I I I ii ^< ^< 7hi I I 5! ^^ Umisas Citp public Xlftrarp This Volume is for REFERENCE USE ONLY ^^li^HS^Sl !SlM!MIfi From the collection of the ^ m 0 PreTinger V JLJibrary t P San Francisco, California 2008 »- . ' • i <.. c^.s^ TM¥ BELL SYSTEM TECHNICAL JOURNAL A JOURNAL DEVOTED TO THE ■:v SCIENTIFIC AND ENGINEERING ^N^-' ASPECTS OF ELECTRICAL COMMUNICATION ADVISORY BOARD S. Brackev F. R. Kappel M. J. Kelly EDITORIAL COMMITTEE E. L GREENf, Chairman A. J. BuscH F. R. Lack W. H. DoHERTY J. W. McRae G. D. Edwards W. H. Nunn J. B. FisK H. I. RoMNEs R. K. Honaman H. V. Schmidt EDITORIAL STAFF Philip C. Jones, Editor M. E. Strieby, Managing Editor R. L. Shepherd, Production Editor INDEX VOLUME XXXI 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY NEW YORK HE BELL SYSTEM. Jechnical journal VOTED TO THE SC I E N T I F IG^^^ AND ENGINEERING PEGTS OF ELECTRICAL COMMUNICATION 'LUME XXXI JANUARY 1952 NUMBER 1 The Ferromagnetic Faraday Effect at Microwave Frequencies and its Applications — The Microwave Gyrator c. l. hogan 1 DiaHng Habits of Telephone Customers CHARLES CLOS AND ROGER I. WILKINSON 32 Selective Fading of Microwaves A. B. CRAWFORD AND W. C. JAKES, JR. 68 Propagation Studies at Microwave Frequencies by Means of Very Short Pulses o. e. de lange 91 Properties of Ionic Bombarded Silicon russell s. ohl 104 Mechanical Properties of Polymers at Ultrasonic Frequencies warren p. mason AND H. J. MCSKIMIN 122 Relay Armature Rebound Analysis eric eden sumner 172 Abstracts of Bell System Technical Papers Not PubUshed in This Journal 201 Contributors to This Issue 213 COPYRIGHT 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL PUBLISHED SIX TIMES A YEAR BY THE AMERICAN TELEPHONE AND TELEGRAPH COMPANY 195 BROADWAY, NEW YORK 7, N.Y. CLEO F. CRAIG, President CARROLL O. BICKELHAUPT, Secretary DONALD R. BELCHBR, Treasurer EDITORIAL BOARD F. R. KAPPEL O. E. BUCKLEY H.S.OSBORNE M.J.KELLY J.J.PILLIOD A.B.CLARK R. BOWN D. A. QUARLES F. J. FE E LY PHILIP C.JONES, Editor M. E. STRIEBY, Managing Editor SUBSCRIPTIONS 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. PBINTEO IN V. S.A. THE BELL SYSTEM TECHNICAL JOURNAL VOLUME XXXI JANUARY 1952 NUMBER 1 The Ferromagnetic Faraday Effect at Microwave Frequencies and its Applications The Microwave Gyrator BY C. L. HOGAN A new microwave circuit element dependent on the Faraday rotation of a polarized wave has been developed. The element violates the reciprocity theorem and, because it shares this property with a gyroscope and because it is dependent on gyromagnetic resonance absorption, it has been termed a microwave gyrator. It is a loiv-loss broadband device with many applica- tions. Among these are one-way transmission systems, microwave circula- tors, microwave switches, electrically controlled variable attenuators and modulators. The microwave gyrator has been realized by making use of the Faraday rotation in pieces of ferrite placed in the waveguide. Polder has previously shown, in his analysis of the gyromagnetic resonance phenomenon, that ferromagnetic substances should show appreciable Faraday rotations at microwave frequencies. In the present study, Polder^s analysis has been extended to include a wave being propagated through a ferromagnetic sub- stance with dielectric and magnetic loss, and data are presented which give experimental verification of the theory. In addition an experimental tech- nique is described which may be of some interest in studying the properties of fer rites at microwave frequencies. THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Photograph of the experimental setup shown diagrammatically in Fig. 5. INTRODUCTION In a recent series of articles, Tellegen^ has discussed the possible applications of a new circuit element which he calls a gyrator. He defines the ideal gyrator, in principle, as a passive four-pole element which is described by: (see Fig. 1) Vl = -Si. V2 = Sii (1) Since the coefficients above are of opposite sign, the gyrator violates the theorem of reciprocity. Any network composed of the usual electrical circuit elements — resistors, inductors, capacitors, and transformers — - will satisfy the theorem of reciprocity. In simple terms, this theorem states that if one inserts a voltage at one point in the network and measures the current at some other point, their ratio (called the transfer impedance) will be the same if the positions of voltage and current are interchanged. In the gyrator, however, this transfer impedance for one Lt 1-2 1 V| 1 1 1 1 V2 1 1 1 c Fig. 1 — General four-pole. THE MICROWAVE GYRATOR 3 direction of propagation is the negative of that for the other direction of propagation. Essentially this means that a 180° phase difference exists between the two directions of propagation. For this reason it has been suggested that the element could be more aptly called a directional inverter.- Network synthesis today is based upon the existence of four basic circuit elements: the capacitor, the resistor, the inductor, and the ideal transformer. It is apparent that the introduction of a fifth circuit element, the g3'rator, would lead to considerably improved solutions for many network problems. In fact, Tellegen has shown that the synthesis of resistanceless four-pole networks would be much simplified by its introduction. In addition, McMillan has shown that it would be possible to construct a one-Avay transmission system if a gyrator were available, and Miles has shown that it would be possible by use of a gyrator to construct a network which is equivalent to a Class A vacuum tube amplifier circuit. While the realizable power gain of these gyrator circuits is necessarily always not greater than unity, many other networks in- cluding gyrators are possible which have properties analogous to vacuum tube circuits and some of these may be of practical importance. Since this new element offers such interesting possibilities in network synthesis, a study has recently been made in these Laboratories of possible methods for realizing the gyrator. A gyrator was employed by Bloch^ in his measurement of the mag- netic moment of the proton. Bloch made use of the phenomenon that if two crossed coils with a mutual core are adjusted so that there is zero mutual inductance between them and if a steady magnetic field is appHed perpendicular to the axes of both coils, then an ac voltage ap- plied to one of the coils will induce a voltage in the second due to the gyromagnetic resonance phenomenon. This induction is ordinarily extremely small unless the magnetic field is adjusted so that the exciting frequency coincides with a gyromagnetic resonance frequency of the material which forms the mutual core of the two coils. In Bloch 's ex- periment, the magnetic field was held constant and the exciting fre- quency was adjusted until it coincided with the gyromagnetic resonance frequency of the proton. If they were wound over a paramagnetic or ferromagnetic material, the two crossed coils would form a gyrator when the magnetic field was adjusted so that the frequency of the exciting field coincided with the gyromagnetic resonance of the unpaired electrons. The fact that this structure constituted a gja-ator was first recognized by Tellegen^ and has been discussed by Beljers and Snoek 4 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 ill a paper which gives a very satisfying physical model with which to mterpret gyromagnetic phenomena occurring within ferrites. Physical analysis indicates that the properties of ferromagnetic materials can be explained by assuming that the electron behaves as if it were a negatively charged sphere which is spinning about its own axis wdth a fixed angular momentum. This rotation of charge imparts to the electron a magnetic moment which is a function of the electric charge on the electron, the angular velocity of the electron, and its size. Thus the electron behaves as if it were a spinning magnetic top, whose magnetic moment lies along the axis of rotation, and its behavior can be understood by considering a spinning gyroscope suspended in gimbal rings at a point not coinciding with its center of gra^dty. If a gyroscope, thus supported in a gravitational field, is lifted away from its position of minimum potential energy and then released, it wall not return to the position of minimum energy but \\dll precess about the vertical axis. This is illustrated in Fig. 2 where the spinning gyroscope makes an angle d with the vertical 0, axis. Its equilibrium motion, in the absence of damping, is a precessional motion about the vertical axis with a velocity cop. If the gyTOSCope be regarded as initially hanging vertically downward |\ \ A Y^ \ N / \ s / \ \ 1 1 \ \ 1 \ \ ^ \ 7 < g. ^S X X Fig. 2 (left) — Precessional motion of a gjToscopic pendulum in a gravitational field. Fig. 3 (right) — Precessional motion of a gyroscopic pendulum in a gravita- tional field which oscillates between the directions A and B. THE MICROWAVE GYRATOR 5 as indicated in Fig. 3 and then a gravitational force is suddenly made to act along the y axis so that the net gravitational force acts along A, it is obvious that the gyroscope will begin to preccss about the gravita- tional field direction as indicated by the small dotted circle. However, if after completing a half cycle, the horizontal component of the gravita- tional field is reversed so that now the net gravitational field acts along the vector B, the gyroscope will begin to precess about B as indicated ])y the intermediate size dotted circle. If the horizontal component of the gravitational field is again reversed after the gyroscope completes another half cycle in its precession, the gyroscope will again begin to precess about the direction A and the actual path of the precessional motion will be along the path a-b-c-d. If this process is continued in- definitely, the gyroscope will precess in larger and larger circles around the vertical until the damping becomes large enough to contain the gyroscope in some equilibrium circle (assuming that the damping is large enough to accomplish this). The above model affords a classical picture which can be used quite readily to describe the motion of the electrons in a ferrite. If the ferrite is initially saturated along the z axis by a steady magnetic field, the electrons will come to rest with their magnetic moments lying along the 0 axis, as the gyroscope in Fig. 3. If now an alternating magnetic field is applied along the y axis, the electrons will begin to precess in larger and larger circles about the g axis until they finally reach some equilibrium position under the influence of the magnetic fields and the damping. Thus it is apparent in the gyromagnetic resonance experi- ments described above why an alternating field applied perpendicular to a steady magnetic field in a ferrite will give rise to a varying flux perpendicular to both the steady field and the alternating field. It is also apparent why the alternating flux along the x axis is 90° out of phase with the alternating flux along the y axis. Since precession of the top will always be in the same direction regardless of whether the alter- nating field is applied along the x or y axes, consideration of Fig. 3 makes it apparent how the two crossed coils with ferrite at their center can constitute a gyrator which violates the reciprocity relation in a manner described by Equations (1). To the present time, however, no practical circuit element making use of this phenomenon has been constructed because the coefficient of coupling between the coils is always small, even in the vicinity of the resonant frequency, and also because the losses in the materials available are so high in the vicinity 6 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 of resonance that the insertion loss of such a device would be prohibitively large. McMillan in his original article, showed that a gyrator could be realized by means of mechanically coupled piezo-electric and electro- magnetic transducers. Later, McMillan pointed out that a gyrator could be realized by means of the Hall effect in a square plate of bismuth, as was also predicted by Casimir. Another similar possibility would be an electrical-electrical coupling through a gyroscopic link. A gyrator has been built by W. P. Mason of these Laboratories which makes use of the Hall effect in a crystal of germanium. This gyrator showed an insertion loss somewhat higher than the theoretical loss of 12.3 db. R. 0. Grisdale of these Laboratories suggested that these losses could be greatly reduced if the same Hall effect principle were applied to a vacuum tube which contained four electrodes which could both emit and collect electrons. This device is no longer passive, but such a struc- ture has been built and showed an insertion loss of about 7 db, only slightly higher than the theoretical loss which would be expected from this geometry. In view of the substantial losses found to exist in the earlier forms of gyrator discussed above, a study of other "anti-reciprocal" phenomena which might lead to the realization of a relatively low loss gyrator was undertaken. It has long been known that the Faraday rotation of the plane of polarization in optics is anti-reciprocal. In order to observe the Faraday rotation, polarized electromagnetic waves must be transmitted through a transparent isotropic medium parallel to the direction of the lines of force of a magnetic field. The effect is usually produced by placing the material along the axis of a solenoid. The rotation is "positive" if it is in the direction of the positive electric current which produces the field and "negative" if in the opposite direction. All optically transparent substances show the Faraday rotation. Its anti-reciprocal property distinguishes the Faraday effect from optical rotations caused by birefringent crystals, or by the Cotton- Mouton effect, which are reciprocal. That is, if a plane polarized light- wave is incident upon a birefringent 'crystal in such a manner that the plane of polarization is rotated through an angle 6 in passing through the crystal, then this rotation will be cancelled if the wave is reflected back through the crystal to its source. In the Faraday rotation, however, the angle of rotation is doubled if the wave is reflected back along its path. Hence, if the length of path through the "active" material is adjusted so as to give a 90° original rotation, the beam on being reflected THE MICROWAVE GYRATOR 7 will luiNO its plane of polarization rotated a total of 180° in passing in both directions through the material. Thus, the Faraday rotation in optics affords an anti-reciprocal relation quite analogous to the anti- reciprocal property of the gyrator postulated by Tellegen. Lord Raylcigh^ described a one-way transmission system in optics which makes use of the Faraday rotation. Lord Rayleigh's "one-way" system consisted of two polarizing Nicol prisms (oriented so that their planes of acceptance made an angle of 45° with each other), with the material causing the Faraday rotation placed between them. Thus, light which was passed by the first crystal and whose plane of polariza- tion was rotated 45° would be passed by the second crystal also. But, in the reverse direction, the rotation would be in such a sense that light which was admitted to the system by the second crystal would not be passed by the first. Although Rajdeigh's one-way transmission system can be actually realized, it is experimentally difficult since most substances show ex- tremely small Faraday rotations. In fact, large rotations for transparent substances in the optical region are of the order of one degree per cm path length for an applied magnetic field of 1000 oersteds. To realize a rotation of 45° would require maintaining a field of 1000 oersteds over a distance of approximately one-half meter. The Faraday effect in ferromagnetic substances, however, is unique in that it shows rotations many orders of magnitude greater than the rotations exhibited by any other substances. For instance, K6nig^° reports rotations of 382,000°/cm by passing light through thin layers of magnetized iron. These data, of necessity, however, were taken on extremely thin sections and the total rotation obtained for any specimen did not exceed 10°. In order to obtain appreciable rotations in a device of practical size, it is necessary to obtain a material which shows a rotation at least intermediate be- tween those reported for iron and other ordinary materials. In addition, in order to make effective use of these rotations, the material must be transparent to the radiation which is being used. THEORY OF THE FERROMAGXETIC FARADAY EFFECT Polder^^ has shown in his analysis of the ferromagnetic resonance phenomenon, that a plane electromagnetic wave at microwave fre- quencies should show appreciable Faraday rotation when propagated through a ferromagnetic material which is magnetized in a direction parallel to the direction of propagation of the wave. Polder has neglected both magnetic and dielectric losses in his analysis and although for the ferrites which are of greatest interest, this approximation is quite 8 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 valid, nevertheless the more complete theory is developed below. The exact theory of this phenomenon should, of course, be approached through quantum mechanics, but since the classical theory, in this particular case, gives a result as satisfactory as the quantum theory and since it lends itself more aptly to a fundamental physical inter- pretation of the phenomenon, it is the classical theory which is developed here. Quantum mechanically, Faraday rotations in the optical region are accounted for by the Zeeman splitting of the spectral lines. The classical model which proves quite adequate for the description of ferromagnetic resonance is that illustrated in Figs. 2 and 3 which regards the electrons of the material which contribute to the magnetism as being spinning magnetic tops. The angular momentum of each electron is : \J\ = Uh/2ir) (2) J = Angular momentum of electron (gm cm /sec) h = Planck's constant (6.62 X 10~ erg sec) The magnetic moment which arises due to this rotation is: where : IJLB = Magnetic moment of electron (Bohr magneton) e = Charge on electron (4.80 X 10"'" E.S.U.) m = Mass of electron (9.10 X 10~"^ gm) c = Velocity of light (3 X 10'° cm/sec) The so-called gyromagnetic ratio of the electron is the ratio of these quantities and is given by: ^ = ^24 = ^ ^^) If a steady magnetic field is applied to the sample such that the elec- tron sees an effective field H, then a torque will be applied to the electron which tries to turn the electron so that its magnetic moment lies along the field direction. However, as indicated in Fig. 2, the electron will precess around the field direction until damping forces dissipate the energy of precession. The equation of motion of the electron is: ,,Xi/=^^. =T ^ (5) THE MICROWAVE GYRATOR 9 The equation of motion of the magnetization per unit volume can thus be written: 4^=yMXH (6) where: M = Magnetization of medium H = Macroscopic internal magnetic field The above equation, however, does not include damping. The damping force, regardless of its origin, must be so introduced into the above equation that it tends to cause the electron's axis of rotation to line up with the field direction. It has been shown by Yager, Gait, Merritt and Wood " that the shape of the resonance absorption line can be accounted for if the damping term is introduced in the following way: ^ = tM X i/ - ^ [M X (M X ^)] (7) The vector M X (M X H) is simply a vector which is in the proper direction to act as a damping force (torque) and the coefficient is chosen so as to give the correct units along with the parameter, a, which must be determined experimentally and which gives the magnitude of the damping torque. Equation 7 then is the equation of motion of the magnetization of an arbitrarily shaped l^ody under the action of an arbitrary internal field, H. In the appendix, it is shown that if a steady magnetic field, Ha , is applied along the s axis and then a small alternating field is ap- plied in an arbitrary direction to a sample which is infinite in size, the equation relating the resulting alternating flux density, b, and the ap- plied alternating field, h is : bx = M^x — jKhy by = jKhx + iihy (8) be = hs where M = M'-iM" (9) K = K' - jK" (10) Equations which give n and K in terms of the applied magnetic field and fundamental atomic constants are given in the appendix. 10 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Equations (8) are easily interpreted in terms of the spinning gyroscope model of Fig. 3. If magnetic losses had been ignored (i.e. a = 0) then both fjL and K would have been real. Under this condition, it is seen that if an alternating field, hy , is applied along the y axis, then an alternating flux, hy , is created along the y axis which is in phase with hy , and an alternating flux, hx , is created which is 90° out of phase with hy . Reci- procity between the x and y directions would demand that both terms containing jK should have the same sign. Thus, Equations (8) give a quantitative expression for the results which were previously qualita- tively deduced by means of the electronic model illustrated in Figs. 2 and 3. If a waveguide is filled with a ferromagnetic material such as a ferrite and if then a steady magnetic field is applied along the axis of the waveguide, it is necessary in order to describe this wave to find a solu- tion to Maxwell's equations which is consistent with Equations (8) and in which b, h, E and D are all proportional to exp [jut — Vb\. This problem is not solved exactly. However, in the appendix a solution is obtained for an infinite plane wave. It is found that the ferromagnetic medium can support only a positive or a negative* circularly polarized wave or a combination of both. It is also shown in the appendix that the propagation constants for these two circularly polarized waves are dif- ferent and are given by the following expressions: and \ = -^ V(m + K)[z] (11) r- =^~y/{n-K)[z] (12) where V± — Propagation constant CO = Angular frequency of wave c = Velocity of light in unbounded space (3 X 10 cm/sec) e = Complex dielectric constant of medium In Equations (11) and (12) it is apparent that the effective perme- ability of the medium to a positive circularly polarized wave, for in- * The usual notation is used here, where the positive component is the com- ponent which is rotating in the direction of the positive electric current which creates the steady longitudinal field. THE MICROWAVK GVKATOR 11 stance, is given by the expression (n + K), and not by the usual perme- ability, bx/hx = iJL. It is also apparent that the quantity fi -\- K can vary over wide limits in the vicinity of the ferromagnetic resonance. For this reason, care must be taken in interpreting permeability data for ferromagnetic materials which now occur in the literature and which were obtained by means of impedance measurements at microwave frequencies, since the above equations indicate that this method does not measure the same quantity that is measured at low frequencies by means of a toroidal sample overwoiuid with two coils. The low fre- quenc}' measiu'ement of permeability obviously measures the quantity which is designated as m in Ecjuation (8). If Equations (11) and (12) are solved for the attenuation constants, a± , and the phase constants, jS± , the following results are obtained: ^^ ^co ^/{^' ±K')e' < /|/ (1 + tan 5,„[4 tan 8d + tan 5^(1 + tan- 8d)] + tan- 8d — 1 — tan 8m tan 8d and /3± = where : tan 8m = (13) (m' ± K')e' (1 + tan 8m[4: tan 8d + tan 5„(1 + tan- 8d)[ + tan ^ 8d + 1 + tan 8m tan 8d (14) (The + sign must be used for a positive circularly polarized wave; the negative sign for the negative circularly polarized wave.) tan 8d = —r = dielectric loss tangent e = e' — je" = complex dielectric constant 12 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 It is almost impossible to get a feeling for what these equations mean with respect to a wave travelling through the medium, especially since ju and K are given by equations which are almost as difficult to per- ceive. An appreciation of these equations can be obtained however, by reference to Fig. 4 which gives qualitatively the behavior predicted by these expressions. Essentially, a and (3 are functions of two variables. These are co, the frequency of the wave, and Ha , the applied magnetic field. In Fig, 4, the index of refraction and attenuation of the positive circularly polarized component are given relative to these values for X > 111 (T Q< z a. - H •a m INDEX OF REFRACTION OF POSITIVE COMPONENT^ ^\ ABSORPTION OF ^, POSITIVE COMPONENT /\ 1 / V \ \ / A / \ / \ \ \ \ \ / \ \ \ >» 1 1 \ \ / y^ ' MAGNETIC FIELD AT RESONANCE \, / WITH FO CYLIN ^ SAMF 3RICAL =LE SYMM ETRY V APPLIED MAGNETIC FIELD (ARBITRARY UNITS) Fig. 4 — Index of refraction and absorption of a positive circularly polarized wave relative to the same quantities for a negative circularly polarized wave being propagated through a magnetized medium. the negative component. Hence, both the index of refraction and attenuation of the negative component are represented by the abscissa of the graph. In Fig. 4 these quantities are plotted as a function of the apphed magnetic field for a wave of a fixed frequency. Many of the properties of the medium are clearly displayed in this graph. In partic- ular, as the field necessary for ferromagnetic resonance is approached, the attenuation of the positive component becomes larger and larger. Eventually this component will be substantially completely absorbed and only the negative circularly polarized component will be propagated. Hence it should be possible to establish a circularly polarized wave in a waveguide simply by passing the dominant mode through a ferromag- netic material which is subjected to a longitudinal magnetic field of the • proper amplitude. However, there will be an absorption of one-half of the power being propagated. If Fig. 4 had been plotted as a function THE MICROWAVE GYRATOR 13 of the frequency of the wave for a fixed magnetic field, a similar set of curves would have resulted. This set would indicate the frequency dependence of the Faraday rotation. If the frequency of the wave is far removed from the resonance frequency, the difference between the indices of refraction of the positive and negative component is not frequency dependent. However, near resonance, this difference is a very rapidly varying function of the frequency. It is to be remembered that these equations were derived for an infinite plane wave. However, it would be expected that these equations would describe quite accurately the propagation of the dominant mode in a waveguide. The approxima- tion would, of course, be better when the cut-off wavelength was much greater than the unbounded wavelength. This condition is met when the waveguide is filled with ferrite and for these cases quantitative agreement is obtained. The above analysis shows that if a dominant mode wave (plane polarized) is incident upon a ferromagnetic material which is magnetized along the length of the waveguide, the wave will split into positive and negative circularly polarized waves whose phase constants are given by Equation (14).) Since the two circular components travel with different velocities in the medium, they will upon emerging from it unite to form a plane polarized wave whose plane of polarization has been rotated with respect to the incident polarization. The angle of rotation of the polariza- tion is given by: 0 = ^ [^_ - /3+] (15) where : I = path length through ferromagnetic material (cm) In order to evaluate Equation (15), it must be combined with Equation (14). However, a few approximations are valid in Equation (14) which make it much simpler. In particular many ferrites exist for which the magnetic losses are extremely small as long as the internal field within the body is kept small so that the frequency of the wave does not ap- proach the ferromagnetic resonance frequency. This field can be kept small if the magnetic field is not raised above the point necessary to saturate the ferrite. Kittel has shown that for a finite body the effective internal magnetic field that determines the resonant frequency is given by: Hi = [Ha -h (iVx - N.)M.][Ha + (Ny - N.)M.] 14 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 where He is in oersteds. The ferromagnetic resonance frequency is given by: /res = 2.^He Vl + cc^ megacyclcs (16) It is easily shown that the following formula is approximately valid for a ferromagnetic body with a circular or square cross-section at saturation. Where: n = true dc permeability at saturation A'^ = demagnetizing factor in x and y directions. If an average value of 1000 is assumed for the dc permeability, then He can be readily computed for various shapes. For a thin disc : AT = 0 A^. = 47r and ^' 1000 If a thin disc saturates at 1500 gauss, then: He = 1.5 oersteds /res ^4.2 megacycles For a long thin pencil: N, = 0 N = 4t and He = lOOOHa For this case the body could be saturated with a field of about 1.5 oersteds, so: He = 1500 oersteds and /reg = 4200 megacycles If, for this case, the resonance frequency is so close to the operating frequency that losses due to ferromagnetic resonance become pro- THE MICROWAVE GYRATOR 15 hibitive, it is wise to then raise the appHcd field to some high value, so that the resonance freciuency will fall well above the operating fre- ({uency. Thus, for many cases of interest it is possible by various means to place the ferromagnetic resonance absorption frequency sufficiently far from the operating fre(}uen(;y so that magnetic losses due to this phenomenon are negligible.* The data accumulated to date indicate that the major component of the magnetic losses at microwave fre- quencies is due to this phenomenon. Only in a few cases have data been taken which have indicated that other factors, such as domain wall relaxation, contribute to the magnetic loss at microwave frequencies. If then, the magnetic field is controlled so that the ferromagnetic resonance absorption is negligible, Equations (13) and (14) can be simplified to: ^^^.^^^pl ^yj + j^„,,__i (18) and : (19) is: ""//SZZ v^7^:r (20) which can be written as: «± and fc = " 4/'^^' V7^:r (21) where /x' and K' are given in the appendix. If Equation (21) is now inserted into Equation (15) a formula for rotation is obtained which is valid within the limits of the above ap- proximations. If in addition, the frequency of the wave is sufficiently greater than the resonance frequency, so that: CjOr.s « CO (22) then Equation (15) takes the particularly simple form: P _ CO f ~ 2c i/'^T/'+'-^V- 47rM^7 (23) Most ferrites saturate at 2,000 gauss or less. Hence, for a frequency of * This is not always possible, for some ferrites, in the polycrystalline state, exhibit extremely broad ferromagnetic resonance absorption lines and it is diffi- cult to operate at anj' frequency without apprecialilo absorption. 16 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 9,000 megacycles, 4Filf.7 ^ 2000 X 17.6 X 10^ _ ^ ^ . . CO - 9000 X 27r X 10« ' '^^ ^ ^ Hence, the following approximation will be valid to within 5 per cent. With this approximation, Equation (23) reduces to: Equation (25) is quite remarkable. Not only does it predict large rotations, but it also predicts that, within the above approximations the rotation will not depend upon the frequency of the incident radia- tion. For the assumed values, 8' = 15 e" = 0 47rM, = 1000, Equation (25) predicts rotations of, - = 65°/cm. DESCRIPTION OF EQUIPMENT AND MEASURING TECHNIQUES The Faraday rotation has been measured in a large number of ferrites in order to verify the above theory and in an effort to improve the characteristics of the microwave gyrator. A diagram of the experimental equipment is given in Fig. 5, and a diagram of the test chamber in which the rotations were measured is given in Fig. 6. In the test chamber, two rectangular waveguides are separated by a circular waveguide, the proper nonreflective transitions being made at each end of the circular section, which is about twelve inches long. One rectangular guide is supported so that it can be rotated about the longitudinal axis of the system. The dominant TEw mode is excited in one rectangular guide, and by means of the smooth transition this goes over into the dominant TEn mode in the circular guide. The rectangular guide on the opposite end will accept only that component of the polarization which coincides with the TEio mode in that guide, the other component being reflected THE MICROWAVE GYRATOR 17 at the transition. Absorbing vanes, inserted in the circular section, absorb this reflected component. The circular guide is placed in a solenoid to establish an axial magnetic field along its length. The ferrite cylinders to be measured were placed at the mid-section of the circular guide. When a cylinder was used which did not fill the cross-section of the guide, it was supported along the axis of the guide by means of a hollow polystyrene cylinder which did fill the guide. In addition to measuring the Faraday rotation, measurements of insertion loss were made by determining the power transmitted under identical conditions with the ferrite cylinder removed, and the ellipticity of the transmitted wave was determined by measuring the power trans- mitted when the rectangular guide on the detector side was rotated to both positions of maximum and minimum transmission. Power trans- mission measurements could be repeated within 0.2 db. Measurements of the angle of rotation of the plane of polarization could be repeated within 1° except in the region close to the gyromagnetic resonance where rotations were large and ellipticity so great that it was difficult to decide the positions of maximum and minimum transmission. These errors increased up to the point where the transmitted wave was circularly polarized where it was impossible to measure the angle of rotation. EXPERIMENTAL RESULTS Equation (25) indicates that the rotation per unit path length through the ferromagnetic material is proportional to the magnetization of the CATHODE -RAY OSCILLOSCOPE XTAL DETECTOR (2K VARIABLE ATTENUATOR J^ SIGNAL OSCILLATOR ^_1 -/^ m BEAT OSCILLATOR VARIABLE ATTENUATOR TEST CHAMBER ^ AMPLIFIER DETECTOR (60 MC) VARIABLE I F ATTENUATOR METER VARIABLE ATTENUATOR VARIABLE ATTENUATOR VARIABLE ATTENUATOR XTAL CONVERTOR Fig. 5 — Experimental equipment set-up used to measure Farada}' rotations. 18 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 sample and is not dependent directly on the applied magnetic field. Fig. 7 shows the dependence of rotation upon magnetization for a sample of manganese zinc ferrite, and indicates that after the sample is saturated, the rotation is sensibly independent of the applied magnetic field. In addition, the complex dielectric constant and the saturation magnetization of this sample were measured. From these the rotation per centimeter path can be computed from the above theory using ROTATABLE SECTION FERRITE WINDING- STATIONARY PROTRACTOR RADIAL VANE TO ABSORB VERTICALLY POLARIZED WAVES TAPERED TRANSITIONS TO REDUCE REFLECTIONS RADIAL VANE TO ABSORB HORIZONTALLY POLARIZED WAVES Fig. 6 — Detail of test chamber in which rotations were measured. 120 a. 13 "oj Z\L zl Ol- < UJ &^ cctr UJ Q < n n 0 / f / / /\ / N 500 2500 1000 1500 2000 MAGNETIC FIELD IN OERSTEDS Fig. 7 — Angle of rotation versus applied magnetic field for a thin disc of man- ganese zinc ferrite. THE MK^ROWAVK GYRATOR 19 E(|uati()n (25). For a particular sanipl(> of nianganesc zinc ferrite, the following measurements were made: (Sample No. 1) c' = 17 c" = 24 47r3/sat = 1500 gauss Using this data, eciuation (25) predicts: - = 121.27cm. It is seen in Fig. 6, that the actual measured rotation at saturation is approximately 123°/cm. Hence an extremely good agreement with theory has been obtained for this particular sample. Equation (25) also indicates that the rotation per unit path length should be sensibly independent of frecjuency within the above ap- proximations. The data are shown in Fig. 8. However, it will be noticed that the frequency difference between these two sets of data is relatively small (3 per cent), and the cumulative experimental error in measuring angles is such that it is difficult to state that the rotation is closer than 1° between the two sets of data. This represents a possible differ- ence of 5 per cent in the rotation for a change of 3 per cent in the fre- quency. Thus, even though these preliminary data support Equation (25), it cannot be accepted as conclusive evidence until more measure- ments can be made over a wider band width. 24 O 9200 MC D 8930 MC ^ .^ -^ ^ ^ ^ ^ ^ ^ 0 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 APPLIED MAGNETIC FIELD IN OERSTEDS Fig. 8 — Dependence of Faraday rotation ujjon freciuency. 20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 The loss characteristics of different ferrites as a function of the ap- phed magnetic field differed distinctly from each other. Some ferrites, such as manganese zinc ferrite showed extremely high loss which was associated with the imaginary part of the dielectric constant. This loss was not affected by the application of a magnetic field but remained substantially constant as the field was applied. However, as the field approached that necessary for ferromagnetic resonance, the total power absorbed by the ferrite increased, since the positive circularly polarized component was almost completely absorbed by the sample. In fact by measuring the ellipticity of the transmitted wave, it is possible to com- pute the difference between the absorption of the positive and negative circularly polarized components. This has been done for Sample No. 1 and the result is indicated in Fig. 9. If the curve were continued to higher fields, it would represent the shape of the ferromagnetic resonance absorption line. Some ferrites, such as Ferramic G, showed an almost zero dielectric loss but on the other hand caused an extremely large absorption at 9000 megacycles due to magnetic losses. The major contributions to magnetic loss at this frequency should be either losses associated with a domain wall relaxation or ferromagnetic resonance absorption due to anisotropy fields. Unequivocal data can be obtained by the above techniques to identify which loss is predominant. If the loss were due to domain wall relaxation (or resonance) it would absorb both the negative and positive circularly polarized components equally. Thus as the magnetic field was apphed and as the ferrite became saturated, the losses in both components should decrease as the domain walls disappeared. However, OQ CO ^ UJ Q. B 3 "S, s 03 Material Dimension (cm) < S o C o ^ Pi m 03 o h:i c .2 u tc 1— 1 '2 -a * .1.3 3 1 BTL 0.447 X 2.28 0 0 10.0 >50 MnjZni_5Fe204 (length X dia.) 245 15.6 10.3 >50 490 33.5 10.0 23.2 735 58.2 9.2 15.0 980 81.6 9.1 12.1 1225 107 9.2 10.9 1470 120 10 10.4 1715 125 11 9.3 1960 123 11.2 9.0 2206 121 11.3 7.7 2450 123 11.4 6.6 2695 — - 12.4 5.0 2940 — 13.0 3.7 3185 — 3.0 3675 — 1.4 2 BTL 0 0 0.8 >40 NiaZni_iFe204 1.36 X 2.28 245 25 1.9 =40 490 44 2.7 =40 735 56 2.9 =40 980 61 2.7 40 1225 68 2.8 1715 82 3.3 1960 85 4.9 2450 118 7.3 0.8 3 Ferramic A 2.54 X 0.635 0 0 1.1 >50 0.7 245 34.9 0.8 0.3 490 43.7 0.8 0.3 735 48.3 0.8 0.3 980 51.1 1.0 0.4 1225 54.0 1.1 — 1715 57.0 1.1 — 1960 60.0 1.9 — 2450 63.0 3.0 35 — 2695 64.2 3.7 — — 4 Ferramic G 1.77 X 2.28 0 0 23.2 »30 245 38 21.4 23.0 490 77 16.7 7.6 735 124 12.4 2.1 980 157 9.9 1.4 1225 170 7.7 0.7 1470 180 6.0 0.7 3430 c.p. 7.1 0.0 * Data given is the difference in db between the major and minor comi)onents of the elliptically polarized transmitted wave. 24 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 CONSTANTLY VERTICALLY POLARIZED WAVE I HORIZONTALLY POLARIZED ROTATING 90° COUNTER-CLOCKWISE VERTICALLY POLARIZED Fig. 11 — The microwave gyrator with diagrams which help to explain its operation. THE MICROWAVE GYRATOR 25 The "active" element of the device, the fenite cylinder, has been termed a "Faraday Plate." As was pointed out earlier, the fundamcMital property of the gyrator is the 180° phase difference introtluced Ix^tween the two directions of propagation through it. Thus the gyrator may be thought of as a four terminal circuit element having no phase shift for one direction of trans- mission, and having a 180° phase shift for the opposite direction of transmission. A convenient circuit symbol for the gyrator, which indi- cates this property, is shown in Fig. 12. If the rectangular waveguides on each side of the Faraday Plate are rotated about their common axis so as to make an angle of 45° with 77- Fig. 12 — Circuit symbol for gyrator. -b FARADAY PLATE POLARIZATION CIRCULATOR CIRCUIT SYMBOL FOR POLARIZATION CIRCULATOR Fig. 13 — Schematic diagram of polarization circulator. each other, then a one-way transmission system can be created which is similar to Lord Rayleigh's one-way transmission system of optics, but with the important difference that this one-way transmission system does not depend upon frequency but is broad band. This one-way trans- mission system can be used, for example, to isolate the generator or detector from the waveguide in microwave systems. In this application it has the great advantage over the attenuators which are presently used for this purpose in that it can be made practically lossless for the direction of propagation which is desired but the reflected wave will be completely absorbed and hence more complete isolation can be effected. A more complex and more usefid circuit element, than this simple one-way transmission property would at first indicate, is obtained by adding a second connection on each side of the 45° Faraday Plate. It is suggested that this device be called a polarization circulator. Thus, the 20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 polarization circulator actually has four output branches corresponding to the two different polarizations at each end. The polarizations of the four output branches are indicated in Fig. 13. It is noticed that power sent into the polarization circulator with polarization a is turned into polarization h, also h is turned into c, c is turned into d, and d is turned into yninus a. This property is indicated very clearly by the circuit sym- bol suggested in Fig. 13, the phase inversion between arms d and a being indicated by the minus sign between the d and a arms. Another one-way transmission system can be created by combining the gyrator with two-normal hybrids. This combination is indicated in Fig. 14. Since this device has all of the fundamental properties of the CIRCULATOR Fig. 14 — Schematic diagram of circulator. CIRCUIT SYMBOL FOR CIRCULATOR polarization circulator with the exception of the phase inversion be- tween arms d and a it is suggested that it be called a "circulator" and the circuit symbol suggested w^hich indicates its properties is also given in Fig. 14. This list of applications is obviously not complete since it includes only the fundamental elements from which innumerable specific applica- tions can be made. In addition to the applications discussed above, which depend upon the anti-reciprocal property of the element for their operation there are several simple applications which are based only upon the fact that the amount of rotation can be controlled externally by adjusting the magnetic field. Among these uses are electrically controlled attenuators, modulators, and microwave switches. ACKNOWLEDGMENTS The author is indebted to a rmmber of persons for aid in developing this circuit element. In particular, he wishes to thank A. G. Fox for THE MICROWAVE GYRATOR 27 permission to use his terminology and circuit symbols, and also for the many discussions concerning the properties and uses of these microwave circuit elements. The author is also indebted to S. E. Miller for help in designing the microwave elements and to J. K. Gait for many discus- sions concerning the theoretical aspects of this paper. The author also wishes to extend his appreciation to J. L. Davis whose able technical assistance made possible the accumulation of much of the data presented in this paper. APPENDIX The equation of motion of the magnetization of a ferromagnetic material is: ^ = yOi xm- 1^1 [M X {M X m (1) at \M I where H = internal magnetic field (oersteds) 47ril/ = magnetization of medium (gauss) a = parameter which measures the magnitude of the damping force on the precessing dipole moment of the sample 7 = gyromagnetic ratio of the electron (7 = (/e/2mc where g is the Lande g factor for the electron). If a ferromagnetic material is subjected to a steady magnetic field, Ha , along the z axis and if then an alternating field is applied in an arbitrary direction, Equation (1) must be solved in order to find the behavior of the magnetization of the material. To solve this problem, the following notation is introduced: 4TrM^ = magnetization of medium in absence of alternating field Ha = externally applied steady magnetic field (oersteds) hx , hy , h, = components of applied alternating magnetic field nix , niy , m. = alternating components of magnetization hi , hi , Hi = components of internal magnetic field hi = hx — Nxtrix K = hy - NyVly H\ = Ha + h - A'zCl/. + nu) Nx , Ny , Nz = demagnetizing factors of body. 28 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Hence the magnetic field, H, occurring in Equation (1) is defined by: H = hSi + hlj + hZ and M = md + rriyj + {Mz + mz)k In solving Equation (1), an exponential, exp [jwt], time dependence is assumed for the alternating magnetic field and magnetization, and if the following assumption is made: hx , hy , Ih « Ha it is easily shown that the alternating components of the magnetization of the medium are given by (neglecting terms of the second order in small quantities) : nix = m„ = y'Hi\l + a') - co^ + j[2o:yaHi] [y'MzHiil + a) + jyaMz<^]K-\-jyM^K (2) where : Since y'Hi\l + a') - c' + j[2o:ycxH:] m^ = 0 b = h' -\- 47rw, (3) it is possible by means of Equations (2) and (3) to find the relation between the alternating flux density b and the internal alternating field h\ If the ferromagnetic body is considered as being infinite, the internal fields and applied fields are equal. Hence, for this case: hx = fiK - jKhy by = jKK + t^hy (4) b^ = Ih where: fi = n — Jli K = K' - jK" THE MICIIOWAVIO GYRATOH 29 and: , , ^ [YHl{\ + a') .- co-][47r.y.7'//a(l + «')] + SirM^Va'Ha n = I -\ K' = K" = // M = 47ril/,7c ^ >- • < \ ' is \ 8^ \ • \ < CO 8^ in O o , : • n ^ < Miaa 3NOi nvia aaivoiaNi 3hi nvhi u3iv3Ho SAviaa a3b3iNnoDN3 ivhi stivd is31 3N0i nvia do iN3o asd DIALING HABITS OF TELEPHONE CUSTOMERS 37 rcspoiuLs to Figs. 2(a) and 2(h). Fig. (i is for the coin class of service with 34 line finders and corresponds to Figs. 2(c) and 2(d). Plotted on Figs. 3 to 6 are theoretical fitting dial tone tester delay- curves, curves A, determined by meai'is of the following formulae: 1. The generalized trunking formula for determining the proportion of calls that encounter congestion, i.e., find all line finders busy. \ \ V . OBSERVED RESULT FOR HALF HOUR STUDY PERIOD THEORETICAL DIAL TONE TESTER DELAY FITTING CURVE % s. UJ _l CD < \ \ < 5 \ OlU \ : LL UJ \ i Z _J V >- < O LU •Jz mO 1 o < "X s. \ •^•\ <^\ ^ V V ^ < V)^ ' Qq z OUJ cog o_, „ < -O Q "-' a "^ Uv ii \ V ^.0 a: 1X1 ID '7'=' >- < Oo O UJ oz UJQ 0,is: PoOO = Kc)po(>t) = m exp (-d/jH) (9) The delay distribution for a test call which finds one call waiting ahead of it is : PiOO = [1 + c/j - (c/j) exp (-t/H)] exp (-d/jH) (10) The probability is /(c + 1) that a call made at random will find all 44 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 line finders busy and one call waiting. Hence the weighted delay distri- bution, Pi(>t), is: PiOt) = Kc + 1)[1 + c/j - (c/j) exp (-t/H)] exp {-d/jH) (11) In the general case pni>t) is given by the following formula: PnOt) = F„+i(0 exp (t/H) (12) where F„+i(0 is given by Conny Palm.® The over-all delay distribution is then : POO = PoOt) + PiOO + P2OO + (13) By making appropriate substitutions and summing the result, ex- pression (13) becomes: POO = V(o) c! ■ aexp i-t/H) a' exp (-21/ H) c+i ^ (c+j)(c + 2j)^ " exp [-ct/jH + (a/j)[i - exp (-t/H)]] (14) Expression (14) is equivalent to that of Riordan involving two in- complete gamma functions as follows : P(>t) = p(>o)^t^/i>(«/i)^^P(-^/-^)l y(c/j, a/j) where the incomplete gamma function, y(N,x) = f x'^-'i Jo dx (15) (16) The theoretical dial tone tester delay curves shown on Figs. 1(a) to 1(d), 2(a) to 2(d), and 3 to 6 were computed from expression (14), using the following values of j and H for the classes of service studied, these values being determined in a manner explained later: Class of Service j factor E MRI MR 2-party FRI Coin 6.6 5.8 6.5 2.1 24 seconds 42 seconds 27 seconds 74 seconds On Figs. 1(a), 1(c), 2(a), and 2(c), which show the per cent of dial tone tests encountering delays greater than three seconds for various amounts * Equation 53, loc. cit. DIALING HABITS OF TELEPHONE CUSTOMERS 45 of load carried, it may be noted that most of the theoretical dial tone tester delay curves are in close agreement with the observed data, with a tendency perhaps to be slightly high. On Figs. 1(b), 1(d), 2(b), and 2(d), which are for dial tone delays greater than ten seconds, it may be noted that the theoretical curves have a slightly stronger tendency to lie on the high side of the observed data. On Figs. 3 to 6 the theoretical dial tone tester delay, curves A, again lie in the proximity of the curves of the observed data, with a tendency to lie higher than these latter curves, especially at the ends where the dial tone delays are greatest. Among the factors which account for this discrepancy are: 1. A feature is present in panel line finder circuits for momentarily releasing trip circuits with waiting calls to prevent the orphaning of calls under certain trouble conditions. The release occurs after a call has been waiting from 5 to 12 seconds and reoccurs every 7 seconds thereafter. When such a release occurs the call yields whatever Avaiting preference it may have had to a subsequently placed call which is not 5^et affected by such a release. The dial tone test calls did not wait beyond 12 seconds. Hence for these test calls there was only one possi- bility of such a release and for many of them the release occurred near the end of their waiting period. Hence they were more likely to gain preference over other calls than to lose their preference. 2. Subscribers while waiting for dial tone frequently become impatient and proceed to flash (move their switchhook up and down). While flashing, a subscriber may lose preference to a subsequently placed test call (the latter of course does not flash). 3. Many subscribers fail to observe dial tone and proceed to dial. During such dialing, a subscriber may lose preference to a subsequently placed test call. 4. Line finders serve a large proportion of call attempts of short holding time whose presence may militate against the occurrence of the longer delays. In connection with the measurement of the j factor, the following proportions of call attempts and average holding times were noted on which no dialing occurred or where no more than two digits were dialed. Class of Service Proportion of Attempts with No Peg Counts Average Holding Time MRI MR 2-Pty. FRI Coin 35.2%* 33.3%* 25.1% 9.8% 4.4 seconds* 5.8 seconds* 5.8 seconds 8.3 seconds Partly estimated 40 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 The individual contributions of these four factors to discrepancies between theory and observation are not easy to assess. The first three explain a tendency for test calls to get ahead of calls already waiting for dial tone. On Figs. 3 to 6, inclusive, additional theoretical dial tone delay curves, curves B, for the case where a dial tone tester always gets first in line are shown. Even these curves tend to lie above the curves of the observed data on Figs. 4 and 5 where ten line finders were available; they more nearly agree with the observed data on Fig. 3 where twenty line finders were available, and they lie below the observed data on Fig. 6 where 34 line finders were available. This is an indication of the fact that with higher traffic loads (which occurred on the larger line finder groups) a test call will encounter more competition from other calls and therefore will have a lesser chance of gaining precedence over all of the other calls. The fourth factor indicates that the call attempts served on line finders consist of two distinct holding time universes and not just one, as was assumed in the development of the dial tone tester formula. The effect of the presence of both a short and long holding time universe of calls would be to introduce a change of slope in the delay curves which may be seen in Figs. 3 to 6 to be at about t = 4 seconds. There is reason to believe that the same cause may have been responsible for the tendency of the observed delay curves to fall away from the theoretical at the lower levels of load carried. Due to the reasons given above and to the fact that the dial tone delay observations were made by the test call method, the above re- sults may not directly describe service from the customer's point of view. Conny Palm has developed the following formula which gives a slightly different measure of customers' dial tone service. It indicates the proportion of calls which have neither received dial tone nor have dropped out at time t. POO = POO) yWJAa/j)e.p i-t/H)] ^^p (_^/^) ^^„) y(c/j, a/j) Curves for this formula are shown plotted on Fig. 1(a) and at C on Figs. 3 to 6. They are quite close in many cases to the observed dial tone tester results. It would appear that a sufficiently good estimate of the customer's dial tone service, whatever its precise definition, can be obtained by the dial tone tester method. Recently revised tables for the capacity of step-by-step line finders have been published for Bell System use based on Palm's formula using a factor of j = 5. This was selected as being slightly conservative for DIALING HABITS OF TELEPHONE CUSTOMERS 47 most applications after reviewing the above Sterling-3 results and other line finder data collected in step-by-step offices. MEASUREMENT OF THE j FACTOR BY CLASSES OF SERVICE As indicated pi-eviously, the data recorded on the tapes showed the states of being bus}^ or idle and of changes in these states for line finders and the associated trip circuits. A fully equipped line finder group of 400 lines has ten trip circuits each of which serves two sub-groups of twenty subscriber lines in the following manner. When a line originates a call its line relay is operated. This causes a ground to appear on a lead which is common to all twenty line relays in the sub-group and starts a line finder hunting for the calling subscriber's line. As soon as this hunt is completed the cutoff relay associated with the calling Une operates and disconnects the line relay, removing the ground (unless, of course, another line in the sub-group has originated a call in the meantime). During periods of overload when line finders are not im- mediately available, the ground due to a single subscriber will persist until : 1. A line finder is obtained, or 2. The subscriber abandons the attempt, or 3. The subscriber receives an incoming call which operates the cut- off relay. The twenty leads from the trip circuit sub-groups were brought out to the pen recorder and a record taken of the grounds that occurred on each lead. Except for the possibility that more than one subscriber is waiting for service at the same time on a given trip circuit sub-group, the record of the occurrences of the grounds gives a substantially ac- curate'^ record of the demands for service and of the number of calls waiting for service. Hence- an analysis of the events occurring on the trip circuit sub-groups and on the line finders as recorded on the tapes gives a means for determining H. The quantity H was introduced in equation (2) in the term ^"""^^""VC^ + 1) (18) For convenience in the ensuing discussion this term will be replaced by ' To obtain absolute accuracy would require the use of a pen recorder with one pen for each of the 400 subscribers served on a line finder group plus one for each line finder. 48 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 the equivalent expression: Ny='j^J{z-\-y) (19) where A''^ = The average number of waiting calls that drop out per unit of time during the state {z + y). y = The number of waiting calls. z = The number of Hne finders occupied with calls. 1/H = A measure of the rate at which calls tend to abandon waiting. f(z -\- y) = The proportion of time that the state {z + y) exists. On the tapes we can measure f(z + y) and count A'"^^. Hence H can be determined. The result is a statistical quantity subject to many- chance factors. In the actual analysis of a tape, the composite average value of H was determined for all possible observed states where calls were waiting. By an analogous process, the composite average value of h for all possible observed states where calls were being served by line finders was determined. Also as a side computation, a composite average value of h' for calls that were served by line finders but for which no peg counts were scored was determined. This value of h' is included in h on the basis that data for engineering line finders consist of estimated calls based on peg counts and of holding times which in- clude an allowance for these short holding time calls. The average values of H, h and of the j factor for the four classes of service studied are given in Table III. The results for H by individual half hours and by various percentages of dial tone delays greater than three secolids are shown on Figs. 7(a) to 7(d) respectively for the four classes of service. On some of these figures an upward bulge may be noted in the center. This is not con- sidered to be characteristic of the habits of the subscribers but is the overall effect resulting from a number of arbitrary mles followed in making the analysis in order to simplify the work and to offset par- Table III Message rate individual Message rate two-party Flat rate individual. . . . Coin Average Values in Seconds 24 42 27 74 159 243 176 153 j = h/H 6.6 5.8 6.5 2.1 DIALING HABITS OF TELEPHONE CUSTOMERS 49 o40 z30 <30 . MESSAGE RATE- INDIVIDUAL . . AVERAGE • • • (a) MESSAGE RATE- TWO- PARTY . . • 1 AVERAGE • • (b) 50 • FLAT RATE-INDIVIDUAL 40 • . ^ ^ • AVERAGE 20 10 0 *" (C) COIN 150 100 a * , * - . averageJ 50 . • 0 (d) 30 40 50 60 70 80 90 0 10 20 30 40 PER CENT DIAL TONE DELAYS GREATER THAN 3 SECONDS Fig. 7 — Average customer waiting time (H). 50 60 70 tially the effect of occasionally having two or more calls waiting on one trip circuit sub-group. The rules and the reasons for them will be de- scribed with the aid of Fig. 8, which shows a hypothetical section of one of the tapes. The rules were as follows: 1 . Initial Overlap Referring to Fig. 8, at ti a subscriber has initiated a request for service. At t-y a line finder rises to serve the subscriber. At tz the sub- scriber receives service. This case is typical of a subscriber receiving prompt dial tone service. The span from ^i to ti was difficult to measure accurately because, for the usual case, it was about the same as the maximum error due to misalignment of the recorder pens. It w^as not measured unless the com- bined span from ti to ts exceeded one second. The span from ^2 to ts involves an overlap, it represents a period when a line finder is busy hunting for the terminal of the subscriber who originated the request for service. It also represents a period when a subscriber is waiting for service. In the analysis this span was treated as a case where a line finder was busy with a call and not as a call wait- ing for service. If the span from ^2 to ^3 and all similar cases had been treated as calls waiting for service and if in addition all spans from ti to ti which were not measured had also been treated as calls waiting for service, the average values for H would have increased slightly for each class of service. 50 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 2. Three Second Rule for the Bridging of Calls Referring to Fig. 8, again, at h a request for service is originated on trip circuit 5 and at h this request is withdrawn. At tw apparently a new request for service is initiated which is then withdrawn at ^n. From manual service observations it is found that subscribers often flash when dial tone is slow. A few pens were used to observe individual subscribers, and Fig. 9 shows a case where a subscriber made several flashes when his tone was slow. When a subscriber flashes it appears as though he 20 19 18 Q. 17 OtE 14- muj 13 ^Z 10 3z 9 CTuj 8 OQ- 7 - I H 4 3 2 1 10 tr 9 UJ 8 "" 7 Z2 j > fTl k i ^ i i i • 1 ir 1 — 1 1 \ ' 1 III 1 to 1)12^3 t4 ts te ty tg tg 15 SECONDS t,2 Fig. 8 — Section of a hypothetical tape showing activities on trip circuit sub- groups and on line finders. were making several bids for service. Actually he is making only one real bid. On the trip circuit sub-group pens it was generally impossible to distinguish between flashes and requests for service by two or more subscribers. To resolve this problem many observations of subscriber lines recorded on the tapes were examined from which it was concluded that no great error would result if a break in the demand for service on a trip circuit sub-group of less than three seconds were considered as a flash and was to be bridged, and a break greater than three seconds was to be considered as the termination of one call attempt and the start of another DIALING HABITS OF TELEPHONE CUSTOMERS 51 S. Treatment of Cases Where Two or More Calls Were Found to he Waiting on One Trip Circuit Sub-Group The oecurrence of several calls \vaitiiif>; on one trip circuit was occasion- ally noted in the analysis. Referring to Fig. 8, a case is shown on trip circuit sub-group 12. At ^9 a line finder is seized. Trip circuit sub-group 18 shows that a subscriber is dialing before tone. The appearance of dial pulses on this trip circuit indicates that only one subscriber is demanding service otherwise the dialing would not show. Trip circuit sub-group 12 however appears to have two or more recjuests for service. One of these requests for service began at ti. The start of the second recjuest occurred somewhere between ^4 and t%, perhaps half-way be- tween. At tg, one of the requests was served by a line finder. To sim- plify the handling of such cases, the assumption was made that the first attempt started at ^4 and ended at ^9 and the second request started SUBSCRIBER FLASHED J ><. f 1 ? -LINE RELAY OPERATED '^ LINE RELAY RELEASED DIAL TONE -" (SUBSCRIBER TOOK (SUBSCRIBER DEPRESSED RECEIVED I RECEIVER OFF HOOK) SWITCHHOOK) I |<_ 21.6 SECONDS -*H Fig. 9 — Example of a customer flashing for dial tone (Tape made October 18). at /g. The effect of this is to understate by an indeterminate amount the average value of H for each class of service. This understatement should be noticeable for the higher degrees of overload because the occurrence of several calls on one trip circuit sub-group is then most likely to occur. The effect of several calls simultaneously waiting in a trip circuit sub-group and of one or more calls dropping out is to overstate the magnitude of H. For instance if two simultaneous call attempts of five seconds each overlap for one second and both attempts are abandoned, the apparent average waiting period is nine seconds, whereas it should be five seconds. It is believed that the above three rules tend to create understatements which roughly balance this type of overstatement. DISTRIBUTION CURVES OF SIMULTANEOUS CALLS The detailed analysis of the tapes provided distributions of simul- taneous calls. For each class of service studied these distributions can be compared with theoretical chstributions derived from the generalized trunking formula using the j factors developed in the analysis. Several 52 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 such comparisons are shown on Figs. 10 and 11. The agreement is quite good in most cases. SUBSCRIBER DIALING HABITS AS OBSERVED WITH A MONITORING CIRCUIT ON A SENDER WITH INDUCED DIAL TONE DELAYS^ As a separate study a series of tests was made by means of a moni- toring circuit on one of the senders serving in common the subscribers in the Sterhng-3 and Main 2 central offices, for the purpose of obtaining further information on subscriber diahng characteristics under overload conditions. A large amount of data was collected on the time intervals from the seizure of the sender to the first action taken by subscribers when encountering dial tone delays, the latter being introduced under the control of the observer. The monitoring circuit was wired to a particular sender in a group of 100 serving all classes of subscribers. When the circuit was in use, the only irregularity introduced was that the dial tone could be delayed even though the sender was actually available to the subscriber. The delay did not affect the sender in its functions if the subscriber elected to dial before tone. The sender monitoring circuit provided the following four features: 1. A receiver was bridged across the tip and ring leads in the sender so that an observer could hear certain actions taken by a subscriber connected to the sender. The sender was of course disconnected before conversation. 2. The observer was able to preselect one of several intervals by which dial tone was delayed on successive calls served by the sender. This was accomplished with a capacitance-resistance-vacuum tube cir- cuit. 3. By means of a timer which started when the sender was seized, the observer was enabled to note elapsed time intervals to the occurrence of the various actions of the subscribers. The reading of the time of the first action of a subscriber had to be made when the second hand was in motion, which introduced certain errors later to be discussed. 4. By means of colored lamps the observer was able to classify all calls observed as being message rate, flat rate or coin. During the sender dial tone delay tests, observations were made only during the afternoons when the flow of traffic was light and the prob- ability of a subscriber obtaining a delay before reaching the sender was a minimum. * Based on an unpublished report by W. A. Reenstra. DIALING HABITS OF TELEPHONE CUSTOMERS 53 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 OBSERVED THEORETICAL ^ 1 1 ''X' ». \ 1 1 1 \ \ » \ 1 1 1 I \ \ \ / A i » \ 4 MRI 1 6.88 ERLANGS i CARRIED ON —J 10 LINE FINDERS \ V. i y(TWO HALF -HOURS ir OF DATA ) \\ V \ \ <.-- 1 1 / / >. 1 t 1 1 \ \ \ 1 1 1 1 // // ^ \ 1 1 l1 7.93 ERLANGS ' / CARRIED ON 1 11 LINE FINDERS 1 1 J (three half-hours\\ OF DATA) A _^^. "/ \ .^^ A \ A / 1 V 1 "^ \ \ 1 1 1 MR 3 ERL ^rrie NE F /f 8.9( C/ 1? 1 1 ANGS V 3 ON ^ NDERS \i_ i X' TWO half -hours V OF DATA) V J ^ 1 r^' 1 1/ r \ f- \ \ 1 \ 1 1 1 \\ 44 — \\ \\ \\ li ji 6.96 ERLANGS CARRIED ON 10 LINE FINDER TWO HALF-HOU OF DATA) \\ 5 V\ ^S \\ 4 V- A A V 1 \ \ \\ j /] FRI ERL/i RRIED // / 1 6.61 CA NGS ON \ / / 1 (THREE HALF- HOURS OF DATA \ /- ^ ^ <--. A . A / / / / / / / / / / h ' if 1 1 ni 1 8.15 ERLANGS ' CARRIED ON LINE FINDERS (TWO HALF - OURS OF DATA) / If "^ / ^ \ A ^ »^ 10 12 14 16 NUMBER OF SIMULTANEOUS CALLS Fig. 10 — Distributions of simultaneous calls. 54 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 0.18 J 0.16 ) ; 0.14 ) I 0.12 > 5 0.10 jO.08 i 5 0.06 ) •0.04 0.02 0 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 OBSERVED THEORETICAL MR I 14 48 ERLANGS CARRIED ON 20 LINE FINDERS (2 HALF HOURS OF DATA) / \/ \ I V M A ' V 1 1 \ / > \ \ V JVIR 2 PARTY 16.84 ERLANGS CARRIED ON 19 LINE FINDERS (2 HALF HOURS OF DATA) ,4 ' / 1 I 1 ', 1 I 1 / i ■^ f \ ■'^_ COIN 24.38 ERLANGS CARRIED ON 30 LINE FINDERS (2 HALF HOURS OF DATA) A- A r \ 1 1 t A / I \ ^'/ / \ S. MR I 18.23 ERLANGS CARRIED ON , 20 LINE FINDERS A / * / 1 1 1 — U— 11 11 11 -U — ll ll ll I (2 \ HALh HOURS OF DATA) j ll 1 1 f — A \ MR 2 PARTY 18.52 ERLANGS CARRIED ON 21 LINE FINDERS (2 HALF HOURS OF DATA) ^ ? \ \ 1 1 1 , \ 1 \ \ / V COIN 28.33 ERLANGS CARRIED ON (2H 34 L 1ALF INE HOU FINDE ^S 0 ERS - DAT A) 7 k / 1 \ / •\ r \ -/ \ 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 NUMBER OF SIMULTANEOUS CALLS Fig. 11 — Distributions of simultaneous calls. DIALING HABITS OF TELEPHONE CUSTOMERS 55 The observer was provided with a means for introducing either no delay or one of four values of delay 2, 5, 10 or 15 seconds into the sender dial tone circuit. The observer took 50 observations using a particular value of dial tone delay and then shifted to another so that no particu- lar value of delay would become evident to the customers during an afternoon's test. Each group of 50 observations comprised a mixture of message rate, coin and flat rate calls in the approximate proportions of 13 to 6 to 1, representing the respective volumes of traffic from these classes of service during the afternoon periods. It was not possible to distinguish PBX lines or two-party lines from the bulk of the message rate data nor PBX lines in the flat rate data, although to a limited extent the observer could identify PBX dialing by the generally faster pulsing. The coin data represent both public and semi-public customers. Fig. 12 is a diagram for explaining the results shown on Figs. 13, 14 and 15 for the message rate, flat rate and coin classes of service, respec- tively as obtained with the sender monitoring circuit. Fig. 12 was ob- tained by the application of fitting curves to those message rate data of Fig. 13 for which a dial tone delay of five seconds was introduced by the observer. In the interval from t = 0 to t = 5 seconds, three curves A, B and C represent the per cent of subscribers still waiting at time t for dial tone. Curve A and its extension beyond t = 5 seconds represents the action of subscribers who would dial their calls before tone if dial tone were delayed indefinitely. Curve B and its extension beyond t = 5 ^z gg O60 < UJQ Oz ccxu Q-< I 20 x' — ~0\' _ — 7-SURVIVOR CURVE DISCONNECTING^S. ^ BEFORE TONE IF DIAlX. TONE WERE DELAYED INDEFINITELY 1 1 A DIALING BEFORE TONE IF DIAL TONE WERE v.,^ DELAYED INDEF- ^V. / INITELY ' 1 D 1 A L 1 N G^Vy '*""*"C' - disconnectingN^N^ AFTER TONE \\x,.,,^___^ 1 -.1,1 1 i 16 0 2 4 6 8 10 12 14 TIME,t, IN SECONDS AFTER SENDER SEIZURE Fig. 12 — Explanatorj' chart for sender monitoring observations; dial tone at T = 5 seconds. 56 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 seconds represents actions of subscribers who would disconnect if dial tone were delayed indefinitely. Curve C and its extension is the sum of the other two. Curve D in the region beyond t = 5 seconds represents the actions of subscribers who disconnect after tone, curve E represents the actions of subscribers who will dial their calls after tone and curve F represents the sum of the lower curves. Of interest is the fact that for an interval of about two seconds following dial tone (at i = 5 seconds), the observed total survivor curve F lies above the extended portion of CO z _) > q: o < H =5 °r (Du.< (0 (\J ■* <0 CVJ 20> ui CM m 00 01 y ^ 3 III CO O) W (Jl oo > 1 Z 10 i i CD O ~ H o (MiT) om < ^ _I_ n u 1 III w w (/) • UJ w 1 1 ; 5 ^ -IJ — Z) c3 ISl > 111 0) in r« ^ (D OC O - UJ Q fall 7 C UJ 10 o ^rr .-^ ^ UJ r^ t- n < n -2 z o o a; -a 005 ^ « ■^ 3kNll Aa NOIiOV NSMVi iON 3AVH OHM SdaaiyOSQnS do iN30 a3d DIALING HABITS OF TELEPHONE CUSTOMERS 57 the li^-potlictical surxivor curve C for inlinitel}- delayed dial lone. This indicates that most of the subscribers who would have abandoned their attempts durinjj; this in(er\'al abruptly changed their minds and then consumed a noticeable interval of time after hearing tone before start- ing to dial. Thus, as might be anticipated, the subscribers exhibit a reaction time. Fig. 13 shows the results in terms of survivor curves that were ob- served for the message rate class of service. Five sets of curves are ■^ 3\^i\L A9 NOIiOV N3>('(M\-er. The signal then sul't'ers attenuation due to i'e(l(>cti()n of |)art of tlu^ energy from th(> diicct path. Widely separated fi'e(|nencies aie affected in like fashion and the outputs of anteinias sjjaced for dix'ersity I'eception tend to be in agreement although th(> fine structure fachng is usually different. On neither of the experimental transmission paths is there a I'geular ground-i-eflected component of any conseciuence. Due to the roughness of the giound and the presence of vegetation, the effective reflection coef- ficient is of the order of 0.2 for eithei- path. (Ji'ound reflections thus play Fig. 3 — Possible ray paths involved in severe failing, (a) Multiple path trans- mission, (b) Attenuation by reflection from an elevated layer, (c) Abnormal water reflection on the Murray Hill-Crawford Ilill j^ath. (d) Substandard con- ditions on the Southard Hill-Crawford Hill path. 72 THE BELL SYSTEM TECHNICAL JOI'RNAL, JANUARY 1952 no significant part in the fading picture with the exception of the situa- tion illustrated in Fig. 3(c). Occasionally on the Murray Hill path, conditions of atmospheric refraction are such that a strong signal com- ponent is received by virtue of reflection from the water surface of Raritan Bay. Under normal conditions, the geometry of the path does not permit such a reflection. Normally the dielectric constant of the atmosphere decreases with height above ground so that the ray path usually has a curvature in the same direction as the earth cur^^ature. However, it is possible for the dielectric constant of the atmosphere to increase with height above ground (sub-standard conditions) so that the ray path has a curvature opposite that of the earth. This results in the condition illustrated in Fig. 3(d) where the limiting or tangent ray does not reach the receiver and only a weak signal is received by virtue of diffraction. Widely sepa- rated frequencies and vertically spaced antennas are affected alike as regards the average signal level but not the fine structure fading. This effect has been observed only on the Southard Hill-Crawford Hill path which has small clearance to begin with. It has been observed on several nights in late summer or early autumn after a radiation type ground fog has formed in the late evening and usually persists until the fog is dispelled by winds or by the morning sun. There are, of course, times when the transmission conditions are con- siderably more complicated than those described above. Some of these apparently are due to a combination of the situations illustrated in Fig. 3 while others may be the result of an atmospheric focussing or trapping phenomenon. In addition to the various phenomena just described, which, fortunately, occur rather infrequently, there are numberous occasions Avhen the signal varies plus and minus a few decibels relative to the free space level. It has not been possible actually to demonstrate the mecha- nism responsible but it seems most likely that these smaller variations are due to non-linear dielectric constant gradients which give the atmos- phere the properties of a convergent or divergent lens. An important result of the observations made to date is the con- viction that the severe fades, signal excursions to levels 30 decibels or more below the free space field, were all caused by wave interference. It appears that, as the average signal level is depres.sed by any mechanism, it becomes more and more vulnerable to the effects of extra signal components of small amplitude that often may be present but go un- noticed when the signal is near normal levels. Thus, while the average signal level during the conditions illustrated in Figs. 3(b) and 3(d) may be no more than 15 to 20 decibels below the normal daytime level, there SELECTIVE FADING OF MICROWAVES 73 is usually superimposed a fine structure fading in which short duration fades to levels as much as 45 decibels below free space have been ob- served. For this reason it is desirable to avoid paths having small clear- ance over intervening terrain and also paths which have a permanent ground reflection of sufficient magnitude to depress the signal to critical levels when, due to variable atmospheric refraction, the direct and re- flected components are in phase opposition. The following sections describe the angle-of-arrival and frequency- sweep experiments on which much of the preceding discussion was based. ANGLE-OF-ARRIVAL OBSERVATIONS A photograph of the Cra^vford Hill receiving site is shown in Fig. 4. The building housing the receiving equipment and the associated an- tennas are mounted on a framework which can be rotated on a concrete track, permitting investigation of the transmission characteristics of either path. The parabolic antennas on the tower are used for continuous recording of 4195 megacycle signals. The long object at the left of the Fig. 4 — The Crawford Hill receiving site. 74 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 picture is the metal-lens antenna used for making the angle-of-arrival observations. Its half -power beamwiclth is 0.12 degree at the operating frequency of 24,000 mc. The focal length of the lens is 48 feet and its feed is located in the little cupola on top of the building. The feed is held fixed, while the lens is moved vertically by a motor-driven mecha- nism; thus the antenna beam also moves vertically. The antenna scans a total angle of two degrees in ten seconds. It is fed by a 24,000-mc radar set which is gated to receive only the pulses reflected from a corner reflector located at the distant terminal of the transmission path. The spot on the radar cathode ray tube moves vertically in synchronism with the scanning antenna, and the horizontal deflection is proportional to the amplitude of the pulse received from the corner reflector. The display thus shows amplitude of the various incoming signal components as a function of their angles of arrival. The antenna installation on Southard Hill is shown in Fig. 5. At the left is the transmitting paraboloid for the 4195-mc continuous wave transmitter, the radar corner reflector is in the center, and on the right is the horn-reflector antenna used in the frequency-sweep experiments described below. Similar equipment is located at the Murray Hill termi- nus. The corner reflector is 5.5 feet on a side, and at 24,000-mc has sufficient gain to override reflections from other nearby objects, and thus becomes easily identifiable on the radar screen. The radar oscilloscope for typical propagation conditions is shown in Fig. 6. These pictures were obtained by leaving the camera shutter open during the ten-second interval required for the antenna beam to scan through the angular range of 2°. All of these representative photographs were taken on the Murray Hill-Crawford Hill path although similar results were obtained on the Southard Hill-Crawford Hill path with the exception of Fig. 6(f). The normal daytime transmission is shown in Fig. 6(a) to consist of a single path arriving at an angle of —0.2° with respect to a fixed reference angle. The horizontal lines represent intervals of 0.1°, so that changes of 0.05° can be estimated. The other pictures in Fig. 6 were all taken during fading conditions. Figs. 6(b) and 6(c) are good examples of the multiple-path condition shown in Fig. 3(a) in which the individual components are almost equal in amplitude and Avell separated in angle. In Fig. 6(b) there are two components arriving at angles of 0.1° and 0.6° above the normal line-of- sight while in 6(c) there are three components with angles of 0.05°, 0.35° and 0.7° above the normal angle. The position and amplitude of the signal components may change radically in a matter of minutes, and often there is no component that can be identified as the "normal" one. SELECTIVE FADING OF MK Ki )\\ A V KS Fig. 5— The Southard Hill transmitting site. 76 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Fig. 6 — Representative photographs of angle-of -arrival observations on the Murray Hill-Crawford Hill path, (a) Normal da}', (b) Two elevated paths. Sept. 8, 1950; 9:23 p.m. (c) Three elevated paths. Aug. 27, 1950; 1:11 a.m. (d) Multiple paths. August 26, 1950; 11:00 p.m. (e) Wide angle "fill-in". Aug. 26, 1950; 11:04 p.m. (f) Abnormal water reflection. Sept. 8, 1950; 11:28 p.m. During these multiple-path conditions, the recordings of the 4195-mc transmission generally show the broad maxima and sharp minima charac- teristic of wave interference. Figure 6(d) shows a case in which the various paths are not completely separated while Fig. 6(e) (taken four minutes later) shows that energy is being received almost without variation over a vertical angle of 0.4°. This may represent a number of ray paths which would be separable by a narrower-beam antenna, or it may indicate a focussing or trapping phenomenon. Often when the type of transmission illustrated by 6(e) is present, the recorded 4195-mc signal may be as much as 12 to 15 decibels above the free space levels. SELECTIVE FADING OF MICROWAVES 77 Fig. ()(0 illustrates the case of abnormal reflection from the water of Raritan Bay on the Murray Hill path as indicated in Fig. 3(c). Here the "normal" signal component is arriving at 0.1° above the line-of -sight while another component, almost equal in amplitude, is arriving at the \-ery bottom of the scan, about 0.8° below the line-of-sight. It is quite probable that there have been times when this component was present hut was outside the range of the scanning antenna. The mechanisms discussed in connection with Fig. 3(b) and 3(d) can- not be demonstrated by photographs such as those just presented al- though the angle-of -arrival radar was instrumental in furnishing the clues to the phenomena. Due to the two-way attenuation of the radar- corner reflector technique, the signal at these times rapidly falls below the noise level of the receiver. For the same reason, it is not possible to detect the extra signal components of small amplitude which were postu- lated to account for the very deep fades sometimes observed under these transmission conditions. FREQUENCY-SWEEP OBSERVATIONS Since most of the fading is due to interference between waves which travel over different paths of, presumably, different lengths it was realized that the fading was likely to be frequency selective. Just how selective would depend on the relative lengths of the individual trans- mission paths. The usual methods for determining path length differ- ences are to use short pulses, or to sweep the freciuency. Since it was likely that the path-length differences would be measured in feet rather than yards, very short pulses or a wide frequency-sweep were required. An oscillator^ was available whose frequency could be swept over the licensed band of 500 mc between 3700 mc and 4200 mc. The frequency- sweep experiment was set up on the Murray Hill-Crawford Hill path for the summer of 1949. The following summer, the milli-microsecond pulse transmission tests described in the companion paper were con- ducted over the same path. As might be expected, simultaneous observa- tions showed good agreement between the two methods. The frequency of the transmitter, located at Murray Hill, is swept over a 450-mc band centered at 3950 mc at a 60-cycle rate. At the receiver, a similar oscillator is used for the beating oscillator except that its frequency is swept linearly through the same frequency band in one ^ This oscillator was developed b}- M. E. Hines and is described in his paper published in the Bell System Technical Journal, Vol. 29, Oct. 19.50. It uses a 416A close-spaced triode in a wave-guide cavity. The frequency is changed by means of a plunger which is capacity-coupled to the plate of the tube and which is actu- ated by a modified loud speaker unit. 78 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 second. Since the intermediate fretiuency amplifier of the receiver is only 350 kc wide, (centered at 600 kc) narrow pulses are generated each time the frequency of the transmitter crosses the freciuency to which the receiver is tuned. These intermediate frequency pulses are displayed vertically on a cathode ray tube. The horizontal trace is synchronized with the one-second sweep rate of the beating oscillator. The normal daytime frequency-sweep pattern is shown in Fig. 7(a). The vertical scale is linear in amplitude and the horizontal scale is almost linear in freciuency, with frecjuency decreasing from left to right. Visible at the extreme left is the signal used for continuous recording. Since there is only one transmission path involved, the amplitude of the re- ceived signal is nearly constant over the 450-mc band. If another signal were present which had travelled over a path of different length, the two signals would add when the frequency is such that the path length difference is an even multiple of half-wavelengths and subtract when the path length difference is an odd multiple of half-wavelengths. Simple calculation shows that if the path length difference is one foot, the frequencies at which the signals add and subtract are separated about 500 mc. Thus the limit of resolution for the frequency-sweep experiment is a little more than one foot. Photographs taken on a night when the angle-of -arrival radar indicated two almost equal components separated about 0.4 degrees in angle are shown in Fig. 7(b). The time interval between the two pictures is 30 seconds, during which the minimum had shifted about 150 mc. The pictures can be interpreted as simple two-path transmission with an indicated path difference of about two feet and an amplitude ratio of 0.7 to 1. On this night the minimum shifted back and forth across the frequency band — sometimes slowly and sometimes rapidly. At times the position of the minimum might remain fixed but its depth would change. Photographs taken on a night when there were abnormal reflections from the water of Raritan Bay are shown in Fig. 7(c). There are evidently two main components with path difference of about six feet, with a small third component causing the slight decrease in amplitude of the peaks from left to right. These pictures were taken 9 minutes apart, but this type of pattern was observed over a period of about three hours on this night. Usually the frequency sweep patterns are considerably more compli- cated than the ones shown so far. Fig. 7(d) shows two photographs which indicate that at least three signal components and perhaps more were present. The time interval between the two pictures was about 30 seconds. SELECTIVE FADING OF MICROWAVES 79 Fig. 7 — Rei^rpsentative freciuency-sweep patterns oljserved on the Murray Hill-Crawford Hill path. (Summer 1949.) (a) Normal day. (h) Two components with a path difference of two feet, (c) Two main components with a i)ath differ- ence of about six feet plus a small third component, (d) Multiple component pattern. 80 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 SYNTHESIS OF FREQUENCY-SWEEP PATTERNS To aid in the interpretation of the compHcated frequency sweep pat- terns, a computor of the analogue type was built. This apparatus com- bines four signal components, three of which are variable in delay and amplitude, and presents the result on a cathode ray tube in the same form as the actual frequency sweep patterns. Thus a particular pattern can be synthesized on the computor and the number of components, together with their path differences and relative amplitudes, read directly from the computor dials. This is accomplished by generating four 600-kc signals, three of which are phase modulated at 60 cycles per second. The total phase deviation and relative signal amplitude are variable. The four signals are then summed and displayed in vertical deflection on a cathode ray tube having a 60-cycle horizontal sweep. The synthesis of the patterns of Figures 7(b) and 7(c) are shown in Fig. 8. The upper synthesized pattern is simply a combination of two components with relative amplitudes of 0.7 and 1 and a path difference of two feet. The lower pattern consists of the reference component with unity amplitude, a second component with an amplitude of 0.5 and a path difference of 5.7 feet, and a third component with an amplitude of 0.2 and path difference of 0.8 feet. The similarity between the actual and synthesized patterns is obvious. Fig. 8 — Synthesis of the frequency -sweep patterns of Figs. 7(b) and 7(c). Fig. 9 — Synthesis of complicated frequency-sweep patterns using four com- ponents. See Table I for values of the relative amplitudes and path differences. 81 82 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Examples of attempts to sjnithcsize some more complicated frec{uency sweep patterns, taken on the night of Angust 2, 1949, are shown in Fig. 9. Four components were reciuired in each case, with the path differences and delays being summarized in Table I. Although the pic- tures all appear very different, in general major changes were required only in component Xo. 2 to go from one pattern from the next, as Table I shows. All the remaining components had to be very carefully trimmed in both amplitude and delay to get good synthesis (especially in the case of Fig. 9(d), but these changes were relatively small. CONCLUDING REMARKS The special experiments just described have led to the conclusion, expressed earlier, that the severe fading observed on the two test paths is the result of multiple-path transmission in which several components may be involved. These components may arrive at the receiver at various angles up to three quarters of a degree above the normal daytime angle- of-arri\'al and, in the case of abnormal water reflection on the Murray Hill path, as much as 0.8 degree below the normal angle. The path differences among these components may \'ary from a fraction of a foot to about ten feet. The long-delaj^ components are usually small in ampli- tude. In all cases where observations were made during periods of excep- tionally high signal levels, say 10 to 15 decibels above free space level, the frequency-sweep patterns were substantially flat, suggesting a focus- sing or trapping phenomenon. The frecjuencj'-sweep patterns were also flat on those nights when the signal excursions were only a few decibels above and below the normal daytime level. However, the severe fades Table I Fig. 9 (a) Fig. 9(b) Fig. 9(c) Fig. 9(d) Fig. 9(e) Component No. 1 (Reference) Amplitude Path diff. (ft.) 1 0 1 0 1 0 1 0 1 0 Component No. 2 Amplitude Path diff. (ft.) 0.9 1.1 1.2 0.5 0.7 1.7 1.1 0.5 0.4 1.1 Component No. 3 Amplitude Path diff. (ft.) 0.2 5.2 0.1 5.7 0.2 5.6 0.2 5.7 0.2 5.7 Component No. 4 Amplitude Path diff. (ft.) 0.05 9.2 0.1 9.2 0.15 11.0 0.1 9.3 0.1 8.7 Tim e 12:08^ AM 12:09 AM 12:18 AM 12:24 AM 12:25 AM SELKCTIVE FADING OP IMKKO WAVES 83 rs k ~3C^ "^H imn f ^ 1 ' ■ V-^P^^H fc/"C«;a%- p4 ^ ^ J_^jjj ^3 :^^ .^^1 ^^^^^^^^^I^^^^^^^^^H 1 ^H Br^^^^B L J0 ~ JE ^^^^^l[^^H HiA«s*;t«i ^'~'m ^H ^ 1 d_K^^^B k^^H HHHh I -Jji "*^ • '^^B ^Hlr^^^'I^^^H F -^ ' ~ ' 19 K <^,^^l.^fll ^^' ^^^^^ ! ,^1 r^H fctt««ty B|Aw6j;^^ f^ ^jfljl ^^^^^i^^^T^H ^^ ^ - 1 MiB**^^B i ^^H ^^RHpi^pr^Hl L ^ *- - i^l ^1 .^^^^s^^^^l ^SS0«^ * tr i^w^^ ^^H|^^^^HB|g|^|^^^^^| p ^^^^j 1 /I i^H ^^•~ *^^M ^^^■[ji^^itt^^H ^'zJm ^^^T^j^^T|_^H ' 1^1 i^^i HHH^H ^ -J^ ^ -r .j^B K^^fl mwOij_sr' 'f -JB K^ISlJJ jP^flH M RH L^^eI K2 ■9 ^^^^^^H ^^^^^j^ 1 r^ B**^^B ^9 ^^^^^^ 1 ril*^B L3bI ll 9 l^>S t; Mi ^ ^M^^HM^^I ^^^r^^^ y9 rNffl| M H ■baM^^'^^BflH ^HiJMi itts^H ^^^^^B^H ^^^^^^ ^5lJ QM li H ^^SBi L. ^IwhPM^B ^^^p^— < foB Hh ■■fl OXI c 9 ^^s^mH^I ^^^B^^v ^^tB PiSl R Mjl L^^^^J^H ^B^ ^^lii^^l ^^Hj H l^^^^B DM B l^w^^H ^^^^^>^ [3^1 ^1 P 9 Fig. 10— A seciuciice of ;ingle-of -arrival and frequcnc-j'-sweep patterns taken at ten-second intervals. 84 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 (T , OO LL in ■ S ■ : :?0) ,— -^ 'S 0(\J 1 ^--HL; > -CD ±n 1- h"'' u-n < ■ ■> <, X 1- '■^ e 1 3 O i^'^i / 1 "1 — r lO O "O O IT) OiO r oj oj roo ' lilt 91391030 S-139ID3a 51391330 SELECTIVE FADING OF MICROWAVES 85 !- O CO f^ iJ - C _ — 3 ht -r.2^ cc -> 3 :" ,^ •--V<*._: bC ^ O g g 30gS=^ c-bO := o c J, ^Q cl^g-2 -!3 rt 3 w •= ?:X! cu 3 7 3 (U ^ O, o -^ aj 3 .^^o ^_^ °- c-^ cc O C " «^ o g 2S-S^ o " 3 c^ t, =- 5 rt _ O -—■-Go CO Jc-^ siaaioaa S-139l03a '-' "^ H '^ '3 '"' rt' 3 3 S ' — ci 'T- o C^ 86 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 — = ^ o .= 111 -^ - o —■ - ' > <) -^ . 2 — -^ ^ P=- ; : — - o 1- HI (n ^•) -> < V J -3 ~) 111 z J -> -^ — —z ; —^ E- V — < 5 — = - - - r = ; q: n I = : ^ < - - — r Q. ^ - -I = =. _l — = — X -z 1 a q: — : ir < -. [- o U- i .= L 5 "i E: < U - -= r 1 m — — _r UJ _ z_ - : I ' : Q i ' < I - 1- ^ _i ■- 3 o < -1 -^ i_ in —^ - o :=i 9T39l03a S13903a SELECTIVE FADING OF MICROWAVES 87 O UJ Q > O Z t- o o ~ '-. - 1- Q- — OJ = iTi =; := r ^- =^ U - < ^ — — .=_ ^==- =— -) = ^=^ OJ ^=^ =^ z — — — J^ E^ '— ] ~" < ^ 2 ^ I Q. _) a: D- < _l I or a < cc i o LL % < (T O m UJ _l U- _l X >- < en u. z D < o 5 2-c o^ 0) is t- ->-> fl) -t-' hf) Li C^ c •7- a> 0 r/1 0 m tc CJ J M j3 CO 2 ^ -3 oS m:.2 L >3 88 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 of short duration sometimes observed when the average signal level was depressed by the mechanisms of Figs. 3(b) or 3(d) were found to be fre- quency selective. Some of the studies described in this paper were made with vertically polarized waves and some with horizontally polarized waves; at times, 45° polarization was used. In so far as it was possible to determine, the propagation characteristics of both paths were independent of the polari- zation used. No meteorological soundings were made in connection with this work. Considering the rapid changes usually observed with the angle-of -arrival and frequency sweep apparatus, it is doubtful that meteorological meas- urements made in the usual manner would show much correlation with the radio observations except, perhaps, in a general way. The sequence of pictures in Fig. 10 is included to show how the angle-of-arrival and 0.8 0.6 0.5 0.4 0.3 o o H 0.1 I ^ -) 0.08 a III 0.06 < 0.05 ^ -I 0.04 O 0.01 £ 0.008 0.006 0.005 - \ A- JUNE, JULY, AUG. & SEPT. - \ B - APRIL, MAY, OCT. & NOV. C - DEC, JAN., FEB. & MARCH \ , \ \\ \ \ c> \ \ \ k k 1 1 \ \ \ \ \ \ \ \ \ \ s. \ \ \ V - \ \ - \ \ \ \ 0 -5 -to -15 -20 -25 -30 -35 -40 SIGNAL LEVEL RELATIVE TO NORMAL DAYTIME VALUE IN DECIBELS Fig. 13 — Time distribution curves of the signal levels observed on the Murray Hill-Crawford Hill path. Data of 1947, 1948, 1949 and 1950. SELECTIVE FADING OF MICROWAVES 89 frequeiicy-swcep patterns change with time. Tliese pictui'es were taken at lO-second iiiter\'als. On this occasion there was good correlation l)e- Iween the angle-of-arrival and free [uency -sweep data. Sucli was not always the case, howe^'er, and considering the wide difference in opeiat- ing frequencies, 24,000 mc and 4000 inc, instantaneous correlation should not necessarily be expected. Although all the studies described in this paper were made on the two local paths, the results are compatible with pi'opagation measurements made by another group in the Laboratories during a survey for the ti'anscontinental radio relay system. ACKNOWLEDGEMENTS The authors wish to acknowledge the contributions of W. M. Sharpless who, for some time, was associated with this work; also to acknowledge 1.0 0.8 0.6 0.5 0.4 0.3 O O 3 0.08 CO 0.06 - , \ A— JUNE, JULY, AUG. & SEPT. - \\ B - APRIL, MAY, OCT. & NOV. C- DEC, JAN., FEB. & MARCH \ \ \ \ \ \ \ \\ \ ^ ^ - V \\ - \ \ \ \ V \ \ \ \ \ \ w \ \ \ \ - \ \ A - \ \\ \ \\ I 0-05 _l 0.04 < z 0 0.03 to 1 0.02 l- LL O I- z HI U 0.01 S 0.008 CL 0.006 0.005 0 -5 -10 -15 -20 -25 -30 -35 -40 SIGNAL LEVEL RELATIVE TO NORMAL DAYTIME VALUE IN DECIBELS Fig. 14 — Time distribution curves of the signal levels observed on the Southard Hill-Crawford Hill path. Data of 1947, 1948, 1949 and 1950. 90 THE IJELL SYSTEM TECHNIf'AL JOURNAL, JANITARY 1952 the full time assistance of R. A. Desmond and the part time assistance of L. K. Lowry and S. E. Reed. All the work was done under the guidance of Dr. H. T. Friis. APPENDIX This appencUx is included to illustrate some of the characteristics of the propagation as shown by the recordings of 4195-mc signal levels. Fig. 11 is a reproduction of some typical records obtained during severe fading periods. Fig. 11(a) is an example of transmission during the sub- standard conditions illustrated by the ray diagram of Fig. 3(d). Fig. 11(b) is typical of multiple-path type fading in which the signal com- ponents arrive from elevated angles as shown in Fig. 3(a), while Fig. 1 1(c) was recorded on a night when, for a time, there were abnormal reflections from the water of Raritan Bay on the Murray Hill path, see Figs. 3(c), G(f) and 7(e). The records of Fig. 11(d) show how the outputs of two similar antennas, spaced vertically about 30 feet, differ in regard to the deep fades of short duration. The chart of Fig. 12 shows how the fading varies with the time of year. On this chart, the vertical lines represent the extremes in signal level observed during the twenty four hour period from noon to noon. The large signal variations are concentrated mainly in the summer months. The time distribution of the signal levels recorded on the Miu'ray Hill-Crawford Hill path are shown in Fig. 13. Each of the cur\'es is for a four-month period: the period of least fading, December, January, February and March; the period of most fading, June, July, August and September; and the in-between period consisting of April, May, October and November. Data obtained in the years 1947, 1948, 1949 and 1950 are included. Fig. 14 shows similar data for the Southard Hill- Crawford Hill path. The hump in the time distribution cur\'e for the months of April, May, October and November is due to substandard conditions, illustrated by the ray diagram of Fig. 3(d) and the typical record of Fig. 11(a), which affected transmission on this path during several nights in October, particularly in the years 1947 and 1950. When it occurred, this type of ti'ansmission usually persisted for a period of several hours. Propai>ati()n Studies at Microwave Frequencies by Means of Very Short Pulses BY O. E. DE LAXGE (Maimscript roceived March 27, 1951) Microwave pulses with a duration of about 0.003 microseconds were trans- mitted over a 22-mile path from Murray Hill, .V. ./., to Holmdel, iV. ./., in order to deterrninc the effects of the transmission medium upon siich puhes. During '\fading'' periods multi-path transmission effects with path differences as great as 7 feet were observed, as well as some other effects. A microwave frequency of 4000 megacycles was employed. INTRODUCTION This experiment was set up with two main purposes in \\e\\: First, a.s a means of studying microwave propagation, especially with regard to multi-path transmission effects and second, to determine the effect of a transmission path upon the shapes of very short pulses, particularly to learn what restrictions might be imposed upon minimum pulse length or spacing between pulses by distortions produced in the transmission medium. In regard to multi-path transmission the pulse method seems to be the most straightforward way of studying such effects. For example, if there is transmission by more than one path, and if the pulses are suffi- ciently short in comparison to the path length differences involved, then there will be received a separate pulse for each path. Under these condi- tions the number of paths in\'olved, path length differences and other information become directly evident. If pulse duration is too great with respect to the path differences involved, the pulses received via the various paths will o^'erlap in time and the resultant multi-path effect will be pulse distortion rather than reception of indi\4dual pulses. This situa- tion is much more difficult to anatyze. TRANSMISSION PATH The transmission path is the same as that used by A. B. Crawford fen* microwave propagation studies by means of the frecjuency sweep method, 91 92 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 i.e., the path from Murray Hill, N. J., to Crawford's Hill (near Holmdel), N. J.^ The path length is approximately 22 miles, and is partly over water and partly over rough land terrain. The frequency sweep studies had indicated that the path differences involved in multi-path fading were of the order of one or two to about seven feet. In terms of delay times this means differences of about 1 to 7 millimicroseconds. In order to resolve the paths when the path differences were only one or two feet, we should have liked to have pulses of about 1 millimicrosecond duration. Because of the difficulties involved in generating, amphfying 40 PU ANTEh DO MC'. LSES jna\ / CONTINUOUS WAVE REFLEX OSCILLATOR (4000 MC) . PULSE MODULATOR \ J /- ^ 1 , TRAVELING ' WAVE AMPLIFIER BASE-BAND PULSE GENERATOR MONITOR RE (lOMC EPETITI PULSE DN RAT E) Fig. 1 — -Transmitting equipment. and detecting such short pulses, we accepted pulses which, when dis- played on our final indicating equipment, had a length of 3 millimicro- seconds at half ampUtude. (About 6 millimicroseconds at the base.) In free space this pulse would be just about 6 feet long at the base. TRANSMITTING EQUIPMENT The transmitter was mounted on top of a 100-foot tower at Murray Hill. As can be seen from Fig. 1, it consisted of a c-w reflex oscillator operating at 4000 megacycles, a baseband pulse generator, a modulator, or gate, for modulating these pulses on the microwave carrier, a single stage traveling-wave amplifier and finally a horn antenna. Approximately one watt of power was obtained from the transmitter at the peaks of the pulses. The antenna area was 25 square feet and its gain 32 db above that of a dipole. A pulse repetition frequency of 10 mc was employed. ' A. B. Crawford and W. C. Jakes, Jr., "Selective Fading of Microwaves," Bell System Tech. J., 31, Jan. 1952, pp. 68-90. PROPAGATION STUDIES AT MICUOWAVE FHIOQUENCIKS 93 RECEIVING EQUIPMENT The ret'oi\-ing antenna, a large horn, was mounted between two poles guyed for support. It had an apertiu'e of about 90 square feet and a gain of approximately 38 db over a dipole. The recei\'er circuit is sliown in Fig. 2. About GO db of gain at 4000 mc was provided by either two or three stages of t raveling- wa^■e tube amplifier depending upon the gain of the particular tubes used. It was necessary to provide very good shielding and also careful filtering of all power leads to eliminate the tendency for this amplifier to sing. The amplifier fed two crystal detectors through a hybrid tee junction. Each detector employed a silicon crystal of the IN23B type. Two incUcator circuits are shown in Fig. 2. These circuits are very similar except that one employed a vertical amplifier coupled to a Dumont 5XP2 CRO tube, whereas in the second the baseband output of the crystal was fed dii'ectly onto the deflection system of a tra^•eling-wave type of CRO tube. The latter CRO tube, which has been described by J. R. Pierce in the November, 1949, issue of Electronics, has a very high deflection sensitivity and is used with a microscope to enlarge its trace; hence, no amplification was required between it and its driving crystal. The deflection system of this tube has a bandwidth of 500 to 1000 mc. The micro-oscilloscope was provided primarily for photographing pulses by means of a 35-mm camera attached to the microscope. (Exposure time was 5 to 15 seconds. The time recorded for each picture corresponds 4000 MC PULSES WAVEGUIDE HYBRID JUNCTION \ C TRAVELING WAVE AMPLIFIERS BASE-BAND PULSES INDICATOR NO. 1 CRYSTAL DETECTOR : TERMINATION NARROW-BAND CRYSTAL AMPLIFIER DETECTOR Fig. 2 — Receiving equipment. 94 THE BELL SYSTEM TECHXICAL JOURNAL, JANUARY 1952 to that at the end of exposure.) A second microscope made it possible to view and to photograph the screen of the tube simultaneously. The general procedure was to obser^'e continuously during periods of dis- turbed transmission, taking pictures at regular intervals of 5 to 10 min- utes. When conditions were seen to be changing rapidly, pictures were taken much more frecjuently. The large oscilloscope with its vertical amplifier had a bandwidth of about 150 mc and hence caused some deterioration of the pulse. It, however, was less tiring than the small scope, especially for long periods of observation and was watched to follow the general trend of events. It was capable of resoh'ing the pulses resulting from two-path transmission when the path cUfferences were large. The sweep circuits for the two indicator oscilloscopes were practically identical. The horizontal sweep \^oltage for each consisted of the linear portion of a sine wave which was generated by a c-w oscillator operating at one third of the pulse repetition frequency of 10 mc. Each oscillator was synchronized with the incoming pulses by means of a 10-mc voltage deri\'ed by amplifying the pulse energy through a narrow band amplifier. This circuit provided very satisfactory s\aichronization e^'en during the times when signal amplitude was so low as to produce a xevy poor signal- to-noise ratio. Timing markers were provided on each roll of film by periodically photographing a series of pulses spaced by an inter\'al of 9 milhmicroseconds. RESULTS OF THE EXPERIMENT The picture at the left of Fig. 3 shows the transmitted pulse. The right-hand picture shows the recei^•ed pulse under what were considered to be normal transmission conditions. It is seen that, except for the addi- tion of noise and widening of the pulse due to passage through the ampU- fiers and other equipment, the pulse shape is unaffected. The time cali- bration on this and the following photographs are in millimicroseconds, each mark representing one millimicrosecond (0.001 yns). During the summer of 1950, when this experiment was in progress, A Fig.3 — (Left) transmitted pulse (right) received pulse — normal transmission. PROPAGATION STUDIES AT MICRO WAVK FREQITENTIF.S 95 SEPT 8 1950 n : 0 5 - !2,0iP.I| SEPT 8 1950 1 ) 0 7 ~ 2 0 P K ^^^^^^^■H SEPT 8 1950 ^^^^^ ^^^ Fi"-. 4 — Keceived i)ulses — disturbed transmission. there was comparatively little fading over the path in question at micro- wave frequencies. There were, however, a few nights of considerable activity and some interesting results were obtained. The series of pictures on Fig. 4 show one good example of multi-path transmission \\-here the path length difference was great enough to pro- duce complete resolution of pulses. At 11:05-20 there are two pulses, each 7 millimicroseconds wide at the base and with their peaks just 7 millimicroseconds apart; in other words, the path difference was just sufficient to produce two pulses with no overlap. The pulse at the left is presumably coming by the main path and that at the right from some second path resulting from bending of the rays caused by atmospheric effects. At 1 1 : 07-20 the second path appears to have shortened, resulting in a path difference of only about 5 millimicroseconds. This may actually have been due to a change of length of the second path or it may have been due to distortion of the second pulse by energy coming by way of a third path. The pictures taken at 11:08-20 and 11:10-50 show evidence of transmission by a third path. In the first of these, for example, the width of the disturbance at the base line indicates the presence of the two original pulses spaced 7 millimicroseconds apart but the midpoint of the two no longer falls to the base line as was the case in the first picture. This could be accounted for by the presence of a third pulse coming over a path whose length was somewhere between that of the other two. Con- ditions obtaining at 11 : 10-50 could also be accounted for by the presence of pulses from three paths, that is, energy coming by way of a third path might cancel part of one pulse and at the same time add to the other. This could account for the fact that the spurious pulse is larger than the 96 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Fig. 5 — Received pulses — disturbed transmission. normal one. It is also possible that more than thi'ee paths were involved. On a number of other occasions the pulse coming by way of the second path appeared to be of greater amplitude than the one coming by the main path. This same effect has been obser^'ed by ]\Ir. Crawford and his colleagues on the angle of arrival equipment. Information obtained from the above set of pictures shows that for a time-division multiplex system using the length of pulse used here (7 millimicroseconds at the base) and operating over this path, pulses would have to be spaced a minimum of about 14 millimicroseconds apart if it were desired to avoid cUstortion at all times. If verj^ much shorter pulses were used the spacing might be reduced to 9 or 10 millimicro- seconds. However, the 7-foot path difference indicated by these pictures is about the maximimi ever observed and occurs rather infrequently so that if somewhat closer spacings were employed troubles would result only a small percentage of the time. The next series of pictures, Fig. 5, taken July 8, show an example of a more common type of multi-path transmission. Here the path dif- ference is apparently less than for the last series. At 11 : 15 there are two distinct pulses with an apparent path difference of about six feet (6 milli- microseconds) if judged from the spacing between the peaks of the pulses. However, from the length of the disturbance at the base line, which we consider a better criterion, the path difference was more nearly four feet At 11:22 distortion of the trailing edge of the pulse was the only indica- tion of a second path. For the pictures taken at 11:29 and 11:44 the path difference is sufficiently small that there is almost complete can- cellation of pulses, only the leading portion of each pulse being present. PKorAGATlON STUDIES AT MICKOW AVE FI11:qUENCIES 97 On the 11:44 picture there is just a trace of ii second pulse. Tlic next set of pictures (Fig;, (i) were taken a little over an hour later on the same nij;ht and show about the same conditions, that is, pulse amplitude and shape change and other evidence of the presence of a second pulse delayed alxmt 2 to 3 millimicroseconds. On the night of ()ctoh(n- l2, fading, which was apparently due to ti'ans- mission by way of two paths with little path difference, was observed. Some of the results are shown on Fig. 7. At 7:49 two distinct pulses are evitlent, there l)eing (i millimicroseconds btMween tiieir peaks. One might conclude from this that there was a scM'ond path about (> feet longei' than the main path but the total lengtli of the disturbance along the base line and the shapes of the pulses imhcate that the actual path difference was about 2 to :3 feet. Apparently we had here two pulses of r-f energy overlapping in time and in\()l\ing a large number of fre(iiiencies. These pulses are capable of interfering with each other in a rather complicated manner, it being possible for some fre([uencies tt) add and others to cancel at the same time, depending upon their relative phases. Phase relationships of course de- pend upon freciuency and path length differences. As a result pulses may be distorted and ha\'e their peaks shifted about by a considerable amoimt. We must, therefore, realize that the first pietiu'e of Fig. 7 does not really represent two distinct pulses as appears to be the case, but actually shows the resultant interference pattern of two o\'erlapping pulses. Since a change of path difference of only about one and one-half inches is enough to produce a 180° change in relati\'e phase at 4000 mc, it is not I'ig. 6 — Received pulses (list urhed t lansinission 98 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Fig. 7 — Received pulses — disturbed transmission. at all surprising that pulse shapes and amplitudes change very rapidly at times. Looking again at the photograph, Fig. 7, we see that at 7:54-05 there was a complete fade as far as our system is concerned. To produce this degree of cancellation the path difference must have been very small though still sufficient to give a relative phase angle of 180° at the radio frequencies involved. At 8:00-15 and 8:08-00 pulse distortion is the most noticeable effect of the "fading," the pulses being considerably shorter than their normal value. Pictures, not shown here, taken between 7:54 and 8:08 show definite evidence of two-path transmission with a path difference of 2 to 3 feet; therefore the pulses of 8:00-15 and 8:08-00 are probably also the result of two-path effects. The first two pictiu'es of Fig. 8 show another form of pulse distortion observed on a number of occasions. Here the pulse is flattened out on top probably due to energy coming in over a second path differing in length by only one or two feet from the main path. Each time this type of pulse was observed a check was made to be sure that the flattening Avas not dtie to overload in our equipment. The pictures presented up to now have all shown comparatively slow changes of conditions. Very rapid changes were, however, quite common. In many cases pulse shape or amplitude changed considerably during the 5 to 15 second exposure time ordinarily used. The picture taken at 2:20-45 A.M. on August 27 is one example of such a rapid change, there being two definite sets of conditions shown on the one photograph. The remaining picture on Fig. 8 shows the pulses used for obtaining time calibration of the system. These pulses were spaced 9 millimicroseconds apart and by adjust- PROPAGATION STUDIES AT MICROWAVK FREQUENCIES 99 ing sweep expansion so that succeeding pulses fell on proper parts of the scale and by keeping this expansion constant, it was possible to ob- tain a calibration. TWO-PATH SIMULATOR .\s an aid to interpreting the results obtained from the above experi- ment, particularly when the two pulses overlap and interfere, a circuit was set up in the laboratory to simulate two-path transmission. The equipment, as shown on Fig. 9, consisted of a wave guide hybrid junction with the r-f pulse energj^ being fed into the E plane arm. To each side arm was connected a ^•ariable attenuator in series with a few feet of wave guide fitted with a short circuiting plunger. Waves reflected from these two plungers recombine in the H plane arm where the detector is located. There are two separate paths through the hybrid as follows: (1) Input, side arm A, reflecting plunger A, side arm A to output. (2) Input, side arm B, reflecting plunger B, side arm B to output. By adjusting the attenuator in either branch the amplitude of the signal transmitted bj" way of that branch could be adjusted. In the same way by adjusting the position of the reflecting plunger in either branch the distance traveled b}^ a signal in traversing that branch could be varied. If the path lengths were made the same and the amplitudes adjusted to be equal there would be perfect cancellation due to a phase turn-over in the hybrid junction and hence no output from the detector. If one plimger were now left fixed in the above position and the other mo\-ed by a quarter wavelength (to produce a total shift of half wavelength or 180°) AUG 2 6 1950 1l:03-2SP.M. ^UG. 2 6 1950 1l : 0 9 - 4 5 P.M Fig. 8 — Received pulses and calibrating pulses. 100 THE BELL SYSTEM TECHNICAL, JANUARY 1952 the output signal would be at maximum amplitude due to addition of energy coming from the two paths. This amplitude is, of course, twice that of the signal from one branch alone. In the experiment the plunger in one branch was left fixed and the attenuator in that branch left set at zero. The path through this branch then represented the normal transmission path for an actual system. The path through the other branch could be made to correspond to spurious paths ha^ing different amounts of delay and attenuation simply by adjusting the position of the reflecting plunger and the setting of the attenuator. A series of photo- graphs were taken of pulses resulting from these different amounts of delay and attenuation. The first three pictures of Fig. 10 were taken with the path lengths exactly equal. When the amplitudes were also equal there was complete cancellation. As the signal in one branch was attenuated the amplitude of the resultant pulse increased until it became equal to that of the original pulse as shown in the third pictiu'e. Increasing one path by one- half wavelength brought the signals from the two branches into phase and they added up to double amplitude as seen in the fourth picture. It should be pointed out that although in our experiment we changed delay by 0.36 millimicroseconds in going from the fii'st minimum to the first maximimi, in free space a change of delay of only 0.125 millimicroseconds SIDE ARM A--> \/W^ ATT E N U ATOR E PLANE ARM J Nk. H PLANE ARM PLUNGER A PULSE INPUT OUTPUT TO DETECTOR -/\/V\/ AT T E N U ATO R SIDE ARM B--> PLUNGER B Fig. 9 — Apparatus to simulate two-path transmission. PROPAGATION STUDIES AT MKUOW AVI-; FUlXjlTKNCIKS 101 Fig. 10 — Simulated two-path transmission. would be rec[iiired. The discrepancy lies in the large ratio between the phase \'elocity and group velocity in the wave guide used whereas in free space this ratio is, of course, ec^ual to unity. In free space the amount of delay rec^uired to go from a maximum to a minimum signal corresponds to a change of path difference of only about one and one-half inches. With only this slight shift rec^uired to change conditions from those shown In' the first picture of Fig. 10 to those shown by the last, it is not at all siu'prising that the received signal is very unstable during time of multi- path transmission. Fig. 11 shows the effect of changing relative phases in 90° steps while keeping the amplitudes equal to each other. It is seen that even with the carriers in direct opposition cancellation is far from complete due to the •"^ ^NwijiiW ,«>''^*'*S»,.»- mummmmi"'"*' ■" n '' j '— '^ rv ~ =^HASE Q' API Fig. 11 — Simulated two-path transmission. 102 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 ,/V\ — m^ ^'^ "^— III! Fig. 12 — Simulated two-path transmission. relative delay between the tAvo component pulses. Flat topped pulses seem to be characteristic of conditions where the two carriers are in phase quadrature and about equal in amplitude. Fig. 12 shows a set of conditions with a constant delay difference of 2 millimicroseconds (corresponding to a path difference in free space of about 2 feet). For the pulses shown on Fig. 13 there was a constant delay difference of 7.34 millimicroseconds, enough to provide complete separa- tion of the pulses. The carriers were in phase opposition but with this amount of separation there is no overlap of pulses and the results would ha^'e l)een the same if the phase had been aiding. Any increase of delay Fig. 13 — Simulated two-path transmission. PROPAGATION STUDIES AT MICROWAVE FREQUENCIES 103 beyoiul this point results only in moving tiic pulses fartiier apart and has no effect upon pulse shape or amplitude. The experimental set up just described proved to l)e somewhat un- satisfactory since it was not possible with it to produce phase opposition between the two carriers without having zero delay difference between the two paths or a difference of at least 0.