the antenna is fitted with a polyethylene lens 30 inches in diameter. The antenna has a gain of about 51 db and a beam width of about 0.5 degrees in the middle of the 5- to 6-mm wave band. This narrow beam, together wath well-elevated reflectors, essentially eliminated ground reflections from the measurements. %iM -^ ..«S.JS Fig. 7 — Conical horn-lens antenna and mount. 914 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 ID r 14 r 1 13 p f A »^~x 12 3 Y / / y ' / / / 1 i f 1 1 It 6 J 1 5 / r 1 1 11 1 f a 4 ^ '/ ^^m^" 0 ^ r 48 49 50 51 52 53 54 55 56 57 58 59 FREQUENCY IN KILOMEGACYCLES PER SECOND 60 61 Fig. 8 — Calculated and measured absorption by air at sea level. The dots represent the experimental data; the vertical lines indicate the spread in the meas- ured values. Curves A and B are calculated curves of oxygen absorption using line-breadth constants of 600 and 1200 mc, respectively^, and a temperature of 293° K. (Courtesy of T. F. Rogers, Air Force Cambridge Research Center.) RESULTS The data to be discussed are shown in Fig. 8; they were taken at Hohn- del, N. J., during the months of December, 1954, and January, 1955, on days when the temperature was between 25 and 40 degrees Fahrenheit ; the absolute humidity was less than 5 grams/meter^ during the measure- ments. It is believed, therefore, that the resonance of the oxygen mole- cule is the main contributor to the absorption. The spread in llie measurements is indicated by vertical lines through the average values. Each point represents an average of six or more meas- m'oments taken on different days. In the range 49 to 54.5 kmc, (5.5 to ATMOSPHERIC ATTENUATION AT MILLIMETER WAVELENGTHS 915 '6 1 1 1 1 1 1 1 1 1 1 rr — I 1 i 1 1 r "I r ._ SEA LEVEL _---8 KILOMETERS ,--11 KILOMETERS .--32 KILOMETERS 50 52 54 56 58 60 62 64 66 FREQUENCY IN KILOMEGACYCLES PER SECOND 68 70 Fig. 9 — Calculated curves of oxygen absorption at various altitudes for a line-breadth constant of 600 megacycles and a temperature of 293° K. (Courtesy of T. F. Rogers, Air Force Cambridge Research Center.) 6.1 mm) the measurements were highly consistent, due mainly to the longer path that was used. Errors in the absolute values of the absorption are estimated not to exceed ±0.05 db/km in the 49 to 54.5 kmc region, ±0.25 db/km in the 55.5 to 59 kmc region. The errors in absolute absorp- tion are governed mainly by the structural and thermo-mechanical sta- bility of the corner reflectors. -I o Q 0 -2 -4 -6 10 16 /^ \ \ A~^ A / \ y I /^ J n / iry \ / \ a/ V \i\ V \ a/ \j>A y " v/v V/~v^ r 3:20 3:30 3:40 3:50 4:00 TIME OF DAY. P.M. 4:10 4:20 4:30 Fig. 10 — Attenuation of 6.0-mm radiation caused by a light rain. Round-trip path length = 2.72 kilometers Average rainfall rate = 5 millimeters per hour 916 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G Table II Approximate Optical Visibility (miles) Attenuation due to Land Fog DB/KM 0.06 0.13 0.22 In Fig. 8, measured values are compared with the theory of Van Vleck as calculated by T. F. Rogers using line-breadth constants of 600 mc and 1200 mc per atmosphere. The fit with the 600-mc curve is good from 49 to 55.5 kmc, but discrepancies are evident between 56.5 and 59 kmc. For completeness, Rogers' calculations for the absorption at higher alti- tudes are reproduced in Fig. 9. A few continuous recordings of rain attenuation have been made at a wavelength of 6.0 mm; a record taken during a light rain is shown in Fig. 10. The median value of the signal is —6.7 db which corresponds to an attenuation of 2.5 db/km for this 5 mm per hour rainfall. During more intensive rainfalls, short-term attenuations in excess of 25 db/km have been observed. On one occasion, it was possible to measure attenuation by land fog. The measurements given in Table II were made at a wavelength of 6.0 mm. No information regarding water content or drop size was available for this fog. CONCLUSION A frequency-modulation, two-way transmission technique has proven reliable for measurement of atmospheric attenuation at millimeter wave- lengths. Prerequisite to the success of the method are corner reflectors with good mechanical, thermal and electrical stability. The frequency-modulation method has been demonstrated by absorp- tion measurements in the free atmosphere in the 5.1- to 6.1-mm band. The data thus obtained are in good agreement with Van Vleck 's theory of oxygen absorption; the line-breadth constant appropriate to the meas- urements lies between 600 and 800 mc per atmosphere. REFERENCES 1. J. H. Van Vleck, Phys. Rev., 71, pp. 413 ff, 1947. 2. R. Beringer, Phys. Rev., 70, p. 53, 1946. R. S. Anderson, W. V. Smith and W. Gordy, Phys. Rev. 87, p. 561, 1952. J. O. Artman and J. P. Gordon, Phj^s. Rev., 96, p. 1237, 1954. 3. R. H. Dicke, R. Beringer, R. L. Kyhl, A. B. Vane, Phys. Rev., 70, p. 340, 1946. G. E. Mueller, Proc. I.R.E., 34, p. 181, 1946. H. R. Lament, Phys. Rev., 74, p. 353, 1948. A New Interpretation of Information Rate By J. L. KELLY, JR. (Manuscript received March 21, 1956) 7/ the input symbols to a communication channel represent the outcomes of a chance event on which hets are available at odds consistent with their probabilities (i.e., "fair'' odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially. The maximum exponential rate of growth of the gambler's capital is equal to the rate of transmission of information over the channel. This result is generalized to include the case of arbitrary odds. Thus we find a situation in which the transmission rate is significant even though no coding is contemplated. Previously this quantity was given significance only by a theorem of Shannon's which asserted that, with suit- able encoding, binary digits coidd be transmitted over the channel at this rate with an arbitrarily small probability of error. INTRODUCTION Shannon defines the rate of transmission over a noisy communication channel in terms of various probabilities. This definition is given sig- nificance by a theorem which asserts that binary digits may be encoded and transmitted over the channel at this rate with arbitrarily small probability of error. Many workers in the field of communication theory have felt a desire to attach significance to the rate of transmission in cases where no coding was contemplated. Some have even proceeded on the assumption that such a significance did, in fact, exist. For ex- ample, in systems where no coding was desirable or even possible (such as radar), detectors have been designed by the criterion of maximum transmission rate or, what is the same thing, minimum equivocation. Without further analysis such a procedure is unjustified. The problem then remains of attaching a value measure to a communi- ^ C. E. Shannon, A Mathematical Theory of Communication, B.S.T.J., 27, pp. 379-423, 623-656, Oct., 1948. 917 918 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G cation system in which errors are being made at a non-negligible rate, i.e., Avhere optimum coding is not being used. In its most general formu- lation this problem seems to have but one solution. A cost function must be defined on pairs of symbols which tell how bad it is to receive a cer- tain symbol when a specified signal is transmitted. Furthermore, this cost function must be such that its expected value has significance, i.e., a system must be preferable to another if its average cost is less. The utility theoiy of Von Neumann shows us one way to obtain such a cost function. Generally this cost function would depend on things external to the system and not on the probabilities which describe the system, so that its average value could not be identified with the rate as defined by Shannon. The cost function approach is, of course, not limited to studies of com- munication systems, but can actually be used to analyze nearly any branch of human endeavor. The author believes that it is too general to shed any light on the specific problems of communication theory. The distinguishing feature of a communication system is that the ultimate receiver (thought of here as a person) is in a position to profit from any knowledge of the input symbols or even from a better estimate of their probabilities. A cost function, if it is supposed to apply to a communica- tion system, must somehow reflect this feature. The point here is that an arbitrary combination of a statistical transducer (i.e., a channel) and a cost function does not necessarily constitute a communication system. In fact (not knowing the exact definition of a communication system on which the above statements are tacitly based) the author would not know how to test such an arbitrary combination to see if it were a com- munication system. What can be done, however, is to take some real-life situation which seems to possess the essential features of a communication problem, and to analyze it without the introduction of an arbitrary cost function. The situation which will be chosen here is one in which a gambler uses knowledge of the received symbols of a communication channel in order to make profitable bets on the transmitted symbols. THE GAMBLER WITH A PRIVATE WIRE Let us consider a communication channel which is used to transmit the results of a chance situation before those results become common knowledge, so that a gambler may still place bets at the original odds. Consider first the case of a noiseless binary channel, which might be ^ Von Neumann and Morgenstein, Theory of Games and Economic Behavior, Princeton Univ. Press, 2nd Edition, 1947. A NEW INTERPRETATION OF INFORMATION RATE 919 used, for example, to transmit the results of a series of baseball games between two equally matched teams. The gambler could obtain even money bets even though he already knew the result of each game. The amount of money he could make \\'ould depend only on how much he chose to bet. How much would he bet? Probably all he had since he would win with certainty. In this case his capital would grow expo- nentially and after N bets he would have 2^ times his original bankroll. This exponential growth of capital is not uncommon in economics. In fact, if the binary digits in the above channel were arriving at the rate of one per week, the sequence of bets would have the value of an invest- ment paying 100 per cent interest per week compounded weekly. We will make use of a quantity G called the exponential rate of growth of the gambler's capital, where G = Urn -^ log ^ iVH.00 iV V 0 where Vn is the gambler's capital after A'' bets, Vo is his starting capital, and the logarithm is to the base two. In the above example (j = 1. Consider the case now of a noisy binary channel, where each trans- mitted symbol has probability, p, or error and q of correct transmission. Now the gambler could still bet his entire capital each time, and, in fact, this would maximize the expected value of his capital, (Fjv), which in this case would be given by (F;v) = (2qfVo This would 1)6 little comfort, however, since when A^ was large he would probably be broke and, in fact, would be broke with probability one if he continued indefinitely. Let us, instead, assume that he bets a frac- tion, (, of his capital each time. Then v^ = (1 + (y\i -^fvo where W aiid L are the number of wins and losses in the N bets. Then G = Lim ^logd -f o+^iog(i -i) = g log (1 -f /') -1- p log (1 — i) with probability one Let us maximize G with respect to /. The maximum value with respect to the Yi of a quantity of the form Z = ^ Xi log Yi , subject to the constraint ^ Yi = Y, is obtained by putting Y Yi = j^ Xi , 920 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 where X = ^ Xi . This may be shown directly from the convexity of the logarithm. and Thus we put (1 + ^) = 2q (1 - -f) = 2p G^max = 1 + P log p + g log g = R which is the rate of transmission as defined by Shannon. One might still argue that the gambler should bet all his money (make ^ = 1) in order to maximize his expected win after N times. It is surely true that if the game were to be stopped after N bets the answer to this question would depend on the relative values (to the gambler) of being broke or possessing a fortune. If we compare the fates of two gamblers, however, playing a nonterminating game, the one which uses the value € found above will, with probability one, eventually get ahead and stay ahead of one using any other i. At any rate, we will assume that the gambler will always bet so as to maximize G. THE GENERAL CASE Let us now consider the case in which the channel has several input symbols, not necessarily equally likely, which represent the outcome of chance events. We will use the following notation: p{s) the probability that the transmitted symbol is the s'th one. p(r/s) the conditional probability^ that the received symbol is the r'th on the hypothesis that the transmitted symbol is the s'th one. p(s, r) the joint probability of the s'th transmitted and r'th received symbol. q{r) received symbol probability. q(s/r) conditional probability of transmitted symbol on hypothesis of received symbol, a, the odds paid on the occurrence of the s'th transmitted symbol, i.e., as is the number of dollars returned for a one-dollar bet (including that one dollar), a(s/r) the fraction of the gambler's capital that he decides to bet on the occurrence of the s'th transmitted symbol after observing the r'th received symbol ^ A NEW INTERPRETATION OF INFORMATION RATE 921 Only the case of independent transmitted symbols and noise will be considered. We will consider first the case of "fair" odds, i.e., 1 Ois = p{s) In any sort of parimutual betting there is a tendency for the odds to be fair (ignoring the "track take"). To see this first note that if there is no "track take" Ei = i since all the money collected is paid out to the winner. Next note that if p(s) for some s a bettor could insure a profit by making repeated bets on the s* outcome. The extra betting which would result would lower a., . The same feedback mechanism probably takes place in more compli- cated betting situations, such as stock market speculation. There is no loss in generality in assuming that Z a(s/r) = 1 s i.e., the gambler bets his total capital regardless of the received symbol. Since he can effectively hold back money by placing canceling bets. Now r,s where Wsr is the number of times that the transmitted symbol is s and the received, symbol is r. Log ^ = X) ^V^r log oisa{s/r) vl " "^ G = Urn ^ log -f/ = X) P(^> ^) log oLsa{s/r) N^x I\ Vq ts with probability one. Since 1 oil = ' Pis) 922 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G here G = H P(s, r) log a(s/r) - J2v(s, r) log a(s/r) + H{X) rs where H{X) is the source rate as defined by Shannon. The first term is maximized by putting ^kP{k, r) q{r) Then (7max = H(X) — H{X/Y), which is the rate of transmission de- fined by Shannon. WHEN THE ODDS ARE NOT FAIR Consider the case where there is no track take, i.e., but where as is not necessarily 1 V{s) It is still permissible to set ^s a{s/r) = 1 since the gambler can effec- tively hold back any amount of money by betting it in proportion to the I /as . Equation (1) now can be written G = ^ P(s, r) log a(s/r) -f- J2 Pi^) log a. . rs s G is still maximized by placing a(s/r) = q{s/r) and G^max = -H{X/Y) + Y. Pis) log as s = H(a) - HiX/Y) where H{a) = X pis) log as Several interesting facts emerge here (a) In this case G is maximized as before by putting a{s/r) ^ qis/r). That is, the gambler ignores the posted odds in placing his bets! A NEW INTERPRETATION OF INFORMATION RATE 923 (b) Since the minimum value of H{a) subject to s as obtains when a. = p(s) and H(X) = H(a), any deviation from fair odds helps the gambler. (c) Since the gambler's exponential gain would be H{a) — H(X) if he had no inside information, we can interpret R = H{X) — H{X/Y) as the increase of Gmax due to the communication channel. When there is no channel, i.e., H{X/Y) = H{X), Gmax is minimized (at zero) by set- ting 1 as = — Ps This gives further meaning to the concept "fair odds." WHEN THERE IS A "TRACK TAKE" In the case there is a "track take" the situation is more complicated. It can no longer be assumed that ^s a{s/r) = 1. The gambler cannot make canceling bets since he loses a percentage to the track. Let br = 1 — X)s ais/r), i.e., the fraction not bet when the received symbol is the r one. Then the quantity to be maximized is G = 11 p(s, r) log [br + aMs/r)], (2) rs subject to the constraints br+ E«(sA) = 1. In maximizing (2) it is sufficient to maximize the terms involving a particular value of r and to do this separately for each value of r smce both in (2) and in the associated constraints, terms involving different r's are independent. That is, we must maximize terms of the type Gr = q(r)^ q(s/r) log [6, + asa(s/r)] s subject to the constraint 924 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Actually, each of these terms is the same form as that of the gambler's exponential gain where there is no channel (? = X; p(s) log [b + a.a(s)]. (3) a We will maximize (3) and interpret the results either as a typical term in the general problem or as the total exponential gain in the case of no communication channel. Let us designate by X the set of indices, s, for which a(s) > 0, and by X' the set for which a(s) = 0. Now at the desired maximum p(s)as dG da{s) b + a(s)ai log e = k for seX dG y-^ p{s) , , -1- = Z^ -, — . \\ — log e = k dG p(s)as , ^ J r .f — T-r = ^ / log e :^ /c for SfX da{s) b ^ ~ where /c is a constant. The equations yield k = log e, b = b ais) = pis) - 1 -P 1 - a- for seX as where p = Xxp(s), a- = ^x (1/as), and the inequalities yield p(s)as ^ b = 1 -p for SfX We will see that the conditions (X < 1 p(s)as > p(s)as ^ 1 P 1 - a 1 -P 1 - 1 Ft de- creases with t until p(t + l)at+i < Ft or at ^ 1. If the former occurs, i.e., p(t + l)oit+i < Ft , then i^^+i > Ft and the fraction increases until cr< ^ 1. In any case the desired value of t is the one which gives Ft its minimum positive value, or if there is more than one such value of /, the smallest. The maximizing process may be summed up as follows: (a) Permute indices so that p(s)as ^ p(s + l)Qrg+i (b) Set h equal to the minimum positive value of -I t < - — where Pt = ILp (s), at = ^ — i- — (Tt 1 1 ffj (c) Set a(s) = p(s) — b/as or zero, whichever is larger. (The a(s) will sum to 1 — h.) The desired maximum G will then be (rmax = Z) P(s) log p(s)as + (1 - Pt) log 1 -Pt I - CTt where t is the smallest index which gives 1 -Pt 1 - 1 some bets might be made for which p(s)as < 1, i.e., the expected gain is negative. This violates the criterion of the classical gambler who never bets on such an event. CONCLUSION The gambler introduced here follows an essentially different criterion from the classical gambler. At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with 92(i THE BELL SYSTEM TECHNICAL .lOlKXAL, JCLY 195G the value function Avhich he attached to his money, but merely with the fact that it is the logarithm A\hic'h is additive in repeated bets and to which the law of large numbers applies. Suppose the situation were different; for example, suppose the gambler's wife allowed him to bet one dollar each week but not to reinvest his winnings. He should then maximize his expectation (expected value of capital) on each bet. He would bet all his available capital (one dollar) on the event j-ielding the highest expectation. With probability one he would get ahead of any- one dividing his money differently. It should be noted that we have only shown that our gambler's capital will surpass, with probability one, that of any gambler apportioning his money different!}^ from ours but still m a fixed way for each received sjanbol, independent of time or past events. Theorems remain to be proved showing in what sense, if any, our strategy is superior to others involving a{s/r) which are not constant. Although the model adopted here is draAvn from the real-life situation of gambling it is possible that it could apph' to certain other economic situations. The essential requirements for the validity of the theory are the possibilit}' of reinvestment of profits and the abilit}^ to control or vary the amount of money invested or bet in different categories. The "channel" of the theory might correspond to a real communication channel or simply to the totality of inside information available to the investor. Let us summarize briefly the results of this paper. Tf a gambler places bets on the input symbol to a comnumication channel and l)ets his money in the same proportion each time a particular symbol is receiA'cd his, capital will grow (or shrink) exponentially. If the odds are consistent with the probabilities of occvu'rence of the transmitted symbols (i.e., equal to their reciprocals), the maximum value of this exponential rate of growth will be equal to the rate of transmission of information. If the odds are not fair, i.e., not consistent with the transmitted symbol proba- bilities but consistent with some other set of probabilities, the maximum exponential rate of growth will be larger than it would have been with no channel by an amount equal to the rate of transmission of information. In case there is a "track take" similar results are obtained, but the formulae involved are more complex and have less direct information theoretic interpretations. ACNOWLEDGMENTS I am indebted to R. E. Graham and C. E. Shannon for their assist- ance in the preparation of this paper. Automatic Testing of Transmission and Operational Functions of Intertoll Trunks By H. H. FELDER, A. J. PASCARELLA and H. F. SHOFFSTALL (Manuscript received October 19, 1955) Conditions brought about by nationwide dialing increase intertoll trunk maintenance problems substantiaUy. Under this switching plan with full automatic alternate routing there is a considerable increase in the amount of multiswitched business, and as many as eight intertoll trunks in tandem are permissible. In addition, operator checks of transmission on the connec- tions are lost on most calls. These factors iynpose more severe limitations on transmission loss variations in the individual trunks and throw on the maintenance forces additional burdens of detecting defects in the distance dialing network. New methods of analyzing transmission performance to locate the points where maintenance effort will be most effective continue to be studied. The automatic testing arrangements described in this paper enable the main- tenance forces to collect over-all transmission loss data quickly and with a minimum of effort. They also facilitate the collection of such data on groups of trunks in a form to make statistical analyses easier. The use of these testing arrangements will permit the maintenance forces to keep a closer watch on intertoll trunk performance and will assist in disclosing trouble patterns. INTRODUCTION The advent of nationwide dialing, especiall}' with full automatic alternate routing, has presented additional problems in the maintenance of intertoll trunks. Transmission reciuirements are more rigorous, the intertoll trunk connections are more complex, and certain irregularities in the performance of the distance dialing network are difficult to detect. Automatic test equipment has been provided to aid and increase the efficiency of over-all testing. This equipment is capable of automatically 927 928 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 testing the operational (signaling and supervisory) functions of dial-type intertoU trunks, and of making two-way transmission loss measurements and a noise check at each end. The test results may be recorded at the originating end by means of a Teletypewriter. Automatic trunk testing has been used for many years in the local plant for checking the signaling and supervisory features of interoffice trunks. The automatic intertoll trunk testing equipment serves a similar function with respect to these operational features of the intertoll trunks. Because published material is available on automatic operational test- ing,* these features will not be discussed in detail in this paper; more emphasis is given to the transmission testing features which are new. MAINTENANCE ARRANGEMENTS FOR INTERTOLL TRUNKS Except in the very small offices, intertoll trunks usually have a test jack appearance in the toll testboard for maintenance purposes. Cord ended testing equipment in the toll testboard positions enables the attendants to perform various operational tests and to make transmis- sion loss, balance, noise or crosstalk measurements. Facilities are pro- vided for communication with distant offices and with intermediate points where carrier or repeater equipment may be located. Testing of carrier or repeater equipment as individual components or systems is an important aspect of the trunk maintenance problem but is beyond the scope of the present paper. The maintenance of intertoll trunk net losses close to their specified values is currently a most important transmission problem. Various aspects of the problem are discussed in a companion paper, f Although the manual testing equipment mentioned above is vital to trunk net loss maintenance, the need for reduction in time and effort required to make measurements has led to the provision of semi-auto- matic testing arrangements. These arrangements permit a testboard attendant to check transmission in the incoming direction by dialing code 102 over a trunk. The trunk is connected to a source of one milli- watt test power at the far end and a measurement of the received power indicates the net loss. The equivalent of a semi-automatic two-way test may be obtained by making a code 102 test in each direction. If com- plete information on the test results is desired by one testboard at- tendant, the attendant at the other end of the trunk must report back his results. * R. C. Nance, Automatic Intertoll Trunk Testing, Bell Labs. Record, Dec, 1954. t H. H. Felder and E. N. Little, Intertoll Trunk Net Loss Maintenance Under Operator Distance and Direct Distance Dialing, page 955 of this issue. AUTOMATIC TESTING OF INTEETOLL TRUNKS 929 In both the manual and semi-automatic methods of measurement, the results must be recorded manually. For statistical analysis of trunk transmission performance in terms of "bias" and "distribution grade", as discussed in the companion paper,* deviations of the measured losses from the respective specified losses must be computed and summarized manually. The automatic testing equipment described in this paper has been developed as an additional maintenance tool. It will not supplant exist- ing arrangements discussed above but rather is intended to increase the capabilities of plant personnel to do an effective maintenance job. The following features of the equipment contribute particularly to this end: 1. Large numbers of trunks can be tested and the results recorded without the continuous attention of a testboard attendant. 2. The attendant is informed by an alarm whenever the loss of a trunk deviates excessively from the specified value. 3. Computation and summarizing of net loss deviations into class intervals are done automatically, thus facilitating statistical analysis of trunk performance. 4. Data can be collected quickly in large volume for indicating the performance of groups of trunks. Confusion occurring with manual measurements because of changing conditions with time is reduced. 5. Stability of an individual trunk may be checked by a series of repetitive tests. 6. Semi-automatic two-way trunk tests can be made by one attendant when required. To do an equivalent job entirely by manual methods would require an appreciable increase in the amount of manual test equipment and in the number of test personnel. A comparison of the times required for operational and transmission tests by manual, semi-automatic and auto- matic methods is shown in Fig. 1. The time shown for the code 102 test does not include coordination time required if information on test re- sults in both directions is required at one end. GENERAL DESCRIPTION OF AUTOMATIC TESTING EQUIPMENT Automatic intertoll trunk testing requires automatic equipment at both ends of the trunk.At the originating or control end, an automatic test circuit sets up the test call and controls the various test features. In the distant offices, test lines reached through the switching train provide appropriate automatic test terminations. The automatic equipment for * H. H. Felder and E. N. Little, Intertoll Trunk Net Loss Maintenance Under Operator Distance and Direct Distance Dialing, page 955 of this issue. 930 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 use at the control end, adapted for transmission testing, is presently available only for No. 4 type toll switching offices. Fig. 2 is a block schematic of the arrangement for automatic intertoll trunk testing, including transmission tests. In the originating No. 4 toll crossbar oflicc an automatic outgoing intertoll trunk test circuit is used which consists of an automatic outgoing intertoll trunk test frame and one or more associated test connector frames. These frames have been provided in all No. 4 type offices and perform the functions of setting up MANUAL (2 MEN) NEAR END OFFICE FAR END " OFFICE If) 4 lij i3 CIRCUIT OPERATION TESTS TEST BOARD TEST BOARD 5 H 0 1 - 2 -WAY TRANSMISSION MEASUREMENTS ■■ 3.5 8.0 MAN MINUTES PER TEST SEMI-AUTOMATIC {l MAN) TEST BOARD 103 OR 102 OR 104 TEST LINE (CODE 104 TEST INCLUDES NOISE CHECK) CODE TEST / ' s 102 104 a? 0.9 1.8 MAN MINUTES PER TEST FULLY AUTOMATIC 1 ) 1- (CI (/1 T UJ 1- < I I fM T T I o UJ z _ u. t F o >5h ^ <-iz £ > UJ UJ O (\J v- 9: y Z3 ~ O cr lu u5-iz Z 5 UJ UJ O (vJvtrO: Bgfs Q z LU CC < lU z q: o UJ o IL U- O o z z (J q: o UJ S z rr u < 05"- < _i >UJ uJz UJ^J -■ < tru _IZ ou I- UJlO I- UJ ZK o Icn ^?P1^^ ot£uj|a iozj m ID 03 3 c3 (I O t-i +3 c a f-r bC Jl 3 < a -J I- o y^-- |- .0 o^< ^"z t ^ 1 "J a fr t-^tn -£ ^ > > > =) 3 III u I- < o I- < LU CO Z UJ I- D- < ^ _i en -J m ^1 til CL LU O >> UJ o UJ, 10 H Q CC 5ujiu Ct Z I < u. \ 0. UJ « C\J Q. UJ 940 AUTOMATIC TESTING OF INTERTOLL TRUNKS 947 noise condition, or a 120 TPM flashing signal if the far-end has registered a high noise condition. The near end is thus advised of the results of the noise check at the far end. The test frame, on receipt of this signal, causes the Teletypewriter to complete the record and then breaks down the connection and advances to the next trunk. The amplifier, which precedes the amplifier-rectifier includes a net- work which provides FlA noise weighting during the noise check. The amplifier-rectifier is adjusted in the noise checking condition (that is, when its gain is increased) so that a noise indication will be given when the noise exceeds about 35 or 40 or 45 dba. Since this test is intended only as a rough check to detect any abnormal noise condition, the noise rejection limit used in any given office will be governed by the types of intertoll trunk facilities terminating in that office. Xo correction is made for the measured loss of the trunk at the time of the noise check, hence the noise is checked at the receiving switchboard level. For the usual types of noise the results of the noise check agree roughly with those which would be obtained by an average observer using a 2-B Noise Measuring Set for a similar "go-no go" type of check. As is evident from the previous description, each end is expected to complete the various steps of its functions within allotted time intervals. Timing intervals at the far end ai'e controlled by a multivibrator circuit. Timing at the near end is controlled by a similar multivibrator in the intertoll trunk test frame. To insure that the test circuits always perform as they should and that the timing circuits are functioning properl}^, checks are built into the circuits so that anything which pre- vents the successful completion of a 2-way measurement on schedule causes the automatic outgoing intertoll trunk test frame at the near end to stop, hold the trunk busy and sound an alarm while awaiting attention of the attendant. The transmission measuring and noise checking circuit at the far end will, however, release itself from the test line so that it will be free to handle other calls. Semi- Automatic Test One-milliwatt test power supply outlets ha\'e been provided in toll offices for some time for making a one-way transmission measurement freciuently referred to as a code 102 test. A test board attendant can reach the one milliwatt test power supply l)y pulsing forward code 102 or bj^ requesting an operator at the distant end of a manual trunk for a con- nection. The test power is applied at the distant end for about 10 seconds diu-ing which time the attendant measures the loss in the receiving (far-to-near) direction. This is a fairly fast semi-automatic test luit. of 948 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 course, has the disadvantage that it is a one-way test and cannot be used for all purposes. In order to provide a semi-automatic two-way test, the far-end equip- ment is arranged so that a test board attendant can make a code 104 measurement unassisted. This measurement is carried out in 3 steps as shown in the lower portion of Fig. 6. Step 1 The attendant connects a test cord to the test jack of the intertoll trunk and pulses forward code 104 using his test position dial or key set. When the far end is ready, it returns an off-hook signal which retires the test cord supervisory lamp. He then connects the other end of the cord to the one milliwatt test power supply. The far end then adjusts the receiving and transmitting pads in the same way as for a full auto- matic test. After about 3 seconds the attendant disconnects the test power and at that time observes the cord supervisory lamp ; a single flash indicates that the far end was unsuccessful and is requesting a second trial. If the supervisory lamp remains steadily dark he connects the cord to the receive jack of his transmission measuring circuit to prepare for Step 2. Step 2 The far end will pause about 2 seconds after the attendant removes the test power to give him time to prepare for Step 2. During this pause the far end will not receive a short spurt of test power as in the case of a full automatic test. Consequently, after the 2 second interval the far end will return one milliwatt for 10 seconds on a semiautomatic test to give the attendant time to complete a measurement. The received power is read directly on the meter of the transmission measuring circuit and is the loss in the far-to-near direction. When the far end removes the test power, the meter reading drops back to the position of no current (in- finite loss) and at that time the attendant observes the cord lamp. A single flash at this time is an "add 10" signal and indicates that 10 db should be added to the next measurement. A steady dark lamp indicates that the next measurement should be recorded without correction. Step 3 After about 2 seconds delay to give the attendant time to record the first measurement, the far end again returns 1 milliwatt, this time through the transmitting pad set up in Step 1 . The meter now reads the AUTOMATIC TESTING OF INTERTOLL TRUNKS 949 loss of the trunk plus the loss of the transmitting pad at the far end. Since the transmitting pad loss equals the trunk loss in the near-to-far direction, the difference between the measurements in Step 3 and Step 2 is the trunk loss in the near-to-far direction. After about 10 seconds the far end removes the test power and starts the noise check in the same way as if this were a full automatic test. When the far end removes the test power after Step 3, the attendant leaves the connection intact until the cord supervisory lamp lights to indicate completion of the noise check at the far end. A flashing lamp indicates that the noise at the far end exceeds the prescribed limit and a steadily lighted lamp indicates the noise at the far end is below this value. A noise test at the near end may be made by the attendant if he judges, after a listening test, that a noise test is desirable. For this test he uses the standard noise measuring equipment. PRESENTATION OF TEST RESULTS When making operational tests and a Teletypewriter is not being used, troubles are registered by means of an audible alarm and accompanying display lamps. When making transmission loss measurements, however, a complete record of the measurements on all trunks tested, both good and bad is frequently needed. A Teletypewriter then becomes a practical necessity; otherwise the attendant would be required to supervise the automatic equipment continuously and to record, from a lamp display or similar indication, the results of each measurement as it was made. Having provided the Teletypewriter for transmission testing, its ability to print letters to represent trouble indications is utilized to avoid halt- ing the progress of the tests when operational troubles are experienced, except when completely inoperative conditions are encountered. Computer Circuit As mentioned earlier intertoll trunk transmission performance is rated in terms of bias and distribution grade which are calculated from the deviations of the measured losses of the intertoll trunks from their specified values. For such calculations the maintenance forces are, there- fore, more interested in the deviations than they are in actual measured losses. Accordingly, the automatic transmission test and control circuit at the near-end has a computer built into it which will compute the deviation for each measurement so that the deviation can be recorded by the Teletypewriter. The computer is a bi-quinary relay type adder similar to those used 950 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 195G for other purposes in the telephone plant, for example, in the computer of the automatic message accounting system. It obtains the specified net loss of the trunk being tested from the class relay which remains operated throughout the test. When a computation is to be made of the deviation in the far-to-near direction, for example, the control circuit extends to the adder a number of leads from the contacts of the far-near pad control relays. Some combination of the 9 far-near pad control relays remains operated after the far-near pad adjustment is finished and therefore some combination of the leads extended to the adder will be closed. These leads furnish to the adder the measured loss of the intertoll trunk in the far-to-near direction. The adder then subtracts the specified loss from the measured loss and presents the answer together with the proper sign, -|- or — , to the teletypewriter for a printed record. The deviation in the near-to-far direction is computed in the same manner by extending corresponding leads from the near-far pad control relays to the adder at the proper time. Deviation Registers In determining bias and distribution grade by the method discussed in the companion article,* the deviations from specified net loss are cal- culated for each measurement. These deviations are grouped together in 0.5 db increments from +8 db to —8 db, all deviations exceeding +7.8 db or —7.8 db being considered as -(-8.0 db and —8.0 db respectively. For example, all deviations of -|-0.3 db to -f-0.7 db, inclusive are con- sidered to be -f 0.5 db and are so tallied on the data, or stroke, sheet. To assist in this work the automatic test equipment includes thirty- three manually resettable counters corresponding to the 0.5 db incre- ments from -f-8.0 db to —8.0 db inclusive. Just prior to a transmission test cycle all these counters are reset to zero. At the time a deviation computation is made, the computer also causes the proper counter to register one count. After the test run on a group of trunks, the counter readings can be transcribed directly as the final tally on the stroke sheet and may be used to determine the bias and distribution grade. A "total tests" coimter keeps a tally of all the computations. At the end of the test run the total count serves as a check of the total count of the other 33 counters. Check for Excessive Deviations In addition to obtaining data for the calculation of bias and distribu- tion grade, the maintenance forces would also like to know promptly * H. H. Felder and E. N. Little, Intertoll Net Loss Maintenance Under Opera- tor Distance and Direct Distance Dialing, page 955 of this issue. AUTOMATIC TESTING OF INTERTOLL TRUNKS 951 when the loss of an intertoll trunk deviates an abnormal amount from its specified value. The maintenance practices currently require that, Avhenever an intertoll trimk is found to have a deviation of ±5 db or more in either direction, the trunk should be removed from service im- mediately and the cause of the abnormal deviation corrected. Accord- ingly, the computer circuit includes an alarm feature which sounds an alarm to attract the immediate attention of the attendant whenever the computed deviation is ±5.0 db or greater. The maintenance forces may also like to know promptly about trunks with wide deviations but which are not so bad as to recjuire immediate remo\-al from service. For this purpose the computer also includes a limit checking feature. This can be set, by means of optional wiring, to detect deviations in excess of dz3.0 db, dz4.0 db or ±5.0 db. Whenever a deviation exceeds the limit for which the computer is wired, this feature performs as follows: (1) When the Teletypewriter is not in operation the test frame stops and sounds an alarm. (2) When the Teletypewriter is recording all measurements, the letter U is added in a separate column at the end of the test record. The letter stands out on the record to j^ermit fiuick spotting of trunks wdth abnormal deviations. (3) By means of a control key, a transmission test record can be printed only for those trunks whose deviation exceeds the computer checking limit or which are "noisy" at either end. Teletypewriter Record The Teletypewriter is put into operation by means of a key on the test frame. When this key is normal, no records are printed. Under this condition a trouble causes the test frame to stop and sound an alarm. When the Teletypewriter is operating it prints various records and a mi- nor operational trouble may result only in a record, without an alarm. Each record occupies a separate line on the tape. Each line starts wdth the four-digit trunk identification number in the first column. Fig. 7 shows a short specimen of the the Teletypewriter record. When the pass busy key on the test frame is in its nonoperated posi- tion, the Teletypewriter will print the trunk identification number, fol- lowed by the letter B, for each trunk passed over without test because it was busy. This is done on both operational and transmission test cycles. When the pass busy key is operated no record is made of busy trunks passed without test. During operational tests no record is printed for trunks which are 952 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 satisfactory. Troubles during either operational or transmission test cycles result in a record of the trunk identification number followed by a cue letter in a separate column denoting the nature of the trouble. This may be a single line record or a double line record for a repeat test on the same trunk as previously discussed. For example, in Fig. 7, the letter Y in the second line indicates that on trunk 1267 the far end was unable to complete its transmission measurement successfully. The letter A in lines 3 and 4 indicates that the test frame was unable to establish a con- nection over trunk 1293 on either its first or second attempt. The record of transmission tests is printed in several columns. Reading from left to right (see Fig. 7) these are (1) trunk identification number, (2) speci- fied net loss, (3) deviation in the far-to-near direction together with the sign, and (4) deviation in the near-to-far direction together with the sign. In columns 2, 3, and 4 the decimal points are omitted and the ten's digits are omitted when they are zero (0). Column 5 will contain an N if the far end is "noisy" or the letter U if the deviation in the far-to- near direction exceeds the computer check limits, preference being given to N if both conditions occur on the same trunk. Likewise column 6 contains an N if the near end is "noisy" or a U if the deviation in the near-to-far direction exceeds the computer check limits. Transmission test cycles will, of course, include a trouble record whenever an opera- tional trouble is encountered or whenever the transmission test cannot be completed successfully. z o 3 z h- O o g < Ul p 1- 1- (J ;?! <^ 1- z a. < LL 5 in a:o < 7 > 1- z UJ ^ LU 8 3 cr 1- Q cr y o t >- u < > cr Z^ ^i 1 3 a CD i/l LU LU Q Z LU < Qu: cncr FULLY-AUTOMATIC CODE 104 I NO. 4 OFFICE FULLY-AUTOMATIC CODE 104 FAR-END TEST CIRCUIT SxS CROSSBAR TANDEM NO. 5 CROSSBAR \ \ \ ^^ SEMI-AUTOMATIC CODE 104 _- OR CODE 102 I I PC ORJTC I -SEMI-AUTOMATIC CODE 102' SxS CROSSBAR TANDEM NO. 5 CROSSBAR note: it is assumed that code 102 mw supply circuits will be available at all offices and can be used, or that test board to test board measurements can be made when desired LEGEND RC = REGIONAL CENTER SC = SECTIONAL CENTER PC = PRIMARY CENTER TC = TOLL CENTER TP = TOLL POINT Fig. 8 — Typical layout for automatic testing. 954 THE BELL SYSTEM TEC?IXICAL JOURNAL, JULY 1956 trunks from other toll offices can be tested on a semi-automatic basis from the toll test board in the distant office. Fig. 8 shows a possible application of automatic test circuits. In such an application, all No. 4 type toll crossbar offices would have both near- end and far-end equipments. Other offices would have far-end equipment only when they have a sufficient number of direct trunks to No. 4 type offices to justify its use. The several types of tests which would be pos- sible are indicated in the illustration. It can be seen that a well distributed number of near-end and far-end test circuits will make it possible to test automatically a large percentage of the intertoll trunks throughout the country. This is particularly true in the more populous sections, where the concentration of trunks results in the probability of toll centers having trunks to more than one office furnished with near-end equipment. ACKNOWLEDGMENTS Automatic intertoll trunk testing arrangements, including transmis- sion tests, are the result of the ideas, efforts and experiences of many people concerned with intertoll switching and maintenance problems throughout the Bell System. Mr. L. L. Glezen and Mr. L. F. Howard deserve particular mention in this regard. Specific credit should also be given to Mr. B. McKim and Mr. T. H. Neely for the basic scheme of two-way transmission measurements and accuracy checks and to Mr. C. C. Fleming for the design of the amplifier and amplifier-rectifier. Appreciation is given to various departments of the American Tele- phone and Telegraph Company for their assistance during the develop- ment and trial of this equipment. Mention should also be made of the hearty cooperation and aid given by the A.T. & T. and Associated Company plant forces during the field trial of automatic transmission testing. « Intertoll Trunk Net Loss Maintenance under Operator Distance and Direct Distance Dialing By H. H. FEEDER and E. N. LITTLE (Manuscript received March 15, 1956) Nearly all of the components of an intertoll trunk contribute in some degree to its variations in transmission loss. Automatic transmission regulating de- vices in carrier systems and in many voice-frequency systems control in- herent variations in the intertoll trunk plant. These variations in transmis- sion come mainly from unavoidable causes such as temperature changes. The success of these devices depends on how precisely the trunk is lined up and the manner in which the maintenance adjustments are made. When the na- tionwide dialing plan with automatic alternate routing is in full swing, main- tenance requirements will be more severe because of the material increase in switched business and the number of possible links in tandem, and because operator checks will not be obtained on most calls. Therefore, the maintenance forces will have to keep closer watch on intertoll trunk transmission perform- ance and insure that the necessary adjustments are made in the right places. This article discusses some of the maintenance techniques now used and sug- gests fields for further study. TABLE OF CONTENTS Page Introduction 956 The Prolilem of Net Loss Maintenance 956 Effect of Switching Plans 957 Manual Operation 957 Dial Operation 958 Effect of Carrier Operation 960 Table 1 960 Quantitative Aspects of the Problem 962 Table II 963 Use of Transmission Loss Data 964 Procedure for Analyzing Measurements 965 Effectiveness of Over-all Trunk Test and Analysis 969 Simple Layouts 969 Complex Lajouts 970 Need for Education 971 Summary and Conclusions 972 955 956 THE BELL SYSTEM TECHNICAL JOURNAL, JLUY 1956 INTRODUCTION Currently there are over 230,000,000 long distance calls made in the Bell System per month. They range from relatively simple connections involving a single intercity trunk to complex connections involving sev- eral intercity trunks in tandem, perhaps totaling 4,000 miles in length. In each case there is a toll connecting trunk at each end. Almost half of this traffic involves distances over 30 miles. The transmission engineer's problem is how to provide uniformly good and dependable transmission so that every one of these calls will be satisfactory to the customers in- volved. To accomplish this requires among other things that: 1. The design loss of every trunk must be the lowest permissible from the standpoint of echo, singing, crosstalk and noise. 2. The actual loss of every trunk must be kept close to the design loss at all times. Meeting the first requirement is a matter of system design and circuit layout engineering. The factors involved have been covered in a previous article.^ Meeting the second requirement is an important function of the maintenance forces and is discussed in this article. THE PROBLEM OF NET LOSS MAINTENANCE The transition from manual operation under the ''general toll switch- ing plan" to dial operation under the "nationwide dialing plan"^- ^ is re- quiring material changes in intertoll trunk design and also in techniques for maintaining these trunks. While precise maintenance is becoming in- j creasingly necessary, it is also becoming more difficult to achieve. There are three important reasons for this. First, the nationwide dialing plan increases both the possible number of trunks used in tandem for a given call and the variety of the connec- tions in which any particular trunk may be used. This increases' the chances of impairment due to deviations from assigned loss in indi- vidual trunks since these deviations may combine unfavorably in multi- switched connections. To minimize this, the transmission stability of the individual trunk links must be better than under the old plan. Second, more and more of the trunks are being put on carrier because * H. R. Huntley, Transmission Design of Intertoll Telephone Trunks, B. S.T.J. , Sept. 1953. 2 H. S. Osborne, A General Switching Plan for Telephone Toll Service, B. S.T.J. , July, 1930. ' A. B. Clark and H. S. Osborne, Automatic Switching for Nationwide Tele- phone Service, B.S.T.J., Sept., 1952. * J. J. Pilliod, Fundamental Plans for Toll Telephone Plant, B.S.T.J., Sept. 1952. INTERTOLL TRUNK NET LOSS MAINTENANCE 957 it is the best solution to the transmission and economic problems. How- ('\er, carrier involves many more variable elements and requires higher precision of adjustment than voice-frequency systems need. These in- crease the difficulty of maintaining trunk losses close to design values on a day-by-day basis. Third, as operator distance and direct distance dialing grow, there is constantly diminishing opportunity for operators to detect and change unsatisfactory connections or to report unsatisfactory transmission con- ditions to the appropriate testboards for action. Thus the maintenance problem is in two parts: 1. How can we reduce departures from design standards even in the [face of increasing complexity of plant? 2. What substitute can we find for operator detection of troubles, and can we find even better means of detection? The ways in which switching plans and the use of carrier reflect upon Ithe problem of trunk net loss maintenance is discussed in more detail in the follo\\dng sections. EFFECT OF SWITCHING PLANS Manual Operation For many years long distance traffic has been handled on a manual ba- sis under the "general toll switching plan" illustrated in Fig. 1. Between two points indicated by toll centers, TC and TC", it was theoretically possible to get as many as five trunks in tandem. This rarely occurred be- RC' Po'(>: I"- TC ■& .'I .1. I ! / ^^. RC" "O PO" \ "-A ■B TC" TC = Toll Center PO = Primary Outlet RC = Regional Center Fig. 1 — General toll switching plan — manual operation. 958 THE BP:LL system technical journal, JULY 195G cause handliug .switched connections manuully was so complicated and expensive that direct trunks were provided wherever they were econom- ical and alternate routes were assigned and used sparingly. The result was that the manual switching plan was characterized by a minimum of switching. Under manual operation, operators passed information over every trunk in the connection, as w^ell as over the completed connection, before it was turned over to the customers. If anything was radically wrong with a trunk, the operators recognized it and substituted another trunk. When this was necessary, they could report the defective trunk to the appropri- ate testboard for action. Under these conditions, if trunk losses wandered appreciably from their specified values, the consequences were seldom serious. Dial Operation With dial operation, not only is the plan more complex (as shown on Fig. 2), with an abundance of alternate routes, but intertoll trunk switch- ing is so fast and reliable that the number of switching points has little effect on speed of service. Thus the dial operating plan can take full ad- vantage of alternate routing and the use of trunks in tandem \\\\\ occur much more freciuently than with manual operation. TC = Toll Center PC = Primary Center SC = Sectional Center RC = Regional Center Fig. 2 — Nationwide dialing plan — dial operation. INTERTOLL TRUNK NET LOSS MAINTENANCE 959 Here again the alternate routing follows a definite plan.^ As shown in iMg. 2, a call from toll center, TC, to toll center, TC", will follow the direct route, if it is available and not busy. A second choice will be via a higher ranking office in the chain from TC" to the regional center, RC". A third choice may be available to a still higher ranking office. Thus, if the originating office cannot use its direct route, the call will be advanced over the alternate routes according to a predetermined pattern. If all other alternatives fail, the call will follow the heavy solid line 7-link route shown on Fig. 2, or, in special cases, the 8-link route via RC". These attempts involve many operations but the automatic equipment completes them cjuickly. This makes it feasible to provide small, high usage, direct trunk groups between two points, with the realization that in busy periods alternate routes can handle the overflow^ traffic with neg- ligible time delay. Thus over a good part of the day, the direct trunks or first choice trunks will handle the traffic. In the busy periods, use of alternate routes with a mnnber of links in tandem vdW be a frequent occurrence. Therefore, it is important to have losses on alternate routes not greatly different from those on direct routes so customers will not experience noticeable contrasts. Operators will seldom talk to each other over the complete connection, and even less over the individual trunks. Only on person-to-person or col- lect calls will they talk even to the called party. On station-to-station calls they merely dial or key up the desired number and rely on super- visory signals to disclose the progress of the call. On operator dialed calls, the operator may sometimes pick up the in- tertoU trunk in her switchboard multiple, but in many cases she will reach it over a tandem trunk. In the former case she can identify the in- tertoll trunk forming the first link in the connection but assistance would be needed at intermediate testboards to identify succeeding trunks. In the latter case, testman assistance would be required at the originating office in order to identify even the first trunk of the connection. In either case the need for holding the customer's line during identification, to avoid breaking down the connections makes such means of identifica- tion impracticable Avith presently available techniques. On direct distance dialed calls there are no operators involved and pres- ent means of identification of trunks in trouble after the connection has ])een established are even more impracticable. This is because the calling party must release the connection before he can report a trouble, thus destroying any possibility of trunk identification. 6 R. I. Wilkinson, Theory for Toll Traffic Engineering in the U. S. A., B.S.T.J., March, 1956. 960 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Thus under dial operation there is a need for better trunk stability. Therefore, a greater burden is placed on the plant forces to locate unsatis- factory trunks so that proper maintenance action can be taken before customers experience difficulty. EFFECT OF CARRIER OPERATION Carrier is the principal transmission instrumentality which makes it possible to go ahead A\dth natiomvide dialing with assurance that people can talk satisfactorily over the complex connections set up by the switch- ing sj^stems. But it brings with it formidable problems of maintenance. The high attenuation per mile of the hue conductors at carrier frequencies increases the number of variable elements as well as the precision with which they must be adjusted. The interrelation between the elements adds to the complication. Table I illustrates this by giving some figures comparing 100 miles of a voice-frequency cable trunk with 100 miles of a typical trunk on K car- rier, which is \\idely used on cable facilities. The figures apply in both cases to one direction of transmission. The ten-to-one ratio in the number of electron tubes represents a greater chance of trouble developing in the carrier trunk due to aging or failure of electron tubes. In the carrier trunks there are more automatic adjustable features. For instance, in a typical K2 carrier system there are five flat gain regulators and one twist regulator in one twist section of approximately 100 miles, against a single regulator in a voice-frequency trunk 100 miles long. These regulators are depended upon to keep the loss variations to tolerable amounts. Any malfunction can have a serious effect on trvmk loss. Furthermore, they must be adjusted to the desired regulating range and therefore they are points at which maladjustments may be made. The channels of any one carrier system or of a 12-channel group are commonly routed by the circuit layout engineers to a number of terminal Table I Total Conductor Loss -db Gain Required to Reduce to Via Net Loss -db Percentage of Line Loss Represented by a 2 db Variation Number of Electron Tubes Number of Amplifiers Number of Automatic Regulators V-f K2 Carrier Trunk Trunk 35 378 31 377.4 5.7 0.53 3 28 3 7 1 6 INTERTOLL TRUNK NET LOSS MAINTENANCE 961 ALPHA BETA GAMMA ' ' --\ r — — ._ ^ — 1 j — — - - T, A j 1 c 1 T, 1^ -* ^- ^ 1 ^_ V f 1 ■X- T2 T3 \ / • \ / < T3 B / \ 1 D T2 \ -* ■x-- / ,, , \ ^ J > T4 T4 * TO THIS OR OTHER DESTINATIONS Fig. 3 — Typical carrier channel assignments. points even though circuit requirements to a given point are sufficient to utilize 12 or more channels. This is done to minimize the chances that all of the trunks between two points will be interrupted by a system failure. A simple case is illustrated by Fig. 3 which shows trunks between Alpha and Gamma connected at an intermediate point, Beta, in such a manner that a failure in any one of the systems A, B, C, or D will affect only half the trunks. This routing problem, however, complicates the maintenance problem. For example, if trunk Tl were found to have excess loss in the Alpha- Gamma direction it could be corrected by raising the channel gain at Gamma. On the other hand, a correct diagnosis might have disclosed that the trouble was due to a repeater in system A. If this were the case, merely compensating for the excess loss in Tl by changing the channel gain would still leave all other trunks associated with system A in trouble. Later on, if the repeater difficulty were corrected, and no further action were taken, the net loss of Tl would then be too low. Thus, the flexibility which is so desirable to minimize interruptions of whole circuit groups leads to a difficult problem in the administration of trunk loss adjustment and maintenance. Furthermore, because of the larger numbers and greater dispersion of trunks and terminal points, the situation in the actual telephone plant is much more complex than in the above example. Also, the diagnosis of trouble conditions is made more difficult by the normal variations of channel losses in the carrier systems and consequently of the trunk losses about their design values. This can 962 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 be better appreciated when some quantitative aspects of the problem are considered. QUANTITATIVE ASPECTS OF THE PROBLEM When the nationwide dial switching plan began to take shape some 8 or 10 3'ears ago, intensiA-e study of the transmission maintenance prob- lem was undertaken. The existing situation was examined to determine whether or not the plant ^^'ould continue to be satisfactory under the changed conditions. This was done by analyzing the results of many thousands of transmission measurements which had been made on a rou- tine basis in toll test rooms all over the Bell System. Both the measured and the assigned losses were available so the differences between them could be derived and analyzed statistically. Although the distribution of differences expressed in db for an office did not necessarily follow precisely a normal probability law, the distri- butions were close enough to normal law so that they could be treated as normal. The results were similar throughout the System. The differences within an office were random as also were the means of the differeJnces from office to office. However, the means tended to be biased in the direc- tion of excess loss. The performance of trunks in multi-link connections which would be set up by the switching machines could therefore be esti- mated with reasonable accuracy. In the statistical analysis of measure- ments on the group of trunks, the performance was expressed in terms of "distribution grade" and "bias." In telephone transmission mainte- nance terminology, bias is the algebraic average of the measured trans- mission departures in db from individual specified net losses for the group of trunks. The distribution grade is the standard deviation of the differ- ences between measured and specified trunk losses about this bias value. The distribution grades found in these studies were about as follows: For trunks under 500 miles — about 1.8 db. For longer trunks — about 2.5 db. Table II illustrates the effects of the distribution grades on connections involving various combinations of these trunk links, assuming that bias can be neglected. The design loss objective for a 4-link connection, say 1,000 miles long, is abovit 7 or 8 db (including 2 db of connecting trunk or pad loss at each end), ^rable II shows that, in an appreciable percentage of the 4-link con- nections in\'olving the above type of plant, the \'ariations can he ex- pected to exceed the design loss. Variations of this magnitude can result in transmission impairment d\w to (H'ho, hollownoss, singing, crosstalk, INTERTOLL TRUNK NET LOSS MAINTENANCE 963 Table II Number of IntertoU Trunks in the Connection . Distribution Grade in db Per cent of Connections Departing from Average ±2 db or more ±4 db or more ±8 db or more 2.5 42 11 0. 4.4 65 36 7 5.0 69 42 11 8* 5.6 73 47 15 Includes two trunks over 500 miles long. noise or low volume. Furthermore, undesirable contrast may be encoun- tered on successive calls between the same two telephones. The results of the study as well as experience with the beginning of automatic alternate routing show that the performance of the existing trunk plant must be improved. Three immediate objectives have been set: 1. Reduction of distribution grades to about }^ of the values men- tioned above, i.e., about 1.0 db. 2. Maintenance of office bias within ±0.25 db. 3. Removal from ser\'ice of individual trunks differing widely from their design losses (in the order of 4 or 5 db). To achieve these objecti^'es requires effort along four lines. First, systems should be designed to have sufficient stability once they are adjusted. This involves the inclusion of stable circuit elements and the provision of automatic regulating devices to compensate for unavoidable transmission variations arising from natural causes. These features have been applied to existing systems within limits imposed by economic con- siderations and the state of the art. Further extension of these features will be required in the future in order to meet the above objectives. Second, before a trunk is placed in service, each of its component parts and the over-all trunk should be adjusted to give the correct loss. From the transmission maintenance point of view, it is extremely important for each trunk to start out with all of its adjustments correctly made. Third, existing and incipient troubles, and deterioration or maladjust- ment of components, must be detected and corrected by routine mainte- nance of indi\'idual systems used in making up trunks. Such activity must make up for the inability to design systems to have the desired stal)ility. Fourth, significant departures from trunk design losses must be de- tected by over-all transmission measurements, and must be corrected be- 964 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 fore service reactions occur. Such measurements will also be of aid in de- termining the effectiveness of efforts along the first and third lines. As discussed earlier, the presence of the operator on every call was of material assistance in the detection of unsatisfactory trunks. On operator or direct distance dailed calls, there will be little or no operator conversa- tion over the intertoll trunk connection. As a substitute, the maintenance forces may need to make more frequent checks of the transmission per- formance of the trunks unless the stability of individual systems and components of systems can be improved. Manual methods have been used by the maintenance forces in the past to measure trunk losses. Semi- automatic measuring methods have been developed to reduce the time and effort required. In many cases the necessary number of measurements will be economical only when made by automatic devices. One form of such gear is described in a companion paper. ^ The ability to measure over-all trunk losses simply and frequently is of direct aid in detecting when loss deviations exceed maximum toler- ances. Such measurements in themselves, however, are insufficient to detect incipient troubles or to indicate the component part responsible for unsatisfactory transmission. An attempt has been made to achieve these objectives by using statistical analysis of the measured data as an aid to diagnosis. The following sections discuss the application of such analysis. Use of Transmission Loss Data It has been shown that considerable variation can be expected in trunk losses even in the absence of trouble conditions. For any given group of trunks selected for analysis, the performance is described by the distri- bution grade and the bias. If a group of trunks is found to have bias, it is usually an indication of some assignable cause. One such cause might be a change in gain of an amplifier common to the group. Another cause might be improper gain adjustment for channel units of a carrier termi- nal associated with the group. If a group of trunks is found to have a greater distribution grade than the distribution grade for all the trunks in the office, this may indicate excessive instability in a component part common to the trunks in the group. If analysis of all the trunks terminating in an office shows a higher distribution grade than is usually fomid in similar offices, the fault may be due to maintenance routines being inadequately or improperly ap- plied. * H. H. Felder, A. J. Pascarella and H. F. Shoffstall, Automatic Testing of Transmission and Operational Functions of Intertoll Trunks, page 927 of this issue. INTERTOLL TRUNK NET LOSS MAINTENANCE 965 Statistical analyses must thus be made of data for small groups as well as for large groups of trunks. Furthermore, the groups which arc studied must have elements or factors in common in order for the statis- tics to have significance. Analyses of periodic measurements of losses for the same trunk or groups of similar trunks can likewise indicate signifi- cant changes in performance. As yet, the problems of properly selecting the trunks to be analyzed and of correlating the results of the analyses with particular system ele- ments needing maintenance attention have been solved only partially. In addition to the need for proper procedures, there is the need for thor- ough training of maintenance personnel. The complexity of the telephone plant today is increasing the importance of all maintenance personnel ha\dng a thorough knowledge of how individual systems function and how the performance of the various system elements reacts upon over- all trunk performance. Procedure for Analyzing Measurements In an effort to facilitate the application of statistical analysis of trunk performance by plant personnel, a special data sheet and associated templates have been devised. These are shown in Figs. 4, 5, and 6. The method of analysis gives only approximate results but has been found to be sufficiently accurate for reasonably large amounts of data. It is simple, rapid and easily comprehended by the plant personnel. The procedure to be followed consists first of subtracting the specified loss from the measured loss for each of the trunks under study. A stroke is placed on the chart for each of the resulting deviations at the intersection of the appropriate classification and tally lines. For example, the first deviation between —3.25 db and —3.75 db would be stroked on the horizontal line for that band, just to the left of the vertical line for tally 1 (See Fig. 4). The second deviation in that band would be stroked just to the left of the tally 2 line. This is continued until all the deviations have been recorded. The last stroke in each j^^ db band indicates the number of deviations found having values within that band. As shown on Fig. 4, for the analysis by the template method this value is written in the first column, marked "Line Tots. (A)." These values are added and should equal the total number of measurements in the study (533 in the example). Next, the column "Cum. to 3^^" is filled out. Beginning at the top line, totals are accumulated to the point where adding the next line total will result in a value exceeding 3-^ the grand total of measurements (266 in the example). Similarly a value is obtained accumulating the totals from the bottom. In Fig. 4 these values are 246 and 166, respectively. 966 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 P 1 1 -F 1 1 1 1 1 1 1 1 1 1 \ 1 1 1 1 1 \ 1 1 T t I 1 1 f 1 I i o rrnpmqmp<»)or?ocnomom_rooc»>Ofop':PP;^P'^P'3SSS Xi ! 1 1 . , . i i ^ ^ J 'ui6ih6in6tfii>>^ ,h d3 ^ m i3 rf*<^^Ks^^S?S:c;SS<»-»»--- > 1 ^.o<-''^^-5?i5:3:Hfs$SSfv,.>.-oO^ ^ HI9— 2-fi-ii-ii-i-f-f-^-yfM*f -i-»ft*fl*H'Z+-^"'a 1 1 1^ ' ' Is r o o to 2 z 1 S S 1 II in «/3 ■65 i Is \ c o I J5 \ S 1 i o IP II E 5 z ^ 1 = 1 ^ 1 m — 1 — — 1 — 7 8 greater than 7b Therefore, the BIAS- iOate In Fig. 5 is enter line on the located at the -l/4 _ 3 absclsaa marking he chart* + X II •51 • II - !i -S - s -s -S m es OO* to' ^ ' CM 2 at 1 eo r^ «0 m 1 . ■ — -= N re m II — > — » 1 II -J - 1 : = ■ . ' — p *fcr\ «3 j « o -1 1 H £ O rH ...J 1 == - -^ ''F + • II IS is t; = 1 « E w £ = o a,s + ?f 1 ■ t 1 1 ! ■ X ■ + y " ■ 1 \ ■ ■ ■ . .. . , • t • ' f ■ y 1 — T ■ " ' + ■ " i : n h n i q X» 1 1 1 3 ? - c ■sj C " ' n ii SI r 3 t>j r Ci C n u ■- r 3 - n tj -. c si C -- c si e n u S( r ri n 1 u T U 3 fi S u 3 i -> ^ n u SI f o u 6 f -> u > ki^ t/) tJ"* xit tj^ «/> tD U^ li^ t/^ U^ ti^ W> Ut IJ^ U*> «A t/1 tn ^ ^ r^ csi r^ ti'i u^ i/^ ti-> u^ fsl — -^ I I I ^- — «s( cs^ I I i ■*• -t- •*■ •¥ IIP u( $$01 )3M p4!|p9ds snuiH sso*) pajnse)^ t/i u^ M^ Lr> (it u^ fsj r^ cs> 1-^ c>j r^ tiS u~> (O ix> r^ f^ + ♦♦■♦•■•■ + INTEKTOLL TRUNK NET LOSS MAINTENANCE 967 Fig. 5 — Combined template on stroke chart. By use of this information the approximate bias is determined. The scale of the bias values on the stroke sheet is shown in ^^ db steps along the left-hand edge of the "Line Tots. (A)" column, and bias is deter- mined to the nearest }^-i db. If the two cumulative totals differ from each other by less than 25 per cent of the larger value, an arrow indicating the bias is placed midway between the two class lines representing these cumulative totals. Its value is read on the bias scale. If the two cumula- tive totals differ from each other by 25 per cent or more of the larger value, an arrow indicating the bias is placed ^?^ of the distance between the two class lines representing these cumulative totals and nearer the larger value. In Fig. 4, since 246 minus 166 (80) is greater than 25 per cent of 246 (61.5), the arrow is placed ^4 of the way from the line repre- senting 166 toward the line representing 246; i.e., at the — 3>^ db point on the bias scale. A second arrow is placed at the corresponding point on the tally 1 line. As shown in Fig. 5, a combined template is then placed over the chart so that the center line of the template coincides with the two arrows. Along the center line of the template there is a scale indicating numbers of measurements from 50 throvigh 700. The template is moved horizon- tally so that the point on the scale corresponding to the grand total of measurements (533 in the example) is placed on the 1 tally line. En\^elope 9G8 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 — r- <5 >«■" w »■ <-( 3- 5 r^ o ■« 00 r-- o5 ~» T O ^ . ro o m o **> p ro p o iri o m eg CM m ON) d m o en o d « r\< ^ m p lb oi roonpoiofoorop cviibdindioNoiib s :i p 37" CO rn csi I I I eg -i J L o St'? !o a ^ 9 O 0 *■ i 2' S ^ -»' ■H 'T * "^ ^ W) o 0 1^ iti ' tr> p iS p tA p" dOo^^cgcNiro J — . -^ t t t t r ~o K 5 S S iK S S~3 »»^if)ir)ibibr>^r -O A m A -O 8 m b m B Fig. 4 — Space division switching. CROSSBAR SWITCHING NETWORK Dn ^B. ^ [} X )( COMMON CONTROL (MARKER) ): )? NETWORK CONNECTOR Fig. 5 — Typical common control of a crossbar switching network. ELECTRONICS IN TELEPHONE SWITCHING SYSTEMS 997 point element will be actuated if the link to which it connects is idle. h]ventually all available paths between input and output will be marked. Means must be provided for sustaining only one of the possible idle paths. Here the memory property of the crosspoint device takes over to hold the path until it is released by release marks or removal of the sustaining voltages. So it may be seen that in space division networks the memory requirements must be satisfied the same as in electrome- chanical networks. Multiplexing and carrier transmission systems^ employ time and fre- quency division but the physical terminals at both ends of a channel for which the facilities are derived have a one-to-one correspondence Avhich can only be changed manually. In a switching system means must be provided to change automatically the input-output relations as required for each call. Here the need arises for a changeable memory for associating a given time or fref[uency slot to a particular call at any given time. At some other time these points in time or frequency must be capable of being assigned automatically to different inputs and out- puts. For the period that they are assigned, some form of memory must record this assignment and this memory is consulted continuously or periodically for the duration of the call. With time division switching this new concept in the use of memory in a switching network appears most clearly, see Fig. 7(a). To associate an input with an output during a time slot the memory must be con- sulted which associates the particular input with the particular output. To effect the connection during a time slot the input and output must be selected A memory is consulted to operate simultaneously high speed GAS TUBE SWITCHING NETWORK Fig. 6 — Typical "End Marking" control of a gas tube switching network. 998 THE RET.L SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 selectors for both the input and output. Each selector receives informa- tion from a memory which actuates crosspoints to associate the input or output with the common transmission medium. The information from the memory which controls the selection process is known as an "address". The crosspoint is non-locking since it must open when the selector receives its next address. The individual memory of crosspoints for space division networks has thus been changed by time division to changeable memory, usually in the form of a coded address associated with each time slot. Furthermore since the successive addresses actuate the same selectors and hence may be held in a common high speed device, electronic bulk memory is ideally suited for this task. The memory must be changeable to allow for different associations of input to output at different times. In frequency division the control characteristics of the interconnect- ing network require a modulation frequency to be assigned each simul- taneous conversation to be applied within the bandwidth of the com- mon medium. As shown in Fig. 7(b) the application of the modulation frequencies requires a separate selector for each input and output. These I o- 20- 1 -O A -OB COMMON MEDIUM Fig. 7(a) — Time division switching. selectors are nothing more than space division switching networks and therefore require memory in the switching devices whether they are electromechanical or electronic. In addition to memory for associations within the switching network, selecting means are also needed to activate a terminal to be chosen in space division (e.g.. Fig. 6), to place address information in the proper time slot in time division switching or to set the frequency applying switching network in frequency division. CONTEOL The control of the switching system provides the facilities for receiv- ing, interpreting and acting on the information placed into it. In par- ELECTRONICS IN TELEPHONE SWITCHING SYSTEMS 999 20 Fig. 7(b) — Frequency division switching. ticular this is the address of the output desired. A service request de- tector (SR-D) is provided for each hne or trunk. In electromechanical systems these logic and information gathering functions are performed by relays or electromechanical switches. In order to keep up with the flow of information from a large number of customers, a number of register circuits must be provided to perform the same function simultaneously on different calls. Here information is being gathered on a "space division" basis and therefore a control switching network may be visualized as depicted in Fig. 8. The regis- ters designated R-M constitute the memory used to store the input in- formation as it is being received in a sequential manner from lines and trunks. As in the case of the conversation switching network, a space division control switching network has been used in electromechanical systems because the speed of these devices is not adequate to accom- modate the rate at which information flows into the system. It is inter- esting to note in passing that in the step-by-step system the control and conversation switching networks are coincident. In the No. 5 crossbar system^" the same network is used for both control and con- \ersation on call originations but when so used the functions are not coincident, that is, the network is used for either control or conversa- tion. In other common control systems, separate control networks known as "register or sender links" are employed. When using relays to receive the information pulsed into the office l)y customers or operators a plurality of register circuits are needed. The number of the registers required is determined by the time required to actuate the calling device and for it to pulse in the information. The 1000 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 registering function has two parts, one to detect or receive the informa- tion and the second to store it until a sufficient amount has been re- ceived for processing. The processing function is usually allotted to other circuits such as the markers in Crossbar systems. SR-D CONVERSATION SWITCHING NETWORK CONTROL SWITCHING NETWORK R-M R-M R-M Fig. 8 — Control access. Since the input of information to a switching system is usually limited to two conductors, a serial form of signaling is used. It would seem only natural that if a detector were fast enough it could function to receive the serial information in several simultaneously active inputs. Relays are not fast enough to do this, but high speed time sharing electronic devices have been designed to perform this information gathering func- tion. Since it is a time sharing arrangement it is analogous to the time division switching. A time division control access as shown in Fig. 8 and 9 requires memory to control the time division switching function. Time sharing when applied to the gathering of information in telephone switching systems has been called "scanning". The individual register memories are still in parallel form because of the relatively long time required for sufficient information to be received before processing may start. Higher speed means for placing information into switching sys- tems such as preset keysets is one way of reducing, if not eliminating, this need for parallel register storage in the switching system prior to processing. However, with this type of device one merely transfers the location of the storage from the central office to the customer's telephone set. The fundamental limitation is the rate at which a human being is able to transfer information from his brain into some physical repre- sentation. Lower cost memory is a practical means for improving this portion ELECTRONICS IN TELEPHONE SWITCHING SYSTEMS 1001 1 M M M -^ iR-m fR-ivn 1 Fig. 9 — Time division control access with separate functional memory. of the switching system. Many small low cost relay registers have been designed and placed into service.^" Electronics, however, offers memory at one tenth, or less, of the cost per bit if used in large quantities with a common memory access control. New low cost bulk electronic mem- ories are now available to be used in this manner. As shown in Fig. 10 the memor^y for the control of the time division control access network and the register memory may be combined in the bulk memory. o- \o [> — "^ BULK MEMORY . t Fig. 10 — Time division control access with bulk memory. Memory appears in the control portion of a switching system in many ways. Some are obvious and others are more subtle. Fig. 11 shows a typical electromechanical switching system, much like No. 5 crossbar and attempts to indicate various memory functions. First there is active memory designated A such as the call information storage A2 whether in a register, sender or marker during processing. There is also certain pertinent call information storage associated with trunk circuits such as a "no charge class" on outgoing calls or the ringing code used on incoming calls. Another type of active memory Ai has been mentioned in connection with switching networks to remember the input-output associations. In most electromechanical systems active memory has been emplemented with relaj^s or switches. Another form of memory is also employed in all telephone switching systems and much effort has been devoted to devising improved means for effecting this memory. This memory is of the type that is not changed with each call but is of a more permanent nature. Examples of this type of memory, which may be called passive memorj^, designated P, (Fig. 11), are the translations required in common control sytems to obtain certain flexibility between the assignment of lines to the switch- ing network and their directory listing. These translations between 1002 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 equipment numbers (network location) and directory numbers are required to direct incoming calls to the proper terminals (such as the number group frame in No. 5 crossbar, Fig. 12) and to provide on originating calls information for charging purposes (such as the AMA "Dimond" ring translator,!^ pig 13) Each of these translators for a 10,000 line office represent about 10^ bits of information. Another use for passive memory is to translate central office codes into routing in- formation. In local central offices this is also done by cross-connections as shown in Fig. 14. Another form of passive memory is the punched card or tape. These have been used widely in telephone accounting systems. A step toward electronic memory is the card translator which provides routing in- formation in the crossbar toll switching system^^ (see Fig. 15). Here the cards represent passive memory and are selected and read by a com- bination of electromechanical action and light beam sensing with phototransistor detectors. One such device equipped with 1,000 cards represents the storage of approximately 10^ bits of information. In all of the above types of passive memory limitations in the speed are involved in the choice of devices used within the memory or the access to it. This is one of the reaons these translators are subdivided so that the various portions may be used in parallel in order to satisfy the total information processing needs of the office. A discussion of passive memory would not be complete without one further illustration, Fig. 16. This is a wiring side view of a typical relay circuit in the information processing portion of a switching system. It o- EN SWITCHING NETWORK A| MARKER A2-P1 DN-^EN P2 REGISTER A = ACTIVE MEMORY P = PASSIVE MEMORY ON Fig. 11 — Memory in typical electromechanical switching system. ELECTRONICS IN TELEPHONE SWITCHING SYSTEMS 1003 Fig. 12 — No. 5 number group. could be any other unit, for example, a trunk circuit. The principal point is that each wire on such a unit is remembering some passive f| relationship between the active portions of the circuit, such as relays. This is the memory of the contact and coil interrelationships as con- ceived by the designer and based on the requirements of what the cir- cuit is required to accomplish. It is the program of what the central office must do at each step of every type of call. Modern digital com- puters have been built with the ability to store programs in bulk memories for the solutions of the various types of problems put to them. It is conceivable that the program of a telephone contral office may also I >e stored in bulk memories to eliminate the need for much of the fixed wiring such as appears in relay call processing circuits. 1004 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Fig. 13 — AMA translator. ELECTRONICS IN TELEPHONE SWITCHING SYSTEMS 1005 Fig. 14 — No. 5 route relay frame. The form of memory available in electronics is considerably different from that which has been previously available. Electronic memory has been characterized as "common medium" or "bulk" memory. A single device is used capable of storing more than a single bit of information which is the limit of most relays or other devices capable of operating in a bistable manner. A number of different types of electronic bulk memories have been devised for digital processing. They differ appre- 1006 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Fig. 15 — No. 4A card translator. cig.bly in physical form, each taking advantage of the phenomenon of some different area of the physical sciences — electrostatic, electromag- netic, optic. Magnetic tapes" and drums^^ (Fig. 17), cores^^ (Fig. 18), electrostatic storage in tubes^^' ^"^ (Fig. 19) and ferroelectrics^^' 2" (Fig. 20) and photographic storage^' (Fig. 21) are available. Several properites of these memory devices are of interest. Being electronic, the speed with which stored information may be read is of primary interest. This is known as "access speed". Another property of these common medium memory systems or devices is the ability to change what has been written. If the changes can be made rapidly enough they may be used in electronic systems in much the same man- ner as relays are used in electromechanical systems to process informa- tion. If the change must be made relatively infrequently, such as changing photographic plates, they may be used as substitutes for the type of memory in these systems which are provided by cross connec- tions and wiring. The required fixed or semipermanent electronic mem- ory may be characterized primarily b}^ a high reading speed, large capacity, and the ability to hold stored information even during pro- ELECTRONICS IN TELEPHONE SAVITCHING SYSTEMS 1007 iiFi J + > A - ,-, n 1- < > •> J > > -> 1 1* D — -^ J -H J i — i WITH DIODES WITH TRANSISTORS Fig. 26 — A logic function with diodes or transistors. 1016 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195G by judiciously introducing redundancy into the equipment, the chance of simultaneous failures of any two identical parts should be extremely improbable.^- With automatic trouble locating, the maintenance forces will not be reciuired to have a thorough understanding of the device characteristics and the circuitry used. Centralized repair of defective units as in modern telephone transmission systems^^ and perhaps even expendability of defective units are a distinct possibility. As a result of some of these maintenance considerations it is quite likely that equipment in the future, besides being smaller and more com- pact, will appear more generally in enclosed low cabinets rather than exposed frames. The administrative control may be from consoles rather than vertical panels. More attention will be paid to appearance. The appurtenances, such as ladders required for high frames in electro- mechanical systems, may be eliminated. Another change in concept which may come with electronics in tele- phone switching is the form of the power supply. Present day telephone systems use a centralized single voltage dc distribution system with reserve battery. The wide variety of devices and associated voltages, and the need for close regulation in some portions of electronic systems make a reliable ac distribution system with individual power rectifiers at the point of use appear quite attractive. To insure reliability of service the ac distribution must be continuous and not dependent directly upon the commercial sources. There is no question that reliability is imperati\'e if electronic switch- ing systems are to survive among electromechanical systems which have achieved a high degree of reliability over a long period of years. The device reliability of the first electronic system may not be com- parable since some of the components of the electronic switching sys- tems will not in their initial applications be as reliable as the least reli- able component in our present day systems. Reliability will be earned and this will probably require considerable effort. Even if initially some devices employed in electronic systems do not measure up to the present high standard which has been set, continuity of high (luality service is a must. It is, therefore, necessary to design a system which will mask the shortcomings of any individual electronic component. ^^ As their reliability is proven an optimum balance will be sought between system redundancy and component quality. Telephone engineers familiar only with the high degree of reliability of present day apparatus will have to accommodate themselves to the characteristics of new electronic de- vices. ELECTRONICS IN TELEPHENE SWITCHING SYSTEMS 1017 REFERENCES 1. J. R. Eckert, A. Survey of Digital Computer Memory Systems, I.R.E. Pro- ceedings, 41, pp. 1393-1406, Oct., 1953. 2. T. H. Flowers, Electronic Telephone Exchanges, Proceedings I.E.E., 99, Part I, pp. 181-201, 1952. 3. U. S. Patent 2,387,018. 4. U. S. Patent 2,490,833. 5. U. S. Patent 2,408,462. 6. U. S. Patent 2,379,221. 7. W. A. Depp, M. A. Townsend, Cold Cathode Tubes for Audio Frequency Signaling, B.S.T.J., 32, pp. 1371-1391, Nov., 1953. 8. Tone Ringer May Replace Telephone Bell, Bell Laboratories Record, pp. 116-117, March, 1956. 9. W. R. Bennett, Time Division Multiplex Systems. B. S.T.J. , 20, p. 199, 1941. 10. F. A. Korn, J. G. Ferguson, No. 5 Crossbar Dial Telephone Switching System, Elec. Eng., 69, pp. 679-684, Aug., 1950. 11. J. W. Dehn, R. E. Hersev, Recent New Features of the No. 5 Crossbar Switch- ing System, A.I.E.E. Paper No. 55-580. 12. T. L. Dimond, No. 5 Crossbar AMA Translator, Bell Laboratories Record, p. 62, Feb., 1951. 13. L. N. Hampton, J. B. Newsom, The Card Translator for Nationwide Dialing, B.S.T.J., 32, pp. 1037-1098, Sept., 1953. 14. Review of Input and Output Equipment LTsed in Computing Systems. A.LE.E. Special Publication S53. 15. Cohen, A. A., Magnetic Drum for Digital Information Processing Systems, Mathematical Aids to Computation. 4, pp. 31-39, Jan., 1950. 16. M. E. Hines, M. Chruney, J. A. McCarthy, Digital Memory in Barrier Grid Storage Tubes, B.S.T.J., 43, p. 1241, Nov., 1955. 17. M. Knoll, B. Kazan, Storage Tubes and Their Basic Principles, John Wilev & Sons, 1952. IS. M. K. Haj-nes, Multidimensional Magnetic Memory Selection System. Trans- actions of the I.R.E. , Professional Group on Electronic Computers, pp. 25-29, Dec, 1952. 19. D. A. Buck, Ferroelectrics for Digital Information Storage and Switching, Report R212, M.I.T. Digital Computer Laboratories, June, 1952. 20. J. R. Anderson, Ferroelectric Materials as Storage Elements for Digital Com- puters and Switching Svstems, Communications and Electronics, pp. 395- 401, Jan., 1953. 21. G. W. King, G. W. Bi-own, L. N. Ridenour, Photographic Techniques for In- formation Storage, Proc. I.R.E., pp. 1421-1428, Oct., 1953. 22. Staff of Harvard Computation Laboratorj-, Synthesis of Electronic Computing and Control Circuits, Vol. 37 of Annals of Harvard Computation Labora- tory, 1951. 23. B. J. Yokelson, W. Ulrich, Engineering ^Multistage Diode Logic Circuits, Communications and Electronics , pp. 466-474, Sept., 1955. 24. M. Karnaugh, Pulse Switching Circuits Using Magnetic Cores, Proc. I.R.E., 43, pp. 576-584, May, 1955. 25. R. H. Beter, W. E. Bradley, R. B. Brown, M. Rubinoff, Surface Barrier Tran- sistor Switching Circuits, I.R.E. Convention Record, Part 4, pp. 139-145, 1955. 26. R. L. Trent, Two Transistor Binary Counter, Electronics, 25, pp. 100-101, July, 1952. 27. A. E. Anderson, Transistors in Switching Circuits, Proc. I.R.E., 40, pp. 1541- 1558, Nov., 1952. 28. J. H. Felker, Regenerative Amplifier for Digital Computer Applications, Proc. I.R.E., 40, pp. 1584-1956, Nov., 1592. 29. J. H. Felker, Typical Block Diagrams for a Transistor Digital Computer, Communications and Electronics, pp. 175-182, July, 1952. 1018 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMEER 1956 30. Promising Electronic Components — Diode Amplifiers, Radio Electronics p. 45, Nov., 1954. ' 31. A. A. Lawson, Mass Production of Electronic Subassemblies, Electrical Manu- i facturing, 54, p. 134, Oct., 1954. j 32. C. J. Crevelens, Increasing Reliability by the Use of Redundant Circuits Proc. I.R.E., pp. 509-515, April, 1956. 33. A. L. Bonner, Servicing Center for Short-Haul Carrier System, Communica- tions and Electronics, pp. 388-396, Sept., 1954. ^■^^ ^\^}",-- I^aggett, E. S. Rich, Diagnostic Programs and Marginal Checking in Whirlwind I Computer, I.R.E. Convention Record, Part 7, pp. 48-54 1953 35. MAID Service for Computer Circuits, Automatic Control, p. 23, Aug 1955 % Combined Measurements of Field Effect, Surface Photo-Voltage and Photoconductivity By W. H. BRATTAIN and C. G. B. GARRETT (Manuscript received May 10, 1956) Combined measurements have been made of surface recombination veloc- ity, surface photo-voltage, and the modulation of surface conductance and surface recombination velocity by an external field, on etched germanium surfaces. Two samples, cut from an n-type and a p-type crystal of known body properties, were used, the samples being exposed to the Brattain- Bardeen cycle of gaseous ambients. The results are interpreted in terms of the properties of the surface space-charge region and of the fast surface states. It is found that the surface barrier height, measured with respect to the Fermi level, varies from —0.13 to -\-0.13 volts, and that the surface recombination velocity varies over about a factor of ten in this range. From the measurements, values are found for the dependence of charge trapped in fast surface states on barrier height and on the steady-state carrier con- centration within the semiconductor. I. INTRODUCTION This and the succeeding paper are concerned with studies of the properties of fast surface states on etched germanium surfaces. The ex- periments involve simultaneous measurement of a number of different physical surface properties. The theory, which will be presented in the second paper, interprets the results in terms of a distribution of fast surface states in the energy gap. The distribution function, and the cross-sections for transitions from the states into the conduction and valence bands, may then be deduced from the experimental results. Early experiments^ on contact potential of germanium, and on the change of contact potential with light, indicated that there are two kinds of surface charge associated with a germanium surface, over and above the holes and electrons that are distributed through the surface space-charge region. One kind of surface charge, usually called "charge 1019 1020 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 in fast traps" can follow a change in the space-charge region very fast in comparison with the light-chopping time used in that work (Koo sec); the other kind, imagined to be more closely connected with ad- sorbed chemical material, can only change rather slowly. In a previous paper by the authors it was pointed out that the Brattain-Bardeen experiments, taken by themselves, do not furnish unambiguous infor- mation concerning the distribution of these "fast" traps, but that such information might be obtained by performing, simultaneously, other measurements on the germanium surface. More recently Brown and Montgomery^' ^ have provided a valuable tool in their studies of large- signal field effect; they point out that if, under given chemical conditions, it is possible to apply a field, normal to the surface, large enough to force the surface potential to the minimum in surface conductivity; then it becomes possible to determine the initial surface potential ab- solutely (provided certain considerations as to the mobility of the carriers near the surface are valid). This paper concerns studies of a number of physical properties that depend on the distribution and other characteristics of the surface traps or "fast" states. Measurements are reported of (i) the change of conductivity of a sample with field; (ii) the photoconductivity; (iii) the change of photoconductivity with field; (iv) the filament life- time; and (v) the surface photo-voltage. Measurements were made in a series of gaseous ambients, first described by Brattain and Bardeen. Evidence is presented to the effect that the variation in gas ambient changes only the "slow" states, leaving the distribution and other properties of the traps substantially unaffected. From measurements (i) to (iii) it is possible to construct the whole field-effect curve (con- ductance versus surface charge), even though the fields used were in general not large enough to reach the minimum in conductance. Using the field effect data, values for the surface potential Y in units of kT/e could be obtained at each point, and also of the quantity (d'Ls/dY)s=o , where 2s is the charge in surface traps, and the suffix 5 = 0 implies zero illumination. From measurements (ii) and (iv), the sin'face recoml)ination velocity s could be deduced. (A more detailed study of photoconductivity in relation to surface recombination \'elocity will be reported at a later date.) Combined with the field effect data, this enables one to deduce the relation between s and Y. Measurements of the surface photo-voltage may be presented in terms of the quantity dY/d8, where 5 is ec|ual to Ap/ui , Ap being the density of added carrier-pairs in the body of the material, and Ui the intrinsic carrier density. The quantity dY/d8 is closely related to the ratio of the COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1021 change in surface potential produced by illumination of the surface to the change in the quasi-Fermi level for minority carriers. By measuring dY/db rather than dY/dL, discussed in Reference 2, the surface re- combination velocity is eliminated from the surface photo-voltage data: the limiting values of dY/dd, after correction for the Dember effect, ought to be (po/ni) and — (ni/po) , no matter what the surface recombina- tion velocity may be. By combining this information with the field-effect data, one can de- duce the quantity (5Ss/55)r • This and the previous differential, deduced directly from the field-effect data, completely define the dependence of charge in surface traps on the two independent parameters Y and 8 — that is, the dependence on chemical environment and on the bulk non-equilibrium carrier level. The further interpretation of the cjuantities (dI,s/dY)s=o , (52s/55)r and s in terms of the distribution of surface traps is postponed to the succeeding paper. Here it is sufficient to say that the results are con- sistent with the assumption that the traps responsible for surface re- combination are also those pertinent to the field effect and surface photovoltage experiments. Then the ciuantity (d2s/^F)a=o depends only on an integral over the distribution in energy of traps; (31,^/88) y depends also on the ratios of cross-sections for transitions to the valence and conduction bands; and s depends in addition on the geometric mean cross-sections. II. OUTLINE OF THE EXPERIMENT The experiment is carried out with a slice of germanium, 0.025 cm thick, which is supported in such a way that there is a gap 0.025 cm wide between the slice and a metal plate. Substantially ohmic contacts are attached to the ends of the slice. Three kinds of experiment are now carried out: (i) The conductance of the slice is modulated by illuminating it with a short flash of light; the subsequent decaj^ of photoconductivity with time is studied, and the time-constant of the exponential tail measured. (ii) A sinusoidally varying potential difference of about 500 volts peak-to-peak is applied between the metal plate and the germanium. Facilities are available for measuring the changes in conductance pro- duced by the field. The sample is also illuminated with light chopped at a frequency different from that of the applied field. One measures: (a) the magnitude of the peak-to-peak conductance change in the dark; (b) the same in the presence of the light; and (c) the change in con- 1022 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 ductance, at zero field, produced by the light. The applied field is suffi- ciently small for the dark field effect and the apparent field effect in the presence of light to be substantially linear. (iii) The metal plate, disconnected from the high voltage supply, is connected to a high-impedance detector; chopped light is shone on the germanium, and the change in contact potential produced at the sur- face opposite the metal plate by illumination of the sample measured, and compared with the photoconductivity. The interpretation of the field effect data has been given by Brown and Montgomery^' ^ and by the authors." The surface conductance AG is equal to enp{Tp + 6r„),2 where Tp and r„ are surface excesses of holes and electrons, and are, in equilibrium (i.e., in the absence of light) functions of the surface potential Y and of the body type and resistivity. The minimum in the surface conductance curve occurs at a particular value of Y, so that, if a field effect experiment allows passage through this minimum, values of Y may be obtained.* In our experiments, measurements were made in a series of different chemical environments, and the minimum in surface conductance did not, in general, occur within the range of field employed. However, it was found to be possible to piece together the complete surface con- ductance curve (AG versus surface charge) by making use of simulta- neous measurements of the photoconductance and the change in photo- conductance with field. (See Section VI.) From the surface conductance curve, one may deduce the fraction of the surface charge (whether in- duced electrically, by application of a field, or chemically, by changing the environment) which goes into the fast surface states or traps. ^' ^ There is, indeed, an assumption here, to the effect that the distribution of traps is unaffected by a change in the chemical environment. The justification for this is the observation of Brown and Montgomery* that it was possible to superpose overlapping large signal field effect curves obtained in different environments. There is also evidence for the validity of this assumption from the self-consistency of the procedure used (see Section V and Fig. 4). The photoconductivity measurements have been interpreted on the following basis. Illumination of the sample will do two things: it will change the surface excesses Tp and r„ ,^ and it \\ill also change the * The question of the mobility of carriers near the surface should be mentioned liere. For extreme values of Y, the mobility of the carriers tliat are constrained to move in the narrow surface well is reduced. Values for this reduction in mobility have been calculated by Schrieffer.^ However, for values of }' near zero the Schrief- fer correction is small, and at somewhat larger (positive or negative) values AG is increasing so fast that the error in Y introduced by ignoring the Schrieffer cor- rection is small. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1023 steady-state carrier density deep inside the sample. If the sample is thin in comparison with the body diffusion length and with (D/s), as was the case in our experiments, the added carrier density Ap will be. almost uniform throughout the thickness t of the sample, and one can easily convince oneself that the photoconductance arising from this cause is of the order of it/£) times larger than that arising from the changes in the surface excesses, where £ is a Debye length for the I material. This being the case, the photoconductivity may be considered I to be a bulk rather than a surface effect, the surface entering only i through the surface recombination velocity s. Under the conditions of the present work the magnitude of the photoconductivity w^as in fact inversely proportional to s, as was verified in a separate set of experi- I ments. Surface recombination is of interest in that this also calls for I "fast" trapping centers on the surface; in fact any trap contributing \ to the field effect experiment may be a recombination centre, if the i cross-sections are right. The questions as to whether the recombination I centres and the "fast states" affecting the field effect are the same, or I not, is taken up in the succeeding paper. I The surface photo-voltage, like the field effect, is affected both by I changes in the surface excesses and by changes in Ss , the charge in sur- I face traps. In the experiments, the change in contact potential in a cer- I tain light (usually chosen so that the change i s small in comparison [ with kT/e) is compared with the change in conductance produced by the same light. From the latter one may calculate 8 (defined as Ap/ui) directly. The change in contact potential, measured in units of kT/e, is I taken to be equal to AY. Thus the surface photo-voltage experiment I measures the quantity (dY/d8), the differential being taken at constant surface charge. By a slight generalization of the argument previously given by the authors," one can show that: dy ^ _ (d/d8)Y{Tp - rj + id2jd8)Y (.. (18 (d/dVUTp - r„) + (dXs/dVh ^ ^ Now the first terms in the numerator and denominator on the right- I hand side are determinate functions of Y, and so are known ; the quantity I {d'2s/dY)B may be deduced from the field-effect measurements, so that I the only remaining ciuantity, (53,, /(95) r , niay be deduced from the I measurements of surface photo-voltage. In concluding this section, a word as to the meaning to be attached to (dT^s/dS) Y is in order. The sign of this quantity depends, roughly speak- ing, on whether the traps in question (i.e., those near the Fermi level under the conditions of the experiment) are in better contact with the 1024 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 conduction or the valence band. This in turn depends both on the surface potential and on the ratio of cross-sections for transitions to the two .bands. For F « — 1, one expects (52^/(95) y/(62s/ay)5 to have the value — X~'; for F » +1, the vakie +X. These limiting values may be deduced by a somewhat general argument. At some intermediate value of surface potential, the above ratio must change sign. If the distribution of surface states in energy is known from the field effect measurements, then the value of F at which the above ratio changes sign determines the ratio of cross-sections for those traps which are close to the Fermi level for that value of F. By repeating the experiment for samples of differing bulk resistivity, it is then possible to determine whether the same ratio holds for the states at some dif- ferent position in the energy gap. III. EXPERIMENTAL DETAILS Fig. 1 shows the experimental arrangements. The sample of ger- manium, of dimensions shown, was prepared by cutting, sandblasting, etching in CP4 and washing in distilled water. The exposed faces were approximately (100). The end contacts were made by sandblasting and soldering. The slots A, A' in the ceramic were incorporated in order to GERMANIUM GOLD GOLD --GERMANIUM BINDING POSTS- Fig. 1 — Experimental arrangement. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1025 reduce the high field that would otherwise be present near the edges of the ceramic. The gold electrode was deposited by evaporation through a mask. Connections from the gold and from the ends of the sample were made to binding posts passing through the ceramic block. The ceramic block was set into a metal box, divided into two com- partments. In the upper compartment, which contained the sample, there were inlet and outlet tu}:)es, to allow the gas to be changed. The lower compartment contained electrical components, which were thereby protected to a large extent from the changes in gas in the upper compartment. Facilities were available for the type of cycle of gas en- vironment described by Brattain and Bardeen,^ which cycle was found by them to produce reversible cyclic changes in surface potential. In the top of the box was a window, through which light could be shone onto the germanium either from a chopped or a flash source. The electrical circuit is shown in Fig. 2. The condenser Ci is that formed between the germanium and the gold, and has a capacity of about 2 ijlijlF. Impedances Zi and Zo form a Wagner ground, which has to be balanced first. Then, by adjusting resistance Ri and condenser C2 , one may obtain a balance in the case that there is no dc flowing through the sample. A current (determined by the battery B and the HORIZONTAL VERTICAL ELECTROMETER - TUBE PRE-AMPLIFIER :!^ B Fig. 2 — Electrical circuit. 1026 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 resistance Rz)is now switched in, and the resulting off-balance (repre- senting the field modulation of conductivity) presented on the vertical plates of an oscilloscope. The supply voltage is connected, via the high.j bleeder resistance Ri , to the horizontal plates. The frequency of the oscillator was chosen to be 25 cyc/sec, a value sufficiently low to obviate lifetime difficulties; the peak-to-peak swing was generally 500 volts. During a field-effect measurement, the sample was also illuminated with light chopped at 90 cyc/sec. This had the result of causing to be presented on the oscilloscope screen a pattern such as that shown in Fig. 3. The lower tilted line represents the (dark) field effect curve; the vertical separation represents the photoconductivity, as modified by the applied field. Measurements were made of the mean vertical separation, and of the slopes of the upper and lower lines (by reading gain settings). During a surface photo-voltage measurement, the gold electrode was disconnected from the high-voltage supply, and connected to a high- impedance detector, similar to that used in the work of Brattain and Bardeen.^ A value for the chopped light intensity was chosen to give a contact potential change that was generally not more than 5 mV. A simultaneous measurement of the photoconductivity was also made. The gas cycle was similar to that described by Brattain and Bar- deen.^ Some variations were made in it to try to spread out the rate of change with time so that the data could be obtained without large gaps. The cycle used was: (i) sparked oxygen 1 min, (ii) dry O2 , (iii) mixture of dry and wet O2 , (iv) wet O2 , (v) wet No , (vi) a mixture of dry and wet N2 , (vii) dry O2 , (viii) dry O2 , triple flow, and (ix) ozone normal flow. The normal rate of gas flow was about 2 liters per minute; the wet gas was obtained by bubbling through water (probably about 90 per Fig. 3 — Picture of field effect-photoconductivity pattern, as observed on oscilloscope. Dark curve at the bottom. , COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1027 cent r.h.) and the mixture of dry and wet was obtained by letting ap- proximately one-half the gas flow bubble through H2O. In carrying out the experiment, it was found convenient to carry out alternately a com- plete cycle of field effect and surface photo-voltage measurements. The values of the photoconductivity at equivalent points in successive cycles could be compared, in order to check that no systematic error was intro- duced by this procedure. In addition to the foregoing, the folloAving measurements Avere made: 1. All dimensions were determined. 2. The resistivity of the sample was found, and also the body life- time, on another specimen cut from the same crystal. 3. The amplitude of the voltage swing was measured. 4. The amplifiers in the field effect circuit were calibrated. 5. The capacity of the germanium-gold condenser was determined (by a substitutional method). The value obtained was larger than that calculated from the parallel-plate formula, because of the edge effects. 6. A standard square-wave voltage was introduced into the surface photo-voltage circuit, in order to calibrate the high-impedance detector. 7. At several points in the cycle, the fundamental mode lifetime of the sample was determined by the photoconductivity decay method. This calibrated the 90 cyc/sec photoconductivity measurements, without the necessity for a knowledge of the light intensity. IV. RESULTS Measurements were made on two samples: one n-type, 22.6 ohm cm (X = 0.345), the other p-type, 8.1 ohm cm (X = 17.7). The body life- time for both samples was greater than 10~ sec, so that for slices of the thickness used (0.025 cm. or less), and for values of s in the range en- countered, body recombination may be ignored. Results of typical field-effect runs for the two samples are indicated in Tables I and II. The first column in each table gives the time in minutes from the beginning of the cycle at which the measurements were made. The second column shows the "effective mobility," dAG/d'2, obtained from the observed (dark) field effect signal voltage AFi (see Fig. 3) by use of the formula: jueff = wh^AVi/Ipo^CVapp , where w is the width of the slice, t the thickness, / the dc flowing through it, po the re- sistivity, C the capacity of the germanium-gold condenser, and Fapp the voltage applied across it. The third column shows the mean value of 8{ = Ap/ni), obtained from the mean photoconductivity signal voltage 1028 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195G Table I — 22,6 ohm cm n-TYPE Cycle 12. Relative Light Intensity 0.082 Time min. cm^ s ^ cm2 cm '^" volt sec ""^ volt sec sec 0 Sparked O2 1 Changed to dry C >o 1.5 334 7.85 X 10-2 520 90 2.5 344 6.9 X 10-2 585 103 5.5 344 6.2 X 10-2 595 114 6.5 344 6.07 X 10-2 595 117 7.0 Changed to mixture of dry & wet O2 7.5 136 4.4 X 10-2 520 161 8.5 84 4.02 X 10-2 440 177 9.0 52 3.60 X 10-2 270 196 10.0 Changed to full wet O2 11.0 -660 2.26 X 10-2 -440 314 11.5 -890 1.74 X 10 2 -780 408 12.0 -960 1.67 X 10-2 -910 425 13.0 Changed to full wet N2 18.0 Changed to mixture of dry & wet No 19.5 -1150 1.67 X 10-2 -1060 425 20.5 -1050 1.74 X 10-2 -960 408 22.5 -990 1.83 X 10-2 -890 390 23.0 Changed to dry C )., 23.5 -430 2.98 X 10-2 -220 238 23.8 -290 . 3.2 X 10-2 0 222 24.0 -84 3.81 X 10-2 240 186 24.5 31 4.3 X 10-2 310 165 26.5 146 4.7 X 10-2 410 151 27.0 Tripled flow of dry O2 27.5 220 5.1 X 10-2 450 139 29.5 260 5.7 X 10-2 510 124 31.5 280 6.2 X 10-2 510 114 35.0 Changed to ozone 35.5 310 6.8 X 10-2 510 104 37.5 320 8.2 X 10-2 490 87 AF2 by use of the formula: 8 = wtpiAVn/Kmpo', where p, is the intrinsic resistivity and fm the length of the illuminated part of the slice. The fourth column shows the apparent effective mobility in the presence of light,* obtained from the field effect signal voltage AF3 in the presence of light, using the same formula as that giving iJen . The last column shows the surface recombination velocity, which is proportional to 8" for fixed light intensity, the constant of proportionality being deter- mined by comparison with measurements of the fundamental mode lifetime. The results of typical surface photo-voltago nms are sho\\n in Tables * One must be careful to avoid thinking of Mtif* as a true field clfect mobilit.y, since it is really a sum of two cjuite different components: the true held effect mobility Meff , and a term, proportional to thickness of the slice, arising from the jjhotoconductivity. ?! COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1029 Table II • — 8.1 ohm cm ^-type Cycle 5. Relative Light Intensity 0.25 Time min. cm2 s , cm2 cm Me f f 1 ^ Meff ■ fi volt sec volt sec sec 0 Started sparked O2 1.0 Changed to dry O2 1.5 307 4.1 X 10-2 490 503 3.5 318 3.2 X 10-2 490 660 6.0 Changed to mixture of wet & dry O2 7.5 273 1.4 X 10-2 376 1480 9.5 239 1.3 X 10-= 320 1580 11.0 Changed to wet C )2 11.5 94 1.2 X 10-2 -194 1690 12.5 -200 1.1 X 10-2 -230 1820 15.5 -216 1.2 X 10-2 -285 1690 17.0 Changed to wet N2 IS. 5 -352 1 1.6 X 10-2 -570 1310 22.0 Changed to mixture of wet & dry O2 25.0 -80 1.1 X 10-2 -137 1820 26.5 0 1.2 X 10-2 31 1690 27.5 3.3 1.3 X 10-2 58 1630 28. 0 Changed to dry C )■! 29.0 193 1.9 X 10-2 330 1070 29.5 239 2.2 X 10-2 400 1000 30.5 250 2.4 X 10 2 420 873 33.0 Tripled flow of dry O2 33.5 296 3.2 X 10 2 500 645 fc 34.5 296 3.7 X 10-2 525 560 ■ -^6.5 296 4.2 X 10-2 570 490 ■ 37.5 296 4.6 X 10-2 570 455 ■^ 38.0 Changed to ozone 42.5 330 6.4 X 10 2 535 323 III and 1\'. Values of 8 were obtained from the photoconductivity signal, as before, taking the actual ilhiminated length as the length of the sample. In making use of the standard square-wave calibration for the surface photo-voltage measurement (Section III), it is necessary to allow for the fact that the measured capacity involves the whole length of the sample, plus end and side fringing effects, whereas the surface photo-voltage measurements im'ohcs only the illuminated length, plus the fringe effect at the sides. The penultimate column in Tables III and 1\ shows the ratio of the change in contact potential, measured in units of (kT/e), to the added- carrier parameter 5, which was deduced from the photoconductivity. This is not j^et, however, the true surface photo- voltage function {dY/d8), since the observed change in contact potential includes also the Dember potential AF/^"^ which occurs between the illuminated and non-illuminated parts of the body of the semiconductor. The last column in Tables III and lY shows the true values of (dY/d8), obtained by sub- 1080 THE HELL SYSTEM TECHNICAL JOURNAL, SErTEMBER ]mC) Table III — 22.6 ohm cm ti-type Cycle 7 Time mins. Relative Light Intensity « ACP volts /3ACP 6 dV ds Starting condition wet N2 6.5 2.25 0.36 6.5 X 10-3 0.7 -0.10 11.5 2.25 0.34 1.0 0.115 -0.045 12.0 Changed to mixture wet and dry N2 12.5 2.25 0.32 2.2 0.27 0.10 ! 13.0 2.25 0.34 3.5 0.40 0.23 13.5 2.25 0.35 4.6 0.51 0.34 14.5 2.25 0.38 6.8 0.70 0.53 15.5 0.56 0.10 3.1 1.2 1.03 17.5 0.56 0.11 3.7 1.33 1.16 18.0 Changed to dry O2 18.5 0.56 0.16 5.7 1.4 1.2 : 19.5 0.14 0.06 3.4 2.2 2.0 22.0 Changed to dry O2 triple flow 24.5 0.14 0.082 6.1 2.9 2.7 Table IV — -8.1 ohm cm p-type Cy'cle 8 Time mins. Relative Light Intensity ACP volts /3ACP S dV dS~ Starting condition wet N2 0 Changed to mixture wet and dry No 0.5 0.14 0.011 -3.5 X 10-3 -12.5 -12.6 1.0 0.14 0.0088 -2.1 -9.6 -9.7 5.0 Changed to mixture wet and dry O2 5.5 0.56 0.0275 -1.8 -2.6 -2.7 6.0 0.56 0.03 -1.45 -1.9 -2.0 7.5 0.56 0.0325 -1.16 -1.4 -1.5 9.5 0.56 0.035 -1.08 -1.2 -1.3 10.0 Changed to dry O2 10.5 0.56 0.044 -0.71 -0.63 -0.69 11.5 0.56 0.055 -0.49 -0.35 -0.41 14.0 Changed to dry O2 tripl 3 flow 14.5 0.56 0.0625 -0.32 -0.20 -0.26 16.5 2.25 0.28 -0.72 -0.10 -0.16 20.5 2.25 0.33 -0.42 -0.05 -0.11 30.0 Changed to ozone 31.5 2.25 0.47 +0.47 +0.039 -0.023 32.5 2.25 0.53 + 1.4 +0.103 +0.041 tractiiig from (8A c.p./5) a Dember potential correction, given by {b — 1)/(X + b\'^). (The boundaries of the illuminated region were sufficiently distant from the contacts for this formula to apply.) Tables III and IV include only data from the second half of the cycle (wet N2 -^ ozone), since the rate of change of A c.p. during that part of the first half in which dry oxygen was replaced by wet oxygen was too fast to follow. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1031 The reproducibility of all the data from cycle to cycle was good. One surprising result is that the surface recombination velocity assumed its maximum value close to the "wet nitrogen" extreme for both p-type and n-type. This behavior is quite different from that reported by Brattain and Bardeen/ who found s to be constant within 20 per cent throughout the range and Stephenson and Keyes,* who found a maxi- mum value sometimes at one end, sometimes at the other, and some- times in the middle. There is quite good agreement on the other hand, with the results of Many et al.^^"\ who report a maximum in s near the wet end of the cycle. The result of Brattain and Bardeen is not under- stood at the present time, and is probably wrong. The differences between the present work and that of Stephenson and Keyes may be associated with differences in surface preparation. V. ANALYSIS OF THE RESULTS From now onwards we shall express all experimental and calculated c[uantities in terms of the following dimensionless ratios: Xs = :2s/eni£, S = Ss + Tp - r„ (2) AG — AG/enilJLp£, /leff = J"eff/Mp , /"eff* = MeffV^P where AG is the surface conductance, £ the Debye length for intrinsic germanium (1.4 X 10~ cm), and /Xp is the mobility for holes (1800 cm v" sec~^). Tables V and VI show values of the quantities we shall need, as functions of the surface potential F, calculated from the theoretical considerations of Garrett and Brattain." The surface conductance, and the differentials in the fifth and sixth columns, are evaluated for 8 = 0. Table V — 22.6 ohm cm n-TYPE Y F- InX Vp - r„ AG /a(Tp-f„)\ V dY Js -4.1 /diTp - r„)\ V ds )y 3 4.1 -10.3 17.5 -1.3 2 3.1 -7.0 10.6 -2.6 -0.8 1 2.1 -4.9 6.2 -1.8 -0.4 0 1.1 -3.4 3.3 -1.3 0.0 -1 0.1 -2.3 1.45 -1.1 0.5 -2 -0.9 -1.2 0.36 -1.1 1.3 -3 -1.9 0.0 0.0 -1.4 2.7 -4 -2.9 1.7 0.65 -2.1 5.2 -5 -3.9 4.4 2.65 -3.5 9.4 -6 -4.9 8.9 6.8 -5.8 16.3 -7 -5.9 16.4 14.4 -9.5 27.7 1032 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Table VI — 8.1 ohm cm p-type Y F - InX ip - r„ AG /afrp - r„) \ {d{Tp - Vn)\ 8 5.1 -8 9.8 -5.5 -87 7 4.1 -4 3.4 -3.1 -42 6 3.1 -2 0.8 -1.9 -19 5 2.1 0 0.0 -1.45 -8.2 4 1.1 1 0.25 -1.3 -3.4 3 0.1 2 1.3 -1.45 -1.5 2 -0.9 4 2.8 -1.75 -0.62 1 -1.9 6 4.8 -2.4 -0.21 0 -2.9 9 7.4 -3.4 0.0 -1 -3.9 12 10.9 -4.3 0.15 -2 -4.9 18 16.4 -6.4 0.31 -3 -5.9 26 25.0 -10.0 0.53 The first problem is the constructing, from the experimental results, of the curve relating AG and S. The experiments provide a series of pic tures like Fig. 3, each one corresponding to a different chemical environ- ment, and so to a different Y. At each of two succeeding pictures of this sort one knows (i) the vertical displacement (photoconductivity) be- tween the dark and light field effect curves; and (ii) the mean difference in the dark and light slopes, and hence the rate of change of photocon- ductivity with applied field, and therefore with S. The problem is to de- duce the horizontal displacement (in 2) between the two pictures. A corrrection must first be made for the fact that the ambient changes 2 uniformly on both surfaces, whereas the applied field induces charge only on the lower surface, plus fringing effects.* The correction is applied by taking the difference in slopes (lUei* — /Jen), and multiplying this by (2/1.27), where the number 1.27 is deduced for the given geometry from the standard edge-effect formula. This having been done, it is now pos- sible to take the revised pictures and piece them together to form tA\() smooth curves (Fig. 4). The process of assembling such a diagram de- termines the horizontal and vertical distances, and therefore the change of 2 and AG, between successive experiments. This argument may be given analytically as follows. First notice that the photoconductivity ^•oltage in the absence of field (AT% in Fig. 3) is proportional to (1/s). The application of a voltage between the gold and the germanium induces some charge density 2 at each point on the germanium surface, 2 being (due to fringing effects) a complicated function of position. At each point (1/s) is changed by an amount 2[(i(l/s)/(/2]. This causes the photocondu(!tivity in the presence of field We are indebted to W. L. Brown for l^ringing this to our attention. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1033 24 — 22- 20- 18 - 16 - 14 - G 12 10 8 6 4 2 \ \ \ / J L I I I I iLUdi • • • J L J L -50 -40 -30 -20 ■10 0 St 10 20 30 40 50 Fig. 4 — Construction of the curve relating AG (surface conductivity, in units of ejupMjcC) and s (surface charge, in units of en,£). to differ from that in zero field, and gives rise to the voltage difference (AFs — AFi) shown in Fig. 3. Expressing this difference in terms of the difference (jUeff* — Meff) between the apparent and true effective mobilities in the presence of light (see Section IV), one finds: ^.f/(l/s) /CuniA iMeff* ~ Meff) — di: 2w (3) where K is the constant of proportionality between (1/s) and the photo- I conductivity signal AVo , and C'unit is the capacity per unit of the ger- manium-gold condenser in the illuminated region, which is 1.27 times [ the parallel-plate formula. From a series of measurements of (^eff* — fJeu) I and AFo it is now possible to obtain S by graphical integration: S = 2/2 ^ unit dAV-2 eni£>/ \/po'C/ \ 2w / J ijl^.u* — Meff (4) This and the giuplucul method are of course e(iuivalent. it is worth- while emphasizing again that either technitiue depends for its validity I on the fact that the distribution of fast states is unaffected by the gas I changes in the Brattain-Bardeen cycle, as shown in the experiments of jl Brown and Montgomery.^ If, however, the assumption were too far from the truth, the fitting of both slopes in Fig. 4 would be impossible. The only 1034 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 18 16 14 12 10 p3 1 \ 1-2 \ \ -1 y I / V,0 y / \-2_' ./ -50 ■50 -25 0 St -25 25 50 75 100 125 28 26 24 22 20 18 16 14 12 10 150 Fig. 5 — Curves showing AG (surface conductivity, in units of eij.pni£) and S surface charge, in units of en,£) for the 22.6 ohm-cm sample (upper curve) and for the 8.1 ohm-cm sample (lower curve). Values of Y, deduced from the surface conductivity, are indicated on the curves. place at which fitting was at all difficult was at the extreme wet end. For most of the range, therefore, the method is at least internally consistent. Fig. 5 shows the result of carrying out this procedure for the n and p- type samples. The data were averaged over a number of runs. The num- bers appearing on the curves represent values of Y, obtained by reference to Tables V and VI. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1035 From Fig. 5 one may now calculate* the changes occurring in Xs , the (reduced) charge in fast states, since Fp — r„ may be read from Tables V and VI, and Ss - S - (f^ - f,). Fig. 6 shows (d2s/dY)i as a function of 1" — In X, calculated from the experimental results in this way. [The reason for plotting against Y — In X instead of Y is that this quantity represents the difference, in imits of (kT/e), between the electrostatic potential at the surface and the Fermi level. In this way the effects of difference from sample to sample in the position of the Fermi level in the interior are eliminated.] Notice that the measurements of {dT^s/dY)s for the two samples have the same general shape, and that the turning points of the two curves occur at about the same value of dY -30 Fig. 6 — Differential charge in fast states versus surface potential. The graphs show {dZs/dY) plotted against F — In X. Dots: p-type; circles: n-type. Atypical result of Brown and Montgomery, using 28 ohm-cm p-type germanium, is also shown. 1036 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 (F — 111 X). Fig. 7 shows the variation of surface rerombiiiation velocity with F — hi X, using the experimental photoconductivity data and values of y read from Fig. 5. The values of s have been divided by (X + X~'), as indicated, since s/(X + X" ) is expected to be the same, at a given value of ()' — In X), for all samples, so long as the distribution of fast states is the same. The agreement shown in Fig. 7 is probably closer than would be expected in the light of the experimental accuracy. Fig. 8 shows the observed dependence of dY/d8 on (F — In X) for both samples, using the data of Tables III and IV, and using the photocon- ducti^'ity to determine, from Fig. 7, the value of F at each point. On the figiu'e the expected limiting values ( — X and X~ ) are shown for both samples. Of the four asymptotes, the higher limit of (dY/d8) for the At-type sample is satisfactorily reached for large negative values of Y; the experimental values for the p-type sample appear to be approaching the expected limit for large positive values of F, while the information regarding the approach to the two lower limits is too fragmentary to do more than show that the order of magnitude is as expected. Now taking the data shown in Fig. 8, making use of (1) and the calculations given in Tables III and IV, one calculates (82^/88) Y/(d2,/dY)s . The values so found are plotted against Y in Fig. 9. Fig. 6, 7 and 9, showing the ob- served variation of {dXs/dY)s , s and {d2s/d8)Y/id2s/dY)s with F, furnish a complete description of the properties of the fast states at the 200 CO 100 80 60 k 50 + -^40 30 20 10 o o o o ._5 1 — o : • • o o ^2 ? .o • o • o -2 -1 0 1 Y-mx Fig. 7 — Surface recombination velocity versus surface potential. The curves show s/(X + X~') plotted against F — In X. Dots: p-type; circles: n-type. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1037 10 dY drf" 4 10-' e 10 y ^ ,»— — - ^/^ r /^- - P^ • — -2 8 «5. » » /. r • c \ - / \ - / \ 0 - -wl /' \ ® - / •^ £3" a 3 <» 1» 4 / O a, / i - / a ' - / 1 - "~~~"" ^s / » / 5^ - 1 ^ \ / / \l -4 -2 -1 0 1 Y-ln\ Fig. 8 — Surface photo-voltage (change in contact potential in relation to added carrier concentration). dY/d8 is shown plotted against F — In X. Dots: p- fype; circles: 7i-type. Data from different runs are distinguished by modifications to these symbols. The left-hand branches denote absolute magnitudes, since the ratio is negative there. At the extreme left hand of the diagram, the fast states near to the Fermi level are in good contact with the valence band: at the extreme right hand, to the conduction band. The theoretical asymptotes (X~i to the left and X to the right) are also indicated. temperature studied. This is the basic information which any theoreti- cal treatment must explain. In the succeeding paper this matter is dis- cussed from the point of view of the statistics of a distribution of fast states, and information on the cross sections, as well as on the distribu- tion itself, is derived from the data just presented. 1038 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 10 w > 4 10 W 10" 10-2 - y ^ - - / / ^- ^ u — 8--. ,^^ ■o^. / / / ^^ V / < N. - / \ - / i \ - / \ \ - / - - - - -6 -5 -3 -2-1 0 1 Y-ln\ Fig. 9 — The function (a2s/S5)y/(9S,/ay)j plotted against F - In X. Dots: p-type; circles: n-type. VI. FURTHER COMMENTS The development given in the previous section has concerned particu- larly the properties of the fast states. As to the slow states, the experi- ments are much less informati^'e. The variations of 1^ with gas are generally consistent with the variations of contact potential previously reported/ although the total range in Y (±0.13 volt) is smaller by about a factor of 2 than that in contact potential found in the previous work. One must say that roughly half the change of contact potential is in Y B , (i.e., 8Y) and half in Vd , the potential drop across the ion layer. COMBINED MEASUREMENTS ON ETCHED GERMANIUM SURFACES 1039 It may be seen from the figures that it is the quantity (F — hi X), lather than Y, which appears to be characteristic of the point in the cycle reached. This property of a semiconductor surface, and possible reasons therefore, have often been discussed in the literature." The total range of surface potential is illustrated in Fig. 10, which is drawn to scale, and also shows sundry other points of interest found in the present research. The potential diagrams for n-type and p-type are drawn with the Fermi levels aligned, to show the relation between the property (F — In X) = const, and the frequently observed smallness of the contact potential difference between n and p-type germanium. As to the reproducibility and accuracy of the work presented here, the following points may be of interest: (i) The measurements were re- peated on another n-type sample of nearly the same resistivity as the one reported here, but cut from a different crystal. The results on this sample were indistinguishable, within the experimental error, from those found on the first n-type sample, (ii) If the sample was re-etched in pre- cisely the same way as before, and the experiments repeated, the re- sults were in good agreement with those obtained before. However, variations in the etching procedure sometimes gave quite different re- X MAXIMUM IN S o ZERO OF dv/dcT □ INVERSION POINT p-TYPE SAMPLE n-TYPE SAMPLE Fig. 10 — The shapes of the surface space-charge regions for the p-type and /i-type samples in the extremes of gaseous environment. The two surfaces are to the center of the figure. The solid curves show the center of the gap (intrinsic Fermi level) plotted against distance, in units of an intrinsic Debj-e length. Also shown are the positions of the zeros of (dY/dS), the maxima of s, and the minima of surface conductivitj'. 1040 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 suits. Wc hope to discuss this at a future date, (iii) The accuracy of the measurements is not high. Some of the more directly-derivable quanti- ties, such as s, should be known to 5 per cent, but a quantity like (d'2s/d8)/(dZs/dY), which is only obtained after a long and elabo- rate calculation involving a number of corrections, is perhaps uncertain to 30 per cent. ^ VII. CONCLUSIONS This paper has presented results of combined measurements of field effect, photoconductivity, change of photoconductivity with field, fila- ment lifetime and surface photo-voltage, on slices of germanium. From the measurements, the surface potential Y has been found at each point, and the variations of the quantities (dZs/dV), s and {dI,,/d8)/(dI,s/dY) with Y determined. It is a pleasure to record our thanks to W. L. Brown, for comments on field effect techniques and many stimulating discussions, to H. R. Moore, mIio constructed the high-voltage power supply, and to A. A. Studna, who assisted in the experiments. We are also grateful to C\ Herring for comments on the text. BIBLIOGRAPHY 1. W. H. Brattain and J. Bardeen, Surface Properties of Germanium, B. S.T.J. , 32, pp. 1-41, Jan. 1953. 2. C. G. B. Garrett and W. H. Brattain, Physical Theory of Semiconductor Sur- faces, Phys. Rev., 99, pp. 376-387, July 15, 1955. 3. W. L. Brown, Surface Potential and Surface Charge Distribution from Semi- conductor Field Effect Measurements, Phys. Rev., 98, p. 1565, June 1, 1955. 4. H. C. Montgomery and W. L. Brown, Field-Induced Conductivity Changes in Germanium, Pliys. Rev., 103, Aug. 15, 1956. 5. J. R. Schrieffer, Effective Carrier Mobility in Surface Charge Layers, Phj's. Rev., 97, pp. 641-646, Feb. 1, 1955. 6. C. G. B. Garrett and W. H. Brattain, Interfacial Photo-Effects in Germanium at Room Temperature, Proc. of the Conference on Photo Conductivity, Nov., 1954, Wiley, in press. 7. W. H. Brattain and C. G. B. Garrett, Surface Properties of Germanium and Silicon, Ann. N. Y. Acad, of Science, 58, pp. 951-958, Sept., 1954. 8. D. T. Stevenson and R. J. Keyes, ]\Ieasurements of Surface Recombination Velocity at Germanium Surfaces, Physica, 20, pp. 1041-1046, Nov., 1954. 9. J. Clerk Maxwell, Electricity and Magnetism, 3rd Edition, 1, p. 310, Clarendon Press, 1904. K). W. van Roosbroeck, Theory of Photomagnetoelectric Effect in Semiconduc- tors, Phys. Rev., 101, pp. 1713-1725, March 15, 1956. 11. J. Bardeen and S. R. Morrison, Surface Barriers and Surface Conduction, Physica, 20, p. 873, 1954. 12. 1']. Harnik, A. Many, Y. Margoninski and E. Alexander, Correlation Between Surface Recombination Velocity and Surface Conductivity in Germanium, Phys. Rev., 101, pp. 1434-1435, Feb. 15, 1956. Distribution and Cross- Sections of Fast States on Germanium Surfaces By C. G. B. GARRETT and W. IL BRATTAIN (Manuscript recieved May 10, 1956) A theoretical treatment uf the Jield effect, tiurface photo-voltage and surface recombination phenomena has been carried out, starting with the Hall- Shockley-Read model and generalizing to the case of a continuous trap dis- tribution. The theory is applied to the experimental results given in the previous paper. One concludes that the distribution of fast surface states is such that the density is loivest near the centre of the gap, increasing sharply as the accessible limits of surface potential are approached. From the sur- face photo-voltage measurements one obtains an estimate of 150 for the ra- tio (a-p/an) of the cross-sections for transitions into a state from the valence and conduction bands, showing that the fast states are largely acceptor-type. On the assumption that surface recombination takes place through the fast states, the cross-sectioris are found to be: dp '-^6 X 10"^ cm and o-„ -^ 4 X 10"'' cm~. I. INTRODUCTION The existence of traps, or "fast" states, on a semiconductor surface, becomes apparent from three physical experiments: measurements of field effect, of surface photovoltage,' and of surface recombination ve- locity s. Results of combined measurements of these three quantities on etched surfaces of p- and r?-type germanium have been presented in the preceding paper. ^ The present paper is concerned with the conclu- sions which may be drawn from these experiments as to the distribution in energy of these surface traps, and the distribution of cross-sections for transitions between the traps and the conduction and valence bands. The statistics of trapping at a surface level has been developed by Brattain and Bardeen^ and by Stevenson and Keyes,^ following the work on body trapping centers of Half and of Shockley and Read. It is known that surface traps are numerous on a mechanically dam- aged surface or on a surface that has been bombarded but not annealed; 1041 1042 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 and that on an etched surface their density is comparatively low. It is also known that the available results cannot be accounted for by a single level, or even two levels, so that one is evidently dealing either with a large number of discrete states or a continuous spectrum. A given trap- ping centre is completely described by specifying: (i) whether it is donor- like (either neutral or positive) or acceptor-like (neutral or negative); (ii) its position in energy; and (iii) the values for the constants Cp and Cn (related to cross-sections) occurring in the Shockley-Read theory. In this paper we shall deduce what we can about these quantities, using the experimental results previously presented. At the outset it must be admitted that it is by no means certain that the same set of surface states appear in the field-effect experiment and give rise to surface recombination. However, (i) it is found that such sur- face treatments as increase s also reduce the effective mobility in the field-effect experiment; (ii) any surface trap must be able to act as a recombination centre, unless one of the quantities Cp and C„ is zero; and (iii) the capture cross-sections obtained by assuming that the field- effect traps are in fact recombination centres are, as we shall see below, eminently reasonable. As to the nature of the surface traps, not too much can be said at the moment. The lack of sensitivity to the cycle of chemical environment used argues against their being associated with easily desorbable surface atoms; the intrinsically short time constants (Section 5) suggest that they are on or very close to the germanium surface. The possibility that the surface traps are Tamm levels remains; or they could be corners or dislocations. However, the reproducibility with w hich a given value of s may be obtained by a given chemical treatment of a given sample, followed by exposure to a given ambient, suggests that there is nothing accidental about their occurrence. II. STATISTICS OF A DISTRIBUTION OF SURFACE TRAPS We start by quoting results from the work of Shockley and Read and Stevenson and Keyes'' on the occupancy factor ft and the flow U of minority carriers (per unit area) into a set of traps having a single energy level and statistical weight unity: ft = (Cnfi. + Cpp,)/[Cn(n. + ni) + Cp(p, + p,)] (1) U = CnCp(p,ns - ni)/\(\{n. + m) + C,(p.. + pOl , (2) where the symbols ha\^e the following mc^anings: ns , Ps — densities of electrons and iioles present at the surface DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1043 >h , pi — values which the equilibrium electron and hole densities at the surface would have if the Fermi level coincided with the trapping level Cn = NtVTnCTn ', Cp = NiVrpCTp , where iV^ stands for density of traps per unit area, Vm is the thermal speed for electrons and Vtp that for holes, and a„ and a,, are the cross-sections for transitions between the traps and the conduction and valence bands respectively. If we introduce the surface potential Y and the c^uantity 5, defined as (Ap/'Hi), where Ap is the added carrier density in the body of the semi- conductor, we may write: ris = X~^/iie^(l + X5) Ps = Xn;e~^(l + \~^8) where X = po/ni , po being the e(iuilibriiun hole concentration in the body of the semiconductor. We further introduce the notation: 7ii = iiier" pi = nj-e" (4) (Cp/CnY = X The quantity v thus represents the energy difference, measured in units of (kT/e), between the trapping level and the centre of the gap;* and is positive for states below, negative for those above, this le^'el. The parameter x ^vill be most directly associated m ith whether the state is donor-like or acceptor-like. If it is donor-like (neutral or positive), a transition involving an electron in the conduction band will be aided by Coulomb attraction whereas one involving a hole will not; so one would expect X « 1- For an acceptor-like trap, (neutral or negative) the con- trary holds, and one expects x ^ 1- Using (4), the occupancy factor (1) becomes . ^ X~'X-Va + X3) + xe' ' X-'\-'e^l + X5) + x-'e-" + xXe-'Xl + ^''8) + xe" (5) = iX~*e~*''e*'' sech ii {Y + v) - h (n X] for 5 = 0 Note that, in thermodynamic ec[uilibrium, the occupancy factor does not depend in any way on the cross-sections, whereas for 5 5^ 0 it does, through the ratio x- * Strictly speaking, one should say "position of the Fermi level for intrinsic semiconductor" instead of "centre of the gap." These will fail to coincide if the effective masses of holes and electrons are unequal, as they certainly are in germanium. 1044 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Similarly, the flow of carrier-pairs to the surface (2) becomes: U = (6) x-iX-ie^(l + X5) x-'e-" + xXe"''(l + X'^S) xe" wliich, for 6-^0, tends to the linear law U = sniS, where s, the surface recombination velocity, is given by: s/{:VTnVTpf'' = NtSt where St ={\ + X'')ian<7,y''/2\ch(p + fnx) +ch(Y - (n\ - fnx)] (7) The surface density Sg of trapped charge is given by: = Nd\ (8) 2., where ft is the occupancy factor, given by (5). Now let us turn to the question of a distribution of surface traps through the energy v. Suppose that the density of states having v lying lietween v and v -\- dp is N(v) dv, expressed in units (ni£). Then the total surface recombination velocity arising from all traps, and the total trapped surface charge density, are given by : s/ivrnVrpY" = ni£ J St{p)N{v) dv (9) 2, = \ Jt{v)N{v)dp (10) where St{v) and /«(i') are explicit functions of v, given by (5) and (7). The limits of the integrals in (9) and (10) are the values of v correspond- ing to the conduction and valence band edges; however, as we shall see, it is often possible to replace these limits by ± «= . In summing up the contributions in the way represented l\v (9), we ha\'e implicitly ignored the possibility of inter-trap transitions, suppos- ing that the popidation of each trap depends only on the rates of ex- change of charge with the conduction and valence bands, and is inde- pendent of the population of any other trap of differing energy. What kind of function do we expect N{v) to be? Brattain and Bardeen' postulated that N{v) was of the form of two delta-functions, correspond- ing to discrete trapping levels high and low in the band. This assumption is not cousislciit with the observed facts in ri'gard to field cITi^-l, surface DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1045 photo-voltage, or surface recombination velocit}'. The general difficult}' is that the obser\'ed cjuantities usually vary less rapidly with surface potential than one would expect. It is possible to fit the field-effect obser- vations of Brown and Montgomer}'" with a larger number of discrete levels, but this would call for a "sharpening up" of the trapped charge distribution as the temperature is lowered, and this appears to be con- trary to what is observed.* It is always possible that the surface is patchy, in w^hich case almost any variation with mean surface potential could be explained. The simplest assumption, however, seems to be that N{v) is a rather smoothly-varying function. All we need assume for the moment is that it is everywhere finite, continuous and differentiable. We may then differentiate equation (10) with respect to Y and 5 under the integral sign, and get {d^s/dY)^ and (5Ss/55)f, the cjuantities for which experimental measurements were reported in the previous paper :^ i-^ = [- \dYji J 4 N{p) ch ch\h{v -f Y) - \tn X] N{v){h{\-' + \)m{v - Y) -f i ^n X + In x] + \{\~' - X)) civ 4.ch\h{v +Y) ~\ (n X] (11) (12) Notice that the expression in brackets in the numerator of (12) gener- ally has the value X~ or —X, except near the point v = Y — fn\ — 2fnx- This is indicative of the fact that, whatever the exact form of N(v), the ratio of — (32s/35)y/(3Ss/(9F)5 tends to these limiting values (X^^ and —X) for sufficientlj^ large negative and positive Y respectively. It may be verified from (7), (11) and (12) that {dXs/dY)^ , found from the field effect experiment, depends only on N(v) ; (d'Zs 88) y , found from the surface photo- voltage, depends on N{v) and x; while s, the surface recombination velocity, depends in addition on the geometric mean cross-section (anapY''. Both x and (a-„(7p) '"^ might themselves, of course, be functions of p. Thus relations (7), (11) and (12) are integral eciuations, from which the three unknown functions of v may in principle be de- duced from the experimental results. (Equation 11 , in fact, may be solved explicitly. P. A. Wolff'^ has shown, how^ever, that, to determine N{v) unambiguously, it is necessary to know (52^/(9 F)j for all values of Y in the range ± ^ .) The foregoing considerations apply to "small-signal" measurements. * There are some changes with temperature, but not what one would expect if there were only discrete surface states. 1046 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 z >- w >- in 3.0 2.5 2.0 1.5 1.0 0.5 -0.5 -1.0 -1.5 >w, ^ N \ N \ • -4 DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1017 8.1 ohm-cm p-type: = 9.7c/; [0.31 (F - (n X) - 0.5] (14) for 2 > (F - (n X) > -4 For values of (F — (n X) less than —4, it appears that Ss is changing more rapidly with F than is indicated by (13) and (14). We shall comment on this point later. Excluding this region, we note that in both cases the variation with F is everywhere slow in comparison with e^, and proceed on the assumption that N{v) is a function of v that varies everywhere slowly in comparison with c" . Then (11) indicates that there is one fairly sharp maximum in the integrand in the range ± « , occurring at that value of V which coincides with the Fermi level: V ^ -F + (n X (15) The integral in (11) could be evaluated in series about this point (method of steepest descents). The zero-order approximation is got by replacing i sech' \h{v + F) - \(n X] by 6(i^ + F - Cn X). Later we shall proceed to an exact solution, and we shall find that this delta-function approximation is not too bad. From (11) we now find: -f F - (ri X) dv = N{-Y + (n. X) (15) This mathematical procedure will be seen to be eciuivalent to identify- ing {d'Ls/dY)i with the density of states at the point in the gap which coincides with the Fermi-level at the surface. Using (13) and (14), one gets: 22.6 ohm-cm n-type: N{v) = 4.5 chiOMv + 0.8) (16) 8.1 ohm-cm p-type: N{v) = 9.7 chiQMv + 0.5) (17) As we shall see in the next section, the exact solutions differ from (16) and (17) only in the coefficients preceding the hyperbolic cosines. Turning to the surface photo-voltage measurements, we take (12) and again replace I sech' [^{v + }') - \tn XJ by h{v + Y - In X) 1048 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Using (15), one gets: ~ (dXs/dY)B = i(X-' + X) th(-Y + fn\+ (n x) + hO^~' - X) (18) This procedure, inaccurate as it is, has the advantage that no particu- lar assumption need be made concerning the functional dependence of X on V, it being understood that x in (18) has the value holding for v = — Y -\- (n X. In particular, if }^o is that ^•alue of Y at which the ratio -(a2s/a5)y/(a2,/a}')5 changes sign, /"wxo = To - Cn X + t}r\{\ - X"')/(X + X~')] (19) From the experimental data, one linds, for the /^-tj-pe sample, In xo '^ 2.4 (at V = —3.5); for the p-type sample, (n xn ^' 1.0 (at v = 1.9). In \-iew of the approximations made, these estimates would not be expected to be more precise than ± 1 to 2 units. Notice that both \alues are positive, and that the difference between them is small in compari- son with the difference in v. This suggests that we start afresh with the assumption that x is independent of v, and woi'k out the surface photo- voltage integral exactly. This is done in the next section. IV. EXACT TREATMENT FOR THE CASE N{v) = A ch {qv + B) , AVITH CON- STANT CROSS-SECTIONS The results of the previous section suggest the procedure of assuming that N{v) is of the functional form given by (16) and (17), and evaluat- ing the integrals (9), (11) and (12) exactly. The integral for {dliJdY), (11), depends only on the form of N{v) and ma}^ be eA'aluated at once. To get idfijdb), (12), one must know how x depends on v. On the basis of the work of the previous section, we shall suppose that x is in- dependent of V. (Properly, we need only assume that x varies with v more slowly than e^ Since the function th[\{v — Y) -f ^Cn X + (n x] has one of the values ±1 everywhere except close to j' = Y — (n X — 2Cn X, and since the denominator of (12) has a sharp minimum at V = — Y -\- (n X, it follows that the region in which (3Ss/d5)y changes sign will be governed mainly by the value of x at ^ = — (n x) To get s [(9), using (7)], one must also assume something about the geometric mean cross- section, {an(T,^ ". In the absence of any information on this score, we shall assume that (o-„a-p)' " also is independent of v, and see how the com- puted variation of s with Y compares with the experimental results. DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1049 We assume: N(v) = A ch (qv + B) (20) and substitute in (11), (12) and (7). In view of the sharp maximum in the integrands of these expressions, it is permissible to set the limits which should correspond to the edges of the gap or of the state distribu- tion equal to ± <» . The integrals are conveniently evaluated by the con- tour method (see Appendix 1) and yield the following results: ( -^ j = Attq cosec TQ ch [B — q{Y — /n X)] /aS,\ ^ _ATrq cosec vq ch [B - q{Y - (n X)] X where 'y = }' - (n X - (n X (S, = B - qtnx (21) (22) (23) {VrnVrp)"-' (24) = I (X -f X~^)(o-„(Tp)^'^ni£ 2x A sh qy ch (B cosec irq cosech 'y Comparing (21) with (15), we see that the delta-function approxima- tion is in error to the extent that it replaces irq cosec irq by 1. With the value of q found experimentally, this is not too bad; we can now, how- ever, by fitting the right-hand side of (21) to the experimental facts, (13) and (14), obtain exact solutions for A''(j'): 22.6 ohm-cm n-type iY(j;) = 3.6 chiQMu + 0.8) (for u < 4) 8.1 ohm-cm p-type (25) N(v) = 8.3 chiOMp + 0.5) (for p < 4) The question arises as to whether this solution for the distribution is unique. We have already pointed out that the mathematical methods fail if the distribution is discontinuous. It seems that (25) represents the only solution that is slowl3^-^'arying, in the sense used in the previous section; its correctness could presumably be checked by carrying out experiments at different temperatures. For v > 4:, the abo\-e expressions 1050 THK BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 do not fit the observed facts, because, for F — ^n X < — 4, the charge in fast states is found to change more rapidly than is given by the empirical expressions in (13) and (14). The behaviour in this region is perhaps in- dicative of the existence of a discrete trapping level just beyond the range of v which can be explored by our techniques. The observations (see Fig. 6 of preceding paper^) can be described by postulating, in addi- tion to the continuous distribution of states given above, a level of den- sity about 10 ^ cm" , situated at j^ = 6, or a higher density still further from the center of the gap. Statz et al,^' using the "channel" techniques, which are valuable for exploring the more remote parts of the gap, have proposed a level of density '^ 10 cm~^, situated at about 0.14 volts be- low the center of the gap (v = 5.5) : this is not in disagreement with the foregoing. In order to compare (22) with the experimental data derived from the surface photo-voltage, it is necessary to choose a value for x- Fig- 2 shows the comparison with the results presented in the preceding paper. On the vertical axis, the values of (dl^s/d8)/(d'2s/dY) plotted have been divided by (X + X" ), in order to show the n and p-type results on the same scale. (Note that the limiting values of this quantity should be X/(X -|- X"^) and — X~V(^ + ^~^)j so that the vertical distance between the limiting values should be 1, independent of X). The theoretical curves have been drawn with the value Inx = 2.5, in order to give best fit between theory and experiment at the points at which the ordinate changes sign. (It may be seen from the form of (22) that, with the actual value of the other parameters, the main effect of adopting a different value of in x would be to shift the theoretical curve horizontally, while a change of X shifts it vertically without in either case greatly modifying its shape). The fit between theory and experiment is not quite as good as could be expected, even taking into account the rather low accuracy of the measurements. The variation of (6Ss/55)/(6Ss/6F) with Y found experimentally seems to be rather slower than the theory would lead one to expect. The main points to make are : (i) the difference in Y between the zeros for the two samples (5.4 ± 1) is about what it should be (4.8) on the assumption that in X is the same for both samples and of the order of unity; and (ii) paying attention mainly to the zeros, the estimate (nx — 2.5 is likely to be good to ±1. Now let us consider the surface recombination velocity. Here we are on somewhat shakier ground, in that, in deriving (24), we have had to assume not only that x is independent of v, but (o-„crp)^'^ also. First we note from (24) that the maximum value of s should occur at F — (n X = in x- Comparing with the experimental results given in the preceding paper, DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1051 10 W to 10 to 1.0 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 1.0 / ^^ / / •/ 1 y V ,__,s^ > ./ ^ — k / / Vi / ■ >^ y J — y -6 -2 0 2 Y-Ln\ Fig. 2 Experiment and theory for 95 )K^- X + X- Solid lines theory; circles and dots, with smooth curves through the points, repre- sent experimental results for n and p-type samples, respectively. we see maxima at F — ^w X = 2.0 for the p-type sample, and 3.5 for the n-type sample. Both these values are within the limits to (n x given in the previous paragraph, thus confirming the estimate made there. Fig. 3 shows a comparison between the experimental results and (24). The graph has been fitted horizontally, by setting (n x = 2.5, as found above ; vertically, to agree with the mean value at that point. The agreement with experiment is reasonable, although again, just as in Fig. 2, the ex- perimental variation of s with ( Y — in X) is rather slower than one would expect. The fact that the experimental values, both of surface photo-voltage and of surface recombination velocity, vary more slowly than expected, is susceptible of a number of interpretations: (i) The deduced distribu- tion of fast states might be wrong. However, the most likely alternative distributions — isolated levels, or a completely uniform distribution — 1052 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 give (in at least some ranges of F) a more rapid instead of a smoother variation of these quantities so long as the surface is homogeneous, (ii) The estimates of the changes in Y might be too large. It is unlikely that our calibration is sufficiently in error, and other workers have obtained results comparable to ours. The only possibility would be that the mo- bility of carriers near the surface is larger (instead of smaller, as found by Schrieffer) than inside • — which seems cjuite out of the question, (iii) The ratio of capture cross-sections varies with v. This, however, w^ould only be in the right direction if one were to assume that the ratio x 'in- creases with the height of the level in the gap — i.e., that the high states behave like acceptors, and the low ones like donors. While not quite impossible, this is an unlikely result, (iv) The surface is patchy. It is probable that a range of variation of two to four times (kT/e) in surface potential would be sufficient to account for the observed slow variation of surface photo-voltage and recombination \'elocity with mean surface potential. We ha\'e refrained from detailed calculations of patch effects, on the grounds that, without detailed knowledge of the magnitude and distribution of the patches, it would be possible to construct a model that could indeed fit the facts, but one w^ould have little confidence in the result. The possibility of patches warns us to view with caution the exact distribution function deduced for the fast states. It would still be conceivable, for example, that one has but two discrete states, as originally proposed by Brattain and Bardeen," and that the apparent existence of a band of states in the middle of the gap arises from the fact that there are always some parts of the surface at which the Fermi level is close to one or other of these states. Fortunately the conclusions as to the cross-sections are not too sensitive to the exact distribution function assumed. Using the mean of the two coefficients in (25), substituting //,• = 2.5 X 10'^ cnr^ £ - 1.4 X 10"'' cm, {vrnVrp)"' = 1.0 X 10' cm/sec, in (24), and using the experimental result (see Fig. 3) that s,nax/(X -f X~\) = 1.2 X 10" cm/sec, one obtains ((Tp(T,y~ = 5 X 10~' cm'. Now setting (ap/a„) = x" '^ c' ^^ 150, one gets for the separate cross-sections: o-p = <) X 10"'' cm" an = l X 10"'' cm' There values appear lo he emiiicntly reasonable. Burton et al, " who studied re('()inl)ination through body centres associated with nickel and copper ill germanium, found cr^ > 4 X 10 '"^ cm", o-,, = 8 X 10"'^ cm* for nickel, and a„ = 1 X 10 '%„ = 1 X 10"'' for copper. The fact that DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1053 200 m 100 90 80 70 ~ 60 u LU ■sn (/) 5 40 o u 30 + ,< 20 10 8 7 -6 .V o • o o -> X 1 • \ • / / > V ^ 0 • / • 0 • / 0 • / • • / / f / / -5 -4 -3 -2 -1 0 Y-lnX, Fig. 3 — Experiment and theory for surface recombination. Solid curve theory circles and dots for n and p-type samples, respectively. our estimates for ap and o-„ appear to be of the expected order of magni- iitudes lends strong support to the view that identifies the traps appear- ing in the field-effect and surface photo-voltage experiments with those responsible for surface recombination. The result that (o-p/cr„) = 150 is good evidence that the fast states are acceptor-like. This statement must be restricted to the range \ v \ < 4; the states that are outside this range might be of either type. Also one might allow a rather small fraction of the states near the middle to be donor- type, without serious trouble; but the experimental results compel one to believe that most of the fast states within 0.1 volts or so of the centre of the gap are acceptor-like. V. TRAPPING KINETICS The foregoing considerations have concerned the steady-state solution to the siu'face trapping problem. If the experimental constraints are changed sufficiently rapidly, however, there may be effects arising from the finite time required for the charge in surface states to adapt itself 1054 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 to the new conditions.' This section will concern itself with the trapping time constants (which are not directly related to the rate of recombina- tion of minority carriers). One case of trapping kinetics has been discussed by Haynes and Horn- beck.