(i() millimicroseconds. For the length of pulse used this latter amount of delay difference is sufficient to pre\ent anything like complete cancellation of pulses. In fact the amplitudes of the two i-esultant peaks are only about 12 db below the peak amplitude of the original pulse. From this we know that for the natural path any fade which appeared to be complete must have resulted from path dif- ferences of less than 0.60 feet, in fact from differences of less than about one-half foot. SUMMARY The pulse experiment results indicate that over one particular path at least there is, at times, transmission of microwaves by at least two, and probably more than two, paths. Path differences involved are from a fraction of a foot up to about seven feet, differences of less than about three feet being the most common. These results agree with those ob- tained by other methods. These multi-path effects result in Imd distortion of very short pulses and even in the presence of entirely separate spurious pulses. These effects put a definite lower limit on pulselength and spacing between pulses in a pulse transmission system. The limit depends upon the amount of distortion which can be tolerated and also upon the per- centage of time such distortion can be accepted. No statistical data were recorded. With the lalioratory eciuipment for simulating transmission over two paths, many of the waveforms obtained over the natural path could be duplicated. There were times, however, when the waveforms received by way oi the natural path were too complicated to be explained by trans- mission by as few at two paths. ACKNOWLEDGEMENTS Space does not allow giving credit to all of the many people who con- tri})Uted to the success of this project. A. F. Dietrich made the mechanical layouts for most of the ecjuipment and supervised its construction as well as assisting in taking data and in many other ways. I also wish to thank J. C. Schelleng and W. 'SI. Goodall for guidance and suggestions. CI. M. Eberhardt furnished the traveling- wave-amplifier circuits. Properties of Ionic Bombarded Silicon BY RUSSELL S. OHL (Manuscript received August 23, 1951) This paper deals with a new and very interesting technique by which the properties oi silicon surfaces are altered very materially by bombardment with ions of such gases as hydrogen, helium, nitrogen and argon. The change in rectifying properties has been of special interest but there have been con- sidered also changes in the structural features of the material itself. The effects of bombardment on the rectifying properties are illustrated by a series of characteristic curves systematically arranged to bring out the effects of the several variables of experiment such, for example, as ion velocity, in- tensity of bombarding current, length of time of bombardment, kind of gas, and the temperature of the specimen during bombardment. The effect of bombardment on materials contaminated with impurities is also illustrated. It is of particular practical importance that silicon contaminated with boron to the point where it shows relatively little rectification can be modified by bombardment to make it even better than most unbombarded materials. Some years ago, the writer discovered that the electrical properties of silicon surfaces could be greatly modified by bombardment with positive ions. The ions in fjuestion were generated in a low pressure discharge in some gas, like hydrogen, helium or nitrogen, and after passing through a perforated cathode were accelerated to a suitable velocity before imping- ing on the surface to be treated. This scheme may be contrasted with other methods subseciuently reported for treating germanium^ in which high-velocity ions were derived from radioactive sources. Preliminary results of the present research were described in a paper entitled Silicon Transistors, by W. J. Pietenpol and the writer, presented at an Elec- tronics Conference held at the University of Michigan, June 22, 1950. Since that time exploration has continued with a \'iew both to learning about basic principles and about possible practical applications. Editorial Note — Since the resurgence of interest in point-contact rectifiers, considerable research has been carried on into the characteristics of silicon and germanium. The author of this paper was a pioneer in this new field of study, as evidenced, for example, by Patent No. 2,378,944, applied for on July 26, 1939, and Patent No. 2,402,839, applied for on March 27, 1941. More recent work has been descrilied in a large numl)or of text books and technical papers such as Electrons and Holes in Semi-Condnclors l)y William Shockley, D. \'an Xostrand, 1950, and numerous papers by Lark-Horowitz published mostly in Physical Review. The work described in the accompanying paper is a continuation of this long research. 1 Brattain and Pearson, Phys. Rev., 80, Dec. 1950. 104 PKOPEirnES OF lOXIC BOMBARDED SILICON lOo The present paper gives the I'esiilts of some more recent experiments made with im|iro\e(l eciiiipment. Also descrihed biiefly are some related experiments in which sihcon is boml)arded with alpha particles derived from radioactive pt)lonium. The overall results of this work indicate rather clearly that witli suitable variations of l)ombarding x'oltage, target temperature and time of exposure as well as impiuit}^ content in the base material, it is possible to pi'epare to specification silicon surfaces having a wide range of properties. From the materials so treated it has l)een possible to construct improved forms of signal rectifiers, harmonic gener- ators, transistors, modulators, gating devices and also photo-electric cells. It is particularly significant that the voltage range over which these newer devices can be operated has been greatly extended, thus making them useful in places not previously regarded as possible. Since these new surfaces appear to be readily reproducible in large numbers and since they are electricallj- tough, chemically stable and show no unsatis- factory temperature or aging effects, it would appear that bombarding technicjues should have considerable practical value. This paper is concerned mainly with the practical aspect of ion bom- bardment of silicon, namely its effect on the voltage current character- istics at low frequencies. Ef[ually important, perhaps, are its theoretical aspects, particulai'ly with regard to the interpretation of the rather pronounced changes in the properties in light of presently-accepted views of solid-state physics. These aspects are not covered in this paper. METHOD The bombardment process referred to above consists of exposing the silicon surface to ions that have previously been accelerated to energies in the range from about 100 electron volts to about 30 kilo-electron-volts. A recent setup is illustrated in Fig. 1. The electrons from a tungsten cathode are accelerated toward a grid which is at a positi\'e potential with respect to the cathode. Many of the electrons pass a short distance beyond the grid and return for ultimate capture. Ionization due mainly to the impacts of electrons with gas molecules takes place in this turn- around region, producing amongst other things positive particles. Elec- trodes are so proportioned that this ionization is fairly uniform over the grid area. The silicon specimen to be bombarded is made negative with respect to the filament. This accelerates the positively charged particles toward the target. The latter rests on a graphite plate heated by a coil below, carrying high-frecjuency currents. A thermocouple with suitable con- nections to the exterior makes possible an adefjuate measurement of 10() THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 temperature. The apparatus will accommodate circular surfaces as large as 1| inches diameter. The gases from which ions are derived are ad- mitted through the gas inlet. Thus far experiments have been made with hydrogen, helium, nitrogen and argon. The bombarding voltages have as already noted, been varied from 100 to 30,000 volts and the surface temperatures have been varied from about 20°C to 400°C. The effects of these several variables will be discussed more fully below. SAMPLE PREPARATION The material to be bombarded is usually prepared in batches of about 300 grams in fused silica crucibles roughly cylindrical in shape.- After solidifying, the cast is ground to a cylinder approximately 1^ inches TUNGSTEN CATHODE GAS NLET Fig. 1- TO PUMP -Bombardment apparatus. 2Scaff, Theuerer, and Schumacher, A.I.M.M.E., 185, pp. 382-392, 11)4!). I'ROl'ERTIES OF lOXIC 1U)M!{A KDKU SILICON 107 diameter. This has the effect of removiiif>; some of the coiitaminatiiig impurities derived from the crucihle as well as providing samples of convenient size. This U-inch cylinder is then sliced transversely into thin wafers which sul)se(iuently are polishetl on one side. Except as other- wise noted the material coxeretl by this jjaper had an impurity content of about 0.1 per cent. The exception will be found in the data of column (a) of Fig. 8. BOMHARDMENT PROCEDURE The wafer, as prepared abo\e, is placed in the bombarding apparatus with the rough face contacting the graphite support. The vacuum cham- ber is sealed by placing the ion generator in position and the whole assembly is evacuated. The sample is then heated to the proper tempera- ture and the desired kind of gas is admitted, the pressure being estimated from the ion current. When stable conditions prevail, the accelerating voltage is applied to the target and the bombardment is carried out for the proper length of time. A convenient current density is 5 micro- amperes per sciuare centimeter of target area. The target area of our present apparatus, including the silicon and a portion of the graphite support, being 20 square centimeters, the ion current is generally around 100 microamperes. The dosage is sometimes specified in microcoulombs. After l)ombardment, the sample is removed from the apparatus and the rough surface is covered with a tiiin layer of evaporated rhodium. For most of the tests outlined below the 1^-inch diameter wafers were cut into |-inch sciuares, a size convenient for testing. GRAPHICAL REPRESENTATION In considering the merits of non-linear materials such as silicon, per- haps the simplest and most useful characteristic is the voltage-current relation. If this is plotted to a linear scale, it results in a smooth curve of the general form shown in Fig. 2a. Specific curves obtained in practice may depart widely from that shown but in general, all may be regarded as made up of two semiparabolas, one in the first fiuadrant and one in the third, joined by a nearly horizontal straight line. For present pur- poses, we shall further simplify this idealized characteristic by consider- ing it as made up of three straight lines. The first, AB, is associated with the reverse voltage current characteristic. The third, CD, is associated with the forward voltage current characteristic. These two character- istics are joined by the nearly horizontal line, BC. The slopes of these three lines correspond to resistances. The section BC for example, corre- 108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 sponds to a region in which resistance is very high. The points B and C are particularly important for they represent points of inflection where the resistance undergoes rapid change and the material is departing most markedly from Ohm's Law. Ideally they should be sharp but in practice there is usually considerable curvature. Though either inflection point could presumably be used in detection processes, the point to the right of the origin is for practical reasons, usually preferred. Point B defines a voltage Eb at which substantial backward currents flow. It is referred to simply as the reverse voltage. In a similar way, point C defines a, forward voltage Ep- The distance between B and C {Eb + Ep) will be referred to as the inflection interval. The difference in these quantities {Eb — Ef) is also of interest. One-half of this voltage difference is re- ferred to as the self-biasing voltage. It is a significant cjuantity readily measured in practice by noting the d-c voltage across a large condenser placed in series with the crystal and a supply of 60 cycles AC. For de- tectors, point C should preferably be close to the origin and Ep should (a) I /" ■* Eb > / (Rp) A ^' /^ E ( < « » > Ef L- a/ LOG I Fig. 2 — Idealized characteristic curves. PROPERTIES OF lOMC liOMBAKDED SILICON 109 1)0 small. For coi'tain kinds of voltage limiters, Ep should l)e large. In oilher ease the iuflectiou interval should be large. In an alternate graphical representation, see Fig. 2b, voltage-current data are plotted to a log-log scale. This form of representation is of ])articnlar value when large ranges of data are to be shown. It is also of value in determining the resistance (Rp) at small voltages. Corresponding points on the two curves shown in Fig. 2 are identifiable by the letters A 1^, C, antl D. Ciu'ves of both kinds are used interchangeably to show the effects of the several variables of the experiment. EFFECT OF CONTACT PRESSURE In point contact rectifiers,^ pressure is of considerable importance. Usually the best pressure is a compromise between good electrical charac- teristics, usually obtainable only with light pressures, and good stability usually obtainable with higher pressures. Experiments have been per- formed with a range of contact pressures both on bombarded and unbom- barded materials. In general, the results are highly variable, particularly in the case of unbombarded material. From this wide range of data, however, two characteristics have been selected that may be regarded as typical for 10-gram and 60-gram pressure. They are shown in Fig. 3 for silicon taken from nearby portions of the same sample. Significant points on these several curves may be compared with their idealized counterparts shown in Fig. 2. Although the samples chosen show some- what more than the usual intrinsic resistance typical of p-type silicon, the etfects of contact pressure are nevertheless regarded as representative. As indicated in Figs. 3a and 3b, the effect of increased contact pressure,^ particularly in the case of unbombarded material, is of reducing the low voltage resistance, Rp, see Fig. 2b. The more desirable higher resistance is obtainable only with light contact, a condition unfavorable for high mechanical stability. In the case of bombarded material, the effect of contact pressure is less important. Thus it is possible in this case to incorporate in the design higher contact pressures and obtain thereby higher stabilities. For purposes of this paper a contact force of 10 grams has been accepted as standard. In addition to showing the effect of contact pressure, Fig. 3 shows some overall effects of bombardment. It will be noted, for example, that the effect of bombardment, see Fig. 3b, has been that of shifting the plots of Fig. 3a to the left by several orders of magnitude. Thus the resistance (Rp) is increased by a factor of more than 10,000. It is to be noted also 3 Scaff and Ohl, Bell System Tech. J., 26, Jan. 1947. ^ Realy contact force. 110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 1 1 (a) REVERSE -^ ^C^ A ^ / I r iT FORWARD-^ V Y r U'' ^^ J^ r w [<^^ ^ M \~ (\ > f 1 — t-r— '^ -t\ ^ ^-^ REVERSE (b) .-^-; ir; o ^ Ai 1 ''-^^ [ :^=^ ■ "^ r^ ==-- FO =iW 1.RD -? ^pL^^ ^ i^ w 'J V ,0-10 ,0-9 10-'^ 10"6 10-5 ,0" CURRENT IN AMPERES 10"3 10-2 to 10 GRAMS 1 i '0 < _i -I 5 0 2 1 5 -10 - cr D \ •> -20 i 60 GRAMS 1 cr A 1 1 -15 -10 -5 0 5 10 -15 -10 -5 0 5 10 VOLTAGE (d) Fig. 3— Characteristic curves, (u) and (c) unbombarded silicon, (b) and (d) silicon bombarded with 30-kv helium ions. PROPERTIKW ()p^ IONIC HOMHAKDKD .SILICON 111 from a comparison of Fi^. 3a with Fi^. 31) that at the one volt level, the ratio of forward to reverse currents for the unhomharded case is about twenty, whereas that for the boml)ard('(l case, is more than 1(),()0(). At oth(M' levels the difference is even <2;i'(^J>ter. Referring particularly to Figs. 3c and 3d, it will be seen that one effect of bombardmcMit is that of separating the two significant points of inflection B and C. That is, the inflection interval has been notably incn^ased. This increase is the result of a small increase in the forward voltage and a very substantial increase in the reverse voltage. EFFECT OF TYPE OF GAS Four high purity gases were tested as ion sources, namely, hydrogen, helium, nitrogen and argon, having atomic weights respectively of 1, 4, in 7'u\ 10 — -ce. 1-^ 5 - / n:< 0 / tr 1 /- i 1 / ^5 -5 -10 1 - / / 1 1 - i J 1 1 1 -5 0 5 -10 -5 0 5 (a) UNTREATED (b) HYDROGEN - / f 1 1 f, 1 -15 -10 -5 0 VOLTAGE (c) HELIUM -10 -5 0 ; (d) NITROGEN -10 -5 0 5 (e) ARGON ,0-7 2 4 6 a,Q-6 2 4 6 8,^-5 2 4 6 8,^-4 2 4 6 8,^-3 2 4 6 8,0-2 2 CURRENT IN AMPERES Fig. 4 — Characteristics stiowing eti'ect of various gases all with a l)ombarciing potential of 30 kv. 112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 14 and 40. While all four gases worked very well, helium was the easiest to handle. In the course of the tests, identically prepared samples each |-inch square, taken from a high-piiiity silicon melt, were bombarded with ions formed in the particular gas under test. Particular conditions known to be good for producing good rectifier units were adopted as standard for these tests. They corresponded to a total bombarding charge of 600 microcoulombs per sq. cm., a surface temperature of 395°C a contact force of about 10 grams and a bombarding potential of 30 kv. That the effect of bombardment varies with different gases is seen at a glance from the characteristic current-voltage curves shown in Fig. 4. Figs. 4b to 4e in particular indicate that as compared with an untreated sample. Fig. 4a, the effect of bombardment is in general that already noted of separating the two significant points of inflection, B and C. A rather substantial increase in the forward voltage appears in the case of argon as compared with hydrogen, helium and nitrogen. In contrast with a small increase in the forward voltage resulting from the bom- bardment of helium, there is a very substantial increase in the backward voltage. Though substantial for all four gases, the effect of bombardment is largest for helium with progressively smaller effects noted respectively for argon, hydrogen and nitrogen. A particular characteristic of helium bombardment, as compared with argon, not readily appreciated from a linear scale, is shown in Fig. 4f. It will be noted that at the one volt level, the ratio of forward to reverse current for helium is about 130 whereas for argon it is about 25. At other levels the difference is even greater. At the moment helium is regarded as a preferred source of ions. The log-log current-voltage curves show as before that the lowest voltage at which substantial forward currents flow occurs in helium, while the highest forward voltages occur for argon. In a similar way the voltages for substantial reverse currents are highest for helium and lowest for argon. The sharp break in the reverse current characteristic, evident in these cases, has been observed so generally that it is now accepted as typical of bombarded surfaces. EFFECTS OF TEMPERATURE Investigations have been made of the properties of silicon surfaces as affected by the temperature at which bombardment was carried out. This has been done not only for surfaces used as rectifiers but surfaces used as transistors and as photo-electric cells as well. In the case of recti- fiers, a procedure was adopted similar to that used in the previous tests. Measurements were made at five different temperatures ranging from PROPERTIES OF IONIC hOMHAHDED SILICON 113 - J 1 1 z ^^ V- 10 z UJ ^ 0 - / -5 ^ 1 r. 1 - y ' -«— 25=" C y /^ , ^ 1 ;^ - / 1 1 -«— I40°C / '', , 1 - ^ < , - 1 r ■*— 225° C / 1 1 1 1 1 ~y^ - i /, ■*— SOO'C y 1 1 1 t - I ( - i f 1 1 -*— 395° C i /: 0 -5 -5 0 5 -10 -5 0 5 -10 -5 0 5 -15 -10 -5 0 5 10 VOLTAGE Fig. 5 — Characteristics showing effects of voltage and temperature variation. 25°C to 395°C each at accelerating voltages of 1 kv, 3 kv, 10 kv and 30 kv using helium gas. The data so obtained were useful not only for study- ing the effect of temperature but useful in the studies of the effect of ion \Tlocity as well. The latter will l)e discussed in the following section. The results of the above measurements are plotted in Fig. o. They are further summarized in Fig. (ja. The latter figure, in particular, indi- cates that as rectifiers, there is little choice of surface temperature be- tween about 250°C and 400°C. It has been found, however, that for temperatures below about 250°C the point contact seems to be more vulnerable to electrical shock. 114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 EFFECT OF ION VELOCITY The effect of ion velocity (bombarding voltage) has been investigated for several types of silicon. The effects vary with the different types. Typical results are those given in Fig. 5 already referred to. It will be noted from a comparison of the tlata for a particular temperature, say 300°C, that the principal effect of increased ion velocity is that of increas- ing the reverse voltage. Values of these reverse voltages Eb and also the (a) r 30KV / lOKV 1 ^-'"^ 1 — 3KV .--^ lJ 1 KV o — 0 60 100 150 200 250 300 350 400 450 600 TARGET TEMPERATURE IN DEGREES CENTIGRADE < 0.6 1 (b)| > V 1 ! 1 ^ Y J V^ Zl 1 i y /^ / 1 y (. /Eb >' ,' v' ^' y / /' / ' 1 1.6 2.0 2.6 3.0 3.5 4,0 LOG-f (bombarding VOLTAGE) Fig. voltage voltage 6 — Summary of data of Fig. 5. (a) effect of temperature and bombarding on self biasing voltage; (b) effect of bombarding voltage on reverse PROPERTIES OF IONIC BOMBARDED SILICON 115 Table I — Effect of Bombardment Voltage on Eb and Ef 30 KV 10 KV 3KV 1 KV No Bombardment Surfa ce Temp. Deg. C. Eb Ef Eb Ef Eb Ef Eb Ef Eb Ef 395 16.7 1.3 14.0 1.5 10.0 1.5 7.1 1.0 300 16.5 1.5 12.5 1.2 10.0 2.0 6.5 1.0 225 18.5 3.6 12.5 1.5 9.6 1.5 5.5 0.3 140 17.5 2.5 9.8 1.5 6.7 2.5 7.0 1.2 Mean 17.3 2.0 12.1 1.4 9.1 1.9 6.5 0.9 2.4 0.5 Mean E'b 14.9 9.7 6.7 4.1 corresponding forward voltages Ef have been scaled from the above drawings and have been tabulated below. Since they vary only slightly with surface temperature, only their mean values are regarded as signifi- cant. Mean values of Eb are plotted in Fig. 6b. In order to isolate further the effects of bombardment we have sub- tracted from the mean values of Eb value of Eb for untreated sihcoii. Thus the curve marked Eb represents the improvement in backward voltage that has accrued from bombardment alone. This is also tabu- lated as the mean Eb in Table I. EFFECT OF TOTAL CHARGE Tests hav( been made to determine the effect of time of bombard- ment on the rectifying properties of silicon surfaces. In these tests, specimens taken from neighboring regions of the same melt were ex- posed for progressively longer periods all at the same bombarding po- tential of 30 kv and the same rate and density of application, 5 micro- amperes per square centimeter. Representative current-voltage charac- teristics are plotted in Figs. 7a and 7b for two neighboring regions. The results are summarized in Fig. 7c. The latter show a rather rapid improvement of back voltage Eb with total charge up to perhaps 50 microcoulombs per square centimeter.'^ Thereafter the improvement is small. For purposes of comparison, there is plotted as a vertical line a value of bombarding charge that would account theoretically for one positive ion in each unit crystallographic cell on the surface layer. This suggests that when all surface cells have been penetrated bj' a single ion, no marked increase in back voltage can be effected. * Specifying results in terms of microcoulombs implies that bombarding ef- fects are independent of the rate of application. This is known to be true only between factor limits of | and 2 of the boml)arding current. IIG THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 ___^ 01 Q Z ~ o u UJ in o -^ lAJ ^3 ' agviiOA vDvg oz rt rti III _C 2^ .^- -^ n O =3 00 en Q- s O y in 5 -d O cS S3a3d^Mv^^l^^ ni iNaadno PEOPERTIES OF lOMC liOMBAUDEU SILICON 117 KFFECT OF MATERIAL COMPOSITION ^Tlius far discussions have ('(Mit(M-(Hl around a sinj>;lc type of high purity material that was I'egai'ded as repr(\s('iitative. It is of iiit(M'est to exuiniue tiie etfect on otluM' inat(M'ials particuiaily those in which impurities have been added. For this ])urj)ose tests were made on compai'ahle samples from four sources all bombarded foi- two minutes with o microamperes of current and each with li\e r(>pres(>ntati\'e bombardinji; Noltages. The results are illustrated by the curves shown in V'lg. ;her percentajijes of imjiuritics beginning with (a) on the left as a material ha\ing an impurity content believed to ))e less than 0.01 per cent. The impurity content of (b) is not known accurately except that it lies between (a) and (c). The material repre- sented l\y column (c) was protluced by adding 0.02 per cent boron** to a material illustrated in column (a). The last column (d) was produced by adding 0.1 per cent boron to the material illustrated in column (b). It is to be noted from Fig. 8 that marked changes in the voltage- current characteristic may be effected by l)oml)ardment for all degrees of the impurity content shown. It is especially interesting that in columns (c) and (d) corresponding to materials contaminated with boron to the point where nonlinearity is almost absent, rectification can not only be restored but indeed the product may be made better than the best imbombarded material . EFFECTS OF ALPHA-PARTICLE BOMBARDMENT The close relationship between helium ions such as generated above and alpha particles such as emanate from radioactive materials suggests that the latter may be used for the bombardment of silicon surfaces. A few experiments of this kind have been made with results that are not only interesting l)ut possibly useful. For these tests, four sources of alpha particles were obtained. They consisted of fV i'^fli scjuare pieces of nickel on which had been plated a thin coating of polonium followed by a covering of gold. The initial strength was 4 milliciu-ies per square centimeter. The half-life of polonium is 140 days. The process of liombardment consisted simply of placing the polished surface of a standard silicon scjuare against the layer of gold and examin- ing the same periodically. Tests of four samples were carried out simul- taneously. The results are given in Table II. The data for Sample No. 1 departs so markedly fi'om the mean that it may l)e disregarded. Since ^ Boron is a particularly active a^ont in etlcctitig ctiaiigos in the properties of silicon. 118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 -1.0 -0.5 0 0.5 1.0 -1.0 -0.5' 0 -0.5 - 1 r i , ,/ - 1 ^/' - 1 , /' -200 -150-100 -50 - 1 . — i — J — 1 i -f (a) (b) (c) (d) Fig. 8 — Effect of impurity content, (a) hyper-purit}' silicon, (b) liigh-j)urity silicon, (c) hyper-purity silicon plus 0.02 per cent boron, (d) high-puritj- silicon plus 0.1 per cent boron. PROPERTIES OF IONIC BOMBARDED SILICON 119 tlie data for self-bias are tlie result of a direct measurement while those for forward and reverse voltage are transcribed from a cathode-ray plot, they are perhaps the most significant. Samples 1 to 3 represent specimens of increasing degrees of purity. These alpha-particle bombardment experiments indicate rather defi- nitely that results may be obtained similar to those obtained from bombardment with gaseous ions and, like the ion bombardment, they tend to produce high resistance surfaces. Table II Sample Number Days of Exposure 1 <1 2 3a 3b 4 Self Bias-Volts 6 180 60 4 Reverse Voltage <1 20 380 130 4 Forward Voltage <0.5 <0.5 <1 <1 13 Self Bias -Volts <1 160 200 190 13 Reverse Voltage 1 360 440 420 13 Forward Voltage <0.5 <0.5 <1 <1 39 Self Bias-Volts <1 60 130 100 39 Reverse Voltage 0.5 480 500 500 39 Forward Voltage <0.2 350 180 180 56 Self Bias-Volts <1 70 150 110 56 a Reverse Voltage 0.5 440 560 560 56 Forward Voltage 0.3 320 320 240 66 Self Bias-Volts 100 85 66 Reverse Voltage 560 520 66 Forward Voltage 200 320 MECHANICAL EFFECTS OF BOMBARDMENT The marked changes in the electrical properties of silicon imposed by bombardment stronglj^ suggest that bombardment may also impose a corresponding change in the lattice structure and that this might be detected by suitable optical methods. Attempts were made at an early date to detect such changes. To this end a mask of nichrome ribbon 5 mils wide and 1 mil thick was laid over a sample of silicon during bom- l)ardment. An optical examination of the surface showed that after bombardment in the case of helium the surface on either side of the mask was elevated whereas in the case of argon it was depressed. This result has since been confirmed by one of the authors's colleagues, Dr. F. W. Reynolds, who has found that in cases of prolonged bom- bardment by helium the adjacent surfaces may be elevated by as much as 225 Angstroms'^ while in the cases of prolonged bombardment by ^ One Angstrom is 10~* cm. 120 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 argon the adjacent surfaces may be depressed by as much as 130 Ang- stroms. Further investigation of this phenomenon is under way. STABILITY OF BOMBARDED SURFACES No extended test has yet been made of the stabihty of bom})arded surfaces but resuUs extenchng over more than two years are enccxu'aging. Similarly, rectifiers for the millimeter wavelength range, mounted with- out the usual protective impregnation, show little or no change at the end of a year. In a few instances bombarded surfaces have been subjected to rather severe tests with results that suggest that under normal conditions they may be even more stable than surfaces activated by more conventional methods. For example, surfaces contaminated while cutting or while cementing them to their mountings have subsequently been cleaned with solvents such as alcohol and are substantially the same before and after treatment. In other cases, they have been heated in a flame to soldering temperatures with no appreciable effects. Even in the very severe case where the bombarded piece was heated to a cherry red and the superficially oxidized layer was removed wdth hydrofluoric acid the effects of bombardment were still evident. There was, however, con- siderable reduction in the tolerable reverse voltage ."'There is nothing in our experience to date to suggest that bombarded surfaces treated in accordance with the simple straightforward methods outlined above, are in any wise temporary in character. CONCLUSIONS The expei'iments reported above have shown that rather pronounced changes in the electrical properties of silicon may be produced merely by bombarding the polished surface with positive ions. The ratio of forward to reverse currents, for example, which for the usual untreated silicon is seldom more than a few hundred, can be made more than 10,000. Experiments show that the effect depends to some extent on the type of ion gas used, helium being a preferred medium. The effect depends also on the volocity of the bombarding particles, the total bombarding charge and to a lesser extent on the temperature of the specimen tluring bombardment, (lood results are obtained from bombarding potentials of 30 kv with current densities of 5 microamperes per square centimeter for periods of one or two minutes. The temperature should preferable be about 300°C. Ordinarily the properties of siliccju are materiall}' affected by impiuity PROPERTIES OF IONIC BOMBARDED SILICON 121 content. In tlie case of bombarded silicon the effect is much less. More particularly it is possible to contaminate •.silicon with impurities such as boron to the point where its rectifying properties are almost com- pletely lost and by bombardment it is possible to convert the crystal into a very useful rectifier. It is possible to produce results similar to the above by exposing the crystal to radioacti^^e polonium. Bombarded materials appear to be relatively stable. The writer wishes to express his appreciation of the encouragement and help of Dr. G. C. South worth in the preparation of this paper, to A. J. jNIohr, Jr., for his able assistance in the experimental work, and to numerous associates in Bell Telephone Laboratories for their assist- ance in preparing materials and in making special tests for which the author was not adequately equipped. Mechanical Properties of Polymers at Ultrasonic Frequencies BY WARREN. P. MASON AND H. J. McSKIMIN (Manuscript received October 25, 1951) Since the mechanical properties of solid polymer materials are largely de- pendent on the motions that segments of the polymer chains can undergo, to understand these properties one must use measuring techniques which can determine these motions. One of the most promising methods is to measure the reaction of polymer materials to longitudinal and shear waves over a frequency spectrum wide enough to determine the relaxation frequencies due to thermal motions of the principle elements of the chain. The presence of relaxations is indicated by a dispersion in the velocity and attenuation constants of the material, or a dispersion in the characteristic impedance of the material if the attenuation is too high to allow velocity measurements. A number of different types of measuring methods are described in this paper which make possible propagation and impedance measurements not only in solid polymers, but also in liquid polymers and in solutions of polymer molecules in typical solvents. When these techniques are applied to long chain polymers in dilute solu- tions, the three relaxations observed correspond to motions occurring in isolated molecules since as the dilution increases, the molecules seldom touch. The lowest relaxation corresponds to a configurational relaxation of the molecule as a whole, the highest relaxation corresponds to the twisting of the shortest segment — containing about 40 repeating units — while the intermediate re- laxation corresponds to a transient entanglement of chain segments. All three types of relaxations are present in pure polymer liquids but are spread out over a frequency range due to the perturbing effect of near neighbors of adjacent chains. The high frequency shortest chain relaxation can be traced in solid polymers of the linear chain type such as polyethylene and nylon and produces rubber-like response to mechanical shocks of very short duration. I. INTRODUCTION The mechanical properties of sohcl polymer materials are largely deter- mined b}^ what motions, parts or segments of the polymer chains can undergo. Toughness, mechanical impact strength and ultimate elongation depend on the facility with which the polymer molecule can be displaced. 122 AIKCIIA.NK'AL I'UOI'KUTIKS OF POLYM KHS 123 If only a small motion of the polymer chain can occui" within tlu^ time of the measurement, the material has high elastic stiffness coefiicients and acts similar to a rigid solid. On the other hand, if significant segments of the polymer chain can move at the fi'e(iii(»ncy of measurement, the elastic stiffness is much lower and rul)l)er-like behavior results. An inter- mediate case, which oceiu's when the significant motion of the polymer molecule is near the relaxation time at the frequency of measurement, is that of a damping material such as butyl rubloer. Even "long time" ([ualities of plastics such as creep, stress relaxation and reco\'ery depend on the integrated displacements of rapidly oscillating segments of the chain. One of the most promising methods for investigating these motions is to determine the reaction of mechanical waves on the polymer materials over a wide spectrum of wavelengths, eventually going to frequencies comparable with those of thermal \dbrations of significant groups or segments in the macromolecules. If one wishes to understand the origins of these motions it is necessary to measure the molecules in the form of liquids or solutions since then the segments of the molecule are less restrained by their neighbors and can perform all the possible ^'ibrations. Polymer liquids are also interest- ing in themselves as sources of damping material. To apply these results to rubbers and solitl materials, one then has to measure the mocUfications of the poljoner chain motion caused by the close approach of near neigh- bors, b}'^ measuring the mechanical properties of these materials. By using chfferent types of technitiues, these processes can be applied to molecules in solution, to liciuid polymers and to solid polymers. The principal types of methods used for liquids are the torsional crystal, the torsional wave propagation system and the shear wave reflectance method, all of which are described in Section II. For solids an optical method and an ultrasonic method are. described in Section V. All of these methods involve displacements of 10~^ cm or less so that non-linear effects are negligible. All of these methods depend on setting up shear or longitudinal waves in the medium and observing either the velocity and attenuation of the wave, or the reaction of the medium back on the properties of the transducer. If the attenuation of a wave in the medium under consid- eration is low enough to permit the wave parameters, i.e., the velocity and attenuation per wavelength to be determined, the relaxation of some significant part of the polymer molecule is determined by the dis- persion of the wave properties which occur, as shown by Fig. lA, in the form of an increase in velocity and a maximum in the attenuation per 124 THE BELL SYSTEM TECHNICAL JOTJRNAL, JANUARY 1952 — t -P ■^^ >- 1- o o _l UJ > X"'" / / / / / / < O y y a. < a 1 1 / /^ N ATTENUATION 1 v'2 / \ 1 1 _^ / -<— HALF WIDTH— >- N V \ 1 i ^ X ■V — LOG ,0 OF SOUND FREQUENCY, f Fig. lA — Velocity and attenuation for a medium with one relaxation frequencj-. wavelength curve. If the variation in this relaxation mechanism is studied as a function of temperature and chain length, the type of seg- ment may be determined. If, however, the attenuation of the medium is so high that its wave properties cannot be determined, some informa- tion can still be obtained by determining the loading, or mechanical impedance, that such a wave exerts on the driving crystal or trans- ducer. If all the relaxations occur in the stress-strain relation, it can be shown that there is a reciprocal relation between the propagation con- stant r = A + JjB, and the characteristic impedance per square centi- meter Zo given by the equation Zor = (i^ + jX){A + m = icop (1) where A is the attenuation and B the phase shift per centimeter, R the mechanical resistance and X the mechanical reactance per square centi- meter, CO is 27r times the frequency and p the density of the medium. A typical two relaxation mechanism is shown by the curves of Fig. IB. By assuming values for the stiffness and dissipation factors and fitting a theoretical curve to the measured values, the relaxation frequency or frequencies can be determined. 1 All the relaxation mechanisms discussed in this paper are represented in terms of equivalent parallel electric circuits in which the resistance terms repre- sent viscosities and the inverse of capacities represent shear elastic stiffnesses. In mechanical terms these correspond to a series of Maxwell models as discussed in a paper by Baker and Heiss to be published in the next issue. MECHANICAL PROPERTIES OF POLYMERS 125 The most information about the motions of isolated polymer chains can be obtained by investigating the properties of polymer solutions. This follows from the fact that in pure polymer liquids, and in solids, the mechanical properties are mainly determined by interactions be- tween chains on account of the close packing of the chains. If, however, one dissolves the polymer molecules in a solvent, the inter-chain and intra-chain reactions can be separated as the dilution increases. When the polymer is in the order of one percent of the solvent, the chains on the average touch very seldom and the mechanical properties of the solution are determined by the properties of single molecules. As dis- cussed in Section III, three types of chain segment motion have been isolated, (1) a configurational relaxation of the chain as a whole, (2) a position change of the shortest segment and (3) twisting of the shortest chain segment. Above the frequency of relaxation of this chain segment the joints of the polymer molecule become frozen and the chain be- comes very stiff. These shortest chain relaxations occur also in pure polymer liquids, in rubbers and in non rigid solids with linear chain segments such as polyethylene. In pure liquids a lower frequency quasi- 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 LOG,o FREQUENCY F'ig. IB — Mechanical impedance loading for a medium with two relaxation frequencies. 12G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 configurational relaxation also occurs for chain lengths greater than 60 elements but for chain lengths less than 40 elements this type of relaxa- tion disappears. From the difference between the high frequency shear elasticities measured for polyethylene and nylon 6-6 and those measured for static pulls, it appears that there may be lower frequency relaxations in these materials as well. II. METHODS OF MEASUREMENT FOR SOLUTIONS AND PURE POLYMER LIQUIDS To measure the mechanical properties of such dilute solutions, shear waves have been used since for longitudinal waves the added stiffness caused by the dissolved polymers is very small compared to the stiffness of the solvent alone. The velocity and attenuation of a longitudinal wave are given by the equations 0 = 4/^±^^ ; A = 24! [^ + 2,1 (2) where A and /x are the Lame elastic constants, / the frequency, p the density, v the sound velocity, x the compressional viscosity and 77 the shear viscosity. Since for a one percent solution of polyisobutylene in cyclohexane the shear elasticity does not exceed 90,000 dynes/cm^, whereas the value of X is in the order of 2 X 10^" dynes/cm", it is obvious that the longitudinal velocity would have to be measured to an accuracy of 1 part in 100,000 before the presence of polymer molecules could be ascertained. Attenuation measurements give some information on the added viscosity due to the chain molecules but since longitudinal attenu- ations are not easily measured below 1 megacycle, the most interesting frequency range is missed. A pure shear wave in a viscous liquid is propagated according to the equation- y = yoe-V^(' + '> (3) where v is the transverse particle velocity, p the density, / the frequency, r] the shear viscosity, j = y/ — l and z the distance. For typical sol- vents, the attenuation is so high that wave motion cannot be measured. However the viscous wave produces an impedance loading on a crystal generating such a wave which can be measured by the change in the resonant frequency and the change in the resistance at resonance. The mechanical impedance per s(iuare centimeter caused by such a viscous MKCHANICAL lMi( )1'KUT1KS OK I'OLYMKliS 127 wave is equal to" Z.) = Vrfrip (1 + ./■) = /^u + ./A' ,w (4) This causes a change in resistance, and a change in tVcMnioiicy in a crystal generating a shear wave in the licjuid ecjnal to A/?,. = /vi/e.u; A/' = -A',X,, (5) where A'l anil K^ are constants of the crystal which can be oljtained approximately from the dimensions and piezoelectric constants of the crystal but which are more acciu"ately obtained by calibration in known li(iuids. The constants A'l and Ko vary slightly with temperature and should be calibrated over a temperature range. The first instrument to use a vibrational method for measuring vis- cosity was the vibrating wire method of Phillipoff. In this method a wire was \'ibrated in a lifiuid and the damping rate was used as a measure of the viscosity. Another method also applicable in the low freiiuency range is the transducer method of Ferry. In this method wires are ^•ibrated by electromagnetic transducers and the resistance and reac- tance drag on the wires are measured by the change in the electrical resistance and reactance of the transducer. From the constants of the transducer, the equivalent viscosity and stiffness of the licjuid can be measured. In the medium freciuency range a torsional crystaf method was devised by one of the writers which has been applied in the freciuency range from 10 to 150 kc. The torsional crystal is shown by Fig. 2. For these types of measurements the crystal usually is made of cjuartz with four electrodes of gold evaporated on the surface. Four wires are soldered on the surface and serve as supports as well as electrodes. The motion is all tangential to the surface and tests at Bell Laboratories and at the Franklin Institute, where a precision study of the torsional crystal has been made, have shown no observable longitudinal waves from the crystal surface. The process of measurement consists in measuring the * W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, 1950, p. 340. 3 W. Phillipoff, Physik. Zeits, 35, 1934, pp. 884-900. ■• T. L. Smith, J. I). Ferry and F. W. Schemp, "Measurement of the Mechanical Properties of Polvmer Solutions hv Electromagnetic Transducers," J. App. Phys., 20, Xo. 2, Feb. 1949, pp. 144-153. * W. P. Mason, "Measurement of the Viscosity and Shear Inelasticity of Liquids hv Means of a Torsionallv Vibrating Crystal," A.S.M.E., 69, May 1947, pp. 359- 367. ^ P. E. Rouse, Jr., E. D. Bailey, and J. A. Minkin, "Praetors Affecting the Precision of Viscosity Measurements with the Torsional Crystal," Laboratories of the Franklin Inst., Report 204S, presented to Am. Petroleum Inst., May 4, 1950. 128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 resonant frequency and resonant resistance of the crystal in a vacuum, then introducing the solution to be measured, the change in the resonant resistance ARe and the change in resonant frequency A/ are determined by an electrical bridge. Several short cuts are possible if the mechanical impedance is not too high. By measuring the capacity at a frequency considerably higher than the crystal frequency, the resistance at reso- nance and A/ can be obtained by changing the frequency and resistance until a balance is obtained leaving the capacity unchanged. This method has been used to measure viscosity, and a recent precision study at the Franklin Institute has shown that it agrees with other methods to an accuracy of well under a per cent. The torsional quartz crystal has been successfully used to measure liquids having a viscosity up to 10 poise, but above this viscosity the electrical resistance gets so high that it is hard to measure it since it is shunted by the much smaller reactance of the static capacitance of the crystal. A crystal of higher electromechanical coupling such as am- Fig. 2 — Cell and 80-kc cn-stal for shear viscosity and elasticitj^ measurements of liquids. MECHANICAL PROPERTIES OF POLYMERS 129 Fig. 3 — Photograph of torsional ciystai ami I'c^d monium dihydrogen phosphate (ADP) will cause the electrical resistance component to be smaller in comparison to the reactance of the static capacitance and hence can be used to measure higher viscosities. How- ever, since wires cannot be soldered to the surface but must be glued, the crystal is much more fragile than quartz and its use has been aban- doned in favor of another method which makes use of the phase and attenuation change in a torsional wave in a rod caused by the surround- ing liciuid whose properties are to be measured. This method, devised by one of the writers, consists in sending a short train of torsional waves, periodically repeated, down a glass or metal rod. As shown by Fig. 3, the torsional wave is generated by a torsional quartz crystal soldered or glued to the end of the rod. These waves tra\'el to the free end of the rod and are reflected back to the crystal where they are detected, amplified, and displayed on a cathode ray ^ This method is described by H. J. McSkimin, in a paper before the Ac- oustical Soc. of Am. in October, 1951. 130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 oscilloscope. Echoes due to end to end reflections also appear, being attenuated by normal acoustic losses until they are undetectable by the time the next pulse is applied. With only air surrounding the rod, a phase reference and amplitude reference are obtained for the first received wave (or subsequent echoes if greater sensitivity is desired). The rod is then immersed a definite length in the litiuid to be measured, as shown in Fig. 4, with, a resulting phase retardation and amplitude reduction. These are measured by em- ploying the experimental circuit shown by Fig. 5. In order to allow the Fig. 4 — Photograph of complete torsional wave measuring equipment. MECIIAXICAL PROPftETIES OF POLYMERS 131 use of one crystal for both receiving and transmitting, the crystal is put in the bridge circuit of Fig. 5 where a resistance and capacity are used to bahuice out the transmitted pulse so that it will not overload the amplifier. The relatively weak voltages generated by the incoming acoustic waves pass through directly. The gate circuit provides pulses of radio freciuencj^ voltage at repetition rates in the range of 20 to 100 per second with a synchronizing voltage supplied to the oscilloscope for the horizontal sweep. The frequency range of the device is from 20 to 200 kc. Both glass and nickel-iron rods were used, the latter having a very low frequency-temperature coefficient. With a 100-kc quartz REFERENCE BUFFER AMPLIFER ATTENUATOR TO AMPLIFIER AND OSCILLO- SCOPE Fig. 5 — Experimental pulsing circuit for measuring torsional impedance of liquids. torsional crystal, a rod length of 21 inches and diameter of 0.2 inch w^ere used. The entire crystal-rod assembly is placed inside a glass tempera- ture control unit, as shown by Fig. 4, through which water can be cir- culated to provide temperatures in the range 0°C to 80°C. The test liquid is placed either directly into the inner bore of this water jacket, or in another tube which can be inserted from the bottom to surround the rod up to a fixed mark. In use, both phase and attenuator settings were adjusted to balance the first received pulse against the continuous wave component passing through the attenuator. Cancellation for the duration of the pulse was visuall}^ indicated on the oscilloscope. A plot of balance phase and level is made as a function of the temperature. AVhen the liquid is introduced an attenuation change AA and a phase change AjB are required to 132 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 re-balance the circuit. These are measured by the amount of attenuation in nepers (1 neper = 8.68 db) and the number of radians phase shift required to re-establish a balance. An alternate method of measuring phase shift is to measure the change in frequency required to re-estab- hsh balance. If this method is used the phase shift change of the overall circuit with frequency has to be calibrated for the uncovered rod by ^^LIQUID ^v SOLDERED ,' "~ JOINTS- Fig. 6 — High frequency shear reflection method for measuring shear imped- ances of liquids. noting the frequencies for which 360° phase shifts (as measured by balance) occur in the circuit. It is shown in the appendix that the torsional impedance of the liquid per square centimeter is given by where p is the density, and Wo the sound velocity in the rod, a is the radius and I the covered length of the rod and A^ and LB are respec- tively the change in attenuation in nepers and the change in phase shift in radians to re-establish balance. If a very viscous liquid is used it may be necessary to correct for the fact that the torsional impedance may differ from the plane wave impedance as discussed in the appendix. This device can measure liquids having dynamic viscosities from 10 poise to 1,000 poise with an accuracy of the order of 10 per cent. The frequency range covered may be from 20 to 200 kc depending on the size of the crystal used to drive the rod. Hence it supplements the torsional crystal method for very viscous liquids. At frequencies above 500 kc, the torsional crystal becomes too small to be used practically and recourse is had to a high frequency pulsing method.* As shown by Fig. 6, shear waves are set up in a fused quartz 8 W. P. Mason, W. O. Baker, H. J. McSkimin and J. H. Heiss, "Measurements of the Shear Elasticity and Viscosity of Liquids bv Means of Ultrasonic Shear Waves," Phys. Rev., 75, No. 6, March 15, 1949, pp. 936-946. See also H. T. O'Neil, "Refraction and Reflection of Plane Shear Waves in Viscoelastic Media," Phys. Rev., 75, No. 6, March 15, 1949, pp. 928-936. MECHANICAL rUOl'lORTIlOS OF POLYMERS 133 rod by means of a Y-cut or AT cut crystals soldered to a silver paste layer baked on the fused quartz surface. The particle motion of the shear wave is parallel to the large reflecting surface and hence only shear waves are reflected from this surface. These impinge on a sec^ond shear crystal which is connected to an amplifier and oscillograph. Since the attenuation in fused quartz is so low, a long series of reflected pulses appear on the oscillograph. When a liquid, whose shear properties are to be measured, is placed on the fused ciuartz surface, this causes a change in the amplitude and phase of the reflected wave. By using the balance method shown by Fig. 7, in which two identical fused quartz rods are used, one of which has a liquid layer and the other does not, and by using a phase shifting network and an attenuator to balance out TO PULSED OSCILLATOR TO AMPLIFIER Fig. 7 — Method for obtaining resistance and reactance terms for high fre- cjuency shear reflection method. pulses, the shear impedance of the liquid can be determined. If R is the loss per reflection expressed as a current ratio, 6 the change in phase angle recjuired to rebalance the circuit and (p the angle between the wave normal and the reflecting surface, it can be shown* that the shear im- pedance of the liquid is Z M = R\i -\- jX M — Zq cos (p "1 - R'' + 2jR sin 6 .1 + 7^2 _f_ 2R cos d_ (7) where Zq is the impedance pv for shear waves in the quartz. This is equal to Zq = 2.20 X 3.76 X 10^ = 8.27 X lO' mechanical ohms Since this impedance is much larger than that of the liquids that are to be measured, the .sensitivity is increased by making (p large. In practice (f was taken as 80°. This method is applicable from 3 mc up to 100 mc 134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 and complements the other methods. Fig. 8 shows a photograph of the equipment. III. MEASUREMENTS OF POLYMERS IN SOLUTION When such methods are apphed to a polymer solution, it is found that the resistance and reactance components are no longer equal but the resistance is invariably larger than the reactance. This indicates the presence of a shear elasticity in the solution. If the molecules have a single relaxation frequency, it has been found that the shear properties of the liciuid can be represented by a stress-strain equation of the type T = r]A (8) dS + 1 di 1 1 dS + HbS "^'Tt where T is the shearing stress, S the shearing strain, t/^ the solvent viscosity, i]b a molecular viscosity of some particular motion of the chain which disappears when the reactance of the chain stiffness hb of this motion is low enough so that the motion can follow the applied shearing stress at the frequency of the measurement. When this type of mechanism is present in the liquid, it has been shown^ that the impedance the liquid presents to the crystals is PUbVb + j Zo = Rm -{- jXm — ccpVaVb + {.Va -r Vb) CO (9) ■I , -2. I Vb -r mb/w Fig. 9 shows a plot of the resistance and reactance components of an assumed solution having a single relaxation frequency, and a viscosity 30 times the solvent viscosity. At very low freciuencies, the resistance and reactance follow that of a solution, but for frequencies comparable with the relaxation frequency, the resistance becomes larger than the reactance while for very high frequencies the two come together on a line determined by the solvent viscosity. If there is more than one relaxation frequency, the resistance and reactance may coalesce for several intermediate stages. A continuous distribution w^ould give a definite relation between the frequency dependence of resistance and reactance. The torsional crystal and the shear wave reflection method have been applied to long chains of polyisobutylene dissolved in various sol- 8 W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. van Nostrand, 1950, p. 353. MECHANICAL I'HOPKRTIIOS OF POLYMKRS 135 Fig. 8 — Dual reflecting block assembly for measuring high frequency shear impedances of liquids. vents with concentrations ranging from zero per cent to 10 per cent. Polyisobiitylene is a polymer molecule having the chemical formula "CH3 H C- C- CH3 H Non-planar zigzag segments can be expected in the liquid state. Fig. 10 shows measured curves for 20 kc of the resistance and reactance for solutions of viscosity average molecular weight of 3,930,000 in cyclo- hexane. Four values of concentration were used and two temperatures were measured. For pure cyclohexane, the resistance and reactance 136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 0 I 23456789 RATIO OF FREQUENCY TO RELAXATION FREQUENCY Fig. 9 — Resistance and reactance components of a solution having a single relaxation frequency and a solution viscosity 30 times the solvent viscosity. S40 <20 < u 10 I/) 5 ^-^ .£r^ > -^"^ ^ ^ J R^ ^* J^' ^^ t>^ f X -o— "&" -Cr — ^ X ^ ^ 7.5°C 50° C 0.1 0.2 0.3 0.4 GRAMS PER 0.5 0.6 100 CC OF 0.7 0.8 SOLUTION Fig. 10 — Shear resistance and reactance components of a solution of polyiso- butylene (molecular weight of 3,930,000) in cj'clohexane plotted as a function of grams per cc of solution. MECHANICAL PROPKHTIES OF POLYMERS 137 components are equal but as llu> pcicontage of polyisobutylene is in- creased, the resistance iucreas(\s more rapidly tluin the reactance. By solving equation (9) for 77,1 , rj/j and m/j in terms of R antl A' measured at one fre(|uency and 77,1 + >?/; the solntion viscosity, we find dip Va fJ-B = C0p(7J.i + TJb) — 2RX {R~ — X") wriB 2RX Vb = (va + Vb) — Va (10) cop(rj.4 + Vb) Applying these formulae to the measured results, the curves of Fig. 11 result. The shear elasticity is directly proportional to the concentration, the viscosity tja is only slightly larger than the solvent viscosity while the main part of the measured viscosity resides in tjb the viscosity asso- ciated with chain motion. Fig. 12 shows these three quantities for a one per cent solution measured as a function of temperature. The apparent 3.93x10^ MOLECULAR WEIGHT, 7.5°C x' f / / u-y / / / / / / / / / / / / v.^ / / / f / / / / / /■ J / ^' / ^'^ 7?A 0- kir' 0 40 0 36 0.32 0 28 If) LU CO 0 lb 0 0 O.OS 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 GRAMS PER 100 CC OF SOLUTION Fig. 11 — Shear stiffness, series viscosity tja and molecular viscosit}^ tjb for poh'isobutylene (molecular weight of 3,930,000) in cyclohe.xane plotted as a function of grams per 100 cc of solution. 138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 lO^xiS lolO " 2 a >: 9 5^8 \ \ \ \, X 120-X \ X ^ \ V 80-X \ s. S \ ^ -o 120-X ^ ~ =5 b icy 80 -X o ?! 4 •-9 *^^ 3 °f 2 2 ;?5 I- o < o 1 0 10 20 30 40 50 60 0 10 20 30 40 50 60 TEMPERATURE IN DEGREES CENTIGRADE Fig. 12 — Shear stiffness, series viscosity and molecular viscosity plotted as a function of temperature for two molecular weight solutions of polyisobutylene in cyclohexane. I20-X=3.93 MOLECULAR X io6 WEIGHT 80-X= 1.17X10° MOLECULAR WEIGHT ^ \ \ .120 -X \ \, \ \ \| *o. 80 -X ■ ^ — 0.3 - '? "^ poise IS'C 25°C 35°C so-c Liquid M V u V M " •ft 1 A 0.482 X 10' 1.07 0.4 X 109 0.68 0.35 X 109 0.39 0.1 X 109 0.13 B 1.08 X 109 2.64 0.8 X 109 1.39 0.58 X 109 0.7 0.17 X 109 0.293 C 1.65 X 109 5.12 1.2 X 109 2.57 0.885 X 109 1.99 0.27 X 109 0.881 D 2.71 X 109 15.7 2.0 X 109 8.1 1.67 X 109 4.15 0.98 X 109 1.77 E 3.81 X lOs 40 3.0 X 109 20 2.17 X 109 10.6 1.2 X 109 4.62 F 5.48 X 109 73 4.22 X 109 38.3 2.8 X 109 20.2 1.6 X 109 8.8 G 5.9 X 109 107 4.78 X 109 51 3.74 X 109 29.5 2.5 X 109 13.2 H 6.9 X 10» 119 5.55 X 109 65.8 4.22 X 109 31.5 2.9 X 109 14.8 Measurements at 14 megacycles A 0.38 X 109 0.975 0.3 X 109 0.6 0.22 X 109 0.35 0.12 X 109 0.15 B 0.475 X 109 2.53 0.35 X 109 1.4 0.25 X 109 0.74 0.18 X 109 o-.-^s C 0.68 X 109 3.86 0.61 X 109 3.4 0.3 X 109 1.62 0.25 X 109 0.56 D 2.32 X 109 16.3 1.7 X 109 10 0.94 X 109 4.75 0.39 X 10» 1.86 E 3.8 X 109 56.2 2.8 X 109 24.25 2.0 X 109 12.3 0.855 X 109 5.7 F 4.75 X 109 90.4 3.6 X 109 48.3 2.65 X 109 26.6 1.47 X 109 10.8 G 6.03 X 109 136.5 4.6 X 109 81.4 3.24 X 109 39.6 2.0 X 109 15 H 6.64 X io» 160 5.3 X 109 93.5 3.98 X 109 54.2 2.3 X 109 18.5 Measurement at 4.5 megacycles A 0.34 X 10« 1.18 0.19 X 109 0.64 0.18 X 109 0.34 0.09 X 109 0.17 B 0.55 X 109 2.65 0.21 X 109 1.33 0.2 X 109 0.68 0.1 X 109 0.20 C D 1.57 X 109 24 0.96 X 109 11.7 0.72 X 109 5.52 0.34 X 10' 2.1 E 2.79 X 109 76 1.9 X 109 37.4 1.43 X 109 16.5 0.9 X 10» 5.15 F 3.57 X 109 124 2.5 X 109 61.2 1.86 X 109 31 1.04 X io» 12.8 G 4.4 X 109 176 3.4 X 109 98.5 2.6 X 109 54 1.6 X 109 22.6 H 5.65 X 109 186 4.3 X 109 124 2.8 X 109 76 1.8 X 109 28.8 141 142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 response to mechanical shocks of very short duration. The lowest fre- quency configurational relaxation is spread over a wide spectrum of relaxation times in pure liquids. Measurements" of these and other chains in various solvents have also been made and the results are discussed, from a chemical point of view, in a companion paper by W. O. Baker and J. H. Heiss. It is shown that the stifTnesses vary with the polymer chain and the solvent used. IV. MEASUREMENTS OF PURE LIQUID POLYMERS A. Shear Wave Measurements in Liquid Polymers Similar shear wave measurements have been made for pure polyiso- butylene liciuids of molecular weights from 904 to 10,380 (i.e. from 16 chain elements to 186 chain elements), by the techniques described in Section 11. Some of these results have been discussed in reference (8) but the much more comprehensive measurements made since require some revisions of the original conclusions. The easiest data to interpret are the high-frequency data obtained by the shear wave reflectance method. The data of Table I give measure- ments of 8 liquids varying m average molecular weight from 900 to 10,380, at three frequencies and four temperatures. If we plot for ex- ample the Maxwell shear stiffness and viscosity for the three frequencies and for 25°C as a function of the number of chain elements (here a chain element is taken as two adjacent carbon atoms one of which has two methyl groups attached and the other two hydrogens) the 4.5-mc meas- urements are shown by the triangles of Fig. 15. The 14 megacycle measurements are shown by the circles and the 24-mc measurements by the squares. An attempt was made to fit these measurements with a two relaxa- tion mechanism shown by the figure with two stiffnesses which are taken to be independent of the molecular weight and equal respectively to 1.2 X 10* dynes/cm^ and 6 X 10* dynes /cm^ The best fit is obtained by taking the two viscosities rji and 772 equal and these are adjusted for the different molecular weights in such a manner as to best fit the experi- mental curve. A fair agreement is obtained except for the range from 60 to 90 chain elements where the two relaxation model gives too rapid an increase of stiffness with increase in the number of chain elements and at the high molecular weight viscosity range where the viscosity shows a dispersion in values but the model does not. The sum of the two vis- '1 These results on the mechanical impedance of long chain molecules in sol- vents have been presented at the Xllth International Congress of Pure and Applied Chemistry by W. O. Baker, W. P. Mason and J. H. Heiss, Sept. 13, 1951. MECHANICAL PROrEKTIES OF POLYMERS 143 cosities -qi and 772 assumed as a function of molecular weight is shown by Fig. 16. The log of the viscosity starts proportional to the molecular weight but above a molecular weight of 2,400 the increase is very slow and becomes asymptotic to a value of 240 poises. An ecjuation which fits the increase in viscosity with molecular weight is T' 11-8 tanh Z/2370 Vd = Ac \\here Z is the molecular weight. The solid line shows a plot of this curve and the circles are the assumed values to obtain a best fit to the meas- MOLECULAR WEIGHT 4000 6000 8000 40 80 120 160 200 NUMBER OF CHAIN ELEMENTS ^ Fig. 15 — Mea.surpd values of high frequency shear viscosity and ehisticity for 25°C and three fretjuencies plotted against molecular weight. Solid lines are best fit obtained by a two relaxation mechanism having the element values shown by the figure. 144 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 ured values. This equation indicates that when Z = 2,370 or 43 chain elements the viscosity increases only a small amount more by a chain articulation effect and hence in this high frequency range we are dealing with a chain length of about 40 elements or 80 carbon atoms. This is checked also by a comparison of the static and dynamic viscosity. The total dynamic viscosity due to the two relaxation mechanisms compared to the static viscosity does not differ markedly until the number of chain elements is more than 40. Above this value other motions than that of the shortest chain segment can take place and can add to the dynamic viscosity. The static viscosity fits an equation of the same sort, but the indicated chain length for the viscous motion is about -f- times that of the shortest segment. When a similar process is carried out over the temperature range the equations of Fig. 16 are obtained. The static viscosity has an activation energy of 16 kilocalories per mole, while the dynamic viscosity has an activation energy of about 11.2 kilocalories per mole. The relaxation frequencies for the two components are plotted as a function of the number of chain segments by the solid lines of Fig. 17. lO''^ 2000 MOLECULAR WEIGHT 4000 6000 8000 10,000 12,000 6 4 2 6 4 - 1 1 1 1 1 1 . 1 ■s y A 7?s = = 1.2X io-i3e"-8TANH 3,20 e-^ - ji k / / 2 6 4 2 10 6 4 2 / V ^ 7?D = 1 .5 X 10-" e"-8 TANH 33^,0 g "rT - / / I . P V f - I 6 4 2 10-' - M 80 120 160 NUMBER OF CHAIN ELEMENTS Fig. 16 — Triangles are measured static viscosities and circles are dynamic viscosities plotted as a function of molecular weight. Solid lines are a plot of the equations given for static and dj^namic viscosities. MECHANICAL PUOI'KUTIKS OF POLYMERS 145 For loiiji; chain segments the values become asymptotic to 8 X 10 cycles and 1()(),()()() cycles which are not far from the two highest relaxation frcciuencics obtained from the solution measurements of Section III. Hence it appears likely that these relaxations are due to the "entangle- ment" motion and the twisting motion of the shortest chain segment. The increased activation energy is due to the fact that more energy has to be applied to the chain segment to break it loose from its equilibrium position when it is surrounded by adjacent polyisobutylene molecides than when it is surrounded by cyclohexane molecules. The stiffness of t he chain is due more to the slope of the potential well than to any in- t rinsic chain stiffness as is shown by Fig. 18, which shows the two stiff- nesses as a function of temperature. These values are obtained by fitting the 15°, 35° and 50° data in a similar manner to that used for the 25° in10° MOLECULAR 2000 4000 WEIGHT 6000 8000 10 000 6 4 2 6 4 2 6 4 2 10^ - 1 1 1 1 1 i \ \ \ r \ V^ \T \ \ \ . \ - v\ Ay '^ \ A \ \\ \ \ \ ^1' ^ W- k" ~^-- — - - \ \ V 6 4 2 6 4 \_ \ \ ^^ \ \ n\ \ - \ \ >y ^^ \ Nc 2 \ _ 10^ - 6 4 2 10^ — 40 60 80 100 120 140 160 NUMBER OF CHAIN SEGMENTS Fig. 17 — Solid Unes are mean values of relaxation frequencies for the two mechanisms plotted against molecular weight. Dotted lines indicate limits of regions assumed to obtain a better fit to the measured values. 146 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 data, and the activation energy of 11.2 kilocalories per mole is obtained in a similar manner. Due to the closeness of the surrounding polyisobutylene molecules, one would expect that the relaxation frequencies would not have discrete values but would be spread about the center value in some sort of a Gaussian distribution. If we approximate this by representing each region by two relaxations, one-half, and the other twice the frequency of the mean value, as shown by the dotted lines of Fig. 17, the agree- ment with the measured values of Fig. 15 is considerably better as shown by Fig. 19. A wider distribution yet is indicated. For molecular weights greater than 2,000. the shortest chain segment viscosity begins to diverge from the static viscosity indicating that there are other relaxations for these longer chains. Some data for the three longest chain polymers, F, G and H have been obtained by the torsional rod method and the results are given in Table II. These data are plotted on Fig. 20 as a ratio of dynamic to static viscosity plotted a < 109 tl08 - - ""^ ■ HIGH FREQUENCY SHEAR STIFFNESS ' / - - - - ^ ''ENTANGLEMENT" STIFFNESS - ^v,,..,^^ - < t- - ^^"^ - 20 25 30 35 40 45 TEMPERATURE IN DEGREES CENTIGRADE Fig. 18 — Variations of high frequency shear stiffness and stiffness plotted as a function of the temperature. 'entanglement" MECHANICAL PROPERTIES OF POLYMERS 147 against the frequency times the static viscosity. All the viscosity data can be represented within the experimental error by a single curve, but the stiffness curves appear to reciuire different curves for different tem- peratures. On analyzing the data in terms of a distribution of relaxation frequencies, a single curve for all temperatures could be obtained if the stiffness of each mechanism were independent of the temperature and the relaxation frequency were inversely proportional to the static vis- cosity, i.e., had an activation energy variation equal to that for the static viscosity for each relaxation mechanism. This condition holds MOLECULAR WEIGHT 0 2000 4000 6000 8000 10,000 12,0 200 1 1 1 ' ' ' 1 1 VISCOSITY -^ 100 - jr _o 60 40 20 10 - / y^^ D - M - 1 f ll ~ U 6 - l_ _st- 1EAR STIF =fne: 5S_ - / ^. >^- ^ 4 J-- — // ' A i ' 1/ / Y\ \ 2 L f 1 III , /n > 1 o / \ 1.0 m / - J f MEASURED POINTS - 0.6 --J h A 4.5 MEGACYCLES o 14 MEGACYCLES ! / " 7/ D 24 MEGACYCLES 04 ri ' / / - o ^ 0.2 y 40 80 120 160 NUMBER OF CHAIN ELEMENTS 10'° lO 109 Fig. 19— Curves showing better fit to e.xperimental data obtained by assuming the relaxation regions shown by Fig. 17. 148 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 'o P-. 2 S ^ '^ u B^ o o o o S a. XXXX r-H (M (>) IM OC OO u (:■ C«5tX) -*CO '^ CO CO (M o o o o 3. XXXX CO 00 CO •* C<1 !M CO '^ u ir r^ CO lO Tt< o o o o a. XXXX ■* t^ ■* ^^ TjH Tf ic r^ dodo" o to B- lO iC o -^ "^IC^^ CI 1— 1 1— 1 t— 1 a. oooo XXXX 00 CO 00 O OOr-H u ^ OOOO t^ iC C^J CO !M CM C^ r^ OOOO cs a. XXXX tT t^ Ci o ,-1 -H ,-H CO U s- OOOO o lO r^ 00 cOiO "* CO a. oooo XXXX 05 00 o 00 (M CO'*"! > c : D o lO O (N O i Ph X X OO CO ^ 00 »o IC Tt< ^ XXX K Pi o o o XXX CM CO CO XXX O 00 CO £fc^ fcfc^ erg MECHANICAL PHOPKRTIES OF POLYMERS U9 quite well for frequencies much lower than the relaxation frequencies of the smallest chain segment, but as the frc(iuency approaches these relaxation frequencies, the stitYness of these polymers increases as the temperature decreases. A fair approximation to these measured values is obtained by assum- ing one more "contigurational" relaxation frequency in addition to the two smallest segment relaxations discussed previously. Fig. 21 shows calculations of the ratio of dynamic to static viscosity and the shear stiffness for 65°C and 25°C. The lowest relaxation frequency is assumed to have a stiffness of G.3 X 10*^ dynes/cm^ and a viscosity of 20 poises at 65°C. For 25°C the stillness of 2 X 10 dynes/cm is assumed and an activation energy of 17.3 kilocalories gives the component a viscosity of 607 poises at 25°C, and a relaxation frequency of 5,250 cycles. The average value of IG kilocalories for the static viscosity is a result of the sum of the variation due to the two components. Although the agree- ment can be improved by assuming distributions of relaxation fre- quencies centered around these three primary frequencies, there does not seem to be much doubt of the existence of these primary relaxation in iDtr Zuj Q lU _2 7~ - - - • 15°C A 25° C D 35° C A 45° C 0 55° C X 65° C ^ •' <1 A- G A I- G A ^-^ 3 G - .rr^ - U - /V" --C ^ 5 ^^ r - x-<: G G ^ H < H H X" i-(n 1.0 Jil - ^^3^ - -^ Sj 7^ i- - H "^ G G "i-..,.,,^ 01 1 1 1 1 1 PRODUCT OF FREQUENCY TIMES STATIC VISCOSITY Fig. 20 — Plot of ratio of dynamic to static viscosity and the corresponding intermediate frequency shear stiffnesses as a function of temperature and prod- uct of frequenc}' times static viscosity. 150 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 109 ~ CALCULATED VALUES 65°C -• 1 CALCULATED VALUES Zi'Q O MEASURED VALUES 65°C A MEASURED VALUES 25°C y^ \^ y / / /> / ^ y a O QsSJ-- 10' yt 1.0 > ■v is X a. a S " o & a s- CU> oooooo XXXXXX 10.55 6.41 4.11 2.69 1.82 1.25 CO O COiOiO O C5 CO 00 ^ lO ^ (M '^lO t-00 oooooo d o o o o o XXXXXX CO -H t^ CO -H -r O C-. CO C: (M lO C5 t^ CO Id lO ^ 14.65 13.45 12.4 11.05 9.16 7.51 oooooo XXXXXX iC C5 lO LO CO O — I re 'M CO' -r CO x o; -M OOOOOO oooooo XXXXXX t^coio-*coeo lO—i -^ CO O 00 CS|(M t^ lO —CO O O 00 O 00 00 Tf lO 00 GO (M 00 t^ Tj< (M ,— I ,— ( t^ CO o i-o ^ t^ t^ l^ CO CO CO lO 00 00 GO GO 00 GO o o o d d o doo od xxxxx CO kC CO o C^ iC CO 'M ^ -f r-H o o o GO CO 00 ) LO ^ .-H --t< ^ -H O] CO --f< ooooo ddddd ooooo 1— 1 1— 1 1— 1 1— 1 1— I XXXXX CO ^ ■* CO X -r CO iC ^ t^ -* (M CO T^ o o o 05 X X !M OO OO .-I X CO X lO CO CO "—I ooooo ooooo O O d CO t^ O (M coo "^l -f ^ CO CO (M xxxxx ddddd o o o o o o o XXXXXXX lO -M X ■M IQ CO o cr> -r X -r oa eO"3 (M ^ o o o -ndd -' ^66 X CO O —1 oo o d 0.0316 0.0464 0.0756 0.125 o do o E:^ El. S. o o o XXXXXXX >c X (M r- -^ CSI X o s;ss t- -t C^ -H -'-'-' coo CO(N O t / ^ 9^ A ^A < y y^ 'y .' 1 /^ ^ .^ ^^ ^,i y !>"' .^ .^ ' ^ .** ,^' ^ ^> e — ,-' --""^ ^ ^' ■s^c<^ t"^ ,^- '"' .^ " .— -- V 30^ a/*^ , ^^ f" .^' ■J ==1 ■-r- "•"^^ ,'POLYISOBUTYLENE N ■> " -», ^~£^,i V. X^ \ ■< •<. --> ■"^^ > V \. N \ ^> ^"^ \ > ^ V, — POLYPROPYLENE . \j; SEBACATE 1 >^. \\ POLYBUTADIENE ^ ^^ ?rs,^ \: N > N N. :^ ^ •;rPOLYPROPYLENE ^ ^ r^ o**^ t. 10"* PRODUCT OF FREQUENCY TIMES STATIC VISCOSITY Fig. 22 — Ratio of dynamic to static viscosity and the shear stiffness for four polj'mer liquids plotted against product of frequency and static viscosity. the 4.8 kilocalories for polyisobutylene. This presumably indicates that there is more of a difference between the viscosity flow segment and the shortest chain segment in these materials than in polyisobutylene. Since no measurements are available over a range of molecular weights, no direct evidence has been obtained for the various chain lengths. B. Longitudinal Wave Measurements in Liquid Polymers Since the increase in shear elasticity for the highest relaxation fre- quency is so large, it should also appear in longitudinal wave measure- ments. Fig. 23 shows a calculation for the 5590 molecular weight liquid of the longitudinal velocity assuming that the Lame X elastic constant is independent of frequency and that all the variation occurs in the shear 154 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 O 1.85 ^ " - - y X^ • / ( / y / / > y / CALCULATED FROM SHEAR MEASUREMENTS O FROM LONGITUDINAL MEASUREMENTS y y „^-^ O 1.55 1 2 3 4 5 6 7 8 10 20 30 40 50 70 100 FREQUENCY IN MEGACYCLES PER SECOND Fig. 23 — Relation between measured longitudinal velocity for polyisobutylene of molecular weight 5590 and that calculated from shear stiffness measurements assuming the Lam6 X elastic constant is independent of frequenc}-. constant as determined by the shear measurements. The points are velocities measured for longitudinal waves and as can be seen, the meas- urements agree closely with the calculated values. A slightly better agreement would be obtained if X increased by a small amount as the frequency increased. As discussed in the next section there is some experimental evidence for an increase in X in nylon 6-6 and in poly- ethylene. The question also arises as to how much of the attenuation is due to shear mechanisms and how much due to pure compressional effects. From longitudinal velocity and attenuation measurements at 30°C for the polymers E, F and G of Table I, the values of X + 2ju and x + 2r? can be determined and are shown by Table IV. The values of m and t] can be obtained from Table I by interpolation and are given in columns Table IV Polymer X + 2m dynes/cm2 X +2, poise M dynes/cm^ \ dynes/cm^ 5 megacycles E F G 1.92 X 101" 2.26 2.38 62 107 170 0.17 X 101" 0.23 X 101" 0.31 X 101" 26 46 75 1.60 X 101" 1.70 X 101" 1.76 X 101" 10 15 20 8 megacycles E F G 2.01 X 101" 2.26 2.56 50 93 155 0.20 X 101" 0.27 X 101" 0.34 X 101" 22 41 69 1.61 X 101" 1.72 X 101" 1.88 X 101" 6 11 17 MECHANICAL PROPERTIES OF POLYMERS 1.55 4 and 5. Columns 6 and 7 show the values of X and x, the compressional components. A definite longitudinal compressional viscosity is indica- ted which howe\'er is some\\hat smaller than the shear viscosity 77. V. MEASUREMENTS FOR SOLID POLYMERS Recently two new methods have been devised for accurately measur- ing the properties of solid plastics in the ultrasonic frequency range 24 22 20 16 lU I- 16 LU K 14 Z (U ^ 12 IT lU a. o ? 8 3 6 4 2 0 BUTYL RUBBER 7 y / /NEOPRENE TENITE n 7 ILS [ H GRADE / / / / / / / / / LpC RUBBER / / NEOPRENE Gn/o 1 _r* ^ 1 / r ."'" y ^ / /•' .^ ^ / / ,<« I if* y / c Y 0— _ K^ 1 bw* / J. ^Alucite -M-^^ y^ POLYSTYRENE 1 — ■ t^ » E c S ?:< _-...-*-**"T \^ 60 80 0.4 0.6 0.8 1.0 2.0 4.0 FREQUENCY IN MEGACYCLES PER SECOND Fig. 24 — Normal loss in db per centimeter measured as a function of frequency for several rubbers and plastics. SCREEN ^-\|/- SCAL SCALE Fig. 25 — Debye-Sears cell for making sound waves visible. 156 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Fig. 26 — Examples of refraction and focusing effects for sound waves. and these have shown relaxations in such plastics as polyethylene and nylon 6-6. The simplest method for measuring one of the properties for longitudinal waves, i.e., the attenuation, is to measure the change in loss between two transducers in a liquid such as water, caused by insert- ing a sheet of the material. This process, used during the last war, results in the losses in db per centimeter for several rubbers and plastics, shown by Fig. 24. Indications of relaxation mechanisms are given by the rubbers and the plastic tenite II which is a cellulose acetate butyrate. The first fairly accurate method for measuring longitudinal sound MECHANICAL PROPERTIES OF POLYMERS 157 velocities'^ in plastics was the method of observing the focusing efTect of a cylindrical lens made of the plastic. Sound waves can be made visible by the Debye-Sears technique of using a sound wave as a phase diffraction grating. Here light from a slit Si is made parallel by the lens L2 and passes through the cell parallel to the wave fronts of the sound waves as shown by Fig. 25. The compressed parts of the medium retard the light waves more than the rarefied parts do and hence the medium acts as a phase diffraction grating. If a second slit So is used which is small enough to pass only the zero order, a light valve action is obtained which modulates the light according to the sound wave intensity. If now the lens Lj is used which focuses on the median plane of the tank, a picture of the sound beam is obtained as shown on Fig. 26. The bottom figure shows the focusing effect of a plastic and from the focal distance d and the radius of curvature r of the lense, one can cal- culate the velocity in a plastic compared to the velocity in the water bv the formula Vv = -/(-S (11) This method gives velocities good to from 2 to 5 per cent depending on the attenuation in the lens. G. W. Willard" has devised recently a more accurate method for measuring sound velocities as shown schematically by Fig. 27. Here a plastic to be measured is placed half way across the sound beam in the liquid and light is sent along the wave front occurring in both the plastic and the liquid. If the waves are in phase the retardation in the two light gratings, corresponding to sound propagation in both media, add up and for a slit selecting the zero order the darkest pattern occurs on the photographic plate. If the two waves are just out of phase, the retarda- tion is reversed in the two media and the lightest part occurs. With this relation it can be shown^^ that the spacing d of light and dark lines '2 W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, 1950, p. 404. It was used in this country by G. W. Willard as early as 1940. It was also used in Germany by J. Schaefer "Eine Neue Method Zur Messung der Ultraschallwellen in Festkorpern." Diss Strassburg, 1942. By making the front surface part of a cylinder, Schaefer also measured the shear velocity in a solid. 13 G." W. Willard, /. Acous. Soc. Ayn., 23, Jan. 1951, pp. 83-94. The origin of this multiple path interference method goes back to the work of R. Bar (Helvetia Physica Physica Acta Bd 13 page 61 (1940)) who attached a piezoelectric crj^stal to a bar with a 45° end section and set up transverse and longitudinal waves, in the bar. These waves produced longitudinal waves in a surrounding liquid and by observing the interference pattern between them, the longitudinal and shear constants could be determined for an isotropic medium. Willard's method as described above is much more direct and is capable of higher accuracies. 158 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 is related to the wavelength in the hquid X^ and the wavelength in the sohd Xs, by 1 = - _ A d Xf Xs (12) This corresponds to a velocity in the solid compared to a velocity in the liquid given by Vs = Vt Vl/fd (13) where / is the frequenc3^ Fig. 28 shows a photograph of a series of lines in a transparent plastic and a transparent plastic in the form of a wedge. It is seen that beyond the edge of the plastic there is a dark interference band for each one in the transparent plastic. This phenomenon is caused by the refraction of the sound wave that has traversed the plastic and the dark lines are lines of equal phase of the two waves in the liquid. The angle of the dark lines is half the refraction angle. Hence the velocity can also be determined by counting the number of dark bands in the liquid beyond the plastic. This makes it possible to measure the veloci- i i i I I \2 ■*\. d-- PHOTOGRAPHIC PLATE Vl Xs -Ad Fig. 27 — Optical method for measuring sound velocities of plastics by com- paring their velocity with that of a liquid such as water. MECHANICAL PROPERTIES OF POLYMERS 159 Fig. 28 — Photographs of interforcncc pattcrius from sound waves in liciuids and phistics. ties ill opaque plastics. The accuracy of the method is better than 1 per cent if the attenuation is low enough to give a number of interference lines. For plastics of high internal loss, the method becomes somewhat inaccurate. Typical measurements using this system are shown in Table V. Small changes in chemical composition and plasticizer content are shown up as can be seen from the table. Of particular interest is the difference between nylon 6-6 and polyethylene. Chemically as shown by Fig. 29, the two are identical except for the dipoles occurring for every 6 units of the ethylene chain. These dipoles have the effect of bonding adjacent layers together and result in a higher shearing modulus. By attaching shear vibrating crystals to a right angled prism, as shown by the lower part of Fig. 28, with the direction of motion of the crystals parallel to the transmitting face, and setting up shear standing waves between the two crystals, the shear properties of the plastic can be measured. Longitudinal waves are generated in the liquid which inter- fere with one another and cause dark bands perpendicular to the plastic 160 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Table V Measured longitudinal velocity and attenuation at 25°C and 2.5 mc. Longitudinal attenuation in DB/cm at 25°C and 2.5 mc except as noted. Material Dural, 17 ST Brass, half hard Polystyrene . . . Plexiglas Tenite II, (cellulose acetate butyrate), 2% plasticizer Tenite II, 13% plasticizer Poly vinal formal Polyvinlidene chloride Poly N-butyl methacrylate Poly I-butyl methacrylate Neoprene Polyethylene Nylon 6-6 (3-30 megacycles) Nylon 6-10 (3-30 megacycles) Long Shear velocity X 10-s velocity 10-6 A DB/cm cm/sec cm/sec 6.5 3.12 4.7 2.11 — 2.35 1.12 2 2.68 5 2.08 9 2.02 10 2.68 10 2.4 18 1.96 5 2.08 6 1.61 20 2.0 4.7 Pi-ii 2.68 1.0 PiB 2.56 1.0 Fis Density 2.7 1.05 1.18 1.23 1.21 1.24 1.71 1.05 1.05 .99 .90 1.11 1.11 surface. By determining the spacing of these lines the velocity of the shear waves can be determined. Another method has also been developed which is more applicable for high loss materials. This is a pulsing method and is a modification of the method proposed by one of the authors for measuring the properties of small crystal specimens." Here longitudinal or shear crystals are sol- dered to the fused quartz rod as shown by Fig. 30 and a sample to be measured is placed between these by means of a liquid such as poly- isobutylene which has a high shear elasticity. If the specimen has a small attenuation, this can be measured by taking the difference in the amplitude of successive reflections. If the specimen has a high loss, this does not work and another method has been used which consists in sending a pulse from both crystals.^^ One crystal is then used to receive and it receives the wave sent through the sample and the wave reflected from the fused quartz-sample interface. By adjusting the amplitude until these two are equal and the frequency or phase of one channel until the waves cancel, a ratio of amplitudes and a frequency of half wavelength are accurately determined. From these the velocity and attenuation can be calculated. This method has been applied to measuring the longitudinal and shear velocities of polyethylene and 6-6 nylon. The polyethylene was of "equi- librium" crystallinity and average molecular weight corresponding to " H. J. McSkimin, "Ultrasonic Measurement Techniques Applicable to Small Solid Specimens," J. Acoust. Soc. Am., 22, No. 4, July 1950, pp. 413-418. 16 H. J. McSkimin, J. Acoust. Soc. Am., 23, No. 4, pp. 429-435. MECHANICAL PROPERTIES OF POLYMERS 161 an intrinsic viscosity in xylene of [n] = 0.89 at 85°C. Fig. 31 shows the longitudinal velocity of polyethylene plotted as a function of frequency and temperature. The velocity rises with frequency and a dispersion is indicated. This is confirmed by the attenuation per wavelength curve for two different frequencies plotted as a function of temperature, Fig. 32. A definite dispersion is seen to occur with an activation energy of about 12 ± 2 kilocalories per mole. This could occur in either the X constant or the shearing constants ju, but the data of Figs. 33 and 34 show definitely that it occurs in the shear component. Fig. 33 shows the shear velocity for four temperatures plotted as a function of fre- quency. This can be fitted for 30°C with a single relaxation mechanism having a relaxation frequency of 8 megacycles. To agree with the meas- ured attenuation and velocity, there has to be a spreading of the single relaxation over a range as also occurs in liquids. The indicated shear stiffness below this relaxation frequency is 2.6 X 10 dynes/cm . Some H H^ H^ H2 II H2 H2 Hg H2 N CCCCCCCC ' \ / \/ \/ \/ \/\/\/\/\ / C CCNCCCCC Hj H^ •'5 <' Hp Ho Ho H ^2 "2 "2 2 "2 "2 "2 / HN \ c=o>.../ \ / < /C=o ^NH / \ / / HN "A. < A. \ NH, C / y >< Fig. 29 — Spatial structure of nylon 6-6. 162 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195 data on the zero frequency shear modulus is obtained from the Young's modulus for a static pull which is from 30,000 to 50,000 pounds/square inch. Since the Young's modulus is three times the shearing modulus, the zero frequency shearing modulus should not exceed 1.1 X 10 dynes/ cm . Hence one may expect that other relaxations will occur at lower frequencies. Fig. 34 shows the attenuation per wavelength for shear waves. The solid line for 30°C represents the calculated attenuation per wavelength for the model assumed. If all the dissipation were due to shear mech- anisms, the calculated attenuations would occur as shown by the 30°C PULSED OSCILLATOR A,B,C,D AMPLIFIER BUFFERS ATTENUATOR I V .SPECIMEN r Fig. 30 — Ultrasonic pulse method for measuring the velocities and attenua- tions of highlj' attenuating plastics. 10^x2.4 0°C o— ^ — ^ 10 "O— -^ U- o- 1 20 .... ^ ) 0- 30 _ •c \^ 40 .^—J- l_ -J>'"" O" 50°C .-- •-0' — o 1 12 16 20 24 28 32 36 FREQUENCY IN MEGACYCLES PER SECOND Fig. 31 — Velocity of longitudinal waves in polyethylene plotted as a function of temperature and frequency. MECHAXICAL PROPERTIES OF POLYMERS 163 0.19 0.09 . — ' ^25 \^C^ k^ / ^ / / ^ N tf / / X \ y^ • 8,26 ^ "v^MC \ ^, \ \, \ k \ \ \ 10 70 20 30 40 50 60 TEMPERATURE IN DEGREES CENTIGRADE Fig. 32 — Attenuation per wavelength for longitudinal waves in polyethj'lene plotted as a function of temperature and frequencj-. FOR 30' 10^X0.9 5§ liJij Uiij fo = 8 X 10^ CPS 77, = 24 POISE /Z, = 1.2 X 10^ DYNES/Cm2 IX\ = 4.75x10^ DYNES/ Cm2 jlz = 2.6x 10^ DYNES/Cm2 >Ui,= 1.3x108 dyNES/CM2 O^C — ^} O"""" o- 10 J 20 0 0-" 30 — -^ 40°C_ — 0- "■"" a" 8 12 16 20 24 28 32 FREQUENCY IN MEGACYCLES PER SECOND Fig. 33 — Shear velocity of polyethj-lene as a function of frequency and temper- ature. Equivalent circuit shows elements necessary to account for the velocity and attenuation changes at 30°C. 5 t.O .? 0.2 Q. *^^""'-°X / y^ 20 / O 10 o 1 0°C "^ / — -o / 8 12 16 20 24 28 32 FREQUENCY IN MEGACYCLES PER SECOND Fig. 34 — Attenuation per wavelength for shear waves in polyethylene. 10^x2.9 >2.7 u o UJ > 2.6 0°C ^ — — ■""" x^— (•^X 10 - __x _x. ^x- •' "^ — X— •x— X— 20 -X- — ^■x- — JX. 30 -X— ' . — — 40 X— " — __ _v- • X— — -X- :^x J2> , — — — ' 10 12 14 16 18 20 22 24 26 28 30 32 FREQUENCY IN MEGACYCLES PER SECOND Fig. 35 — ^Velocity of longitudinal waves in nylon 6-6 plotted as a function of temperature and frequency. 0.18 0.08 8 10 12 14 16 18 20 22 24 26 FREQUENCY IN MEGACYCLES PER SECOND Fig. 36 — Attenuation per wavelength for longitudinal waves in n3'lon 6-6 plotted as a function of temperature and frequency. 164 MECHANICAL PROPERTIES OF POLYMERS 165 points of Fig. 32, Most of the loss is accounted for by shear mechanisms, but it appears that some compressional mechanisms may also be present. The mechanism causing the relaxation in the megacycle range for polyethylene appears to be the same as for polyisobutylene, namely the relaxation of the shortest chain segment that is free to move. The chain segment acting appears to be longer than six chain units for similar measurements of nylon 6-6 show no relaxations in this frequency range. Fig. 35 shows the longitudinal velocity and Fig. 36 the attenuation per wavelength for longitudinal waves. Since the attenuation per wave- length is still increasing for nylon 6-6 at 25 megacycles a still shorter chain segment may be operating for this material. The shear velocity and attenuation per wavelength for nylon 6-6 are shown by Figs. 37 and 38. Fig. 39 shows the shear stiffness of polyethylene and nylon 6-6 plotted 10* X 1.3 1.2 61-1 — *- > -i 1 — V _ -10° C ' """' <^— - »— X' — 10_ _20_ ____ __ < — — - — X- -X. — -~" .X— 30 40 _50 <•-"" x«— ___^ . <— ^ — — ■ - — ••) — -^ 10 12 14 16 18 20 22 24 26 28 FREQUENCY IN MEGACYCLES PER SECOND Fig. 37 — Velocity of shear waves in nylon 6-6 plotted as a function of temper- ature and frequency. 0.34 0.32 ^^ ^ _ 1 50°C 1 .^ ^ -x— 40 30- 20 y ^ ^ , — ■^■ --^^ = ^=V| i^^ 2--^ ^ r 10 12 14 16 18 20 22 24 26 FREQUENCY IN MEGACYCLES PER SECOND Fig. 38 — Attenuation per wavelength for shear waves in nylon 6-6 plotted as a function of temperature and frequency. 166 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 10'°X2.0 UJ 1.0 ^ 1- >IU 0.90 t2 0.80 !^^, 0.70 wn ^n-^ 0.40 X "-III Oq. 0.30 111(0 _)in ^z TEMPERATURE IN DEGREES CENTIGRADE 50 40 30 20 10 0 Z^ V 6-6 NYLON AU - 0.75 KILOCALORIES PER MOLE c^ / POLYETHYLENE AU FOR STRAIGHT PART = 2.72 KILOCALORIES PER MOLE 3.0 3.8 3.9 XlO-3 1 3.2 3.3 3.4 3.5 3.6 3.7 INVERSE OF ABSOLUTE TEMPERATURE Fig. 39 — Shear elasticity of polyethylene and nylon 6-6 plotted as a function of temperature and frequency. 10 20 30 40 50 TEMPERATURE IN DEGREES CENTIGRADE Fig. 40 — Value of Lame X elastic constant for poh-ethylene and nylon 6-6 plotted as a function of frequencj^ and temperature. MECHANKWL PKOIMORTIES OF POLYMERS 167 against \/T where T is tlie absolute temperature. Both are plotted for 8 me and 30 mc. The dispersion in both materials is evident. Below 30°C^ the shear elasticity of polyethylene varies exponentially with the temperature with an activation energy of 2.72 kilocalories per mole. Above this temperatiu'e a deviation occurs due to the approach to the melting temperature. Nylon has a smaller variation with temperature. Comparing the longitudinal and shear wave measurements one can calculate the Lame X elastic constant and this is shown plotted on Fig. 40 for both polyethylene and nylon 6-6 as a function of temperature for 12 14 16 1S 20 22 24 26 FREQUENCY IN MEGACYCLES PER SECOND Fig. 41 — Equivalent shear and compressional viscosities for polyethylene and nylon 6-6 plotted as a function of frequency for a temperature of 25°C. two frequencies. The dispersion of X for polyethylene is small but is more prominent in nylon 6-6. This correlates with the larger compressional viscosity component present for nylon 6-6 which as shown from Fig. 41 is as large as the shear viscosity. According to the structural rearrange- ment theory of compressional viscosity due to Debye, compressional viscosity can enter when some part of the chain can rearrange from one stable state to another stable state as a function of pressure. This re- arrangement occurs across a potential barrier and hence requires a finite amount of time to occur. This lag in the rearrangement results in a compressional viscosity and as the frequency is increased, a frequency is found for which the motion can no longer occur in the time of a single '« P. Debye, Z. Elektrochem., 45, 1939, p. 174. 168 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 cycle and the X constant increases. It appears from these measurements that the dipole binding present in nylon 6-6 allows a greater structural rearrangement under pressure than can occur for polyethylene which has only linear chains. VI. CONCLUSIONS Measurements in dilute solutions, in pure polymer liquids and in non rigid solid polymers have all shown the presence of a shortest segment whose relaxation leads to a crystalline type of elasticity. In dilute polymer solutions the presence of a configurational type of relaxation and an entanglement relaxation of the shortest chain segment have been shown. For pure poljTner liquids a quasi-configurational type of relaxation has been found for chain lengths greater than 60 segments, but for chain lengths less than 40 segments this type of relaxation dis- appears. From the difference between the high frequency shear elastici- ties measured for polyethylene and nylon 6-6 and the static measure- ment of Young's modulus, it appears that there may be other relaxations in these materials for lower frequency ranges. For pure polyisobutylene and for nylon 6-6 there appear to be struc- tural changes induced by pressure which account for a compressional viscosity and a dispersion in the X elastic constant. This effect is smaller for polyethylene. APPENDIX — EFFECT OF LIQLTEDS ON THE PROPAGATION OF SHEAR W^WES IN RODS For radially symmetric rods, the tangential particle displacement Ue in the rod is given by Ue = Jiikr)e''''-'' (1) where k' = p^ + e' (lA) All other displacements are zero. In this equation waves are considered to be travelling in the +2 direction with a propagation constant 6 = A -\- jB, where A is the attenuation in nepers per cm and B the phase shift in radians per centimeter, ju is the shear stiffness which may be complex to take account of the dissipation within the rod. MECHANICAL I'UUl'ERTIES OF POLYMERS 169 From the clofiniug relations for the stress strain equation Tr8 = I^Sre = 11 dr and the tangential particle velocity due/dl, one can calculate the im- pedance Z per square cm. of cylindrical surface at r = a. This relation is Ue CO 'Jo(ka) _ 2/ J\{ka) ka (2) Since only the first mode is excited, parameters can be adjusted to keep k quite small, i.e. {ka < .2) and equation (2) can be simplified by using power series expansions for the Bessel functions. Neglecting higher order terms this results in Z = ~^'^^^' (3) To evaluate the impedance of the liquid surrounding the rod, the tor- sional wave is first propagated along the length of the rod without the liquid, i.e. with Z = 0. Then from equation (3) A; = 0 and from equa- tion (lA) ■pw = Uo + jBo)' (4) where Ac and Bo are respectively the attenuation and phase shift in the rod alone. With the small loss in metal and glass rods Ao can be taken equal to zero and Bo = co/Vm/p = '^Ao (5) where Vo is the velocity of propagation in the rod alone. When the liquid surrounds the rod, however, k' = P^-^9' = -Uo-i- jBof + U + jBY (6) For the usual case where {B + Bo) » (A + Ao), equation (6) approxi- mates k' = {B -V Bo) (- ^B + iAA) = 2Bo (- A5 + j^A) where A5 is the increase in phase shift per centimeter and AA the in- crease in attenuation per cm, both directly measurable quantities. The 170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 final working equation is then given by Z ='^X 2B,[^A + j^B] = ^ (AA + JAB) = ^ (AA + jAB) (7) Since A.4 and Afi are the attenuation and phase shift changes per unit length, then if f is the length of the rod, covered the total attenuation and phase shift changes will be AA and A5 multiplied by 2f. Hence if A.4o and ABo are the measured attenuation and phase changes, the impedance Z becomes Z = '^^ (AAo + jA£o) (8) This derivation neglects the change of phase occurring at the inter- section between the rod having no liquid and the rod surrounded by the liquid, but it can be shown that this is small and moreover, the change in the wave on leaving is equal and opposite to that occurring on entering and hence this correction cancels out. However, if the liquid is viscous enough there is a correction due to the fact that the measured impedance of equation (8) is for a cylindrical surface, whereas the desired impedance is the characteristic plane wave impedance. Obviously if the radius of cur\'ature is sufficiently large no cori-ection to equation (8) need be made. To obtain a suitable criterion one may con- sider waves propagated into the litiuid from the surface of the rod and solve for the impedance per scjuare cm. of the cylindrical surface. This neglects the variation with z, but since the wavelength along the rod is quite large, little error results from neglecting variations with z. An outgoing cylindrical w^ave in the medium may be represented by n, = [MkrY - jY,(kr)y^' = H['\kry e'"' (9) where the primes refer to the wave in the liquid and / 2 /2 2 / (k) = ^ = '^ or ka = -^r- (10) M Zk Zk where Zu = \/mV is the plane wave impedance of the liquid. The shearing stress TtB = p-'Sre = P-' dlle Ue dr r MECHANICAL PROPERTIKS OF POLYMERS 171 and the tangential velocitj'^ may be obtained as before and the complex impedance over the cylindrical snrface determined to be Z = ■M dHr\kay _ HV\kr)' dr r /j^H'i'^kr)' (11) Noting that for plane waves Zk = \/ p \i , one may eliminate \i fi-om (11) and evalnate Z at r = a with the result ^ = i {kaY {ka)'_ Z, (12) The above equation can be used to obtain a solution for Zk in terms of the measured value of the cylindrical impedance Z for any set of para- meters that may apply. Except for very heavy loading of the rod, how- ever, the results of eciuation (8) may be used directly with little error. For example calculations indicate that for | ka | ' = 10, the multiplier of Zk of equation (12) is approximately 1.07 Z — 7.4° for phase angle of {ka)' of Zo For {ka)' < 10, the correction multiplier rapidly becomes important. This same correction is applicable for the torsional crystal, but since this is only used for dynamic viscosities less than 10 poises, a correction is seldom necessary. Relay Armature Rebound Analysis BY ERIC EDEN SUMNER (Manuscript received October 25, 1951) Rebound of mechanical structures subsequent to impinging on stops gen- erally has deleterious effects on their performance and shoidd, therefore, be minimized. A considerable reduction in rebound cayi often be obtained by introducing additional degrees of freedom to the structure. A mathematical treatise of the dynamics of rebound motion of systems representing idealized relay armatures is presented. Normalized differential equations of motion and their solutions for the "free^' and "impact" inter- vals are derived for systems having one, two, and three degrees of freedom, allowing the rebound behavior of a specific system to be calculated. The equa- tions of series of rebounds, and possible combinations of such series are con- sidered next for systems having one and two degrees of freedom. The field of possible rebound ma.vima is mapped for a practical range of mass distribu- tion constants, coeficients of restitution, and force ratios. A sufficiently broad optimum design region is indicated. The results of this analysis have been checked closely on a model and have led to appreciable reduction of armature rebound in relay designs. I. INTRODUCTION In numerous types of mechanisms it is desirable to arrest the motion of a member at a particular point and to maintain it in this position. One of the simplest means of accomplishing this is to allow the moving member to impinge on a fixed member (stop) and to provide forces to tension it against this stop. Because the member to be arrested possesses kinetic energy and because the stop cannot generally absorb all of this energy, the moving member will rebound from the stop. The rebound motion generally deteriorates the performance of the mechanism and should be minimized. Investigation of this phenomenon has been stimulated by the armature rebound problem in relay operation, where rebound from the front stop* tends to reclose contacts and must therefore be compensated for by additional (waste) travel, resulting in deleterious effects on speed and * Among relay designers the front stop has been generally referred to as "back- stop". In this paper the terms front stop and heel stop have been used through- out for easier identification. 172 RELAY AHMATIKK REBOUND ANALYSIS 173 magnetic characteristics. Analysis in this paper will be directed towards relay armature systems, but it is also applicable to rebound in similar mechanisms. II. STATEMENT OF PROBLEM Analysis will be restricted to planar motion of armature systems having one, two, and three degrees of freedom as depicted in Figs. 1, 2, and 3. Generally one stop must be provided for each degree of free- tlom, although in the three-degree-of-freedom system of Fig. 3, two of the stops have been combined. Applied forces Fi, Fi , Fz , have been chosen so as to be most easily correlated with actual relay designs. The initial condition in all cases will be a pure rotation about the "heel" just prior to a "zero" impact at the "front" of the armature. The "zero" impact will be followed by rebound motion and impacts at the various stops eventually bringing the armature to rest. The object FRONT STOPV FRONT STOP Fig. 1 — Solidly hinged armature — one degree of freedom. Fig. 2 — Loosely hinged armature — two degrees of freedom. ^CORE AND RETURN PATH STRUCTURE [a) } CENTER OF GRAVITY OF ARMATURE (b) Fig. 3— Armature system — three degrees of freedom. 174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 will be to minimize rebound motion at the front, since this is usually near the point actuating the relay contacts. The basic problem is then to find the response of the armature subject to aperiodic but well defined impulses, which are functions of the positions and velocities of the system. III. ASSUMPTIONS In order to facilitate the solution of this problem, the following- modifying assumptions are made: (1) As mentioned in the previous section, analysis is restricted to planar motion. (2) The armature is assumed to be a rigid body. (3) Stops are assumed to be very stiff, massless springs capable of energy absorption during impact with the armature. The associated coefficient of restitution is assumed constant. Core and stop vibration are neglected. (4) The tensioning forces Fi , F2 , F3 are assumed to be constant forces. (This is fairly closely true for moderate rebound amplitudes of practical relay structures.) (5) All displacements are small relative to the dimensions of the system and in particular the angular displacement 6 is sufficiently small so that cos 6 = 1 smd = 6 IV. DERIVATION OF EQUATIONS OF MOTION The derivation of the equations of motion resolves itself into the solution of two different types of intervals: (1) Free Interval: This is the period during which the armature is not in contact with any of its stops and only the tensioning forces are acting. (2) Impact Interval: During such intervals the armature is in contact with at least one of the stops. The stiffness of the latter is assumed so high that the tensioning forces during this interval may be neglected. The three-degree-of-freedom case will be considered first and the others subsequently deduced from it by allowing some of the constants to approach zero. RELAY ARMATURE REBOUND ANALYSIS 175 A. Free Interval The motion of the armature will be described by the displacement at the stoji points: Xi , x^ , Xi * Let m be the mass and R the radius of gyration of the armature about the center of gravity. The latter is located by the dimensions l\R, loR, and fji relative to the stop points, i.e., the points on the armature which contact the stops in the rest position (Fig. 3). The equations of motion arc derived in Appendix I and are put into dimensionless form: yi _ 1 ~ 2 2/2 2/3 _ 1 ~ 2 ^^31 + C,, ^^^^ + C33 ^^^^J + ijw + ho + 2/30 + 2/10 + 2/20 + 2/30 (1) where: Vi = Xi_ XaT XaTn Fi .2 Xi XaTn Xq f (2) Xa is the front velocity ii, just prior to the "zero" impact, and Cn = in + 1) Cn = C31 = fil3 C22 = (f2 + 1) C:2 = Cn = (1 ^23 — [^ ~\~ 1) C2Z = C32 = —12^3 (3) 2/10 , 2/20 , 2/30 , are the initial velocities and yw , 2/20 , 2/30 the initial dis- placements for the free interval in question. The equations of motion for a two-degree-of-freedom system are obtained, if F3 = 0. Then for the two coordinates of interest: 2/1 2/2 = Cn + Cn (fX { - ) + 2/10 Cu + C22 (fX + 2/1 + 2/20 (4) A summar}' of all notations used in this papar is given in Appendix IV. 176 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 For a one-degree-of-freedom system iji = C21 + C22 f -?rM = 0? whence '" = ^h-S](;J + *-«(0 + - (5) B. Impact Interval The change of velocity at point "i" due to an impact at "i" is, by definition of the coefficient of restitution "k'\ Ail = — (1 + ki)Xi It is assumed here that the action of the stops are true impacts, i.e., the changes in velocity take place while there is negligible motion of the body. The velocity changes then occur as instantaneous rotation about the conjugate axis, leading to the general relation for an impact at point "i": VjOn = Vjein-l) + K jiij ie{n-\) (6) The first subscript indicates the coordinate, the second subscript indicates the beginning (0) or the end (e) of the free interval described by the third subscript. The impact transfer coefficient K^ relating a velocity change at point "j" to an impact at point "i": Kh = - ^' (1 + A;,) (7) Equations (1) through (7) allow any one specific case to be mapped, if the mass distribution and force ratio are kno^^^l. A sample of such mapping of rebound motion for a rectangular two-degree-of-freedom armature appears in Fig. 4. v. ANALYSIS OF REBOUND PATTERN — ONE-DEGREE-OF-FREEDOM SYSTEM The rebound pattern for the one-degree-of-freedom sj^stem — as de- rived in Appendix II — consists of an infinite series of parabolic arcs of diminishing amplitudes. The structure comes to rest after a finite time interval. The maximum rebound occurs during the fii'st bounce and equals ^ = - 1 ® RELAY ARMATURE REn(3UND ANALYSIS 177 Avliere C = Cxi - -^ The system returns to rest at t T (7(1 - Ic) (9) (10) VI. ANALYSIS OF REBOUND PATTERN — TWO-DEGREE-OF-FREEDOM SYSTEM The reason for choosing a t\vo-degree-of-freedom system over a one- degree-of -freedom system would be, in keeping with the philosophy of DD 0.02 ^7- /- X FRONT _^^ ^^ c \ \ \ \ / / '^-".— — * '""" "'• — \ N y • ■ HEEL >. - — • ^ FRONT ^ . - F2 = -2 \ \ \ \ s HEEL N / ^ FRONT HEEL ^^ ' V Fz F, " = -3 \ \ \ / / y- — N^ y HEEL HITS FIRST ^-4 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 NORMALIZED TIME W Fig. 4 — Front and heel motion of plate tj'pe armature. 178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 this treatment, to reduce Fi , the greatest excursion at the front. In order to simphfy mapping, this maximum excursion will V)e expressed as 2CYi , the ratio of ]'i to Y^ as given by Efiuation (8) for the case of A; = 1. Thus 2CYi is the ratio of the greatest excursion of the two- degree-of-freedom system under consideration to the greatest excursion of the corresponding perfectly elastic one-degree-of freedom system. We first introduce two basic constants which are functions of the mass distribution relative to the stop locations: M.J -^ (11) This constant represents a mechanical coupling coefficient. As Mij = M ji , the two-degree-of -freedom system under consideration here has only one such non-trivial constant ilf 12 . The second constant represents a force transformation factor from the j" coordinate to the "z" coordinate: ii •}} Pij = ^ (12) In the analysis of the two-degree-of -freedom system only P12 is important. If there is to be any heel motion, the "zero" impact at the front must impart a positive velocity to the heel. By Equations (6), (7), and (12), this recjuires that P12 be negative, which in turn implies that (ik > 1. For the limiting case of Hiii = 1, P\i = il/12 = 0 and no coupling exists between the heel and the front. Physically this means that the two stops are the centers of percussion of each other and the system will act as a simple hinge. With the above foundation, it is possible to analyze the patterns of motion and maximum rebound amplitudes. A. Motion Immediately Following "Zero" Impact After the "zero" impact at the front, both front and heel will lift off in accordance wdth impact Equation (6) and continue to move in ac- cordance with the free interval Equations (4). Whether the next impact occurs at the front or the heel depends on their respective periods, ti and ^2 : (13) tl ^ mJ h t2 1 + Pnf 1 + ki RELAY ARMATURE REBOUND ANALYSIS 179 where : / = Fi A large value of ixjli will result in a series of heel impacts and the heel will come to rest while the front is still displaced from the stop. This will be called a complete heel series. A small value of txlh results in a similar complete front series. If hlU is near unity, a limited number of impacts on one end are followed by an impact on the other end, etc. An analysis of front and heel series follows: B. Front Series If ti/t2 < 1 a series of front impacts occurs. The impact velocities at the front are ?/i„ = 1, k, k'\ ■ ■ ■ , k" The corresponding time intervals are „ zki 2iki 2/c '" " T' "X' ■ ■ ■ ' X where A = (Cu + Cuf) During this time, the heel velocity and displacement are given by '23 (1-t) (15) 2/20(n+l) — y-lOn + i/20(n+l) = U'lOn + - Pud + A-i) IJiOn '2B A ijm + ?/20« yion (16) where B = (7i2 + C22/ The velocity and displacement at the heel after a given number of front impacts are obtained by a summation of Equations (16). For a com- plete front series w — > co ^ and Vicoo = 2ki A{l-\- k^r L A 2/2.00 1 "25A-1 1 - kii A A ~^" \ -. r - Pl2(l + k,)\ 1 (17) 180 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 In addition it is useful to set down energy equations in order to simplify evaluation of greatest rebound for the various groups of re- bound patterns. The kinetic energy function T is evaluated in Appendix I. A potential energy term V — the work done against Fi and F2 from the equihbrium position — is introduced. If To is the total energy of the system prior to the "zero" impact, then T + F .2 , Mi2 .2 2ilfi2 . . —^ — = ?/i + p^ y^ - ~-f^ y^y^ -2(7(2/1+ /2/2) The energy loss due to n front impacts is + V -A (T + \ To = (1 - A-f) (1 - M^,)yU For a complete front series n — > 00 , and -A n^) = (1 - Mu)yl (18) (19) (20) If a complete front series follows the "zero" impact, yieo = 1 and -A T -\- V = (1 - Mn) (21) After completion of this "initial" front series, the system maintains only one degree of freedom (rotation about the front) until a heel impact occurs. By setting |/i = ?/i = ?/2 = 0 in (21) we obtain the heel impact approach velocity yo = P12 . Apparently energy loss due to n front impacts is a function of 71/ 12 , ki , and the approach velocity of the first impact. C. Heel Series An analysis similar to the above can be made for partial and com- plete heel series following the "zero" impact. This is demonstrated in Appendix III, yielding, for ki = /c2* ^Pi2(l + kf [APn B yuoo = ?/u« 1 + k - ky '2APu B k{l k{\ - k) 1 + k ■k) - Muk (1 + k) - Mr2(l + k) (22) The more general form ki 9^ k2 can be obtained as indicated in Appendix III. RELAY ARMATURE REBOUND ANALYSIS 181 The energy relationships for heel series are -(^0 = (1 — /02 ; - — -^f^ y^eo KJ'O) For a complete series n — > oo , and JT +V\ Mio(l - M12) .2 ,„,x If a complete heel series follows the "zero" impact, ?/2eo = ^12(1 + k\), and -A (^^^) = ^^12(1 - i^^i2)(l + k,f {2r^) Finally, for the special case where a complete heel series follows an initial complete front series y^eo = Pn , and -A (J^^YT^) = MAMn - 1) (26) It is to be noted that the energy loss due to a partial heel series is a function of il/12 , Pn , ki , and the approach velocity of the first impact, but that the equation for a complete heel series does not contain ki . Finally, a complete initial heel series is a function of only ilf 12 and ki . D. Complete Mapping of Problem Equations (1) through (26) make it possible to completely map the two-degree-of -freedom rebound problem. The relative maximum ampli- tude 2CYi and the rebound pattern will be determined. Examination of the necessary equations, show that 2CFi is in all cases a function of four parameters: fci , /c2 , M12 and P12/. Of these, k2 enters only if a partial heel series occurs prior to the time of maximum rebound. If it is assumed that for this limited group of cases k^ = ki = k, the number of parameters is reduced to three: k, Mn , Pnj. In Figs. 5 to 10, 2CFi is plotted against Pnj for the most useful range of 1/8 < M12 < 1/2.5, 0.3 < fc < 0.6 and 0 < P12/ < 10. As P12/ is increased from zero to infinity (corresponding to an increase in the heel tension F^, the rebound pattern goes through some or all of five regions. The criterion for location in any one region is based upon the parameter 1 -4- — f ^ _ ^ mJ _ k (1 + k) f ^~ l + Pnf'k—k- ^^^^ 182 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Region I — Complete initial front series for 1 < Q < l/k. Within this region, if M12 > —^ — ; — ^^ j. > the maximum rebound occurs during the 1 +P12/ first bounce and 2CFi = (1 - Mn)h' (28) 1 + P12/ If the maximum rebound occurs later, it must occur during a com- plete heel series which follows the initial complete front series. From Equations (21) and (26) 2C7i = M'n (29) By comparing Equations (28) and (29), the critical requirement 0.30 5 0.12 ^^'- ^^ M,2=^3 / / r /K=0.6 / / _^*»" ■"■" 0.6 ^^^' i / ro.b ij — -< / f 0.3 0.4_ JITIAL REBOUND MAXIMUM LATER REBOUND MAXIMUM UPPER LIMIT TO REBOUND 0 UPPER LIMIT TO INFINITE SERIES OF FRONT IMPACTS A SIMULTANEOUS SECOND IMPACT X LOWER LIMIT TO INFINITE SERIES OF REAR IMPACTS D FRONT VELOCITY EQUALS ZERO AT CONCLUSION OF INFINITE SERIES OF REAR IMPACTS 01 23456789 10 P,2f Fig. 5— Relative maximum rebound amplitude for M12 = 1/2.5. RELAY ARMATURE REBOUND ANALYSIS 183 for the latter case is that P^f > (1 - il/12) k — 1 . It should be noted that while the first rebound maximum, shown in solid lines on Figs. 5 to 10, is always realized, the later rebound given by (29) is an upper limit — shown in dashed lines — and is not always realized. In the dashed re- gions, phasing is extremely critical ^nd small variations in the param- eters may cause large variations in maximum rebound. From an engineering standpoint these regions are essentially undesirable. Region II — Partial initial front series for This region is one of critical phasing, and attention is limited to special cases leading to maximum rebound. These cases occur when a > u fvjo.22 §0.20 -I Q. 2 0.18 0.16 < 0.08 _i LU a 0.06 0.04 0.02 0 .^ \ M,2 = i K=0.6^'' y \ \ ,'-' \ \ 1 \ \ I J < \ ^^- ^* '"* I / \ / \ / ^^^ ,-'• ■"o.s \ \ \ / w / / **** \ \ \ / / / \ \ \ \ V / _, 0.4_ -- -' \ -VJ LJ -i >-'- — 0.3 \ \ \ \ \ \ \ \ \ \ 5 Pia^ Fig. 6 — Relative maximum rebound amplitude for M12 = 1/3. 184 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 heel impact immediately follows the last front impact of the series. These cases occur at Q = 1 - fc" k — k"" (30) and lead to rebound ampUtudes 2CFi = Mi2 + (1 - Mi2)[/c'" - Mx2(l - /c")'] (31) In Figs. 5 to 10, these special points are plotted and connected with straight dotted lines, which therefore indicate upper limits to rebound. Fig. 7 — Relative maximum rebound amplitude for Mio = 1/4. RELAY ARMATURE REBOUND ANALYSIS 185 Region III — Partial initial heel series for 1 + k Q > (1 + k) k{l - k) -\- Muk(l + ky and 2CYi = (1 - ilfi2)fc- 1 + P12/ (34) For the second group the front velocity is still negative when the heel comes to rest from which point on the system acts as a one-degree-of- 0.30 > o CVJ 0.22 D 0.20 _l Q. 5 0.18 < 1 0.16 O m S 0.14 5 D 2 0.12 X < ^ 0.10 y, M,2 = i \ w \\ K=0.6 \ \ \ Y \ , 0.5 \ y \ \ V D— — 0.4 0.3 0.04 0.02 0 01 23 4. 56789 10 Fig. 10 — Relative maximum rebound amplitude for Mr. = 1/8. 188 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 freedom system until the next front impact. The requirement for this group is that n > 2(1 + k) ^ k(l - k) -{- Mi2(l + ky and the maximum rebound is given by 2CFi = ilfi2 -(1 - ilfi2)[(l - k') + i¥i2(l + /v)'] (35) It is to be noted that in the upper part of Group 1 the ampHtude in- creases with successive heel impacts. This can be explored through the use of Equation (22). For simplicity of mapping, however, the hmit given by Equation (35) has been extended back from the lower boundary of Group 2 until it intersects the line marking the first rebound ampli- tude of Group 1. In Figs. 5 to 10 the respective regions have been identified by means of the symbols indicated below: Region I from P12/ = 0 to 0 Region II from 0 to A Region III from A to X Region IV, Group 1 from X to D Region IV, Group 2 from D toPi^ E. Discussion of Rebound Charts Aside from quantitative data contained in Figs. 5 to 10, the following general trends are of interest: For values of ilf 12 > j, and the values of k under consideration, most of the useful range of Pnf involves critical phasing and the rebound maxima are relatively high. For values of ^ < ilfi2 < l, consistently controllable rebound ampli- tude may be obtained. For values of Mu < i rebound increases again and the structure approaches the one-degree-of -freedom case. VII. ANALYSIS OF REBOUND PATTERNS — THREE-DEGREES-OF-FREEDOM SYSTEM Rebound pattern analysis as in Parts V and VI has so far not been performed for the three-degree-of -freedom system, partly because of complexity, and partly because for the system of Fig. 3 friction at the hinging stop will greatly influence the motion. RELAY ARMATURE REBOUND ANALYSIS 189 Ho\ve\-er, it is felt that the approach and notation of the analysis presented here is sufhciently j>;eneral to allow extension of the rebound pattern anah'sis to the tlirce-degree-of-freedom case. At any rate, with the assumption of the magnitudes of frictional forces, the basic equations of Part n" may be used to plot any particular case. VIII. ARMATURE REBOUND MODEL In order to verify the formal analysis presented in Parts III, IV and Y, a large model of a two-degree-of -freedom system was constructed. It consisted essentially of a large bar constrained to move in a plane, biased against two stops, and to the ends of which writing pens were attached. As rebound conditions were simulated by releasing the bar against its stops, chart paper moved at right angles to the bar motion and thus produced a record of end displacement versus time. By changing spring members and attaching masses to the bar, it was possible to vary the mass distribution and the biasing forces. The results obtained closely agreed with those suggested by the analysis. The maximum rebound amplitudes were generally somewhat lower probabl}^ due to frictional effects. IX. RELAY DESIGN CRITERIA RESULTING FROM ARMATURE REBOUND ANALYSIS A. Limitation of Analysis The assumptions which this analysis is subject to have been described under Part II. As applied to relays and probably the majority of mechani- cal structures, one assumption is most frequently and severely violated. The stops, which have been assumed to be stiff springs associated with a definite coefficient of restitution are, in practice, massive bodies which dissipate energy through excitation of high frequency modes of vibration. Accordingly, the assumption that the stops are at rest is violated, particularly if the mechanism is subject to repetitive (pulsing) im- pacts and the stop vibration does not decay greatly in the repetition period . However, mechanisms designed in accordance with this analysis have performed well even under moderate pulsing conditions if the sensitive phasing region was avoided. In addition, every effort should be made to reduce the amount and duration of stop and mounting structure vibra- tion by making them stiff, massive, and dissipative, if possible. 190 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 B. Design Criteria 1. Type of Armature Structure. The selection of the number of degrees-of-freedom for an armature structure depends on the expected coefficient of rebound as well as practical considerations. It can be shown without great difficulty that for very low coefficients of rebound the one-degree-of -freedom system is preferable. This is quite obvious when one considers the limiting value of /c = 0. In this case the one-degree-of-freedom system will have no rebound w^hatsoever, while the two-degree-of-freedom system has a heel bounce followed by re- bound at the front. The value of k below which the one degree system is preferable varies with the mass distribution relative to the stop points, being 0.18 for a rectangular plate armature with stops located at its edges. Experience indicates that k in most practical relays and similar mechanical structures varies from 0.3 to 0.6. Hence the two-degree-of- freedom system is superior in all practical cases to the solidly hinged armature. As far as three and higher degree-of-freedom systems are concerned, it may be said that generally the greater the number of modes resulting in impacts, the quicker the rebound energy can be diverted and dis- sipated and the lower theoretical rebound values can be obtained. This consideration would favor systems containing many degrees of freedom. However, while multi-degree-of-freedom systems can reach very low rebound values, their motion (phasing) must be very closely controlled or they may prove to be inferior to simpler systems particularly under vibratory (pulsing) operation. It is this difficulty which makes it appear that the two-degree-of-freedom system offers the best promise with the three-degree system also quite promising. By the same reasoning, additional spurious rocking modes should be minimized. 2. Armature Mass. The armature mass should be as low as possible. This will minimize stop and structure vibration. In addition, in relay applications light armatures tend to increase magnetic "drag" losses of the armature during the release motion. 3. Stops and Mounting Structure. As discussed before, it is desirable to reduce the amount and duration of stop and mounting structure vibration. 4. Coefficient of Restitution. The coefficient of restitution should be kept low. Stops having low stiffness should, therefore, be avoided. RELAY AliMATURK HIOHOUND ANALYSIS 191 5. Biasing Forces. Fi should be kept as high as practicable. For proper energy loss during impacts, the motion between impacts must occur outside the region of the compression, i.e., the armature and stop must separate. Therefore, because all practical stops have a finite stiffness, the biasing forces (Fi , F2 , etc.) should produce a static deflection less than say, arbitrarily, 5 per cent of the maximum expected reboiuid amplitude. 6. Design Parameters for Two-Degree-of -Freedom Systems. As clearly indicated in Figs. 5 to 10 for the practical range of coef- ficients of restitution, most consistently good results are obtained with a coupling factor Mio = tV to |. This factor is most easily adjusted by correct placement of the front stop. For the above range of il/12 the force ratio F2/F1 should be such as to make the product P12 ^ > 4 M12 = i > 3 ilf 12 = i > 3 Mn = 1 (Note: For a rectangular armature structure with the stops placed at its edges M12 = I, Pn = h and force ratios in the neighborhood of 8 are desirable.) X. ACKNOWLEDGMENT The analytical treatment presented in this paper contains contri- butions by E. L. Norton, R. L. Peek, Jr., and the wTiter. Appendix I DERIVATION OF BASIC EQUATIONS OF MOTION THREE-DEGREE-OF-FREEDOM SYSTEM (1) Free Interval The motion of the armature will be described by the displacement at the stop points, Xi , X2 , .T3 . Let m be the mass and R the radius of gyra- tion of the armature about the center of gravity. The latter is located by the dimensions fiK, iiR, and ^R relative to the stop points (Fig. 3). 192 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 The rotation and displacement of the center of gravity is then + Xz Xh = ix2 - Xi) h + f2 Xv = Xi + (,X2 — X\) Ix h + h X'l — Xi (a) Rill + i2) From this the kinetic energy may be computed T = ^m{xl + xl) + ^mR'd' xlil + f2 + it) + xM + (1 + 1) + xl{(, + IzY + 2{ll + /2)'^ 2xMl\(2 - tl - 1) - 2xz(z{l\ + (2){xi - ±2) 2(^1 + 4)^ (b) Applying LaGrange's Equation to the above, the equations of motion are obtained: 1 (^\ _ ^ = p dt \dqr/ dqr F\ ^ xM + f3 + 1] + :g2[fif2 - fs - 1] - ^^^3(4 + I2) ^ m {i\ + 12)' F2 ^ x^2 - fl - 1] + X2[l\ + fs + 1] + xzhih + ^2) /^3 -xihik + 4) + ^y'sOi + ii) + xsCfi + ^2)' \ (c) w (i'l + l2f The Equations (3) may be solved for xx , X2 , xz and the results inte" grated, yielding Xx = ^- [ClxFx + Cl2i^2 + CxzFzlt' + x,ot + Xio 2m X2 = ^ [C2xFi + C22F2 + C2zFz\f + X20^ + X20 Xz = ^ [CziF, + C32F2 + CzzFzli' + xz,t + Xz, 2m (d) RELAY ARMATURE REBOUND ANALYSIS 193 where Cn = (fi + 1) Cu = C31 = fif3 C22 = (^2 + 1) Cn = C21 = (1 - (iQ C33 = (d + 1) C23 = C32 = -(ojz (3) Xw , i;2o , 2:30 are the initial velocities, .Xio , X20 , 2:30 the initial displace- ments for the free interval in question. Interpretation of the analytic results is simplified by the introduction of normalization. Let Xa be Xi just before the "zero" impact and define Vi — . ■> Xx XaT XaVl Xain Vi = d ^ Xi Xa (9 Fi (2) Dividing Equations (d) by XaT yields the normalized equations of motion: ^1 = 1 2 2/2 = 1 2 2/3 = 1 2 621 + 622-^ +623^ J C31 + C32^+C33^j + 2/10 ( - ) + 2/10 + 2/20 ( - ) + 2/20 -) + 2/30 ( - ) + 2/30 (1) (2) Impact Interval The change of velocity at point "i" due to an impact at "f" is, by definition of the coefficient of restitution " fc" : Axi = — (1 + ki)xi (e) Since this velocity change occurs as rotation about the conjugate point as an instant center of rotation, the impact relationships may be written, for an impact at point "1", Ail = — (1 + ki)xi AX2 = — (1 + ki)xi (f - ••') 194 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 /I 1 7 \ • (^1^2 — 1) W2 /, , , \ . = (1 + ^-l)^! ,,2 , ... = - 77- (1 + ^l)^l (fl +1) (-11 Ax3 = — (1 — A-Orci — = -(l + /cO:^i-^= -^^(1 + A:0ii fi + 1) ^11 Similarly it can be shown that impacts at points (2) and (3) follow the same pattern. The general impact relations for impact at point "i" are then 2/iOn = 2/je(n-l) + Kjiyie{n-l) (6) The first subscript indicates the coordinate, and the second subscript indicates the beginning (0) or end (e) of the free interval denoted by the third subscript. The impact transfer coefficient Kji relating a velocity change at point "f to an impact at point ''i": Kji = - ^ (1 + fci) (7) Appendix II ANALYSIS OF REBOUND PATTERNS — ONE-DEGREE-OF-FREEDOM SYSTEM The equation of motion of this system is yin = hCt'' + ywnt' + 2/lOn (/) where C = Cn- ^ (9) r and is measured from the start of the particular interval of free motion in question. The impact relationship is 2/lOn = —k\yie{n-l) The motion consists of a series of parabolic arcs having periods of 2yio/C in general, or 2/C, 2k/ C, 2fcVC, • • •, 2A;'*~VC. The time elapsed RELAY ARMATTHE REBOUND ANALYSIS 195 is a convergent series and approaches, for a complete series: 2 . . -7 Lim-d + k + /v- + n— »M O k"] C(l - k) (10) The maximum rebound amplitude in any interval is — yio„/2C. The maximum excursion occurs during the first bounce at t' = 1/C and equals — k /2C. Appendix III ANALYSIS OF REBOUND PATTERNS^ — T\VO-DEGREE-OF-FREEDOM SYSTEM The equations of motion of this system are yi = ^At'^ + yiont' + yion y2 = ^Bt'^ + yiOnt' + 2/20n where A = Cn ^ Cuf B = Cu + C722/ (g) , _ t measured from the start of the par- 7" ticular free interval in question. A. Complete Front Series At the beginning of a front series 1/1 = 0 yi = 2/leO y2 = 2/2eO 2/2 = heO (h) (i) In a manner analogous to that for the one-degree-of-freedom system each front impact reduces 2/1 to —kiyi . Therefore, after the n*'' impact, in. yion — —i^iyieo and the time elapsed in the n^^ interval is In = —r- yuo (J) 19G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 At the heel, from (g), the heel velocity preceding the n*'^ impact is 2/2e(„-:) = ?/20(n-l) + Bt' (k) The velocity change during the (n — 1) interval is then equal to BTn-i . From Equations (6), (7) and (12), the change in velocity during the n*'' impact is — Pi2(l + ki)ki'~^yiea • The total change of ?/2 between impacts is then 2/20n - 2/20(n-l) = BTn-1 " Pl2(l + ki)kl~^yieO Similarly in preceding intervals: yiO(n-l) — y20(n-2) = BTn-2 " -Pl2(l + ki)ki~^ yuo ^202 — ?/2oi = BTi - Pi2(l + ki)kiyeo ^201 — 2/2eO = - Pl2(l + kl)yieO By addition of the above y20n — y2e0 ^ Bi^Tm- Pud + h) Z k". n-1 E m=0 yuo = -^ yuo 2Z ki — Pnil + ki)yuo 22 k" ■A. m=l n-1 E TO=0 The summations may be evaluated, yielding ^2B ki - ki y20n y2e0 = - Pl2(l + ki) 1 - ki' yuo (1) A I - ki "' ' ^' 1 - ki To evaluate the displacements at the heel, Equation (g) yields y20n — 1/20 (n-1) = ^20(n-l) 7^n-l + ^BT n-\ Adding these expressions for intervals 0 to n; the total change in 1/2 is n— 1 n— 1 2/20n — 2/201 = ^ y20mT„, + ^^ 23 ^T^ m=l m=l 2h{i - kr') . . ~ yieOy2eO A{1 - h) r2Biki - 2fcr+^ + ki 2Pi2fci(i - /cr - kr'' + /ci"-')"" (m) ^(1 - hy yuo RELAY ARMATURE REBOUND ANALYSIS 197 Expressions for an initial soiics may be obtained by setting ijieo = 1 y-ieo = y-2eo = 0, and, finally, for an initial cc^mplete series m — > oo and hence A;"" -^ 0, and Equations (1) and (m) become 1/2C00 = ?/2eco = 2/vi ^(1 + hY L A Bk^ P ' —r- — Jrn 1 1 - fci '2Bh Pi2(l + A-O (17) B. Complete Heel Series For heel series, Equations (1) and (m) may be used by interchanging the initial velocities, accelerations, and impact transfer coefficients for those relating to heel motion : yion — yieo = '2 A k-2 - k2 Mioil + itz) 1 - kf yion — yioi = B 1 - k2 2Ulj-_kr') B{1 - ko. 1 - ko 1/2 eO yieOy2eO + '2A{kl - 2k^^' + A-D _ 2Mi2A:2(l - k^ - kr' + A-^~^)' 52(1 - kl) BPu(l - kl) (n) (o) .2 'i/2f0 An initial heel series occurs when the heel strikes first after the "zero" impact. The first heel impact then occurs Ti = 2Pn/B(l + A;i) after the zero impact and the initial conditions are ?/2el = Pl2(l + A-i) 9 4 P,, 2/1.1 = -A-i +ATr = ^p" (1 + A-i) - A-: yui = -k,T, + \ATl = ?^Ml + A;i) B (1 + A^i) - A-i Substitution of the above into {n) yields yion = — A:i + 1 + A-, 1 - h 2 _ 2AFi (1 - ^2)" -ilfl2(l + A:2)(l - /v2") (P) The corresponding expression for 7/ion is quite involved. For the special case of A: = A;i = k2 198 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 l/lOn — APi,{l + kf B APl2 \ B k' fc(i - r) kj 1 - F Mnil - k")(k - k") (1 - ky If the initial series is a complete series, n -^ =o and (q) ?/lOo t/leoo APn(i-\-ky B{1 - ky L B 'APi2 k(l - k) 1 + k - Mnk 1 + k '2APi2 kil - k) B (1 +/c) - Muil + k) (22) C. Partial Front Series The worst rebound occurs when heel and front impacts occur nearly simultaneously, with the front hitting first. From Equation (m) for an initial front series, this requires that B APu = Q = 1 - A;" k - k- (30) After n front impacts conditions are given by Equations (14) and (19) with yi = tjt = 0, and = 1 - (1 - Mx2)(l - kn = k'" + ,„ , Mnyl 2Mi2A:"r/2 Ph This may be solved for y-z = Pi2(l — k"). The maximum front excur- sion now possible is that for a complete series of heel impacts. The above value of ?/2 in Equation (24) yields 2CYi = Mu + (1 - Mi2)[/c'" - ilfi2(l - fc")'] (31) D. Partial Heel Series The worst rebound occurs again when heel and front impacts occur nearly simultaneously, with the front hitting first. From Equation (9) for an initial heel series, this requires that B APn = Q = 1 k{l - k) 1 + k + k{\ - k^)Mi (32) RELAY AHMATUllE REHOUXD AXALYiilS 1 'pe crystals, potassium niobate and sodium niobate. Potas- sium niobate is orthorhombic at room temperature, changing to tetragonal at about 225°C. and cubic near 435°C. Sodium niobate is orthorhombic at room temperature, changing to tetragonal at about 370°C. and to cubic at about 640°C. The second part of the paper discusses relations among the structures of the ABO3 compounds. Subjective Sharpness of Additive Color Pictures * M. W. Baldwin^ Proc. I.R.E., 39, pp. 1173-1176, Oct., 1951. This is a report on the first numerical results to come from a laboratory experiment on the subjective sharpness of additive three-color pictures. The sharpness factor is isolated by using out-of-focus projection (of slides) instead of actual television transmission. An observer's acuity for defocus is greatest for the green component and least for the blue component, in an additive three-color picture. When the same picture is reproduced in monochrome (white, red, green, or blue) at the same brightness, the observer's acuity for defocus is equal to that found for the green component. Frequency-Modulation Terminal Equipment for the Transcontinental Relay System * J. G. Chaffee^ and J. B. Maggio^ Elec. Eng., 70, pp. 880-883, Oct., 1951. To meet the exacting requirements of the new transcontinental microwave relay system, specially designed frequencj'-modulation terminal equipment was * A reprint of thi.s article may be obtained on request. iBell Tel. Labs. 208 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 constructed. The terminal transmitter converts either message or television signals to a frequency-modulated signal centered on 70 mc and the terminal frequency-modulation receiver recovers these signals, thus providing a link between the relay system and other telephone facilities. Observer Reaction to Low-Frequency Interference in Television Pictures* A. D. FowLERi. p^oc. I.R.E., 39, pp. 1332-1336, Oct., 1951. This paper presents results of tests to determine how much low-frequency interference can be tolerated in black-and-white television pictures. Various levels of single low-frequency interference were superimposed on a locally trans- mitted television picture. Observers viewed the picture and rate the disturbing effect of each level of the interference. Ratings were made in terms of preworded comments ranging from "not perceptible" to "unusable." Interfering frequencies from 48 to 90 cycles per second were employed. Just visible interference appears as a flicker. The rate of flicker is the differ- ence between interfering and 60-cycle field frequencies. The most disturbing interference produced a flicker rate of 5 or 6 cycles per second. To be tolerated, peak-to-peak amplitude of this interference had to be 54 db weaker than the peak-to-peak amplitude of the television signal (including synchronizing pulse). For flicker rates of 0.5 and 12 cycles per second, the amount of interference which could be tolerated was larger by 14 and 3 db, respectively. Arcing at Electrical Contacts on Closure. Part II. The Initiation of an Arc* L. H. Germeri. Jl. Appl. Phys., 22, pp. 1133-1139, Sept., 1951. The capacity of the plates of an oscilloscope charged to 35 or 40 volts is dis- charged repeatedly by approaching electrodes of carbon, active silver, and inac- tive silver. Facts about the discharges, which are arcs of very short duration, are inferred from resulting open circuit potentials and calculated electrode separations. The separation at the first arc varies in different experiments but corresponds on the average to a nominal electric field of 0.6 X 10* volts/cm for carbon or active silver and to 2 X 10^ volts/cm for inactive silver. Each arc is initiated by a very small number of field emission electrons. The hypothesis that a single electron may perhaps be sufficient is consistent with observations at later stages of each closure when the electrodes are closer and the field much higher. The earlier observation, that the potential across a short arc is constant and independent of current, is not true if the arc time is sufficiently short. For active silver a time comparable with 2 X 10~^ sec is required to establish the steady arc voltage characteristic of later stages of arcs which last longer than this. The initial time during which the potential is decreasing toward its final steady value is 100 times the transit time of a silver ion across the gap. * A reprint of this article may be obtained on request. iBell Tel. Labs. ABSTRACTS OF TECHxNICAL ARTICLES 209 Computation of Control Limits for p-Charts Whe7i the Samples Vary in Size. 11. L. JoNES^ Ind. Quality Control, 8, pp. 26-27, Sept., 1951. The Design of Switching Circuits. W. Keister^, A. E. Ritchie^, and S. H. Washburn^. N. Y., Van Nostrand, 1951. 556 pp. (Bell Telephone Laboratories Series) . This is the first published textbook in its field. It presents, first, the funda- mental design principles of switching circuits composed of discrete-valued switching elements. Most of the discussion concerns two-valued elements, with greatest emphasis placed on electromagnetic relays. Chapters cover basic circuit paths, the logical interpretation of retiuirements, and the techniques of con- structing networks to fulfill these requirements. The symbolic methods of Boolean algebra and its application to the design of combinational and sequen- tial circuits is covered. Later chapters cover various unifunctional circuits such as selecting, connecting, translating, counting, and lockout. Final chapters dis- cuss methods of sjoithesising unifunctional cncuit building blocks into larger circuits and systems. Measurement of the Elastic Constants of Silicon Single Crystals and Their Thermal Coefficients. H.J. McSkimin^ W. L. Bond^ E. Buehler\ and G. K. Teal^. Letter to the Editor. Phys. Rev., 83, p. 1080, Sept. 1, 1951. Interest in the properties of sihcon single crystals arising from their use as semiconductors has led us to make measurements of the elastic constants of two single crystals. Measurements of velocities of propagation for both shear and longitudinal waves were made in the crystals as described in a recent paper by McSkimin. Frequencies in the range 8-12 mc/sec were used. The three independent elastic constants were evaluated, a density of 2.331 (measured by pycrometer) being used. Data and formulas used are summarized in Table 1. Two crystals were measured — as indicated — with data obtained from the larger one being used to determine the elastic constants. Check meas- urements were made for the smaller crystal; and despite the less accurate "pulse overlap" technique used for two of the measurements, velocity agreement to within 0.15 per cent was obtained. Both crj'stals were of a high degree of crystalline perfection as shown by etching and X-ray tests. Domain Wall Relaxation in Nickel. W. P. Mason^ Letter to the Editor. Phys. Rev., 83, pp. 683-684, Aug. 1, 1951. Phase Transitions in Ferroelectrics. B. Matthias^ National Research Council, Comm. on Solids. Phase Transformations in Solids. Ed. by R. Smoluchowski, J. E. Mayer, W. A. Weyl. N. Y., Wiley, 1951. 660 pp. iBell Tel. Labs. 5 111. Bell Tel. Co 210 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 Under the name ferroelectrics are classified those materials which exhibit dielectric anomahes phenomenologically similar to the magnetic l^eliavior of the ferromagnetics. Perhaps it would have been more logical to use the term Rochelle-electrics, thus emphasizing the similarity in the dielectric behavior to that of Rochelle salt. In this paper the known ferroelectrics are listed first, and then there follows a discussion of the various theories which have been created to explain them. Data on Random-Noise Requirements jar Theater Television * P. Mertzi. Jl. S.M.P.T.E., 57, pp. 89-107, Aug., 1951. Provisional evaluation of permissible random noise for theater television is considered from several sources of information. These cover broadcast television experience and the graininess in motion picture film; the recjuirements deduced from the various sources generally agree. For broadcast television, a frequency weighting and hmit on weighted noise power have been used. The finer picture detail of theater television presumes a lower permissible random noise. Changes in weighting curve are discussed. A hmit figure of noise is suggested, which is comparable to graininess effects in motion pictures, though slightly more severe than present published performance on camera tubes. A Spatial Harinonic Traveling-Wave Amplifier for Six Millimeters Wavelength.* S. Millman^ Proc. I.R.E., 39, pp. 1035-1043, Sept., 1951. This paper describes a travelmg-wave amplifier in which the electron beam interacts with a spatial harmonic of an electromagnetic wave propagating along an array of resonator slots. The result is a considerable reduction in operating beam voltage for a given physical separation of the circuit elements. This type of amphfier operating at about 1,200 volts has yielded net power gains of about 18 db in the 6-mm wavelength region. A magnetic field of about 1,600 gauss is sufficient for proper beam focusing. Aside from small variations of gain with frequency that is caused by internal reflections, the bandwidth is of the order of 3 per cent. Form of Transient Currents in Tomnsend Discharges with Metastahles* J. P. MoLNARi. Phys. Rev., 83, pp. 933-940, Sept. 1, 1951. The form of the current is calculated for a Townsend discharge stimulated by a pulsed light beam, with particular reference to the current component initiated by metastable effects. The calculation is directed particularl}^ to the development of methods for quantitative interpretation of current patterns observed experimentally. Studies of y-Processes of Electron Emission Employing Pulsed Town- send Discharges on a Millisecond Time Scale.* J. P. Molnar\ Phys. Rev., 83, pp. 940-952, Sept. 1, 1951. * A reprint of this article may be obtained on request. > Bell Tel. Labs. ABSTRACTS OF TECHNICAL ARTICLES 211 The relative amounts of electron emission from the cathode in a Townsend discharge caused by ions, photons, and metastables have been studied experi- mentally for several cathodes in argon, using pulsed-light stimulation of the discharge. The current initiated by metastables exhibits a slow build-up and deca}', thus permitting easy separation fi-om the faster rising effects of gas ionization and electron emission by photons and ions. Time constant studies of the slow component yielded a diffusion constant for metastable argon atoms of 45 cm^ sec~i at one millimeter pressure. The efficiencies of electron emission by metastables and ions was found to be closely the same, while the quantum yield for photon emission was found to be generally smaller. Electrical Properties of aFe-yOz and aFe-yOz Containing Titanium.* F. J. MoRiN^ Phys. Rev., 83, pp. 1005-1010, Sept. 1, 1951. Electrical conductivity. Hall effect, and Seebeck effect have been measured on two sets of polycrystalline samples of aFe203 and aFe203 containing from 0.05 to 1.0 atomic per cent titanium (n-t3^pe impurity). One set of samples con- tained 0.6 atomic per cent excess of iron (n-type impurity), the second set con- tained 0.6 atomic per cent deficienc}' of iron (p-type impurity). The conductivity of pure aFeoOs is independent of this amount of stoichio- metric deviation. The slope of the log conductivity vs reciprocal temperature plot is 1.17 ev and the intercept at 1/T -= 0 is 2.1 X 10* ohm-i cm-i. Room temperature conductivity varies from — lO"^"* ohm"' cm~i (extrapolated) for pure aFe203 to 0.3 ohm~i cm"' for aFe203 containing 1.0 atomic per cent titanium. The measured Hall ^'oltages seem to result entirely from magnetization of the samples, which are weakly ferromagnetic, and disappear above the ferro- magnetic Curie temperature. The temperature variations of the Fermi level are determined from Seebeck data. The temperature variations of carrier concentration are determined from Fermi level and of mobility from carrier concentration and conductivity for some samples. Carrier concentration results indicate that each added titanium ion donates approximately one electron to the conduction process. jNIobilities are found to be less than 2.0 cmVvolt sec, suggesting that conduction involves electrons in the d level of iron. Acceptance Inspection of Purchased Material.* J. E. Palmer' and E. G. D. PatersonK Ind. Quality Control, 8, pp. 15-19, Sept., 1951. This paper describes some of the principles and procedures employed in the inspection of purchased material in the form of components or finished products. The authors' experience has been largely with procedures used in the Bell Sys- tem, and the illustrations have therefore been drawn from this source. It is felt, however, that considerations leading to the choice of specific inspection tech- * A reprint of this article may be obtained on request. 1 Bell Tel. Labs. 3 W. E. Co. 212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 niques will be generally applicable even though the number and volume of items purchased and the number of supphers involved may in some cases differ widel}'. In this presentation, stress has been placed on a discussion of the broader gauge factors underlying the engineering planning of inspection procedures rather than on specific sampling and control techniques. Analysis of Audio-Frequency Atmospherics * R. K. Potter^ Proc I.R.E., 39, pp. 1067-1069, Sept., 1951. Sound portrayal techniques used in studies of speech and noise reveal the structure of atmospheric disturbances well known to long-wave radio and ocean- cable engineers as "whistlers," "swishes," and tweeks." It is suggested that renewed investigation of these effects, using modern analyzing tools, might yield information of considerable scientific interest. Reflection of Electromagnetic Waves from Slightly Rough Surfaces* S. 0. RiceK Communications on Pure and Applied Math., 4, pp. 351-378, Aug., 1951. Color Television and Colorimetry* W. T. Wintringham^ Proc. I.R.E., 39, pp. 1135-1172, Oct., 1951. The high lights of the history of color measurement and of color photography are reviewed. Following this introduction, the principles of modern three-color colorimetry are developed from a hypothetical experiment in color matcliing. The conventional theory of "perfect color reproduction" by color television is built up from colorimetric background. Some of the difficulties to be expected in applying colorimetry to color television are brought out. Finally, there is some discussion which tends to show that colorimetry maj'' not be a sufficiently powerful tool to provide answers to all of the questions which wiU arise in the reproduction of scenes in color b}' television. The advantage of colorimetrj^ as a background is indicated, however. * A reprint of this article may be obtained upon request. 1 Bell Tel. Labs. Contributors to this Issue Charles Clos, C.E., New York University, 1927; New York Tele- j)hone Company, plant extension engineering, valuation and depreciation matters, intercompany settlements and tandem and toll fundamental plans, 1927-47. Pratt Institute, Evening School, Mathematics Instructor, 1946-49. Bell Telephone Laboratories, studies on development planning for local and toll switching systems and research in switching probability, 1947-. Member of A.I.E.E., New York Electrical Society, Mathematical Association of America, A.A.A.S., American Statistical Association, Iota Alpha, and Tau Beta Pi. A. B. Crawford, B.S. in E.E., Ohio State University, 1928; Bell Telephone Laboratories, 1928-. As a member of the Radio Research Department, he has been concerned with ultra short wave apparatus, measuring techniques, and propagation, and with microwave apparatus, measuring techniques, and propagation, as well as microwave radar and microwave antenna research. Member of I.R.E., Sigma Xi, Tau Beta Pi, Eta Kappa Nu, and Pi Mu Epsilon. O. E. De Lange, B.S., University of Utah, 1930; M.A., Columbia University, 1937. Bell Telephone Laboratories, 1930-. Mr. De Lange has been engaged in radio research, including studies on high-frequency trans- mitters and receivers, frequency modulation, radar, broad-band systems, and pulse systems. Associate member of the I.R.E. C. L. HoGAN, B.S. in Ch.E., Montana State College, 1942; M.S. in Physics, Lehigh University, 1947; Ph.D. in Physics, Lehigh, 1950. Anaconda Copper Mining Co., Great Falls, Montana, 1942-43. U. S. Navy, 1943-46. Instructor in Physics, Lehigh, 1947-50. Bell Telephone Laboratories, 1950-. Dr. Hogan has engaged in development work on boro-carbon resistors and microwave gyrators. Gold medal award for "Outstanding Engineer in Graduating Class," Montana State Col- lege, 1942. Letter of Merit from Chief of Naval Operations for work done in establishing and maintaining the acoustical torpedo shop at Pearl Harbor, 1944-46. Member of American Physical Society, Sigma Xi, Tau Beta Pi, and Phi Kappa Phi. W. C. Jakes, Jr., B.S.E.E., Northwestern University, 1944; M.S., Northwestern University, 1947; Ph.D., Northwestern University, 1949; 213 214 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1952 U.S. Navy, Airborne Radar Maintenance, 1944-46; Bell Telephone Labo- ratories, 1949-. Dr. Jakes has been engaged in microwave antenna and propagation studies. Member of I.R.E., Sigma Xi, Pi Mu Epsilon, and Eta Kappa Nu. W. P. Mason, B.S. m E.E., University of Kansas, 1921; M.A., Ph.D., Columbia, 1928. Bell Telephone Laboratories, 1921-. Dr. Mason has been engaged principally in investigating the properties and applications of piezoelectric crystals, in the study of ultrasonics, and in mechanics. Fellow of the American Physical Society, Acoustical Society of America and Institute of Radio Engineers and member of Sigma Xi and Tau Beta Pi. H. J. McSkimin, B.S., University of Illinois, 1937; M.S., New York University, 1940. Bell Telephone Laboratories, 1937-. Here he has worked chiefly on crystal filters, piezoelectric elements, ADP crystals, studies of the acoustic properties of liquids and solids. Fellow of Acoustical Society of America and Member of Eta Kappa Nu and Sigma Xi. R. S. Ohl, B.S. in Electro-Chemical Engineering, Pennsylvania State College, 1918; U. S. Army, 1918 (2nd Lieutenant, Signal Corps); Vacuum tube development, Westinghouse Lamp Company, 1919-21; Instructor in Physics, University of Colorado, 1921-22. Department of Develop- ment and Research, American Telephone and Telegraph Company, 1922- 27; Bell Telephone Laboratories, 1927-. Mr. Ohl has been engaged in various exploratory phases of radio research, the results of which have led to numerous patents. For the past ten or more years he has been working on some of the problems encountered in the use of millimeter radio waves. Member of American Physical Society and Alpha Chi Sigma and Senior Member of the I.R.E. E. E. Sumner, B.M.E., Cooper Union, 1948, holding Schweinburg Scholarship throughout entire college curriculum; Instructor of Physics, Cooper Union, 1947-48; Non-resident instructor of Massachusetts Institute of Technology, Probahility and Statistics — Applications to Sampling and Quality Control, summer, 1950; Bell Telephone Labora- tories, 1948-. Mr. Sumner was given rotational assignments in apparatus, switching, and television transmission development and switching re- search, and has worked on a number of projects, including the card translator, the magnetic drum, the video transmission evaluator, and the vibrating reed selector. He is currently engaged in the development of wire-spring relay. Member of Tau Beta Pi and Pi Tau Sigma. CONTRIBUTORS TO THIS ISSUE 215 Roger I. Wilkinson, B.S. in E.E., 1924, Professional Engineer (hon- orary), 1950, Iowa State College, 1924; Northwestern liell Telephone Company, 1920-21; American Telephone and Telegraph Company, 1924- 34; Bell Telephone Laboratories, 1934-43 and 1946-. U. S. War Depart- ment, ^^'ashingt()n and South Pacific, 1943-45. Mr. Wilkinson has been engaged in applications of the mathematical theory of probability to telephone problems. IMedal for Merit, 1946. Member of A.S.E.E.; A.S.A.; Institute of Mathematical Statistics; American Society for Quality Con- trol; Associate Member of A.I.E.E.; and Member of Eta Kappa Nu; Tau Beta Pi; Phi Kappa Phi; and Pi Mu Epsilon. PHE BELL SYSTEM nicm ourna \^^^ AP / E > O T E D TO THE SCIENTIFIC ^W>^ AND ENGINEERING SPECTS OF ELECTRICAL COMMUNICATION OLUME XXXI MARCH 1952 NUMBER 2 k Introduction to Formal Realizability Theory — I BROCKWAY MC MILLAN 217 An Application of Boolean Algebra to Switching Circuit Design BOBEBT E. STAEHLER 280 Interaction of Polymers and Mechanical Waves W. O. BAKER AND J. H. HEISS 306 The Reliability of Telephone Traffic Load Measurements by Switch Counts W. S. HAYWARD, JR. 357 Network Representation of Transcendental Impedance Functions M. K. ZINN 378 Abstracts of Bell System Technical Papers Not Published in This Journal 405 Contributors to This Issue 409 COPYRIGHT 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE 15ELL SYSTEM TECHNICAL JOURNAL PUBLISHED SIX TIMES A ^ K A K IJ V T H E A M E U I G A N TELEPHONE AND T K L E (; li A P 11 COMPANY 195 B R O A D WAY, NEW Y O K K 7, N. Y. CLEO F. CRAIG, President CARROLL O. BICKELHAUPT, Secretary DONALD R. BELCHER, Treasurer EDITORIAL BOARD F. R. KAPPEL O. E. BUCKLEY H. S. OS B O R N E M. J. K E L LY J. J. P 1 L L I O D A. B. C L A R K R. BOWN D. A. QUARLES F. J. FE E LY PHILIP C.JONES, Editor M. E. STRIEBY, Managing Editor SUBSCRIPTIONS Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 1 1 cents per copy. PRINTED IN U.S.A. THE BELL SYSTEM TECHNICAL JOURNAL (copyright 1952, American telephone and teleqkaph company) VOLUIME XXXI MAI{( II 1952 NUMBER 2 Introduction to Formal Realizability Theory — I By BROCKWAY McMILLAN (Manuscript received October 15, 1951) This paper offers a general approach to (he realizability theory of net- worlxs with many accessible terminals. The rnethods developed are applied to give a complete characterization of all finite passive networks. I. SUMMARY 1.0 A principal result of this paper is to characterize those matrices Z(p), functions of the frequency parameter p, which can be realized as open-circuit impedance matrices of finite passive networks. This char- acterization is provided by the following theorem: 1.1 Theorem:* Let Z{p) be an n X n matrix whose elements are Zrs(p), 1 < >', -'^ ^ n, where (i) Each Zrsip) is a rational function (ii) Zrsip) = Zrsip) (the bar denotes complex conjugate) (iii) Zrsip) = Zsrip) (i\') For each set of real constants ki , • ■ • , /.'„ , the function ^zip) = 2 Zrsip) krks r,s = l has a non-negative real part whenever Re(p) > 0. Then there exists a finite passive network, a 2n-pole, which has the impedance matrix Zip). * Presented to the American Mathematical Society, April 17, 1948. Abstract 260, Bulletin of the A .M.S. No. 54, July, 194S. 217 218 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Conversely, if a finite passive 2n-pole has an impedance matrix Z{p), that matrix has the properties (i), (ii), (iii), (iv). A formally identical dual theorem holds for open-circuit admittance matrices Y(p). 1.2 A general realizability theorem, applicable to and characterizing completely all finite passive networks, whether having impedance ma- trices or not, is also proved. 1.3 An effort is made to lay a foundation adequate for the realizability theory of both active and passive multi-terminal devices. To this end, a large part of the paper is devoted to the scrutiny of fundamental properties of networks. II. INTRODUCTION AND FOREWORD 2.0 Network theory provides direct means for associating with an electrical network a mathematical description which characterizes the behavior of that network. Typically, this results in shifting engineering attention from a detailed, possibly quite intricate, electrical structure to a mathematical entity which succinctly describes the relevant be- havior of that structure. An essential feature of this shift in focus is emphasized by the word "relevant": only those terminals of the net- work which are directly relevant to the problem at hand are considered in the mathematical description. Design work can then be done in terms of constructs relating explicitly to these accessible terminals, the effect of the internal structure being felt only by imphcation. The physical origins of these mathematical constructs, and the im- plications of the internal structure upon them, cannot however be en- tirely forgotten, for they have mathematical consequences which are not always immediately evident. Until he knows these limitations — imposed upon him by the physical nature or the necessary structural form of the networks he is designing — a design engineer cannot make free use of the mathematical tools that network theory has provided. We give the name "realizability theory" to that part of network theory which aims at the isolation and understanding of those broad limitations upon network performance, i.e., upon the mathematical constructs which describe that performance — which are imposed by limitations on the network structure. One may also include in the province of realizability theory some of the converse questions: the study of those structural features common to all networks whose per- formance is limited in some specified way. Reahzability theory would have little content were it not that "per- FORMAL REAIJZABILITY THEORY — I 219 formance" here must be construed to mean performance as viewed from the accessible terminals only. Were all branch currents and node poten- tials in a network available to observation, a mathematical statement of performance would be equivalent to stating the full system of dif- ferential equations governing these quantities, i.e., equivalent to giving the detailed network diagram. 2.1 With a few important exceptions, the converse kind of problem in realizability theory docs not lead to a strict implication from fimctional limitations to structural features, because the held of equivalent struc- tures for a specified performance is very broad. Typically, it is only by imposing some general a priori limitations on structure that fiu-ther conclusions can be firmly drawn from a functional limitation. In study- ing this kind of problem one is rapidly led from those basic issues which are clearly part of realizability theory toward general, diflRcult, and usually unsolved problems of network synthesis. One cannot, and should not, draw a sharp boundarj'^ here, but Nature so far has provided a fairly definite one for us, in that most of these problems have proved too difficult of solution. 2.2 Th^ direct realizability problems, the passage from structural prop- erties to functional properties, have been somewhat more tractable. Here, again, there is no clear dividing line between general realizability theory and the sort of design theory in which, for example, one specifies a particular filter structure depending on a limited number of param- eters and examines the performance of the structure as a function of these parameters. There is an extensive literature at or near this latter level of generality, most of it relating to filters or filter-like structures (e.g., interstage couplers in amplifiers). At a more basic level, the limitations on a network's structure which are commonly met in practice are of the following kinds: a. Limitations on the kind of elements appearing, e.g., to passive networks, networks without coupled coils, networks whose elements have specified parasitics, etc; b. Limitations on the general form of the network diagram, e.g., to ladder or lattice structures, without limitation to a specified number of elements or parameters. Here the problems are varied and difficult. We survey briefly the l)resent status of some of them. 2.3 Networks with two accessible terminals, two-poles, are basic in network technology. Fortunately, also, two-poles are unique among networks in that there is always a simple way to describe their perform- 220 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 ance. Except for the trivial limiting case of an open circuit, every two- pole has a well-defined impedance, Z{p), a function of the complex frequency parameter p, which describes its performance in a way which is by now well understood. Dually, except for the Hmiting case of a short circuit, every two-pole has a well-defined admittance function F(p). Even the limiting cases are tractable: every open circuit has the admittance function Y{p) = 0 and every short circuit the impedance function Z{p) = 0. In other words, by exercising his option to speak in terms either of impedance or of admittance, one can always describe the performance of a two-pole by using a single function of frequency. The descriptive simplicity and practical importance of two-poles led early to a fairly complete realizability theory for them. In 1924 R. M. Foster^ gave a function-theoretic characterization of the impedance functions of finite passive two-poles containing only reactances. The corresponding problem for two-poles which are not at all limited as to structure, beyond being finite and passive, was solved by 0. Brune in 1931. The effects of various structural limitations have since been studied by several writers (cf. Darlington, Bott and Duffin ). 2.4 Technology, and the promptings of conscience, have meanwhile urged the study of devices with more than two accessible terminals. Here, however, Nature has been less kind, in that no uniquely simple method is available for describing the performance of such devices as viewed from their terminals. Indeed, basic network theory has been remiss here, in not even mak- ing available a mode of description which is generally applicable — whether simple or not. W. Cauer^ showed that, when one admits ideal transformers among his network components, it is sufficient to study networks which are natural and direct generalizations of two-poles, namely, 27i-poles,* for arbitrary values of n. The corresponding natural generalization of the impedance function Z(p) of a two-pole is the impedance matrix of a 2w-pole: just as one multiplies a scalar current by a scalar impedance to get a scalar voltage, one multiplies a vector current by an impedance matrix to get a vector voltage. 2.41 Not all descriptive difficulties are resolved, however, by consider- ing 2n-poles and their impedance or admittance matrices. For the moment, a simple example will suffice to show this: the 2 X 2-pole which consists simply of one pair of short-circuited terminals and one pair of * Defined in Cauer,* and also later here. FOiniAL KKAL1ZA15ILITY TIllOOUY — I 221 open-circuited terminals is a finite passive 2n-pole (n = 2) which has neither an impedance matrix nor an admittance matrix. 2.42 When one ehminates this kind of descriptive difficulty by fixing his attention only upon 2/;-pol(\s for which an impedance matrix (or, dually, an admittance matrix) is available, the general realizability prol^lem for finite passive devices is solved. A partial solution, for the case n = 2, was published by C. M. Gewertz in 1933. The solution (Theorem 1.1) of the problem for a general value of n has been an- nounced recently by three authors, independently: Y. Oono, in 194G,* the present author, in 1948, t and M. Bayard,' in 1949. The problem for reactive 2n-poles is much simpler and was solved by Cauer, in 1931. 2.5 Intermediate between the fairly specific problems of filter theory on the one hand and the general realizability theory of multi-terminal devices on the other, lies the study of four-poles as transducers. There is a considerable literature on the realization of transfer functions or transfer ipipedances under various structural limitations. The basic and extensive work of Bode^ on active systems belongs also in this category since it is avowedly limited to single-loop structures. 2.6 Beyond the important result that, by sufficiently elaborate circum- ventions, one may avoid the use of transformers in the synthesis of any two-pole, (Bott and Duffin ) little in general is known about networks which do not have transformers. 2.7 The present paper is an essay in the realizability theory of devices with many accessible terminals. Ideal transformers are admitted as network elements; indeed, their use is essential. This fact is indicated by the adjective "formal" appearing in the title. The availability of ideal transformers makes it possible to exploit the simplification noted by Cauer and to consider only networks which are 2w-poles in his sense. The aim of the paper, therefore, is to set a foundation for realizability theory which is completely general within this framework. 2.71 The first problem is that of description. We observed above an example — entirely trivial — of a passive four-pole which had neither an impedance nor an admittance matrix. Unfortunately, opportunities * Date of Japanese publication. This reference, and manuscript of Oono'"- ", were sent by Professor Oono in a personal communication to R. L. Dietzold, who showed them to me in December, 1948, while an early draft of the present paper was in preparation. The recent (1950) American repul)licat ion ;eiiei-al i-e;i]izaliiiil\' theorems for si nictures containiiifi; xacuum lubes wilh fr(Mluency-in(le))en(l(Mit li-anscouductances, \a('uum luhes with non- \anishiiiu; transit times, unilateral devices with si)ecified parasitics, etc. 2.7") Actually, the i)ostulates as we have fi;i\en them are certainly not adeiiuate for such an ambitious program. Exigencies of the presentation ha\e dictated a number of condensations and compromises. It is hoped that the basic ideas are still e\'ident even if not isolated indix'idually in separate and entirely independent postulates. In any event, it is the author's firm belief that the presentation as given is at least illustrative of the kind of approach, and the level of mathematical detail, which will be needed if one is ever to provide a truly adequate realizability theory: a theory which will cover, for example, the broad range of active linear systems which present-day technology allows us to consider. 2.8 Apart from the network theoretic concepts, which must be evalu- ated by their effectiveness in solving problems — an assessment which is by no means yet complete — this paper is strongly marked b}^ an idio- syncracy of its author: a consistent and insistent use of geometric ideas and terminology. This is based on the personal experience that linear algebra achieves logical unity and a freedom from encumbering notation when viewed in this way. A general reference covering most of the linear algebra (geometrj^) rec^uired here is P. R. Halmos' elegant monograph^. 2.9 For a proof solely of 1.1, which has already been three times proved in the literature, ' ' this paper provides an apparatus which is too cimibersome. There is even a sense in which 1.1 alone provides a charac- terization of all finite passive devices, for it seems to be generally ac- cepted that, by the use of ideal transformers, any finite passive network can be represented as a network which has an impedance matrix to which is adjoined suitable ideal transformers. Therefore we cannot claim that, in using this cumbrous apparatus to characterize all finite passive 2/;-poles (including the degenerate ones), we have offered anything not already provided by a simpler proof of 1.1. Three things may be said in rebuttal. First, we have already empha- sized that the apparatus here exhibited was designed for more problems than that to which it is here ai)plied. It is presented in the belief that it will prove of further use. Second, even in the study of passive networks, it has schemed to the author helpful to look at the manifold things which are not passive net- 224 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 works. One gets then a clearer view of the unique position occupied by passive devices among all linear S3^stems. Third, there is a kind of semantic issue here: the assertion that any finite passive network (sic) can be put in such a form that 1.1 applies seems to this author to give a kind of circular characterization of such devices. A characterization which did not itself involve the concept of a network seems more satisfying. Logically, there is no circle here, but this is a fact requiring proof. A careful reading of this paper will show that it provides a proof. This particular subtlety does not of itself justify the lengths to which we have gone. It is, however, no longer a subtlety if one wishes to consider devices which do not have a representation in terms of something non-degenerate to which ideal transformers have been added. 2.91 The present Part I of the paper is so organized that at the end of Section 8 the reader is in possession of all of its principal results and its basic ideas. The remaining Sections, 9 through 20, may then be regarded as an Appendix containing the details of proofs. Indeed, Part II will be largely devoted to further details of proof, though there will be there one important idea not mentioned, save casually, in Part I — the idea of degree for a matrix. In Sections 4 through 11, technical paragraphs have been distinguished from explanatory or heuristic ones by starring the paragraph numeral. Part II of the paper contains the bulk of the proof of 1.1. This proof is modelled after that of Brune for the realizability of two-poles. One familiar with the Brune process will probably find Part II readable without extensive reference to Part I. Let the reader be warned that the Brune process is not a practical one for realizing networks because of its critical dependence upon a difficult minimization and balancing operation. The same criticism applies to the generalized Brune process of Part II. The Brune process is of theoretical importance because it does realize a network with the minimum number of reactive elements. These facts will be brought to light in Part II. The proofs of Oono and Bayard are different from ours. That of Oono again follows the Brune model. III. INTRODUCTION TO PART I 3.0 We keep before us first the problem of finding a mathematical de- scription applicable to and characterizing the behavior of all finite pas- FORMAL REALIZABILITY THEORY — I 225 sive networks. Second, wq seek to make mathematically precise those ideas which appear to form the basis of general realizability theory. Sections 4 through 7 introduce the immediate mathematical machinery for this. Section 8 states the fundamental realizability theorem and outlines its proof. At this point the reader has had an introduction to the results of the paper. The remainder of the paper is then devoted to the technical details of proof. Beginning with Section 12, the de\'ice of starring the technical passages will be dropped. 3.1 Cauer distinguished precisely the class of networks called 2ri-poles from the class of all multi-terminal n(>t works. He also showed that, by the use of ideal transformers, any multi-terminal network is eciuivalent to a network which is a 2/(-pole (for some n) in his sense. We shall in Section 4 define a class of objects to be called general 2n-poles. This class includes all electrical networks which are 2n-poles in Cauer's sense. Its definition abstracts the significant properties isolated by Cauer. For the study, alone, of finite passive networks, this definition is uimecessary, since one can in fact so put the arguments as to deal only with 2N-poles which are finite passive networks, and therefore to deal only with concepts already defined in Cauer . The somewhat physical notion of a general 2/i-pole is a convenient backdrop against which to display the important physical properties of finite passive networks, and, indeed, of networks in general. Having it available, we use it throughout the realizability arguments. IV. DEFINITION OF GENERAL 2n-P0LE 4.0* Network theory establishes a correspondence between oriented linear graphs and systems of differential equations. With each node of the graph is associated a potential En = EniO and with each oriented branch a current h = hit). These potentials and currents are constrained, first by Krichoff's laws, and second by differential eciuations which de- pend upon the nature of the branches but not upon the topology of the graph. 4.01* A finite passive network is one whose graph has the following properties : (i) There are finitely many nodes, 1,2, • • • , A''. (ii) There are finitely many branches, \, 2, ■ • • , B. Technical paragraph as explaiiKMl in Section 2.91, 226 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 (iii) Let the 6-th branch begin at node nt and end at rib . Let Vb = Enf, — E„l . Then for each b, one of Vb = Rbh, Rb > 0 (a) h = Cb~, Cb>0 (b) at Vb = ^ Lbb' -TT (c) 6' at holds, where the matrix Lb,b', is real, symmetric, and semi-defi- nite. Cauer has shown how an ideal transformer can be defined as the limiting case of a finite passive network. It is indeed no more nor less ideal than an open circuit {Rb = °o or Cb = 0) or a short circuit (Rb = 0 or Cb = 00 ). 4.02 We seldom deal with networks in the detail which is implicit in (iii) above. We are usually interested in the external characteristics, so to speak, of such networks as viewed from a relatively small number of terminals (nodes). These multi-terminal devices, however, we continue to incorporate into larger network diagrams. It is usually clear how Kirchoff's laws are to be applied in these cases, and what the differential equations of the resulting system are. We are obliged, however, to make these matters precise before we can deal intelligently with the most general physical properties of networks. 4.1 We have seen the two kinds of constaint that a multi-terminal de- vice imposes on the branch currents and node voltages in a network in which it is incorporated: the topological ones contained in Kirchoff's laws and the dynamical ones described by differential equations. Cor- respondingly, there are two aspects to the concept of general 2n-pole. 4.11* In its relation to Kirchoff's laws, a general 2n-pole is indicated as an object with n pairs of terminals {Tr , Tr), 1 < r < n. Each terminal can be made a node in an arbitrary finite diagram constructed out of network elements and other general 2m-poles, with arbitrary values of m. This diagram is not an oriented linear graph, so we have no basis for the use of Kirchoff's laws. From it, however, we construct an ori- ented linear graph, called the ideal graph of the diagram, by the follow- ing rule: The nodes of the ideal graph are those of the original diagram. Every * Technical paragraph as explained in Section 2.91. FORMAL REALIZABILITY THEORY — I 097 oriented branch of the original diagram is repeated in the ideal graph, similarly situated and oriented. Between those nodes which, in the original diagram, were the (7V , Tr) of a 2n-pole N, is drawn a branch iSr , called the r-th ideal branch of N, oriented from 7V to 7\ ■ This is done for each such terminal pair. Kirchoff's laws now apply to this ideal graph. 4.12* Consider a particular 2/i-pole N. Let Er be the potential of Tr , Er that of Tr . Define Vr{t) = Er- E'r. Then Vrit) is the voltage across the ideal branch I3r so oriented that Vr{i) > 0 when Tr is positive relative to 7V . Let kr{t) represent the cur- rent entering Tr . Then Av(0 = Ir{t), the current in /3r , so kr{l) is also the current lea\-ing Tr . This is the force of the notion of ideal branch and the fact which distinguishes a network which is a 2Ai-pole from an arbitrary network with 2n terminals. 4.13 For example, the network at (a) of Fig. 1 is not a 2 X 2 pole because its currents are not constrained to meet the ideal branch requirement. The addition of ideal transformers in either of the ways shown in (b) or (c) of the figure converts it to a 2 X 2 pole. Of course in a circuit in which the currents are constrained externally — as they would be, for T,c^— ^WV ^WV ■^Tj T.'o- (a) ■OT,' T,o. —I I — ^A^^ — r — VA — I r T,-^-rii ■oT, Jn^,- fC) Fig. 1 — Conversions of a four pole to a 2 X 2 pole. * Technical paragraph as explained in Section 2.91. 228 THE BELL SYSTEM TECHNICAL JOURNAL, VL^RCH 1952 example, when the 2X2 pole is dri\en by separate generators in the two external meshes — these trans^formers can be eUminated. The definition of 2r<-pole requires however that in even,' context the ideal branch con- cept is vahd. 4.2* The second aspect of the concept of general 2r<-pole is that it im- poses some kind of constraint — other than that imphed by 4.11 and KirchofF's laws — upon the voltages across and currents in its ideal branches. Define the s%Tnbols and £ = ijt^ = [I'lit), i'-.\t), " , P-(0] A- = A-(0 = [hit), hit), • • • , Ut)] as the n-tuples. respectively, of voltages across (Tr , Tr) and currents into Tr , I < r < n. These are added and multiphed by scalars bj' the usual rules of vector algebra. If v and k represent simidianeous values of voltage and current in the 27j-pole N — i.e., values satisfying all the constraints — then we say that N admits the pair [v, k]. We say that N admits r if there is a /: such that N admits the pair [r, k]. This k is said to correspond to v. Dually, N admits k if there is a V (corresponding to k) such that N admits [r, /:]. The constraints imposed by a general 2n-pole N on voltages and cur- rents are completeh' described by the totality of pairs [r, k] which N admits. We shall define a general 2n-pole, therefore, as (i) a collection of n oriented ideal branches, as in 4.11, and (ii) a list of pairs [r, k] of voltages and currents admitted in these branches. Hereafter we shall usuallj' drop the adjective "'general." 4.21 The definition we have jiLSt given is, in a way, a postulational form of an r< -dimensional Thevenin's theorem. It postulates that a 2n-i>ole is a thingt which, as far as the outside world is concerned, can be represented b\' a collection of two-poles ,3, , 1 < r < n, among which there e.xists a comphcated agreement as to what currents and voltages will be admitted. 4.22 The passive networks (b ) and (c ) of Fig. 1 define 2X2 poles, be- cause they satisfy 2.01 and clearly permit a complete specification of the admissible pairs [r, k]. Any equivalent network would also specify * Technical paragraph as explained in Section 2.91. t In fact, at this level of generality, any thing. FORMAL REVLIZABILITY THEORY— I 229 the same 2X2 pole, because — by its very equivalence — it would admit the same pairs. The closest association we can make between a 2n-pole and a network, then, is to identify the 2n-pole with an equivalence class of networks. 4.23 The completely symmetric role played by voltages and currents in this definition of general 2/<-pole will make it possible to take early advantage of the well-known duality principle of network theorj'. We shall do so freely. ■4.3* We shall call a 2n-pole physically realizable if its admissible pairs [v, k] are the solutions of a .system of differential equations obtained from a finite passive network, admitting the limiting elements: ideal transformers, open circuits, and short circuits. V. PHYSICAL PROPERTIES OF XETWORK.S 5.0 There are clearly a great many properties of finite passive networks which are not \'et possessed by the general 2n-poles now introduced. It is instructive to examine these properties physicallj'. 5.1 In the fin«t place, the d>'namical coiLStraints (a), (h), and (c) of 4.01 are expressed by linear, time invariant, differential equations. Accord- ingly, the 2n-poles of network theory' are: 5.11 Linear, in that the class of admissible pairs [v, k] is a linear space; 5.12 Time invariant, admitting \N-ith each [v(t), k(t)] also all [v(t -\- t), kit -{- t)] for aribtrary r. 5.2 In the second place, a physical network N cannot predict the future, i.e., it cannot respond before it is excited. This can be formalized in terms of the pairs [v, k] admitted by N, but to do so would require some digression. The rea.sons \\ill be .seen under 5.7 below. 5.3 We ha\e already mentioned the constraints imposed on voltages and currents in a network by the topologj' of the network, through the medium of Kirchoff's laws. The.se constraints have three important properties : 5.31 They are workless, since they are impased by resistanceless connections, leaklass nodes, and, in the formal theor>', by ideal transformers. 5.32 Though it .seems scarceh' necessar\' to .say it, they are the onh' workless constraints. All other constraints are djmamical and have powers or energies associated with them. * Technical paragraph as e.xplained in Section 2.91. 230 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 5.33 They are frequency independent, that is, holonomic in the sense of dynamics. 5.4 The workless and the dynamical constraints in a physical network are all defined by relations with real coefficients. The space of admissible pairs is then a real linear space. 5.5 The positivities specified in 4.01 are characteristic of passive sys- tems. They correspond to the fact that the power dissipation and the stored energies are all positive. 5.6 By definition, finite passive networks contain finitely many lumped elements. Correspondingly, their resonances and anti-resonances are finite in number. 5.7 We are accumstomed to dealing with networks which have, in addi- tion to the properties listed above, a kind of non-degeneracy, in that the list of admissible pairs [v, k] satisfies: 5.71 At least one of v or k can be specified arbitrarily — any real function is admitted; 5.72 When the free number of [v, k] is specified, the other is uniquely determined. For these non-degenerate networks, the property 5.2 above is easily formalized: if, say, k is determined by v, then v\t) = v\t) for t < k implies k\t) = k'it) for t • can be specified arbitrarily, and the resulting voltage amplitudes v are then fixed by /,■ and />, by ((>). Z(p) is called the impedance matrix of the 2n-pole N/, . It is also sometimes called the open-circuit, impedance matrix, because each Zrsip) is, by (7), the voltage amplitude across (7V , Tr) when the current amplitudes at all terminals save (7\ , T.,) are zero — i.e., when all pairs sa\-e the s-th are on open circuit. (■).:U Dually, the pairs [v, Y(p)v] defined by an admittance matrix Y'(p) as v ranges over V define a linear time inxariant 2n-pole which is non-degenerate. VII. WORK AND ENERGY 7.0* A linear correspondence satisfying Pi and P2 is something which abstracts the properties of linearity and time invariance. Most of the remaining properties of physical networks involve the mention of work or energy. These concepts enter our picture by way of the scalar product ((', /.•) between a voltage /i-tuple (1) and a current /i-tuple (2), of 0.11. This scalar product is defined by {V,k) = Zvrkr. (1) r=l 7.01 If p = ice, one easily calculates from (3) and (4) of 6.11 that 2ne{v,k) -y^rn^[ Z Vrm-rit) dt. That is, when p = iu, the real part of 2(v, A) measures the average total power dissipated by the system of currents Av(/) against the driving voltages Vr{t). When p is not a pure imaginary, the interpretation of the scalar product {v, k) is not so clearly physical as this. The reader will ulti- mately observe that our significant statements about such products can all be reduced to statements applicable when p = ico, i.e., when the power interpretation is valid. 7.1* An important concept in what follows is that of the annihilator of a linear manifold (Halmos^, par. 16). Let Vi C V be a linear manifold. * Technical paragraph as explained in Section 2.91. 234 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Then its annihilator (Vi)" is the set of all k such that t'eVi implies (v, k) — 0. (Vi) is a linear manifold in K. Dually, given Ki C K, (Ki)" is the linear manifold of all reV such that /ceKi implies (v, k) = 0. The annihilator concept is the analog in our general geometric frame- work of the idea of orthogonalit,y. It clearly suggests a connection with workless constraints. 7.2* The complex conjugate of an /i-tuple v (or /.•) is defined in the obvious way: if V = [vi , • ■ ■ , Vn] then V = [Vi , ■ • ■ , Vn]. This conjugation operation clearlj^ has the properties I = ^ (2) a^ -{- br] = a^ -\- hfj where a and h are scalars and ^ and rj are (consistently) elements of V or K. Furthermore, at once from (1) of 7.0, (v, k) = (v, k). (3) 7.21* A linear manifold will be called real if it contains, with any n-tuple also the conjugate of that n- tuple. 7.22* A real manifold is spanned by real 7i-tuples. This will be proved in the Appendix, Section 20. 7.23* The annihilator of a real manifold is real. For let Ki be real and k , • • • , k^ he real n-tuples which span Ki . Then if ve (Ki) every (v, k') = 0, and conversely. But then also (v, k') = {v, k') = 0 = 0, so i;6(Ki)''. * Technical paragraph as explained in Section 2.91. FORMAL REALIZAHILITY THEORY — I 235 7.3* (livcMi a linoar corrcspoiuhMico />, \v(^ make sovcnil definitions: y ,.(])) is tlie sot of all reY such that there is a /,• with [v, ^eL(p). KiX'p) is the set of all I.eK such that there is a v with |/', L-\eL('p). V;,(i(/j) is the set of veViXp) such that [v, 0]eL(p). Klo(p) is tlie set of keKiXp) «uch that [0, k]eL(p). 7.31* The postulate P2 implies that for each p^Tl , V/.(p), K/ip), '^Loip) and K/.o(p) are all linear manifolds. 7.32 Vl(p), for example, is the set of veY such that Nz, admits v at frequency p. 7.4* We now postulate P3. There exist fixed linear manifolds Vl C V, K,, C K such that (A) For every peT, , W,(p) = V, = (KUp))' (I) For every peT, , K,{p) = K. = (Y,oip)Y. 7.41* We may henceforth write Vlo , K^o , for Ym{p), Klo(p), knowing that, under P3 V.0 = (K^)°, 7.42 Linear correspondences satisfying P3 abstract the properties men- tioned in 5.3. The equalities Vl(p) = Vl , Kl(p) = K/, guarantee the frequency-independence of the workless constraints. The equalities Vl(p) = (Klo(p))", K/.(p) = (Vlo(p))° in a sense guarantee that the only constraints imposed upon admissible currents and voltages (as opposed to constraints relating currents and voltages) are those which arise from open or short circuits, i.e., are workless. 7.43 An illustrative conseriuence of P3, for example, is that if L satisfies P3 and if N^ is such that all of the current amplitudes can be specified arbitrarily, then indeed the voltages are determined by the currents. This will appear as a consequence of 8.1. It is a very general theorem about networks of a kind that this author, at least, has not heretofore encountered. * Technical paragraph as e.xplained in Section 2.91. 236 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 7.5* Continuing toward realizability, we introduce P4. If peTL , then peTz.. If [v, k]eL{p), then [v, k]eL(p). This postulate embodies most of the reahty properties of networks. It has as an immediate consequence the 7.51* Lemma: If L satisfies PI, P2, P3, and P4, then all of are real. Proof: By P4, veY l{v) = V^ implies veW tip) = Vl . Hence Vl is real. Then K^o = (Vl)° is real, and dually. 7.6* The three remaining postulates on L refer to scalar products. They are concerned with the energy questions related to passivity, rather than with the workless constraint questions. P5. If [w, j]eL{p) and [v, k]eL{p), and if (A) u and v are real, or if (I) j and k are real, then {u, k) = (v,j). 7.61 This is the property which provides the reciprocity law. In its presence, the relations in P3 may be weakened to V.(p) = Vz, 3 (K,o(p))", KM = K, 3 (V.o(p))'. This fact will appear as a consequence of the lemma of Section 12. 7.7* Lemma: A consequence of P2 and P3(A) is that if [v, kr]eL{p), r = 1,2, then for any ueY^ , (u, ki) = {u, ko). For by P2 w^e have that [v - V, A-i - A-.,] = [0, A-i - k.2\eL(p), hence A;i — k^eKLo . Then however, by P3(A), ueYL imphes ue(K.ij)) , so * Technical [laragraph as explained in Section 2.91. FORMAL RF.ALIZABILITV THEORY— I 237 that 0 = (//, A-i - A-,) = {u, /m) - (a, /C2). (^IvD. A dual result follows from P3(I). 7.71* The result of 7.7 above means that the scalar product (v, k) is fixed by r alone when we know that [v, k]€L{p). This means that, given reV,. , there is a unique function Fi.(p) defined for peVi, by FM = (v, k) where [v, k]eL{p). Duall.y, 'hip) = (v, k) is defined for each fixed keKi, . 7.72* (PO.) The complement of F/, is finite and (I) For each veV^ , Fy(p) is rational (A) For each keK^ , Jk{p) is rational. 7.73* (P7.) (A) Re(p) > 0 implies Re(F,(p)) > 0 (I) Re(p) > 0 implies Re(.A.(p)) > 0. VIII. THE FUNDAMENTAL REALIZABILITY THEOREM 8.0* We can now state our fundamental realizability theorem: If a linear correspondence L satisfies PI, • • • , P7, the associated 2/i-pole Nz, is physically realizable. Conversely, given a physically realizable 2/i-pole N, the associated linear correspondence satisfies PI, • • • , P7. 8.01 Actually, the postulates PI, • • • , P7 are not unique nor even en- tirely independent. Many changes may be rung on them. We indicated one above. At the expense of apparent asymmetry, the (A) or (I) por- tions, in various combinations, can be deleted or weakened. We shall not pursue this subject further at this point, but must come back to it in Section 12. 8.02 We close this Section by outlining the proof of 8.0. The details are then contained in the remainder of the paper. 8.03 The proof that PI through P7 are necessary for physical realiza- bility will be a direct one: it will be show^n that, considered individually, each network branch and each ideal transformer satisfies the postulates. * Technical paragraph as exj)hiine(i in Section 2.91. 238 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 By an application of Kron's method (described by Synge ), it will then be shown that the imposition of Kirchoff's laws preserves the postulates. This work is most efficiently performed after the full machinery of the sufficiency proofs is available, and will be done in Section 19. 8.04 The sufficiency of PI through P7 can be deduced — and we will do so — from the lemmas to be quoted below. Apart from Section 19 on necessity, the remainder of the paper is devoted to the proofs of these lemmas. 8.1* Lemma: If L is a linear correspondence satisfying PI, P2, P3, and P4, then there exists a fixed real nonsingular matrix W such that 8.11 The list L wip) of all pairsf [W-\ W'k], where [v, k]eL(p), describes a linear correspondence Lw satisfjdng PI, P2, P3, and P4. 8.12 The 2n-pole Nir (= ^lw) associated with Lw consists of (i) Some number r of open-circuited terminal pairs (Ti , Ti), ■ • • , (ii) Some number s of short-circuited terminal pairs (Tns+i , Tn-s+i), (iii) A set of w = n — r — s terminal pairs (T'r+i , Tr+i), • - • , \J- r+m ) -t r+m) • 8.13 Either m = 0, or the terminal pairs in (iii) are those of a 2??i-pole Ni which has a nonsingular impedance matrix Zi(p). This lemma, and the following, will be proved in 13.2. 8.2* Lemma: If L satisfies Po, P6, and P7, then Zi(p) is a positive real| matrix, that is, Zi(p) satisfies (i), • • • , (iv) of 1.1. 8.3* Lemma: If a 2m-pole Ni has a positive real impedance matrix, then Ni is physically realizable. This is the sufficiency half of the matrix realizability theorem 1.1. Part II will be devoted to its proof. 8.4* Lemma: If N n- is physically realizable, then N can be constructed from it by the use of ideal transformers. This is Cauer's Transformation Theorem" about which we shall say more in Section 9. * Technical paragraph as explained in Section 2.91. t W'^ and W are respectively the reciprocal and the transpose of W. X Gewertz's terminolog}-*, by now traditional. FORMAL REALIZABILITY THEORY — I 239 8.5* Tlie sufficiency half of 8.0 is now clear. By 8.2 and 8.8, Ni is physi- call.y realizable. Clearly then N w is, simply by the adjunction of the necessary open and short circuits. Finally N is by Cauer's theorem, 8.4. 8.0* We can see now how to prove the necessity of positive reality for the realizability of a positive real matrix Z(p). This is the necessit}^ half of the matrix theorem 1.1. Let Z(p) be the matrix of a realizable N. Then N has an associated linear correspondence L satisfying PI, • • • , P7, by tiie necessity half of 8.0. The pairs of L are the pairs [Z{p)k, k] generated as A; ranges over all /^-tuples. By definition, then, the pairs of L w are [W~'Z{p)k, W'k]. As k ranges over all n-tuples, the nonsingularity of W implies that W'k does also. Let U = W . Then the pairs above are the same as [UZ{p)U'k,k] as k ranges over all n-tuples. Hence Lw has the impedance matrix UZ{p)U', where U = W~ is real and nonsingular. Because Lw has an impedance matrix, r = 0 in 8.12. Now by 8.1 and 8.2, Zi{p) is positive real and the matrix UZ{p)U' of L w is just Zi{p) bordered by s rows and columns of zeros. It is then easy to see that UZ{p)U' is positive real, and finally also that Z(p) is. These last two facts will be proved formally in Section 16. IX. cauer's transformation theorem 9.0 Cauer's transformation theorem is the cornerstone of formal reali- zability theory. In one form, the theorem reads: 9.1* Let Z{p) be the impedance matrix of a physically realizable 2/i-pole N. Let U be a real, constant, nonsingular matrix. Then UZ{p)U' (1) is again the impedance matrix of a physically realizable 2n-pole, Ny . Nr can be constructed from N by the use of ideal transformers. 9.2* A superficial generalization of this theorem can be obtained at once from Cauer's proof. It asserts that if N is physically realizable and is described by the linear correspondence L, then there is a physically realizable 2n-pole N »■ , obtainable from N by the use of ideal trans- * Technical paragraph as explained in Section 2.91. 240 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 formers, which is described by the Hnear correspondence Lw whose pairs at each p are the pairs [W'v, W'k], (2) where [v, k]eL{p). We refer to Cauer for the proof. It is straightforward. 9.21 We shall use the second form (9.2) of Cauer's theorem in our reahzation process. Notice that it is in a sense a "physical" theorem, about the way one physical network is related to another. It is used in this way: we shall always solve a realizability problem by finding some network N which is easily realized, and then a W such that N „ , which is now realizable, provides a solution to the given problem. 9.22* We shall call the 2n-pole N w a Cauer equivalent of N. 9.3 Although Cauer's theorem will be applied, in a sense, only a posteriori, its effect is fundamental. For it implies that formal physical realizability is a property of matrices which is invariant under the operation (1) or a property of correspondences which is in^'ariant under (2). There is an extensive classical literature on the properties of matrices invariant under operations like that of (1), and the effect of Cauer's theorem is to make these results all available to formal realizability theory. 9.31* It is worth observing here that we are already well set up to use Cauer's theorem: Lemma: If L is a linear correspondence satisfying Pi, • • • , P7, then the correspondence Lw of 9.2 also satisfies PI, • • • , P7. Proof: Let M = L^ . Pi and P2 for M are obvious, with Tm = Tz. . By definition of M, V.u(p) = W-'Y,{p) = W-'Y, KAp) = W'KM = W'K, Yuoip) = W-'YUp) = W-'Y,o Km(p) = W'KUp) = TF'K^o where W~^S for a manifold S consists of all n-tuples W~ v, where veS. Hence in P3, yap) = v., = ir^^v. K.,(p) =K,, = W'K, for fixed manifolds Y m , Kn as defined. * Technical paragraph as explained in Section 2.91. FORMAL REALIZAHILITY THEORY — I 241 No\v if reV/,11 , thou (r, k) = 0 for every /.eK^ = (V/.o)'. Then, liowever, by direel calculalioii from l;5ection 7.0, (ir"V, w*k) = 0, wlicrc ir* is the adjoint, i.e. tran,si)()se(l conjugate matrix of IT. But because IT is real, IT* = IT'. Hence if reV^i , then (W^'v, k) = 0 for every Aeir'K^ = K.^ . Hence K.„ = (ir-'V;.,,)" = (YUp))'. By this and its dual, P3 is completed for M . The reamining postulates for M follow from those for L by the simple equality (v, k) = (W'\, W'k) already established, combined with r,,/ = r^ . 9.32 For fi.xed Z(p), the matrices (1), as U ranges over a group, form an ecjui valence class. Classical matrix theory treats of such equivalence classes. This author's predilection is to regard this theory from a geo- metrical point of view. In part this prejudice may be justified by the ease with which that slightly more general object, a linear corre- spondence, can be treated by geometrical methods. In any event we shall begin our program of proofs with a lirief introduction to the geometrical approach. X. GEOMETRICAL PRELIMINARIES 10.0* We now wish to consider V and K as complex n-dimensional linear spacesj respectively of voltage vectors v and current vectors k. The distinction here is in point of view. A vector v is regarded as an absolute geometrical object; an n-tuple [v] = [ai , • • • , a„] is regarded as a set of coordinates for the vector v, relative to some coordinate basis. Given a fixed coordinate basis, there is a one-to-one correspondence between vectors v and the n-tuples [v] which represent them in that basis, a correspondence which preserves the operations of vector algebra. * Technical paragraph as e.\plained in Section 2.91. t For a reference concerning the ideas in this section, see Halmos^ Chapters I and II. 242 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 10.01 The effect of attaching a geometric identity to vectors, rather than to M-tuples, is to make it possible to {;hoose coordinate bases freely and as convenient, without elaborate constructions or even interpretations. We can then discuss properties of n-tuples (and other objects, e.g. matrices) which are invariant under the kind of operations exemplified by (1) and (2) of Section 9 as properties of a single geometric object, rather than as properties shared by an extensive class of concrete ob- jects which are converted into each other by the group of operations. 10.1 This change in point of view need not change formally anything we have said to date; it simply erects a conceptual superstructure, or pro- vides a conceptual foundation, depending on the reader's personal attitude. We shall support this statement by going through the important ideas of Sections 4, 6, and 7 and examining their geometrical meanings or counterparts. It is convenient to consider first and at some length the notions of scalar product and complex conjugate. The geometric struc- ture will then be complete enough to permit a rapid survey of the remaining ideas. 10.11* The geometrical counterpart of the scalar product introduced in 7.0 is a numerically valued function a = a{v, k) of two vector variables. Its first argument v ranges over V and its second argument k ranges over K. The function (x{v, k) is linear in v and conjugate linear in k: (x{au -\- hv, k) = aa{u, k) -f ha{v, k), a(v, ak -jr bC) - daiv, k) + baiv, (). We denote this function (j{v, k) by the simple bracket notation {v, k). 10.12 With this scalar product, the geometry of V and K is that of a space K and the space K* = V of conjugate linear functionals over K. This is analogous to the real geometry of space and conjugate space discussed at length in Halmos^. In fact, in the introduction to Chapter III of Halmos^ the modifications introduced by the conjugate linearity of {v, k) over K are treated in detail. 10.13* Because of its importance, we quote here a paraphrase of the results covered in Halmos^, par. 12. (i) If f{v) is any numerically valued homogeneous linear function of I'cV, then there is a unique vector A-/eK such that f(v) = {v, kf) for all veY. * Technical paragraph as explained in Section 2.91. FORMAL REALIZAlJlLirV THEORY — I 243 (ii) If g(k) is any niimerically values hom()j>;(Mio()us ('f)ii,iuj>;aie-liiioar function of A'eK (i.e., if g{k) is linear in A:) tiien there is a unique ?'„eV such that for all keK. 10.2* The annihilator (Vi)" of a manifold Vi C V is, as in 7.1, the set of all A-6K such that reVi implies (v, A:) = 0. 10.21* It is shown in Ilalmos-' that, to each basis v , • • • , v" in V there exists a unique dual basis A- , • • • , A-" in K such that {V, k') = brs , (2) where 5^ is the Kronecker symbol: 5,5 = 0 if r ?^ s, 5rr = 1, 1 < ^, s < n. 10.22 If [v] = [oi , • • • , a„] (3) [A-] = [^1, ■■■ .hn] are the ?)-tuples representing v and A; relative to a pair of dual bases, then it is easily computed from (1) and (2) that {v, k) = i; aA. (4) r=l Therefore the concrete scalar product of 7.0 is indeed the geometric scalar product here considered, when we restrict our pairs of bases in V and K alwaj^s to be dual in the sense of (2). 10.23* We shall use the words "coordinate frame" or simply "frame" to denote a pair of dual bases in V and K. Any basis in V (or K) specifies a frame by the imiqueness result quoted above. 10.24 We shall henceforth deal always with coordinate frames, in fact, ultimately, real coordinate frames, rather than arbitrary pairs of bases. This means in classical language that we are considering as "geometrical ]5roperties" all properties which are preser\'ed under the group of linear transformations which leave the bilinear form (4) invariant. The properties related to physical realizability will turn out to be invariant only under the subgroup of real linear transformations pre- serving (4). * Technical paragraph as explained in Section 2.91. 244 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 10.3* Conjugation is an operation which to each vtW associates a vector V uniquely determined by v with the properties V = V, (5) {au + hv) = au + hv, where a and h are any complex numbers and f7, h their conjugates. 10.31* Given any such conjugation operation in V, and given any A;tK, define a function gkiv) by g,{v) = (^) (6) for yeV. Then gk(v) is linear in v, by (5) above and (1) of 10.11. There- fore, by 10.13, there is a unique vector keK such that Ukiv) = {v, k). (7) 10.32* Directly from (1) of 10.11 and (6) above, if./ = ah + h(, then Qjiv) = agk{v) + hgi{v). From (7), therefore (y,j) = a{v,k) + h{v,0 for all yeV. Comparing this with (1) of 10.11, we see that ] = ok + hL (8) The second item of (5) above then holds for vectors A:eK. That % = k follows easily: We have from (6) and (7), written for the vector k, that (^) = {v, k). (9) We also have, by writing (G) and (7) for vectors v and k that (y, k) = (y, k). Taking complex conjugates of these two numbers, and using v = v from (5), we have (y, k) = (^). (10) Then (9) and (10), which hold for all ytV, identify k and k. by 10.13. 10.34* We have now showed in (5), (8) and (10) that this complex conjugate satisfies the formal properties of the conjugate for n-tuples introduced in 7.2. Technical paragraph as exphiined in Section 2.91. FORMAL REALI/.AHII.ITY THIOOKY — I 245 10.35. The abstract scalar product of 10.11 turned out in the end to be no more than the concrete one of 7.0 when we restrict our attention to //-tuples derived from vectors by the use of coordinate frames. In a similar way, it is not iiard to show that there always exists a coordinate frame in which the abstract conjugation now introduced has the form of 7.2. This will be done in the Appendix (20.2). 10.36* Our need for writing out the components of vectors has now almost vanished. Henceforth we shall use subscripts to denote particular vectors, e.g. Vi , rather than components. 10.4* A vector will be called real if it is equal to its own (!onjugate. A manifold will be called real if it contains with each vector also the conjugate of that vector. V and K are then real. A basis will be called real if it is made up of real vectors, and a frame will be called real if its bases are real. Any frame in terms of which our conjugation operation takes the form of 7.2 is real by definition because its basis vectors in that frame have components which are 0 or 1. The vector 0 is real, similarly. 10.41* The basis dual to a real basis is real, for if \Vr, ks) = Ors , then by (10) of 10.3 and the hypothesis that Vr = Vr , we have (iV , ks) = 8rs = 8rs so the ks satisfy the same equations as the k. The uniqueness of the basis dual to i'l , • • • , Vr then proves that ks = ks , I < s < n. 10.42* Any vector v can be written where Vi and ro are real. Namely vi = 2 (^ + ^^■ V2 = -^.{v - v). 10.5* It is shown in Halmos'', par. 34, that if reV, AeK are represented by [i'], [k] in some coordinate frame, and by [r]i , [A']i in some other frame, then there is a nonsingular matrix [IT], which (a) depends only upon the * Technical paragraph as explained in Section 2.91. 246 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 (11) two frames, and (b) relates these n-tuples as follows: Ml = [wr'[v], [k]i = [W]*[kl It is easy to show that if [W] has real elements, so that [IF]* = [W]', then the two frames involved above are either both real, or else neither is real. Also, conversely, if both frames are real, then necessarily the [W] of (11) has real elements and [IF]* = [IF]'. 10.6* Some further important geometrical notions must be mentioned before we proceed. If Vi and V2 are disjoint linear manifolds in V — i.e. linear manifolds having in common only the single vector 0 — we write Vi e V2 for the linear manifold consisting of all vectors v = ^1 + ^2, where VieYi , i = 1, 2. The circle around the plus sign is used to denote the disjointness of Vi and V2 . It is shown in Halmos^, par. 19, that if V = Vi © V2 then (12) (13) K = K: © K2 , where Ki = (¥2)°, K2 = (Vi)" and the dimension of Kj is equal to that of Vi , i = 1, 2. We call (13) the decomposition dual to (12). We some- times write Ki = Vf to denote the Ki dual to Vi in the decomposition (13). It is shown in Halmos^, loc. cit., that there exists a basis V\ , • ■ • , Vn in V and its dual ki , ■ ofVi, ki, kr+l , Furthermore, if fi , • • • kn in K such that, if r is the dimension is a basis for Vi is a basis for V2 is a basis for Ki is a basis for K2 (14) Vn is any basis in V satisfying the first half of (14), its dual basis satisfies the second half, and dually. We shall show in the Appendix that if any one of Vi , V2 , Ki , or * Technical paragraph as explained in Section 2.91. FORMAL REALIZAIilLITY THEORY — I 247 K2 is real, then they all are, and that in this case the bases (14) can be chosen to be real. Similar considerations apply to decompositions into more summands: if V = Vi e V2 e • • • 0 v^ then K = Ki e K2 © • • • © K„ , where V* = K, = n v; = (EV; XI. GEOMETRICAL CORRESPONDENCES 11.0 With the geometr}" of V and K now in hand, we consider the geometric aspects of our network theoretic concepts. The definition in Section 4 of general 2/z-pole describes a concrete thing and stands unaltered in our geometric view. The definitions in 6.11 of the terminology typified by "N admits [v, k] at frequency p" are unchanged except that we should now explicitly indicate that we are discussing concrete n-tuples of complex numbers by placing brackets around the vector symbols, thus: [v], [k]. In other words, a 2n-pole is described by a concrete relation between /i-tuples. 11.1* All of the postulates PI, • • • , P7 are stated in a language which now has been given an absolute geometric meaning. In this meaning, PI and P2 describe a geometrical linear correspondence between vectors veW and AeK. This is the geometric counterpart of the concrete notion of a linear correspondence between n-tuples. 11.11 An impedance matrix, as in 6.3, describes a particularly tightly knit linear correspondence, namely a linear function from K to V. The geometrical counterpart is an impedance operator which for each p is by definition a linear homogeneous function which assigns to each vector A'eK a unique v = Z{p)keW. That is: an operator is afunctional relationship between vectors and as such has a geometric identity. 11.12 It is easy to prove f that, given an impedance operator Z{p), and given any coordinate bases in V and K respectively, there is a matrix [Z(p)], with elements Zrs{p), 1 < r, s < n, such that relative to these bases the coordinates k^ of a vector k and the coordinates tv of V = Z{p)k are related by (7) of 6.3. We call [Z(p)] the matrix of Z{p) * Technical paragraph as explained in Section 2.91. t Cf. Halmos'', par. 26. 248 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 relative to the given pair of bases. A strong analog of this observation is contained in the following lemma. 11.13* Lemma: (i) Let L be a geometrical linear correspondence. Fix any real coordinate frame and let [L] be the linear correspondence whose paired n-tuples are [M, M, where [v, k]eLip). (ii) Alternatively, let [L] be a (concrete) linear correspondence be- tween w-tuples. Interpret the n-tuples related by [L] as representing vectors in some real coordinate frame. Let L be the geometrical cor- respondence whose pairs, expressed as n-tuples in this frame, are those of the concrete correspondence [L]. In either case, (i) or (ii), the geometric correspondence L satisfies the geometric postulates PI , • •• , P7 if and only if the concrete corre- spondence [L] satisfies the concrete forms of these postulates. The proof of this lemma consists essentially in reading the postulates carefully. We shall not reproduce it. 11.2 Our position is now this: We have on the one hand geometrical objects, vectors v, k, operators Z{p), Yip), and geometrical correspond- ences L. On the other hand, we have concrete n-tuples [v], [k], matrices [Z{p)], [Y{p)], and linear correspondences [L]. Given any pair of bases in V and K, in particular, given any coordinate frame, each geometric object generates a corresponding concrete object which represents it relative to those bases or that frame. Conversely, given a concrete ob- ject [^], we can choose a frame in V and K and find that geometric object ^ whose coordinates in the chosen frame are given by [^]. 11.21* The concrete object, linear correspondence, defines a linear time- invariant 2n-pole by 6.21. To complete the picture, we might say that a geometrical ^^, x-esponden«.e L defines a Cauer class of 2/i-poles. 11.22* This terminology is motivated by the following observation: if [L] and [L]i are hnear correspondences representing L in two distinct real frames, then there exists a real nonsingular matrix [W] relating the M, [kML](p) and the [Ml , [k]MLUp) * Technical paragraph as explained in Section 2.91. FORMAL REALIZABILITY THEORY — I 249 by the formulas of 10.5. This means that [L] and [L]i are related like the [L] and [L „ ] of 9.2. The 2/i-pole associated with [L]i therefore is a Cauer equivalent of that associated with [L]. 11.23 The observation of 11.22, combined with (ii) of 11.13, gives an alternative proof of 9.31. This proof is deceptively free of calculation, but of course the calculations are concealed in the extensive geometrical developments of Section 10, many of which are there offered on faith. XII. THE FUNDAMENTAL LEMMA 12.0 This section is devoted to the statement, and the proof in part, of a lemma which, on the face of it, looks like an exercise in manipulating the postulates. In fact, the content of the lemma, and most of the details of its proof, are essential in what follows. To postpone them would force us into needless duplication of effort. Lemma: Let L be a geometrical linear correspondence satisfying "1, P2, P4, Po(I), P6(I), P7(I) and the following w^eak form of P3(I): P3'(I): If per, , then K,(p) = K, 3 (V,.o(p))". Then there is a frequency domain t'l C Tl , differing from Pz, by a finite set, such that L satisfies all of the postulates for peVL- The statement of the dual result is evident and will be omitted. The proof that L satisfies P3 will be given in this section. Verification of the remaining postulates will follow in paragraph 16.6. We assume that the properties of positive real (PR) functions are known. They are summarized for later use in Section 15. We make occasional advance references thereto. To the proof: 12.01 First, Kl is a real manifold and for peTL K. C (V.o(p))". (1) This, with P3'(I), gives P3(I) for L. Proof: Kl is real, as in 7.51. Consider no>v^ a pePz, c, ^tl a vcVlo(p) ; then [v, 0]eL{p). Consider any real jtK^ ; then there is a utV^ip) such that [u, j]eL(p). Now 0 and j are real. Hence by P5(I) (v, j) = {u, 0) = 0. Therefore any real jeKt has a vanishing scalar product with every reYLoip). Since Kl is real, it is spanned by real j and (1) follows. 12.1 By the dual of 7.7, if we know that [v, k]eL{p), 250 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 then the value of {v, k) is determined by k. This makes it possible to state P6(I) and P7(I) for L (we take P6(I) to include the hypothesis that Tl has a finite complement). 12.11 If keKi^ , then Jk{p) is PR. Proof: if k is real then JM = (v, k) = (v, k), (2) where, of course, [v, k]€L{p). Then however [v, k]eL{p), by P4. Hence by 12.1, (2) gives us Jk{p) = J kip). From this and P6(I), P7(I) we conclude that Jk{p) is PR for any real fceKx, . Now, given any keKi, , we have keK-L by 12.01. Then k = ki -\- ik-2 where ki and k-i are real and in K^ , since Kl is a linear manifold (see 10.42). Let [Vr , kr]eL(p), r = 1, 2. Then we have (P2) [vi + iv2 , k]eL{p). Then '^kip) = {vi , ki) + (vo , k-i) + i{vi , k'2) — i{v2 , h). Now by P5(I), {vi , k-i) = {ih , ki). Hence J,(p) = (v, , /m) + (v, , b^ (3) for any peTL . Since each summand in (3) is a PR function, it follows that Jk(p) is PR for any AeK. 12.12 Let Ki be the set of all vectors /ceKz, such that Jk(p) = 0 for every peT^ . Notice that we do not assert that Ki is a linear manifold. If A:eKi then keK^ and (3) above applies. Then (vi , ki) + (v, , k.2) = 0 and, using this and the PR property of each summand, we conclude that ki and k2 are in Ki . FORMAL REALIZABILITY THEORY — I 251 12.13 We wish now to show that Ki C K/,o(/)). Consider a real jeK^ and a real keKi . Let (4) Hp),jhL(p), Hp), k]eL(p). Then, given any real X, by P2 [u{p) -^Xiip),j -{-U]eL{p). Then, because keKi , {u + Xv,j + XA-) = ((/,./) + \{v,j) + X(w, k). Since j and k are real, by P5(I) this can be written (w + Xr, ./ + X/o) = (uj) +2\{v,j). (5) Choose any pi such that Re(pi) > 0. Then P7(I) implies that the left side of (5) has a non-negative real part at p = pi . The right side, by suitable choice of X, can have any chosen real part unless Re(Kpi),i) = 0. (6) Hence P7(I) implies (6). Now (v{p), j) is a rational function, by P6(I) applied to the other members of (5). By (6), this rational function has a vanishing real part throughout the right half p-plane. Hence it is an imaginary constant: (v(p)J) ^ iO" (7) Then iv(p),j) = (.v(p),j) = -ia. (8) But [v(p), k]eL(p), so [v(p), k]eL(p) by P4. Since also [v(p), k]eL(p), by 12.1, we have from (8) that {v(p),j) = -ia. Comparing this with (7) written for p, we have a = 0 and iv(p)J) = 0 for peT,. (9) Now v(p) was determined by (4) wherein k is real. For any /ceKi , k = ki + ik2 , where ki and /jo are real and in Ki (12.11). A correspond- ing v(p) satisfying (4) can be written v(p) = vi{p) + iViip), (10) 252 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 by P2, where [vr(p), kr]€L(p), r = 1, 2. Then (9) holds for each of Vi(p), Vi{p) and therefore also for the v{p) of (10). We have showed now that for any peFz, and any AeKi , the vip) of (4) has a vanishing scalar product with every real jeKz, . Since Kl is spanned by real j, v{p)e{}Lj)' = W,o. (11) 12.14 By (11), [v(p), 0]eL{p). Comparing this with (4), and applying P2, [v(p) - v{p), k-0] = [0, k]eL(p). Since k is now any vector in Ki, we have Kl C K^o(p) C K^ (12) for every^peTi, . 12.15 We can now also show that Vi,(p) C (Ki)°. We return to 12.13 and read (9) thereof as originally derived for real j and k. Applying P5(I), we have from (9) that (m(p), k) = 0 for peVr. . (13) By the argument immediately following (9), (13) also holds for any fceKi , provided j is real. As in 12.11 any jeK^ can be written j = jx + iji , where ji and ^2 are real, and the corresponding u{p) = ui(p) + iihip) where [Ur{p), jV]eL(p). Therefore, finally, (13) holds for any u(p) satisfying (4) — i.e., any w(?>)eV/.(p) — and any keKi . Therefore V.(p) ^ (KO" (14) for any peT^ . 12.2 We now fix our attention on a specific real Po^Tl 12.21 By P4, if [v, k]eLipo) we have also [v, k]eL{po) = L{po). In particular, Kim(po) is real. FORMAL REALIZAUILITY THEORY — I 253 12.22 Wc civn now sliow that Ki is a real linear manifold. Consider a real keKi^ilh). Tlien [0, A-]eL(/;n) and by 12.1 Jdlh) = 0. Then l)y 12.1 1 (and ir).r2), Jk{p) = 0, so A'eKi . Since K,j){pn) is spanned by real k (12.21), we have K;,,(pn) C K, . Comparing- this with (12) gives us K,o(po) = Ki . (15) Since K/,n(/)n) is a real linear manifold by definition and 12.21, we see that Ki is. 12.3 Let us now write, by (12) and (15), K, = K.> e Ki (16) where K^ is an arbitrary fixed manifold disjoint from Ki and with it spanning K^ . All three manifolds are real (12.21, (15), 10.6). Choose a K;i disjoint from K/, such that K = K:i e K. e Ki . (17) Let the decomposition of V dual to (17) be (10.6) V = V3 e Vo e Vi . Then V.3 = (Ko 0 KO" = (Kj" = V,.o by 12.01. Hence V = V;.n e V2 ® Vi . (18) By (14) and the definitions, V.0 C Ydp) C V.0 © V2 . (19) 12.31 Consider a real p„ . Then by P3'(I), (15) and (16) we have Kz.o(po) Q K,(pn) C Ko © Kz,o(po). (20) This is now an expression dual to (19). We shall prove next that, given any keKL{po)^^2{= K2), there is a unique VkeYLipojC^^i such that [v, , k]eL(p,). (21) Dually, given any veV L(pn)(^y2 , there is a unique /v,.eK/,(P())nK2 such that [r, k„]eL{po). 254 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 The proof is a standard one in algebra and depends only upon P2, (19), and (20). Proof: Given A;eKi,(po)nK2 , there is some yeVz,(po) such that [v, k]eL(po). (22) By (19), then, V = Vo -\- V2 where VqcVlo , i^2eV2 . Then [vo , 0]eL(po) SO, applying P2 to this and (22), [v - vo,k - 0] = [V2 , k]eL(po). (23) Hence i'2eVi(po)nV2 and Vk = v^ satisfies (21), Suppose now v^€Nl{v^)^^2 and [vz , k]eL{po). Then using this with (23) and P2 [V2 — Vz , 0]eL(po). Hence {v2 — Vi)eWLo . Now VL(po)nV2 is a linear manifold and contains V2 , Vs . Hence (^2 - y3)6Vz.onv^(po)nv2 = 0. Therefore v^ = Vz . The dual argument completes the proof. 12.32 The argument actually exhibited in 12.31 uses only P2 and (19), hence the Vk of (21) is unique whether or not po is real. Indeed, this is true even when AcKl . 12.33 The result of 12.31 establishes a bi-unique linear mapping between K2 and VL(7Jo)nV2 . Hence these two manifolds are of the same dimen- sion. Since K2 and Vo = K* are of the same dimension by construction, it follows that Vz,(po)nV2 = V2 and, by (19), that Vj,(7>o) = V^o e V2 . 12.4 Let us now introduce a real frame in V and K which provides real bases in Ki , K2 , K3 and in Vi , V2 , Vlo of (17) and (18). Let ki , • • • ,km FORMAL REALIZABILITY THEORY — I 255 ])(' the basis vectors spanning; Ko . By 12.32, there are iinif|uc vectors "lip), ■ ■ ■ , Umiv) i» V2 such that Let?^i , • • • , Vm be the (real) basis xcclors in Vjdual to the Ai , ■ • • , /,„, : (iV , A\) = drs 1 < '• < s. (24) Since the i(r(p) are all in V.. we hav(> for each pel\ Ws(p) = S ars{p)vr (25) r = l wIkm'c the coefficients Orsip) are calculated by (24) to l)e orsip) = Mp), a-,.). (2G) 12.41 Because the Av are real, P5(I) implies that Usrip) = (Urip), ks) = Mp), kr) = ttrsip) . (27) By the reasoning just following (8) and by the uniqueness of the Ws(p)€V2 , since V2 is real, we have Us(p) = Ws(p). Then arsip) = (Usip), kr) = (Usip), kr) = ttrsip). 12.42 We have by P2 that [urip) + \K.ip), kr + Xks]eLip), (28) for any X. The identity illr + Vs , kr + ks) - (Wr " 1h , K - A',) = 2(w, , A-.) + 2(W3 , kr) (29) holds in fact for any vectors Ur , u^ , Av , A-^ . Using (27), (28) and P6(I), it exhibits Orsip) as a rational function. 12.5 Consider the »i X m matrix [Ziip)] whose elements are the arsip). the s-th column of this matrix consists of the components of «.,(p). The rank of the matrix is by definition the dimension of the space spanned by these columns. 12.51 Now the rank of [Ziip)] can be expressed in terms of the vanish- ing or not of its various minor determinants. There are finitely many such minors and each is a rational function. Each is either identically zero or else Nanishes at only finitely many points. Hence the rank of [Ziip)], except at these finitely many points, and at the p in the comple- 256 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 ment of Tl , is a constant. We call this constant the nominal rank of 12.52 Let Tl consist of all peT,^ where [Zi(p)] has its nominal rank. Then r^, has a finite complement. By the reality result of 12.41, if peTi then peT^ ■ It is clear that at any peFt the rank of [Zi(p)] does not exceed its nominal rank. 12.53 By construction, the vectors Wi(p), • • • , Um(p) all lie in Vi:.(p)nV2 . By the reasoning of 12.33, at any real poeTL they span V2 . Hence the nominal rank of [Zi{p)] is m. Therefore, for any perl , [Zi(p)] has rank m and the Hi(p), ■ ■ • , Umip), lying in V2 , still span V2 . Therefore for all perl v^(p)nv2 = Vo . By (19), then, v,(p) = Vz.0 e Vo = V,, , (30) a fixed manifold, for all peT^. 12.54 It is clear by its construction (cf. Halmos'', par. 26) that [Zi{p)] describes the mapping of 12.32 from K2 to V2 = VL(p)nV2 by [iv] = [Z^(p)][k]. Here the m-tuples [vk] and [A] are the components of Vk and k relative to the bases now available in V2 and K2 . 12.55 We repeat Ki C K,o(p) C K, = Ki e Ko . (12) Fix a peTL and a A:€Kz.o(p)nK2 . Then [0, k]eL{p). Since 0€V2 , it follows from 12.54 that [Zi(p)] annihilates A:. Suppose m 9^ 0. Since the rank of [Zi(p)] is m, it follows that A- = 0. Hence for peFi, Kz.o(p)nK2 = 0. By (12), then, (31) K,o(p) = Ki = K^ , a fixed manifold. This, with the result of 12.53, proves that L satisfies P3(A), when m 5^ 0. If m = 0 then V. = 0, K2 = 0 and (31) follows from (12) and (16). 12.56 [Zi(p)] is of dimension m and rank m for any ptV^ . Therefore FORMAL REALIZABILITY THEORY — ^I 257 the correspondeiiee of 12.32 and 12.54 between V2 and K> is bi-uniqiie for any peF/, . This extends 12.31 to any peF;, . 12.57 If m - 0, i.e., if V, = K, = 0, then V,,, = (K,,,,)' and the fact that L satisfies all the i)()stulates is trix'ial l)eeause all scalar prtxhicts (r, A') for reV;, = Vm and A'eK/, = Klo are zcn-o. If )n ^ 0, we have yet to show that L satisfies P5(A), P(KA), P7(A). 12.6 8in('(> now L satisfies P3, 7.7 as gi\'en is ai^plicable and we find (with 12.1) that if peV',. and then {v, k) is fixed by either v or /,-. Furthermore, (V, k) = (V + To , k + /,•„) for any roeV/ji , koeKio . 12.61 If perl and [v, k]eL(p), then reV;, , A'eK;. . By (30), (31), and (16), therefore, there exist roeVz.n , koeKu< such that u = v — VoeV^ , j = (k - ko)eK.2 . Then by P2 [>',j]eUp). (32) By 12.6, theji, any value assumed by a scalar product {v, k) with [r, l:]eL{p) is also assumed by a product (n, j), where (32) holds and ;/eV, , jeK. . XIII. SUFFICIEXCY OF THE POSTULATES 13.0 We suppose that L satisfies the postulates of 12.0. Then the results of Section 12 are applicable. The ones of first importance are contained in the facts from (15), (30) and (31), that V;. = Vz.0 e V, , K, = Ko e K,o , where the choice of Ko was governed only by the requirement that the second of these formulae hold. 13.01 Considering K2 and V2 as separate spaces, V2 = K2 by 10.6. Let .1/ be the geometrical linear correspondence between them with frequency domain T,, and pairs described by 12.31 and 12.56 (or 12.54). That is, as vectors in V2 and K2 [V, k]eM^) 258 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 if and only if, as vectors in V and K, [v, k]eL{p). 13.02 In the real frame of 12.4 let us renumber the basis vectors so that Vi, • • ■ ,Vr span Vio , Vr+l , • ■ , Vr+m Span V2 , Vr+m+i , ' • • ,Vn Span Vi . Then h , - • - , kr+m span K2 , kr+m+i , • • • , kn Span KlO • We say that such a frame reduces L. 13.1 Let us now interpret the s-th components of [v] and [k] in this frame respectively as the voltage across and the current in an ideal branch j8s of a 2/2-pole N, 1 < s < w. By construction, the vectors feV/, in this frame have components ctr+m+i = • • • = fln = 0, sluce Vi , • • • , iv+m spau V/, . At the same time, the components 6r+m+i , • • • , ^n of [A;] may be chosen arbitrarily without altering the fact that [[y], [A']]e[L](p) because of 12.06. Therefore, the ideal branches jSr+m+i , • • • , /3„ can each be realized physically by a short circuit. In a dual way, since /Cr+x , • • • , /Cn span K^ , any kiK.^ has components 61 , • • • , &r all zero in our chosen frame. Furthermore, the components tti , • • • , ttr of [v\ can be chosen at will. Hence the ideal branches /3i , • • • , jSr can each be realized physically by an open circuit. Let Ni now be the 2m-pole whose ideal branches are /3r+i , • • • , ^r+m • Let the pairs [[v], [k]] admitted by Ni at each peV^ be the [[v], [k]], where [v, k]eAI(p) (13.01). The representation just found for N shows that N is physically realizable if and only if Ni is. 13.11 The matrix [Zi{p)] of 12.54 is the impedance matrix of the 2m- pole Ni . 13.12 We now show that [Zi{p)] is a positive real matrix. The displayed formulae of 12.41 show (ii) and (iii) of 1.1, and 12.42 shows (i). Now suppose that [v, k]eM{p). Then, as vectors in V and K, [v, k]eL(p) by definition of M{p). Then, however, if k is fixed Jk{p) = {v, k) FORMAL Ri: VIJZAIULITY THEORY 1 2o9 is a PR function (12.11). "Regarding v and k in Vo and K2 let Then by (1) of 7.0 m m (v, ^O = 2 S as,(p)bi+rhs+r ( = 1 s = l and this has a non-negative real part if Ke(p) > 0. This is (iv) of 1.1- 13.2 AVe can now prove the lemmas 8.1 and 8.2. Given a linear cor- respondence [L] which satisfies PI, • • • , P7 by 11.13 we can interpret [L] as the concrete correspondence representing a geometrical cor- respondence L in some chosen real frame, and L satisfies PI, • • • , P7. Then b}^ the results in 13.01-13.12 there exists a real frame in which the representative [L]i of L has the properties claimed in 8.1 and 8.2 for L w . But we saw in 11.22 that [L] and [L]i are related by a real matrix W like the L and Lw of Section 8. Q.E.D. 13.21 With the proofs of 8.1 and 8.2 we have reduced the sufficiency claimed for PI, • • • , P7 in 8.0 to the sufficiency of positive reality of [Z(p)] claimed in 1.1, by the argument outlined in 8.5. XIV. OPERATOR-VALUED FUNCTIONS OF p The next three sections are directed principally toward the proof of the matrix theorem of 1.1. They do however, contribute to 12.10 and to the necessity proof. 14.0 We continue to use the geometric language. The reader who re- gards this as unduly pedantic is free to place a concrete interpretation upon every argument, for all of the arguments are either frankly based on matrix representations or upon the three identities: 14.01 {Zj, k) = (Z*k, j) for all j, /ceK. 1402 Zk = (Zk) for all keK. 14.03 Z' = (Z)* = (Z*) 14.04 These identities are obvious for matrices using 7.0 and 7.2. Geometrically, the first and second define Z* and Z, and the third defines Z' in two ways. The equivalence of these two ways is a theorem based on (10) of 10.33. 14.05 The symbol Z will always denote an impedance (operator, matrix, scalar), and Y will always denote an admittance. An impedance oper- 260 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 ates from K to V, an admittance dually. The operators in Halmos^ are physically dimensionless, in that they operate, e.g., from V to V. This difference is scarcely noticeable. We shall regularly omit the duals to concepts or proofs given in terms of impedances. In doing so, we adopt the rule that the dual to an expression (Zk, k) is (v, Yv). 14.1 An operator is called symmetric if Z = Z'. Such operators have three useful special properties : 14.11 If Z is symmetric and ^" and A; are real, then (Zj, k) = {Zj, k) ^ iiZrkJ) = (Z'kJ) = iZkJ) by (10) of 10.33, 14.02, 14.01, 14.03, and hypothesis. 14.12 Let k = ki-\- ik^ , where ki and A:2 are real (10.42). If Z is symmetric then (Zk, k) = (Zk, , k,) + iZk2 , k,), for, by 14.11, (ZA-i , ih) = -i{Zki , k.) = -i(Zk2 , ki) = -{Z(ik,),h). (Cf. the similar identity in 12.11.) 14.13 The symmetric operator Z is completely defined by the quadratic form {Zk, k) (1) as a function of real keK. For 14.11 permits the formula (29) of 12.42 in any real frame, where Us = Zks . The matrix elements of [Z{p)] in that frame are then defined by that formula in terms of values of(l) for real k. The form (1) specifies any Z (symmetric or not) if k is allowed to range over all of K (Halmos , par. 53). 14.2 Let Z(p) now be an impedance operator depending on p. We say that po 9^ 0 implies Re(/(p)) > 0. The property (i) of being rational is of course on a quite different level of ideas from the other properties, but it saves words later to in- clude it specifically in the meaning of positive real. We abbreviate the words positive real to PR. 15.01 The open region of the complex plane consisting of all finite p such that Re(p) > 0 — the right half plane — we denote by T+ . 15.1 Brune, loc. cit., established a number of properties of PR functions f(p) which will be useful to us here: 15.11 f{p) has no poles in r+ . 15.12 If Re(/(p)) = 0 for some peT+ , then/(p) = 0 for all p. 15.13 If it exists, -~ is PR. f(p) _ _ 15.14 If /(p) has a pole at p = pn ,it has one at p = po . 15.15 If /(p) has a pole at p = icco , that pole is simple and lip) = -^^2 r + hip), where r > 0, and fi{p) is PR. * Properly, T^o is a residue only when m = 1. There is no convenient name available for general m. 262 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 15.16 If /(p) has a pole at p = «> , that pole is simple and fip) = pr + flip), where r > 0, and fi(p) is PR. 15.17 We shall use all of these in the next section, save 15.13. Our aim is to prove properties analogous to 15.11, • • • , 15.16 for PR matrices and operators. The reader familiar with the Brune process for realization of a 2-pole will remember the importance of the properties 15.11, • • • , 15.16 for the success of that process. Correspondingly, we must establish the analogs of these properties to implement the general Brune process for 2ri-poles. XVI. POSITIVE REAL OPERATORS 16.0 An operator Z{p) from K to V will be called positive real (PR) if in some real coordinate frame the matrix [Z(p)] is a PR matrix in the sense of 1.1 — that is (i) [^(p)] has rational elements Zrs(p) (ii) Zrsip) = ZrsiP) (iii) Zrsip) = Zsrip) (iv) For any real /ceK and any peT+ ReiZip)k, k) >0. We intend in this section to establish for PR operators the properties listed below. By subtracting 0.9 from the designation of each property one obtains the designation of the analogous property of a PR scalar function, stated earlier. 16.01 Zip) has no poles in r+ . 16.02 If Re(Z(p)/c, k) = 0 for some peT+ , then Zip)k = 0 for all p. 16.03 If it exists, Z~\p) = Yip) is PR. 16.04 If Zip) has a pole at p = po , it has one at p = po . 16.05 If Zip) has a pole at p = tcoo , that pole is simple* and 2p Zip) = -T-x^ R + Ziip), p -r Wo where R is real, symmetric, and semi-definite, not zero, and Ziip) is PR. * i.e., of order one. FORMAL REALIZARILITY THEORY — -I 203 KkOC) If Zip) has a polo at ?j = =o , that i^olo is simple and Zip) = ph' + Z,ip) whoro R = ir = R, h' > 0 and Z^ip) is I'K. J 0.07 There is property of rational scalar functions f{p), whether PR or not, that is essential in the Brune theory: the existence of a finite integer, the degree of /. Each step in the Brune reduction of f(p) leaves an unreduced portion which is of lower degree than the function upon which the step was performed. The finiteness of the original degree of / then guarantees the termination of the process in finitely many steps. There exists jjilso for rational matrices (and operators) a concept of degree. This degree plays the same role in the general Brune process for 2 /i -poles as the degree of a scalar function does in the process for 2-poles. To define this degree and develop its properties requires an excursion into classical algebra. Since we shall not need these ideas until Part II we defer further discussion of them to that part. 10.1 If Z(p) is PR it follows at once that the matrix [Z{p)] is PR in any real frame. Proof: Two such matrices are related by [Zip)], = [U][Z(p)][UY where U is real, by 11.22 and the argument in 8.0. The PR properties of [Z(p)] are obviously preserved by this operation. 10.11 If Z(p) is PR, then Zip) = Z'ip) = Z*(p) = Zip). Proof: Use 10.0 and 14.03 in a real frame. 10.12 If Zip) is PR, then for any given A'eK the function Juip) = iZ(p)k, k) is a PR scalar function. It follows that the limitation in (iv) of 10.0 to real k is a simplification, not a restriction. Proof: J kip) is independent of coordinate representation. By use of a real frame, (i) of 10.0 implies (i) of 15.0. Bv 14.01 and 10.11 Jkiv) = {Z*ip)k, k) = iZip)k, k) = J,ip). This is (ii) of 15.0. For any k, 14.12 and (iv) of 10.0 imply (iii) of 15.0. 264 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 16.13 Conversely to 16.12, if Z{p) is symmetric and Jk(p) is PR for every real k, then Z(p) is PR, and J kip) is PR for all k. Proof: J kip) is rational so (i) of 16.0 holds in any frame by 14.13. Clearly (iv) of 16.0 holds. Now for real k, by (10) of 10.33 and 14.02 Jkip) = Jkip) = iZip)k, k). Hence Zip) = Zip) by 14.13. This is (ii) of 16.0, and (iii) there holds by hypothesis. 16.2 Proof of 16.01: By 15.11 and 16.12, J,(p) has no poles in r+ . This is 16.01 by the definition 14.3 of pole. 16.21 Corollary: Any PR Zip) can be considered as defined throughout r+ : for any k, Jkip) is defined throughout r+ by 16.2. For each p, as a function of A', Jkip) defines Zip) (14.13). 16.3 Pz-oo/o/ 16.03: In any frame [Z"'(p)] = [Zip)]'' = [r(p)] consists of rational elements, by direct calculation of the inverse matrix. In a real frame [F(p)] = [Z'^p)] is symmetric and real for real p by the same argument (both facts are also deducible geometrically). Hence we have the duals of (i), (ii) and (iii) of 16.0 for Yip). Clearly Yip) is defined throughout r+ . Now suppose that for some veW and some po€T+ we have Re(y, Yipo)v) < 0. Then there is a keK such that v = Zipo)k. Therefore Re[Zipo)k, k) = l\eiZipo)k, k) < 0. Since this is impossible, we have the dual of (iv) of 16.0 for Yip) and Yip) is PR. 16.4 Proof of 16.04: This is immediate from 15.14, 14.3, and 16.12. 16.5 Proofs of 16.05 and 16.06: Suppose Zip) has a pole at p = io:o . Then iZip)k, k) does and that pole is simple by 15.15 and 16.12. Then by 14.3 we can write Zip) = L^ 7?o + Zoip) P — tOii) where Zoip) is regular at p = iwo . Now Zoip) has a pole at p = —iojo by 16.5, so a similar argument gives Zip) - ~ Ro + ^ . /?! + Zrip), (1) p — iwo p -\- two FORMAL REALIZAHILITY THEORY 1 205 whoi-e Ziip) has no pole at tcoo or — tcoo . The symmetry of Z and hnear iii 0 for all A-. Hence Ro — Ri ^ R (say) and R is semi-definite. Also, (Zi(p)A-, A) appears as the residue fi{p) in 15.15 and is therefore PR. Then Zi{p) is PR by 16.13. With Ro and Ri identified, (1) above is the expansion given in 16.05. We have now proved all of 16.05 save the reality of R. But p -\- OOo is PR, by 10.13, hence is real for real p. Therefore R is real. The proof of 16.06 is similar. 1 (').(■) To prove 16.02 we appear to digress somewhat, by first com- pleting the proof of the fundamental lemma of 12.0. It was established in Section 13 that the matrix [Zi{p)] describing M(p) in the chosen basis is PR. The case in which it is nonsingular (i.e., m 9^ 0, cf. 12.56, 12.57) remains to be examined. 16.61 If [Ziip)] is nonsingular then its inverse is PR (16.3). Then for any veW2 , (v, k) = (v, Y{p)v) (2) is PR (16.12 dual). By 12.01, for any ?/eVL , the values of the function F,Xp) are the values of (2) for some reV2 . Hence Fuip) is PR. This is PO(A) and P7(A) for L. 10. ()2 To settle P5 for L in 12.0, consider peT^ and [i',k]dAp), [uJ]eL{p), where u and v are real. Then, say, V = Vo + Vi , where t'oeV^o , VieV-z . But then V = V = Vo -{- h 266 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 and, because Vm and V2 are real, vo = Vo , h = Vi , and these vectors are real. Using similar reasoning for w, (vj) = ivi , YiP^i), (>', k) = {u, , Y(p)v,), (3) by 12.61. The equality {u, k) = (v, j) now follows from (3) and the duals of 16.11, 14.11. Hence we have P5(A) for L and 12.0 is proved. 