^ A general treatment of surface trapping kinetics is necessarily quite involved, and will be taken up in a future paper. Here we shall restrict ourselves to giving an elementary argument relating to the high-fre- quency field effect experiment of Montgomery. To simplify the discus- sion, we assume that the surface in question is of the "super" type; i.e., the surface excess of the bulk majority carrier is large and positive. At time i = 0, a large field is suddenly applied normal to the surface; the induced charge appears initially as a change in the surface excess of the bulk majority carrier; as time elapses, charge transfer between the space- charge region and the fast states takes place, until equilibrium with the fast states has been re-established. What time constant characterizes this process? Take electrons as the majority carrier. Then the flow of electrons into the fast states must equal the rate of decrease of the surface excess of electrons. For a single level one may write: Un = NtVTn(rn[(l " ft)ns - Ml] (26) = -r„ For a continuous distribution of levels, one can say that only those levels within a few times (kT/e) of the Fermi level at the surface will be effective, so that one may regard the distribution as being equivalent to a single state with rii = rii exp (Y — In X), which will be about half full. We assume further that the density of fast states is sufficient for the changes in r„ to be large in comparison with those in ft , as is reasonable, having regard to the relative magnitudes of the measured values of (dI,s/dY)i found in the present research, and of (dTp/dY)5 and {dT„/dY)s . Thus/< may be treated as a constant in equation (26). Further, we may set Hs = 4r„ /nj£ , as may be proved from considerations on the space- charge region.' Solving (26) with these conditions, one finds, for the transient change in r„ between the initial and the quasi-equilibrium state : Ar„ cc M - th-] (27) where r = \e-''&/[NtVTn(rnV2 Vftil - ft)] DISTRIBUTION AND CROSS-SECTIONS OF GERMANIUM SURFACES 1055 I I To clarify the order of magnitude of time constant invoh'ed, let us substitute £ '^ 10" cm, Nt ~ 10^^ cm~^, tv,. ~ 10^ cm/sec, cr„ ~ 10"^^ cm ,/t '^ 0.5, Ae~ '~ 1. This gives r -^ 10~' sec, which suggests that one would be unlikely to run into trapping time effects in the field-effect ex- periment at frequencies less than 10 Mcyc/sec. This conclusion is con- sonant with the findings of Montgomery. Appendix 1 evaluation of the integrals in section 4 The integrals occurring in Section 4, giving the experimentally acces- sible quantities (d2s/dY), (dXs/d8) and s in terms of the surface trap distribution and cross-sections, are conveniently evaluated by contour integration. In view of the general applicability of this method in deal- ing with integrals of the sort that arise from such a distribution of traps, we include here a short note on the precedure used. The integrals needed are : .+00 /T-ou ch{cx + g) sech" x dx 00 /+00 th{x -\- b) ch(cx -{• g) sech^ x dx 00 -L +00 h chicx -\- g) con- e/la; -\- chk To evaluate /i , we evaluate / ch(cz + g) sech^ z dz around the tour shown in Fig. 4. The contributions from the parts z = ±R vanish in the limit R -^ oo , so that the integral has the value : /+00 /.+00 ch(cx -\r g) sech'^ x dx — i sin ctt / 00 •'—00 sh(cx + g) sech'^ x dx Fig. 4 — Evaluation of 7i . 1056 THE BELL SYSTEM TECHXICAL JOURNAL, SEPTEMBER 1956 The integrand has one pole Avithin the contour, at x = ^iw, at which the residue is — c(cos ^cr sh g -\- i sin ^cir ch g). Multiplying by 2x1 and equating the real part to that in the above expression, one obtains: /i = xc cosec \cir ch y The same contour is used in evaluating lo ; there are now poles at z = ^/tt and at z = \iir — b, and one obtains: 1-2 = TTC coth b ch g cosec ^ctt — 27r cosec ^CTT cosech" b sh ^bc ch{}/'2bc — g) To evaluate h , one integrates / [ch{cz + g)/(chz + chh-)] dz around the contour shown in Fig. 5. There are poles at iw ± k. Proceeding as before, one finds: I3 = 2ir sli ck ch g cosec ire cosech k Appendix 2 limitation of surface recombination arising from the space- charge barrier The ([uestion of the resistance to How of carriers to the surface arising from the change in potential across the space-charge layer has been discussed by Brattain and Bardeen. Here we shall recalculate this effect by a better method, which again shows that, \\ithin the range of surface potential studied, the effect of this resistance on the surface recombina- tion velocity is for etched surfaces ciuite negligible. Let Ip and /„ be the hole and electron (particle) currents towards the surface, and let x be the distance in a direction perpendicular to the sur- face, measuring .r positive outwards. Then the gradient of the fiuasi- Fermi levels (pp and ^^\ TIME Fig. 1 — (a), a perfectly regenerated pulse train; (b) showing the effect of low- frequency cutoff; (c), showing (a) after passing over equalized line; (d), showing (b) after passing over equalized line; (e), effect of (d) minus (b); (f), inverted pedestal timing wave; (g), composite wave at input to repeater, namely, (d) minus (b)plus (f). 1062 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 ferred to as "zero wander." In a regenerative repeater the trigger poten- tial is tied to the zero level by a constant bias. Zero wander then will produce a changing bias which reduces the signal to noise margins of the repeater, or in some cases even prevents regeneration. Suppose, for example, a transmission line is equalized so the ideal pulse train shown on Fig. 1(a) will appear as Fig. 1(c) after being transmitted over the line. The individual pulses have widened until the envelope of a sequence of consecutive pulses shows as a ripple with a much smaller amplitude than the individual pulse. If the pulse train distorted by low frequency cutoff shown on Fig. 1(b) is transmitted over this line its output will appear similar to that shown on Fig. 1(d). The portion of the signal where the peak amplitude Hes below the trigger threshold will not be regener- ated. 1.2 Compensatio7i for Low-Frequency Distortion In the past many circuits have been devised to prevent zero wander, but none have been completely satisfactory. The repeater described in this paper effectively eliminates zero wander in a string of consecutive repeaters by means of a new and simple method. This may be better un- derstood by referring to Fig. 2. Here are represented two successive re- peaters of a transmission system. These repeaters have what appears as a conventional negative feedback loop consisting of a pair of resistors, R. The function performed by this feedback loop bears little if any resem- blance to the negative feedback of linear amplifiers and is referred to as "Quantized feedback" in this paper.* Suppose an isolated pulse of amplitude P,„ is regenerated in repeater M and is applied to the line through its output transformer. The low freciuency cutoff" of this transformer will produce a transient response to the regenerated pulse as given in (1). A spectrum analysis of the transient tail shows that most of its energy occurs in the lower portion of the pass band of the equalized line. Consequently, it will be transmitted over the line to the next repeater with little if any frequency or phase distortion, but will be attenuated by a factor a. This transient at the input of the following repeater may be expressed as Tm - akMPMe~'' (2) where t is again measured from the end of the pulse. Suppose the re- generation of the pulse at the output of repeater N is delayed by time ti * A paper by Rajko Tomovich entitled "Quantized Feedback" was published in the I.R.E. Transactions on Circuit Theory. There are some fundamental dilTer- ences in the meaning of the term, quantized feedback, as used in these papers. TRANSISTOR BINARY PULSE REGENERATOR LINE INPUT K Z>^ LINE EQUALIZER REGENERATIVE REPEATER M R AAAr A^A^■ R LINE Da R AAA- R REGENERATIVE REPEATER N SIGNAL OUT , >c 1063 — I LINE Fig. 2 — Block diagram of a section of equalized line and its terminating regenerative repeaters. compared to the pulse at the input of the repeater. The transient re- sponse of the regenerated pulse after passing through its output trans- former f will be Tiv = fcivPive"'^'-'^^ (3) 7 T^ '''l ~~^^ KN^Ne e (4) If the transient (4) is attenuated by factor 3 and added in opposite phase to Tm through the feedback loop at the input of the repeater, their sum is T,r BT.y = ak-MPMc''' - 8kNPNe"'e~'" -bl. bt\ = e-'Xak,,PM - Bk^Pj.e"'') This can be made equal to zero if akufPyi = Bk^PiijC (5) (0) (7) which is accomplished by adjusting the value of d which represents the feedback attenuation introduced by resistances R. If the regenerated t It is assumed that the electrical characteristics of the output transformers of all the repeaters are identical. In this case the damping coefficients will be identical for all the regenerated outputs. 1064 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 output pulses of .1/ and N are identical, then Pm = Fj^ and Icm = Id^ and eq. (6) becomes Tm - QTs = e-^'huPMioc - 8e"') (8) This expression can be made equal to zero if 8 = ae"^'' (9) By this means zero wander produced in one repeater can be eliminated at the input of the next repeater. The low frecjuency distortion of one repeater corrects for the corresponding distortion produced in the pre- vious repeater. If the electrical characteristics of uU the repeater output transformers are identical it is possible to completely remove the effects of the tran- sient tails due to low frequency cutoff.* It is important however that t\ should not be so large that the feedback pulse occupies the next timing interval. W. R. Bennett has shown that a similar cancellation of tran- sients can be accomplished for more complicated types of low frequency cutoff characteristics. In this case the transient tails ^^ ill be the sum of a number of exponentials having different amplitudes and damping co- efficients. Here the ciuantized feedback must be provided by multiple loops, of greater complexity. It may be disturbing at first to observe the resultant sum of the incom- ing signal and feedback as shown on Fig. 1(e). It should be noted how- ever that the signal is not changed in any way until the repeater has triggered the regenerated pulse, and at the next time slot the tails have been cancelled, so that when the next pulse arrives it too will begin at the zero axis. Tails may also be produced by high fre(|uency phase-loss characteristics. These however, may be removed by proper equalization. 1.3 Timing In a Regenerative Repeater The binary regenerative repeater must not only regenerate the shape and amplitude of each individual pulse but it must also keep them in proper time seciuence with other signal pulses. To accomplish this a suit- able timing wave must be provided. This timing wave may be trans- mitted over separate pairs of wires or it may be derived from the signal. In the past it has been connnon to obtain a sine wave of the repetition * It can be shown that, with reasonable differences in damping coefficients, quantized f(!edback will fjreatly reduce interyyml)()l interference even when con- sidei'ing a single pulse. If the coiil ril)utions from all the transients of an infinite train of random pulses are summed, the resvUt;int interference is further reduced and can be considered negligible. TRANSISTOR BINARY PULSE REGENERATOR 1065 I frequency by exciting a high Q filter circuit from the received pulse train. [Short timing pips generated from this wave are used to time the regen- erated output pulses precisely. This procedure is far too involved to be used in a simple repeater. If less precision in timing is acceptable it may be accomplished with a minimum of circuitry by use of a sinusoidal wave derived from the repeater output. This is referred to in this paper as "self timing." Self timing prohibits the use of short timing pips derived from the i-egenerator output. In this case most of the timing control would be exercised by the filter circuit and little, if any, by the input signal. The direct use of the sinusoidal output of this filter provides suflficient control by the input signal with only a small penalty due to less precise timing.* Self timing also sets certain requirements on the regenerator. If the tim- ing wave is derived from an independent source it can be added to the signal in such a way as to act as a pedestal, lifting the signal above the tiigger level. In such a circuit neither the signal nor the timing wave alone can trigger the regenerator. If the timing wave is derived from the output it is obvious that the signal alone must be able to trigger the regenerator, since the generation of a timing wave depends upon the sig- nal triggering the regenerator. A timing wave derived by filtering the output of a random pattern of binary pulses will also have a varying amplitude which could cause variations in repeater noise margins. It is apparent then that self timing output cannot be used as a pedestal in a regenerator. All these objections can be overcome by the use of "inverted pedestal" timing. Inverted pedestal timing is produced by tying the peaks of the timing wave having the same polarity as the signal pulses to a fixed level by means of a diode. This is illustrated on Fig. 1(f). The timing wave is added to the signal at the input so the sum of the signal, feedback and timing looks somewhat like the wave on Fig. 1(g). The effect of the inverted pedestal timing is to inhibit triggering except in the time interval near the peaks of the timing wave. This permits the signal to trigger the re- generator without a timing wave, yet allows timing control to be exer- cised as the amplitude of the timing wave builds up. With sinusoidal timing, noise often causes the regenerator to trigger either early or late, introducing a phase shift in the regenerated ouput which will be reflected in the timing wave. Since the timing wave is derived from the code pat- tern by a relatively high Q tuned circuit, the phase distortion of the tim- ing wave from a shift of a single pulse will be small. With a random dis- * E. D. Sunde, Self-timing Regenerative Repeaters (paper being prepared for publication). 106G THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 tribiition of noise the resultant phase shift of the timing wave will be negligible. If the interference has low frequency components, the phase shift of the timing wave may be appreciable but these are slow and con- sequently will not seriously effect the performance of the regenerator. 2.0 DESCRIPTION OF REPEATER CIRCUIT The circuit diagram shown on Fig. 3 will aid in understanding the op- eration of the repeater. The incoming signal after being transmitted over the equalized line is applied through the input transformer Ti to the emitter of transistor (1). The function of this transistor is to provide gain to the incoming signal. This amplified signal is applied to the emitter of transistor (2) through the blocking condenser C2 . The second transis- tor functions in a single shot blocking oscillator circuit being biased in the "off" condition through the resistance i?2 . When the positive signal ex- ceeds the trigger threshold, a pulse is regenerated by the blocking oscil- lator. During the pulse period a large emitter current flows through Di in the conducting direction. T^ is the output transformer while trans- former T3 provides the essential positive feedback for the blocking oscilla- tor. L, R, -'TW VW N BOOTSTRAP TIMING TUNED TO 672 K.C INPUT QUANTIZED FEEDBACK AAA- Fig. 3 — Circuit diagram of the regenerative repeater. TRANSISTOR BINARY PULSE REGENERATOR 10G7 2.1 Inhibiting in Blocking Oscillator The secondary of Ts is connected between the transistor base and ground with the diode D2 and resistor R^ in series across it. The combina- tion of diode and resistance across T^ serves a very important function, the inhibiting of multiple triggering on a single input pulse. During the interval in which the pulse is regenerated a negative potential is applied between the base and ground. A current h flows through the base of the transistor, the diode Do being poled to restrict the flow of current in 7?3 . At the end of the pulse the current h in T^ drops suddenly to a low value. This current change in the inductive winding of T3 induces a relatively large potential across the base of the blocking oscillator. The impedance of D2 becomes low and current flows in Rz and T3 . The potential across T-s decays exponentially and with proper circuit values will take the form of a damped cosine wave. E = Eoe~"' cos wo^ (10) I where t is the time measured from the peak of the pulse. The values of a and coo can be adjusted by varying the inductance the transformer and the capacity and resistance connected across it. E should become sub- stantially zero at or near the next timing interval. The damping coeffici- ent a should be sufficiently large to prevent an appreciable negative ex- cursion of E since this will reduce the effective bias on the repeater and consecjuently its noise margins. This will be further discussed in the sec- tion on the measurements of errors. 2.2 Quantized Feedback The quantized feedback is provided by coupling the input and output transformers by means of resistances R. The fed back pulse must be in the opposite phase compared to the input signal. 2.3 Timing Wave Circuit The timing wave is derived by means of the parallel resonant tank cir- cuit L2C5 which is tuned to the signal repetition frecjuency. The regen- erated pulses are applied to this network through the relatively large resistance 7^4 . The amoimt of energy added to the network by each pulse as well as the amount dissipated in it is a function of Q. The higher the Q the smaller will be the variations of timing wave amplitude as the aver- age pulse density of the signal train changes. This does not mean that the highest Q will be the most desirable for increased Q means larger, 1068 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 more expensive coils. Higher Q's also produce greater variations in im- pedance and phase with small changes of resonant frequency which re- (luirc much closer control of inductance and capacity with temperature. In the circuit described here the Q has a value of about 100 and its opera- tion is quite satisfactory'. The tank circuit is coupled through the small condenser Cz to the diode Dz . This diode ties the positive peaks of the timing wave to ground as is reciuired for inverted pedestal timing. The network N pro^'ides the timing delay needed for optimum repeater per- formance. 2.4 DC Compensation in Timing Wave The timing wave amplitude from the tank circuit is insufficient to allow it to be applied directly to the emitter of the blocking oscillator. Conse- quently in the interest of circuit simplicity the signal amplifier is used for the timing wave as well. To avoid the complications introduced by dc coupled circuits when close bias tolerances must be maintained, the amplifier was coupled to the blocking oscillator by condenser C2 . This presents a problem as to how to neutralize the charge the dc component of the timing wave builds up on C2 . The means by which this is accom- plished can be more easily understood by referring to Fig. 4. In this figure the time constant of the feedback loop RoCiRi , is made large so that substantially equal charges are added to Ci by each regen- erated pulse. In the timing loop this is also nearly true even though noise INPUT T yf Cp REGENERATIVE REPEATER R, :C, .^ A A _. V V ^ X X c T Roy r^i x'' OUTPUT Fig. 4 QUANTIZED FEEDBACK Method for maintaining the dc values of timing wave. TRANSISTOR BINARY PULSE REGENERATOR 1069 may change the phase of indi^•idllal pulses. The change of amplitude of the sinusoidal timing wave in one pulse period will be AAr = Ar[l - e-''"^""] (11) w here Q = wL/R and tm is the timing interval. In a similar manner the \ariation of the amplitude of the voltage across Ci will be AAc = AcW - e-'-'/^i^^] (12) If now 7?i and Ci are adjusted until TV 1 Q RiC, (13) and Ro varied initil the amplitude Ac is eriual to the a^•erage value of At , the charge on the interstage coupling condenser should be effectively neutralized at all times. Since both loops are made up of passive elements with common inputs and outputs a single adjustment should suffice even though the pulse amplitude, width, or signal pulse density may vary. In the repeater circuit shown on Fig. 3 this neutralizing principle is used but is more difficult to see. When a pulse is regenerated, a large emitter current flows in Di , which produces a sharp negative voltage spike. This voltage adds a charge to C2 which tends to neutralize the one the timing wave adds to it. The time constant of C2 and its associated circuit may be made to equal the decrement of the tank circuit and the two amplitudes made equal by adjusting the level of the timing wave. By this means effective dc transmission of the timing wave is achieved through capacity coupling. 2.5 Line Equalization The line equalizer is not essentially a part of the repeater itself. It is however so intimately connected with the repeater it is logical that they be considered together. One of the important ecjualizer requirements is simplicity, another, that the impedance seen from the repeater input shall be substantially constant over a relatively large frequency range. This latter requirement comes from the need of transmitting the feed- back pulse around the feedback loop to the emitter of the first transistor without too much distortion. The equalizer is not used to equalize the low frecjuency losses of transformers but only the frequency characteris- tic of the line. The eciualization must be such that the individual pulses are allowed to widen but not enough to cause inter-symbol interference. 1070 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 A gaiLssian shaped pulse at the output of the Hue is one of the most eco- noniical to use and can have a maximum span of on(! timing interval at its base. Howe^'er, in this case the envelope of a long consecutive se- (juence of such pulses will show substantially no ripple. It can be readily seen that in such a seciuence the onl}- timing control exercised by the input upon the timing wnve comes from the first pulse. In the interest of better timing and consecjuently better repeater performance one should be content with narrower pulses at the repeater input. The resulting rip- ple of the envelope of a consecutive pulse sequence allo^^"S each incoming pulse some control over the repeater timing. 3.0 REPEATER PERFORMANCE To check the performance of the regenerative repeaters a binary code generator was built having a nominal pulse repetition rate of 672 kc producing an eight digit code. Any code combination from the possible 256 can be selected or the code automatically changed at periodic inter- vals reproducing all possible codes in orderly sequence. Random codes may also be generated by making the absence or presence of a pulse 52 1 EQUALIZER = LUS "-y ^ 1.75 MILES CABLE A ^^ \ 2.3 MILES CNB 19 GAUGE CABLE .^ ^' 1 1 1 1 1 1 1 1 1 1 1 8 10 20 40 60 80 100 200 FREQUENCY IN KILOCYCLES PER SECOND 400 600 1000 Fig. 6 — Effect of changing the length of 19 gauge line with fixed equalization. 4.1 Description of Error Detecting and Counting Circuit An error detecting and counting circuit was built to count insertion and omission errors. This circuit (block diagram, Fig. 7) is a coincidence detector in which each pulse or space of the repeater input signal is compared to its corresponding regenerated output. As long as the two sources are the same, i.e., having corresponding pulses or spaces, there is no output from the detector. If the two differ the detector produces an output pulse which may be caused to actuate the counting circuit. The code generator as has already been described produces a number of different types of signal codes. The output of the code generator is transmitted over 0.56 miles of equalized 32 gauge cable to the regenerative repeater under test. Inter- ference is introduced at the repeater input when desired. A portion of the code generator output is differentiated and passed over a delay cable whose delay is substantially that of the section of 32 gauge line over which the signal is transmitted. This delayed signal is regenerated without error by the single shot blocking oscillator A, The width of the blocking oscillator pulses are adjusted to be about half of the total timing interval. The width of the pulses from the regenerative repeater 1076 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 are likewise widened to a corresponding width by blocking oscillator B. Unfortunately a variable phase shift is introduced in the repeater output by interference and by variations in the timing wave amplitude and phase. This variable phase shift prevents perfect coincidence between the outputs of blocking oscillators A and B. An example of phase "jitter" caused by interference is shown on Plate V(a). To overcome this a sharp sampling pip; as shown on the same plate, is provided to enable the detection of the narrow region of coincidence between the two signals. These pips are generated from the repeater timing wave, hence they follow the timing wave phase variations. The regenerated signal pulses also follow the timing wave phase. If the sampling pulse is positioned to fall in the center of the regenerated pulses, it will tend to maintain that position as the timing wave changes. The gates require a signal pulse and sampling pip to be present simultaneously before there can be an output. This output, then, will have substantially the same shape and position as the sampling pip. When a signal pulse is simultaneously applied to each gate the two outputs can be made to cancel when added in opposite phase as is done in Ti . If however there is a pulse on one gate and a blank on the other, an output pulse will be produced. The polarity of this pulse will depend upon which gate contains the signal pulse. Since the decade counter is PULSE CODE GENERATOR ■w-v DELAY LINE BLOCKING OSCILLATOR (A) V\AP AMPLIFIER DIFFEREN- TIATOR iULJL "and" GATE (A) VvV MAI SAMPLING BLOCKING OSCILLATOR BLOCKING OSCILLATOR (B) J] n. "and" GATE (B) OMISSION ERROR OUTPUT TO CABLE "AND NEXT REPEATER 11_L TI JLJ_ POLARITY REVERSING SWITCH JL BLOCKING OSCILLATOR (D) JL DECADE COUNTER Fig. 7 — Block diagram of error detecting circuit. TRANSISTOR BINARY PULSE REGENERATOR 1077 (a) (b) Plate V — (a) with 1, repeater output; 2, jitter on output pulse; 3, sampling pulse, (b) with 1, signal pulse at repeater input; 2, 672-kc timing pips; 3, interfer- ence input. triggered by pulses of one polarity, the reversing switch permits the independent measuring of different types of errors. The counter used in this study has 9 decades capable of counting and recording (10^ — 1) errors at 10^ counts per second. 4.2 Discussion of Impulse Noise Generator A study of the noise in cable pairs leading from a central office indicate that impulse noise will cause much of the expected interference on pulse systems. In order to simulate the effect of this type of interference, a generator was built which produces uniformly shaped pulses over a wide range of rates. The polarity of these pulses can be reversed and their amplitude varied continuously from zero to a value exceeding the peaks of the signal pulses. These impulses were introduced into the center of a transmission cable through a high impedance. Plate V(b) shows photographs comparing the impulse with a signal pulse. The repetition rate for the impulse interference used in this investigation was lOVsec, which is low compared to the nominal pulse repetition rate of the signal (6.72 X lOVsec). With the relatively large separation be- tween interfering impulses, there is no measurable interaction between errors produced in the repeater. At the same time the impulse rate is high enough to get an excellent statistical distribution in the 10 second interval used in these measurements. 1078 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 4.3 Production of Impulse Errors — Nomenclature and Discussion To expedite the discussion of impulse errors, the following system of nomenclature is used. Any impulse having the same polarity as the signal pulse is designated as "plus." Those having the opposite polarity are "minus." Two types of errors are produced. First, a spurious pulse may be added to the regenerated signal; this is called an "insertion" error. Second, a signal pulse may be removed, which is called an "omis- sion" error. A "plus insertion" error is a spurious pulse introduced bj^ an impulse having the same polarity as the signal. A "plus omission" error on the other hand is pulse omitted because of a pulse of same polarity as the signal. A "minus omission" error is a pulse omitted be- cause of an impulse having a polarity opposite to that of the signal. A positive pulse, if large enough, can produce a spurious pulse at any instant of time not already occupied by a pulse. The only require- ment for the production of such a pulse is that the sum of the impulse and timing wave exceed the trigger level.* On the other hand, a nega- tive impulse cannot produce a spurious pulse but can only cause a signal pulse to be omitted. If a pulse is to be omitted the sum of its amplitude, the timing wave and the impulse must not exceed the trigger level. It would be expected that the number of plus insertion errors will exceed the minus omission errors. This follows from the fact that a spurious pulse may be produced at any point not already occupied by a pulse. On the other hand if a signal pulse is to be omitted the negative impulse must occur in the time interval occupied by the signal pulse. A positive impulse is indirectly responsible for the positive omission error. When a spurious pulse is produced a short interval of time ahead of a signal pulse, the latter may be removed by the inhibiting reaction of the spurious pulse. There is no apparent way in which a minus insertion error can be produced. This is confirmed by the fact that no error of this type was observed in this investigation. Thus we have three types of errors produced: plus insertion, minus omission and plus omission. 4.4 Results of Impulse Interference Measurements Preliminary measurements of errors as functions of impulse amplitude were made using random code. These measured values, shown on Fig. 8 exhibit many of the expected characteristics. For example the insertion errors are more numerous than the omission and the threshold of the plus omission errors is considerably higher than those of the other two. * The trigger level is normally considered to be the negative dc bias applied to the emitter of the blocking oscillator. There are however other components of the bias that will be discussed later. TRANSISTOR BINARY PULSE REGENERATOR 1079 If) / UJ X 0) / _1 J fi 2 25 ^ / / 5 / ^ / PLUS /o/ o INSERTION/ /' z ERRORS/ / / / fe20 A / / / a. LU m 5 Z) / ^ / / MINUS / / OMISSION z f / / ERRORS, s" c/ ^^ — ^"^ / / " '""'^ o / ^^^^ 1— / / y^ > o / ^ / / • 1- 2 10 / / / / / f A' f / P / / / / / / / 1 / A^' •■^ UJ Q- 1 1 / r^ CALCULATED X o A MEASURED f) 1 1 / / < tr ^ o \ / y 9^' CE 7 / PLUS UJ A / / / OMISSION ^.5. , ^ /J ERRORS ^x-^'- '■^' I ^ ^x^^'^ 0 wr / v „ X — "—^ "T"*'^ 40 50 60 70 80 90 100 IMPULSE AMPLITUDE AS PER CENT OF SIGNAL AMPLITUDE Fig. 8 — Repeater errors as a function of interference amplitude. On the other hand there are some deviations from the simple theory of a perfect regenerator such as the low common threshold value of the plus insertion and minus omission errors. Some of the differences can be attributed to the extremely sensitive method of measuring errors. Here the maladjustments of timing tank circuit, quantized feedback ampli- tude as well as other factors which cannot be readily detected by other means are reflected as sources of error. However with care these errors can be made small and the measured values should follow the theoretical values reasonably well. Most variations from theoretical values are due to changes in the effective bias caused by intersymbol crosstalk. This can be demon- strated by measurements made using set codes. In all these codes the number of pulses equaled the number of blanks but combinations varied from one to another. On Fig. 9 the omission errors are plotted for a fixed impulse amplitude as a function of the nimiber of pulses which 1080 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 1-4 CODE; 1 CODE COMBINATIONS ' juiriji 2 jirL.._.._n_rL.._.. 3 iirL.A..A.._.. - _rL..A..jL._n... 1 2 3 4 5 6 NUMBER OF PULSE - BLANI<, COMBINATIONS IN EACH CODE GROUP Fig. 9 — Repeater errors as a function of pulse distribution in code. are followed by a space in the particular code. The codes used for various points on the abscissa are shown on the graph. The omission error curves plotted in this manner are linear. These data demonstrate that the presence of a pulse modifies the trigger level in the next timing interval. This is largely due to the negative excursion of the damped cosine volt- age from base to ground in the blocking oscillator. On Fig. 10(a) is shown the circuit of the single shot blocking oscillator used in the repeater. With no timing an incoming signal must overcome bias V dc to trigger the repeater. The solid curve on Fig. 10(b) shows the dc bias with the timing wave added at the blocking oscillator emitter. Fig. 10(c) shows the base voltage when a pulse is produced in the first timing inter- val. The pulse begins at U and ends at U . As previously mentioned the sudden rise of the base and collector impedance coupled with the fall of the current in the transformer windings, produces an inductive voltage surge across transformer Tz at h . The decay of this voltage surge can be controlled by the inductance of the transformer and the damping resistor Rf, . This positive decay voltage across the base will inhibit the blocking oscillator from triggering. It is essential that this decay be adjusted so it will inhibit triggering until the following time slot. If TRANSISTOR BINARY PULSE REGENERATOR 1081 the decay transient is a damped oscillation and the base voltage passes through zero at the next normal triggering time, sufficient damping must be provided so the negative excursion is negUgible. The dashed line shows how the effective bias at the emitter is modified by this voltage across the base. Fig. 1 1 shows the measured values of plus insertion and minus omis- sion errors for two set codes. These are plotted as functions of impulse amplitude. The first code has alternate pulses and blanks while the second consists of pairs of pulses separated by pairs of blanks. With these two curves the error threshold values may be determined from REPEATER BLOCKING OSCILLATOR OUTPUT NO. 2 PLUS I INSERTION I THRESHOLD I TIME TIME Fig. 10 — (a) Circuit diagram of blocking oscillator showing various compo- nents of the effective bias, (b) The effective bias as a function of time, (c) Inhibit- ing voltage Vb produced by a regenerated pulse. 1082 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 30 UJ CODE rL.Ji..Ji..Ji nR....jin_._. CALCULATED MEASURED o X 45 50 55 60 65 70 75 80 IMPULSE AMPLITUDE AS PER CENT OF SIGNAL AMPLITUDE Fig. 11 — Calculated and measured repeater errors for two set codes. 1 ST NEGATIVE OMISSION THRESHOLD ^ PULSE HEIGHT 2 ND POSITIVE INSERTION THRESHOLD = C=Vd 1ST POSITIVE INSERTION THRESHOLD = b Fig. 12 — Bias levels used in calculating repeater errors. the points of discontinuity. Fig. 12 illustrates these various error thresh- olds with reference to a signal pulse. Theoretical curves were plotted using these values and the observed values of timing and signal aniph- tudes as shown on Fig. 11. It can be seen that very good agreement exists between the measured and computed values. The separate lower thresholds for insertion and omission errors may TRANSISTOR BINARY PULSE REGENERATOR 1083 be explained from Fig. 10(b). These are caused by the phase shift intro- duced by the inhibiting voltage to the effective bias compared to that of the timing wave. The omission thresholds are determined chiefly by the maximum signal amplitude. On the other hand the insertion thresh- olds are determined by the point of maximum trigger bias. There exists then two separate threshold values for a timing interval which follows a regenerated pulse. These values can be measured from points "a" and "b" on Fig. 10(b). 4.5 Result of Sinusoidal Interference Measurements On Fig. 13 are shown the errors produced by sinusoidal interference. Here a 110-kc sine wave is added to the signal and the various types 10 1.0 a: o (£ UJ to I- 3 m UJ 2 10" li. o 10" OJ u a. 10" 10" ^^ TOTAL errors yP^ <** ^ V"^ x'' / / /' / / l/l 7 l/l ///omission /// errors 1 1 1 HI pi 1 ' 1 1 'I ' 1 INSERTION ^ ERRORS '' / 1 1 1 f 50 55 60 65 70 75 80 PEAK-TO-PEAK AMPLITUDE OF INTERFERENCE X 100% 85 Fig. 13 - interference PEAK AMPLITUDE OF SIGNAL Repeater errors as a function of interferences level for sinusoidal 1084 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 of errors counted. Random code was used in this case and the repeater bias was adjusted to provide equal omission and insertion thresholds. The threshold for this particular case occurred when the peak to peak sinusoidal interference was 63 per cent of the signal amplitude. This is lower than the theoretical maximum which with a constant bias centered at the half amplitude point, would be 100 per cent of the peak to peak signal amplitude. For the bias conditions illustrated on Fig. 12, this percentage would be 86 per cent for the positive insertion threshold and 88 per cent for the minus omission. This becomes apparent when the negative and positive excursions of the interfering sine wave are con- sidered as minus and positive impulses respectively. The remaining loss in the interference margins can easily be due to maladjustments of tim- ing, quantized feedback or inhibiting. When the frequency of the sinusoidal interference is varied, the number of errors for a constant interference voltage at the blocking os- cillator emitter does not change appreciably. However, the input trans- former and condenser coupling introduce a substantial frequency charac- teristic. This reduces considerably the errors caused by power line crosstalk. One of the striking things about the sinusoidal interference errors is the rate at which they increase above the threshold. For ex- ample, a change of 1 per cent of the interference amplitude can triple or quadruple the total number of errors. 5.0 SUMMARY New techniques and devices now" make it possible to build practical regenerative repeaters for use in digital transmission. Such a repeater which is suitable for a 12-channel, 7-digit PCM system, is discussed. Simple, inexpensive devices are used to eliminate the effects of distortion due to low frequency cutoff and to provide self timing for the circuit. Experimental evidence is presented which shows the repeater to func- tion as expected. ACKNOWLEDGEMENTS I am deeply indebted to J. V. Scattaglia for his aid in tliis project and to the pioneering work of A. J. Rack on quantized feedback which was of great help in the development of this regenerative repeater. I also wish to thank W. R. Bennett, C. B. Feldman and Gordon Raisbeck for their aid and many valuable suggestions. Transistor Pulse Regenerative Amplifiers By F. H. TENDICK, JR. (Manuscript received April 5, 1956) A pulse regenerative amplifier is a histate circuit which introduces gain (ind pidse reshaping in a pulse transmission or digital data processing system.. Frequently it is used also to retime the ptdses which constitute the flow of information in such systems. The small size, r-eliahility , and low power consumption of the transistor have led naturally to the use of the transistor as the active element in the amplifier. It is the purpose of this paper to describe some of the techniques that are pertinent to the design of .synchronized regenerative amplifiers operating at a pulse repetition rate of the order of one megacycle per second. An illustrative design of an amp- lifier for use in a specific digital computer is presented. 1. INTRODUCTION A basic building block of many modern digital data processing or transmission systems is a pulse regenerative amplifier. The particular high speed transistor regenerative amplifiers to be discussed in this paper are intended for use in systems where the logic operations on the digit pulses are performed by passive circuits and the amplifiers are inserted at appropriate intervals to amplify, reshape, and retime the pulses. The design of these amplifiers for any specified system involves a knowledge of the environment of the amplifier in the system, a study of possible functional circuits which are combined to form an amplifier circuit, and the selection of a combination of these functional circuits to achieve the desired amplifier performance. Although a study of the functional circuits constitutes the major portion of this paper, the design of an amplifier for a particular digital computfM' is presented to illustrate the general design procedure. One important way in which these amplifiers differ from many pulse amplifiers is that they must function properly under adverse conditions. That is, instead of merely expecting superior performance most of the 1085 1080 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 time under relatively special operating conditions, consistently good performance is demanded at all times, even with wide variations of cir- cuit parameters and operating conditions (as, for example, a twenty-to- one variation in the required output current). Therefore, various circuit possibilities will be examined from the standpoint of reliable per- formance. When the switching and mathematical operations of a digital data processing system are accomplished by a network of passive logic cir- cuits with amplifiers interspersed to overcome circuit losses,^- ^ the environment of an amplifier is generally as indicated in Fig. 1. The signal information that passes from one logic network to another is represented in a code by a group of discrete pulses. Due to the nature of this digital information, utmost reliability of each amplifier is an important requirement that greatly influences the amplifier design. Since the position of a pulse in time or place determines its significance to the system, it is necessary that each pulse be identically amplified and that noise or extraneous disturbances do not cause false output pulses from an amplifier. The effect of an error or a failure in operation is different for different systems and in a given system depends upon the time or place of the failure. In some computers a single mistake will invalidate an entire computation cycle, while a permanent failure of even a single amplifier will cause complete system failure in almost any digital machine. Experience with the type of amplifier under discussion indicates that failure rates of less than a tenth of one percent per thou- sand hours are attainable. Jf LOGIC LOGIC t! LOGIC DELAY u it LOGIC U LOGIC It LOGIC LOGIC n DELAY Fig. 1 — Typical environment of an am])lifier. TRANSISTOE PULSE REGENERATIVE A.MPLIFIERS 1081 This goal of reliable circuit operation can be realized if the amplifiers have : a. Simple circuitry with a minimum number of parts. b. The ability to operate with wide variations of signal level. c. Ample margins against crosstalk and noise. d. Low sensitiveness to changes in component values. e. Low power dissipation to realize long component life. f . Sufficient gain margins with system variations. Although these features are desirable in any circuit, they are often subordinated in order to obtain special performance, usually at the ex- pense of reliability. In the amplifiers under discussion these features rep- resent the primary design goal. As is so often true, some compromises usually must be made to obtain a suitable balance of these features in a particular design. It is sometimes possible to accept an increase in power consumption for other desired performance. However, because of the large number of amplifiers em- ployed, low power operation is desirable in order to reduce the physi- cal size and weight of a system. In this paper considerable emphasis is placed on efficient low power circuits which do not require critical com- ponents. A convenient way to study regenerative amplifiers is to consider an amplifier as a small system. The following functional breakdown has- been found useful: a. Transistor properties. b. Feedback circuits. c. Input trigger circuits. d. Output coupling circuits. e. Synchronizing circuits. The block diagram of an amplifier then might take the form shown in TIMING SIGNAL INPUT SYNCHRONIZING CIRCUIT FEEDBACK CIRCUIT ' ' 1 SIGNAL INPUT TRIGGER CIRCUIT TRANSISTOR OUTPUT CIRCUIT OUTP ~ "*" Fig. 2 — Regenerative amplifier block diagram. 1088 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Fig. 2. In the following sections the relation between each of the above functional features and amplifier performance is discussed, various circuit configurations to achieve each function are investigated, and the inter- actions between the functional circuits are examined. The design of any particular ampUfier then consists of a suitable selection of a transistor and functional circuits to achieve the desired amplifier performance. 2. TRANSISTOR PROPERTIES In a regenerative amplifier the transistor operates as a switch with power gain. The "on" and "off" state usually are characterized, re- spectively, by high and low collector current levels, and changes of state are initiated by applied control signals. The performance items of interest are the power dissipation in the two states, the speed with which the transistor changes state, the amount of power gain available, and the attainable margins against false operation. The transistor parameters related to these items, as discussed below, are listed in Table I with typical values for several classes of transistors. Desirable and satisfactory values have been indicated in italics. The power dissipated in a transistor in the "off" state is proportional to Ico , the collector current with the emitter open circuited, and to the collector supply voltage. This is wasted power and, since the minimum collector supply voltage usually is dictated by other considerations, a low Ico current is desirable to reduce standby power. Point contact units are relatively poor in this respect. In junction imits the 7,0 power is almost negligible compared to other circuit standby power. The power dissipated in a transistor in the "on" state is proportional to the saturation voltage between the collector and the common terminal. Table I — Transistor Switching Properties Switching Features Ico at Vc = lOv Collector to emitter satura- tion voltage at Ic = 10 ma. fa cut-off Base resistance Collector capacitance at Vo = lOV Collector breakdown volt- age Punch through voltage llmitter breakdown voltage Ratio of alpha at le = 10 /xa to alpha at lo = 1 ma . . . Point Contact Transistors (Low Resistivity Ge) 1500 Ma 0.8 V 15 VIC SO oJmis 0.5 UUF 40 V no punch through 40 V 3 Junction Triode Transistors Ge Grown 5 fxa 0.5 V 2 mc 500 ohms 10 UUF 100 V 100 V 5 V 0.8 Ge Alloy 5fxa 0.05 V 4 mc 100 ohms 20 UUF 35 V 35 V 35 V 0.8 Si Grown 0.01 iM 4 V 4 mc 500 ohms 10 UUF 100 V 100 V 1 V 0.6 TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1089 Again, this represents wasted power, but also important is the fact that it places an upper limit on the output power available from the transistor. Hence, it is desirable to have as low a saturation voltage as possible. Alloy junction transistors are especially good in this respect. The speed with which a transistor changes state is principally a func- tion of the alpha cut-off frequency (which should be high), base re- sistance, and collector capacitance (both of which should be low).*'* Both the rise and fall times of the transistor response are greatly in- fluenced by the associated circuitry; generally a blocking oscillator circuit yields the fastest response. The amount of effective power gain available from a regenerative amplifier is influenced by two transistor properties. One property is the breakdown voltage, which may be the collector to base breakdown volt- age or the collector to emitter punch through voltage (whichever is lower). This limits the output power by limiting the collector supply voltage. The other factor is the variation of alpha with emitter current, especially at low emitter currents. The minimum average emitter current required to initiate self-sustaining positive feedback determines the minimum input power. Point contact units are especially good in this respect in that alpha may approach ten at emitter currents as low as five microamperes. Junction units are poor since alpha generally de- creases rapidly at emitter currents below one hundred microamperes. Even though the attainable margins against false operation are largely a matter of circuit design, two transistor properties occasionally become important. In point contact units trouble with lock up in the "on" state may occur due to internal base resistance. Although this property of base resistance is exploited in negative resistance feedback circuits, it is un- desirable in circuits where the feedback is obtained by external coupling. In grown junction units the emitter to base reverse breakdown voltage may limit the voltage margin against false triggering caused by noise or crosstalk. Normally it is desirable to have a one or two volt margin. From the above discussion it can be seen that no one type of transistor is outstanding in all features. The choice of which unit to use in a specific amplifier depends upon the repetition rate, gain, and power requirements desired of the amplifier. Although the point contact type has the best overall performance of the types shown in Table I, it is quite possible that new types (such as PNIP or diffused triodes^^) and improved de- signs of the present types will change the picture. 3. FEEDBACK CIRCUITS The use of positive feedback in an amplifier results in high gain and short rise time. If the input circuit is isolated from the feedback loop by 1090 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 a diode or large resistor, these effects are enhanced and the shape, dura- tion, and ampHtude of the output signal become independent of the input signal. These results are possible because once the circuit has been triggered and the feedback loop gain is greater than unity, the response proceeds independently of input conditions and is determined solely by the transistor and circuit parameters. By definition a regenerative amplifier must have positive feedback sufficient to cause instability during the transition period between the "off" and "on" states. When investigating various circuits, it is neces- sary to eliminate circuits which are never unstable when a pulse is applied to the input circuit. If the circuit is unstable under either of the conditions shown in Fig. 3, sufficient instability is possible. However, if the circuit is stable, linear, and either the small signal open circuit voltage gain or the short circuit current gain is less than unity or nega- tive at all frequencies, it is impossible to have instability. These latter conditions for instability often can be easily checked by inspection with- out tedious computation or experimentation. This use of positive feedback requires that attention be given to its control. To be useful, the amplifier must be stable in one state and at least quasi-stable in the other state. The change from instability in the transition period to stability in the end states is accomplished by a non- linear change in the gain or impedance of some element in the feedback loop. Usually the "off" state is made stable by causing the voltage and current conditions in the input circuit to reverse bias the transistor in- put. The "on" state may be made stable (or quasi-stable when there are reactive coupling elements in the loop) in several ways. For example, the transistor may be permitted to saturate when the desired pulse voltage is reached; a "catching" diode may be used to clip the pulse voltage at an appropriate level; or a current switch may be used to FEEDBACK TRANSISTOR FEEDBACK TRANSISTOR Voc ef'v iSC (a) OPEN-CIRCUIT LOOP VOLTAGE (W SHORT-CIRCUIT LOOP CURRENT Fig. 3 — A check for instability. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1091 introduce an impedance in the feedback loop at a predetermined current level. The degree of stability of the amplifier in the "on" state may be thought of as the amount of power required to initiate the transition to the "off" state. During the early portion of the output pulse duration the degree of stability should be large, but near the end of the pulse duration it should be relatively small to make turn-off easier. Also, the degree of stability should not change over the range of output loading expected for the amplifier and should be effected without excessive wastage of pulse or supply power. These conditions are difficult to fulfill when the range of output load current may be as large as 20 to 1 . Three methods of obtaining positive feedback in transistor circuits will now be considered: (a) negative resistance feedback; (b) capacitor coupled feedback; and (c) transformer coupled feedback. Of these, transformer coupled feedback appears to be the best for most applica- tions. It will be assumed that the type of feedback under discussion is the dominant or only type present; circuits employing more than one feedback mechanism generally violate the premise of simple circuitry and will not be discussed. 3.1 Negative Resistance Feedback With the advent of point contact transistors a novel form of negative resistance was offered to circuit designers for use in positive feedback applications.^ This negative resistance property occurs when the current gain of a transistor is greater than unity and the emitter and base small signal currents are in phase.* At first sight this property appears to lead to attractively simple regenerative amplifiers. However, as systems become more complex and, consequently, amplifier requirements more severe, the original simplicity often is lost due to the additional circuitry required to control the negative resistance. An example, shown in Fig. 4, is similar to a regenerative amplifier described by J. H. Felker.^ The functional circuits are indicated by dashed outlines. This amplifier operates at a one megacycle pulse repetition rate with one-half microsecond, three volt pulses. It is capable of driving from one to six similar amplifiers. The output pulse rise time is 0.05 microsecond, the average dc standby power is 33 milliwatts, only a few components operate at as much as half of maximum ratings, and the supply voltage marginsf are greater than ±15 per cent. Seven hundred of these ampli- * Although point contact transistors are noted for this property, certain types of junction transistors also exhibit it. For example, see Reference 7. t Supply voltage margins, the amount by which the supply voltage may be 1092 THE BELL SYSTEM TECHNICAL JOUKXAL, SEPTEMBER 1950 fiers operated in a system for over 17,000 hours with a faihire rate of] slightly less than 0.07 per cent per thousand hours. These features, however, are obtained at the expense of relative com-] plex circuitry. This negative resistance type of high speed regenerative amplifier has the following inherent limitations. 1. The degree of stability in the "on" state depends critically on the collector current. In the example a dummy load must be strapped in when the amplifier drives less than four logic circuits. 2. A steering diode (D3) and a timing circuit diode (Dl) have critical reverse recovery time^ specifications.* 3. The requirements on transistor parameters (primarily the dynamic alpha versus emitter current and base resistance characteristics) are relatively critical. 4. A relatively large amount of synchronizing power is required. 5. With transformer output coupling (as discussed in Section 5.1) a large amount of the total standby power is absorbed by a circuit required to protect the transistor in case the timing voltage fails (In the example 21 milliwatts, or 64 per cent of the standby power, is absorbed by R3.) INPUT TRIGGER CIRCUIT !+6V FEEDBACK CIRCUIT TRANSISTOR OUTPUTS 20V PEAK-TO-PEAK 1 MC SINE WAVE SYNCHRONIZING CIRCUIT OUTPUT COUPLING CIRCUIT DUMMY LOAD Fig. 4 — Negative resistance feedback amplifier. varied without causing an operational failure, are an indication of the sensitivity of the amplifier to changes in component values. * At lower pulse repetition rates this property may not be critical. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1093 The use of an inductor, instead of a resistance, in the base lead does not appear to mitigate the hmitations. * 3.2 Capacitor Coupled Feedback A second method of obtaining positive feedback is by external coupling through a capacitor or capacitor-resistor network. This method is sel- dom used for the principal feedback for reasons to be mentioned. Oc- casionally, in conjunction with some other type of feedback, it may be used to provide additional feedback during the rise time of an amplifier. Since the voltage and current gain of a capacitor can not exceed unity, the open circuit voltage gain and the short circuit current gain of the rest of the loop (Fig. 3) must be greater than unity for instability. This criterion indicates that capacitor feedback is limited to point con- tact, or other transistors with an alpha greater than unity, or to a junc- tion transistor in the common emitter configuration.* A circuit with capacitor feedback around a short-circuit stable point contact transistor might take the form shown in Fig. 5. Although this type of circuit has the merit of simplicity, it has the following limitations : 1. The initial feedback current is highly dependent upon the incre- mental output load impedance. This may result in a failure to trigger when the load approximates a short circuit, as in the case of diode gates or a large stray capacitance. 2. The degree of stability in the ''on" state is critically dependent on the load current and the collector supply voltage. Variations in either may cause a foreshortened output pulse or require an excessive timing signal current for turn-off. FEEDBACK CIRCUIT R V, \AAr TRIGGER CURRENT FROM INPUT CIRCUIT ■ INPUT TRIGGER CIRCUIT AAA- c SINE WAVE TIMING VOLTAGE SYNCHRO- NIZING CIRCUIT n:i .^^^UTPU T ^ TRANSISTOR I OUTPUT COUPLING CIRCUIT Fig. 5 — RC feedback amplifier. An inverting transformer is necessary with the junction transistor. 1094; THE BELL SYSTEM TECHNICAL JOvJRNAL, SEPTEMBER 1956 3. The necessity of a feedback circuit time constant equal to or shorter than the output pulse length results in a relatively low output power efficiency. Due to the above considerations, capacitor feedback appears to be the least attractive type of feedback. 3.3 Transformer Coupled Feedback A transformer appears to be the most convenient and versatile com- ponent for feedback coupHng in a regenerative amplifier. The pertienent features* of a transformer are: 1. Current or voltage gain (impedance matching.) This feature per- mits full use of the power gain of the transistor, even if such gain be in the form of voltage or current gain only. 2. Bias isolation between circuit parts and the possibility of supplying dc voltage bias without the use of additional elements. 3. Phase inversion, if desired. All of these featvu-es, conveniently combined in a transformer, provide great design freedom to meet specified circuit objectives. Since positive feedback is possible with any type of transistor (with power gain, of course), the choices of transistor and connection are determined by other circuit requirements. The use of transformer coupled feedback yields the familiar blocking oscillator circuit. An important feature of this circuit is the fast rise time that is obtainable. Linvill and Mattson^ have shown that a junction - transistor with an alpha cutoff frequency of two megacycles may exhibit a rise time of 0.1 microsecond in an unloaded blocking oscillator with collector to emitter coupling. Fig. 6 (a). It can be shown that the same response may be expected with collector to base or base to emitter coupling, provided that the transformer turns ratio is modified. Figs. 6 (b) and 6 (c) . When the circuit is providing useful output power into a load, a slightly different turns ratio would be used for optimum rise ■ time, which may be appreciably slower than in the unloaded case. How- ever, it should be noted that the foregoing gives no information about the initial response of the circuit from the time that the input trigger is applied until the output reaches ten per cent of its final value. In some instances this initial time, which is a complicated function of the trans- istor non-linearities, may be comparable to the output rise time. In a blocking oscillator circuit with a fixed output load, the degree of stability in the "on" state decreases with time. The reason is that the * The operation of a transformer over the non-linear portion of its magnetiza- tion characteristic is outside the scope of this paper. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 109 O voltage across the coupling transformer, which is approximately constant during the pulse duration, causes an increasing magnetizing current to be subtracted from the initial feedback. When the feedback current can no longer support the required output current, the circuit turns off. In a synchronized amplifier the value of the feedback transformer mutual inductance may be specified to give the desired degree of stability at the end of the predetermined pulse length. Thus, the least stable condition occurs at the end of the pulse duration and is under the circuit designer's control. At other times during the pulse duration the circuit is more stable, which reduces the possibility of premature turn-off. Other considerations, such as stability variations with output current, power dissipation, and output voltage regulation, depend upon whether the output load is in series or in shunt with the feedback loop. Therefore, these considerations are discussed in connection with output coupling in n+i: 1 (a) COMMON BASE (b) COMMON EMMITER n+1 (C) COMMON COLLECTOR P + CVc te (d) ASSUMED TRANSISTOR EQUIVALENT CIRCUIT Lg = LEAKAGE INDUCTANCE OF TRANSFORMER n = TURNS RATIO FOR COMMON EMMITER CONNECTION CCq = LOW FREQUENCY VALUE OF COMMON BASE SHORT CIRCUIT CURRENT GAIN CVq = CUTOFF RADIAN FREQUENCY OF Ot Fig. 6 — Transformer coupled blocking oscillator circuits. 109G THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Section 5.2. For constant voltage, variable current loads, transformer coupled feedback with the output load in series with the feedback loop results in low power dissipation, relatively small degree of stability variations versus output current variations, and non-critical com- ponents. The possible limitations are that transformers generally are more expensive than other passive components and are not as readily available in a variety of stock values. 4. INPUT TRIGGER CIRCUITS The primary function of the input trigger circuit is to initiate the tran- sition from the "off" to the "on" state when there is an input signal. At all other times the input circuit must provide a threshold or margin against false triggering due to noise or spurious disturbances. Although the input circuit must supply sufficient energy to establish regeneration, it is unnecessary and undesirable that any additional energy be supplied. To do so reduces the gain of the amplifier, since gain may be defined as the ratio of the output power to the input power during one cycle of operation. Because regeneration makes the input and output power independent of each other, any reduction in input power results in greater amplifier gain. In an amplifier with external feedback coupling it is possible, but not always practical, to have the input circuit trigger the transistor at the collector, base, or emitter terminal. The collector terminal seldom is selected because then the input circuit must supply energy to the output load as well as to the transistor. Also, the base is usually not used (ex- cept occasionally with negative resistance feedback) because extra com- ponents are required to steer the triggering energy into the transistor and it is diffiult to apply a timing signal.* However, the following dis- cussion and the dc input characteristic of Fig. 7 (a) are equally valid for triggering at the base or emitter terminal of junction or point contact transistors which are short-circuit stable. One of the simplest types of triggering circuits is shown in Fig. 7 (b). The voltage and current increments assumed necessary to initiate regen- eration are designated Vt and /, . Therefore, the required input signal voltage Vs and current /« are : Vs ^ V, + IJix (1) ^ Ri\ , Vt , Fi - V2 '■^'■['+m) + w. + '^^ '■■'^ * Also, for junction tninsistors, about twice as much energy is required to trig- ger at the base as at the emitter.' TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1097 The purpose of diode Di is to provide a low impedance current threshold, the amount of current given by the last term of (2) . This type of thresh- old is especially effective for preventing false operation from electro- statically induced crosstalk. Also, it allows a faster rate of discharge of stray capacity on the input terminal at the end of the input pulse period. Although the circuit of Fig. 7 (b) is attractively simple, it is undesir- ably sensitive to variations in signal voltage. An increase in the input pulse voltage causes excessive triggering current and a decrease may easily result in failure to trigger. Since the circuit must be designed to operate reliably with the smallest expected input pulse, it is wasteful of input power with the average amplitude of input pulse. '1 h-it A f ^1 — V, (a) TRANSISTOR INPUT CHARACTERISTIC .Sr\ J, ± D1 I V, R1 TRIGGER CURRENT R2 (b) SIMPLE INPUT CIRCUIT V^ ,1^ V, D2 R2 I+V3 'R3 It TRIGGER CURRENT V ' r'l A 1 ^ s, T- / _j V, (C) ONE TERMINAL AND-TYPE INPUT CIRCUIT Fig, 7 — Input trigger circuits. 1098 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 The single terminal AND-type circuit^ ■ ^° Fig. 7 (c) has the desirable characteristics of the previous circuit, and is relatively insensitive to input signal variations. In this circuit the input pulse switches the cur- rent through R3 into the transistor input and then encounters the rela- tively high resistance R2, as compared to the parallel resistance of R2 and El in Fig. 7 (b). The blocking action of D2 thus reduces variations in the input signal current. However, R2, R3, F3 and V2 cannot be increased without limit to reduce the variations; the dc power dissipated in R2 and R3 would become excessive. Another advantage of the AND-type circuit is that several inputs may be paralleled with a common R3 to provide an AND logic function as well as an input trigger function. This feature, when desired, saves com- ponents and does not reduce the gain of the amplifier. When both the input circuit and the feedback circuit terminate at the same transistor input terminal, as is usually the case, some additional components are generally required to prevent one circuit from shunting the other circuit. To steer the trigger current into the transistor, a diode may be placed in the feedback path so that the diode is reverse biased except when there is feedback current. Similarly, a diode or a resistor may be placed in the input circuit so as to prevent the feedback current from flowing into the input circuit.* Although the discussion has assumed positive polarity input pulses, the remarks apply equally well to negative pulses if the polarity of the diodes and the supply voltages are reversed. It is recognized that the preceding remarks assume that the minimum triggering energy is known and that a step function of current or voltage is the optimum form of the triggering energy. Actually, until a study is made of the circuit and transistor parameters (including the non-linear aspects) that affect the initial triggering before the feedback is estab- lished, the design of an optimum input trigger circuit will remain an experimental art. Experience with the AND-type input circuit has indi- cated that appreciably more current is required to trigger junction luiits than point contact units. 5. OUTPUT COUPLING CIRCUITS In addition to the obvious function of efficient power transfer from the amplifier to a load, the output coupling circuit is a convenient point at which to perform other functions, as for example, dc level restoration i * This precaution is not necessary if the transistor input exhibits appreciablr negative resistance. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1099 [and pulse inversion. In a system of logic circuits interspersed with ampli- fiers at regular intervals, it is apparent that the dc level at similar points, such as the outputs of the amplifiers, must be identical if the amplifiers are to be interchangeable. Without some circuit or element to restore the dc level, the levels along the transmission path will monotonically de- crease* due to the dc voltage loss through the logic circuits and across the transistor in the amplifier. The output circuit is one point where restoration of the dc level may be readily combined with other functions.! In the following two sections three methods of output coupling are discussed and the interaction between the output and feedback circuits is considered. 5.1 Output Coupling Elements Three types of coupling circuits are RC, transformer, and diode cou- pling. Each of these methods permits the dc level of the signal pulses to be corrected to a predetermined level. However, the restoration,! efl&ciency, and versatility characteristics of each circuit are quite different. Although RC coupling is common in linear amplifiers, it is seldom used in transistor pulse amplifiers that operate at duty cycles near 50 per cent. The reason is that the time constants encountered do not permit both proper restoration of the capacitor and high efficiency of the output circuit. As indicated in Fig. 8 (a), the transistor is a low impedance in TRANSISTOR TRANSISTOR ON R1 yOFF (SMALL) ".RS -—\ I >(LARGE) ■R3 (a) RC COUPLING (b) TRANSFORMER COUPLING Fig. 8 — Reactive output coupling circuits. * Decrease for positive pulses; increase for negative pulses. t An exception, to be discussed, is diode output coupling where it is occasionally more convenient to correct the dc level in the input of the logic circuits or the amplifier. t This refers to restoration of a reactive element (i.e., the return to a quiescent state) and is not to be confused with restoration of the dc level of a circuit. llOO THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 the "on" state and a high impedance in the "off" state. Since C must be relatively large to make the voltage drop across it small during the pulse duration, R3 must be equal to or smaller than Rl for satisfactory restora- tion (50% duty cycle assumed). But then the current transmission efficiency of the coupling network is less than 50 per cent because gener- ally Rl is smaller than the input resistance of the driven circuits during the pulse duration. Unless the pulse length is only a small fraction of the pulse repetition period, it is seldom possible to effect a suitable compro- mise. Also, it might be noted that variations of Ico current, which flows through R3, cause variations in the output pulse amplitude. Finally it is not possible to obtain pulse inversion. A transformer coupled circuit. Fig. 8 (b), works efficiently with a transistor. Diode D2 isolates the transformer from the load and inter- lead stray capacitance during the interdigit period* so that the restora- tion time of the transformer is controlled by the value of R3. The restora- tion time is approximately proportional to the mutual inductance divided by the total shunting resistance. Diode Dl prevents R3 from shunting down the output during the pulse duration, thus permitting high output efficiency and proper restoration. f As noted in Section 2, the maximum output power from the transistor is determined by the maximum collector voltage (as set by breakdown or punch-through) and the maximum collector current consistent with the permissible dissipation in the transistor. Usually this maximum voltage exceeds the desired amplifier output voltage and, occasionally, the maximum collector current is insufficient; in such instances a voltage step down is desirable. When the transistor is not required to operate at maximum power dissipation, it often is advantageous to balance the "off" and "on" power dissipation. An increase in the collector supply voltage increases the "off" power and decreases the "on" power (by decreasing the required collector current for the same output power). Thus the collector voltage may be adjusted to give the lowest total power dissipation consistent with the average duty cycle of the amplifier. The transformer turns ratio is specified to match the optimum collector voltage to the desired output voltage. Furthermore, Ico variations have negligible effect on the output voltage amplitude and pulse inversion (if desired, for example, for inhibition) is possible. For these reasons trans- former coupling appears to give optimimi output coupling performance. * The minimum time interval between the end of one pulse and the beginning of a succeeding pulse; for a 50 per cent duty cycle the interdigit period is equal to the pulse duration. t Occasionally it is possible to specify the collector impedance, the transformer losses, and the reverse impedance of D2 so that Dl and R3 are not necessary. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1101 A third method of couphng, which is attractive for systems using only AND- and OR -type logic, utilizes the reverse characteristic of a break- down diode, Fig. 9 (a). The interesting feature of this diode is the sharp transition between the high and low incremental resistance regions of the reverse characteristic. With this diode it is possible to shift dc levels by an amount ec^ual to the rcA^erse breakdown voltage of the diode, as indi- cated in Fig. 9 (b). In the ciuiescent state D2 operates in the breakdown region and Dl serves to clamp the collector voltage at — F3 ; during the pulse duration D2 operates in the high resistance portion of its reverse characteristic. If the driven circuit has a ^'oltage threshold, like the transistor threshold in Fig. 7 (a), less than — F^ + Vb + Vs and Vs <|Fb|, the circuit operates like a normal AND-type circuit except for the dc level change. For this reason it is convenient with AND-OR logic circuits to include only Dl and R2 in the output circuit of the amplifier and use D2 and Rl as the AXD input elements in the logic circuits. The principal advantages of diode coupling are simplicity and the lack of an energy storage element. The limitations are that there is no opportunity to match transistor and output conditions, variations in -V ♦--BREAKDOWN REGION -*• ' +v- RRFAKDOWN ^| VOLTAGE 3W Vb I 1 ; ' (a) BREAKDOWN DIODE V-I CHARACTERISTIC -V. If^ Dl BREAKDOWN DIODE N I W l+v, R1 -V3) OUTPUT D2 (-V3 + VB) R2 -V3 -V2 (b) COUPLING CIRCUIT UTILIZING A BREAKDOWN DIODE Fig. 9 — Direct output coupling circuit. 1102 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 diode breakdown voltage reduce amplifier margins, and pulse inversion is not possible. For these reasons diode coupling has limited utility, but is attractive for some applications. 5.2 Connection of Output and Feedback Circuits The performance of the amplifier is greatly affected by the method used to connect the output circuit and the feedback circuit together at the | output of the transistor. Should these two circuits be connected in a shunt or a series fashion? Performance features, such as rise time, suf- ficient output voltage, degree of stabihty versus load current variations, and power dissipation directly depend upon this choice. With transformer output coupling, the choice always exists; with other types of output coupling the choice may or may not exist, depending upon the type of feedback coupling. The following discussion is in terms of transformer coupled output and feedback circuits and the general conclusions may be extended to other cases. Y\ . t-- OUTPUT OF AMPLIFIER _!_ ■ LOGIC CIRCUIT . Vs 1 NUMBER 1 , i '^ LOGIC CIRCUIT Vs NUMBER n (a) CONNECTION OF AMPLIFIER LOAD ■nl. I T (b) V-I CHARACTERISTIC OF AMPLIFIER LOAD Fig. 10 — Output load characteristic. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1103 The principal factor that influences the choice of the output-feedback connection is the nature of the output load of the ampHfier. In the majority of computer and switching systems the ampUfier must drive a multiplicity of paralleled load circuits, as indicated in Fig. 10 (a). The input characteristic of each load circuit is assumed to be of the threshold type, like the AND-type input characteristic of Fig. 7 (c), which results in the amplifier load characteristic of Fig. 10 (b). During the initial por- tion of the rise time of the output pulse the incremental impedance is almost zero and during the remainder of the pulse duration it is rela- tively large. Due to the voltage threshold nature of the load, the ampli- fier load variations are current variations at a constant voltage. The minimum current is encountered in the system position where the ampli- fier drives the smallest number of logic circuits, often a single logic cir- cuit; the maximum current is hmited by the maximum output power of the amplifier. Although a desirable ratio of maximum to minimum cur- rent may be as high as 20 : 1 , the amplifier is expected to exhibit optimum performance at any load current within this range. The shunt connection of the output and feedback circuits is illus- trated in Fig. 11.12 Windings l:wi constitute the feedback couphng and 1 : Hi the output coupling. The two circuits shunt each other in the sense that the ratio of the feedback to the output current is determined by the ratio of the impedance of these circuits as modified by the turns ratio of the transformer. INPUT TRIGGER CIRCUIT -TL OUTPUT SINE WAVE TIMING VOLTAGE SYNCHRONIZING CIRCUIT OUTPUT COUPLING CIRCUIT Fig. 11 — Shunt connection of output and feedback. 1104 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 There are two limitations associated with this output-feedback con- nection. In the first place there is the possibility of insufficient output voltage, slow rise time, or complete faihu'e of regeneration. This is caused b}' the shunt effect of the output load which places an almost zero initial incremental impedance across the feedback path. In order to overcome this limitation a current switch (R5 and DO in Fig. 11) is used to obtain a low initial feedback impedance and the output diode (D4) is reverse biased so that the initial load impedance is large. The price paid is the undesirable power dissipation in the current switch. ]\Iore- over, stray capacity across the output terminal or a load current that exceeds the design value may still result in a long rise time, low output voltage, or regeneration failure. The series connection of the output and feedback circuits is shown in Fig. 12. In this connection the output load is in series with the feedback loop. Thus, the transistor output current, feedback current, and output load current are all proportional to each other. This situation assures regeneration regardless of output load current variations. The regeneration cycle of the series type amplifier is as follows. In the quiescent state diode D2 is reverse biased by VI to prevent false triggering. After the arrival of an input signal, the timing signal voltage goes positive and steers the trigger current into the transistor. Xo TRIGGER CURRENT R1 INPUT TRIGGER CIRCUIT FEEDBACK CIRCUIT D2 1^ Dt TIMING SIGNAL SYNCHRONIZING CIRCUIT Lp TRANSISTOR D6 I I I I V4 »l t _rL OUTPUT I I OUTPUT COUPLING Fig. 12 — Series connection of output and feedback. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1105 appreciable output current flows until the voltage across transformer Tl is sufficient to forward bias diode D2. Then both the feedback and output current build up simultaneously and rapidly since the turns ratio l:ni of Tl is selected to give a feedback loop gain greater than unity. When the sum of the voltages across the primaries of the feedback and output transformers almost equals the collector supply voltage, the transistor saturates and stabilizes the feedback loop. At the end of the pulse dura- tion the timing signal voltage goes negative and robs current from the feedback loop, thus forcing the transistor out of saturation and causing the amphfier to turn off. Because the feedback current is proportional to the output current during the rise time, the amplifier can deliver any value of load current up to the current corresponding to the maximum allowable collector current. Also, assuming that the leakage inductances of the transformers are small, a large stray capacitance across the output terminal does not appreciably degrade the rise time. Since a current switch is not neces- sary, the standby power dissipation in the feedback loop is negligible. These are the outstanding features of the series connection. Two important performance considerations of the series type amplifier are the change in the degree of stability versus load current variations and the action of the amplifier when the timing signal fails. Both of these items may be controlled by the selection of suitable values for the turns ratio and the primary inductance of the feedback and output transform- ers.