16.7 We now prove an important Lemma: Let Z{p) be a PR operator from K to V. Let Tl be the set of p where Z(p) is defined and has a rank equal to its nominal rank. Let L be the correspondence with domain r^ and pairs [Z{p)k, k], keK, . Then L satisfies PI, • • • , P7. Proof: L satisfies PI and P2 (6.3). Tl satisfies P4 by the argument of 12.52. Then L satisfies P4, for by 16.11 Z{p)k = Zip)k. L satisfies P5(I) by 14.11 and 16.11. r^ satisfies P6 by 12.52. Then L satisfies P6(I) and P7(I) by 16.12. The fundamental lemma, 12.0, now proves that L satisfies all the postulates. 16.71 We call a correspondence satisfying all the postulates PR. 16.72 Proof of 16.02: Suppose Re{Z{po)k, k) = 0 for some po€T+ . Because this function of p is PR (16.12) we have Jk(p) = {Z{p)k, k) ^ 0. Hence A;eKi = Kz.o (12.12, 12.55). Hence [0, k]eL(p) for every peVL. That is Z{p)k = 0 for peTL . 16.73 Corollary: If Z{po)k = 0 for some poeT+ , then Zip)k = 0. For the hypothesis here implies that of 16.72. This is the analog of 15.12; the result of 16.02 is stronger. 16.8 An important consequence of 16.7 is the Lemma*: If Z(p) is PR and of rank m, then there exists a real coordi- nate frame in which the matrix [Z(p)] is an m X m nonsingular PR matrix [Zi(p)] bordered by zeros. Proved by Cauer^. FORMAL REALIZABILITY THEORY — I 267 Proof: Consider the PR correspondence L defined by Zij)). Then Vi,o = 0, because Z{:p)Q = 0 for every peT/, . Consider the real frame of 13.02. [Z{p)] in this frame takes any of Av+m+i ,•••,/>■» into 0 because these span K^o . Within Ko , [/^(p)] must describe the same operation as the [Zi{p)] of 12.54. Because [Z{y)] is symmetric the lemma follows. XVII. THE JUXTAPOSITION OF CORRESPONDENCES 17.0 Tliis section and the next will consider ways of constructing new correspondences from old. This will provide the basis of the necessity proof of Section 19. 17.01 It is obvious that if two physical networks are set side by side and their accessible terminals regarded as the terminals of a single larger network, that enlarged network is again a physical network. This is the gist of the present section. 17.1 Suppose that V = Vi e V2 , K = Ki e K2 , where K^ = V* and all spaces are real (10.6). Let Ei project on V along V2 (Halmos", par. 33) and E2 = I — Ei project on V2 along Vi . Then E* projects on Ki along Kj , j ^ i (Halmos^, loc. cit.). It is easily verified that Ei = Ei , Ei = Ei , from the analog of 14.02 for dimensionless operators. Considering Vi and K, as separate spaces, let Li be a geometrical linear correspondence between them with frequency domain r» , i = 1, 2. Consider the correspondence L between V and K defined by (i) The frequency domain r^ = rinr2 (ii) [v, k]eL(p) if and only if [Eiv, Eik]eLi{p), i = 1, 2. In (ii), of course, we regard EiV and Eik as elements of Vi , Kj . 17.11 L so defined is called the juxtaposition of Li and L2 . 17.2 Lemma: L is PR if and only if each of Li and L2 is PR. 17.21 Proof of ''if": It is clear that L satisfies PI and P2. Further notation is now simplified if we put Li = M, L-i = N. Consider the manifolds v,/ e v^ , v.,/0 e Va-o , k,, e k^ , k.^o e k^o, where V m C Vi is the manifold of voltages admitted by Li = M con- sidered as a correspondence between Vi and Ki , and W uo the manifold 268 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 of voltages yeVi such that [v, 0]€Li(p) for all peFi . Dual definitions for Km, K a/0 , and symmetrical ones for Vat , • • • , Katq need not be repeated. It is clear from these definitions that the four manifolds above are, in the order listed, the manifolds Vl , Vlo , Kl , Klo for L. Now, for example, (Kz,o)° = (Kmo e K^o)" = (K,,o)»n(K^o)'' by 10.6. This last manifold, in V, is (V„ 0 V.) (1 (V^ 0 Vi), byP3 for M and A^, and by 10.6. But by direct calculation (Vm 0 V2)n(v^ 0 Vi) = Vm 0 v^ = v^ . The dual of this result then completes P3 for L. P4 for L is immediate because the Ei and Ei are real. The duality of the decompositions of V and K implies the identity (v, k) = {Eyv, Etk) + {E.yv, E*k) (that is E1E2 = E2E1 = 0, and dually. This is Halmos^, par. 33). All of P5, P6, and P7 for L follow at once from this identity. 17.22 The "only if" of 17.2 is a special case of the result of Section 18. Its proof will be deferred to 18.4. 17.23 It is obvious that the notion of juxtaposition and the lemma of 17.2 extend to juxtapositions of more than two correspondences. 17.3 Even without the "only if" part of 17.2, we have enough for the following characterization of PR correspondences: Theorem: A correspondence L is PR if and only if it is the juxtaposi- tion of (i) a correspondence defined by a nonsingular PR matrix between a Vi and a Ki = Vi , (ii) a correspondence consisting of short circuits: that is of pairs [0, k] for all keK-y and all p, (iii) a correspondence consisting of open circuits: that is, of pairs [v, 0] for all f eVs and all p. Proof: If L is PR, the decomposition indicated is that of 13.1, 13.11, 13.12. If L is the juxtaposition indicated, then it is PR by 16.6 and the "if" in 17.1, provided the short and open circuits are PR correspondences. The verification of the postulates for these latter is easy and will be omitted. FORMAL REALIZABILITY THEORY — ^I 2G9 17.31 The labor of coii.sidoring PR con-espondences instead of matrices lias yielded the disappointingly simple resnlt of 17.3. We have already been warned of this, however, by our knowledge of the properties of physical networks (2.9). XVIII. THE OPERATION OF RESTRICTION 18.0 In addition to juxtaposition, which is an operation on correspond- ences clearly motivated by physical considerations, there is an operation, Ikm'c called restriction, which has important use in the next section. There the physical meaning of the operation will become clear. 18.1 Let V and K = V* be a pair of dual spaces. Let U and J = U* be another pair. Suppose that C is a given fixed linear operation from J to K: given any je], there is a unique k(j)eK, written Hj) = Cj, such that if Av = QV , r = 1,2, then Giki + aoA-o = C(aiji + Uzji) for any complex scalars ai , ao . 18.11 Let {v, k)i denote the scalar product between V and K, and (;/, j)2 that between U and J. Given C, and any veY, let us find that unique vector u{v)e\J for which (uiv),jh = (v,Cj), (1) for every je]. That such a vector u{v) exists and is unique follows from 10.13 when we notice that the right-hand side of (1) defines a function conjugate linear in J. Now for fixed j, the right-hand side of (1) is linear in i\ hence so also is the left side. That is, there is a linear operation C* from V to U such that u{v) = C*v. The following chart illustrates the situation : V K c*[ ]c u J 18.12 We suppose now that C takes real j into real k, i.e., that C is real. Then by (1) {C*v,jh = (C*v,j), = {v,C% = {v,Cj\. 270 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 By comparison with (1), we have C*v = C*v. Hence C* also takes real vectors into real vectors and is real. 18.2 Now let L be a PR correspondence between V and K. We define one, say AI, between U and J, as follows: For each peFi, , let M{p) consist of all pairs [u, j] such that u = C*v and [v, Cj]eL{p). This definition can be illustrated l^y enlarging the chart of 18.11 : c*l ]c The u's corresponding to jej can be constructed by going around through C, L and C*. This then defines a direct mapping from J to U. 18.21 We call the M defined by 18.2 a restriction of L, since its pairs are images under C* and CT^ (which is not defined over all of K) of a restricted set of pairs drawn from L. 18.22 Clearly there is a dual operation defined by an operator D from U to V. We might distinguish the operation of 18.2 by calling it a current restriction, its dual by calling it a voltage restriction, 18.23 The restriction Af of L is defined by lists M(p) which exist for any peFi, . The frequency domain of M has not yet been specified, however. 18.3 Theorem: If L is PR, then there is a frequency domain T m ior M such that M is PR. Proof: PI and P2 for M are evident at once, for any peT^ . The remainder of the proof is divided among 18.31, • • • , 18.37 below. 18.31 For P3, let Jm be all jej such that CjcKl . Then, given jcJm , for each peTi, there is a y such that [v, Cj]eL(p), whence [C\j]eM(p). FORMAL REALIZABILITY THEORY — I 271 Therefore ] m(p), the space of currents admitted by M at frequency p, coincides with the fixed J.u at each peFi . Clearly J a/ is a real linear manifold. 18.32 Consider now U.uo(p): if [u, 0]eM(p), then there is a t; such that u = C*v and [v, CO] = [v, 0]eL(p). Hence veWwip) = Vlo for each peYt . Therefore, for each pcTi, , VUp) CC*V.o. (2) Now suppose, conversely, that peT^ and t'eVio = 'Vlo(p)- Then [v, 0]eL{p). Now 0 = CO, so [v, CO]eL{p). Hence [C*v, 0]eM(p), so C*velJ Mo(p). This proves the inequality opposite to that of (2), so for peTi, U.vo(p) = C*V,o = Umo, (3) a fixed space. 18.33 Now consider (U.^o)". If ie(U,/o)", then (w,i)2 = 0 for everj^ weXJ^/o . That is, by (3), iC*v,j), = (v,Cj), = 0 for every vcVlo . Therefore Cjiiyuof = Kz, , and je] m by 18.31. That is, we have proved and, combining 18.31 with this and (3), ]^f{p) = J.U 2 (U.vo(?>))" = (U.v,o)". (4) This is the weak form P3'(I) of 12.0 for M. It is as far as we can go with P3 at the moment. 18.34 Consider P4. If for peTL we have [u, JhMip) then [v, Cj]eL(p) and u = C*v. But then [v, Cj]eL(p) and ii = C*v, by 18.12. Then however [uJ]eM(p) by definition of M. This is P4. 272 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 18.35 Consider P5 (I): if [UrJrVMiv), where jr is real, r = 1, 2, then (Wr , ^)l = {C*Vr , js)l = {Vr , Cjs)l , (5) where [tv, Cjr]eL{p). Since Cjr is real {vi , Cj-i)i = {v-i , Cji)i by P5(I) for L. This with (5) for r ^ s proves P5(I) for .1/. 18.36 Fix a jtjjif and for each peW a uip) such that Hp),j]eM{p). Then u(p) = C*v(p) and Hp), Cj]eL{p), for some v{p). Then as in (5) above (u{p),j)2 = ivip),Cj)r. P6(I) and P7(I) for L then imply that P6(I) and P7(I) hold for M, using Tl for r.„ in P6. 18.37 We now have M satisfying the hypotheses of 12.0. Therefore there is a T m such that M satisfies all the postulates. This is 18.3. 18.4 Proof of "only if" in 17.2: Suppose that L between V and K is the juxtaposition of Li between Vi and Ki , L2 between V2 and Ko . Let, say, U = Vi and J = Ki . Let C be the identity map from Ki to K: if jej = Ki , then Cj is just j considered as a vector in K. Then C is real. It is easily computed that C* is Ei . Consider the restriction M oi L based on this C. Its pairs for P€TmQ:Tl are all the pairs [u, j] such that j = E*jeKL and u = Ev, where [v,j]eL{p). (6) But then [u,j] = [Ev,E*j] and this is in Li{p) by (6) and the definition of juxtaposition. Therefore the list M(p) is contained in Li(p). Suppose that [u,j]eLi{p). We have [0, 0]eL2(p) so by P2 and the defi- FORMAL REALIZARILITY THEORY — I 273 nit ion of juxtaposition I^ut tluMi/ = PJ*j, u = Eu, and In- (k^tinition of M Tlioroforo for every peT m , Mip) = L\{p). Therefore there is a fre- (luency domain (F.v/) for Li such that Li is ni. XIX THE NECESSITY PROOF 19.0 Fortunately for this section, those parts of network tlieory which we recfuire have recently been very succinctly stated by J. L. Synge '. We shall paraphrase them here, referring the reader to the source " for details of definition. 19.01 First, we observe that in Cauer's definition'', which w(> shall repeat in detail below, an ideal transformer with m windings is a 2w-pole whose terminal pairs are the termini of the respective windings. A system of m coupled coils is a 2w-pole with similarly defined terminal pairs. 19.02 Given a 2n-pole N which is a finite passive network, let us adjoin ideal transformers as in Figure 1(b). We then draw the ideal graph of this network. Adjoin to the graph ideal generator branches 71 , • ■ • , 7„ , jr between 7V and TI , I < r < n. Let 0r be the ideal branch repre- senting the transformer winding between Tr and Tr , I < r < n. Enu- merate the remaining branches of the graph /3„+i , ■ ■ ■ , (3b . 19.03 The branch 7^ is in a mesh with (3r and no other branches. Let us call this the r-th external mesh. Any basic set of meshes must include each of these. 19.04 Let fi , • ■ • , fn be the currents in the generator branches, Ai , • • • , kb the currents in the branches I3i , ■ ■ • , ^b and Let Wi , ■ ■ ■ , Wn be the voltages across the generator branches, t'l , • • • , ^6 the currents in the /3i , ■ • • , f5b and [w] = [Wi , ■ ■ ■ , Wn , Vi , ■ ■ • , Vb], [v] = [Vi , • • • , Vb]. 19.05 Let us choose a basic set of meshes, let ji , • • • , js be the respec- tive mesh currents, and [J] = [jl, '•• , js]- 274 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Let [it] = [ui , ■■- , u,] be the s-tiiple of mesh voltages. We suppose that j'l , • • • ,jn,Ui, ■ • • ,Un refer respectively to the n external meshes. (Cf. 19.03.) 19.06 The results of Synge^" can now be stated as follows: There exists a real constant matrix [Ci] of s columns and h -\- n rows (having, in fact, elements which are +1, —1, or 0) such that for any [j] [C] = [Ci][j] (1) is a set of branch currents satisfying Kirchoff's node law, and for anj^ [iv] M = [C.]'{w] (2) is a set of mesh voltages satisfying Kirchoff's mesh law. Furthermore, given any [I] which satisfies the node law, there is a [j\ such that (1) holds. * 19.07 If we interpret the [t], [j], etc., as representations in real bases then [Ci] is real and [d]' = [Ci]*. 19.08 The matrix [Ci] has the form [Ci] = w^here [C2] is an 71 X n diagonal matrix (having diagonal elements ±1, in fact). Proof: By construction, ji , • • • , j„ are mesh currents in the external meshes. These are then equal, save for sign, to the currents ^1 , • • • , /„ in the generator branches. 19.09 By 19.08, (1), and the definitions in 19.04, [k] = [C][j], [v] = [C]'[v], and by 19.07, [C]' = [C\*. 19.1 Let us suppose that we have enumerated the branches |S„+i , • • • , /Sft in 19.02 in such a way that iS„+i , • • • , ^c are all the two poles in the graph, iSc+i , • • • , /3d are all the branches containing coils Avhich are magnetically coupled, and ^d+i , ■ ■ ■ , (3b the remaining ideal branches of ideal transformers. Let [Zd{p)] be the (d — n) X (d — n) impedance matrix relating the voltages across the branches /3„+i , • • • , /3d to the currents in them when C2 0 0 c FORMAL REALIZAHILITY THIOOKY 1 275 we consider the individual two-polos and the system of coupled coils as separate unconnected networks. Then [Z,i{p)\ is composed of a (c — n) X (c — n) diagonal mat rix in 1 he upper left field and a (d — c) X {d — c) matrix in the lower right, with zeros elsewhere. 19.11 The diagonal part of [Z,i(p)] has elements drawn from the follow- ing list : (i) /(?>) = P (ii) f{p) = 8p (iii) f(p) = \p where p, 8, \ are non-negative constants, possibly different for each branch. 19.12 It is shown in texts on electromagnetic theory that the matrix representing a system of coupled coils is of the form p[G], where [G] is a real, constant, symmetric, and semi-definite matrix. The lower right field of [Zd{p)] is then such a matrix. 19.13 It is obvious from this description that [Zd{p)] is PR. It therefore describes a PR correspondence between (d — n)-tuples of current and voltage. 19.2 We must at last consider ideal transformers in detail. Let Vi and Ki be m-dimensional spaces represented as aggregates of m-tuples. Let pi , P2 , • • • , p,n be m real numbers. Let Vt consist of all m-tuples [a] = [oi , • ■ • , am]eYi such that fll (22 (I'm Pi P2 Pm We interpret these relations as follows: (a) If any pr = 0, then a^ = 0 (b) If any two pr , Ps are not zero, then ttr _ tts Pr p. (c) If only one pr ^ 0, then Ur is arbitrary. Let Kr consist of all 7M-tuples [b] = [bi , • • • , 6,„]eKi such that pJh + P262 + • • • + Prnbm = 0. Vr and Kr are linear manifolds. 276 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Let [Lt] be the concrete linear correspondence defined by the list \Lt]{p) which consists for each complex p of all pairs [[a], [b]] where [aleVr , [6]eKr . The correspondence described by [Lt] is what Cauer defines as an ideal transformer. He shows, loc. cit., how it can be defined as the limiting case of a physical transformer. There is also a dual kind of device, described by a correspondence admitting all [6]eKi for which ^ = ^- = . . . = ^ Xl X2 Xm and all [a]eVi for which Xiai + • • • + X,„a,„ = 0. This also is an ideal transformer obtainable as a limiting case of a physical one. 19.21 The correspondence Lt is PR. Proof: We observe that Vt = (K^)", for let [aleVr , [^jeKr , and let t be the common value of the Ur/pr ■ Then (a, 6) = Sa,6, = ^2p,6, = ^(2pA) = 0. The postulates are now all easily proved. We omit the details. 19.3 Let V and K be 5-dimensional spaces. We interpret the 6-tuples [v] and [A:] of 19.04 as representing vectors yeV, keK in a real frame. Let L be the correspondence between V and K formed by juxtaposing (i) the correspondence described by [Zdip)] relating components with indices in the range n + 1 to d, (ii) the several correspondences described by ideal transformers, relating components with indices in the ranges 1 to n and rf + 1 to b. L is PR because it is the juxtaposition of PR correspondences. 19.31 Let U and J be s — n-dimensional spaces. We interpret the [(/] and [j] of 19.04 as representing luV, je] in a real frame. 19.32 Let C be the operation from J to K whose matrix in oiu' chosen frames is [C]. Then C* operates from V to U with the matrix [C]* = [Cy. By these definitions, C is real. Let M be the correspond- ence between U and J obtained by restricting L with C. Then there is a frequency domain T m such that M is PR (18.3). 19.4 By 19.09, [M] in our chosen frame is the correspondence estab- lished between mesh currents and mesh voltages by the network of the FORMAL UKAI.IZAUILITV THlOOltY 1 277 2n-pole N. When this network operates as a 2/t-pole, the only mesh voltages which are not zero are tiiose relating to the external meshes, since there are no internal sources of x'ollage. We must now account for this. 19.41 Let Y-> , K2 be /i-dimensional spaces. Choose a real frame and let 1) be the operation which takes [ai, ■ ■ ■ , dnUV' (3) into [n,, ■■■ , «,. ,(),•••, OleU (4) in the frame of 19.31. Then /) is real and J)* in the chosen frames takes [bi , • • • , hhJ (o) into [hi, ■■■ , K]eK, . (6) 19.42 We interpret the n-tuples (3) and (6) as voltages and currents in the external meshes of N. Their relations to (4) and (5) are con- sistent with this interpretation. Let us restrict M by D, to get a correspondence Mi between V2 and Ko . In our chosen frame, the passage to [Mi] corresponds, by (3) and (4) of 19.41, to considering mesh voltages in N which vanish for every internal mesh, and, correspondingly letting the mesh currents adjust themselves to this situation. We of course observe only the external mesh currents (6). 19.43 M was PR. So, therefore is il/i (18.3 dual). Since [Mi] is the correspondence established by the physically realizable 2n-pole N, the necessity of PI, • ■ • , P7 for formal realizability is established. XX. APPENDIX TO PART I 20.0 We must prove 7.22 and those assertions of 10.0 which are not covered in Halmos'. These concern reality. 20.1 Let Vi be a real manifold and V = Vi e V2 , K = Ki e Ko where Ki = (V.)", etc. The basis (14) of lO.G exists by Halmos^ par. 19. We show that it can be chosen to be real. We have linearity independent vectors Vl , • • • , Vr , Vr + \ , • • • , Vn , 278 THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1952 where the first r span Vi , the last n — r, Yo. Let Vs = Ifs + iWs , 1 < S < 71, where u, , uh are real (10.42). Since Vi is real and a linear manifold, ^is = K^s + Vs)eYi , 1 < s < /•, and, similarlj^, WseVi , I < s < r. Among the 2/i real vectors the first 2r are in Vi , and they span Vi because the i'^ , 1 < s < r, can be constructed from them. The whole list (1) spans V, because from it all the Vs , I < s < n, can be constinicted. Since the VseV-i do not use in then- construction any of the first 2r vectors (1), it follows that the last 2(n — r) vectors in that hst must contain a set spanning V2 . The realit}' of the vectors (1) then establishes the existence of a real basis, say, v[ , ■ • • ,Vr , Vr+l , • • • ,Vn (2) which provides a basis in Vi and V2 . 20.11 We now have 7.22. The unique dual basis h' ... h' to (2) is real by 10.41. Hence all of Vi , V2 , Ki , K2 are real. The proof of 10.6 is then complete. 20.2 If in a real basis (2) (dropping primes) V = ail'i + a2l'2 + • • • + dnVn , that is, if [v] = [ai , • • • , Gn], then by (5) of 10.3 V = diVi + • • • + dnVn , hence [v] = [ai , • • • , Qn]. The geometrical conjugation of 10.3 is therefore simply the concrete one of 7.2 in any real basis. This proves the remark of 10.35. FORMAL REAIIZAHILITY THEORY — I 279 BIBLIOGRAPHY 1. M. Bayard, "Synthese des R^seaux Passifs a un Nonibre Quelconque de Paires di> IV)rnes Connaissant Lcurs Matrices d'Impedance ou d'Admit- taiu'P," Bulletin, Societe Frnncaisc des Electriciens, 9, 6 series, Sept. 1949. 2. (). Hruiie, Jour. Malh. and Phys., M.I.T., 10, Oct. 1931, pp. 191-235. 3. W. Cauer, Kin Rcaktanztheoron, Sitzunysberichte Preuss. Akad. Wissenschafl, Heft 30 32, 1931. 4. W. Cauer, "Die Verwirkliohuiig voii Wechselstromswiderstiindeii vorge- schriebener Frequenzalthanjii^keit," Archiv fur Elektrolechnik, 17, 1926. 5. W. Cauer, "Ideale Traiisfdrinatoren und Lineare Transformationen," Elek- trische Xachrichten-Technik, 9, May, 1932. 6. S. Darlinjitou, Journal of Mathematics and Physics, M.I.T. 18, No. 4, Sept. j)p. 257-353. 7. R. M. Foster, Bell System Tech. J., April, 1924, pp. 259-267. S. C. M. Gewertz, Xetwork Synthesis, Baltimore, 1933. 9. P. R. Halmos, Finite Dimensional Vector Spaces, Princeton, 1942. 10. Y. ()ono, "Synthesis of a Finite 2n-Terminal Network by a Group of Net- works Each of Which Contain.s Only One Ohmic Resistance," Jour. Inst. Elec. Comm. Eng. of Japan, March, 1946. Reprinted in English in the Jour. Math, and Phys., M.I.T., 29, Apr., 1950. 11. Y. Oono, "Synthesis of a Finite 2n-Terminal Network as the Extension of Brune's Theory of Two-Terminal Network Synthesis," Jour. Inst. Elec. Comm. Eng. of Japan, Aug., 1948. 12. J. L. Svnse, "The Fundamental Theorem of Electrical Networks," Quarterly of Applied Mathematics, 9, No. 2, July, 1951. 13. R. Bott, and R. J. Duffin, "Impedance Synthesis Without the Use of Trans- formers," Jour. Appl. Phys., 20, Aug., 1949, p. 816. 14. II. W. Bode, Network Analysis and Feedback Amplifier Design, New York, 1945. An Application of Boolean Algebra to Switching Circuit Design BY ROBERT E. STAEHLER (Manuscript received January 10, 1952) This paper discusses the application of switching (Boolean) algebra to the development of an all-relay dial pulse counting and translating circuit em- ploying the minimum number of relays. An attempt is made to outline what appears to be the most promising method of obtaining beneficial residts from, the use of the algebra in the design of practical switching circuits. INTRODUCTION The demands made upon telephone switching systems in regard to im- provements in handhng capacity, speed, flexibihty and economy are con- tinually increasing. In order to meet design objectives enabling the fulfillment of these demands, switching circuits have of necessitj^ become more and more complex and intricate. As certain types of relay switch- ing circuits increase in complexity, the problem of control and output contact network design becomes more and more laborious and time con- suming. This is especially true in those circuits in which an attempt has been made to achieve the ultimate in efficiency and economy in that the number of relays used therein approaches the absolute minimum neces- sary to provide the required number of distinct output combinations. In this type of near-minimum combinational or sequential relay circuit there are numerous parallel control and output contact paths which thread through the same relays repeatedly, thereby causing the indi- vidual relay contact loads to become relatively large. Thus the designer's problem becomes that of first de^'eloping a workable control and output contact network and then manipulating and minimizing contacts within that network so that the maximum number of contacts used on any one relay is within that permissible on any commercially available relay having the necessary speed characteristics. Even in those combinational and sequential relay circuits which ai'e not near-minimum and therefore probably have fairly light individual relay contact loads, there are, of course, advantages to be gained by using the least number of contacts possible. Although the initial cost per additional contact (assuming that a few added contacts per relay will not impair the relay speed or space characteristics to an extent that the circuit recjuirements are not met) is almost negligible, there are other 280 BOOLEAN ALGEBRA AXD CIKCUIT DESIGN 281 (M'ononiie stn'ings possible. Since each contact miust he connected to the remainder of the contact network, minimizing contacts and conseciuently soldered connections means a sa\in,ii; in wiring time and labor. Further- more, if the designer will manipulate the contacts so that the relays can be chosen from a comparatiA'cIy few standardized codes, which are in large demand, it is possibk^ to a\()i(l the expensive stockpiling of mmier- ous special designs having only a limited demand. In addition, using the least numl)er of contacts minimizes the focal points of most relay cir- cuit failures which are the contacts themselves (i.e., dirty or worn con- tacts). It might also be noted at this point that electronic combinational or secjuential circuits usually require electronic gatmg networks to perform functions which are completely analogous to those of relay contact net- works. Hence, the same problem of minimization exists. However, in electronic circuits, gate minimization is even more advantageous since the cost per additional electronic gate is much higher than the cost per additional relay contact. It is rather obvious that the multiplicity of paths in most combina- tional and seciuential circuits can cause their design to become an ex- tremely difficult and time consuming problem if the contact paths are developed with the aforementioned considerations in mind. The circuit designer's usual approach to the solution of such contact minimization and manipulation problems is that of inspection. The method of inspection presupposes a background of considerable experi- ence in that the designer must recognize certain contact network arrange- ments that may allow further rearrangements and thereby he must mentally develop his ow^i rules. In order to check on any of his manipu- lations he must repeatedly redraw the network during this inspection design process. It is evident that this is often a long and tedious method and, depending on the skill of the designer, may or may not result in an optimum or even adequate solution. Suitable contact network arrangements often appear only after con- sideration of several alternative schemes and the rearrangements of the network interconnections of these schemes. Realization of this m kes it ciuite evident that any means of obtaining and comparing these various schemes quickly and with a mathematical accuracy which does not require continuous checking of network paths permits a more rapid and complete exploration of the particular problem. Switching algebra, first codified by C. E. Shannon', is the systematic application of G. Boole's^ 1 C E. Shannon, A Symbolic Analijsis of Relay and Suntching Circuits, Trans. AIEK, 57, 1938. ^- G. Boole, The Malhetnalical Analysis of Logic (Cambridge 1847) and An In- vestigation of the Laws of Thought (London 1854). 282 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 "Algebra of Logic" to switching circuits and is just such a means. It is a too] which can be used to investigate the complex combinational and sequential networks to determine satisfactory contact arrangements or reject unsatisfactory ones with a minimum of time and effort. It should be emphasized, however, that as with any tool, satisfactory results de- pend upon the judgment, ingenuity and logical reasoning of the user. Furthermore, as A\dll be evident from the following development, switch- ing (Boolean) algebra in its present state is not to be considered entirely selfsufhcient but, for the most beneficial results, should be applied, when warranted, in conjunction with inspection techniques so that the latter may fill in any limitations in the algebra techniques which have not been completely systematized as yet due to the newness of this field. The problem of solving the contact requirements of a minimum relay dial pulse counting and translating circuit recently developed as a com- ponent of the originating register of the No. 5 Crossbar System will be used as a means of illustrating the practical use now" being made of switching algebra and of indicating exactly where the application of the algebra enters the design problem. BASIC DIAL PULSE COUNTER REQUIREMENTS The primary function of the originating register is to receive pulse signals representing digits from a telephone dial or similar calling device and to store a record of the digits in a form suitable for use by an external circuit. The dial pulse counting and translating circuit, an integral part of the originating register, is oriented with respect to other parts of the register by the block diagram of Fig. 1. The L relay is the pulse detecting relay. When the subscriber's switchhook contact is closed due to the lifting of the phone, the originating register is connected to the line and the L relay is operated. Thereafter it follows the breaks and makes of the subscriber's dial and feeds these repeated dial pulses into the counter. After the pulses are counted they are translated to a new code. In switch- ing systems it is advantageous to translate from the basic dial ten pulse decimal code to a "two out of five" self-checking code. In this latter code any single error within the circuit will result in either one or three relays operated in the associated storage circuit rather than two and thus an error can readily be detected. The output of the translator is fed via a steering circuit to the register or storage circuit. The slow release RA relay is the pulse train detecting relay which holds between the indi- vidual pulses of a digit and releases only at the end of the pulse train. When it releases it activates the translating circuit and thereby transfers the translated code information to the storage circuit. The RAl relay BOOLEAN ALGEBRA AND CIR' UIT Di^SIGN 283 ill operating torminatos tho output from the translator and simultane- ously releases the relays in the counter to prepare it for the next digit. Speeifte requirements imposed by the originating register circuit neces- sitate the counting of one to eleven pulses; the use of a driving source consisting of a single break-make (or transfer) contact with ground on the ai-mature spring; and outputs as follows: 1. Count of 1 tiirough 10: ground on two of the 0, 1, 2, 4, 7 output leads ill the combination corresponding to the count. 2. Count of 10: ground on the ZO lead. 3. Count of 1 1 : ground on the 0 lead only (this is a trouble-detecting feature"). In adchtion, the design of the steering and register-storage circuit reciuires that no output leads be connected together until the second pulse is received. Furthermore, each relay is limited to a combination of simple make and break contacts not exceeding a total of twelve. This uti- lizes the maximum number of springs obtainable on presently available rela3\s and also avoids the larger armatiu'e gaps imposed by transfers which would result in a reduction in the relay speed of operation. Speed requirements also do not permit the use of shunt release in the circuit operation. COUNTER RELAY COILS AND ASSOCIATED CONTACTS FOR PRODUCING REQUIRED OPERATING SEQUENCE TRANSLATOR CONTACTS ON COUNTER RELAYS FOR TRANSLATING RELAY SETTINGS TO DESIRED OUTPUT CODE FORM r' r' r r r' - VIA STEERING CIRCUIT TO 4 > REGISTER OR STORAGE CIRCUIT 'b_ Fig. 1 — The schematic of a portion of a dial pulse register circuit for counting decimal code pulses and translating them to "two out of five" signals. (In the symbolism used in the illustrations a cross indicates a "make" contact and a vertical bar indicates a "break" contact.) 284 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 THEORY OF A MINIMUM RELAY COUNTER The counting circuit under consideration does not contemplate the use of any circuit elements other than relays that react to the beginning or end of a pulse. Therefore it must establish a distinct combination of relays operated or released during and between successive pulses. The minimum number of ordinary "two-position" relays, R, reciuired to count P pulses can be obtained from the expressions (1) 2P < 2^^ if the counter is to lock up during, or recycle after, the last pulse or (2) 2P < 2'^ — 1 if the counter is to lock up after the last pulse. The usual counting circuit used for determining the number of pulses in a dial train is required to count ten pulses, however there are certain advantages in regard to trouble indications if the counter counts eleven pulses. In either case the minimum number of relays necessary, accord- ing to the preceding formulae, is five. It should be noted that the ease with which this minimum number can be attained depends upon whether the input is derived from a single, double or transfer contact source. DETERMINATION OF OPERATING SEQUENCE Having determined that the minimum number of relays necessary is five, the first step in design is to develop an operating sequence pattern from the resulting 2^ or 32 possible relay combinations. These combina- tions may be utilized in any order deemed desirable to obtain the 23 dis- tinct combinations needed to differentiate between eleven pulses (22 for the eleven makes and breaks plus an all-relays-normal combination). In this phase of the design switching algebra is not involved. The optimum sequence to meet a particular set of requirements can only be determined by repeated trials guided by an intimate knowledge of objectives. Initial studies, made by Joseph Michal, of various possible sequence patterns for a five relay circuit, including those having a three relay "ring" followed by tw^o auxiliary relays and those having a two relay pulse divider followed by three auxiliary relays, resulted in the conclu- tion that the latter approach was the most fruitful. The secjuence pat- tern adopted is shown in detail in Table I. The pattern is extended through 12 pulses, and it can be seen that the nature of the sequence is such that this employs all 32 combinations of the 5 relays. Several of these are transient and occur during part of a pulse or inter-pulse inter- val. Examination of the tail end of the sequence indicates that it will be simpler to design on the basis of a full 12 pulses than attempt to block at the end of the 11 pulses specified by the I'cquirements. If trouble con- BOOLEAN ALGEBRA AND CIRCUIT DESIGN 285 Table I Sequence of Operation Pulsing Relay L Counting Relays Relay Combination Two out of A B c D E Five Code Seizure 0 1 1 1 1 1 1 1st i)ulse 1 0 0 0 1 0 1 1 2 3 0,1 2nd pulse 1 1 0 1 1 1 0 0 1 1 0 0 4 5 6 0,2 3ril pulse 1 0 0 0 0 0 1 0 0 0 0 0 0 7 8 9 1,2 4th i)ulse 1 0 1 1 0 1 0 0 0 0 10 11 0,4 5th pulse 1 1 0 0 0 0 1 1 0 0 0 0 0 12 13 14 1,4 6th i)ulse 1 0 0 1 1 1 0 1 1 0 0 0 0 15 16 17 2,4 7th pulse 1 0 0 0 1 0 0 0 0 0 18 19 0,7 8th pulse 1 1 0 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 20 21 22 1,7 9th pulse 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 23 24 25 2,7 10th pulse 1 0 1 1 0 1 0 0 0 0 26 27 4, 7-ZO 11th pulse 1 1 0 0 0 0 1 1 0 0 1 1 0 0 0 28 29 30 0 12th pulse 1 0 1 1 0 1 1 1 0 0 31 32 0 Total of 2^ = 32 combinations used. Note: 0 is used to indicate that the relay listed at the head of the column is operated, and 1 is used to indicate that the relay is released. ditions introduce pulses beyond 12, the circuit will without difficulty recycle through combinations corresponding to pulses 11 and 12. Table I also indicates the leads which must be grounded in order to provide the translations to the "two out of five" and "single lead" codes. The characteristics of this circuit may be summarized as follows: It contains only five relays which is the absolute minimum necessary. It 286 THE BELL SYSTEM TECHMCAL JOURNAL, MARCH 1952 uses all 32 of its available combinations. Its control and translating job is complex enough to indicate the need for a considerable nimi})er of con- tacts and hence the need for extensive contact manipulation to minimize and distribute these contacts. It is apparent that a great deal of time would be necessary to accom- plish this manipulation by inspection methods, therel^y indicating the need for an additional tool such as switching algebra to assist the de- signers in this task. ALGEBRAIC METHODS APPLIED TO CONTROL CIRCUIT The seciuence of operations of Table I is used as the starting point in the application of the algebra. The exact calculations necessarj^ to develop the control and translating circuit by this means are shown in detail later. However, the indi\'idual steps in the solution might well be outlined here. First, the design of the control and translating networks will be regarded as separate problems. In theory these can be integrated together, but the resultant network is likely to be so complex that under- standing and maintenance of the circuit would suffer. Each of the two netw^orks can be individually considered as a multi-terminal network of the single input type. That is, the control network is an associated set of contacts which connects a single ground input to the ^^dndings of five relays, and the translating network is an associated set of contacts w'hich connects a single ground input to the six output leads. Since switching algebra is directly applicable to two-terminal networks rather than multi- terminal networks, the approach to this particular problem is of neces- sity somewhat indirect. The most satisfactory method of attack is to develop first a two- terminal network for each of the output paths of the multi-terminal network under consideration. The two-terminal networks can be ex- pressed algebraically and manipulated into their simplest form by means of the switching algebra theorems to be given later. The individual net- works can then be inspected carefully, either in algebraic or circuit form, with the objective of combining them in the most advantageous fashion. It will be found, in general, that the simplest network configurations do not readily combine and that further manipulation is necessary to obtain an economical circuit. It is at this point that the algebra achieves its greatest utility, since its application permits the simple and rapid chang- ing of a given two-terminal network into a large variety of different forms with mathematical assurance that circuit eciuivalence is main- tained. Inspection of the networks in the several forms provides clues BOOLEAN ALGKHHA AND C'lllCUIT DESIGN 287 as to tho profei'able combiiiiiijj; forms and oftcMi iiulicajcs additional maiiii)ulations tliat ini^lit l)(> desii'ahlo. Hiis network (l(>\(>Iopin('nt is a conihinat ion of mathematics and into- jii'ation l)y inspection. It is cliai'actcrizcd by repeated trials of altei'nati\-e foi'nis and at no sta^e is there any definite assurance* that the optimum ciicuit lias h(HMi attaiiUMJ. Ilo\ve\-er, tiie ease of manipulation proxided t)y the ali>;el)ra f>reatly eiihanc(\s the pi-ohahility of desi<;iiinrly connect outputs together. The alg(>bra usually offers means of introducing one or two additional contacts which permit combining networks and y(*t eliminate the adx'erse effects of the sneak paths. The abox'c pi-ocedui'e will now be cai'ried out in detail with the switch- ing algebra theorems that ai'e used in all the following algebraic manipu- lations noted at the margin by the number which corresponds to the numl)er of the theorem in the complete listing in Table II. This table is Table II Switching (Boolean) Algebra* Definitions Postulates Addition (+) = ,LY/) = Series Multiplication ^.) = OR = Parallel Circuit States 0 = Closed Circuit 1 = Open Circuit (1) X = 0 or X = 1, where X is a con- tact or a network. (2a) 0-0=0 (21)) 1 + 1 = 1 (3a) 1-1 = 1 (3b) 0 + 0 = 0 (4a) 1-0 = 0-1 = 0 (4b) 0 + 1 = 1+0=1 Theorems (la) X + )' = r + X (11)) X)' = 1-x (2a) X + Y + Z = (X + r) + z = X + {Y + Z) (21) ) XYZ = (XY)Z = X(l'Z) (3a) XI- + XZ = X(Y + Z) (3b) (X + }-)(X + Z) = X + YZ (4a) X + X = X (4b) XX = X (5a) X + XI' = X (5b) X(X + )■) = X (6a) (X)' = X' (61)) (A'')' = X (7a) (X + }• + Z+ ••• ■)' = X'Y'Z'- --- (7b) (x-r-z- •••)' = X' + F' + Z' + ••• (8a) X' + X = 1 (8b) X'X = 0 (9a) 0 + X = X (9b) 1-X = X (10a) 1 + X = 1 (10b) 0-X = 0 (11a) (X + }'')}' = XY (lib) XY' + Y = X + Y (12a) (X + F)(X' + Z){Y + Z) = (X + Y){X' + Z) (12b) XZ + X'Y + YZ = XZ + X'}' (13) (X + F)(X' + Z) = XZ + X'l' (14a) /(X) = A-/(X)a = i.a' = o + A'-/(X)a=o. a' = i (14b) /(X) = [A +/(X)a.o.a' = .] [A' +/(X)a=,.a' = o] * Reprinted from The Design of Switching Circuits by Keister, Ritchie and Washburn with the permission of D. Van Nostrand Co., Inc. 288 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 taken from The Design of Switching Circuits by Keister, Ritchie and Washburn*. The development of the algebraic expressions from the sequence of operations table will be in exact parallel to the methods suggested in the aforementioned text. . The symbolism adopted in the following development is l)asically that oi using the notation A for all the make contacts on the A relay, and A' for all the break contacts on the .4 relay. Contacts or groups of contacts in series are related by the symbol of addition (+) and contacts or groups of contacts in parallel are related by the symbol for multiplication (•) which may or may not be explicitly written, as in ordinary algebra. Therefore (.4 + B') symbolizes a series contact path that is closed when the A relay is operated and the B relay is released, while {AB') symbolizes the parallel contact path that is closed when either A is operated or B is released. S\^dtching algebra includes only two numerical values, 0 and 1, with the quantity 0 assigned to represent a closed path and 1 to represent an open path. For the tabular notation of Table I, 0 is used to indicate that the relay listed at the head of the column is oper- ated and 1 is used to indicate that the relay is released. As stated earlier, the present application of switching algebra utilizes the sequence of operation chart of Table I. The operate and release combinations for controlling the A, B, C, D and E relays can be selected from this table by observing where each relay to be controlled changes state. For example, the operate combination for relay D is relay combi- nation 8 and the release combination for relay D is relay combination 24. It is not necessary to include the contacts of a relay in its own operate and release combinations. Note that the A and B relays which serve as a pulse divider can be controlled solely by the L relay and contacts on A and B without reference to C, D, E. However the C, D and E relays are internally controlled by all five counting relays. The development of all these control paths uses the following abbreviations: g{X) = operating combinations for the X relay r(X) = releasing combinations for the X relay h(X) = holding combinations for the X relay X = make contact on the A' relay Furthermore as expressed by theorem (6a and 6b) the negative of a contact network X is defined as a network which is a closed path under all conditions for which X is open, and is open under those conditions * D. Van Nostrand, 1951. The Bell Telephone Laboratories Series. BOOLEAN ALGEBRA AND CIRCUIT DESIGN 289 for Avhich .Y is closed. Hence h(X) may be o])taiiie(l from r{X) by iiotiiifi; I hat li(X) is the negative of r(X). Thei'efore the entire control i)ath of any relay can be expressed geiuMally as f(X) = ' + E')(A' + B + D + E) r(C) = [(A + 5' + L> + E')iA + B' + D' + E)] h{C) = [(A + 5' + /) + E'){A + 5' + £>' + E)]' = (A'BD'E + A'BDE') (7a, 7b) and /(C) = [(.4' + 5 + £»' + ^OC^-i' + 5 + L> + j^)] [C + A'BD'E + A'5Z)£'] = [A' + B + (Z)' + ^')(£> + £")] [C + A'B{D' + 7?')(/) + E)] (3a, 3b, 13) = [.4' + 5 + (i)' + ^')(^ + ^^)][^' + ^^'B] [C + (/)' + E'){D + £")] (3b) = [{A' + 5)C + {D' + £')(/>> + i^)]K' + -i'5] (3b) The schematic circuit which the above represents is shown in Fig. 3a. Circuits of this type which use certain contacts more than once can some- ■i ^ 1 ) ^ .V 1 db -X — X- L B -X- (a) : — -v 1 L -X X- (b) 'b_ (A) ADDED TO COVER TRANSFER TIME OF (L) RELAY CONTACTS (B) ■5VFR (- J I'Hl HllhL (c) Fig. 2 — Pulse divider of counting circuit. BOOLEAN ALGEBRA AND CIRCUIT DE.SIGN 291 times bo drawn in bridge form with a consequent saving of contacts. One method is to manipulate the expression into a form whicli is known to be the series-parallel {M[ui\alent of a bridge. However, following usual algebraic procedures it is often difficult to recognize where this is possible. In the present case a method developed by (J. K. Frost (not yet published) was used effectively. This resulted in the l)ridge circuit of Fig. 3b which has the series-parallel eciuivalent : /((') = [(' + {D' -f E'){D + E)][A' + 7? + (L>' + E'){1) + K)] [C + A']{C + B] By use of theorem (3b) this is seen to be ecjuivalent to the pi-evious expression for /(('). For the D relay which only operates and releases once in the entire cycle g{D) = (.1 + R + C + E') r{D) = {A + B + C + E) h{D) = {A -^ B + C + E)' = A'B'C'E' and /(D) = (A + 5 + C + E'){D -f A'B'C'E'). A B D E J~ X" J~ [a) Q D + E- r (b) 'i (c) rtn, b. (D) '^ J- a (d) D-- E-- x-^ J C ABE ; — i— X * X— X— (- (E) HI tL (D) ^"^ X" cHll^ (e) !-" 'i Fig. 3 — Internal control of counting circuit. 292 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 By noting that A + B -\- C is the negative of A'B'C, this can be reduced to f{D) = (.4 + 5 + C + E')(D + A'B'C) (3b, 12a) Howevev, this transformation introduces a hazard caused by the transit time of .4 relaj^ contacts in passing from relay combination 9 to 10. Therefore the original expression will be used for relay D. The control path is sho^^Tl in Fig. 3c. For the E relay which operates and locks only once in the cycle giE) = (A' + B' + C' + D) h(E) = E and f(E) = (A' -\- B' -\- C -\- D)(E + E) (A' -{- B' + C + D)E (4a) This control path is shown in Fig. 3d. Apart from the problem of developing the rec}uired contact network, the practical problem of what operating power must be given to the relays in order to meet speed requirements must be dealt with. Since the use of low resistance windings in series with protective external resistors is called for to obtain the speed required, it appears that the use of two windings per relay might prove advantageous. By operating on the low resistance winding while locking on the high resistance wind- ing, the current drain may be reduced (thereby saving a fuse) and furthermore some code reduction may be made possible as showni later. If double windings are used, two of the external series resistors may be eliminated by combining the control network so as to make certain that only one of the low resistance windings on the C, D, or E relays is energized at any one time. This W'ould permit the use of one common external resistance with the aforementioned relays instead of three. Keeping these practical considerations in mind, further savings may be made by combining the control circuits as shown in Fig. 3e. Although there is in this circuit a possibility of contact stagger on the A relay contacts causing the C and D low resistance windings to be energized at the same time, this will not be harmful since, when the stagger occurs, both relays are firmh^ locked operated by their high resistance holding windings. TRANSLATING CIRCUIT The translating circuit is particularly adaptable to switcliing algebra manipulation. Table I show^s the combinations which prevail at the end BOOLEAN ALGEBRA AND CIRCUIT DESIGN 293 of each piilso and the necessary "code" leads that must be si'oinuled at th(>se times. Reference to the block diagram of Fig. 1 shows that the output of the translator is not activated until the slow release RA relay- releases after the last pulse of a digit has been received. Therefore the A relay can be eliminated from these combinations since at the end of every pulse the .4 and B relays are either both operated or both released and hence only one is needed to indicate the condition of both. Table ITT lists the munerous combinations which must close a ground path through to each of the five code leads and the ZO lead. At the conclusion of the algel)raic manipulation, the A and B contacts may be redistributed e\'cnly since they perform interchangeable functions in translation. The objective in the design of the translating circuit is to obtain the most economical contact network subject to a spring distribution that Table III Translation Output Lead Counting Relays Decimal Pulse Grounded B c D E 0 0 1 1 1 1 1 0 1 1 2 1 0 0 1 4 0 1 0 0 7 0 1 1 0 11 1 1 1 0 12 1 0 1 1 1 1 0 0 0 1 3 0 1 0 1 5 1 0 0 0 8 2 1 0 1 1 2 0 0 0 1 3 1 1 0 0 6 0 0 1 0 9 4 1 0 0 1 4 0 1 0 1 5 1 1 0 0 6 1 0 1 0 10 7 0 1 0 0 7 1 0 0 0 8 0 0 1 0 9 1 0 1 0 10 ZO 1 0 1 0 10 Inconsequential combinations that may be used for simplification 0 0 1 1 1 1 0 1 0 0 0 0 1 1 1 1 294 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 fits in with the control contact network. Again, this is a multi-terminal network problem and the procedure is to design two-terminal networks that combine most readily. Since it is impractical to illustrate all the repeated trials that led to the final design, each network will be designed separately with the understanding that some of the steps are imposed by the form of all networks viewed collectively. The procedure adopted for developing the "0" lead network is as follows. First set up the miniature table repeating the portion of Table III that corresponds to the "0" lead. These parallel combinations should then be manipulated algebraically to obtain the greatest simplification possible. It is rather easy to apply some of the algebraic rules by observ- ing the condition of the relay in the several combinations in the table. A simple "shorthand" rule to follow is: if in the table of combinations describing a particular two terminal network, all possible combinations of certain relays appear in conjunction with a single combination of other relaj^s, the network contacts on the former relays may be neglected. In other words when 2" different combinations of any number of variables m, are identical in all but n columns, contacts on the corresponding n relays are not required. This procedure is carried out below. B C D E 0 1 1 1 1 0 1 1- 1 0 0 1 0 1 0 0 0 1 1 0 1 1 1 0 (B + c + DO (B' + c + E') CB + c + E) (C + D' + E) Thus we have the follo^^^ng algebraic expression for the "0" lead, which can be simplified as shown. (B + C + D')(B' -f- C + E')iB + (" -j- E)iC' + /)' + E) [C + (B + D'){B + E)(D' + E)]{B' + C + E') (3b) [C + (E + BD')(B -f D')](B' + C -\- E') (3b) This is shown on Fig. 4a. A somewhat different manipulation of the equation permits placing the network in the bridge form of Fig. 4b. The algebraic equation, given below, can easily be shown to be the equivalent of the original. [E + C + B{B' + D')](B' + C + E')(B + C -f D') BOOLEAN ALGEBRA AND CIKCUIT DESIGN 295 111 ('(M'taiu cases the use of theorem (14b), iioi-mally employed to reduee I he contacts of a particular relay to a siiifi;le make and break, can pro- duce simplifications difficult to accomplish ()tiiei\vis(>. This is shown below, with the theorem applicnl with respect to i-elay /.' since E tended otherwise to be hea^•ily loaded. (li + (" + D'){ir + r + K')iB + (" + K)((" + ly + K) [E + (B + C + I)'){B' + (' + \)(B + (" + 0)((" + jy + ())] [A" + (7^ + C + iy){B' + (' + 0)(/i + (_" + 1)(C" + // + 1)] (14b) (K + (" + Bfy)[E' + (B' + C)(B + C + 7^')] (9a, 10a, 3b, 9b, 5b) By modifying the first factor of the final expression in accordance with theorem 1 la, this equation can be put in bridge form as shown on Fig. 4c. [E + (" + B(B' + iy)W + (/^' + (')(B + (" + D')] The above equation uses the same contacts as the previous expression, and although the right hand member is in a slightly different form, the expression is equivalent to the one obtained earlier. When it is known that output conditions are inconsequential for some relay combinations, these inconseciuential relay combinations may be combined with valid combinations to eliminate contacts in the network. Inconsequential means that the output during these particular combina- tions does not affect the proper functioning of the circuit. Four such combinations are listed in Table III. Only those inconsequential combi- nations which will combine readily with the actual coml)inations, thereby resulting in a reduction in the nvmiber of contacts, are to be used. Al- though the use of all the all-relays-released condition may be helpful in certain cases, it will not be used in the circuit under consideration since its use makes the recjuirement that no tie shall exist between output leads until the second pulse is received hard to meet. With this in mind the "0" lead network is again examined. Note the use of another "shorthand" rule which states that if a part of the 2" possible combinations is used in closing a path, the negative of the unused part of the 2" possible combinations is e(iuivalent to the original com- binations. Thus if in the case at hand three of the possible four combina- tions of the B and C relays occur in series with the same combination of the D and E relays, the expression used is that for the series })ath of the D and E relays plus the negative of the missing combination of the B 296 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 and C relays. In the following tabulation the combinations below the horizontal line are inconsequential. B C D E (B + D + E) The expression becomes: (C + E' + B'D'){C' + D' + BE){B + D + E) [£; + (C + 1 + B'D'){C' + D' + BO){B + /) + 0)] [^' + (C + 0 + B'D'){C' + JD' + B1)(B + Z) + 1)] (14b on E) [E + (C + D')(B + D)W + (C + B'D')iC' + D' + B)] (9a, 9b, 10a, 10b) [E + (C + D')(B + D)][E' + CD' + C^ + 5'Z)'] (8b, 9a, 4b, 5a) [E + (C + jD')(5 + i))][A^' + BC + 5'Z>'] (12b) [E + (C + i)')(5 + D)][E' + (5 + Z)')(B' + C)] (13) Fig. 4d shows the schematic of the above expression. It is possible to put this in a bridge form without other changes because of the manner in which the front and back contacts of D are related to the other con- tacts. Comparison of all the circuits of Fig. 4 indicates that they all use the same number of contacts although final decision should be post- poned until all the output circuits are obtained and the ease of combina- tion of the different circuits can be compared. The procedure for determining the remaining code leads is carried out on the following pages. BOOLEAN ALGEBRA AND CIKCUIT DESIGN 297 'V lead— B C D E P(B + E') (C + D + E) i-osiiltin<>; in [K + (' + D][E' + B] which is shown on Fig. oa. It will later be found advantageoii.s, in combining, to include the B' term in the first factor, giving the expression: {E -\- B' -^ a + D)(E' + B) shown on Fig. 5b "^" lead— B C D E (C + D' + E') hence or (C + ly + E'){B' + C + D){B + C) [C + B{D' + E')\[C' + B' + D] For later ease in combining, this is changed to: [C + B{B' + // + E')][C' + B' + D\ shown on Fig. oc. (3b) (11a) 298 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 //' lead— B C D E hence {D -\- E' + B'C'){B' + C + D)(5' + C + £>' + E) [D + {B' + C'){B'C' + E')][D' + B' -^ C -\- E] (3b) which is shown on Fig. 5d. "7" lead— B C D E hence or (5 + £> + E)(C + E) E + C{B + D) (3b) which is shown on Fig. 5e. "ZO" lead— B C D E 10 10 hence one has {B' -\- C + Z)' + E) which is shown in Fig. 5f. BOOLEAN ALGKHUA AND t'lKCUIT DKSIGN 299 Tho final contact savings arc acliic\'(Ml l)y comlnning the \'ari<)ns out- |)ut j)atlis. The c()ml)inc(l translation circnit that appears (o he as i-c- (Inccd as possible is shown in V\'j^. (1. Note that certain toiins ol' the in(li\-i(lual output paths comhine nior(> i-eadily than others. Foi- example Fig. 4c and ')b coml)ine moi'c readily with th(> reniainiiifj; i)aths than Fig. 41) and oa. Xot(> also tliat sometimes it is not the most reduced loim of the indix'idual output paths that pei-mits efficient combining. Tiiis is - I ^-CP "0" LEAD (a) X "0" LEAD (b) "0" LEAD (c) — (d) Fig. 4 — The "0" leail of tlu> translating circuit. J-p ^ 1 - E C D B "l" LEAD (a) JT T^ — '"T E B c D l-X 1 X x-i (b) "l" LEAD -rn H A- (c) '2"LEAD xn "4" LEAD B D B c E H \ X X- (d) X ^ B D^ _ " 7 " L E A D X C ? ,^ "ZO" LEAD -X I X - — (e) Fig. 5— The "1, 2, 4, 7, and ZO" leads of the translating circuit. 300 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 exemplified by the use of Fig. 5b rather than Fig. 5a. Although various forms of all of the output leads were tested for efficient coml)ination only the form used is shown for outputs other than the "0" and "1" leads. It is essential to scrutinize the final network for possible sneak paths. Sometimes to avoid these sneak paths it is necessary to add one contact on one relay to allow savings on others. Here again the inspection tech- niques go hand in hand with switching algebra and the need for both is obvious. The algebra obtains the various forms which are capable of B n 1 RA E B DC I h— I \—* X > I i f— l- — E A I X— •— — (- RAl "o" LEAD -X- Rf>' "7" LEAD P E " zo" LEAD X ~~ D RAl "," LEAD RAl "4.. LEAD "2" LEAD Fig. 6 — Combined translating circuit. different degrees of combination very quickly and efficiently. The inspec- tion method is then necessary for the actual combination of these forms. The additional RA relay contact is necessary to assist in avoiding in- terconnections between the output leads until after the second pulse is received. The final assignments of either .4 or B relay contacts are chosen to equalize the load on these relays. THE COMPLETED CIRCUIT The final form of the counting and translating circuit is shown on Fig. 7. The relays are all doul)le wound to gain the l)enefits of current drain reduction. One additional advantage of using double windings is the relay code reduction made possible since now only two codes are necessary. One code serves the A relay and one other code serves the 1500LEAX ALGEHUA AND CIUCUIT DESIGN 301 /?, r, D and E relays. In comparison to this total of five relays and two codes the cii-cuit in present iis(> in the latest crossbar system re(niii-es ten relays and s(>\'en codes. AN ALTERNATE DESIGN OF THE I'ULSK DIVIDKK To ilhisti-ate th(> application of aiji;cl)ra whei'e the ai)pai'atus conlem- plated i:)uts less premium on contact minimization i)iit more on stand- ardization and winding minimization, certain modifications of the l)roposed circuit are considercMl. In the (>\-ent that new apparatus developments make possible the construction of relays that meet the necessary speed reciuirements even thou,i>;h windint;- impedance is increased, it appears possible (if the pulse (li\-ider is ledesigned) to use only one code having a single winding for all fi\-e relays. The use of added contacts might be allowable if the new type of re-lay carries more springs than the present relay. The redesign of the pulse divider to use single windings can be accom- plished i)y manipulation of the basic algebraic expressions derived earlier for the pulse divider. Thus for the A relay /(.4) = {U + B'){A + LB') = [U + B'][A + B'{L + B)] (11a) By attempting to manipulate the B relay control circuit into the same form, one obtains f{B) = (L + A){B + L'A) ^ (L + A){L' + B){A + B) (3b) = [U + B][A + BL] (3b) = [U + B][A + B{L + B')] (11a) The schematics represented by the above algebraic expressions are shown in Fig. 8a and 8b. The circuit of Fig. 8c is obtained by combining the first two circuits so that only a single transfer is needed on the L relay. Note however that it is necessary to make the lower tw^o B trans- fers have continuity action to insure proper functioning. Fig. 8d shows the pulse divider drawn in conventional form. CONCLUSION As far as is known, the dial pulse (-ounting and translating circuit described herein requires fewer relays than any other circuit with similar -7 UJ uj 30 3 U OCT. ' I 3-^1 303 304 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 functions at present employed in Bell System standard switching equip- ment. The previous dial pulse counter used in the latest crossbar system required a total of ten relays. Thus the present design represents a considerable saving in cost and space. To a certain extent this result can be ascribed to the use of switching algebra during the circuit develop- ment. Relay circuits designed on the basis of utilizing a large proportion of the possible combinations permitted by the component relays usuallj'" require heavy spring pile-ups. Since general purpose relays are limited in the number of springs which they can carry, this type of circuit usually (A) T^i'^ (<3) (b) (B) TJ^I.hL J7 X"^ (A) B A 'b. jx-l ^ - B A — X— »— X — 'b_ (B) -^ (d) t Fig. 8 entails considerable design effort to make most effective use of the avail- able springs. Application of switching algebra to this aspect of the design problem can often provide crucial assistance. It is recognized that switching algebra, in its present state of develop- ment, does not permit complete mathematical statement and manipula- tion of multi-terminal networks as represented by the counting and translating circuits. It does provide, however, facilities in manipulating two-terminal networks into a variety of forms from which can be selected those that combine most readily. This can result not only in a saving of time, but also in improved circuits which might not be realized by other design techniques. Unfortunately the algebra in its present state does not indicate when the optimum circuit has been attained. To some extent this is caused by apparatus or circuit considerations to which, since it is BOOLEAX ALGEBRA AND CIRCUIT DESIGN 305 concerned solely with contact networks, the alge])ra does not apply. Thus, there is still considerable room left for the ingenuity and judgment of the switching circuit designer. As a result of the experience in designing the dial pulse counter and translator, certain observations on the use of the algebra are believed to be valid. Although switching algebra may be used in the design of the simplest circuits, the most noticeable benefits are obtained by the appli- cation of the algebra to the design of those circuits in which the conti-ol and output paths are complex and interrelated. The particular minimum relay counting and translating circuit under discussion is an excellent example of this type of circuit. A secondary advantage of the algebra is its compact notation and its value as an efficient circuit "bookkeeping" method. ACKNOWLEDGMENTS The author is indebted to Joseph Michal who made invaluable contri- butions as co-designer of this circuit and also verified all the algebraic manii)ulations contained in this paper. The author also wishes to ac- knowledge the suggestion made by G. R. Frost as to the possibility of using the bridge network in the control circuit. This work was carried on under the supervision of L. J. Stacy and F. K. Low whose many valuable suggestions were incorporated into this development. Interaction of Polymers and Mechanical Waves BY W. O. BAKER AND J. H. HEISS (Manuscript received October 19, 1951) New techniques of Mason, McSkimin, Hopkins and co-workers for gener- ation of shear waves over the frequency range 2 X 10 to 2.4 X 10 cps have been used to study mechanical properties of chain polymers. Polymer solids, melts and dilute solutions, representing the main states in which plastics and rubbers are fabricated or used, were explored to find the char- acteristic relaxation times, rigidities and viscosities of various chemical structures. Polyisobutylene, hevea rubber, polydimethyl siloxane, vinyl chlo- ride-acetate copolymers and plasticized nitrocellulose were compared with polyethylene and polyamides as examples of the range of solid properties encountered. As melts, several polyisobutylene s, polybutadiene, polypropylene, poly- propylene sebacate and poly-a-methyl styrene were investigated as models for varying degrees of chain substitution. Chain rigidity in, for instance, polyisobutylene, seemed to reflect visco-elastic over-all configurational changes up through the kilocycle range, but nearest neighbor interactions took over in the megacycle region, leading to fnodidi of 10 dynes/ cm even for syrupy fluids. In dilute solution, polyisobutylene, polystyrene, natural rubber and poly- butadiene microgel exhibited characteristic dynamic viscosities and rigidi- ties depending linearly on concentration. Presumably, this reflects mechan- ical properties of isolated chains. Some possible models were suggested for the frequency dependence of such properties. INTRODUCTION The "equilibrium" mechanics of polymers, the giant molecules of plastics and rubbers, have been quite elegantly de\'eloped in the range of high strains ("kinetic theory" of elasticity — Meyer, et al.). However, the molecular displacements as these strains, and, indeed, much smaller ones, occur, are little understood. Nevertheless, it is essential to know about detailed motions in connecting chemical structure with physical properties. Only in this way can there be obtained from the chemical industry compositions which will serve properly in telephone apparatus. 306 INTKRAC'TION OF I'OL^MKKS AM) M lOCHANICAL WAVKS 'M)7 (^tlior studies have treated one way of getting at these mecliaiiisnis l)y relatiiit>; stress rehixation, creep, viscosity, etc. to a dislrihnt ion of molecular I'elnxatioii lini(\s (;ui(l cn('ru;\' hni'riers), as oriuiiiatcd \)\ Kuhii."' Anotiier approach is to sti'ain pt»lyniers with pci'io(hc \va\'es o\'er a \-ei\v wi(h> spectrum of \\a\'elen>>;ths, exciitually j2;oinj>; to fre- (|uencies conijiai'ahle with those of the thei'mal xihrations of sij>;meiits in \\\v macromolecailes. The result iiifz; dispei'sion or resonance phenomena can then l)e examined. Hence a mechanical radi- ation ti(>ld can interact with the masses of elementary structural units, as the usual elect romafz;netic field interacts with atomic and f^roup charges. In genei-al, direct int(M-i)retations of this kind must be done witii shear wa\'es, and, at least, not onlji with loii<),itu(liiial or ultrasoni(' waves. This kind of study is now proceeding using waves generated and fol- h)W(Hl by piezoelectric crystals connected in as actual electromechanical circuit elements (A. M. Nicolson, 1919). Recent schemes of Mason and co-workers cover the frequency range from 10 X 10 to 60 X 10 cps, as reported m the paper by Mason and McSkimin in the last issue, while a tuning fork method used by I. L. Hopkins has been applied to "soft" polymers (rubbers) over the range 10" to 10 cps (the general range of J. D. Ferry's work at Wisconsin on concentrated polymer solutions). The relation of these studies to the scientific and technical exploita- tion of plastics and rubbers is in knowing what a particular chemical composition does to strength, stiffness, ease of molding, impact tough- ness, etc. That is, are there qualities of the interaction of saturated aliphatic groups that make polyethylene or polyisobutylene have some glass-like as well as liquid-like, or rubbery, nature even at room tem- perature? If so, conditions causing brittle failures must be watched for. How is the storage of molecular strains in injection molded plastics re- duced by increasing molding temperature (when the kinetic theory stiff- ness per chain actually increases)? These and many similar problems may be generalized under the headings below; in each case the chemical structure of the macromolecule appears to be reflected in relaxation times which combine in different ways to give flow^ or rigidity, toughness or brittleness. Extrusion and Mohlituj Non-Newtonian flow leading to "fi'ozen-in" stresses, subseciuent dis- tortion and irregular shapes of plastics and rul)bers, implies energy 308 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 storage in the sheared molecules. The dynamite shear studies will confirm this. Also dispersion of carbon l)lack and othei' i)igments is restrained by elastic qualities of "iiciuid" polymers (i.e., instead of "mixing", com- pounds just micr()sco})ically deform and later re-form.) Likewise, the efficiency of compounding' and extrusion** depend on how c^uickly the molecules relax after straining. Impact Strength, Brittleness and Tenacity Toughness, mechanical shock resistance, ultimate elongation and strength reflect the facility with which the polymer molecules can be dis- placed without breaking the piece. Thus, they accommodate to the stress by motions presumably similar to those described above. (The situation is complicated when crystallites are also displaced. ) In any case, time sensitivity in the range 10~^ sec upward exists. ^'^' ^^ The discussion by Morey^ is a valuable survey of these ideas, and explicitly notes the significance of multiple relaxation processes on damping of shock waves. Evidence of the relation of simple changes in chemical structure to the principle relaxation times efTective in these physical properties of plastics and rubbers is thus another part of the dynamics studies. The "brittle point", or volume-temperature transition of amorphous polymers, ' apparently reflects du-ectly the correspondence of the time of experiment with dominant relaxation time of the polymer. ' A few measurements on plasticized polymethyl methacrylate (from which, however, no actual rigidities were calculated) indeed indicate abrupt stiffening as a function of frequency at a given temperature. However, the changes measured were too small and indefinite to indicate any particular molecular relax- ation. Other work ' with plasticized polymers is nevertheless concordant with the current findings that molecular relaxations and not long range order determine embrittlement. The converse of this is, of course, that as some "transition" is approached, hysteresis, heat build up, flex crack- ing and fatigue are greatest. Creep, Stress Relaxation and Recovery Even these "long time" qualities of plastics, such as found in cold flow, apparently result from integrated displacements of rapidly oscil- lating segments of the chains. A most interesting analysis of stress relaxation in rubbers employs Kuhn's suggestion of a particular distribu- tion of relaxation times. The present point is that, again, these relaxa- tion times reflect processes which should appear directly in reaction of the polymer with high frequency shear waives. INTERACTION OF POLYMERS AND MECHANICAL WAVES 309 From those aspects al)()\-e the cmreiit results of (lyuamics studies will he re\'ie\ve(l. POLYMER solids: over-ai,l ,m kcua ntcs Solid polymers will denote ruhhers and plastics in the state iu which they are technically used. This is usually their most complex form, with inter- and intra-molecular factors undistin<>;uished. Thus, se])aration and identification of the main relaxation processes are difficult or impossible. IIo\ve\'er, it is interesting to consider typical values of modulus and \-isc()sity as related to chemical structure, in the I'ange of fr(v|nencies corresponding to extrusion rates, and stresses in actual use. These values of dynamic modulus and viscosity are distinct from the usual cjuantities in the literature. The usual expressions are for longi- tudinal (sound) waves, and give dynamic Young's modulus E* = Er - iE. E-2 measures the out of phase part of the force-displacement relation, and Eo = w- ("effective viscosity coefficient"). Now, the general elastic constants are X + 2/i, with X = Lame's constant and (jl = shear modu- lus. Here. X -f- 2m = /C + l/x, with K = bulk modulus. Alternately, SK 3X + 2m X + M X +M However, in general the present results lead to the simpler shear modu- lus M- Further the energy losses studied are expressible directly as the usual shear viscosity Previous comprehensive studies of the dynamics of rubbers over sig- nificant frequency ranges have yielded loss factors either written as E-2 El (see above),'* or as a function of the shear viscosity based on Stoke's assumption that the compressional (dilatational) viscosity is zero.^" But as Nolle'^ and Ivey, Mrowca and Guth^° clearly re(;ognize, recent work has strongly manifested the presence of compressional vis- cosity in simple liquids"' as well as polymeric ones."" " Hence, the pres- ent understanding relating molecular structure to viscosity, plasticity and visco-elasticity is unsuitable for interpreting mechanical wave mo- tion more complex than in shear, unless shear constants are also known. 310 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 This sums up to mean that the chemical interpretation of basic poly- mer mechanics recjuires shear wave measurements. Nevertheless, fas- cinating evidence of the existence of fine-structure rehixations in polymer solid has come from longitudinal wave investigations. ' " ' ' " ' Also, the pioneering shear wave studies of Ferry and collaborators" ' " on con- centrated solutions of polymers have suggested intrinsic relaxations of the chain molecules in a highly plasticized "semi-solid" state. The more simplified findings cited below will be seen to luiify ap- proaches in this field. Comment must first be made, however, on formu- lation of experimental results in dynamics of polymers. Expression of Dynamic Properties Alternate and equi^'alent expressions have been thoroughly surveyed;" all represent combinations of either Maxwell (series) springs and pistons (elasticity and viscosity) or Voigt (parallel) springs and pistons. Obvi- ously, there is no physical separation of elastic and viscous elements in a polymer molecule, so the irrelevance of the detail of the model need not be emphasized. However, the models lead to convenient formulation of relaxation times which dielectric studies, in particular, have shown have clear connections with chemical structure. In this chapter, some- times one and sometimes the other model, or combination, will be used, with the symbols shown on the next page. Other symbols are sometimes used, but should be easily identified in terms of the above. Rubbers and Soft Plastics In Table I, the shear moduli of rigidity, n, and of viscosity, ij.', are shown as calculated for the Kelvin-Voigt model, for polymers having the indicated units of structure. The freciuencies are from a few hundred to a few thousand cycles, hence, in the range of much technical use, (flexing of tires '~300 cps) and rates of shear during processing. ' Data are from a general study by I. L. Hopkins of the Bell Laboratories, based on a tuning fork transducer introduced by Rorden and Grieco. The strains employed were always small, in the range 0.3 to 1.5 per cent; ^l and /x' were essentially independent of strain, except for some loaded rubber stocks. The /x values clearly trace the magnitudes to be expected in going from the most typical rubber (hevea) to the semi- rigid plastics (vinyl chloiide-acetate copolymer and plasticized cellulose nitrate). As anticipated from steady-stress observations the "plastics" have /I > 10' dynes /cm". Increase of ijl with fre(iuency is also greater as INTERACTION OF POLYMKHS AND M KCIIAXK AL WAVKS 31 the ''plastics" range is approached; a relaxalioii region is implied. Figs. 1 to 4 show the dispersion of rigidity with fretiuency in moi-(> detail. Especially striking in Figs. 1 and '2 is the small lemperature dependence (at least between 27° and ()()°C) of /x- Jiecanse of experimental uncer- tainty, M cannot be said to be actually higher at the higher temperatures in accord with straight kinetic theory, but at least it is strongly tencHng that way, as also noted for lower freciuencies studies on natural rubber. Nothing like this appears for the plastics; in plasticizcd nitrocellulose the 100-cycle rigidity decreases lO-fold from 27° to GG°C. This is, then, the second general dynamic (|uality which reflects the low van der Waals' (dipole, dispersion and induction) forces in hevea rubber and polydi- methyl siloxane, as well as their intrachain flexibility. Interchain forces in polyisobutylene (Butyl rubber) are low too, but barriers to flexibility because of sterically hindered-CHs groups come in. Table I and Fig. 3 KELVIN-VOIGT MAXWELL I da Jt IdS S fj. dt T] »S = 5oe " = aSoc ^ (7 = strain *S = stress t = time T = relaxation time T = retardation time jj. = G = modulus yu' = 17 = viscosity dt t] S M< (T = - ( I — e '' ) = (Toe For const. S, da S dt = - ov r] = s_ da dt There is same stress on each ele- ment ; the total strain = sum of single strains. T = - M There is same strain in each ele- ment; the total stress = sum of single stresses. 312 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH Table I 1952 Polvmer Unit Shear Modulus, 11, dynes/cm' 27°C Shear Viscosity Poises, m', 27°C 100 cycles 5000 cycles 100 cycles 5000 cycles Hevea rubber 3 X 10« 5.5 X 106 350 40 CH3 1 1 — CH2— C=CH— CH2— Polydimethyl siloxane 0.7 X lOe 1 X 106 300 30 CH3 1 1 — 0— Si— CH3 Polyisobutylene 5 X 10« 30 X 106 8,000 1,500 CH3 1 1 — CH2— C— 1 1 CHa Polyvinyl chloride ('^92%) -acetate (~8%) plasticized by ~36% di- octj'lphthalate 13 X 10« 80 X 106 25,000 2,000 — CH2 — CH — and CI — CHo— CH— 0 II 1 II 0— C— CH3 Cellulose nitrate 60 X 10« 250 X 106 80,000 4,500 CH2ONO2 / CH 0 / \ — C CH— 0— \ / CH CH ONO2 ONO2 and ^^25 \vt. % Camphor plasti- cizer. INTERACTION OF POLYMERS AND MECHANICAL WAVES 313 1000 800 600 400 300 200 100 80 60 40 30 20 10 SHEAR MODULUS ■— =J ,-» *- m 66° Z ^ ■=3r^F==^ J.-5U- L;-=^' ^=H 27°C a \ ^v 1 ^ > ^ V. V s, ) V ^ ^ISCOSITY \>^ « K, ^^ n2J°C \ ^S \ o 66°C^ \ N K 10 X 8 10^ 0.2 0.1 100 200 400 600 1000 2000 4000 6000 10,000 FREQUENCY IN CYCLES PER SECOND Fig. 1 — Viscosity and shear modulus of hevea rubber (cross-linked). 400 O liiO 100 90 80 70' 60 50 30 100 V N ^. /ISCOSITY iHEAR MODULUS X x- > ^ k 66°C, ■ bv> ■ — ■ 27°C J^ -■v ■V- — — — -"■ A ' ■ ."^^^"^^ ' ^ X ■ -^-' — ' X S V V^7°C N s ^"^ <^ • ^ 4.0x10* 200 300 FREQUENCY 400 500 600 800 1000 N CYCLES PER SECOND 1.0 0.9 0.8 0.7 0.6 0.3 2000 .0 Z 0.8 >■ 1- Q 0.6 ID a. a. 04 < lU I 0.3 CO Fig. 2 — Viscosity and shear modulus of polydimelhyl silo.xane (cross-linked). 314 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 10,000 8000' 6000 4000 3000 w 2000 000 800 400 200 100 1 VISCOSITY SHEAR MODULUS - V \ \, \ \^ oN ^ > .cv y 27 =■0 V ^ ^ s < > ^ V ^ ( 1 a o 5- - 27 °C ^ ^ ^ 6€ o ^ v.^^ • *" ^ ■^ ^ ■ _^ , -a-^ H ^ rQ — * r^ ,-- 1— ' — V. ^,^- ° >^ • • - 56 ° - 100 200 I 10,000 400 600 1000 2000 4000 6000 FREQUENCY IN CYCLES PER SECOND Fig. 3 — Viscosity and shear modulus of polyisobutyleue (cross-linked Butyl). emphasize that thinking about the mechanics of a particular chemical structure must inchide the spatial relationships of groups within the chains, as well as between them. The dynamic viscosities in Table I are also in accord with the se- quence of structures. Their frequency dispersion again connotes varying relaxation processes. Natural rubber's low inner friction, for both com- pressional and shear waves is famous in its low hysteresis heating. (This unique property is geopolitically crucial, because adequate truck and bus tires cannot yet be made of any other rubber.) Indeed, it is striking that at 100 cycles, a piece of gum rubber has a local viscosity of only 350 poises. The silicone rubber gum also has high elastic efficiency, and its temperature coefficient of viscosity is very low (see Fig. 2), like the thermal coefficients for familiar silicone liquids. It is exciting to speculate in Figs. 1 and 2, whether more precise measurements which Hopkins is now undertaking will confirm the apparently negative tem- perature coefficients of viscosity at some frequencies. "Kinetic theorj'- INTFRACTIOX OF POLYMERS. AND MIX'HAXICAL WAVES 315 visoosit.v" arising from transfer of momentum among thermally agitated chain segments, does not seem to have been considered in the theory of perfect rubbers. As in gases, it would require an increase of viscosity with temperature. In polyisobutylene, however, the dynamic viscosity leaps upward in both magnitude and temperature dependence. It should be emphasized that this is, again, for a cross-linked (Butyl) gum — an infinite network like the hevea gum, with presumably infinite macroscopic viscosity. The striking thing is that this internal viscosity is not greatly dependent on the network, at the degrees of "cure" used in rubber technology. For instance, recent studies over the frequency range 20-600 cycles, on high molecular weight, M^ = 1.2 X 10^ linear polyisobutylene," give, at 2o°C and 100 cycles, n' = 4800 poises, although the steady flow viscosity of this polymer at this temperature is greater than 3 X 10^ poises.^* Then, the infinite network (Tjstea.iy now -^ °o ) Butyl polymer of Fig. 3 has at 27°C and 100 cycles n' = 8000 poises. At 1000 cycles agreement ajjpears to be about the same, and is tolerable considering the several 100,000 80,000^ 60 ,000 40,000 30,000 VI 20,000 10,000 8000 4000 3000 2000 1 VISCOSITY SHEAR MODULUS- o \, ^ y \ \, 66° X a^ ) \ S S >i '< D <1^ c -.^ V, ,^1 '^ ^ "^^ ^°C f- ^— -—-tf^ ^^ o o \ "^ ^ . \ • \ N N V V V ■>. • • ^C — • 100 80 1000 800 w 200 rvj 2 5 100 ^ z 60 z 4 9 40 9 100 200 400 600 1000 2000 4000 6000 10.000 FREQUENCY irj CYCLES PER SECOND Fig. 4 — Vi-scosity and .shear iukIuIus of j)lasticizc(l cellulose nitrate. 316 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 per cent of compounding ingvedients in the Butyl g.\\m, and possibility of a small fraction of low mcjlecular weight uncured jjolymer in it. Also, Avide variations in the degree of cure of Butyl gums were studied without large changes in jjl'. In this regard, the particular sample of Fig. 3 had an e(iuilibrium swelling ratio (= volume swollen polymer in cyclohexane at 25°C/volume insoluble part of dry vulcanizate) of 4.84. This indicates an Mc value (average molecular weight between cross- links) of < 20,000. ■ Actually many of the dynamic properties can prob- ably be fovnul in individual chain units or segments even smaller than this. This is a significant point in engineering applications where plastics may be cured to reduce creep but where it is desired to retain typical "chain" properties to increase impact toughness. That is, usually some optimum condition for this compromise can be found. The later section on lifiuids will suggest that physical properites typically associated with chain polymers can mdeed reside in even shorter chain sections than the ilf c's observed in usual gum rubbers. Filled Polymers Marked effects of carbon black and other pigments are of course familiar hi both steady and alternating mechanics of rubbers.^ . - . «. Brief comment on their influence on dynamic shear properties and thus relaxation mechanisms involved may be directed toward plastics, also, however. Thus, technologically it would be desirable to load thermo- plastics with considerable volumes of "inert" fillers, just as is done with rubber. But, almost invariably strength and toughness decline, instead of improvmg, as in the rubber case. A reason for this appears in inves- tigations by Hopkins when carbon black (a standard type of reinforcing black) was added to Butyl rubber of the sort described in Table I. It is that stiffness seems to rise more rapidly than internal viscosity — i.e., a given strain results in proportionately higher stress than the accompany- ing internal viscosity provides means for dissipating the stress (as on impact). Hence, the brittleness which fillers normally engender hi ther- moplastics may represent this change in /x vs. m' balance. Table II illus- Table II Wt. Per Cent of Carbon Black Shear Modulus, /i, dynes/cm^ at 27°C Shear Viscosity, m', Poises, at 27°C in Butyl Stock 100 cycles 5000 cycles 100 cycles 5000 cycles 15.2 28.6 8 X 10« 45 X 106 60 X 106 150 X 106 11,000 35,000 2,000 3,000 IXTEHACTIOX OF roLVMKKS AND MKCIIAMCAL WAVKS 817 tratos some values for Butyl compouncls. Tlio .swelling ratio (SR) for tiie coinpouiid eontaining 28/) wt. per ceiil filler has di-opped to 3.2, implying also considerable I'cduclions in .1/,. (since tlicoiclicnlly {I SR.y'^ = 77- — 2hMc^'^). Thus, the apparent ciiain segment between M c cross-links is shorter than in tlie nnfilled stock (the two wei'e cui-ed to give closely similar degrees of primaiy valence cross-linkage) and corre- spondingly the steady-pull modulus is higher. Yet, the internal friction, while also higher, seems to reflect iuav relaxations from intei'aclion with th(> tiller, and total shock-absoi'bing power has declined. MicrocrystaUine Polymers The i)receding studies at comparatively low frec}uencies indicated (1) magnitudes of shear rigidity and internal viscosity characterizing rub- bers and soft plastics. By familiar shifts of temperature oi- frequency, the}' would also apply to polymers known as hard, amorphous plastics at room temperature such as polystyrene and polymethyl methacrylate. (2) Dispersion of /x and m' ^^ith frequency affirm that the intrinsic or fine structure relaxations have times <10~ to 10^ sec, and so refer to chemical luiits much smaller than the average molecules in the usual technical rubbers and plastics. A way to get at what sizes and habits these units might have will be by investigation of low molecular weight polymer liquids. But, while still in the section on solids, it is recalled that microcrystalline polymers such as polyethylene, polyesters (Teryl- ene), polyamides (nylons), cellulose esters, polyvinylidene chloride, poly- acrylonitrile etc., have mechanical properties dominated by their crystal- line-amorphous ratios. ' ' ' The amorphous \-olumes are clearly those which donate the flexibility, toughness and shock-resistance of these plastics and textile fibers. ' An interesthig point is, how "viscous" are the disordered chain segments? In an over-all sense, all kinds of dissipa- tion including crystallite friction, analogous to grain friction in metals, scattering of longitudinal waves, and stiffening by low temperatures can occui- in these polyphase systems. Thus, effects of chain orientation as well as lateral order (crystallinity) have been detected in dynamics studies.' ' ' The intrinsically li(iuid-like or amorphous components of this behaviour — and the things which will correlate most simply with dipole concentration and other chemical features — arc most accessible to study at very high fretiuencics. For, in these polymer solids, unlike tiie essentially continuous and homogeneous amoiphous ones first dis- c-ussed, the mechanics reflect small regions lun'ing widely di\'ergent properties. Thus, methods developed by H. J. AlcSkimin of Bell Tele- 318 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 phone Laboratories (described in the last issue), have been used to probe for elemental reactions at the upper end of the frecjuencies presently available. Both longitudinal and shear waves were used. In polyethylene, a wavelength for the shear waxes was 0.0074 cm., at / = 8.55 X 10^ cycles, and in polyhexamethylene adipamidc (the usual G-6 textile ny- lon), the shear w^avelength was 0.0125 cm., for/ = 8.67 X 10^ cycles, all at 25°C. The important consequence of these experiments so far has been that, despite the small strains involved, the viscosity appears to be a "poly- mer" viscosity, rather than an inner friction involving just a few liquid- like atoms per unit. Thus, polyethylene of "equilibrium" crystallinity and average molecular weight corresponding to an intrinsic viscosity in xylene of [r/] = 0.89 (at 85°C), was measured over the range from 0 to 50°C. The results from both longitudinal and shear wave measurements Table III Temp., °C. /, cycles/sec viscosity Poises m' X' + 2m' y 0 0 30 30 8 X 106 25 X 106 8 X 106 25 X 106 15 5 15 5 38 14 34 13 8 4 4 3 are given in Table III. These viscosities are expressed in this case for a Kelvin-Voigt model, of rigidity and viscosity in parallel. The rigidities associated with these viscosities are about 3 X 10 dynes/ cm , or not far from the value under steady pull of about 1 X 10^ Now this suggests that the rigid plastic polyethylene retains, even under mechanical impulse of microsecond duration, a shock-absorbing capacity reflected in a shear viscosity of 5-15 poises, and a compres- sional viscosity of 3-8 poises. The former, /x', may roughly correspond to the liquid viscosity of a paraffin-like chain of from 50 to 65 c-atoms in length. Thus, the dynamics measurements seem to relate to basic premises of polymer structure. These are that the amorphous regions (whose existence is shown quite independently by x-ray scattering, den- sity, heat-capacity, etc.) indeed take up and dissipate sudden stresses which the microcrystallites, despite their great strength, would be too brittle to sustain. These results give hope that further probing of the dynamics of liquid- like elements in rigid plastics will eventually lead to precise adjustment INTERACTION OF POLYMERS AND MECHANICAL WAVES 319 of molecular wciglit, chemical structure (degree of branching in poly- ctliylene), crystallinity, etc. These (luantities, when fitted to a given pattern of /i, X, n' and X' at i)rop('r freiiueiicie.s would yield plastics of optinnnn sei-N-iceahility under the multitude^ of stresses encoimtered in US(\ A similar li(iuid-like structure-even where the (crystalline) rigidity is much higher and mobile chain segments smaller — apparently occurs in l)olyaniides. Presumably the hydrogen l)onding and dipole interactions are \'ery imperfect in the disordered regions, and there the chain inter- action is reminiscent of polyethylene. For instance, in polyhexamethyl- ene adipamide, measurements in the 8 to 30 megacycle range do indicate that the Lam6 elastic constant X is al)out o.t) X lO' dynes/cm", but only about 3 X 10 for polyethylene. This reflects over-all stifTness dominated by crystallites. Nevertheless, the compressional viscosity, X' is 17-6 poises (going from 8 to 30 mc) for the polyamide, but only 5-2 poises for polyethylene. Of course, since there is dispersion in both cases, these relative magnitudes might be quite different at some other fre- (juency or temperature (all above are at 25°C). Yet it remains that the nylon, despite its hardness, also has a liquid-like component more vis- cous than that of polyethylene. Similar relations appear in the shear ^•iscosities, m', also determined for these two systems. For the 6-6 poly- amide, fx' goes from 19 to 7 poises over the 8 to 30 mc interval while polyethylene changes from 15 to 5. These quantities indicate again, as with the polyethylene, that "polymer liciuids" rather than just a few small groups of atoms are the important mechanical elements even at frec|uencies of 10 . Now polystyrene, an amorphous polymer, also has rigidities of about 10'° dynes/cm^ but the m' and X' values at room tem- perature are far below 5 to 20 poises, and glass-like brittleness (although not so bad as silica glass) results. So far, then, the two characteristic extremes of polymer mechanics have been discussed: (1) the soft rubbers, whose dynamics at low kilo- cycle frequencies imply, at ordinary temperatures, predominantly over- lapping combinations of relaxation processes whose relaxation elements involve many segments per molecular chain; and (2) the hard, micro- crystalline plastics, whose behaviour is predominated by relaxation proc- esses having times of 10~ to 10~ sec because the longer period (slower) displacements have been relaxed out at the temperatures of normal use. (Likewise, interconvertability by temperature between these two ex- tremes is presumed. Also, a certain correspondence between dielectric and dynamic relaxations in these classes is indicated. "") Next, it is in- 320 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 teresting to see what are the simplest structures (particularly in terms of molecular weight) yielding these effects. In other words, what kind of liquid really exhibits "polymer mechanics?" No detailed answer to this can be given l^elow, but results on some polymer lic|uids of low average molecular weight will indicate that the mechanisms in rubbers and plas- tics are probably more general than previously supposed. POLYMER LIQUID MECHANICS By techniques described in detail elsewhere,-^ -^ a series of polyisobutyl- ene liquids have been investigated. These polymers were made by ionic catalyzed mass polymerization at reduced, but not very low, tempera- tiu'es. While no great care to purify the monomer was used, such poly- merizations require fair purity to go at all. Seemingly, the resulting lici- uids do represent a polymer homologous series, although head-to-tail setiuence of the monomer units, some single ethyl rather than paired methyl side groups, etc., may differ slightly from the higher molecular weight forms in Butyl rubber and polyisobutylene gum. Whatever are these details, it appears that the polymers represent a linear hydro- carbon chain, with essentially two methyl groups on every other chain atom: CH, -CH. • C CH3 CH, CH2-C- CH. 3JDP-1 By contrast, polyethylene, with the nominal chain CH2 — and to a lesser extent polystyrene, — CH2CH— , -CH2CH2CH0. have chains in which rotation about the bonds is less sterically hindered. The final section, on isolated polymer chains (in dilute solution), will consider this aspect further. However, some results will be reported be- low on a low molecular weight poly-a-methyl styrene, which may be considered structurally a cross between the rubber, polyisobutylene, and IXTKKACTION OF I'OI.V.M KKS AM) MKCIIAXK AL WAVES 321 the plastic, polyst yi'ene, — CHo-C— . ()tli('r stu(li(>s in |)i'()<>i'(\ss on licinid polyhuladicnc, i<()l>'is()i)r('n(', poly- propylene, and ])olypropyl('n(' sehacatc tVoni which t'nrthci' inlorniation ahout intra-chain stiffness may l)e derixed, will also be noted. PropcM'lies of the polyisobutylenes studied are summarized in Table I\', some additional molecular weights in this range appear as extra pomts in some of the hioh-fre(iuency graphs. The molecular weights Af, are "inti'insic viscosity" averages "' '"'' and, with reasonable estimations Mn ot the ,rv~ ratio, check with cryoscopic number average, Mn , values on such materials, which are in tui'u listed in the table as expressed by melt \iscosity relations of Fox and Flory. '" These molecular weights repre- Table IV Polyiso- butylene DPr, M, Mn 25°C viscosity Poises Ma.xwell Voigt Maxwell Freq. Cycles Polymer Poises Poises A 10 565 318 0.37 3 X 10^ 0.6 0.6 14 X 106 A" 30 1660 697 39.6 6.2 X 106 1.7 X 109 16.5 7.9 18.8 10.0 2 X 104 14 X 106 B 45 2520 1070 216 3 X lO" 15.2 24.2 14 X 106 C 56 3350 1720 737 3.6 X 109 20.2 47.9 14 X 106 D 74 4170 2530 1840 4.5 X 109 23.4 78.9 14 X 106 E 147 8240 4850 4600 5.3 X 109 27.2 92.3 14 X 106 sent i-easonable a\'erages rather than absolute values for these hetero- geneous polymers. The DP values are just the number of isolnitylene units per average chain. The r? values are the steady flow viscosities at low rates of shear — usually determined by a falling ball. Rigidit]) atid Viscosity Mmjnilndes The properties of these li([uids I'anging from polymer A liaving only forty times the viscosity of water to E, which begins to approach fluid- ities of technical polymer melts (polyamidcs, for instance), were exploi'ed 322 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 in the kilocycle range with shear waves generated by torsional crystals and in the megacycle region by shear waves with tiie reflectance method and by longitudinal (ultrasonic) waves from a pulse i^ropagation tech- nique. The results have been expressed in two ways. Fu'st, hi earlier reports, " a trend corresponding with experiment was given by two Maxwell elements arranged in parallel. This result is too simple com- pared to the distributions of relaxation times pre\'iously proposed for high molecular weight polymers to reproduce detailed observation. Nevertheless, perhaps because of the smaller molecules involved, there seems to be clear indication that two principal relaxations predominate the mechanical reactions of these liciuids over the range of frequencies of present interest, 10" to 10' cps. For example, for polymer D, these are: First Relaxation Second Relaxation /c '^ 4 X 103 cycles M ~ 4 X 10" dynes/cm2 /c ~ 5 X 106 cycles M ~ 6 X 109 dynes/cm2 (In accounting for the second main relaxation, a hysteresis component had to be introduced whose significance has been suggested. ') Second, specific values of shear rigidity /x (Maxwell) and m (Voigt), shear viscosity n' (Maxwell) and m' (^ oigt) as well as the constants for related compressional wave systems, X -|- 2/x (elastic) and X' + 2^' (viscous) have been calculated for particular frequencies. Unlike in the first way of expression, these latter quantities are all highly frequency dependent. However, they describe conditions at various frequencies of interest, and are thus often worthwhile. Both ways of looking at the data lead, as implied by the figures above, to the proposal that typical polymer stiffness (shear rigidity of '^10 dynes/cm") is present at M, '^ 1600, ^\ith DPr, '^ 30, or an average chain length of about 60 carbon atoms. This appears when the straining is done in 10~ to 10~ sec. In the 10~ to 10~ sec range, rigidity occurs for even an average chain length of 20 atoms as shown in Table IV. STRUCTURAL FACTOR IN LIQUID MECHANICS The mam relaxations in the kilocycle range in polyisobutylene liquids seem to lead to quasi-configurational elasticity. This is where the kinetic theory tendency for a most probable separation of chain ends is retarded by viscous interaction of segments between and within the chains. Hence, the middle dashed curves of Fig. 5, showing shear elasticity for some of the polymers of Table IV, decrease e.xponentially with increasing tem- INTERACTION OF POLYMERS AND MECHANICAL WAVES 323 pcrature. While pure kinetic theory elasticity would give a modulus increasing linearly with increasing temperature, these systems, like all practical rubbers and plastics, actually grow softer with rising tempera- ture when deformed dynamically. It is striking, nevertheless, that a modulus of '^lO' dynes/cm" seems characteristic of the visco-elastic energy storing of these simple polymer structures. As noted below this is 10 less than the crystal-like, close-packed, stiffness found for these same molecular frequencies above their second principal relaxation time. TEMPERATURE IN DEGREES CENTIGRADE 70 60 50 40 30 20 10 0 200 XtO^ 3.2 i/T X 10^ Fig. 5 — Sheiir elasticities and viscosities of pol^yisohutyleiie li(iui(ls assuming a Ma.wvell model with relaxation frequencies 10^-10<. (Lower dashed curves for fre- iiuency-dependent models.) 324 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Thus, it seems that the former, 10 , modulus is typical of the struc- tural arrangements in polymers said to be above their second order transition temperatures^ ' ^ while the second, 10^, modulus reflects in- teractions below the freezing-in. These conclusions obtain regardless of the particular expression of the data. But, for comparison, curves are shown on Fig. 5 for a polyiso- butylene A" in which the dynamic modulus n at 20 kc is computed for both Maxwell and Voigt elements. The two points denoting the steady flow viscosity of polymer A" rank it with respect to the others in the series. Apparently even very fluid polymer melts, chaui molecule plasticizers, and small segments of long molecules must be expected to show appre- ciable rigidity when stressed rapidly. Referring to the introduction it is reasonable that rough extrusions, frozen-in molding stresses and the like are so easily produced. The lines of Fig. 5 are not, of course, implied to be linear over any considerable temperature range. In the region rep- resented, the temperature coefficient for viscous flow is about 16 kcal for the B, C and D liquids (about 12 for A). This agrees roughly with the steady flow values found for very high molecular Aveight polyiso- butylene. ' "'^ The temperature coefficient for the rigidity is less, as would be expected, since the whole center of gravity of the chain need not be displaced, but only local segments. This quasi-configurational elasticity is increased by molecular weight (although kinetic theory elasticity of chain segments in a network is de- creased by increasing segment length). The log m vs density at 25°C plotted in Fig. 6 indicates that the chief influence is the number of chains per cc, since the points for all the molecular weights now lie on a single line. It should be repeated that the elasticity modulus plotted, H, is again for a roughly frequency-independent or "absolute" model."' " The same is true for the three solid lines on Fig. 6, showing ju in the second, or 10 cycle, relaxation range. Here effects of detailed liquid structure come out; the three average molecular weights no longer lie so nearly on a single line. This elasticity is presumably from the crystal- like interaction of nearest-neighbor segments. If temperature is adjusted so that densities are the same, it is seen that the lower average molecular weight liquid has the higher elasticity modulus. This difference is not large, and should not be interpreted as showing an equal segment inter- action, for a polymer of lower specific volume (B compared to D). Rather, it emphasizes in this relaxation range, approaching the "glass" behaviour, that the relaxation rate is vastly more temperatiu'o dei^endent than the specific volume change alone, and structural variations in the INTERACTION OF POLYMERS AND MECHANICAL WAVES 325 > D O - 1 / / I O POLYMER B D POLYMER C A POLYMER D / / / 5 '° - / ~ LU - / - < D - / 1 - to "* liJ Q. - - liJ z Q i 1 > « - i^e . 7/ - u , - 7 / V - 1- ^ lO < - o/ / jj / - UJ < LU - / /j / - 01 2 < Z ° in6 / / / / f < n - / - 0) fc - / - z O 4 - / f - 10 111, — , Z ul >- OJ Q > CC lO'O ? 3 >- t Q H o <~— UJ cr LU cc I- V 111 u (J ZoJ 111 (r 2 < Od cc J) ^a. 0.86 DENSITY 0.88 OF LIQUID 0.92 Fig. 6- ties. -Principal shear elasticities of pol3'isobut3'lenes as related to their densi- packing of segments in the lif|iiid (coordination number, etc.) become important. ">So/r' and^^ Hard" Liquid States; Second Order Transitions A recent noteworthy study of vokime-temperatiu'e and viscosity- temperature changes in polystyrene (with a note on polyisol)utylene) brings out many points in common with ideas of polymer licjuid struc- ture indicated by the dynamics work."' Particularly, the fact that according to steady state measurements, the "local configurational ar- rangement of the polymer segments" below Ty remains fixed accords with the postulations from dynamics work. That is, above the second main relaxation, it seems to be just the interactions in these fixed ar- rangements which cause the glassy (or "crystalline") dynamic modulus of 10 to 10 dynes/cm . Further, the point that Tg is not an isoviscous state for polymers ' agrees with the dynamics result that macroscopic 326 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 viscosity of the polymer has relatively little to do with the actual values of dynamic viscosities. These would be at frequencies where the response of the polymer liquid to the mechanical field is determined only by motions within the local fixed arrangements mentioned above. Fig. 7 illustrates this, where on one scale the macroscopic viscosity is plotted according to the familiar log-log relation with molecular weight. Two extremes of average molecular weight, M,, and M„ are used for the liquids, to show that the molecular weight distribution does not alter the general conclusions. (M, is an upper limit weight average figure.) On the other scale, the dynamic viscosity /x', ii^ this case for a single element frequency-dependentYoigt model, shows low values and marked curvature. These betoken the relaxation in which molecular weight, through its effect on free volume and other structural factors, is signif- 2.0 Z 0.8 < 8 0-^ ^^=-- =a^ X y ^^' i/ ^ / / / / / / / / / / LOG Mr? LOG Mn / / / ,^ 4 55 2 ,.^ y"^ tx •— ""^^ / ^< ^'^ / / / / / ''n / / / / / / / / O 2.4 2.6 2.8 3.0_ 3.2 3.4 3.6 3.B 4.0 LOG M;^ AND LOG Mn Fig. 7 — Comparison of steady flow and dynamic viscosities (at 25°C and 8 mc) of polyisobutj'lene liquids of different molecular weights. INTERACTION OF POLYMERS AND MECHANICAL WAVES 327 icaiit for displacements suporficially quite different from those in macro- scopic viscosit}'. The compressional viscosity, X' is also plotted, for the same model, in Fig. 7. It is, within experimental erior, zero for polymer A', as deter- mined by shear and compressional wave studies at 8 mc frecjuency." This is a rare case, then, where the attenuation of sound waves through a liquid has been quantitatively accounted for by the shear viscosity. But, as soon as the a\'erage molecular weight rises to 1000 or so, X' comes ui clearly, and the new mechanism for dissipating compressional or dilatational stresses is de\'eloped. As this presumably represents di- rectl}^ free volume or coordination number changes in liquid struc- ture, ' " its detailed study near Tg , and in connection with brittle ])oints of rubbers, may eventually be especially fruitful. Another depiction of influence of average molecular weight in these licjuids on dynamic viscosities occurs in Fig. 8. Here, the Xc curve is 2 4 6 8 _ 10 I2xl03 MOLECULAR WEIGHT, M;^ Fig. 8 — Dynamic viscosities of i)olyisol)Utyloiic liciuid.s as function of average molecular weight (25°C). 328 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 for, again, a crude model attempting to show compressional viscosity over the whole frequency range, while the other viscosities are Voigt expressions at 8 (or 14) mc. Extremes of molecular weight averages are shown. Comparison of the "soft" or quasi-conhgurational rigidities, expressed, like the n of Fig. 5 as relatively frequency independent Hc , w'ith the "hard" or glassy rigidities is given in Fig. 9. The X and /x values are for the Voigt model at 8 mc. The graph does not show the bend-over of the "soft", He , curve with molecular weight, but that happens more grad- ually. The "hard" rigidities X and h quite readily show this inflection. As before, the relaxing segments must be < 100 chain atoms, according to the behaviour of the molecules at room temperature. Concerning influence of molecular weight on engineering "brittle points" of such importance in rubber technology, the present studies agree with earlier proposals. Thus, although the Tg or v-T second order transition point always decreases with decreasing molecular weight, LLI LU < < ^ <^^ 1 600 If) >- D Z _l < Q !5 1200 i o ^ I 1 tu 2 rv ?- 1 FROM KC 1 (RELAXATION! ' MODEL ' 1 1 /^c(Mn") ,'/"c X(Mnl fi^ \^_ ^- -o -A K i"; t^ / 1 1 1 / 4/ / 1 UL 1 1 1 1 i I 1 i J 1 / 1 / 'I 1 I I 1 y 7 MOLECULAR WEIGHT, M/^ 12 X 103 Fig. 9 — Dynamic rigidities of polyisohutylene liquids as function of average molecular weight (25°C). IXTKUACTIOX OF I'OLVM KI{S .\ND M i:( 11 A.\ 1< AL WAVKS 329 the l)rittle point tends to iiicroa.so as the molcculo.s j>;ot smallci'. Tliis wa.s supposod to !)(> lu'causc the ultiinalc cloiijial ion (doubtless due to \isco- elastie or (|uasi-c'()iifij>;iirational elasticity' and not kinetic theory ela.s- ticity as sonietini(>s said) declined with chain lena;th, so that the speci- men hroki' at lower and lowci' sti'ains, " e\-en though it was not I'eally in the glassy state. Now, the results above"' '' demonstrate that the shear modulus at a given temperature does fall off for a\erage molecular weights of polyisobutylene b(>low '^5000. Hence, the mechanical em- brittlement of the low molecular weight samples is not because they are stiffer, l>ut because they are weaker. pc)LY-cv-mi:thyl styrene: a "plastic" liquid Most of the li(iuid studies have been on polyisobutylene pol3''mers, made "hard" or "soft" by temperature or frequency, but under use Table V Poly-a-methyl 7; Poises m', 14 mc, Poises /I, 14 mc dynes/cm^ styrene Maxwell Voigt Maxwell Voigt I II 242 4340 111.4 502.7 23.6 14.3 5.1 X lO'' 7.6 X 10^ 4 X 10^ 7.4 X 109 conditions, considered rubbery. If one of the methyls in polyisobutylene is replaced by a phenyl, poly-a-methyl styrene, a hard plastic is pro- duced. Low molecular weight polymers of this composition are, however, liciuids at room temperature. Hence, it is interesting to compare their reaction to mechanical waves with that of polyisobutylene liquids of similar macroscopic viscosity. Table V lists a few properties at 25°C. Polymer I has roughly the steady flow 7/ of polyisobutylene B. Also, the ^'oigt 11' at 14 mc is similar: 15.2 poises compared to the 23.6 poises of the poly-a-methyl st^^'ene. However, the Voigt /x is already 50 times higher for the phenyl substituted chain. (This shows the shift of the second relaxation range, of course, where nearest neighbor interactions rule.) Even more striking, the temperature coefficient of a /x^ , calculated as the second principal shear viscosity in a frequency independent model," as before, is about 24 kcal, compared to about 12 for Hc for polyisobutylene in its similar model. Thus, although there was no ap- iwrent difference between the mechanical properties of these two poly- mer li([uids in their room temperature state, their dynamics diverged remarkabl3\ This was when they were studied with shear waves whose 330 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 frequencies approached the glass-into-rubber relaxation times. Clearly, again, mdividual interaction of chain-like chemical units and not any micellar or other special aggregation of them, predominates polymer mechanics. It still remains, however, to sepai'ale interactions of the basic; units within and between chains. Most likely, the model plastic vs rubber h quids just discussed differ in the high frequency region substantially only because of m/er-chain forces between phenyl vs methyl groups. However, especially in the high-frequency region, questions of intra- chain structure, such as the steric hindrance of adjacent pairs of methyl groups in polyisobutylene, restricted rotations about bonds, etc., come in. Obviously where configurational or quasi-configurational displace- ments are important, as in all cases of elongation >20 per cent (this is certainly an upper limit), flexibility of single chains needs to be under- stood. This is built deeply into chemical structure; plasticizers pre- sumably may change over-all configuration as well as modify interaction, but they are impotent to vary flexibility. Accordmgly, problems of rub- ber, usable in the Arctic, and of wire and cable insulation bendable at low temperatures always come back to whether the polymer chain bonds have free rotation. Some examples of the combinations of effects within and between chams can indeed be shown m several other polymer liquids which are rubber models. This influence of small changes m chemical structure is compactly illustrated by comparing a few other hydrocarbon polymer liquids with polyisobutylene. Also, rather dilute dipolar groups have been introduced in the linear polyester liquid polypropylene sebacate, whose structure is otherwise like that of hydrocarbons.'*^'' In Table VI, liquids of the given structure with some (unknown) distribution of molecular weights, were studied with shear waves at 77 and 142 kc at a temperature where each had the same steady flow viscosity. The figure chosen was 700 poises, and the temperature range required to adjust to it in the series was 10.9° to 85°C, meaning that the liquids had comparable consistencies at ordinary temperatures. Despite these similarities under steady stress, the retardation times, t', vary three-fold, with the highly substituted hydrocarbon chains, polyisobutylene and polypropylene, the highest. Despite the intermo- lecular action of the dipoles in polypropylene sebacate, the low polymer has a short retardation time, although its "brittle point" with decreas- ing temperature is far above that of polybutadiene or even polyiso- butylene. Presumably the flexibility around C — 0 — C bonds rather compensates for mcreased dipole interaction. Where both low polarity INTKKACTIOX OF TOLYMKHS AND MECHANICAL WAVES 331 r- r- r- r- III 1 u ^ u oio o XXX X K o !0 >o r- rt T-H 00 „ ,, s ooo o t— 1 1-H 1-H i"H --^ O ^ og XXX X Q> « eoic^ t^ X 01 f-H m lO lO o ^-t r% B-.S O (M l--^ '^t' 00 !U d, s 30 r~ -x r- "o 1 S — 1 rM rt .-1 •H Si XXX X C^ !» t^ CO (M (M t--i .-1 S CO^ B-.M r^^' W3 o o OCCKM -5tH ■^ CL, o r^ t^ r- t^ CO s III 1 OOO o (O 1 II XXX X •-^ -H (M GO t^ 5^ V (M C^ s „ I* to o to ^ s ooo o "o o 1— ( rt rt r-l Oh g T3 XXX X -o^ -t< O CO I© M ?3 d to O CO 1— 1 *r^ '5 •-• i-H > m !M 0 o CO(M Oh dl 1-H « « OOO o 'c^ a o dl XXX X s> .M >. lO lO lO 9 f^ •a COCd"iO OS J^ S --1 «D(N t^ > (^.a locot^ (m" o COC^ .-1 a Ph HJ n 0) ooo o < p.a ooo o o t^ t^ t- t^ r^QO t^ -H >OTh CO (M 00(M t^ -* i OOCXDOO O d ' ' .-I o Tf< O Oi t^ UOIOO '-H (M OOr-H -* , (U a; 0) -r- n c a >) 4J lu a; kl ^ T.:5T.« SJ s J Z-rJ S-d >, «2 - ~ r =3 "o (£ 332 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 aiul chain flexibility obtain, as in polybutadiene and the siUcones, dy- namic properties apparently accord with brittle points hi implying small temperature coefficients of relaxation times. In fact, the temperature coefficient for dynamic viscosity of jjolybutadiene is only al)out 1.5 kcal, whereas a comparable figure for polyisobutylene and polypropylene is 12 kcal. The frequency range in which the structural comparisons above were made, resides, as discussed earlier, ui the zone of configurational visco- elasticity. That is, over-all shape changes, rather than just nearest neighbor interactions, are predominant even at these comparatively short average chain lengths. Now, other recent studies of polyisobutyl- ene liquids, at 5 to 100 cps frequency, exhibit no rigidity at 25°C and above, although they become non-Newtonian rapidly as temperature is Table VII. Shear Dynamics of Polyisobutylene A". (Mv = 1660; 77,25° = 39.6 Poises) T °C Freq., cps Voigt Maxwel M dynes/cm^ 7) Poises 11 dynes/cm^ jj poises 25 25 27 27 27 266 1601 25300 41390 53060 3.8 X 103 4.8 X 10^ 5.4 X 10^ 1.5 X 10" 1.5 X 10« 39.2 38.4 19.9 19.9 18.1 1.2 X 106 3.1 X 106 1.9 X 10^ 1.9 X 10^ 2.6 X 10" 39.3 39.0 20.5 21.5 19.3 reduced. ' The questions are, where does the configurational elasticity drop out, as frequency is reduced at 25°C; and does it seem reasonable that this dispersion correlates with a shift in frequencies at lower tem- peratures. Partial answers are given by very recent studies of I. L. Hopkins of Bell Telephone Laboratories. He has eciuipped the tuning fork vibrator described earlier with two parallel vanes filled in between wdth a film of polymer liquid. Pure shear properties can be derived from the response of this system. Table VU lists a few typical figures obtained on polyisobutylene polymer A". These indeed show that the kilocycle relaxation zone (some new data by McSkimin's torsional pulse method are given for it) extends smoothly down to where dynamic and steady stress viscosities are equal. Seemingly there are no new "extra long time" relaxation mechanisms; probably the slow relaxation times some- times indicated for high molecular weight rubbers are just displacements of this configurational relaxation to long times because of high molecu- lar weight and internal viscosity. By contrast to the conclusions associated with the data of Table VII, INTERACTION OF I'OLV.M i:i{S AND M KC 11 ANICA L WAVES 333 some observations at low fretiuencies on isoviscous i>roperties of polyiso- l)ut\ienes A" and C indicate nenrW identical retardation times. Thus A" at '2o°C and (' at ()1°C have 77., = 39. () poises. The 77 vahies at 2()() and 1(100 cps arc also al)ont 39 poises, jjl at 'iOO cps is 3800 dynes per cm" tor l)()th li(iuids, and at 1(500 cps is from 3..") X 10* to 4.8 X 10* dynes per cm". Ill the filial section, mechanical waves have been used to explore dilute pol>'mcr solutions, to see how isolated molecules behave, free of interaction with cacii oth(>r. DIHTH I'OLYMKR SOLUTIONS Physical Principles in McdsKroncnIs Precise information on dynamics of solutions aiiproachiii<>; iniinite dilution (and thus complete separation of the polymer chains) is desired here. Again these must be shear dynamics; bulk rigidity of orduiary liciuids is so high that a few polymer molecules added cause little effect. Dilution is emphasized because even at 1 per cent by volume, high polymer molecule coils frequently interact, especially in "good" solvents. Thus, several workers have detected shear rigidity in polymer solutions, in one case for polymethyl methacrylate of average molecular weight 320,000, at 1 per cent concentration in o-dichlorobenzene. Very low frequencies used ('^10 cycles) there and in an earlier study suggest, however, that even here, appreciable entangling of the molecules created a temporary network such as studied b}^ Ferry."'' "'' Such was certainly present in the 5 to 18 per cent solutions of cellulose acetate m dioxane measured in one of the earliest observations of shear rigidity in polymer solutions. "" Accordingly, shice strictly linear, and hence non-interacting, mechanics are sought for the macromolecules in dilute solution, careful evaluation of experiments is essential. Since already it appears that important over-all (quasi-configurational) relaxations occur for, say, polyisobutyl- ene in the kilocycle range, and it is suspected that not all of the inter- actions involved are between chains, the torsional crystal technicjues are attractive. The absolute viscosity of these solutions is very low, so the ammonium dihydrogen pho.sphate crystal whose piezoelectric (lual- ities are appropriate for polymer licjuids in the circuits pre\'iously noted"' ' ' is advantageously replaced by quartz. Detailed electromechanical behaviour of such crystals in the pure liquids cyclohexane and benzene is of first concern. The electric field applied to electrodes on the suspended crystal produces mechanical 334 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 torsion generating pure shear waves. These waves may be modified by the environment around the crystal (vacuum, gas, liquid, solid) and react back. Thereby a mechanical I'esistance, Rm , and a mechanical reactance, X m , are imposed on the electrical properties of the crystal element in the circuit. This connection comes out as: ARe = KiRm A/= -K,Xm, where ARe is the increase m measured electrical resistance of the crystal element in the medium compared to in vacuum (or practically in dry air or nitrogen). The decrease in resonant frequency of the crystal ele- ment under these conditions is A/. Thus Ki and K2 are electromechanical constants, which fundamentally may be calculated from the dimensions and piezoelectric constants of the crystals. Now, in simple, Newtonian liquids, Rm = Xm = 'Virfrfp Thus, by carefully measuring ARe (or A/) on a liquid of accurately known density p and viscosity 17, at a given frequency /, and a given temperature, the constants Ki and K2 may be evaluated without as- sumptions and approximations of deriving them. Their constancy will then reflect the electromechanical stability of the system. Their be- haviour under various conditions will be illustrated below. One further point is that when a liquid or solution does exhibit shear rigidity, or, in other words, if the single large molecules in a dilute solution are able to store energy, then Rm > Xm ■ Hence, in this case, the observed quantities ARe , and especially A/ require particular precision. In this regard, typical magnitudes of change of /r between dry air and pure cyclohexane, at various temperatures, appear in Fig. 10. Questions often arise as to the arbitrariness of suspension of the radi- ating crystal, by the fine supporting and lead wires. The effects with the plain wires, in the solid curves of Fig. 10, are somewhat, but not radically, changed when a metal bead is put on, heavily loading vibra- tions in the wires, as shown by the dashed curves. In Fig. 11, a some- what larger influence of the loaded support wires is shown for the Re values, but both curves, by their smoothness and shape over a tem- perature range where the thermal expansion and other elastic constants of the metal support wires are quite different from those of the quartz crystal, affirm reliability of mounting and electromechanical coupling. Fig. 10 shows, even for an 80-kc crystal, that A/ for an organic liquid IXTKHACTION OF POLYMERS AXD MECIIAMCAL WAVES 335 78,930 78,920 > O o u Z UJ 78,910 3^ 78,900 - U: 78,890 78,870 AIR ,0»^-M ^^>r£^^=^ 8=5 ^2^^ &s^ "^^^Y^ >r2r^ P^^ BEFORE AFTER .- «=^ CYCLOHEXANE ^ 1 ,^-;^:::::=^^ ^^=^ r^ ^ ^ '^ 45 50 0 5 10 15 20 25 30 35 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 10 — Temperature variation of resonant frequency before and after add- ing weights to mounting wires. (or dilute polymer solution) is bothersomely small. An excellent oscillator at 20 kc can hardly be expected to drift less than ±2 cycles, but at 20 kc the A/ like that between the sets of curves on Fig. 10 might be only 10 cycles, so 20 to 35 per cent error could come in. Hence, a different scheme for measurement of ju than that in earlier systems ' was evolved. The tenth harmonic of the (say 80 kc) resonant frequency was beat against the 79th harmonic of a controlled standard 10-kc frequency. An interpolation oscillator accurately readable to 1 cycle then supplies the many hundred (roughly 1000) difference between these two high 90,000 70,000 (^ 60,000 m 2400 ^ h^^ — - BEFORE - AFTER A***- CRYSTAL IN ^ CYCLOHEXANE (80 KC) A^ L w r "^^ '^'^J kj 10 15 20 25 30 35 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 11 — Temperature variation of resistance at resonance before and after adding weights to mounting wires. 336 THE BELL SYSTEM TECHNICAL JOURNAL. MARCH 1952 __^__^i^;^___^_a===^ 2000 < 15 20 TEMPERATURE 25 30 35 DEGREES CENTIGRADE Fig. 12 — Temperature variation of crystal constants A'l and iv^ at 80 kc be- fore and after adding weights to mounting wires. harmonics. In this way, and m about 30 sec a balance can be conven- iently achieved and the requii"ed ten-fold gam in accuracy attained. By these means, and with best literature values of viscosity and density (which were checked m the laboratory at several temperatures) for purified solvents, curves for Ki and K2 were obtamed as exhibited in Fig. 12 for 80 kc. Behaviour of Ki at different frequencies over a tem- 1500 ^20KC -=:i p OAD CYCLOHEXANE • ▲■ BENZENE 40 KC n — ,80 KC, a— ^; 1 J 1 'i-v tJ Jn—— ' 45 50 0 5 10 15 20 25 30 35 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 13 — Temperature variation of crystal constant Ki over a frequency range with benzene and cvclohexane as standard fluids. INTERACTION OF POLYMERS AND MECHANICAL WAVES 337 perature range is shown in Fig. 13, and of A'o , in Fig. 14. Fig. 14 })rings out the significant point that in the present arrangement, wliere the osciUating crystal is innnei'sed in the li(iui(l stuched, the (liclcrlr/c pvop- erties of the hciuid are important. Apparently the dielectric losses e\'eii of thes(> pm'iHed hydrocarbons are dilTerent enough so that K> at 80 kc is (|uit(> dilfereiit for IxMizene and cyclohexaiie. (I)iel(M'tric studies ha\'e previously indicated difliculty in preparing benzene ha\'ing theoi'etically expected loss.) It is also possible that slight differences in wetting the crystal cause Ko to vary with the liijuid used. The A'l and /vo valu{\s determined for all the various conditions above were then used und(M- these conditions for measurements on the polymer 10 15 20 25 30 35 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 14 — Temperature variation of crystal constant K2 over a frequency range witli l)enzene and cj^clohexane as standard fluids. solutions in the kilocycle range. In the megacycle range, the balanced shear wave reflectance technique gave satisfactory results over certain concentration zones which could fairly well be extrapolated to high dilutions. Thus, over the whole spectrum, there seems to be no doubt al)out the reality of the effects described below. That is, their magnitude far exceeds experimental uncertainty, as demonstrated in this section. POLYISOBUTYLENE SOLTTIOXS; DYNAMICS OF SEPARATE CHAINS Solutions of i)olyisobutylcne of M, = 1.2 X 10*' from about 0.1 to 1.0 wt. per cent concentration in cyclohexane yield R m and Xm curves as shown in Fig. lo. The points coincide for the i)ure soh'ent, as they 338 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 40 5 O 30 U 25 I 20 pes^ TMiS^ ^^ ' „.^ .o-- -^r- REACTANCE ( u 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CONCENTRATION IN GRAMS ( PER 100 ML. OF SOLUTION) Fig. 15 — Electromechanical interaction of solutions of poh-isobutylene (M, = 1.18 X 10^) in cj'clohexane with crj-stal vibrating torsionall}- at 20 kc. should for a liciuid having only viscosity. But, apparently as soon as any polymer chains are added, the curves diN'erge. A stiffness coming from separate chain molecules is being displayed. ^° Qualitatively, theoretical expectations of Kuhn ' " and others seem justified, at least that there is a relaxation mechanism for isolated chains. The usual question of how best to express the dynamical results arises. The procedure of earlier sections for polymer solids and liquids will be followed. In general, a frequency dependent modified Maxwell element as sketched on Fig. 16 will be used. However, a frequency-independent analysis has also been carried out for one sample system, and, from this, basic mechanical constants of single "average" molecules are obtained, if it is reasonable to relate the mechanical models for the liquid con- tinuum to the discrete chains dissolved in it. Fig. 17 shows typical results from the simple scheme of Fig. 16, where the pure solvent viscosity, rjA , has been considered to be in parallel with a Maxwell element. The total shear rigidity of the solution (at a given concentration) is represented by fXB . The viscosity of the polymer molecule coils in solution with the solvent streaming through them is i Mb = (r2-x2) CO 7)2 a;/?77s-2RX Mb\ 2RX (r2-x2)' v.\h v.\h r)A = top COfi (0/37)^- 2RX Fig. 16 — Relations for calculation of shear stiffness and viscosity of dilute polymer solutions. INTERACTION OF POLYMERS AND MECIIANK'AL WAVES 339 taken to be tjb . Tims, tlie stead}' flow viscosity, r/, = r;,, + Vn . Also, V.i + Vb = Vr Ol Vb V'P Va n.i under steady flow, or, allernati\ely, approximately a "dynamic intrinsic viscosity" LVa C. can be written for any given freqneney. The curves in Fig. 17 are frequency dependent, however, although it turns out that t?/, is only slightly so. Nevertheless, the considerable rise of 7? -1 al)o\'e the piu'e solvent viscosity, as the concentration is increased, indicates other mechanisms are being lumped into tja . As usual, some 0.03 O 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ^ CONCENTRATION IN GRAMS (PER 100 ML. OF SOLUTION) Fig. 17 — Rigidity and viscosities of polj'isobutj'lene (M, = 1.18 X 10* clohexane, at 25°C and 20 kc. in cy- extensive distribution of relaxation times is probably responsible. How- ever, from the chemical point of view, it is best to see if some principal mechanisms related to known structures can be identified. If so, they could be associated with new ideas about the details of polymer intrinsic viscosities, as well as the form of isolated molecules. ' ' ' First, the frequency dependence of the hb of the model of Fig. 16 is as shown on Fig. 18. Striking regions of dispersion appear, although more points are needed to define the 10 cycle zone. Actually, many sets of data have been obtained in the 10 cycle zone. Recently, an immersed (luartz tuning fork has given the approximate value shown for 2300 cycles. The experiments of Fig. 18 were on a polyisobutylene having M, = 3.9 X 10 , dissolved in cyclohexane. Values of 77.4 and tjh were, of course, also obtained. The results were then analj'zed for a system of 340 THE BELL SYSTEM TECHNICAL JOVRXAL. MARCH 1952 90,000 1 2 4 6 i: 00,000 -: 100 400 looo io,cc: FP.z;_5:.CY IN KILOCYCLES PER SECOND Fig. IS — Frequency dependence of shear stiffne^ of 1 per cent solutions of polj'isobutylene in cyclohexane at 25"C. Fig. 19 — Schematic diagram of possible sources of rigidity of single chain molecules in solution. IXTKKACTIO.N OF I'OL^M KHS AM) M KCHA.NK ' AL WAVKS 311 tlirco ^Maxwell eloiiKMits in i)arall('l witli again, as in Fig. 10, a sohcnt \is('()sity. this tinio calUMl tji (truly ahsolulc sohxMit x'iscosity) in pai'allcl with tlicni. 'I'hc ^u/, cui'xc ot Fig. IS, running through the ol)S('r\'('(l points could then be calculated, by some special ti'ial methods, with which Messrs. H. T. O'Xcnl and (). J. Zol)el of Bell Telephone Labora- tories kiii(ll>- helped. This analysis, identifying three principal relaxation regions for the motions of polyisobutylene chains in cyclohexane, gave for a 1 per cent solution (taken as linear part of concentration curx'e and hence e(iui\'a- lent to high dilution). Principal rigidities H2 — 890 dynes cm" fjL-.i = 3,190 dynes cm' M4 = 84,000 dynes/cm" Principal viscosities r?i = 0.0082 poise (pui'c cyclohexane) 7]2 = 0.255 poise 173 = 0.000 poise r?4 = 0.004 poise Principal relaxation frequencies j'2 = 550 cycles f, = 8.45 X 10' cycles fi = 3.52 X 10' cycles Tentatively, these mechanisms may be schematically described as on Fig. 19. Here, the polymer coil, subjected to shear waves hi dilute solu- tion, exhibits rigidities ^2 , Ms and ^4 , all shoAvn on different scales. m2 is the configurational elasticity because of actual changes in root mean scjuare separation of chain ends, as from R to R'. It is retarded by vis- cous drag through the solvent, 772 , which is presumably the main source of characteristically high t?^ of chain polymer solutions. The relaxation frecjuency for this mechanism is low — a few hundred cycles. It may come in significantly in work on more concentrated solutions at low frequencies,"*' "' ' "' where chain entanglement is nevertheless the dom- inant factor. ju:! is when segments of the same chain in the molecular coil tempoi'ai-ily entangle with each other. Striking evidence has recently been given liy Fox and Flor}^^* that because of mutual interference, the theoretical random flight configuration of a chain gives very much too small a 342 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 molecular coil volume, Ve . This suggests that thermal agitation tending toward a smaller Ve , and excluded volume or repulsions forcing a large one, will cause collisons or entanglements which might last long enough to give a van der Waals cross-bond as denoted by crosses on the ms sketch. (The actual forces in these would somewhat resemble those be- tween different molecules in the concentrated solutions of Ferry .^*' ") This mechanism has the reasonable (based on Ferry's and others' work) relaxation frequency of 8.45 X 10 . A small viscosity, 773 , may comprise friction of slippage at the entanglement points, with both the polymer and associated solvent molecules. In Fig. 19, M4 is a relatiA'ely high stiffness presumed to be some average hindrance to rotation of one segment with respect to another. In the sketch, close-packed spheres representing methjd groups in polyiso- butylene are portrayed. Their force fields overlap more in some places than others, in the meandering of the chain to form the molecular coil (of course, some tail-to-tail structures may be important here; they have all been shown head-to-tail in the sketch). Thus, this total internal steric restraint on chain flexibility, with a relaxation frequency of 3.5 X 10 , contributes greatly to the large dispersion of rigidity in the megacycle range noted in Fig. 18. The related viscosity, 774 , is again low. There is no doubt a considerable distribution of relaxation character- istics associated with each and all of these mechanisms. Physical Properties Per Molecule Since the viscosities and rigidities in the dilute solutions indeed seem to be additive with the number of molecules present, values of these properties, for the hypothetical mechanisms, can be expressed per aver- age chain. Of course, the measured quantities are expressed as constants per cc of solution, but it may be useful to think of in terms of one aver- age chain in each cc. Then, the shear deformation of this cham could be denoted by a force constant. The associated viscosities remain, however, dependent on solvent surroundings. Thus, for the polyisobutylene of M„ = 3.9 X 10 , in cyclohexane solution, at 25°C the molecular quan- are: Ih] = 17 X 10"'' d>Tie cm [rj,] = 1.6 X 10"'' poise [fz] = G X 10"'^ dyne cm [773] = 3.9 X 10"'^ poise [/4] = 16 X 10"" dyne cm [774] = 2.4 X 10~'' poise In the section on polj^mer liquids, the high-frequency modulus ^ was attributed to a nearest-neighbor glass or crystal-like mteraction (since the actual values were indeed typical of the hardest organic solids). INTERACTION OF POLYMERS AND MECHANICAL WAVES 343 However, in polyisobutyleiie (and to some degree in poly-a-methyl styrene), it is espeeiall}' difficult to distinguish inter-chain from inira- cliain crowding of metliyl groups. Tfius, while average center-to-center separation of methyls is '^4 A in adjacent chains, it is <2.5 A within chains, in polyisobutylene. This crowding is apparently strong; tlic ob- served AHp^n is only 12.8 kcal per mole instead of the 19.2 expected.^"^' ''' The energy of steric hindrance thus amounts to almost half of the actual heat of polymerization. It is reasonable that a large part of the hardness of a mass of polyisobutylene chains, such as in the liquids, should there- fore reflect the same mechanism as that for ^4 (Fig- 19) in the dilute solutions. A rough check on this can be made. A polyisobutylene having considerably lower molecular weight than 3.9 X 10*^ and thus inter- mediate between the "liquid" and "solid" ranges, had a Maxwell shear modulus in the megacycle region (14 mc) of /x = 5.3 X 10 , at 25°C. The number of molecules/cc, with individual [/4] given above, necessary to give the observed density of this polymer was multiplied by [/4], giving M = 2.8 X 10^ dynes/cm . Accordingly, about half of the observed high frequency rigidity of polyisobutylene, at 25°C, may be calculated from a "molecular constant" embodymg mtra-chain stiffness. jMuch more refined and detailed treatments are requu-ed to generalize these "molecular constants" which are after all, as shown below, de- pendent on using a thermodynamically "inert" solvent. However, much as structurally significant dipole moments can be derived from measure- ments in dilute solutions, it seems hopeful that macromolecular me- chanics can be so elucidated. Also additional structures, such as poly- propylene and polj^dimethyl siloxane compared to polyisobutylene, are currently being studied. Temperature Variation Some further behaviour at different temperatures and solubihties of separate chains in dilute solution may now be considered against this background of possible mechanisms. Practically, these studies will bear on processing and properties, lacquers, paints, and casting solutions of polymers, as well as on the other qualities outlined in the introduction. Results may be conveniently discussed in terms of the modified Max- well single element, with factors va , Vb , and mb (Fig. 16). Mostly, the kilocycle range, reflecting molecular coil changes, will be of interest. For comparison, it may be noted that at 20 kc, the polyisobutylene whose fiB = 1061 dynes/cm in 1 per cent solution in cyclohexane receives 889 dynes, cm" of this from /xo , the retarded configurational mechanism; 169 344 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 j\ 1 000 \ ■\ ^ • 20 KC O 40 KC A 80 KC ^^ ^^^ ^ 7^ T : ^ '~~~~ ■~~~ ■ — : — • 10 45 50 15 20 25 30 35 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 20 — Temperature variation of rigidity of 1 per cent solution of polyiso- butylene (M„ = 1.18 X lO^) in cyclohexane. dynes/cm" from the entangled segments stiffness, ms ; and only 3 dynes/ cm^ from the intra-cham stiffness, m . Thus, chain configuration me- chanics, including the associated viscosities, can be well enough thought of in the foUowmg paragraphs, in terms of mb , ^b and r\A ■ The exponential decrease of m with temperature familiar for polymer solids and liquids is much suppressed in the /xs vs T curve of Fig. 20. While the in , internal rotation, mechanism for single poljasobutylene molecules probably has a considerable activation energy, that for the JU2 , configurational, rigidity should be very small. Then, without re- tardation, the mtrinsic chain modulus would rise with rismg tempera- ture. These influences seem to combine to give the modest declme of /xb appearing m Fig. 20. If these rigidities are plotted against 1/T, the temperature coefficient is 2.3 kcal. This is much less than the familiar values for the stiffening of rubbery solids, and emphasizes that inter- chain action reigns then. Solvent Variation Effects of solvents of different (mostly positive) heats of mixing on state of polyisobutylene molecules in solution have been nicely estab- lished by Fox and Flory. Especially, this work has clarified principal factors in the intrinsic viscosity expression w = ri 100 = luM 1/2 3 INTERACTION OF POLYMERS AND MECHANICAL WAVES 345 Here, F. = effective volume per molecule (and hence as determined by chain configuration), M = moic'cular weight, a represents change in linear extcMil of molecule l)ec:iuse of mutual interference of segments and s? expresses [\\v hydrodynamics int(M'action of solvent and molecular coil (including \'aryiiig degrees of "straining tiu'ough" the coil). ' '*■' '' ' Interpretation of the mechanical properties of chains in tlilute solution, with reference to the rough concepts of Fig. K), arouses particular in- terest in the factor a'\ For a high molecular weight polyisobutylene, intrinsic xiscosily theory""'^ indicated that a the ratio for volume of actual coil dividetl by volume for ideal random flight coil was 3.81 in cyclohexane but only 1.42 in benzene, both at 30°C. This strikuig alter- ation in equilibrium chain configuration, a variable which is not readily introduced into polymer liciuids or solids, appears in the inherent vis- cosity vs c curves in cyclohexane, Fig. 21, and benzene. Fig. 22. The lai'ge difference in [r}] at 25°C, 6.00 in cyclohexane vs '^l.S in benzene, indeed emphasizes the different soh'ent powers. " Likewise, the large in- crease of [77] with temperature in Fig. 22 accents the poor solvent qual- ities of benzene. Too, empirically, polymer molecules which are either tight coils or are actually chemically cross-linked to form microgel mole- cules characteristically show positive slopes of inherent viscosity vs c plots. Accordingly, all this evidence for large changes in the conforma- tion of chain molecules in "good" vs "poor" solvents should show up in dynamics of dilute solutions. Also, technically, Ciiiite different physical properties are found for polymer-plasticizer compounds where compat- ibility is high (good solvent) than where it is low (poor solvent). Here, 6.00 f \ \ \ \ k \ k \ ^ --. ^ ^^-> -^^ 0.250 0.500 0.750 1.000 CONCENTRATION IN GRAMS PER 100 ML. OF SOLUTION Fig. 21 — Inherent viscositv of ])olvisol)ut vlenc (.1/^ = 3.87 X lO") in cyclo- hexane at 25°C. 346 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 3.500 50°C '^ 35°C • — - ( J ._-- 25°C ^ 1.500 ;=— — 0.250 0.500 0.750 CONCENTRATION IN GRAMS PER 100 ML. OF SOLUTION Fig. 22 — Inherent viscosity of polyisobutylene (Af, = 3.87 X 10'') in benzene, at various temperatures. more flexible compositions are often produced with low compatibility plasticizers — indeed, sometimes with those on the verge of phase sepa- ration than with those with highly favorable heats of solution. This would mean that the bad solvents would compress the chains so that they would be more easily strained than if they were in a "free cham" or even extended configuration. If single chain, visco-elastic stiffnesses are acting this way, the dynamic mb would then actually decline as heat of mixing become more positive. 2 1200 Fig. 23- 20 kc. 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000 CONCENTRATION IN GRAMS PER 100 ML OF SOLUTION -Rigidity of polyisobutylenes in cyclohe.xane and benzene at 25°C and INTERACTION OF POLYMERS AND MECHANICAL WAVES 347 This seems to take place, as indicated by the lower compared to the upper curve on Fig. 23. Here, the /xs of the usual modified Ma.wvell model, at 20 kc, is plotted against c for the polyisobutylene of M, = 3.9 X 10 . Also, the middle curve shows hb for a polymer of about a tiiird of this molecular weight; while there is a small reduction in hb with il/, in this range, it is much less than the reduction caused by tightening up the polymer coil. Tiie M« N'alues per average molecule, [fa], fall from 18 X 10~ ' dyne cm in cyclohexane to 7 X 10 ' in benzene. (Of course, [Jb] for the inter- mediate molecular weight polymer in cyclohexane is only 5 X 10~ be- cause so many more molecules are present in solution.) The temperature dependence of /xb ^ilso becomes nearly zero at least o^•er the narrow range from 25 to 50°C, in benzene compared to cyclo- hexane. This seems to accord with the indications pre\'iously, from Fig. 20, for a lower molecular weight polymer, that different mechanisms are competing. These may be the configurational, with fi cc T, and relaxa- tion, witli M \'arying in some complicated way with T. Thus [ry] increases markedly with T, and presumably denotes an expandmg molecular coil tending toward the "normal" configuration in cyclohexane. At the same time, the relaxation processes with rising temperature tend to cause the decrease in yus typical of the upper, solid, curves on Fig. 24. In engineer- ing use, often times poorly compatible plasticizers give compounds which stiffen more gradually with temperature than do "solvent" plas- ticized ones. For similar reasons, the dynamic molecular coil viscosity, rji? , ought to vary less with temperature in thermodynamically poor than in good solvents. This is indeed seen in Fig. 25. On the other hand, tia for the modified ^laxwell element has been described as the solvent viscosity with segment hindrance and restricted rotation terms from the polymer molecules lumped in with it. These latter terms are presumably little affected by over-all configiu'ation (n-i term; the ms mechanism will be somewhat affected, but not the m4 , on Fig. 19). Thus, tja should have comparable temperature dependence in both good and bad solvents, as seems to be indicated by Fig. 26. Microgel Molecule Solutions The statistical coil of linear polymer molecules may be replaced by a chemically fixed, cross-linked network in microgel molecules. ' These may be made completely rigid, like Einstein spheres, or highly swell- able. The latter are hj^brids between rigid spheres and coiled chains. In 348 THE BELL SYSTEM TECHNICAL JOURNAL, MAR:H 1952 5 2600 ^v. "\ CYCLOHEXANE BENZENE ^- ^ O 40 KC A 80 KC V v,,^ T ^^^" ^ ^ ^^ ^ • — 1 4 1' — ^ ^ ; :rz:: ^ --_ ~--~":;; ( >— -^ ^ 200 5 10 15 20 25 30 35 40 45 50 TEMPERATURE IN DEGREES CENTIGRADE Fig. 24— Temi)erature variation of rigidity of 1 per cent solution of polj-iso- butylene (Af, = 3.87 X 10'') in cyclohe.xane and benzene. a \ CYCLOHEXANE BENZENE D 20 KC O 40 KC A 80 KC X V =^ ^^ ^ ^=^ "^ ^ ^, ■-HJ &-= ~ ^^^^ .^^ " „. ==^ 16 20 24 28 32 36 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 25 — Temperature variation of r/ij for 1 per cent solution of polyisobut ylene (M, = 3.87 X 10'') in cyclohexane and l)enzene. INTERACTION OF POLYMERS AND MECHANICAL WAVES 349 synthetic rubber, they confer uniciue flow properties, causing the excel- lent proccssibility of GR-S (>(). However, dynamic lenacity, such as in flex crack ^lowth, is (Ungraded \)\ their ])rcs('iicc. Now presumably the excellent extrusion ciualities of synthetic rubber composed of from 00 to 80 per cent microgel molecules are because of their individual shear stiffness. Thus, if a wire coating, for instance, is extruded at high rates of shear, chain molecules are deformed, and store energy just as dis- cussed in the earlier sections on li(iuids. After emerghig from the extru- sion die, they relax, and cause the gross retraction, shrinkage and rough- ness shown in the wire insulation of the upper photograph of Fig. 27. A polymer with about 70 per cent microgel molecules gi\es the smooth covering shown in the lower specimen of Fig. 27. Here, the shearing stresses of extrusion seem insufficient to distort the tiny networks of the microgel molecule; in any case, the covering does not roughen or relax. Similar effects have been found for microgel plastics. Neverthe- less, unlike gross or macro gelation, the whole melt can flow. On this basis, dilute solutions of microgel molecules ought to hidicate high shear rigidity per molecule. The mechanism /i3 of Fig. 19, in which now the junction pomts are not temporary, but are primary valence cross-links, should be predominant. Fig. 28 shows, for a polybutadiene microgel in cyclohexane," that hb has mdeed risen, compared to equal ui 0.020 o o y 0.012 •-V CYCLOHEXANE BENZENE ^^ ^ ^. O 40 KC A 80 KC ^ ^ :-^ K ^*^*^« ! H -^"^ brr. ~~-: -• 5 10 15 20 25 30 35 40 45 50 TEMPERATURE IN DEGREES CENTIGRADE _Fig. 26 — Temperature variation of ?? i for 1 per cent solution of polyisobutylene {Ml = 3.87 X 10") in cyclohexane and l)enzene. 350 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Fig. 27 — Effect of microgel molecules in synthetic rubber on smoothness of extruded wire insulation. Rough covering is fi-om high-speed extrusion of GR-S without microgel. weights of chain molecules. Further, accompanying the extremely high average molecular weight of the microgel (18.6 X 10 ), the [/s] per average molecule is 42 X 10~ " dyne cm or about twenty-five times that of the polyisobutylene with M, = 3.9 X 10 . Also, the temperature coefficient for ms of polybutadiene microgel is low. Of course, polybutadiene, as chams or as microgel molecule segments, has many double bonds. These will surely influence the m , or internal rotation mechanism. Further work remams to show just what is their effect in the microgel case. But, it is interesting to compare ms values for Hevea rubber chains with those for, say, polyisobutylene, which has only single bonds in the chain. =3. ..OJ ^2 10 Q 2000 5? D 30 35 40 45 50 55 TEMPERATURE IN DEGREES CENTIGRADE Fig. 28— Rigidity of 0.5 per cent solution in cyclohexane of polybutadiene microgel {Mw = 18.6 X lO^) at 20kc. INTERACTION' OF POLYMERS AND MECHANICAL WAVES 351 Ilci'ca Rubber Solutions The compari.soii of etiual weight coiiceutratioiis of natural rubber m cyclohexane ^vith polyisobutylene in cyclohexane is surprising: IIe\ea rul)bcr M„ = .23 X 10' nu = 1350 dynes/cm", 1 per cent sohition (eorr.). Polyisobutylene M, = 1.2 X lO'' /x^ = 1000 dynes/cm , 1 per cent solution (corr.). Both results are at 20 kc. The higher value for natural rubber may be l)ecause of the double bonds causmg stiffening of the cham. On the other hand, maybe easy rotation around single bonds raises the ms part. Cer- tainly the VISCOUS retardation ivithin natural rubber chains is very low, as noted in the section on solids. However, its interaction with, or con- figuration, in cyclohexane may be peculiar. The [j'b] per average mole- cule is, however, low, being 15 X 10~ dyne cm at 25°C. Polystyrene Solutions Much work, on light scattering and other properties, has mdicated appreciable intra-chain stiffness for polystyrene, "* but still much freedom compared to polyisobutylene. ^ However, this work, as well as AHp^n of 17 kcal compared to '^lO kcal calculated for no steric hindrance, sug- gests comparati\-ely small restraints on ideal flexibility. This needs to be checked by a frequency analysis of dilute solution mechanics, but poly- styrene seems to be a reasonable example of "plastic" behaviour at room temperatiu'e because of interaction between the chains. (It is recalled that, earlier, a-methyl styrene polymer was cited as plastic model show- ing both intra- and inter-chain stiffness. Unlike in polystyrene, the intra-chain factor shows up in a AHp^n of 9-10 kcal, a third less than that calculated if there were no steric hindrance.) Thus, no evidence of unusual stiffness appears in Fig. 29, when, indeed, the mb values are considerably lower, for eriual weight concentrations, than those for nat- ural rubber. The highly milled rubber studied had ikf, very nearly that of M, = 0.234 X lO'^ of the polystyrene, so the [Jb] per a\'erage poly- styrene chain, 4.5 X 10~ dyne cm is less than a third that of the rub- l)er. No wonder that at high temperatures, where the phenyl group interaction between chains is much reduced, polystyrene makes a good rubber. Also, hi Fig. 29 are shown data for a polymer of Mr, = 1.2 X 10 , made in emulsion and having [tj] = 4.350 in b(>nzcne at 25°C. 352 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 The polystyrene solutions discussed above were in benzene, a good solvent. Here, the situation is converse to that for polyisobutylene ; for polystyrene, cyclohexane is a poor solvent and benzene, good. Hence, if the previous interpretation of reduced single chain quasi-confignrational (fj-'i) stiffness is general for solvents of more endothermic mixing, the "plastic" molecule polystyrene shoukl show it in cyclohexane. This is indeed evident in Fig. 30, showing one of the same polystyrenes of Fig. 29, measured at 20 kc (normalized to 1 per cent concentration). Also, on Fig. 30 are shown the inherent viscosity (practically, the intrinsic vis- cosity, in this case) and the absolute viscosity of the 1 per cent solution 2 1200 (0 ^ 1000 A Mt; = 334,000 O M;^ = 1,204,000 / y \ y J y 25°C /^ ^^ 1 y [^ 60°C .--'^ ( K ^ / r 0 (0 20 30 40 50 60 70 80 90 100 FREQUENCY IN KILOCYCLES PER SECOND Fig. 29 — Change of shear stiffness, ^xb , with frequency, for 1 per cent solutions of polystyrene in benzene. under steady flow, rjs . These are all plotted against temperature down to phase separation, at about 26° to 27°C. The marked positive slope of the ^nrjr/c curve denotes the large con- traction in molecular coil volume preceding phase separation or msolu- bility. The absolute viscosity, rjs , however, rises with declining tem- perature because it is dommated by solvent viscosity, but when the polymer phase comes out, rjs abruptly falls off. The Mb values are consistent with this steady flow behaviour, except that the rise of hb at the tiu'bidity point seems to be because a layer of swollen polymer-rich phase forms on the torsional crystal surface. This condition is seen in Fig. 30 to coincide nicely with the abrupt changes in steady flow \'iscosity. The slight maximum m the mb curve at about 35°C may not be real. 353 INTKUACTIOX OF 1>( )LVM KUS AND M KCl I A M( ' A L WAVES It does come near tlie point of niinnnuni interaction tor tlie wliole sys- {vn\. In any ease, as discussed Ix'toic, the a\'era^(' temperature coefficient of fji/i in the poor solvent is very low compared to th(> jiood so1v(Mi1. The \alues of Mb iii't' roughly ^ to ^ those in benzene. GENERAL THEORY OF SINGLE CHAIN MECHANICS; KTIIN AM) KIHKWOOD As noted before, much of the present undei-standing of stress-strain properties of polymer chains, in dilute solutions, liciuitls or solids, has come from W. Kuhn's long intei(\st in this sul).iect. Many supplementary contributions have been stimulated by Kuhn's work, and new points of view have been introduced by others. For instance, I'ecently new and different proposals have V)een made about the flow birefringence and non-Xewtonian viscosity of solutions of deformable spheres. These ideas could be tested on suitable microgel solutions. Recently, moreo\-er, an especially significant general theory of visco- elastic beha\-iour of pol>'mer in solution has been constructed by Kii'k- wood.'' It explicitly considers the hydrodynamic conditions leading to the rigidity now observed for high-frequency shear waves. It formulates definitely the configurational changes of isolated chains in solution when strained in shear. As this theory is advanced to forms where simpler calculations can be made, it may answer many of the cjuestions raised by the new experiments on single chain properties. 14.0 0.8 13.0 0.7 12.0 0.6 9.0 0.3 0.1 r'"". I \ ,^ ^____ ^: h^ C (^ ■ \ ^ '^ 1 ^>^ \^ L *^ /^B ■ ZbY 30 35 40 45 50 55 visible ^temperature in degrees centigrade turbidity; ^ 5 If) o (0 yj Fig. 30 — Temperature dependence of absolute viscosity, 17.S , inherent viscosity, (nnr/c, and shear stiffness, mb of, 1 per cent snhitinn nf jiolyst yrene in eyclohexane down through turbidity point. 354 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 CONCLUSIONS To leave some impression of the elemental chemical structures which move around when wood, rubber, plastics, textiles and finishes are used mechanically — that has been the aim of this study. Polymer \'iscosities have been found in a variety of "solids"; rigidities have been demon- strated for very fluid "liquids" and solutions. Studies of these solid and licjuid extremes have given some chemical reality to the classical sprhig and dashpot models. Existence of compressional viscosity has been shown for polymer liquids and sohds. It may comprise a new quality for mvestigation of polymer structure. At present, too little is known of its origin to mter- pret further the effects of mtense ultrasonic Eradiation of polymer solu- tions. Experiments of Schmid and co-workers'" early indicated degrada- tion of molecular weight of polystyrene, so irradiated, but whether this is chemical, from local heating m the solvent, or actual physical coupling with the wave field, is still unsettled. However, these workers also con- sidered a compressional stiffness of the polj^mer molecules m the solu- tions, and showed that if there was coupling, it was not mertial (by dissolving polystyrene in solvents of exactly the same density, no reduc- tion in effect was observed). A point of general interest arises here; impact fractures of plastics presumably actually fracture some primary valence bonds. This is certamly true for many therm oset materials, and probably for cham compounds. Hence, if the detailed mechanism of how compressional waves move and perhaps rupture pohTner segments were known, information on the baflflmg problems of ultimate strength would be gained. The observations above on dependence of X and X' on molec- ular weight and structure provide only the barest start on this but a new goal is in view. Too, basic questions of how rapidlj" molecules being formed in a polymerization equilibrate in temperature with then* sur- roundings are elucidated by compressional wave propagation constants. For mstance, absolute rate measurements on velocity of chain growth cannot be said to be isothermal if they seem to be faster than the ther- mal relaxation time§ which the ultrasonic measurements indicate can be '-'lO"'^ to 10~^ sec. Likewise, more thorough understanding of velocity and dispersion of compressional waves in polymer solutions would clear up anomalies in velocity measurements for a wide variety of pohoners," some of which have been tentatively attributed to cham branching. INTERACTION OF POLYMERS AND MECHANICAL WAVES 355 ACKNOWLEDGEMENT Besides the extensi\'e collaboration of W. P. Mason and H. J. Mc- Skimin, we should like to attest to the help of T. G. Kinsley. BIBLIOGRAPHY 1. Meyer, von Susich and Valko, Kolloid-Z., 59, j). 208 (1932); Meyer and Ferri, Helv. chim. Acta, 18, p. 570 (,1935). 2. Meyer and yan der Wyk, J. Poli/mer Sci., 1, p. 49 (1946). 3. Kuhn, Zeit. phijsik. Chem., B42, p. 1 (1939). 4. Simha, /. AppL Phijs., 13, p. 201 (1942). 5. Gilmore and Spencer, Mod. Plastics, 27, p. 143 (Apr. 1950); Ibid., 27, (Dec. 1950). 6. Mooney, J. Colloid Sci., 2, p. 69 (1947). 7. Vila, I'nd. Eng. Chem., 36, p. 1113 (1944). S. Dexter and Dienes, J. Colloid Sci., 5, p. 228 (1950). 9. Hopkins. Baker and Howard, J. Appl. Phi/s., 21, p. 206 (1950). 10. Haward, Trans. Far. Soc, 34, p. 267 (1943). 11. Morev, Ind. Eng. Chem., 37, p. 255 (1945). 12. Jenckel, Zeit. f. Elektrochem., 43, p. 796 (1937). 13. Boyer and Spencer, J. Appl. Phi/s., 16, p. 594 (1945). 14. Richards, J. Chem. Phijs., 4, p. 449 (1936). 15. Baker, India Rubber World, 110, p. 543 (1944). 16. Alexandroy and Lazurkin, Acta Phi/sicochimica USSR, 12, p. 647 (1940). 17. Boyer and Spencer, J. Polymer Sci., 2, p. 157 (1947). An apt survey of much study of jilasticizers and mechanical properties. 18. Andrews, Hofman-Bang and Tobolsky, J. Polt/mer Sci., 3, p. 669 (1948); Dunell and Tobolsky, Textile Res. J., 19, p. 63 (1949); Brown and Tobol- sky, J. Polymer Sci., 6, p. 165 (1951). 19. Nolle, J. Polymer Sci., 15, p. 1 (1950); J. Appl. Phys., 19, p. 753 (1948). 20. Ivey, Mrowca and Guth, ./. Appl. Phijs., 20, p. 486 (1949). 21. Hall, Phys. Rev., 71, p. 318 (1947). 22. Mason, Baker, McSkimin and Heiss, Phys. Rev., 73, p. 1074, p. 1873 (1948). 23. Mason, Baker, McSkimin and Heiss, Ibid., 75, p. 936 (1949). 24. Sack, Motz and Work, ./. Appl. Phys., 18, ]). 451 (1947). 25. Lyons and Prettyman, J. Appl. Phys., 19, p. 473 (1948). 26. Ballou and Smith, J. Appl. Phys., 20, p. 493 (1949). 27. Ferry, Sawyer and Ashworth, J. Polymer Sci., 2, p. 593 (1947) for general review. 28. Ferry, Jour. Res., NBS., 41, p. 53 (1948). 29. Alfrey and Doty, J. Appl. Phys., 16, p. 700 (1945). 30. Leaderman, J. Colloid Sci., 4, p. 193 (1949). 31. Kelsey and Dillon, ./. Appl. Phys., 15, p. 352 (1944). 32. Wilson and Smith, Ind. Eng. Chem., 41, p. 770 (1949). 33. Hopkins, Trans. A.S.M.E., 73, p. 195 (1951). 34. Rorden and Grieco, /. Appl. Phys., 22, p. 842 (1951). 35. Gehman, Woodford and Stambaugh, Ind. Eng. Chem., 33, p. 1032 (1941). 36. Dillon, Prettvman and Hall, J. Appl. Phys., 15, p. 309 (1944). 37. Marvin, Fitzgerald and Ferry, J. Appl. Phys., 21, p. 197 (1950). 38. Fox and Flory, J. Am. Chem. Soc, 70, p. 2384 (1948). 39. Flory, Ind. Eng. Chem., 38, p. 417 (1946). 40. Baker, Chapter 8 in High Polymers, edited by Twiss, Reinhold Publishing Corp., New York, 1945. 41. Work, Textile Res. Journal, 19, p. .381 (1949). 41a. Tuckett, Trans. Far. Soc, 40, p. 448 (1944); Wiirstlin, Kolloid-Z., 120, p. 84 (1951). 42. Flory, /. Am. Chem. Soc, 65, p. 372 (1943). 356 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 42a. Fox and Florv, J. Phys. and Colloid Chem., 55, p. 221 (1951). 43. Fox and Florv, J. Appl. Phi/s., 21, p. 581 (1950). 44. Debve, Z. Elektrochem., 45, p. 174 (1939). 45. Gait", Phijs. Rev., 73(2), p. 1460 (194S). 45a. Baker, Rubber Chem. Tech., 18, p. 632 (1945). 45b. Harper, Markowitz and DeWitt, Abstracts, XII Int. Congress Chemistry, p. 274 (1951). 46. Van Wazer and Goldberg, /. Appl. Phi/s., 18, p. 207 (1947). 47. Kendall, Rheol. Bull., 12, p. 26 (1941). 48. W. Philipoff, Pht/sik. Z., 35, p. 884, p. 900 (1934). 49. Mason, Trans. A.S.M.E., 69, p. 359 (1947). 50. Baker, Mason and Heiss, Bull. Am. Phys. Soc., 24, p. 29 (1949). 51. Kuhn and Kuhn, Helv. chim. Acta, 29, p. 609, p. 830 (1946); /. Colloid Sci., 3, p. 11 (1948). 52. Florv and Fox, ./. Am. Chem. Soc., 73, p. 1904 (1951). 53. Fox," Fox and Florv, Ibid., 73, p. 1901 (1951). 54. Fox and Florv, Ibid., 73, p. 1909 (1951). 55. Fox and Florv, Ibid., 73, p. 1915 (1951). 56. Kuhn and Grim, J. Polymer Sci., 1, p. 183 (1946). 57. Carver and Van Wazer, /. Phys. and Colloid Chem., 51, j). 751 (1947). 58. Fox and Florv, Ibid., 53, p. 197 (1949). 59. Evans and Tvrrall, /. Poli/mer Sci., 2, p. 387 (1947). 60. Roberts, Jour. Res., XBS., 44, p. 221 (1950). 61. Kuhn, Kolloid-Z., 68, p. 2 (1934); Kuhn and Kuhn, Helv. chim. Acta, 26, p. 1394 (1943). 62. Huggins, J. Phys. Chem., 43, p. 439 (1939). 63. Del)ve, Phys. Rev., 71, p. 486 (1947). 64. Brinkman, Appl. Sci. Res., Al, p. 27 (1947). 65. Huggins, J. Appl. Phi/s., 10, p. 700 (1939). 66. Alfrev, Bartovics and Mark, J. Am. Chem. Soc, 64, p. 1557 (1942). 67. Baker, Ind. Eng. Chem., 41, p. 511 (1949). 68. Bover, /. Appl. Phi/s., 20, p. 540 (1949). 69. Zinim, J. Chem. Phi/s., 16, p. 1099 (1948); Outer, Carr and Zimm, Ibid., 18, p. 830 (1950). 69a. Kunst, Rec. trav. chim., 69, p. 125 (1950). 70. Cerf, Compt. rend., 226, p. 1586 (1948); Ibid., 227, p. 1221 (1948). 71. Kirkwood, Rev. trav. chim. Pai/s-Bas, 68, p. 649 (1949). 72. Schmid and Rommel, Z. Phi/sik. Chem., A85, p. 97 (1939). 73. SchmidandBeuttenmuller,Z.£'/eA-//oc/!ew..49, p.325 (1943) ; 50, p. 209 (194-!). 74. Natta and Baccaredda, Gazz. chim. Ital., 79, p. 364 (1949). The Reliability of Telephone Traffic Load Measurements by Switch Counts BV W. S. IIAY\VAI{1), Jli. (Maiiuscii|il rcccivcMl ( )cl()l)i'i' 15, 1051) The .s'(r//r7(. coiinl mcllwd of Iclcplwuc lidjlic mcafiurcmeitl /.s suhjcci to sam- pling errors. The nature of these errors is discussed and formidas are de- rived which describe the extent of the errors under normaHy encountered traffic conditions. IXTRODUCTION Of prime importance to the telephone traffic engineer is the deter- mination of the busy season busy hour load carried by groups of trunks or other circuits of a telephone switching system. Three direct methods of measuring such loads are found in the field today. These are: a. Peg Count and Holding Time Method The number of calls carried by the circuit group during the observa- tion period is counted. This number multiplied by the average holding time per call (in hundreds of seconds) and divided by the length of the observation period (in hours) gives an estimate of the group load in units of hundred-call-seconds per hour (CCS). The major drawback to this peg count method is that it requires a separate determination of the average holding time per call for the group under observation. R. I. Wilkinson^ has analyzed the sources of errors of holding time measure- ments. In addition, correlation between load and holding time introduces an error which has not been studied. b. Switch Count Method At fixed intervals the circuit group is scanned and the nvnnl)cr of busy circuits is counted. The total number of busy conditions cijunted divided by the number of scans is, then, an estimate of the load on the group in units of average simultaneous calls or erlangs*. This estimate is generally converted to CCS (1 erlang = 3G CCS) by ti-affic engineers since the * The name "erlang" for average simultaneous call was adopted at a plenary meeting of the CCIF at Montreux in October, 1946. 357 358 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 load entries of most traffic tables are in terms of CCS. For theoretical studies the erlang is a more convenient unit and wdll be retained here. c. Continuous Method The busy condition of each circuit is represented by a fixed increment of electrical current through an ampere-hour meter. The instantaneous current is then analogous to the calls simultaneously present so that the meter, which integrates the current, may be cahbrated to indicate hun- di'ed-call-seconds or erlang-hours directly. Although this method is po- tentially the most accurate, practical difficulties have limited its use. In addition to these direct methods, there are several methods of in- direct load measurement which, relying more heavily on traffic theory, make use of partial load indications, such as duration of group busy or the number of calls finding the group busy. Such measurements are less reliable than the direct measurements particularly when apphed to un- derloaded groups. This paper is concerned with the reliability of switch count load meas- urements since this method appears to have prospects of considerably wider adoption in the future. Main emphasis will be placed, both quali- tatively and by the appHcation of error formulas, on the relative effects of various measurement and traffic parameters on the accuracy of s'^itch count measurements. Where long derivations of formulas are required they are deferred to the Appendix. SOURCES OF ERROR As has been described, switch count measurements yield the average number of calls found present when a group of circuits is scanned at fixed intervals during an observation period. Usually only that period of the day during which the load is greatest is of interest to the traffic engineer. Because the load during such periods also fluctuates from day to day, measurements of the loads for several days must be averaged to provide a useful load estimate. There are two main sources of error, therefore, in sudtch count esti- mates of telephone traffic loads: 1. Each individual count of busy circuits is separated from the next by a time interval during which changes in load are not detected. Con- sequently, the load indicated by measurement may differ appreciably from the actual load carried. This difference can be decreased by de- creasing the interval between scans. RELIABILITY OF TRAFFIC MEASUREMENTS 359 2. Even if the load carried during a measurement period were known very accurately, it is still only a sample of the many loads that might be offered l)y the same source of traffic inider statist ically identical con- ditions. Therefore, the average of several load readings may be expected to be somewhat in error as an estimate of the true average of the traffic source. The latter aWU be referred to as the source load to distinguish it from the carried load. Mechanical and human errors are likely to be present as well but, since they are not inherent in the switch count method, they will be neglected here. SWITCH COUNT ERROR As shown in the Appendix, for periods of observation w^hich are rela- ti^•ely long with respect to average holding time made on traffic with certain assumed characteristics, the average error of switch counts in estimating traffic load carried in the same period is zero. The coefficient of variation of the error, which is the standard deviation of the error expressed in per cent of the traffic load carried, is given by: V:, = 100 rctnh( ^ ) - 2 t (p) a' NT /r\ _ 1 + e "^ _ , , ,. , . ,. r ctnhl - ) = = hyperbolic cotangent of V2/ 1 - e-"- 2 re = T/i > 20 where /■ = ratio of scan interval to holding time I = average hokhng time a' = carried load in erlangs c = number of switch counts T = length of observation period N = number of observation periods and where the following assumptions are made: a. Calls originate individually and collectively at random.! b. Holding times are exponentially distributed. c. Congestion loss from the group is neghgible. * I have recently learned that these carried load formulas have been published hy ConnyFalm mTekniska Medelandenf ran Kungl. T'elegrafstyrelsen, 1941. nr. 7-9. t See T. C. Fry, Probability and lis Engineering Uses, D. van Nostrand Co. Inc., New York, p. 216, for a definition of this condition. 360 THE BELL SYSTEM TECHNICAL JOURXAL, MARCH 1952 As shown ill the Appendix, this formula simplifies, when r < 2, to T = M / 1 ^ c'tV CWNTl ^^^ where c T = rate of scan in cycles per time unit. From equation (2) it is apparent that if the scan interval is of the order of a holding time, the error of an estimate of ti-aiSc carried is inversely proportional to the rate of scan and inversely proportional to the sfjuare root of avei'age load, holding time and hours of observation. For example, take the case where switch counts are made dui'ing the busy hour, five minutes apart on a trunk group carrying calls with an average holding time of 3 minutes and an average load of 5 erlangs (180 CCS). What is the error in the estimated load carried if the readings for ten days are averaged? (As- sume conditions (a), (b) and (c) are met.) We have A^ = 10 observation periods T = 1 hour t = 1/20 hour a' = 5 erlangs c = 12 scans per observation period re — T t = 20 average holding times per observation period From equation (2) since T / = 20 and r = T ci = 1.7 T' - iQQ,/ 1 -0150/ * ^ " 12^ r 6-O-10-M/20 " -^''/^ If, as proposed in the Appendix, it is assumed that the error has a normal distribution, there is 90 per cent assurance that observed values ■s\'ill fall within 1.64 F^ , or in the example within 3.52 per cent, of the true average*. Note that this error limit would be halved if the rate of scan were doubled or if four times as many hours of observation were taken. The coefficient of variation of the SAAdtch count error for constant values of T/t as a function of r is plotted on Fig. 1 for one observation period of a one erlang load. For loads other than one erlang the coefficient of variation is found by dividing by \/a'N- Thus in the example we have, using the dotted curve. * This assumes that a sufficient number of observations are taken so that a priori information maj- be neglected in making an estimate of the universe. HKLlMilLirV OF TUAFFK' M IvVSUKFMKNTS :^(;i Of more accuralch' usiiii>; tlic solid ciiiAc 1() The (MTor of usiiiu; (Mjuatioii ('2) is seen to 1h' ii(\<;Ii,ii:il)l(' for most jjuiposcs o\-(Mi when T I is less than 20. The pi'ohahilit y of an ()l)s(M-\'ation oc- cuiiiiii witliiii a .^ivcii miinlH'r of slaiulai'd deviatioii.s is widely published for the iioiinal curve. A few \'alues are y:i\-en below: 0.6745 1.44 1.64 2.00 3.00 Pi Probability of exceeding ± jw or ± zP' 0.50 n.S5 ().«)() 0.9545 0.9973 Fig. 2 is a plot for 40 obser\-ations of measured load vs carried load. Each observation was made for a half hour period on a panel hne finder group with switch counts made at the start and middle of the period. I t.o ;o.8 ) • 0.6 . 0.5 iO.4 - / // y ;'/ // - 6 / .'^ '/ // / / / f/ / ' y / / // // V / / 1 / / // / ■'/ / / // // 7 / // / / / ' / rc = / / /Y 20// /. / / / / / y / - y / y ^ ///> / \ - J / >■ ■y'l / 'A Y ^ Y ' ^ Y / Y APPROXIMATE FORMULA /^ i / /A ' . / Y VxVaN / 100 V ^' nctnh (|-)-2 / • '/ * / /) f / / / , ^J / y / 4 / / re A // 7 ^ / y/ / / / / \Cr FORMULA 1/ < y / .( / ^ 100 III, ^[r.ctnh(^)-2](rc-Kr + 2b)+(Kr2-2Kr+2b') ,1 1 1 ""r 1 1 , 1 , 1 8 t 0 l 5 ^ \ . 3 ( 5 { 3 1 0 2 0 3 0 4 0 6 0 8 0 IC )0 20 VvVdN , COEFFICIENT OF VARIATION OF ERROR IN ESTIMATING A ONE ERLANG CARRIED LOAD IN ONE OBSERVATION PERIOD Fi^;. 1 — Accurtifv of switch count estimate of load actually carried. 362 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 26 V- 24 Z D O O 22 20 ^ 12 < £ 10 z o _l fR 6 / / / ' / • =1 OBSERVATION PERIOD T = 1/2 HOUR t = 3 MINUTES L = 15 MINUTES C = 2 SCANS crx = 0.615Va' FROM FIG. 1 / / / / / » / / / / / / / > • y y y J / / / » / r / y y y / / / / • » / • y y y y / / / / / • / « • y • y' +20-/ / / / y y y / / / / ' / • y y y y / / ^''» '-20- / / / / y / / / / /? »„ / y y y / • • / 0 2 4 6 8 10 12 14 16 18 20 22 24 26 CARRIED LOAD IN ERLANGS ESTIMATED BY 60 SWITCH COUNTS Fig. 2 — Accuracy of switch count estimate of true average load. 28 This is compared with the average of switch counts made every 30 sec- onds which has a relatively negligible error. The average holding time per call for the group was 176 seconds. The accuracy of only two counts is surprisingly good and the observations are seen to he satisfac- torily between the 2o- limits. ERROR OF TRAFFIC IN A GIVEN PERIOD AS AN ESTIMATE OF THE SOURCE LOAD The average traffic carried in two different periods but generated by the same traffic source is subject to statistical variation. As a result, any measurement of load, even if measurement errors are eliminated, is only a sample of the wide range of traffic loads that might have been gen- erated by the same source of traffic under identical circumstances. KELIABILITY OF TRAFFIC MEASl'REMIONTS 3()3 ■I. Uiordiiu has shown' that the standard tlcviaticjn of the average traffic load for any one period is given by ~-'(^- 1 + «-"'') (3) \vli(M-(' a = the average soui'ce load / = a\'erage liolding tunc per call T = lcn_ti;lh of ohsciA'ation period (Assumptions are as liefoi'c with an additional one that all periods are in statistical equilibrium) When ? T « 1 this reduces to the form also given by F. W. Rabe^ or expressed in a per cent of the average I' (4) V, = 100 ^^ (a) Wlien A^ periods of length T are obser\'ed the coefficient of variation is reduced further to: In the example of the previous section, A^ = 10 T = 1 t = 1/20 a = 5 -' - /i^: = ^-% COMBINATION OF ERRORS Evidently if switch count readings are used to estimate the average which may be expected in other periods, the two errors described above should both be taken into account. The errors are probably correlated but this correlation is weak and at present no method of allowing for it 364: THE BELL SYSTEM TECHXICAL J(jrRXAL, MARCH 1952 is evident. Such a refinement would probably change the etiuation for standard deviation only slightly from that derived for the intlependent case; therefore independence will be assumed. The standard deviation of the sum of two independent variables is the square root of the sum oi the squares of the component standai'd deviations: 0". = Val + al 0) rctnh! ta' 2ta (Q\ S'f ^ NT a . Assuming -— is approximatelv iniity, that is, that carried load is approxi- a mately eciual to source load, Vs = 100 a/^ rctnhf ^) (9) anT \2 In the example given, K = 4 There is, then, 90 per cent assurance that the source average is within 1.64 X 4.96 = 8.1 per cent of the observed average. Note that doubling the switch count rate (which halves the switch count error) reduces the total error only to 7.6 per cent (about 6.7 per cent improvement), while doubling the number of hours of observations reduces the error to 5.9 per cent (about 30 per cent improvement). Plots of the coefficient of variation of a one hour observation of a one erlang load versus scan rate for various average holding times are given in Fig. 3 for a wide range of holding times. The coefficient of variation of error in estimating other loads may be found from Fig. 3 by dividing the unit load coefficient by -s/aNT. In the example, the unit load coefficient is found, by entering Fig. 3 with l = 3 minutes and rate of scan = c/T = 12/1 scan cycles per hour, to be 35.0 per cent. Dividing by \/5-l-10 gives a coefficient of variation of 4.96 per cent as before. It is evident from Fig. 3 that in- creasing scan rates is not a universal way to improve the accuracy of source load estimates. CHOICE OF SCAN RATES What then governs the choice of scan rate? Evidently increasing the rate increases the accuracy of carried load estimates to any point de- ]{i;mability of tuaffic mkasurements 3G5 sired. This is fai' tVom true if source load is l)(>iiij>; (vstimated. If the cost of maivinj>; a scan is constant, inci-easinji; the number of observation periods and decreasing the scan rate will iinproxc accui'acy of source load estimates M'ithout changini;- measurement costs. Tiie mmiber of hours axailahle for measuiina;, of course, limits this procedure, while the increase in accuracy becomes nej>;lioible as r be(H)mes large. On the other hand, if the cost of each obsei'vation is only slightly affected by the cost of making additional scans, a high scan rate might be justified. In ai)iilying the abo\-e relationships to traffic measurements, the usual cjuestion raised by the traffic engineer will be cither how many hours of data need he take to l)e reasonably- suic of his estimate or, conversely, how sure is he of an estimate based on available data. Assuming as before that the error distribution is normal, the per cent plus or minus error limits within which a proportion, P^ , of the estimates will fall is given by zVs ; the value of z corresponding to any selected P; may be found from tables of the normal probability distribution. "Reasonably sure" is often taken to mean that there is 90 per cent assurance that the error does not exceed 5 per cent. When P^ is 0.90, z is 1.64, so that under this condition 1.647s = 0.05, or Vs = 0.0305. Given scan rate and holding time, Vs is proportional to l/s/aNT accord- ing to equation (9) or Figure 3. When Vs is held constant, aNT is con- stant so that the plot of log NT against log a is linear, as shown in Figs. 4 and 5. The number of hours needed to meet any chosen reliability 100 80 s. - s \ ^ CALL HOLDING TIME \ ^ ^^' - \ v^ ^v'^ - \ v*o.\Po. - V N [* - N S ^ \^ \ ^ N. - \, N. N - MEASURING INTERVAL 120 SECONDS N s,^ V S - \ s S \, - N S ^ \ 1 _J- _L 1 1 _L ± 1 1 1 1 \ ^ \ 1 2 4 6 8 10 20 40 60 100 200 400 1000 2000 4000 10,000 LOAD IN HUNDRED-CALL-SECONDS PER HOUR Fig. 4- — Hours of measurement required for 90 per cent assurance that error in estimating source load does not exceed plus or minus 5 per cent when measuring interval is 120 seconds. requirements may then be read directly from such graphs. In the second type of question, z, NT, scan rate and holding time are fixed so that zVs is proportional to l/\/a. Plotting log zVs against log s/a again gives a linear plot as sho^\ai on Fig. 6. In the numerical example above, the limits of error corresponding to 90 per cent assurance may be read from Fig. 6 which is plotted for the appropriate assurance, average holding time and scan interval. Reading the error limits at the point where the 10 hours measured line crosses 180 CCS (5 erlangs) gives ±8.1 per cent as before. Fig. 5 may be entered to find the total number of hours required to reduce this error to 5 per cent. Reading at the point where the 180 second holding time line crosses 180 CCS gives 26 hours. QUALITATIVE EXTENSION OF THEORETICAL APPROACH The original traffic assumptions made in deriving the theoretical re- sults above are: a. Calls originate collectively and individually at random. RELIABILITY OF TKAFFIC MKASUHKMKNTS 367 LOAD IN ERLANGS 2 4 6 8 (0 800 600 *{? 200 100 80 2 60 O 1 1 1 1 1 Mill 1 1 1 1 Mill 1 1 1 1 II 1 II 1 — ^ N - s - N s ^. - -^ . CALL HOLDING TIME -^ [^' 0 - ^^ W - to. - ^ ^v - -^ ^. ^ ^ - ^N s. - MEASURING INTERVAL 300 SECONDS \ \^ s - \ ^ ^. - ^ \ 1 1 1 1 1 1 1 1 ^ ^ 1 2 1 5 3 1 0 2 0 4 0 6 0 1( )0 2( )0 4( DO 10 00 40 00 10,0 LOAD IN HUNDRED-CALL-SECONDS PER HOUR Fig. 5 — Hours of measurement required for 90 per cent assurance that error in estimating source load does not exceed plus or minus 5 per cent when measuring interval is 300 seconds. h50 "J 30 oc S^20 z O 10 in i ^ i 3 f) 3 7^ 1 0.06 0. 1 1 1 1 ! 1 1 0.2 LOAD IN 0.4 0.6 1.0 2 1 1 1 I 1 1 1 1 1 ERLANGS 4 6 8 10 1 1 1 M 1 1 1 2C 40 1 1 1 30 100 Mill 200 1 ^>>k ^c-« .^ V. ^^1 1 1 1 1 "^•v^* O' ^; .>., *^^^> ^.^f ^ ^ NUMBER OF HOURS ^^1 MEASURED ^^^- § k2 "^ ^ ^ \ ^ - ^-|: N> "V, V^ "^^ b^ '>v ^X - 1 s ^ "N. -"^^ >4? ^ ^ V ■\^ - CALL HOLDING TIME 160 SECONDS MEASURING INTERVAL 300 SECONDS V ^5^ V ^ ^ \, V, - 4> ^ ^ ^ ^ V V V 1 1 1 1 1 1 1 1 1 1 i 1 ^ ^: > 6 8 10 20 40 60 100 200 400 1000 2000 4000 LOAD IN HUNDRED-CALL-SECONDS PER HOUR 10,000 Fig. 6 — Limits of error reached with 90 per cent assurance in estimating source load. 368 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 b. Holding times are exponentially distributed. c. Congestion loss from the group is negligible. d. Observation periods are in statistical equilibrium. How do departures from these assumptions affect the reliability of usage measurements? a. Holding Time Distribution Experience in application of delay and loss formulas has shown that theories based on exponential holding times are often applicable to other holding time distribution cases which have a wide range. However, for a constant holding time distribution special theories often are called for. The average and standard deviation of smtch count estimates of carried load when holding time is constant, are given in part 2 of the Appendix. It is shown there that for estimates of carried load, X = 0 r > 1 T^, = 100 /j/~ (r - 1) (10) r < 1 minimum T'^ = 0 (/' = 1, |, ^, j, etc.) maximum T^x = 100 - / 1 is the most hkely to be met. For all values of r greater than one, the error given by formula (1) for exponential holding times is some- what greater than the error given by formula (10) for constant holding times, so use of formula (1) for the constant holding time case is con- servative. For values of r less than 1, the error is an oscillating function of r. The coefficient of variation varies from zero to 23 per cent above that for exponential holding times. Where r may not be accurately known the formula for exponential holding times again seems appropriate. In making estimates of the source load when the holding time is con- stant, if r > 1, each scan is uncorrected with any other, since no call can be counted twice, and may be considered a random sample of traffic. There are a total of Nc scans which have an average scan of a and standard deviation Va- The average error in estimating a is, therefore: s = 0 with coefficient of variation '■• = ^°° V^c = ^™ l/^ ('2' RELIABILITY OF TRAFFIC MEASl^REMENTS 3G9 Equation (12) may also be deri\'0(l with the ))1'(»('(m1ui-(' used for eciuation (9) using al = al -{- al . Foi' ^•aluo,s of /' large enougli to make ctnhf - j = 1 e(iuation (12) is approached by equation (9). For smaller values of /• (but with r still great(M- than 1), 1% for constant holding times is less than 1',. for exponciilial holding times. When r = 1, there is no carried load error. For \'alues of r less than 1, the coefficient of varia- tion of error in estimating source load average will vary from depending on the exact \'alue of r. It is interesting to note that Vs for r = 0.5 is the same as for r — 1.125. b. Loss The effect of loss in the group depends upon the disposition of the lost calls. In general, accuracy in measuring carried load increases with in- creased loss because under these circumstances fewer load changes occur l)(Hween scans. This is evident in the extreme case of a group which is 100 per cent loaded; a single switch count gives a correct reading for any length period. Obviously load readings at 100 per cent occupancy are not very useful in estimating offered loads since the amount of lost load cannot even be guessed at. However, in the cases of lost calls held (Poisson) or cleared (Erlang B), the offered load may be estimated from the carried load (less and less accurately as occupancy increases) and in the case of lost calls delayed the offered and carried loads are likely to be the same even at high occupancies. With high loss, therefore, estimates of source load are subject to errors not considered in deriving ecjuation (99); however, switch count error in estimating carried load will be materially less than predicted by equation (1). c. Random Call Originaiion On trunk groups which are alternate routes, calls may no longer be considered as oi-iginating at random. The resultant grouping of call orig- inations will tend to dcci-ease the accuracy of switch count measui'ements in estimating carried load; however, there is a corresponding decrease in accui'acy in estimating the source load from the carried load so that ac- curacy in estimating carried load may be less worthwhile. 370 THE BELL SYSTEM TECHNia\L JOURNAL, MARCH 1952 d. Statistical Equilibrium Statistical equilibrium may be thought of as the absence of trends in subscriber calling rates or holding times with the passage of time. The effect of trends on switch count accuracy in measuring carried load is very small except where the changes in traffic level are frequent and abrupt with respect to the scan frequency. Such traffic behavior is rare. Trends within the busy hour compUcate the problem of estimating the average source load. However, it can be shown that if the trends are small (in the order of 10 per cent to 20 per cent) little error is introduced by assuming that no trend exists. Large trends (in the order of 100 per cent), however, may indicate that the traffic source is so unstable that more hours of traffic data should be taken in order to insure that the sample is representative. Trends from day to da}^ do not affect the source load estimates in the same way as \\dthin hour trends. The source loads are seldom exactly the same on any two days although in most offices a load pattern is repeated from week to week. The traffic engineer may be interested in the average source load of either a typical week day in the busy season or, some- times, of the average of the two highest days in the week. As long as the source load of each particular day remains close to the average for that day of the week, the general average for several different days of the week, mil be known with about the same accuracy as if they had all come from a common source. If, however, there is no stable pattern in the source load, a third error in estimating the average is generated. There is some difficulty in determining whether or not variations in load, as indicated by measurements, are due to sampling variations or to an unstable source. Quahty control methods might be used to detect in- stability but gathering and processing sufficient data for such an analysis might prove uneconomical. In general, if a traffic engineer feels that his source load is unstable he \d\\ need more hours of data than indicated by formula (9) to meet a given criterion of reUabihty. CONCLUSIONS A theoretical approach to the problem of the accuracy of switch count measurements in estimating carried load and average source load has been explored.' It is believed that the assumptions made are satisfied sufficiently often in practice to enable fairly ■s^^de application of the re- sults of this exploration to traffic measurements. However, it should be kept in mind that where the assumptions are clearly not valid, special allowances mil need to be made. In any case, the confidence placed in RELIABILITY OF TRAFFIC MEASUREMENTS 371 usage measurements by a traffic engineer is a function of his experience and judgment. It is hoped that tlie lesuhs of this study will add to the knowledge essential to sound (lallic ciigincciing. APPENDIX DERIVATION OF SAVITCH COUNT ERROR IN ESTIMATING CARRIED LOAD — WITH EXPONENTIAL HOLDING TIMES This derivation is based on a similar deiix'ation by R. I. Wilkinson'. However, since load rather than holding time is of interest here, the em- phasis has been somewhat shifted. Assume^ that switch count measurements are being taken on traffic with : a. Calls originated indi\'idually and collectively at random b. Exponentially distributed holding times c. Negligible loss Let i = interval between scans t = average holding time a' = traffic carried, in erlangs T = length of observation pei'iod r - ^ = number of holding times in a scan interval t T c = . = number scans in observation period I T re = . = number of holding times in observation period N = number of observation periods. Consider that the observation period begins with the first scan and ends i time units after the last scan. It is desired to find the error in esti- mating the true load carried by averaging the number of circuits found busy on each scan. Following Wilkinson's method we will first estimate the error of the switch count method in measuring the contribution of a single call to total usage and then modify it to take account of n calls. Calls of two types must be considered, those originating outside the in- terval and extending into it, Type I, and those originating within the interval, Type II. Both types may be subdivided depending on whether or not they extend beyond the end of the observation period. These are 372 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 indicated in Fig. 7. Only that part of a call which falls within the obser- x'ation poi-iod contrilxites to the usage of that period. First the error made by switch counts in measuring liiis contribution will be deri\'ed. Type I Consider a call which is already in progress at the start of the ol)ser- vation period. Its duration beyond that point, according to theory, will be exponentially distributed about an average of i. If this duration, t, is between 0 and i, the call will be counted once (a measured contribution of i erlang hours) and a positive error of x = i — t Avill be made. The same error will be made if t = 2i — x so that the call (C-2)(C-I) C Fig. 7 — Graphical indication of the two types of calls with their two sub- divisions. is counted twice and so forth. Summing all the ways of making an error X, we have : P(x) dx = f{i - x) -\- fi2i - x) + • • • f(ci - x) (1) where f{i — x) is the probability of t ^ i — x and m^le-r- dx Calls lasting beyond ci neither start nor end in the observation period so that their contribution is measured without error. For these: P(0) = P(t > ci) = e~ (2) Therefore : I — x Px>oix) dx = ^ c t dx + k 2i — X ci t dx + + r" dx (3) HKLlAlilLITV OF TUAKFIC MEASUREMENTS 373 Letting i ' ' n=o 1 — e-'' I\^,{.v) (Ix = (''erKdij (4) The moiiKMit o(x) dx = c—l -• ni— n=0 i rfx i - x^ I -^ n=l ^ (8) (ix = 2/j 7v = ^ e "'^ as before n=0 c-1 — nr ne = 71 = 0 i K - cb i + y.\{K-'y:^-^^ + ^ Px>o(x) dx = e V c I - e-" y dy '-'r)['' cl rc IK - cb (9) dy The moment generating function of this pair of equations is the sum of their separate m.g.f.'s: AL i{a) = [ Py 1 Xi = —t X2 = i — t 1 V.= ^^]/^(r- 1) (19) 2. /• < 1 :\IiiL 1' = 0 Max. V. = 100 I ^/^ (20) ^100 /:3:: c/T y 4a'NTt Equation (20) compares favoralily with the exponential holding time coefficient of variation of error of c/T y Qa'NTi REFERENCES 1. R. I. Wilkinson, "The RelialiilitN' of Holding Time Measurements," Bell Si/slem Tech. ./., 20, pp. 365-404, October, 1941 . 2. J. Riordan, "Telephone Traffic Time Averages," Bell Si/stem Tech. J., 30, j^p. 1129-1144, October, 1951. .3. F. W. Rabe, "Variations of Telephone Traffic," Electrical Cointnunicalinns, 26, pp. 243-24S, 1948. The following books contain descriptions of the use of generating functions in solving probability problems: 4. A. C. .\ilken. Statistical Mathematics, Oliver and Ho\-d, Ltd., Edinl)urgli, 1947, pp. 16-23. 5. W. Feller, An Introduction to Probability Theori/ and Its Applications, John Wiley & Sons, Inc., New York, 1950, Chap. 11. Network Representation of Transcendental Impedance Functions BY M. K. ZINN (Manuscript received November 5, 1951) The purpose of the paper is to show that the admittance or impedance of certain continuous structures, such as, for example, a finite length of trans- mission line of any sort, or resonant cavity, can be represented exactly at all frequencies hy a network comprising lumps of constant resistance R, inductance L, conductance G and capacitance C. The network will contain an infinite number of branches, in general, although a finite number may be used if it is desired to represent only certain modes. The procedure is based upon a proposition known to students of function theory as "Mittag-Leffler's theorem," which amounts, roughly, to an ex- tension of rational functions to apply to transcendental functions of the type encountered in the theory of continuous structures. Several illustrative examples of the network synthesis are given. GENERAL Students of network theory are familiar with the fact that the im- pedance at a pair of terminals in a linear network comprising a finite number of resistors, inductors and capacitors, connected in any manner, is a rational function of the frequency having, in general, the fractional form of one polynomial divided by another. Thej'' are also familiar with the partial fraction rule whereby the function can be broken up into a series of elementary fractions, each of which exhibits one of the poles of the original function. This form is sometimes useful in the problem of network synthesis, where the impedance function is given and the ob- ject is to find a network having this impedance. The purpose of the present paper is to show how a similar procedure can be carried out for certain transcendental impedance functions per- taining to structures having distributed constants, such as, for example, a resonant cavity or a piece of transmission line. The method employ's a well-known proposition of function theory, which is usually referred to as Mittag-Leffler's theorem. This theorem provides a tool for breaking up a transcendental meromorphic function into an infinite series of simple fractions in much the same way as the partial fraction rule is used to break up a rational meromorphic function. The series representation 378 NETWORKS FOR IMrKDAXCE FUNCTIONS 379 provides a means of (letormiiiin<2; a netwoik of resistors, inductors and capacitors that will have an inii)e(huicc (M|ual to ihe s|-)(>cifie(l ti-anscen- (lental impedance function. 'I'liis pi-occss will he I'cfci-red to as ol)tainiii.u; a ''network representation" of the function. If the g;i\-en function is the impedance of some continuous (i.e., non-lumped) electric structure, the result ^^'ill be an equivalent network for the structure. For other pur- poses, such as, possibly, analogue methods of computing, the given func- tion may not arise from any electrical structure. In either case, the net- work representations to Ix^ deiixcd are possible only if the function satisfies certain i-esti'ictions, which are stated in the section immediately following. The discussion is confined to transcendental impedance functions he- cause of the technological interest in the electromagnetic structures with which they are associated and because they have not received as much attention as rational functions in the literature dealing with network synthesis. The problem with which this paper is concerned can then be stated as follows: gi\-en, a transcendental impedance function satisfying certain conditions: to determine a network comprising elements of constant resistance, inductance and capacitance whose driving-point im- pedance function, at a pair of terminals, will equal the given function at all frequencies, real and complex (except at the poles). For illustration of the procediu'e, three examples are given. The first is the impedance of a short-circuited or open-circuited transmission line in which the distributed primary constants, R, L, G and C are assumed to be invariable \\ith frequency. The second and third examples are the impedances of resonant cavities driven in two different modes. In these examples the variation of resistance with frequency, due to "skin-effect," is taken into account. IMPEDANCE FUNCTIONS The functions inider discussion will he referred to as "impedance functions" with the understanding that the term is meant to include "admittance functions" as well. By reason of the duahty principle that runs through all electric circuit theory, any general proposition devel- oped f(^r one must apply to the other. The functional designation, F(p), will be used to denote either an impedance or an admittance function. When a distinction is necessary, the impedance will he designated by Z(p) and the admittance by Y(p). The independent compk^x variable p is the generalized radian frequency. (For sustained sinusoidal currents and voltages, p = ico = ^rif where / is the real frequency.) For the applications contemplated, F(p) is a transcendental mero- 380 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 morphic function, which term implies that the function is given by the ratio of two entire functions, one or both of which is transcendental, and that the singularities of the function are ordinary poles, except for the point at infinity, which is an essentially singular point. In order to realize the particular network developments to be given, it will be supposed that the function satisfies the further restrictions given below: (1) All the poles lie in the left half of the p-plane ^^'ith none on the imaginary axis. (2) F{p) = F(p). (The superbar denotes the complex conjugate of the unbarred symbol.) (3) Real part [F(ii>))] > 0 for all real values of o). These three conditions are necessary to insure that the function is the impedance of a possible linear, passive electric circuit structure. Inter- preted physically in terms of this possible equivalent structure, the first condition specifies that the structure shall be stable ; that is, every natu- ral mode of oscillation dies away exponentially. The second condition specifies that the natural oscillations are real functions of time. The third condition specifies that if a sinusoidal current flows at the driving- point terminals of the equivalent structure, the average real power de- livered to it will be positive. Since these three conditions, or their equiva- lents, are frequently mentioned in discussions of network theory, it is assumed that they are understood without more detailed explanation. In addition to the above restrictions on the form of the impedance function, the following two conditions, while not necessary, will be im- posed to limit the scope of the discussion: (4) All the poles of F(p) are simple. (5) F(p) = 0(1), exactly, as | p | — > m everywhere except at the poles. Condition (4), while limiting the scope of the exposition required, does not restrict the application of the results in any important way, because most impedance functions for which a network representation may be required have only simple poles. Condition (5) implies that as p increases along any straight line drawn through the origin and not passing through any pole of F(p), the modu- lus of F{p) either approaches a limit or oscillates between finite Hmits. The physical implication of this condition is that the response of the network as a function of time to a suddenly applied cause begins ^vith a discontinuity of the same degree as that of the cause. For example, the current response of the network to an applied step of voltage begins with a finite discontinuity. This behavior is a characteristic of continuous (non-lumped) electromagnetic structures, which furnish the principal application of the network developments to be described. networks for impedance functions 381 mittag-leffler's theorem^ Lot the polos of tlio given function F(p) bo Pi , P2, Vs ' ' ' , where 0 < I Pi I < I P2 I < I p. I • • • and let the residues at the poles be Ai ,A2,Az • ■ • , respectively. Suppose that it is possible to draw a sequence of closed contours, r„ , such that C„ encloses pi , p2 , • • ■ Pn but no other poles and such that the minimum distance of C„ from the origin tends to infinity witli /;. Suppose also that F{p) satisfies conditions (2), (4) and (5) above. Then Mittag-Leffler's theorem gives the follomng series development for F{p) : F{p) = F(0) + Limit i: ('_A_ + ^ (1) AT-oo „=-Ar \P — Pn Pn J The notation here used employs the con^'ention that p_„ = Pn and A_„ = A„, since, by virtue of condition (2), the poles occur in conjugate complex pairs. The value, n = 0, then allows for a pole on the negative real axis. Given any suitable function, the procedure is to determine its value for p = 0 and the location of its poles. The residues are next determined by An = Limit (p - pn)F{p). V-*Pn Then the Mittag-Leffler expansion can be written down at once. network representation In the series (1) the terms occur in pairs with conjugate complex poles and residues. The object is to obtain a network representation of each such pair of terms. If F(p) is taken as an admittance, the branches rep- resenting the pairs of terms will all be connected in parallel; if F(p) is taken as an impedance, they \vill all be connected in series. Methods for obtaining a network representation for a rational func- tion, such as the one comprising a pair of terms in the series (1), are well known. It is only necessarj^ to describe certain procedures of particular application to the present problem. Brune has stated that the necessary and sufficient condition for a network representation of a rational func- tion of p to be realizable is that it be a "positive real function," that is, a function that is real for real values of p and whose real part is positive, 382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 or zero, when the real part of p is positive, or zero. In view of conditions (1) and (2) above, only one test^^ need be applied to each pair of terms of the series (1): the sum of a pair of terms will be a positive real func- tion if, and only if, the real part of their sum is greater than, or equal to, zero for all purely imaginary values of p. The general term pair for which a network representation is sought is F,Xp) = -^^ + -^^ + ^^ + i^ = P^Xp) - PniO) (2) P — Pn P — Vn Vn Vn Evidently two cases can be distinguished at the outset, depending upon whether P„(0) is positive or negative. If P„(0) is positive, the network branch, in order to be realizable, should be designed to represent Pniv)- The left-over negative term, — P„(0), then can be absorbed in the posi- tive first term, F(0), of the series (1); more will be said of this later. If, on the other hand, P„(0) is negative, the network branch should repre- sent the whole term, P„(p) — P«(0). This procedure insures that the real part of the branch impedance will be positive, or zero, at zero and infinite frequencies. To guarantee that the resistance is positive at all other frequencies requires further tests now to be specified. Let the real and imaginary coefficients of the poles and residues of the n^^ term be Pn = -dn + i^n , Pn = —an - i^n An = ttn -\- ihn , An = an - ihn (With this notation, «„ and /3„ are always positive ; a„ and 6„ can be either positive or negative.) Then (dropping the subscripts) p. . _ 2{aa — hl3) + 2ap ^^ ~ a^ -f ^2 _^ 2ap + p2 „.p,. ., 2{aa - h^)(a' -f /3') + 2o:\aa + 6/3) ,^. R[P{tco)\ = ., , ., — (3) [a- + («-)- + 2(,}-{a- - I3-) -fco* P(0) = R[P{io^) - P(0)] = 2{aa - 5/3) «- + ^- -2{aa - Sab(3 - 3aa;g^ + 5/j^)co' - 2{aa - hjSW (a2 + ^'')[(a'' + /32)2 + 2(a2 - /32)a;2 + co*] The necessary and sufficient conditions for the real part of a rational function of p to be positive, or zero, for purely imaginary values of p are that the function be positive for p — > ±t=o and have no imaginary roots of odd multiphcity. When this test is applied to the functions P(p) and NETWORKS FOR I.MI'KDANCK FrXCTlOXS 383 P(p) — P(0), as given by (3), the following coiiditions are ohtaiiied: P(p) will be a positive real funetion if, and oiilij ij\ aa - bi3 > 0; i.e. 7^(0) > 0 (4) and aa + bid > 0 P{p) — P{0) will be a positive real funetion if, (uid onltj if, aa - bis < 0; i.e. P(0) < 0 (5) and aa' - 3abi3 - SaajS' + biS' < 0. If all terms of the series satisfy one or the other of these conditions, network branches can be devised to represent all the terms and all the R, L, G, C elements of the branches will be positive. In case all the terms are of the type where P„(0) is positive, so that the network branches are made to represent Pnip), the left-over constant terms can be collected and added to the first term, F{0), of the series. This collection of terms then must be represented by a final branch of pure resistance, or conductance, of value, FiO) - Z PM n=o If the sum of the variable terms approaches zero for p —^ ±zoo, the final constant term supplies the high frecjuency resistance of the func- tion F(p) and since this must be positive, if condition (3) is satisfied, the final resistive element will be positive. If the series converges non-uni- formly, the sum of the variable terms can have a value other than zero as p ^ ±/oo in spite of the fact that every term approaches zero indi- vidually. In that case (see example 1) all or part of the high frequency resistance may be supplied by the sum of the variable terms. In case all the terms are of the type where Pn(0) is negative, so that the network branches are made to represent the sum, P„(p) — Pn(0), of the variable and constant terms and the series is uniformly conver- gent, all the high frecjuency resistance is provided by the branches rep- resenting these terms. The first term, F(0) then supplies the dc re- sistance, which is positive by condition (3). Non-uniform convergence can modify this division of high- and low-freciuency resistance, however. Cases can arise in which the series contains terms of both types. In such a case the dc resistance, or high frecjuency resistance, or both, of 384 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 the given function might be less than the sum of the variable terms for these frequencies, with the result that the final resistance branch would be negative for either the series or parallel type of network development. To make the procedure as concrete as possible, particular forms of networks are described in the section following with, explicit formulas for computing their elements. NETWORK FORMULAS Simple forms of network branches are shoAMi in Figs. 1 and 2. Those of Fig. 1, referred to as branches of "the first kind" are suitable for con- nection in parallel where the given function F(p) is an admittance, F(p), w^hile networks of "the second kind," shown in Fig. 2, are suitable for connection in series to represent an impedance, F(p) = Z(p). The net- works of Figs, la and 2a apply where the value P„(0) of the general term is positive, while Figs, lb and 2b apply where P„(0) is negative. Figs. 3 and 4 illustrate, respectively, networks of the types of Figs, la and 2a Cn Rn Lr Ln J_ Gn Rn Cn '-WV-' _1_ Gn ;a) (b) Fig. 1 — General branches of the first kind. Fig. la (use where F(p) = Y{p) and F„(0) > 0) Ln = 2a„ jr = — (cLnOCn — hn^n) Rn 1 {ana„ + 6„/3„) Ln dn Go = F(0) - E F„(0) Ln — 1 Fig. lb (use where F(p) = Y(p) and F„(0) < 0) ^l{al + l3lY(al + //„) 2M^ M^ LnCn Plial + bl) ttnOLn — bn^n CrnLn — — iin^n M N Go = F(0) (6) NETWORKS FOR IMPEDANCE FUNCTIONS 385 coniuH'tcd to form the ('()in{)k'tiHl network witli Ihe liiuil non-reactive hranch, (/„ or />'„ , in place. Formulas for the network elements are obtained by ecjuating the j)oles and residues of the network impedance function to the ^iven poles and residues of the general term of the series. Since botli pok^s and residues occur in conjugate complex pairs, and since equality of real and imagi- nary parts is involved, there are four equations, wliich are necessary and sufficient to determine the four constants, R, L, G, C, of the network. The formulas that are obtained by solving these equations are given beneath Figs. 1 and 2. The values given for Go and Ro in each case assume that all the terms of the series are of the type specified foi- that case. ^" Ln OKKT -wv -\AA — OKKI^ ' — V^ Rn Cn (a) (b) Fig. 2 — General branches of the second kind. Fig. 2a (use where F(p) = Z{p) and Z„(o) > 0) C -± LnLn \a^ Fig. 2b (use where F(p) = Z{p) and Zn(o) < 0) 2ilf3 ^ = 1 Ln ttn ^ = 1 (ttnan - bnl3n) {ttnan + hn^n) 00 Ro = Z{Q>) - E ZM 1 ttnan — b„^n PC = — GnLin — M (7) N Ro = Z{0) where M = Gni^l - al) + 2a„/3„6„ N = —ttnan + 3Q!„Mn + SttnOinlS n " ?>n/3„ 386 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 111 the case of the parallel-type iietAVorks (Figs, la and lb), p„ = — an + il^H is a pole of the admittance, F(p), and A„ = a„ + ih^ is the corresponding resicUie. In the case of the series-type network, the same symbols represent a pole and residue of the impedance. Zip). The networks specified by Figs. 2a and 2b are duals of the networks of Figs, la and lb, respectively, and are obtained from the latter merely by replacing L„ by C„ , Rn by Gn , and vice versa. The formulas are intended to apply to complex poles. They can be applied to real poles by taking 6„ and /?„ equal to zero and doubling the residue, a„ , but this procedui'e is uiniecessary, because the network rep- Fig. 3 — Network of the first kind (branches la) C, C2 \[ Ro — ^^A — I R, L, \^A •— • * R2 U Fig. 4 — Network of the second kind (branches 2a). resent ation of a real pole can be found readily enough by inspection of the impedance terms involved. (See Example 1.) The above discussion is intended to sketch a general picture of the procedure. Indi\'idua] cases may involve considerable detail that can be understood more readilv bv reference to the next section. APPLICATIONS Example la: A transmission line with its far terminals short-circuited affords a simple illustration of the equivalent network theory. Let it be assumed that the parameters, R, L, G and C of the line are constants. In the more advanced examples to follow, the \'ariation of these parame- ters with frequency for a particular kind of line will be taken into con- sideration. NETAVORKS FOR IMPEDANCE FUNCTIONS 387 The iinpedancc^ of the short-circuited lino (Fig. 5) is Z = Zo tanii r (i-o; where Zo is the characteristic impedance and T is the total i)r()pagati()ii constant of the hnc. We have (R + pLY \G + pCj .G + pC, T = m + pL)(G + pC)f" (1-1) (1-2) R, L, G and G bein"; gix'cn for the total lou/lh of line. To obtain a development in terms of network bi'anches of the kind shown in Fig. 1, we consider the admittance function, }' = I'o coth r (1-3) where F = 1/Z and I'o = 1/Zo . Our first task is to find the poles of this function and the residues. Since the complex freciuency variable p occurs R,L,G,C 2=^ Fig. 5 — Short-circuited transmission line. under square roots in both Zo and T, it might be suspected, offhand, that the singularities of the function are branch points rather than poles. Such is not the case, however. There are no branch points and all the poles are simple. The singularities of Y are to lie found among the zeros of tanh F, which occur at r = iwn, n = 0, ± 1, ± 2, ± 3, • • • To determine them, we soh^e r' = (/? + pL){G + pG) = -ttV and find these roots: Pn = —Oin-\- i^n , P-n = Pn = " ^n " t^n where (1-4) (1-5) «„ = G R 2G "^ 2L [Tr'n- / G R /3« = LC \2C 2L [n > 0) (1-6) 388 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 For n = 0, the above would give R G But if we let T -^ 0, so that tanh T — > F, we find that only the point, —R/L, is a singularity of Y] the other point, —G/C, is a regular point. Therefore Y has only one real singularity. To find the nature of the singularities of F, we next calculate Limit V - Vn p-*pn LZiiip) tanh r(p) and find that at each p„ the limit exists and has the value = An (1-7) 1 ^"""l) Zo(p„)r'(p„) L ^nL where r'(p„) = -— r(p), evaluated at p = Pn ■ The fact that this limit dp exists shows that all the singularities are simple poles. The values of ^„ are then the residues at these poles. When we now apply formulas (6) to determine the elements in the general branch of the equivalent network of Fig. la, we obtain, for n> I, T = - 1 ^ T^ ^ = ? Rn ^ R / qv " " 2' LnCn ~ LC Cn~ C L„ " V ^'^^ The network then comprises an infinite number of such branches in parallel. Each branch has the same elements 2?„ and L„ , equal, respec- tively, to half the total resistance and inductance of the transmission line, but the elements Gn and C„ decrease from one branch to the next in inverse proportion to the squares of the integers. The Q of the n^^ branch, which can be regarded as the Q of the asso- ciated resonance of the short-circuited line, is Q_ Wn tOn C*)„ n — where 2a„ Gn Rn G R (1-10) Cn Ln C L - ./I ^n _ ,/^ G' (, 11X '" ~ y LnCn CI ~ V LC~ C' ^ ^ NETWORKS FOR IMPEDANCE FUNCTIONS 389 Thus, for small dissipation, the resonances would became sharper in di- rect proportion to the frequency (if the parameters R, L, G, C, were invariable with frequency, as assumed). The above described branches of the ecjuivalent network account only for the complex poles (n > 1) of the admittance function. Two more branches remain to be calculated. One is for the real pole (n = 0), which occurs at po = —R/L, with residue, Aq = - . The required branch for Li this pole is Ao 1 p — po R + pL The other is the final conductance branch, which is calculated as follows : Go = F(0) + f: 4:? = a/^ coth VGR - i n=-«, Pn y R R (1-13) - 2(? E nt^oo TT^n^ + GR 0 so that, for this example, the conductance branch vanishes. The network is drawn in Fig. 6. A series type of network, as shown in Fig. 7, can be determined by Fig. 6 — Network of the first kind equivalent to the short-circuited line of Fig. 5. (Ro=o) Vv\ G •-^AA/ — —ymu-^ 2R 2L ^'iif — vw G i— Vv^ — ^smy-^ 2R 2L Fig. 7 — Network of the second kind ecjuivalent to the short-circuited line of Fig. 5. 390 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 similar means. Since, however, it is a dual of the parallel network of Fig. 9 for the open-circuited line, next to be discussed, it can be drawn im- mediately, without further calculation, once the latter has been found. Example lb: We now calculate a network for the same line with its far terminals open (Fig. 8). To obtain a network of the first kind, vvith branches in parallel, we deal with the admittance function. Y = Yo tanh r (1-14) The singularities of Y are found among the zeros of coth r, which occur at r = iirin + i), n = 0, ± 1, ± 2, ± 3, • • (1-15) The points p = —R/L and —G/C are both regular points tliis time. { — G/C is a zero of F.) The singularities are simple poles, as before, with residues, ^ (1-16) An — Zoip„)T'{p^) as before. The network branches for the complex poles are therefore obtained merely by putting n + | in place of the n in all formulas of the short- circuit network. There is no branch corresponding to the branch R -\- pL of the other network and the conductance branch is again found to be zero. The complete parallel network is drawn in Fig. 9 and the series net- work, in Fig. 10. It will be observed that the series network of Fig. 10 is the dual of the R,L,G,C Z=- Fig. 8 — Open-circuited transmission line. Fig. 9 — Network of the first kind equivalent to the open-circuited line of Fig. 8. NETWORKS FOR IMl'KDANCE FUNCTIONS 391 parallel network of Fig. 6 and the series network of Fig. 7 is the dual of ilu^ parallel network of Fig. 9. These dual i-elationships are of course a result of the fact that the impedance of an open-circuited line is the dual of the impedance of the same line when short-circuited. Example 2: Short-circuited Concentric Line (or Toroidal Cavity with E Radial). The preceding example considered a fictitious transmission line of invariable parameters, R, L, G, C, having a perfect short circuit at one end. The present example has to do essentially with the same problem but considers it from a more practical point of view. The vari- ation of R and L with frequency is taken into account and the impedance of the "short-circuit" is no longer neglected. Let the line be the piece of coaxial cable plugged at both ends with conducting material as illustrated in Fig. 1 1 . Considered from an alter- native point of view, our line is now a toroidal cavity oscillating in the (Ro=o) 2. G -VAr 2. G Fig. 10- Fig. 8. -L ^R_ 2L_ -Network of the second kind eciuivalent to the open-circuited line of 2R 2L 2 mode where the electric force E is directed radially and the magnetic force H lies in planes at right angles to the axis. If w^e assume the cavity to be excited, or "driven," from one end,* the impedance that is effective in defining the selective characteristic of the cavity with respect to fre- quency is the total impedance at that end, that is, the sum of the im- pedance Zi , viewed into the cavity, and the impedance, Z2 , of the ad- jacent end-plug. Therefore, w^e have to deal with the impedance, Z = Zi + Z2. (2-1) By "impedance" is here meant the same thing that one considers in look- ing at the problem from the point of view of transmission line theory, namely, the complex ratio, for exponential oscillations, of the voltage between the inside and outside cylindrical surfaces to the total current * For determining the "natural frequencies" of oscillation of the cavity, it is immaterial at what point along it the imi)edance is taken; the total impedance at every point has the same roots. The impedance is, nevertheless, not the same at all points so that the behavior of the cavity, when driven, will depend to some extent on the driving point. 392 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 flowing axially in the inner conductor at the same point. The zeros of Z define the natural frequencies of oscillation of the cavity and their asso- ciated damping constants, or Q's. Our task is to develop an equivalent network for this Z. We have Z = Z\ -\- Zi = ^0 1 + pe -iyh + 1 + P 1 - pe-2T'. ' 1 - p (2-2) -VA- hR, 2 ^W^ 2 he / 3_ _3_> (-1^) ^(-i53; + i4' -^^A^ 3 3 '2h(fn ^^J^ M 2g ^nv 27T C = 27r6o :=3(108) 10-'* 367r FOR AIR IN M. K.S. UNITS (e.g.) fJ-Q^ATT do-"') , 60 = H=Hq^ g = 5.8(10''), FOR COPPER IN M. K.S. UNITS (n=1 FOR FUNDAMENTAL MODE) Fig. ll^Toroidal cavity, E radial. NKTWOUKS FOR IMPKDANCK FUNCTIONS 393 where Zo = A' + pL P = 7 = (72 + pLy'\G + pcy" Z. + Zo Z.= ^ log r? /o(; the derivative of the func^tion, I — p'e^^'''', with i-esi)ect to /;, e\aluated at pi„ . This now takes account of the actual impedance of the end-plugs. The values of the zeros, so obtained, are Pn = - OCn + i^n , P-n = Pn = " «» " i^n where (2-15) ^" = """ (^ + i ~ i where d,* is the real part of o-q,, . That is, 8n = {mniJ-g/'2) where Trny (^on — -7— • As an incidental matter pf interest, the above gives the Q of the cavity at any resonance, namely Qn = ^ = d8n ^-^, (2-16) h For example, the dimensions, a = .5 cm., 6 = 1.0 cm., h = .5 cm. pro- vide a (•a\ity that resonates at about 30,000 megacycles. Then the Q's at the first tiu'ee I'esonances would be as follows: n Won 27r Q 1 30,000 X 10' 4250 2 (;0,00() X 10' 6010 3 90,000 X 10' 7360 * For an\^ frequency, 5 = (a;yu(//2) '- is sometimes referred to as the "skin depth" because it is the deptfi of metal at which tlie current density falls to 1/c times its value at the surface of the metal. 396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 The importance of including the effect of the end-plugs in determining Q is shown by the fact that, if they were assumed to have zero impedance, Q at the first resonance would be 12,120 instead of 4250. To determine the residues at the poles, we write (1 - p)(l - pe-'-'') _ Fip) ^ - ^° 2(1 - p^e-^y^) G(P) ^^^^ and then the residue at a simple pole p„ is An = ^!p{ (2-18) This limit is found to exist, showing that the poles are, in fact, simple. The value found for the residue, ^„ , is An = an -\- ihn, A-n = An = ttn — iK _ HpWQn /, _ 1 \\ ""^ " "^1^ V 2(i5„ 2Un) (2-19) ^ ^^oo^, / 1 1 \ irn \2dbn 2h8nJ When formulas (6) are applied to determine the elements of the tuned branches of the equivalent network of the first kind, the results are, for the n*^ branch, _ Koir7l / 1 1 2won \ 2d8n 2hbn, 1 2/1,1 2 = coon 1 + "Tir — Gn _ WOn Ln " \d8n 2h8n In terms of the R, L and C of the piece of coaxial line, the elements of the n^^ branch are as follows: ^" ~ 2 V 2d5„ ^ 2h8n 2 \ 2/i/ (2-21) " 7r2n2 V 2d8n 2h8n/ p _ COOnC / 3 , 3 " ~ ^^%^\ mn 2h8n NETWORKS FOR IMPEDANCE FUNCTIONS 397 where The network is shown in Fig. 11. It will 1)(> foiuul that a "leakage" element, Gn , appears in the equiva- lent network, although the air dielectric in the cavity was assumed to have no leakage (G = 0). This element arises from the end-[)lugs and is necessary to account for the dissipation in tlu^m. To obtain a network exactly equivalent to the cavity at all frequen- cies, we should add a branch corresponding to n = 0, as was done in example 1. This branch would make the equivalence hold do^vn to and including zero frequency. But, inasmuch as the approximations that have been made hold only for the high frequencies, where the resonances oc- cur, it would be inconsistent to add this branch. What has been arrived at, then, is a partial network representation that gives a close approxi- mation to the impedance of the cavity at high frequencies, only. Example 3: Toroidal Cavity with E Axial. For further illustration, we consider another mode of oscillation of the short-circuited concentric transmission line investigated in the previous example. This time it is assumed that the radial electric force vanishes while the axial electric force between the end-plugs exists. The magnetic force is directed in circles concentric mth the cylindrical central conductor, as before. This situation is illustrated in Fig. 12, which is the same as Fig. 11, except for the new disposition of the £"- vector. For the new mode of oscillation, where the wave is a cylindrical one propagated back and forth between the inner and outer conducting cylinders, the oscillatory space is naturally thought of as a "toroidal cavity," while, in the previous example, where the wave was propagated axially back and forth between the terminal discs, the space was called a "concentric line." Actually, the cavity itself has the same geometric form in the two cases. A pi'actical distinction may exist, however, in that the axial mode of oscillation could be more easily excited in a cavity whose axial length is large compared to its radius, while the cylindrical mode would arise more easily in a flat "pillbox" cavity whose radius is large compared to its axial dimension. The approach to the problem will be that of transmission line theory, as before. This time, the "line" comprises two circular discs between 398 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 which the pyhndrical wave is propagated. The series impedance and shunt admittance of such a line are functions of the radius and so will be designated Z(r) and Y{r), respectively. Their values are gi\'en below: Z{r) Yir) 2ri + icciih 2^ ico2Treo h (3-1) (3-2) These formulas take into account the losses in the flat walls but assume the conductance of the air between them to be negligible. Losses in the inner and outer "short-circuiting" cylinders will be taken into account by the boundary conditions. Z^2n2 I 2h£fn 2(b-a)(fn (b-a)L r , , I c ^AA, "^m^ (b-a)RA 3h >> 2 " 2fb-a)^ P_ 1 if^W ^~7raV 2g 'on>" b-a L^- //oH C = _ 277"afo 3 (108) SEE FIGURE 11 FOR /Jo,€Q,/J,g (n = I FOR FUNDAMENTAL MODE) Fig. 12 — Toroidal cavity, E axial. NETWORKS FOR IMPEDANCE FUNCTIONS 399 Tf T^ is the voltage between the flat faces of the cavity at a radius r and 1 the total current in the lower face at this radius, we have '^ = -IZ(r) ar ^= -VYir) ar (3-3) By differentiating, d'V _ jdZ rydl _ (i dZ\dV But dr- dr dr \Z dr / dr ZY = (2, + i.,h) '^ = 7^ h which is a s(iuared propagation constant, independenl of r, and 1 rfZ ^ _1 Z dr r Therefore, i^+-f- y-v = 0 (3-5) dr^ r dr is the differential ecjuation for the voltage. The usual solution of this equation is a linear combination of Io(yr) and Ko{yr) but since, in this case, the arguments will be almost purely imaginary, it is more con- venient to employ the pair of functions, Jo( — iyr) and Noi — iyr). The solution for the voltage between the upper and lower surfaces at radius r is V(r) = AJ,{-iyr) + BN,{-iyr) (3-6) and, from this, the total radial current in the lower surface, at that radius, is /(/•) = -i '^ = -iY,{r)[AJ,{-iyr) + BN,{-iyr)] (3-7) Z dr where }'„(/•) = l,'Zo(/-) = [Y{r)/Z{r)r The impedance at the inner radius a, looking outward, is then Ziia) = J-—- = tZoia) -r^j- — -. — . , p„ , — -. — -^ (3-8) I {a) AJx{ — iya) -f BNi{ — iya) 400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 The total impedance at a (inward + outward) for which we require an equivalent network is Z = Z,ia) + Za where Za is the impedance of the central plug to axial current, viz., „ r]h /o(o-a) . . 27ra /i(o-a) To evaluate the constants A and J5, the following boundary conditions are imposed at radii a and h: at a: F = F(a), a given voltage at h: V = I{b)Z, where Zb is the impedance of the other "short circuit," comprising the outer cylindrical wall. It is given by „ 7]h Koiab) , . ZtO K\{ab) Except for ignored small deviations of the field around the corners of the cavity, the above expressions are exact. The process of finding the singularities of Z by successive approximations results in expressions that are too long to wTite down here. To obtain results sufficiently com- pact for engineering use, we resort to the following asymptotic approxi- mations for the Bessel functions: Jo{z) ~ { — I cos (2 — 7r/4) J^{z) '-'(—) cos {z - 37r/4) N^{z) ^ (^— j sin (2 - 7r/4) (3-11) Nr{z) ^ f- j sin (2 - 37r/4) Uz) ^ Koiz) ^ /i(2) ' K,{z) Also, with an error on the order of 10"*, Zo{r) - ^ = Ko(r) = l///o(r) 2irr NETWORKS FOR IMPEDANCE FUNCTIONS 401 These substitutions result in the following asymptotic formula for the total impedance Z at radius a 27? , ./ J?2\ — cos fcx + 1 1 1 + ~2 ) sm fcrc Z = KM "^ ^. "^ (3-12) cos kx -\ sin kx where A' = - — 1 and x = —iya. a To find an equivalent network of the first kind to represent Z, we deal with the admittance, Y = 1/Z. It is instructive and saves much work to put Y in the form of exponential functions, with the substitution V — Vo P = — r — which is the reflection coefficient at both inside and outside cylindrical surfaces of the cavity. By this means we obtain y - HM ^' ;(f^;^p (3-13) This is now identical in form to the formula (2-13) of example 2, where the £'-vector was radially, instead of axially, directed. In fact, since ikx = y(b — a) and comparison ^vith the similar formulas of example 2 shows that all the results of that example can be made to apply to the present one merely by changing the dimensional parameters as follows : Example 2 {E radial) Example 3 (E axial) h goes into h — a ^ 2ah log (h/a) '^~ a+h goes into h The first result of interest is the value of Q, which is Qn (3-14) b — a 402 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 where, as before, 5„ is the "skin depth" e([ual to the real part of o-„ . That is, , _ /cOOnMg To gain an idea of numerical magnitudes, consider the same cavity used in example 2. The dimensions are, as before, h ^ .5 cm,, b — a = .5 cm. For the square cross-section chosen, the first resonance again occurs at 30,000 megacycles, very nearly, and we can make the following direct comparison of the Q's for the two modes of oscillation: n a)0ii/2?r Qn Ex. 2 (E radial) Ex. 3 (£ axial) 1 2 3 30,000 X 10« 60,000 X 10« 90,000 X 10« 4250 6010 7360 4370 6180 7560 Due to the asymptotic approximations used, the results for example 3 are not as accurate as those for example 2; the two sets of results show only that the Q of the cavity is substantially the same for the two differ- ent modes of oscillation. The poles of Y are given by 1 P-n Pn = —Oin i^n Oin = Won 2hbn "^ ih /3„ = COOn 1 + and the residues are An = ttn + ibn , an = irn Hoiajwon 1 2)^ - a)8j {b - a)8nj (3-15) A_n — An — dn tOn bn = irn 1 - 1 1 2h8n + 2(b - a) 5 J 1 (3-16) _2h8n 2(b - a)8n_ Applying formulas (6) gives the following values for the n*"" branch of NETWORKS FOR IMPEDANCE FUNCTIONS tho network of the fii'st kind: TT/l 403 2coo« L 2/(,5„ 2(6 — a)8n_ 1 Won 2 Cm (6 - a)8n^ C„ 2(6 Rn y— — a"()„ ■Lin - o)5„ "1 + Jl8n 2(6 3 - a)5„_ (3-17) ill nil of which wo« = Tnv/{b — a) and v = l/(Moeo)"" = 3(10'^) meters per second. The results can be put in the same form as those obtained for the other cavity mode, dealt with in example 2, by employing the "primaiy con- stants" of the cylindrical transmission line, viz.: R{a) = ~ wa L2<, J lira G{a) = 0 C(a) - ,^ \\\ terms of these constants, the elements of the n*'' branch of the eriiiiva- lent network of the first kind are ^^ ^ (6 - a)L{a) (^^ _^ J_ _^ 1 Rn = ^^ - ;^^^"^ I 1 + Cn = 2(6 - a)C(a) TT-n' (13-8) 2h8n 2{h - a) 8n Sh 2(6 - a) fl - ^ + ' \ 2/i5,. ^ 2(6 - a)8n „ ^ oionC(a) / _ J3_ 3 \ TT^n^Sn V 2/i5„ ^ 2(6 - a)8,J The network is shown in Fig. 12. As in the preceding example, a leakage element arises, in spite of the fact that we assumed initiall}^ that go of the air in the cavity is zero. This element accounts for the losses in the inner and outer cylindrical walls. 404 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 A number of people with whom the above material has been discussed have given helpful comments and criticisms. I wish to acknowledge my debt in tliis respect to H. Nyquist, S. A. Schelkunoff, R. M. Foster, S. (). Rice, J. Riordan and W. H. Wise. BIBLIOGRAPHY 1. R. M. Foster, "A Reactance Theorem," Bell System Tech. J., 1924, 3, p. 259; also, "Theorems on the Driving Point Impedance of Two-Mesh Circuits," Bell System Tech. J., 1924, 3, p. 651. 2. W. Cauer, "Die Verwirklichung von Wechselstromwiclerst stJinden vorge- schriebener Frequenzabhangigkeit," Archiv fur Electrot., 1926-27, 17, p. 355. 3. G. A. Campbell and R. M. Foster, "Fourier Integrals for Practical Applica- tions," Bell System Tech. J., Oct. 1928, pp. 639-707; Bell System Monograph B-584. 4. T. C. Fry, "The Use of Continued Fractions in the Design of Electric Net- works," Bull, of Am. Math. Soc, 1929, 35, p. 463. 5. O. Brune, "Synthesis of a Finite Two-terminal Network Whose Driving-point Impedance Is a Prescribed Function of Frequency," Journal of Math, and Physics (M.I.T.), Oct. 1931, I, p. 191. 6. Sidney Darlington, Journal of Math, and Physics {M.I.T.), Sept. 1939, 4, pp. 257-353. 7. E. C. Titchmarsh, "The Theory of Functions," Oxford Univ. Press, 2nd ed., 1939, p. 110. 5. Gustav Doetsch, Theorie und Anwendung der Laplacetransformation, 1st Am. Ed. 1943, p. 139. 9. S. A. Schelkunoff, "Representation of Impedance Functions in Terms of Resonant Frequencies," Proc. of the I.Ii E., 1944, 32, No. 2, p. 83. 10. H. W. Bode, Network Analysis and Feedback Amplifier Design, D. Van Nos- trand Co., 1945. 11. P. I. Richards, "A Special Class of Functions with Positive Real Part in a Half-plane," Duke Math. J., 1947, 14, pp. 777-786. Also "General Impedance Function Theory," Quart. Appl Math., 1948, 6, pp. 21-29. 12. E. A. Guillemin, The Mathematics of Circuit Analysis, John Wiley and Sons, Inc., New York, 1950, pp. 409-422. Abstracts of Bell System Technical Papers Not Published in This Journal Universal Equalizer Chart. D. A. Alsberg\ Electronics, 24, pp. 132, 134, Nov., 1951. Modification of famiUur Siuith chart consolidates on one time-saving j^lot all positive-value solutions to the two general equations for series, shunt, and hridged-T audio o(|ualizers. Limits on the Energy of the Antif err o magnetic Ground State. P. W. Anderson'. Phys. Rev., 83. p. 1260, Sept. 15, 1951. Post-War Achievements of Bell Laboratories, I. O. E. Buckley . Bell Tel. Mag., 30, pp. 1G3-173, Autumn, 1951. Filamentary Growths on Metal Surfaces — ^'Whiskers". K. G. Compton\ A. Mendizza\ and S. M. Arnold\ Corrosion, 7, pp. 327-334, Oct., 1951. (Monograph 1885). Filamentary growths have been found on metal surfaces of some of the parts used in telephone communications equipment, particularly on parts shielded from free circulation of air. The growths are of the same character as those known as "whiskers," which developed between the leaves of cadmium plated variable air condensers and caused considerable trouble in military equipment during the carl}- i)art of World War II. An investigation has been under way in an attempt to determine the mechanism of growth of the whiskers, found not only on cadmium plated parts but also on other metals. This paper summarizes the findings to date as revealed by the study of approximately one thousand test specimens of different metals, solid and plated, exposed under various environ- mental conditions. The study is being extended in the light of the findings which have developed during the course of the work. An Unattended Broad-Band Microwave Repeater for the TD-2 Radio Relay System. R. W. Friis' and K. D. Smith\ Elec. Eng., 70, pp. 976- 981, Nov., 1951. (Similar article in The Bell System Technical Journal, * Certain of these papers are available as Bell System Monographs and may be obtained on reciuest to the Publacation Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14, N. Y. For papers availaljlc in this form, the monograph number is given in parentheses following the date of publication, and this number should be given in all requests. 'B. T. L. 405 400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 October, 1951, Part II, entitled The DT-2 Microwave Radio System reprinted as Monograph 1921). To meet the stringent requirements of the 4,000-mile transcontinental micro- wave relay system, a number of new developments had to be included in the design of the repeater stations. The circuits of these uuattended stations, and how the}' are maintained, are the subject of this article'. The Bell System's Part in Defending the Nation. F. R. Kappel". Bell Tel. Mag., 30, pp. 141-152, Autumn, 1951. Quickly and accurately checks performance of private or common-carrier p-m or f-m mobile telephone transmitters, such as those used in 30 to 44 -mc highway and 152 to 17o-mc urban service. Measures r-f power output, audio sensitivity, signal-to-noise ratio and harmonic distortion and gives speech intelligibility check in few minutes. Mobile Transmitter Testing Set. G. J. Kent . Electronics, 24, pp. 106- 109, Nov., 1951. A New Electrolysis Switch for Underground Lead Sheath Cable Drainage Systems. V. B. Pike\ Corrosion, 7, p. 1, Oct., 1951. A High Temperature Stage for the Polarizing Microscope. E. A. Wood . Am. Mineral, 36, pp. 768-772, Sept.-Oct., 1951. A Precise Sweep-Frequency Method of Vector Impedance Measurement. D. A. Alsberg'. Proc. I.R.E., 39, pp. 1393-1400, Nov., 1951. (Mono- graph 1911). The impedance of a two-terminal network is defined completely by the inser- tion loss and phase shift it produces when inserted between known sending and receiving impedances. Recent advances in precise wide-band phase and trans- mission measuring circuits have permitted practical use of this principle. Reac- tive and resistive impedance components are read directly from a simple graphi- cal chart in which frequency is not a parameter. The basic principle described promises attractive possibilities in many cases of impedance measurements where present methods are inadequate. Electron-Vibration Interactions and Superconductivity. J. Bardeen . Revs. Modern Phys., 23, pp. 261-270, July, 1951. (Monograph 1912). The Copper Oxide Rectifier. W. H. Brattain\ Revs. Modern Phys., 23, pp. 203-212, July, 1951. It is shown that the conductivity in the ohmic part of the cuprous oxide laj^er can be explained with the usual band picture of semiconductors onh* by assum- 2 A. T. & T. Co. 3W. E. Co. AHSTUACTS OF TKCIIXK AL AUTICLKS 407 ing the presence of some donor-type impurities in addition to the usual acceptor type. The energy (hfference between the acceptors and the fdled l)and is 0.3 electron volt, and the total numl)er of impurity atoms is al)out 10" to 10"'' i)er cm^, the number of donors being less than but of the same ordei' as the numl)er of acceptors. One finds that the density of ion chaige in the rectifying layer is of the same order of magnitude as the difference between the donors and acceptors found from the conductivity. The field at the copper-cuprous oxide interface is about 2 x 10^ volts/cm; the height of the potential at the surface as compared with the oxide interior is about 0.5 volt; and the thickness of the space charge la^'er about 5.0 x 10~* cm. The diffusion equation for flow of current through this space charge region can be integrated to give the current in terms of tlie field at the interface and the applied potential across the space charge layer. Two currents are involved, one from the .semi-conductor to the metal (/,) and one from the metal to the semiconductor (/,„) which is similar to a thermionic emis- sion curient into the semiconductor. The net current is, of course, / = /,„ — I>. One can get this "emission" current (/„.) by dividing the true current by the factor 1 - exp { — eVa/kT), where Va is the applied potential. This emission current depends on the absolute temperature and on the field at the copper-cuprous oxide interface. At high fields the logarithm of the current is proportional to the square root of the field, and at low fields the current decreases more rapidly indicating a i)atchy surface having small areas of low potential maximum from which all the emission comes when the field is large. Effect of Packaging on Corrosion of Zinc Plated Equipment. K. G. C'ompton\ S. M. Arnold^ and A. Mendizza . Corrosion, 7, pp. 365- 372, Nov., 1951. Physics as a Science and an Art. K. K. Darrow\ Phys. Today, 4, pp. 6-11, Nov., 1951. (Monograph 1914). The last of six invited papers presented on October 25th during the symposium on "physics todaj'" which keynoted the 20th Anniver-sar}- Meeting of the American Institute of Physics in Chicago. Ionization hy Electron Impact in CO, N2, NO, and O2. H. D. Hagstrum . Revs. Modern Phys., 23, pp. 185-203, July, 1951. (Monograph 1916). Ionization b}' electron impact in diatomic gases has- been studied in this work with a mass spectrometer designed to measure 7ti/e, appearance potential, and initial kinetic energy for each ion observed. Results have been obtained for the gases CO, N2 NO, and O2 with some confirmatory work in H2. Discussion is included of the nature and identification of dissociative ionization processes and of the retarding potential and appearance potential measurements. Values of important quantities such as the dissociation energies of CO, N2 , and NO; the sublimation energy of C ; the electron affinity of 0 ; and the excitation energy of 0~ are determined again by electron impact in this work. > B. T. L. 408 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 Equivalent Temperature of an Electron Beam. M. E. Hines ., Letter to the Editor., J. Appl Phys., 22, pp. 1385-1386, Nov., 1951. Bell System Cable Sheath Problems and Designs. F. W. Horn^ and R. B. Ramsey'., Elec. Eng., 70, pp. 1070-1075, Dec, 1951. (Monograph 1917). Engineering Planning. H. S. Osborne^. J. Eng. Educ., 42, pp. 121- 125, Nov., 1951. Acceptance Inspection of Purchased Material. J. E. Palmer^ and E. G. D. Paterson\ Ind. Quality Control, 8, pp. 23-27, Nov., 1951. A Note on the Partial Differential Equations Describing Steady Current Flow in Intrinsic Semiconductors. R. C. Prim . Letter to the Editor. /. Appl. Phys., 22, pp. 1388-1389, Nov., 1951. General Theory of Syynmetric Biconical Antennas. S. A. ScHELKUNorF\ J. Appl. Phys., 22, pp. 1330-1332, Nov., 1951. (Monograph 1922). This paper presents the input admittance of a symmetric biconical antenna of an arbitrary angle as the limit of a certain sequence of functions. The first term of this sequence approaches the exact expressions for the input admittance as the cone angle approaches either zero or 90°. For this reason our conjecture is that this term represents a good first approximation for all angles. Artificial Dielectrics for Microwaves. W. M. Sharpless\ Proc. I. R. E., 39, pp. 1389-1393, Nov., 1951. (Monograph 1923). This paper presents a procedure for measuring the dielectric properties of metal-loaded artificial dielectrics in the microwave region by the use of the short-circuited hne method. Formulas, based on transmission-line theory, are included and serve as guides in predicting the approximate dielectric properties of certain loading configurations. 