* In order to prevent burnout of the transistor in the event that the timing signal fails, the amount of excess feedback current must decrease during the pulse duration. Due to the low impedance of the feedback loop, this condition may be approximately f stated in terms of the pri- mary inductances as: Vi niLi > a (3) where Vsat is the collector saturation voltage and Li and L2 are the primary inductances of Ti and T2 respectively. The degree of stability in the series amplifier at the end of the pulse duration is proportional to the output load current. This situation may be seen more clearly if a "catching" diode (D6 in Fig. 12) is added to the * If the transistor is not short circuit stable, it is also usually necessary to use a small resistance in series with the emitter. t The principal approximation is that alpha is constant versus collector current. The value of alpha at the end of the pulse duration is a conservative value. 1106 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 circuit to prevent saturation in the transistor,* Because the feedback loop gain, as determined by the alpha of the transistor and the turns ratio 111 , must be greater than unity for regeneration reasons, there will be current flow through D6 during the pulse duration. This current is proportional to the degree of stability. An increase Aiout in the output current causes an increase of Az, = ^^^^'-^ (1) in the collector current. Therefore, the current in D6 increases by an amount equal to AZdb = ^-1 \_ni riiAiont (5) This variation in the degree of stability may be reduced by selecting a/rii close to unity and reducing no . However, since it is desirable to have a/rii much larger than unity for short rise time and since any reduc- tion in n2 increases the Ico standby power, f a compromise is necessary. 6. SYNCHRONIZING CIRCUITS The majority of modern digital data processing systems employ coin- cidence gate circuits to perform the logical functions. In order to insure that digit pulses will coincide at the inputs to the logic circuits, it is con- venient to synchronize the amplifiers. Usually a master oscillator, or "clock," produces the timing signals that are distributed to the ampli- fiers. The function of the synchronizing circuit in the amplifier is to turn on and to turn off the amplifier at predetermined time intervals in re- sponse to the clock signal. In a regenerative amplifier there is always a small delay from the time triggering commences until the full output pulse is developed. Then there are variations in the transmission time to other amplifiers. For these reasons the clock signal must lag the input signal to the amplifier in order to maintain control of turn-on and to obtain a uniform pulse length from all amplifiers. Generally the time lag is one-fourth of the * In an actual amplifier D6 is not required if the transistor saturation voltage is relatively constant versus collector current and the pulse fall time is not ad- versely affected by minority carrier storage in the transistor. Often the inductive "kick" of the transformers and the regenerative feedback are sufficient to make the minority carrier storage effect negligible. If D6 is used, its reverse recovery time may adversely affect the pulse fall time, thus nullifying its usefulness. t The Ico standby power is proportional to V2 , which, for a given output volt- age, is inversely proportional to no . TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1107 repetition period and, in such a case, the clock signal is made available in four phases. Although the clock signal may have any one of a number of forms, a sine or a square wave are the most common forms. Usually a sine wave is preferred because it is simpler to distribute to a large number of amplifiers. Exceptions occur in cases where exceptionally precise timing is necessary, or the use of a square wave requires considerably less clock power. In the following discussion of where to synchronize, a square wave will do as well or better than the assumed sine wave. With either signal it is desirable to keep the clock power to a minimum. If the synchronizing circuit is to be effective, the clock signal must be capable of accomplishing the following actions: a. It must be able to hold the transistor in the "off" state in the pres- ence of trigger current in order to control turn-on. b. At the turn-on time it must rapidly inject the trigger current into the transistor. c. At the turn-off time it must alter the conditions in the feedback loop in such a manner that the transistor turns off promptly. In other words the synchronizing circuit must act like an inhibit logic circuit with the clock signal appearing as the inhibit signal during the interdigital period. It is recognized that there are many amplifier configurations and several ways to synchronize each configuration. Generally it is preferable to synchronize at only one input terminal of the transistor or at only one point in the feedback circuit. A relatively complete discussion can be given with the aid of the following four examples. A circuit that employs negative resistance feedback, such as in Fig. 4, requires a relatively large amount of clock power for synchronization. Because a capacitor (C2) is required on the emitter foi regeneration, ^ the clock signal must be applied to the base of the transistor to control turn-on accurately. As far as turn-off is concerned, another clock signal might be applied to the emitter or to the current gate in the feedback circuit. However, this would result in additional components, a second clock signal 180° out of phase with the base clock signal, and approxi- mately the same required clock power as if the base clock signal alone were used. Turn-off at the emitter is impractical due to the negative re- sistance characteristic. The power that the clock signal on the base must furnish is made up of two parts. One part is the average standby power that is absorbed every time the clock voltage is positive. It is composed of the Ico power supplied to the transistor plus the power dissipated in in Rl and R2. R2 and D2 serve to reduce the clock current in Rl and 1108 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 R2, but the maximum value of R2 is limited by stray capacitance from the transistor base to ground.* The average clock standby power for this circuit (with a 10 volt peak clock voltage) is approximately 13 milli- watts. The second part of the clock power occurs at turn-off when the clock must supply approximately the full "on" state collector current. In this design the clock supplies about 20 milliamperes of current for 0.1 microsecond at voltages up to about 6 volts peak before the transistor turns off. Therefore, a negative resistance feedback circuit usually re- quires a relatively large amount of standby clock power continuously and a high peak clock power at turn-off. Also it should be noted that diode Dl must have a short reverse recovery time in order to prevent false triggering during the negative portion of the clock cycle. A second example of synchronization is shown in Fig. 11. Here the clock signal is introduced in the feedback circuit to control turn-off. It is also applied to R2 in the input circuit so as to control turn-on. In this circuit most of the clock power is dissipated in R5 and R6 when the clock voltage is positive during the output pulse time slot (whether or not an output pulse is produced). Necessarily, this power is relatively large be- cause the clock must supply the full amount of feedback current. Also, it is necessary to clip the positive peak of the clock voltage in order to prevent false triggering via R2 when there is no input pulse. A square wave clock signal would eliminate the need for R6 and D7, but would not change the power in R5. The average clock power in a typical circuit of this type is approximately 20 milliwatts, which is relatively large. The principal advantage of this method is that diode reverse recovery time is not a problem. A third method of synchronization is to apply a square wave clock signal (a sine wave is not suitable in this case) between the base of the transistor and ground (for example, assume in Fig. 11 that R2 and R5 are returned directly to V6 and that the base of the transistor is the clock terminal instead of ground). Before turn-on the clock voltage must be more positive than the trigger voltage on the emitter. At turn- on the clock voltage drops rapidly to ground potential and triggering takes place. During the pulse duration the base current of the transistor is supplied by the clock source. At turn-off the clock voltage must rise rapidly several volts until D6 conducts and robs current from the feed- back loop. The clock power required by this method is relatively large (order of 20 milliwatts) for point contact transistors because the base current of such units is large. In a junction transistor with alpha close * The capacitance causes the base voltage to lag the clock voltage at turn-on if R2 is large, which degrades the timing. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1109 to unity the base current is small and the required clock power may be as low as 3 milliwatts. However, it should be noted that this method of synchronization applies only to amplifiers with a gated feedback circuit (such as R5 and D6 in Fig. 11). In other circuits (Fig. 12, for example), a clock voltage applied to the base terminal of the transistor may never be able to turn off the transistor (the feedback current may actually increase instead of decrease). Thus, this method of synchronization is limited and is a low power method only when used with junction tran- sistors. A fourth synchronization method, which avoids the limitations cited in the previous examples, is illustrated in Fig. 12. The timing circuit is simply diode Dl. The operation of the circuit, which is like an inhibit logic circuit, is as follows. When trigger current commences, the clock voltage is negative and Dl conducts the trigger current away from the emitter terminal. As the clock voltage rises positiveward, the emitter voltage follows until it reaches the threshold voltage of the transistor, usually ground potential. Then the trigger current flows into the tran- sistor which turns on. As the clock voltage continues positiveward the emitter conduction clamps the emitter voltage so that Dl opens and the clock does not shunt the feedback path during the pulse duration. At the end of the pulse duration the clock voltage goes negativeward through ground potential and Dl becomes conducting. This action robs current from the feedback loop, thus causing the transistor to turn off. If no input pulse is present, Dl is always non-conducting and any small re- verse leakage current is drained off through El (which is returned to voltage VI). Because diode Dl is always non-conducting when no input pulse is present, the standby clock power is essentially zero. During a pulsing cycle the clock conducts only a small current before turn-on and only instantaneously at a low voltage at turn-off. Hence, the required clock power is usually less than two milliwatts. It is important to note that the amplitude of the negative peak of the clock voltage usually should not be more negative than the quiescent bias voltage on the emitter. If it should be, Dl will conduct and, due to minority carrier storage, may cause false triggering when the clock volt- age goes positive. The current through Dl at turn-off might have the same effect in the succeeding cycle except that the flyback voltage of the transformers during the interdigit period removes the minority car- riers from both Dl and D2. Since D2 carries a larger current for a longer period than Dl, the carriers are cleared from Dl first. It is then reverse- biased for almost one-half the repetition period before there is any chance 1110 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 of false triggering. Hence, diode reverse recovery time is not a problem. However, Dl should have a short forward recovery time in order that : turn-off will occur rapidly. One possible limitation of this synchronization method is that a low impedance clock source is necessary. This is usually not difficult to ob- tain with a resonant circuit in the output of the clock signal source. Offsetting this point are the advantages of low clock power, esentially zero standby clock power, only one additional component, and no criti- cal component tolerances. 7. ILLUSTRATIVE DESIGN In the preceding sections the features of various configurations for the functional circuits of an amplifier have been described. The following discussion illustrates the application of these ideas to an amplifier design for use in a digital computer system. It is intended that the descrip- tion of the design philosophy be sufficient to permit its application to other systems. In the computer under consideration the amplifier is to be combined with a single level, diode logic circuit to form a logic network. The logic networks, together with delay lines, will be connected in appropriate arrays to perform the logic functions of the sytem, such as addition, multiplication, etc. Digital information is to be represented by one-half microsecond pulses and the amplifiers are to be synchronized at a one megacycle pulse repetition rate by a four phase sine wave master oscil- lator. Other system requirements are mentioned in connection with the selection of the corresponding functional circuit. Since the amplifier is considered as a small system of functional cir- cuits, it is necessary, as in most system designs, to re-examine, and pos- sibly change, circuit choices as the design progresses. However, for the sake of clarity, the following discussion omits the re-examination and frequently refers to the final schematic shown in Fig. 13. The first step in the design is to select the feedback configuration most suitable to the computer requirements. For this computer the dc and clock power are to be minimized and the amplifier should be able to drive from 1 to 12 logic networks. Miniaturization of the computer implies that there may be an appreciable amount of stray capacity across the amplifier output. These considerations suggest transformer coupled feed- back connected in series with an output circuit. Since both positive and negative output pulses are to be required (one polarity for AND and OM logic and the other polarity for inhil)ition), transformer output cou- pling is indicated. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS nil The next basic selection is the choice of an appropriate transistor. In this computer it is expected that pulses will occur in only about one third or less of the pulse time slots due to the nature of the digital in- formation. In order to minimize the dc standby power an alloy junction transistor is a logical choice for this application because of the low Ico current. However, even with a junction unit possessing an alpha cut- off frequency of eight megacycles, it is difficult if not impossible to ob- tain acceptable gain and rise time with the desired output load current at a one megacycle repetition rate. If the rise time is improved by in- creasing the trigger current, the gain is decreased. The principal cause of the poor "gain-bandwidth" appears to be the depletion layer capaci- tance.^^ The difficulty can be overcome by selecting a point contact transistor. A particular germanium transistor coded GA-52996* appears to be suitable and has the following pertinent characteristics: a. Collector capacitance less than 0.5 uuf, b. Alpha cut-off frequency in excess of 80 mc, c. Base resistance less than 100 ohms. Since the alpha of this unit is greater than 2 at collector currents of the order of 10 ma, the common base connection will yield the greatest current gain. The disadvantage of a point contact unit, of course, is the Ico current. For this reason the amplifier will have to be designed to use the smallest possible collector supply voltage. The point contact transistor, due to its high cut-off frequency relative to the amplifier pulse repetition rate and its high alpha at small emitter T~L INPUT I -8V I I +6V 3 VOLT PEAK i I MEGACYCLE SINE WAVE VOLTAGE I-2V _rL -w- OUTPUT Fig. 13 — Illustrative design * This is a relatively special unit especially suited for high speed switching applications. 1112 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195G currents,* permits the use of a simple input circuit. The AND type input circuit is suitable and desirable for another reason. When AND type logic is added to the amplifier, it may be paralleled with the basic input circuit and the input sensitivity of the complete network will be the same as for the amplifier alone. Other logic circuits will be added to an ampli- fier in a manner similar to that described by Felker- so that the input sensitivity will be reduced at most by the voltage drop across one series diode (approximately 0.3 volts). The input pulse voltage and current requirements depend upon the voltage threshold necessary to prevent false operation and the minimum trigger current for reliable regeneration. A test of several sample tran- sistors indicates that approximately 0.3-ma emitter current is required to trigger the transistor with an estimated collector supplj^ voltage of 10 volts. The emitter breakpointf voltage is found to vaTy between — 0.25 and +0.25 volts. To allow for aging variations of the transistor and of R2, it seems reasonable to use a 6-volt source and R2 ec^ual to 9090 ohms, which results in a trigger current a little more than twice the required minimum. Previous experience with computers of this type in- dicates that a 2-voIt threshold will be sufficient to prevent false trigger- ing. Thus, the secondary winding of the feedback transformer is returned to —2 volts and Rl is chosen to give a quiescent emitter voltage of —2 volts. With these considerations and an estimated voltage drop across R3, the input pulse amplitude is calculated to be 2.3 volts and 0.9 ma. Allowing 0.3 volts for a series logic diode, the minimum output voltage and current of the amplifier are 2.6 volts and 0.9 ma per driven network. The selection of the collector supply voltage and the turns ratio of T2 depends upon the dc power dissipation due to Ico current and output voltage regulation versus collector current. For this transistor a unity turns ratio appears to represent a reasonable compromise. Then, by estimating the voltage drops across Tl, T2, and the transistor, it is found that a collector supply voltage of — 8 volts is suflScient to produce an output pulse voltage about 0.5 volt greater than the required miminum. The next step is the selection of the turns ratio of Tl and the primary inductances of both Tl and T2. The two considerations involved are sufficient feedback with the minimum output current (the worst case with respect to feedback) and the maximum collector dissipation in the event that the clock fails. By means of the formulas and assumptions indicated in section 5, primary inductance values of 0.4 mh for Tl and * Usually a > 4 for ie = 0.5 ma. t The transition point of the emitter diode from cut-ofT to conduction. TRANSISTOR PULSE REGENERATIVE AMPLIFIERS 1113 0.2 mh for T2 together with a turns ratio of 1.4 for Tl are selected. Since the GA-52996 transistor is not quite short circuit staple, a 50-ohm resistor is added in series with the emitter. The excess emitter current at the end of the pulse duration is greater than 2 ma, thus assuring suffi- cient stability, and, if the clock fails, the amplifier will turn off by itself in approximately 7 jusec, at which time the instantaneous collector dissi- pation will be approximately 240 mw (considered to be a safe instan- taneous dissipation for this transistor). For low clock power and circuit simplicity the single diode synchroniz- ing circuit is chosen. Although a peak clock voltage of 2 volts would nor- mally be used (this value corresponds to the quiescent emitter bias volt- age) it is found that the clock may be varied between 1 volt and 6 volts peak without a failure occurring. Therefore, the nominal clock voltage is set at a centered value of 3 volts peak. The dc level of the clock voltage is 0 volts, which approximately corresponds to the emitter break point voltage of the transistor. This concludes the basic selections in the de- sign procedure. The power dissipated in the amplifier is quite modest. In the quiescent state the amplifier absorbs only 0.2 mw average clock power and 30 mw dc power (this would be only 10 mw if the I co power w'ere negligible). When the amplifier is pulsing every microsecond the dc power is 50 mw and the averge clock power is 2 mw. Since the amplifier is so conser- vative of power, it is possible to use 4,000 networks in a computer and require less than 200 watts dc power. One indication of the component sensitivity of a pulse amplifier is the magnitude of the supply voltage margins. In this amplifier the supply voltages may be varied, one at a time, over ±12 per cent of the nominal values before a failure occurs. Generally margins of this magnitude under the worst conditions are considered sufficient to guarantee against fail- ures caused by aging, or to insure that such failures will be indicated by routine checks before they occur. It is interesting to note that in a tem- perature test the amplifier continued to operate properly over a tempera- ture range from —20 to +80°C. Even at -f75°C the supply voltage margins were 10 per cent or better. 8. SUMMARY A method of analysis and design procedure have been presented in which a transistor regenerative amplifier is considered as an intercon- nected system of functional circuits. Each functional circuit may be evaluated or chosen in terms of the requirements of the complete digital system in which the amplifier is to be used. In general no particular cir- 1 lU-i THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 cuit or collection of circuits can result in an amplifier suitable for use in every type of digital system. The use of an AND type input circuit, transformer coupled output and feedback circuits, and an inhibit type synchronizing circuit appear to be an optimum set of functional circuits to make up an amplifier for use in a synchronous digital computer system emplojdng passive logic circuits. An illustrative design is presented for such an amplifier which operates at a pulse repetition rate of 1 mc, uses 12 components (none of which are especially critical), requires an average of 40-mw dc power and 1-mw clock power, is capable of driving from 1 to 12 similar amplifiers, and has voltage margins in excess of 12 per cent. Although the design philosophy was developed for this type of amplifier, it is believed that much of the philosophy is applicable to regenerative amplifiers for use in other digital data processing systems. 9. ACKNOWLEDGEMENT The final design and the performance data of the illustrative amplifier are due to L. C. Thomas and H. E. Coonce. The author also wishes to express his appreciation for the many helpful and stimulating discus- sions with other colleagues, especially A. J. Grossman, T. R. Finch, J. H. Felker, and J. R. Harris. REFERENCES 1. S. Greenwald, et al., SEAC, Proc. I.R.E., Oct., 1953. 2. J. H. Felker, Regenerative Amplifier for Digital Computer Applications, Proc. I.R.E., Nov., 1952. 3. J. L. Moll, Large-Signal Transient Response of Junction Transistors, Proc. I.R.E., Dec, 1954. 4. J. G. Linvill and R. H. Mattson, Junction Transistor Blocking Oscillators, Proc. I.R.E., Nov., 1955. 5. A. E. Anderson, Transistors in Switching Circuits, B. S.T.J. , Nov., 1952. 6. T. E. Firle, et. al., Recovery Time Measurements on Point-Contact Germa- nium Diodes, Proc. I.R.E., May, 1955. 7. S. L. Miller and J. J. Ebers, Alloyed Junction Avalanche Transistors, B.S. T.J., Sept., 1955. 8. J. J. Ebers and S. L. Miller, Design of Alloyed Junction Germanium Transis- tors for High Speed Switching, B.S.T.J., July, 1955. 9. T. C. Chen, Diode Coincidence and Mixing Circuits in Digital Computation, Proc. I.R.E., May, 1950. 10. L. W. Hussey, Semiconductor Diode Gates, B.S.T.J., Sept., 1953. 11. J. M. Early, Design Theory of Junction Transistors, B.S.T.J., Nov., 1953. 12. Q. W. Simkins and J. H. Vogelsong, Transistor Amplifiers for Use in a Digital Computer, Proc. I.R.E., Jan., 1956. 13. M. Tanenbaum and D. E. Thomas, Diffused Emitter and Base Silicon Tran- sistors, B.S.T.J., Jan., 1956. Observed 5-6 mm Attenuation for the Circular Electric Wave in Small and Medium-Sized Pipes By A. P. KING (Manuscript received March 20, 1956) At frequencies in the 50-60 kmc region the use of circular electric wave transmission can provide lower transmission losses than the dominant mode, even in relatively small pipes. The performance of two sizes of waveguide was investigated. In the small size (Kg" ^•^- X Me" wall) the measured TEoi attenuation was approxi- mately 5 db/100 ft and is appreciably less than that of the dominant mode. The measured attenuation for the medium sized (%" I.D. X }^i'^ wall) waveguide was 0.5 dh/100 ft which is about one-fourth that for the dominant mode. This paper also considers briefly some of the spurious mode conversion- reconversion effects over the transmission band and their reduction when spurious mode filters are distributed along the line. Allowance has been made for the added losses due to oxygen absorption when air is present. INTRODUCTION Since 5.4-mm dominant-mode rectangular waveguide has attenuations of the order of 60 db/100 ft, another transmission technique is required in applications which involve appreciable line lengths. Losses may be reduced by the use of oversize waveguide ; some earlier work with domi- nant mode transmission in slightly oversize round waveguide (two or three propagating modes) has been reported.^ The possibility of still lower losses exists with circular electric wave transmission in an over- size round waveguide. Miller and Beck- have computed the theoretical relative transmission losses of the TEoi and TEn modes as functions of ' A. P. King, Dominant Wave Transnaission Characteristics of a Multimode Round Waveguide, Proc. I.R.E., 40, pp 966-969, Aug., 1952. 2 S. E. Miller and A. C. Beck, Low Loss Waveguide Transmission, Proc. I.R.E., 41, pp 348-358, March, 1953. 1115 1116 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 guide size and frequency. At 5.4 mm, a J^e" I-I^- waveguide has an appreciably lower attenuation with the circular electric mode than with the dominant mode. A %" I.D. guide has a circular electric attenuation approximately one-fourth that of the dominant mode in the same pipe. It is the purpose of this paper to present some experimental results which have been observed with circular electric wave transmission in the 5-6 mm wavelength region. The attenuation for three different hnes and the transmission variations due to moding effects are reported. Al- lowance for the loss due to oxygen absorption has been included. DESCRIPTION OF THE TEST LINES The TEoi mode attenuation measurements were made on approxi- mately straight runs of line ranging from about 100 to 200 feet in length. The copper pipe comprising these lines is believed to conform to the best tolerances and internal smoothness which are current manufacturing practice for waveguide tubing. The relative tolerances and their effect upon transmission are considered in a later section. Three kinds of copper line were measured: a waveguide of oxygen-free copper, one line of low phosphorous-deoxidized copper and one line of steel with a 20-mil low phosphorous-deoxidized copper inner lining. The oxygen-free high-con- ductivity-copper with its higher conductivity and somewhat greater ductility was chosen to provide comparative performance data with the low phosphorous-deoxidized copper which is commonly used in wave- guide manufacture. A waveguide whose outer wall is constructed of steel to provide the necessary strength and wall thickness to support a very thin copper inner wall has the advantage that such waveguide re- quires less copper. This composite wall tubing was obtained to ascertain whether the tolerances and the nature of the inner surface would yield transmission data comparable to solid copper waveguide. The lines Avere supported on brackets which were accurately aligned and spaced at 6-ft intervals. Although the brackets provided for an accurately straight line, the manufactured pipe was not perfectly straight but, in some samples, varied as much as %" in a 12-ft length. Installing the pipe on the brackets tended to straighten the line and reduce these variations to about half this amount. A general view of the lines is shown in the photograph of Fig. 1. The sections of waveguide were joined together with a more or less conventional threaded coupling, but with one very important difference. The threads, which are cut at the ends of each section, are cut relative to center of the inside diameter and not the outside diameter. This is achieved by employing a precision pilot to provide a center for the cut- i 5-6 MM ATTENUATION FOR THE CIRCULAR ELECTRIC WAVE 1117 ^''■m m 9^ m #% Fig. 1 — General view of the circular waveguide lines and the millimeter wave measuring equipment. ting die. Since the internal diameter is made as precise as possible, the variations of outside diameter become a function of the tolerances of both the internal diameter and wall thickness and cannot be as precise as the inside of the pipe. Any thread cut relative to the outside diameter as in regular plumbing practice, will not, in general, be concentric to the 1118 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 inside wall. To avoid an offset at the joint it is therefore important that the thread be centered relative to the inside diameter. After a section was threaded the ends were faced off to make the ends square and thus avoid any tilt between sections when the ends are butted together. Of the two sizes tested the smaller diameter (Ke" I-D. X Ke" wall) was chosen to provide a moderate line loss, while limiting the number of propagating modes. In the band of interest (5.2-5.7 mm) the theoreti- cal TEoi wave attenuation is about 4 db/100 ft. The number of modes which can be supported at X = 5.2 mm is limited to 12 modes and to only one of the circular electric modes. The higher order TEon modes are beyond cut-off. These features limit the number of spurious modes and simplify the mode filtering problem. Furthermore, in this smaller sized waveguide, the associated components which may set up TEon waves, for example conical tapers, need not be as long proportionately as in larger waveguides. The %6" I-I^- guide has the advantage of smaller size, lower cost and greater ease of transmitting TEoi through specially constructed bends. The attenuation of this smaller diameter guide is large enough that system requirements will usually restrict its usage to lengths of line of a hundred feet or so. The larger size (J^'' I.D. X 3^" wall) is exactly twice the diameter of the small size discussed in the preceding paragraph but has only one- tenth the attenuation, or about 0.4 db/100 ft. The low loss of this larger size becomes more attractive for runs as long as several hundred feet. This diameter guide will, of course, support more modes, 50 at X = 5.2 mm; four of which are circular electric modes — • TEoi , TE02 , TE03 and TE04 . Some of the disadvantages which accompany the increased diameter are: (1) greater care must be taken as to line straightness, (2) longer conical tapers are required when converting from one guide diam- eter to another, and (3) longer mode filters are required since the desired mode-filtering attenuations vary inversely with the filter diameter at a given frequency. Flexible spaced-disk lines employed as uniform bends for TEoi transmission require much greater bending radii than bends in the smaller diameter guide if the bend loss is to be kept proportionately low. This problem is considered in some detail in another paper.^ With reasonable care the accumulative effect of these foregoing factors can be held to a reasonably low value. Expressed in terms of the ratio of measured to theoretical attenuation the values are, on the average, about 10 per cent higher in the %" I.D. waveguide than in the J4.6" I.D. waveguide. A. P. King, forthcoming paper on bends. 5-6 MM ATTENUATION FOR THE CIRCULAR ELECTRIC WAVE 1119 »» ' • " K>if .^^. .^W- -^P- .I^P. Fig. 2 — Waveguide portion of millimeter wave measuring set. MEASURING PROCEDURE With straight runs of round, TEoi waveguide lines whose length lies in the 100-200 ft range, it is convenient to make attenuation measure- ments on a round trip basis. This method has the advantage of conven- ience in that the attenuation can be measured directly by using a wave- guide switch but has the disadvantage of requiring a careful impedance match of the measuring equipment to the line. Fig. 1 shows an overall view of the lines; Fig. 2 shows the arrangement of the 5-6 mm measuring set, and Fig. 3 shows a block diagram of the set-up employed. This measuring set makes use of two klystrons developed by these laboratories.* The double detection receiver features a separate beating oscillator klystron which is frequency modulated and a narrow band (1.7 mc at 60 mc) IF ampHfier. The resulting IF pulses are detected wuth a peak detector and then amphfied to provide the usual meter indication. This method with its circuitry has been developed by W. C. Jakes and D. H. Ring,^ and provides a greater amplitude stability than is possible with a cw beating oscillator. In the waveguide schematic of Fig. 3 about a tenth of the power is ^ E. D. Reed, A Tunable, Low Voltage Reflex Klystron for Operation in the 50-60 Kmc Band, B.S.T.J., 34, p. 563, May 1955. * W. C. Jakes and D. H. Ring, unpublished work. 1120 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 taken from the signal oscillator to provide monitoring and wavemeter indication. The remaining power, after suitable padding, is fed into a 3-db directional coupler or hj^brid junction 2. This junction is employed as a waveguide bridge so that, when arms A and B are properly termi- ' nated, no power flows in receiving arm C. Any reflection in line A will, WAVE METER 0 MONITOR SIGNAL OSCILLATOR X •-10DB , Q oDB v_y I ^4- 2>/ ^3DB COUPLER WAVEGUIDE /-^{HYBRID JUNCTION) SWITCH / y^ ROUND TUNER TAPER ^^ <• WAVEGUIDE LINE X TE TEo, TEo, Fig. 3 — Schematic of measuring equipment. ADJUSTING KNOB -RG 98/u Fig. 4 — Structure of impedance matching tuner. 5-6 MM ATTENUATION FOR THE CIRCULAR ELECTRIC WAVE 1121 Fig. 5 — Structure of waveguide switch. however, produce a power flow in the arm C to the balanced converter of the receiver and an indication in the output meter. So far this set is similar to a setup for measuring the round trip loss in a terminated waveguide system. The impedance of the TE?o ^ TEoi wave trans- ducer,^ taper section and mode filter connected as shown in the section A-D of Fig. 3 can be matched to the rectangular waveguide at A by an appropriate adjustment of the dielectric post tuner^ Ti whose structure is shown in Fig. 4. Under these conditions a conical taper termination placed in the round waveguide at D will again produce a balance and again no power will flow in arm C. A ^vaveguide switch whose structure is shown in Fig. 5 is connected between the point D and the line under test. A movable short at the far end of the line completes the set-up. With the impedances matched as described above, the only reflection which reaches the receiver wdll be from the far end of the line when the switch S is open or, when shorted, from the switch itself. The round- trip attenuation is the difference in attenuation measured for the two positions of the switch. By means of a movable short at the far end of the line, the line length can be varied to produce mode conversion and mode reconversion effects, and the resultant variation in TEoi mode transmission can be observed. This phenomena is described in some de- tail elsewhere.^ " Reference 2, page 354, Fig. 14. ■ C. F. Edwards, U.S. Patent 2,563,591, Aug. 7, 1951. The millimeter tuner employs an adjustable dielectric post in place of a metallic tuning screw described in the patent. * Reference 2, pp 356, 357. 1122 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 LOSSES DUE TO OXYGEN ABSORPTION In addition to the losses which result from imperfect conductivity, surface effects, and mode conversions, there is a very appreciable loss due to oxygen absorption when the guide is open to the atmosphere. In a waveguide the loss due to O2 absorption is: where A is the absorption due to oxygen in the atmosphere = X/Xc = free space wavelength = cut-off wavelength V X Xc= ^ = k 3.83 -.: ._ d = internal diameter of waveguide ^ k = Bessel root for TEoi mode = 3.832 The loss due to absorption of oxygen which is present in the at- mosphere (at approximately sea level) was obtained from the experi- mental data of D, C. Hogg.^ The added loss produced by the presence 0.5 o o CO o O o I- UJ D Q (/) W o _l in < a. o z 0.4 0.3 0.2 0.1 \ \ V V ■h- \ v\ \ 2,.c- >\ \ \ \ \ \ \ \ X ^^ =-- 5.0 5.2 5.4 5.6 WAVELENGTH IN MM 5.8 Fig. 6 — TEoi transmission loss in waveguides due to oxygen absorption. ' A. B, Crawford and D. C. Hogg, Measurement of Atmospheric Attenuation at Millimeter Wavelengths, B.S.T.J., 35, pp. 907-917, July, 1956. 5-6 MM ATTENUATION FOR THE CIRCULAR ELECTRIC WAVE 1123 of oxygen in the waveguide in terms of (1) is plotted in Fig. 6. It will be noted that this loss becomes very appreciable at the short wave- length end of the band. At X = 5.2 mm this loss is in the 0.3-0.4 db/100 ft range. For the larger size waveguide hue (J^" I.D.) the loss due to O2 is approximately equal to the theoretical wall losses; for the smaller size lines this amounts to about a tenth the wall loss. At the other end of the millimeter band the O2 losses are very small, being in the 0.02 - 0.03 db/100 ft range at X = 5.7 mm. The relative effects of theoretical wall and expected oxygen ab- sorption losses are shown plotted in Fig. 7. For the two sizes of wave- guide the upper dashed curve represents the combined effect of these two factors and the lower solid line curve is the theoretical attenuation of the TEoi mode in empty pipe. The shaded area indicates the increase which is the result of oxygen absorption. In order to minimize the transmission losses in any practical system it becomes desirable to exclude the presence of oxygen from the hne, for example, by introducing an atmosphere of dry nitrogen. Since the ex- 5.0 4.5 I- UJ ID O o a. Ill Q. HI m o UJ o 4.0 3.5 3.0 WAVEGUIDE +O2 LOSS rO J LOS S -3--' ^ \ V/^ >2 7^ ■^.^^ / ^ y^ ^ THEORETICAL WAVEGUIDE LOSS 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 WAVELENGTH IN MILLIMETERS (a) TEq, loss IN 7/16" I.D COPPER 0.8 Z O w z < a. 0.7 0.6 0.5 0.4 0.3 0.2 THEORETICAL WAVEGUIDE LOSS 5.0 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 WAVELENGTH IN MILLIMETERS (.b) TEq, loss in ys" I.D. COPPER Fig. 7 — TEoi transmission losses. 1124 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 tn a. D OJ <. o O Q- cc^-'ai - Q. LINE RLTE JOINT T LIN 1 to q Z LUl"- - <^ M < gO'O I N -i 9 z LU OJ z « I k \ ^ y _/> \ -J \l 7 ' < Q \ \ 1- UJ \ \ Ol cr N.\ QC CO \' S. iJJ < > /^I UJ /^ ^ X o t-|^ CABLE TO BRUSHES Fig. 3 — Test fixture, loaded. 1138 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 test exceeds a predetermined value. This causes another relay to lock up and remember the faihu'e until the network reaches the reject location. In a typical transmission test position (Position 10, 35 or 36) a fixed- voltage, swept-frequency signal, 300 to 3,500 c.p.s., is impressed across two terminals of the network. The three tests are for transmission and short and long hne sidetone with suitable terminations connected as in actual use. In each case the signal from two output terminals should be less than or greater than a specified value. This signal is amplified and fed Fig. 4 — Test fixture, unloaded. AUTOMATIC TESTING IN TELEPHONE MANUFACTURE 1139 to a sensitrol relay, which is mechanically biased in amount and sense to correspond to the limit. If the sensitrol operates it prevents rejection. In the three capacitance test positions 12, 13, and 14, capacitors in the networks are connected into a 60 c.p.s. comparison bridge. The out- put signal from the bridge is amplified and rectified, and impressed on a balanced dc amplifier which drives a sensitrol relay. If the bridge is out of balance (that is if the capacitance is greater or less than nominal) cur- rent flows in the relay, but always in the same sense. If the current in the relay exceeds an amount corresponding to either capacitance limit, re- jection occurs. Determination of which capacitance limit was violated is done manually in a separate analysis of defects. It may be observed also that any rejection at the capacitance positions could have been caused by a loss unbalance of the bridge. If the conductance of the test capacitor were such as to cause this it would so appear in the separate analysis, mentioned above. The effect of any ordinary conductance deviation at 60 c.p.s. is neghgible. Quality is protected by the fact that a conductance deviation could not cause an out-of -limit capacitor to be accepted. Considerable pains are taken at each capacitance test position to pre- vent damage to the equipment from various kinds of mishaps. The sen- sitive winding of the sensitrol is short circuited at all times except for about 0.2 second when the actual test is performed. This prevents dam- age and erroneous rejections that would otherwise be caused by switching Table II — Sequence of Events in Network Test Machine CAM ROTATION — (TWELFTHS OF A POSITION) POSITION PROCESS C ) 2 4 6 8 10 12 1 TO 6 LOAD 1 7 BURNOUT DISCHGl 8 AC BKDN. TEST- 'P 7KV) O - - Ill - -» UJ - * 9 DC BKDN. TEST- 10 TRANSMISSION 1 1 , TEST- II DISCHARGE 1 12 CAPACITANCE 1 ^CIRCUIT SETUP-^ lESTJ^ M iMORY 4 13 2 II M 11 14 3 ■1 H II 15 BURNOUT -CHA R6E- DISCHGl 1 16 TO 31 (1 MINUTE) CHARGE FOR LEAKAGE TEST - C^Anuc 32 LEAKAGE 1 TEST 33 2 II 34 3 II 35 TRANSMISSION 2 II 36 3 M 37 CONTINUITY >- TEST 4 CIRCUITS AT ONCE - ^ 38 UNLOAD —UNLOAD 39 RESET 40 LOAD 1 MANUALLY OPERATED TEST KEY X PIN JACKS PIN PLUGS CONTACT FIXTURE / ^ — ^ ^ TEST CABLE-" /Cl CIRCUIT TERMINALS PORTION OF RELAY CKT UNDER TEST Fig. 2 — Simplified circuit sketch for manual test operation. WATCHING 1 RELAYS SIGNAL RELAYS CONTACT CROSS CONNECTING FIXTURE DEVICE / CIRCUIT > /TERMINALS I I f TEST CABLE PORTION OF RELAY CKT UNDER TEST Fig. 3 — Simplified circuit sketch for automatic test operation. AUTOAIATIC TESTING OF RELAY SAVITCHING CIRCUITS 1159 system is at best a slow and laborious one which is subject to human error. Wages for testers are determined not primarily on their ability to operate keys and check the indications of lamps but on their skill in analyzing and clearing trouble conditions. If some quick and automatic means could be devised to make the initial cross connection setup, apply the potentials in the proper sequence under control of some programing device and check the circuit responses at each step a real advance in speeding up tests and reducing human errors would be accomplished Such an automatic set ideally should have improved response indications to aid the the tester in locating circuit troubles when the test set stops on the failure of meeting any test requirement. THE AUTOMATIC TEST SET The key and visual lamp indicating functions of the manual test set can be replaced by relays in an automatic test set which perform these operations if they are under control of suitable programing and advanc- ing circuits as shown in Fig. 3. Here the "signal" relays operate through the contacts of the relay under test and their operating positions are checked by the "watching" relays whose contact closures must match those of the signal relays. The series path through the contacts of all signal and watching relays is called a chain lead. The program circuit establishes the positions of the watching relays to meet the expected con- ditions prior to operating the key relay and then any lack of continuity through the chain lead caused by failure to satisfy test conditions halts the progress of the tests under control of the advancing circuit. At this point additional contacts (not shown) on the signal and watching relays may be used to light signal lamps to convey information to the tester as to which portion of the circuit failed to operate properly. For quick setup a pre-wired multi-contact adapter plug may be used as a cro?s-connection device to permit establishing the proper test con- nections to the unit under test. One will be required for each type of relay circuit to be tested. These, together with some means whereby the sequential operation of the programing circuit can be controlled, constitute the essential features of an elementary automatic relay switch- ing circuit test set. How these basic features can be extended into prac- tical embodiments will be explored further below. THE CARD-O-MATIC TEST SET Key equipment relay units are small switching circuits used as cir- cuit building blocks to provide the desired optional features in conjunc- tion with the key boxes or key-in-base telephones often seen in small 1160 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195G Fig. 4 — Card-0-Matic test set. AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1161 )usines.s offices to furnish the flexibiHty needed in answering and trans- ferring calls. These s3^stems are used where the number of telephones served does not warrant the use of a regular PBX switchboard. These circuits are relatively simple but their large scale pi'oduction warrants the use of high speed automatic test sets to perform the test functions and to indicate circuit trouble. Fig. 4 shows the operating position of the Card-0-Matic* test set which was developed to test such unit assemblies. The keys shown are used to initiate and control the automatic operation of the test set and in trouble shooting. They are not to be confused with those that per- form the actual testing functions described previouslj^ for the manual test set. The lamps pro^'ide indications of the progress of the tests and of the positions of the watching relays which are also needed to aid in determining the point of circuit failure. The meter type relay in the upper left corner of the operating panel provides a sensiti\'e checking de\'ice for audio freciuency tests through the \'oice transmission circuits. The telephone dial affords a simple means of generating any recjuired number of pulses for operating stepping selectors on some types of units. The terminal field in the lower front of the cabinet gi\^es the tester access to the circuit terminals of both the unit under test and the test set for his use in analyzing and locating faults. The upper cabinet was a later addition and contains the multi-contact rela3\s needed to permit testing units with more than one circuit. The row of push buttons are used to select the circuit to be tested. Fig. 5 is a rear ^'iew of the set that shows the perforated insulating card from which the set derives its name. The coded card controls the sequence of test operations and is hung on pins over the field of 1,000 spring plungers (20 X 50) as a part of the setup operation for a particu- lar relay unit. Closing the door and screwing up the hand wheel, which is necessary to provide the force required to depress the plungers, will ground those which coincide with holes in that particular card. Cross-connection setup of the test leads is achieved by the use of a plug-board such as is commonly used for quick change over on perforated card type business machines. Fig. 6 shows the plug board being inserted into the transport mechanism. The relatively large number of terminals are retjuired because each of 60 test leads must be capable of being patched in to an equi\'alent number of terminals on a maximum of ten different circuits. Not all of our test sets are equipped with the upper cabinet since most key units have only one circuit and on these a simpler * Patent No. 2,329,491. 1162 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Fig. 5 — Rear view of Card-0-Matic test set showing insertion of perforated card. cross connection fixture is plugged into the location where the lower end of the cable joining the two cabinets is shown terminated in Fig. 5. A side view of the test set is shown in Fig. 7 to give an indication of the amount of switching e([uipinent and wiring necessary for an auto- matic test set of this sort. The set is powered from a r20-\'olt 60-cy('le AUTOMATIC TESTING OF RELAY SWITCHING CIRCNITS 1163 source from which are derived the 24-volt dc, 90-volt 20-cycle ringing current and 600-cycle audio tone supplies that are required. The test circuit features mchide tone transmission checking, dial pulsing, 90-volt 20-cycle ringing and ground and batter}^ supplied either directly or under relay control. Other battery and ground relays are available for checking the response of the circuit under test. These test features have been sufficient to perform operation tests on most relay units associated with key telephone systems. The test cycle is fast and the twenty test steps can be performed in approximately ten seconds. The lamp indications given when the test is interrupted by an open-circuited chain lead, convey information to the tester as to which test step is involved and when any pairs of signal and watch relays fail Fig. 6 — Insertion of cross-connection plug board into Card-0-Matic test set. IKU TIIK BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBEE 1956 Fig, 7 — Interior of Card-0-Matic test set. AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1165 to match each other. Simplified circuit sketches which show the inter- connection of test set and wired unit circuits are provided to enal:)le the 1 ester to determine quickly the cause of the failure. I Th(^ C'ard-(^-Matic test set, while performing admirably on the rela- tively simple relay circuits within its range and capabilities, falls down on the more complicated relay switching circuits used in telephone central offices for several reasons. The most important of these are: 1. A fixed cycle within a maximum of twenty steps with any one coded card. 2. No provision for alternate or optional circuit conditions on a card. 3. The only power supplies provided to operate relays are negative 24-volt dc and 90-volt 20-cycle ringing whereas telephone office units frequently also require negative 48-volt and positive 130-volt dc as well as positive or negative biased ringing currents for party line ringing. 4. The increase of either test steps or features would increase the size of the perforated card beyond a practical size. THE TAPE-O-MATIC TEST SET The experience gained in the design and successful operation of the Card-0-Matic test set led naturally to the exploration of ways and means whereby a more versatile and comprehensive set could be devised. The five hole coded perforated teletype tape was selected as a cheap and flexible programing device. It afforded a means of providing a test cycle of any required length and, since the perforating and reading mechanisms were already available, it appeared to be nearly ideal for its purpose. Consideration was given to the following desirable features all of which were incorporated in the design of the new set: 1. Provision for cross-connecting (under control of the coded tape) any test set circuit to any terminal of the circuit under test for as long as necessary and then disconnecting for reuse in later testing steps if required. This A\'ould greatly extend the range and capabilities of the set. 2. Provision for several power voltage sources which could be selected as required to meet the normal telephone office voltage requirements of the unit under test. 3. Provision for alternate or optional tests to be coded into the tape to meet the various circuit arrangements that may be wired into the unit as required by the Telephone Company who is our customer. Such optional test arrangements could be applied by the test set under the control of keys to be operated by the tester as part of the setup at the start of the tests. 1166 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 4. Provision for stopping the test cycle to enable the tester to per- form manual operations such as inserting a test plug in a jack on the unit or insulating relay contacts in order to isolate portions of the cir- cuit for test simplification and to obtain a more detailed test. 5. Provision of improved lamp indications to aid the tester in clearing wiring faults or in locating defective apparatus. These would include the necessary information as to which test set circuits are connected to which unit terminals as well as which relays of the wired unit should be operated at that stage of the test cycle. 6. Provision for connecting several terminals of the unit under test together as a means of providing circuit continuity where required. 7. Provision for measuring resistance values of circuit components. 8. Provision for insertion of various resistors in battery or ground leads to control currents to desired values. 9. Provision for checking voice transmission paths through non- metallic circuits such as transformers or capacitors. 10. Provision for measuring circuit operating times in steps of ap- proximately 100 milliseconds. 11. Provision for sending and receiving dial pulses. 12. Provision for a single code for releasing all test connections and conditions previously established by the coded tape as a means of quick disconnect. This is in addition to the release of individual connections mentioned in (1) above. 13. Provision for audible and visual indications of completion of a successful test cycle. Through the use of two letters (each of which has its own combination of the five holes) for each signal it was possible to obtain the over 500 codes required to control all test and switching functions even though the teletype keyboard has only 32 keys. The only Teletype transmitter (tape reader) available when the test set was first designed operated at a speed of 368 operations per minute and was arranged for sequential read out on two wires by means of a commutator. Conversion to five wire operation and removing the commutator permitted reading each row of holes simultaneously. The gearing was also changed to permit 600 operations per minute but even so the hole reading contact dwell time was increased from approximately 20 milliseconds to 70 milliseconds for more reliable operation with ordinary telephone relays. The machine which was designated as the Tape-0-Matic* test set, is shown in Fig. 8 in operation on a typical wired relay unit mounted in its shipping frame. The contact fixture is attached to the unit terminal * Patent No. 2,328,750. AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1167 Fig. 8 — Tape-0-Matic test set in operation". strip and cabled to a gang plug which in turn is plugged into a receptacle behind the operator. These leads are extended through a duct to the metal enclosure at the base of the set for entry to the test set proper. The coded tape is dropped into the receptacle at the side of the key shelf to which it returns after its traverse through the reader. A row of circuit breakers on the end of the key shelf control the application of 1168 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 and provide protection for the various power supplies. Two of these supplies are mounted on the top of the set. The rows of vertical push button keys on the key shelf afford the tester a means of determining (for trouble shooting purposes) the asso- ciation (through lamp display signals) of the wired unit circuit terminals with those of the test set and the corresponding test voltages which are connected at that particular stage of the test. The lamp display panel also indicates which test set circuits are in use and through fast or slow (0.5 or 1 second) flashes whether the fault thus indicated is the result of a failure to meet either an expected condition or the occurrence of an unexpected condition. This feature is illustrated in Fig. 9 which shows one link of the chain leads which extend through all pairs of signal and watching relays for the check of satisfaction of all test conditions and the application of steady or interrupted ground to the associated test feature lamp. The operating condition of all test set key relays as pre- viously established by the tape is also indicated by the display lamps. xA.nother type of information obtained from the lamp display panel which is valuable to the tester in trouble clearing is the indication of the particular unit relays which should be operated at that part of the test cycle. By checking the lamps against the operated or non-operated position of the relays he can frequently localize the fault in a minimum of time. As mentioned above an important part of the test set flexibility is the ability of the tester to set up the test set to test only those optional circuit arrangments which are provided in any particular unit ordered X FAST GROUND PULSES TO CKT. UNDER _ TEST ASSOCIATED FEATURE SIGNAL LIGHT ( cr TO PRECEDI WATCH RELAY NG^ CHAIN LEAD ' TO BATT OR GRO.ASREQ'D. "j-t.^" SIG RELAY c-^ SLOW GROUND PULSES •- TO GROUND WHEN REQUIRED FOR EXPECTED OPERATION HI ^ CHAIN T»^^ 0 SUCCEEDING SIGNAL RELAY "watch" relay Fig. 9 — Chain circuit showing watching rehiy function. AUTOMATIC TESTING OF KEL.VY S^^ ITCHING CIKCUITS 1169 Fig. 10 — Lower portion of lamp display panel. by the customer. Failure to provide this would result in fixed test cycles and many more tapes, which might be similar but varying only in regard to the options, would have to be prepared. Figure 10 shows the lower portion of the lamp display panel with the push-pull option keys on the bottom row. Directly above are the manual operation keys with their associated lamps which the tester must operate to cause the test set to resume the testing cycle after it has stopped for him to perform a manual operation. A side view of the interior of the set is shown in Fig. 11. Two bays each facing the opposite direction from the other are housed within the cabinet and are used for mounting the crossbar switches and telephone type relays which are the principal circuit components. Two doors on 1170 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195G . M-f! x-t I ■! I .( •[ .r i J- a* ■ > • I • 1 • r ( I f • > ' I n j . ji -> -I t I ( ! I i t .r ^ M'-mf fi--«~^ f^** wfifr -wm ■*• '|r||M|l||||4|fj 'i^rrrriTllHfiiil ft "» Fig. 11 — Interior of Tape-0-Matic test set. each side give convenient access to all wiring and apparatus for mainte- nance purposes. A fairly large portion of the mounting space is occupied hy the cross- bar switches which perform the functions of interconnecting the circuit terminals of the unit under test and those of the test set. They also connect the proper voltages to these circuits. The switching plan Fig. 12 AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1171 shows in abbreviated diagramatic form that the unit terminals 0-99 appear on the horizontal inputs of the two 10 X 20 and one 10 X 10 switches that comprise the primary group. The horizontal multiple of these switches are split so that each section runs through five verticals to afford connection to each of the hundred unit terminals. The vertical outputs of the primary switches are connected to the horizontal inputs of the two 10 X 20 secondary switches. The horizontal multiple of these switches are split so that each section runs through eight verticals. The verticals of the secondary switches are linked to the horizontals of the two 10 X 20 tertiary switches which have their tru- tjj fr, 90 GROUP 0 '00 GROUP 0 9 Q GROUP 0 ,8 I I I 1 ■o— p A09 THROUGH - GROUPS - 1-8 crui uji- \ THROUGH -GROUPS - 1-3 THROUGH -GROUPS - 1 8. 2 99 GROUP 9 '09 -o — GROUP 4 0 0 0 0 GROUP 3 ,30 39 PRIMARY SWITCHES 2- 10 X 20 1 -10 X 10 SECONDARY SWITCHES 2 -10 X 20 TERTIARY SWITCHES 2 - 10 X 20 i| QUJ 0.1- 28 THROUGH FEATURES 0-9 -^-O- 00 09 THROUGH FEATURES 10-29 THROUGH FEATURES 30-39 CORRESPONDING LEVELS MULTIPLED 30 39 TERMINATING SWITCHES ■ 10X 20 Fig. 12 — Switching plan. 1172 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 multiple split into groups of ten. The 40 verticals of the latter are con- nected directly to the 40 test set feature circuits designated 0-39. Two additional 10 X 20 crossbar switches perform the function of connecting any of the five power or five multiple terminations to any of the forty test set features. These terminations are comprised of 5 loops and one each of ground, negative 24 volts, negative 48 volts, 90- volt 20-cycle ringing current and positive 130 volts. Thus it can be seen that, through proper operation of the primary, secondary, tertiary and terminating crossbar switch cross points, a path can be established from any circuit terminal to any test set feature and supplied with any of the available power or loop terminations. It is 7 CROSSBAR SWITCHES RELAY SWITCHING TYPE CIRCUIT UNDER TEST MAX. 99 LEADS AUTOMATIC SWITCHING OF CIRCUIT TERMINALS TO DESIRED TEST FEATURES QUICK CONNECT CONTACT FIXTURES TELETYPE TRANSMITTER MODIFIED FOR 5-WIRE OPERATION 40 LEADS CONNECT «. DISCONNECT SIGNALS TRANSMITTED CODES 5 LEADS TEST FEATURES CIRCUITS APPLY DIFFERENT TESTS AND CHECKING CONDITIONS 40 LEADS TEST FEATURE SIGNALS TAPE DECODING CIRCUITS 10 KEYS OPTIONAL FEATURES CIRCUITS PERMITS VARIATION OF TEST LISTED ON STANDARD TAPES TEST SIGNALS AND CHECK RESPONSE 2 CROSSBAR SWITCHES CONTROL CIRCUITS A SPECIAL CODE AT END OF EACH TEST STEP PERMITS THESE CIRCUITS TO TRANS- FER CONTROL OF THE TRANSMITTER TO THE CHAIN LEAD RESET KEY INDEXES TAPE i^ TO STARTING POSITION i TERMINATION SWITCHING OF POWER 8. SIGNALS t130VDC GROUND 90V PO'^ RING 5 LOOPS -48 V DC -24 V DC TO TEST FEATURES CIRCUITS TERMINATION SWITCHING SIGNALS TROUBLE INDICATING CIRCUITS 1. LAMPS INDICATE PROGRESS OF TEST AND ANY FAILURES 2. INDICATES FAILING CIRCUITS 3. SHOWS POSITION OF RELAYS IN TESTED CIRCUITS (-—CHAIN LEAD CHECKS POSITION OF ALL TEST FEATURE RELAYS WITH WATCH- ING RELAYS. IF CHAIN IS CLOSED, TAPE IS INDEXED TO NEXT TEST CODE POSITION. IF CHAIN IS OPEN, CONTROL CIRCUIT WILL SWITCH TO TROUBLE INDICATING CIRCUITS START KEY STARTS AUTOMATIC PROGRESSION OF TAPE Fig. 13 — Block schematic. AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1173 also apparent that several paths can be found that will satisfy any one switched connection. Paths are assigned in sequence by a series relay loop circuit. The entry point on this circuit is changed periodically to distribute wear on the relays and switch cross points. Although only one lead for the switched circuit is shown for each cross point in Fig. 12 there are actually four leads through corresponding pairs of contacts through each cross point. The remaining leads are associated with the holding and signalling functions of the switch. The block schematic (Fig. 13) shows the principal functions which must be included in an automatic test set of this sort. A somewhat more detailed schematic is presented in Fig. 14 in order to show the functions of the forty test features 0-39. These are tabulated in Table I. The coding of the two letter combinations in the tape must follow a defuiite sequence in order that the machine may recognize and act on the information it receives. This sequence is as follows: 1. Code FW to stop the tape at the end of the reset cycle after which tests will proceed w^hen the start button is pressed. This is the first code on all tapes. 2. Codes to set up crossbar switches to connect each circuit terminal to its proper test set terminal and the proper termination. Knock down or release codes may also be sent. 3. Codes to operate or release "Kej^" relays. These relays are shown without windings in Fig. 14. 4. Codes to operate or release the watching relaj^s associated with the "Signal" relays which are shown with windings in Fig. 14. 5. Codes to operate or release relays controlling the lamps associated with relays in the circuit under test to aid in trouble shooting. 6. Codes to delay the timing out interval up to a maximum of ten seconds. 7. Code FJ which checks the matching of all signal and watching relays through the chain circuit for satisfaction of all test conditions being applied. In addition to the above, additional codes can be inserted after each FJ test signal to stop progress of the test to permit the tester to perform some required manual operation. After completion of this step he presses a button associated with that operation and the test proceeds. Option codes can also be inserted at the beginning and end of each testing step to permit bypassing of that part of the tape if the corresponding option keys are operated at the beginning of the test. A common knock down code FR can be inserted at any time to release all connections and re- lays for a quick disconnect and make all test set features available for 1174 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 03810038 SV SNOIiVNIWagj. Oi S3H0J.1MS dvassodD naHi a.03d SV 66-0 uNwai j-ind Avnsy oi S3hoiims yvgssoyo Auvaaai qnv oas 'iHd ndHi AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1175 reuse. A final code SC must be put in every tape to operate the OK lamp and gong if a successful test cycle has been performed or conversely to indicate that the tape should be re-run if trouble has been found and cleared during the test cycle to be certain that no new faults have been introduced. Preparation for testing a particular wired relay unit requires only the selection of a test cable one end of which is equipped with a suitable contact fixture for attachment to the unit terminal strip and the other with a gang plug for connection to the set. The proper tape is selected from a nearby file cabinet and inserted in the gate of the tape reader as shown in Fig. 15. The tape is stored in a cardboard carton 3^^ X 4 Table I Feature Numbers 1 and 2 3 and 4 5 through 19 20 21 and 22 23 and 24 25 through 34 35 36 and 37 38 and 39 Description of Functions High sensitivity relay circuit. Simulates 1,800-ohm sleeve circuit for busy test and general continuity through high resistance ciruits. Simulates the distant tip and ring terminations of a subscriber or exchange trunk. Provides for ringing, tone receival, dial pulse sending, line resistance, high-low or reverse battery su- pervision, pad control, continuity, and resistance verifica- tion. Auxiliary tip and ring circuit for holding, checking continu- ity, receival of tone on four wire or hybrid coil circuits. Loss range of less than 0.5 db, 0.5 to 1.5 db, 1.5 to 6 db and 6 to 15 db can be checked. Direct connections for supplying any of the ten terminating conditions. Simulates low or medium resistance sleeve circuits for margi- nal tests. Simulates the local tip and ring terminations of a switch- board or trunk circuit. Provides for ringing and dialing re- ceival, high-low reverse battery supervision, transmission pad control, tone transmission, continuity and resistance check by balance. An auxiliary tip and ring circuit for holding, checking con- tinuity, tone transmission on four-wire hybrid coil circuits. Low sensitivity relay circuits for general continuity checking. A circuit for checking balance on the (M) lead of composite or simplex signalling circuits and for checking receival of none, one or two pulses. Medium sensitivity relay circuits for continuity checking. Direct connections for supplying any of the ten terminations. 1176 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Fig. 15 — Perforated tape being inserted in reader. inches in size, the label of Avhich carries all pertinent information required for setup of the option keys, preliminary tests and manual operations during test. A separate 12 conductor cable equipped with individual test clips permits connection to internal parts of the circuit if needed for adequate tests. No other information than that on the box label, the circuit schematic and the lamp panel display is needed by the tester to operate the test set and to analyze and locate circuit faults when they occur. With the tape inserted, the test connections established and am'- preliminary operations pei-formed the tester has only to push the RESET button to index the tape to the initial perforation on the tape and the START button to initiate the test cycle. The set will continue to operate until either a circuit trouble is encountered or a manual operation must be performed. After a defect has been repaired, the automatic progres- sion of the tape is again started by the momentary depression of the STEP button. When a manual operation is performed the tape is re- started by the momentary depression of the red button associated with the lighted manual operation lamp signal. AUTOMATIC TESTING OF RELAY SWITCHING CIRCUITS 1177 TEST-TIME PER CIRCUIT HANDLING-TIME PER CIRCUIT iv'AsSl SETUP-TIME PER CIRCUIT START-UP-TIME PER CIRCUIT LOCATING-TIME PER DEFECT MANUAL TAPE MANUAL TAPE Fig. 16 — Comparison of manual and Tape-0-Matic test operation times. As might be expected the easy setup, automatic testing and superior trouble indicating features of the Tape-0-Matic test set have materially improved the quality and reduced the testing time and effort required for wired relay units as compared to the older manually operated sets. The aA'erage time per circuit for six representative units are shown graphically in Fig. 16. One time consuming operation on manual testing is the start up time allowance for reading and understanding the written test instructions which has no counterpart in the Tape-0-Matic tests and this alone represents a sizeable gain. The handling time of the unit itself is the only operation which is not reduced in automatic testing. HISTORY The initial Card-0-Matic test set was installed in 1938 in the Western Electric, Kearny, New Jersey plant. Post war and subsequent expansions of production levels have necessitated construction of six more sets of improved design of the type described earlier in this article. The first three Tape-0-Matic test sets were built in 1942 for the Wired Relay Unit Shop and additional sets have since been constructed to bring the number to twenty-six including six that are used in testing trunk units in the Toll Crossbar Shop. They have performed admirably with few changes from the initial design. They have been used to test well over a million wired units with a minimum of maintenance. This 1178 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 may be accounted for, in part, by the fact that most of the component parts are telephone type apparatus designed for heavy duty use. A maintenance feature is the use of 18 specially coded tapes which, together with a properly strapped input plug, permit the maintenance technician to obtain indications on the lamp display panel of the per- formance of the set. Nearly three thousand tapes have been coded to date. Of these ap- proximately two thousand are in active use on the many types of wired relay units made at the Kearny plant. More tapes are being added weekly as the Bell System telephone plant grows in size and complexity. CONCLUSION Automatic testing of wired relay switching circuits has been success- fully applied to the manufacture of these equipments at the Kearny, New Jersey, plant of the Western Electric Company for a number of years. Even though the total production is large, manufacture is essen- tially of a job lot nature due to large number of types made and is further compounded by the optional circuit arrangments that may be ordered. The solution to the problem was found through provision of flexibility in programing and cross connection leading to quick setup, rapid testing and improved transmittal of essential information to the tester to aid him in clearing circuit faults. Automatic Machine for Testing Capacitors and Resistance-Capacitance Networks By C. C. COLE and H. R. SHILLINGTON (Manuscript received May 8, 1956) The modern telephone system consists of a variety of electrical components connected as a complex network. Each year, millions of relays, capacitors, resistors, fuses, protectors, and other forms of apparatus are made for use in telephone equipment for the Bell System. Each piece of apparatus must meet its design requirements, if the system is to function properly. This article describes an automatic machine developed hy the Western Electric Company for testing paper capacitors and resistance-capacitance networks used in central office switching equipment. INTRODUCTION The capacitors discussed in this article are the ordinary broad Hmit units made ^^dth windings of paper and metal foil, packaged in a metal case. They include both single and double units in a package, connected to two, three, or four terminals. The networks consist of a capacitor of this same type connected in series with a resistor. The testing requirements for capacitors include dielectric strength, capacitance, and insulation resistance. These same tests plus impedance measurements are specified for networks. In general, requirements of the kind involved here could be adequately verified by statistical sam- pling inspection. However, in equipment as complex as automatic tele- phone switching frames, even the minor number of dielectric failures that would elude a properly designed sampling inspection would result in an intolerable expense in the assembly and wiring operations. While engineering considerations thus called for a detailed inspection for di- electric breakdown, it was recognized that detailed inspection of the other electrical requirements could be obtained at no additional expense for labor with automatic testing machines. 1179 1180 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 DESIGN CONSIDERATIONS In the development of this machine, the designer was faced with the same problems that obtain in the conception and design of any unit of complex equipment. These included the economic feasibility, reliability, simplicity, and versatility of such a machine. Economic Feasihilitij This can be determined by comparing the cost of performing the opera- tions to be made by the proposed machine with the cost by alternative methods. Estimates indicated that the cost of the machines could be re- covered within two years by the saving in labor that would be effected. Reliability Reliability has two connotations, (1) freedom from interruptions of production because of mechanical or electrical failure and (2) consistent reproducible performance. A rugged mechanical design combined with the use of the most reliable electrical components available is necessary. In addition, safeguards are required to protect the equipment from me- chanical or electrical damage. To achieve consistent reproducible per- formance, it is important that testing circuits of adequate stability be used. Besides, it was recognized that each circuit should be so ar- ranged that in case of a circuit failure, there would be immediate and positive action by the machine to prevent acceptance of defective prod- uct. All circuits are designed to provide positive acceptance. This means that the machine must take action to accept each item of product at each test position. In the case of the dielectric strength tests, a self- checking feature is included. Fig. 1 — Types of capacitors and networks tested. AUTOMATIC MACHINE FOR TESTING CAPACITORS AND NETAVORKS 1181 SimTplicity This type of equipment is operated by non-technical personnel. To 'minimize the possibility of improper operation of the equipment, it is important that adjustments and judgment decisions by the operator be minimized. From a production standpoint, it is important that the ma- chine be designed to permit quick changes to handle the variety of prod- uct to be tested. All "set-ups" are made by the operator and the switch- ing of circuits and changing of contact fixtures are simply and easily done. Versatililij The product tested by this machine includes a variety of physical sizes and terminal arrangements with a wide range of electrical test requirements (Fig. 1). a. Physical Sizes. The aluminum containers for this type of capacitors and R.C. Networks all have the same nominal length and width but are made in three different thicknesses. b. Terminals. The product is made with terminals of two different lenths, two different spacings, and four different patterns connected in eight combinations. It is necessary to provide contact fixtures and switching facilities to handle all of these combinations. c. Electrical Tests (1) Dielectric strength tests are made between terminals, and between terminals and can, on single unit packages. Two-unit packages require an additional test between units. (2) Capacitance: The capacitance of the product to be tested ranges from 0.02 mf to 5.0 mf or any combination within this range in one- or two-unit packages with no series resistance in the case of capacitors, but with a series resistor from 100 ohms to 1,000 ohms in the case of net- works. This problem is discussed in more detail in the description of the capacitance test circuit. (3) Insulation Resistance : The minimum requirements vary from 375 megohms to 3,000 megohms. (4) Impedance: The RC networks have impedance requirements at 15 kc that range from 100 ohms to 1,000 ohms. MECHANICAL ASPECTS OF TESTING MACHINE Packaging of the product precludes a magazine type of feed because the variety of terminal combinations associated with two-unit packages necessitates orientation in the contact fixtures that can not be done by 1182 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 1. HANDWHEEL FOR POSITIONING TEST FIXTURES. 2. ROTARY FEED MECHANISM. 3. PRODUCT PASSING ALL TESTS EJECTED FROM FIXTURE. 4. INSULATION RESISTANCE TEST PANEL AND TERMINAL COMBINATION "setup" SWITCHES. 5. CABINET HOUSING TEST CIRCUITS. 6. CONTAINERS FOR REJECTED PRODUCT. Fig. 2 — Testing machine in operation. mechanical means. A turret type construction is used to permit one operator to perform both the loading and unloading operations. Fig. 2 shows this machine in operation. The networks or capacitors are fed into the fixtures by an operator and as the turret carries the fix- tures past the feed mechanism, rollers on the feed mechanism are syn- chronized with the fixtures and the roller forces the unit under test into the contact fixture against a spring loaded plunger to make contact with the fixture contact springs. Also, synchronized wdth the feed mechanism is the closing of the gripper hook on the bottom end of the can contain- ing the unit under test. AUTOMATIC MACHINE FOR TESTING CAPACITORS AND NETWORKS 1183 REJECT CHUTE UPPER FIXTURE FOR SHORT TERMINAL PRODUCT. PRODUCT ON TEST GRIPPER HOOK PIN FOR SYNCHRONIZING ROTARY FEED MECHANISM. gripper h00;< follower arm and roller. "acceptance" SOLENOID plunger "acceptance" solenoid. Fig. 3 — View of rejection and acceptance mechanisms. commutator brush assembly and associated wiring. UPPER CONTACT FIXTURE FOR SHORT TERMINAL PRODUCT. OVERLOAD SLIP CLUTCH AND OVERLOAD SHUT-OFF SWITCH. Fig. 4 — View of turret. 1184 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 195G The acceptance or rejection of a unit under test at any one of the six' test positions depends on whether the test on the unit energizes the i "acceptance" solenoid associated with that test position. The gripper hook, which locks the unit under test in the contact fixture, is con- nected to a release shaft, follower arm, and roller (see Fig. 3). The roller rides in a track in which the plunger of each "acceptance" solenoid hes, unless removed by energizing the solenoid from its associated test cir-i cuit. In the case of a defective unit, the acceptance solenoid is not ener- ■ gized and the roller in passing over the plunger of the "acceptance" » solenoid trips the gripper hook and the spring loaded plunger in the 'H contact fixture ejects the defective unit. Units that pass all tests are ejected on a turntable to the left of the operator from which they are stacked in handling trays by the operator. The turret assembly includes the test fixtures, the gripper hooks and associated release shaft, follower arm and I'oller, and the brush assembly DIELECTRIC STRENGTH CONTROL PANEL. RESISTANCE STANDARDS FOR IMPEDANCE TEST CIRCUIT. SENSITROL RELAY FOR IMPEDANCE TEST CIRCUIT. 5 KILO-CYCLE OSCILLATOR. POWER SUPPLIES AND SWITCHING PANELS. Fig. 5 — Control panels for dielectric strength and impedance tests. AUTOMATIC MACHINE FOR TESTING CAPACITORS AND NETWORKS 1185 connected to the test fixtures. The commutator is stationary and its segments are connected to the test circuit through permanent wiring. Fig. 4 shows the turret. Each fixture has two sections, one above the other, with the contacts wired in parallel. The lower section is designed for making contact to stud mounted units with long terminals and the upper section for strap mounted units with short terminals. To change the machine "set-up" from one fixture to the other, the turret assembly i.s raised or lowered by means of the hand wheel, shown on Fig. 2, lo- cated at the right of the operator. This feature was incorporated in this machine to facilitate rapid "set-up" which is essential for testing small lots. An overload clutch is incorporated in the driving mechanism to prevent mechanical damage to the machine in case of a "jam". Fig. 5 shows the control panels for dielectric strength and impedance and Fig. 6 shows the control panels for the capacitance circuits. * 'JSi. 