1 B. T .L. ■^ W. E. Co. Contributors to this Issue W. O. Baker, B.S., Washington College, Maryland, 1935; Ph.D., Princeton University, 1938; Bell Telephone Laboratories, 1939-. Dr. Baker has carried on investigations of the molecular structure and physi- cal properties of polymers, particularly the fundamental constitution of synthetic rubbers and plastics. Harvard Fellowship, 1936-37 and Proctor Fellowship, 1938-39. Member of American Chemical Society, American Physical Society, and American Society for Testing Materials. J. H. Heiss, B.S. in Ch.E., Newark College of Engineering, 1942; Bell Telephone Laboratories, 1934-. Mr. Heiss has devoted his time to study- ing experimental wire coating procedures and the test methods involved, the experimental production of high poljoners (plastics) and the exami- nation of their physical properties, and the properties of high polymers in solution. Member of American Chemical Society. W. S. Hayward, Jr., A.B., Harvard University, 1943; S.M., Harvard Graduate School of Engineering, 1947; Aircraft Radio Laboratory, June- December 1943; U. S. Navy, Aviation Electronics Officer, 1944-46; Bell Telephone Laboratories, 1947-. Mr. Hayward has taught telephone smtching circuit design at the Laboratories and is currently making probability studies of telephone traffic. Brockway McMillan, B.S., Massachusetts Institute of Technology, 1936; Ph.D., Massachusetts Institute of Technology, 1939; Instructor of jNIathematics, Massachusetts Institute of Technology, 1936-39; Proctor Fellow and Henry B. Fine Instructor in Mathematics, Princeton University, 1939-42; U.S.N.R., 1942-46, studying exterior ballistics of guns and rockets; Los Alamos Laboratory, Spring 1946; Bell Telephone Laboratories, 1946-. Dr. McMillan has been engaged in mathematical research and consultation work. Member of American Mathematical Society, Institute of Mathematical Statistics, and A.A.A.S. R. E. Staehler, B.E.E., College of the City of New York, 1947; M.E.E., Polytechnic Institute of Brooklyn, 1948; U. S. Army 1943-46, Conmiunications Officer; Instructor in Electrical Engineering, Polytechnic Institute of Brooklyn, Fall, 1950; Bell Telephone Laboratories, 1948-. After completing the Communications Development Training Program, 409 410 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 with rotational assignments in the transmission, switching, and apparatus departments, Mr. Staehler became a member of a switching group concerned with both local and toll signaling. He is working at present on a new voice-frequency toll signaling development. Member of Tau Beta Pi and Eta Kappa Nu. M. K. ZiNN, B.S. in E.E., Purdue University, 1918; U. S. Army, 1918-19. Amer. Tel. and Tel., 1919-1934; Bell Telephone Laboratories, 1934-. As a transmission engineer, Mr. Zinn has been concerned with land and submarine loaded cables, telephone instruments, buried cable and submarine cable with amplifiers and special problems. Member of A. I. E. E. and Tau Beta Pi. HE BELLSYSTEM Jechnical journal E VOTED TO THE SC I E N T I FIC^W^ AND ENGINEERING • ."^ ,1 SPECTS OF ELECTRICAL COMMUNICATION ' /^ . ^^^« . , — OLUME XXXI MAY 1952 "%^UMBER 3 - Present Status of Transistor Development I J. A. MORTON 411 An Experimental Electronically Controlled Automatic Switching System w. a. malthaner and h. earle vaughan 443 New Techniques for Measuring Forces and Wear in Telephone Switching Apparatus WARREN p. MASON AND SAMUEL D, WHITE 469 A Comparison of Signalling Alphabets e. n. gilbert 504 Principal Strains in Cable Sheaths and Other Buckled Surfaces I. L. HOPKINS 523 A New Recording Medium for Transcribed Message Services JAMES Z. MENARD 530 Introduction to Formal Realizability Theory-II BROCKWAY MCMILLAN 541 Abstracts of Bell System Technical Papers not Published in this Journal 601 Contributors to This Issue 608 COPYRIGHT 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL PUBLISHED SIX TIMES A YEAR BY THE AMERICAN TELEPHONE AND TELEGRAPH COMPANY 195 B R O AD WAY, NEW YORK 7, N. Y. CLEO F. CRAIG, President CARROLL O. BICKELHAUPT, Secretary DONALD R. BELCHER, Treasurer EDITORIAL BOARD F. R. KAPPEL O. E. BUCKLEY H.S.OSBORNE M.J.KELLY J. J. PILLIOD A.B.CLARK R, BOWN D. A. QUARLES F. J. FE ELY PHILIP C.JONES, Editor M. E. STRIEBY, Managing Editor SUBSCRIPTIONS Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 1 1 cents per copy. PRINTED IN U.S.A. The Bell Sysleiu Teeliiiieal Journal Volume XXXI May 1952 Number 3 COPYRIGHT 1952, AMERICAN TELEPHONE AND TELEGRAPH COMPANY Present Status of Transistor Development By J. A. MORTON (Manuscript received March 17, 1952) The invention of the transistor provided a simple, apparently rugged device that could amplify — an ability in which the vacuum tube had long held a monopoly. As with most new electron devices, however, a number of extremely practical limitations had to be overcome before the transistor could be re- garded as a practical circuit element. In particular: the reproducibility of units was poor — units intended to be alike were not interchangeable in circuits; the reliability was poor — in an uncomfortably large fraction of units made, the characteristics changed suddenly and. inexplicably; and the "designability" was poor — it was difflcidt to make devices to the wide range of desirable characteristics needed in modem communications functions. This paper describes the progress that has been made in reducing these limitations and extending the range of performance and usefulness of tran- sistors in communications systems. The conclusion is drawn that for some system functions, particidarly those requiring extreme miniaturization in space and power as w^ll as reliability with respect to life and ruggedness, transistors promise important advantages. IXTRODUCTIOX When the transistor was announced not quite four years ago,- it was felt that a new departure in communication techniques had come into view. Here was a mechanically simple device Avhich could perform many of the amplification functions over which the electron tube had long held a near monopoly. The device was small, required no heater power, and was potentially very rugged; moreover, it consisted of materials which might be expected to last indefinitely long, and it did not appear to be too complicated to make. However, as might be expected for a newly invented electron device, the practical realization of these promises still required the overcoming 411 412 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 of a number of obstacles. While the operation of the first devices was well understood in a general way, several items were limiting and puz- zling, for example: a — Units intended to be alike varied considerably from each other — the reproducibility was had. b — In an uncomfortably large fraction of the exploratory devices, the properties changed suddenly and inexplicably with time and tem- perature, whereas other units exhibited extremely stable characteristics with regard to time — the reliability was poor. c — It was difficult to use the theory and then existing undeveloped technology to develop and design devices to a varied range of electrical characteristics needed for different circuit functions. Performance char- acteristics were limited with respect to gain, noise figure, frequency range and power— the designability was poor. Before the transistor could be regarded as a practical circuit element, it was necessary to find out the causes of these limitations, to under- stand the theory and develop the technology further in order to produce and control more desirable characteristics. Over the past two years measurable progress has been made in reduc- ing, but not eliminating, the three listed limitations. These advances have been obtained through an improved understand- ing, improved processes and very importantly through improved ger- manium materials. As a result: a — the beginnings of method have evolved in the use of the theory to explain and predict the electrical network characteristics of transistors in terms of physical structure and material properties. b — It is now possible to evaluate some of the effects and physical meaning of empirically derived processes and thereby to devise better methods subject to control. Previously, inhomogeneities in the material properties masked the dependence of the transistor electrical properties even on bulk properties (such as resistivity) as well as on processing effects. c — ^As a result, on an exploratory development level, it is now possible to make transistors in the laboratory to several sets of prescribed char- acteristics with usable tolerances and satisfactory yields. d — Such transistors are greatly improved over the old ones in so far as life and ruggedness are concerned, and some reduction in temperature dependence has been achieved. However, it is not to be inferred that all reliability problems are solved. e — It has become possible in the laboratory to explore experimentally some of the consequences of the theory with the result that point con- PRESENT STATUS OF TRANSISTOR DEVELOPMENT 413 tact devices with new ranges of performance are indicated. Even more importantly, new p-n junction devices have been built in the laboratory anil those junction devices have indicated an extension in several per- formance characteristics. f — By having interchangeable and reliahle devices with a wider range of characteristics, it has become possil)le to carry on exploratory circuit and sj'stem applications on a more realistic basis. Such applications effort is, in turn, stimulating the development of new devices towards new characteristics needed by these circuit and system studies. It is tlie purpose of the remainder of tliis paper to give an over all but brief summary of recent progress made at Bell Telephone Laboratories in reducing the above-mentioned limitations on reproducibility, relia- bility and performance. Since a fair number of types of devices are cur- rently under development, each with different characteristics to be op- timized, the data will be presented as a sort of montage of characteristics of several different types of devices. It is not to be inferred that any one type of transistor combines all of the virtues any more than such a situation exists in the electron tube art. Moreover, it will be impossible in a paper of practical length to present complete detailed characteristics on all or even several of these devices under development; nor would it be appropriate since most of these data are on devices currently under development. Rather, what is desired, is a summary of progress across the board to give the reader an integrated and up-to-date picture of the current state of transistor electronics. REPRODUCIBILITY STATUS Description of Transistors Before quantitative data comparing the characteristics of past with present transistors are presented, it will be useful to briefly review physical descriptions of the various types of transistors to be discussed. Fig. 1 shows a cutaway view of the now familiar point-contact cartridge type transistor. All of the early transistors were of this general construc- tion and the characteristics of a particular one, called the Type A\ will be used as a reference against which to measure results now obtainable with new types under current development. Fig. 2 is a semi-schematic picture of the physical operation of such a device. Pressing down upon the surface of a small die of ?i-type germanium are two rectifying metal electrodes, one labelled E for emitter, the other C for collector. A third electrode, the base, is a large area ohmic contact to the underside of the die of germanium. The emitter and collector electrodes obtain their 414 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 ^ Fig. 1 — The type A transistor structure. rectifying properties as a result of the 'p-n barrier (indicated by the dotted hnes) existing at the interface between the n-type bulk material and small p-type inserts under each point. When the collector is biased with a moderately large negative voltage (in the reverse direction) so that the collector barrier has relatively high impedance, a small amount of reverse current flows from the collector to the base in the form of electrons as indicated by the small black circles. Now, if the emitter is biased a few tenths of a volt positively in the forward direction, a cur- rent of holes (indicated by the small open circles) is injected from the Fig. 2 — Schematic diagram of a point-contact transistor. PRESENT STATUS OF TRANSISTOR DEVELOPMENT 415 Fig. 3 — The M1689 point-contact transistor is typical of those used iu minia- ture packaged circuit functions. emitter region into the n-type material. These holes are swept along to the collector under the influence of the field initially set up by the original collector electron current — thus adding a controlled increment of collector current. Because of their positive charge these holes can lower the potential barrier to electron flow from collector to base and thus allow several electrons to flow in the collector circuit for every hole entering the collector barrier region. This ratio of collector current change to emitter current change for fixed collector voltage is called alpha, the current gain. In point-contact transistors alpha may be larger than unit3^ Since the collector current flows through a high im- pedance when the emitter current is injected through a low impedance, voltage amplification is obtained as well. Some of the new transistors are point-contact transistors similar in physical appearance to the type A. However, their electrical character- istics will be shown to be significantly improved not over the old type A only insofar as reproducibility and reliability are concerned, but also as to range of performance. For use in miniature packaged circuit functions, the point contact transistor has been miniaturized to contain only its bare essentials. Fig. 3 is a photograph of a so-called "bead" transistor (compared to a paper clip for size) and several of the current development types are being made in this form. In Fig. 4 is shown the family of static characteristics representative of the Ml 689 bead type transistor. Note in particular the collector 416 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 w \ y \ w \ \v V \ s \ w \\ A \ 1 \\ \ \ N oS V w V > ^. \\ \ \ V dS \ \ ' \ ° o\ (\J \ \ \ \ s ^ II ' \ I o in ^ IWV ^ ll\\\\ o 1- ^LLI II (J Wy^ o. \ luuv l\\u\ \ TO\\ v \.. ^ \\ \\ «»1 v^ wVv Nov \ \ \, \\, \rV IvTX- s. Vvv 1 ^^ ^\ \ II -^ o t— I ■^ ^ ^ ^ ^-- ^Sii >t- UJ >o a. 5^< o Otr y "-< in I o I ^^ CD K o k ^ ^- 1 ■^ o ^"^ .^ 1 q --. o ^-- 6 II u IT) d -1 q - 1 o 1 II - , 33V1"I0A d31im3 gsvinoA aoioaniOD PRESENT STATUS OF TRANSISTOR DEVELOPMENT ir family whicli gixcs tlic (l('{)(MhkMicc of collector volta|i;(' upon oollector cuncMit with emitter eiirreiit ms parameter. These characteristics may he thought of as the dual to the phitc family of a triocle.'- The slope of these curves is very nearly the small-signal ac (u)llcctor impedance of the transistor.* For a fixed collector N'oltage of —20 volts, when the emitter current is changed from zero to one milliampere, note that the collector current correspondingly changes slightly more than two niilliampcrcs, indicating a current gain, alpha, of slightly more than t wo. Xewest memhei' of the transistor family ret^entl}' d(>scril)ed by Shock- ley, Sparks, Teal, Wallace and Pietenpol is the n-p-n junction tran- sistor. ' Fig. 5 is a schematic diagram of such a structure. In the center of a bar of single crystal n-type germanium there is formed a thin layer SINGLE-CRYSTAL /-"'GERMANIUM BAR COLLECTOR T p-TYPE Fig. 5 — The n-p-n junction transistor PRIMARY EMITTER CONTROLLED ELECTRON CURRENT COLLECTOR JUNCTION B! SMALL RESIDUAL COLLECTOR REVERSE CURRENT 'not CONTROLLED BY emitter) Fig. 6 — Schematic diagram of a junction transistor. * As shown by Rj-der and Kircher,' the ac collector impedance, Tc = R22 — R12, where R-v. is the open-circuited output imj)edance and Rij is the open-circuit feed- hack impedance. Usually, R22 3> R12. 418 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 of 2?-type germanium as part of the same single crystal. Ohmic non- rectifying contacts are securely fastened to the three regions as shown, one being labelled emitter, one base and one collector. In many simple respects, except for change in conductivity type from jp-n-'p in the point- contact (see Fig. 2) to n-p-n in the junction type, the essential behavior is similar. As shown in Fig. 6, if the collector junction is biased in the reverse direction, i.e., electrode C biased positively with respect to electrode B, only a small residual back current of holes and electrons will diffuse across the collector barrier as indicated. However, unlike the point- contact device, this reverse current will be very much smaller and rela- tively independent of the collector voltage because the reverse impedance of such bulk barriers is so many times higher than that of the barriers produced near the surface in point-contact transistors. Now again, if the emitter barrier is biased in the forward direction, a few tenths of a volt negative with respect to the base is adequate, then a relatively large forward current of electrons will diffuse from the electron-rich n-type emitter body across the reduced. emitter barrier into the base region. If the base region is adequately thin so that the injected electrons do not recombine in the p-type base region (either in bulk or on the surface), practically all of the injected emitter current can diffuse to the collector barrier; there they are swept through the collector barrier field and collected as an increment of controlled collector current. Hence, again, since the electrons were injected through the low forward impedance and collected through the very high reverse impedance of bulk type p-n barriers, very high voltage amplification will result. No current gain is possible in such a simple bulk structure and the maximum attainable value of alpha is unity. However, because the bulk barriers are so much better rectifiers than the point surface barriers, the ratio of collector reverse impedance to emitter forward impedance is many times greater, more than enough to offset the point-contact higher alpha; thus, the junction unit may have much larger gain per stage. ' ' Fig. 7 is a photo- graph of a developmental model of such a junction transistor called the M1752. The upper part of Fig. 8 is a collector family of static characteristics for the M1752 n-'p-n junction transistor. By way of comparison to those of the point contact family, note the much higher reverse impedance of the collector barrier (relatively independent of collector voltage) and the correspondingly smaller collector currents when the emitter current is zero. In fact. Fig. 9 is an expanded plot of the lower left rectangle of the collector family of Fig. 8. The almost ideal straight-line character PRESENT STATUS OF TRANSISTOR DEVELOPMENT 419 Fig. 7 — The M1752 junction transistor. and regular spacing of these curves persists down to voltages as low as 0.1 volt and currents of a few microamperes. Thus, essentially linear Class A amplification is possible for as little collector power as a few microwatts. Constant collector power dissipation curves of 10, 50 and 100 microwatts are shown dotted for reference. Reproducibility of Linear Characteristics In describing progress in the reproducibility of those transistor char- acteristics pertinent to small-signal linear applications, one possible method is to give the statistical averages and dispersions in the linear open-circuit impedances of the transistor as defined by Messrs. Ryder and Kircher.^ Such a procedure, of course, implies a state of statistical control in the processes leading to a reasonably well behaved normal dis- tribution for which averages and control limits can be defined. This situation can be said to be in effect for most transistors under current development. However, for the old type A unit, control simply was not in evidence; so that in quoting figures on type A's, ranges for commensurate fractions of the total family will be given. In order that symbols and terminology 420 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 will be clear, it will be useful to review briefly the method of defining the linear characteristics of all transistors. In Fig. 10 is shown a gener- alized network representing the transistor in which the input terminals are emitter-base and the output terminals are collector-base. Then, over a sufficiently small region of the static characteristics, the linear re- lations between the incremental emitter and collector voltages and currents may be represented by the pair of linear equations shown.^ 0 MA. \ -1.0 \ -1.5 -2.0 V -2.5 \ -3.0 \ -3.5 -4.0 -4.5j -5.0, \ S \l J / lie- 0 MA. " -0.5 j -1.0 ' 1 'i -1 -2.5, 1 / / / / -3.=y -4.0i -4.5 ^^ / / -so, 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Ic IN MILLIAMPERES Fig. 8 — Static characteristics of the M1752 junction transistor. PRESENT STATUS OF TRANSISTOR DEVELOPAIKXT ■421 / (AMP / / -25 ' >-50 '' f-lb -100 / -125 -150 '-175 -0.08 0 20 40 60 80 100 120 140 160 180 200 Ic IN MICROAMPERES Fig. 9 — Expanded plot of the microwatt region of the static characteristics of the M1752 transistor. 422 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 The coefficients are simply the open-circuit driving point and trans- fer impedances of the transistor, or the slopes of the appropriate static characteristics at fixed dc operating currents. These equations may be represented by any one of a large number of equivalent cir- cuits of which the one shown in Fig. 11 is perhaps currently most useful. In this circuit Vg is very nearly the ac forward impedance of the emit- ter barrier, Vc is very nearly the ac reverse impedance of the collector barrier, Vb is the feedback impedance of the bulk germanium common to both, and a is the circuit current gain representing carrier collec- tion and multiplication if any. It turns out this is very nearly equal to the current multiplication factor a of the collector barrier mentioned before. Average values of these elements for the type A transistor are given in Fig. 11. In Fig. 12 are given the ranges of these parameters for the type A as of September, 1949, and the control limits* for the same characteristics for new point-contact transistors now under de- velopment. For September, 1949, the ranges are taken about the average values shown in Fig. 11 for the type A transistor. The control limits given for the present situation apply to a number of different types of point contact transistors so that the present average values of these -^ ^ Vet H Vg = LfZ|, + LcZ,2 V^ = Lg Z21 + Lq Z22 Fig. 10 — The general linear transistor. equivalent circuit elements depend upon the type of transistor con- sidered. In Fig. 13 are given the average values of the characteristics of the M1729 point-contact video amplifier transistor which bears the closest resemblance to the older type A transistor. By way of contrast are given some typical values of the elements for the M1752 junction transistor which is not yet far enough along in its development to have design centers fixed nor reliable dispersion figures available. As Ryder and Kircher have shown, ^ transistors in the grounded-base connection may be short-circuit unstable if a > 1 and ri, is too large, * A.S.T.M. Manual, "Quality Control of Materials," Jan. 1951, Part III, pp. 55- 114. PRESENT STATUS OF TRANSISTOR DEVELOPMENT 423 since /•(, appears as a positive lecdback olomoiit. ll\c cuinc in Fiji;. 14 is a plot of the short-circuit stabiUty contour when r, and /v have the nominal values of 700 and 20,000 ohms. Transistors haviiifi; (i and /•,, sufficiently large to place their representative points al)ove this contour will l)e short-circuit unstable, i.e., they will oscillate when short-cir- cuited. Those having an a — rt, point below the stability contour will be unconditionally stable under any termination conditions. The large unshaded rectangle bounds those values of a and vo , which were repre- Tf = 250 OHMS rZ = 250 OHMS r^ - 20,000 OHMS a= 2 Fig. 11^ — Equivalent circuit and average element values of the type A transistor. ELEMENT RANGE SEPTEMBER 1949 RANGE JANUARY 1952 a 4 : t ± 20 % Tc 7 : 1 ±30°/o ^e 3 : 1 ±20% ^b 7 : 1 ±25% Fig. 12 — Reproducibilitj' of point-contact linear characteristics. TYPE M 1729 M 1752 ^e 120 25 ^b 75 250 ^c 15,000 5 XIO^ a 2.5 0.95 Fig. 13 — Average characteristics of the M1729 and typical characteristics of tlie M1752 transistors. 424 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 sentative of the type A transistor in September, 1949. It is apparent that the circuit user of" type A units had approximately a 50 per cent chance of obtaining a short-circuit unstable unit from a large family of type A units. The smaller shaded rectangle bounds the values of a and Vb now realized in the M1729 transistor presently under development. Not only has the spread in characteristics been greatly reduced as shown, but also the design centers have been moved to a region for which all members of the M1729 family are unconditionally stable. It is of interest to note that spreads of the order of ±20 to ±25 per cent are of the same magnitude as those dispersions now existing amongst the characteristics of presently available well-controlled electron tubes. These kinds of data on reproducibility of the linear equivalent circuit element values hold for practically all classes of point-contact devices Fig. 14 — Stability contour and ranges of a and rt . now under development for cw transmission service. While it is too early to prove that such a situation pertains as well to junction tran- sistors, there is every reason to expect similar results after a suitable development period. Reproducibility of Large-Signal Characteristics for Pulse Application When electron devices are employed for large-signal applications, particularly those of switching and computing, it is well known that the characteristics must be controlled over a very broad range of variables from cutoff to saturation. In September, 1949, very little attempt was made to control such pulse use characteristics. In the intervening time, transistor circuit studies have proceeded to the point where it is possible to define certain necessary large scale transistor characteristics which, if met, permit such transistors to be used interchangeably and repro- ducibly in a variety of pulse circuit functions such as binary counters, PRESENT STATUS OF TRANSISTOR DEVELOPMENT 425 hit registers, regenerative pulse amplifiers, pulse delay amplifiers, gated amplifiers and pulse generators. Moreover, it has been possible to meet these requirements on a developmental \v\v\ with good yields in at least three types of point-contact switching transistors. The scope of this paper will not permit a detailed accounting of the technical features of this situation and such an account will be forthcoming in future papers on these particular studies. However, a brief description of some of the more important i)ulse characteristics and their tolerances is certainly l)ertinent. In practically all of the transistor pulse handling circuits examined to date, one characteristic common to all is the ability of the transistor, by N'irtue of its current gain, to present various types of two-state negative resistance characteristics at any one or all of its pairs of terminals. A typical simple circuit and corresponding characteristic is shown in Fig. 15 for the emitter-ground terminals when a sufficiently large value of resistance is inserted in the base to make the circuit unstable. In region I where the emitter is negative, the input resistance is essentially the reverse characteristic of the emitter as a simple diode. In region II as the emitter goes positive, alpha, the current gain rises rapidly above luiity. If Rb is sufficiently large and alpha, the current gain, is greater REGION I (CUT-OFF) REGION HI (SATURATION) Fig. 15 — Emitter-ground negative resistance rircuit and characteristic. 426 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 than unity the emitter to ground voltage will begin to fall because of the larger collector current increments driving the voltage of the node .¥ negative more rapidly than the emitter current drop through Ve would normally carry it. This transition point is called the peak point. If then a{ri) + /?&) is sufficiently large, in this sense, the input resistance may be negative in this region II. When the internal node voltage has fallen to a value near that of the collector terminal the "valley point" has been reached. At this point, the emitted hole current has reduced the collector impedance to a minimum value beyond which a is essentially zero; the transistor is said to be saturated. From this point on the in- put impedance again becomes positive and is determined almost entirely by the base and emitter impedances. By terminating the emitter- ground terminals in various ways with resistor-capacitor-bias com- binations, such a network can be made to perform monostable, astable or bistable functions. Under such conditions, the emitter current and correspondingly the collector current switch back and forth between cutoff and saturation values. For example, in Fig. 16 is shown a value of emitter bias and load resistance such that there are three possible eciuilibrium values of emitter current and voltage. It may be shown that the two intersections in regions I and III are stable whereas that in region II is unstable. Hence, if the stable equilibrium is originally in I, a small positive pulse Ap applied to the emitter will be enough to switch from stable point I to stable point II and conversely, — A^. will carry it from the high current point to the low current point. The circuit designer is interested in reproducing in a given circuit (with different transistors of the same type) the follo^vdng points of the characteristic: a — The off impedance of the emitter — he desires that this be greater than a certain minimum. b — The peak point Vep — he desires that this be smaller than a certain maximum. c — The value of the negative resistance — he desires that tliis be greater than a certain minimum. d — The valley point Ves , Ls — he desires that these be greater than certain minima, and e — The slope in region III — he desires that this be smaller than a certain maximum so that he ma}^ control it by external means. It may be shown that these conditions can be satisfied for useful circuits by specifying certain maximum and minimum boundaries on the static characteristics. Fig. 17 is an idealized set of input or emitter characteristics. By specifying a minimum value for the reverse resistance PRESENT STATUS OF TRANSISTOR DEVELOPMENT 427 ill region 1, couditiou (a) above is satisfied. By specifying a maximum slope in region II and III, condition (e) is satisfied. Now refer to the idealized collector family in Fig. 18; by specifying a maximum value to Tf;), it is possible to insure condition (d) and by specifjnnga minimum value for r.,<„ condition (b) can be satisfied. Finally, in Fig. lU 1)}^ de- Fig. 16 — Bistable circuit and characteristics showing trigger voltage requirements. \^>^ ^^ \(^^ ^^'^'^ .^'"''^^^^ ^^^^ ^^ ^'''''^^^ 0 ^ ^--^'^^^^ ^^^^Js- • ^ ^ REGION I-- / 1 ■* REGIONS n, m- Fig. 17 — Idealized emitter characteristics — slope = Ru 428 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 manding that alpha, as a function of /« , go through a transition from a negUgible vahie (at small negative le) to a value well in excess of unity (at a correspondingly small positive value of /«) and maintain its value well in excess of unity at large values of /« , conditions (b) and (c) can be met. In Fig. 20 are given the characteristic specifications which must be met by the M1689 bead type switching transistor now under develop- ment. With these kinds of limits, circuit users find it possible to inter- change such M1689 units in various pulse circuits and obtain ovei-all circuit behavior reproducible to the order of about dz2 db. Ic= -5.5 MA ic= riw -2MA 1 ■Vc Ic r^ Leryr^^ Vci VC3 ^^ -n / / ''' 7 7 / / /''^''° i 1, /, / / / /--- Vc=- J J y ./,/ o/ ' Fig. 18 — Idealized collector characteristics. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 EMITTER CURRENT, If , IN MILLIAMPERES Fig. 19 — Effective alpha characteristic. PRESPJNT STATUS OF TRANSISTOR DEVKLOPMENT 429 TEST CONDITIONS MINIMUM MAXIMUM Tco-OFF COLLECTOR DC RESISTANCE Vc = - 35 V DC Ig = OMA DC 17,500 OHMS Vc, -ON COLLECTOR VOLTAGE Ic = -2MA DC If = I MA DC -3V DC Vc3-0N COLLECTOR VOLTAGE I(. = - 5.5 MA DC If - 3 MA DC -4V DC OFF EMITTER RESISTANCE Vc = -10 V DC 50,000 OHMS ON EMITTER RESISTANCE R„ V(- = -10 V DC If = 1 MA DC 800 OHMS ai Vc = - 30 V DC If = 1.0 MA DC 1.5 32 Vc = -30 V DC If = + 0.05 MA DC 2.0 ^3 Vc = - 30 V DC If = - 0.1 MA DC 0.3 Rl2 - OPEN CIRCUIT FEEDBACK RESISTANCE Vc = -10 V DC If = +1 MA DC 500 OHMS R21 - OPEN CIRCUIT FORWARD RESISTANCE Vc = -10 V DC If = +1 MA DC 15,000 OHMS R22 - OPEN CIRCUIT OUTPUT RESISTANCE Vc = -10 V DC If = +1MA DC 10,000 OHMS Fig. 20 — Tentative characteristics for the M1689 switching transistor. RELIABILITY FIGURE OF MERIT SEPTEMBER 1949 JANUARY 1952 AVERAGE LIFE = 10,000 HOURS > 70,000 HOURS EQUIVALENT TEMPERATURE COEFFICIENT OF Tc -1% PER DEG C -V4»/o PER DEG C SHOCK ? > 20,000 G VIBRATION ? 20-5000 CPS NEGLIGIBLE TO 100 G Fig. 21 — -Reliabilit}' status. RELIABILITY STATUS Life Reliabilit}^ figures of merit are not too well defined for electron tubes and the same situation certainly holds at present for transistors. How- ever, insofar as these quantities can be presently defined, Fig. 21 shows 430 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 a comparison between the present status and that in September, 1949. Estimates of the half-hfe of a statistical family of devices are at best arbitrary and necessarily amount to extrapolations of survival curves assuming that a known survival law will continue to hold.* In Septem- ber, 1949, life tests on type A units had been in effect some 4000 hours. With the assumption of an exponential survival law, it was not possible, on the basis of a 4000 hour test, to estimate the slope sufficiently accu- rately to warrant a half -life estimate in excess of 10,000 hours. These same type A units have now run on life test for approximately 20,000 hours. With the more reliable estimate of survival slope now possible, the half-life is now estimated to be somewhat in excess of 70,000 hours. It should be emphasized, however, that these are type A units of more than two years ago made with inferior materials and processes. It is believed that those units under current development, being made with new materials and processes, are superior; but, of course, life tests are only a few thousand hours old. Although these new data are encouraging, it is still too early to extrapolate the data such a long way. Temperature Effects Transistors like other semiconductor devices are more sensitive to temperature variations than electron tubes. In terms of the linear equivalent circuit elements, the collector impedance, re , and the current gain, a are the most sensitive. Over the range from — 40°C to 80°C the other elements are relatively much less sensitive. For type A transistors these temperature variations in r<; and a are shown in Fig. 22. While these curves are definitely not linear, an average temperature coefficient for fc of about — 1 per cent per degree was estimated for the purpose of easy tabulation and comparison in Fig. 21. Thus, for the early type A, Vc fell off to about 20 to 30 per cent of its room temperature value when the temperature was raised to -f 80° C; at the same time a increased from 20 to 30 per cent over the same temperature range. Today, this variation has been reduced by a factor of about four for re in most point-contact types, the variations in the current gain being relatively unchanged. Fig. 23 illustrates the tem- perature dependence of re and a for the M1729 transistor now under development. Again, for purposes of easy comparison in Fig. 21, the actual dependence of Fig. 23 was approximated by a linear variation and * Estimates of life, of course, depend upon definitions of "death". For these experiments, the transistors were operated as Class A amplifiers. A transistor is said to have failed when its Class A gain has fallen 3 db or more below its starting value. PRESENT STATUS OF TRANSISTOR DKVKLOl'M HXT i;-;i only the slope given in Fig. 21. For linear applications such as the grounded base amplifier, the Class A power gain is approximately pro- portional to a'r^ ; hence the gain of such an amplifier will stay essentially constant within a db or two oxer the temperature range from — 40°C to +80°C. For pulse applications, and of importance to dc biasing with point-contact transistors, is the fact that the dc collector current (for fixed emitter current and collector voltage) will change at about the ;ioo , 90 I I 80 t I 70 i : 60 i I • 50 i 40 30 20 CURRENT GAIN a^ , 1/ ^ \ \ V \ \ COLLECTOR RESISTANCE \ '' \ \, \ \ 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 IC 0 TEMPERATURE IN DEGREES CENTIGRADE Fig. 22 — Collector resistance and a versus temperature for type A transistor rCjIOO ^ 80 CURRENT GAIN a > /' ^X X y^ ^«^ COLLECTOR'S RESISTANCE s 20 30 40 50 60 70 80 90 TEMPERATURE IN DEGREES CENTIGRADE Fig. 23 — Collector resistance and a versus temperature for type M1729 transistor. 432 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 same rate as does Tc , the small signal collector impedance. Similar im- provements have been made in these variations for switching transistors and Fig. 24 is a series of graphs showing how the M1689 bead type switching transistor changes the pulse characteristics defined in Fig. 20 with respect to temperature. For those switching functions examined to date, it is believed that these data mean rehable operation to as high as +70°C m most applications and perhaps as high as +80°C in others. --0- — ,15.- ,^. :> "" Te 3 2 70 60 z U50 z < ■^30 in LJ a. X 103 ^ — o Jjn, --- — o .— ~^"" ■"'o..^ — o Tc ■*^v 20 10 "■>)^ 40 70 50 60 70 80 20 30 40 50 60 TEMPERATURE IN DEGREES CENTIGRADE Fig. 24 — Temperature behavior of the M1689 transistor. In junction transistors the laws of temperature variation are not so well established, the device being in a much earlier stage of development. Preliminary data indicate smaller variations in the small signal pa- rameters such as a and Vc . On the other hand, variations in the dc cur- rent, particularly Ico , are many times greater, of the order of 10 per cent per degree centigrade.* The only saving grace here is the fact that Ic« is normally very much less than the actual operating value of Ic . In summary, it may be said that while significant improvements have been made in temperature dependence to the point where many appli- cations appear feasible, it is not to be inferred that the temperature limitation is completely overcome. Much more development work of de\'ice, circuit and system nature is required to bring this aspect of reliable operation to a completely satisfying solution. Shock and Vibration With regard to mechanical ruggedness, current point-contact tran- sistors have been shock tested up to 20,000 g with no change in their leo is the collector current at zero emitter current. PRESENT STATUS OF TRANSISTOR DEVELOPMENT 433 electrical characteristics. Vibration of point-contact and junction tran- sistors o\(M- tlic tro(|uency range from 20 to 5000 cps at accelerations of lOOg produces no detectable modulation of any of the transistor elec- trical characteristics, i.e., such modulalion, if it exists, is far loelow the inherent noise level. At a few spot frequencies in the audio range, vi- bration tests up to lOOOg accelerations similarly failed to produce dis- cernible modulation of the transistor characteristics. MINIATURIZATION FIGURE OF MERIT TYPE A SEPTEMBER 1949 JANUARY 1952 NEW DEVELOPMENT TYPE VOLUME Vso IN 3 '/2000 IN 3 POINT- M1689 V500 IN 3 JUNCTION -M1752 MINIMUM COLLECTOR VOLTAGE FOR CLASS A OPERATION 30 V 2V POINT- M1768, M1734 0.2 V JUNCTION -M1752 MINIMUM COLLECTOR POWER FOR CLASS A OPERATION 50MW 2MW POINT- Mt768 I0//W JUNCTION -M 1752 CLASS A EFFICIENCY 20% 35°/o POINT- M1768, M1729 49% JUNCTION-M1752 Fig. 25 — Miniaturization in space and power drain. MINIATURIZATION STATUS Space Requirements In smallness of size, the transistor is entering new fields previously inaccessible to electron devices. The cartridge structure (see Fig. 25), such as the type A, has a volume of 5V cubic inch, compared to about -g^ cubic inch for a sub-miniature tube and about 1 cubic inch for a minia- ture tube. Under current development, the M1689 bead point-contact transistor has substantially similar electrical characteristics to the M1698* cartridge switching unit but occupies only about -2^00 cubic inch. The M17.52 junction bead transistor has a volume of approximately 5^ cubic inch but this may be reduced to the same order as the point-contact bead if necessary. For further substantial size reductions in equipment, the next move must comprise the passive components. It should be pointed out that the low voltages, low power drain, and correspondingly lower ec[uipment temperatures should make possible further reductions in passive component size. * The M1698 transi.stor is a cartridge tyjje point-contact transistor witli elec- trical characteristics designed for switching and pulse applications. This unit is ])roving u.seful in the laboratory development of new circuits or in cases where miniature j)ackages are unnccessarj'. 434 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Power Requirements The transistor, of course, has the inherent advantage of requiring no heater power; moreover, significant advances have been made in the past two years in reducing the collector voltage and power required for prac- tical operation. Consider the minimiun collector voltage for which the small-signal Class A gain is still within 3 to 6 db of its full value. In September, 1949, the type A transistor could give useful gains at col- lector voltages as low as 30 volts. Today, several point-contact devices (M1768 and M1734) perform well with collector voltages as low as 2 to 6 volts even for relatively high-freciuenc}^ operation. One junction tran- sistor, the j\I1752, can deliver useful gains at collector voltages as low as 0.2 to 1.0 volt. Under these same conditions, the minimum collector power for useful gains may be as low as 2-10 mw for point-contact devices and as low as 10 to 100 /xw in the case of the junction transistors.* Class A efficiencies have been raised for the point -contact devices to as high as 30-35 per cent and for junction transistors this may be as high as 49 per cent out of a maximum possible 50 per cent. Class B and C efficiencies are correspondingly close to their theoretical limiting values. PERFORMANCE STATUS Exact electrical performance specifications for the transistor depend, of course, upon the intended applications and the type of transistor being developed for such an application. These types are beginning to be specified; and in fact, they are already so numerous that mention of only a few salient features of some of them will be attempted. Bear in mind, as was pointed out before, that no one transistor combines all the \drtues any more than does any one tube type. Fig. 26 attempts to compare the progress made in se^'eral important performance merit figures by development of several point-contact and junction types during the last two years. Again the reference performance is that of the type A as of September, 1949. Some switching and transmission applications need transistors having high current gain. By gouig to a point-junction structure, useful values of alpha as high as 50 are now possible with laboratory models. For straight transmission applications, the single stage gain of point- contact types (Ml 768, M1729) has been increased to 20-24 db, whereas for the M1752 junction type the single stage gain may be as high as 45-50 db. * In some special cases, depending upon the application, practical operation may be obtained for as little as 0.1 to 1.0 microwatt. PRESENT STATUS OF TRANSISTOR DEVELOPMENT 435 PERFORMANCE FIGURE OF MERIT TYPE A SEPTEMBER 1949 JANUARY 1952 NEW DEVELOPMENT TYPE a -CURRENT GAIN 5X SOX JUNCTION SINGLE STAGE CLASS A GAIN 18 DB 22 DB POINT- M 1729, M176e 45 DB JUNCTION-M1752 NOISE FIGURE AT tOOO CPS 60 DB 45 DB POINT-M1768 10 DB JUNCTION-MI752 FREQUENCY RESPONSE SMC 7-lOMC POINT- MI729 20-50MC POINT- M 1734 CLASS A POWER OUTPUT 0.5 WATT 2 WATTS JUNCTION SWITCHING CHARACTERISTICS NONE GOOD POINT-M169e,Ml689 MI734 FEEDBACK RESISTANCE 250 OHMS 70 OHMS POINT-M1729 'I'fpT PHOTOCURRENT °^"^ RATIO 2:1 20 ;i JUNCTION-MI740 Fig. 26 — Performance progress. For high-sensitivity low-noise appHcations, the point-contact devices have been improved to have noise figures of only about 40-45 db, whereas the Ml 752 n-p-n transistor has been shown to have noise figures in the 10 20 db range. All such noise figures are specified at 1000 cps and it should be remembered that they vary inversely with frequency at the rate of about 11 db per decade change in frequency. For video, I.F., and high-speed switching applications, measurable improvement has been attained in the frequency response. For video amplifiers up to about 7 mc, the M1729 point-contact transistor is capable of about 18-20 db gain per stage. For high-freciuency oscillators and microsecond pulse switching, the M1734 point-contact transistor is under development. Preliminary models of 24 mc I.F. amplifiers using the ^11734 have been constructed in the laboratory, these amplifiers having a gain of some 18-24 db per stage and a band-width of several megacycles. However, more work needs to be done on the ]\11734 to reduce its feedback resistance. For pulse-handling functions, such M1734 units work very nicely as pulse generators and amplifiers of | micro- second pulses, requiring only 6-8 volts of collector voltage and 12-20 mw of collector power per stage. The amplified pulses can have ampli- 436 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 tudes as large, as 4-5 volts out of a total collector ^'oltage of 6 volts and rise times as little as 0.01-0.02 microsccoiid. By increasing the thermal dissipation limits of junction transistors, the Class A power output has been raised to 2 watts in lal)oratory models. This, however, does not represent an intrinsic upper limit but rather a design objective for a particular application. Characteristics suitable for switching are now available in the ]M1698, M1689 and M173-4: point-contact types, as previously described, but this is a continually evolving process and more work certainly remains to be done. At present it is possible to operate telephone relays requiring as much as 50 to 100 ma with Mir)89 and M1698 point-contact tran- sistors. New junction-type phototransistors^ represent a marked advance over the earlier point -contact tj^pe.® While their quantum efficiencies are not as high as those of the point-contact types, nevertheless the light/dark current ratios are greatly improved and the collector impedance has been raised 10-100 times thus making possible much greater output voltages for the same light flux. SOME SELECTED APPLICATIONS Data Transmission Packages To determine the feasibility of applying transistors in the form of miniature packaged circuit functions, several of the major system func- tions of a pulse code data transmission system have been studied. This investigation has been undertaken under the auspices of a joint services engineering contract administered by the Signal Corps. It was desired that these studies should lead to the feasibility develop- ment of unitized functional packages combining features of miniaturiza- tion, reliability and lower power drain. Accordingly, it was necessary to carry on in an integrated fashion activities in the fields of system, circuit and device development to achieve these ends. In particular, circuit and system means have been developed to perform with tran- sistors the functions of encoding, translation, counting, registering and serial addition. The M1728 junction diode, M1740 junction photocell and Ml 689 bead switching transistor are direct outgrowths of this program and are the devices used in the circuit packages. At this point, the major system functions shown in Fig. 27 have been achieved with interchangeable transistors. These major system functions are in turn built up of some seven types of smaller functional packages listed in Fig. 28. The end result of this exploratory development can be PRESENT STATUS OF TRANSISTOR DEVELOPMENT 437 said to have domonstrated the featsihility of siicli a data transmission system in the sense that a workable (though not yet optimal) system can l)e synthesized from reprodneible transistor-circuit packages wliich have been produced at reasonabl{> yi(>lds and with reasonalile (thouj^h not yet complete) service reliability. I'^urther d(>vel(){)ment woi'k woiild be needed in all phases to make such a system of ))ackages suitable for field use. It is estimated tliat the present laboratory model requires about one-tenth the space and powcM- re([uired to do the same job with present tul)e art. Fig. 29 is a pliotograph of a transistor bit-rc^gister package and Fig. 30 is another i)hot()gi'apli of such packages sliowing lioth sides of tlie \-arious types employed.* Actual final jjackages would 1. 4 DIGIT REVERSIBLE BINARY COUNTER 2. 6 DIGIT ANGULAR POSITION ENCODER 3. 6 DIGIT GRAY -BINARY TRANSLATOR 4. 5 DIGIT SHIFT REGISTER 5. 2 WORD SERIAL ADDER Fig. 27 — System functions tested. DEVELOPMENT PACKAGE TYPE PACKAGE FUNCTION DEVELOPMENT TRANSISTOR, DIODE TYPES USED M 1731-1 REGENERATIVE GATE M 1689 M 1727 M 1732-1 M 1736 M 1790 BIT REGISTER M 1689 M 1727 M 1734 M 1733-1 M 1792 PULSE AMPLIFIER M 1689 M 1735-1 M 1747-1 M 1748-1 M 1751 -1 M 1751-2 M 1751-3 DIODE GATE M 1727 400 A M 1745-1 M 1791 BINARY COUNTER M1689 400 A M 1749-1 PHOTOCELL READOUT M1740 M 1746-1 DELAY AMPLIFIER M1689 Fig. 28 — Development tran.sistor — circuit packages. * The Auto-Assembh' Process used in the construction of these packages is a Signal Corps Development. 438 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Jill register puekugc. probably not use such clear plastics and Fig. 31 shows some packages in which the plastic has been loaded with silica to increase its strength and thermal conductivity. The assembly in Fig. 31 consists of a six-digit position encoder at the left, followed by six regenerative pulse amplifiers which in turn feed a six-digit combined translator-shift register. N-P-N Transistor Audio Amplifier and Oscillator* To the right in Fig. 32 is shown a transformer-coupled audio amplifier employing two M1752 junction transistors. This amplifier has a pass band from 100-20,000 cps and a power gain of approximately 90 db. Its gain is relatively independent of collector voltage from 1-20 volts, * The material of this section represents a summary of some work by Wallace and Pietenpol described more completely in Ref. 4. PRESENT STATUS OF TRANSISTOli l)i;\ KI.Ol'.M K\l' 439 Fig. 31 — Laboratory model of encoder-transistor-register using transistor packages. 440 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Fig. 32— Packaged oscillator and amplifier using junction transistors. only the available undistorted power output increasing as the voltage is increased. At a collector voltage of 1.5 volts it draws a collector current of approximately 0.5 ma per unit for a total power drain of 1.5 milli- watts. Under these conditions it will deliver Class A power output of about 0.7 milliwatt. The noise figure of such an amplifier has been measured to be in the range from 10-15 db at 1000 cps depending upon the operating biases. To the left of Fig. 32 is shown a small transistor audio oscillator having a single M1752 transistor, a transformer and one condenser. To see just how little power was the minimum necessary to produce stable oscilla- tions such an oscillator was tried at increasingly lower collector supply voltages. It was found that stable oscillations could be maintained down to collector supply voltages as low as 55 millivolts and collector current as low as 1.5 microamperes for a total drain of 0.09 microwatt. SUMMARY With respect to reproducibility and interchangeability, transistors now under development appear to be the equal of commercial vacuum tubes. With regard to reliability, transistors apparently have longer life and greater mechanical ruggedness to withstand shock and vibration than most vacuum tubes. With regard to temperature effects, transistors are inferior to tubes and present upper limits of operation are 70-80°C for most applications. This restriction is often reduced in importance by the lower power consumption which results in low equipment self- heating. This, however, is the outstanding reliability defect of transistors. PRESENT STATUS OF TRANSISTOR DEVELOPMENT 441 With regard to miniaturization, \\\v comparison figures are so great as to speak for themselves. ()))eration with a few milliwatts is always feasible and in some cases operation at a few microwatts is also possil)le. With regard to performance range, it is heheved that the above results imply the following tentative conclusions: In pulse systcniis (up to 1-2 mc repetition rates) transistors should be considered seriously in comparison to tubes, since they provide essen- tially equal functional performance and have marked superiority in miniature space and power. Bear in mind that in some reliability figures they are superior whereas in the matter of temperature dependence they are inferior to tubes. In CW transmission at low freriuencies (<1 mc) essentially the same conclusions are indicated, primarily because of junction transistors. In the range from 1-100 mc, tubes are currently superior in every functional performance figure (except perhaps noise and bandwidth) so that for transistors to be considered for such applications, much greater premium must be placed on miniaturization and reliability than for the first two applications areas. Thus, it might be assumed that, even though there are many out- standing development problems of a circuit and device nature to be solved, it is appropriate for circuit engineers to explore serioush' the application possibilities of transistors — not only in the hope of building better systems, but also to influence transistor development towards those most important systems for which their intrinsic potentialities best fit them. It should not be inferred that all important limitations have been eliminated — nor, on the other hand, that the full range of performance possibilities have been explored. If one remembers the history of engineering research and development in older related fields, it seems apparent that a relatively short time has elapsed since the invention of the first point-contact transistor. Already, new properties and new tj'^pes of devices are under study and some have been achieved in the laboratory. It therefore is possible, and certainly stimulating, to infer that more than a single new component is involved; that much more lies ahead than in the past; that, indeed we may be entering a new field of technology, i.e., "transistor electronics". ACKNOWLEDGMENTS It was stated earlier that these advances in the development of tran- sistors have resulted from improved understanding, materials and proces- ses. These improvements have been made through the efforts of a large 442 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 number of workers in physical research, chemical and metallurgical research and transistor development. In reality, these colleagues are the authors of this paper; and it is to them the writer owes full and appreciative credit for the material that has made possible this report of progress in transistor electronics. REFERENCES 1. R. M. Ryder, R. J. Kircher, "Some Circuit Aspects of the Transistor", Bell System Tech. J., 28, p. 367, 1949. 2. R. L. Wallace, G. Raisbeck, "Duality as a Guide in Transistor Circuit Design", Bell System Tech. J., 30, p. 381, 1951. 3. W. Shocklev, M. Sparks, G. K. Teal, "p-/i Transistors", Phys. Rev., 83, p. 151, 1951. 4. R. L. Wallace, W. J. Pietenpol, "Some Circuit Properties and Applications of n-p-n Transistors", Bell System Tech. J., 30, p. 530, 1951. 5. W. J. Pietenpol, "p-n Junction Rectifier and Photocell", Phys. Rev., 82, No. 1, pp. 122-121, Apr. 1, 1951. 6. J. N. Shine, "The Phototransistor", Bell Laboratories Record, 28, No. 8, pp. 337-342, 1950. An Experimental Electronically Controlled Antomatic Switching System By W. A. MALTllANKR AND H. IvARLE VAUGHAN (Manuscript received February 15, 1952) An automatic telephone switching system, built as a laboratory experiment, is described in which electronic techniques, high speed relays and a sub- scriber telephone with a preset dialing mechanism were employed. One-at-a- time operation within the office was made possible by these fast tools; that is, only a single control circuit was provided for each function. This ex- perimental system, although not commercially economical, showed that an advantageous reduction in the number of control and connector circuits is made possible by this method of operation. IXTRODUCTIOX This paper describes a laboratory experiment in automatic telephone switching systems. The in\'estigation was conducted at the research level to gather valuable information and circuit techniques from a labo- ratory trial and not to evolve a system economically competitive with existing systems since the area of investigation is always broader and the results more general in character when the work is imfettered by economic restraints. Indeed, the results are not economically com- petitive. Purposes of the investigation were to determine what advantages may be derived from faster operation, largely through the use of electronic technicjues, and to introduce and test some previously imexplored philos- ophies in switching and signaling. Some of the basic tools employed were dry-reed relays, mercury relays, multi-element cold cathode gas tubes, cold cathode gas diodes, and thermionic electron tubes. An experimental subscriber's telephone set, incorpoi'atiiig a preset dial mechani.sm with circuits for generating dialing signals of a new form, together with suitable signal receivers for the central office was designed as well as a novel type of switching network with its control circuits. A basic aim of the experiment was one-at-a-time operation within the central office. 443 444 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 BACKGROUND AND OBJECTIVES In many recent designs of dial telephone central offices, especially those in use in large urban areas, the subscriber's dial does not control directly the setting of switches leading toward the desired destination as was the case in early dial systems. Instead the information is received first by a register circuit which is selected from a group of such register cir- cuits and is connected to the calling subscriber's line on the origination of a call. The register cooperates with other complex circuits to ascertain the location of idle trunks to the called subscriber's office and possible routes through the switching network to these trunks, and to control the selection and use of one such path to this called office. In the called office another register circuit, frequently of a type different from that into which the subscriber originally dialed, is selected from a group of such circuits and the directory number of the called subscriber is transmitted to it from the register-sender circuit in the calling office. In the ter- minating office the procedure of locating and testing the called line and switching paths to it, and of establishing a connection over one of these paths is accomplished through the use of additional control cir- cuits. These various circuits which are used in setting up a conversa- tional path are called common control circuits. Each type of common control circuit is provided in sufficient number to halidle the expected traffic. The number required is, of course, related to speed of operation since the shorter the holding time of a circuit, i.e., the length of time a circuit takes to complete its functions for one call, the more calls such a circuit can complete in a given time. The holding time of a control circuit is, in turn, dependent upon the operating speed of the equipment controlled. Furthermore, control circuits of the same type, if more than one of a given type is required, will have added to their normal functioning time during busy traffic periods a delay time interval since they must not interfere with each other's actions in the controlled equipment. Common control circuits, such as dial pulse regis- ters, which receive information directly from subscribers must be engi- neered on the basis of an average holding time which allows for the variable reaction times, hesitations, partial usages and other personal idiosyncrasies of subscribers. Present designs of automatic central offices require a number of each type of control circuit and auxiliary circuits for selecting and connecting the control circuits as required in the opera- tion of the system. These control circuits and connectors embrace a considerable fraction of the space and cost of such an office. Dr. T. C. Fry, at the time he was Director of Switching Research AUTOMATIC SWITCHING SYSTEM 445 at the Bell Telephone Laboratories, suggested that a program be started to explore the possibilities of a new system whicli would ixMiuiic only a single control circuit of each type. This would reciuiio that each group of functions assigned to a common contiol ciicuit \)e pciformod on a one- call-at-a-time basis. It might be accomplished in a fi'csh approach to system design employing recent developments in high speed components. High speed in the common control units alone would not be sufficient. 1 1 would also be necessary to have fast switches since the operating time of a switching network is part of the holding time of the control circuit which operates the network. Similarly, since the signaling time is part of the holding time of the control circuit which receives and registers the signals, some form of liigh speed signaling would also be required. Further, the subscribers should have no direct control of the holding time of any common control unit. It was hoped that a great reduction in the number of common control circuits and connectors would result in a reduction in the size and cost of a central office even if the individual control circuits were somewhat more expensive. Furthermore, a speed permitting one-at-a-time operation would result automatically in faster service for the subscriber. Consideration of the various factors of one-at-a-time operation was undertaken by the members of the Switching Research Department and possible system components evolved. Primary elements of inherently high speed, such as cold cathode gas tubes, thermionic electron tubes, dry-reed relays and mercury relays, were immediately adopted for the system. A network of high-speed switches with its high-speed control circuits was designed. A pre-set dialing device in the subscriber telephone set ^vith transmission of high-speed dialing signals to the central office under control of common equipment in the office was selected as a means of eliminating the direct influence of subscribers on control circuit holding time. A code of high-speed signals, suitable for transmission over all existing types of local telephone facilities, with means for the pre-selec- tion and controlled generation of telephone numbers w^as designed into the subset. Such a subset is necessarily complex since it becomes a form of manually operated register wath all digits of a number stored before transmission to the central office. Circuits to control the generation of subset signals from the central office and receiver circuits to decode and register the signals were constructed. These parts were then combined in the design of the Electronicallj'' Controlled Automatic Switching System, ECASS. A skeletonized labo- ratory version was built and tested to investigate the feasibility of com- bining the circuit elements and techniques, and to prove the operability 446 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 ^ Fig. 1 — Cold-cathode gas tubes — pentode, diode and octode. of such a system. System operation is described in this paper after a more detailed discussion of the components mentioned above. COMPONENTS Cold cathode tubes, usually diode or triode types, have found wide- spread application in the past but the gas tubes used in the ECASS system were developed to have special characteristics for switching use. The three types of cold cathode gas tubes used were: a diode, a screen grid pentode and a multi-purpose octode. Fig. 1 shows a photograph of each type and Fig. 2 gives a schematic drawing of the internal elements. These tubes were developed by W. A. Depp and R. L. Vance. The diode ANODE MAIN ANODE MAIN ANODE CATHODE A- 1627 DIODE SCREEN , . GRID Pf^- STAR' CATHODE Fig. 2 — Schematics of cold-cathode gas tubes. AUTOMATIC SWITCHING SYSTEM 447 i.s usctl at many points throughout the .switching network, the screen- grid pentode in the patli selection pi'ocessc^s in tlie switching network, and the octode tor miscenan(M)us i)ui"poses in the hn(\ trunk, niniilxM' group and other circuits. The dry-reed switch, which is usinl as the contact element in many fast, rehiys as well as in the metallic talking path through the oflice, is shown in Fig. 3. This switch consists of two permalloy I'o Is sealed in opjjosite ends of a small glass tube which is filled with an inei't gas. The o\'er- lapping ends of the nxls normally ha\'e a gap l)etw(>en them and the ai)i)lication of a magnetic field coaxial with the rcH'ds will cause them to pull togethei' and close a metallic path from one rod or reed to the other through I'hodium plating at the contacting ends. The dry-reed switch has an extremely small operate and I'elease time, and because of the gas sealed and permanently adjusted construction prox'ides a highly reliable dirt -free contact for low current applications. The dry-reed switch and relays employing it were developed by W. 13. EUwood. Mercury contact relays, also of a sealed and permanently adjusted construction, are used where fast operation at heavier currents is required. A sectional drawing Fig. 3 — Glass-sealed dry-reed switch. of a mercury contact relay is shown in Fig. 4. These relays were developed by J. T. L. Brown and C. E. Pollard. Dry-reed relays and mercury relays are described in Electrical Engineering, Vol. 66, pp. 1104-1109, Novem- ber, 1947, and in Bell System. Monograph, 1516. THE PRE-SET SUBSCRIBER'S TELEPHONE 111 order to eliminate direct control of any common equipment by the subscriber and thereby to reduce the holding time of the dialed in- formation receiving circuits and the associated subscriber-connecting circuits, the experimental pre-set dial telephone set shown in Fig. 5 was designed for this system by K. S. Dunlap, H. E. Hill and D. B. Parkin- son. Eight selector finger wheels are grouped on a common shaft with only their edges visible across the front of the telephone housing. Each finger wheel is provided with ten indentations along its exposed peri- phery. Each indentation is designated by an engraved number or group of letters conforming to the telephone directory numbering system and each indentation is of suitable configuration to permit a subscriber's 448 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 finger to engage and move the wheel in either direction to one of ten detented positions. All of the wheels may be returned to their normal "zero" position simultaneously by depressing the release button on the front right corner of the housing. To place a call the subscriber positions each of the wheels so that the desired number may be read across the wheels on the line of indentations immediately above the lower edge of the enclosing frame. The first three wheels are set to the code of the called office and the next five to the called line directory nimiber with the last of these being used for the party letter, if required. A number is preset in this manner before the handset is removed from its cradle across the back of the housing. With this method of operation the num- ber may be rapidly and completely transmitted to the central office when its receiving circuit has been connected to the line. CONTACT leads: FRONT -- BACK UPPER POLE PIECE -^^^ contacts: ARMATURE ' FRONT BACK ARMATURE- LOWER POLE PIECE COIL- TUBULAR STEM " WAX FILLING \_ FELT WASHER WELD-'- ■--MEDIUM OCTAL BASE Fig. 4 — Mercury contact relay. AUTOMATIC SWITCHING SYSTEM 440 Fig. 5 — Pre-set pulse-position-dialing telephone set. As shown in Fig. 6, which is a schematic of the mechanism and circuit of this telephone set, the handset when resting in its supporting cradle depresses the switchhook pins and causes two bell cranks to operate two sets of switchhook contact assemblies. One of these contact assemblies is controlled solely by the position of the handset while the other con- tacts are controlled jointly by the handset and by a magnetic locking device. This magnetic locking device consists of a permanent magnet yoke wliich holds the contacts in the position shown after the removal of the handset from its cradle until direct current of the correct polarity is allowed to flow in the windings of a latch magnet. These two sets of switchhook contacts jointly control the connection of any of three subdivisions of the apparatus in the telephone set to the line to the central office. If the handset is removed from its cradle to originate a call, the free set of switchhook contacts releases to complete a circuit through the latched set of contacts to the signaling equipment of the station. In this signaling condition the voice transmission ecjuip- ment remains disconnected from the circuit; thus, interference and transmission losses caused by voice transmission equipment are avoided during signaling. Upon completion of signaling direct current is pro- vided from the central office to trip the latched switchhook contacts. SIEPP|R_ SWITCH COMMON WINDING LATCH MAGNET AND SIMPLEX COILS SUBSCRIBER SELECTOR SWITCH (NUMBER PRESET= MAIN 2-5790W) Fig. 6 — Pulse-position-dialing subset schematic. 450 AUTOMATIC SWncilIN'G SYSTEM 451 With both sets of switchhook coiitticts now released the usual trans- mitter, receiver and induction coil ai-rangement I'or transmission of voice currents is connected to the tel(>phone line and all of the station signaling eciuipment, including the tripping windings of the latch mag- net, is disconnected from the circuit, fnterference and transmission losses caused b}^ signaling eiiuijjment are thus avoided dui'ing conversa- tion. When the handset is resting on its cradle between calls with both sets of switchhook contacts operated, the usual ringer and ringer con- denser are connected across the line for responding to incoming calls. I'pon removal of the handset in answer to such an incoming call, direct current is provided from the central office to trip the latched switchhook contacts and thereby the set is placed immediately in the talking con- dition. PULSE POSITION DIALING SIGNALS Before describing further the operation of this telephone set, it will be necessary to explain briefly the dialing signals generated by it and used in the system. From the subscriber's telephone set eight digits are transmitted for a complete local area directory number and the transmission is repeated as many times as necessary for the functioning of the central office equipment. In order to indicate the starting point of the transmission of a complete called number, a time interval of two digits duration during which no signals are transmitted is provided at the beginning of each transmission. Each digit interval is 0.01 seconds; therefore, a time interval of 0.1 second is reriuired for the no-signal or blank period and the eight digit number. These signals, as shown in the wave form-time diagrams of Fig. 7, consist of two pulses per digit: a start pulse of 1 millisecond duration and a stop pulse of 1 millisecond duration, each pulse approximately a single cycle of a 1,000-cycle per second sine Avave. The time interval between a start pulse and its following stop pulse is the measure of the associated digit value. The start pulses are generated at intervals of 0.01 seconds, or 10 milliseconds, and one stop pulse is generated some time during the 3.2 to 6.8 millisecond interval after each start pulse. In order to provide sufficient margins to permit reliable signaling over a wide variety of transmission facilities 3.2 milliseconds are allowed for the decay of each pulse and the pulses themselves occupy a section of the voice-frequency spectrum transmitted by practically all communi- cation facilities. The possible starting times of stop pulses representing 452 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 l1 ■* 00 ^ o a. (0 tu -I -1 ""■""■• ^^^ (0 . '~ Q. l-iti ^ D 0_| 5§ \ '~* -_i_jt AUTOMATIC SWITCHING SYSTEM 453 digits of successive magnitudes differ by 0.4 milliseconds. Thus, digit 1 is represented by a start pulse followed l\v a stop pul.se 3.2 milliseconds later; digit 2 is represented by a start pulse followed by a stop pulse 3.6 niilli.seconds later; and so on. It will l)e observed that the stop pulse for the digit 0 is 6.8 milhseconds after its start pulse and 3.2 milliseconds before the next succeeding start pulse. Thus, there is provided an in- crement of time of 3.2 milliseconds for the decay of the start pulse, increments of 0.4 milliseconds each for the generation of a pulse at any one of the ten times necessary to represent the various digits, and a last increment of 3.2 milliseconds to permit a stop pulse to decay should it occur at the end of the ninth increment of time. Referring again to Figure 6, the signaling pulses are generated by the eleven pulse transformers shown. These saturation-type transformers are assigned, one for each of the numerals 0 to 9 and one for the start pulse. The excitation for the signaling apparatus is a constant amplitude 50-cycle current of sinusoidal wave form transmitted from the central office on a simplex circuit consisting of the two fine wires to the set with ground return.* The currents from the line wires pass into the signaling apparatus through the windings of the latch magnet. These latch mag- net windings thus serve also as a simplexing coil and since the excitation magneto-motive-forces in the two windings are mutually opposing there is no reaction on the latch itself. From the simplex coil the excitation current flows through a stepper switch and its shunting phase shifter to a phase splitting network in which the current is converted to a two phase source with its two cur- rents 90 degrees out of phase. Each of the pulse generating transformers has a single secondary and two primary windings. The primary windings of the transformers are serially interconnected and connected with the two phases of the excitation current so that one phase is applied to one primarj^ winding of each transformer and so that the other phase is applied to the other primary winding of each transformer. The secondary windings are connected across the line through the pre-set selector, con- tacts of the stepping switch and a series capacitor. The secondary wind- ing of the pulse transformer for the start pulse is in a lead common to all the stop pulse secondaries. The magnetic core of each pulse transformer is designed to be satu- rated except for very small values of ampere-turns, and a voltage pulse * The time interval spacings of signal pulses given in this section and in the following section on the signal receiver are based on a SO-cj^cle control current. The system operated satisfactorily on 50 cycles. However, in most of the labora- tory tests a control current of 45 cycles per second was used since a stable source of this frequency is readily derived from commercial 60-cycle power sources. 454 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 is generated in the secondary winding of each transformer when the flux is changed from saturation at one polarity to saturation at the other polarity. The flux genei'uted in the core of each transformer depends upon the number of turns in the two primary windings and upon the current flowing in each winding. In order to assure that all pulses be substantially alike as to wave form and amplitude it is necessary that the total miximum ampere-turns on each core be ec^ual. In order to cause each transformer to generate a pulse at a suitable time during each half-cycle of the excitation current the total ampere-turns driving flux through the transformer cores must be controlled so that the flux in each transformer is zero at the time assigned to the pulse which that transformer serves to generate. These conditions determine the number of turns and the polarity of each winding when the angular position of the desired pulse is fixed in relation to each half-cycle of the basic ex- citation current. Since the magnetic flux in each transformer is reduced to zero two times during each cycle of excitation current, it follows that a combina- tion of two pulses representing a digit must occur during each half-cycle of the excitation current and that each combination of two pulses rep- resenting a digit is of opposite polarity to the preceeding two pulses. The capacitor through which the pulse generating transformer secon- dary windings are connected to the line is so proportioned to the im- pedances of these windings and to the impedance of the line that each half-cycle pulse as generated by a transformer is applied to the line as a single complete cycle of alternating current of about 1 millisecond duration. A selector switch, which is the internal mechanism connected with the finger wheels pre-set by the subscriber, serves to interconnect the transformer pulse windings with the line through the stepper switch. Thus, pulses representing any of the digits 0 to 9 may be impressed across the telephone line as any desired part of a complete telephone number in accordance with the setting of the selector switch. The stepper switch employs ten relays of the glass-sealed dry-reed type and each of the relays has an individual coil surrounding two nor- mally open reed contacts. The reeds are polarized by a permanent mag- net of sufficient strength to hold the reed contacts closed but not strong enough to close them until assisted by current of the correct polarity through the winding. A reverse current through the winding is required to release the contacts. In addition a common winding is provided which surrounds all of the reeds in such a manner that when a current of suffi- cient magnitude is passed through the winding the reeds of a predeter- AUTOMATIC SWITCHING SYSTEM 455 mined delay will be closed and the reeds of uU the other relays will be opened. This action is produced by reversing the individual winding and bias magnet of the single relay which is to be operated by the cur- rent through the common stepper winding. The preliminary setting of the stepper to insure correct operation is provided on each origination of a call by the discharge current from the ringer capacitor through the common winding of the stepper. The ringer capacitor is charged from the central office between calls. One reed in each of the relays is employed to coiniect successive brushes of the digit selector switch with the line while the other reed in each relay in conjunction with two diode rectifiers per relay winding is employed to control the operation of the stepper. The stepping opera- tion may be explained by reference to Fig. 6 as follows: The stepper is shown with the reeds for the sixth step closed. When the 50-cycle excita- tion current makes the terminal common to the individual stepper coils positive with respect to the terminal common to the stepphig control contacts, current flows through the upper reed contact of the sixth step, a diode rectifier and the winding of the seventh step relay causing its reeds to close. With the seventh set of reeds closed current flows through a diode rectifier and the winding of the sixth step relay causing its reeds to open. The stepper will remain in this position until the re- versal of excitation current a half-cycle later at which time a circuit through an oppositely poled diode rectifier will cause the operation of the relay for the eighth step followed by the release of the relay for the seventh step. The phase of excitation current through the stepper is so adjusted by the shunt phase shifting network that the stepper relays operate and release during the 3.2-millisecond guard interval preceding a start pulse. This prevents mutilation of the signal pulses. The stepping circuit is made reentrant so that the pre-set number will be transmitted repeatedly so long as excitation current is provided. With the chosen 50-cycle excitation the complete transmission of eight digits and a two-digit silent interval takes only 0.1 second. This results in a short holding time for the central office receiving circuit and the repetitive signaling feature permits repeated trials in case of signal mutilation as well as direct dialing from the subscriber's tele- phone set to distant offices rather than some form of relayed signaling from registers in the subscriber's own office. SIGNAL RECEIVER A simplified block diagram of an experimental receiver for the pulse- position signals used in this system is shown in Fig. 8. The receivers 456 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 were designed by N. D. Newby and the authors of this paper. The sig- nals after passing through a bandpass filter are amplified to a standard level by a circuit incoporating backward acting automatic volume con- trol. The arrival of each signal pulse is detected by a threshold device. Since the minimum time interval between the generation of a pulse and the next succeeding pulse is 3.2 milliseconds, the threshold device is arranged to disable itself upon the detection of a pulse for about 3 milliseconds. This prevents false operations of the detector either by tail transients resulting from distortion of a pulse in the transmission medium or by noise occuring in this interval. When the silent or blank interval which exists between the complete transmission of a number and its next repetition is recognized by the ^TPPDIMC AND DIGIT ABSORPTION CONTROL r SIGNAL 1 INPUT BAND- PASS FILTER AUTOMATIC VOLUME- CONTROL AMPLIFIER ^ SIGNAL DETECTOR ■■ > START CIRCUIT , ' DIGIT- REGISTERS AND CHECKING CIRCUIT ' V. TIME- DECODER '■ » 1 RESET Fig. 8 — Pulse-position-dialing receiver. start circuit attached to the detector, the time-decoder circuit is en- abled as well as the steering and digit absorbing circuit. The time de- coder subsequently measures the length of time between each detected start pulse and the following detected stop pulse, and energizes the corresponding digit value leads into the registers. The steering circuit enables a separate set of register elements for the storage of each decoded digit which is to be used by its associated circuits and withholds such enablement through its digit absorbing features for digits wliich are not of immediate intei-est. The steering circuit also enables a check circuit associated with the registers. Several features of the signaling code permit a check to be made that the received signals are in accordance mth the code. The number transmission cycle has been already described but a brief restatement is made here to emphasize the checkable features : The first pulse following AUTOMATIC SWITCHING SYSTEM 457 the blank interval is a start pulse ami eight start i)ulses at luiitorin 0.01- second time-interval spacing occiu- between l)laiik intervals. One and only one stop pulse occurs between start pulses. The total number of signal pulses between blank intei'vals is sixteen. The cluM-k (;ircuit uti- lizes one or more of these properties to insure Ihat no signal pulses have been lost during transmission and that no extraneous pulses have been detected. If the actions of the check circuit indicate that an error in transmission has occurred, the receiver circuits are completely reset for another trial. THE SWITCHING NETWORK To meet the objective of a single common control circuit for the opera- tion of the switching network, wliich provides the selectable paths between any subscriber and any trunk, it was necessary to have the switches in the network considerably faster than any of present com- mercial design. The laboratory model of the switching network and its associated path selecting equipment employing cold cathode gas tubes and dry-reed relays was developed by E. Bruce and S. T. Brewer. In addition to liigh operating speed this switching arrangement has certain other desirable properties: The idle path testing and selection functions are incorporated in the internal controls of the network. Busy sections of the network are automatically isolated from the sections tested for subsequent calls. Selection of a trunk within a trunk group, as well as path selection through the network, may be accomplished by the internal controls of the network if the trunks of a group are assigned one trunk per frame. Selection of an idle trunk and an idle switch path in com- bination reduces blocking. These internal selection controls eliminate many of the connector contacts that would otherwise be required be- tween the switches and external common control circuits. The switching network consists of line frames and trunk frames with each frame divided into primary and secondary switches. Each primary line and trunk switch has a number of vertical input columns across the switch to which are connected line or trunk circuits respectively and a number of horizontal output rows across the switch. At the intersection of each row and column of a switch is a relay consisting of an operating coil and three dry-reed make contacts. By analogy to the crossbar sys- tem which employs a somewhat similar rectangular array of rows and columns per switch and a similar primary-secondary path distribution, a switch intersection is called a crosspoint and a switch relay is called a crosspoint relay. In the crosspoint relay two of the contacts are used 458 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 to connect the talking conductors associated with the particular column to the talking conductors associated with the particular row. A cold- cathode gas diode is also associated with each crosspoint relay, and this diode in series with the winding of the relay is connected between the control lead of the particular column and the control lead of the particu- lar row. The third contact of the crosspoint relay is used to short-circuit the associated gas diode. A typical crosspoint is shown schematically in Fig. 9. The use of these crosspoint gas diodes in the control leads facili- tates the identification and selection of idle paths through the switching network and the short-circuiting of the diodes at operated crosspoints facilitates the holding of an established connection through the net- work at a lower power level than required for initial operation and the maintenance of a busy indication along an established connection during the path selection processes of subsequent calls. Dry-reed contact relays, rather than a more conventional type, are used in the crosspoints to provide the operating speed required for single control circuit operation. Each secondary switch is a similar rectangular array except that the horizontal rows are used as input terminals and the vertical columns as switch outlets. Within a frame the horizontal outputs of the primary switches are interconnected with the horizontal inputs of the secondary VERTICALS (control) -- >o (talking) Fig. 9 — Reed-diode switch — ^crosspoint connection. AUTOMATIC SWITCHING SYSTEM 459 switches so as to pro\'ido one path from each primary switch to each secondary switch. Connections are made between the secondary hne frame switches and the secondary trunk frame switches to provide talking paths between each Hne frame and each trunk frame. A direct metalUc connection is made for the two talking conductors of each path but the control lead from each secondary line switch outlet is connected to an individual control circuit, called a junctor, and the control lead from a secondary trunk switch outlet associated with the same talking path is connected to the same control circuit or junctor. The size of the switches on each type of frame and the number of frames in each particular office will be determined by the number of subscribers and other offices connecting to this office and the calling habits of the subscribers served. The operation of the switching network may be explained by reference to Fig. 10 which shows the control lead diagram of a skeletonized switch- ing network of a large size office. This figure shows two line frames, each of which has two primary switches and two secondary switches. Three vertical inlets are provided on each primary switch and two ver- tical outlets on each secondary switch. The figure also shows two trunk frames, each of which has two primary switches and two secondary switches. The trunk switches provide two vertical trunk inlets on the primary switches and two vertical outlets on the secondary switches. Eight junctors are required as indicated. This switching network' then serves to interconnect twelve subscribers with eight trunk appearances. This is the actual size built in the experimental model. As shown in Fig. 10 each control lead path between a primary and a secondary switch on both the line and trunk frames is connected through a high value of resistance to a — 45-volt power supply. In addition each control lead path from a secondary switch terminates in a similar re- sistor connected to a — 105-volt power supply. In a junctor involved in an established connection, such as junctor 5 of Fig. 10, the control leads connect to a —24- volt source through low resistance relay wind- ings. A talking path is shown as fully established between line C on line frame 2 and trunk D on trunk frame 2. This connection is held by the current flowing from the — 24-volt source in junctor 5 through the opera- ted reed crosspoints in the line frame to a ground in the line circuit and in the same manner through the operated reed crosspoints in the trunk frame to a ground in the trunk circuit. The —24-volt potential^on the junctor leads and the resulting — 12-volt potential on the primary-to- secondary switch link leads are effective path busy indications for sub- sequent path selection operations in the network. rvj 10 h- I o > < z 00 .-^ + ' 460 AUTOMATIC SWITCHING SYSTEM 4GI If a talking path is now desired between line A on line frame 1 in Fig. 10 and trunk B on trunk frame 2, a +80-volt power source is connected to the control leads at these points. These applied voltages are called "marks" and originate in a number group circuit. The +80-volt mark at line A in conjunction with the — 45-volts supplied to the primary- secondary switch links causes the cold cathode gas diodes of the line A vertical to fire and conduct at low current. The substantially constant voltage-drop characteristic of gas diodes causes the voltage on the two horizontal outlets of this primary switch to shift to +20 volts thereby "marking" one input lead on each secondary switch of this line frame. These -f20-volt marks in conjunction with the —105 volts supplied from the junctors causes the gas diodes between the marked secondary switch inlets and the junctor outlets to fire, to conduct at low current and thereby to mark the associated junctors with —40 volts again by virtue of the gas diode characteristic. As indicated by the shaded diodes in Fig. 10 a mark on line A results in marks on junctors 1, 2, 3 & 4 and thus reveals all the idle paths from hne A through the line frame. In a similar manner the -f 80- volt mark applied to trunk B results in ihe firing of the diodes along the idle paths from this trunk to junctors 2, 4 and 7. The path to junctor 5, which is in use on the connection between line C and trunk D, is not marked in this case. The —24 volts presented by junctor 5 on its trunk control lead is not sufficient when combined ^vith the +20-volt mark on the trunk primary-secondary link which leads to this junctor to fire the associated crosspoint diode. For tliis desired connection there are two possible paths, either through junctor 2 or through junctor 4, as indicated by the —40- volt marks existing on both the line and trunk sides of these junctors. Selection between these paths is automatically accomplished by use of a lockout circuit which is common to all junctors serving the same line frame. It is kno^vn that if a conduction path through a negative resistance gas tube is provided with a load impedance of proper value which is common to a similar conduction path through one or more other similar gas tubes, only one tube will ionize and remain ionized even if firing potentials are applied to several tubes either simultaneously or in sequence. Such a circuit employing two or more gas tubes with a common load impedance fun(^tions as a lockout circuit. The phenomenon is due to the region of negative resistance in the characteristics of the gas tube through which the tube current passes in the range between the breakdown and sus- taining voltages. In this region as the current through a tube increases, the voltage across the tube decreases, tending to prevent other tubes with the common load from firing. To reduce the possibility that two 462 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 tubes fired simulatneously will then travel through this unstable region exactly together, an inductive element is used in the common load cir- cuit. This increases the time interval required to traverse the unstable region thereby permitting differences between tubes to result in lockout. In each junctor a five-element cold-cathode gas tube is used for path detection and selection. One control element of this tube is marked from the line side and other control element from the trunk side of the junctor if this junctor is usable in the call being set-up. The main anode is con- nected, together with those of the other junctors of the same line frame, in a lockout circuit so that only the gas tube in one junctor can conduct in its main gap. The junctor in which the gas tube does conduct in the main gap is the selected junctor and the switching network path associ- ated with it is the selected path. Assume that junctor 2 is so selected. It first shorts out the resistors in its — 105-volt supply leads. This permits a higher value of current to flow through the gas diodes along the selected path and causes the operation of the reed contacts associated with the crosspoint relay windings which are in series with the diodes. The control lead contact at each of these crosspoints, as shown along the selected path in Fig. 10, shorts out the gas diodes. With the diodes shorted out a further increase in the current operates relays in series with this con- trol lead path in the line and junctor circuits. These relays cause the — 105-volt supplies in the associated junctor, junctor 2 in this case, to be replaced by the —24- volt sources and the +80- volt marks on the line and trunk terminals to be replaced by ground. This shift of power sources permits the gas diodes along paths marked but not selected for this call to extinguish but holds at a low power level the crosspoint relays along the selected path. With all diodes extinguished the switching network is ready for the next path selection operation. Removal of the ground at the trunk end of an established connection, at the end of conversation, results in complete release of the associated operated crosspoints and junctor. With a central office traffic rate during busy hours of 50,000 calls per hour, 50 milliseconds is the maximum allowable holding time for a single common control ciruit at 70 per cent usage. A single control circuit, even during its busiest periods, should not be in use more than about 70 per cent of the time. If the usage is increased beyond this point the delays which other circuits encounter in attempting to use the common control circuit increase very rapidly. This produces the same effect as increased control circuit holding time. The holding time of the control circuit for the switching network determines the traffic capacity of the switching arrangement if only a AUTOMATIC SWITCHING SYSTEM 403 single control circuit is provided. The control circuit holding time, in turn, consists of three parts: operate and release times of connector relays, line testing and "marking" times, and the operate time of the switches and junctors. The average liolding time for the control circuit of the switching network for the system described was about 40 milli- seconds. This is considerably shorter than the maximum 50 milliseconds permissible under the heavy traffic conditions of the preceding para- graph. SYSTEM OPERATION An experimental skeletonized ECASS constructed for laboratory tests is shown in Fig. 11. The equipment is located on these frames from left to right as follows: Frame No. 1, line and originating actuator circuits, switching network and controls; Frame No. 2, trunk, outward actuator and number group circuits; Frame No. 3, originating receiver circuits; Frame No. 4, powder supplies; and Frame No. 5, terminating receiver circuits. Without further detailed description of the various component cir- cuits the successful placing of a call through the system may now be traced by reference to the block diagram of Fig. 12. 1 r©'''^~^ ^^ l>.^ "^ N. "^ , L^ ^' x \ ^ \, \ %. \ \ \ 1 k k \ \ ^ \ > \ c^ k \ \o V >5 ^. N \ \c N V \ \ \ \ 1 V \, \ \ \ \ 1 \ 1 \ \ \ 1 1 \ the effect of additives. The data of Fig. 2 show^ the effect of firing tem- perature on the initial open circuit voltage for sample disks 0.775 cm in diameter and approximately 0.15 cm thick. The variation of voltage with thickness and area was taken account of by multiplying the meas- ured voltage by the area and di\dding by the thickness. The open circuit voltage was measured by using the circuit of Fig. 3. A barium titanate cylinder and metal horn described in a previous paper," vibrating at 18,000 cycles, strikes the sample a blow at its central position. The voltage generated is applied to the input of a high resistance tube similar to the one shown by Fig. 1, and then actuates a cathode ray tube. The voltage corresponding to the height of the peak is cali- brated by putting a known voltage in series with the ceramic across a small resistance R and hence the magnitude of the open circuit voltage can be quantitatively determined. The value of the mechanical blow applied to the polarized ceramic can be adjusted by controlling the drive on the ceramic cylinder. With the feed back circuit described in the MEASURING FORCES AND WEAR IX SAVIT(^HIXG APPARATITS 475 previoius paper," this value can be held veiy eoiistant and can be con- trolled by controlling the bias on the limiting device. The magnitude of tlie force can be determined by comparing the voltage with tiuit ob- tained l)y suddenly lifting a weight off liie ceramic and has been adjusted to e([ual 500 grams. The voltages shown then correspond to the open circuit voltages generatcMl b>- api)Iying 500 grams to a point at the center of the ceramic. As shown in the appendix, tiie effect of applying a force at a point in a ceramic is not the same as that caused by distril)uting the force uni- formly over the surface due to the fact that radial strains are generated and the.se act through the radial piezoelectric constant to reduce the \'alue generated })y the thickness piezoelectric constant. It is shown that I he point application of stress generates only 40 per cent as much as would BARIUM TITANATE (CLAMPED) CATHODE RAY OSCILLATOR 4//F Fig. 3 — Circuit used to measure open circuit voltages for barium titanate samples. 476 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 be generated in a disk with the stress appHed uniformly. With this factor the open circuit voltage per unit of force — which determines the effective gus piezoelectric constant of the ceramic — agrees well with that obtained by other methods of measurement. For the most desirable ceramic obtained for the 1325°C l)ai:ing temperature the value of ^33 equals 2 ^33 = 3.25 X 10"' ---^ r in c.g.s. units = 0.98 X 10"' stat-coulomb (1) meters . , ' ., — ni m.k.s. units JN ewton The dielectric constant £ of this material is about 1500 so that the d33 piezoelectric constant is d33 = 9J^ = 390 X 10-« «^^^-^»^l»"^bs ^ ^3Q ^ ^^-u coulombs ^^^ 47r dyne Newton Since for some applications in this paper, high voltage gradients of opposite sign to the poling voltage are applied to the ceramic, it is a matter of importance to find out whether the ceramic will become de- poled by the action of this voltage. To test out this feature the circuit of Fig. 3 is equipped with a high voltage generator, which is applied to the ceramic through 10-megohm resistors and the high voltage is kept out of the measuring circuit by 4-microfarad condensers. The procedure was to apply a negative voltage for 3 minutes, then to recalibrate the voltage due to impact. This was repeated with a higher voltage each time until the range was covered. The curves of Fig. 2 show that there is an optimum baking tempera- ture for a large coercive field. Above this temperature larger sized crys- tals grow in the ceramic and the coercive field decreases markedly. It is thought that the smaller crystal size corresponds to a more strained condition in the individual crystallites and it requires a higher field to overcome the mechanical bias and change the direction of the ferroelec- tric axis. A similar condition has been found by x-ray techniques for single crystals where it has been found impossible to make a single domain out of a multidomain crystal by the application of a field, if the crystal is too highly strained. The effects of additives are also very marked on the properties of the polarized ceramics. It has previously been reported' that the addition of 4 per cent of lead titanate to the commercial barium titanate increases the coercive field. This is confirmed by the curves of Fig. 4 which show MEAST'HIN'd FORCKS AND WllAK IN S\\ ir< 1 1 1 .\( 1 Al'l-AUATrS h/ the open circuit voltage for a 4 per cent lead litanate l)ariinn litanate ceramic for various baking temperatures and negali\-e biasing xoltages. The optimum tempera! ui'e for a small grain size structure is lowered about 50°C by the addition of the lead titanate. As can be seen the co- ercive field is considerably increased and it apjx'ars safe to use a negative field of (1000 volis per centimeter without any depolarization. In Section 1\' a system is described for which an ac voltage of this magnitude was successfully used for many days with no change in sensitivity of the (•(M'amic. The open circuit i)iezoelectric constant for the optimum ce- ramic of Fig. 4 is Qco ^ in-s stat-coulombs -2 meters ,„v g33 = 3.82 X 10 T = l.lo X 10 — — -— (3) dyne JNemon Since the dielectric constant is about 1000, the effective ^33 piezoelectric constant is about ./33 = 310 X 10- ^tat coulombs ^ ^^^ ^ ^^-^ coulombs ^^^ dyne JNewton Another property of interest is the stability of the piezoelectric proper- ties of the ceramic o\'er a long period of time. While no very good com- parisons have been made between the various baking conditions and between barium titanate with and without additions, some long time measurements have been made on four samples of the optimum 4 per cent lead titanate used in the transducer of Fig. 3. Over a period of two years during which they have been continuously used in a calibrated oscillator, the calibration has not changed noticeably, i.e. less than 5 per cent. On accoimt of the superior voltage and time stability of the lead titanate, barium titanate mixture, all of the elements used have had this composition. Two types of units have been used for force measurements, one type that responds to normal forces and tlie other to tangential forces. The type responding to normal forces as shown by Fig. 5 is poled in the thick- ness direction which is also the direction in which the force is applied. The sensitivities for forces applied at points are given by the values of Fig. 4. For example for typical units having the dimensions 0.1 cm by 0.1 cm in cross section and 0.05-cm thick will produce an open circuit \'oltage of 2.7 volts for 100 grams applied to the ceramic. Such ceramics have been used in measuring the dynamic forces when various parts of the relay close or open. Fig. 6(a)'° shows the voltage generated when the two relay contacts come together. The dynamic stress is somewhat higher than the static stress and varies with time due to mechanical 478 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0 I 2 3 4 5 6 7 8 9 10 11 12 13 DEPOLING FIELD IN KILOVOLTS PER CENTIMETER Fig. 4 — Effect of 4 per cent lead titanate on the open circuit voltages generated for 500 grams force, and for depoling voltages. vibrations of the relay structure. Fig. 6(b) shows the forces produced by opening the contacts. The large spikes are due to wire vibrations. By using such ceramics in various parts of the relay the points of high stress can be located. The second type of structure which responds to tangential forces is poled as shown by Fig. 5 so that the poling direction lies along the direc- tion for which the tangential force is applied and perpendicular to the DIRECTION OF POLING AND FORCE ~-^^ ^ ' —s::i :!:n ^ ^ ^ ^x ^^ T -o— •X [ p — j C"^ N >^ ^. ^ '« \ \ N^ "^ \ \i \ k^ \ > \ ^ P > \ 4 \ i N 5. \ \ \ \ \ i L > %. ^\ NORMAL TANGENTIAL Fig. 5 — ^Methods for polarizing barium titanate to respond to normal and tan- gential forces. MEASURING FORCES AND WEAK IN S\\ ITCIIIXG Al'l'AKATLS 47*.) CONTACT MAKE Z 30 - /^i^yi^^Kw CONTACT BREAK 2 3 4 5 TIME IN MILLISECONDS Fig. 6 — Oscillograph tracings of forces generated in make and break opera- tions. direction of the electrodes. In this process, the crystal is first poled, after which the poling electrodes are ground or etched off and electrodes per- pendicular to the poling direction are put on by using a polimerizing cement in which silver dust is mixed. The cement serves not only as an electrode but also holds the ceramic in the desired place. Fig. 7 shows an arrangement used for studying frictional forces. A small ceramic 0.1 l)y 0.1 cm in cross-sectional area is glued to a metal base while a thin specimen of the material whose frictional forces are to be studied is glued to the top surface. The forces cavised by a wire drawn over the surface are transmitted to the crystal and generate a voltage which ap- pears on the oscillograph. Pictures of such force generated voltages are TANGENTIAL--' BARIUM TITANATE CERAMIC -SAMPLE Fig. 7 — Experimental arrangement for studj'ing frictional forces. 480 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 shown by Fig. 9 of the next section and are discussed there. The sensitiv- ity of this type of unit is higher than that for the normal force measuring unit. As shown in the appendix, the voltage generated is independent of the area of application and is about 9.7 volts for a unit the same size as discussed above which gave 2.7 volts for 100 grams applied at a point. By placing weights on the upper surfaces both types of units can be used as accelerameters. They are cemented to the surface whose ac- celeration is to be measured and the force applied is equal to half the mass of the ceramic plus the added mass times the acceleration. By put- ting weights on the shear pickup ceramic types, tangential accelerations can be measured in the direction of the poling. By using three such accelerameters, the normal and two tangential components of accelera- tion of any surface can be measured. III. METHODS FOR INVESTIGATING CAUSES OF WEAR Wear in various parts of a relay is the limiting factor when a very large number of relay operations are desired. This wear opens up the spacing between contacts and causes the relay to lose its adjustment over a course of time. A. Force Measurements and Wear Caused by Normal Forces Since the forces operating on a material can be divided into normal and tangential forces, it appears desirable to separately determine the effects of each. Normal forces were produced by using the barium ti- tanate, metal horn detail of Fig. 3. With a steel ball on the end of the metal horn, and a barium titanate specimen glued to the pivoted arm, the peak forces in grams are plotted against the volts used to drive the titanate unit for various static forces in Fig. 8. The pattern of voltage is approximately a rectified sine wave, since the ball is out of contact with the measuring titanate a part of each cycle. To observe the wear caused by normal forces a piece of material to be studied was glued to the pivoted arm on top of the barium titanate and the force was adjusted to the required value. For forces in the order of those measured in relays no wear at all was observed over a period of 18 hours which corresponds to a billion impacts, since the number per second is 18,000. For larger impulsive forces, it was found that the result of 60-million impacts against an insulator such as a phenolic was to produce a pit only a few tenths of a mil inch deep by a plastic flow. Since no wear of the type involved in relays was observed it was concluded that practically all of the wear was produced by tangential forces. MEASrUlNd FORCES AXU WEAR IN' SWITCHIXG APPARATUS 481 DISPLACEMENT IN MIL-INCHES 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 1200 1)00 1000 900 5 800 < cr. "^ 700 z Lil 600 O i \ 1 1 1 1 X ^ STATIC FORCE IN GRAMS = 90^ L^ / / / / / 30_ — m o u- 500 < liJ 400 c 300 200 100 0 / ^^ r /^ ^ / / 10 _ , — - / ______^ ^^ — ' ' /- 15 20 25 10 35 40 45 50 55 60 VOLTS ON DRIVE Fig. 8 — Variation of normal impulse forces with drive voltages for thrcse values of static force. B. TangcitU'al Force and Wear Measurements To study the effect of tangential forces in producing wear, the trans- ducer was mounted horizontally and the steel ball was replaced by a wire such as are used in some relays. The length of the wire was made short enough so that no lateral vibrations were generated and the motion was strictly tangential. Samples to be studied as shown by Fig. 7 were mounted on top of shear type ceramics which were glued to the pivoted arm in such a way that they responded to tangential forces applied perpendicular to the arm. When a piece of A phenolic (which is a paper filled phenolic) was placed on top of the ceramic a series of oscillograph pictures were taken when the total displacement of the wire varied from 0.05 mil inch to 2.0 mil inches and the steady weight on the wire was 40 grams (0.0885 pound). These pictures are shown in Fig. 9. For amplitudes under 0.075 mil inch, the force is a good sine wave which increases with amplitude luitil the maximum force equals the product of normal force times the co- efficient of friction. The force in this region is essentially elastic as is shown bv the fact that the maximum force occurs at the time when 482 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 the maximum displacement of the wire takes place. Above this ampli- tude the wire begins to slip on the plastic and for a travel of 0.3 mil inch there are indications that the point of contact between the wire and the plastic has changed from one position to another. This agrees with the idea that friction is due to a definite bonding between points of contact of the two materials which is broken by their relative motion. New points of contacts are then made and a stick-slip process occurs. At 0.5-mil-inch motion a number of small contacts occur during the LENGTH OF TRAVEL IN MIL INCHES 0.05 0.15 '^rtM 0.75 1.0 Fig. 9 — Tangential forces measured for an 18,000 cycle oscillatory motion whose total displacements in mil inches are shown by the values given. MEASURIN'O FOIU'KS AXD \\I:a1{ I\ SWri'CHIXCi API" AHATTS 483 __< VOLUME OF WEAR -^ ^ ^^^ - — -•"' A c- DEPTH OF WEAR Y / 1 5000 0) 2000° 1000 2 0.4 as 0.6 BILLIONS OF CYCLES Y'\^. K) — Typical wear curve for A phenol fibre plotted as a function of the nunil)er of cycles. traA'el. Since the picture is a trace of ati oscillograph pattern which is being repeated 18,000 times a second and since a t\vo second exposure is reciuired to produce the picture it is obvious that the wire goes back and forth over the same points for a large number of times. Most of the energy is lost in producing elastic vibrations in the points of contact. These oscillations are produced by the bending of the areas of contact by the bonding force between them and by the motion. When the bond is broken the plastic forming the point is free to vibrate and the elastic energy goes into mechanical vibrations and eventually into heat. Since a pattern such as that for the 0.5-mil inch or the 0.75-mil inch displace- ment lasts inichanged for a number of minutes, it is obvious that very little of the energy goes into breaking the plastic points of contact and producing wear. This is confirmed by a rough calculation given later which shows that only about 1 part in lO'"* of the energy goes into pro- ducing wear. For displacements above a mil-inch motion it appears that groups of point contacts are broken at one time, and the pattern changes rather rapidly indicating that there is more wear at these amplitudes. Over a two-second interval the pattern is changing fast enough so that sharp pictures are not obtained. Quantitative values of wear for \'arious materials were obtained by nnining the barium titanate unit for various periods of time, different lengths of strokes and different normal forces. Fig. 10 shows a typical 484 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 QJ 5000 '-' 3000 5 2000 ? y STATIC FORCE y^ IN GRAMS = 30/"^ ^ y X ^ 0 ^ y J2— - J \^ ^ o 0.6 0.8 1.0 1.2 1.4 AMPLITUDE OF MOTION IN MIL-INCHES Fig. 11 — Total wear for one billion cycles plotted against the length of stroke for two normal loads. wear curve obtained for A phenolic (a paper filled phenolic) plotted as a function of the number of cycles. This wear was obtained by drawing a 0.025-inch nickel silver wire for a distance of 2.0 mil inches over the surface of the bar. The bar was \ inch wide. The normal force used was 30 grams (0.0665 pound). The wear was measured from the depth cut in the material and from this since the wire was round, the total volume of wear in cubic mil inches could be calculated. The rate of wear was faster at the start but approached a limiting rate mth a large number of cycles. A number of different lengths of stroke were employed and for the A phenolic the total wear for a billion operations is shown plotted by Fig. 11. The wear is approximately proportional to the slide but ex- trapolating down to small motions it appears that there is a threshold of motion below which the wear is very small. The values indicated are close to the no gross slide regions found from the force curves of Fig. 9 for both forces shown in Fig. 11. To check that the wear was definitely less in the no gross slide region an amplitude of motion of 0.075 mil inch for a normal force of 50 grams (0.11 pound) was run for a billion operations. The wear observed was so small that it could not be measured quantitatively, confirming the lower rate of wear in the elastic region. MKASrHlXti FOKCKS AND WI'AK I\ SWlTClll XC AlM'AKATrS 485 AiiolluM- type of wcivr inoa.surenunit has also been employed. As shown l)y Fiji. 12 Ihe motor is a modification of the Western Electric W re- corder, which was oi'i^inally desi,<;ned for cuttiujj; "hill and dale" phono- j>;raph records/ The moviiiii; syst(>m of this recorder consists of two coils (a drive coil and feedback coil) and a stylus, all rij>;idly coupled and coaxial. The dri\(> coil is s(>cui-ed b) Ihe base of a cone shaped vibrating Fig. 12 — Low frequency wear measuring device. 