'ii. ij;; CAPACITANCE STANDARDS FOR PADDING TEST CIRCUIT ON UNIT NCI -CAPACITANCE STANDARD SERIES-PARALLEL AND RANGE SELECTOR SWITCHES. CAPACITANCE STANDARDS FOR PADDING TEST CIRCUIT ON UNIT N0.2 Fig. 6 — Control panels for capacitance circuits. < L z LU liJ3 Ct LU — OrCTP 1 <^ u < ;o "^ o q'~ tr CO O V O _l < I < -J z tr t ^ (r Q _| Z ONilVDIQNI NM0aviV3aQ AaVQNODaS cr lU O O I- o o > LU tr Hi O _l 'J Z (/) cc -1 tui- OJ ' I- ^ ■S- ^ ( 1 1 UJ I 'a q:<; 1186 > AUTOMATIC MACHINE FOR TESTING CAPACITORS AND NETWORKS 1187 ELECTRICAL ASPECTS OF TESTING MACHINES Tests are applied to the product in sequence during one revolution of the turret. 1. Dielectric strength test between terminals and can, and between terminals and studs. 2. Dielectric strength test between units in the same can when the can contains two units. 3. Dielectric strength test between terminals of each unit. 4. Impedance test. 5. Capacitance test. 6. Insulation resistance test. Dielectric Strength Test Circuit Operation Since the three dielectric strength tests are made on similar circuits, the operation of one of these circuits is described using the nomenclature and circuit designations shown in Fig. 7. A graphic interpretation of the circuit operations shown in Fig. 7 is given in Fig. 8. The "heart" of each circuit is a calibrated current sensitive relay K2 that operates on minute values of current resulting when a defective unit under test attempts to charge on the "test" commutator position. CAPACITOR ATTACHED TO INITIAL ICHARGE COMMUTATOR SEGMENT I 3 SECONDS TEST CAPACITOR ® DEFECTIVE PRODUCT ; CAPACITOR CHARGED ACCEPTABLE /Os PRODUCT W K2 K3 K4 o- CAPACITOR ATTACHED TO TEST SEGMENT OPENS K5 OPERATORS PATH (PRODUCT REJECTED) K2- K3- I-l LAMP TEST CIRCUIT NORMAL -> RESETS TEST CIRCUIT T ■^ 2 SECONDS v;s2 CAM SWITCH --K5 2| SECONDS S4 CAM SWITCH K9 CAPACITOR DISCHARGED --K7,K11 ACCEPTANCE SOLENOID, PRODUCT ACCEPTED wCAMMED TIME ■SWITCH :k6 K2 K3 --*^^1 %^\ I-l LAMP Fig. 8 — Sequence chart for dielectric strength test circuit operation. 1188 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 Two commutator segments are required to make a dielectric strength test. These segments are known as "initial charge" and "test". After the unit under test has been charged at the test voltage for three sec- onds on the "initial charge" segment, it passes to the "test" segment in' which the unit is again connected to the test voltage through relay K2,' current limiting and calibrating resistors R3 and R4 and the contacts; on the preset terminal selecting relays KIO. ■. One of the two conditions (under heading A and B below) may be^ encountered in making this test and the circuit operation for each will be discussed separately. A. Circuit Operation for AcceptaUe Product. An acceptable product retains the charge received on the "initial charge" segments and when! this unit reaches the "test" segment, no further charging current of a- magnitude great enough to operate relay K2 will flow through the unit. Two seconds after the unit under test has been connected to the test segment, a cammed timing switch S4 closes to operate discharge relay K9 to discharge the unit under test to ground through R7. The "self-check-" ing" feature mentioned earlier in this article under "Design Considera- : tions" functions as follows: After the unit under test has been on the "test" segment for approximately 2% seconds, a cammed timing switch ; (not shown) closes the memory test relay K6 which in turn closes the \ "go" calibration indicator relay K5 and the "A" contacts on this relay; grounds the high voltage test circuit through resistor R6. This resistor | is of such a value as to permit sufficient current to operate relay K2. i The contacts on relay K2 are not adequate to carry much current, so ' an auxiliary relay K3 is closed through contacts "A" on relay K2. Contacts "B" on relay K3 closes the indicator light circuit II and oper- ates relay Kll and the acceptance solenoid K7. Contacts "A" on the same relay lock relay Kll. The circuit is reset for the next vmit to be tested by momentarily opening the reset cammed switch S2. Relay Kll was added to the circuit to eliminate a "sneak circuit" that occurred occasionally following the reset when relay K5 opened faster than relay K3. This would result in relay K4 operating to reject the next unit tested. Relay Kl is controlled by switch SI operated by the manual control Tl on the test voltage power supply. The function of this relay is to add calibrating resistor R4 to the test circuit for voltages above 1,000 volts. Resistor R5, relay K8, and switch S3 control the manual calibrating "No Go" circuit for breakdown indicating relay K2. B. Circuit Operation for Defective Product. Defective product will not retain the charge it received on the "initial charge" segment and when it reaches the "test" segment, current will flow through the breakdown AUTOMATIC MACHINE FOR TESTING CAPACITORS AND NETWORKS 1189 indicating relay K2 in an attempt to charge the defective unit, l)ut this current will close relay K2 which in turn closes relay K3. This completes the circuit through the "B" contacts of relays K3, K5, and Kll to close memory relay K4. The closure of relay K4 prevents the memory test I relay KG from closing the "go" calibration indicator relay K5, thereby heaving contact "C" open on relay K5 and no power is applied to the "acceptance solenoid" K7 circuit, which rejects the unit under test. IMPEDANCE — • TEST CIRCUIT OPERATION The impedance test is made with a 15-kc circuit (see Fig. 9). One arm of the circuit, composed of resistor R12 paralleled by capacitor C5 and the imit under test, is compared with another arm, composed of resistor Pvll, paralleled by capacitor C4 and either one of two resistance boxes, R13 and R14 respectively, representing maximum and minimum im- pedance limits. The detector consists of a balanced diode V2 with a 1-0-1 microampere sensitrol relay K24 connected between the diode cathodes. If the impedance of the unit imder test falls within the limits for which the resistance boxes were set, the acceptance solenoid will be energized to accept the unit under test. A product outside the preset limits is re- jected because the acceptance solenoid is not energized. The circuit operation is discussed for the following four conditions under A, B, C, and D. A. Impedance Test on Dual Unit Capacitors This test is made on capacitors to prevent shipment of resistance- capacitance networks mislabeled as capacitors. Fig. 9 shows dual unit networks connected to the test terminals. Capacitors to be tested are connected to these same terminals. The greater than minimum test cutout relay K18 is preset closed by the switching circuit K23. The cammed memory reset timing switch S14 (normally closed) is opened momentarily to clear relay K19, K20, and K21 at the start of the test. The sensitrol relay reset switch S16 is cammed shut momentarily to reset the contactor on the sensitrol relay K24. With relay K2(3 open, the "less than maximum" resistance box R13 is connected to the test cir- cuit. If unit "A" of the dual unit capacitor under test is acceptable product, the contactor on sensitrol relay K24 will close on contact "A", which applies power to close and lock test No. 1 "less than maximum" memory relay K19. Cam operated switch S13 applies power to close relay K26 to connect the "greater than minimum" resistance box R14 into the test circuit. This resistance box is set on zero ohms when capaci- —nm^ — aiON3"10S ,3Nlld3DDV„ S2M CO f\ -2- O < 5 O 20< Ul '" ^ , n 1^ ^ J; < 5 > I- Ul X ,f , liJ in 1/1 < !i ct Ly uj 5 ^ 1 — I 1 -' ( I u UJ CO I/) UJ O CC Z aj5 OCt- 't— UJ .(HI I (M I Avn3d aaAOHoiiMS xoB aavQNVis tr c/) < ~ a. tr O 5-1 en Sciences. Served on Panel for Basic Research of Research and Develop- ' ment Board, 1947-49, and Scientific Advisory Board of Army Air Force, 1946-47. C. G. B. Garrett, B.A., Cambridge University (Trinity College), 1946; M.A., Cambridge, 1950; Ph.D., Cambridge, 1950. Instructor in Physics, Harvard University, 1950-52. Bell Telephone Laboratories, 1952-. Before coming to the Laboratories, Dr. Garrett's principal re- search was in the field of low-temperature physics. At the Laboratories he has been engaged in research and exploratory development on semi- conductor surfaces and, for the past year, has supervised a group work- ing in this field. He is the author of "Magnetic Cooling" (Harvard CONTRIBUTOES TO THIS ISSUE 1235 University Press, 1954). Senior Scholar of Trinity College, Cambridge, 1945. Twisden Student of Trinity College, 1949. Fellow of Physical Society (London). Member of American Physical Society. L. D. Hansen, B.S., Montana State College, 1924; Western Electric Company, 1924-. Mr. Hansen joined the Equipment Engineering Or- 'ganization at the Hawthorne Plant of The Western Electric Company in Chicago in 1924 where he was engaged in preparation of telephone central office specifications. He transferred to the Kearny, N. J., Plant in 1928 where he was promoted to section chief in 1929. He transferred to the Engineer of Manufacture Organization in 1930 and worked on carrier and repeater test development and methods until 1941 when he was promoted to Department Chief in charge of wired switching ap- paratus and equipment test set development and methods. William C. Jakes, Jr., B.S.E.E., Northwestern University, 1944; M.S., Northwestern, 1947; Ph.D., Northwestern, 1948. Bell Telephone Laboratories, 1949-. Dr. Jakes is engaged in microwave antenna and propagation studies and holds a patent in microwave antennas. He is the author of chapter in antenna engineering handbook (McGraw-Hill). Member of Sigma Xi, Pi Mu Epsilon, Eta Kappa Nu, LR.E. and Phi Delta Theta. Amos E. Joel, Jr., B.S., Massachusetts Institute of Technology, 1940; M.S., M.I.T., 1942; Bell Telephone Laboratories, 1940-. Mr. Joel is Switching Systems Development Engineer responsible for coordinating the exploratory development of a trial electronic switching system. Prior to his present position he worked on relay engineering, crossbar test laboratory, fundamental development studies, circuits for relay com- puters, preparation of a text and teaching switching design, designing j AMA computer circuits and making fundamental engineering studies on new switching systems. He holds some forty patents. Member of A.I.E.E., LR.E., Sigma Xi and Association for Computing Machinery. Archie P. King, B.S., California Listitute of Technology, 1927. After three years with the Seismological Laboratory of the Carnegie Institu- tion of Washington, Mr. King joined Bell Telephone Laboratories in 1930. Since then he has been engaged in ultra-high-frequency radio re- search at the Holmdel Laboratory, particularly with waveguides. For the I last ten years Mr. King has concentrated his efforts on waveguide trans- mission and waveguide transducers and components for low-loss circular 1236 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1956 electric wave transmission. He holds at least a score of patents in the waveguide field. Mr. King was cited by the Navy for his World War II radar contributions. He is a Senior Member of the I.R.E. and is a Mem- ber of the American Physical Society. D. T. RoBB, B.S., University of Chicago, 1927; Western Electric Company, 1927-. Mr. Robb has been concerned with measurement and testing problems throughout his career. In the electrical laboratory ■ at Hawthorne Works, Chicago, he specialized in ac standardization. Later he worked on the development of shop test methods and test sets. ' In 1944 he transferred to take charge of radar test engineering at the Eleventh Avenue Plant of Western Electric in New York City. In 1946 he supervised the engineering of the standards laboratory at Chatham Road Plant in Winston Salem, N. C. Currently, he has charge of trans- mission test set development and test set design at Kearny Works, N. J. Harry R. Shillington, B.S. in E.E. Iowa State College, 1937; Long Lines Department of the American Telephone and Telegraph Company, 1928-1932; Western Electric Company, 1937-. Mr. Shillington's first assignment was that of product engineering on panel dial equipment. During World War II and the Korean War he was engaged in test engi- neering on various defense projects. He is presently concerned with the development of special test facilities for telephone apparatus. Member of Eta Kappa Nu and Tau Beta Pi. Friedolf M. Smits, Dipl.Phys. and Dr.Rer.Nat., University of Frei- burg, Germany, 1950; research assistant, Physikalisches Institut, Uni- versity of Freiburg, 1950-54; Bell Telephone Laboratories, 1954-. As a member of the Solid State Electronics Research Department of the : Laboratories, Dr. Smits has been concerned with diffusion studies of germanium and silicon for semiconductor device applications. He is a member of the American Physical Society and the German Physical Society. Frank H. Tendick, Jr., B.S.E.E., 1951, University of Michigan; Bell Telephone Laboratories, 1951-. Mr. Tendick was first engaged ini work pertaining to the synthesis of networks employed in the L3 coaxial i cable system. Later he engaged in the design of transistor networks for digital computers. More recently, he has been associated with exploratory studies of submarine cable systems. He is a member of the I.R.E. Mr. CONTRIBUTORS TO THIS ISSUE 1237 Tendick also belongs to four honor societies, Tau Beta Pi, Eta Kappa iNu, Sigma Xi and Phi Kappa Phi. Leishman R. Wrathall, B.S., 1927, University of Utah. Mr. Wrathall did another year of graduate work at the University of Utah and joined Bell Telephone Laboratories in 1929. For many years he was primarily concerned with studies of the characteristics of non-linear coils and ca- pacitors. During World War II non-linear coils were used extensively in radar systems, and his work in this field was intensified. Later he was occupied with general circuit research. He is now engaged in studies of conductor problems, particularly digital repeaters, as a member of the Transmission Research Department at Murray Hill. ! HE BELL SY S^ E M / ecnnicm louma^ OTED TO THE SC I E N T I FIC^^^ AND ENGINEERING •ECTS OF ELECTRICAL COMMUNICATION UME XXXV NOVEMBER 1956 NUMBER 6 Ft Nobel Prize in Physics Awarded to Transistor Inventors i Theory of the Swept Intrinsic Structure w. t. bead, jbT. 1239 A Medium Power Traveling- Wave Tube for 6,000-Mc Radio Relay J. p. LAico, H. L. Mcdowell and c. r. moster 1285 Helix Waveguide s. p. morgan and j. a. young 1347 Wafer-Type Millimeter Wave Rectifiers w. m. sharpless 1385 Frequency Conversion by Means of a Nonlinear Admittance C. F. EDWARDS 1403 Minimization of Boolean Functions e. j. mccluskey, jr. 1417 Detection of Group Invariance or Total Symmetry of a Boolean Function e. j. mccluskey, jr. 1445 Bell System Technical Papers Not Published in This Journal 1454 Recent Bell System Monographs 1461 Contributors to This Issue 1465 COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD A. B, GOETZE, President, Western Electric Company M. J. KELLY, President, BeU Telephone Laboratories E. J. McNEELY, Execviivc Vice President, American Telephone and Telegraph Company EDITORIAL COMMITTEE B. McMillan, Chairman S, E. BRILLHART E. I.GREEN A. J. BUSCH R. K. HONAMAN L. R. COOK H. R. HUNTLEY A. C. DICKIE80N F. R. LACK R. L. DIETZOLD J. R. PIERCE K. E. GOULD G. N. THAYER EDITORIAL STAFF J. D. TEBO, Editor R. L. SHEPHERD, Production Editor THE BELL SYSTEM TECHNICAL JOURNAL is published six times a year by the American Telephone and Telegraph Company, 195 Broadway, New York 7, N. Y. F. R. Kappel, President; S. Whitney Landon, Secretary; John J. Scan- Ion, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed in U. S. A. Nobel Prize in Physics Awarded to Transistor Inventors The Swedish Royal Academy of Sciences announced on November 1 that a Nobel Prize in Physics, most highly coveted award in the world of physics, had been awarded jointly to Dr. Walter H. Brattain of the Laboratories Physical Research Department, with Dr. John Bardeen and Dr. William Shockley, both former members of the Laboratories. The prize was awarded for ''investigations on semiconductors and the discovery of the transistor effect." This marks the second time that Avork done at the Laboratories has been recognized by a Nobel Prize. The previous recipient Avas Dr. C. J. Davisson who shared in the 1937 prize for his discovery of electron dif- fraction as a result of experiments carried out with Dr. L. H, Germer, also of the Laboratories. Each of the three Avinners of this year's prize Avill receive a gold medal, a diploma and a share of the $38,633 prize money. When he Avas notified that he Avas one of these Avinners, Dr. Brattain said, "I certainly ap- preciate the honor. It is a great satisfaction to have done something in life and to haA^e been recognized for it in this Avay. HoAvever, much of my good fortune comes from being in the right place, at the right time, and having the right sort of people to Avork Avith." The principle of transistor action Avas discovered as a result of funda- mental research directed toAA^ard gaining a better understanding of the surface properties of semiconductors. Following World War II, intensiA^e programs on the properties of germanium and silicon AA'ere undertaken at the Laboratories under the direction of William Shockley and S. 0. Morgan. One group in this program engaged in a study of the body properties of semi-conductors, and another on the surface properties. Dr. John Bardeen served as theoretical physicist and R. B. Gibney as chemist for both groups. These iuA'estigations, Avhich resulted in the in- \'ention of the transistor, made extensiA^e use of knoAvledge and tech- niques developed by scientists here and elscAvhere, particularly by mem- bers of the Laboratories — R. S. Ohl, J. H. Scaff and H. C. Theuerer. Since the transistor Avas announced, little more than eight years ago, it has become increasingly important in Avhat has been called the "neAv 11 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 195G The Nobel Prize winners in an historic photograph taken in 1948 when the annonncement of the invention of the transistor icas made. Left to right, John Bardccn, William Shockley and Walter H. Brattain. electronics age." As new transistors and related semiconductor devices are developed and improved, the possible fields of application for these devices increase to such an extent that they may truly be said to have "revolutionized the electronics art." The invention of the transistor, basis for the Nobel Prize award,' represents an outstanding example of the combination of research team- work and individual achievement in the Bell System that has meant so; much to the rapid development of modern communications systems. Dr. Brattain received a B.S. degree from Whitman College in 1924, an M.A. degree from the University of Oregon in 1926, and a Ph.D. degree from the University of Minnesota in 1928. He joined Bell Telephone Laboratories in 1929, and his early work was in the field of thermionics, particularly the study of electron emission from hot surfaces. He also studied frequency standards, magnetometers and infra-red phenomena. NOBEL PRIZE IN PHYSICS 111 Subsequently, Mr. Brattain engaged in the study of electrical con- ductivity and rectification phenomena in semiconductors. During World War II, he was associated with the National Defense Research Com- mittee at Columbia Fni\'ersity ^\■here he worked on magnetic detection of submarines. Mr. Brattain has received honorary Doctor of Science degrees from Whitman College, Union College and Portland University. His many awards include the John Scott Medal and the Stuart Ballantine INIedal, both of which he received jointl^y with John Bardeen. Mr. Brattain is a Fellow of the American Academy of Arts and Sciences. Dr. Bardeen received the B.S. in E.E. and M.S. in E.E. degrees from the University of Wisconsin in 1928 and 1929 respectively, and his Ph.D. degree in Mathematics and Physics from Princeton University in 1930. After serving as an Assistant Professor of Physics at the Uni- versity of Minnesota from 1938 to 1941, he worked with the Naval Ord- nance Laboratory as a physicist during World War II. In 1945 he joined the Laboratories as a research physicist, and was primarily concerned Clinton J . JJuvisson Previous Laburatories Nobel Laureate In December, J 937, Di'. Clinton J. Davisson of the Laboratories was awarded the Xobel Prize in Piiysics for his discovery of electron tliffrac- tion and the wave properties of electrons. He shared the jDrize with Professor G. P. Thompson of London, who worked in the same field, though there was little in common between their techniques. Dr. Davisson's work on electron diffraction started as an at- tempt to understand the characteristics of secondary emission in multi- grid electron tubes. In this work he discovered patterns of emission from the surface of single crystals of nickel. By studying these patterns, Dr. Davisson, with Dr. L. H. Germer and their associates, proved that reflected electrons have the properties of trains of waves. Dr. Davisson was awarded the B.S. degree in physics from the Univer- sity of Chicago in 1908 and the Ph.D. degree from Princeton in 191L From September, 1911, until June, 1917, he was an instructor m physics at the Carnegie Institute of Technology, coming to the Laboratories on a wartime leave of absence. He found the climate of the Laboratories conducive to basic research, however, and remained until his retirement in 1946. Besides his work on electron diffraction. Dr. Davisson did much significant work in a varietj^ of fields, particularly electron optics, mag- netrons, and crystal physics. iv THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 with theoretical problems in solid state physics, including studies of semiconductor materials. Mr. Bardeen, whose honors include an honorary Doctor of Science degree from Union College, the Stuart Ballantine Medal, the John Scott Medal, and the Buckley Prize, is a member of the National Acad- emy of Sciences. He joined the University of Ilhnois in 1951. Dr. Shockley received a B.Sc. degree from the California Institute of Technology in 1932, and a Ph.D. degree from the Massachusetts In- stitute of Technology in 1936. He joined the staff of Bell Telephone Laboratories in 1936. In addition to his many contributions to solid state physics and semiconductors, Mr. Shockley has worked on electron tube and electron multiplier design, studies of various physical phe- nomena in alloys, radar development and magnetism. His many awards include an honorary degree from the University of Pennsylvania, the Morris Liebmann Memorial Prize, the Buckley Prize, the Comstock Prize and membership in the National Academy of Sciences. Dr. Shockley left the Laboratories to form the Shockley Semi- conductor Laboratory at Beckman Instruments, Inc., in 1955. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXV NOVEMBER 1956 number 6 Copyright 1966, American Telephone and Telegraph Company Theory of the Swept Intrinsic Structure By. W. T. READ, JR. (Manuscript received March 4, 1956) The electric field and the hole and electron concentrations are found for reverse biased junctions in which one side is either intrinsic (!) or so weakly doped that the space charge of the carriers cannot he neglected. The analysis takes account of spare charge, drift, diffusion and non linear recombination. A number of figures illustrate the penetration of the electric fii eld into a PIN structure with increasing bias for various lengths of the I region. For the junction between a highly doped and a weakly doped region, the reverse cur- rent increases as the square root of the voltage at high voltages; and the space charge in the weakly doped region approaches a constant value that depends on the fixed charge and the intrinsic carrier concentration. The mathematics is greatly simplified by expressing the equations in terms of the electric field and the sum of the hole and electron densities. i I. INTRODUCTION Applications have been suggested for semiconductor structures having j both extrinsic and intrinsic regions. Examples are the "swept intrinsic" structure, in which a region of high resistivity is set up by an electric [ field that sweeps out the mobile carriers, and the analogue transistors, ! where the intrinsic region is analogous to the vacuum in a vacuum tube. However, the junction between an intrinsic region and an N or P region 1239 1240 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 is considerably less well understood than the simple NP junction. Most of the assumptions that make the NP case relatively simple to deal with do not apply to junctions where one side is intrinsic. Specifically, the space charge is that of the mobile carriers; thus the flow and electrostatic problems cannot be separated as they can in PN junction under reverse bias. The following sections analyze the iV-intrinsic - P structure under reverse bias. For a given material with fairly highly doped extrinsic regions, the problem is defined by the length of the intrinsic region and the applied voltage. Taking the intrinsic region infinitely long gives the solution for a simple A^-intrinsic or P-intrinsic structure. The results are given and plotted in terms of the electric field distribution. From this the potential, space charge and carrier concentrations can be found; so also can the current-voltage curve. The final section considers the case where the middle layer contains some fixed charge but where the carrier charge cannot be neglected. Qualitative Discussion of an N-intrinsic-P Structure Consider an A^-intrinsic-P structure where the intrinsic, or /, region is considerably wider than the zero bias, or built-in, space charge regions at the junctions, so that there is normal intrinsic material between the junctions. The field distribution at zero bias can be found exactly from the zero-current analysis of Prim.' Throughout the intrinsic region, hole and electron pairs are always being thermally generated and recombining at a rate determined by the density and properties of the traps, or recom- bination centers. Under zero bias the rates of generation and recombina- tion are everywhere equal. Suppose now a reverse bias is applied causing holes to flow to the right and electrons to the left. Some of the carriers generated in the intrinsic region will be swept out before recombining. | This depletes the carrier concentration in the intrinsic region and hence raises the resistivity. It also produces a space charge extending into the intrinsic layer. The electrons are displaced to the left and the holes, to ■ the right. Thus the space charge opposes the penetration of the field into the intrinsic region; that is, the negative charge of the electrons on i the left and positive charge of the holes on the right gives a field distribution with a minimum somewhere in the interior of the intrinsic region and maxima at the NI and IP junctions. If holes and electrons had equal mobilities, the field distribution would be symmetrical with a minimum in the center of the intrinsic region. Likewise, the total carrier 1 R. C. Prim, B. S. T. J., 32, p. 665, May, 1953. THEORY OF THE SWEPT INTRINSIC STRUCTURE 1241 [concentration (holes plus electrons) would be symmetrical with a maxi- [mum in the center. As the applied bias is increased the hole and electron distributions are further displaced relative to one another and the space charge increases. Finally, at high enough biases, so many of the carriers are swept out immediately after being generated that few carriers are left in the intrinsic region. Now the space charge decreases with increas- ing bias until there is negligible space charge, and a relatively large and constant electric field extends through the intrinsic region from junction to junction. This may happen at biases that are still much too low to appreciably affect the high fields right at the junction or in the extrinsic layers, which remain approximately as they were for zero bias. The current will increase with voltage until the total number of carriers in the intrinsic region becomes small compared to its normal value. After that, there is negligible further increase of current with voltage. All the carriers generated in the intrinsic region are being sw^ept out before recombining. In general, the current will saturate while the minimum field in the intrinsic region is still small compared to the average field. Comparison with the NP Structure The analysis is more difficult than in a simple reverse-biased NP structure. In the NP case there is a well defined space charge region in I which carrier concentration is negligible compared to the fixed charge of i the chemical impurities; so the field and potential distributions are easily found from the known distribution of fixed charge. Outside of the space i charge region are the diffusion regions in which the minority carrier con- j centration rises from a low value at the edge of the space charge region ; to its normal value deep in the extrinsic region. However, there is no , space charge in this region because the majority carrier concentration, I by a very small percentage variation, can compensate for the large per- I centage variation in minority carrier density. The minority carriers flow by diffusion. Since the disturbance in carrier density is small compared to the majority density, the recombination follows a simple linear law (being directly proportional to the excess of minority carriers). Thus the minority carrier distribution is found by solving the simple diffusion equation with linear recombination. None of these simplifications extend to the NIP or NI or IP structure. There is, in the intrinsic region, no fixed charge; hence the space charge is that of the carriers. There is no majority carrier concentration to maintain electrical neutrality outside of a limited space charge region. 1242 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 It is necessary to take account of (1) space charge, (2) carrier drift, (3) carrier diffusion and (4) recombination according to a nonlinear bi- molecular law. Of these four, only space charge and recombination are never simultaneously important in practical cases. Nevertheless certain simplifications can be made if the problem is formulated so as to take advantage of them. The field and carrier distributions in the intrinsic region are found by joining two solutions: one solution is for charge neutrality; the other, which we shall call the no-recombination solution is for the case where the recombination rate is negligible compared to the rate of thermal generation of hole electron pairs. We shall show that in practical cases the ranges of validity of the two solutions overlap; that is, wherever recombination is important, we have charge neutrality. Prim's Zero-Current Approximation Prim* derived the field distribution in a reverse biased NIP structure; on the assumption that the hole and electron currents are negligibly small differences between their drift and diffusion terms, as in the zero- bias case. He showed that the average diffusion current is large compared to the average current. However, as it turns out, this is misleading.: Throughout almost all of the intrinsic region (where the voltage drop occurs in practical cases) the diffusion current is comparable to or smaller than the total current. The larger average diffusion current comes from the extremely large diffusion current in the small regions of high space charge at the junctions. Prim's analysis, in effect, neglects the space charge of the carriers generated in the intrinsic region. These may be neglected in calculating the field distribution if the intrinsic region is sufficiently narrow or the reverse bias sufficiently high. In the appendix we derive the limits within which Prim's calculation of the field and potential will be valid. The range will increase with both the Debye length and the diffusion length in the intrinsic material. However, in cases of practical interest the zero-current approximation may lead to serious errors in the field distribution and give a misleading idea of the penetration of the field into the intrinsic region. The present, more general analysis, reduces to Prim's near the junctions where the zero- current assumption remains valid. The zero current approximation was, of course, not intended to give the hole and electron distributions in the intrinsic region or to evaluate the effects of interacting drift, diffusion and recombination. Ibid. THEORY OF THE SWEPT INTRINSIC STRUCTURE 1243 Outline of the Following Sections Sections II through V deal with the ideal ease of equal hole and elec- tron mobilities. Here the problem is somewhat simplified and the physics easier to visualize because of the resulting symmetry. In Section VI, the general case of arbitrary mobilities is solved by an extension of the methods developed for solving the ideal case. The technique is to deal not with the hole and electron flow densities but with two linear com- binations of hole and electron flow densities that have a simple form. Section II deals with the basic relations and in particular the formula for recombination in an intrinsic region for large disturbances in carrier density. The nature and range of validity of the various approximations are discussed. Section III derives the field distribution in regions where recombination is small compared to pair generation. Section IV treats the recombination region and the smooth joining of the recombination and no-recombination solutions. Section V considers the role of chffusion in current flow and the situation at the junctions where the field and carrier concentration abruptly become large. The change in form of the solution near the junctions is shown to be represented by a basic in- stability in the governing differential equation. Section VI extends the results to the general case of unequal mobilities. Section VII deals with the still more general case where there is some fixed charge in the "in- trinsic" region. If the density of excess chemical impurities is small com- pared to the intrinsic carrier density, the solution remains unchanged in the range where recombination is important. In the no-recombination region the solution is given b}'' a simple first order differentiatial equation which can be solved in closed form in the range where the carrier flow is by drift. The fixed charge may have a dominant effect on the space charge even when the excess density of chemical impurities is small com- pared to the density n, of electrons in intrinsic material. Consider, for example, a junction between an extrinsic P region and a weakly doped n region having an excess density N = Nd — Na oi donors. In the limit, as the reverse bias is increased and the space charge penetrates many difi"usion lengths into the n region, the field distribution becomes linear, corresponding to a constant charge density equal to m + Vn^ + 8 n.-^jeV^i'] where Li is the diffusion length in the weakly doped n type region and £ is the Debye length for intrinsic material. For germanium at room tem- perature £,/Li is the order of 10~^ Thus, in this example, a donor density as low as lO" cm~^ will have an appreciable effect on the space charge. 1244 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 II. BASIC RELATIONS The problem can be stated in terms of the hole density p, the electron density Ji, and the electric field E and their derivatives. Let the distance .T be measured in the direction from N to P. The field will be taken as positive when a hole tends to drift in the -{-x direction. The field in- creases in going in the -fx direction when the space charge is positive. Poisson's equation for intrinsic material is f = a(p - «) (2.1) where the constant a has the dimensions of volt cm and is given in terms of the electronic charge q and the dielectric constant k by ■iirq a = — - K For germanium a = 1.17 X 10~ volt cm. The hole and electron flow densities Jp and J„ are^ Jp = nEv - d'^ = ^,Je -—^Inv ax \ q dx Jn = —hi nEn + D -^ j = —bun E -\ -Inn q dx (2.2) where n and D = n kT/q are the hole mobility and diffusion constant re- spectively, k is Boltzmann's constant (8.63 X 10~^ cv per °C) and T is the absolute temperature. The ratio b of electron mobility to hole mo- bility we take to be unity. This makes the problem symmetrical in n and p and consequently easier to understand. Section \T will extend the re- sults to the general case of arbitrary b. Charge and Particle Flow For some purposes it helps to express the flow not in terms of Jp and Jn but rather in terms of the current density / and the flow density J = Jp -\- Jn oi particles, or carriers. The current density / = q{Jp — Jn). Each carrier, hole or electron, gives a positive contribution to J if it goes in the +.r direction and a negative contribution if it goes in the —X direction. In other words, J is the net flow of carriers regardless of their charge sign. The current / is constant throughout the intrinsic ^ See, for example, Electrons and Holes in Semiconductors, by W. Shockley. D. Van Nostrand Co., New York, 1950. THEORY OF THE SWEPT INTRINSIC STRUCTURE 1245 region. Particle flow is away from the center of the intrinsic region. Carriers are generated in the intrinsic region and flow out at the two ends, the electrons going out on the N side and holes on the F side. Thus J is positive near the IP junction and negative near the NI junction. From the definitions of / and J and equations (2.2) - = nE{p -\- n) - D -J- {-p - 11) q ax J = m£'(p - n) - D^(p-\-n) (2.3) It is convenient to express the equations in terms of E and a dimen- sionless variable s = n + p 2ni (2.4) I which measures how "swept" the region is. In normal intrinsic material s = 1. In a completely swept region s = 0; at the junctions with highly '' extrinsic material s ^ I. Using Poisson's equation to express p — n in terms of E, equations (2.3) become r J, qD ctE 1 = asE - ■ — -— a ax- J = d_ dx 2a - 27uDs (2.5) where a = 2 /x n,g is the conductivity of intrinsic material. The particle flow J is thus seen to be the gradient of a flow potential that depends only on E and s. Equations (2.5) can be written in the form [ a\ sE - £■ drE dx^ (2.6) (2.7) where £ — \/kT/2aniq is the Debye length in intrinsic material and V2kT q& (2.8) is a field characteristic of the material and temperature. Specifically Ex is \/2 times the field required to give a voltage drop kT/q in a Debye 1246 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 length. For germanium at room temperature £ = 6.8 10~* cm and Ei = 383 volts per cm. Both / and / are the sum of a drift term and a diffusion term. For charge neutrahty, where p — nis small compared to p + n, both charge diffusion and particle drift can be neglected. We shall see later that, except right at the junctions, charge diffusion is negligible. The Equations of Continuity The two equations of continuity are I (.ttJ p (J/fJ fi dx dx = g - r (2.9) where g is the rate of pair generation and r the rate of recombination. In terms of / and /, these become I ^ = 0 (2.10) or / = constant and ^ = 2(^ - r) (2.11) which says that the gradient of particle flow is equal to the net rate of particle generation, that is, twice the net rate of pair generation. To complete the statement of the problem it remains to express g and r in terms of n and p. Generation and Recombination The direct generation and recombination of holes and electrons follows the mass action law, in which g — r is proportional to w/ — np. The con- stant of proportionality can be defined in terms of a lifetime t as fol- lows: Let 8p = 871 <$C Ui be a small disturbance in carrier density. Then defining T(g — r) = —8n, we see that the proportionality constant in the mass action law is (2niT)~ . So , _ , = !^ipi!£ (2.12) and the generation rate g = ^ (2.13) is independent of carrier concentration. THEORY OF THE SWEPT INTRINSIC STRUCTURE 1247 In actual semiconducting materials, recombination is not direct. Rather it occurs through a trap, or recombination center. The statistics of indirect recombination has been treated by Shockley and Read^ for a recombination center having an arbitrary energy level &i somewhere in the energy gap. At any temperature the trap level can be expressed by the values rii and pi which n and p would have if, at that temperature, the Fermi level were at the trap level. Shockley and Read showed that, at a given temperature, the lifetime for small disturbances in carrier density is a maximum in intrinsic material. It drops to limiting values T„o and Tpo in highly extrinsic n and p material, respectively. The formula for gr — r in terms of n and p is g - r = — T— - — . , ^. . r (2.14) Tpo(n + ni) + Tnoip + Pi) For our purposes it is more convenient to define the hfetime r not by '''(d ~ f) = — 6n « Wi , but rather as the proportionality factor in the mass action law. Then r is not necessarily constant independent of carrier density. From (2.12) and (2.14) Tpo(n + Wi) + T„o(p + P\) i^ . re. r = ~ (2.15) We shall be interested in the hfetime in the region where 7i and p are equal to or less than n, . r decreases as n and p decrease; that is, t is less in a swept region than in normal intrinsic material. Let r = Tj for 7i = p = 7ii and T = To for n = p = 0. The total range of variation of r is by a factor of II = 1 + ^^'(t'po + rpo) ,^ ^g. TO PlTnO + rilTpO Let the energy levels be measured relative to the intrinsic level, and define a level 8o by \ TpO Then if &t = &o , niTpo = piTno • Now eq. (2.16) becomes So = kT\n .., TpO ^^^sech(^i^) (2.17) Thus the variation in r increases as the ratio of Tno to Zpo deviates from unity and as the trap level moves away from the level 8o . 3 W. Shockley and W. T. Read, Jr., Phys. Rev., 87, p. 835, 1952. 1248 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 The data of Burton, Hull, Morin, and Severien* shows that a typical value of the ratio of Tpo and r„o is about 10. This means that the varia- tion in r with carrier concentration will be less than 10 per cent provided St is about -ikT from So . In what follows we shall assume that this is so. Then we have the mass action law (2.12) with r a constant, which could be measured by one of the standard technicjues involving small dis- turbances in carrier density. The general case of variable t is considered briefly at the end of Section IV. Outline of the Solution To conclude this section, we discuss briefly the form of the equations and the solution in different parts of the intrinsic region. First consider (2.6) for the current in the ideal case of equal mobilities. In Sections III and V we shall show that throughout almost all of the intrinsic region the current flows mainly by pure drift so we can take I = asE. The reason for this is as follows. The quantity £ is so small that the diffusion term remains negligible unless the second derivative of E becomes large — so large in fact that the E versus x curve bends sharply upward and both the drift and diffusion terms become large compared to the current /. This is the situation at the junction where / is the small chfference be- tween large drift and diffusion terms. Thus (2.6) has two limiting forms: (1) Except at the junctions the current is almost pure drift so 7 = asE is a good approximation. In Section III we derive an upper limit for the error introduced by this approximation and show how the approximate solution can be corrected to take account of the diffusion term. (2) At the junction, the drift term becomes important and the current rapidly becomes a small difference between its drift and diffusion terms and the solution approaches the zero current solution, for which sE = £^ (fE/dx^. In Section V we derive an approximate solution that joins onto the I = asE solution near the junction and then turns con- tinously and rapidly into the zero current solution. We shall call this the junction solution. The abrupt change in the solution from (1) to (2) near the junction is shown to be related to a basic instability in the differential equation. This makes it impractical to solve the equations on a machine. When the applied bias is large compared to the built-in voltage drop, the junction region will be of relatively little interest so the I = asE solution can be used throughout. In the region where / = > y / ^ y /^ / '< ^ A=0.01 -<- ^ y — 0.02 0.04 , 0.06 x/2Ll 0.08 0.10 Fig. 2 — Field Distributions for L = 0.2Li 1254 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 As A decreases and becomes negative the cubic approaches the form E' = Eo' + Ei (3.6) where Eq = —AEi is the minimum value of E". This form of the solu- tion will be valid when the minimum E is large compared to {lEi/a). As Eq increases, the voltage increases and the curve becomes flatter. This is because the increasing field sweeps the carriers out and reduces the space charge; so the drop in field decreases. If (3.4) for E/Ei versus x/2Li is extended to indefinitely large values of x/2Li , it approaches the straight line of slope 1 going through the origin. Since E is always positive the curve is above this straight line at X = 0. li A is negative the curve is always above the straight line and always concave upward. If A is positive, the curve crosses the straight 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 ^ y y ^^ ^ / A = -1 ^^ ^ ^^ // // / / / / 0 / / / ' > /o.i y / / /, / /'/3 / / / / / f / / / / / Ia=2/3 y / / / ^ J _^ / 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1,0 X/2Ll Fig. 3 — Field Distributions for L = 2L,- THEORY OF THE SWEPT INTRINSIC STRUCTURE 1255 line at E/Ei = I/aEiA and thereafter remains under it approaching it from below. For positive A the curvature, which is upward near the ori- gin, changes to downward at about x/2Li = -y/A. The carrier concentrations n and p can be found from the E versus x curves with the aid of Poisson's equation p — n = 1/a dE/dx and the definition s = (w + p)/2ni with s = I/aE. These relations and (3.4) give p — n _ X p + n L 1 1 + IE,'' (3.7) From (3.4) and (3.7) we may distinguish the following two regions on the cubic: (1) When E^/Ei is smaller than I/aEi (which as we have seen is usually smaller than unity), the E versus x curve is concave upward, the hole and electron concentrations are almost equal (charge neutrality) and the particle flow is by diffusion. (2) When E^/E-^ is greater than I/aEi , in general there is space charge and the particle flow, like the charge flow, is by drift. The curve is concave downward for positive A. Figure 6, which we will discuss in Section IV, shows the field and car- rier distributions for L = 2Li and A = 0.665 plotted on a logarithmic 2.8 2.4 ^ / /// 2.0 / // // / 1- / y/ // / 1.2 A 2 ~~ 3 / // / / / 0.8 0.4 . . n / / / V / / y^ 2 / 0 ^ 3 r 0 0.4 0.8 1.2 1.6 2.0 2.4 2 8 3 2 X/2Ll Fig. 4 — Field Distributions for L = GL,- . 1256 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 scale to show the behavior at low values of field and carrier density. In the region of no-recombination the field distribution is indistinguish- able from that for A = ^, which is plotted in Fig. 3 on a linear scale. In the region where recombination is important the solution is found from the assimiption of charge neutrality as will be discussed in Section IV. The cubic and charge neutrality solutions are each shown dashed outside of their respective ranges of validity. For A = 0.665 the half length of the intrinsic region is 2.098 X 2Li . Thus the length of the intrinsic re- gion is more than twice the effective length 2L in which current is generated. The effective length will be discussed in more detail in Section IV and it will be shown that the effective length 2L of current generation is equal to the twice the distance from the IP junction to the minimum on the cubic. As explained earlier, it is convenient to take x = 0 at the mini- mum on the cubic. Inirinsic-Exirinsic Junction Under Large Bias Consider the limiting case of an intrinsic-extrinsic junction as the bias is increased and the space charge penetrates many diffusion lengths into the intrinsic material. Then the field distribution approaches the straight line E/Ei = x/2Li . This, by Poisson's equation, means that there is a constant charge density of Ni where 2aLi Li Thus in the limit, the field in the intrinsic region approaches that in a completely swept extrinsic region having a fixed charge density of Ni . In germanium at room temperature Ni is about 4.10^" cm~^ As the field approaches the limiting form, the voltage V approaches EiL^/iLi . Thus the limiting form of the current voltage curve is aEi L, y 2EiL So in the limit the current varies as the square root of the voltage. Typical values for germanium at room temperature are a-Ei = 7 amps cm"""', £/Li = 10"^ and 2EiLi = 50 volts. Equivalent Generation Length for an Lntrinsic-Extrinsic Junction It should be noted that for an IP structure the current is the same as for an NIP structure with an infinite / i-egion, or at least an / region that is long compared to the distance of penetration of the space charge. THEORY OF THE SWEPT INTRINSIC STRUCTURE 1257 Thus the equivalent length of current generation is 2L even though the current is actually being generated in an effective length L. The reason is that for an NIP structure the holes entering the right hand half of the / region were generated in the left hand side. For an IP structure the holes entering the space charge regions from the left were injected at the external left hand contact to the / region. Applied Voltage In all cases the voltage can be found from the area under the E versus X curve. In Figs. 2 to 4 the area under the curves gives the voltage ac- curately; recombination becomes important only where the field is so low as to have a negligible effect on the total voltage drop. Correction of the Cubic To conclude this section we consider the error introduced by using the assumption / = asE. For simplicity take Ei as the unit field, 2Li as the unit length and aEi as the unit current. Then the cubic becomes E" — I IE = x" — A. The corresponding exact solution is E' — s = x^ — A where the relation between s and E is given by equation (2.6) which in dimensionless form is ^'%-^B-I (3.8) where £ is of the order of 10~ . Let bE and bs represent the difference between the cubic and the cor- rect solution at a giv^en x. Assume that bE and its second derivative are small compared to E and its second derivative respectively. Then bs = 2EbE and on the correct solution sE - I = (2lf -f I/E)bE. So (3.8) becomes bE ( £' \d'-E E \2E^ + // dx'- (3.9) To obtain a first approximation to bE/E we use the cubic to evaluate d E/dx . It is convenient to express the results in terms of a dimension- less variable z = E/I^'^, or if E and / are measured in conventional units z = E{a/Eilf'\ Then (3.9) becomes bE __ 1 (L,£\-" ( z \- , ( X Y ^'(1 - z) (3 ^Q^ E 2\U / \z^ + hJ \2LJ {z^ + i)^ iV'hen the lengths are in conventional units. 1258 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 The first term has a maximum value of 0.35 (L,£/L^)^'^ at 2 = 0.6 and the second term a maximum value of 0.18 at 2 = 0.5 and x = L. The dashed curve in Fig. 2 for ^ = ,01 is the corrected cubic. For the other curves in Fig. 2, the correction is smaller. For the curves in Figs. 3 and 4 the correction is too small to show. Limits on the Solution We now show that 8E as derived above is not only a first approxima- tion but also upper limit on the correction necessary to take account of charge diffusion. That is, an exact solution to (3.8) lies between the cubic and the corrected cubic. Consider the region where the second derivative of E is positive so that the perturbed curve lies above the cubic as in Fig. 2. On the cubic we have s£' — 7 = 0. As we move upward from the cubic and toward the dashed curve, sE — I increases. The value oi sE — I on the dashed curve just equals the value of £' d'E/dx' on the cubic. However, the dashed curve has a smaller second derivative than the cubic. Thus in moving upward from the cubic toward the dashed curve sE — I increases from zero and £' d E/dx , which is positive, decreases; on the dashed curve sE — I is actually greater. Therefore there is a curve lying just under the dashed curve where the two sides of (3.8) are equal. The same argu- ment applied to the region where the second derivative is negative shows that the equation is satisfied by a curve lying just above the first per- turbation of the cubic. Where the curvature changes sign, the cubic is correct. It should be emphasized again that the neglect of the diffusion term in the current is justified only for the ideal case of equal hole and electron mobilities. For unequal mobilities both drift and diffusion will be im- portant in current flow. However, as we will discuss in section 5, we can simplify the problem of unequal mobilities by defining a fictitious current that has the same form as / in (2.6) and (3.8). IV. RECOMBINATION As discussed in Section III, when the voltage for a given current is re- duced, s increases and near x = 0 becomes comparable to unity. Then recombination becomes important and the cubic solution breaks down, or rather joins onto a solution that takes account of recombination. When recombination is important the center Xi of the intrinsic region is no longer at the .r = 0 point on the cubic but to the left of it. That is, if we want the same current with continually decreasing voltage, we even- THEORY OF THE SWEPT INTRINSIC STRUCTURE 1259 tually get to the point where a longer intrinsic region is required. Finally for a given current we reach a minimum voltage which corresponds to an infinite length of intrinsic region. Another way of saying this is that, when recombination becomes important, the length L defined in terms of the current by / = qg2L = qrii/rL is no longer the half length of the intrinsic region. Equivalent Generation Length We shall continue to define L by / = qnilrh. Thus L is an equivalent, or effective, half length of current generation and not the half length of the intrinsic region. By definition L is the length such that the amount of generation alone in the length L is equal to the net amount of genera- tion (generation minus recombination) in the total half length of the intrinsic region. Hence gL = [ \g - r)dx (4.1) where Xi is at the center of the intrinsic region and Xp at the IP junction. We shall for the most part deal with reverse biases of at least a few kT/g, in which case recombination is negligible at the junctions. Then the exact solution becomes the no- recombination solution before reaching the junc- tions. We shall continue to take x = 0 at the point dE/dx = ds/dx = 0 on the no-recombination solution which the exact solution approaches as recombination becomes negligible. Simplifying Assumptions The general differential equation with recombination will be the same as for no-recombination except that g — r replaces g. From (3.1) and (3.2) From (2.12) and (2.13) and Poisson's equation r = !^ = A^ + pY _ (^ - P)^ = s' - 9 (— —\ a '\) g n? V 2n, / (2n,) ^ \E, dx ) "-^"^^ The following analysis will be based on the assumption of charge neu- trality. That is we neglect terms m p — n in comparison with those in p -\- n.\n particular this means: (1) The charge flows by drift so / = asE. This is the same assumption 1260 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 made in the no-recombination case. It will be an even better approxima- tion in the recombination region, where the second derivative of E is less. (2) The particle flow is by diffusion. That is, E'^/E-C can be neglected in comparison with s. (3) The ratio of recombination rate r to generation rate g is propor- tional to ^ — r; that is ^ — r = ^(1 — s~). All of these simplifying assumptions can be justified by substituting the resulting solution into the original expressions and showing that the neglected terms are small when recombination is important. If the solution is substituted into (4.3) and (2.6) the neglected terms will turn out to be negligible — and therefore assumptions (1) and (3), justified — when s^ is large compared to £/L, . Assumption (2) follows from (1) and the fact that IjaEx is small compared to unity. Assumptions (2) and (3) may also be justified by the discussion fol- lowing (3.7) in the following way: Where recombination is important s must be near unity. So the cubic will begin to break down when s = II e;-) (5.18) Setting V = V j and E = Ej in (5.18) gives the total voltage drop in the region where the junction solution holds. Let AF be the difference between Fy — Fo and the built in voltage drop at the junction. Then substituting (5.6) with Ec — Eq into (5.18) and subtracting the built in drop we have for AF, AT = ^ 9 L (n En Eo_ s p -i (5.19) 1270 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 5 o f< 0.5 0.2 O.I 0.05 '! 0.01 0.1 1.0 20 10 100 Fig. 7 — Variation of (L — Xo)/( with Zo . I/Eo is equal to the value of s on the cubic at x = L. For positive values of A the maximum value of E^/I is L/I = l/\/2£ as can be seen from the cubic. In germanium at room temperature <£ is about 10~ (for 2Li = unit length) so the reverse bias produces an additional voltage drop in the junction region equal to about IkT/q. For negative values of A the additional voltage drop near the junction would be higher. Comparing (5.3) and (5.13) we see that the junction solution reduces to the zero bias solution when £"" is large compared to Eo" + 2. In this case both solutions have the simple form (5.20) and Vi- V Q E (5.21) Case of Eo Small Compared to I Now from (5.7) and (5.8) with xo = L = I\/2£, we have £' 2Fj + {E- Eof (^ + ^' (5.22) THEORY OF THE SWEPT IXTRINSIC STRUCTURE 1271 Again there are t^^■o o\-erIapping ranges where the solution has a simple form : Range 1. Here E' is small compared to 21 /Ea . This will be so even when E becomes large compared to Eq . Setting Ci = 2Eo/I and y = E — Eo in equation (5.22) and integrating gives X ^0 — ^ /\/ ~~r~ E, r^"^" dy I X Vci^ + If (5.23) ^ ,/Eo . , -1 /E - Eo\ and V - Vo IT /9F (5.24) = ^ y Y (Vci^ + (^ - E,r- - ri) + 2Eo(x - .To) Range 2. Here E is large compared to Eq . It follows from Eq « I that E is also large compared to Ci . Setting ci = 21 /Eq we have •^'- dE L - X = V2£ / f Jr e E VWT~c? Joining (5.21) and (5.23) where they overlap we have in range (2) X — Xo = £ a/ ~ hi I ^3 'Mf. E C2 + \/c.^ + E' (5.26) Putting X = L and /i" = Ej in (5.26) gives the length f, — .r„ in which the junction solution holds. If Ej is lai'ge compared io c.> , then ^=y/|(«i (5.27) where as before Zo = Eo/I^'^ and Z is given by (5.16). Fig. 7 is a plot of (L — Xo)/l versus zo . The two approximations (5.15) and (5.27) for Zn « 1 and Zo ^ I respectively are shown dashed. Both become inaccu- rate as they are extended toward zo = I. The point at ^o = 1 was ob- tained graphically. Each approximation is in error by about 28 per cent here. The error will decrease as each approximation is (wtendod away from 00 = 1 toward its range of validity. The voltage in Range 2 is given by 1272 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 195G Vj — V = Sinn — ^ — sinh - (o.28> q L C2 C2J or again joining (5.28) to the solutions in Range 1 we have in Range 2 V - 7o = -- sinh"' - + 2Eo{x - Xo) (5.29)1 q C2 The total voltage drop in the junction can be found by setting V = Vy' and E = Ej in (5.29). The term 2Eo(L - xo) will be negligible. Wheni E'^ is large compared to Co" + 2 the junction solution reduces to the zero current solution as can be seen by comparing (5.3) and (5.25). Then the solution has the simple form (5.20) and (5.21). Ej will always be large compared to C2 . (Ef is appi'oximately Sp/e and Co" = 2so where 1 So is the value of s where the junction solution joins the cubic.) Thus the difference AV between Vj — T^j and the built in voltage is AF = --fn^ (5.30) q I Example. Fig. 8 shows the field distribution near the IP junction for the case L = 2Li and /I = f , for which the intrinsic region is in- finitely long. The field distribution near the junction, however, will he indistinguishable from that for A = 0.6G5, or ,% = 0.95, for which the; intrinsic region is about twice the effective length of current generation. We have taken Ej = 30, which corresponds to an excess acceptor den- sity P = 4.7 X 10' Ui in the P region. Over the range where the junc- tion solution holds the cubic gives an almost constant field E = En = Ec . The junction solution goes from the cubic to the zero bias solution in a , distance of the order of the Debye length. The sum of the built in volt- age and the voltage derived from the cubic differ from the correct voltage by the order of £Ei or about kT/q. The total voltage is about 0.3 EiLi , which would be about 11 volts in germanium at room temperature. VI. GENERAL CASE, UNEQUAL MOBILITIES This Section deals with the general case where the ratio of the hole and electron mobilities is arbitrary. The procedure is similar to that used in the preceding Sections. Many of the results for 6 = 1 are useful in the present, more general, case. We shall deal first with the no-recom- bination case and again find that E is given by a cubic. However, the field distribution is no longer symmetrical and the coefficient of the I/E term in the cubic is a linear function of x instead of a constant. The differential equation foi' .s in the recombination region remains un- THEORY OF THE SWEPT INTRINSIC STRUCTURE 30 20 10 8 6 5 1273 _E_ E, 1.0 0.8 0.6 0.5 0.4 0.3 0.2 0.1 E = E J - - ll h / li 1 1 1 - X = Xo // / ^ _^^^ ^ 1 ; / -o— SS=^. . ——— -T" f -( E-Ec-ho CUBIC 1 1 1 1 t /zero / BIAS / / 1 1 / / / / X-L Fig. 8 ■ — Field Distribution near the IP Junction for L = 2L; and A = f. changed. It is no longer so that charge diffusion can be neglected except near the junctions. However, there is a linear combination of Jp and J„ in which the diffusion term is negligible except near the junctions. Basic Relations The equations are the two continuity (2.9) and Poisson's (2.1). The formulas for g — r remain unchanged, since they involve only the statistics of recombination and are independent of mobility. The hole and electron currents are given by (2.2) with b arbitrary. Eciuation (2.2) for Jp in terms of E, p and n remains unchanged. Now ./„/6 has the same 1274 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 form as J„ had for the b = 1 case. It is therefore desirable to deal with the fictitious carrier flow J p + J„/b and the ficfitious current q{Jp — Jn/b) since these have the same form in terms of E and s = (w + p)/2ni as J and / had for b = 1. Thus bj 1 + 6" L"" ™ dx ' dx^] Es - £'~\ (6.2) where Ei and £ have the same meaning as before and the conductivit}' of intrinsic material is noAv a = qniij.(l + b). As before D and fj. are re- spectively the diffusion constant and mobility for holes. Equations (6.1) and (6.2) reduce respecti\Tly to (2.7) for ./ and (2.6) for I = cj{Jp — /„) where 6=1. When the flow is by pure diffusion, the holes and electrons diffuse "in parallel" so the effective diffusion constant is the reciprocal of the average of the reciprocal hole and electron diffusion constants. Hence the effective diffusion length is given by Lf = Dr ^^ (6.3) We continue to let 2L = I/qg be the effective length of current genera- tion; again it is the actual length for the no recombination case. Let x,, and Xp be the coordinates of the AU and IP junctions respectively. Since the problem is not symmetrical we will not take a' = 0 in the center of the intrinsic I'ogion even for the no-recombination case. No-Recoinbiualiun Case Setting r = 0 we can immediately integrate the continuity equa+ ' dJp _ dJn _ dx dx subject to the boundary conditions: at the iV/ junction, x = rc„ , Jp = 0, Jn = ~Uq at the IP junction, x = Xp , Jp = I/(j, Jn = 0 The result is Jp = g(x — .r„) and J„ = g{x — Xp). This agrees with / = q(Jp — J„) = "^qgL since 2L = Xp — x„ is the length of the intrinsic region, which, for no-recombjnation, is also the effective length of cur- I THEORY OF THE SWEPT INTRINSIC STRUCTURE 1275 i| fent generation. It will be convenient to choose a; = 0 so that .t„ = —Xp/h. Then the origin is nearer to the NI junction for 6 > 1. Now from this and the boundary conditions (6.4) and I = 2qgL we have the positions of the junctions: L 1 + 6' L 1 + 6 ^^'^^ A.S before, the junctions are at .r = ± L for 6=1. We can now find the fictitious carrier flow Jp + J„/6 and the fictitious current q{Jp — Jn/i>) as functions of x. ■fp+T= (M^) !'^ (6.6) where the dimensionless parameter j8 = (6^ — l)/46. Thus the fictitious current q(Jp — J„/b) is equal to the actual current times a linear func- tion of X. This function is always positive and varies from a minimum of 1/6 to a maximum of 1. Combining (6.6) with (6.1) and integrating gives the equation that we had before. Now, however, E is not a minimum at the same point where s is a maximum. As before, when recombination is negligible throughout all of the intrinsic region, A determines the voltage; and, when recombination is important over part of the region, A determines both the voltage and the length of the intrinsic region Xp — Xn > 2L = \/o ' ibining (6.7) with (6.2) gives (6.9) which is similar to the previous (3.6) except that / is nuiltiplied by the factor 1 + j3.r/L, which ^'aries from 1 + 1/6 to 1 + 6. The same argu- ments used in Section V apply here and show that the second term in brackets (the diffusion term) can be neglected except near the junctions. In other words, although / is always part drift and pai't diflusion, 7(1 + ^x/L) is approximately pure drift except at the junctions. Eliminating s between (6.9) and (6.8) and neglecting the diffusion 127G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 l! term in (6.9) gives the cubic equation ' for the field distribution. In germanium, where b = 2.1, /? = 0.406, Xp = 1.35L and x„ = — 0.65L. The coefficient of I/aEi therefore varies from 1.47 to 3.10, or by a factor of a Httle more than 2. This will introduce some asymmetry into the E versus x curve in the low field region where the fictitious car- rier flow Jp + Jn/b is by diffusion. It is evident that, as the voltage in-i creases, the field versus x curve becomes increasingly symmetrical about the x = 0 point; so the effect of having b 9^ 1 is simply to shift the field distribution along the x axis. Recombination -s The arguments of section 4 again apply. Where recombination is im- portant, n — p is small compared to n -\- p, so g — r = g(l — s^). The diffusion term dominates in the fictitious particle flow Jp + Jn/b; that is, E^/Ei is small compared to s, so (6.1) becomes •^= -2n,D^ 0 ax Jp +-^= -^mD"^ The continuity equations give So again we have (fs^^ (1 - /) dx^ 2Li2 (6.11) The solution joins the no recombination solution where s = A — {x/2Li)". Therefore A is again related to Sq , the maximum s, by ^ = :j So(l — si/Z) and the s versus x curve is given by (4.8) and is symmetrica] about the point where s is a maximum. When the recombination solu- tion joins onto no-recombination solutions, there will be a difi'orent no-recombination solution on each side of the recombination region. The junctions will be at the points Xp and .r„ on the respective no-recom- bination solutions. The length of the intrinsic region will not be Xp — Xn = 2L since the x = 0 points are different on the two no-recombination solutions and are separated by a region of maximimi recombination. THEORY OF THE SWEPT INTRINSIC STRUCTURE 1277 To find E when s is known we express the current I = qiJp — Jn) in terms of s and E. Since w — p is small compared to n + p, we set /; = p = sUi in (2.2) and obtain I = [ j^ 1 - h kT dsl ,„ ,„v Thus the current contains both a drift and a diffusion term. This is to be expected for unequal mobilities. When holes and electrons have the same concentration gradient, the electrons, which have the higher dif- fusion constant, diffuse faster than the holes; hence the diffusion gives a net current. It is seen that in the recombination region the total carrier concentration has a symmetrical distribution about the point where it is a maximum but the field remains unsymmetrical. Junction Solution When (Eo/Eif is large compared to I/cEi the junction solution is independent of 6; so the solution obtained in Section V is valid. In all cases the junction solution can be found using the method of Section V. The effect of h will be small over most of the range where the junction solution holds because the concentration of one type of carrier will be negligible. To be exact, / in (5.8) should be multiplied by the factor (1 + ^Xo/L), which can be taken to be (1 + b)/2b at the NI junction and (1 -f b)/2 at the IP junction. Instead of equation (5.7) we have as can be seen by differentiating (6.10) with Ei = 2Li = o- =1. VII. EFFECT OF FIXED CHARGE This section will deal briefly with the case where there is some fixed charge but where the carrier charge cannot be neglected. For no recom- bination, the field distribution is given by a first order differential equa- tion. Solutions in closed form are obtained for the case of pure drift flow. For recombination and charge neutrality the solution in Section IV is valid provided the fixed charge is small compared to Ui . We have seen that at large fields the E versus x curve becomes linear, correspond- ing to a fixed charge density of A''; where Ni = \/2n{£/L,- . Thus the fixed charge may have a dominant effect on the space charge while having a negligible effect on the solution in the range where recombi- nation is important. 1278 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Let the density of fixed charge he N = Nd — Na = excess density of donors over acceptors. N may be either positive or negative. In what follows we shall assume that N is positive. So the structure is NvP where v means weakly doped n-type. Equations (2.2) for the hole and electron currents remain unchanged. Poisson's equation becomes ^= aip-n-\- N) (7.1) ax We shall deal with the general case of arbitrary mobilities. As in Section VI it is convenient to deal with a fictitious current q(Jp — Jn/b) and a fictitious particle flow Jp + Jn/b. The extra term in (7.1) drops out by differentiation when (7.1) is substituted into the equation for Jp —J„/'b so (6.2) remains unchanged. However, instead of (6.1) we have So the fictitious particle flow is no longer the gradient of a potential involving only E and s. No Recovibination As in Section VI the continuity equations can be immediately inte- grated to give (6.6) and (6.7). Again / is given by (6.9) where the dif- fusion term on the right can be neglected except at the junctions; so again we have asE = 1(1 -\- ^x/L). Substituting this into (7.2) and com- bining (7.2) and (6.6) gives a first order differential equation for E versus X. It is convenient to again use dimensionless quantities with Ei , 2L, and aEi as the units of field, length and current respectively. Then the differential eriuation becomes !l dx where = 2(.'c + aE) (7.3) I N and as before Ni = \/2ni£/Li , which is around 4 X 10'" in germanium at room temperature. The solution of (7.3) contains one arbitrary con- stant (which corresponds to A in the V = 0 case). The lower limit on the constant is determined by the necessity of joining onto a recombina- tion solution \\hcre s approached unity. The positions of .r„ and Xp of the Nv and vP junctions respectively are given by (6.5). THEORY OF THE SWEPT INTRINSIC STRUCTURE 1279 In the region of low fields where E^ is comparable to or less than I, (7.3) would have to be solved graphically or on a machine. At higher fields the equation is easily integrated as discussed below. Case of Pure Drift When the flow is entirely by drift, E^ » / and (7.3) becomes 5^ = £ + " (^-^^ which is made integrable by the substitution E = yx. A family of solu- tions for positive E throughout the v region is {E - a,xT{E + a.xT- = Eo"'^"' (7.5) where 2ai = \/4 + «' + « and 2a2 = -vZ-i -\- a- — a and Eo is the value of E at x = 0. For an intrinsic region N = a = 0 and (7.5) reduces to E^ = Eq' + x^, which is the same as (3.9) for negative .4. Fig. 9 shows several curves for \'arious values of Eo . These remain above, and at 5 large distances approach, the asymptotic solutions E = aix on the I right of the origin and E = —a2X on the left. These curves differ from the corresponding curves for an intrinsic region in that the straight line I asymptotes now have slopes of ai and — oo instead of ±1. Toward the P I side the slope is greater because the positive change qN of the excess do- I nors is added to the charge of holes. Toward the A^ side of the v re- ' gion the slope is reduced because A^ compensates to some extent for the I electron charge. As a increases and the v region becomes more n type, the solution approaches that for a simple NP junction, where E = ax on the A^ side. Another set of solutions of (7.4) are given by (ai.r - EY^aox + E)"' = ai^'aa^V (7.6) Several of these are shown in Fig. 9. They remain below the linear asymptotes and go through zero field at x = ±:Xc . Actually these will join onto solutions of the more general equation (7.3) when E becomes small and the diffusion term becomes important. Rccomhinaiion. When the fixed charge density is small compared to the intrinsic hole and electron density the treatment of recombination in Section IV remains \'alid. The recombination solution joins onto a solu- tion of (7.3) at small fields. When N is comparable to ??» the recombina- tion solution is difficult even with the assumption of charge neutrality. 1280 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 E. E, J a' Xf J y' / X ' y ^J^ti \ -3.0 -2.5 -2.0 -1.5 -1.0 0.5 0 0.5 1.0 1.5 2.0 2.5 3.0 X/2LL Fig. 9 — Field Distribution in the Range of Pure Drift for a fixed charge N = N{ ,ora = 1 . ACKNOWLEDGEMENTS The author wishes to thank Miss M. M. Segrich for doing the exten- sive computations and plotting the curves, and Miss M. C. Gray for help with the calculations leading to Fig. 7. APPENDIX A Prim's Zero-Current Approximation Prim's analysis is based on the assumption that the hole and electron currents are negligibly small differences between their drift and diffusion terms. Setting Jp = /„ = 0 then gives n and p as functions of the po- tential, which is found by substituting n and p into Poisson's equation and solving subject to the boundary conditions at the junctions. These conditions involve the applied bias and the majority carrier densities in the extrinsic regions. Since the current is assumed to vanish, the phe- nomena of carrier generation and recombination do not enter the problem and the results are independent of carrier mobility. The results will be exact when there is no applied voltage; the potential drop across the unit is then the built-in potential. In this appendix we use an internal consistency check to see for what values of applied bias the analysis THEORY OF THE SWEPT INTRINSIC STRUCTURE 1281 breaks down. First we find where the carrier concentration is in error by finding the bias at which the minimum drift current as calculated from qn(n + p)E becomes equal to the total current, as found from the excess of generation over recombination in the intrinsic region. We then go on to find where the error in carrier concentration gives a sufficient error in space charge to affect the calculation of electric field. As we shall see, the zero-current approximation gives too low a carrier concentration in the interior of the intrinsic region. This will lead to serious errors in the field distribution only if the space charge of the carriers is important. When the bias is sufficiently high or the intrinsic region sufficiently narrow that the intrinsic region is swept so clean that the carrier space charge is, in fact, negligible, it will not matter that the calculated carrier density is too low, even by orders of magnitude. In such cases, the electric field is constant throughout most of the intrinsic region. In the following we shall, for simplicity, take 6=1 and assume that the extrinsic regions are ecjually doped so that the problem is symmetri- cal. Carrier Density We now find where, on the zero current assumption, the drift current becomes equal to the total current. This involves knowing only the carrier concentrations and the field Ei in the center of the intrinsic region, where the drift current qyi,{n -f- p)Ei is a minimum. By symmetry n and y are equal here and n = p = 7ii exp ( — qVa/2kT) where Va is the applied bias. The minimum field Ei is given by the total voltage drop V and the field penetration parameter rj, which is the ratio of the minimum field to the average field. Thus r) = 2LEi/V where 2L is the width of the intrinsic regions. The difference between V and Va is the built-in voltage {2kT/q)/ln{N /rii) where N is the majority carrier concentration in the extrinsic regions. We now have for the drift current in the center of the intrinsic region ,.(,. + p)£ = , ,0(1;)^ exp (-1^) (Al) We next find the total current from the excess of generation over re- combination in the intrinsic region. From the zero current assumption, np = Ui exp ( — qVa/kT) is constant throughout the intrinsic region. Hence g — r is constant. So the current / = q(g — r)2L = qL{ni — np)/ TUi is qLrii 1 — exp (A2) 1282 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Equating this to the drift current (Al) in the center of the intrinsic region gives The error in carrier concentration is less for narrower intrinsic regions and lower biases. Thus (A3) gives a curve of L versus Va such that the zero current solution gives a good approximation to carrier concentration for points in the VaL plane lying well below the curve. As expected, for zero bias, the solution is good for any value of L. However, for a bias of \ several kT/q, the solution for carrier concentration breaks down unless ^ L is a very small fraction of a diffusion length. Carrier Space Charge. In Prim's analysis the carrier space charge is so low throughout most of the intrinsic region that the field remains approximately constant and equal to Ei . However there must be enough carriers present that the drift currents of holes and electrons can remove the carriers as fast as they are generated. In this section we ask where the space charge of the necessary carriers becomes large enough that its effect on the field can no longer be neglected. Let i^E be the change in field due to the space charge in the intrinsic region (not counting the high field regions near the junctions). Unless LE is small compared to Ei the neglect of carrier space charge will not be justified. We shall find the ratio of AE to Ei . If the field is to be approximately constant, then the hole and electron concentrations can easily be found from the hole and electrons currents. We shall deal with applied biases of at least a few kT/q, for which recombination is negligible and the total current I = qg2L = qUiLlr. Since g — r = g\s> constant, the hole and electron currents are linear in X and, for constant field, are proportional to the hole and electron con- centrations respectively. Thus the net space charge of the moving carriers q{'p — n) is proportional to x and varies from zero in the center i of the intrinsic region to qp near the IP junction, where n is small compared to p and the current flows by hole drift, so / = q^ipEi . Thus the maximum charge is I/iiEi and the total positive charge of the car- riers on the P side of the center is IL/2iJ.Ei . This gives a drop in field „ _ alL _ arii kT L " 2qiJ.Ei ~ 'YqEiL} THEORY OF THE SWEPT II "JTRINSI Dividing by Ei = 7]V/2L gives AE L' (kTV Ei £'L,' \mvj 1283 (A4) Setting AE equal to some fraction, say 20 per cent of E^ gives a family of curves for V versus L with ?? as a parameter. Prim has plotted such curves in Fig. 11 of his paper. His curves will be good approximations when V for a given L and 77 lies above the V given by (A4). Prim's results are expressed in terms of the parameters U = qV/2kT and L = 2L/£e where £e is the Debye length in the extrinsic material. £e is given by the same formula as £ except that A'' replaces n» . Substituting these into (A4) and setting AE = Ei/5 gives L = 3.57 — '■ r,U (A5) ni£ Prim's U versus L curves will be accurate up to the point where they intersect the corresponding curves from (A5). For germanium a reason- able value of NLi/ni£ is about 10 . This says that Prim's curves go bad at about L = 10 , which would be about 2.1 X 10~^ cm in germanium at 300°C. Branching of the V versus L Curves An effect which does not emerge from the zero-current analysis is that V may have several values for the same L and 7/. In other words the V versus L curve for given ?? will have more than one branch. Specifi- cally, there will be a single V versus L curve up to a certain L at which the curve splits into three branches that diverge as L increases. This may be seen as follows: Consider an intrinsic region that is long compared to the diffusion length. Suppose a bias is applied that is low enough not to appreciably affect the space charge and potential drop at the junc- tions. A current will flow and a proportional, ohmic voltage drop will be developed across the intrinsic region. If the intrinsic region is long enough, this ohmic voltage may become large compared to the built-in voltage before the voltage drop at the junctions has changed appre- ciably. In this range the field penetration parameter will be rising from zero to about unity as V increases from the built-in voltage and ap- proaches the ohmic voltage. As the voltage continues to increase, the space charge begins to penetrate the intrinsic region and a majority of the voltage drop comes in the space charge regions. Let L be the ef- fective length of current generation. When L is larger than a diffusion 1284 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 length but small compared to the length of the intrinsic region, then the voltage drop at the ends of the intrinsic region will be proportional to L while the current, and consequently the minimum field, will be propor- tional to L. Thus r? will be proportional to 1/L and will decrease as V increases and the region becomes more swept. Finally the two space charge regions meet; then ?? rises again with V and approaches unity. Hence, for a given -q and length of intrinsic region, there will be three different values of V. For lower L the dip in the i] versus V curve will be less, and there will be only one V for some values of 77. Since -q starts from zero at the built-in voltage and approaches unity for infinite volt- age, there must be either one or three values of V for every r?. Thus when the V versus L curve (or in Prim's notation the U versus L curve) branches, it branches at once into three curves. Prim's plot gives the upper branch in cases where all three are present. A Medium Power Traveling-Wave Tube for 6,000-Mc Radio Relay By J. P. LAICO, H. L. McDOWELL and C. R. MOSTER (Manuscript received May 15, 1956) This paper discusses a traveling-wave amplifier which gives 30 dh of gain at 5 watts output in the 5,925- to 6,425-nic common carrier hand. A descrip- tion of the tube and detailed performance data are given. TABLE OF CONTENTS Page I. Introduction 1285 II. Design Considerations 1288 III. Description of the Tube 1291 3.1 General Description 1291 3.2 Tlie Electron Gun and Electron Beam Focusing 1295 3.3 The Helix 1302 3.4 The Collector 1311 IV. Performance Characteristics 1314 4.1 Method of Approach 1314 4.2 Operation Under Nominal Conditions 1315 4.3 Operation Over an Extended Range 1325 4.4 Noise Performance 1333 4.5 Intermodulation 1336 V. Life Tests 1342 VI. Acknowledgements 1343 I. INTRODUCTION During the past ten years traveling-wave tubes have received con- siderable attention in vacuum tube laboratories, both in this country and abroad. So far their use in operating systems has been somewhat limited, the most notable exceptions being in radio relay service in France, Great Britain, and Japan. However, it appears that sufficient progress in both tube and system design has been made so that traveling-wave tubes may see widespread application in the near future. This paper describes an experimental helix type traveling-wave tube representative of a class which may see extensive use as a power amplifier in radio relay systems. The tube is designated as the Bell Laboratories type MI789. Stated briefly, the performance characteristics under nominal operating conditions arc: 1285 1286 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Frequency Range 5,925-6,425 mc Power Output 5 watts Gain at 5 watts output 31-35 db Noise Figure < 30 db The tube is designed for use with w-aveguide input and output circuits. The input voltage standing wave ratio (VSWR) is less than 1.1 and the output VSWR is less than 1.4 over the 500-mc band when the tube is delivering 5 watts of output. Fig. 1 shows a photograph of an MI789 and of an experimental permanent-magnet focusing circuit. In developing this tube we have endeavored to produce an amplifier which could be considered "practical" for use in a transcontinental radio relay system. Because such an application requires a high degree of reliability and refinement in performance, the tube was rather con- servatively designed. This made it possible to obtain the desired gain and power output without difficulty. On the other hand, the contem- Fig. 1 — The M1789 traveling-wave tube and an experimental permanent mag- netic circuit used to focus it. The circuit contains two specially shaped bar mag- nets l)et\veen which the tube is mounted. The magnetic flux density obtained is 600 gauss, and the overall circuit weight is about 25 pounds. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1287 plated system application made it necessary to investigate in detail the problems associated with band flatness, matching, noise output, certain signal distortions, reproducibility, and long life. The solution of some of these problems required the development of a precisely constructed helix assembly in which the helix winding is bonded to ceramic support rods by glaze. Others required the initiation of a life test program. Early results indicate that life exceeding 10,000 hours can be obtained. This, in no small measure, is a result of a dc potential profile which minimizes the ion bombardment of the cathode. Since power consumed by focusing solenoids seriously degrades the o\'erall efficiency of a traveling-wave amplifier, permanent magnet focusing cir- cuits such as the one shown in Fig. 1 have been designed. Finally, to further improve efficiency, a collector which can be operated at abcut half the helix voltage was developed. The major difficulties encountered in the course of the MI789 develop- ment were: excessive noise output, ripples in the gain-frequency char- acteristic, and lack of reproducibility of gain. There is evidence that a growing noise current wave on the electron stream was the source of the high noise output. This phenomenon has been observed by a number of experimenters but is not yet fully explained. By allowing a small amount of the magnetic focusing flux to link the cathode, the growing noise wave was eliminated, and the noise reduced to a reasonable level for a power amplifier. Reflections caused by slight non-uniformities in the helix pitch were the source of the gain ripples. Precise helix winding techniques re- duced these reflections so that the ripples are now less than ±0.1 db. The lack of reproducibility in gain was caused by variations in helix attenuation. Here, too, careful construction techniques alleviated the problem so that in a recent group of tubes the range of gain variation at five watts output was ±2 db. We have divided this paper into four main parts. The next section discusses some of the factors affecting the design of the traveling-w-ave tube. (We will henceforth use the abbreviation TWT.) Section III describes the tube itself. Certain performance data are included there when closely related to a particular portion of the tube. Section IV considers the rf performance in detail. There comparisons are made lietween the performance predicted from TWT theory and that actually observed. Finally Section V summarizes our life test experience. This paper is written primarily for workers in the vacuum tube field and assumes knowledge of TWT theory. However, we believe that readers interested in TWT's from an application standpoint may also benefit from the discussion of the rf performance in Section IV. Much of that section can be understood w'ithout detailed knowledge of TWT's. 1288 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 II. DESIGN CONSIDERATIONS While TWT theory served as a general guide in the development of the MI789, a number of important tube parameters had to be determined either by experimentation or by judgement based on past experience. The most important of these were : Saturation power output 12 watts Mean helix diameter 90 mils 7a --^ 1.6 Magnetic flux density 600 gauss Cathode current density '^ 200 ma/cm^ These quantities and the requirement of 30-db gain at five watts output largely determined the TWT design. The saturation output of 12 watts was found necessary to obtain the desired linearity at five watts output and the 7a value of 1.6 to obtain the flattest frequency response over the desired band. The choice of helix diameter and magnetic flux density represented a compromise. For the highest gain per unit length, best efficiency, and lowest operating voltage, a small helix diameter was called for. On the other hand, a large helix diameter was desirable in order to ease the problem of beam focusing and to facilitate the design of a light-weight permanent magnet focusing circuit. In particular, the design of such a circuit can be greatly simplified if the field strength required is less than the coercive force of available magnetic materials. This allows the use of straight bar magnets instead of much heavier horseshoe magnets. More- over, the size and weight of the magnetic circuit is minimized by employ- ing a high energy product material. These considerations led us to choose a flux density of 600 gauss, thereby permitting us to design a magnetic circuit using Alnico bar magnets. To obtain long tube life we felt it desirable to limit the helix intercep- tion to about one per cent of the beam current. On the basis of past results we estimated that this could be done with a magnetic flux density 2.6 times the Brillouin value for a beam entirely filling the helix. With this restriction, Fig. 2 shows how the TWT design is affected by varying the helix diameter. A choice of 600 gauss is seen to result in a mean helix diameter of 90 mils. In the selection of cathode current density, a compromise between long life and ease of focusing had to be made. To obtain long life, the current density should be minimized. However, this calls for a highly convergent gun which in turn complicates the focusing problem. We decided to use a sprayed oxide cathode operating at about 200 ma/cm^. Experience with the Western Electric 41 6B microwave triode had shown bOUO 5000 4000 3000 2000 1000 0 / t / / / y / y 30 25 20 15 10 5 0 \ \i N \. v,^^ ^ 120 100 80 60 40 20 0 \ > \ \ s. ' ^ ^ » in 120 UJ a. LU CL < 100 80 60 40 20 12 10 5 z z LU tr cr D O 5 < UJ CD (/) LU I u z z X _l LU I LL O I I- 15 Z UJ _l 1200 1/1 01 < 1000 40 60 60 100 120 140 160 >- (3 ul z UJ Q X D _l U. u I- m z o < 5 800 600 400 200 40 60 80 100 120 140 160 MEAN HELIX DIAMETER IN MILS Fig. 2 — Alternate designs for the M1789. These curves are an estimate of how the TWT design would be affected by changing the helix diameter. They represent essentially a scaling of the M1789 design. In all cases the expected maximum power output is 12 watts and the low-level gain is 33 db. The line at 90 mils mean diameter in the curves represents the present M1789 design. In these calculations it was assumed that : a. 7a = 1.6 b. power output = 2.1 CIoVo = 12 watts c. the magnetic flux density is 2.6 times the Brillouin flux density for a beam entirely filling the helix. d. the ratio of wire diameter to pitch is 0.34. e. the dielectric loading factor is 0.79. f . the ratio of effective beam diameter to mean helix diameter is 0.5. 1289 1290 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table I — Summary of M1789 Design I. Helix Dimensions Mean Diameter Inside Diameter Wire Diameter Turns per Inch Pitch Wire Diameter/Pitch Active Length II. Voltages and Currents Electrode III. IV, V. 90 mils 80 mils 10 mils 34 29.4 mils 0.34 5^ inches Voltage (volts) Cathode 0 Beam Forming Electrode 0 Accelerator 2600 Helix 2400 Collector 1200 Heater Power TWT Parameters at Midband (6175 mc) 6 watts Current (ma) 40 0 < 0.1 <0.4 >39.5 ka C QC 1.58 0.148 0.058 0.29 30 As defined by Tien' N (number of X's on helix) Dielectric Loading factor 0.79 Impedance Reduction factor 0.4 Electron Gun Gun type — Converging Pierce Gun Cathode type — Spraj'ed oxide Cathode Current Density 213 ma/cm^ (for/x = 40 ma) Cathode diameter — 192 mils Convergence half angle 12° 40'_ Cathode radius of curvature (r^) 438 mils Anode radius of curvature (/•„) 190 mils It^c/n, 2.3 Pervernce 0.3 X 10~^ amps/volts^'z VVa/Tk = 1.61 for Tk = 720°C At the beam minimum in absence of magnetic field: rmin (from Pierce^") from Danielson, Rosenfeld & Saloom^ r9i Tt/a- a Brillouin flux density for 80 mil helix ID Actual focusing flux density required Beam transmission from cathode to collector at 5 watts output 11.5 mils 0.220 20.5 mils 3.50 4.80 mils 240 gauss 600 gauss 99% RF Performance Frequency range 5925-6425 mc Saturation power output 12 watts Nominal power output 5 watts Gain at 5 watts 31-35 db Noise figure <30 db Input VSWR <1.1 "I impedance match to WR 159 / waveguide Output VSWR (at 5 watts) <1.4 For an explanation of symbols see page 1345. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1291 that tube life in excess of 10,000 hours was possible with such a cathode. Moreover, an electron gun of the required convergence (about 13° half angle) could be designed using standard techniques. The various dimensions, parameters, voltages and currents involved in the design of the MI789 are summarized in Table I. For the sake of completeness, some rf performance data are also included. III. DESCRIPTION OF THE TUBE 3.1 General Description This section describes the mechanical structure of the MI789 and presents some performance data closely associated with particular portions of the tube. The overall rf performance is reserved for considera- tion in the next section. In the MI789 we have tried to achieve a design which could be easily modified for experimental purposes and which would also be adapted to quantity production. To assist in obtaining low gas pressure, a rather "open" structure is used, thereby minimizing the pumping impedance. In addition, all parts are designed to withstand comparatively high temperatures during outgassing, both when the tube is pumped and, in the case of the helix and gun assemblies, during a vacuum firing treatment prior to final assembly. Fig. 3 shows an MI789 and its subassemblies. Fig. 4 shows a simplified drawing of the whole tube and Fig. 5 shows how the tube is mounted with respect to the perma- nent magnet circuit and to the waveguides. The permanent magnets are shown schematically in Fig. 5. In actual practice they are shaped so as to produce a uniform field between the pole pieces. The means of doing this was discussed by M. S. Glass at the Second Annual Meeting of the I.R.E. Professional Group on Electron Devices, Washington, D. C., October 26, 1956. Control of Positive Ions Our experience with previous TWT's has indicated that an improve- ment in life by as much as a factor of ten is obtained by arranging the dc potential profile so that positive ion bombardment of the cathode is minimized. This improvement has been observed even in tubes in which all reasonable steps have been taken toward minimizing the residual gas pressure. From Table I it is seen that the relative values of ac- celerator, helix, and collector voltage are arranged to drain positive ions formed in the helix region toward the collector. These ions are thereby kept from reaching the cathode. Spurious ion modulation which can result from accumulation of ions in the helix is also prevented.^ a: o \~ u bj -J _j o u n ^ U "J 1 CD ;i 5 Id 10 to < l ac O {- o bj -J _J X o 3 ! yj X X 3 lU X i i Z ^ D O >- o: < 1- z D o UJ CL O _J UJ > Z LJ CO D H h- ^ D H UJ CO UJ fv. J a. s 5 5 o u XI o >> t, 03 a -^ -13 O 0) G _^ ■+^'>-. bC n u G CU OJ CJ iH >> go g^ G . O)^ -*J '-' in ^ CO ." O ^ « S'^ G G g^^ G 0) X 0) QJ ' — I ^ '•^ ^ Oj ~ IS X o3 to +^ M rt_0 G _G _Q QJ 02 -5 m G o G OJ N o; " o £ ^ "' -G ~ CO ,^~ G «« T^ C C3 O s g ^ ^ Qj _-: — -e o =: 5 03 ^ CO "" ~ *J -T" J- a; ra CO o "^ ^ _, ^ S G ^' -1^ 00.2 G --G ^ C CO HIg i-"^ I ^ ^^ M G 2 -^ • - C _c O] 1^ a; ^ ^ m -J CD ■^" V en o F^ T* _D '^ ^ *-< X 1^ o w o ":3 oj CO 111 ■r; cu c-"i^ 5 OJ c o ,£2 c3 to cl C W2 bC CO a>

> in a^^.ii a O „ :^^ G it *^ CO tH 03 ^ S 0) S =^ S ^ B a^-3 -*j tH 'is a; Oi-r; c3 o5 ri < t^i t4-i QJ ^ --=- O ^ ,ii! ^ -(J aj _g -2 = 2^^ ^ ^C3 CO - - « $ ^ 02 D O > a '" ^ ^ ^ 1^ " 3 -Co CO a; -^^ e 'So (-1 03 O o3 a; •-! ^^ rH S-i C !-, « fl rt .^"^ a:" 02 C o c3 c3 o3 Hi G Ml 03 ^ C 13 ^ ^ c G 02 ;::; 02 G a S G 9i'a3 03 3 o J 03 ^ 03 -^J Ij-- P MJ -H* 03 02 03 ^ s 03 02 r^ 03 !h 03 0) -G 03 G o3 03 ^H d ;h ^ ^ _, 02 O •+J 03 C3 •^ CO rt fe 0) " ■^-G 02 b» CO 03 M^ V; o3 •i-H /-^ *"• _ X ?2 MS p' 03 o3 CO C 03 5 ■" M 03 c3 03 .-, -G M-> 03 P "^ 13 t« O O-G CZ3 ^£ >j 02 T3 03 yG G (2 M a; o3 o 03 G rS-G =" =^ G O 03 'X 02 G +-■ G I U M ^ ^"os' CO h'r G 1^ " O iJ 03 o ;3 G 03 fl 0;£3 03 <^ 03 o o B o. ■S 02^ 5:g^ -■^ G 03 1/2 tZ3 o3 ^ M ;-< +J 03 G^ 03 -^^ S=" §° O 03 02 '^ 03=5-- a, 1- +2 O GyG — . 03 4^ «^ 03 > s § 03 "o M " « G c3 ->^ TS 03 „. c^ M-r- o3 (-» s-( _, -7^ G 02 — 03 *-^ rn G ^ .ij >> 03 C3 C3 03 -T3 -D 03 03 G ti • r-l 03 _Q (.'J 03 o 03-73 03 CO c3 -u o 03 03 03 -G -a G X C3 r1 1— < D4 c3 C0«<-1 & s 1293 1294 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 rr \— /^l^ — ^ T-'^f^— rrr^ ■■■,-■ ■>..Ul- ■ LU O ;: LU : \ <^< 5=4 z°- : ■ < ^ ■ _l . s f=? /■ ■■ 1 UJ,:,: ^:.>, O ,0. rr- ^#^ ^ -vZ CD ■;;;.;■■■;; ;:;:;-: .:.....:....,..:;.:.^^t III ]■■ '^'~ f^^:- Z^ i 1 UJ -o -• T ^ ^ m — ~J c '■ ' '■ 'i "-l , lllli rnTi lll.tr &— T h G u \ . 'vvvvvvVvvV/VV - t\ \ , ////////// //////// ^ ^ /////////////. , /' 1 \ 's '/////// J ////////////// / . ; \ //////// '//yx// /////// N '' \ ^ //////// nu.f \ ;; ^m ■ V '!a ^ •. ^ \z. ^ 1 E yA\ cc UJ \ UJ ^//yyyy// CL Q_l ////yyyy < ZQ. //////// 1- O /yyy//vv z <8 ^w I z>- //////// u LUb V//V/V/ yvY.'^ 'vvy//// (3 HO y/jr///// /Z^/^/yyyyyyy < z< //////// ovy'^oY/yy// 2 -::^r: :v;-P5;':^::: 1 T , -LU. '^'1 ' 1 f;^ MAGNE SUPPC ri 1 i ^ \ \ 1 \ 11 1 s ^ ■:K 1 LU :::■::■:: ■: 1 Ti " 1 ^ ' ^ ^Vlv-zm ■■:::;:■.■::- ^ I?."ill \^,_^^^JJ^)^ ^^^-*— .^^:j>i*-^ ^\^ j^y • V ii. ^j^*'''*'^***^*-^ — __^~j Si O O I CO bjO*" O^ IS bC tH o^ ?f 5 oi ci^ js cc X 0^ ^ fl -^ gj fl d 03 bil OJ "q H=o 2 >H »^ a. o "3 c ^ — ■*-';n r * «, b£^ OJ bX) ii C fl c3 K •- CO 03. a; q; 2 '^ ^ aj--2 TO o-.S S ^ o H o t>. s- C) CO is !- H o S a> S u tp M CO C G a; fii Ifi 2 'S- b£ S ° r <« to 5 o o Oi CO .-. r^ CD O -TI - Qj bJ} O bC g ■ti G o3 cc G t, " CO -^ 5i ^, -iJ o tS ~- ^ ^ m p bC c CO TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1295 The effect that ions can have on cathode life was clearly demonstrated in a TWT which was in many aspects a prototype of the MI789. This tube operated with the accelerator, helix and collector at successively higher voltages, with consequent ion draining toward the cathode. Severe ion bombardment of the cathode brought about failure of most of these tubes in from 500 to 2,000 hours. In contrast to this the average life of the M1789 is in excess of 10,000 hours in spite of a cathode current density about twice that in the prototype tube. Moreover, failure of the M1789 comes about from exhaustion of coating material rather than as a result of ion bombardment. During the course of the work of the proto- type tube, an experiment was performed to determine how much the ion bombardment would be affected by changing the potential dif- ference between tube electrodes. In this experiment a small hole was drilled in the center of the cathode and an ion current monitoring elec- trode placed behind it. The ion monitor current was then investigated as a function of electrode voltages. Fig. 6 shows the results. We see that comparatively small potential differences are adequate to control the flow of positive ions. 3.2 The Electron Gun and Electron Beam Focusing The electron gun used in the MI789 is a converging Pierce gun. The values of the gun parameters are summarized in Table I. Included are both the original parameters introduced by Pierce as well as those defined in a recent paper by Danielson, Rosenfeld and Saloom- in which the effects of thermal velocities are considered. Fig. 7 shows a drawing of the electrically significant contours of the J\II789 gun. Fig. 8 shows the completed electron gun assembly. The method of constructing the gun is a modification of a procedure used in oscilloscope and television picture tubes. The electrodes are drawn parts made of molybdenum or, in the case of the cathode, of nickel. They are supported by rods which are in turn svipported from a ceramic platform to which these rods are glazed. The whole gun structure is supported from the end of the helix by the helix connector detail. Since this part must operate at helix potential, it is insulated from the remainder of the gun by a ceramic cylinder which is glazed both to it and to the accelerator. To obtain good focusing, the cathode must be accurately aligned with respect to the other electrodes. However, it must be omitted from the gun during the glazing process and during a subsequent vacuum out- gassing because the cathode coating cannot withstand the temperatures involved. To insure proper placement of the cathode in the gun assembly 1296 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 ION COLLECTOR -20 VOLTS BEAM-FORMING ^ ELECTRODE 0 VOLTS COLLECTOR 1600,1800 OR 2000 VOLTS ^^S9W \ 1 \\ / 1 / 1 1 1 \ /(;= 18( DO V f 1 O.IO 0.09 0.08 0.07 0.06 0.05 0.04 0.03 /j .^^ 1 \ .-^"' r 1 '"2000 V ^ ^ A a^ 0.02 — • — r 16C 1 " )0V . 1 1 , .- -300 -200 -100 0 100 POTENTIAL DIFFERENCE BETWEEN HELIX AND ACCELERATOR IN VOLTS 200 Fig. 6 — Effect of electrode voltages on ion bombardment of the cathode in a prototype of the M1789. In this e.xperiment the helix voltage was varied while the positive ion current to a monitor electrode behind a hole in the cathode was meas- ured. Curves are shown for the collector voltage greater than, equal to, and less than the accelerator voltage. During this experiment the accelerator voltage was held constant at 1800 volts with a resulting beam current of 40 ma. The experi- ment was performed on a continuously pumped sj'stem with the pressure main- tained at 2 X 10-' mm Hg. The helix ID was 80 mils, the cathode diameter 300 mils, and the cathode hole diameter 20 mils. These curves show that the ion bombardment of the cathode can be reduced by as much as a factor of 20 by prop- erly arranging the voltage profile. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1297 at a later stage, an alignment cylinder is included in the gun at the time of glazing (outer cathode alignment cylinder in Fig. 8) . When the gun is ready to receive the cathode, the subassembly shown in Fig. 9 is slid into the outer alignment cylinder. The cathode to beam forming electrode spacing is set using a toolmakers microscope, and welds are made be- tween the inner and outer aligmnent cylinders. Initially, we thought that the cathode should l)e completely shielded from the magnetic field, and that the field should be introduced in the region between the accelerator and the point at which the beam would reach its minimum diameter in the absence of magnetic field. This ar- // /ra=l9i / / / / ACCELERATOR ; / > y//////////////////////////////^///^//y CATHODE \er r^ =192 >j (COATED DIAMETER) Fig. 7 — The electricall.y significant contours of the M1789 gun. All dimensions are in mils. These contours were determined using an electrolytic tank and follow- ing the procedure originated by Pierce. The measured potential at the beam boun- dary in the tank was made to match the calculated value within ±j per cent of the accelerator voltage to within 10 mils of the anode plane. The aperture in the accelerator was made sufficiently large so that substantially no beam current is intercepted on it. The significant parameters of this gun are: P = 0.3 X 10-« amps/volts3/2 reli+^ ^ > 5^ o3 o a3 e 2j a^g ^.2§ ao S "1^ 3 ^ 03 C G c3 _C cS ■r' 03 -»^ Si . 03 ;;3 13 03 X -^ fl-e CO Cl P 03 O 03 a b +- .2 • CO . (/3 . S o •-H O 03 cc 03 03 6 03 X' C ^ 03 O 03 bC --^03 C -iJ « 03— H oj.i 5 S g^^ 5P2 oj a 3 S^ o 032 g .2 03 03 ^ 03" c o 0 03-dH-=.>M _ "^^ 03 tC r- S ^ flj -^ '"^ 03 P O N « .^ M o3 bJD.S m 1h . X 03 O (^ r, Oi _o •—73-0 '^ tH O 03^ •r ra x: c3 c ° o TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1305 of about ±25 volts. It is not difficult to hold the average pitch variations to less than d=l per cent. The loading, however, is a more difficult prob- lem for not only must the dielectric properties of the support rods and of the glaze material be closely controlled, but attention must also be paid to the size and density of the glaze fillets. The gain of the tube is affected by the amount of loss in the helix attenuator. For the particular loss distribution used in the MI789 a variation of ±5 db out of a total attenuation of 70 db results in a gain variation of about ±1 db. The helix attenuator depends to a large extent on a conducting "bridge" between helix turns and therefore the amount of attenuation is sensitive to the size and the surface condition of the glaze fillets. Thus, the glazing process must be in good control in order to minimize variations in both gain and operating voltage. With our present techniques, we are able to hold the voltage for maximum gain to within ±50 ^'olts of the nominal value. The gain is held to ±2 db — about half of the spread we believe to be caused by variations in loss distribution and about half by differ- ences in beam size. Fig. 15 — Enlarged photograph of part of an M1789 helix. Two of the ceramic support rods can be seen. The other is directly opposite the camera behind the lielix and is out of focus. The fillets of glaze which bind the helix to the rods can he seen along the upper rod. This section of helix was free from applied loss. 1306 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Helix-to-Waveguide Matching In the helix-to-waveguide transducer the hehx passes through the center of the broad face of the waveguide and energy is coupled between helix and waveguide by an antenna and matching taper. A capacitive coupler on the helix and an rf choke on the waveguide place an effective ground plane at the waveguide end of the antenna. The rf choke also assists in minimizing leakage of rf power. Details of this transducer are shown in Figs. 5 and 14. 600 Hi Q < a. \- z o Fig. 17 — Pitch deviations and internal reflections in an early M1789 TWT. ■ The ordinate of the pitch deviation curve is the difference between the measured spacing between heli.v turns and the nominal value, which for this particular helix was 25 mils. (The tube operated at 1,600 volts.) Each point represents a helix turn. It is seen that the pitch deviations are periodic in nature, repeating about every 0.450 inch. The internal reflections were measured by matching the TWT with beam off ' at each individual frequency with a tuner to a VSWR of less than 1.01 (return loss greater than 40 db). The beam was then turned on and the resulting reflection taken as an approximate measure of the internal reflection. There appeared to be no appreciable change in the helix-to-waveguide transducer reflection as a result of turning th(! beam on. Evidence for this is the fact that when the beam was turned on with the lielix voltage adjusted so that the TWT did not amplify, there was little change in the reflection. The peaks of the internal reflection curve occur at five, six and seven half wave- lengths i)er i)eriod of the helix pitch deviations, indicating that the reflections from each period arc adding in phase at these frequencies. At the 5,800-mc peak the return loss is positive. This indicates a reflected signal larger than the incident signal. Shorting the TWT output caused the tube to oscillate at this frequency. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1309 the gain fluctuations will be about 0.25 db, the amplitude of the phase I fluctuations will be about 0.9 degree and the periodicity of the fluctua- tions will be about six mc. This effect may be eliminated by using an I isolator between the TWT and the antenna to eliminate the echo signal. I In addition to echo signals that occur between the TWT and the 1 antenna there are echoes which occur wholly within the TWT as a result of a reflection of the signal from the output transducer and a second reflection from some point along the helix. Thus even if a TWT is operat- ing into a matched load it may have ripples in gain or phase characteris- tics. These ripples may be controlled by minimizing the internal re- flections. In the MI789 they are less than ±0.1 db in gain and one-half degree in phase. Their periodicity is about 100 mc. In addition to causing transmission distortions, internal reflections can seriously reduce the margin of a TWT against oscillation. Outside of the frequency band of interest, the helix-to-waveguide transducer may be a poor match or the TWT may even be operating into a short circuit in the form of a reflection type bandpass filter. At such fre- quencies, the internal reflections must not be large enough so that an echo between transducer or filter and an internal reflection point will see any net gain, or else the TWT will oscillate. With many types of helix winding equipment, variations in helix pitch are periodic in nature. This causes the helix to exhibit a filter-like behavior with respect to internal reflections. At frequencies at which the period of the pitch variations is an integral number of half-wave lengths, the resultant reflections from each individual period will add in phase, thereby causing the helix to be strongly reflecting at these fre- quencies. This effect can perhaps best be illustrated by considering some results obtained in an early stage of the MI789 development. Fig. 17 shows measurements of the spacing between turns of an early helix. Also shown is the return loss as a function of frequency that a signal incident on the output of an operating TWT would see as a result of internal reflections alone. Helix-to-waveguide transducer reflections were eliminated with waveguide tuners during this experiment. The deviations in helix pitch from nominal are rather large and are markedly periodic in nature. The resulting internal reflections show strong peaks at fre- quencies corresponding to five, six and seven half-wavelengths per period of the pitch deviations. In the present M1789 this situation has been considerably improved b}^ increased precision in helix winding and by insuring that the re- maining periodicity does not produce a major reflection peak in the band. Fig. 18 shows pitch measurements and internal reflections for a recently constructed tube. 1310 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 (6 cc ~ 30 /^^ \ \ \, / \ / \ s^ ^ J\ VJ / v„ 5.0 5.5 FREQUENCY IN 6.0 6.5 KILOMEGACYCLES PER SECOND 7.0 7.5 01 Z o > ai o -1 t -2 NOMINAL TPI = 34 \ / k^ kj ■V S. .^\v- A, /\ A , ^,^/XM ^ \ ^ \.^ ^0^ s. V V Twy*- 1 V \jf V^ DISTANCE ALONG HELIX (EACH POINT REPRESENTS ONE TURN) Fig. 18 — Pitch deviations and internal reflections in a recent M1789 TWT. By precise helix winding techniques the pitch deviations have been reduced by a i factor of about 10 over those occurring in early tubes. The resulting internal re- flections have been improved by about 25 db although there is still a residual periodicity remaining. For return losses greater than about 25 db, we begin to see internal reflections originating from the edge of the heli.\ attenuator. At these values of return loss, the measurements also begin to be in appreciable error as a result of the residual transducer reflections. Helix Attenuator Attenuation is applied to the helix by spraying aquadag directly on the heUx assembly and then baking it. The result is a deposit of carbon on the ceramic rods and on the glaze fillets. The attenuation is held between 65 and 80 db and is distributed as shown in Fig. 19. Evidently most of the loss is caused by a conducting bridge which is built up between helix turns. This was indicated by one experiment in which we cleaned the deposit off the rods of a helix by rubbing them with emery paper. Only the carbon directly between helix turns then remained. This de- creased the total attenuation by less than 20 per cent. Having the helix glazed to the support rods is apparently necessary in order to get good contact between the winding and the carbon "bridge." We have been able to obtain about four times as much loss per unit length with glazed hehces as with non-glazed ones. Using our method of applying attenua- tion we can add in excess of 80 db/inch to a glazed helix. The ability to obtain such high rates of attenuation allows us to concentrate the loss along the helix thereby minimizing the TWT length. The machine used for spraying aquadag on the helix is shown in Fig. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1311 20. A glass cylinder and photocell arrangement is used to monitor the amount of carbon deposited. In this manner the attenuation added is made independent of both the aquadag mixture and the nozzle setting of the spray gun. This machine has been checked alone by using it to spray glass slides which are then made into attenuator vanes. Over a two-year period we have found that a gi\'en light transmission through the monitor slide results in the same vane attenuation within ±2 db out of 40 db. After a helix has been sprayed, it is vacuum fired at 800°C for thirty minutes and then the loss is measured. About 60 per cent of the helices fall within the desired range of 65-80 db. The principal cause of the differences in attenuation is believed to be variation in the condition of the glaze fillets. Helices not meeting specifications are sprayed and fired a second time (after cleaning off excess acpadag if necessary) . This second treatment, brings the attenuation of almost all helices to within the desired range. 3.4 The Collector It is desirable to operate the collector at the lowest possible voltage in order to minimize the dc power input to the TWT. This increases the overall efficiency and simplifies the collector cooling problem. On the -J- lUU u z S 75 Q. to i 50 u LU Q 2 25 10 f) o -J 0 1 j \ \ \ 0.5 1.0 1.5 2.0 2.5 3.0 3.5 DISTANCE FROM INPUT HELIX INPUT 4.0 4.5 5.0 HEL 5.5 t X OUTPUT Fig. 19 — Distribution of helix attenuation. The attenuation pattern has a gradually slanting edge facing the output to provide a smooth transition into the loss for any signals traveling backwards toward the input. Reflections of these signals must be \evy small since the reflected signals will be amplified in the process of returning toward the output. Cold measurements (i.e., measurements on the heli.x without electron beam) made by moving a sliding termination inside the helix, indicate that the return loss from the attenuator output is better than 45 db, the limiting sensitivity of our measurement. The input side of the helix attenuator is also tapered to minimize reflections but this taper is much sharper than that on the output side because there is comparatively little gain lietween input and attenuator. Cold measurements with a sliding termination showed a return loss for this taper of about 40 db. (Surprisingly, even a sharp edge pro- duces a reflection with a return loss of almost 30 db.) 1312 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 other hand, if there is appreciable potential difference between helix and collector, we must insure that few secondary or reflected electrons return from the collector to bombard the helix and accelerator, or else we may overheat these electrodes. Fig. 21 shows a drawing of the col-- lector used in the M1789. It takes the form of a long hollow cylinder shielded from the magnetic field. Inside of the collector the beam is allowed to gradually diverge and the electrons strike the walls at a graz- ing angle. This design reduces secondary electrons returned from the collector to almost negligible proportions. ^CYLINDRICAL ' GLASS SLIDE PHOTOCELL Fig. 20 — Schematic diagram of the machine used for .spra.ying aquadag attenu- ation on the helix. In this machine the helix is rotated rapidly to insure uniform exposure to the spray. At the same time the masking drum rotates at a slower speed and the spra}- gun traverses back and forth along the masking drum. The drum therefore acts as a revolving shutter between the helix and the spray gun and its degree of opening serves to control the amount of aquadag reaching the helix. From a knowledge of the rate of attenuation increase as a function of the amount of carbon deposited (empirically determined) the shape of the drum open- ing can he calculated so as to give any desired attenuation pattern. The spray gun also passes over a glass cylinder at one end of the masking drum so that it receives a sample of the aquadag spray. A photocell is used to monitor light transmitted through the cylinder. Before starting to spray, the glass is cleaned and the photocell reading is taken as 100 j)er cent light transmission. The helix is then sprayed until the light transmission has decreased to the proper value. The photoelectric monitoring techniciue makes the attenuation added in- sensitive to the aquadag composition and to the spray gun nozzle opening. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1313 O QJ 03 a. LU CL (a) / A A / 40 35 30 25 GAIN IN DECIBFt S -r^ 38-t] f=^ A H h r^i S — 2400 / VOLT^/ ^ V K yi 600/1 1100 A / / 150 100 50 0 -50 1 (c) INVE 1 I 1 =tSION 2400 VOLTS y A 1 COMF 'RESS ON A y I 1 1 1 t ^_^ k ir^ V ^600 / 2" '00 EXPA 1 u SISiON 'i' — H r^-^ U-^ 15.0 Z 12,5 o_i S.v, 10.0 Z' Q- U1 I LU QUJ HCC I 15 5liJ 7.5 5.0 2.5 (e) 2700 /\ VOLTSy^ ^ \ K \ / 2600) \ / V \ \ 1 r^ ^ A \2400 15.0 zCi I LU OLU I e> 51U 12.5 10.0 7.5 5.0 2.5 -15 -10 -5 0 5 10 15 POWER INPUT IN DBM 20 25 0 28 (f) / 2700 / VOLTSy^ 2( 5ooyC P^ Pi -^ ^ ^ 1 ^,-1 30 32 34 36 38 POWER OUTPUT IN DBM Fig. 23 — See opposite page for caption TRAVELING WAVE TUBE FOR G,000-MC RADIO RELAY 1317 The maximum output at saturation is obtained at a higher helix voltage as is common in TWT's. The helix voltage also affects the shape of the input-output curves — linear operation being maintained to higher output levels at higher helix voltages. As a measure of the efficiency of electronic interaction in a TWT, we use an "electronic efficiency" which is defined as the ratio of the rf out- put power to the beam power (product of helix voltage and beam cur- rent). The "over-all efficiency" we define as the ratio of the rf output power to the total dc power (exclusive of heater power) delivered to the tube. With the collector operated at 1,200 volts, it is about twice the electronic efficiency. For the M1789, maximum efficiency occurs at the saturation level with a helix voltage of 2,600 volts. The electronic and over-all efficiencies there are equal to about 14 per cent and 28 per cent, respectively. The curves of Figs. 23(a) and (b) were taken with sufficient time al- lowed for the tube to stabilize at each power level. If the TWT is driven to a high output level after having been operated for several minutes ' with no input signal, the output will be somewhat greater than is shown ' in the curves. It will gradually decrease until it reaches a stable level in a period of about two minutes. This "fade" is caused by an increase in the intrinsic attenuation of the helix near the output end. The increase is a result of heating from rf power dissipation. At maximum output the fade is about 0.6 db (about 15 per cent decrease in output power). At the five-watt output level the fade is about 0.1 db (about 2 per cent .( — — ■ ' Fig. 33 — See opposite page j -c: (a) Output power as a function of input power. Both ordinate and abscissa are in dbm (db with respect to a reference level of one milliwatt). A straight line at 45° represents a constant gain. A gain scale is included along the top of the figure. For this tube a helix voltage of 2,400 volts gives maximum gain at low signal levels and a voltage of about 2,600 gives maximum output at saturation. (b) Gain as a function of output power. This is an alternate way of presenting the information shown in (a). (c) Compression as a function of input power. Three regions are shown in the figure. The "compression" region is that in which there is less than one db change in output level for a db change in input level. The "expansion" region is that in which there is more than one db change in output level for a db change in input level. The "inversion" region is that in which the output level decreases when the input level increases (or vice versa). It occurs for input levels greater than that necessarj^ to drive the TWT to saturation. In this region the change in out- put is of opposite sign to the change in input. Using the definition in the text this gives rise to compression values in excess of 100 per cent. (d) Compression as a function of output power. (e) Conversion of amplitude modulation to phase modulation as a function of input power. This conversion arises because the electrical length of the TWT is a function of the input level. The effect can cause rather serious difficulties in cer- tain types of low index FM systems. (f) Conversion of amplitude modulation to phase modulation as a function of output power. 1318 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 decrease in output power). We will present some additional data on this effect in Section 4.3. Distortion of the Modulation Envelope The curves of Figs. 23(a) and (b) tell what happens when a single frequency carrier signal is passed through the TWT. In addition we would like to know the effect on modulation which may be present on the signal. In particular, it is desirable to know the compression of the envelope of an AM signal and the amount of phase modulation generated in the output signal as a result of amplitude modulation of the input signal, (an effect commonly known as A]\I-to-PM conversion). As a measure of compression of an AM signal the quantity per cent com- pression will be used. This is defined as % Compression AV,/V,_ 100 where Vo is the voltage of the output wave, Vi is the voltage of the input wave, and AYo is the change in output voltage for a small change AVi in the input voltage. When AF/F is small it can be expressed in db as 8.68 AF/F = AF/F in db. From this it follows that % Compression 1 =- > ni do APi 100 where APo is the change in output power for a change APi in input power, and the two powers are measured on a db scale. When the per cent compression is zero the TWT is operating as a linear amplifier; when it is 100 per cent the TWT is operating as a limiter. From the above expression it may appear that the per cent compres- sion could be determined directly from the slopes of the input-output curves. This would be the case were it not for fading effects. Since there is fading, however, the slope for rapid input level changes is different at high levels from the slope of the static curves. Thus it is necessary to determine compression from the resulting effect on an AM signal. The electrical length of a TWT operated in the non-linear region is 1 1 > some extent dependent on the input level. Therefore, an AM signal ap- plied to the input of the TWT will produce phase modulation (PM) of the output signal. This effect ma}^ be of particular concern when a TWT operating at high output levels is used to amplify a low-index FM signal. If such a signal contains residual amplitude modulation, the TWT generates phase modulation with phase deviation proportional to the input amplitude variation. Under certain circumstances this can cause TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1319 severe interference with the signal being transmitted. We wU discuss a particular example after consideration of the compression and AM-to- PM conversion characteristics of the M1789. As in the case of compression, we must measure AM-to-PM conversion dynamically. This is necessary because point-by-point measurements of the shift in output phase as input level is changed include a component of phase shift caused by changes in temperature of the ceramic support rods and a consequent change in their dielectric constant. However, this thermal effect does not follow AM rates of interest and therefore does not produce AM-to-PM conversion. Fig. 24 shows a simplified block diagram of the test set used to measure compression and AM-to-PM conversion. This equipment amplitude modulates the input signal to the TWT under test by a known amount and detects the AM in the output signal with a crystal monitor and the PM with a phase bridge. A more complete discussion of this measurement is given by Augustine and Slocum.^ Compression is given as a function of power input in Fig. 23(c) and as a function of power output in Fig. 23(d). We see that compression sets in more suddenly at higher helix voltages. Above about 2,500 volts REFERENCE PHASE PHASE SHIFTER SIGNAL SOURCE .1 H AMPLITUDE MODULATOR HYBRID JUNCTION PHASE BRIDGE OSCILLO- SCOPE TRAVELING- WAVE TUBE Fig. 24 — Simplified block diagram of test set used to measure compression and conversion of amplitude to phase modulation. A ferrite modulator introduces one db of 60 cps amplitude modulation into the test signal. The 60 cps rate is much higher than that which can be followed by thermal changes in the TWT. Half of the modulated signal serves as input to the TWT under test and half serves as a reference phase for a phase detector. The signals at the phase detector input are maintained equal and at constant level and nominally in phase quadrature. The detector is essentially a bridge circuit, the output of which is a dc voltage propor- tional to the phase difference of the two inputs. When operated with inputs in quadrature it is not sensitive to amplitude changes of as much as two db in either or both inputs. Phase modulation introduced by the amplitude modulator appears at both inputs and thus does not produce an indication. The output of the de- tector is therefore a direct measure of the phase modulation created in the TWT. Compression is determined by comparing the percentage amplitude modulation at the input and output crystal monitors. 1320 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 35.0 Z g m a 12.50 UJ > 2 q 11.25 2 Q. 10.00 I o 5 < 5.00"- 32.5 30.0 I 8.75 - CD 27.5 O 7.50 h '=' 25.0 Z 6.25 I- Z 22.5 < 20.0 17.5 15.0 2200 / r (b) GAIN^ • r*"" "^ < 1 1 1 c Y N \, 1 CONVERSION k compression\ N i J H 1 f\ I r 1 ^*v.^ 1 > \ ' 1 N ^ 2300 2400 2500 2600 HELIX VOLTAGE IN VOLTS 2700 100 9 in LU 75 Q: Q. 50 8 I- 25 5 O o5 2800 II- Fig. 25 — Gain, compression and amplitude to phase conversion as a function of heli.x voltage with the output power maintained constant at a level of five watts (a) and ten watts (b). there is expansion for some values of power input. Figs. 23 (e) and 23(f) give the AM-to-PM conversion, as functions of input and output power respectively. These data indicate that the conversion is very much less if the tube is operated at lower helix voltages. For example, the con- version at the saturation level of the 2, 700- volt curve is about 2^ times that for the 2, 400- volt curve. A final method of plotting gain, compression, and AM-to-PM con- version data is shown in Fig. 25. The abcissa here is the helix voltage. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1321 For these measurements power output was held constant by adjusting input level at each voltage. The figure shows that as helix voltage is increased, the compression decreases but the AM-to-PM conversion increases. The choice of a helix voltage at which to operate the tube must therefore represent a compromise between these quantities. Phase Modulation Sensitivity The equipment of Fig. 24 was also used to measure the phase modula- tion sensitivity of various electrodes by omitting the amplitude modula- OSCILLO- SCOPE SWEPT VIDEO SIGNAL SOURCE 0-10 MC FM TRANSMITTING TERMINAL o FM RECEIVING TERMINAL -■Z. ^ X SOURCE OF ONE DB OF AM TRAVELING- WAVE TUBE UJ O UJ Q Q. I- O UJ > < -J UJ -2 -3 -4 -5 ADDING PHASE,,^ .--- ^^ "^ OPPOSING .^.PHASE ^^ ^^ ^ ^ 234 56789 VIDEO FREQUENCY IN MEGACYCLES PER SECOND 10 Fig. 26 — Example of frequency response shaping caused bj' AM-to-PM con- version. This figure shows the calcuhited frequency response viewed between FM terminals for the system shown in the block diagram. Curves are given for the case in which the phase modulation generated in the TWT both adds to and sub- tracts from that of the transmitted signal. Inclusion of a limiter at point A would result in a flat frequency response. 1322 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 -20 -15 0 5 10 INPUT IN DBM Fig. 27 — Output power as a function of input power at various beam currents. Tliese curves were all taken with the helix voltage adjusted to give the maximum gain at low signal levels. At low beam currents (<20 ma) there is insufficient gain between the attenuator and the output so that at these currents the attenuator section is limiting the power output. This accounts for some of the difference in shape of the curves near maximum output. tor and introducing small changes in electrode voltages. The modulation sensitivity of the helix is about two degrees per volt and that of the accelerator about 0.1 degree per volt with the TWT operating under nominal conditions. Significance of AM-to-PM Conversion Let us return briefly to a discussion of some consequences of AM-to- PM conversion. As an example, we will consider the case of a low -index FM signal. Assume the frequency deviation is ±5 mc peak to peak. This gives a phase deviation of ±0.5 radian for a 10 mc modulating signal. These values are typical of what might be found in a radio relay system. Let us also assume that there is a residual amplitude modulation of one db (about 13 per cent) in this signal and suppose further that the signal is amplified by a TWT having a value of AM-to-PM conversion of 10 degrees per db. The phase modulation thus created in the TWT can either add to or subtract from that of the original FM signal, thus chang- ing its modulation index. At low modulation signal frequencies the phase deviation of the FM signal will be large compared to that of the PM interference and the interference will be of little consequence. At high modulation signal frequencies the phase deviation of the original FM and of the interfering PM signals will be comparable and the interference TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1323 can considerably change the net phase deviation of the overall signal. For the example we are considering the frequency responses in Fig. 26 show what would be seen at the output FM terminal. Curves are given both for the PM interference adding to and subtracting from the original FM signal. We see that a gain-frequency slope of about 4 db over 10 mc is introduced by AM-to-PM conversion. To prevent such an effect, a limiter should be used prior to the TWT in applications of this nature so as to remove the offending AM from the input signal. The fact that compression and amplitude-to-phase conversion vary with input level means that in addition to the first order distortion just described, higher order distortions of the modulation envelope will occur. If, for example, the input signal is amplitude modulated at fre- quency /i , the output modulation envelope will contain amplitude and phase modulation both at /i and at harmonics of /i . The amount of higher order distortion can be estimated by expanding the compression and amplitude-to-phase conversion curves as a function of power input in a Taylor series about the operating point. Such an expansion shows that the greater the slope of these curves the greater will be the higher order distortions. 40 35 30 25 20 15 10 0 .X A y^" A ."'■' HELIX VOLTAGE J w Y / \A V > / V ;ain / 7 / / 2450 2400 l5LU io o>, 2350 li.^ 2300 2250 IU_| I< 2200 10 20 30 40 50 60 BEAM CURRENT IN MILLIAMPERES 70 Fig. 28 — Low-level gain as a function of beam current. The helix voltage was adjusted for maximum gain at each current. 1324 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Reproducibility The curves presented in this section are all for the same tube, one which is representative of a group of 50 which were built at the conclusion of the Ml 789 development program. The tubes in this group had char- acteristics falling within the following ranges. The numbers represent the range containing 90 per cent of the tubes tested. Accelerator Voltage for 40 ma 2,500-2,700 Helix Voltage for maximum low-level gain 2,350-2,450 Low-level gain 33-37 db Gain at 5 watts output 31-35 db Maximum power output J 40.5-42 dbm \(1 1.2-15.8 watts) / / 45 / y / / ^ 1 ,70 MA 40 / ^ r^ n"'cx^6o >v K / ^ N k /) \^ f 7 ^ r^^^° \ \ 35 / / y^ / ^^s-l Nl \ // / Y / / ^^^0 N \ k ^ 30 LU m u UJ a 25 z / V/i r^ / / ^>^ i N L N \ lj[/ / / / '--^ k \ \ ^ M ^ / / ^ k \ 1 K \ \ z < O 20 15 10 X 1 K \ \ \ ^ \^ \ L WA Y 1 1 1 1 1 i \ A y i/^ 1 1 >^o 1 \ \ \ \ T m/ 1 1 1 L N i *^ y 5 juj/l A ^ \ \ \ 1 \ \ ^ / 1 N <5 MA N 1 \ \ \l 0 /f \ \ \ 2000 2100 2200 2300 2400 2500 2600 2700 HELIX VOLTAGE IN VOLTS 2800 2900 3000 Fig. 29 — Low-level gain as a function of helix voltage for various beam cur- rents. The dotted line represents the locus of the maxima of the curves. TRAVELING WAVE TUBE FOR O-OOO-MC RADIO RELAY 1325 4.5 5.0 5.5 6.0 6.5 7.0 7.5 FREQUENCY IN KILOM EGACYCLES PER SECOND 8.0 Fig. 30 — Low-level gain and helix voltage for maximum gain as functions of frequenc}' for several beam currents. The TWT was matched to the waveguide (with tuners where necessary outside of the 5,925 to 6,425-mc range) at each fre- (luency. The solid curves show the gain-frequency characteristic with the helix voltage adjusted for maximum gain at 6,000 mc for each beam current and then held constant as frequency was changed. Experimental points correspond to this condition. The dotted curves show how the characteristics change when helix voltage is optimized at each frequency. The optimum helix voltage increases by about 100 volts in going from 6,000 down to 4,500 mc because of slight dispersion in the phase velocity of the helix. 4.3 Operation Over an Extended Range We now turn to a consideration of tj^pical Ml 789 characteristics over an extended range of beam current, frequency, and magnetic field.* We shall concentrate on two items, the low-level gain and the maximum power output. From variations in these quantities the complete compres- sion ctirves can be roughly deduced. This situation is illustrated in Fig. 27 which sho^vs output as a function of input at different beam currents. While the shapes of these curves are slightly different, for the most part they can be derived from the 40-ma curve by shifting it along the abcissa * The characteristics of the tuV)e used for the low-level gain measurements in this Section were slightly different from those of the tube used for the maxi- mum output measurements and both were slightly different from those of the tube used for the measurements of Section 4.2. All tubes, however, had charac- teristics falling within the ranges listed above. 1326 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 ( by the amount the low-level gain changes, and along the ordinate by the amount the maximum output changes as beam current is varied. A similar procedure can be followed for variations Avith frequency and magnetic field. In all figures in this Section, parameters not being pur- posely varied were held at the nominal values given on page 1315. Low-level Gain Fig. 28 shows the variation in low-level gain with beam current and Fig. 29 shows its variation with helix voltage for several different beam currents. Fig. 30 shows the variation with frequency and Fig. 31 the variation with magnetic field. X Hz lUO 0.8 ujq: 0.4 - Ol 2400 I- O > 2380 H 2360 OJ I 2340 1 6 JO 01 _l UJ m 36 c r^ ~ . o UJ Q Z ■" 34 Z < 19 ^ 32 500 550 600 650 MAGNETIC FLUX DENSITY IN GAUSS 700 750 Fig. .31 — Low-level gain, helix voltage for maximvim gain and helix intercep- tion at low signal level as functions of magnetic flux density. These measurements were made using different strength permanent magnet circuits. The gain varies with magnetic flux density mainly as a result of its effect on l)eam size and there- fore on the degree of coupling l)etween electron stream and lielix. The helix voltage varies because of the effect of beam size on QC and therefore on the ratio of the optimum gain voltage to the helix synchronous voltage. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1327 70 65 60 55 50 45 IT) LU 40 OJ o UJ Q 35 z z 30 < 15 25 20 10 o- ■-0 EXPERIMENTAL , r- ALCULATED b/b =0.6 f / / / / } /. V ^0.4 / ''A / / / 0 ^ A / / / ^/ y t // f /' 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 r/3-(MA)'/^ Fig. 32 — Measured and calculated low-level gain as a function of the one-third power of beam current. The parameter b/a is the ratio of effective beam diameter to mean helix diameter. 40.0 37.5 35.0 32.5 Q 30.0 Z 27.5 < 25.0 22.5 20.0 ^ "^ ^ 1 = 0.6 ^ ^ [>»•''" ? — ^—1 '" >- — .. 3 .J ? r""- \ ,^''' 5'-' y" ^, '■--, -. V^ "\ \ ^ ^v s. ^^ X 4 = 0.4^ \ ^v '^^ N, CALCULATED EXPERIMENTAL N \ o — — O \ •v 4.5 5.0 5.5 6.0 6.5 7.0 7.5 FREQUENCY IN KILOMEGACYCLES PER SECOND 8.0 Fig. 33 — Measured and calculated frequency response for a current of 40 ma. 1328 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 The observed gain compares well with that calculated from low-level TWT theory provided that we properly consider the effect of the helix attenuator and provided that we assume a hi a of one-half. The method we have used in calculating the Ml 789 gain is discussed further in Appendix I. Fig. 32 compares the measured and calculated gain as a function of beam current and Fig. 33 compares them as a function of frequency. Fig. 34 shows measured and calculated ratios of voltage for maximum gain to synchronous voltage as a function of beam current. In all these figures calculations are shown for several values of the ratio of effective beam diameter to mean helix diameter (6/a). We see that the effective value of 6/a appears to be about one-half. On the basis of measurements made by probing the beam of a scaled up version of a 1.26 1.24 1.22 1.20 1.18 1.16 1.14 V/Vs 1.12 1.10 1.08 1.06 1.04 1.02 1.00 y <• ^/^ y b/a = o.2^ y y / / 0.4^ ^ -^ / ^ \ ( ** -•"^ / ^^ fe ,""" -"^ '^ / ^ x^ <^ "^ 0^8^ , 2500 liJ 2400 I 3 o r V FORE FADE TER FADE HELIX VOLTAGE SET FOR MAXIMUM OUTPUT, / / • ^ - AF / A \ ( / y, / \ A / / / / A t ^ > r • V y ^X / / • A /^^ HELIX VOLTAGE ^ SET FOR A / f .^' r MAXIMUM GAIN AT LOW LEVEL A {' /' Y ^ A / J, v> ^^ ^'^ .^ #^^ 2300 2200 ^^ ^ ^ "^ ,y X^^OLTAGE FOR MAXIMUM OUTPUT y' A ) __, ' ^ > > VOL TAGE FOR MAXIMUM LOW LEVEL GAIN ^ 10 15 20 25 30 35 40 45 50 55 60 65 70 BEAM CURRENT IN MILLIAMPERES Fig. 35 — Maximum power output and heli.x voltage as functions of beam cur- rent. Curves are shown for before and after fading, and for tlie helix voltage ad- justed for the ma.ximum gain at low-level and for maximum output. 1330 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 FREQUENCY IN KILOMEGACYCLES PER SECOND Fig. 36 — Maximum power output after fading as a function of frequency for several beam currents; in (a) with the helix voltage adjusted for maximum gain at low-level and in (b) with the helix voltage adjusted for maximum power output. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1331 b Z ^- Q. UJ U a. LU 1- 3 z X _i ^2 t- Z UJ u \ k 1 \ \ \ \ \ \ tr ' UJ Q. ^ P 0 500 550 600 650 700 MAGNETIC FLUX DENSITY IN GAUSS 750 Fig. 37 — Maximum power output after fading, voltage for maximum output, and helix interception at maximum output as functions of magnetic flux density. These measurements were made using magnetic circuits charged to different strengths. Helix interception above about one per cent is undesirable if long tube life is required. 1332 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 focusing system similar to that employed in the M1789, we estimate the actual beam diameter (for 99 per cent of the current) to be about 65 mils (Jb/a = 0.7). However, the current density distribution is peaked at the center of the beam because of the effect of thermal velocities of the electrons. Thus an effective h/a of 0.5 is not unreasonable. Maximum Power Output ' Fig. 35 shows the maximum power output as a function of beam cur- rent both immediately after rf drive is applied and after the tube has had time to stabilize. We see that at high rf power outputs the fading 3.0 2.5 iJ 2.0 > u z ly 1.5 o u. H] '-0 0.5 0 4.0 3.5 3.0 ^2.5 U z y 2.0 u IL li 1.5 1.0 0.5 THEORETICAL 60 MA 20 MA O 6 a 4 r 0 MA 0 MA L— 3 -^ '^ \; — -< — ^ 1 t — 20 MA I \ (a) ^^ 60 MA — — — S y***"*.^ THEORETI 60 MA CAL — — 40 MA 20 MA ^ ) 1 }■ -r:' J . 1^ F^— P "" I 20MA i I k I 1 ' ' ' ' (b) 4.0 4.5 5.0 5.5 6.0 6.5 FREQUENCY IN KILOMEGACYCLES PER SECOND 7.0 7.5 Fig. 38 — Ratio of electronic efficiency to gain parameter C as a function of frequency. The efficiencies used for this comparison are all before fading. The dot- ted line.s are estimated from the Tien theory corrected for the intrinsic loss of the helix. The curves in (a) are for the case of the heli.x voltage adjusted for the maxi- mum low-level gain and those in (b) for the case of the helix voltage adjusted for maximum power output. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1333 becomes very serious and eventually limits the TWT output to about 30 watts. If it were necessary to reduce this fading, the envelope shrinking technicjue illustrated in Fig. 16 could be used. The maximum power output after fading is shown as a function of frequency for several beam currents in Fig. 36 and as a function of magnetic flux density in Fig. 37. The theory of the high level behavior of a TWT** predicts that the ratio of electronic efficiency (i.e., E = power output/beam power) to the gain parameter C should be a function of C, QC and 7b (where h is the beam diameter). However, with the range of parameters encountered in the M1789, the variation in E/C should be small. Fig. 38(a) shows E/C as a function of frequency when the TWT is operating at the voltage for maximum gain at low signal levels. Fig. 38(b) shows the maximum value of E/C obtainable at elevated helix voltage. In both figures we show the efficiency as estimated using the results of Tien^ corrected for the effect of intrinsic loss following the procedure of Cutler and Brangaccio.^ All etticiencies in these two figures are the electronic efficiency before fading. It would be quite difficult to compare the efficiency after fading with theory because the intrinsic attenuation in this case varies along the helix in an unknown manner so that we cannot properly take it into account. From the figures we see that the calculated value of E/C at 6,000 mc and 40 ma is not far from the experimental value but the ex- perimental points show more variation with frequency than is predicted by theory. The low efficiency at 20 ma results from the fact that there is insufficient gain between the helix attenuator and the output. As a result, the TWT "overloads in the attenuation." 4.4 Noise Perjormance A new and important noise phenomenon was observed in the course of the Ml 789 development. It was found that the noise figure is strongly dependent on the magnetic flux linking the cathode and on the rf output level of the TWT. For example, with the TWT operating near maximum output and with a cathode completely shielded from the magnetic field, noise figures of about 50 db were observed. By allowing 20 gauss at the cathode, the noise figure was reduced to 30 db. Fig. 39 shows the noise figure as a function of magnetic flux density at the cathode for several values of rf power output. We see that there is a peak of noise figure roughly symmetrical about zero flux at the cathode, and that the magni- tude of this peak is considerably increased by operating the TWT at high output levels. Some additional observed properties of the noise peak are: (1) The magnitude depends on the synchronous voltage of the helix. For a 1,600-volt helix it is about 10 db higher than shown in Fig. 39 and 1334 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 for a 2,600-volt helix it is about 5 db lower. The noise figure for 25 gauss at the cathode remains constant, however. (2) There appears to be a threshold level of about 15-ma beam current below which the peak does not occur. Between 15 and 25 ma the peak increases. Above 25 ma it is roughly constant in magnitude. (3) The peak can be considerably reduced by intercepting some of the edge electrons before they reach the helix region. For this discussion it has been necessary to extend the concept of noise figure to the case of non-linear operation of the TWT. Essentially this noise figure is defined by the means we use to determine it. A block diagram of the equipment is shown in Fig. 40. The outputs of a calibrated broad band noise source and a signal oscillator are combined and used for the input to the TWT under test. The noise output from the TWT is passed through a filter tuned about 100 mc away from the signal so as to reject the carrier. It is then detected by a receiver tuned to the filter frequency. The noise figure is measured by turning the noise source off and on, noting the change in receiver output level and calculating the noise figure in the conventional manner. This procedure reduces to an ordinary noise figure measurement in the absence of input signal. There are other ways that could be used to measure noise figure of a non-linear amplifier. A method more closely related to the use of the 50 45 -20 -15 -10 -5 0 5 10 15 MAGNETIC FLUX DENSITY AT THE CATHODE IN GAUSS 20 Fig. 39 — Noise figure as a function of magnetic flux density at the cathode for several values of rf power output. The flux density was varied by using an in- ductive heater through which ac current was passed. The present ]\I1789 uses 19 gauss at the cathode, all of which is obtained from the focusing magnet — the heater now being non-inductive. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1335 TWT in an FM radio relay was investigated briefly. In this measurement an FM receiver tuned to the carrier frequency was used to detect the noise modulation present in the TWT output. The noise figure was deter- mined in the usual manner from the ratio of receiver outputs with the noise source turned off and on. When the TWT was operated in the linear region, this measurement gave the same result that our first method did. With the TWT operated in the non-linear region it gave a value within a few db of that obtained from the first method. The cause of the high noise output observed for low magnetic flux densities at the cathode is at the present time not clearly understood. Fried at MIT and Ashkin and Rigrod at Bell Laboratories have all probed the beam formed by guns of the M1789 type and have found certain anomalous efl"ects. Normally one would expect to find a standing wave of noise current along the electron beam. For the M1789 gun they find instead that after about two minima of the standing wave pattern, the noise current on the beam begins to grow and continues to do so until a saturation value is reached. The noise current at this saturation LOW NOISE TRAVELING- WAVE TUBE SIGNAL POWER MONITOR NOISE LAMP SIGNAL SOURCE RECEIVER y/C FILTER HYBRID H GK-r SIGNAL 6000 MC X TRAVELING- WAVE TUBE UNDER TEST RECEIVER LOCAL OSCILLATOR 6170 MC FILTER 6080 BAND 6100 20 MC Fig. 40 — Block diagram of noise measuring equipment. Tiie noise source con- sists of a fluorescent lamp the output of which is amplified by a low-noise TWT so as to bring the noise level to about 35 db above kTB at the M1789 input. The out- put from the M1789 is passed through a 20-mc bandpass filter which eliminates both the single frequency test signal and the noise in the image band of the re- ceiver. The noise figure is measured by noting the difference in noise level at the receiver output with the noise source off and on, in a manner similar to that used in a conventional noise figure measurement. 1336 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 value may be considerably higher than the original average noise level. As is the case with the noise figure in the M1789, the growing noise current has been found to be very sensitive to magnetic field at the cathode. By allowing sufficient field to link the cathode, the growing noise current can be eliminated leaving the normal noise current standing wave pattern on the beam. This phenomenon is not peculiar to the M1789 gun. It has been observed by various workers at MIT^ and else- where on other guns producing beams with comparable current densities. A satisfactory explanation for it has not, at the time of this writing, been arrived at. It seems safe to say, however, that the growing noise current on the beam is the source of the high noise figures obtained in the M1789 when the cathode is completely shielded from the magnetic field. 4.5 Inter modulation It has been found that certain intermodulation effects in the Ml 789 can be predicted from a knowledge of the compression and AM-to-PM conversion. Alternatively, these effects can be used to determine com- pression and AM-to-PM conversion. The procedure to be described has the advantage of being simple to implement as compared with the phase bridge arrangement of Fig. 24. UNIT VECTOR--. .^ — ■ ROTATING AT / ANGULAR / VELOCITY 1^ 2 7rAf \ UNIT AMPLITUDE -Af — >| FREQUENCY (a) (b) 4^ 4* 4^ AM VECTORS PM VECTORS (C) Fig. 41 (a) Spectrum of input signal to amplifier. (b) Vector diagram of two input signals and the resultant signal (R) in a frame of reference rotating at an angular velocity 2irAf . Dotted line is the locus of the re- sultant signal. (c) The rotating vector of the proceeding diagram can be broken down into a set of two vectors representing amplitude modulation and a set of two vectors rep- resenting frequency or phase modulation. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1337 Intermodulation effects are ordinarily complicated and results are jvery hard to predict from single frequency measurements on an amplifier, i'or a TWT, however, one case — that in which two signals of very [different amplitude are passed through the tube — can be treated simply, IConsider an input to a TWT consisting of two signals at frequencies l/i and /i + A/ with the signal at/i being very much larger in amplitude. The composite signal applied to the amplifier will then be a signal at frequency /i which is amplitude and phase modulated at a rate A/ in an amount proportional to the relative magnitudes of the two signals. This can be represented vectorially as shown in Fig. 41(a) and b. In this figure the amplitude of the signal /i + A/ has been normalized to unity. "A" thus represents the ratio of the larger to the smaller signal. The locus of the resultant signal is shown by the dotted line. The single rotating vector can be considered as the sum of vectors at /i + A/ and /i — A/ as shown in Fig. 41(c). One set of vectors produces PM and the other AM. The AM and PM vectors cancel at /i — A/ and add at /i + A/. Suppose this signal is put through an amplifier operating in com- pression. For the time being let us assume this amplifier has no AM-to- PM conversion. The compression in the amplifier will operate on the AM sidebands of the signal but will leave the PM sidebands unaffected. Let us define the quantity c as a measure of compression in the amplifier by ' = '- AVW< ^'^ where Vo is the output voltage, Vi input voltage, and AFo is the change in output voltage for a change AVi in the input voltage. This quantity is the per cent compression used in Section 4.2 divided by 100. If the signal in Fig. 41 is put through the amplifier while it is in compression, and the level of the signal at /i is subsequently brought back to amplitude A, we would then expect to have the situation shown in Fig. 42. Each AM sideband component has been multiplied by the factor (1-c). The locus of the composite signal is now elliptical. Let Si and S2 be the magnitude of the sidebands at /i + A/ and /i — A/ respectively. From Fig. 42 it is seen that ^1 = K + Hd - c) =1- c/2 (2) S9 = y2- Hil - c) = c/2 (3) When c = 0, the amplifier is operating in the linear region and *Si = 1, 1338 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 ^2 = 0. This is the condition in Fig. 41. When the amphfier is operating \ as a perfect Umiter, c = 1 and Si = S2 = 0.5. Thus, in this case, the side- 1 band *Si is down 6 db from its value when the amplifier is operating in the linear region. When there is conversion of AM-to-PM in the amplifier, the situation becomes somewhat more complex. Suppose an AM signal is fed into the amplifier and that its voltage is given by V = Vi{l -\- a sin wj) sin Uct where coc and oom are the carrier and modulating radian frequencies and V\ and a are constants. The outputs will be given by V = KVi[l + «(1 — c) sin oo,nt] sin {coct + kpa sin co^O (5) Here K is the amplification, c is the compression factor and kp is a factor which is a measure of the AM-to-PM conversion. It is seen that kp is the output phase change for a given fractional input change a. Thus rCp — A^ a (6) where AO is the phase change in radians caused by a fractional input change a. Later on it will be desired to express kp in terms of degrees phase shift per db change in input amplitude. To express a in db we i(i-c) ^ AM VECTORS 4^ PM VECTORS (a) (4) ' Fig. 42 (a) After passing through an amplifier in compression tlie AM sidebands are reduced in amplitude but the PM sidebands are unaffected. The lower two side- bands which represent a signal at frequency fi — Af no longer cancel and so there is a net signal at that frequency. (b) The locus of the resultant signal now assumes an elliptical shape. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1339 lust evaluate 20 logio ( 1 + a). The quantity loge (1 + a) can be ex- )anded in a series to give loge (1 + a) =a — -a-\--a+ • • • . A.S long as a <3C 1, we can approximate it by taking only the first term of the above expression. Converting to the base ten and converting Ad Prom radians as it appears in (6) to degrees, we find that k. 0.152 Ad (in degrees) A input level (in db) (7) Now let us consider the case in which the signal of Fig. 41 is put through an amplifier having AM-to-PM conversion. Fig. 43 shows the vector picture of the resulting signal after the level of the signal at /i has been brought back to amplitude A. In this case the original PM sidebands and the compressed AM sidebands are the same as in Fig. 42, but there is now an additional set of PM sidebands as a result of the AM- to-PM conversion. Since the peak deviation of output phase due to this latter set of sidebands comes when the instantaneous amplitude is either a maximum or a minimum, they are 90 degrees out of phase with the other two sets of sidebands. From Fig. 43 it is seen that we can write PM VECTORS GENERATED BY AMPLIFIER AM VECTORS PM VECTORS (a) o j » o ) » ys. Fig. 43 (a) After passing through an amplifier having both compression and amplitude to phase conversion, the AM vectors are reduced in magnitude and a new set of PM vectors have appeared. (b) The locus of the resultant signal of the vectors shown above is elliptical but the axis is tilted with respect to vector A. 1340 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 1.2 a z < in LU Q 3 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -15 32 = 2400 VOLTS ::--Y -ZS'-'^^ .^ -p- / -^ /S2 = 2600 VOLTS •10 0 5 10 POWER INPUT IN DBM 15 20 25 Fig. 44 — Relative side band amplitudes Si and S2 for the M1789 as a function of power input for two values of helix voltage. for the sideband amplitudes &\ and >S^2 at/i + A/and/i — A/ respectively ^" = [M + M(i - c)f + I-/,. -12 = (1 - c/2f + (^ S2' = [3^ - Vzil - c)f + "A^ i = {c/2r + Solving for c and kp we obtain c = 1 - {Si - S2) K = 2[s.' - (i^ .s: 2\ 2- 1/2 (8) ; (9) (10) (11) Thus we see that from a measurement of the amplitudes Si and S2 the values of c and kp can be determined. To check the validity of this approach to intermodulation, we deter- mined the values of compression and AM-to-PM conversion for an M1789 from an intermodulation measurement and compared them with values obtained using the phase bridge set-up described in Section 4.2. In the intermodulation measurement the two signals were 100 mc apart J TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1341 150 125 I/) LU 100 CE a. o o 75 50 25 r 7 ^^ kJ D ,^ ^y/^ ^600 VOLTS / 2400 VOLTS^ ^ / J ^ y / ^ — A>-^ ^ J2 r-^ -tf^ x^ -10 -5 5 10 POWER INPUT IN DBM 15 20 25 Fig. 45 — Compression as a function of input level for two values of helix volt- age. Triangles represent data obtained with the test set of Fig. 24. Circles and squares represent data obtained by the two signal intermodulation measurement. in frequency and 30 db different in level. From measurements of signal strength at the various frequencies involved, the magnitudes of S>\ and *S2 were determined with the results shown in Fig. 44. From these re- sults the values of c and h^ were calculated and then converted to % compression and degrees per db in order to compare with the results of _l 10.00 UJ m o 8.75 111 Q rr 7.50 LU Q. 01 6.25 LU LU ct (J 5.00 LU O z 3.75 z n (0 2.50 cr LU > 7 1.25 O O ? 0 Q- O 1- -1.25 5 < -2.60 D 2600 VOLTS/ r \ / \ / 1 J^ /- n k t \ A ^ \ A V f 2400 VOLTs\ D -- ^ !> ' ■J i -15 -10 -5 0 5 10 POWER INPUT IN DBM 15 20 25 Fig. 46 — Conversioia of amplitude modulation to phase modulation as a func- tion of input level for two values of heli.x voltage. Triangles represent data ob- tained with the test set of Fig. 24. Circles and squares represent data obtained by the two signal intermodulation measurement. 1342 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 20 cr 18 O ^ 16 u. o (0 14 Q Z ^ 12 O f 10 UJ o z I- < a o TUBE TUBES FAILURES TEST — ^ — — — — — — Fig. 47 — Life test results. The open bars indicate tubes that have failed; the solid bars tubes that were operating as of May 1, 1956. These tubes were operated with cathode temperatures between 720° and 760°C. Figs. 23(c) and 23(e). The latter curves are repeated as Figs, 45 and 46 with the experimental points calculated from Si and S2 shown. It is seen that the results of the two types of measurements compare remark- ably well considering that the calculations of c and kp both require the subtraction of nearly equal quantities. Thus we may conclude that our method of considering the intermodulation is substantially correct and that we can obtain compression and AM-to-PM conversion from an intermodulation measurement . V. LIFE TESTS We feel that sufficient data have been accumulated to indicate that tube life in excess of 10,000 hours can be expected. Fig. 47 summarizes our life test experience. All tube failures were caused by cathode failure and these were evidently the result of exhaustion of coating. End of life for these tubes comes comparatively suddenly i.e., in a few hundred hours after the cathode current begins to drop. At this time the emission becomes non-uniform over the cathode surface with consequent beam defocusing and helix interception. This in turn causes gas to be released into the tube which then accelerates the cathode failure through cathode poisoning. The rf performance remained good over the tube life — the gain and output power actually increasing slightly near the end of life as the beam started to defocus. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1343 VI. ACKNOWLEDGMENTS The M1789 TWT is the outcome of an intensive efTort which has included many individuals in addition to the authors. R. Angle, J. S. Gellatly, E. G. Olson, and R. G. Voss all have contributed to the me- chanical design of the tube and to its reduction to practice. R. W. DeVido has materially assisted with the electrical testing. M. G. Bodmer and J. F. Riley have been responsible for setting up the life test program and J. C. Irwin and J. A. Saloom contributed importantly to the design work on the electron gun. P. P. Cioffi and M. S. Glass have been largely responsible for the design of the magnetic circuits and P. I. Sandsmark for the helix-to-waveguide transducers. D. 0. Melroy studied the effects of positive ions and performed the experiments on ion bombard- ment referred to in Section III. D. R. Jordan contributed to the studies on noise. In addition to the above, the authors would like to thank E. D. Reed for his very helpful criticism of this manuscript. Appendix I — Gain Calculations The gain calculations for the M1789 follow the procedure outlined by Pierce"^ with some minor modifications. The steps involved in the gain calculations for the loss free region of the helix are as follows: - (1) The experimental synchronous voltage is used to determine ya and the dielectric loading factor as defined by Tien.^ (2) From 7a the value of helix impedance K is obtained from Ap- pendix VI of Pierce.^ (3) The value of K is corrected using Tien's^ results and C is then calculated in the usual manner. (4) The number of wavelengths Ni per inch of helix is obtained using the experimentally determined (from synchronous voltage) wave- length. (5) The value of cog/w is determined. In this calculation the curves for cop/cog from Watkins^ are employed. (6) QC is determined from QC = (7) From QC, B is determined from Fig. 8.10 of Pierce'^ and the gain BCNi in the loss free region is calculated. In calculating the effect of the attenuator section, we have had to make some rather gross assumptions. Fortunately, it turns out that the 134-4 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 gain in the attenuator is a small fraction of the total gain in the tube so that the over-all gain is not particularly sensitive to the means we use for treating the attenuator. Essentially what we have done is to con- sider the high loss part of the attenuator as a severed helix region and the low loss part of the attenuator as a lossy helix region. Fig. 48 shows the value of the growing wave parameter as a function of the loss parameter d for various values of QC as calculated from theory. Because of discontinuity losses to the growing wave as it propagates in a region of gradually increasing loss, the actual gain will be less than that calculated from Fig. 48. Some rather crude probe measurements have indicated that the effective x vs. d curve can be approximated by a straight line through the d = 0 and d = 1 points — the dotted line in Fig. 48. Since the helix is effectively severed by the high loss portion of the attenuator we must subtract some discontinuity loss from the gain in the attenuator region. The effective drift length in the severed region is unknown so this discontinuity loss cannot be accurately calculated from the low-level theory. The discussion in chapter nine of Pierce^ indicates that an average value of about 6 db is reasonable. An alternate method of treating the attenuator was also tried. In this calculation, the x vs. d curves in Fig. 48 were assumed to be correct to 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 \ N, \s v v^ ^-^^ V ^^•. """--- QC = 0 ^ S^^ V^ ^sN ^> ' ^ % ....,,0^5 ^^N — ^ 0.5 — V ^N 0.4 0.8 1.2 1.6 2.0 d 2.4 2.8 3.2 3.6 4.0 Fig. 48 — Curves of growing wave parameter x as a function of loss parameter d showing approximation (dotted lines) used in gain calculations for the M1789. TRAVELING WAVE TUBE FOR 6,000-MC RADIO RELAY 1345 (/ = 1. The region for which d > 1 was considered as a severed helix region with 6-db discontinuity loss. Calculations using this procedure gave total gains for the TWT within a couple of db of the first method. The remaining steps in calculating the gain of the TWT are therefore: (8) The quantity a is determined from the slope of the dotted lines in Fig. 48. (9) The length of helix, 4 in the attenuator for which x > 0 is determined by using Fig. 48. (10) The total attenuation L, in the section of the attenuator effective in producing gain is calculated. (11) The initial loss parameter A is obtained from Fig. 94 of Pierce J (12) The gain is calculated from Gain = A -6dh +aL + BCNi (3.5 + /«) where the six db is the discontinuity loss in the attenuator section and the 3.5 inches is the length of loss free helix. Glossary of Symbols a loss factor from Pierce^ A discontinuity loss parameter at input of helix from Pierce^ B magnetic flux density or the space charge parameter from Pierce Bb Brillouin flux density for a beam entirely filling the helix C gain parameter from Pierce^ a helix radius b beam radius d loss parameter from Pierce'^ / frequency Ik cathode current la accelerator current Ih helix current Ic collector current k 2ir/Xo where Xo is the free space wavelength fe length of helix attenuator in which gain is possible L loss in the part of the attenuator section which is capable of pro- ducing gain. A'' number of wavelengths in TWT Ni number of wavelengths on the helix per inch QC space charge parameter from Pierce^ Ta anode radius of curvature of gun Tc cathode radius of curvature of gun Tmin minimum beam radius from Pierce'" 1346 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Tc cathode radius r95 radius at the beam minimum through which 95 per cent of t current flows (7 standard deviation of electron trajectory Tk cathode temperature Va accelerator voltage Vn helix voltage Vc collector voltage X growng wave parameter from Pierce CO radian frequency oic carrier radian frequency ojm modulating signal radian frequency ojp radian plasma frequency cog corrected radian plasma frequency c compression factor kp AM-to-PM conversion factor 7 radial propagation constant References 1. Cutler, C. C, Spurious Modulation of Electron Beams, Proc. I.R.E., 44, , pp. 61-64, Jan., 1956. 2. Danielson, W. E., Rosenfeld, J. L., and Saloom, J. A., A Detailed Analysis of " Beam Formation with Electron Guns of the Pierce Type, B. S.T.J. 35, pp. 375-420, March, 1956. 3. Augustine, C. F., and Slocum, A., 6KMC Phase Measurement System For Traveling-Wave Tubes, I.R.E. Trans. PGI-4, Oct., 1955. 4. Tien, P. K., A Large Signal Theory of Traveling-Wave Amplifiers, B.S.T.J., 35, pp. 349-374, March, 1956. 5. Brangaccio, D. J., and Cutler, C. C, Factors Affecting Traveling-Wave Tube Power Capacity, I.R.E. Trans. PGED-3, June, 1953. 6. Smullin, L. D., and Fried, C, Microwave Noise Measurements on Electron Beams, I.R.E. Trans., PGED-4, Dec, 1954. 7. Pierce, J. R., Traveling-Wave Tubes, D. Van Nostrand, Inc., 1950. 8. Tien, F*. K., Traveling-Wave Tube Helix Impedance, Proc. I.R.E., 41, pp. 1617-1623, Nov., 1953. 9. Watkins, D. A., Traveling-Wave Tube Noise Figure, Proc. I.R.E., 40, pp. 65-70, Jan., 1952. 10. Pierce, J. R., Theory and Design of Electron Beams, D. Van Nostrand, Inc., 1949. Helix Waveguide By S. P. MORGAN and J. A. YOUNG (Manuscript received July 23, 1956) Helix waveguide, composed of closely wound turns of insulated copper wire covered with a lossy jacket, shows great promise for use as a communi- cation medium. The properties of this type of waveguide have been investi- gated using the sheath helix model. Modes whose wall currents follow the highly conducting helix have attenuation constants which are essentially the same as for copper pipe. The other modes have very large attenuation constants which depend upon the helix pitch angle and the electrical proper- ties of the jacket. Approximate formidas are given for the propagation con- stants of the lossy modes. The circular electric mode important for long- distance communication has low loss for zero-pitch helices. The propagation constants of sotne of the lossy modes in helix waveguide of zero pitch have been calculated numerically, as functions of the jacket parameters and the guide size, in regions where the approximate formulas are no longer valid. Under certain conditions the attenuation constant of a particular mode may pass through a maximum as the jacket conductivity is varied. Glossary of symbols a Inner radius of waveguide h = 13 — ia Complex phase constant n Angular mode index p Denotes p„„, or pnm' according to context Pnm in^^ zero of Jn{x) Pnm' w*^ zero of Jn{x) r, d, z Right-handed cylindrical coordinates a Attenuation constant /? Phase constant /?o = 27r/Xo = wifxoeoY'^ Free-space phase constant €o Permittivity of interior medium € Permittivity of exterior medium e e/eo 1347 e" 13-48 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 .. r 2 / / • //\ 7 2il/2 §2 L<^ Mo€o(.e — te ) — ll \ Xo Free-space wavelength Xc = 2-Kaf'p Cutoff wavelength juo Permeability of interior and exterior media V = Xo/Xc = p\o/2ira Cutoff ratio [e - I + V - te ) ^ + IT] e' - ie" n Electric Hertz vector n* Magnetic Hertz vector 0" Conductivity of exterior medium rj/ Pitch angle of helix CO Angular frequency e" Harmonic time dependence assumed throughout J nix) Bessel function of the first kind Jn(x) dJn{x)/dx Hn'^ix) Hankel function of the second kind Hn^'^'ix) dHS-\x)/dx MKS rationalized units are employed throughout. Superscripts i and e are used to indicate the interior and exterior regions. I. INTRODUCTION AND SUMMARY Propagation of the lowest circular electric mode (TEoi) in cylindrical pipe waveguide holds great promise for low-loss long distance communi- cation.^' ^ For example, the TEoi mode has a theoretical heat loss of 2 db/mile in waveguide of diameter 6 inches at a frequency of 5.5 kmc/s, and the loss decreases with increasing frequency. Increased transmission bandwidth, reduced delay distortion, and reduced waveguide size for a given attenuation are factors favoring use of the highest practical fre- quency of operation. An increased number of freely propagating modes and smaller mechanical tolerances are the associated penalties. Any deviation of the waveguide from a straight circular cylinder gives rise to signal distortions because of mode conversion-reconversion effects. One solution to mode conversion-reconversion problems is to obtain a waveguide having the desired low attenuation properties of the TEoi mode in metallic cylindrical waveguide and very large attenuation for all other modes, the unwanted modes.^' ^ The low loss of the circular electric modes in ordinary round guide is the result of having only cir- 1 S. E. Miller, B.S.T.J., 33, pp. 1209-1265, 1954. 2 S. E. Miller and A. C. Beck, Proc. I.R.E., 41, pp. 348-358, 1953. 3 S. E. Miller, Proc. I.R.E., 40, pp. 1104-1113, 1952. HELIX WAVEGUIDE 1349 cumferential current flow at the boundary wall. All other modes in round guide have a longitudinal current present at the wall. Thus the desired attenuation properties can be obtained by providing a highly conducting circumferential path and a resistive longitudinal path for the wall cur- rents. This is done in the spaced-disk line by sandwiching lossy layers between coaxially arranged annular copper disks. ^ Another possibility which has been suggested is a helix having a small pitch. Helix waveguide, formed by winding insulated wire on a removable mandrel and coating the helix with lossy material, has been made at the Holmdel Radio Research Laboratory. Wires of various cross sections and sizes have been used to wind helices varying from 3^ to 5 inches in diameter, which have been tested at frequencies from 9 to 60 kmc/s. Pitch angles of from nearly 0° (wire in a plane perpendicular to the axis of propagation) to 90° (wire parallel to the axis of propagation) have been used. The helices having the highest attenuation for the unwanted modes while maintaining low loss for the TEoi mode are those wound with the smallest pitch from insulated wire of diameter 10 to 3 mils (American Wire Gauge Nos. 30 to 40). The high attenuation properties for unwanted modes also depend markedly on the electrical properties of the jacket surrounding the helix. In this paper the normal modes of helix waveguide are determined using the sheath helix approximation, a mathematical model in which the helical winding is replaced by an anisotropic conducting sheath. A brief formulation of the boundary value problem leads to an equation which determines the propagation constants of modes in the helix guide. Since the equation is not easy to solve numerically, approximations are presented which show the effects of the pitch angle, the diameter, the conductivity and dielectric constant of the jacket, and the wavelength, when the conductivity of the jacket is sufficiently high. By proper choice of the pitch angle and, in some instances, of the polarization, a helix waveguide can be made to propagate any mode of ordinary round guide, with an attenuation constant which should be essentially the same as in solid copper pipe. The pitch is chosen so that the wall currents associated with the desired mode follow the direction of the conducting wires. The losses to the other modes are in general much higher, and are determined by both the pitch angle and the jacket material. Special attention is given in the present work to the limiting case of a helix of zero pitch, since the attenuation constant of the TEoi mode will be smallest when the pitch angle is as small as possible. To explore the ^ Reference 3, p. 1111. 1350 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 region where the approximate formulas for the propagation constants of the lossy modes break down, some numerical results have been ob tained for helices of zero pitch using an IBM 650 magnetic drum calcu lator. Tables and curves are given showing the propagation constants of various modes in such a waveguide, as functions of the electrical proper- ties of the jacket and for three different ratios of radius/ wavelength. In many cases it is found that the attenuation constant of a given mode passes through a maximum as the jacket conductivity is varied, the other parameters remaining fixed. The numerical calculations indicate that it is possible to get unwanted mode attenuations several hundred to several hundred thousand times greater than the TEoi attenuation for the size Avaveguide that looks most promising for low-loss communi- cation. US J," Fig. 1 — Schematic diagrams of the helical sheath and the helical sheath de- veloped, showing the unit vectors and the periodicity. HELIX WAVEGUIDE 1351 II. SHEATH HELIX BOUNDARY VALUE PROBLEM Ordinary cylindrical waveguide consists of a circular cylinder of radius a, infinite length, and zero (or very small) conductivity, imbedded in an infinite* homogeneous conducting medium. The sheath helix waveguide has the same configuration plus the additional property that at radius a dividing the tAvo media, there is an anisotropic conducting sheath which conducts perfectly in the helical direction and does not conduct in the perpendicular direction. The attenuation and phase constants are deter- mined by solving Maxwell's equations in cylindrical coordinates and matching the electric and magnetic fields at the wall of the guide. The helix of radius a and pitch angle \J/ = tan~^ s/2ira is shown in the upper part of Fig. 1. The developed helix as viewed from the inside when cut by a plane of constant 6 and unrolled is shown in the lower part of the illustration. A new set of unit vectors e^ and Cj. parallel and perpen- dicular respectively to the helix direction is introduced. These are re- lated to er , ee , and Cz by er X e\\ — ex e\\ = ez sin t^ + ee cos rp fij. = ez cos \p — e$ sin xj/ The boundary conditions at r = a are K = E{ = 0 Ej = E/ where the superscript i refers to the interior region, 0 ^ r ^ a, and the superscript e refers to the exterior region, a '^ r -^ co . An equivalent set of boundary conditions in terms of the original unit vectors is E; tan ^p + Ee' = 0 E," tan rp + Ee' = Q (1) e: = e: H; tan ,A -f He' = Hf tan ^p + /// We are looking for solutions which are similar to the modes of or- * The assumption of an infinite external medium is made to simplify the mathe- matics. The results will be the same as for a finite conducting jacket which is thick enough so that the fields at its outer surface are negligible. 1352 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 dinary waveguide, i.e., "fast" modes as contrasted with the well-known "slow" modes used in traveling- wave tubes. ^' ^ To solve the problem we follow the procedure set up by Stratton^ for the ordinary cylindrical waveguide boundary problem. The fields E and H are derived from an electric Hertz vector IT and a magnetic Hertz vector 11* by ^ = vxvxn- ^■w/xV X n* H= {a + iwe)'^ X ii + V X V xn* (2) where (3) (4) n = ezU, n* = Ln* and, assuming a time dependence exp (iwt), 00 ni V~^ i 7" / V N —ihz—inB z = 2^ anJn{tir)e n=— 00 00 ne V"* ejr (2)/ 5, \ —ihz—inB n=— 00 00 n*i V~^ 7 i T / <, \ —ihz—inB z = 2^ OnJn{hr)e n=— 00 Jl= — 00 In these expressions 5-2 2 7,2 f 1 = CO )Ltoeo — h 5-2 2 / / ■ ,/\ 7 2 e — ie" = e/eo — ia/weo where the interior region is assumed to have permittivity eo and perme- ability ^0 , while the exterior region has permittivity e, permeability no , and conductivity a. The superscripts i and e refer to the interior and ex- terior regions respectively, and the a's and 6's are amplitude coefficients. 6 J. R. Pierce, Proc. I.R.E., 35, pp. 111-123, 1947. * S. Sensiper, Electromagnetic Wave Propagation on Helical Conductors, Sc.D. thesis, M.I.T., 1951. In Appendix B of this reference, Sensiper shows that when the interior and exterior media are the same, only slow waves will exist except in special cases. Fast guided waves become possible if the conductivity of the exterior medium is sufficiently high. 'J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941, pp. 524-527. Note that Stratton uses the time dependence exp (—icot). HELIX WAVEGUIDE 1353 Attention is restricted to waves traveling in the positive ^-direction, which are represented by the factor exp { — ihz), where /i ( = /3 — ia) is the complex phase constant. However it is necessary to consider both right and left circularly polarized waves; this accounts for the use of both positive and negative values of n. Substitution of (2), (3), and (4) into the boundary conditions (1) leads to the following set of equations: V 2 , , hn fi tan \l/ — — a Jn(^ia)an + i(^iJ.(ihJn'{tia)hn = 0 ^2 tan yp — — a (5) •to;eofiJ«'(fia)a„* + . 2 . , hn fi tan yp — — a Jn(^ia)hn (2)', + (o- + icoe)^2Hn ' {^2a)an [ > 2 + , hn ti tan \l/ — — a Hr.''\ha)h: = 0 If the conductivity of the exterior region is infinite, it is possible to satisfy the boundary conditions with only one of the amplitude coeffi- cients different from zero; for example hn = a«' = 6„* = 0 a„ 0 or dn — CLn = bn = 0 bj 9^ 0 Jni^xO) = 0 Jn'iria) = 0 The first case corresponds to TM modes and the second to TE modes in a perfectly conducting circular guide. Linearlj^ polarized modes may be represented as combinations of terms in a,/ and a-n\ or bn and 6_„*. If the exterior region is not perfectly conducting, one can still find solutions having the fields confined to the interior region by propedy choosing the angle of the perfectly conducting helical sheath. For exam- ple, it is easy to verify that equations (5) are satisfied under the follow- ing conditions: ttn an = bn = 0 bj 9^ 0 tan yp = hn Jn'i^ia) = 0 1354 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 li n 9^ 0, these conditions correspond to circularly polarized TE„ waves, in which the wall currents follow the direction of the conducting sheath. If n = 0, then i^ = 0, and one has TEom modes with circum- ferential currents only. \ The equations can also be satisfied with bn = ttn = bn = 0 ttn 9^ 0 yl^ = 90° Jnitia) = 0 corresponding to the TM„m modes (either circularly or linearly polarized) of a perfectly conducting pipe, which are associated with longitudinal wall currents only. In the general case when the jacket is not perfectly conducting and the helix pitch angle is not restricted to special values, it is necessary to solve (5) simultaneously for the field amplitudes. The equations admit a nontrivial solution if and only if the determinant of the coefficients of the a's and b's vanishes. The transcendental equation which results from equating the determinant of the coefficients to zero is f: f 1 tan V — r— ) r it^ \ ~ ^ Moeo = t. (6) (2)// f 2 tan 1^ - -— I , — CO MoeoCe - te ) , - f2ay H^V'{^2a) Hr.^"\^2a) J The solution of this equation determines the propagation constant ih and therefore the attenuation and phase constants a and /3. When ih has been obtained, it is a straightforward matter to determine the a and b coefficients from equations (5) and the electric and magnetic fields from (2), (3), and (4). "It is well known^ that the only pure TE or TM modes that can exist in a circular waveguide with walls of finite conductivity are the circularly symmetric TEom and TMom modes. The other modes are all mixed modes Avhose fields are not transverse with respect to either the electric or the magnetic vector. In general the modes of helix waveguide are also mixed modes, and no entirely satisfactory scheme for labeling them has been proposed. In the present paper we shall call the modes TE„m or TM„m according to the limits which they approach as the jacket conductivity becomes infinite, even though they are no longer transverse and their 8 Reference 7, p. 526. HELIX WAVEGUIDE 1355 field patterns may be quite different when the jacket is lossy. This sys- tem is not completely unambiguous, because as will appear in Section IV the mode designations thus obtained are not always unique. However it is a satisfactory way to identify the modes so long as the jacket con- ductivity is high enough for the loss to be treated as a perturbation. Approximations derived on this basis are presented in the next section. III. APPROXIMATE EXPRESSIONS FOR PROPAGATION CONSTANTS If the jacket were perfectly conducting, the helix waveguide modes would be the same as in an ideal circular waveguide, with propagation constants given by where V = Xo/Xc = p\o/2Tra p = ??i*^ zero of Jnix) for TM„m mode, or rn^^ zero of Jn(x) for TE„m mode If the jacket conductivity is sufficiently large, approximate solutions of (6) may be found by replacing Hn'\^2a) and Hn'^'i^iO) with their asymptotic expressions, and expanding Jni^ia) or Jn'(^ia) in a Taylor series near a particular zero. This calculation is carried out in the ap- pendix. The propagation constant may be written in the form ih = a + i{^nm + A|S) where to first order the perturbation terms are TM„„ modes a + m = ,\ ^ \„ rXT-^r-, (7a) a(l — v-y^ 1 -\- tan^ \p TE„TO modes a 4- i\B - ^ + ''^ ^V" [tan ^ - n(l - vyVyvf . , . ^'^^~a(l -, 2)1/2 ^^^7^2 1 -f tan^ ^ ^'^^ and ? + ^> = (e' - ie'T'" e = e/eo , e = cr/coeo The approximations made in deriving (7) are discussed in the appen- 1356 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 dix. In practice, the range of validity of these expressions is usually limited by the criterion /1 2\l/2 ^^^"^^ \a + iA^\«l (8) V The numerical calculations described in Section IV indicate that the approximations are good so long as the left-hand side of (8) is less than about 0.1, and that they break down a little sooner for TE modes than for TM modes. Inspection of (7) reveals three cases of particular interest, namely ^ = 0°,\p = tan~^ w(l — v^f^/pp, and i/- = 90°. These cases, which were mentioned in Section II and are discussed again below, correspond to preferential propagation of certain modes, in which the wall currents follow the direction of the conducting helix. The preferred modes have zero attenuation in the present treatment because the helical sheath is assumed to be perfectly conducting. In practical helices wound from insulated copper wire the loss should be only slightly greater than in round copper pipe of the same diameter. The slight increase (of magni- tude 10 per cent to 30 per cent) is due to the slightly nonuniform cur- rent distribution in the wires, an effect that can be kept small by keeping the gaps between the wires of the helix small. In general the attenuation constants of modes whose wall currents do not follow the helix are orders of magnitude larger than the attenuation constants of the preferred modes. iA = 0° The circular electric (TEom) modes have attenuation constants sub- stantially the same as in solid copper pipe. The additional TEom loss if the pitch angle is not quite zero is proportional to tan^ \{/. This added loss can be made very small by using fine wire for winding the helix. The losses for the unwanted modes can be made large by a proper choice of jacket material. When ^ = 0, equations (7) yield TM„„j modes a(l — v^y^ TE„m modes a + iA/3 = i^ 'J- -^^ (^ + iv) (9b) a p^ — n^ HELIX WAVEGUIDE 1357 It may be of interest to compare the attenuation constants given by (9) with the results obtained by calculating the power dissipated in the walls of a pipe' which has different resistances in the circumferential and longitudinal directions. If the wall resistance for circumferential currents is represented by Re and for longitudinal currents by Rz , the expressions for a are TM nm modes Rz a TE„TO modes a = {(xo/eoY'-aa - I'V Rev' + Rz{n/v)\l - v') p' (Mo/€o)^/-a(l - i/'Y'^ p2 _ ^2 The results for ordinary metallic pipe are obtained by setting Re = Rz — R = (co^o/2cr) [f Re = 0, the expressions above agree with (9), inasmuch as I = R(eo/ixo) '" when the jacket conductivity is large. 4/ — tan~^ n(l — v')^''/vv, n ^ 0 For this value of rp the circularly polarized TE„^ mode which varies as exp(—in9) has low attenuation. (We assume 7i 9^ 0, since the case n = 0 has been treated above.) One of the properties of helix waveguide is the difference in propagation between right and left circularly polarized TE„m modes. By properly designing the helix angle for the frequency, mode, and size of guide, the loss to one of the polarizations can be made very low. If the jacket is lossy enough the attenuation of the other polarization should be quite high. Thus only one of the circularly polar- ized modes should be propagated through a long pipe. Such a helix has features analogous to the optical properties of levulose and dextrose solutions, which distinguish between left and right circularly polarized light. Let an be the attenuation constant of the mode which varies as exp{ — i7i9), and a_„ the attenuation constant of the mode which varies ^ S. A. Schelkunoff, Electromagnetic Waves, van Nostrand, New York, 1943, pp. 385-387. 1358 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 as exp (-{-ind). Then from (7b), for any pitch angle t/', ^ p' v' [tan ^ + 7i(l - vflvvf a -n = - i CXn = a -n — a„ = 4 a p2 _ ^2 (1 - v'yi' 1 + tan2 yf, ^ p V [tan \p — n{l — v^Y' /pvf ap^ - n'' {1 - v2)i/2 1 + tan- rp ^ np V tan ip ap"^ — 'n? 1 -\- tan^ yp The mode which varies as exp( — zn0) has lower loss if \p and n have the same sign. The TM„m attenuation constants are independent of polarization and are given by (7a). yp = 90° These "helices," with wires parallel to the axis of the waveguide, should propagate TM„m modes with losses approximately the same as in copper pipe. For the TE„„i modes, (7b) gives TEnm modes 2 2 a + iA(3 = " -T^—2 (^ + ^■'?) a(l — j'-)^'- p^ — v} IV. NUMERICAL SOLUTIONS FOR ZERO-PITCH HELICES The main interest in helix waveguide is for small pitch angles where the TEoi attenuation is very low. The propagation constants of various lossy modes in helix guides of zero pitch have been calculated by solving the characteristic equation (6) numerically. These calculations will now be described. Equation (6) is first simplified by setting yp — Q and replacing the Hankel functions with their asymptotic expressions. The condition for validity of the asymptotic expressions, namely I r2a I » I {^n - l)/8 I is well satisfied in all cases to be treated here. Equation (6) may then be rearranged in the dimensionless form Fni^a) = i^oaf [{nhafJn\Ua) - (/5ca)'(fia)V„''(ria)] - i{^,af [{nhaf -f (^oa)^(e' - Z6")(^a)V/(fia)/n(fia) (10) = 0 There is no difference between the propagation constants of right and HELIX WAVEGUIDE 1359 left circularly polarized waves when xp = 0. Using the relationships ha = KM' + (M' (e' - ie" - l)f\ Im^a < 0 ha = {%af - (rla)T'^ Im /la < 0 it is clear that Fni^a) is an even function of ^a, involving the parame- ters Pott (= 27ra/Xo), e', e", and n. When specific values have been assigned to /3oa, e', and e", roots of (10) can be found numerically by the straightforward procedure of evaluating Fni^a) at a regular network of points in the plane of the complex variable ^a, plotting the families of curves Re F„ = 0 and Im Fn = 0, and reading off the values of ^a corresponding to the inter- sections of curves of the two families. The procedure just outlined has been applied to the cases n = 0 and n = 1. When n = 0 one can take out of Fo(fia) the factor Jo'(fia), whose roots correspond to the TEom modes; the roots of the other factor are the TMom-limit modes. When n = 1 the function Fi(ha) does not factor, and its roots correspond to both TEi^-limit and TMi^-limit modes. If the jacket conductivity is high it is easy to identify the various limit modes, and a given mode can be traced continuously if the conductivity is decreased in sufficently small steps. The numerical calculations were set up, more or less arbitrarily, to cover the region 0 ^ Re ^a ^ 10, —10 ^ Im ^a ^ 10, for each set of parameter values. A few plots of Re Fn and Im Fn made it apparent that for propagating modes the roots in this region are all in the first quadrant and usually near the real axis. The entire process of solution was then programmed by Mrs. F. M. Laurent for automatic execution on an IBM 650 magnetic drum calculator. The calculator first evaluated Fni^ci) at a network of points spaced half a unit apart in both directions, then examined the sign changes of Re F„ and Im Fn around each ele- mentary square. If it appeared that a particular square might contain a root of Fn , the values of Fn at the four corner points were fitted by an interpolating cubic polynomial ° which was then solved. If the cubic had a root inside the given square, this was recorded as an approximate root of Fn . The normalized propagation constant iha = aa -{- i(3a was also recorded for each root. The calculated roots ^a and the normalized propagation constants are summarized in Tables 1(a) to 1(f), which relate to the following cases: Table 1(a)— /3oa = 29.554, e' = 4, e" variable Table 1(b) —/3oa = 29.554, e' = 100, e" variable Table 1(c) —/3oa = 29.554, e = e", both variable 10 A. N. Lowan and H. E. Salzer, Jour. Math, and Phys., 23, p. 157, 1944. 1360 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table 1(d) — M = 12.930, e' = 4, e" variable Table 1(e) — fi.a = 12.930, e' = e", both variable Table 1(f) — fi^a = 6.465, e' = 4, e" variable j The three values of ^oa correspond to waveguides of diameter 2 inches, I inch, and ye inch at Xo = 5.4 mm. The jacket materials (mostly carbon- loaded resins) which have been tested to date show a range of relative permittivities roughly from 4 to 100. There is some indication that the permittivity of a carbon-loaded resin increases as its conductivity in- creases; this suggested consideration of the case e = e . The tables cover the range from e" = 1000 down to e" = 1 at small enough intervals so that the general course of each mode can be followed. It is worth noting that at 5.4 mm a resistivity (l/o-) of 1 ohm cm cor- responds to e' =32. Copper at this frequency has an e of approxi- mately 2 X 10^ In general the tables include the modes derived from Fo(^ia) whose limits are TMoi , TM02 , and TM03 , and the modes derived from /^i(fia) whose limits are TEu , TMn , TE12 , TM12 , and TE13 (except that in the i^-inch guide TM03, TM12 , and TE13 are cut off). Some results are given for the TMis-limit mode, namely those which satisfy the arbitrary criterion Re fia ^ 10; but these results are incomplete because for large e" the corresponding root of Fi(^ia) approaches 10.173. Furthermore for small values of e" the attenuation constants of a few of the TM-limit modes become quite large and the corresponding values of ^la move far away from the origin. Since our object was to make a general survey rather than to investigate any particular mode exhaustively, we did not attempt to pursue these modes outside the region originally proposed for study. The results of the IBM calculations are recorded in Table I to three decimal places. Since the roots f la were obtained by cubic interpolation in a square of side 0.5, the last place is not entirely reliable; but spot checks on a few of the roots by successive approximations indicate that it is probably not off by more than one or two units. The propagation constants of some of the relatively low-loss modes (especially TE12 and TE13 , whose wall currents are largely circumferential) were calculated from the approximate formulas,* as noted in the tables. The attenuation (Text continued on page 1375) * The formulas used were (A9) and (AlO) of the appendix, which are slightly more accurate than (7) of the text. Table 1(a) — 2-INCH Guide at Xq = c ).4 MM (/3oa = 29.554) WITH e' = 4 AND t" Variable Limit Mode €" fia aa + t/3o TMoi 00 2.405 29.4561 1000 2.154 + 0.384i 0.028 + 29.4781 250 2.094 + 0.974i 0.069 + 29.4961 100 2.408 + 1.679i 0.137 + 29.5041 90 2.482 + 1.772i 0.149 + 29.5031 SO 2.579 + 1.878i 0.164 + 29.5021 64 2.804 + 2.083i 0.198 + 29.4951 40 3.519 + 2.547i 0.304 + 29.4561 25 4.604 + 3.165i 0.496 + 29.3691 16 5.870 + 3.763i 0.756 + 29.2191 10 7.564 + 4.131i 1.082 + 28.8871 8 8.464 + 4.158i 1.229 + 28.6461 TM02 00 5.520 29.0341 1000 5.399 + 0.127i 0.024 + 29.0571 250 5.274 + 0.268i 0.049 + 29.0811 100 5.109 + 0.445i 0.078 + 29.1131 90 5.081 + 0.472i 0.082 + 29.1181 80 5.047 + 0.504i 0.087 + 29.1251 64 4.968 + 0.569i 0.097 + 29.1391 40 4.716 + 0.701i 0.113 + 29.1841 25 4.375 + 0.677i 0.101 + 29.2371 16 4.172 + 0.551i 0.079 + 29.2641 10 4.047 + 0.448i 0.062 + 29.2791 8 4.004 + 0.412i 0.056 + 29.2851 4 3.905 + 0.344i 0.046 + 29.2971 1 3.820 + 0.310i 0.040 + 29.3081 TM03 00 8.654 28.2591 1000 8.577 + 0.078i 0.024 + 28.2821 250 8.500 + 0.1601 0.048 + 28.3061 100 8.408 + 0.260i 0.077 + 28.3341 90 8.395 + 0.275i 0.081 + 28.3381 80 8.378 + 0.293i 0.086 + 28.3431 64 8.344 + 0.330i 0.097 + 28.3541 40 8.253 + 0.424i 0.123 + 28.3821 25 8.125 -t- 0.545i 0.156 + 28.4211 16 7.943 + 0.678i 0.189 + 28.4751 10 7.658 + 0.779i 0.209 + 28.5561 8 7.511 + 0.780i 0.205 + 28.5951 4 7.200 + 0.693i 0.174 + 28.6731 1 6.986 + 0.612i 0.149 + 28.7241 TEii 00 1.841 29.4971 1000 1.703 + 0.234i 0.014 + 29.5061 250 1.764 + 0.630i 0.038 + 29.5081 100 2.465 + 0.963i 0.081 + 29.4671 90 2.660 + 0.748i 0.068 + 29.4441 80 2.633 + 0.604i 0.054 + 29.4431 64 2.594 + 0.464i 0.041 + 29.4441 40 2.546 + 0.312i 0.027 + 29.4461 25 2.508 + 0.226i 0.019 + 29.4481 16 2.481 + 0.176i 0.015 + 29.4501 10 2.455 + 0.140i 0.012 + 29.4521 8 2.445 + 0.129i 0.011 + 29.4531 4 2.418 + 0.106i 0.009 + 29.4551 1 2.394 + 0.095i 0.008 + 29.4571 1361 Table 1(a) — Continued Limit Mode t" rw aa + i/3o TMn 00 3.832 29.305i 1000 3.652 + 0.197i 0.024 + 29.328i 250 3.457 + 0.440i 0.052 + 29.355i 100 2.978 + 0.880i 0.089 + 29.417i 90 2.821 + 1.215i 0.116 + 29.445i 80 2.945 + 1.476i 0.148 + 29.444i 64 3.146 + 1.868i 0.200 + 29.446i 40 3.728 + 2.564i 0.325 + 29.432i 25 4.659 + 3.175i 0.504 + 29.361i 16 5.921 + 3.727i 0.756 + 29.204i 10 7.613 + 4.135i 1.090 + 28.875i 8 8.487 + 4.153i 1.231 + 28.639i TE.2 00 5.331 29.069i 1000 0.0008 + 29.070i* 250 0.0016 + 29.071i* 100 0.0026 + 29.072i* 64 0.0033 + 29.072i* 40 0.0042 + 29.073i* 25 0.0055 + 29.074i* 10 0.0092 + 29.075i* 4 5.297 f 0.072i 0.013 + 29.076i 1 5.322 + 0.096i 0.018 + 29.071i TMi2 00 7.016 28.710i 1000 6.918 + 0.099i 0.024 + 28.733i 250 6.821 + 0.203i 0.048 + 28.757i 100 6.701 + 0.330i 0.077 + 28.786i 90 6.683 + 0.349i 0.081 + 28.791i 80 6.660 + 0.372i 0.0S6 + 28.796i 64 6.612 + 0.419i 0.096 + 28.808i 40 6.475 + 0.535i 0.120 +28.8411 25 6.253 + 0.655i 0.142 +28.8931 16 5.965 + 0.682i 0.141 +28.9541 10 5.719 + 0.590i 0.116 +29.0021 8 5.641 + 0.541i 0.105 +29.0161 4 5.471 + 0.419i 0.079 + 29.0471 1 5.317 + 0.347i 0.063 + 29.0741 TEi3 00 8.536 28.2951 1000 0.0003 + 28.2951* 250 0.0006 + 28.2951* 100 0.0010 + 28.2961* 64 0.0012 + 28.2961* 40 0.0016 + 28.2961* 25 0.0020 + 28.2961* 10 0.0034 + 28.2971* 4 0.0050 + 28.2961* 1 0.0058 + 28.2951* TM„ 00 10.173 27.7481 100 9.963 4- 0.219i 0.078 + 27.8251 90 9.952 + 0.231i 0.083 + 27.8291 80 9.938 + 0.246i 0.088 + 27.8341 64 9.911 + 0.277i 0.098 + 27.8451 40 9.840 + 0.356i 0.126 + 27.8701 25 9.746 + 0.460i 0.101 +27.9051 16 9.625 + 0.591i 0.204 + 27.9501 10 9.433 + 0.757i 0.255 + 28.0201 8 9.305 + 0.837i 0.278 + 28.0651 4 8.836 + 0.898i 0.281 + 28.2181 1 8.485 + 0.781i 0.234 + 28.3221 Approximate formula. 1362 HELIX WAVEGUIDE 1363 Table 1(b) — 2-inch Guide at Xo = 5.4 mm (/Soa = 29.554) WITH e' = 100 AND e" Variable Limit Mode t" fia aa + i^a TMe. 00 2.405 29.456i 1000 2.178 + 0.391i 0.029 + 29.4761 250 2.291 + 0.885i 0.069 + 29.4791 100 2.677 + 1.062i 0.097 + 29.4521 80 2.764 + 1.047i 0.098 + 29.4431 64 2.834 + 1.0191 0.098 + 29.4361 40 2.928 + 0.950i 0.094 + 29.4241 25 2.973 + 0.893i 0.090 + 29.4181 10 3.004 + 0.831i 0.085 + 29.4131 4 3.013 + 0.806i 0.083 + 29.4111 1 3.016 + 0.793i 0.081 + 29.4111 TMo2 00 5.520 29.0341 1000 5.406 + 0.133i 0.025 + 29.0561 250 5.339 + 0.298i 0.055 + 29.0691 100 5.372 + 0.473i 0.087 + 29.0661 SO 5.398 + 0.508i 0.094 + 29.0621 64 5.429 + 0.535i 0.100 + 29.0561 40 5.492 + 0.566i 0.107 + 29.0451 25 5.540 + 0.573i 0.109 + 29.0361 10 5.589 + 0.569i 0.109 + 29.0271 4 5.608 + 0.563i 0.109 +29.0231 1 5.617 + 0.560i 0.108 + 29.0211 TMo3 00 8.654 28.2591 1000 8.581 + 0.082i 0.025 + 28.2811 250 8.537 + 0.179i 0.054 + 28.2951 100 8.548 + 0.279i 0.084 + 28.2921 80 8.561 + 0.300i 0.091 + 28.2891 64 8.575 + 0.317i 0.096 + 28.2851 40 8.606 + 0.339i 0.103 +28.2761 25 8.630 + 0.348i 0.106 + 28.2681 10 8.658 + 0.352i 0.108 + 28.2601 4 8.669 + 0.352i 0.108 + 28.2571 1 8.675 + 0.351i 0.108 + 28.2551 TEn 00 1.841 29.4971 1000 1.719 + 0.236i 0.014 + 29.5051 250 1.871 + 0.504i 0.032 + 29.4991 100 2.132 + 0.484i 0.035 + 29.4811 80 2.161 + 0.451i 0.033 + 29.4791 64 2.178 + 0.420i 0.031 + 29.4771 40 2.191 + 0.372i 0.028 + 29.4751 25 2.192 + 0.343i 0.026 + 29.4751 10 2.190 + 0.316i 0.023 + 29.4751 4 2.188 + 0.306i 0.023 + 29.4751 1 2.187 + 0.301i 0.022 + 29.4751 1364 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table 1(b) — Continued Limit Mode «" fia aa + i^a TMa 00 3.832 29.305i 1000 3.663 + 0.204 I 0.026 + 29.3271 250 3.579 + 0.485 I 0.059 + 29.341i 100 3.715 + 0.788 I 0.100 + 29.331i 80 3.787 + 0.826 I 0.107 + 29.322i 64 3.856 + 0.843 1 0.111 + 29.3141 40 3.969 + 0.836 I 0.113 + 29.299i 25 4.043 + 0.817 I 0.113 + 29.288i 10 4.100 + 0.777 I 0.109 + 29.279i 4 4.119 + 0.759 I 0.107 +29.2761 1 4.128 + 0.749 I 0.106 + 29.2741 TE12 00 5.331 29.0691 1000 0.0008 + 29.070i* 250 0.0018 + 29.0711* 100 0.0028 + 29.0711* 64 0.0032 + 29.0701* 40 0.0034 + 29.0701* 25 0.0035 + 29.0701* 10 0.0036 + 29.0701* 4 0.0036 + 29.0701* 1 0.0036 + 29.0691* TM12 00 7.016 28.7101 1000 6.923 + 0.103 I 0.025 + 28.7321 1 250 6.868 + 0.226 I 0.054 + 28.746i \ 100 6.885 + 0.355 I 0.085 + 28.7431 1 80 6.902 + 0.381 I 0.092 + 28.7401 { 64 6.922 + 0.403 1 0.097 + 28.7351 ^ 40 6.965 + 0.429 I 0.104 + 28.7251 25 7.000 + 0.440 I 0.107 +28.7171 ; 10 7.037 + 0.443 I 0.109 + 28.7081 4 7.051 + 0.441 L 0.108 + 28.7041 1 7.058 + 0.440 I 0.108 + 28.7031 TEx3 00 8.536 28.2951 1000 0.0003 + 28.2951* ! 250 0.0007 + 28.2951* 100 0.0010 + 28.2951* 64 0.0012 + 28.2951* 40 0.0013 + 28.2951* 25 0.0013 + 28.2951* 10 0.0013 + 28.2951* 4 0.0013 + 28.2951* 1 0.0013 + 28.2951* Approximate formula. Table 1(c) — 2-INCH Guide at Xq = 5 WITH e = e .4 MM (I3,a = 29.554) Limit Mode e' and e" fia aa + i0a TMoi CO 1000 250 100 64 40 32 25 16 12 10 4 2 1 2.405 2.338 + 0.341i 2.418 + 0.707i 2.677 + 1.062i 2.925 + 1.226i 3.309 + 1.324i 3.540 + 1.299i 3.787 + 1.162i 3.946 + 0.800i 3.950 + 0.647i 3.946 + 0.573i 3.905 + 0.344i 3.869 + 0.252i 3.820 + 0.185i 29.4561 0.027 + 29.4641 0.058 + 29.4641 0.097 + 29.4521 0.122 + 29.4351 0.149 + 29.3991 0.156 + 29.3711 0.150 + 29.3341 0.108 + 29.3011 0.087 + 29.2961 0.077 + 29.2951 0.046 + 29.2971 0.033 + 29.3011 0.024 + 29.3071 TM02 00 1000 250 100 64 40 32 25 16 12 10 5.520 5.469 + 0.136i 5.423 + 0.282i 5.372 + 0.473i 5.337 + 0.624i 5.294 + 0.874i 5.279 + 1.0611 5.319 + 1.367i 5.852 + 1.969i 6.472 + 2.1781 7.026 + 2.1981 29.0341 0.026 + 29.0441 0.053 + 29.0541 0.087 + 29.0661 0.115 + 29.0751 0.159 +29.0901 0.193 + 29.0991 0.250 + 29.1051 0.397 + 29.0391 0.4S7 + 28.9231 0.536 + 28.7961 TM03 00 1000 250 100 64 40 32 25 16 12 10 4 2 1 8.654 8.620 + 0.0851 8.587 + 0.1731 8.548 + 0.2791 8.521 + 0.3551 8.483 + 0.4611 8.458 + 0.5261 8.425 + 0.6111 8.330 + 0.8241 8.206 + 1.0371 8.034 + 1.2401 7.200 + 0.6931 7.098 + 0.4831 6.998 + 0.3491 28.2591 0.026 + 28.2691 0.052 + 28.2801 0.084 + 28.2921 0.107 + 28.3021 0.138 + 28.3151 0.157 +28.3231 0.182 + 28.3351 0.242 + 28.3691 0.300 + 28.4131 0.350 + 28.4711 0.174 + 28.6731 0.120 + 28.6941 0.085 + 28.7161 TEu 1000 250 100 64 40 32 25 16 12 10 4 2 1 1.841 1.810 + 0.1901 1.911 + 0.3841 2.132 + 0.4841 2.270 + 0.4531 2.365 + 0.3661 2.389 + 0.3241 2.406 + 0.2811 2.420 + 0.2191 2.424 + 0.1871 2.424 + 0.1691 2.418 + 0.1061 2.409 + 0.0781 2.394 + 0.0561 29.4971 0.012 + 29.4991 0.025 + 29.4951 0.035 + 29.4811 0.035 + 29.4701 0.029 + 29.4621 0.026 + 29.4591 0.023 + 29.4571 0.018 + 29.4561 0.015 + 29.4551 0.014 + 29.4551 0.009 + 29.4551 0.006 + 29.4561 0.005 + 29.4571 1365 Table 1(c) — Continued 'V Limit Mode «' and e" ria aa + i/3a I TMu oo 3.832 29.3051 1000 3.759 + 0.203i 0.026 + 29.3151 250 3.714 + 0.439i 0.056 + 29.3231 100 3.715 + 0.788i 0.100 + 29.3311 64 3.797 + 1.070i 0.139 + 29.3291 40 4.080 + 1.400i 0.195 + 29.3051 32 4.276 + 1.550i 0.226 + 29.2851 25 4.586 + 1.661i 0.260 + 29.2451 16 5.359 + 1.579i 0.291 +29.1091 1 12 5.587 + 1.043i 0.201 + 29.0411 10 5.560 + 0.859i 0.164 + 29.0401 4 5.471 + 0.419i 0.079 + 29.0471 2 5.438 + 0.249i 0.047 + 29.0511 1 5.444 + 0.131i 0.025 + 29.0491 TEi2 1000 250 100 64 40 25 10 5.331 29.0691 0.0009 + 29.0701* 0.0018 + 29.0701* 0.0028 + 29.0711* 0.0035 + 29.0711* 0.0044 + 29.0711* 0.0055 + 29.0721* 0.0087 + 29.0731* 4 5.297 + 0.072i 0.013 + 29.0761 2 5.272 + 0.108i 0.020 + 29.0801 1 5.198 + 0.132i 0.023 + 29.0941 TMio oo 7.016 28.7101 1000 6.971 + 0.107i 0.026 + 28.7211 250 6.931 + 0.217i 0.052 + 28.7311 100 6.885 + 0.355i 0.085 + 28.7431 64 6.852 + 0.457i 0.109 + 28.7531 40 6.801 + 0.6101 0.144 + 28.7681 32 6.768 + 0.708i 0.167 + 28.7781 25 6.720 + 0.850i 0.198 + 28.7931 16 6.562 + 1.359i 0.309 + 28.8501 12 6.869 + 2.095i 0.499 + 28.8251 10 7.322 + 2. 3741 0.605 + 28.7371 TE,3 00 1000 250 100 64 40 25 10 4 1 8.536 28.2951 0.0003 + 28.2951* 0.0007 + 28.2951* 0.0010 + 28.2951* 0.0013 + 28.2951* 0.0016 + 28.2951* 0.0021 + 28.2951* 0.0032 + 28.2961* 0.0050 + 28.2961* 0.0094 + 28.2951* TMu 00 10.173 27.7481 25 9.981 + 0.4971 0.178 + 27.8231 16 9.910 + 0.6521 0.232 + 27.8521 12 9.841 + 0.7851 0.277 + 27.8801 10 9.776 + 0.8931 0.313 + 27.9071 4 8.836 + 0.8981 0.281 + 28.2181 2 8.656 + 0.5961 0.183 + 28.2651 1 8.523 + 0.4091 0.123 + 28.3021 Approximate formula. 1366 HELIX WAVEGUIDE 1367 Table 1(d)— |-inch Guide at Xq = 5.4 mm {^oa = 12.930) WITH e' = 4 AND e" VARIABLE Limit Mode e" no aa + ij3o TMoi 1000 250 100 64 40 25 10 6.4 4.0 2.5 1.0 2.405 2.286 + 0.1401 2.183 + 0.3241 2.113 + 0.5951 2.114 + 0.8001 2.185 + 1.0721 2.377 + 1.3691 3.212 + 1.6991 3.694 + 1.4401 3.765 + 1.0291 3.700 + 0.8531 3.624 + 0.7331 12.7041 0.025 + 12.7271 0.056 + 12.7491 0.098 + 12.7711 0.132 + 12.7821 0.183 + 12.7901 0.255 + 12.7861 0.431 + 12.6471 0.426 + 12.4821 0.312 + 12.4161 0.254 + 12.4211 0.214 + 12.4351 TMo2 1000 250 100 64 40 25 10 6.4 4.0 2.5 1.0 5.520 5.468 + 0.0541 5.416 + 0.1111 5.356 + 0.1831 5.317 + 0.2351 5.266 + 0.3081 5.206 + 0.4101 5.073 + 0.7721 5.095 + 1.137i 5.486 + 1.4291 5.818 + 1.3791 6.041 + 1.1881 11.6921 0.025 + 11.7171 0.051 + 11.7421 (V083 + 11.7701 0.106 + 11.7891 0.137 + 11.8141 0.180 + 11.8441 0.328 + 11.9231 0.485 + 11.9481 0.664 + 11.8141 0.689 + 11.6501 0.624 + 11.5111 TMo3 CO 1000 250 100 64 40 25 10 6.4 4.0 2.5 1.0 8.654 8.620 + 0.0341 8.587 + 0.0691 8.550 + 0.1111 8.525 + 0.1411 8.494 + 0.1831 8.459 + 0.2391 8.393 + 0.4111 8.386 + 0.5321 8.426 + 0.6681 8.515 + 0.7691 8.676 + 0.8241 9.6071 0.030 + 9.6371 0.061 + 9.6671 0.098 + 9.7011 0.124 + 9.7231 0.160 + 9.7521 0.207 + 9.7851 0.350 + 9.8511 0.452 + 9.8661 0.571 + 9.8471 0.669 + 9.7841 0.741 + 9.6511 TEu 00 1000 250 100 64 40 25 10 6.4 4.0 2.5 1.0 1.841 1.767 + 0.0741 1.717 + 0.1911 1.706 + 0.3681 1.734 + 0.5001 1.857 + 0.6561 2.126 + 0.7731 2.436 + 0.4111 2.413 + 0.3161 2.386 + 0.2621 2.364 + 0.2341 2.341 + 0.2121 12.7981 0.010 + 12.8091 0.026 + 12.8171 0.049 + 12.8221 0.068 + 12.8231 0.095 + 12.8131 0.129 + 12.7781 0.079 + 12.7061 0.060 + 12.7071 0.049 + 12.7111 0.043 + 12.7141 0.039 + 12.7181 1368 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table 1(d) — Continued Limit Mode f" ria aa + ij3a TMii 00 1000 250 100 64 40 25 10 6.4 4.0 2.5 1.0 3.832 3.750 + 0.081i 3.676 + 0.171i 3.588 + 0.290i 3.530 + 0.382i 3.447 + 0.516i 3.329 + 0.757i 3.749 + 1.664i 4.275 + 1.750i 4.701 + 1.553i 4.843 + 1.274i 4.844 + 1.031i 12.349i 0.025 + 12.375i 0.051 + 12.398i 0.084 + 12.426i 0.108 + 12.445i 0.143 + 12.474i 0.201 + 12.519i 0.499 + 12.496i 0.606 + 12.343i 0.600 + 12.160i 0.511 + 12.067i 0.415 + 12.040i TE12 1000 250 100 64 40 25 10 4 1 5.331 11.780i 0.0007 + 11.780i* 0.0015 + 11.781i* 0.0024 + 11.782i* 0.0030 + 11.782i* 0.0039 + 11.783i* 0.0051 + 11.784i* 0.0085 + 11.785i* 0.0125 + 11.784i* 0.0146 + 11.781i* TMu 00 1000 250 100 64 40 25 10 6.4 4.0 2.5 1.0 7.016 6.972 + 0.043i 6.930 + 0.087i 6.883 + 0.141i 6.853 + 0.179i 6.814 + 0.233i 6.769 + 0.305i 6.679 + 0.541i 6.670 + 0.718i 6.755 + 0.935i 6.942 + l.Oeii 7.193 + 1.054i 10.861i 0.027 + 10.889i 0.055 + 10.917i 0.088 + 10.947i 0.112 + 10.967i 0.144 + 10.992i 0.187 + 11.023i 0.326 + 11.090i 0.431 + 11.109i 0.570 + 11.080i 0.671 + 10.981i 0.700 + 10.819i TEi, CO 1000 250 100 64 40 25 10 4 1 8.536 9.712i 0.0002+ 9.712i* 0.0005 + 9.712i* 0.0008 + 9.712i* 0.0010 + 9.713i* 0.0012 + 9.713i* 0.0016 + 9.713i* 0.0027 + 9.713i* 0.0040 + 9.713i* 0.0048 + 9.712i* TM,3 CO 10 6.4 4.0 10.173 9.949 + 0.340i 9.943 + 0.436i 9.970 + 0.543i 7.980i 0.409 + 8.276i 0.523 + 8.293i 0.655 + 8.277i Approximate formula. HELIX WAVEGUIDE 1369 Table 1(e) — |-inch Guide at Xq = 5.4 mm (jSoa = 12.930) with e = € Limit Mode e' and e" fia aa + t/3a TMoi 00 2.405 12.704i 1000 2.360 + 0.141i 0.026 + 12.714i 250 2.339 + 0.295i 0.054 + 12.720i 100 2.351 + 0.482i 0.089 + 12.724i 64 2.382 + 0.608i 0.114 + 12.724i 40 2.450 + 0.766i 0.148 + 12.720i 25 2.573 + 0.942i 0.191 + 12.708i 10 3.052 + 1.244i 0.301 + 12.630i 4 3.765 + 1.029i 0.312 + 12.416i 2 3.841 + 0.653i 0.203 + 12.366i 1 3.768 + 0.438i 0.133 + 12.378i TMo2 00 5.520 11.692i 1000 5.497 + 0.058i 0.027 + 11.704i 250 5.475 + 0.118i 0.055 + 11.715i 100 5.451 + 0.190i 0.088 + 11.727i 64 5.435 + 0.241i 0.111 + 11.735i 40 5.416 + 0.310i 0.143 + 11.746i 25 5.393 + 0.402i 0.184 + 11.760i 10 5.338 + 0.701i 0.317 + 11.802i 4 5.486 + 1.429i 0.664 + 11.814i 2 6.389 + 1.780i 0.996 + 11.425i 1 6.901 + 1.040i 0.652 + 11.003i TMo3 00 8.654 9.607i 1000 8.639 + 0.0.37i 0.033 + 9.621i 250 8.624 + 0.074i 0.067 + 9.635i 100 8.607 + 0.118i 0.105 + 9.650i 64 8.596 + 0.148i 0.132 + 9.661i 40 8.581 + 0.189i 0.168 + 9.675i 25 8.563 + 0.241i 0.213 + 9.694i 10 8.512 + 0.393i 0.344 + 9.747i 4 8.426 + 0.668i 0.571 + 9.847i 2 8.320 + 1.094i 0.910 + 9.999i 1 8.812 + 1.915i 1.721 + 9.806i TEii oo 1.841 12.798i 1000 1.810 + 0.072i 0.010 + 12.803i 250 1.807 + 0.161i 0.023 + 12.804i 100 1.833 + 0.265i 0.038 + 12.802i 64 1.870 + 0.330i 0.048 + 12.799i 40 1.939 + 0.401i 0.061 + 12.790i 25 2.047 + 0.459i 0.074 + 12.776i 10 2.295 + 0.414i 0.075 + 12.732i 4 2.386 + 0.262i 0.049 + 12.711i 2 2.389 + 0.186i 0.035 + 12.709i 1 2.369 + 0.129i 0.024 + 12.712i 1370 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table 1(e) — Continued Limit Mode e' and «" fio aa + ij3a TMu 00 3.832 12.349i 1000 3.794 + 0.086i 0.026 + 12.361i 250 3.766 + 0.176i 0.054 + 12.371i 100 3.739 + 0.288i 0.087 + 12.381i 64 3.725 + 0.369i 0.111 + 12.388i 40 3.711 + 0.485i 0.145 + 12.396i 25 3.708 + 0.651i 0.195 + 12.406i 10 3.893 + 1.161i 0.365 + 12.390i 4 4.701 + 1.553i 0.600 + 12.160i 2 5.319 + 1.062i 0.477 + 11.843i 1 5.241 + 0.614i 0.272 + 11.840i TE12 00 5.331 11.780i 1000 0.0008 + 11.780i* 250 0.0016 + 11.780i* 100 0.0026 + 11.781i* 64 0.0032 + 11.781i* 40 0.0041 + 11.781i* 25 0.0051 + 11.782i* 10 0.0081 + 11.783i* 4 0.0125 + 11.784i* 1 0.0236 + 11.782i* TM12 00 7.016 10.861i 1000 6.996 + 0.047i 0.030 + 10.874i 250 6.976 + 0.094i 0.060 + 10.887i 100 6.955 + 0.149i 0.095 + 10.902i 64 6.942 + 0.187i 0.119 + lO.Olli 40 6.924 + 0.238i 0.151 + 10.923i 25 6.903 + 0.305i 0.192 + 10.939i 10 6.841 + 0.509i 0.317 + 10.988i 4 6.755 + 0.935i 0.570 + ll.OSOi 2 7.053 + 1.730i 1.106 + 11.030i 1 8.138 + 1.672i 1.325 + 10.272i TE,3 00 8.536 9.712i 1000 0.0003 + 9.712i* 250 0.0005 + 9. 7121* 100 0.0008 + 9.712i* 64 0.0010 + 9.712i* 40 0.0013 + 9.712i* 25 0.0016 + 9.712i* 10 0.0025 + 9.713i* 4 0.0040 + 9.713i* 1 0.0076 + 9. 7131* TM13 00 10.173 7. 9801 4 9.970 + 0.543i 0.655 + 8. 2771 2 9.863 + 0.826i 0.963 + 8. 4571 1 9.698 + 1.418i 1.561 + 8. 8081 Approximate formula. HELIX WAVEGUIDE 1371 ^1 Table 1(f)— t^ -INCH Guide at Xq = 5.4 mm %a = 6.465) with e' = 4 AND e" Variable Limit Mode t" fia aa + i/3o Moi 1000 250 100 64 40 25 10 4 1 2.405 2.287 + 0.141i 2.228 + 0.244i 2.197 + 0.324i 2.170 + 0.439i 2.169 + 0.594i 2.355 + 0.943i 2.740 + 1.040i 2.961 + 0.878i e.ooii 0.024 + 6.0251* 0.053 + 6.0491 0.090 + 6.0741 0.117 + 6.0901 0.156 + 6.1071 0.210 + 6.1231 0.364 + 6.1051 0.478 + 5.9661 0.446 + 5.8301 rMo2 00 1000 250 100 64 40 25 10 4 1 5.520 5.468 + 0.054i 5.439 + 0.088i 5.420 + 0.112i 5.396 + 0.146i 5.370 + 0.191i 5.327 + 0.328i 5.369 + 0.512i 5.539 + 0.614i 3.3651 0.043 + 3.4081* 0.086 + 3. 4501 0.137 + 3.4991 0.172 + 3.5301 0.221 + 3.5701 0.284 + 3.6161 0.471 + 3.7071 0.740 + 3.7121 0.965 + 3.5241 TEn 00 1000 250 100 64 40 25 10 4 1 1.841 1.772 + 0.069i 1.744 + 0.129i 1.731 + 0.176i 1.726 + 0.244i 1.744 + 0.334i 1.925 + 0.493i 2.121 + 0.425i 2.152 + 0.319i 6.1971 0.009 + 6.2061* 0.020 + 6.2181 0.036 + 6.2271 0.049 + 6.2311 0.068 + 6.2351 0.093 + 6.2351 0.153 + 6.1931 0.147 + 6.1231 0.112 + 6.1061 TMii 00 1000 250 100 64 40 25 10 4 1 3.832 3.751 + 0.082i 3.710 + 0.134i 3.683 + 0.173i 3.650 + 0.227i 3.615 + 0.303i 3.581 + 0.546i 3.763 + 0.810i 4.038 + 0.816i 5.2071 0.028 + 5.2351* 0.058 + 5.2661 0.094 + 5.2971 0.119 + 5.3171 0.155 + 5.3431 0.204 + 5.3721 0.360 + 5.4221 0.570 + 5.3501 0.639 + 5.1541 TE12 00 1000 250 100 64 40 25 10 4 1 5.331 3.6571 0.0005 + 3.6571* 0.0009 + 3.6581* 0.0015 + 3.6581* 0.0019 + 3.6591* 0.0024 + 3.6591* 0.0031 + 3.6591* 0.0052 + 3.6601* 0.0079 + 3. 6601* 0.0097 + 3.6581* Approximate formula. 29.6 oca 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 29.4 29.2 29.0 /3a 28.8 26,6 28.4 2 8.2 28.0 1000 TEii TMqi 1000 K") 1000 TE 12 1000 TE 13 1000 "^ TM„ TMc TM,2 TM, 03 W~^, (b) /3oa =29.554 f'=100 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 aa Fig. 2(a) and (b) Fig. 2 — Plots of phase constant versus attenuation constant for modes in various helix waveguides. Representative values of «" are shown on the curves. , 1372 HELIX WAVEGUIDE 1373 0.04 0.08 O.I aa 0.20 0.24 0.28 0.32 0.36 0.40 11.4 yOa 11.0 10.6 10.2 9.8 9.4 TM 12 TMr 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 aa Fig. 2(c) and (d) 1374 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 13. 12.6 12.2 11.8 J 11 .4 caa 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 /3a 11.0 10.6 10.2 9.8 9.4 6.4 6.0 5.6 5.2 /3a 4.8 4.4 4.0 3.6 3.2 .TE,2 01 TM -TM11 TM02 0.1 0.2 0.3 0.4 0.5 aa (f) /3oa =6.465 f'=4 0.6 0.7 0.8 0.9 1.0 Fig. 2(e) and (f) HELIX WAVEGUIDE 1375 constants calculated from the approximate formulas are given to four decimal places, i.e., usually two significant figures. The contents of Table I are displayed graphically in Figs. 2(a) through (f), which show plots of (3a vs aa for all modes except TM13 . Repre- sentative values of e are indicated on the curves. Note that the scales are different for the different guide sizes, and that the jSa-scale is com- pressed in all cases. If aa and jSa were plotted on the same scale, the curves would make an initial angle of 45° with the aa-axis when e = constant, or 22.5° when e — e". Figs. 3(a) to (f) show the normalized attenuation constants aa of various modes plotted against e" on a log-log scale. In Fig. 3(b) the curves for all TM modes would be similar to the two shown, and in Fig. 3(d) the TM03 curve is like TM12 . Although for some modes the attenuation constant increases steadily as the conductivity decreases over the range of our calculations, in many cases the attenuation passes through a maximum and then decreases as the conductivity is further decreased. This phenomenon will be discussed in Section V. It may be noticed that in some instances the limit modes are not unique. For example, Tables 1(a), with e' = 4, and 1(c), with e' = e", for the large guide have in common the case e' = 4, e" = 4. For this case consider the circular magnetic mode corresponding to fia = 3.905 -1- 0.344t. If e' is constant (=4) while e" tends to infinity, this mode approaches the TM02 mode in a perfectly conducting guide; but if e' and e tend to infinity while remaining equal to each other, the same mode approaches TMoi in a perfectly conducting guide. Presumably the TMoi-limit mode in the former case coincides with the TMo2-limit mode in the latter case ; but the value of f la for this mode is outside the range of our calculations at e' = e = 4. A similar interchange occurs between the TMii-limit and TMi2-limit modes in the large guide, depending on whether e' is constant or e' tends to infinity with e". There is no evidence of any such phenomenon in the smaller guide of Tables 1(d) and 1(e); but the fact that it can occur means that the limit-mode designations of modes in a lossy waveguide are not entirely unambiguous. The phen- omenon is not due to the presence of the helix, since a helix of zero pitch has no effect on circular magnetic modes. Finally it is of interest to compare the propagation constants given by the approximate formula with those obtained by numerical solution of the characteristic equation. A reasonably typical case is provided by the TMo2-limit mode in a 2-inch guide at Xo = 5.4 mm with e = 4, as in Table 1(a). Exact and approximate results for ^a vs aa and aa vs e" are plotted in Fig. 4. As the conductivity decreases, the attenuation con- 2.0 I .0 0.5 0.2 0.10 0.05 aa 0.02 0.010 0.005 0:002 o.ooto 0.0005 0.0002 1.0 0.5 0.2 0. I 0 0.05 (a) -/3oa = 29.564 f'=4 /" J / / TMoy ^"^^^ TMo3 ^ ■V \ ■^ K.^J^^ ^ N^ TMo2 ^ \ ■ T^ ^ ^ir^ y^ /- ^ ^,3 ^ ^ ^ ^ ^ ^ ^ y^ aa 0.02 0.010 0.005 0.002 0.0010 0.0005 0.0002 2 1.0 0.5 0.2 0.10 0.05 (C) r^ V3oa = 29.55 4 TMo2 ^ k vTMoj ^ ^ ^ s k ^ 9^ V \J\rMi, ^ ^ X^ ^ 1 _^_^ /" " \ ^ ^ ^ ' V < ^ 'TE,2 ^ ^ ,^ ^ TE,3 ^ ^ ^ «a 0.02 0.010 0.005 0.002 0-0010 0.0005 0.0002 /\ (e) f'=f" ^J^^ .yi ^ \ ^ X ^^ ^ r^ J ^ ^ ^ ^^ ■^ > y ^ -> ^.^■^^ E,2 y^ ^ ^ TE„ / TE,2 ^ / TE,3 y ^ (d) TMo2^ TM|2 /3oa =12.930 f'= 4 .^'^ ^ K. TK rMoi_ ^ ^^ ^ y N ^ y TE^" ~ /^ ^ -y* TE,2 ^ ^ X ^^ ^ ,^ TE,3 ^ ,^ ^ y ^ ^ y^ y ^ (f) -/3oa = 6.465 f'= 4 TMo2 .<< TMll y^ ^ 'y- TMoi ,^ y t>^ —- j:eh_ . '^^ \y y ^ ^ > y y ^y y ^ ^ -TEiT -X' y y y^ 1000 200 100 50 20 10 5 2 1 1000 200 100 50 20 10 5 2 1 e" e" Fig. 3 — Attenuation constant as a function of jacket conductivity for modes in various helix waveguides. HELIX WAVEGUIDE 1377 29.32 29.28 29.24 29.20 /3a 29. 16 29.1 2 29.08 29.04 29.00 It k N \ APPROXIMATE \ ■ ) .' ,-'-' ^^^ . — - O — 10 ^-N. ,oJ / y \ N \ / Y 0 \ \ \ \ \ 100^ •noc 0 (a) l< 0 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 0.36 0.40 0.44 aa 0.5 0.4 0.2 0.10 0.08 0.06 0.04 0.02 - EXACT APPROXIMATE ,.'-' ^-' --'• ,--- * ,.-"' x' - << ^ > Ss,^^ - X<^ - ^ f"' "" "--. ^ ^ ^ ^ (b) I 1 1 1 1 \ 1 1 1 1 1 1000 600 400 200 100 60 40 20 10 8 6 4 2 1 e" Fig. 4 — Comparison of exact and approximate formula.s for the propagation constant of a typical mode (TM02- limit in a guide with/3oa = 29.554 and «' = 4). staiit first becomes larger, in all cases, than predicted by the approximate formula. For still lower conductivities the attenuation constant may pass through a maximum, as in the present example, and decrease again. The existence of a maximum in the attenuation vs conductivity curve is not indicated by the approximate formula. 1378 THE BELL SYSTEM TECHNICAL JOUKNAL, NOVEMBER 1956 V. DISCUSSION OF RESULTS «ai The dimensioiiless results of Section IV may easily be scaled to any >. \\ desired operating wavelength, and the attenuation constants and guides wavelengths expressed in conventional units. If Xo is the free-space wave- length in centimeters, then the guide diameter d in inches, the attenua- tion constant a in db/meter, and the guide wavelength X^ in centimeters are given by the following formulas: din = 0.12532 (M(Xo)cn. «db/m = vAgjcm — 5457.5 (aa) (|8oa)(Xo)cm (i8oa)(Xo)cm ,< ta Table II lists the guide diameters and the conversion factors for a and Xg for the three values of /Soa used in Section IV, at frequencies corre- sponding to free-space wavelengths of 3.33 and 0.54 cm. The table also lists the number of propagating modes in a perfectly conducting guide as a function of jSoa (different polarizations are not counted separately). When helix waveguide is used to reduce mode conversions, an im- portant parameter is the ratio of the attenuation constant of any given unwanted mode to the attenuation constant of the TEoi mode. The theoretical attenuation constants of the TEoi mode at Xo = 5.4 mm in copper guides of various sizes are listed below: Diameter aa adb/m ^! 2" 2.77 X 10"' 9.47 X 10"' nti 8 1.50 X 10"' 1.17 X 10"' i 7 // 16 7.11 X 10"' 1.11 X 10"' : 1l Table II — Conversion Factors for Attenuation Constants and Guide Wavelengths in Various Waveguides \ Propa- gating modes Xo = 3.33 cm Xo = 0.54 cm Poa Diameter (inches) a db/meter \g cm Diameter (inches) a db/meter \g cm 29.554 12.930 6.465 227 44 12 12.33 5.40 2.70 55.5 aa 127 aa 253 aa 98.41/^a 43.06/^a 21.53//3a 2.000 0.875 0.4375 342 aa 782 aa 1563 aa 15.959/;8a 6.982//3a 3.491//3a HELIX WAVEGUIDE 1379 Referring to the values of aa listed in Table I, we see that the un- Iwanted mode attenuations can be made to exceed the TEoi attenuation [by factors of from several hundred to several hundred thousand in the [large helix guide. The attenuation ratios are somewhat smaller in the [smaller guide sizes. The attenuation versus conductivity plots of Fig. 3 show that for I many of the modes there is a value of jacket conductivity, depending on ■the mode, the value of ;Soa, and the jacket permittivity, which maximizes the attenuation constant. Since one is accustomed to think of the at- tenuation constant of a waveguide as an increasing function of frequency for all sufficiently high frec^uencies (except for circular electric waves), or as an increasing function of wall resistance, it is worth while to see why one should really expect the attenuation constant to pass through a maximum as the frequency is increased indefinitely in an ordinary metallic guide, or as the wall resistance is increased at a fixed frequency. The argument runs as follows: Guided waves inside a cylindrical pipe may be expressed as bundles of plane waves repeatedly reflected from the cylindrical boundary." The angle which the wave normals make with the guide axis decreases as the frequency increases farther above cutoff; and the complementary angle, which is the angle of incidence of the waves upon the boundary, ap- proaches 90°. If the walls are imperfectly conducting, the guided wave is attenuated because the reflection coefficient of the component waves at the boundary is less than unity. The theory of reflection at an imper- fectly conducting surface shows that the reflection coefficient of a plane wave polarized with its electric vector in the plane of incidence first decreases with increasing angle of incidence, then passes through a deep minimum, and finally increases to unity at strictly grazing incidence. ^^ For a metallic reflector, the angle of incidence corresponding to minimum reflection is very near 90°. Inasmuch as all modes in circular guide except for the circular electric family have a component of E in the plane of incidence (the plane 6 = constant), one would expect the attenuation constant of each mode to pass through a maximum at a sufficiently high frequency. For example, the TMoi mode in a 2-inch copper guide should have maximum attenuation at a free-space wavelength in the neighbor- hood of 0.1 mm (100 microns), assuming the dc value for the conductivity of copper. To find the actual maximum, of course, would require the solution of a transcendental equation as in Section IV. The circular electric waves all have E normal to the plane of incidence. " Reference 9, pp. 411-412. 12 Reference 7, pp. 507-509. 1380 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 For this polarization the reflection coefficient increases steadily from its value at normal incidence to unity at grazing incidence. Thus one has an optical interpretation of the anomalous attenuation-frequency behavior of circular electric waves. If instead of varying the frequency one imagines the wall resistance varied at a fixed frequency, he can easily convince himself that there usually exists a finite value of resistance which maximizes the attenua- tion constant of a given mode. An idealized illustrative example has been worked out by Schelkunoff. He considers the propagation of transverse magnetic waves between parallel resistance sheets, and shows that if the sheets are far enough apart the attenuation constant increases from zero to a maximum and then falls again to zero, as the wall resistance is made to increase from zero to infinity. It may be instructive to consider that maximum power is dissipated in the lossy walls when their impedance is matched as well as possible to the wave impedance, looking normal to the walls, of the fields inside the guide. In conclusion we mention a couple of theoretical questions which are suggested by the numerical results of Section IV. (1) Limit modes. It has been seen that the limit which a given lossy mode approaches as the jacket conductivity becomes infinite may not be unique. Can rules be given for determining limit modes when the manner in which | e' — ?'e | approaches infinity is specified? (2) Behavior of modes as a — ^ 0. It is known^^ that the number of true guided waves (i.e., exponentially propagating waves whose fields vanish at large radial distances from the guide axis) possible in a cylindrical waveguide is finite if the conductivity of the exterior medium is finite. The number is enormously large if the exterior medium is a metal; but the modes presumably disappear one by one as the conductivity is de- creased. If the conductivity of the exterior medium is low enough and if its permittivity is not less than the permittivity of the interior medium, no true guided waves can exist. At what values of conductivity do the first few modes appear in a guide of given size, and how do their propa- gation constants behave at very low conductivities? The complete theory of lossy -wall waveguide would appear to present quite a challenge to the applied mathematician. Fortunately the en- gineering usefulness of helix waveguide does not depend upon getting immediate answers to such difficult analytical questions. 13 Reference 9, pp. 484-489. " G. M. Roe, The Theory of Acoustic and Electromagnetic Wave Guides and Cavity Resonators, Ph.D. thesis, U. of Minn., 1947, Section 2. HELIX WAVEGUIDE 1381 APPENDIX APPROXIMATE SOLUTION OF THE CHARACTERISTIC EQUATION The characteristic equation (6) of the heUx guide may be written in the dimensionless form [i^a tan ^ - -^ ^^.^ - (fta) -j-^^ fia tan ^ - -^ ] J^,,^ . — (M' (Al) If I e' — «" I is sufficiently large, the right side of the equation is large and either J„(fia) or Jn'itio,) is near zero. Let p denote a particular root of Jn or Jn', then to zero order, Tia = p ha = finma = i8oa(l - v')''' (A2) Ua = iSoaie' - U" - 1 - vy where V = p/^oa Henceforth assume that I r2a 1 » I (4n' - l)/8 I (A3a) and I r2a I » I n I (A3b) It is convenient to postulate both inequalities, even though the first is more restrictive than the second unless \ n \ = lor|nl =2. If (A3a) is satisfied, the Hankel functions may be replaced by the first terms of their asymptotic expressions, and Eq. (Al) becomes A , , 7ihaY Jniha) /« n2 ^n'(fia) ijia 2 Ua tan ^p — —- j + (M'U - 'i^") 1382 THE BELL SYSTEM TECHNICAL JOURN.IL, NOVEMBER 1956 It follows from (A3b), using the zero-order approximations (A2), that 1 nha/r2a \ « \ I3,a{e' - ie'f'^ \ so the characteristic equation finally takes the approximate form m (A4) f2a [(fsa tan 4^)' + (MV - ie')] Now let f itt = p + .r, I a; I <^tan lA t€ where ^ + z?? is given by (AS). In view of (A5), the condition that | a; | <3C 1 is equivalent to ^^ I aa + lA^a \ « 1 (AS) (A9) (AlO) (All) In all the numerical cases treated in the present paper, the approximate formulas agree well with the exact ones provided that the left side of (All) is not greater than about 0.1. A condition Avhich is usually satisfied in practice, although not strictly a consequence of the assumptions (A3) or (All), is 1 «1 ^e 1384 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 | This final approximation leads to the simple equations (7a) and (7b) of Section III, namely: TMnm modes a + iA/3 a(l - vY'''[l + tanV] TE„;„ modes . . . ^ ^ (^ + iy) v'V [tan yp - n{l - vf^/yvf " ^ ^^ a(l - v'^yi'' {f- - ^2) [1 + tanV] where i Wafer-Type Millimeter Wave Rectifiers* By W. M. SHARPLESS (Manuscript received June 18, 1956) A wafer-type silicon point-contact rectifier and holder designed pri- marily for use as the first detector in millimeter wave receivers are described. Measurements made on a pilot production group of one hundred wafer rectifier units yielded the following average performance data at a wave- length of 5.4 millimeters: conversion loss, 7£ dh; noise ratio, 2.2; interme- diate frequency output impedance 34O ohms. Methods of estimating the values of the circuit parameters of a point-contact rectifier are given in an Appendix. INTRODUCTION Point-contact rectifiers for millimeter waves have been in experi- mental use for several years. These units, for the most part, have been coaxial cartridges which were inserted in a fixed position, usually cen- tered, in the waveguide. Impedance matching was accomplished by means of a series of matching screws preceding the rectifier and an adjust- able waveguide piston following the rectifier. Tuning screws are gener- ally undesirable l^ecause of the possibility of losses, narrow band widths and instability. It is the purpose of this paper to describe a new type millimeter-wave rectifier and holder which were designed to eliminate the need for tuning screws and to provide a readily interchangeable rectifier of the flat wafer type. This wafer contains a short section of waveguide across which the point contact rectifier is mounted. The necessary low frequency output terminal (and the rectified current connection) together with the high- frequency bypass capacitor, are also contained within each wafer. The basic idea of the wafer-type rectifier is that the unit can be inserted in its holder and moved transversely to the waveguide to obtain a resistive match to the guide ; the reactive component of the rectifier impedance is then tuned out by an adjustable waveguide plunger behind the rectifier. * This work was supported in part by Contract Nonr-687(00) with the Office of Naval Research, Department of the Navy. 1385 1386 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 The wafer unit and holder were developed primarily for use as the first converter in double detection receivers operating in the 4- to 7-mil- limeter wavelength range. In order to check the practicability of the design and to supply rectifiers for laboratory use, a pilot production group of one hundred units was processed and measured. Performance data obtained with this group are presented. A balanced converter using wafer rectifiers is also described. Methods of estimating the values of the various circuit parameters of a point-contact rectifier are outlined in an appendix. These calculations proved useful in the design of the wafer unit and in predicting the broad- band performance of the converter. DESCRIPTION OF WAFER UNIT AND HOLDER Fig. 1 is a drawing of the wafer type rectifier. The unit is made from stock steel iV-inch thick and is gold plated after the milling, drilling and soldering operations are completed. To allow for the transverse impe- dance matching adjustment, the section of waveguide contained in the wafer is made wider than the RG98U input guide to the holder. By making the wafer thin {-^ inch), the short sections of unused guide on either side will remain "cut-off" over the operating range of the recti- fiers. The silicon end of the rectifier consists of a copper pin on which the silicon is press mounted, the assembly held in place with Araldite ce- ment which also serves as the insulating material for a quarter-wave- length long high frequency bypass capacitor. The pin serving as the inter- mediate frequency and direct current output lead is also cemented in place with Araldite cement. A soft solder connection is made between this pin and the pin holding the silicon wafer. A nickel pin with a conical end on which a pointed tungsten contact spring is welded is pressed into place from the opposite side of the guide at the time of final assembly. DC AND IF OUTPUT 0.031 "x 0.234^ WAVEGUIDE 0.063" "BRIGHT GOLD" STEEL WAFER BORON- doped/ SILICON -CONTACT SPRING Fig. 1 — Millimeter-wave wafer unit. WAFER TYPE MILLIMETER WAVE RECTIFIERS 1387 ?^^^^^^^^ "ARALDITE" BONDING RESIN WELD 0.014" SILICON SQUARES 0.0065" THICK 0.0009" DIA TUNGSTEN WIRE WAVEGUIDE Fig. 2 — Millimeter-wave point-contact assembly. The region of the wafer unit containing the silicon and point contact is shown in Fig. 2. The methods used in preparing the silicon wafer and the spring contact point are similar in many respects to the standard techniques used in the manufacture of rectifiers for longer wavelengths. Some modifications and refinements in technique are called for by a decrease in size and the increased frequency of operation. A single-crystal ingot, grown from high purity DuPont silicon doped with 0.02 per cent boron, furnishes the material for the silicon squares used in the wafer unit. Slices cut from the ingot are polished and heat treated. Gold is evaporated on the back surface and the slices are diced into squares approximately 0.014-inch square and 0.0065-inch thick. These squares are pressed into indentations formed in the ends of the 0.030-inch copper pins which have previously been tin-plated. The rods are then cemented in place in the wafer. The spring contact points are made of pure tungsten wire that has been sized to 0.9 mil in diameter by an electrolytic etching process. A short length of this wire is spot welded on the conical end of the 0.031 -inch nickel rod. The wire is then bent into 1388 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Fig. 3 — ■ Micro-photograph showing successive stages in the formation of the contact spring. The posts are 3*0 inch in diameter. (a) (b) Fig. 4 — Cathode-raj^ oscilloscope display of wafer unit static characteristic: (a) before and (b) after tapping. the "S" configuration in a forming jig. By an electrolytic process the spring is then cut to the proper length and pointed. The niicro-photo- graphs in Fig. 3 show successive stages in the formation of the contact spring. In the final assembly of the unit the nickel rod with the contact spring is pressed into place until contact is made with the silicon. It is then advanced a half mil to obtain the proper contact pressure. The voltage- current characteristics as viewed at 60 cycles on a cathode-ray oscillo- scope will then appear as shown in Fig. 4(a). The unit is "tapped" into final adjustment. This is done by clamping the unit in a holder and rapping it sharply on the top of a hard wood bench. This procedure re- quires experience as excessive "tapping" will impair the performance of the unit. Usually one vigorous "tap" is sufficient to produce the desired effect and the voltage-current characteristic will appear as WAFER TYPE MILLIMETER WAVE RECTIFIERS 1389 shown in Fig. 4(b). The static characteristic of a typical unit is shown in Fig. 5. The conversion loss of each unit is measured before the end of the nickel rod carrying the contact point is cut off flush with the wafer. In the event that this initial measurement shows that the conversion loss exceeds an arbitrarily chosen upper limit (8.5 db), it is possible at this stage to withdraw the point and replace it with a new one. This pro- cedure, w^hich was necessary on only a few of the units processed, always resulted in an acceptable unit. The final operation is to cut off the pro- truding end of the nickel rod flush with the wafer. A holder designed to use the wafer units is shown in Figs. 6 and 7. At the input end of the converter block is a short waveguide taper section to match from standard RG98U waveguide to the ^-inch high wave- guide used in the wafer unit. As the wafer unit is moved in and out to match the conductance of the crystal to the waveguide, the output pin of the wafer unit slides in a chuck on the inner conductor of the coaxial 6.0 5.5 5.0 4.5 (O 4.0 111 a: ai Q- 3.5 < J 3.0 ?2.5 I 2.0 cr ^ 1.5 1.0 0.5 -0.5 •0.4 -0.2 0 0.2 VOLTS 0.4 0.6 Fig. 5 — • Static characteristic of typical millimeter-wave wafer unit. 1390 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 output jack. The unit may be clamped in position after matching adjust- ments are made by tightening the knurled thumb screw which pushes a cylindrical slug containing an adjustable piston against the wafer unit. The piston is a short septum which slides in shallow grooves in the top and bottom of the ya-inch high waveguide, thus dividing the waveguide into two guides which are beyond cut-off. This septum is made of two pieces of thin beryllium copper bowed in opposite directions so that good contact is made to the sides of the grooves in the top and bottom of the waveguide. Since the piston with its connecting rod is very light in weight and is held firmly in place by the spring action of the bowed septum, no additional locking mechanism need be provided. Since the rectifier is essentially broadband by design, the adjustment of the piston is not critical and is readily made by hand. The piston rod is protected by a cap which is snapped in place over the thumb screw when all tuning adjustments are completed. B n L, 3 "0 or 2 ^ - 1 £ 1^^:: — l| J SECTION B-B SECTION A -A Fig. 6 — Assembly drawing of millimeter-wave converter. WAFER TYPE MILLIMETER WAVE RECTIFIERS 1391 '" ^^H****-^ \ \ Fig. 7 — Explosed view of millimeter-wave converter. With the converter fixed-tuned at 5.4 millimeters, a shift in wave- length to 6.3 millimeters (17 per cent change) produces a mismatch loss of from 1.6 to 4.0 db depending on the rectifier used. PERFORMANCE DATA FOR WAFER-TYPE RECTIFIER UNIT A pilot group of one hundred wafer units was processed and measured. Figs. 8, 9 and 10 are bar graphs of the distribution of the conversion loss L, and noise ratio A^r*, and the 60 megacycle intermediate frequency output impedance Zip , for the hundred rectifiers measured in the * Nu is the ratio of the noise power available from the rectifier to the noise power available from an equivalent resistor at room temperature. 1392 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 mixer of Fig. 7 at a wavelength of 5.4 millimeters. In order that the measurements might be more readily compared with those made on commercially available rectifiers used at longer wavelengths, the avail- able beating oscillator power was maintained at a level of one milliwatt for all measurements.* Further, in the case of the conversion loss, a 35 30 25 20 15 10 6.2 T ^ '"" !1 ^^S"' 1 6.6 7.0 7.4 7.8 8.2 CONVERSION LOSS IN DECIBELS 8.6 Fig. 8 — Conversion loss (L) of 100 wafer units at a wavelength of 5.4 railli- meters with one-milliwatt beating oscillator drive (average 7.2 db). 50 45 40 35 (f) Z 30 D "25 LU 2 20 Z 15 to ' 1 1 1 1 1.2 1.6 2.0 2.4 2.8 3.2 NOISE RATIO Nr times 3.6 4.0 Fig. 9 — Noise Ratio (jVr) for 100 wafer units at a wavelength of 5.4 milli- meters with a one-milliwatt beating oscillator drive (average 2.21 times). * Power levels were determined by the use of a calorimeter. See, A Calorimeter for Power Measurements at Millimeter Wavelengths, I. R. E. Trans., MTT-2, pp. 45-47, Sept., 1954. WAFER TYPE MILLIMETER WAVE RECTIFIERS 1393 40 35 12 30 2 3 u. 25 O N 5"" ■ s, 1 1" £20 m 5 10 ■ J. 5 ^^ ""vwpp 0 150 200 250 300 350 400 450 500 60-MC INTERMEDIATE FREQUENCY IMPEDANCE, Z|p, IN OHMS Fig. 10 — Sixty-megacj'cle intermediate-frequency output impedance (Zif) for 100 wafer units with one milliwatt beating oscillator drive (average 338 ohms) limit of 8.5 db was arbitrarily adopted. This required the readjustment of eleven units, with a new point inserted in each case. No units were rejected because of high noise and none of the hundred units processed was lost. From the bar graphs it may be seen that the wafer units have the average characteristics shown in the accompanying table at a wave- length of 5.4 millimeters.* Conversion Loss L 7 . 2 db (5.3 times) Noise Ratio A^r 2.2 times IF Impedance (60 mc) Zj-p 338 ohms Knowing the noise figure, Nif , of the IF amplifier intended for use with the rectifiers, the overall receiver noise figure, A^rec > may be cal- culated by the following formula (using numerical ratios) : NnKc = L(N,, - 1 + A^if) Assuming an IF amplifier noise figure of 4.0 db (2| times) and the average values of "L" and 'Wr" given above for the millimeter wafer units, we have for the case of a noiseless beating oscillator; ATrec = 5.3 (2.2 - 1 + 2.5) ^ 20 (13 db) * A few wafer units have also been measured at a wavelength of 4.16 millimeters. The conversion losses averaged about 1.6 db greater than those measured at a wavelength of 5.4 millimeters. 1394 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table I — Comparison of Low-Power Characteristics of Cartridge-Type and Wafer-Type Rectifiers Test Conditions power Frequency Beating oscillator level Noise reference resistor Conversion loss Noise ratio Nominal IF impedance range JAN Specifications for Cartridge-Type Rectifiers IN26 23984 mc 1.0 milliwatts 300 ohms 8.5 db (max) 2.5 (ma.x) 300 to 600 ohms IN53 34860 mc 1 .0 milliwatts 300 ohms 8.5 db (max) 2.5 (max) 400 to 800 ohms Performance of Wafer-Type Rectifiers 55500 mc 1.0 milliwatts 300 ohms 8.5 db (max)* 2.2 (average) t 250 to 500 ohms * Limit arbitrarily set on basis of 100 per cent yield as explained in the text, t Limit not set. Actually in more recent production A'^r has averaged 1.7 times. In practice, the beating oscillator noise sidebands can be eliminated by the use of a matched pair of rectifiers in a balanced converter ar- rangement described later. The resulting overall noise figure of 13 db on an average compares quite favorably with the figures obtained at longer wavelengths. In Table I it is seen that a high percentage of the group of one hundred units would be able to pass low-power JAN specifications similar to those set down for the commercially available IN26 and IN53 rectifiers used at longer wavelengths. effect of VARYING THE BEATING OSCILLATOR POWER When the optimum over-all receiver noise figure is desired, it may well turn out that a beating oscillator drive of one milliwatt (correspond- ing to a dc rectified current for different wafers of from jq to Ij milli- amperes) is too large. Fig. 11 shows the effect on the performance of a typical unit as the beating oscillator drive is varied above and below the one milliwatt level as indicated by the change in the dc rectified current. It is seen that the value of N^. tends to increase rapidly for a beating oscillator drive much in excess of one milliwatt; with reduced drive, the over-all noise figure of the receiver, A^'rec for the example taken, im- proves, reaching a minimum value near a rectified current of about ^u m.illiampere corresponding to a drive of about f of a milliwatt. A BALANCED CONVERTER FOR WAFER UNITS A broad-band balanced first converter has been developed which makes use of a pair of wafer- type millimeter-wave rectifiers. This converter WAFER TYPE MILLIMETER WAVE RECTIFIERS 1395 14. Or -I 13.5 o HI 13.0 Q 12.5 d: 2 12.0 LU 5 11.5 11.0 ai to O 10.5 cc ^10.0 LU o ^ 9.5 Q < 9.0 ID I/) O 8.5 _i 2 O 8.0 CO cc HI > 7.5 z o u 7.0 6.5 note: this curve was calculated using an assumed value of 5.5db for the noise figure of the if amplifier "^^ 3.5 (0 -3.0 (/) h ^2.5 2.0 O '-^ t- < °^ 1.0 LU If) O 0.5 STANDARD UNIT NO. 46 • \ \ HI > CC Q O CD 5 111 z o ^^ ^ .^^^ "«^, _ Nrec s N N .Z,F ^> X. \ N N s. ^x^ ^ 'x ^x < > k-- ^^ ^ Nr r -^ ^L — — — — 380 360 340 320 300 280 260 240 >- O 111 o cr I U- (J I- 9 < C, wC ^(^^) R,= 1 + (a)CR)' COL; R2=Rs+Ri + KM Rs + R, Fig. 13 — Point contact rectifier and equivalent circuits. * W. Shockley, Electrons and Holes in Semiconductors, New York: D. Van Nostrand Co., Inc., 1950, p. 284. WAFER TYPE MILLIMETER WAVE RECTIFIERS 1399 itf Rs , assuming a circular contact area, may be calculated from the for- mula, Rs = p/4ri .* For the above example, Rs = 18 ohms. Barrier Resistance The approximate operating value of the barrier resistance, /?, may be determined from a knowledge of the intermediate frequency impedance of a typical rectifier. A. B. Cra^^^ord has sho-wn that the optimum inter- mediate frequency output impedance of a crystal mixer rectifier is a function of the exponent of the static characteristic of the rectifier and the impedance presented to the rectifier at the image and signal fre- quencies. This information is presented in Fig. 12.3-6 in G. C. South- worth's book.f In the millimeter wave case it is a good assumption that the impedances for the signal and image frequencies are equal; for this case and for matched conditions, the magnitude of the high frequency impedance is seen to be a simple multiple of the intermediate frequency impedance Rif • From numerous measurements on mixer rectifiers operating at differ- ent frequencies it is known that the intermediate frequency impedance of an average rectifier is very nearly 400 ohms. We also know from the DC static characteristics of our millimeter wave type rectifiers that the average exponent is about four. With this information, and the curves in Southworth's book, it is found that R ^ Rif/1.5. Thus, the barrier resistance R is about 250 ohms.| Capacitance of Barrier Layer From a knowledge of the point contact area, the barrier layer thick- ness, and the dielectric constant of the silicon, the capacitance of the point contact may be calculated. The radius of the contact point area is the same as that used for the calculation of the spreading resistance. The barrier layer thickness, h, for the heat treated silicon used for millimeter waves has been measured by R. S. Ohl to be about 10' meters. The dielectric constant of sihcon is fr = 13. The capacitance is given by the following formula ^& 2 * J. H. Jeans, Mathematical Theory of Electricity and Magnetism, 5th Ed., Cambridge University Press, 1925. t G. C. Southworth, Principles and Applications of Waveguide Transmission, New York: D. Van Nostrand Co., Inc., 1950. t This resistance cannot be readily measured directly at millimeter waves. 1400 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 For the above case C = 5.7 X 10" farads or l/coC at 5.4 millimeters is about 50 ohms. The accuracy of this capacitance calculation can be verified later when a completed rectifier is measured for its high frequency conversion loss. This is possible because we know the calculated low frequency conversion loss of the rectifier, for the case of zero spreading resistance from Southworth's book. Fig. 12.3-7. For an exponent of four this loss is given as 4.4 db. The additional loss at high frequency due to the capacitance, C, may be calculated (see Equivalent Circuit II) by the formula : Additional Loss 10 logic :?L+_?f db /ti (2) From the text (Fig. 8), it is seen that the average wafer rectifier unit has a conversion loss at 5.4 millimeters of 7.2 db; thus, the difference between the low and high frequency conversion losses is very nearly 3 db. This means that about one-half the signal power is lost in the spreading resistance; hence Ri and i?, are about equal. By transferring back to Equivalent Circuit I, the average value of the capacitance is found to be 4.1 X 10 farads, which is a reasonable check with the calculated value given by (1). Inductance of the Contact Spring The remaining parameter of the equivalent circuit to be determined is the inductance of the contact spring. The value of the equivalent parallel resistance, i?2 , depends on the inductance L (the other param- eters being fixed), or conversely, for a given value of R2 , the appropriate value for L may be calculated from the formula for Equivalent Circuit III. For an off-center match of the rectifier to the waveguide, R2 must equal the guide impedance, Zd , at the rectifier location. Also, for a match, the distance, I, from the rectifier to the waveguide piston must '/////////////////////////////J///////////////////^///////A 'R'. I PISTON V////////////////////////////^///////////////////////>////////. WAVEGUIDE B| J T I b I I I i_ V///y^////y'y'//J///////////////77777y I V///////////f////y/////////y/y7/77/A k . -A Fig. 14 — Mulching circuit for rectifier offset in waveguide. WAFER TYPE MILLIMETER WAVE RECTIFIERS 1401 satisfy the relation, Zd tan 2-Ki/\g = — coLa . (See Fig. 14.) The imped- ance of the guide as a function of d/a is given by, Zd = 2407r - '/RU sni xd a (3) As a compromise between electrical and mechanical requirements, a waveguide height, 6, of -5^ inch was chosen for the wafer unit ; the width of the guide was taken to be the same as RG9SU. For b = 7.88 X 10~*, a = 3.76 X 10"', d/a = | and X = 5.4 X 10~', (3) gives a value of 113 ohms for Zd (and R2). The appropriate value for L then becomes 3.38 X 10"^" henries. An estimate of the size of a contact spring having the inductance given above can be made from the formula below which gives the in- ductance of a straight thin wire of length *S as a function of its sidewise position in the waveguide.