486 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 element which is carried at its base by a diaphragm and at its apex by cantilever springs. These furnish the restoring force and restrict the mo- tion of the moving system to a single degree of freedom, motion parallel to the cone and coil axis. The second or feedback coil is secured to the cone near its apex, at which point the stylus or drive pin is attached. The coils move axially in annular air gaps polarized by a single magnet. In the space between the two coils, copper ring shielding (a shorted turn) is provided to minimize inductive coupling between them. The output of the driving amplifier is supplied to the drive coil, while the feedback coil is connected in proper (negative feedback) phase to the amplifier input. The voltage generated in the feedback coil is proportional to the instantaneous velocity of the moving system, and by virtue of the nega- tive feedback, the amplifier-recorder system becomes a high force, high mechanical impedance generator of mechanical motion, with the veloc- ity very nearly proportional to the input voltage over a large range of frequency and mechanical load. Measurements of the voltage generated in the feedback coil provides a means of monitoring the velocity. Enough power capacity is present in the amplifier so that large changes in the load will not cause changes in the motion. The samples of the materials to be tested for wear resistance are carried by a grooved aluminum beam, one end of which is hinged, the other being driven b}^ the record stylus. The rubbing member, in this case 25-mil nickel silver wires, are tensioned against the test samples as they might be in switching apparatus. The wires can be removed for ob- servation and measurements of the wear, and accurately replaced as the parts are do welled together. Fig. 13 shows a measurement of a number of materials for a normal force of 30 grams (0.0665 pounds) and a slide of 2 mil inches. The A phenolic, which is the same as that tested and recorded in Fig. 10 by the 18,000-cycle barium titanate transducer, produced essentially the same wear showing that the wear is approximately independent of the rapidity of motion for these materials. Nylon showed a rather erratic wear curve due to the fact that it has a low melting point and tends to ball up on the wires. This effect was considerably more pronounced at 18,000 cycles, where a very large indentation was found. Only three materials show low wear at reasonably uniform rates out to a large number of cycles. These are C phenolic, a fabric filled phenolic, the B phenolic, a wood flour filled molding phenolic and the D phenolic, a cotton flock phenolic with graphite added. At lower forces and shorter slides the wear at 10^ cycles is approximately proportional to the force iMEASrUING FOKCKS AM) WIOAK 1 .V S\\ IT( III N'(! ATPAHATUS -1S7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 BILLIONS OF CYCLES Fig. 13 — Typical wear curves for a luinihcr of materials. times the length of slide. Any of these three materials give sufficiently small wear to produce a long relay life, but the best performer under all conditions of force and slide appears to be the D cotton flock filled phenolic with graphite added. In order to determine the causes of wear over a greater range of parameters a number of other materials were run by means of the barium titanate transducer. The wear for 2 mils motion, 30 grams (0.0665 pounds) force, and 10 cycles are shown by Table II. C. Wearing Energy and Causes of Wear A rough estimate of the energy reciuired to break off pieces of the material shows that most of the energy goes into producing heat and \'ery little into wear, i.e., into breaking pieces from the material. To show this let us consider a small cube fixed at one end and with a tan- gential force at the other. The force will cause the top surface to move with respect to the bottom surface as shown by Fig. 14, and a shearing strain S is set up in the material whose value is equal to F = fxS dx dij (5) where dx and dy are the cross section dimensions and fi the shear stiff- ness. In this displacement work is done by the sidewise displacement II equal to W kuF (6) 488 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 But u the displacement is u = du dz dz — S dz (7) and hence the total work done is W = ^nS' dxdydz = Km'S") X volume of material (8) If the force is increased, the shearing strain S increases until it reaches the limiting strain that the material can stand. This limiting strain depends on the material and whether the strain is long repeated so that the material becomes fatigued. For most plastics this limiting strain is in the order of 1 per cent and for most metals the value is less than this. Fig. 14 — Representation of points of contact and their displacements for plastic and wire. Hence the energy to break up one cubic centimeter of material is W = ^fxSl (9) where S^f is the breaking strain. For a plastic having a shear stiffness of M = 2 X 10 dynes/cm and a breaking strain of 0.01, the energj'^ is 10 ergs per cubic centimeter. This rough calculation and the amount of wear observed for various length strokes and forces allow a determmation of the amount of energy going into wear production. The amount of work generated by a dis- placement of 0.002 inches or 0.005 cm with a normal force of 30 grams is W = 0.005 X 30 X 980 X / in ergs (10) where / is the coefficient of friction. Since this is about 0.25 the work per stroke is 37 ergs. Twice this amount results from a complete cycle and for 10 cycles the work done is W = 37 X 2 X 10' = 7.44 X 10'° ergs (11) The volume of wear observed for this condition is about 1 X 10 cubic cm for the A phenolic and hence we find that the part of the energy MKASURING FORChB AM) ^VKAH IN SWITC'IilNG APPARATUS 489 that goes into producing wear is 1 X 10"' X 10' 7.4 X 10' ^_"_ = 1.35 X 10" (12) or about 1 part in 10 . This suggests that the wire goes back and forth o\er the same high points many miUions of times until the material tinall}^ becomes fatigued and breaks off. This view is confirmed by the oscillograph pictures of Fig. 9 which are a stationary pattern for millions of oscillations. According to this picture, the material that will wear the best is the one with the highest limiting shearing strain. If we assume that the Umiting shearing strain is proportional to the limiting elongation strain under repeated vibrations — of which there are tables — the wear for various materials given in Table II agrees roughly with this concept. Table II shows the yield stresses, the Young's moduli, the per cent strains at the yield point and the relative wear at 10 cycles. It will be seen that the materials with the highest yield strain will in general wear longer than those with smaller yield strains. An exception to this rule was nylon which had a large wear even though it has a large ^neld strain. However, nylon has a relatively low softening temperature and a low heat conductivity. Observations showed that the nylon was melted off rather than abraided off. According to this rule gum rubber should wear much better than any other material since it has such a high limiting shearing strain. A run was made with a two mil inch motion on a gum rubber specimen and no observable wear was found. The fact that a rubber tire will outwear a metal tire is also con- firmation of this rule. All the tests showed that the wear on the stainless steel or nickel silver Table II Amount of Wear for Varioits Materials Caused by Sliding a 0.025 Inch Nickel Silver Wire for 2 Mil Inches, 30 Grams Normal Force and 10^ Cycles Wear, Cubic Cm, for 2- Material Yield Stress Dynes/Cm^ Youngs Modulus Dynes/Cm2 Per Cent Yield Strain Mil Motion for 109 Cy- cles and 30-Gram Force Lead Glass .... 2.4 to 2.7 X 109 6.5 X 1011 0.0037 to 0.0041 0.027 Brass 3.7 to 4.6 X 10^ 9 X 10" 0.0041 to 0.0051 0.0075 Stainless Steel. 1.1 to 1.4 X lO'o 2 X 1012 0.0055 to 0.007 0.00075 B Phenolic. . . . 7.2 X 108 6.9 X 1010 0.0105 0.000025 490 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 wire used to produce the wear on the plastic was always very much less than tliat of the plastic. The reason for this as seen from Fig. 14 is that since the displacement for a given force to break the bond between two high points is going to be inversely proportional to the shearing stiff- nesses of the two materials, the disphicement for stainless steel with a shear stiffness of 8 x 10^^ dynes/cm" will be ^ that of the plastic with a shear modulus of 2 x 10^^ dynes/cm^. Hence, the shearing strain for the stainless steel is much further below its limiting strain than is the shearing strain for the plastic. When the stainless steel wire was run against a bar of synthetic sapphire — which has a much higher shear con- stant— the stainless steel wire was soon worn through, while little wear occurred on the sapphire. IV. THEORETICAL AND EXPERIMENTAL INVESTIGAIION OF THE NO GROSS SLIDE REGION Since in the no gross slide region, the shearing strain is less than in the gross slide region, the rate of wear should be considerably less. This is confirmed by direct tests of the wear as shown by Fig. 11, and by supplementary tests. Hence a further experimental and theoretical in- vestigation has been made of this region which is defined by the condi- tion that the tangential force is less than the product of the normal force by the coefficient of friction. If sliding motions can be kept small enough to be in this region, very little wear should occur. Using a shear ceramic for measuring the tangential force, the static load was varied and the motion required to produce no gross slide was determined. Oscillograph figures of the type shown by Fig. 9 were used and when the figure was broadened out as shown by the third figure it was assumed that slide had occurred. Fig. 15, upper curve, shows the total motion in mil inches, plotted against the static force in grams, which will just cause gross slide. The bottom line shows the maximum shearing force in grams. This is slightly lower than the force determined by the coefficient of friction since the force becomes slightly larger as shown by the pictures of Fig. 9, when gross slide occurs. The total displacement for no slide increases as the two-thirds power of the static load. Since neither the wire nor the plastic material is smooth, contact be- tween the two is established at only a few points. To interpret the results obtained above, some calculations due to R. D. Mindlin are used. These deal with the tangential forces and displacements of two lialls pressed together, and are for conditions occurring before gross slide begins. MEASURING FORCES AXD WEAK IX SWITCIIIXG APPARATUS 491 .?, 0.06 ^ ,^ ^ DISPLACEMENT ^ y y / / / SHEAR FORCE __— / ^^^- ,— ' '"' /'■■'" '" - — -*"■ 70 2 < 5 20 3 X ,0< 40 50 60 70 80 STATIC FORCE IN GRAMS 90 100 110 120 Fig. 15 — Maximum total motion for no gross slide plotted against normal force. Low curve shows maximum tangential force. From the Hertz theory of contaets, " the radius of contact a between I\V() spheres is e(itial to y\^-^~-'-^) (13) where r is the radius of the spheres, iV the normal force, mi and cri the shear elastic constant and Poisson's ratio for one sphere and )U2 and a-i the same quantities for the second sphere. If now a tangential force T is applied to one of the spheres directed in the form of a couple, elastic theory shows that the tangential traction is everywhere parallel to the du-ection of the applied force and contours of constant tangential trac- tion are concentric circles. The magnitude of the traction as shown by Fig. 16 rises from one half the average at the center to infinity at the edge of the circle of contact. The displacement of the circle of contact of one sphere with respect to its center is 8/za (14) whei'e a is the radius of the contact area which is given in terms of the normal force by Equation (13). A feature of this solution that requires further study is the infinite traction at the edge of the circle of contact. Presumably the tangential component of traction cannot exceed the product of the coefficient of friction / and the normal component of traction p, which from tlic Hertz 492 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 V si \ \ \ \ 1 \ 1 \ 1 \ 1 \i Xy FOR T/f N = 0.3 / / \ y //■ ^ <^^ . — -- rrr: r:^ h<,^ ' __^ _ V— 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 RATIO OF p/a Fig. 16 — Traction plotted against radius for elastic displacement and modifi- cation introduced by the effect of slip. contact theory is V = 2ira- 1 - r- (15) Mindlin assumes that sKp takes place bet^veen the two surfaces until the tangential traction is equal to -^ - Hi/' for a' < p < a (16) and less than this for all interior points, \vhere in this equation p is the radius vector and a' the inner radius for which slip stops. This corre- sponds to the introduction of a new system of forces and Mindlin has shown that equilibrium is reestablished when the surface tractions are given by Equation (16) when a' < p ^ a and by X, = 27ra2 W' when p < a' (17) Fig. 16 shows this distribution for the case T/fN = 0.3. The inner radius a' is given such a value that the integrated traction over the surface MEASURING FORCES AND WEAR IN SWITCHING APPARATUS 493 equals T and its value is found to be fN (18) The added slip increases the displacement bx and it is shown tiiat the total displacement is equal to 5« = 3/A^ (2 - a) 16)ua 1 - 1 - }N (19) A plot of this curve is shown by tlu^ line OPQ of Fig. 17 and it is evident that the displacement before gross slip occurs is 1.5 times larger than the elastic displacement calculated on the assumption of no slip. These calculations have been extended in a recent paper"* to include the case of a cylically varying force T ^ jN and it is shown that the force displacement curve is a hysteresis type loop whose end points lie TOTAL DISPLACEMENT INCLUDING SLIP -0.8 -0.6 -0.4 -0.2 0 3(2-cr)fN PACTOR TIMES — 7^—^ 0.2 0.4 0.6 0.8 TO GIVE DISPLACEMENT Fig. 17 — Displacement versus force wiieu slip is introduced. Hysteresis curve PRU shows displacement for an oscillating force. 494 THE BELL SYSTEM TECHMCAL JOURNAL, MAY 1952 on the OPQ tnir\'e of Fig. 17 and whose theoretical area W is ^^, ^ 9(2 -a)fN' ^ lOMtt 1+1- ../■n. (20) where during the oscillation the tangential force T varies between the limits ±T*. Slip takes place as before between the radii a and a' given by a' = a A/ I - - or conversely — = 1 (21) Since the distribution of traction over the surface cannot be uniquely derived from elastic theory, the introduction of the slip function is an assumption that has to be justified by experiment. This assumption has been shown to correspond with experiment by employing the experi- mental arrangement show^n by the photograph of Fig. 18. A barium titanate driver shown in more detail in Fig. 19 drives the middle of three glass lenses that are pressed together by a static force applied to the lever system as shown by Fig. 19. The central glass lens has a radius of curvature of 4.85 inches on each side while the other two lenses have the same radius of curvature on the sides touching the middle lens, but are flat on the other two sides and are rigidly attached to the lower platform and upper hinged lever by cement. In order to get sufficiently large Fig. 18 — BiU'iuiu Uluuale driver, i»ick-ui> device and g)a»6 lenses. MEASURING FOKCKS AXD \VKAU IX SWITCHING APPARATUS 495 Fig. 19 — ^The entire experimental arrangement. circles of contact to be easilj^ observed, normal loads on the lens system were 10 and 15 pounds which resulted in contact circles of 0.030 and 0.034 inches diameter. With normal forces up to 15 pounds and two surfaces in contact, tangential forces up to 7.5 pounds are necessary in order to bring the central lens near the sliding point if the coefficient of friction is near one-quarter. This force was obtained by impressing voltages in the order of 3,000 rms volts on the barium titanate lead titanate hollow cylinder. This cylinder is 4f inches long, and has an outside diameter of 1 inch and an inside diameter of ^ inch. The ceramic was poled in a radial direc- tion and the constants of the material were such that a force of 167 pounds could be generated along the length for a clamped driver when a voltage of 3,000 \'olts (4,750 \'olts cm) was used. On the other hand if the driver works against no stress, the expansion in the plated length of 4 inches is 0.7 x 10~ inches. The actual force applied depends on how much the relative slip be- tween the glass lenses amounts to. To measure this force, a poled lead titanate barium titanate disk is placed between the driver and the metallic bracket which clamps the middle lens as shown by Fig. 18. All the force exerted on the lens has to be exerted through the disk and hence the \-oltage generated by the disk is a measure of the force exerted on the middle lens. This voltage is calibrated by attaching a spring load of known constants and measuring the displacement of the load by means of a microscope. Using a 60-cycle driving voltage, a number of sets of disks were run with varying tangential and normal loads and the wear patterns ob- served. Fig. 20 is a photograph (magnified 100 times) for a normal load 496 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Fig. 20 — Wear circles (magnified 100 times). of 10 pounds and a maximum tangential load of 2.04 pounds per lens run for about 3 hours at 60 vibrations per second. The outer area of contact is seen to be 0.03 inches in diameter. The inner area of wear is a circle displaced slightly from a concentric form and has a diameter of 0.0175 inch. If we plot 1 — {a' /of" against the ratio of tangential to normal force, where a' is the inner radius and a the outer radius, as shown by Fig. 21, a pomt at 0.204 and 0.8 is obtained. A number of sets of lenses were run and as shown by Fig. 21 the results can be plotted on a straight line corresponding to a coefficient of friction of 0.25. This value agrees well with other determinations^^ of the coefficient of friction of glass on glass. Hence the assumption of slip between spheres under tangential forces appears to be verified. This type of slip may be re- sponsible for some types of wear, such as in ball bearings, where no gross slide of one surface over another occurs. An attempt was also made to check the area of the loop as determined theoretically by Equation (20). The applied force is measured directly by the barium titanate pickup and the displacement was measured by MK.\8l'RIXG FORCKS .\N'D WK.M? IN' SWITCHING .Vl'l'.V K.VTUS 49 attaching a velocity inicroplionc jjickiip to the transducer. The force voltage was placed on one set of plates of an oscillograph while th(> integrated output from the velocity pickup was placed on the other set. A series of oscillographs were taken for various amplitudes of motion and the pictures are shown by Fig. 22. Since the force and displacement measurements were separately calibrated, the area of the curves in inch pounds could be evaluated and are shown by Fig. 23. For amplitudes of motion near the gross slip amplitude, the area agree well with that calculated from Equation (20) from which the dotted line is obtained. For lower amplitudes the measured area is larger than the calculated area. Possibly a stick-slip process is causing the displacement to lag be- hind the applied force. The measured areas are nearly propoi'tional to the square of the amplitude. The mechanical resistance associated with the stress-strain hysteresis curves of this sort is of the same type that 0.30 0.28 0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0 / O N =: 10.4 LBS A N = 14.1 LBS / / / / / / / / / c ¥ / / / / / / / V JT /. / / A / / / y // / // y /. r / 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 21— Plot of 1 — (o'/o)' against ratio of tangential and normal forces. 498 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Fig. 22 — Force displacement loops. occurs in an assemblage of granular particles such as in a telephone transmitter for which the motion is so small that gross slide does not occur. Since the theoretical displacement of Equation (19) has been verified by the glass lens experiment, we can use it to determine some of the quantities involved in the oscillographs of Fig. 9 and the displacement- normal force curve of Fig. 15. To obtain the relation between the total displacement 8x and the normal force A^, we have to eliminate a from Equation (17) since a is also a function of the normal force as shown by Equation (13). Introducing this equation, and neglecting l//i2 as com- pared to l//ii , since for the wire jU2 is 40 times m of a plastic, 5. = (ff (2 - cr)f 8^yr(l - a) 1 - ~^. Tn 2/3-1 (22) MEASURING FORCES A\D Wl'.AU I\ SWITCHING APPARATTS 499 lIcMice in agrconiont witli the dahi of l''i<;. I."), the (lisplaccnicnt I'oi- no gross slide should viwy as I lie Iwo-thirds power of llic normal force. Anothei- deduction from Iviuatioii (2'2) is that the displacement for no slide should vary as the iiiNcrse two-thirds power of the shear stiff- ness constant pi. For example <;um inibher with ti shear stiffness of 2 x 10 dvnes cm' should give 100 times the displacement of a ])lastic with a shear stiffness of 2 x lO'" dynes em". A rough check of this deduction has l)een made l)y cementing a thin sti'ip of gum rubber on the face of a shear responding ceramic and with a normal force of 'M) grams (O.OOiio pounds), vibi-ating the wire at its full amplitude of 2 mil inches. Over this range the voltage response was sinusoidal indicating that no gross slide took place. This is 33 times as large a motion as occurred for a plastic with an elastic stiffness 1,000 times that of the rubber and verifies the \ariation of 8x with ju. The other experimental (luantity that can be obtained from Equation (22) is the radius /• of the effective contact points of the plastic. If all 400 300 O 30 - c ^ / /; 1 - r 1 - / 1 - / 1 1 - / / 1 1 1 1 1 [^— GROSS SLIP / / / / / 1 T - / 1 - / 1 1 / f 1 ■•-THEORETICAL / 1 / / 1 1 1 1 1 1 1 _L 1 1 10-3 DOUBLE DISPLACEMENT IN INCHES Fig. 23 — Plot of area of force displacement loop against double displacement. 500 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 the force is supported bj^ a single point at a time, then for 8^ = ±3 X 10"' inches = ±7.5 x 10"' cm; A^ = 30 grams = 2.94 x lO' dynes; IJL = 2 X 10 dynes/cm ; (7 = 0.45 and the coefficient of friction/ = 0.25, the value of r becomes 0.008 cm. If the weight were supported equally by n points the radius would be divided by n . Since the sidewise displacement would result in a strain of 0.009 for a single point and 0.036 for two points, the latter strain would be beyond the yield strain for the material. Hence the evidence seems to indicate that a single point supports the major part of the weight at any particular time. While it is difficult to reduce the gross tangential slide of a relay to the values required for the low wear (no gross slide) region, the existence of such a region has considerable importance for other sources of wear in relays, namely long continued vibrations of component parts such as undamped wu'es. The tangential motions caused by such vibrations are small, but since they are repeated many times for each operation, the total integrated wear is considerable. By introducing damping so that the vibrations are quickly brought down to the low wear, no gross slide region, a considerable reduction in wear has been found for relays. Appendix VOLTAGE generated BY COMPRESSIONAL AND TANGENTIAL CERAMICS BY FORCES APPLIED UNIFORMLY OR AT CONCENTRATED POINTS When a stress is applied to a prepolarized barium titanate ceramic it has been shown that the open circuit field generated along the Z axis is given by the equation E, = -2[Qn[8^J^ + 8,J, + 8,J\] + Qvl8,,iTr + T-^ - {8,J, + 8,J,)]] ^^^^ where Si^ , 52„ , 8s^ are the remanent values of polarization introduced along the three axes by the poling process, Ti , T^ , Ts , Ti , T^ , T^ the three extensional stresses and the three shearing stresses, and Qn and Qi2 are the two electrostrictive constants for the ceramic. From the "effective" piezoelectric constants measured for these ceramics we find MEASURING FORCES AND WEAR IN SWITCHING APPARATUS ^)()\ lliat Qu5,„ = 2.2 X 10"'; for pure l);ii'iuni litanate and Qn8,„ = 2.4 X 10 '; ^^12530 = - .8 X 10 cgs units (25) r^i.,5;,,, = -.9 X 10"' cgs units (2()) for 4 jxn- cent lead titanate barium titanate ceramic. If a force F is applied uniformly over the whole siu-face of a small barium titanate unit, then T3 = F/A, where .4 is the area, and all the other stresses are zero. Under these circumstances when tho pcMinanent l)()larization 83^ is along the Z axis (normal poling), tlu^ oi)en circuit potential is E, = F3 2Qnd,,F 2 X 2.4 X 10 ' X F h u Itnl cgs units (27) where h is the thickness and /„. and / the cross-sectional dimensions. To get the number of volts generated this factor is multiplied by 300 and V, = 1.44 X 10~'F tint ^olts (28) where force F is expressed m dynes. However for the data of Figs. 2 and 4, the voltage measured is that for a load applied at the center of the ceramic and for this case the stresses 7\ and T2 of Equation (24) cannot be neglected. The solution'^ for the stresses occurring Avhen a load F is applied at a point on the surface of a semi infinite solid is used to evaluate the corrections caused by the non- uniform load. In cylindrical coordinates the formulae for the three stresses Tzz and Trr and Tee given by Timoshenko are Trr = Te T.. = F (1 - 2a) (1 -3F r- 1 , z -. {r + zy 3r"2(;-" + z'Y 1 I * / 2 I 2\— 1/2 I / 2 , 2\- r- r- (29) z'{r + z') -512 where r is the radial distance from the point of contact, z the distance below the surface and o- = Poisson's ratio. The response of a barium titanate unit in terms of cylindrical co- ordinates has been shown to be for a unit polarized along the z axis E, = -2 [QuKT:. + Q125.3, {Trr + Tee)] (30) 502 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Now since the ceramic is plated, the major sui-face is an equipotential surface and hence E. does not vary with r oi' 6. Hence integrating over the surface of the ceramic, we have for the open circuit field E, / rdrdd=-2 Qii5,„ / J\,r dr d9 Jo Jo Jo Jo + Qu5,, [ f (7V. + Tee)rdrdd Jo Jo Introducing the values of T^^ , Trr and Tee from Equation (29) and per- forming the integrations we find EA = 2 [Qn8,,F + Qi253o (1 + 2a)F] (32) where A is the cross-sectional area of the ceramic. The first term agrees with that for a uniform stress, but the second term shows that we have a correction due to the radial and tangential stresses generated by the application of the force at a point. The amount of correction can be calculated by putting in the values of Qu and o- the Poisson ratio. Recent measurements of the thickness resonance and the resonance of a torsional ceramic have shown that the best values of the Lame elastic constants are X = 5.8 X 10'' dynes/cm'; m = ^ X lO" dynes/cm' (33) With these values, Poisson 's ratio becomes X 5.8 2(X + m) 19.6 = 0.29G (34) For 4 per cent lead titanate barium titanate ceramic, introducing the values given above, the voltage generated by a force applied at a point is about 0.4 of that for a force apphed uniformly, giving ,. 0.575 X lO"' Fit ,, .^.s V z = — volts (3o) This value corresponds reasonably well with the data of Fig. 4. When the remanent polarization is applied along the Y axis and the voltage measured along the Z axis. Equation (24) shows that the open circuit voltage will be E,= -2 (Qr, - Qv^d,J, (36) where Ti = Y^ is the stress in the direction of polarization (F) applied to the surface of the ceramic. Since the single stress Ti is involved, the MEASURING FORCES AND WKAR IX SWITCHIXG APPARATUS 503 open circuit voltage will be iiidci)eiicleut of whether the force is applied uniformly over the surface or at a point. This follows from the fact that E-i is independent of .c and // and hence Ez\ \ dxdii=-2{Qn-Qn)hA / T.dxdy (37) Jo •'O ''0 -^0 Integrating over the surface gives the total force F for the light side and liencc ■fT 2(Qii — Qn)S2JiF . . .„„. Vz = -^ 1 J — "^ '^^^ units (38) 1.98 X 10~%F . ,, = J— in volts For a ceramic 0.1 cm by 0.1 cm in cross-section and 0.05 cm thick a tangential force of 100 grams should generate a voltage of 9.7 volts. REFERENCES 1. L. Vieth and C. F. Wiebusch, "Recent Developments in Hill and Dale Re- corders," J. Soc. Motion Pictures Engrs., Jan., 1938. 2. W. P. Mason and R. F. Wick, "A Barium Titanate Transducer Capable of Large Motion at an Ultrasonic Frequency," J . Acous. Soc. of A., 23, pp. 209-214, Mar., 1951. 3. R. D. Mindlin, W. P. Mason, T. F. Osmer, and H. Deresiewicz, "Effects of an Oscillating Tangential Force on the Contact Surfaces of Elastic Spheres," presented before First National Congress of Applied Mechanics, June 14, 1951. The results of this paper are summarized here. 4. Measurements have been made by T. F. Osmer. 5. This circuit was devised by G. A. Head. 6. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, Chapter XII, 1950. 7. The data of Figs. 2 and 4 were obtained by L. Egerton. 8. Elizabeth A. Wood, "Detwinning Ferroelectric Crystals," Bell System Tech. J., 30, No. 4, Part I, pp. 945-955, Oct., 1951. 9. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, Chap. XII, 1950. 10. The photographs of Fig. 6 were obtained by T. E. Davis. 11. R. D. Mindlin, "Compliance of Elastic Bodies in Contact," J. Appl. Mech., pp. 259-268, September, 1949. 12. A. E. H. Love, Theory of Elasticity, 4th Edition, page 198, Cambridge Univer- sity Press. 13. I. Simon, O. McMahon and R. J. Bowen, "Dry Metallic Friction as a Function of Temperature Between 4.2°K and 600°K.,"" J. App. Phys., 22, pp. 170-184, Feb., 1951. 14. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, Chap. XII, p. 300. 15. S. Timoshenko, Theory of Elasticity, McGraw-Hill Co., p. 311. 16. W. P. Mason, Piezoelectric Crystals and Their Application to Ultrasonics, D. Van Nostrand, Appendi.x A9, p. 490. A Comparison of Signalling Alphabets By E. N. GILBERT (Manuscript received March 24, 1952) Two channels are considered; a discrete channel which can transmit se- quences of binary digits, and a continuous channel which can transmit hand limited signals. The performance of a large number of simple signalling alphabets is computed and it is concluded that one cannot signal at rates near the channel capacity without using very complicated alphabets. INTRODUCTION C. E. Shannon's encoding theorems associate with the channel of a communications system a capacity C. These theorems show that the output of a message source can be encoded for transmission over the channel in such a way that the rate at which errors are made at the re- ceiving end of the system is arbitrarily small provided only that the message source produces information at a rate less than C bits per second. C is the largest rate with this property. Although these theorems cover a wide class of channels there are two channels which can serve as models for most of the channels one meets in practice. These are: 1. The binary channel This channel can transmit only sequences of binary digits 0 and 1 (which might represent hole and no hole in a punched tape; open-line and closed line; pulse and no pulse; etc.) at some definite rate, say one digit per second. There is a probability p (because of noise, or occasional equipment failure) that a transmitted 0 is received as 1 or that a trans- mitted 1 is received as 0. The noise is supposed to affect different digits independently. The cpacity of this channel is C = 1 + p\ogp+ (1 - p) log (1 - p) (1) bits per digit. The log appearing in Equation (1) is log to the base 2; this convention will be used throughout the rest of this paper. 1 C. E. Shannon, "A Mathematical Theory of Communication," Bell System Tech. J., 27, p. 379-423 and pp. 623-656, 1948, theorems 9, 11, and 16 in particular. 504 COMPAUISO.N OF SIGNALLIXG ALPHABETS 505 2. The low-pass Jiltcr The second channel is an ideal low-pass filter which attenuates com- plelel}' all t're(iuencies above a cutoff frequency W cycles per second and which passes frequencies below IT witliout attenuation. The channel is supposed capable of handling only signals with average power P or less. Before the signal emerges from the channel, the channel adds to it a noise signal with average poAver N'. The noise is supposed to be white Gaussian noise limited to the fi-equency band | J' 1 < W. The capacity of this channel is C = TF]og(^l + ^^) (2) l)its per second. Shannon's theorems prove that encoding schemes exist for signalling at rates near C vnih. arbitrarily small rates of errors without actually giving a constructive method for performing the encoding. It is of some interest to compare encoding systems which can easily be devised with these ideal systems. In Part I of this paper some schemes for signalling o\er th(^ binary channel will be compared with ideal systems. In Part II the same will be done for the low-pass filter channel. Part I THE BINARY CHAXXEL 1. Error-Correcting Alphabets Imagine the message source to produce messages which are sequences of letters drawn from an alphabet containing K letters. We suppose that the letters are equally likely and that the letters which the source pro- duces at different times are independent of one another. (If the source given is a finite state source which does not fit this simple description, it can be converted into one which approximately does by a preliminary encoding of the type described in Shannon's Theorem 9.) To transmit the message over the binary channel we construct a new alphabet of 7v letters in which the letters are different sequences of binary digits of some fixed length, say D digits. Then the new alphabet is used as an en- coding of the old one suitable for transmission over the channel. For example, if the source produced sequences of letters from an alphabet of 3 letters, a typical encoding with D = b might convert the message 506 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 into a binary sequence composed of repetitions of the three letters. 00000 11100 and 00111 If K = 2°, the alphabet consists of all binary sequences of length D and hence if any of the digits of a letter is altered b}^ noise the letter Avill be misinterpreted at the receiving end of the channel. If K is somewhat smaller than 2^ it is possible to choose the letters so that certain kinds of errors introduced by the noise do not cause a misinterpretation at the receiver. For example, in the three letter alphabet given above, if only one of the five digits is incorrect there will be just one letter (the correct one) which agrees with the received secjuence in all but one place. More generally if the letters of the alphabet are selected so that each letter differs from every other in at least 2A- + 1 out of the D places, then when A- or fewer errors are made the correct interpretation of the received sequence will be the (uniciue) letter of the alphabet which differs from the recei\'ed seciuence in no more than A' places. An alphabet with this property will be called a A- error correcting alphabet . Error correcting alphabets have the advantage over the random alphabets which Shannon used to prove his encoding theorems that they are uniformly reliable whereas tShannon's alphabets are reliable only in an average sense. That is, Shannon proved that the probability that a letter chosen at random shall be received incorrectly can be made ar- bitrarily small. However, a certain small fraction of the letters of Shan- non's alphabets are allowed a much higher probability of error than the average. This kind of alphabet would be undesii-able in applications such as the signalling of telephone numbers; one would not want to give a few subscribers telephone numbers which are received incorrectly more often than most of the others. It is only conjectured that the rate C can be approached using error correcting alphabets. The alphabets which are to be considered here are all error correcting alphabets. A geometric picture of an alphabet is obtained by regarding the D digits of a sequence as coordinates of a point in Euclidean D dimensional space. The possible received sequences are represented by vertices of the unit cube. A A- error correcting alphabet is represented by a set of vertices, such that each pair of vertices is separated by a distance at least \/2k + 1 Let Ko{D, A) be the largest number of letters which a D dimensional ^ R. W. Hamming, "Error Detecting and Error Correcting Codes," Bell System Tech. J., 29, pp. U7-160, 1950. COMPARISON OF SIGNALLING ALPHABETS 507 k error correcting alphabet can contain. Except wlien k = I, there i.s no general method for constructing an alphabet with /vo(I>, k) letters, nor is A'o(D, />•) known as a function of 1) and A'. Crude upper and lower bounds for Ki,(D, k) are given by the following theorem. Theorem 1. The largest number of letters 7v ()(/->, k) satisfies where N{D, k) = T.Co,r r=0 is the number of sequences of D digils which differ from a given sequence in 0, 1, ' ■ • , or k places. Proof The upper liound is due to R. W. Hamming and is proved by noting that foi' each letter S of a A" error correcting alphabet there are N(D, k) possible received sequences which will be interpreted as meaning S. Hence N(D, k) Ko(D, k) < 2 , the total number of sefiuences. The lower bound is proved by a random construction method. Pick any sequence Si for the first letter. There remain 2 '' — N(D, 2k) se- (luences which differ from »S'i in 2A- + 1 or more places. Pick any one of these 8-2 for the second letter. There remain at least 2 — 2N{D, 2k) sequences which differ from both Si and S-> in 2A' + 1 or more places. As the process is continued, there remain at least 2 — rN(D, 2k) secjuences, which differ in 2A- + 1 or more places from *S'i , • • • , Sr , from which *SV+i is chosen. If there are no choices available after choosing Sk , then 2'' - KN{D, 2k) < 0 so the alphabet (Si , ■ ■ • , Sk) has at least as many letters as the lower bound (3). For all the simple cases (D and A- not very large) investigated so far the upper liound is a better estimate of Ko(D, k) than the lower bound. The upper and lower bounds differ greatly, as may be seen from a quick insjx'ctioii of Table I. For example, in the case of a ten dimensional two error correcting alphabet, the bounds are 2.7 and 18.3. 2. Efficiency Graph The first step in constructing an efficiency graph for comparing alpha- bets is to decide on what constitutes reliable transmission. The criterion used here is that on the average no more than one lettei- in 10 shall be misinterpreted. 508 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Table I Table of 2^/N{D, k) V -^ 1 2 3 4 5 6 7 Z) = 3 2 4 3.2 5 5.3 2 6 9.1 2.9 7 16 4.4 2.9 8 28.4 6.9 2.8 9 51.2 11.1 3.9 2 10 93.1 18.3 5.8 2.7 11 170.7 30.6 8.8 3.6 2 12 315.8 51.8 13.7 5.2 2.6 13 585.2 89.0 21.6 7.5 3.4 2 U 1092.3 154.4 34.9 11.1 4.7 2.5 15 2048 270.8 56.8 16.8 6.6 3.3 2 Missing entries are numbers between 1 and 2. This sort of criterion might be appropriate for a channel transmitting English text. For other messages it is not always appropriate. For ex- ample, if the messages are telephone numbers, one would naturally require that the probability of mistaking a telephone number be small, say less than 10 . If the telephone numbers are L decimal digits long, and if the alphabet has K different letters in it (so that it takes about L log 10/log K letters to make up a telephone number) the probability of making a mistake in a single letter should be required to be less than about 10"' log K L log 10 which gives alphabets with large K an advantage over alphabets with small K. Since the probability that exactly r binary digits out of D shall be received incorrectly is Cu^rV" (1 ~ p) ^ we achieve the required re- liability with a D-dimensional fc-error correcting alphabet provided p satisfies T=k+\ (4) The value of p which makes the inequality hold with the equals sign determines the noisiest channel over which the alphabet can be used safely. Let K be the number of different letters in the alphabet. Then the COMPARISON OF SIGNALLING ALPHABETS 509 rate in bits per digit at wliicli iut'oiniatioii is being lecieved is log K R = D (5) In Equation (5) we have neglected a term which takes account of the information lost due to channel noise. This is legitimate because all but 10" of the letters are received correctly. The worst tolerable probability p of (4) and tlic rate R of Equation (5) determine the noise combating ability of an alphabet. To compare different alphabets one may represent them as points on an efficiency graph of R versus p. Fig. 1 is an efficiency graph on which the values (p, R) for a number of simple error correcting alphabets liave been plotted. Each point on the graph is labelled with the two numbers /;;, D in that order. The alphabets represented were not found by any systema- tic process and are not all proved to be best possible (i.e., to have the largest K) for the stated values of k and D. Fortunately, R depends on K only logarithmically so that it is not likely the points representing the best possible alphabets lie far away from the plotted points. The sohd line represents the curve i^ = (7 = 1 + p log p + (1 - ?;) log (1 - v)- According to Shamion's theorems, all alphabets are represented by points lying below this line. The eiSiciency graph only partially orders the alphabets according to O 0.4 0.2 " 0,1 ,31 • =-. ^^ ■^ - 'V^ N, 1,1 >• N \ \ \ ? e • 2, \ \ \ > \ \ ',S • 1, 3 3,15 • 0 • 2,8 2,5 \ \ N V \ •• 3,11 c 3,7 \ \ \ \ \ 24.50^ ■^ 0.0004 0.001 0.004 0.01 0.02 0.04 0.1 0.2 0.4 0.6 1.0 PROBABILITY OF ERROR IN ANY DIGIT Fig. 1 — Probability of error in a letter is 10~^ 510 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 their invulnerability to noise. For example, it is clear that the alphabet 3, 15 is better than 2, 8. However, without further information about the channel, such as knowledge of p, there is no reasonable way of choos- ing between 3, 15 and 3, 7. 3. Large Alphabets We have been unable to prove that there are error correcting alphabets which signal at rates arbitrarily close to C while maintaining an arbi- trarily small probability of error for any letter. A result in this direction is the following theorem. Theorem 2. Lei any positive e and 8 he given. Given a channel with p < \ there exists an error correcting alphabet which can signal over the channel at a rate exceeding Ro — e where R,= l + 2p log 2p-\- (1 - 2p) log (1 - 2p) bits per digit and for which the probability of error in any letter is less than 8. Proof The probability of error in any letter is the sum on the left of (4) . This is a sum of terms from a binomial distribution which, as is well known, tends to a Gaussian distribution with mean Dp and variance Dp(l — p) for large D. Hence there is a constant ^(5) such that all k error cor- recting alphabets with sufficiently large D have a letter error proba- bility less than 8 provided k>Dp + A(8) (Dp(l - p)y" (6) Let k{D) be the smallest integer which satisfies (6) and consider an alphabet which corrects k(D) errors and contains Ko(D, k(D)) letters. By Equation (5) and the lower bound of Theorem 1, this alphabet signals at a rate R(D) satisfying 1 - ^ log N{D, 2k{D)) < R{D). Since p < i, 2k(D) < D/2 for large D and hence N(D, 2k{D)) < C2k{D) + l)Co.-2HD,. Then an application of Stirling's approximation for factorials shows that as Z) -^ c» 1 -^logiV(A2fc(£>))-^/?o. COMPARISON OF SIGNALLING ALPHABETS .") 1 1 Hence by takiuj^ D large enough one obtain.s an alphal)et witli rate ex- ceeding Ro — e and letter error probability less than 8. The rate Ro appears on the efficiency graph as a dotted line. It has not been shown that no error-coirecting alphabet has a rate exceeding Ro . In fact, one alphal)et which exceeds Ro in rate is easy to construct. If the noise probability p is greater than j, then Ro = 0. The alphabet with just two letters 0 ()()()... 0 and 1 1 1 1 ... 1 will certainly transmit information at a (small) positive rate, and with a 10 probability of errors if D is large enough, as long as p < |. Using a more refined lower bound for Ko(D, k) it might be shown that there are error-correcting alphabets which signal with rates near C. If one repeats the calculation that led to Ro using the upper bound (3) (which seems to be a better estimate of the true Ko(D, k)) instead of the lower bound (3), one is led to the rate C instead of Ro . The condition (4) is more conservative than necessary. The structure of the alphabet may be such that a particular sequence of more than />• errors may occur without causing any error in the final letter. This is illustrated by the following simple example due to Shannon : the alphabet with just two letters 0 0 0 0 0 0 111000 corrects any single error but also corrects certain more serious errors such as receiving 0 0 1111 for 0 0 0 0 0 0. An alphabet designed for practical use would make efficient enough use of the availat)le sequences so that any seciuence of much more than k errors causes an error in the final letter; the random alphabets constructed above probably do not. If this kind of error were properly accounted for, the rate Ro could be improved, perhaps to C. 4. Other Discrete Channels If instead of transmitting just O's and I's the channel can carry more digits 0, 1, 2, • . • , n 512 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 a similar theory can be worked out. The simplest kind of noise in this channel changes a digit into any one of the n other possible numbers with probability p/n. Then the capacity of the channel is C = log (n + 1) + p log ? + (1 - p) log (1 - p). n Error-correcting alphabets for this channel can also be constructed and the criterion (4) for good transmission remains unchanged. The proof of theorem 1 can be repeated with little change using k N{D, k) = E Co. rn r=0 as the number of sequences which can be reached after k or fewer errors [the terms 2° in (1) and (3) are replaced by (n + 1)^ ]. Once more, using the lower bound, one finds an expression for Ro which is the same as the one for C but with p replaced by 2p. Part II THE LOW PASS FILTER 1. Encoding and Detection If f(t) is a signal emerging from a low pass filter (so that its spectrum is confined to the frequency band | i* | < W cycles per second) then f{t) has a special analytic form given by the sampling theorem w=-cc \2[V / -n-{2\\ t — m) Thus the signal is completely determined l)y the sequence of sample values j{m/2W). The average power of the signal /(/) is measured by P = lim -^ [ fit) dt 7"— »«) 2ft ^'« which can be expres.sed in terms of the sample values as follows As in Part I, consider a message source producing a sequence of letters from an alphabet of K equally likely letters. To transmit this informa- tion over the low pass filter we must encode the sequence into a function ^ C. E. Shannon, "Communication in the Presence of Noise," Proc. I. R. E., 37, pp. 10-21, Jan. 1949. COMPARISON' OF SIGXALLIXG ALPIIABI:TS 513 /(/) of the form (7), or in other words into a seciueiice of .suinplc \iihies f{in/2\V). To do this, we construct a new alphabet containing K letters which are different sequences of real iuuul)ers of some fixed length, say D places. When we let the letters of the new alphabet correspond to letters of the old one the message is translated into a sequence of real numbers which we use for the sequence /(m/21F). If the 7v letters of the sequence alphabet are Si'. On , • • • , OiD S2' Q21 , ' ' ' , (f2D the expression (8) for the average power of the function /(/) becomes P = ^{d\ + d\+ ■■■ -{-dl) (9) where D di\ = S « ii • If the D numbers in the sequence 8i are regarded as coordinates of a point in Euclidean D dimensional space, d1 represents the square of the distance from the point representing Si to the origin. AVhen/(0 is transmitted, the received signal will be/(0 + n{t) where n{t) is some (unknown) white Gaussian noise signal. The noise signals n{t) are characterized by the fact that their sample values n{m/2W) are independently distributed according to Gaussian laws. That is, P,„b(„(^.)(/',: < Vj for all j 7^ i). To prove that this choice of U i is best possible consider any other detector such that Ui contains a set V of points in which I'i > Tj . A direct calculation shows that the detector obtained by removing V from Ui and making F part of U j has a smaller probability of error per letter. The set of points equidistant from two given points is a hyperplane. The region Ui of a maximum likelihood detector is a convex region bounded by segments of the hyperplanes Ti = n , Ti = ro , • • • . To compare signalling alphabets under the most favorable possible circumstances, we always compute letter error probabilities assuming that the detector is a maximum likelihood detector. 2. Computation of error probabilities Exact evaluation of the letter error probability integral (11) is im- possible except in a few special cases. Fortunately we are only interested in (11) when a- is small enough in comparison to the size of Ui to make the integral small. Then fairly accurate approximate formulas can be de- rived. Theorem 3. Let Rij be the distance between letter points *S, and Sj . Then 11(1 - Qi,) < Pi = 0>i + • • • + Vk)/K is 'V'ZTr J Tola where 2/-o is the smallest of the /\ (7v — 1) '2 distances Ra and .V is the average o\-er all l(>ttei-s in the ali)iiabet of the numb(n- of lettei' points which are a distance 2/o away. 3. Efflciency graph The efficiency graph to be described was constructed originally to comj)are alphabets for signalling telephone numbers of length equal to t(Mi decimal digits. It was desired that on the average only one telephone number in 10^ should be received incorrectly. As described in Part I section 2, if the telephone numbers are encoded into sequences of letters from an alphabet of K letters, we must reciuire that the average prob- ability of error in any letter be p = 10"' logio K (U) or smaller. Given an alphabet, one can compute with the help of (13) and (14) and a table of the error integral the largest value of the noise power • ttGjS i8,4 -;^> /^ ^ 5 4 0 •lO • (1,16,7 5 ) A f F • /,o 'RIO ••10,9 ' 1 / (3,2,7) • • (3,32 • 32,16 ,15) / / • fi4 ^ 20,19 P / / • ' / • 50,^ •/ 100,99 / 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 R/W = RATE PER UNIT BANDWIDTH IN BITS 5.0 Fig. 2 — Probability is 10"'' that an error is made in a 10 digit decimal number. COMPAKISON OF SIGNALLING ALI'IIABKTS ,") 1 7 bets. All alphabet is considered poor il' its point on the ethcicncy j^rapli Hes far above the ideal curve R/W = C/W = log (1 + Y). 4. The alphabets The alphabets which appear on the efficiency graph arc the following: excess three (XSS): tiie t(Mi sequences of 4 binary digits whicli I'cpre- , sent 3, 4, • • • , and 12 in binary notation; two out of five: tlie ten setincnccs of five binary digits which contain exactly two ones; pulse position (PPIO): the ten sequences of ten binarj^ digits which contain exactly one one; 2" binary: all of 2 sequences of D binary digits. pulse amplitude (PAn) : the 2n + 1 sequences of length 1 consisting of — n, — w + 1, • • • , n. This alphabet gives rise to a sort of quantized amplitude modulation. pulse length (PLn) : the ?i + 1 sequences of n binary digits of the form 11 • • • 10 • • • 0, i.e., a run of ones followed by a run of zeros. Minimizing alphabets (K, D) : The above alphabets are taken from actual practice. They are convenient because, aside from PAn, they require a signal generator with only two amplitude levels. If we ignore ease of generating the signals as a factor, a great many geometric ar- rangements of points suggest themselves as possible good alphabets. The principle by which one arrives at good alphabets may be described as follows. When a D and K have been determined which give the desired information rate R [by Equation (16)] try to arrange the K letter points in D dimensional space in such a way that the distances between pairs of points are all greater than some fixed distance and that the average of the K sc[uared distances to the origin is minimized. By Equations (9) and (13) it is seen that, apart from the small influence of the factor A^, this process must minimize the signal to noise ratio Y required. Ordinarily it is difficult to prove that a configuration is a minimizing one. Even to recognize a configuration which leads to a relative minimum {i.e. a minimum over all nearby configurations) is not always easy. The eight vertices of a cube, for example, do not give a relative minimum. Consequently, most of the alphabets to be described are only conjectured to be "best possible." Each of them satisfies one necessary requirement of minimizing alphabets that the centroid of the point configuration (assuming a unit mass at each letter point) lies at the origin. That this condition is necessary follows from the easily derived identity r2 = n - Ro 518 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 K=3 K=4 K = 5 K=6 K=7 K=8 K=9 K = 10 Fig. 3 — Two dimensional alphabets. = 10 K=13 Fig. 4 — Three dimensional alphabets. COMPARISON" OF 8IGNALLING ALPHA HKTS .") 1 <) when' fi is the rms dislaiu'c from llie origin lo tiic points of u fonfiguru- tioii .1, Ri) is the distance fioni tho origin to the centroid of A, and r-z is the nns distance from the i)oints of A to the centroid of A. In plotting points on the efhciency graph the notation A', 1) is used for the best K-letter D-dimensional alphabet which has been found. The arrangement of points for various A', 2 and A, 3 alphabets is given in Figs. 3 and 4. In these hgures two points are joined by a straight line if the distance between them is 1 (which is the value we have adopted for the minimum allowed separation 2/-()). Although not shown, the origin is always at the centroid of the hgure. To aid interpretation of these diagrams we have included Fig. 5 which demonstrates how all the signals of a typical alphabet can be generated. The functions of time shown in ^^(^'v^'-ri?) s, 2W 2 3 2W 2W Y3 J. L T i_ Z-{6 ITT 2V3 — != "2V6 2V3 Fig. 5 — Generation of the 4,3 code signals. 520 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Fig. 5 are not the code signals themselves but impulse functions which are to be passed through a low pass filter with cutoff at W c.p.s. to form the code signals. The best possible higher dimensional alphabets can be described more easily verbally than pictorially. In four dimensions we have found four alphabets. The 25,4 alphabet consists of the origin and all 24 points in 4 dimen- sional space having two coordinates equal to zero and the remaining two equal to l/\/2 or — l/'s/2 . Each of the 24 points lies a unit distance away from the origin and its 10 other nearest neighbors; they are, in fact, the vertices of a regular solid. This alphabet has an advantage be- yond its high efficiency. The code signals are composed entirely of posi- tive and negative pulses of fixed energy and so should be easier to generate than most of the other codes which appear in this paper. The 800, 4 alphabet is constructed in the following way: Consider a lattice of points throughout the entire 4-dimensional space formed by taking all the linear combinations with integer coefficients of a basic set of four vectors. That is, the lattice points are of the form CiVi + C2V2 + C3V3 + CiVi where Ci , • • • , C4 are integers and the Vi are the four given vectors. In connection with our problem it is of interest to know what lattice, (i.e. what choice of Vi , Vo , Vz , Va) has all lattice points separated at least unit distance from one another and at the same time packs as many points as possible into the space per unit volume. When a solution to this "packing problem" is known, it is clear that a good alphabet can be obtained just by using all the lattice points which are contained inside a hypersphere about the origin as the letter points. Many of the two dimensional alphabets illustrated in the sketches are related in this way to the corresponding tw^o dimensional packing prob- lem (which is solved by letting Vi and v-i be a pair of unit vectors 60° apart) . A solution to the four dimensional packing problem is aff ored by ''■ = ^' Vr "■ " "=^2' °' °' 72 "*='°' vl' V2- "• This lattice contains two points per unit volume (twice as dense as the cubic lattice in which Vi , • • • ,Vi are orthogonal to one another) and each COMPAKISOX OF .SIGNALLING A Ll'llA HKTS 521 point has 18 nearest neighbors. A hyi)ersphei'o of radius 3 a))Out tlie orijiiii has a volume (7r"/2)3 , ahoul 100. Thus it contains about 800 kiltiee points. Take these as the code points of the 800, 4 code. Their average squared distances from the origin can be estimated as ,.3 5 r dr JQ ^3 \ r'dr Jo = I m' = G. A'' in Equation (13) may be estimated at 18; this is conservative because some lattice points outside the spliere arc being counted. The two remaining four dimensional alphabets l)elong to two families of Z)-dimensional alphabets. The 4, 3; 5, 4; • • • ; Z) + 1, -D • • • alphabets are the vertices of the simplest regular solid in /^-dimensional space. For example, 4, 3 is a tetrahedron. Such a solid can be constructed from D + 1 vertices whose coordinates are the first Z) + 1 rows of the scheme 0 0 0 0 0 ••• 10 0 0 0 ••• 2V3 1 2\/3 2V6 1 1 5 2\/3 2\/6 2vl0 1 1 1 2\/3 2Vg 2\/iO The vertices all lie a distance \/D/2(D + 1) from the centroid of the figure. 6, 3; 8, 4; • • • ; 2D, D, • • • are obtained by placing a point wherever any positive or negative coordinate axis intersects the sphere of radius 522 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 1/V^2 about the origin. Tlm.s it follows that G, 3 consists of the ver- tices of an octohedron. Error correcting alphabets {{k, K, I))): The error correcting alphabets discussed in Part I can be converted into good alphabets for this channel by replacing all digits which equalled 0 by —1. Three error correcting alphabets appear on the chart; each is labelled by three numbers signi- tying (/v, K, D). Slepian alphabets (SD): Using group theoretic methods, D. Slepian has attempted to construct families of alphabets which signal at rates approaching C. Although this goal has not yet been reached, families of alphabets depending on the parameter D have been found which approach the ideal curve to within G.2 db and then get worse as D -^ co. In the simplest of these families of alphabets, D = 2m is even and the letters consist of all the 2'"C2m, m sequences containing m zeros, the remaining places being filled by ztl. The best alphabet in this family is the one with D = 24. It lies 6.23 db away from the ideal curve and contains 1.1 x 10^^ letters. The alphabets of this family for D = 10, 24, and 70 appear on the efficiency graph labelled »S10, /S24, and *S70. The conclusion to which one is forced as a result of this investigation is that one cannot signal over a channel with signal to noise level much less than 7 db above the ideal level of Equation (2) without using an un- believably complicated alphabet. No ten digit alphabet tolerates less than 7.7 db more than the ideal signal to noise ratio. It would be interesting to know more about good higher dimensional alphabets. They are very much more difficult to obtain. The regular solids, which provided some good alphabets in 3 and 4 dimensions, pro- vide nothing new in 5 or more dimensions ; there are only three of them and they correspond to our D -f I, D; 2D, D, and 2° binary alphabets. Worse still, the packing problem also becomes unmanageable after dimension 5. ACKNOWLEDGMENT The author wishes to thank R. W. Hamming, L. A. MacColl, B. McMillan, C. E. Shannon, and D. Slepian for many helpful suggestions during the investigation summarized by this paper. Principle Strains in Cable Sheaths and Other Bnckled Surfaces By I. L. HOPKINS (Manuscrii)t Hccoived February 25, 1952) Equations are developed for rujorous determination of magnitudes and direetions of principal strains in plastic deformation, by means of measure- ments of rectangular strain rosettes. Application to the study of telephone cable sheath is described. Ill the course of certain studies of the polyethylene used in the sheath of telephone cable, it was necessary to calculate the magnitudes and direc- tious of the principal strains from data obtained by measurements of the distortion of a square grid which had previously been printed on the surface of the cal)le. The strains were large, rendering useless the usual expressions for analysis of strain rosette data . Such large strains are characteristically sustained for a wide variety of high polymeric ma- terials of increasing importance for wire and cable sheathing as well as other structural uses. In this article the reciuisite formulas are developed. The basic assumptions are: (1) The strains maj^ be large. (2) The strains are uniform over any square of the grid (equivalent to the condition that a square transforms into a parallelogram). (3) The square may be regarded as plane. (4) Two of the principal strains are parallel to the surface. We shall first consider only the two principal strains in the plane of the surface of the cable. Suppose these two strains to be parallel with llie .r and y coordinate axes, respectively, and that one side of the square is aligned, before straining, at the angle 0 willi the .v axis. This is illus- trated in Fig. 1. Let ex = maximum principal strain Cy = minimum principal strain \x = I + ex \ = 1 ->r Cy 1 Cf. for example, Max Frocht, "Photoelasticitj-," 1, p. 37, 1941. 523 524 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 If primes are used to refer to the strained state, K = Xu — Xb - •^a *^d *^c Xh - ~ Xa Xd Xc y'b - 1 / / ' Va _ Vd - Vc Vb - Va yd - yc If Li and L2 are the lengths of the sides of the unstrained square, and Ls and L4 the diagonals, (Li + A LiY = Xlixb - Xaf + 'kliub - yaf {Li + A L2)- = \l{xd - Xcf + \l{yd - ycf whence, if {Xb - Xaf = {yd - ycf = L\ cos'i = LI cos^i (yb — yaf = {xd - Xcf = Li sin^0i = LI sm(j)i ^' + ^^' = L[,^'\^^'=L[,etc. U Li L'i = Xx cos" ^1 + X^ sm" 01 L2 = Xi sin" 01 + Xj, cos" 01 (la) (lb) (2a) (2b) (2c) (3a) (3b) ^b>yb Fig. 1 — Lines ab and cd, before the xy plane is strained by stretching (or com- pressing) in the x and y directions. PRINCIPAL STRAINS IN BUCKLED SURFACES 525 Henceforth, for clarity, suppose the subscript "1" to refer to the longer side of the parallelogram, "2" to the shorter side, "3" to the longer dia- gonal, and "4" to the shorter. *Si = original slope of Li = tan <^i = S2 = original slope of Lo = tan 02 = S[ = tan ct>[ =^Si (4a) Ax S', = tan 0; = ^ ^o (4b) Vb - - Va Xb - - Xa Vd - - Vc *^d »^c Smce 01 - 02 = 90°, 82= -^ (5a) Oi /S2 = — Xj//Xi;Si (5a) By expansion and substitution from Equations (4) and (5), r («' + 1) tan (.i,; - *;) = ^-^ ,^^l' (6) ' - fe) Let ^ = 90° - (0'i - 0o) then 1 - f tan (90° - (01 - 02)) = tan ^ = —-. ^^^^ (7) H'^' + s. which is the shear between Li and Li . si) 2 {Sx + l/^Si) = tan 01 + cot 0i = ^^— (8) sin Z01 and substituting this in equation (7), . 2K\y tan ^p «"^ 201 = -^^^r^ (9) 526 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 whence cos 201 = a/i - 4^'^v tanV (Xj — 'XyY Remembering that and 2, 1 + COS 201 , . cos 01 = (11a) . 2 , 1 - cos 201 , . sm 01 = (lib) and substitutmg Equation (10) in Equation (11), Equation (11) in Equation (3), and then sohdng the quadratic equation thus formed for Xx and X„ , we have ,2 ,2 {L? + l7) ± V (L? + L',r - 4L? L? cos^ ^ ,^.^. Ax, Ay — 2 ^^-^ Referring to Fig. 2, and using the law of cosines, and remembering that Lz is the ratio of the strained to the unstrained length of the diagonal, ^ a • t 2-^3' ~ (^1 + -^2 ) r 1 o \ — cos d = sni \p = j—j (13a) 2/viL/2 whence cos lA = ,r">T"> — " — — ^^^^' Fig. 2 — A parallelogram formed by straining a square. L/, La' and L3' are the ratios of the lengths of the indicated lines to their original lengths. PRINCIPAL STRAINS IN BUCKLED SURFACES 527 This expression, substituted in Equation (12) and reduced, gives Ai, Aj, — \i.^) It may be noted here that a property of the parallelogram, namely, in the notation used here, //f + L? = L'i + L'i (15) makes it immaterial which diagonal is used. This may be readily seen by substituting L3 = Li" -\- L2 — Li in Equation (14). The effect is merely that of substituting L4 for L3 . In Equation (13a), howe\'er, the result is a change in the sign of \l/. As an example of the application of these equations, the measurements of one specimen were : l'i = 2.1 L2 = 1.2 L3 = 2.0 From Equation (14), X' = 4.758, X;, = 2.181, e^, = 1.181 Xj = 1.092 \y = 1.045 By = 0.045 From Equation (13a), sin \p = 0.4266, whence xP = 25.3° tan 1/' = 0.472 From Equation (9), From Equation (4a), sin 2(^1 = 0.587 01 = 18.0° tan 01 = 0.324 tan 01 = 0.1554 01 = 8.83° FromEquation (9), it is obvious that the maximum value of tan ^ occurs at 01 = 45°, and is in this case equal to 0.804. 528 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 This example is illustrated in Fig. 3. The question of direction of the x and y axes is simply settled by draw- ing a line through either of the acute angles of the parallelogram, crossing the parallelogram at an angle <^i with the longer side. This line will be parallel to the x direction, which is, according to the convention, that of greatest strain. So far no mention has been made of strain in the third dimension; that is, a change in thickness of the sheath. In plastic deformation, the volume change is generally negligible. This requires that whence KKK = 1 1 X. = AxAu XmI Fig. 3 — A square and the parallelogram resulting from stretching to length ratios \x = 2.1 Al in the x-direction and Xj, = 1.045 in the y-direction. Table I Principal Strains, per cent Degrees of twist in 3 feet Parallel to Surface Perpendicular to Surface Max. Min. 180 16 06 -19 270 26 09 -27 360 33 14 -34 450 36 20 -39 540 42 19 -41 630 43 22 -43 720 46 24 -45 PRINCIPAL STRAINS IN BUCKLED SURFACES 529 In the example given, X. = 0.439, e, = -0.561 Polyothyloiie sheaths of cabh^ specimens 3 feet long buckled severely over their entire length when the cables were twisted 720° and showed the strains given in Table I at steps up to the final twist . The ratio of maximum to minimum strain parallel to the surface is about 2:1. Tests with a 1: 1 ratio , a more severe condition, have shown that the principal strains at rupture will be of the order of 300 per cent. Therefore it is evident that the strains incidental to the most severe types of handling will not, of themselves, cause rupture of the sheaths. 2 Unpublished memorandum by A. G. Hall. 3 I. L. Hopkins, W. O. Baker and J. B. Howard, J. Appl. Phys., 21, No. 3, pp. 206-213, March, 1950. A New Recording Medium For Transcribed Message Services By JAMES Z. MENARD (Manuscript received March 10, 1952) A magnetic recording medium composed of rubber impregnated loith mag- netic oxide and lubricant is particularly suited to applications requiring the continuous repetition of short transcribed messages. It affords excep- tional life, reliability, and economy in telephone applications, where it is utilized in the form of molded bands stretched, over cylinders of the recording mechanisms. In the Bell System there are several applications requiring the repetition of short voice announcements. Some of the existing applications are weather announcements, intercept of calls to vacant and miassigned numbers, quotations of delays on long-distance calls, and certain leased industrial services, such as stock price quotation. Most of these require continuous repetition of messages between 5 and 60 seconds in length. In some the message remains fixed but in others it is changed at frequent intervals. Magnetic recorders offer particular advantages for services of this nature, because they require a minimum of equipment and operating skill to produce durable records which are instantly reproducible without processing. For several years the Bell System has used a magnetic re- corder employing a loop of Vicalloy tape in the 3A announcement system to furnish weather announcements, and a similar tj-pe of recorder has been used in a leased industrial system at the New York Times. Recently these Laboratories have undertaken the development of transcribed message facilities to meet additional service applications. It was required that the new facilities should provide satisfactory trans- mission quality and afford considerable flexibility in regard to message length, but the paramount requirement was for reliabilit.y and long life. It did not appear practicable to extend the techniques of the Vicalloy tape machine to give the flexibility, convenience of operation and re- 630 NEW RECORDING MEDIUM .j8 I liability (h^sircd in the new tip})li(';i(ion.s, and attention wa« therefore directed to two new elasses of ma^netie reeordinp; media whieh have l)een dcNcloped in recent years. These are the (>lect I'opialed media and the powdered mecha. In recent years magnetic recording metlia iiave been connnercially produced by an electroplating process by the Brush Development C'om- pany of Cleveland, Ohio. Evaluation by Bell Telephone Laboratories shows that such a plating does not easily deteriorate, gives a relatively iiigh signal output and is capable of excellent transmission character- istics. But in order to realize consistently satisfactory transmission, it is necessary to maintain intimate contact between the recording medium and tlu^ magnetic recording and reproducing heads. The expense of ])r()\iding the relatively precise mechanisms necessary to obtain the d(>sired performance objectives suggested the exploration of other media which might simplify this problem. The powdered magnetic media have evolved from German work dating back to about 1932 and from American work since about 1941. In these media the active magnetic material is a finely divided ferro-magnetic powder, usuall}^ iron oxide. This is usually applied with a binder as a surface coating on a tape of plastic or paper, but the Germans at one time produced a tape which was a homogeneous mixture of oxide and plastic. In their present state of development, media of this type offer excellent transmission characteristics and are relatively economical. In the past four or five years they have found widespread commercial appli- cation in the form of coated tape in all fields of recording and transcrip- tion work. Attempts were made to employ commercial types of these coated tapes in various forms of continuous-loop mechanisms, but none met the de- sired recjuirements in regard to life, reliability, and flexibility of oper- ation. An analysis of the experimental results indicated that most failiu'es were due to physical failure of the media as a result of the tension, flexion and abrasion to which they were subjected, but the magnetic records were substantially undeteriorated even wdien physical failure of the supporting base occurred. It became apparent that a specialized recording medium would have to be developed to meet the Bell System requirements for transcriV)ed message services. Development effort was concentrated on the field of powdered media, because these media offered attractive transmission properties and because the expanding commercial importance of this field promised a continuing industrial development and production pro- gram which would provide an economical source of high equality magnetic 532 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 materials. The followdng premises guided the development program: (1) The recording medium should be subjected to the least possible physical manipulation in use to minimize failures. To accomplish this it was decided to develop the recording medium in a form suitable for use on the surface of a rotating cylinder and to use a helical recording track on this surface when the message length required more than one revolu- tion of the cylinder. It was hoped that with this arrangement, physical failure of the recording mediimi would be eliminated, and the service life would be determined by the wear occurring between the medium and the magnetic pole pieces. (2) The recorcUng medium should exhibit some compliance to facilitate intimate contact with the magnetic pole-pieces. (3) The transmission quality should meet present-day telephone stand- ards for transmission of speech. The liigher quality necessary for the recording of music, while desirable, should not be considered a re- quirement. A number of experimental powdered media Avere prepared and tested. These all utilized commercially available iron oxide powder with a coercive force of approximately 250 oersteds, and the samples included coated media, made by dipping, spraying and doctoring the coatmg on various base materials, and impregnated media, prepared by mixing the oxide in the base material and forming the mixture. A medium consistmg primarily of an elastic rubber band impregnated with magnetic particles was found to be particularly suited to appli- cations requiring long life in continuous service. A study of compounding and manufacturing processes for this medium was made by the rubber products group at these Laboratories under the direction of H. Peters, and the compound which evolved consists primaril}^ of sjmthetic rubber loaded with magnetic iron oxide, and containing small amounts of lubri- cants, inliibitors and curing agents. The compound is decidedly rubber- like in character, and is utilized in the form of seamless bands about ^ to I inches thick, w^hich are stretched over the surface of cylinders about 10 per cent larger than the bands. The bands are prepared by thoroughly milling together the following : 100 Parts by weight type GN neoprene 100 Parts by weight magnetic iron oxide 5 Parts by weight zinc oxide 4 Parts by weight magnesium oxide 2 Parts by weight paraffin NEW RECORDING MEDIUM 533 and forming the compound into bands by conventional rul)ber molding and curing techniques. The i-esulting bands show a tensile strength of about 2500 pounds per scjuarc inch, and the elongation before breaking is about 700 per cent. No particularly difficult mainifacturing problems are encountered, and present evidence indicates that satisfactory overall quality control can be achieved by carefully controlling the compounding constituents, the milling and the molding. Several bands which are used in telephone services are shown in Fig. 1 . These bands are utilized in recorder-reproducer mechanisms by stretch- ing them over a cylinder, on which pivoted magnetic pole-pieces trace a cylindrical or a helical track as it rotates. When the bands are first taken from the mold they exhibit a high coefficient of friction. After a few hours enough paraffin migrates to the surface to form a thin, slippery film. If the bands are then put into service the pole-pieces form a polished track and the continuing migra- tion of paraffin maintains the lubricating film between the band and the pole-pieces. If tliis recording medium is used intermittently, the self-lubrication may cause difficulty. The migration of lubricant to the recording surface is continuous, and the lubricant may accumulate on the surface in suffi- cient thickness to impair the contact with the magnetic head if the recording equipment is not operated for several weeks. It may then be Fig. 1 — Typical magnetic rubber bands used in telephone applications. 534 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 necessary to wipe the excess lubricant from the surface to obtain satis- factory operation. The lubricant in this particular recording medium was chosen for operation in central offices and similar locations where temperature ranges are moderate. If extreme temperatures are to be encountered the lubricant problem will have to be re-examined. Continued research in this field should result in improvement in this characteristic. In life tests, five million message repetitions have been obtained with insignificant wear of the band and the magnetic head, and with no measurable deterioration in the level and ciuality of the recording after an initial level drop of about 2 db which occurs during the first few reproductions. The head pressure is a significant factor affecting the life, and in these tests a head pressure of 25 grams was used with a 0.100 inch wide head. This medium represents some compromise in the attainable trans- mission properties to favor the physical properties desired for reliability and long life, but the transmission is entirely adequate for the intended applications. A typical frequency response characteristic is shown in Fig. 2. This is representative of the results obtained when the equipment is main- tained by field personnel. The output level, also indicated by Fig. 2, is from 8 to 12 db below that obtained from commercial coated magnetic tape. This is largely because the concentration of magnetic oxide, on a volume basis, cannot be made as high in the impregnated material as is possible in the coating of conventional tape. This is not a serious dis- advantage, however, as the level is high enough to permit amplification without special precautions in regard to noise. h-^ 10 n > O-l 5 °^5 , \ n i-m ,-'n Q. ^I -5 "n i-< -10 O-L 5,11 -TS Oo _D -20 REPRODUCING SPEED - HEAD- 6 INCHES PER SECOND 0.100 INCHES WIDE GAP- 0.0005 INCHES TURNS - 1800 ^ ^ -^ ^s^ ^ ^ \ ^ \ 1 1 1 1 1 1 1 _L 1 1 1 _L 60 80 100 200 FREQUENCY 400 600 1000 2000 vl CYCLES PER SECOND Fig. 2 — Frequency response of magnetic recording equipment using iron oxide impregnated molded neoprene bands. NEW RECORDING MEDIUM 535 When ring tj^pe magnetic heads are used for recording, these bands exhibit frequency response characteristics quite similar to coated tapes using the same magnetic oxides, although the bands are of homogeneous magnetic material up to | inch thick and the tapes have magnetic coat- ings less than O.OOl inches thick. This is because the field from the record- ing gap becomes ineffective at a distance of about 0.001 inches, and the signal is recorded only on a thin siu'face layer of the medium, regardless of its total thickness. The noise characteristic of this medium is somewhat unusual. It has been shown* that the reproducing process is not restricted to the surface layer of the medium, but that to a first approximation, when the medium has low permeability, the signal from a magnetized element at any depth ill the recording medium will be attenuated with respect to the signal produced by the same element in intimate contact with the reproducing head by the factor: 55 decibels X S where S = distance between magnet and head X = "wavelength" of magnet This indicates, for example, that the signal from a magnetized element at a depth of X/2.75 will be attenuated by only about 20 db and may therefore make significant contribution to the total output. In the Bell System telephone applications, where a transmission band- width of 100 to 4000 cycles per second is required, the belts are run at a speed of about 6 inches per second. The wavelength at 100 cycles per second is then 0.060 inches, and at this frequency significant output can be obtained from a layer about 0.02 inches thick. The desired recording is limited to a layer about 0.001 inches thick, but a layer of about twenty times this thickness may contribute to noise. As a consequence, at low frequencies this medium tends to exhibit higher background noise than do the co.ited tapes. The magnitude of the noise is appreciably affected by the method of erasure. Two methods of erasing a magnetic record are known to the art. These are the saturation erase, in which the magnetic record is exposed to a unidirectional magnetic field of saturation intensity, and the neutral- ization erase in which the magnetic record is exposed to an alternating field which reaches saturation intensity and decreases cyclically to zero * R. L. Wallace, Jr., "Reproduction of Magnetically Recorded Signals," Bell System Tech. J., Oct., 1951. 536 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 over a period of several cycles. It is well known that a neutralization erase results in a residual background noise which may be as much as an order of magnitude below that produced by a saturation erase. The neutralization erase is therefore widely used in tape recording, and is obtained by energizing the erase head with alternating current of a frequency several times the highest signal frequency passed by the recorcUng equipment. With the impregnated recording medium, the recorded signal can be successfully erased by using a conventional ring-type erase head energized with high-frequency current. The field from this type of erasing effec- tively neutralizes the surface layer which contains the recorded signal, but does not penetrate appreciably beyond. Therefore, if precautions are not observed, the lower layers of tliis medium beyond the reach of the erase field may acquire a random cumulative magnetization from switch- ing surges, accidental exposure to magnetized tools and strong fields, and this will be evidenced by a gradual deterioration in the signal to noise ratio at the low-frequency end of the transmission band. The qual- ity, however, remains entirely adequate for commercial telephone use. The foregoing limitations are minimized by an erasing method which has been developed at these Laboratories for applications where it is convenient to erase the entire message in one revolution of the recording cylinder, preparatory to recording a new message. Tliis method employs an erasing structure in the form of an E-shaped stack of magnetic laminations, carrying on the center leg a coil which is energized by low- frequency (60 cycle) alternating current. The lamination stack is ap- proximately the width of the recording medium, and the gaps between the center leg and each side leg are about j inch T\dde. When this struc- ture is spaced about y§- inch from the surface of the recording mediiun traveling at 6 inches per second or less, and is energized by 60-cycle power to produce a maximum field of about 2000 gauss, the entire thickness of the recording medium is subjected to an alternating magnetic field which reaches saturation intensity and over a period of several cycles decreases progressively to zero. This effectively demagnetizes the full tliickness of the recording medium. If the current is switched off with the erase structure in operating position, those elements of mag- netic material wdthin the field at that instant would be subjected to no further reversals and would consequently behave substantiallj'' as if they had been subjected to a direct-current magnetic field of the same in- tensity as the alternating-current field at the time it was interrupted. The section of record medium under the influence of the erase structure at the time it was de-energized would exhibit excessive noise in com- NEW RECORDING MEDIUM 537 parison with the remainder of the record-mecHum which was subjected to the normal aUernatiug-current erase. 'J'liis effect becomes neghgible if the separation between the record medium and the erase structure is increased b}' | inch before the current is interrupted. This is accomi)Hshed by u;?ing a solenoid -actuated moiuiting for the erase structure so ar- ranged that the structure normally is retracted from the erasing position and holds open a switch in the circuit to its coil. When erasure is desired, the solenoid is actuated. This moves the erase structure into operating position and releases the switch to energize the coil. When the erase cycle is completed the solenoid is de-energized, and the erase structure retracts, opening the switch at the end of its travel. Fig. 3 is a sketch SCHEMATIC PHYSICAL ARRANGEMENT Fig. 3 — Method of erasing magnetic recorder. 538 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 5ZJ -40 O-i -\-45 0-65 REPRODUCING SPEED HEAD - 6 INCHES PER SECOND -0.100 INCHES WIDE GAP -0.0005 INCHES TURNS -1800 \ \ ■ . 1 .- 1 1 1 1 1 , _^ 40 10,000 60 80 100 200 400 600 1000 2000 FREQUENCY IN CYCLES PER SECOND Fig. 4 — Typical noise spectrum of i-inch iron oxide impregnated molded neo- prene bands measured in 200 cycle bands after neutralization erase with 60-cycle ac field. showing the apphcation of this erase method to a cyhnder-type machine. This method of erase results in a background noise level measured un- weighted over a 4000-C5^cle band which is at least 45 db below a 1000- cycle signal recorded with 4 per cent total distortion. A typical back- ground noise spectrum is shown in Fig. 4. Fig. 5 — General view of recording machines in 3A announcement system at Cleveland. NEW RECORDING MEDIUM 539 Fig. 6 — Closeup of recording machine in 3A announcement system at Cleveland. The first installation of transcribed message equipment employing this new medium was in the 3 A announcement system at Cleveland, Ohio, to supply weather announcement service. The magnetic recording equipment in this installation is a cylinder- type mechanism with associated recording-reproducing amplifier. The mechanism uses bands Ye inch thick, If inches wide and 7t| inches in diameter, stretched over a cylinder 9 inches in diameter. A single record- reproduce head in a pivoted mounting is cam-controlled to trace a helix on the cylinder. The cam is coupled to the cylinder via a quick-change gear train which gives a choice of a three-turn, a five-turn or an eight- turn helix, and the cylinder is driven from a gear-reducer which allows a choice of two slightly different operating speeds. Six different c.ycle times, ranging from about ten seconds to about 45 seconds, are provided 540 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 by the two operating speeds and the three cam ratios. Approximately 90 per cent of any cycle time is available for recording or reproduction, and the remaining 10 per cent is occupied by the return of the head to the beginning of the helix. The recorded track is 0.100 inches wide, and when an eight-turn helix is used, there is a separation of 0.025 inches ))etween tracks. The previ- ously described low-freciuency alternating current erase is used. The 3A announcement system employs three channels of this recording eciuipment in a complex control circuit which provides facilities for erasing, recording, monitoring and automatic switchover to stand-by channels in event of failure. Figs. 5 and G show the recording equipment in the Cleveland installation. Other ecjuipments using this recording medium have been designed to furnish transcribed message service for intercept of calls to vacant, changed and unassigned numbers, to quote delays on long distance calls, and to furnish stock cjuotation service. Some of these ecj[uipments are now undergoing service trials preparatory to standardization for Bell Sys- tem use. This new recording medium has been developed to provide the maxi- minn attainable life and reliability in applications requiring an enormous number of repetitions of voice messages. Ecjuipment for such applica- tions is usually located in central offices where the temperature range and other operating conditions are fairly w^ell stabilized. These favorable conditions have facilitated the development of a recording medium which has made it possible to design simple and economical magnetic recorders which are sufficiently versatile and reliable to stimulate the use of transcribed message services to an extent hitherto unrealizable. There are a number of potential Bell System applications for tran- scribed message services which do not recjuire an extreme numbei- of message repetitions, but put a premium on low initial cost and trouble- free operation in intermittent service under wide extremes of en\'iron- ment. It may prove desirable to meet the life requirements for applica- tions of this type with a different approach to the lubrication problem, with an unlubi'icated compound, or with a coated medium which would have some transmission advantages. It is expected that finiher work in these fields will produce improved recording media for applications of this nature, to expand the field of use in the telephone plant. Introduction to Formal Realizability Theory — II By BROCKWAY McMILLAN (M:imi8cri|)t r(>c(>iv'■ ^ >h where (i) Each Z,.,(p) is a rational function (ii) ZUP) = Zr.iP) Cm) Zrsip) = Zsrip) (i\-) For each set of real constants A'l , ■ • • , k\ , the function 0. Then there exists a finite passive network, a 2/i-pole, which has the impedance matrix Z(p). A dual result holds for admittance matrices Y{p). 1.2 The converse of this theorem was proved in Part I: that if a finite passive 2/i-pole has an impedance matrix Z(p), then this matrix has properties (i) through (iv) of 1.1. 1.3 We recall that in Part I matrices satisfying the conditions of 1.1 were called positive real (PR). 1.4 The proof of 1.1 is a direct generalization to matrices of the Brune process" for realizing a two-pole impedance function f(p). For this l)roof we shall recjuire from Part I certain specific properties of positi\'e real operators and matrices. These will be summarized in Section 2 be- low. Further than this, the present part is almost independent of Part I, 541 542 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 although in terminology, notation, and method a direct continuation of it. References to sections or paragraphs in Part I will be made thus: (I, 6) or (I, 6.23). 1.5 The distinction emphasized in Part I between operators, as abstract geometrical objects, and matrices as concrete arrays of numbers repre- senting these geometrical objects, is not one which we have now to maintain with any strictness. We shall generally preserve it verbally but not use the bracket notation for matrices introduced in Part I. II. PROPERTIES OF POSITIVE REAL OPERATORS AND MATRICES 2.0 We recall that an impedance operator Z{p) is a linear function from the linear space K of current vectors k to the linear space V of voltage vectors v. A positive real operator Z{p) is one whose matrix in any real coordinate frame is positive real. In Section 16 of Part I the following properties of a PR operator Z(p) were established: 2.01 Zip) has no poles in r+ .* 2.02 If Rc(Z(p)k, k) = 0 for some peT+, then Z(p)k ^ 0 for all p. 2.03 If it exists, Z~'(p) = Y(p) is PR. 2.04 If Z(p) has a pole at p = po , it has one at p = po . 2.05 If Z(p) has a pole at p = fcoo , that pole is simple and Zip) = ^^2 R + Zrip), where R is real, symmetric, semidefinite, and not zero, and Ziip) is PR- 2.06 If Zip) has a pole at p = oo , that pole is simple and Zip) = pR + Z,ip) where R and Ziip) are as in 2.05. 2.07 It was emphasized at several points in Part I that the fact of pos- sessing an impedance matrix, and that of possessing an admittance matrix, are each restrictions on a 2n-pole N. It is readily verified from (I, 6.3) and (I, 6.31) — and, indeed, well known — that if N has both an impedance matrix Zip) and an admittance matrbc Yip), then Yip) = Z-\p). r+ is the open right half plane: all finite p such that Re{p) > 0. FORMAL REALIZABILITY THEORY — II 543 That is, if the impedance matrix of a 2n-pole N is non-singular, then its admit tancp matrix exists, and conversely. 2.08 It was provcnl by C'auer'\ and in (I, 16.8), that if Z(p) is PR and of rank m < n, then there exists a real, constant, non-singular matrix W such that Zip) = W'Z"{p)W (1) where Z"(p) is a non-singular )n X m PR matrix bordered by zeros. 2.09 Properties (i) through (i\') of 1.1 define the PR property for a matrix Z(p). A convenient equivalent definition is that (i) Z(p) is symmetric, (ii) For each A: e K, the function 0. 2.12 If 5(Z) = 0, then Z(p) is a constant — that is, does not depend upon p. 2.13 If Z~\p) exists, then 5(Z) = 5(Z~'). 2.14 If Z(p) = Zi(p) -f Z2(p), where Zi{p) is finite at every pole of Z-iip), and Z2(p) is finite at every pole of Zi(p), then 5(Z) = 5(Zi) -f 5(Z2). 2.15 If Z(p) = f(p)R, where /(p) is a scalar and .R is a constant operator, then 5(Z) = [degree of/] -[rank of 72]. Here the degree of / is the sum y] [order of the pole of f(p) at po] Pa where po runs over all poles oi j{p), including oo. 544 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 2.16 If A and B are constant non-singular matrices, then 8(Z) = 8(AZB). It is evident then that 8(Z) is a geometrical property, being constant over the usual equivalence classes W'Z(p)W or W-'Z{p)W of matrices. Hence we may speak of the degree 5(Z) of an operator Z(p). 2.17 If Z(p) is formed from an m X m matrix Zi(p) by bordering the latter with zeros, then 8{Z) = 8{Z,). 2.18 Concerning the degree 8{Z) we here state a fundamental theorem: Theorem: The 2n-pole whose existence is asserted by 1.1 can be con- structed with 8{Z) reactive elements, and no fewer. The proof of this theorem will be distributed through Sections 4 and 6. In fact, we must even define exactly the phrase "can be constructed with X reactive elements." This will be done in Section 3. 2.2 Lemma: If Zi{p) and Z-iip) are PR operators or matrices, then Z{p) = Zy{p) -f Z,{p) is also PR. If either of Zi{p) or Ziip) is non-singular, then Z{p) is. Proof: Clearly Z{p) is symmetric. By 2.09, therefore, Z{p) is PR if the function {Z{p)h, k) = (ZMk, k) + (Z,(p)k, k) (1) is PR for each A; € K. The right hand side is obviously PR by hypothesis. If either of Zi(p) is non-singular, the function (1) cannot vanish in r^. unless A; = 0 (this is 2.02). Hence in this case Z{p) also is non-singular. 2.21 Clearly 2.2 is independent of the implication, tacit in the notation, that the operators involved are impedances. The lemma holds for PR operators, whether interpreted as operating from K to V (impedances) or from V to K (admittances). % 2.3 In (I, 6.21) and (I, 6.3) it was noted that any n X n impedance matrix Z(p) defines by fiat a general 2w-pole N whose impedance matrix is that Z(p). Such is the generality of the notion of general 2n-pole (I, 4). FORMAL REALIZABILITY THEORY — II 545 Given 2n.-poles Ni and No , with impedance matrices respectively Zi(p) and Zo(p), we know then that there is a general 2/i-pole N whose impedance matrix is Z(p) = Z,{p) + Z,(p). We call this N the series combination of Ni and No . 2.31 Designate the terminal pairs of Ni by (Sr , Sr), those of No b}^ (Tr , Tr), 1 < /■ < n- It is evident that if Ni and No appear in a diagram so connected that (i) Sr is connected to Tr , 1 < r < n; (ii) No other connections are made to these nodes; then the device with terminals Sr , Tr is 'N. This follows at once from Kirchoff's laws applied to the ideal graph (I, 4.11). 2.32 Duallj'', if Ni and N2 have admittance matrices Yi(p), Yiip), then Y(p) = Y,{p) + Y,{p) is the matrix of a 2n-pole N defined as the parallel connection of Ni and N2 . N is the device whose terminal pairs are formed by joining Sr , Tr and also S'r , Tr , 1 < r < n. 2.33 Fig. 1 shows the conventions to be used in indicating 2n-poles (n = 4 in the Figure) with, respectively, impedance matrices and ad- mittance matrices. Fig. 2 then shows the series connection of two im- pedance devices and the parallel connection of two admittance devices. In each case the terminals on the left are those of the composite device. 2.4 The series and parallel connections just described are special ways of combining 2n-poles needed for the generalized Brune process for matrices. They have been introduced here on their merits, as new op- ^■-t; ^--v -— t; IMPEDANCE MATRIX ADMITTANCE MATRIX DEVICES WITH FOUR TERMINAL PAIRS Fig. 1 — Conventions used in representing 2N poles. 546 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 erations. They are, however, expressible in terms of the basic operations of juxtaposition (I, 17) and restriction (I, 18). For example, the series connection of Ni and N2 is formed by first juxtaposing Ni and No , to get a 2 X 2n-poIe N. Let J be the 2n dimen- sional space of 2?i-tuples j = [jl , • • ' , jn , ^1 , • • ■ , 4]. We interpret this j as a 2n-tuple of currents in the 2 X 2/(-pole N, where jr is the current in the r*^ terminal paii* of Ni and 4 that in the r*^ pair of N2 , 1 < r < n. Let K be an n-dimensional space. Given an n-tuple k e K, say k = [m , • • • , Kn\, we define the operator C from K to J by J ^ Ck = [ki , • • ' , Kn , ki , • ' • , kn\. Restricting N by C gives the series combination N of Ni and N2 . The details may easily be supplied by the interested reader. 2.41 Representing the series and parallel connections in terms of juxta- position and restriction makes the lemma, 2.2, an inmaediate conse- quence of the lemma of (I, 17.2) and the theorems of (I, 17.3, 18.3). 2.5 We report here for record a curious property of PR operators which has so far found no application: Lemma: If Z{p) is a PR impedance operator from K to V = K*, and Y{p) any PR admittance operator from V to K, then the operator 1 + Y(p)Z(p) ^ a 1 1 - 1 1 - 1 I , 1 N, N2 N, N2 SERIES CONNECTION OF N, AND N^ PARALLEL CONNECTION OF N, AND N2 Fig. 2 — Series and parallel connection of 2N poles: Series, left, and parallel, right. FORMAL REALIZ ABILITY THEORY — II 547 ill K is non-singular. Dually i + z{v)y{p) in V is non-singular. Proof: Suppose that A; e K is such that (1 + Y{v)Z{v))k = 0 (1) for all p. TluMi 0 = Z(/))(l + Y{v)Z{v))k = Z{p)k -f Z{p)Y(p)Z(p)k for all p. Then, however, iZ(p)k, k) + (Z(p)Y{p)Z{p)k, k) = 0. (2) We may write the second term as {Z*(p)k, Y{p)Z(p)k) = {Z{p)k, Y*(p)Z*{p)k) (3) by (I, 14.0) applied t^^ice. Now Z(p) is PR, in particular real and sym- metric, so Z*ip) = Z*(p) = Z'ip) = Zip). Using a similar calculation with Y(p), the quantity (3) becomes {Z{p)k, Y(p)Z(p)k). (4) For each p e T+ , we have p e r+ and the first term of (2) has a non- negative real part. But for p e r+ , (4) is the conjugate of (v, Y(p)v) (5) where v = Z(p)k. Xow (5) is a PR function of p, hence has a non-nega- Uve real part for p e T+ , for any v. In particular therefore this is true for the V which, at p, makes (5) the conjugate of (4). Therefore (4) has a non-negative real part throughout r+ . It follows from (2) then that Re{Z{p)k, k) = 0 for all p er+. By 2.02, then, Z(p)k = 0. By (1), then Ik = k = 0. Hence (1) implies k = 0. Therefore the operator in (1) has an inverse 548 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 III. A SIMPLE REALIZ ABILITY THEOREM 3.0 The follo\Aing theorem is contained in Cauer^ Since it provides the basic step in our reaUzability process, we shall prove it here. 3.1 Theorem: Let f(p) be any one of the four functions (i) fip) - 1, (ii) fip) = p, (iii) fip) = - V M M = p^ + ,^ ^ wo > 0. Let 72 be a real, constant, symmetric semidefinite n X n matrix of rank r. Then: (A) The matrix Zip) = fip)R is PR and there exists a finite passive 2/i-pole N with the impedance matrix Zip). (B) The 2n-pole N can be realized with ideal transformers and, re- spectively, (i) with r resistors, (ii) with r coils, (iii) with r capacitors, (iv) with r coils and r capacitors. (C) The dual statements to (A) and (B) are true. Proof: That Zip) in PR is easily verified directly. It will follow also from the results of Part I when we exhibit a (finite passi^•e) network whose matrix is Zip). To construct this latter, let Z) be a diagonal matrix such that R = WDW where W is a real, constant, non-singular matrix. That Z) and W always exist is the analog for impedance operators of the result of Hahnos , par. 41, for dimensionless operators. In fact, W can be taken to be orthogonal (Tr~^ = TF', cf. Halmos'*, par. 63). If R is of rank r, T> has r non-vanishing diagonal elements, say rfn , c?22 , • • • , d„ . Since R is semidefinite, each dafip), 1 < i < r, is the impedance of an ob\'iously passive two pole. Call this two-pole M, . Let Mr+i , • • • , Mn be two poles consisting of short cu'cuits. Consider the 2n-pole Ni FORMAL RE ALIZ ABILITY THEORY — II 510 in;i(le hy coimecliiig Ms between Ts and V, , 1 < s < n. This 2/i-polc lias tlie impedance matrix Z.{v) = fip)D. Then Z(v) = f(p)WDW' = WZ,(p)W' is the matrix of a 2/i-pole N which can be obtained from Nj liy the use of ideal transformers. C'learlj' Ni , and therefore N, contains exactly the elements claimed in (B) of the theorem. The dual theorem (C) needs no comment. 3.11 Corollary: The conclusion (A) of 3.1 holds if the hypotheses on j"(p) are replaced by "/(p) is PR." The same method of proof applies but one must use the Brune theory to realize the impedances diif(p), 1 0, either (i) Constructs a physical realization of N^ , or (ii) Constructs a 2nr+i-pole Nr+i such that if N^+i is physically real- izable, then Nr is. To show that this induction actually gives a realization of any PR matrix Zo{p) we must demonstrate that, first, it is effective — i.e. that at any stage N^ at least one of (i) and (ii) is possible. Second, we must show that the process terminates with the construction of a finite net- work. The details of these demonstrations are given in the paragraphs 4.1 et seq. of this section. In the paragraphs 4.01 to 4.07 we describe the logical pattern into which these details are to be fit when they are established. 4.01 There are nine basic operations by which the networks N^ are con- structed. We name the operations here, in order to give a clearer picture of the logic of the process, but their mathematical treatment is deferred to later paragraphs. IP: A PR impedance matrix Zr(p) which has poles on p = ioi is represented as 1 2o P k P -T (^k where Zr+\{p) is PR and has no poles on p = ico. AP: A PR admittance matrix Yr{p) is represented dually: 1 2v P k P -V 0 is the largest a for which Zr+\{p) is PR. Con : The dual to Res. FORMAL REALIZABILITV THEORY — II 551 IB: This is the analog of the step in the Brune process for scalars in which the reactance of a minimum resistance structure is tuned out to create a zero. The details are intricate in the generalization to 2/i-poles. AB: This is the dual operation to IB. F: A 2wr-pole Nr which has a constant PR matrix (admittance or impedance) is realizable at once, by 3.1. The operation F de- notes this realization. To each Nr , one of these nine operations is to be applied. The effect of the last (F) is clearly salutary. That of each of the others is to split off a realizable piece of N^ and leave a 2nr+i-pole N^+i to which again some one of the operations is to be applicable. Exactly which of these operations to apply at any stage depends upon the properties of the Nr in cjuestion. We shall first devise a notation for describing the relevant properties of Nr , and then in 4.04 present a table which summarizes what is to be proved in the paragraphs 4.1 et seq. 4.02 Definition: We say that Z(p) has a zero of its real part at p = v'coo if for some A" e K, k 9^ 0, w^e have [Z(t«o) + Z(-ia)o)]A- = 0. 4.03 Let / be an integer describing a 2n-pole N as follows: 7 = 0 if N has no impedance matrix. / = 1 if N has a non-constant impedance matrix which has no poles on p = ico, and no zeros of its real part on p = t'co. 7 = 2 if N has a non-constant impedance matrix with a zero of its real part on p = z'w, but no poles on p = iw. 7 = 3 if N has an impedance matrix with a pole or poles on p = ico. Let A be an integer describing the admittance category of N in a dual way (e.g., A = 0 if N has no admittance matrix, etc.). Let (7, A) denote the category of 2w-poles N for which the indicated values of both 7 and A are true. Let (7i + U , A, + .42) (1) denote the category of 2n-poles N for which either 7i or I2 is true and, simultaneously, either Ai or Ao is true, with a similar definition for more summands. Then for example the category (1) above consists of the logical union of the following: (h , A,), ih , A,), (I, , .40, (72 , A,). Let C denote the category of 2n-poles N which have a constant ma- trix, impedance or admittance. 552 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 It is clear that any 2n-pole N belongs in C or in exactly one of the six- teen elementary categories whose union is (0 + 1 + 2 + 3, 0+1 + 2 + 3). Table 4.04 shows for each category of Nr , except (0, 0), which opera- tions may be applied, and the possible categories of the resulting N^+i . A 2n-pole N not in (0, 0) has at least one matrix, and if it has two these are of the same degree (2.07, 2.13). We may then denote the degree of whatever matrix N has simply by 5(N). The fourth and fifth columns of Table 4.04 show the relations of 5(Nr) to 5(Nr+i), and of Ur to w^+i . 4,05 Table 4.04 summarizes facts to be proved in 4.1 et seq. Assuming now that the assertions in this table are true, we can show that the inductive procedure is effective. We observe first that the category C and every possible elementary category (/, A) except (0, 0) is contained in at least one of the categories listed in the first column. Hence to any 2n-pole not in (0, 0) there is at least one operation applicable. Further we note that the category (0, 0) does not appear in the third column. Since by hypothesis No is not in (0, 0), it follows by induction that no N^ will be. Therefore the process can stop only by the operation F: completion. Second, we notice that if N^ is not in the category (1, 1), then an applicable operation can be found which actually reduces one of the two numbers 5(Nr), n^ . Furthermore, if Nr is in (1, 1), a sequence of two operations can be found which reduces one of 5(Nr), ih . Therefore before the realization process terminates (with F), (i) There are not more operations chosen from the list IP, AP, IB, AB, than the integer S(No); (ii) There are not more operations chosen from the list ID, AD, than the integer no — 1 (since after these, still ?ir+i > 0) ; Table 4.04 Category of Nr Oper- ation Category of Nr+i «(Nr) - «(Nr + l) fir — nr + 1 (3, 0 + 1 -1- 2 -F 3) (0 -^ 1 -}- 2 + 3, 3) (1 + 2, 0) (0, 1 + 2) (1,1) (1,1) (2, 1 + 2) (1 + 2, 2) C IP AP ID AD Res Con IB AB F (7+(l + 2, 0 + 1 -[- 2 + 3) C+(0 + 1 + 2 -f- 3, 1 + 2) (1 + 2, 1 4- 2 -h 3) (1 + 2 + 3, 1 + 2) (2, 0 -f 1 + 2 + 3) (0 -1- 1 + 2 -1- 3, 2) C+(l +24-3, 0 + 1 + 2 + 3) C+(0 + 1 + 2 + 3, 1 + 2 + 3) >0 >0 0 0 0 0 >0 >0 0 0 >0* >0* 0 0 But rir+i > 0. FORMAL REALIZABILITY THEORY — II 553 (iii) There are not more operations chosen from the list Res, Con, than the integer 5 (No) -{- tio — 1. Finally, then, the process must terminate after at most 25 (No) + 2a?o — 1 operations. 4.06 Besides the data in 4.04, one other fact must be established about each operation: that Nr is physically realizable if Nr+i is. This ^vill be done as we discuss each operation. When it is established, we reason back from the result of operation F, which pro\ddes a physical realiza- tion of some N„. {m < 25(No) + no — 1), through N^-i to Nq = N, and obtain a realization of N in finitely many steps. 4.07 Finally, we shall prove about each step that: If Nr+i can be realized with .iv+i reactive elements, then Nr can be realized with Xr+i 4- S(Nr) - 5(N.+i) reactive elements. This observation will pro\dde the basis for proving the theorem of 2.18. For if N^ is the network with which the process terminates, then by 3.21 N„ can be realized with 5(Nm) reactive elements. Reading back through the construction, each increment of degree that is encountered is balanced by an equal increment in the total number of reactive elements, so that, finally, 5(N) is the total number of reactive elements used. That no construction using fewer reactive elements can succeed will be shown in Section 6. We now turn to IP, ID, Res, and IB, omitting the dual considera- tions. In each case, notation is simplified by writing Z, Y, N, n respec- tively for Zr , Yr ,'Nr , rir , and Zi , Fi , Ni , ni for Zr+i , Fr+i , N^+i , 4.1 Given a 2n-pole N in any category for which / = 3, its impedance matrLx Z(p) exists by hypothesis and has poles on p = tco. These can be removed successively by 2.05 and 2.06, giving Zip) = pR„-\--Ro-i-J2 -2-^2 Rk + Z,{p). (1) p fc=i p -r (^k In this expansion, either of i?o , ^oo may of course be absent, and all the Rk are real, symmetric, constant and semidefinite, for k = 0, 1, • • • , K, 00 . Furthermore, Zi{p) is PR and has no poles on p = ioj, by 2.05, 2.06 and construction. Let Ni be the 2wi-pole whose impedance matrLx is Zi(p). We define IP to be the operation giving Ni from N. Either Ni e C, or / = 1 or 2 554 THE BELL SYSTEM TECHXICAL JOURNAL, MAY 1952 for Ni , since at least Zi(p) exists. Furthermore, by construction Zi(p) is again an n X n matrix, so rii = n. By 2.14 and 2.15, K 8{Z) = rank {RJ + rank (7^0) + 2 E rank (R,) + 5(Zi). (2) k=l Since 5(Z) is finite, this shows that K is finite. Indeed, 2K < 8{Z). Furthermore, 8{Z) > 8{Zi), because a matrix of rank zero is itself zero, and by hypothesis Z(p) has a pole on p = ioi. Therefore we have estab- lished the claims in the first line of the Table 4.04, and by a dual argu- ment those in the second line. We must yet show that if Ni is physically realizable, then N is. Each term in (1), save Zi(p), is the matrix of a physically realizable 2n-pole, by 3.1. There are at most K -{- 2 such terms. The series combination of their respective 2n-poles is therefore physically realizable and N results from the series connection of these and Ni (2.2). Therefore if Ni is real- izable, so is N. Fig. 3 shows the relation of N and Ni under IP, and the dual rela- tion under AP. Here we have shown n = 3. The boxes labelled 0, », • • , K are the devices corresponding to the poles at 0, °° , • • • , *cojc , the terminals on the extreme left are those of N, and Ni is on the right. 4.11 From (2), and (B) of 3.1, we see that the number of reactive elements used in the realization of the network between Ni and N is exactly 5(Z) - 5(Zi) = 5(N) - 5(Nx). Clearly the dual result holds for AP. This verifies 4.07 for IP and AP. 4.2 Consider a 2n-pole N in (1 -|- 2, 0). In particular, then, the imped- ance matrix Z{p) of N exists and is not constant, but Z{p) has no in- verse. Then 2.08 applies, and we have Z(p) = W'Z!{p)W, (1) where W is real, constant, and non-singular, and Zi (p) is a non-singular matrLx Zi(p) bordered by zeros. Let Ni be the 2ni-pole whose impedance matrix is Zi(p). We define ID as the operation which gives Ni from N. Now Hi < n, because Z{p) is singular and Zi(p) is not. Also, Zi(p) is not constant, because Z{p) is not, and 5(Zi) = 5(Z), by 2.17. Therefore ni 9^ 0, also Ni is not in C. Because Zi(p)"^ exists, Ni is in A = 1,2 or 3. Because Z(p) has no poles on p = z'w, neither has Zi(p), so Ni e (1 -f- FORMAL REALIZAIULITV THKOUV — II 2, 1 + 2 + 3). This verifies the statements on the third hue of the Table 4.04, and the fourth by duality. That N is pliysically realizable if Ni is, is the gist of (I, 8.11) and (I, 8.4). We prove it here by noting that Zf (p) is the matrix of a 2n-pole N2 which obtains by adjoining /( — tii > 0 pairs of shorted terminals to Ni . Then (1) shows that N obtains from N2 by the use off deal trans- formers (I, 9.1). Fig. 4 shows in schematic form the effects of the operation II) and AD. In each case, it is emphasized that Ni has a matrix dual to that of N. We have shown n = 5, th = 3. 4.21 No reactive elements are used in this construction, so 4.07 is satisfied. 4.3 Consider now a 2n-pole N in (1, 1). Then its impedance matrix Z(p) is finite for every p = ico, and not constant. Let R(p), Hp), respectively, be the real and imaginary parts of Z(p): 2R(p) = Zip) + Zip) = Zip) + Zip) = Zip) + Z*ip); 2ilip) = Zip) -W) = Zip) - Zip) = Zip) - Z*ip); Zip) = Rip) + Hip). 1 , 1 ! 0 1 J c 1 0 00 K STRUCTURE RESULTING FROM IP N, STRUCTURE RESULTING FROM AP Fig. 3 — Structure resuUing from IP, above and AF, below. 556 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Then R{p) = R*ip), I(p) = I*{p), and both are real and symmetric. If k is any vector, (Z(p)k, k) = (R(p)k, k) + i{I{p)k, k), and the seljf-ad joint property of R and / imply that each scalar product on the right is real. Therefore (1) Re{Z(p)k, k) = {R(p)k, k), Im{Z{p)k, k) = (I(p)k, k). We note that 2il(p) = Z(p) - Z*(p) = Z*(p) - Zip) = -2il(p) so that, in particular, /(tco) is an odd function of co. 4.31 Lemma: Let *S be a given real, constant, symmetric, and positive definite matrix. Then there exists a unique number a > 0 such that (i) The matrix R(icci) — aS is semidefinite for every a;, (ii) For some coo > 0, possibly + «> , Riiuo) — aS is singular. Proof: We first show how the number a would be calculated, and then reduce the claims of the lemma to a well-known and basic theorem in rz rz VI W Ni SCHEMATIC OF OPERATION ID W N, SCHEMATIC OF OPERATION AD Fig. 4 — Schematic of operation ID, left and AD, right. FORMAL REALIZABILTTY THEORY — II 557 the theory of qu:uhatic forms. Fix co and eoiisider the matrix /i'(/a)) - \S as a function of X. Its (letermiuaut, A,(X) = I R{ioo) - \S\, is an n degree polynomial in X with the following two properties: (a) The coefficient of X" in Au(K) is not zero and is indepencknit of co, (l3) The n roots of A„(X) = 0 (2) are real and positive. Now R(i(j:) is rational, hence continuous, and finite for all oj, including oj = °o, by the hypothesis that N is in (1, 1). By (a) above, therefore, each root of (2) is a continuous function of oj on the compact set — <» < CO < . Let a(co) denote the least root of (2). Then a(w) is again l)ounded and continuous for all co. There is, therefore, an coo where a(a)) takes its least value. This is the wo referred to in the lemma, and a = a(ajo). We see that this calculation requires solving an n degree polynomial equation containing a parameter (w), and then minimizing the least root b}' varying the parameter. Though some properties of R{ico) are available to assist in the process, and the choice of S is somewhat free to us, this is scarcely a feasible calculation in practice. Even w^hen one reduces the minimizing problem to finding the roots of a derivative, there remains a prodigious calculation in all but the simplest cases. Since by its definition i?(tco) = R( — io:), we may take coo > 0. The relation (1) above implies that {R{ico)k, /v) > 0 for all real co and all k e K, because Z(p) is PR. That is, R{i(xi) is semi- definite. The hypothesis that Z(p) has no zero of its real part R(j)) on p = ioi then implies that R{iui) is positive definite. All of (i), (ii), {a), and (/3) then follow from well-known properties of definite quadratic forms. They may, for example, all l)e deduced from Halmos , paragraphs 62, 63, and 74, by choosing a coordinate frame in which the operator corresponding to *S above is represented by the unit matrix. A more elegant reduction to the cited results of Halmos can also be constructed. 4.32 Lemma: Given N in (1, 1), we choose any real constant symmetric 558 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 and positive definite matrix aS and find the a described in 4.31. Then the matrix Zi(p) = Z(p) - aS is PR and has a zero of its real part at p = t'coo . Proof: Clearly Zi(p) is symmetric. By 2.09, then, Zi(p) is PR if the function ^i(p) = (Zi(p)A-, /c) = (Z(p)k, k) - a(Sk, k) (3) is PR for each A'. Clearly this function is rational and has no singularities in r^ . It suffices then to show that its real part is non-negative on p = to:. By (1) of 4.3 Re ^i(fco) = (R{io:)k, k) - a(Sk, k) and this is non-negative by (i) of 4.31. That Zi(p) has a zero of its real part at p = iwo is (ii) of 4.31. 4.33 Let Ni be the 2n-pole whose impedance matrix is the Zi(p) of 4.32. We define the operation Res as that which produces Ni from N. It is evident from (3) above that the poles of Zi{p) are exactly those of Z(p), hence 7 = 2 for Ni . Nothing can be said of the admittance matrix for Ni . 5(Zi) = 8{Z) by 2.14 and 2.15, and rii = n by construction. The claims in 4.04 are now established for Res, and dually for Con. The relation Zip) = Z,(p) + aS shows that N is a series combination of Ni and a device with the im- pedance matrix aS. Since a > 0, this latter is a realizable resistance network (3.1). Hence N is realizable if Ni is. 4.34 We observe that no reactive elements are used in the network between Ni and N (2.12, 3.12). This verifies 4.07 for Res and Con. 4.4 We now turn to the piece de resistance of the generalized Brune process, the operation IB and its dual. Consider a 2n-pole N in the category (2, 1 -f- 2) — i.e., its impedance matrix Z(p) exists, is not constant, is non-singular on p = iu, and has a zero of its real part at some p = iiioo . We have for some /c e K such that k 7^ 0, R(io}o)k = 0. (1) Here, R(p) is as defined in 4.3. 4.41 We now assert that we may assume that 0 < ojo , and icoo 5^ 0° in FORMAL REALIZAUILITY THEORY — II 559 (1). Certainly wo may take coo > 0, liocauso /?(zco) = R( — iw). Further- more, by (1), Z(/a;„)/.- = il(ic^,)k. (2) /(/oj), beiiij^ odd, aiul finite evcrywheic on p = /w, must vanisli at co = 0, and at ice = 0° . Ilenee if wu = 0 or /wu = oo , Z(iajo)/'" = 0 and Z(p) is singular on p = /co. This denies oin- hypothesis that N e (2, 1 + 2). 4.42 Let J be the set of all \e('tors k eK such that (1) holds: the luUl space of /^(/coo). Then cleail>' J is a linear manifold. Furthermoi-e, J is real, because, if (1) holds then R{m^)k = R(im)k = R{im)k = 0 = 0 and k also is in J. Relations (1) and (2) hold for all A' e J. 4.43 By its construction, I(iuo) is real and symmetric, but not necessarily definite. There does however exist a real diagonal matrix D and a real non-singular IT such that /(zcoo) = W'DW. Let D+ be the (diagonal) matrix obtained from D by replacing all negative elements of D by zero, and define Z)_ by D = D+ - D^. (3) Then D+ and Z)_ are real, symmetric, and non-negative semidefinite. Define A = o:oW'D+W, B = - W'D^W. ^^^ We have chosen coo > 0, so A and B are both real, symmetric and non- negative. Certainly therefore Z''\p) = Zip) + -A + pB (5) V is PR. Also Z^'\p) has an inverse, because Z{p) has one by hypothesis and 2.2 applies. 4.431 Let I' e V be such that for some /:i c K V = Aki and for some k-i e K V = Bki . Then ?; = 0. 560 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Proof: We may assume that the first r diagonal elements of D are the non-zero elements of Z)+ , the next s those of — 7)_ . By (4), {W'T'v = mD+Wh , {WT'v = - D_Wh . The first of these relations exhibits (W')~^v as an n-tuple with non-zero components at most among the first r, the second as an /i-tuple with non-zero components at most among the last ??- — r. Hence all com- ponents of {Wy^v are zero. Hence v itself is zero. 4.44 Define X(p) = -^ A - pB, ' (6) V and let Nx be the 2w-pole whose impedance matrix is X(p). Nx is not physically realizable, since it is made up of negative reactances. Let N^^^ be the 2n-pole whose impedance matrix is Z^'\p). Then by (5) N obtains from N*~^ and Nx by connecting them in series. We have the following relation holding on p = ^co, but only thereon since it is only there that X{p) is a pure imaginary: 7M - - A + coB CO Z^'^(tco) = i?(ico) + i In particular, at two , Z^'^tcoo) = R{io:,) + r[/(rcoo) - W'D+W -|- W'DJ[V] = R{icoo), by (3) and (4). Since J is the null space of R(i (7) p + too in parallel with a 2n-pole N^^^ which has an admittance matrLx, say Y''\p) = G(p) + Y''\p), (8) where Y^^\p) is finite at p = two . FORMAL REALIZABILITY THEORY — II 561 (2)/ 4.46 Multiplying (8) on cither side by Z"{v), -^.GZ''\p)+Y''\p)Z''\p) = 1 p + Wo (9) = -^,Z''\p)G + Z''\p)Y''\p). Here, to be strictly correct, we should write two separate equations, interpreting 1 as the identity operator in K for, here, the left ecjuality, and as the identity operator in V for the right equality. Multiplying (9) through b}^ p — iuo and letting /; — ^ iwo , we obtain GZ^'\iu:o) = 0 = Z^'\ic^o)G. Here, as in (9), we have condensed two dimensionally incompatible equalities. From this it follows that each of G and Z^^\ioio) has its range in the null space of the other. In particular, therefore, the range of G is contained in J. 4.47 Consider now a v such that Gv = 0. Then, bj^ (7) and (8), V ^ Z''\p)Y''\p)v ^ Z''\p)Y''\p)v so, at lojo > V = Z^'\ic^o)Y^'\io:o)v = Z^'\iiOo)k for some finite vector k = Y^^\io3o)v- Since Z^^^iuo) is finite, v 9^ 0 implies that k 9^ 0. Then, however, v lies in the range of Z^^\icoo). Combining this fact with the result of 4.46, we see that for Gv = 0 it is necessary and sufficient that v lie in the range of Z^"\iwo): the range of Z^^'(zwo) is exactly the null space of G. 4.48 Now in Halmos , par. 37, it is shown that for any dimensionless operator in an n-space the dimensionality of its range space (its rank) and the dimensionality of its null space (its nullity) add up to 71. A similar result and proof hold for operators between V and K. Let m be the dimensionality of J. Then n — w is the rank of Z^^\icoo), and there- fore the dimensionality of the range of Z^^^iwo), and by 4.47 the dimen- sionality of the null space of G. Hence, finally, rank (G) = n — (n — m) = m. By 4.46, therefore, J is exactly the range of G. 4.49 Now N^^\ whose admittance matrix is Y^^\p), might not be ex- 562 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 pected to have an impedance matrix. The following reasoning shows that it does have, however: Consider a y e V for which Y''^\p)v = 0. Then from the right side of (9), with (5), 2v 2 2p- V = 2 , 2 Z{p)Gv + ^- — 2 AGv + -r-r—2 BGv. (10) p + 0)0 p + Wo p + COo We have by hypothesis that Z(p) is finite on p — ico. Therefore w^e may calculate, by letting p -^ 0 in (10), that V = -2 AGv, COo and, by letting p -^ oo in (10), that V = 2BGv. These two equations exhibit v as an element in the range of A and also an element in the range of B. The only possible such y is y = 0, by 4.43. Therefore there is no non-zero v such that Y^^\p)v = 0. Then Z''^\p) = Y''^\p)~^ exists as a PR operator. 4.491 Let Up) =-H + pF (11) P be the matrix whose poles at p = 0 and p = 'x, are those of Z^ (p). That is, let Z''\p) = L(p) + Z''\p), (12) where Z^*^(p) is PR and finite at 0 and oo . Because Z^^\p) is PR, H and F are both real, symmetric, and semidefinite. Let N/, be the 2n-pole whose impedance matrix is L{p), and N^^' the 2/i-pole with matrix Z^(p). In fact, Nl is realizable. N^^^ is the series combination of N^ and N * , by (12). 