* (See Fig. 15.) 2S log, . ird 2a sm — a r2«d \-y X 10 henries (4) For d/a = I and 2r2 = 2.28 X 10 ^ (0.9 X 10 ' inches), the length, S, is found to be about 3.38 X 10" meters or about 0.013 inches. Since the spring must be so very small, the circuit from the base of the spring to the waveguide wall is completed with a large low induc- tance conical post as shown in Fig. 2 of the text. Bandwidth Calculation Having assigned values to all the parameters of the equivalent cir- cuit, it is now possible to calculate the mismatch loss of a fixed-tune Y- a i '/////////////////////////////7777. I fSTRAIGHTA •"V WIRE ] -ZT2 V///////////}//////////////77777yA U-d-J I s I I JL Fig. 15 — Thin wire in waveguide. * Private communication from S. A. Schelkunoff. 1402 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 converter for a given change in operating wavelength. This loss is given by the following formula: Mismatch loss = 10 log 4Zd 10 R2 1 + R2 + + wLo tan 2iri/\g, db (5) For the wafer miit, calculation shows that the rectifier is matched to the waveguide at a wavelength of 5.4 X 10"^ meters for d/a = j and ^ = 3.14 X 10^ . If now the wavelength is changed to 6.3 X 10~^ meters, without re tuning (17 per cent change) the mismatch loss calculated by (5) is 1.6 db. It was stated in the text that a number of wafer units gave measured mismatch losses of from 1.6 to 4.0 db for a 17 per cent change in wavelength without retuning. This is considered to be a reasonable correlation between calculations and measurements. Frequency Conversion by Means of a Nonlinear Admittance C. F. EDWARDS (Manuscript received June 20, 1956) This paper gives a mathematical analysis of a heterodyne conversion transducer in which the nonlinear element is made up of a nonlinear re- sistor and a nonlinear capacitor in parallel. Curves are given, showing the change in admittance and gain as the characteristics of the nonlinear ele- ments are varied. The case where a conjugate match exists at the terminals is treated. It is shown that when the output frequency is greater than the input fre- quency, modulators having substantial gain and bandwidth are possible, but when the output frequency is less than the input frequency, the con- verter loss is greater than unity and is little affected by the nonlinear ca- pacitor. The conditions under which a conjugate match is possible are specified and it is concluded that a nonlinear capacitor alone is the pre- ferred element for modidators and that a nonlinear resistor alone gives the best performance in converters. INTRODUCTION Point contact rectifiers using either silicon or germanium are used as the nonlinear element in microwave modulators to change an inter- mediate frequency signal to an outgoing microwave signal and in re- ceiving converters to change an incoming microwave signal to a lower intermediate frequency. Most point contact rectifiers now in use behave as pure nonlinear resistors as evidenced by the fact that in either of the above uses the conversion loss is the same. In recent experiments with heterodyne conversion transducers* using point contact rectifiers made with ion bombarded silicon this was found to be no longer true. The conversion loss of the modulator was found to be unusually low and * This term is defined in American Standard Definitions of Electrical Terms — ASA C42 — as "a conversion transducer in which the useful output frequency is the sum or difference of the input frequency and an integral multiple of the frequency of another wave". 1403 1404 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 that of the converter was several decibels greater. In one instance the loss in a modulator used to convert a 70 mc signal to one at 11,130 mc was found to be only 2.3 db but when the direction of transmission through it was reversed and it was used as a converter, the loss was 7.8 decibels. h Similar effects were observed several years ago in conversion trans- ducers using welded contact germanium rectifiers. In these early experi- ments substantial converter gain and negative conductance at the inter- mediate frequency terminals were also observed. These results were accounted for by assuming the presence of a nonlinear capacitance at the point contact in parallel with the nonlinear resistance. At that time at- tention was devoted mainly to the behavior of converters where noise is a vital factor. It was found that although the conversion loss could be reduced, the noise temperature increased and no improvement in noise figure resulted. However, the noise temperature requirements in modulators are much less severe and the nonlinear capacitance effect is useful and can substantially improve the performance. THEORY The mathematical analysis given here was undertaken in order to clarify the effect of the nonlinear capacitance in the frequency conversion process and to obtain an estimate of the usefulness of modulators ex- hibiting gain. The analysis is restricted to the simplest case in which signal voltages are allowed to develop across the nonlinear elements at the input and output frequencies only. This is not an unrealistic restric- tion since the conversion transducers used in microwave relay systems have filters associated with them which suppress the modulation products outside the signal band. The final results will be given only for those con- ditions which permit a conjugate match at the input and output of the transducer. The procedure used to obtain expressions for the admittance and gain of conversion transducers utilizing a nonlinear element made up of a nonlinear resistance and a nonlinear capacitance in parallel follows the commonly used method of treating the nonlinear elements as local oscillator controlled linear time varying elements. The current through the nonlinear resistor is a function of the applied voltage. The derivative of this function is the conductance as a function of the applied voltage. Thus when the local oscillator is applied, the conductance varies at the local oscillator frequency and the conductance as a function of time may be obtained. This is periodic and may be expressed as a Fourier series. The conductance is real and if we make the usual assumption that FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1405 it may be expressed as an even function of time, we may write 7 = (1) where coo/27r is the local oscillator frequency /o and the Fourier coeffi- cients Gn are real. Similarly the charge on the nonlinear capacitor is a function of the applied voltage. The derivative of this function is the capacitance as a function of the applied voltage. The application of the local oscillator thus causes the capacitance to vary at the local oscillator frequency so that it also may be expressed as a Fourier series. The ca- pacitance K is real, and assuming it may be expressed as an even function of time, we have + Cae"''""' + Cre~'"'' + Co + Cie^""' + C^e'^"'' + (2) It is assumed that the current and charge functions are single valued and that their derivatives are always positive. When a small signal voltage v is apphed to the nonlinear resistor, the signal current through the resistor is given by yv. When it is applied to the nonlinear capacitor the charge on the capacitor is kv. The total cur- rent i which flows through the two nonlinear elements connected in parallel thus becomes (3) V of course must be small and not affect the value of 7 and k. Fig. 1 shows a heterodyne conversion transducer made up of a non- linear resistor and a nonlinear capacitor in parallel driven by an internal local oscillator, /i is the signal frequency at the terminals 1-2, and 7/1 is the external admittance connected to these terminals. The signal fre- quency at the terminals 3-4 is /2 , and y2 is the external admittance. I, > 1 + v- 3 h* ^ ( ' + yi + V, A B m ys r^ *.« 2 4 L<^>_i y/////////////////////////m///////m//////////////////////,//////^^^^^ Fig. 1 — Heterodyne conversion transducer. 140G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 A, B and C are ideal frequency selective networks whose admittances are zero at /i , f^ and /o respectively, and infinite at all other frequencies. This circuit permits the application of the local oscillator voltage to the nonlinear elements but permits signal voltages to develop across them at /i and f^ only. Similarly, signal currents at frequencies other than /i and /2 encounter no external impedance, so they cannot alter the signal voltage or contribute to the external power. This, of course, assumes that if the nonlinear element is a point contact rectifier the spreading resist- ance normally present is negligible. If /i is a frequency less than half the local oscillator frequency /o (it is generally very much less), the network B can be selected to make /2 either /o + /i , or fo — fi ■ To distinguish between the two cases, we will call the former a noninverting conversion transducer since an increase in one signal frequency causes an increase in the other. The latter will be called an inverting conversion transducer since an increase in one signal frequency results in a decrease in the other. When yi contains the generator and 2/2 the load, the device becomes a modulator. When 2/2 contains the generator and yi the load, it is a converter. The real part of the signal voltage may be written V = Vie''^'' + Vi*e~'"'' + V^e'"'' + ¥2*6-'"'' (4) where V* is the complex conjugate of V and w = 27r/, Similarly, the real part of the signal current may be written • ^ j^^J-it _|. /^*e-i"i« + 72gi"2t ^ /2*e~'"=^' (5) If we multiply equations (1) and (4) and retain only those terms con- taining /i and /2 we obtain, in the case of the non-inverting conversion transducer where /2 = /o + /i , (6) + [GoV,* -f G,V2*]e~'"'' -f [GiFi* + GoV2*]e~'"'' Similarly, if we multiply (2) and (4) we get an expression like (6) with the G's replaced by C's. If we differentiate this expression we get ~ M = jcci [CoVi + CV^le'"'' + MiCiVi + C0V2W'''' at (7) - jcoiiCoFi* -f CiF2*]e-^"^ ' - icoslCiFi* + C,V2*]e ■joi2 t When we perform the addition indicated by (3) and compare the result with (5) we obtain /i = [Go + icoiColFi -f [G, -f jmCi]V2 (8) h = [Gi + ic^CJFi + [Go + JCC2C0W2 FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1407 Going through the same steps for the hivertmg conversion transducer where /2 = /o — /i we obtain /i = [Go + icoiCo]Fi + [G, + iciCilFo* h* = [G, - jwoCiW, + [Go - MC0W2* Equations (8) and (9) are in the form /i = FnFi + F12F2 I2 = ^21'''l ~l~ F22F2 (9) (10) A heterodyne conversion transducer may thus be represented by a linear 4-pole, and the admittance and gain of the 4-pole may be expressed in terms of the admittance coefficients. In Fig. 1 we see that the admit- tance of the 4-pole yi at the terminals 1-2 is eciual to Ii/Vi and the admittance 2/2 connected to terminals 3-4 is — /2/l^ . Putting these in (10) we find YuY,, (11) yi Yn F22 + Vi Similarly the admittance of the 4-pole 2/2' at the terminals 3-4 is 72/ F2 and the admittance yi connected to terminals 1-2 is —Ii/Vi . Putting these in (10) gives 2/2 F 22 i 12^ 21 Yn + 2/1 (12) To compute the gain of the 4-pole when 7/1 contains the generator and y-i the load, it is convenient to assume a current source connected across ?/i . If the current from this source is Jo we have Ii = lo — yiVi . I2 equals —y2V2 as before. Putting these in (10) gives -'0 _ T^ (Fn + 2/l)(F22 + ?/2) F (13) 21 If we let yi = Qi -{- jbi and ?/2 = ^2 + i&2 , the power in the load isF2 ^2 and the power available from the generator is /oV4^i . Therefore the transducer gain ri2 defined as the ratio of the power in y2 to that avail- able from 2/1 becomes F2' Tu =4gig2j-^ = "igig 1 n" 21 (14) F12F21 - (Fn + 2/1) (F22 + 2/2) When ?/2 contains the generator and yi the load, we may proceed in the same way (letting 7o flow in terminal 4) and obtain 2 r2i = 4^-1^2 F 12 F12F21 - (Fu + ?/0(F22 + 2/2) (15) 1408 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 We may now obtain expressions for the admittance and gain of the 4-pole when the nonhnear element consists of a nonUnear resistor and a nonlinear capacitor in parallel. We shall do this for the case where a conjugate match exists at the terminals by letting ?// = ^i* and y^ = 1)1*. Equations (11) and (12) may thus be written (Fii - 2/i*)(F22 + ?/2) = (Fn + VxWii - y2*) = F12F21 (16) When this is multiplied out, letting ¥„,„ = Gmn + jBmn , and the real and imaginary parts set equal as indicated by the first equality we ob- tain Gng2 = G-iigx and giiBn + &i) = QiiB-n + 62). In (8) and (9) it is seen that Gn = (722 = (?o and that ^22 is positive in equations (8) and negative in equations (9). We thus obtain gi = g2 bi + wiCo = 62 ± C02C0 (17) where the upper symbol of the ± sign is used in the noninverting case and the lower symbol in the inverting case. When the real and imaginary parts are set equal as indicated by the second equality in (16) we obtain, using the results in (17), g' = Go' - Gi ± C01C02C1' - B' (18) where g = gi = g2 (19) and B = bl + COiCo = 62 ± COsCo = ± ^ (C02 ± C0l)(7l (20) 2G-0 These results may be put in (14) to obtain the modulator gain. Since a conjugate match exists at the terminals of the 4-pole, this is the maxi- mum available gain. The result is MAO. = ,^::^^% (2.) For the converter, using equation (15) we obtain These results are valid only when a conjugate match exists at the ter- minals. For this to be possible, the right side of (18) must be positive. If it is negative no combination of values of gi and ^2 will result in a match. FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1409 It may be shown that if the slope of the voltage-current characteristic of the nonlinear resistor is always positive, then Gi/Go can never be greater than unity. (Reference 1, p. 410.) It is therefore convenient to normalize the above results with respect to Go . If we let — = p, CO-) COlCl = px, CO' 2C1 Go Go equations (18) through (22) become = X, Gi Go 7T = y, Go _ Gx ^ If ± px- - XIJ (l±p)i^ Go = ± :i±p)f pxz. MAG12 = MAGn = 62 ^ Jf + X- XIJ ±(1 ± p) '-^ ± xz 1 + Go + (1 ±p) xii_ y' + {9xY 1 + Q_ Go + (1±P)^ (23) (24) (25) (26) (27) In these equations, p is less than \ in the noninverting case and less than 1 in the inverting case. Ordinarily it will be very much less than 1. The value of z will be determined by the shape of the nonlinear capacitor characteristic. However z appears only in (25) where it influences the values of the matching susceptances so that it does not affect the con- ductance or gain. While we can be certain that y will have values be- tween 0 and 1, limitations on the value of x will depend on the particular device used. We will therefore assume that x may have any value. EFFECT OF NONLINEAR CAPACITOR We may now examine, in a general way, the manner in A\hich the non- linear capacitor influences the behavior of the 4-pole. Consider first the case where the nonlinear capacitor is absent. It is well known, and can be seen in the above equations by letting Go = Gi = 0, that the non- inverting and inverting cases are alike, that the 4-pole can always be matched and that the gain is the same in both directions and can never be greater than unity. In addition, the matching susceptances are zero and the gain is independent of frequency so that there is no limitation to the bandwidth. When the nonlinear capacitor is added, all but one of these conditions are changed. Equations (8) and (9) show that the non- 1410 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 inverting and inverting cases are different, (24) may become negative so that the 4-pole cannot ahvays be matched and (26) and (27) are dif- ferent so that the gains through the 4-pole are not the same in the two directions. Furthermore, (26) can be greater than unity so that modula- tors may have gain. However, as will be shown, the converter gain given by (27) is still restricted to values less than unity. It is also seen that the matching susceptances are no longer zero and that the gain varies with frequency so that the bandwidth is limited. If we remove the restriction that a conjugate match exists and operate the 4-pole between arbitrary admittances, it may be shown in (11) and (12) that the conductance of the 4-pole may become negative, and in (14) and (15) that the gain may have any value, however large. This is true for both noninverting and inverting modulators and converters. How- ever, we see in (14) and (15) that the ratio of the modulator gain to the converter gain is | F21/F12 1". This is greater than unity, so that for the same operating conditions the modulator gain will be greater than the converter gain. Although increased gain is possible, it is obtained at the expense of reduced bandwidth and increased sensitivity to changes in the terminating admittances, particularly in the case of converters. The present analysis will therefore be restricted to the case where a conjugate match exists. 1.0 0.9 0.8 0.7 0 6 0.5 0.4 0.3 0.2 0.1 \ N^ yv Go 1 \^ O.J \ 0 sV v i \s V^ \ \ ^ ^ x ^.9 V X^ ^ ^ ^ ^^ ^ ^ ^ == = ^^^ 1.0 |-^ / 1.1 y^""^ ^ ■'^ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 X Fig. 2 — Conductance contours of noninverting transducer. FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1411 1.0 0.9 0.8 0 7 0.6 0.5 0.4 I 0.3 0.2 0 1 \ N^ \ V \\ \ \ > s^ \ \^ !bv \ \^ ^ V ^ \ \ \; ^ "^^ 0.3 k \? ES X 07^ 05^ ^ ^-«^^ 0.9\ \ X ' ^ X 7 X 10 11 12 13 14 Fig. 3 — Conductance contours for inverting transducer. CONDUCTANCE AND GAIN VERSUS X AND y By assigning a value to p, curves may be plotted showing how the conductance and gain of the 4-pole change as the characteristics of the nonlinear resistor and nonlinear capacitor are varied. The particular case when /2 is about 160 times /i will be treated. This corresponds, for ex- ample, to an intermediate frequency of 70 mc and a local oscillator f re- fluency of 11,200 mc. Figs. 2 and 3 show the normalized conductance contours as functions of .T and y as given by (24) for the noninverting and inverting cases re- spectively. It wall be seen that in most instances, increasing the value of X causes g/Go to decrease. An exception occurs in the noninverting case (Fig. 2) when y is less than 2-\/p/(p + 1) or 0.157 where it is seen that increasing x causes g/Go to increase. When x and y have values corre- sponding to points above the g/Go = 0 curve, the 4-pole cannot be matched and (23) through (27) are not applicable. However, it will be noted that connecting a resistor across either the nonlinear elements or across the input and output terminals has the effect of increasing Go . By this means the 4-pole can always be reduced to the condition w^here it can be matched. Figs. 4 and 5 show the modulator gain contours as functions of x and y 1412 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 1 .u 0.9 0 8 0.7 0.6 0.5 0.4 0.3 \ V x^ \ \ \ \ \ \ \ \ \i \ \ \ \ \ -5 I \ N \^ ' 0 \^ ^^. "-^. \ ^ I """* — ^^ _, 0.2 0.1 0 ^*-^v^ — — ^. V \, 15DB\ 1 \ 3 4 5 6 7 B X 9 to \\ 12 13 14 Fig. 4 — Gain contours for noninverting modulators. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 \ V \ \ I \ \ \ \ \ \ \ \ \ \ \ \ \ \" \ \ 1 V -5 1 \ \ \ > \ C 3 \ \ X, \ . "^^ '^-'^. ^^ \l( 3 N ^- "^--. '•«*^ A 15DB ^. --^ -s. 3 4 5 6 7 6 9 10 11 12 13 14 X Fig. 5 — Gain contours for inverting modulator. FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 1413 ■--, \. -5DB X \ .^> -10[ 3B ^ ^-< NONINVERTING / ^ -.^" ^^^ --' 'fo / .'-- // r"^ /f ^ / // ■V // / '/ / / / 0 - 2 4 6 a 10 12 14 16 18 20 22 X Fig. 7 — Gain of noninverting modulator, q/Gq = 0.3. computed using values of y which make qIG^ = 0.3 at midband. They are thus near the largest gains obtainable for a given value of x. The matching susceptances were assumed to be a single inductance or capaci- tance connected across the terminating resistors. Co/Ci was arbitrarily assumed to have a value of 2. The procedure used was to compute y, hi/ Go , ho/ Go and the maximum available gain at midband using (24), (25) and (26); hi/Go and hi/Go were then multiplied by the appropriate frequency ratio to obtain the terminating susceptances at 50 and 90 mc and the gain at these frequencies was then computed using (14). Figs. 7 and 8 show that with the simple matching susceptances used, the gain variation across the band increases as the gain increases. For the same midband gain, the variation in the inverting case is somewhat greater than in the noninverting case. The gain is thus limited by the bandwidth requirements . When the gain at 50, 70 and 90 mc is calculated using larger values of g/Go it is found that as g/Go increases the gain variation across the band decreases. In the limit the least variation is obtained when y is FREQUENCY CONVERSION BY A NONLINEAR ADMITTANCE 24 22 20 18 16 1415 14 ^t2 u LU Q 10 Z 8 < (J 2 0 -2 -4 ./ y 90 MC ,/ yA . ^ / :^ ,-'-' y^ -y ^-^ 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 16, 18, 19, 29, 30) (a) I (b) II (c) III XiXiXzX-iXi X^XiXzX-iXi X^XiXzXiXi 02 OOO-OV 0246 00--0V 04 00-00\/ 028 10 O-O-OV 08 0-OOOV 02 16 18 -00-0 0 16 -OOOOa/ 048 12 0--00V 0 0 0 0 0 0 V 2 4 8 16 0 0 0 1 0 v 0 0 1 0 0 V 0 1 0 0 0 V 1 0 0 0 0 a/ 6 10 12 18 0 0 1 1 0 V 0 1 0 1 0 v 0 1 1 0 0 V 1 0 0 1 0 V 4 12 0 - 1 0 0 V 8 10 0 1 0 - 0 v 8 12 0 1 - 0 0 V 7 OOlllV 16 18 100-OV 11 0 1 0 1 1 V 13 01101a/ 67 0011- 14 01110a/ 6 14 0-llOv/ 19 10 0 11a/ 10 U O 1 O 1 - 10 14 0 1 - 1 O V 29 1 1 1 0 1 V 12 13 0 110- 30 11110a/ 12 14 0 1 1 - 0 V 26 00-lOV 26 10 14 O-'lOV 2 10 0-010a/ 46 12 14 0-1-OV 2 18 -0010a/ 8 10 12 14 01--0\/ 4 6 0 0 1 - 0 V 18 19 10 0 1- 13 29 14 30 - 1 1 0 1 - 1 1 1 0 (d) IV XfiXiXzXiXl. 02468 10 12 14 0 0 MINIMIZATION OF BOOLEAN FUNCTIONS 1423 The first step in the revised method for determining prime implicants is to list in a column, such as that shown in Table 11(a), the binary equivalents of the decimal numbers which specify the function. It is expedient to order these binary numbers so that any numbers which contain no I's come first, followed by any numbers containing a single 1, etc. Lines should be drawn to divide the column into groups of binary numbers which contain a given number of I's. The theorem stated above is applied to these binary numbers by comparing each number with all the numbers of the next lower group. Other pairs of numbers need not be considered since any two numbers which are not from adjacent groups must differ in more than one binary digit. For each number w^hich has I's wherever the number (from the next upper group) with which it is being compared has I's, a new character is formed according to the theorem. A check mark is placed next to each number which is used in forming a new character. The new characters are placed in a separate column, such as Table 11(b), which is again divided into groups of char- acters which have the same number of I's. The characters in this new column will each contain one dash. After each number in the first column has been considered, a similar process is carried out for the characters of column two. Two characters from adjacent groups can be combined if they both have their dashes ill the same position and if the character from the lower group has I's wherever the upper character has I's. If any combinations are possible the resulting characters are placed in a third column such as Table 11(c), and the Column II characters from which the new characters are formed are checked. All the characters in this third column will have two dashes. This procedure is repeated and new columns are formed, Table 11(d), until no further combinations are possible. The unchecked characters, which have not entered into any combinations, represent the prime implicants. Each binary character is labeled with the decimal equivalents of the binary numbers which it represents (see note in Example 3.1). These decimal numbers are arranged in increasing arithmetic order. For a character having one dash this corresponds to the order of its formation : When two binary numbers combine, the second number always contains all the i's of the first number and one additional 1 so that the second number is always greater than the first. Characters having two dashes can be formed in two ways. For example, the character (0, 2, 4, 6) can be formed either by combining (0, 2) and (4, 6) or by combining (0, 4) and (2, 6) as given in Table III. Similarly, there are three ways in which a character having three dashes can be formed (in Table II the 0, 2, 4, 1424 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table III — Example of the Two Ways of Forming A Character Having Two Dashes 0 0 0 0 0 2 4 0 0 10 0 10 0 0 2 0 4 0 0-0 0-00 0 2 4 6 (0426 0 - 0 - - 0 -0) 2 6 4 6 0-10 0 1-0 0 110 6, 8, 10, 12, 14 character can be formed from theO, 2, 4, 6, and 8, 10, 12, 14 characters or the 0, 2, 8, 10, and 4, 6, 12, 14 characters or the 0, 4, 8, 12 and 2, 6, 10, 14 characters), four ways in which a character having four dashes can be formed, etc. In general, any character can be formed by combining two characters whose labels form an increasing sequence of decimal numbers when placed together. It is possible to shorten the process of determining prime implicants by not considering the combination of any characters whose labels do not satisfy this requirement. For example, in Table 11(b) the possibility of combining the (0, 4) character with either the (2, 6), (2, 10) or the (2, 18) character need not be considered. If the process is so shortened, it is not sufficient to place check marks next to the two characters from which a new character is formed; each member of all pairs of characters which would produce the same new character w^hen combined must also receive check marks. More simply, when a new character is formed a check mark is placed next to all characters whose labels contain only decimal numbers which occur in the label of the new character. In Table II, when the (0, 2, 4, 6) character is formed by combining the (0, 2) and (4, 6) characters, check marks must be placed next to the (0, 4) and (2, 6) characters as well as the (0, 2) and (4, 6) characters. If the process is not shortened as just described, the fact that a character can be formed in several ways can serve as a check on the accuracy of the process. It is possible to carry out the entire process of determining the prime implicants solely in terms of the decimal labels without actually writing the binary characters. If two binary characters can be combined as de- scribed in this section, then the decimal label of one can be obtained from the decimal label of the other character by adding some power of two (corresponding to the position in which the two characters differ) to each number in the character's label. For example, in Table lib the label of the (4, G) (0 0 1 - 0) character can be obtained by adding 4 = (2^) to the numbers of the label of the (0, 2) (0 0 0 - 0) character. By searching for decimal labels which differ by a power of two, instead of binary char- acters which differ in only one position, the prime implicants can be MINIMIZATION OF BOOLEAN FUNCTIONS 1425 determined as described above without ever actually writing the binary characters. 4 PRIME IMPLICANT TABLES The minimum sum is formed by picking the fewest prime imphcants whose sum will equal one for all rows of the table of combinations for which the transmission is to equal one. In terms of the characters used in Section 3 this means that each number in the decimal specification of the function must appear in the label of at least one character which corresponds to a ms-term (term of the minimum sum). The ms-terms are selected from the prime implicants by means of a prime implicant table,* Table IV. Each column of the prime implicant table corresponds to a row of the table of combinations for which the transmission is to have the value one. The decimal number at the top of each column specifies the corresponding row of the table of combinations. Thus the numbers which appear at the tops of the columns are the same as those which specify the transmission. Each row of the prime implicant table represents a prime implicant. If a prime implicant equals "one" for a given row of the table of combinations, a cross is placed at the inter- section of the corresponding row and column of the prime implicant table. All other positions are left blank. The table can be written directly from the characters obtained in Section 3 by identifying each row of the table with a character and then placing a cross in each column whose number appears in the label of the character. It is convenient to arrange the rows in the order of the number of crosses they contain, with those rows containing the most crosses at the top of the table. Also, horizontal lines should be drawn partitioning the table into groups of rows which contain the same number of crosses, Table IV. If, in selecting the rows which are to correspond to ms-terms, a choice between two equally appropriate rows is required, the row hav- ing more crosses should be selected. The row with more crosses has fewer literals in the corresponding prime implicant. This choice is more obvious when the table is partitioned as suggested above. A minimum sum is determined from the prime implicant table by selecting the fewest rows such that each column has a cross in at least one selected row. The selected rows are called basis roivs, and the prime implicants corresponding to the basis rows are the ms-terms. If any column has only one entry, the row in which this entry occurs must be a basis row. Therefore the fir.st step in selecting the basis rows is to place * This table was first discussed by Quine."' However, no sj'stematic procedure for obtaining a minimum sum from the prime implicant table was presented. 1426 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table IV — Prime Implicant Table for the Transmission of Table II 0 2 4 8 16 6 10 12 18 7 11 13 14 19 29 30 B C D E F G H X X X X X X XXX X X X X X X an asterisk next to each row which contains the sole entry of any cohmm (rows A, B, C, D, E, G, H, in Table IV). A line is then drawn through all rows marked with an asterisk and through all columns in which these rows have entries. This is done because the requirement that these col- umns have entries in at least one basis row is satisfied by selecting the rows marked with an asterisk as basis rows. When this is done for Table IV, all columns are lined out and therefore the rows marked with asterisks are the basis rows for this table. Since no alternative choice of basis rows is possible, there is only one minimum sum for the transmis- sion described in this table. 5 ROW covering In general, after the appropriate rows have been marked with asterisks and the corresponding columns have been lined out, there may remain some columns which are not lined out; for example, column 7 in Table V(b). When this happens, additional rows must be selected and the columns in which these rows have entries must be lined out until all columns of the table are lined out. For Table V(b), the selection of either row B or row F as a basis row will cause column 7 to be lined out. However, row B is the correct choice since it has more crosses than row F. This is an example of the situation which was described earlier in connection with the partitioning of prime implicant tables. Row B is marked with two asterisks to indicate that it is a basis row even though it does not contain the sole entry of any column. The choice of basis rows to supplement the single asterisk rows be- comes more complicated when several columns (such as columns 2, 3, and 6 in Table VI (a)) remain to be lined out. The first step in choosing these supplementary basis rows is to determine whether any pairs of rows exist such that one row has crosses only in columns in which the MINIMIZATION OF BOOLEAN FUNCTIONS 1427 31 Table V — Determination of the Minimum Sum for 5" = E (0. 1. 2, 3, 7, 14, 15, 22, 23, 29, 31) (a) Determination of Prime Implicants 0 0 0 0 0 V 1 2 0 0 0 0 1 v/ 0 0 0 1 0 >/ 3 0 0 0 1 1 V 7 14 22 0 0 1 1 1 v/ 0 1 1 1 0 v/ 1 0 1 1 0 V 15 23 29 0 1 1 1 1 >/ 1 0 1 1 1 v/ 1 1 1 0 1 V 0 0 1 2 X5X4X3X2X1 X5X4X3X2X1 0 0 0 0 - v 0 0 0 - 0 v 0 12 3 7 15 23 31 0 0 0 - - --111 1 3 2 3 0 0 0 - 1 V 0 0 0 1 - v 3 7 0 0-11 1 1 1 1 1 >/ 7 15 0 - 1 1 1 V 7 23 - 0 1 1 1 V 14 15 0 111- 22 23 10 11- 15 31 - 1 1 1 1 V 23 31 1 - 1 1 1 V 29 31 111-1 (b) First Step in Selection of Basis Rows 1 2 3 7 14 22 15 23 29 31 A B C D E F 1 1 i : X i c : I ^ If n ■% r 1 t X * * (c) Minimum Sum r = 2 ((0, 1, 2, 3), (7, 15, 23, 31), (29, 31), (22, 23), (14, 15)] T — Xi'XtXi + X3X2X1 + X6X4X3X1 + X6X4'X3X2 + X6'X4X3X2 other member of the pair has crosses. Crosses in Hned-out cohimns are not considered. In Table VI (a), rows A and B and rows B and C are such pairs of rows since row B has crosses in columns 2, 3, and 6 and row A has a cross in column 6 and row C has crosses in columns 2 and 3. A convenient way to describe this situation is to say that row B covers rows A and C, and to write B3A,BZ)C.If row i is selected as a sup- plementary basis row and row i is covered by row j , which has the same total number of crosses as row i, then it is possible to choose row j as a basis row instead of row i since row j has a cross in each column in which row i has a cross. The next step is to hne out any rows which are covered by other rows in the same partition of the table, rows A and C in Table VI (a). If any 1428 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 195G Table VI — Prime Implicant Tables for 7" = Z (0. 1» 2, 3, 6, 7, 14, 22, 30, 33, 62, 64, 71, 78, 86) (a) Prime Implicant Table with Single Asterisk Rows and Corresponding Columns Lined Out 0 1 2 64 3 6 33 7 14 22 30 71 78 86 62 !i A B C D E F G H I A B C D E F G H I ■ c : X : X X X X X 1 1 1 XXX X 1 V V Y 1 1 * ^ r 1 3 C X 1 ■^ r ^ r 1 1 (b) Prime Implicant Table with Rows which are Covered by Other Rows Lined Out 0 1 2 64 3 6 33 / ' 14 22 30 71 78 86 62 \ 1 1 y V 1 f ■> X X X V J c ^ : c 1 , column now contains only one cross which is not lined out, columns 2 3, and 6 in Table VI (b), two asterisks are placed next to the row in which this cross occurs, row B in Table VI (b), and this row and all columns in which this row has crosses are lined out. The process of draw- ing a line through any row which is covered by another row and selecting each row which contains the only cross in a column is continued until it terminates. Either all columns will be lined out, in which case the rows marked with one or two asterisks are the basis rows, or each column will contain more than one cross and no row will cover another row. The latter situation is discussed in the following section. 6 PRIME implicant TABLES IN CYCLIC FORM If the rows and columns of a table which are not lined out are such that every column has more than one cross and no row covers another row, as in Table VI 1(b), the table will be said to be in cyclic form, or, in short. MINIMIZATION OF BOOLEAN FUNCTIONS 1429 Table VII — Determination of Basis Rows for a Cyclic Prime Implicant Table 'a) Selection of Single Asterisk Rows 0 4 16 12 24 19 28 27 29 31 A. B C D E F G H (c) Selection of Row 1 as a Trial Basis Row (Column 0) X X X X X X X X X X X X X X X I X X X X X 0 4 16 12 24 19 28 27 29 31 A B C D E F G H I J 1 r y-L, y *n — 1 V I X I r 1 LT ^ ■ X ' t 1 ' ** ** (b) Selection of Double Asterisk Rows 0 4 16 12 24 19 28 27 29 31 A B C D E F G H I J (d) Selection of Row 2 as a Trial Basis Row (Column 0) X X X X X X X X X X X X V V ' -'' 1 ' 1 1 1 1 0 4 16 12 24 19 28 27 29 31 A B C D E F G H I J 1 1 r^ \ " 1 ' y , ^ ' I ' ^ : ■J c ■( c . '^ 1 y I 1 * to be cyclic. If any column has crosses in only two rows, at least one of these rows must be included in any set of basis rows. Therefore, the basis rows for a cyclic table can be discovered by first determining whether any column contains only two crosses, and if such a column exists, by then selecting as a trial basis row one of the rows in which the crosses of this column occur. If no column contains only two crosses, then a column which contains three crosses is selected, etc. All columns in which the trial basis row has crosses are lined out and the process of lining out rows which are covered by other rows and selecting each row which contains the only cross of some column is carried out as described above. Either all columns will be lined out or another cyclic table will result. Whenever a cyclic table occurs, another trial row must be se- lected. Eventually all columns will be lined out. However, there is no guarantee that the selected rows are actually basis rows. The possibility exists that a different choice of trial rows would have resulted in fewer selected rows. In general, it is necessary to carry out the procedure of selecting rows several times, choosing different trial rows each time, so 1430 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 CT 0 that all possible combinations of trial rows are considered. The set of fewest selected rows is the actual set of basis rows. '| \\ Table VII illustrates the process of determining basis rows for a cyclic prime implicant table. After rows G and J have been selected u| |et( cyclic table results, Table VII (b). Rows A and B are then chosen as al pair of trial basis rows since column 0 has crosses in only these two rows. , The selection of row A leads to the selection of rows D and E as given in ; Table VII (c). Row A is marked with three asterisks to indicate that it is a trial basis row. Table VII (d) illustrates the fact that the selection! of rows C and F is brought about by the selection of row B. Since bothi sets of selected rows have the same number of rows (5) they are both sets of basis rows. Each set of basis rows corresponds to a different min- imum sum so that there are two minimum sums for this function. Sometimes it is not necessary to determine all minimum sums for the transmission being considered. In such cases, it may be possible to shorten the process of determining basis rows. Since each column must have a cross in some basis row, the total number of crosses in all of the basis rows is equal to or greater than the number of columns. Therefore, the number of columns divided by the greatest number of crosses in any row (or the next highest integer if this ratio is not an integer) is equal to the fewest possible basis rows. For example, in Table VII there are ten columns and two crosses in each row. Therefore, there must be at least 10 divided by 2 or 5 rows in any set of basis rows. The fact that there are only five rows selected in Table VII (c) guaran- tees that the selected rows are basis rows and therefore Table VII (d) is unnecessary if only one minimum sum is required. In general, the process of trying different combinations of trial rows can be stopped as soon as a set of selected rows which contains the fewest possible number of basis rows has been found (providing that it is not necessary to discover all minimum sums) . It should be pointed out that more than the minimum number of basis rows may be required in some cases and in these cases all combinations of trial rows must be considered. A more accurate lower bound on the number of basis rows can be obtained by considering the number of rows which have the most crosses. For example, in Table VI there are 15 columns and 4 crosses, at most, in any row. A lower bound of 4 {—- = 3f ) is a little too optimistic since there are only three rows which contain four crosses. A more realistic lower bound of 5 is obtained by noting that the rows which have 4 crosses can provide crosses in at most 12 columns and that at least two additional rows containing two crosses are necessary to provide crosses in the three remaining col- umns. MINIMIZATION OF BOOLEAN FUNCTIONS 1431 CYCLIC PRIME IMPLICANT TABLES AND GROUP INVARIANCE It is not always necessary to resort to enumeration in order to deter- ne all minimum sums for a cyclic prime implicant table. Often here is a simple relation among the various minimum sums for a trans- nission so that they can all be determined directly from any single ninimum sum by simple interchanges of variables. The process of select- ng basis rows for a cyclic table can be shortened by detecting before- aand that the minimum sums are so related. An example of a transmission for which this is true is given in Table VIII. If the variables a'l and x-2 are interchanged, one of the minimum sums is changed into the other. In the prime implicant table the inter- change of Xi and Xz leads to the interchange of columns 1 and 2, 5 and 6, 9 and 10, 13 and 14, and rows A and B, C and D, E and F, G and H. The transmission itself remains the same after the interchange. In determining the basis rows for the prime imphcant table, Table VIII (d), either row G or row H can be chosen as a trial basis row. If row G is selected the i-set of basis rows will result and if row H is selected the ii-set of basis rows will result. It is unnecessary to carry out the procedure of determining both sets of basis rows. Once the i-set of basis rows is known, the ii-set can be determined directly by interchanging the Xi and X2 variables in the i-set. Thus no enumeration is necessary in order to determine all minimum sums. In general, the procedure for a complex prime implicant table is to determine whether there are any pairs of variables which can be inter- changed without effecting the transmission. If such pairs of variables exist, the corresponding interchanges of pairs of rows are determined. A trial basis row is then selected from a pair of rows which contain the only two crosses of a column and which are interchanged when the varia- bles are permuted. After the set of basis rows has been determined, the other set of basis rows can be obtained by replacing each basis row by the row with which it is interchanged w^hen variables are permuted. If any step of this procedure is not possible, it is necessary to resort to enumeration. In the preceding discussion only simple interchanges of variables have been mentioned. Actually all possible permutations of the contact varia- bles should be considered. It is also possible that priming variables or both priming and permuting them will leave the transmission unchanged. For example, ii T = Xi Xs Xo Xi + x/ Xs x-/ xi , priming all the variables leaves the function unchanged. Also, priming Xi and x^ and then inter- changing X4 and x^ does not change the transmission. The general name for this property is group invariance. This was discussed by Shannon.^ 1432 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 A method for determining the group invariance for a specified trans^ mission is presented in "Detection of Group Invariance or Total Sym; metry of a Boolean Function."* 8 AN APPROXIMATE SOLUTION FOR CYCLIC PRIME IMPLICANT TABLES It has not been possible to prove in general that the procedure pre sented in this section will always result in a minimum sum. However, this procedure should be useful when a reasonable approximation to a minimum sum is sufficient, or when it is possible to devise a proof to! show that the procedure does lead to a minimum sum for a specific trans- mission (such proofs were discussed in Section 6). Since this procedure is much simpler than enumeration, it should generally be tested beforef resorting to enumeration. The first step of the procedure is to select from the prime implicant table a set of rows such that (1) in each column of the table there is a cross from at least one of the selected rows and (2) none of the selected rows can be discarded without destroying property (1). Any set of rows having these properties will be called a consistent row set. Each consistent row set corresponds to a sum of products expression from which no product term can be eliminated directly by any of the theorems of Boolean Algebra. In particular, the consistent row sets having the fewest members correspond to minimum sums. The first step of the procedure to be described here is to select a consistent row-set. This is done by choosing one of the columns, counting the total number of crosses in each row which has a cross in this column, and then selecting the row with the most crosses. If there is more than one such row, the topmost row is arbitrarily selected. The selected row is marked with a check. In Table IX, column 30 was chosen and then row A was selected since rows A and Z each have a cross in column 30, but row A has 4 crosses while row Z has only 2 crosses. The selected row and each column in which it has a cross is then lined out. The process just described is repeated by selecting another column (which is not lined out). Crosses in lined-out columns are not counted in determining the total number of crosses in a row. The procedure is repeated until all columns are lined out. The table is now rearranged so that all of the selected rows are at the top, and a line is drawn to separate the selected rows from the rest. Table X results from always choosing the rightmost column in Table IX. If any column contains only one cross from a selected row, the single selected-row cross is circled. Any selected row which does not have any See page 1445 of this issue. MINIMIZATION OF BOOLEAN FUNCTIONS 1433 Table VIII — Determination of the Minimum Sums for T = J2iO, 1, 2, 5, 6, 7, 9, 10, 11, 13, 14, 15) (a) (c) 0:4X3X2X1 f 0 0 0 0 0 V ' 1 2 0 0 0 1 v 0 0 1 0 v , 5 6 9 ■ 10 0 1 0 1 V 0 1 1 0 V 1 0 0 1 V 1 0 1 0 V 7 0 1 1 1 V 11 1 0 1 1 V 13 1 1 0 1 V 14 1 1 1 0 v 15 1 1 1 1 v (b) 1 X4X3X2X1 0 1 0 0 0- : i 0 2 0 0 - 0 1 5 0 _ 0 1 V 1 9 - 0 0 1 V 2 . 6 0 - 1 0 V 9 10 - 0 1 0 V 5 7 0 1 - 1 V 5 13 - 1 0 1 V 6 7 0 1 1 - V 6 14 - 1 1 0 V 9 11 1 0 -1 V 9 13 1 — 0 1 V ;10 11 1 0 1 - V |10 14 1 - 1 0 V 7 15 _ 1 1 1 V 11 15 1 - 1 1 V 13 15 1 1 -1 V 14 15 1 1 1 - V A B C D E F G H 1 2 5 9 13 6 10 14 X4X3X2X1 --01 --10 5 6 9 10 7 13 15 7 14 15 11 13 15 11 14 15 -1-1 - 1 1 - 1 - - 1 1 - 1 - (d) 0 1 2 5 6 9 10 7 11 13 14 15 X X X X X X X X X X X X X X X X X X X X X X X X X X X X (e) (i) (0, 1) + (2, 6, 10, 14) + (5, 7, 13, 15) + ( 9, 11, 13, 15) (ii) (0, 2) + (1, 5, 9, 13) + (6, 7, 14, 15) + (10, 11, 14, 15) Ti = X4'X3'X2' + X2X1' + X3X1 + X4X1 Tii = X4'x3'xi' + X1X2' + X3X2 + X4X2 of its crosses circled can be discarded without violating the requirement that each column should have at least one cross from a selected row. Rows with no circled entries are discarded (one by one, since removal of one row may require more crosses to be circled) until each selected row contains at least one circled cross. This completes the first step. The se- lected rows now correspond to a first approximation to a minimum sum. A check should be made to determine whether the number of selected rows is equal to the minimum number of basis rows. In Table X there are at most 4 crosses per row and 26 columns so that the minimum num- 1434 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table IX — Table of Prime Implicants for Transmission ^ = Z (0> 1. 2, 4, 5, 6, 7, 8, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30) The selection of row A is shown 0 1 2 4 8 16 5 6 9 18 20 24 7 11 13 14 19 21 25 26 28 15 23 27 29 30 A B — X 1 X C k D X E X F X G X H X I X J X K X L X M X X X N X X X 0 X X X P X X X Q X X X R X X X S X X X X T X X X X U X X X X V X X X X W X X X X X X X X X Y Z X X X X X X X X X X X — X I X X i 2' ber of basis rows is [-^] + 1 = 7. Since the number of selected rows is 9 there is no guarantee that they correspond to a minimum sum. If such an approximation to a minimum sum is not acceptable, then further work is necessary in order to reduce the number of selected rows. For each of the selected rows, a check is made of whether any of the rows in the lower part of the table (non-selected rows) have crosses in all columns in which the selected row has circled crosses. In Table X row E has a circled cross only in column 19; since row Y also has a cross in coluimi 19 rows E and Y are labeled "a". Other pairs of rows which have the same relation are labeled with lower case letters, b, c, d, e in Table X. It is possible to interchange pairs of rows which are labeled Avith the same lower case letter without violating the requirement that each column must contain a cross from at least one selected row. If a non-selected row is labeled with two lower case letters then it may be possible to replace two selected rows by this one non-selected row and thereby reduce the MINIMIZATION OF BOOLEAN FUNCTIONS 1435 ej .O « T3 aj o^ 4j a o ec 05 c^ r^ (M w (M >C r-H on c^ «o o c^ "Z NH >« z (N o 1— 1 1—1 p-^ H tf 05 < 1— I Ph •^ « 1— t w £2 < n H © X © XXX X X X © X ® X X © XX X X X X X X X XX X © XX X ® X © X © XXX X © X X X X X © XX XXX X © X XX X X X XX X X XX © X X ® XX XXX X X X XX X © X X © XX XX © X © X X XXX XXX -jJWfeOH-iWH;::)^ moQW^ jS^OPiO'tf a5>X>^N 1436 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 o CO Q CO < ^ Q (M «t^ ►J 05 w « f-H ^ e«5 a '4. 1-H < ^ t>. \^ ^ •^ W C^ cc o is "^ H 05 X »o H n CD < ^ H I 00 X a IN n < -^ © X © XXX © X X © X © X X © XX X © XX X ® XXX X ® XX X © X © X © X XX X © X © XXX ® XX XXX X © XXX © X X © X X © X X X X X X G XX X X X © XX XX © X (^ X X XXX XXX <*^>^0>-^\^U'^ Wp^WP30QWH3S;z;OPHC?tf a!>Xts5 MINIMIZATION OF BOOLEAN FUNCTIONS 1437 total number of selected rows (a check must be made that the two selected rows being removed do not contain the only two selected-row crosses in a column). In Table X no such interchange is possible. Next a check should be made as to whether two of the labeled non- selected rows can be used to replace three selected rows, etc. In Table X rows Y(a) and J(b) can replace rows E(a), F(b) and K or rows Y(a) and P(d) can replace rows E(a), T(d) and K. The table which results from replacing rows E, F and K by rows Y and J is given in Table XL The number of selected rows is now 8 which is still greater than 7, the minimum number possible. This table actually represents the minimum sum for this transmission even though this cannot be proved rigorously by the procedure being described. If it is assumed that a minimum sum can always be obtained by ex- changing pairs of selected and nonselected rows until it finally becomes possible to replace two or more selected row^s by a single selected row, then it is possible to show directly that the Table XI does represent a minimum sum. The only interchange possible in Table XI is that of rows T and P. If this replacement is made then a table results in which only rows J and F can be interchanged. Interchanging rows J and F does not lead to the possibility of interchanging any new pairs of rows so that this process cannot be carried any further. On the basis of experience with this method it seems that it is not necessary to consider interchanges mvolving more than one non-selected row. Such interchanges have only been necessary in order to obtain al- ternate minimum sums; however, no proof for the fact that they are never required in order to obtain a minimum sum has yet been dis- covered. 9 AN ALTERNATE EXACT PROCEDURE It is possible to represent the prime implicant table in an alternative form such as that given in Table XII (b). From this form not only the minimum sums but also all possible sum of products forms for the trans- mission which correspond to consistent row sets can be obtained sys- tematically. For concreteness, this representation will be explained in terms of Table XII. Since column 0 has crosses only in rows B and C, any consistent row set must contain either row B or row C (or both). Similarly, column 3 requires that any consistent row set must contain either row D or row E (or both). When both columns 0 and 3 are con- sidered they require that any consistent row set must contain either row B or row C (or both) and either row D or row E (or both). This requirement can be expressed symbolically as (B -f C) (D -f E) where 1438 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table XII — Derivation of the Minimum Sums FOR the Transmission T = E (0, 3, 4, 5, 6, 7, 8, 10, 11) (a) Table of Prime Implicants Xi Xi XiX%X\ X3X2X1 Xi'XiXi Xz'x^Xx XiXa'Xi A B C D E F 0 3 4 5 6 7 8 10 11 X X X X X X X X X X X X X X (b) Boolean Representation of Table (B + C)(D + E)(A + B)(A)(A)(A + D)(C)(F)(E + F) (c) Consistent Row Sets (A, C, F, D), (A, C, F, E) T = Xi'xz + xz'xi'xi' + xaz'x2 + a;4'x2a;i T = Xa'X3 + Xs'Xi'Xi' + X^Xz'Xi + Xi'XiXx "or" (non-exclusive) and multiplication signifies addition stands for "and." This expression can be interpreted as a Boolean Algebra expres- sion and the Boolean Algebra theorems used to simplify it. In particular it can be "multiplied out": (B + C) (D + E) = BD + BE + CD + CE This form is equivalent to the statement that columns 0 and 3 require that any consistent row set must contain either rows B and D, or rows B and E, or rows C and D, or rows C and E. The complete requirements for a consistent row set can be obtained directly by providing a factor for each column of the table. Thus for Table XII the requirements for a consistent row set can be written as: (B + C)(D + E)(A + B)(A)(A)(A + D)(C)(F)(E + F) By using the theorems that A-(A + D) = A and A- A = A, this can be simplified to ACF(D + E). Thus the two consistent row sets for this table are A, C, F, D and A, C, F, E and since they both contain the same number of rows, they both represent minimum sums. This is true only because rows D and E contain the same number of crosses. In general, each row should be assigned a weight w = n — \og,2k, where n is the number of variables in the transmission being considered and MINIMIZATION OF BOOLEAN FUNCTIONS 1439 Table XIII — Determination of the Minimum Sums for the Prime Implicant Table of Table VII by Means of THE Boolean Representation (a) Boolean representation of the Prime Implicant Table of Table VI (A+B) (A+C) (B+D) (C+E) (D+F) (G) (E+F+H) (G+I) (H+J) (1+ J) (b) The expression of (a) after multiplying out. (The terms in italic correspond to minimum sums) ADEJG + ACDFJG + ACDHJG + ADEHIG + ACDHIG + ABEFJG + ABEFHIG + BCDEJG + BCDHJG + BCDHIG + BCFJG + BCFHIG -G- A (c) Tree circuit equivalent of (b) J B E---F- ---D ■--H- --H- --I --J --I --J -E--- -B--- -C D- -_F ---H ---J "1 ---E ---H --I --I --J ---H 1 ---J 5 6 5 5 5 5 4 V 5 5 5 5 4 V k is the number of crosses in the row.* To select the minimum sums, the sum of the weights of the rows should be calculated for each row set containing the fewest rows. The row sets having the smallest total weight correspond to minimum sums. If, instead of the minimum sum, the form leading to the two-stage diode-logic circuit requiring fewest diodes is desired, a slightly different procedure is appropriate. To each row set is assigned a total weight equal to the sum of the weights of the rows plus the number of rows in the set. The desired form then corresponds to the row set having the smallest total weight. The procedure for an arbitrary table is analogous. A more compli- cated example is given in Table XIII. In this example the additional * n-log2 k is the number of literals in the prime implicant coriesponding to a row containing k crosses. 1440 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 theorem (A + B)(A + C) = (A + BC) is useful. This example shows that for a general table the expressions described in this Section and the multipHcation process can become very lengthy. However, this pro- cedure is entirely systematic and may be suitable for mechanization. Since the product of factors representation of a prime implicant table is a Boolean expression, it can be interpreted as the transmission of a contact network. Each consistent row set then corresponds to a path through this equivalent network. By sketching the network directly from the product of factors expression, it is possible to avoid the multi- plication process. In particular the network should be sketched in the form of a tree, as in Table XIII (c) and the Boolean Algebra theorems used to simplify it as it is being drawn. For hand calculations, this method is sometimes easier than direct multiplication. I 10 d-TERMS In Section 1 the possibility of having rf-entries in a table of combina- tions was mentioned. Whenever there are combinations of the relay conditions for which the transmission is not specified, f/-entries are placed in the T-column of the corresponding rows of the table of combinations. Table XIV — Determination of the Minimum Sum for the Transmission T = X)(5, 6, 13) + f/(9, 14) Where 9 and 14 are the cI-Terms, (d) (a) Determination of Prime Implicants Xi Xs X2X1 5 0 1 0 1 V 6 0 1 1 0 V 9 1 0 0 1 V 5 13 6 14 9 13 13 1 1 0 1 V (d) 14 1 1 1 0 V (b) Prime Implicant Table 5 6 13 * X X * X X X4X3 ^"2X1 - 1 0 1 - 1 1 0 1-01 (c) Basis rows: (5, 13), (6, 14) (d) T = XaXi'xi + Tso-oa-i' MINIMIZATION OF BOOLEAN FUNCTIONS 1441 The actual values (0 or 1) of these d-entries are chosen so as to simplify the form of the transmission. This section will describe how to modify the method for obtaining a minimum sum when the table of combina- |{ tions contains rf-entries. The p-terms which correspond to rf-entries in the table of combinations will be called d-terms. These d-terms should be included in the list of p-terms which are used to form the prime implicants. See Table XIV. However, in forming the prime implicant table, columns corresponding to the d-terms should not be included. Table XlV(b). The d-terms are used in forming the prime implicants in order to obtain prime impli- cants containing the fewest possible literals. If columns corresponding to the f/-terms were included in forming the prime implicant table this would correspond to setting all the rf-entries in the table of combina- tions equal to 1. This does not necessarily lead to the simplest minimum sum. In the procedure just described, the rf-entries will automatically be set equal to either 0 or 1 so as to produce the simplest minimum sum. For the transmission of Table XIV the 14 d-entry has been set eciual to I and the 9 c^-entry has been set equal to 0. II NON-CANONICAL SPECIFICATIONS A transmission is sometimes specified not by a table of combinations or a canonical expansion, but as a sum of product terms (not necessarily prime implicants). The method described in Section 3 is applicable to such a transmission if the appropriate table of combinations (decimal specification) is first obtained. However, it is possible to modify the procedure to make use of the fact that the transmission is already partly reduced. The first step is to express the transmission in a table of binary characters such as Table XVa. Then each pair of characters is examined to determine whether any different character could have been formed from the characters used in forming the characters of the pair. For example, in Table XV (a) a (1) (00 00 1) was used in forming the (0, 1)(0000-) character and a (3) (000 1 1) was used in forming the (3, 7)(0 0 - 1 1) character. These can be combined to form a new char- acter (1, 3) (000- 1). The new characters formed by this process are listed in another column such as Table XV (b). This process is continued until no new characters are formed. In examining a pair of characters, it is sufficient to determine whether there is only one position where one character has a one and the other character has a zero. If this is true a new character is formed which has a dash in this position and any other position in which both characters have dashes, and has a zero (one) in any position in which either charac- 1442 THE BELL SYSTEM TECHXICAL JOURNAL, NOVEMBER 1956 Table XV — Determination of the Prime Implicants for the Transmission of Table XV Specified as a Sum of Product Terms (a) Specification (b) Characters Derived from (a) Xa Z4 Z3a;2 a:i a;5 X4 3:3 a;2 a^i 0 1 0 0 0 0 - v/ 0 2 0 0 0 - 0 V 3 7 0 0-11 14 15 0 111- 22 23 10 11- 29 31 111-1 1 3 0 0 0 - 1 V 2 3 0 0 0 1 - V 7 15 0 - 1 1 1 V 7 23 - 0 1 1 1 V 15 31 - 1 1 1 1 x/ 23 31 1 - 1 1 1 \' (c) Characters De ivcd from (a) and (b) XiX^XzX-iXi 0 12 3 7 15 23 31 0 0 0 - - - - 1 1 1 ter has a zero (one). In Table X\'a the (0, 1) character has a zero in the .r2-position while the (3, 7) character has a one in the .ro-position. A new character is fornied (1, 3) which has a dash in ihe .<-2-p()sition. This rule for constructing new characters is actually a generalization of the rule used in Section 3 and corresponds to the theorem. .ri.r2 + .r/.rii = XiX-s + .ri'.r;5 + .r2.r3 . Repeated application of this rule will lead to the complete set of prime implicants. As described in Section 3, any character which has all of the numbers of its decimal label appearing in the label of another character should be checked. The unchecked characters then represent the prime implicants. The process described in this section was discussed fi'om a slightly different point of view by Quine.^ 12 summary and conclusions In this paper a method has been presented for writing any transmis- sion as a minimum sum. This method is similar to that of Quine; how- ever, several significant improvements have been made. The notation has been simplified by using the symbols 0, 1 and - instead of primed and unprimed variables. While it is not completeh^ new in itself, this notation is especially appropriate for the arrangement of terms used in determining the prime implicants. Listing the terms in a column which is partitioned so as to place terms containing the same number of 1 's in the same partition reduces materially the labor involved in determining the prime implicants. Such a list retains some of the advantage of the arrangement of squares in the Karnaugh Chart without reciuiring a geometrical representation of an n-dimensional hj^percube. Since the MINIMIZATION OF BOOLEAN FUNCTIONS 1443 l)i-ocodure for determining the i)rinie iniplicants is completely systematic it is capable of being programmed on a digital computer. The arrange- ment of terms introduced here then results in a considerable saving in both time and storage space over previous methods, making it possible to solve larger problems on a given computer. It should be pointed out that this procedure can be programmed on a decimal machine by using the decimal labels instead of the binary characters introduced. A method was presented for choosing the minimum sum terms from the list of prime iniplicants by means of a table of prime implicants. This is again similar to a method presented l:)y Quine. Howe\'er, Quine did not give any systematic procedure for handling cyclic prime impli- cant tables; that is, tables with more than one cross in each column. In this paper a procedure is given for obtaining a minimum sum from a cyclic prime implicant table. In general, this procedure requires enumera- tion of several possible minimum sums. If a transmission has any non- trivial group invariances it may be possible to avoid enumeration or to reduce considerably^ the amount of enumeration necessary. A method for doing this is given. The process of enumeration used for selecting the terms of the mini- mum sum from a cyclic prime implicant table is not completely satis- factory since it can be quite lengthy. In seeking a procedure which does not require enumeration, the method involving the group invariances of a transmission was discovered. This method is an improvement over complete enumeration, but still has two shortcomings. There are trans- missions which have no nontrivial group invariances but which give rise to cyclic prime implicant tables. For such transmissions it is still necessary to resort to enumeration. Other transmissions which do possess nontrivial group invariances still reciuire enumeration after the in- variances have been used to simplify the process of selecting minimum sum terms. More research is necessary to determine some procedure which will not require any enumeration for cyclic prime implicant tables. Perhaps the concept of group invariance can be generalized so as to apply to all transmissions which result in cyclic prime implicant tables. 13 ACKNOW'LEDGEMENTS The author wishes to acknowledge his indebtedness to Professor S. H. Caldwell, Professor D. A. Huffman, Professor W. K. Linvill, and S. H. Unger with whom the author had many stimulating discussions. Thanks are due also to W. J. Cadden, C. Y. Lee, and G. H. Mealy for their helpful comments on the preparation of this paper. 1444 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 This research was supported in part by the Signal Corps; the Office of Scientific Research, Air Research and Development Command; and the Office of Naval Research. BIBLIOGRAPHY 1. Karnaugh, M., The Map Method for Synthesis of Combinational Logic Cir- cuits, Trans. A.I.E.E., 72, Part I pp. 593-598, 1953. 2. Keister, W., Ritchie, A. E., Washburn, S., The Design of Switching Circuits, New York, D. Van Nostrand Company, Inc., 1951. 3. Shannon, C. E., A Sj^mbolic Analysis of Relay and Switching Circuits, Trans. A.I.E.E., 57, pp. 713-723, 1938. 4. Shannon, C. E., The Synthesis of Two-Terminal Switching Circuits, B. S.T.J. , 28, pp. 59-98, 1949. 5. Staff of the Harvard Computation Laboratory, Synthesis of Electronic Com- puting and Control Circuits, Cambridge, Mass., 1951, Harvard University Press. 6. Quine, W. V., The Problem of Simplifying Truth Functions, The American Mathematical Monthly, 59, No. 8, pp." 521-531, Oct., 1952. 7. Quine, W. V., A Wav fo Simplify Truth Functions, The American Mathe- matical Monthly, 62, pp. 627-631, Nov., 1955. Detection of Group Invariance or Total Symmetry of a Boolean Function* By E. J. McCLUSKEY, Jr. (Manuscript received June 26, 1956) A method is presented for determining whether a Boolean function pos- sesses any group invariance; that is, whether there are any permutations or primings of the independent variables which leave the function unchanged. This method is then extended to the detection of functions which are totally symmetric. 1 GROUP INVARIANCE For some Boolean transmission functions (transmissions, for short) it is possible to permute the variables, or prime some of the variables, or both permute and prime variables without changing the transmission. The following material presents a method for determining, for any given transmission, which of these operations (if any) can be carried out with- out changing the transmission. The permutation operations will be represented symbolically as fol- lows: Si2z...nT will represent the transmission T with no variables permuted 8213.. -nT will represent the transmission T with the xi and X2 variables interchanged, etc. Thus *Si432T(.x-i , X2 , xs , X4) = T(xi , Xi , Xs , X2') The symbolic notation for the priming operation will be as follows: Noooo-.-oT will represent the transmission T with no variables primed A^ono. --oT will represent the transmission T with the .r2 and ;i;3 variables primed, etc. Thus NiowT(xi , :r2 , X3 , Xa) = T(xi, X2 , Xs, Xi). The notation for the priming operator can be shortened by replacing the binary subscript on N by its decimal equivalent. Thus N9T is equiv- * This paper is derived from a thesis submitted to the Massachusetts Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Science on April 30, 1956. 1445 . 144G THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 Table I — Transmission Matrices Showing Effect of Interchanging or Priming Variables (a) Tr ansmission IVIatrix (b) Transmission jNIatrix with a-3 and x^ columns interchanged (c) Transmission Ma- trix with entries of the x% and Xi cokimns primed 0 1 2 9 10 11 Xi X% Xz Xi 0 0 0 0 0 0 0 1 0 0 10 10 0 1 10 10 10 11 0 2 1 10 9 11 X\ Xi Xi Xz 0 0 0 0 0 0 10 0 0 0 1 10 10 10 0 1 10 11 3 2 1 10 9 8 X\ X2 Xz' Xi' 0 0 11 0 0 10 0 0 0 1 10 10 10 0 1 10 0 0 alent to NwoiT. The permutation and priming operators can be combined. For example, S2mN3T(xi , X2 , xs , Xi) = T{x2 , Xi , x^, Xi) The symbols SiNj form a mathematical group, ^ hence the term group invariance. The problem considered here is that of determining which A^,- and Sj satisfy the relation NiS/F = T for a given transmission T. Since there are only a finite number of different Ni and Sj operators it is possible in principle to compute NiSjT for all possible NiSj and then select those NiSj for which NiSjJ' = T. If T is a function of n variables, there are n! possible Sj operators and 2" .V, operators so that there are n!2" pos- sible combinations of N'iSj . Actually, if NiSjT = T then NiT must equal SjT''^^ so that it is only necessary to compute all NiT and all Sj7\ For /I = 4, n! = 24 and 2" = 16 so that the number of possibilities to be considered is quite large even for functions of only four variables. It is possible to avoid enumerating all NiT and SjT by taking into account certain characteristics of the transmission being considered. The first step in determining the group invariances of a transmission is the same as that foi finding the prime implicants.* The binary equiva- lents of the decimal numbers which specify the transmission are listed as in Table 1(a). This list of binary numbers will be called the transmis- sion matrix. When two variables are interchanged, the corresponding columns of the transmission matrix are also interchanged, Table 1(b). When a variable is primed, the entries in the corresponding column of the transmission matrix are also primed, 0 replaced by 1 and 1 replaced by 0, Table 1(c). If an NiSj operation leaves a transmission unchanged then the cor- * Minimization of Boolean Functions, see page 1417 of this issue. GROUP INVAKIANCE OR TOTAL SYMMETRY 1447 responding matrix operations will not change the transmission matrix aside from possibly reordering the rows. In other words, it should b^ possible to reorder the rows of the modified transmission matrix to re- gain the original transmission matrix. The matrices of Table 1(a) and (b) are identical except for the interchange of the 1 and 2 and the 9 and 10 rows. It is not possible to make the matrix of Table 1(c) identical with that of Table 1(a) by reordering rows; therefore the operation of priming the x^ and .r4 variables does not leave the transmission T = J] (0, 1, 2, 9, 10, 11) michanged. If interchanging two columns of a matrix does not change the matrix aside from rearranging the rows, then the columns which were inter- changed must both contain the same number of I's (and O's). This must Table II — Partitioning of the Standard Matrix for 2^ = Z (4, 5, 7, 8, 9, 11, 30, 33, 49) (a) Transmission Matrix Xi X2 Xs Xi Xi X6 4 0 0 0 1 0 0 8 0 0 1 0 0 0 5 0 0 0 1 0 1 9 0 0 1 0 0 1 33 1 0 0 0 0 1 7 0 0 0 1 1 1 11 0 0 1 0 1 1 49 1 1 0 0 0 1 30 0 1 1 1 1 0 Number of O's 7 7 5 5 6 3 Number of I's 2 2 4 4 3 6 (b) Standard Matrix for (a) Matrix Weight 1 1 1 Xi Xi Xs Xi Xs xe' 4 0 0 0 1 0 0 8 0 0 1 0 0 0 32 1 0 0 0 0 0 5 0 0 0 1 0 1 6 0 0 0 1 1 0 9 0 0 1 0 0 1 10 0 0 1 0 1 0 48 1 1 0 0 0 0 31 0 1 1 1 1 1 7 7 5 5 6 6 2 2 4 4 3 3 2 2 2 2 2 (c) Second Partitioning of rows for (b) matrix (d) Final Partitioning for (b) matrix Xi Xi 0 0 0 0 a-3 Xi 0 1 1 0 Xi xe' 0 0 0 0 X\ 0 0 X2 0 0 Xz Xi 0 1 1 0 Xi Xe' 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 oooo oooo 0 1 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1448 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 be true since rearranging the rows of a matrix does not change the total jiumber of I's in each column. Similarly, if priming some columns of a matrix leaves the rows unchanged, either each column must have an equal number of I's and O's or else for each primed column which has an unequal number of O's and I's there must be a second primed column which has as many I's as the first primed column has O's and vice versa. Such pairs of columns must also be interchanged to keep the total num- ber of I's in each column invariant. For the matrix of Table 11(a) the only operations that need be considered are either interchanging xi and X2 or Xz and Xt or priming and interchanging x^ and .re . For the present it will be assumed that no columns of the matrix have an equal number of O's and I's. It is possible to determine all permuting and priming operations which leave such a matrix unchanged by con- sidering only permutation operations on a related matrix. This related matrix, called the standard matrix, is formed by priming all the columns of the original matrix which have more I's than O's, the Xq column in the matrix of Table 11(a). Each column of a standard matrix must contain more O's than I's, Table 11(b). The NiSj operations which leave the original matrix unchanged can be determined directly from the oper- ations that leave the corresponding standard matrix unchanged. It is therefore only necessary to consider standard matrices. Since no columns of a standard matrix have an equal number of I's and O's and no columns have more I's than O's it is only necessary to consider permuting operations. The number of I's in a column (or row) will be called the weight of the column (or row). Only columns or rows which have the same weights can be interchanged. The matrix should be partitioned so that all columns (or rows) in the same partition have the same weight. Table 11(b). It is now possible to interchange columns in the same column partition and check whether pairs of rows from the same row partition can then be interchanged to regain the original matrix. This can usually be done by inspection. For example, in Table 11(b) if columns .r4 and .r3 are interchanged, then interchanging rows 4 and 8, 5 and 9, and 6 and 10 will regain the original matrix. The process of inspection can be simplified by carrying the partition- ing further. In the matrix of Table 11(b), row 32 cannot be interchanged with either row 8 or row 4. This is because it is not possible to make row 32 identical with either row 8 or row 4 by interchanging columns .ti and X2 . Row 32 has weight 1 in these columns while rows 8 and 14 both have weight 0. In general, only rows which have the same weight in each submatrix can be interchanged. Permuting columns of the same partition does not change the weight of the rows in the corresponding submatrices. GROUP INVAUIANCE OU TOTAL SYMMETRY 1449 The matrix can therefore be further partitioned by separating the rows into groups of rows which have the same weight in every cokmin parti- lion, Table 11(c). Similar remarks hold for the columns so that it may then be necessary to partition the columns again so that each column in a partition has the same weight in each submatrix, Table 11 (d). Par- titioning the columns may make it necessary to again partition the rows, which in turn may make more column partitioning necessary. This process should l)e carried out until a matrix results in which each row (column) of each submatrix has the same weight. Inspection is then used to determine which row and column permutations will leave the matrix unchangetl. Only permutations among rows or columns in the same partition need be considered. From the matrix of Table 11(d) it can be seen that permuting either columns .r^ and .r4 or columns x^ and x^' will not change the matrix aside from reordering certain rows. This means that interchanging .T3 and X4 or priming and interchanging X5 and x^ in the original transmission will leave the transmission unchanged. Interchanging x^ and .T5 means re- placing X5 by xt and x^ by x^,' which is the same as interchanging x^ and x% and then priming both Xi, and Xq . Thus for the transmission of Table II 0124356-Z = T and A* 000011*^123465-^ = N^Sus^ebT = T. A procedure has been presented for determining the group invariance of any transmission matrix which does not have an equal number of I's and O's in any column. This must now be extended to matrices which do have equal numbers of O's and I's in some columns, Table Ill(a). For such matrices the procedure is to prime appropriate columns so that there are either more O's than I's or the same number of O's and I's in each column, Tal)le Ill(a). This matrix is then partitioned as described above and the permutations which leave the matrix unchanged are de- termined. The matrix of Table Ill(a) is so partitioned. Interchanging Table III — Transmission Matrices FOR T = Z (0, 6, 9, 12) (a) Transniission Matrix (b) Tr with ansmission Matrix Xi and X2 primed 0 Xi X2 0 0 Xz Xi 0 0 0 10 5 12 Xi'X2' 0 0 Xz Xi 0 0 6 9 0 1 1 0 1 0 0 1 1 0 0 1 1 0 0 1 12 1 1 0 0 1 1 0 0 Number of O's Number of I's 2 2 2 2 3 3 1 1 2 2 2 2 3 3 1 1 1450 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1956 both xi and X2 , and .T3 and Xi leave this matrix unchanged so that ^21437" = T. The possibiHty of priming different combinations of the columns which have an equal number of O's and I's must now be con- sidered. Certain of the possible combinations can be excluded before- hand. In Table III (a) the only possibility which must be considered is that of priming both xi and X2 . If only xi or X2 is primed, there will be no row which has all zeros. No permutation of the columns of this matrix (with Xi or .1-2 primed) can produce a row with all zeros. Therefore this matrix cannot possibly be made equal to the original matrix by re- arranging rows and columns. Priming both Xi and X2 must be considered since the 12-row will be converted into a row with all zeros. The opera- tion of priming .Ti and X2 is written symbolical!}^ as A^ioo = A''i2 . In general, if the matrix has a row consisting of all zeros, only those Ni operations for which i is the number of some row in the matrix, need be considered. If the row does not have an all-zero row, only those A'', for which i is not the number of some row need be considered. Similarly, if the matrix has a row consisting of all I's, only those A\- for which there is some row of the matrix which will be convei'ted into an all-one row% need be considered. This is equivalent to considering only those Ni for which some row has a number /c = 2" — 1 — t* where n is the number of columns. If the matrix does not have an all-one row, only those A^, for which no row has a number A: = 2" — 1 — i should be considered. Each priming operation which is not excluded by these rules is applied to the transmission matrix. The matrices so formed are then partitioned as described previously. Any of these matrices that have the same par- titioning as the original matrix are then inspected to see if any row and column permutations will convert them to the original matrix. For the matrix of Table III (a) the operation of priming both Xi and X2 was not excluded. The matrix which results when these columns are primed is shown in Table Ill(b). Inspection of this figure shows that interchange of either Xs and .T4 or Xi and X2 will convert the matrix back to the matrix of Table III (a). Therefore, for the transmission of this table SuizNimT = T and S2i3iNnmT = T. 2 TOTAL SYMMETRY There are certain transmissions whose value depends not on which relays are operated but only on how many relays are operated. For * The number of the row which has all ones is 2" — 1 . If Ni operating on some row, k, is to produce the all-one row, i must have I's wherever k has O's and vice versa. This means that i + k = 2"" - 1 or A; = 2" - 1 - i. GROUP ixvakiaxcf: or total symmetry 1451 Table IV — Transmission Matrix for T = S (3, 5, 6, 7) = S2,z(xi , X2 , .1-3) X3 X2 Xi 3 Oil 5 10 1 6 110 7 111 example, the transmission of Table IV equals 1 whenever two or three relays are operated. For such transmissions any permutation of the variables leaves the transmission unchanged. These transmissions are called totally symmetric. They are usually written in the form, T = Soi , a«---a„X^i , X2 , ••• Xn), whcrc thc transmission is to equal 1 only ^^•ilen exactly ai or a-z or • • • or Um of the variables Xi , x^ • ■ • Xn are equal to one. The transmission of Table IV can be written as 'S2,3(.i"i , x 4, 7, 10, 13) showing that T = Si, 4 {Xi , X-i , Z3 , x/) 1 Xl 0 X2 0 XsZ 0 - 4 1 2 0 0 1 0 4 0 1 0 0 8 1 0 0 0 15 1 1 1 1 3 3 3 3 2 2 2 2 (d) Standard Matrix for T = V (3, 5, 10, 12, 13) showing that it is not totally symmetric Xl' X2' X3 Xi' 0 0 0 0 0 1 0 0 0 1 8 1 0 0 0 7 0 1 1 1 14 1 1 1 0 3 3 3 3 2 2 2 2 Table VI — Determination of Total Symmetry for ^ = Z (0, 3, 5, 10, 12, 15) (a) Transmission Matrix for T{xi , X2 , 3-3 , Xi) Xl X2 Xs 2-4 0 0 0 0 0 3 0 0 1 1 5 0 1 0 1 10 1 0 1 0 12 1 1 0 0 15 1 1 1 1 Number of O's 3 3 3 3 Number of I's 3 3 3 3 (b) Standard Matrix for Til, X2 , Xz , Xi) Xi Xz Xi 1 0 0 0 1 0 0 0 1 Number of O's 2 2 2 Number of I's 1 1 1 T{1, X2 , Xz , Xi) = SiiXi', Xz', Xi) (c) Standard Matrix for T(0, x^ , Xz , Xi) X2 Xz Xi' 0 0 1 0 1 0 1 0 0 Number of O's 2 2 2 Number of I's 1 1 1 T{0, X2 , Xz , Xi) = Sl{X2 , Xz , Xi') = ^2(^2', Xz', Xi) GROUP INVARIANCE OR TOTAL SYMMETRY 1453 numbers of zeros and ones as in Table VI (a). For such a matrix it is not clear which variables should be primed. It is possible to avoid considering all possible primings by "expanding" the transmission about one of the variables by means of the theorem T{xi ,x.2, ■■■ Xn) = XiT{l, X,, ■■■ x„) + x,'T(0, x., , ■ ■ ■ XnY-' and then making use of the relation: *^ai 1 a-> 1 ' ' ' amv^'l ) •^''' ) " * ' "^n) = Xik!)ai_i , a-z—l ) ao— 1 > ' ' ' a„— IV-^'s j ' ' ' X^J 4- XiSa^ , a-, , •••