4.5 Equations (5), (7), (8), and (12) above are statements about matrices in a particular coordinate frame — that frame appropriate to the given N. We can interpret them as operator relations by simple decree. We wish now to draw a circuit diagram illustrating these relations. To do so, Ave introduce a suitable new coordinate frame. Because G(p) is PR and of rank m, we know that a frame can be found in which the matrix for G(p) is an m X m non-singular matrix bordered by zeros (2.08, or (I, 16.8)). By (7) and the result of 4.48, we FORMAL RKALIZAHILITY THEORY — II oGS may take the first m current vectors, /v'l , h , • ■ • , k,^ , specifying this frame, to span J. It follows from the matrix form of G then that the corresponding dual vectors Vi , ■ ■ ■ , v,,, span the range of G — i.e., the null space of Z^"\iwo)- We shall adopt such a frame for the further discussion. Let Ki be the space spaiuied by k^+i , • • • , kn , and Vi that spaimed by i;„!+i , • • • , Vn , ill this frame. Then K = J © Ki (1) V = U © Vi , say, where U = J*, Vi = Kf [Cf. (I, 10.0)]. If M is the name of any given 2n-pole discussed in the paragraphs 4.4 to date, we let M denote the Cauer equivalent of M in this new frame. 4.51 Let N,; be the 2//?-pole whose matrix in the new frame is the in X w non-singular admittance matrix which, when bordered, gives the matrix of the operator G^V) = -rf-2 G. V + Old The 2n-pole whose matrix is (j(p) then obtains by adjoining n-m open circuits to No . The matrix of No operates from U to J and has an inverse. 4.52 Fig. 5 shows a diagram, which n = 5, m = 3, of the manner in which we now have N represented. The terminals on the extreme left are those of N. N is obtained from N by a transformer. The horizontal current paths cut the dotted section A-A at points which may be inter- preted as the terminals of N. Ideal transformers, as in Fig. 1 of I, can be introduced here as needed. Putting them in the diagram merely com- plicates the picture )nnection of Nx are on B-B. N^^\ again, is the parallel connection of a 2n-pole obtained from No by the adjunction of open circuits, and N^^\ The latter has its terminals on C-C. Again, N^^^ is the series connection of Ni, and N^^\ 4.53 Let M.^ be the device between A-A and D-D of Fig. 5. This device has n terminal pairs on A-A and n more on D-D. We may suppose that ideal transformers are attached at each terminal pair as in Fig. 1 of I, since including them in the construction of N Avould not alter its be- havior. Then M.^ is a 2(2w)-pole. M AD is constructed from certain 2r poles (with various r) as indicated in the diagram of Fig. 5. The ideal graph* of this diagram (rather, of *Cf. (I, 4.1). N is the series connection of Nx and N'^\ The terminals of the latter 564 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 the relevant part of it between A-A and D-D) obtains from Fig. 5 by inserting ideal branches — two poles — across each terminal pair of each box, and neglecting the outlines of the boxes. The upper m channels of this ideal graph are then T sections, and the lower n-ni are degenerate T sections with no shunt arm. This ideal graph is shown in Fig. 6. The ideal branches are shown as small boxes. The program of the next few paragraphs is to demonstrate that Mad is a physically realizable 2(2M)-pole. 4.54 Let us designate the terminal pairs of Mad on the section A-A by Ti , Ti , • • • ; Tn , Tn , where the r* pair is the intersection with A-A of the leads to the r^^ terminal pair of N. We designate the pairs on rz rz rz r A '^ B Fig. 5 — Original form for N C ^^ D ^'"' 1 1 1 1 1 1 • • 1 9 r^ ! J!^ T 1 1 ! 1 1 JL 1 - ■ 1 ' ( * -. 1 X 1 I 1 t i ' 1 ^ 1 1 1 ' ° 1 ' ' 0 1 \ 1 1 I 1 • V 1 o dj ! 1 ^. 1 1 1 1 A 1 -. - 1 ' J^ ! 1 X ? 1 1 1 1 1 1 1 B C Fig. 6 — Ideal graph of Mad FORMAL REALIZABILITY THEORY — II 565 D-D by »Si , *Si ; • • • ; Sn , S„ , where here the r^ pair is the intersection Avith D-D of the leads to the r^'' terminal pair of N^*\ In each case we orient the pair T, T' or S, S' so that the primed (negative) terminal is on the lead to the primed terminal of N or N^*'. Let the 2«-tiiplc [tti , a2 , • ■ ■ , ttn , bi , bo , ■ ■ ■ , bn] (2) represent the currents into the terminals of Mad in the order Tl , To , • ■ • , Tn , Si , • • ■ , On • Then we may interpret [«!,•••, dn] (3) as a vector in K expressed in the coordinate frame introduced for Fig. 5, and also [&!,•••, bn] (4) as a vector in K in the same frame. That is, any current vector into Mad can be written as an ordered pair /h , k2 (5) where each k, e K, with the convention that such a pair determines a 2«--tuple (2) from the ?i-tuples (3) of fci and (4) of ki . We shall WTite the ordered pair (5) in the form A-i e ^-2 . (6) Because we have K represented in the special way K = J e Ki , where J is the subspace spanned by n-tuples (3) in which the last n-m components vanish (this is (1) of 4.5) we can further split the 2n-tuple (2) into (ii e A) © (J2 @ ^2), (7) where ji ej, (i eKi , i = 1, 2, and in (6) A-.- = ji @ ii . (8) Formulas dual to those of (2) through (8) of course hold for voltage (2rt)-tuples. Let K^ be the space of current 2n-tuples (2) (or (7)) and V^ the space of voltage (2n)-tuples [ei , 62 ,'••, t'„ , /i ,•••,/„] = (ui © ^i) © (W2 © ^2) 566 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 analogous to (2) and (7), mth the scalar product n n JLerttr+Hfrhr. (9) r=l r=»l It is a common and convenient malpractice in vector algebra to use, for example, the symbol j both for an m-tuple in J and for the ?i-tuple i 0 0 e J e Ki of the form (8). Taking this advantage, we can see that (9) is simply (wi + vi , ii + A) + {U2 + v-2 , J2 4- ^2) (9') where here the parentheses denote scalar products between V and K. The form (9') can also be derived directly from (1), (7), and (I, 10.6). 4.55 We now wish to compute the voltage-current pairs admitted by M.vD • Referring to Fig. 5, we observe that Nx and Nl both have im- pedance matrices (X(p) and L(p) respectively, or, rather, the matrix forms of these in the frame of present interest) finite at all p except p = 0, p = 0° . Each will, therefore, admit any current n-tuple into its terminals, i.e., through its ideal branches, at any but these exceptional frequencies. By construction, Ng has a non-singular admittance matrix and therefore also will admit any current m-tuple into its terminals (2.07), except at most at certain isolated frequencies. It is evident by Kirchoff 's laws applied to Fig. 6 then that M ad will admit an}^ current 2n-tuple of the form (ii e /:) © 0-2 © i-k)) (10) where ji e], i = 1,2, and fc e Ki , except at most at finitely many ex- ceptional frequencies. Conversely, if the current 2/i-tuple specified by (7) is that in Mad , conservation at the absent shunt arms of the lower degenerate T-sections implies that, as elements of K, Ai + h = 0, that is, the current is of the form (10). Hence 2n-tuples of the form (10) span the space of currents admitted by Mad • Let us call this space Km . It is a proper subspace of K" unless ?n = n. 4.56 Let G~^(p) denote the m X m impedance matrix of Ng . Then by (7) of 4.4, interpreted as an operator equation, G-(p)=(i. + !)«-■ (H) where G~^ is a real, constant, symmetric, non-singular m X m matrix. FORMAL REALIZAlilLITV THKOltV — II 567 We can now compute tlie voltage across Mad eorrespoiulinp; to the ciirreut (10). Let w be tli(^ /(-tuple of voltages appeai'iuf;- at the section B-B or C-C' of Fig. (i, with its coiupoiieiits listed in the a|)pro|)riate order. Then we may iiitei|)i'et ir as a xcctor in V, and write it IV = ;/„ © To (12) where »o c U, ro e Vi . Now by KirchofV's current law applied to the slumt arms in the upper channels of Fig. (>, the current into No is ii + h , and therefore «o = G-\p)(j,+j-;). (13) B}^ Kirchoff's voltage law applied to a typical mesh which begins on A-A, goes through N.v to B-B, and then through a shunt arm and i-eturns to A-A, the voltage /(-tuple appearing at A-A is X(p)(ii + k) -f w. Referring to (12), let us use Vo also to denote the vector ^/o © 0 e V, and I'o to denote 0 © Vo € V. Interpreting (13) in this way we get X{p)(j, + k) + G-'(p)in + J2) + Vo (14) as the voltage n-tuple on A-A. A similar calculation gives L{pKJ2 - k) + G-\p)(jr -f i,) + Vo (15) as the voltage n-tuple on D-D. The ordered pair (14), (15) then gives the voltage 2M-tuple corresponding to (10), in the notation analogous to (5). 4.57 X(p), L(p), and G~\p), respectively, are defined in (6) of 4.44, (11) of 4.491, and (11) of 4.5G. Each one is finite except at p = 0 and p = CO , Let Fm be the complex plane from which these two points are deleted. It is now possible to show that the linear correspondence whose pairs, for each p e Fm , are the voltages (14), (15) e V and the ciu'rents (10) € K', satisfies PI through P7 of (/, 6, 7)— that is, is PR (I, 1G.71). 568 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 In the present special circumstances it is almost as easy to study Mad in a slightly different way than this. Since fewer direct references to I are involved, we shall take the alternative path. We first calculate the scalar product between the voltage (14), (15) and an arbitrary current of the form (10), say the current Qh ® () @ (ho ® (-{)) eKli. To do so, we consider the form (9') for such a product. In the first writing, then, this scalar product is (XipXJx + k) + G-\p)(h + h) + vo , h + () + {L(p)(j, - k) + G~\p)(j, + j,) + Vo , h, - ^). Each of these scalar products has three voltages appearing in it. Dis- tributing the products over these voltages, and using the facts that the range of G~^(p) is J and that I'd e Vi = (J)° we get a second form: (X(p)(ii + /.■), Ih + 0 + {G-\p)Ui + J2), h,) + (%, () + (L(p)(j, - k), h, - i) + iG^\pM + Jd, Ih) + (.0, - ^). The terms involving Vo go out and we can collect to (X(p)(:/\ + fc), ih + () + {G-\p){j, + i2), h, + /!,) + (L(p)(j, - fc), h, - I). This is the desired scalar product. 4.58 Let us now consider the {n + m) -tuples [oi , 02 , • • • , a„ , &i , • • • , hr,] = ji @ k @ J2 (17) obtained from (2) by deleting the hm+x , • • • , b« . We still interpret these as currents into the relevant terminals of Mad • We also observe that when the current (17) is given, (2) can be determined, because by (10) a„,+s + hm+s = 0, s = 1, 2, • • • , n - w. Given (17), and therefore (2) or (10), we can determine the voltages (14) and (15), where Vo is an arbitrary element of Vi . Let us agree now always so to choose Vo that the component of (15) in the subspace Vi vanishes. This means that, in (17), we have specified arbitrarily the cur- rents into the left-hand terminals of M ad (on A-A) and into the upper m of the right-hand terminals. We have also agreed that the voltages across the lower n-m terminals on D-D shall be zero, so that (15) is an FORMAL REALI7ABILITY THEORY — II oG9 n-tuple of the form w e 0 (18) where u e U. Regarding (15), Avith this determination of Vo , as simply an m-tuple m (ignoring its last n - m zero components), we see that (17) and the ordered pair (14), (15) are now currents and voltages in a 2(n + m)-pole Mad obtained from Mad by shorting and thereafter ignor- ing the lower n - m terminals on D-D. ■4.59 Now (17) is unrestricted. Given it, the corresponding voltages can be computed from (14) and (15) by determining /'n so that (15) lies in U. Hence Mad has an impedance matrix, since any single valued linear mapping from (17) to voltages can be described by a matrix. Oiu- job is now to show that this matrix comes under 3.1. Before doing this, however, we shall point out that a realization of Mad provides one for Mad . Fig. 7 shows how a 2(2n)-pole equivalent to Mad would be con- structed from Mad • The equivalence is evident almost at once: The pairs of Mad are the currents (17) and the voltages (14) and (15) with a special determination of v^ , where (15) is regarded as an m-tuple. The current (10) is clearly that which flows in the 2(2n,)-pole of Fig. 7 when (17) flows in Mad • Furthermore, regarding (15) as an «-tuple of the form (18), we see that the voltages in Fig. 7 can be obtained from (14), (15) b}'^ adding an arbitrary voltage of the form (0 e v) e (0 e v), where v e Vi of course. This arbitrarj'' added voltage eliminates the special role played by Vq in (14) and (15). Hence therein Vo itself may be considered to be an arbitrary element of Vi , and (14), (15) represent the \oltages in Fig. 7. The pairs admitted by the 2(27?)-pole of Fig. 7 are therefore exactly those admitted by Mad , Q.E.D. 4.60 We have now established that Mad bas an impedance matrLx, say M{p). M{p) operates from an (n + w) space of currents (17) of 4.58 to an (n + ?n) space of voltages (14), (15) of 4.56, where in (15) we prop- erly choose Vo so that the last (n — m) components are zero and can he ignored. Now any impedance matrix Z(p) is completely determined when we know for each two currents m\ and m^ the scalar product {Z(p)m, , m.-) (1) (Cf. Halmos'', par. 53). We shall make this computation for M(p). The 570 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 currents (17) of 4.58 may l)e regarded as elements of the subspace (10) of 4.55. We have called this subspace Km • The voltages (14), (15), with vo chosen to make (15) an n-tuple of the form (18) (4.58), are elements of a subspace Vji of V . It is evident at once that the scalar product between a current (n + m)- tuple (17) and the (n + m)-tuple (14), (15) (vo properly chosen!) is exactly the same as the scalar product between the current (2n)-tuple (10) and the (27i)-tuple formed from the (n + w)-tuple (14), (15) by adjoining (n — m) zeros to expand (15) to an n-tuple of the form (18). n di n [II n ai o M- Mad Fig. 7 = Construction of Mad from Mad*- The solid terminals are those of Mad*, the open circles those of Mad • Now we know that we may regard (15) as an n-tuple of the form (18) by a suitable choice of Vo . But we calculated in 4.57 the scalar product between an arbitrary (2n)-tuple and (14), (15) with an arbitrary vq . The answer was (16) of 4.57. By proper choice of Vo , then, (16) represents the bilinear form (1) above for il/(p). Since (16) is independent* of Vo , it contains in itself the whole of the properties of M{p). 4.61 To show that M{p) is PR, we need show only that M(p) is sym- metric and that is quadratic form (j, = hi and k = ^in (16)) is a PR function of p (2.09). . By their definitions, X(p), L(p), and G~ (p) are all symmetric. Hence if all currents are real, the value of (16) is unchanged by interchanging ji with hi , i = 1, 2, and k with /. Therefore M{p) is symmetric. 4.62 Henceforth we consider the quadratic from * This is the gist of P3 of (I, 7.4). Use of the results of I here would have given a more direct but much less constructive representation of Mad • FORMAL REALIZABILITY THEORY — II 571 (X(p)0\ + ^0,^1 + k)-h (G-'(p)0\ + i.),ii + J2) 2) + {L(v)(j-2-k),j,-k) obtained from (IG). By the definitions of X(p), L(p), and G~ (p) this is a rational function taking real values for real p. Hence we need only show of (2) that its real part is non-negative when Re(p) > 0 to show that it and M(p) are Pll. Referring to (G) and (11) of paragraph 4.4 and (11) of 4.5G for the definitions, we see that (2) can be written - \- Uijx + k),j, + /.•) + "^ iG'\j, + J2), Jl + J2) VL 2 + mj,-k),j2-k)] (3) + P [- (Bij^ + /v), Jl -\-l^)+l (G-'Ul + J2),ji + J2) + {FU2- /v),i2-fc)j. That is, the quadratic form in question has poles, simple ones, only at 0 and 00 , and has no constant term. If we can show that the residues at these poles are non-negative, then it will follow not only that M(p) is PR but that M{p) is of the form - Mo + pM„ V where each of these summands is realizable by 3.1. Unfortunately, there still remains some computation to verify that the residues of (3) are non-negative. 4.62 We first recapitulate some relations obtained earlier; r(2)/ N ^/ N . 1 Z''\p) = Z{p) + -A + pB; (4) V this is (5) of 4.42. '(2)/ \ 2p (3) this is (7) and (8) of 4.45. Z''\p) = -H+pF-^Z''\p)', (6) V this is (11) and (12) of 4.491. 572 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 By their definitions, for i = 2, 3. By hypothesis, Z{p) and Zip)-' = Y(p) are both finite everywhere on p = fco. By its construction, Z''^\p) is finite at p = 0 and p = ^o , 4.63 We claim now that each Y'''\p) is finite at p = 0 and oo , ^ = 2, 3. Proof: We need consider only F^"^(p) since F'^'(p) differs from it by something which vanishes at p = 0 and p = <» ((5) above). Let Y''\p) = Y(p) + ^-E + pQ P where f (p) is finite at p = 0 and p = <» . Since F^'^p) is PR (4.43), E and Q are real and symmetric. Using the form (4) above for Z^^\p), 1 = Z''\p)Y''\p) = Z{p)np) + BE+AQ + p(Z{p)Q + BYip)) + p'BQ (7) + -{Z(p)E + Anp)) + \AE. P P" Multiplying through by p, p, -^ , - and taking limits as p — » 0, 0, <» , <» , p^ p respectively, we obtain AE = ^ z(o)je; + Afco) = 0, (8) BQ = 0, Z(o.)Q + 5f(oo) = 0. We can also write a formula like (7) with the factors in reverse order, and obtain the analogous forms to (8) in which the factors are com- muted. Let us call these commuted relations (8'). Multiply the second relation (8) on the left by E and use the first relation of (8'). We obtain EZ{^)E = 0. (9) Working similarly with the last two relations in (8) and (8'), we get QZ(oo)Q = 0. (10) FORMAL REALIZABILITV rilEORY — II 573 Now let V be an arbitrary voltage in V and let lu = Z(0)Ev. Then w e V, and by (9) the current Ew = 0 for any v. Hence 0 = (v, Ew) = (w, E*v) = (w, E*v) = {Z{0)Ev, E'v) (11) by (I, 7.2, 14.0). Now E is real and symmetric, as noted above. Hence E ^ E — E' . Furthermore, Z(0) is real, so (11) becomes {Z{Q)Eu, Eu) = 0 (12) where m = v is an.y element of V. Now Z(p) is non-singular on p = loj, and its real part is semidefinite there. At p = 0, Z(0) is its own real part, hence semidefinite and non-singular, hence definite. Then (12) implies that En = 0. This being true for all u e VJE = 0. The proof that Q = 0 follows similarly from (10). 4.G4 With Y^' (p) and F^^^(p) simplified at p = 0 and oo^ we can go back and compute 1 = Z"\p)Y'"(p) = (z(p) + iA + ,ij)(^G+r»(p)). ^''^ Of the six terms obtained on expanding this exactly one, namely ^-AY''\p) V is not ob^•iously finite at p = 0, and another, vBY''\v) is not a priori finite at p = oo . We conclude by multiplying through by p and letting p ^ 0, and dually at p = oo ^ that AY^'\0) = 0 = F'''(0)A (14) BY^'\o.) = 0 = y^'^(oo)/?, where the commuted form can be established bj^ a new calculation from 1 = Y'(p)Z~(p), or by taking transposes. 574 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 In a similar way, we compute from 1 = Z''\v)y'\v) = Z''\v)Y''\p) -^^- HY''\v) + vFY''\v) (15) V that HY^^\0) = 0 = Y'-^\Q)H, m ,-1^ (16) FY^'\oo) = 0 = Y^'\oo)F. Now Y^^^ (p) is finite at 0 and =0 , so we may expand it in a power series about either point. Let these be Y''\p) = F"^(0) + pYi'^O) + Oip\ Y^'\p) = Y''\o.) + 1 rl^)(oo) + 0 (i) ^^'^ p \py. Putting the appropriate one of these into (13) and taking a limit at 0 or CO Ave get, by using (14), that 1 1 = -o AG + AY'riO) + Z{0)Y''\0), 1 = 2BG + BYi'\^) + Z(oo)F^'\oo) A relation (18') ^^^th factors commuted is also true. We may also put (17) into (15) and get 1 = Z^'\0)Y^'\0) + HYi'\0), 1 = Z^'\o.)Y^'\o.) + FY['\o.), (18) (19) and also a commuted form (19')- Right multiply the first line of (19) by A and the second by B, and use (14). This gives A = HYi'\0)A, (20) B = FY['\oo)B. Left multiply the first line of (18') by H and the second by F. This gives, by (16), // = -2 HGA + HY['\0)A, 0)0 (21) F = 2FGB + FFl'^(oo)5. Using (20) in (21), we have the relations FORMAL REALIZABILITY THEORY — II i)iO ". HGA = H - A, ^0 (22) 2FGB = F - B. These are fundamental to the evaluation of the residues of (3). Before calculating these residues, we draw a further important conclusion from the formulas just developed. Relation (20) exhibits .1 as a product of II and a possibly singular matrix (viz., Fi''(0).l). Hence rank (.1) < rank {H). But relation (21) shows // as a product of .1 b}" 4 HG + //IT'(O). Wo Hence rank (//) < rank (A). That is, rank (.4) = rank {H), (23) rank (B) = rank (F), the latter being established in the same way. 4.65 The formulas developed in 4.64 are all quite symmetric as between relations obtained at p = °° and those at p = 0. We shall now con- tinue to the evaluation of the residue of (3) at p = oo . The evaluation at p = 0 proceeds in an exactly similar manner. The residue in question is, from (3), - {B{J, + A-), h + A-) + hiG~\h + J2), ii + j^ ^ ^ 24 Here ;i and j-i are any elements of J and k any element of Ki . The range of G is J and the operator G~^ operates from J to U = J*, representing the inverse to the operation G from U to J. Let us define h and eliminate j-i by the relation j, = -III + 2GB(j, + k) - J, . (25) Since the range of (i is J, h e J. The definition analogous to (25) for the other pole of (3) is i2 = 4/1+ 4g.4(Ji+ a-) -,/i. COo OJo 576 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 We shall now say no more about this pole. Putting (25) into (24) we get at once the form -{BU, + k),j, + /,•) + {G-'h + G'GBij, + k), 2h + 2GB(j, + k)) + (2Fh + 2FGB(j, + A-) - Fj, - Fk, 2h + 2GB(j, + k) - ./i - k). Here we cannot at once put G'"'G' = 1, because this is only true in U. We expand in the following way: The first product is left intact, the second is expanded by distributivity into four terms, and in the third we use (22) and expand into five terms by distributivity. The ten re- sulting terms are: - (5(ii + /.■),7i + /0 + 2{G-%h) + 2(G-'GB(jy + k), h) 4- 2(G-'h, GB(j, + A-)) + 2{G-'GB{j, + A-), GB{j, + A-)) + 4(F/i, h) - 2{Bi_n + /■■), h) + 2(F/i, 2GB{j, + A-) - h - /'■) - 2(5(ii + /.•), GB{j, + A-)) + (5(ji + A-), Ji + A-). Enumerate these terms 1,2, • • • , 10 in the order written. We shall show by combining that only 2 and (3 remain. Clearly 1 and 10 cancel. Consider the operator G~^G as we have defined it. If v e V, we can put V = U -{- Vi where u e U, wi e Vi . Then Gv = Gu + Gvi = Gu, because of the matrix form for G in the coordinate system chosen in 4.5. By definition of G~^ (in 4.56), since u e U, G~'Gu = u. Hence, combining the last three relations, G ''Gv = V - vi (26) for any v e V, where i'l is a suitable element of Vi (depending on v of course) . Using (26) in term 3, we get for this term 2(5(ji + A-), h) - 2(ri, h) FORMAT. HIOALIZAHILITY TIIKOIJV II .)/ / for some Vi e Vi . liut /i e J = (Vi)" ((I) of 4.5). Heuco the kocoiuI term here \aiiishes and term 3 cancels term 7. By an exactly similar arj^u- inciit, since GB(J\ + />') e J, we lind that lei'in ."> cancels term <). Consider tei'in I, and wiite it in the form •2{a 'A, /,-,) = 2(((7-')*/m , h) = 2{(a ■)*A^, ,A) =2(r; 'I-^j,). This follows hy (i, 7. '2, 14.0) and the fact that (/ ' is symmetric. Fnt- ting in the definition of /,i , and using the fact that G and B are real, we get 2((r% , h) = 2{G-'GB{J^ + k). Ti) = 2{G^'GBin + k), h). Xow J is ival (4.42) so /i e J. Therefore the reasoning used on term 3 yi(4ds finally 2(7^0, + k), h) as the value of term 4. We now write term 8 as 2{Fh, k.) and transform it to 2(Fh,h), !)>■ the reasoning just used on 4. Putting in what k-^ is, this is 2(2FGB0, ^k) - Fj, - Fk, h). Using the reality of G and B, and (22), this is - 2(B0i + k), Ti). This cancels term 4 and all terms save 2 and (> are accounted for. Fi- nally, then, the residue of (3) at p = » is 2(G-'li, h) + MFh, h). {:21) Since G~ is definite in J and F is semidefinite, this residue is non-neg- ative, and indeed not zero if /i ?^ 0 and /i e J. 4.7 We have established the non-negativity of the residue of (3) at /J = CO. A similar argument (exactly parallel, in fact) wdll establish the same for the residue at p = 0. Each term in the representation Miv) = - J/o + pM^ V 578 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 .* of 4.01 is tlieu realizable by 3.1. Hence Mal> is a realizable reactance 2(n + ?/?)-pole, and so therefore is Mad , as we noted in discussing Figure 7 (4.59). Therefore, if N^^' of Figure 5 is physically realizable, so also is N and therefore N. We denote by Ni the N^^' obtained in this way from N, and define IB as the operation which constructs Ni from N. We must still establish the claims made in 4.04 for IB. No properties of N^^* = Ni have been proved beyond the existence of its impedance matrix, Z*^*(p), but this is all that is claimed in the third column of 4.04. The fifth column is also established. We must now however com- pare the degree of Ni , i.e., of Z^^'(p), with that of Z(p). By 2.13, 2.14 and 2.15 applied to (4), (5), and (0) of 4.G2, 5(Z^'') = 5(Z) + rank (A) + rank (B), d{Z^'^) = 8{Y^'') = 5(F''') + 2 rank (G), 5(F"^) - 6(Z''") = 5(Z*") + rank (H) + rank (F). We know m = rank (G) > 1. Let r = rank (A) + rank (B). Then from (23), and the relations above in order, 5(Z) = 6(Z^") - r = (5(Z^") + 2m) - r = {8(Z^'^) + r) + 2m - r = 8(Z^*^) + 2m. Hence 8{Z) - 5(Z''') = 6(N) - 5(Ni) = 2m > 0. The claims of 4.04 are then established. 4.71 We must yet verify 4.07 for IB. Let 8{M) be the degree of M{p) = i Mo + pM^ . V Then by 3.21, Mad , whose matrix is M{p), can be realized with 8(M) reactive elements. By Figure 7, then Mad can be so realized, and it follows that exactly 8(M) reactive elements are comprised between N and Ni under IB. Now by 2.14 and 2.15, 8(M) = rank (Mo) + rank {MJ. We shall compute the second term. The first is obtained by an exactly parallel calculation. FORMAL REALIZABILITY THEORY — II 579 Using the fact that M{p) is determined by its quadratic form, we see that M^ is the matrix whose form is the residue of that of M{p) at p = CO , This residue was computed in (27) of 4.65 to be 2{G-'h, h) + 4{Fh, h) (1) when the current vector, (17) of 4.58, is ii e A; © J2 , (2) and, (25) of 4.65, 2h =j,-{-j\- 2GB(jr + k). (3) Here ji , ^2 e J and k eKi . Now M„ is an (w + m) X (« + m) matrLx by construction. Then V = n -\- m — rank (M„) (4) is its nuUity, the dimension of its null space. This is proved in Halmos^, par. 37, for dimensionless operators, and a similar proof applies to im- l)edance operators. Now for any symmetric and semidefinite impedance operator Z, the null space of Z is exactly the aggregate of currents k such that the quadratic form {Zk, k) = 0. This may be seen at once by choosing a coordinate frame in which the matrix of Z is diagonal. Since we know from 4.65 that AI^ is symmetric and semidefinite, we can compute v as the dimensionality of the space of vectors (2) above for which (1) vanishes. As noted in 4.65, he], and (1) vanishes if and only if /i = 0, because G~ , as an operator from J to U, is definite (semidefinite and non-singu- lar). Hence v is the maximum number of linearly independent vectors (2) for which, from (3), (1 - 2GB)j\ + j, - 2GBk = 0. (5) The left member of (5) is a vector in J depending linearly and homo- geneously on the vector (2). Hence, regarding J as a subspace of the space J © Ki © J in which (2) lies, the left member of (5) is the value in J © Ki © J of a certain linear operation applied to the vector (2), Let us cull this operator P. The numl)er v, by definition the number of linearly independent vectors (2) for which (5) holds, is the nullity of P. The dimension of P is ?i + m, and its rank is clearly ni because the left member of (5) — a typical element in the range of P — lies in J and by 580 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 suitable choice of j-> can be made to be any element of J. Hence the nullity of P is (n + m) — m = n (Halmos^, par. 37). That is V = n, and, by (4) rank (M^) = m. A parallel argument Avill establish the same result for J/o . Hence 8(M) = 2m = 5(N) - 5(Ni) by a result of 4.7. Therefore Mad and Mad can be reahzed with 6(N) - 5(N,) reactive elements and 4.07 holds for IB. V. THE DEGREE OF A RATIONAL MATRIX 5.0 In this .section we consider arbitraiy n X n matrices Z(p) whose elements are rational functions of the complex variable p. They are treated, generally, as arrays of functions with certain rules for addition, multiplication, and reciprocation, wdthout geometric interpretation. A geometric development is possible, but would be cumbrous. Related ideas may be found, geometrically developed, in Appendix I of Halmos". This section deals w^holly Avith concepts well known in the algebraic theory of matrices over an arbitrary field — in this case the field of rational functions. I have not found, however, any place where the particular developments which seem to be needed here are made suffi- ciently explicitly for reference. Accordingly, the presentation here is somewhat detailed. The particular path of argument followed is only one of many possible; it was chosen to lead easily to results needed in Section 6, and to parallel generally the rest of the paper. This section could be abbreviated somewhat if one restricted himself to PR matrices Z(p). We prefer not to limit the applicability of these results, however, since they may well be useful in non-passive realiza- bility theoiy. 5.01 Definition: If R{p) is a rational function of the form Rip) = (p - p,y"Ri(p), where Ri{p) is finite and not zero at po , and w may be of an}^ sign, we call m the exponent of (p — po) in R(p). The number FORMAL REALlZAHlLirV THEORY II 08 1 r = sup {—m, 0) is called the order of (he polo of R{p) at p^ , even if r = 0. 5.1 Let Z{p) be au n X n matrix wiiose elements Zrs(p) are rational functions of the complex variable />. We write Nrsip) Zrsip) = DrAp)' wiiere .Vrs and Dr.. are n^latixcly piime polynomials. Let ""^/.(p) b(> the hvist common multiple of all Drs{p), (1 < '", s < n), so normalized that tiie coefficient of the highest power of p appearing in ^z(p), (the leading coefficient) is unity. Then ^z(p) is uniquely determined by Z{p). The matrix ^z(p)Z(p) has polynomial elements. Its Smith normal form is a diagonal matrix E(p), E{p) = 'Ey{p) 0 0 Eiiv) 0" Eniv) •0 = A{p)-^,{p)Z{p)B{p), (1) with the following properties: (a) R is the rank of ^z(p)Z(p). (b) Each Ei(p), I < i < R, is a polynomial with unit leading coef- ficient. (c) Each Ei{p) is a factor of Ei+i(p), 1 < i < R — I. (d) A(p) and B(p) are polynomial matrices, each with a constant non- vanishing determinant. (e) Ei(p)E2{p) • • • Ek(p) is the normalized (and therefore unique) highest common factor of all /v-rowed minor determinants of ^z{p)Z(p). These properties of E{p) are developed for example, in Bocher^^, Theorems 2 and 3 of paragraph 91 and Theorem 1 of paragraph 94. A simple variation of this last cited theorem will also pro\'e the following uniqueness lemma. .■3.11 Lemma: If some E (p) satisfies (1) and (a), (b), (c) and (d) above, all written with superscripts on each E, and on A and B, then E^{p) = E{p). Proof: E (p) is equivalent to E(p) in the sense of paragraph 94 of 582 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Bocher^ , for E\v) = A\v)A-\v)E{v)B-\v)B\v). Therefore it is also equivalent in the sense of par. 91 of Bocher \ (for this is Theorem 1 of paragraph 94). Hence the normalized greatest common factor of all fc-rowed minors of E^{p) is the same as that of £'(p), that is, Eiip) • ■ • -E'tCp). But the greatest common factor of all k rowed minors of E^i-p) is E?(p) • • • El{p), because of property (c). In particular then £i(p) = £'S(p), and consequently Ek{v) = ^Jt(p) by induction for I < k < R. Q.E.D. 5.12 Definition: The normal form TF(p) of Z(p) is the matrix '^'^\p)E{p). We write the elements of TF(p) in their lowest terms, W{p) = A{p)Z(p)B(p) = ~ei(p) ^i(p) 0 0 • • • 0 -^iip) 0. ■•0 (2) with the polynomials ek{p), ^k(p) each having unit leading coefficients. 5.13 Theorem: The normal form W(p) of Z{p), as given by (2), has the properties (a'), (b'), (c'), (d'), and (e') listed below. Further- more, any TF°(p), given by (2) wdth superscripts on W, A, B, Ck , and ^;fe(l < k < R), which satisfies (a'), (b'), (c'); and (d') with correspond- ing superscripts, is in fact W{p). (a') R is the rank of Z(p) (b') For each k, 1 < k < R, ek{p) and ^k{p) are relatively prime polynomials with unit leading coefficients, (c') Each ek{p) is a factor of ek+i{p), I < k < R — 1, and each ^i(p) is a factor of ^j-i(p), 2 < j < R. (d') A{p) and B{p) are polynomial matrices each with a constant non-vanishing determinant (eO ^i(p) = ^z(p). Proof: (a') and (d') follow immediately from (a) and (d) of 5.1. (b') is a matter of definition, (c') follows from (c) of 5.1 and the definition, 5.12, since the effect of cancelling common factors in each fraction of the sequence E,{p) E,{p) EAp) ■^z(p)''^z{py '" '^z(p) FORMAL KEALIZAIULITY THEORY II 583 cannot remo\'e from any Ek(p) a factor which was present in carUer Ej{p)(j < A) but was not cancelled therefrom (treat each linear factor of ^z and of Ki as distinct, and each linear factor of ^,^ as distinct Ek(p) to see this easily). Property (e') is best proved by a reductio ad absurdum. We recall that Ei(p) is the highest common factor of all elements of '^yXp)^(p)- Suppose now that Ei{p) contained a factor

). Hence no denominator contains ^ as a factor, but this denies its presence in their least common multiple, ^z(p). The uniqueness of W{p) follows at once from the uniqueness lemma, 0.11. Multiply (2) by ^z(p). Then ^z(p)W\p) = A\p)^z(p)Z(p)B\p) (3) lias diagonal elements of the form '^z(p)ekip) ^kip) ' 1 < k < R. (4) But by (3) and (d'), these are the result of polynomial operations on the polynomial matrix ^z(p)Z(p). Hence the elements (4) are poly- nomials, and each has unit leading coefficient. ■^z(p)TF°(p) then clearly satisfies (a), (b), (c), and (d) of 5.1. Therefore by 5.11, '^z(p)W\p) = Eip) = ^z{p)W(p). Therefore W\p) = W(p). Q.E.D. 5.14 Corollary: W{p) is its own normal form. 5.15 Corollary: Let (p{p) be a rational function and Z^{p) = ) = jk+i , and 71 = r. We write the 7a- in an ordered array S{Z, Po) = [71 , 72 , • • • , 7«]- FORMAL KKALIZ AlilLirV I'llKoKY 11 585 5.25 Dcfniilion: roiisidci- fwo nintri('(>s Z(/)) mid Zi(p), with .S{Z, /h) = [71 , 7.. , ■ • • , 7„|, N(Z, , •/>„) = [71 , 72 , • • • , tVI- V^'v say S(Z, po) > SiZ, , p,) (2) if and only if Ti + 72 + • • • + 7a- > 7i + 72 + • ■ • + 7I for ('\-(My /,■ = 1,2, • • • ,11. We say S{Z, p,) = S(Z, , /;o) (3) if 7/c = Ih for A- = 1, 2, • • • , /). It is easy to see that (3) is equi\^alent to the simul- taneous validity of (2) and the reverse inequality. 5.2G Theorem: Let pn ^ oo be a pole of Z(p). Let F(p) be a rational n X // matrix which is finite at /Ai • Tlien SiZ, po) > SiFZ, po). In paiiiculai', if F(p) is also non-singular at po , then S{Z, Po) = S(FZ, Po). Proof: Let \Pf(p) and rpzip) be the least common denominators of the elements of F{p) and Z(p), respectively. Then the exponent of {p — Po) in ypz(p) is r, while in 4^f(p) it is zero by 5.23. Let —Ck be the exponent of (p — po) in the k^^ diagonal element of the normal form of Z, and — f^ the similar (juantity for FZ. Then ii > f2 > • ■ ■ > f „ , £l > £2 > • ■ • > t'n , by (c') of 5.13. Let 7/,- = sup (fk , 0), yi = sup {i-'k , 0), Then 7^ > Ck- , 7t > ^^ , and S{Z, Po) = [71 , 72 , • • ■ , 7.1, S(FZ,po) -\y[,y'., ■■■ ,y'A. 586 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 By 5.15, the normal form of FZ is (^f4'z) ■ (normal form of i/'fi/'z/''Z). Hence the exponent of (p — />„) in tiie /o* diagonal element of the nor- mal form of ^F^zFZ is r — fk . By a similar argument, the exponent of {jp — po) in the A;*^ diagonal element of the normal form of \pzZ is r — £k . Hence, by (e) of 5.1, (r - f 'x) + • • • + (r - e[) is the exponent of (p — po) in the highest common factor of all ?;-rowed minor determinants of \J/f^zFZ. Similarly (r - £1) + • • • + (r - £b) is the exponent of (p — po) in the highest common factor of all 6-ro\ved minor determinants of i/'zZ. Now yppi^zFZ is a polynomial matrix. A typical 6-rowed minor de- terminant of this matrix is of the form ^Wz Z ^hN, , (4) where the summation is over certain products MbNb of 5-rowed minors Mh of F and 6-rowed minors Kb of Z. For a proof of this, see MacDuffee , Theorem 99.1. The expression (4) is the same as E (^pUU){rl^''zNb) (5) where the factors (rpz^^b) ai'e now 6-rowed minors of ^zZ. If (p is a factor common to all 6-rowed minors of ^zZ, it certainly is a factor conmion to all expressions (4) or (5). Hence the highest common factor of all 6-rowed minor determinants of xpp^pzFZ — i.e., of all expressions (4) or (5), — ^has an exponent for {p — po) no lower than that in the highest common factor of all 6-rowed minor determinants of rpzZ. Hence for any b, (r -€[)+■•■ + (r - £6) > (r - fi) + • • . + (r - f,), or f 1 + • • • + £b > £1 + • • • + f 6 . It follows that 7i + • • • + lb > £i + ■ • • + f 6 . This being true for every h, it is certainly true for every h such that all terms on the right are >0 (cf. (2)). This means that for 6 = 1, and for FORMAL REALIZABILITY THEORY — II 587 every successive 6 > 1 such that ft > 0, 7i + • • • + 7!> > 7i + • • • + 76 • This inequahty is now not altered if non-negative numbers are added to its loft member and zeros to its right member. Hence it holds for all b, 1 < b < n, and S{Z, po) > S(FZ, po). (6) This is the first claim of the theorem. Now if F(p) is non-singular at po , then F (p) is rational, and finite at Po . Hence by what is already proved, S{FZ, Po) > S{F~\FZ), Po). This last array is just S{Z, po). Hence we have (6) and its reverse, and the theorem is proved. 5.27 Theorem: \{ po 9^ °o and z{p) = ZM + z,(p), where Z-^ip) is finite at po , then S(Z, Po) = S{Z, , Po). The proof of this depends upon the foUoAving lemma. 5.28 Lemma: Let Z*{p) be such that at p = po ?^ °° its only elements having poles are on the main diagonal. Let — fi , — £2 , • • • be the ex- ponents of (p — Po) in the diagonal elements of Z*(p), so enumerated that -\- £1 > -\- £2 > •■ ' > + £n ■ Let —ti, —&2, • ■ • , — f „ be the exponents of (p — po) in the successive diagonal elements of the normal form of Z*{p). Then if tb > 0 we have f 1 + • • • + £b > £1+ ' • • + f 6 • Proof: There exist constant non-singular matrices F, G such that FZ*G has the same rows and columns as Z* so permuted that the diag- onal elements of FZ*G are arranged in the order of ascending powers of (p — Po), the highest order pole l)eing in the first position. Since the normal forms of Z* and FZ*G are identical, it suffices to consider Z* it- self to be in this form. Let \p = \pz'{p). Now xpZ* has its diagonal elements in the order of increasing posit i\'e power of (p — po). Furthermore, any off-diagonal element of ypZ* has (p — po)'^ as a factor. 588 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Let h be such that Cb > 0. Any 6-rowed minor of \f/Z* is a sum of products of 6 elements of \f/Z*. That b-rowed minor which has in it a term with a lowest possible exponent of (p — po) is the upper left 6-rowed minor. Even this minor has a term with exponent (r - el) + • • • + (r - ei) (7) for (p — Po), this term being the product of the main diagonal elements. Hence the highest common factor of all 6-rowed minors of \I/Z* has an exponent for (p — po) not less than (7). Hence (r - f i) + • • • + (r - f,) (8) is not less than (7), since this is the exponent of (p — po) in the product of the first h diagonal elements of the normal form of \f/Z*. The in- equality between (8) and (7) is just the conclusion claimed in the lemma. 5.281 Proof of 5.27: Let W(p) = A(p)Z{p)B(p) be the normal form of Z(p). Then W = AZiB + AZoB. (9) If we expand all three terms here in Laurent series about po , the term AZ'iB contributes no negative powers. It follows then from the diagonal form of IF that the matrix Z* = AZxB satisfies the conditions of 5.28. The Ck of that lemma are, from (9), just the exponents of {p — po) in the successive diagonal elements of W, the normal form of Z, and the Ck of 5.28 are the similar fiuantities for the normal form of Z* = AZiB. But the normal form of AZiB is the same as that of Zi (5.16). Therefore in the inequality of 5.28 we may interpret all of the e's as exponents in the respective normal forms of Z and Z\ . Now Zi(p) = Z{p) + (-Z,(p)) and —Z-iij)) is again finite at po • Hence we may conclude by the argu- ment just used that if fj, > 0 also f 1 + • • • + f6 > £l + • • • + f 6 . Hence if either of Ch or £& is non-negative €\-\- ■ ■ ■ -\-Jb=j:i + • • • -\-Jh . FORMAL REALIZABILITV THEORY — II 589 By induction on b, then, Sk = £k for k = 1, 2, etc. until such k tliat both are negative. Therefore 7a = sup (ft ,0) = Tfc = sup (f'k , 0) for all /,'=], 2, •••, n. That is, S(Z, po) = S(Zi , po), Q.E.D. ").29 Theorem: Let Z(p) be such that a.t p = pn 9^ <^ its only elements having poles lie on the main diagonal. Let ai , a-> , • • • , cr„ be the orders of these poles, so enumerated that CTl > O") > • • • > (T„ . Then S(Z, Po) = [o-i , 0-0 , • • • , (T,J. Proof: We write Z(?>) = Z*{p) + Z,(p), where Z*(p) is diagonal, having exactly the diagonal elements of Z(p). By 5.27, S{Z, Po) = S(Z*, Po). Now Z*(p) falls under 5.28, but is diagonal in addition. Li the proof of 5.28, therefore, it is exactly the principal minors of if/Z* which have the lowest exponents for (p — po), since all non-principal minors vanish and have zeros of arbitrary order at p = po • Furthermore, (7) is exactly the least exponent of (p — po) in any 6-rowed minor of xpZ* since the principal minors are simple products. Hence (7) and (8) are equal, for any 6 = 1,2, • • • , n. Therefore the exponents in the normal form of Z* are exactly those of Z* and S{Z, Po) = S{Z*, Po) = kl , (72 , • • • , (tJ. Q.E.D. 5.3. Definition: Let p = 3-(«) = ^ yq -\- 8 be a non-singular bi-rational transformation from the 5-sphere to the 590 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 p-sphere. Denote its inverse by q = T-\v)- Given a rational Z(p) the matrix Z,{q) = Z{T{q)) is rational in q. For any po such that T~ (po) 7^ <» , we define 5.31 Theorem: If po and T~ (po) are both finite, SriZ, Po) = S(Z, Po). Proof: Let Wi{q) be the normal form of Z,(q) = Z{T(q)). We have W,{q) = A(q)Z,{q)B{q). Consider TF2(p) = Pri(r-'(p)) = A{T-\p))Z{p)B(T-\p)). Here the pre- and post factors of Z(p) are rational, finite, and non- singular at Po . Hence by 5.26 S{W, , Po) = S(Z, Po). (1) Let 5o = T~\po). It is then easily computed that the inverse trans- formation T~^(p) takes the form aip - Po) . f. q - qo = T-—^ f-ny a ^ 0. Dip - Po) + 1 Any given diagonal element of Wi{q) is of the form {q - qo)'R(q), where e may have any sign, and R(q) is rational, finite, and not zero at go • The corresponding diagonal element of 11^2 (p) is then (p - p»)' G(p-p.)+i) «■(?'• where Ri(p) = R(T~ (p)), and the factor multiplying (p — po)'' is again FORMAL KEALIZAHILITY THEORY — II 591 finite and not zero at po . The exponents of (p — po) in the elements of \V-i(p) are therefore exactly the exponents oi q — qo in the elements of Wi(q). From 5.29, then S(W2 , Po) = *S:(TFi , qo). This with (1) and the definition 5.3 proves the theorem. 5.32 Definition: Given any po , let p = T(q) be a non-singular bi-rational transformation such that qo = T~\po) 9^ <». We define *S'*(Z, po) by aS*(Z, Po) = UriZ, Po). 5.33 Lemma: S*{Z, po) is independent of the T chosen to define it. Proof: Consider q = T~ (p) and r = U~ (p), each such that po is mapped on a finite point. Then by definition Sr{Z, Po) = S(Z, , qo), Sv(Z, Po) = S{Zi , To), where go = T~\po), To = U~\po), Z,{q) = Z(T(g)), Z,{r) = Z(U(r)). Now r = U~\T(q)) = V(q), say, and /o and qo are finite. Hence by 5.31 Sy(Zo , ro) = S{Z, , To) = Su(Z, Po). (2) But by definition Sy(Z, , n) = S(Z, , V-\ro)) = S(Z, , qo) (3) where But Hence Zz{q) = Z.iViq)) Z.iViq)) = ZiU(U-\T(q)))) = Z(T(q)) = Z,(q). S(Zs , qo) = S(Z, , qo) = St(Z, po). This, with (2) and (3), prov^es the lemma. 5.34 Theorem: Theorems 5.2G, 5.27, and 5.29 hold for S* without the restriction that po be finite. 592 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Proof: Let qo = T~\po) 9^ =° . For 5.26 we have where the eciuahties are by definition and the inequaUty is 5.2G apphed to matrices rational in q, since Fr(q) = F(T(q)) is by hypothesis finite at qo . The remaining conclusion of 5.26 follows similarly. The proofs of 5.27 and 5.29 are equally simple. 5.35 Theorem: If we extend 5.3 to aS* by defining Sl{Z, po) = S*{Z, , T-\po)), then 5.31 holds for S* ^^ith no restrictions on po or T" (po). Proof: By their definitions, S*r(Z, Po) = 5*(Zi , T-\po)) = SciZ, , T-\po)), (4) Avhere V is such that V^ {T^^ {po)) is finite. But S,.{Z, , T'\po)) = S{Z, , U-\T'\po))) (5) where Z,(r) = Z^ixT) = Z{T{U{:r))). Let y{r) = T{U(r)). Then, by definitions, S{Z, , U-\T-\po))) = Sy(Z, Po) = S*(Z, Po), (6) since V~^ipo) = U'^\T^\po)) is finite. The theorem follows from (4), (5), and (6). 5.4 Definition: Let S*(Z, Po) = [Ti , 72 , • • • , Tn]. Define 8{Z, Po) = 7i + 72 + • • • + 7n , 5(Z) = j:hz,po), where the latter summation is over all poles po of Z(p), including po = °° • This 8(Z) is the degree of Z for which we must establish the properties claimed in 2.11 through 2.17. These properties will be demonstrated in 5.41 through 5.45, in numerical order, saving 2.13, which is deferred to 5.46. FORMAL HKALIZABILITY THEORY — II r)<)8 5.41 Clearly 8(Z) is an iutegor and non-nogativo. If 8{Z) = 0, thou every 7 at every p^ is zero. Tlenee no fh , not even oo , is a pole of Z. Hence (>acli el(Mnent of Z(p) is a constant. This estal)lisjies 2.1 1 and 2.12. ."). 42 Snppose Z(p) = Z,(p) + Z,(p) where each Zi(p) is finite at e\'ery pole of the other. The poles of Z(p) are then exactly the poles pi of Zi and those po'^ of Zo . At each pole, 5.27 api)lies in the enIarti;(Ml sense of 5.34, so Hreakinii the sum defining 5(Z) into sums over the /Jo" and po"^ pi'oves that 8{Z) = 6(Zi) + 5(Z2). This is 2.14. 5.43 If Zip) = f(p)R, where R is a constant matrix, then the normal form of Z(p) is f(p) times a diagonal matrix of the same rank as R (5.15). 2.15 then follows at once. 5.44 If Zi(p) = AZ(p)B, where A and B are constant and non-singular, the poles of Zi(p) and Z(p) are the same. At each, 5.26 applies in the enlarged .sense of 5.34. Therefore 5(Zi) = 5(Z). This is 2.16. 5.45 If Zi(p) is Zip) bordered by zeros, they have the same poles. One verifies at once from 5.11 that the normal form of Ziip) is that of Zip) bordered by zeros. Since also ZiiTiq)) is ZiTiq)) bordered by zeros, it follows that S*iZr , po) = S*iZ, po) at every pole, whence 6(Zi) = 5(Z). This is 2.17. 5.46 We must prove that if Zip) is non-singular, then 8iZ) = 8iZ~') Proof: Choose a bi-rational transformation p = Tiq) such that at 594 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 V = T(oo) both of Z(p) and Z~'(p) are finite. Let ZM) = z{r{q)). Then ZX\q) = Z-\T(q)). Let Wi{q) be the normal form of Zi(q), with diagonal elements ek(q) ^kk(q) in lowest terms. Since Zi(q) is of rank n, none of these vanish identically. We first claim that 8(Z) — 5(Zi). The poles po of Z are exactly the points Po = T(go) where go runs over the poles of Z\ . At each pole, S*(Z, po) = S*r(Z, po) = S*(Z, , go) by 5.35. Hence 8(Z, po) = 5(Zi , go) and the result follows by addition. Similarly, then, d{Z~') = 5(Z7'). Next we assert that 8(Zi) is just the degree of the polynomial For 5(Zi , go) is the exponent of (g — go) in this polynomial, and the zeros of this polj^nomial are exactly the poles of Zi{q). We observe that if Tri(g) = A(q)Z^(q)B(q), then WT\q) = B-\q)Zl\q)A-\q). This then is the result of polynomial operations on Z7 (g), and has diagonal elements ^ (1) Clearly by arranging these in reverse order, we have a normal form. This^is 5.13. Hence the functions (1) are the diagonal elements of the normal form of Z7\g). The argument above applied to Z7^(g) then shows that 8{Z^^) is the degree of ei(g) • • • e„(g). FORMAL REALIZABILITY THEORY — II 595 Finally, we note the determinant relation 1 Wr(q) I = I A(ci) II Z,(c,) II B(q) \ = (constant) X I Z,(q) \ , since the determinants of .1 antl B are constant. Now Zi(q) lias no pole at 9 = 00 , hence its determinant is finite there. The same is true of Z7 (q), so indeed Now by direct calculation I "^'W I = I'll ' ' ' 1" n- Since this is finite and not zero at q = oo, the numerator and denom- inator are of the same degree. Hence 8(Z) = 5(Zi) = degree W,) = degree (He,) = d{ZT') = 5(Z~'). VI. THE EXACT COUNT OF REACTIVE ELEMENTS 6.0 We showed in the inductive argument of 4.07 that the Brune proc- ess constructs a realization for a given Z(p) which uses exactly 8{Z) reactive elements. To establish 2.18, we must still show that no net- work with fewer than 8(Z) reactive elements can do this. To prove this, we shall show that if Z(p) is the impedance matrix of a network con- taining X reactive elements, then 8{Z) < X. (1) We shall, in fact, in this Section show somewhat more than (1). The demonstration of (1) requires enough calculation that is as easy to prove the following extension of 2.18. 6.01 Theorem: Given any linear correspondence L, (I, 6.2), which PR, (I, 16.71), there exists a number 8{L) such that (i) The realization process outlined in (I, 8) and 4.07 of this Part constructs with 8{L) reactive elements a network realizing a member of the Cauer class of L. (ii) If L is the correspondence established by the Cauer class of a physical network which contains x reactive elements, then 8{L) < X. The proof is divided among the remaining paragraphs of this Section. We maintain here a strict distinction between geometric objects and their concrete coordinate representations. 596 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 6.02 We observe at once that if a 6(L) exists satisfying (i) and (ii), then it must be unicjue because it is exactly the minimum number of reactive elements required to realize any representative of the Cauer class L. No particular pains then need be taken as we go along to verify that the value of b{L) ai'rived at is in fact independent of the mode of defining it. 6.1 Given a PR geometrical linear correspondence L between V and K, there is a frame which reduces L in the sense of (I, 13.02). In this frame we have the dual decomposition V = Vz.0 e Vo e Vi K = Ki 0 Ko e K,.o in which each subspace is real and spanned by selected basis vectors. Furthermore, K,. = K.. 0 K;.o , Finally, if r is the dimension of V> and K2 , there is an r X r PR matrix [Zxii))] such that, when [v-i , ko] e L(p) and V2 eV-i , hi e K2 , then [v.^ = [Z,{v)]M. Here the r-tuples are those representing v-2. and ki as elements of V2 and K2 in the chosen frame. 6.11 Definition: We define 5(L) by 6(L) = 5([Zx]), where [Zi(p)] is the matrix described above. 6.12 This number b{L) is the number of reactive elements used when the Brune process is applied to realize [Zi(p)]. (This is 4.07). Then, however, by the argument of (I, 8.5), the representative [L] of L in the particular frame in question can be realized by adjoining open and short circuits to a realization of [Zi(p)]. This operation adds no new reactive elements. Neither does the operation of converting [L] to any FORMAL UEALIZAHILITY THKOKY — II ')\)7 Caiier eqiiivalout [L]i I)}' the use of ideal transformers. Therefore the partit'uhir 8(L) we have defined — whieh depends for its definition upon a somewhat arbitrary ciioice of coorchnale frame — satisfies (i) of 6.01. 0.2 Lenuna: Let L be a PR geometrical linear correspondence between K and V, and M another between spaces J and U = J* obtained by restricting L as in (I, 18). Then 8{M) < 8(L). Proof: We use the results and notation of (I, 18). In particular, C is a real constant operator from J to K, ('* its adjoint from V to U, and tJH^ pairs of .V(p) are those pairs such that u = C*v and [r, Cj] e L(p). Choose a frame in V and K which reduces L as in G.l. We recall that J u consists of all vectors j e ] such that Cj e Kl (I, 18.31). Let J2 con- sist of all J e J such that Let J:j consist of all j e ] such that Cj eK,o. Then Jo and J3 are disjoint and both are subspaces of J.w . We can there- fore write ]m = J2 e j:i e J4, after a suitable choice of J4 . We now claim that J3®J4CIJ,,0. (1) For we have if./ e J.u that, luiiquely, j = J2 + jz + ji, \\here ./,• e J,- . Therefore (\} - (\h + cu + Cj, where bj' construction Cj-i e K2 , (ji e K/.0 , and, necessarily, then Cji = 0. If j-2 = 0, therefore, Cj e K^o and [0,Cj]eL{p). where 598 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Therefore [C*0,j] = [0,j]eMip). this proves (1). We can now write Jm = J21 0 J20 e Jo (2) J2 = J21 e J20 , J20 = J2 n J ji/o , (3) Jo = J3 0 J4 , J MO = J20 0 Jo . Choosing an arbitrary J5 disjoint from J m , we can write, using (2) and (3), J = J5 0 J21 0 Jmo, (4) where Jm = J21 © Jjwo • Using the arguments of (I, 12.3), we find that the decomposition of U dual to (4) is, because ilf is PR, U = Umo © U21 0 U3 (5) where Ua, = \Jmo 0 U21 . As in (I, 12.3) we can now introduce a frame appropriate to the de- composition indicated in (4) and (5) and obtain a matrix [Z2i(p)] de- scribing the correspondence between J21 and Uoi . Say this is an m X m matrix, m being the dimension of J21 . We can define 8(M) = 5([Z2i]). Let J2 have dimension ?/ii . By (3), if we border [Zoi(p)] by nii — m rows and columns of zeros, to obtain an nii X Wi matrix [Z^ip)], we can interpret [Z25 per cent) counting efficiencies in some region, and the majority of the remainder had low counting efficiencies. These experiments lend further support to the suggestion that in- * Certain of these papers arc available as Bell System Monographs and may he obtained on request to the Publication Department, Bell Telephono Laboratories, Inc., 463 West Street, Xew York 14, X. Y. For papers available in this form, the monofiraph number is given in jjarentheses following tlic date of publication, and this inunbcr should i)e given in all requests. ' Bell Telephone Laboratories. 601 602 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 homogeneous fields at least partially account for the inhomogeneities in bom- bardment conduction. Serious errors in the normal estimates of range and mobility of electrons or holes in insulators can be introduced by neglecting these field inhomogeneities. Diffusion in Alloys and the Kirkendall Effect. J. Bardeen and C. Herring^ Pp. 87-111. Am. Soc. for Metals. Atom movements; a semi- nar . . . held during the thirty-second National Metal Congress and Exposition, Chicago, Oct. 21-27, 1950. Cleveland, Ohio, Am. Soc. for Metals, 1951. 240 p. Some Roots of an Equation Involving Bessel Functions. B. P. Bogert . Jl. Math. Phijs., 30, pp. 102-105, July, 1951. (Monograph 1903). Creep Test Methods for Determining Cracking Sensitivity of Poly- ethylene Polymers. W. C. Ellis^ and J. D. Cummings\ A.S.T.M. Bull., No. 178, pp. 47-49, Dec, 1951. Conventional creep testing methods for evaluating the cracking sensitivity of polyethylene polymers are described. The tests show that sensitivity to cracking in the presence of an active agent decreases with increasing average molecular weight of the poljmier. For a given stress condition and environment, there appears to be a threshold value of stress and strain for the occurrence of cracking. Observer Reaction to Video Crosstalk. A. D. Fowler . J. Soc. Motion Picture and Television Engrs., 57, pp. 416-424, Nov., 1951. (Monograph 1928). Presented here are results of tests to determine how much video crosstalk can be tolerated in black-and-white television pictures. Experienced observers viewed a television picture and rated the disturbing effects of controlled amounts of crosstalk from another video system. Crosstalk couj)ling was simultated by a network which permitted changes in frequency characteristic as well as in coupling loss. Tolerable limits for crosstalk coupling are derived from the test results. Mass Spectrometric Studies of Molecidar Ions in the Noble Gases. J. A. Hornbeck^ and J. P. Molnar\ Phys. Rev., 84, pp. 621-625, Nov. 15, 1951. Molecular ions of the rare gases (He2"'', Ne2'*', A2"'', Kr2+, and Xe2"'") produced by electron impact at gas pressures from 10^'' to lO"^ mm Hg have been studied with a small mass spectrometer. The ion intensity increased linearly with elec- tron current and with the square of the gas pressure. The form of the ionization versus electron energy curves resembles closely curves of excitation probabilitj' by electron collision. The appearance potentials of the molecular ions were less ^ Bell Telephone Laboratories ABSTRACTS OF TECHNICAL ARTICLES 003 tlian those of the atomic ions by lAttl volts in He, 0.7+°;^ volt in Ne, 0.7^2;^ \()lt in A, OJlo.'j volt in Kr. These results can he interpreted, we l)elieve, only l>y assumiiiii; that the process of formation of the molecular ions observed in this experiment is, usinj^ helium as an eNami)le, an excitation by electron impact, He + G + K.K. — > He* + e, followed by the collision process, He* + He —' He-i^ + e, where He* stands for a helium atom raised to a hi{2;h-lyin}r excited state. Our results differ from those of Arnot and M'Ewen on helium particu- larly in that they reported the appearance potential low enough to permit meta- stable atoms to form molecular ions. The Drift Velocities of Molecular and Atomic Ions in Heli^nn, Neon, and An/on. J. A. IIoun'beck . Phys. Rer., 84, pp. 6ir)-G20, Nov. 15, 1951. Drift velocity measurements as a function of E/po , the ratio of fiekl strength to normalized gas pressure, are presented for atomic and molecular ions of He, Ne, and A in their respective parent gases. Identification of the molecular ions is based upon the time resolution of the apparatus and the dei)endence of ion concentration on pressure, appUed voltage, and gas purit}'. Extrapolation of the low field measurements to zero field yields mobility values for atomic ions, Mu (He+) = 10.8 cmVvolt sec, mo (Ne+) = 4.4, and mo (A+) = 1.63 in good agreement with theor}': Massey and Mohr compute mo (He+) = 11, and Hol- stein gives mo (Ne+) =4.1 and mo (A+) = 1.64. Drift velocitj^ data at low field for the molecular ions agree within experimental error with data of Tyndall and Powell (He), and Munson and Tyndall (Ne and A), which they assigned to atomic ions. A qualitative description in terms of ion-atom interaction forces is given for the observed field variation of the atomic ion drift velocities up to high£'/po . Checking Analogue Computer Solutions. E. Lakatos . Proc. Inst. Radio Engrs., 39, p. 1571, Dec, 1951. E.xperimental Heat Contents of SrO, BaO, CaO, BaCO^ and SrCOs at High Temperatures. Dissociation Pressures of BaCOz and SrCOz. J. J. Lander\ J. Am. Chem. Soc, 73, pp. 5794-5797, Dec, 1951. (Mono- graph 1930). The high temperature heat contents of SrO, BaO, CaO, BaCOs and SrCOs have been measured using the "drop" method. Values have been obtained for the heats of the transitions of the carbonates. The dissociation pressures of the carbonates have been measured to pressures below 0.1 mm and values calcu- lated for lower pressures from the observed heat contents and observed disso- ciation pressures at higher temperatures. Electron-Hole Prodnction in Germanium by Alpha-Particles. K. G. McKay'. Phys. Rev., 84, pp. 829-832, Nov. 15, 1951. The number of electron-hole pairs i)roduced in gei'maniinn by alplia-jiarticle 1 Bell Telephone Laboratories 604 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 bonibaidinent has been determined l)y c(jllecting the internally produced car- riers across a reverse-biased n-p junction. No evidence is found for trapping of carriers in the barrier region. Studies of individual pulses show that the carriers are s\vei)t across the barrier in a time of less than 2 X 10^ sec. The counting efficiency is 100 per cent. Tlie enei'gy lost by an alpha-particle i)or internally produced electron-hole pair is 3.0 ± 0.4 ev. The difference between this and the energy gap is attributed to losses to the lattice b}- the internal cari'iers. It is concluded that lecombination due to columnar ionization is negligible in ger- maniunr. The n-p-n Junction as a Model for Secondary Photoconductivity. K. G. McKay . Phys. Rev., 84, pp. 833-835, Nov. 15, 1951. A germanium n-p-n junction with the p region floating, has been suljjected to alpha-particle bombardment. The transient currents resulting from indi- vidual incident alphas liave been studied. This enables one to stud}' the rate of deca.y of excess holes in the p-region. This decay time appears to increase with applied bias, pass through a maximum, and eventuallj' approach a con- stant value. The total charge flowing across the unit, as a result of the bombard- ment by a single alpha-particle, maj^ become large; quantum yields of greater than 60 haxe been obser\'ed. The unit possesses many of the important charac- teristics of materials which exhibit "secondary photoconductivity." It is con- cluded that various forms of n-p-n barriers must therefore play an important role in such materials and that their understanding can he greatly facilitated by studies of n-p-7i barriers in germanium. Frequency Detection and Speech Formants. E. Peterson . Acoustical Soc. Am., JL, 23, pp. 668-674, Xov., 1951. This study is aimed primarily at evaluating the utility of axis-crossing de- tectors in tracking speech formants. Detectors of the usual tj'pe are found sub- ject to an error, fundamental in nature. To remove this source of error speech is modulated up in frequency as a single sideband before limiting and detecting processes are applied. Experimental results with this carrier type of detector on a small number of speech samples are presented, and compared with spectro- grams. Conclusions are that the average axis-crossing rates cannot be trusted in general to follow specific formants, whether the speech is normal or differen- tiated. But when the formants are sufficientlv localized by frequenc}' selectivity, prospects of tracking the lower formants look promising. Transistor Circuit Design. G. Raisbeck\ Electronics, 24, pp. 128- 132, 134, Dec, 1951. (Monograph 1932). How to derive amplifier, oscillator, modulator and multi-vibrator transistor circuits from known vacuum-tube circuits. Technique, known as duality, is explained in detail and may be applied to any complex vacuum-tube circuit to find the corresponding transistor circuit. Communication Theory — Exposition of Fundamentals. C. E. Shannon . Pp. 44-47. General treatment of the problem of Coding. Pp. 102-104. 1 Bell Telephone Laboratories ABSTRACIS OF TECHNICAL ARTICLES ()().") (Ireat Britain. Mini.stry of Supply. Symposium on Informal ion Tlu^ory. Report of Proceedings held. . . Royal Soc, Burlinslon House, Loud., Sept. 20-29, 1950. On Uir h'ddtion Between the Sound Fields Radiated and Dijfmeted bij Plane Obstaeles. F. :M. Wiener'. ./. Aeoust. Soc. Am., 23, pp. ()97-700, Nov., 1951. In the i)a8t, acoustic ditYiaction and radiation problems have often Ix'en treated sei)anitely, although their intimate connection is clear from theory. In the case of i)lane piston radiators and plane rifj;id scatterers exposed to a perpendicularly incident plane wave, this connection becomes particulaily simple and useful. It is easy to show that the radiated sound field is everywhere the same as the field scattered (diffracted) in the diffraction case, except for a factor of proportionality. It is also shown that the reaction of the medium on the ra- diator, as expressed by the mechanical radiation impedance, is equal to the force per unit incident pressure exerted on the same obstacle, held rij^id as a scatterer, excei)t for a factor of proiK)rtionality. By way of illustration, the foregoing prin- cijiles are applied to the important case of the circular disk. Magnetic Modulators. E. P. Felch\ V. E. Legg\ and F. G. Merrill\ References. Electronics, 25, pp. 113-117, Feb., 1952. Conversion of low-level, low-frequency or dc signals to ac signals capable of lieing amplified b\' conventional means is accomplished by magnetic-amplifier- type device that combines higli efficiencj' and reliability with extreme rugged- ness. Conservation of Nickel. G. R. Gohn\ A.S.T.M., Bull., Xo. 179, p. 32, Jan., 1952. The Mechanism of Electrolytic Rectification. H. E. HaringV Electro- chem. Soc, JL, 99, pp. 30-37, Jan., 1952. (Monograph 1929). An electrochemical theory is proposed for rectification, as exemplified by the tantalum (or aluminum) electrolytic rectifiei- and capacitor. A detailed con- sideration of the mechanism of formation of the oxide film which constitutes the rectification barrier leads to the conclusion that this barrier consists of an electrolytic polarization, in the form of a concentration gradient of excess metal ions, permanenth' fixed or "frozen" in position in an otherwise insulating matrix of electrolytically-formed oxide. The phj'sical structure which has been de- scribed functions as (a) a current-blocking ionic space chai'ge or (b) a current- passing electronic semiconductor, de])ending solely upon the direction of the applied voltage. The movement of electrons only is required. An exi)lanation for breakdown of the l:)arriei' at excessively high voltages is suggested. This explanation may be api)licable to dielectric breakdown of other kinds. ' Boll Telephone Laboratories 006 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 Nullification of ^^pace-Charge Effects in a Converging Electron Beam hy a Magnetic Field. M. E. Hines\ Proc. Inst. Radio Engrs., 40, pp. 61-64, Jan., 1952. (Monograph 1935). This paper presents the conditions necessary for maintaining a uniformly converging conical electron beam in the presence of space charge, It is an ex- tension of the Brillouin focusing condition to conical flow, requiring a con- verging rather than a uniform magnetic field. In this type of electron flow, the diverging effects of space charge are balanced against magnetic reaction forces for reasonably small cone angles of convergence. Though the balance of forces is exact only for infinitesimal angles, it is reasonably accurate for cones of half- angle as great as 10 degrees. The minimum beam size will be limited only by the effects of thermal \'elocities, by gun aberrations, and by the magnetic field ob- tainable. Continuous Motion Picture Projector for Use in Television Film Scan- ning. A. G. Jensen\ R. E. Graham\ and C. F. Mattke\ Bibliography. J. Soc. Motion Picture and Television Engrs., 58, pp. 1-21, Jan., 1952. The projector used for this equipment drives a 35-mm motion picture film at the standard (nonintermittent) speed of 24 frame/sec and produces a tele- vision signal of 525 fines and 30 frames interlaced 2 to 1. The projector utilizes a system of movable plane mirrors mounted on a rotating drum and controlled by a single stationarj- cam. Vertical jitter in the television image is minimized by means of an electronic servo system operating on the film sprocket holes, resulting in a residual vertical motion of about 1/2000 of a picture height. A second electronic servo system is incorporated to suppress flicker. The combina- tion of this scanner and a high-grade monitor is capable of producing a tele- vision picture with a resolution corresponding to about 8 mc and with good tone rendition over a range up to 200 to 1. Low Temperature Polymorphic Transformation in WO3. B. T. Mat- TmAS^ and E. A. Wood'. Phys. Rev., 84, p. 1255, Dec. 15, 1951. The Concentration of Molecules on Internal Surfaces in Ice. E. J. Murphy\ J. Chem. Phys., 19, pp. 1516-1518, Dec, 1951. In this paper the experimental expression for the "local conductivity" of ice is given. This expression has two terms, one of which has already been discussed and brought into close relation with the structure of ice, that is, with its heat of sublimation and its lattice constant. This paper brings out another relation, deriving it from the second term of the experimental expression. It is concluded from an analysis outlined here that the second term of the local conductivity gives the concentration of molecules in "internal surfaces". For the specimen of ice to which this method was applied the concentration of molecules on internal surfaces comes out as 1.03 X 10*' molecules /cc. This is proposed as a new method of studying imperfections (internal surfaces) in dielectric crystals, and one which seems to be well suited to this purpose. It gains its advantages from * Bell Telephone Laboratories ABSTRACTS OF TECIIXirAL ARTICLES 607 the fact that it is not dependent upon the regularity of tlie imperfections, as in x-ray diffraction methods, or u[)on the connectivity of the system of internal surfaces, as in direct current condution. Meditations on Physics Today. J. R. Pierce^ Phy. Today, 5, p. 3, Jan., 1952. Stabilization of Dielectrics Operating Under Direct Current Potential. H. A. Sauer\ D. a. McLean\ and L. EgertonV Ind. Eng. Chem., 44, pp. 135-1-40, Jan., 1952. 1 Bell Telephone Laboratories. Contributors to this Issue E. N. Gilbert, B.S., Queens College, 1943; Ph.D., Massachusetts Institute of Technology, 1948. M.I.T. Radiation Laboratory, 1944-46. Bell Telephone Laboratories, 1948-. Dr. Gilbert's first assignment was in a group studying information theory, and in 1949 he joined a group concerned with switching theory. Member of the American Mathematical Society. I. L. Hopkins, B.S., Massachusetts Institute of Technology, 1927; Bell Telephone Laboratories, 1927-. For eighteen years, Mr. Hopkins designed testing equipment and tested insulating materials. Right after World War II, he tested and developed special-purpose rubber com- pounds, and since 1948 he has been conducting research in the physical properties of polymers. W. A. Malthaner, B.E.E., Rensselaer Polytechnic Institute, 1937. Bell Telephone Laboratories, 1937-. Mr. Malthaner is currently engaged in research on new automatic telephone central-office systems, inter- office signaling systems, and subscriber dialing and supervisory arrange- ments. Until World War II, when he worked on the development of automatic fire control systems and fire control radar, Mr. Malthaner tested and developed central office circuits and switching systems. As- sociate of the American Institute of Electrical Engineers. Member of the Institute of Radio Engineers, Tau Beta Pi and Sigma Xi. Warren P. Mason, B.S. in E.E., University of Kansas, 1921; M.A., Ph.D., Columbia, 1928. Bell Telephone Laboratories, 1921-. Dr. Ma- son has been engaged in investigating the properties and applications of piezoelectric crystals, in the study of ultrasonics, and in mechanics. Fellow of the American Physical Society, Acoustical Society of America and Institute of Radio Engineers and member of Sigma Xi and Tau Beta Pi. Brockway McMillan, B.S., Massachusetts Institute of Technology, 1936; Ph.D., Massachusetts Institute of Technology, 1939; Instructor of Mathematics, Massachusetts Institute of Technology, 1936-39; Procter 608 CONTRIBUTORS TO THIS ISSUE 609 Fellow and Honrv H. Fine Instructor in ^lathematics, Prineeton Uni- versity, 1939 42: ^.S.^M^, 1942 4(), studying exterior l)aUisties of guns and rockets: Los Alamos i-ahoratory, spring 1946; Bell Telephone Laboratories, 1946 . Dr. McMillan has been engaged in mathematical research and consultation work. Member of American Mathematical Society, Institute of Mathematical Statistics, and A.A.A.S. .Iamks Z. Mk.nakd, H.'r>., Arkansas State Teachers College, 1941. V. S. .Vrmy, 1941-46). Bell Telephon<> T^aboratories, 1946-. Mr. Menard has been engaged in the devel()i)ment of magnetic recoi'ding eciuipment and autlio e(iuipment for telephone plant applications. J. A. MoRTox, B.S. in K.E., Wayne University, 1935; M.S., Uni- \ ersity of Michigan, 1936. Bell Telephone Laboratories, 1936-. Mr. Morton is currently in charge of the development of the transistor and other semi-conductoi' devices. In the past he has been concerned with research on coaxial cables, microwave amplifier circuits, radar receivers, and with \acuum tube development. He designed a microwave tube used in the Xew York-San Francisco microwave relay system. Member of the I.R.E., Eta Kappa Xu, Alpha Delta Psi, jNIackenzie Honor Societ}^ Phi Kappa Phi, and Sigma Xi. H. Eahlk \'ArGHAX, B.S. inC.E., Cooper Union, 1933. Bell Telephone Laboratories, 1928-. Since World War II, Mr. Vaughan has been investi- gating switching systems and high speed signaling means. In the past he studied voice operated devices and fundamental effects of speech and noise on voice-frequency signaling systems. During World War II, he was engaged in government projects, conducting research on anti-aircraft computers and fire contr(jl radars. Samuel D. White, B.S. in E.E., Rutgers University, 1927; E.E., Ivutgers University, 1932; Bell Telephone Laboratories, 1927-. Lentil 1939, Mr. White was a member of the acoustical research department. He then entered the switching apparatus development group and is cur- rently .studying some aspects of relay problems. Member of I.R.E., Acoustical Society of America, and Sigma Xi. HE BELL SYSTEM / Jecnmcai lourna^ SCIENTIFIC^^^ AI\ 5PECTS OF ELECTRICA L^ClO M>I^U N F C A T I O N t^u,;:--:- )LUME XXXI 11^1^1952-^^1'* NUMBER4 \ 1 y Thirtieth Anniversary • , 611 Lee de Forest and William Shockley DJscuss Electronics 612 Network Synthesis Using Tchebycheff PolynomiaTSwies SIDNEY DARLINGTON 613 A Carrier Telegraph System for Short-Haul Applications J. L. HYSKO, W. T. REA AND L. C. ROBERTS 666 The Type-0 Carrier System PAUL G. EDWARDS AND L. R. MONTFORT 688 Efficient Coding b, m. Oliver 724 Statistics of Television Signals e. r. kretzmer 751 Experiments with Linear Prediction in Television c. w. harrison 764 Generalized Telegraphist's Equations for Waveguides S. A. SCHELKUNOFF 784 Photoelectric Properties of lonically Bombarded SiUcon EDWIN F. KINGSBURY AND RUSSELL S, OHL 802 Abstracts of Bell System Papers Not Published in this Journal 816 Contributors to this Issue 820 COPYRIGHT 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL PUBLISHED SIX TIMES A YEAR BY THE AMERICAN TELEPHONE AND TELEGRAPH COMPANY 195 BROADWAY, NEW YORK 7, N.Y. CLEO F. CRAIG, President CARROLL O. BICKELHAUPT, Secretary DONALD R. BELCHER, Treasurer EDITORIAL BOARD F. R. KAPPEL O. E. BUCKLEY H. S. O S B O R N E M. J. K E L LY J. J. PILLIOD A.B.CLARK R. BOWN D. A. QUARLES F. J. FE ELY PHILIP C.JONES, Editor M. E. STRIEBY, Managing Editor SUBSCRIPTIONS Subscriptions are accepted at $3.00 per year. Single copies are 75 cents each. The foreign postage is 65 cents per year or 1 1 cents per copy. PRINTED IN U.S.A. The Bell Svsteiii Techiiieal Journal Volume XXXI July 1952 Number 4 COPYRIGHT 1952, A.MLOIUCAN TELEPHONE AND TELEGRAPH COMPANY Thirtieth Anniversary Thirty ycar.s ago this month The Bell System Technical Journal began pubhcation. Suggested l\y Dr. George A. Campbell, it had been luider disciLssion for some years. Dr. K. W. King, ^vho had been one of its most active ad\'oc'ates, became its editor when the staff of the Journal was established. Except for a six-year period following 1928, while he was in England, Dr. King continued as editor until he retired m 1949. By July, 1922, when Xo. 1, ^'ol. 1 of the Journal appeared, research and development was a long established practice in the Bell System. The high-vacuum electronic tube, which had already begun to revolu- tionize electrical communication, was itself a product of Bell System research. Since electrical communication was a still comparatively new field of study, however, its publications were widely scattered. There seemed a need for a magazine that would serve the communication engineers exclusively, and it was largely to meet this need that The Bell Sy'Stem Technical Journal was launched. In the thirty years since that time, the art and science of communica- tion has advanced and ramified beyond anything likely to have been then foreseen. A very substantial part of this increase has originated within the Bell System, and this progress has been reflected in the pages of the Technical Journal. There seems little reason to doul)t that the next three decades will witness an ad\'ance at least comparable with that of the past three, and it is planned to ha\'e the Journal pre- sent the work of the coming years, with perhaps even greater effective- ness than in the past. Abstracts or titles of all Bell System technical and scientific j^apers appearing in other publications are listed in the Journal and reprints of many of these papers are available and may be obtained In' subscribers. In one way or another, therefore. Journal readers have access to essentially all the technical papers published by the Bell System. With this increased coverage, it is hoped that the Journal will proxe increasing!}^ useful to a growing circle of readers. 611 Lee de Forest and William Shockley Discuss Electronics Dr. de Forest is the inventor of the Audion from which the modern vacuum tube w its many forms and types has sprung. Dr. Shockley is the leader of the researcl group at Bell Telephone Laboratories whose members invented the transistor. Stand ing side by side these two men seem to epitomize the basic change in the pattern o our technical life which has taken place during the first half of the present century—'^ the change from the struggling individual inventor to the great industrial scientifi, laboratory as the source of much of our technological advance. Network Synthesis Using Tchebycheff Polynomial Seriest By SIDNEY DARLINGTON (Manuscript received April 17, 1952) A general method is developed for finding functions of frequency which approximate assigned gain or phase characteristics, within the special class of functions which can he realized exactly as the gain or phase of finite networks of linear lumped, elements. The method is based, upon manipula- tions of two Tchebycheff polynomial series, one of which represents the assigned characteristic, and the other the approximating network function. The ivide range of applicability is illustrated with a number of examples. 1. IXTRODUCTION Network synthesis is the opposite of network analysis — namely, the design of a network to have assigned characteristics, as opposed to the evaluation of the characteristics of an assigned network. In general, there are specifications on the internal constitution of the network, as well as requirements relating to its external performance. A common form of the general problem is the design of a finite network of linear lumped elements, to produce an assigned gain or phase characteristic over a prescribed interval of useful frequencies. The present paper re- lates to this particular form. In general, the restrictions on the network are such that the assigned performance cannot be matched exactly. This gives rise to an approxi- mation or interpolation problem. For present purposes, the problem is: to choose a function of frequency w^hich matches the assigned gain or phase to a satisfactory accuracy, from that special class of functions which can be realized exactly with physical finite networks of linear lumped elements. The function of frequency may be defined in terms of network singularities (natural modes and infinite loss points). The t Presented orally, in briefer form, at the 1951 Western Convention of the Institute of Radio Engineers, and at the Sj-mposium on Modern Network Syn- thesis sponsored by the Polytechnic Institute of Brooklyn and The Office of Naval Research, New York City, April, 1952. 613 614 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 interpolation problem may then be regarded as solved when a suitable set of network singularities has been obtained; for quite different tech- niques are used to design actual networks with these singularities. The interpolation problem may be attacked in a number of different ways; and a variety of different techniciues are, in fact, needed to cover the wide diversity of practical applications. The present topic is a fairly general way of attacking the problem, based upon manipulations of two series of Tchebycheff poljniomials. The two series represent expansions of two functions of frequency — one, the ideal assigned gain or phase, the other, the network approximation to the ideal. The interpolation problem may be solved in this way because it is feasible, as ^^'ill be shown, to determine network singularities from arbitrarily^ assigned values of coefficients in the corresponding Tchebycheff polynomial series. The techniques to be described were derived originally from studies of the so-called potential analogy; but they can now be developed most easily without reference thereto. f In a sense they may be regarded as extensions of familiar filter theory, using Tchebycheff polynomials, to more general gain and phase functions. The extensions, however, depend upon a number of new principles. Extensions of the filter theory applied to more general problems have been noted in published papers; but those noted have not used the specific general approach employed here.f The wide applicability of this general approach will be illustrated by specific examples. 2. NETWORK AND TRANSMISSION FUNCTION It will be sufficient for our present purposes to limit the discussion to the general 4-pole shown in Fig. 1. The 4-pole may be active or passive, but it must be a stable finite network of linear lumped elements. E and V are steady state ac voltages, E the driving voltage and T" the response. The gain a and phase |S are here defined as the real and imaginary parts of log V/E. For a finite network of lumped elements, a + ijS always has the fol- lowing form: a + iff = log K ^^-^-'^■■- (1) t Tchebycheff polynomials are related to potential analogue charges on el- lipses, as described in the author's paper "The Potential Analogue Method of Network Synthesis''^ Section 15. X For the most part, they have used the potential analogy, in such a way that Tchebycheff polynomials do not appear at all in general applications. For ex- amples, see methods of Matthaei^, Bashkow', and Kuh^. XETWORK SYN'THESIS 615 The "frequency variable" p represents, of course, ioj. The zeros p^ of the rational fraction are those values of p at which there is infinite loss. The poles p'J are the so-called natural modes,' or values of p at which response V can exist in the absence of driving voltage E. The scale factor K determines the average level of transmission. The zeros, poles, and scale factor together determine the gain and phase completely. For a physical stable network, the zeros and poles must meet certain well known restrictions, which are commonly stated in terms of loca- tions in the complex plane for frequency variable p. Within these re- strictions, the zeros and poles can bo subject to arbitrary choice, say for purposes of network synthesis. eK LINEAR LUMPED ELEMENTS a+ l/3=LOG V/e Fig. 1 — A general 4-pole. The symmetries required by the physical restrictions permit a and /3 to be represented separately as follows:! (p'l' - P^)(P2'^ - p^) ••• 2a = log K «2/3 = log / "2 2n/ "2 2\ {Pl - p ){p-2 - p ) (p[ - p) • • ■ (p'l + p) • ■ (2) ip'i + ?>)••• (pi — p) ■• • These expressions hold at all real frequencies, but only at real fre- quencies. 3. TCHEBYCHEFF POLYNOMIALS It is functions of these special types which we are to synthesize with the help of Tchebycheff polynomials. More generall}^, TchebychefT polynomials come in various forms, and may be analyzed in various ways. For our special purposes, however, they take somewhat special forms (a little different from textbook definitions); and they are best analyzed in quite special ways.J It will be simplest to start with arbi- trary definitions, to be justified later on by demonstrations of usefulness. t The phase eciuation omits a possiljle 180° phase reversal, which is trivial for present purposes. + For discussions of Tchebycheff polynomials from other viewpoints, see Courant and Hilbert^, and also a paper by Lanczos" on trigonometric interpola- tion. 616 THE BELL SYSTEM TECHXICAL JOURXAL, JULY 1952 Actually, the definitions must \'ary ^vith the nature of the useful fre- quency iuter\'al. For the present, however, it will be assumed that the useful interval extends from co = 0 to coc ; or more precisely', from CO = — Wc to +Wc (in accordance with the symmetries of gain and phase functions). Useful intervals which do not include w = 0 require changes in the definitions, which will be taken up in Section 28. For our present purposes, Tchebycheff polynomials Tk may be defined as follows: p = ico = ioic sin <^ Tk = cos /:(/), /: even Tk = i sin k^ "^ ] -(^c\ ^^.'^ -1 J '{ , J-J / \ "^ /I -'^c \ / \ 1^^^ Fig. 2 — Tchebycheff poh-nomials. NETWORK SYNTHESIS ()17 is especially appropriate for general network applications, because the odd ordered polynomials contribute to the imaginary parts of complex network functions — such as ijS in a -\- ib.'\ It is apparent from (3) that the Tchebycheff polynomials become simply Fourier harmonics, if they are plotted against a distorted fre- quency scale — that is, against 0. This means that they must be ortho- gonal, over that particular range of frequencies which corresponds to real values of 4>. From the relation between 4> and co, it is clear that real \'alues of (/) cover the frequency interval between — Wc and -\-Uc , which is our useful interval. In other words, the interval of orthogonality coin- cides witii the useful frec}uency inter\'al. The corresponding interval of p is of course p = —iojc to -\-iuc . If a given fiuiction is plotted against <^, instead of w, it may be ex- panded in a Fourier series. Each term in the series may be replaced by a Tchebycheff polynomial, to obtain an expansion of a given function in terms of polynomials, for the specific useful interval co = — Wc to -\-cx}c . Established techniques are available for expanding experimental, or other numerical data, in Fourier series, as well as actual analytic fiuictions. In Fig. 2, some of the Tchebycheff polynomials are plotted against co. The frequencies — co^ and +coc are also indicated. Frequencies between these limits correspond to real values of the angle variable (p. If this part of the frequency' scale is stretched, in the proper non-uniform way, the various "loops" not only have the same maximum values, but also the same shapes. In other words, they become periodic. More specifically, a stretch which changes the frequency scale into a . 4. TRANSFORMATION OF VARIABLE An alternate to (3) may be obtained by relating a new variable, z, to 4> l^y z = e'* (4) Substituting z in the exponential equi\'alent of sin , in the first equa- tion of (3), gives an alternative definition of z directly in terms of p, namely: t A small change in the definition of 0 would bring the definitions closer to convention, by repkicing both sines l)y cosines (without altering Tk as a function of p). This however, would complicate our later analj'sis. 618 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Substitutions in the exponential ecjiiivalents of the other sine and cosine in (3) give: T, = K-->i) A; even n = \ [z' - 3 ) , k odd (6) Network appUcations depend upon the nature of the relationship be- tween the variable p, and the variable z. The relationship is illustrated in Fig. 3, which indicates corresponding contours in the p and z planes. Since angle 0 is real in the useful interval, z, as defined by (4), has unit magnitude. In equivalent conformal mapping terms, the unit circle in the z plane maps onto a segment of the axis of real frequencies in the y plane — namely the segment extending from p = —iwc to -\-iwc . Hereafter, we shall say merely that the useful interval in the z plane is the unit circle, or | 2 | = 1. The real frec[uency intervals outside the useful interval map onto the imaginary axis in the ^-plane. 2-plane circles with radii other than unity map onto p-plane ellipses, all with foci at p = zb iuic ■ This is reminiscent of filter theory using Tcheby- cheff polynomials, and in fact such a filter may be obtained by spacing 2-plane mappings of natural modes uniformly around such a circle. f p-PLANE 2-PLANE + La;c .p = La;cSiN9!> I REAL p ,-^ ^^ / y N / \ / \ / / 1 / ^- — ^ = 6'*-";^ \ / . -~ \ \ / { / N \ REAL Z \ y \ J ] 1 \ \ \ -""J 1 \ \ / 1 \ ^^^ ^^^ 1 \ ^^~ ■^^ / \ / \ \ / / / \ y ^v Fig. 3 — The complex planes for p and z. t The filter theorj^ is developed in detail in a monograph by Wheeler^, which also includes an extensive bibliography. NETWORK SYN'THESIS 619 O. Z-PLAXE MAl'PIiVGS OF NETWORK SINGULARITIES 2-plane mappings of network singularities are also an essential part of synthesis applications. The mapping z^ of a typical zero or pole p„ is illustrated in Fig. 4. From (5), the analytic relation must be: Po = Wr (7) By its quadratic nature, there must be exactly two values of Zo , corre- sponding to one Pa . The relation is such that replacing z^ by — l/z, leaves p, unchanged; and hence the two values of z„ must be negative reciprocals, each of the other. Thus, one mapping of pa falls outside the unit 2-plane circle, and the other inside. A unique definition of z„ may be obtainetl by rcMiuiring that Za must be the mapping outside the unit circle. Then | 2„ | > 1 by definition, and the complete definition of Zc may be : We p. = Za\ > I (8) This definition is unique provided network singularities pa are excluded from that very special line segment of the real frequency axis which corresponds to the useful frequency interval, — Wc < co < +Wc (where I Za 1 would be exactly 1). We are going to solve the interpolation problem by choosing the z^ first, instead of the p-plane singularities p„ , after formulating the inter- polation problem in suitable 2;-plane terms. For this, however, we must ---tlCJc Pcr^ p-PLANE REAL p -LWc 1 "«. Z-PLANE ,"' \ REAL Z > \ \ / / / / Fig. 4 — Mappings of a network singularity. 620 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 know what further conditions must be imposed upon the Za , so that the corresponding pa will meet the special conditions necessary for physical networks. A simple analysis of the definition (8) of Za , and of the well known restrictions on the p^ , leads to the following assertion; The physical restrictions on z„ are exactly the same as those on pa . It is ob\dous, for example, that conjugate complex z, are necessary for conjugate complex p^ . Also, because \za\ > 1, z„ dominates —l/z„ . Then the sign of Re pa is the same as that of the Re z^ , and pa with negative real parts require Za with negative real parts, and so on. Thus the direct choice of z„ is restricted in exactly the same way as- the choice of pa , except for the additional general requirement \ Za \ > 1. The last condition imposes no important restriction on the corresponding Pt, . Initially, it was adopted to make z^ unique for any p„ (not at a useful real frec[uency); but this condition does also play an essential role in the z-plane formulation of the interpolation problem. 6. NETWORK GAIN AND PHASE IN TERMS OF Z A first step in the 2-plane formulation of the interpolation problem is' the formulation of the network gain and phase functions, (1) and (2), in terms of z. This is most usefully examined as a transformation of func- tional form, rather than as a conformal mapping. The gain and phase function (1) transforms as follows: The analytic relation between p and z is regular in the neighborhood of the singular- ities Pa of the network function. Therefore, there will be similar singu- larities of the transformed function at the ^-plane mappings of p„ ^ which are z, and — l/z, . These singularities, and also suitable behavior at infinity, are exhibited by the following expression for a -{- f/3 as a function of z. XT is used here to designate a product of factors of the type following it.f t The expression is readily confirmed in the following very elementary man- ner: For every factor ( 1 — — j , in (9), there is also a factor (l + — ) • The product of the two maj' be expanded as follows: .--in + i =i '^ ' '•■ (io> NETWORK SYNTHESIS G21 If we define a new scale factor /v^ by 7v , = Kl , we may write (9) as follows: a + ?"/3 = log < ame Rational n/ i.\f\ Function in — 1/0 (13) Similar expressions for the separate gain and phase functions may be derived from (2) : 2cL = log< n(i-z^ JSame Rational \ Function in — 1/2 i2(i = log< z z 1-7 1+7 n — ?n- ' (14) 1 + z 1 - -, z„ Same Rational Function in — 1/z Equation (13) holds at all values of p and z, while (14) holds at all real frequencies. Simplifications of (14) should be noted, good for the useful interval only. When \z\ = 1, l/z = z*. Recalling also that log I X f is 2 log \x\, and similar elementary relations, one obtains from (14) : When 1 2 I = 1, log K\ n(-l) n(i-|- z z 1-7 1 + 77 /? = Phase n — ; n - (15) 1 + z 1 - -7/ Zn Comparison with (5) and (7) gives: ( 1 - - ) ( 1 + — ) = -— (p - p,) \ zj \ z,z/ U^Z, Thus (9) is equivalent to (1) provided K==K n - n - (11) (12) 622 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 7. THE POWER SERIES IN Z Our applications to network synthesis depend upon a correspondence which may be shown to exist between certain functions of z and certain power series in z. The functions of z may be formulated in terms of network singularities. The power series in z may be derived from the Tchebycheff polynomial series in p representing the corresponding gain and phase. The Tchebycheff polynomial expansion of a gain and phase function may be written: « + //3 = Z C^-n (16) If a + ^/3 corresponds to a finite network, it may be represented by the function of z in (13). At the same time, Tk may be represented by the function of z in (6). With these changes, (16) becomes: log< f n(-.4)l n 1 Same Rational Function in — 1/zj (17) ^' + 'V The logarithm of the product of the two rational functions, in z and — 1/z respectively, may be written as the sum of two logarithms. The series in sums of z'' and { — l/zY may be written as the sum of two series. Then log< f n(i-,^)] n(i-z' }+ log Same Rational Function in — 1/z (18) + Z^'^~''' The above expression equates the sum of two similar functions, in z and — 1/z respectively, to the sum of two power series, also respectively in z and —1/z. The theorem on which the synthesis methods are based asserts that the functions and power series in z and — 1/z may be equated separately, throughout the useful interval. That is: XETWORK SYNTHESIS ()23 When U I = 1, n(i-.T)] 0") rSamo Ha.ioual \ ^^J-T^^ [ r unction HI —\'z\ \ z The relation (18) dtu's not, l)y itself, require (19) to be true. (19) fol- lows from (18) if and only if the function of z has a power series expan- sion in^'ol^■ing only i)ositive powers of z, and the function in —1/z has a power series expansion in — l/z, with the same coefficients. This added condition, howe^'er, is readily established for the useful interval. f Combining (19) and (16) yields a most useful relationship connecting the z-plane mappings z^ , of the network singularities />„ , and the coefficients Ct , of the TchebychefT polynomial expansion of a + z/3: Eife' = log A'.- ^ "'^ (20) n 1 In more qualitative terms: The transfornmiion from variable p to variable z converts an expansion in Tchebycheff polynomials in p into an expansion in a power series in z. Thus, b}' working with the Za , in place of the Pa , one may use a power series sort of analysis in calculating, or in choosing, the coeffi- cients C'k in the Tchebycheff polynomial series. The relations (20) refer to the combined gain and phase function. Exactly similar relations can readily be obtained, however, for gain and t As defined in (8), | s„ | > 1. In the useful interval, \ z \ =1. Hence | z/z, | < 1. It follows that log (1 — z/z^) has a power (MacClauren) series expansion in posi- tive powers of z, convergent on and within the circle | 2 | = 1. Finally the first logarithm in (19) may be expressed as a sum of logarithms of this simple type, each of which may be expanded separately. Sul)stituting —l/z for z maps the unit circle onto itself. It follows that the second logarithm in (19) has an expan- sion in positive powers of —l/z, in the useful interval, provided the first loga- rithm has an expansion in positive powers of z; and the coefficients in the two series will be the same. 624 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 phase separately. These may be derived from (14), and take the form: « = X CkTk j 11 I ^ ^'2 J > k, even CkZ = logA, 7 -^ n(i-4.)J E c,z' = log n 1 - z 1 + z ^(T n i^a z Z 1 + "7 2^ 1 - Za (21) > k, odd (The absence of factors | in ^ Ct^*", as compared with (20), reflects the factors 2 associated with a and /3 in (14).) 8. REPRESENTATION OF ASSIGNED GAIN AND PHASE In synthesis problems, the network gain or phase, a or 13, is to ap- proximate an assigned (ideal) gain or phase, say a or ^. To make effec- tive use of the 2-plane analysis, in network synthesis, we need to describe a and jS by relations analogous to (20) and (21), which express a and /3 in 2-plane terms. These relations, while similar to (20) and (21), must take a more general form (since a or ^ need only be approximately the gain or phase of a finite network). For our present purposes, the ap- propriate relations are those noted below. Let a + i^ be any function of p which has the following properties: It must be analytic throughout the useful interval. Further, there are to be no singularities within a (p-plane) distance e of the useful interval, where e is finite (but may be small). Finally, at real frequencies, a and tj8 are to equal respectively the even and odd parts of a + i^. Under the conditions stated, a + i^ may always be expanded in terms of our Tchebycheff polynomials Tk . Let ^ CkTk be the expansion. To obtain a parallel to (20), we may form (arbitrarily) a power series XI hCkz''. Then we may defiyie a function R{z) by identifying log R(z) with the power series. All this adds up to the following, comparable to (20): « + ?^ = Z CkTk E hCkz' = log R{z) (22) NETWORK SYNTHESIS 025 The luuctions of z luive the foUowitijj; properties: Because of tlie mild restrictions, wiiich we have imposed on the singularities of a + /jS, the series zL ^^~' defines a function which is analytic within, and on the circle \z\ = 1. Then R{z), also, is analytic within, and on the circle. Further, R{z) has no zeros anywhere in the same I'cgion. (Kiz), however, may be more general than the rational fraction in (20).) Finally, because of the even and odd symmetries, required of a and i^, (22) may be broken into the following parallels of the equations (21): a = 2_/ CkTk T.Ckz' = \og[R{z)R{-z)] i$ = Z C,T, Z C,z' = 1 R{-z, k, even /,-, odd (23) In some applications, it is possible to express R(z) in closed form. In all applications, it is possible to expand R{z) as a power series, convergent in the region 1 ^ | ^ 1. The same is true of 1/R(z), since there are no zeros in the region. Coefficients of either series (R{z) or 1/R(z)) may readily be calculated by means which we shall examine a little later. For the present we shall say merely that R{z) is a known function, corre- sponding to an assigned a + i^. \). A DESIGN CRITERION When the gain and phase function, a + i,5, is to approximate a + ?'^, the error in the approximation is (a — a) + i(l3 — j8). The error ma}' be expressed in terms of * by taking the difference of corresponding equations in (20), (22). The difference of the logarithms may be ex- pressed as a single logarithm of a ratio. Alternatively, and also more conveniently for our later piu'poses, it may be expressed as the negative of the logarithm of the reciprocal ratio. Ayhen this is done, {a- a) + i((3 - ^) = E (C, - On z E Kc-.- - c<)z' = -log { ^ n 1 n 1 KW (24) Consider the following arbitrary requirement, as a design criterion: The series 2_/ ^kTk is to match exactly the series iL CkTk , through 626 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 terms of order m. If both series have converged to small remainders when k = m, this criterion will surely make a -\- i^ a good approxima- tion to « + ^^^-t 111 terms of the coefficients, the criterion requires: C, = C, , k S m (25) If (25) is applied to the second equation of (24), the power series is zero through terms of order m. In other words, the logarithm, equated to the series, will approximate zero in the power series, or "maximally flat" manner, to order m. The logarithm is zero when the expression in brackets is unity. Further, the logarithm \vi\\ approximate zero in the maximally flat manner when, and only when the bracket approximates unity in the maximally flat manner. Thus a condition which is equivalent to (25) is the following: 1 n(i-7) i \ W .^(^) ^ 1 ^ ^m,.l _^ ^.+2 . . . (26) n(.-|,) This may be represented symbolically by 1 nfi-T^) _ . ^ V ^- / .ii(^) = 1 (27) ^-nd- where = is used to indicate equality through power series terms of order w. When (27) is applied to network sjaithesis, the singularities 2<, , and scale factor Kz are the unknowns, while R{z) is known. If m is equal to the total number of Zo , (27) will determine the network function com- pletely. When jn is smaller, (27) will furnish m + 1 relations between the network parameters (including the zero order condition), which may be combined with specifications of other sorts. Since (27) is equiv- alent to (25), this procedure amounts to the determination of network parameters which will yield assigned values of the coefficients, Ck = Ck , k ^ m, in the Tchebycheff polynomial expansion of a + i(3. Equation (27) applies when both gain and phase are to be approx- imated. For approximation to gain only, or to phase only, similar rela- tions may be derived from (21) and (23). Only even ordered Tchebycheff t When both residues are relatively large, the approximation may still be good, for the remainders maj' be quite similar, and the error will be their differ- ence. In practical applications, this is a not uncommon situation. NETWORK SYNTHESIS 627 polynomials contribute to gain. The following condition turns out to be the eciuivalent of C-2k = C^k , k ^ m: Kl n('-a R{z)R{-z) = 1 (28) where = means approximation in accordance with a power series of e\'en ordered terms, through order 2m. Correspondingly, only odd ordered Tchebycheff polynomials contribute to phase. The following condition is equivalent to C2a_i = C->k-\ ,^^— 1 to ni'. 1 - -77 1 + 1m n — ^ n — ^-#^ = 1 (29) i -f- // 1 — / The remaining sections (except the last) develop in more detail the application of s-plane techniques to more specific synthesis problems, of various sorts. Most of these (but not quite all) are based directly on (27), (28), or (29). The exceptions use a modification of (28), in which the function of z on the left is retained, but with the zeros and poles me adjusted for a different kind of approximation to unity, = but not = . In all cases, unity is approximated with one of the functions appear- ing in (27), (28), (29). It will be convenient to use H{z) to represent the error in the approximation, or departure from unity. When gain only is of interest, the function in (28) is used, and H{z) is defined by: . n (i - f) R{z)R(-z) = 1 + H(z) (30) In developing the specific techniques, we shall start with a very definite, rather special example, in order to illustrate the techniques with specific operations. This will be discussed in considerable detail in Sections 10 through 14. Thereafter we shall examine how these specific operations may be generalized, in a number of different respects. 10. AX INTRODUCTORY EXAMPLE The example which has been chosen for detailed discussion is the equalization of the gain distortion produced by two resistance-capacity 628 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 type cut-offs. The equalization is to be accomplished with a network which has n natural modes, but no finite frequencies of infinite loss. (This is simply one of the arbitrary specifications which define this problem.) The two cut-offs may be due to circuits or devices at two different points in a communication system, which may be represented schemat- ically as in Fig. 5. Their effect can be described in terms of two assigned natural modes. Two assigned modes are assumed, instead of only one, because a single mode would make the problem too simple. Our present purposes will be served well, however, if we simplify the problem by requiring the two assigned modes to be identical, say at p = po . Fig. 5 — A system with two resistance-capacity type cut-offs. The two natural modes would be cancelled completely by two infinite loss points at the same location in the p plane. A network with two infinite loss points, however, is not physically possible unless it has also at least two natural modes ; and the natural modes will have to introduce distortion of their own. Thus no finite network will give perfect equali- zation of unwanted natural modes. Sometimes it is desirable, in practice, to use an equalizer configuration which produces no finite frequencies of infinite loss, the entire equalization being accomplished by a suitable choice of its n modes. Configurations of this sort are illustrated in Fig. 6. Thus, our simple illustrative problem, though chosen to introduce principles, is also of some practical interest. The exclusion of finite frequencies of infinite loss simplifies the repre- Fig. 6 — Configurations which produce no finite frequencies of infinite loss. NETWORK SYNTHESIS 0)29 sentatioii of the network <;;uii (v. In (121 ), [\\c z„ ('orrespond to finite^ fie- ciueiu"i(>s of infinite loss, and aic to We omitted when there are to l)e natural modes only. What is left is the lo<;ai'ithm of the rcM-iprocal of a polynomial, which is of course the u(»f>;ative of the lojj;ai'ithm of the polynomial itself. Tims a may he described as follows, for this particular application: Z Co,^' = -h)s K:U(i -K) 1. Otherwise, the function of z in (31) will have no power series expansion over the useful interval \z\ = 1 ; and (31) will not, in fact, determine the gain a over the useful interval. It turns out, however, that the condition does not give trouble in the synthesis of natural modes, when there are no arbitrary frequencies of infinite loss. This may be demonstrated l)y the argument outlined below. The z„ are zeros of the polynomial in (34), which we have given the special form (36), by applying (35). A function theoretic test f or | 2, | < 1 632 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 makes use of the contour in the complex plane for the polynomial , corre- sponding to the 2;-plane circle \ z \ = 1. (This is like a Nyquist diagram except that the contour for the variable, z, is different.) There will be \ Za\ < 1 if and only if the contour for the polynomial encloses the origin. Now the polynomial in (34), and (35), is merely a special case of the function on the left in (28), and (30). For this special case (30) becomes E K,z"' = 1 + Hiz) (38) The polynomial cannot enclose the origin without passing thrcjugh some negative real value. But this requires an ] H{z) \ > 1, at some point on the contour in question, | ^ | = 1, which happens to be also our useful interval. On the other hand, a — a = 0 when ^ Kkz'''' = 1, and H{z) is in the nature of a correction term, which is small in the useful interval when a — a is small. The conclusion is: There will be no | 2, | < 1 unless the approxima- tion, a to a, is so poor that a — a exceeds several db in the useful inter- val. Besides the requirement \ z„ \ > 1, the z„ must meet physical restric- tions, which we found to be the same as those limiting the natural modes p„ . The Za may be calculated as follows: The z„ are roots of the polynomial in (35), in terms of z~. All the roots in terms of z' are z„ , except the two required roots at zl , which correspond to assigned gain a. Each Za is a square root of a 2^ . There are two possible square roots, however, differing only as to sign. As far as gain a is concerned, either choice of sign is permissible; for a depends only on z^ . For a physical network, however, the choice must be such that Re z^ < 0. This choice is possible if, and only if -y/zi has a finite real part. A pure imaginary Za corresponds to a negative real z„ , and thus negative real roots in terms of z"^ are excluded by physical considerations. Table I lists both zl and z„ for a number of values of n. When n is even, all roots are physical. On the other hand, when n is odd, one root is always non-physical. In a sense, an odd n is not really appropriate for the present illustrative problem, with any physical design. An odd n must necessarily bring in a real natural mode, which merely increases the sort of distortion we are trying to equalize — that is the distortion due to unwanted real modes. The following argument substantiates the suggestion, and also illus- trates manipulations of a sort which are frequently useful in more gen- eral applications: The highest order coefficient in (34), Kn+2 , may be set aside for adjustments to satisfy physical requirements. The rest of NETWORK SYNTHESIS 033 Table I — Z-Plauc Natural i\fodes for Equalization of Two Identical Unwanted Modes n zMz\ V4N^ zj\zo\ 1 -.5000 0 ± i .7071 Non Physical 2 -.3333 ± i .4714 ±(.3492 ± i .6747) -.3492 ± i .6747 3 -.6059 0 ± z .7784 Non Physical -.0720 ± / .6384 ±(.5340 ± i .5977) -.5340 ± i .5977 4 +.1378 ± i .6782 ±(.6441 ± i .5264) -.6441 ± i .5264 -.5378 ± i .3582 ±(.2328 ± / .7695) -.2328 ± i .7695 5 -.6703 0 ± / .8187 Non Physical + .2942 ± i .6684 ±(.7157 ± / .4670) -.7167 ± i .4670 -.3757 ± i .5701 ±(.3918 ± i .7275) -.3918 ± i .7275 the coefficients may then be chosen to eliminate terms from the series 23 {C-ik — C-ikYr-ik , representing a — a, subject to the previous condi- ne (n— l)e tion that two zeros must be z' = zl . Tliis replaces = by = , in (35), and changes (3G) to: E it.^"- = 1 - (n + 1) (l)" + n (^ij 1 i> In (~1 2\2 (39) If n is odd, all the roots z^ can be physical only if iv„+2 is negative. On the other hand, any finite negative it„+2 leads to a larger error, a — a, than K"„+2 = 0. Reducing /C„+2 to zero is the same as reducing the degree of the polynomial by one, which amounts to reducing n by 1, from an odd to the next smaller even integer. In other words, a physical design with an odd number of natural modes is less effective, for the present application, than a simpler network, with the next smaller even number of modes. Note that the z„ in Table I are proportional to Zo . This means that root extraction methods need be used only once for each value of n, after which the roots may be cjuickly adjusted for any value of zq , corresponding to any assigned value of the two identical modes, po . 13. ACCURACY The accuracy of a completed design can be checked by calculating a from the natural modes p„ , and comparing a with a. It is important, however, to have at least some information about accuracy in advance 634 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 of the detailed calculation of the p, . Otherwise, it may be necessary to carry out several designs, in all detail, in order to obtain one satisfactory design. The needed information about accuracy can in fact be obtained from the error function H{z), which we formulated for general gain applica- tions in (30), and for the present application in (38). The analysis which yields (15) may be used to obtain a very similar expression for a — a, in which R{z)R( — z) appears in combination with the rational function of z from (15). It may be expressed in terms of the error func- tion H(z) of (30), as follows: a - a = -log I 1 + H(z) I (40) When H{z) is zero, a — a is zero. When H(z) is small, a — a depends on phase H{z) as much as on \ H{z) \ . When H{z) is a positive real, a — a is negative. When H(z) is imaginary, a — a is very small. When H{z) is a negative real, a — a is positive. When H(z) is complex, | a — a | is always smaller than with a real H(z) of the same magnitude. The last statement may be expressed as follows: - log {1 + 1 H(z) 1} ^ « - a ^ - log {1 - I H(z) \} (41) The left hand relation is an equality when phase H(z) is an even num- ber of IT radians; the right hand side, when it is an odd number of x radians. In the useful interval, where z = e'*, the H(z) corresponding to (36) is as follows: H(z) = -(n + 2) (^y + (n 4- 1) (jj \H{z)\ = _2n+2 ^0 in + 2)zt (42) phase H(z) = tt + (2w + 2)0 -f- phase ^ 1 — 7 — ZlToV^ ^^ * As CO varies from 0 to Wc , 0 varies by - radians. The corresponding phase of H(z) varies by (n -f l)7r radians, which means that H(z) is successively positive real, imaginary, negative real, imaginary, through n + 1 half cycles. This accounts for the oscillatory nature of the a — a curve, illustrated in Fig. 7. The amplitudes of the oscillations are fixed by \ H(z) \ , which varies relatively slowly. Specifically, the two logarithms in (41) determine NETWORK SYNTHESIS G35 envelopes, between which the actual error curve oscillates. These are the dashed lines in Fig. 7. The maximum error, in the useful intcn-val, is determined by the maximum value of the envelopes, i.e.. 2o n + 1 I' (43) This function is plotted in Fig. 8, for various values of n. The abscissae "distortion before equalization" represent distortion relative to the median loss in the useful interval, or one half the total variation in the interval. (This is a function of the top useful frequency coc , relative to the assigned natural mode po ; and (7) makes Wc/po a simple function of 2o .) The figure is convenient for estimating the values of n needed for specific applications. The various ripples in a — a do not all have the same amplitude, (43). For some values of n and 2o , the amplitudes are almost uniform; for others they are quite variable. A measure of the variability in ripple 0.1 0.08 0.06 0.05 -' 0 04 M 0.03 5 0.01 a ^ 0.008 cc t= 0.006 11. < 0.005 O 0.004 O 0.003 0.001 0 / / / 1 / / / / / TZZi / / TZII / / ; / 0=2/ / i \ / 10 — 1 — — J -j — \ / / IT 1 / / zt 1 / / 1 / / 1 / / f 1 1 / / 1 0.2 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 DISTORTION BEFORE EQUALIZATION IN DECIBELS Fig. 8 — Distortion before and after equalization — n natural modes equalizing 2 identical natural modes. 636 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 amplitude, across the useful interval, is: |/^(^)|n,ax. ^ (n + 2)zl + (n + 1) \H{z)Lin. (n+ 2)zl - (n+ 1) 14. APPROXIMATION IN THE TCHEBYCHEFF SENSE (44) The above analysis suggests a way of improving the design deter- mined l)y (37) (or the equiv^alent, (36)). An optimum a — a is commonly one which has the following properties, in the useful interval: A maximum number of ^' ripples/^ all maxima of \ a — a \ equal. (This usually minimizes the largest departure in the useful interval, thereby yielding an "approximation in the Tchebycheff Sense.") Since the variation in phase H(z) determines the number of ripples, while I H(z) 1 determines the amplitudes of the ripples, the abo\'e conditions will be met if H{z) has the following properties, in the useful interval: Phase H(z) as variable as possible, , . \ H(z) I constant. These conditions may be regarded as alternative design criteria, re- placing C-2k = Cofc . They can in fact be applied to our special example, and also to certain other special problems which will be noted later. For more general applications, a suitable H(z) can be defined, but no reasonably simple procedure has yet been found for calculating the re- quired constants. (The difficulties will be particularized in a later section.) For the present example, (38) may be used to replace (34), and hence also the second eciuation of (33), by: E (C2. - Cn)z' = - log [1 + H{z)] (46) (33) requires H{z) to be a polynomial in z', of degree n + 2, with two zeros of [1 + H{z)\ at z' = H . The object is to find an H{z) of this sort, which also satisfies (45), at least to a good approximation. The following H{z) does in fact exhibit the required properties: H{z) = Gz [1 _ J/,.] W') The function is a polynomial because the factor [1 — /""^V^"""*"^] is divisible by (1 — J/z^). The constants J and G are to be chosen to NKTWOIIK SYXTHKSIS ()37 give the required double zero of [1 + H{z)] at z' = zl . One value of J, so determined, is real and of order I'zl . This is the appropriate solution. Then ] ,/" "/'""^ I is "f order l/lo"^^ , when j z | ^ 1. This suggests the following appi'oxiniation in place of (47): H{z) ^ Gz^'^^' f^l^ (48) 1 — J/z- Tiie approximation is at least as good as 1/zl"^* , compared with unity, both in the useful interval and in the neighborhood of the singu- larities 2o , and 2ff . This means that the approximation can be used: in estimating the error a — a (in the useful interval), in calculating ./ and G, and in finding the roots z„ . In the useful interval, \ z \ = 1, and therefore 1/z = z*. Then (1 — J/z') is (1 — Jz~)*; and their ratio has magnitude unity. Thus 1 H(z) I = I G 1 , in the useful interval, to order of l/zl"'^* compared with unity f. With \ J \ < 1, phase H(z) varies over the useful interval to the same extent as the phase of /""^"'.J Fig. 9 illustrates the difference ill a — a, as determined by (42) and (48). These curves, however, are for single values of n and Zo ; and the improvement obtained by using (48) would be different with different values of n or Zo . The values of / and G, determined from (48), and the requirement that [1 + H(z)] must have two zeros at z^ = zl , turn out to be: J = 71 + 1 I 2 1 + -4 + a/(i - _^Y 4- . .^^.... (49) "^""^-g+t /(l- z'oj ^ {n + 2)% 1 1 - J/zl -2n+2 1 7--2 Zo 1 — JZo G = - _^„^^ 2o 1 JZo Xote that this J is in fact smaller than l/zo . 15. GENERALIZATION The several sections preceding describe a quite specific example, as an introduction to synthesis applications. The next several sections de- scribe how the specific methods of the example may be generalized, in several respects. First, the ideal gain, a, is generalized, so that it need not even have the sort of functional form associated with finite networks. Then, the t The I H(z) I determined b}' (42) is constant only to order \/sl . t This is the most we can expect, when we have n singularities, whicli can pre- vent the dominance of only lower order terms, through z^". 638 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 EQUAL RIPPLES r\ 1.0 0.6 C2k= C2k,k54 1 "N, 1 1 1 r\ ' Ai 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 \, 1 i \ .'/ ' N 1 1 1 / \ \ k \ \ \ \ \ \ \ / / / / / / ( / \ 1 \\ \\ \\ 1 1 1 \ \ \1 \ \ 1 1 1 , 1 1 \\ \\ \\ \\ i 1 1 1 1 1 1 \ i \\ \\ \ 1 1 1 1 1 1 J \ nW \\ji i -0.8 1 ; 1 \ ' -1.0 ''J_ 1 \ / -1? 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 9 — Comparison of design procedures — four natural modes equalizing two natural modes at po = — r^ uc. approximating network gain is generalized, by introducing arbitrary frequencies of infinite loss, in addition to the arbitrary natural modes. The methods are also modified for approximation to an assigned phase, instead of gain, or to phase and gain simultaneously. Finally, the anal- ysis is modified to permit useful intervals of the "high-pass" type, or (in the case of gain simulation) of the "band-pass" type. 16. APPROXIMATION TO A GENERAL ASSIGNED GAIN a If we now permit the assigned gain a to be general, in the sense of Section 8, we must return to the formulation: J2C2kz"' = log [R(z)R{-z)] (50) NETWORK SYNTHESIS 639 If we retain simulation with a network which has n natural modes, and no frequencies of infinite loss, we must retain the formulation: a = 2-> C2kT2k z c../^ = -log Ki n (i - 1) , - = 1, ^^^^ n The corresponding formulation of the error is (in place of (33)): OC — a = 22 (^2fc — C2k)T2k Z (C2. - Cu)z"' = -log n(i-|)-^(^)^(-^)] (52) For Cofc = Cik , J<^ ^ m, the following special case of (28) is now required : 1 - -2YR{z)R{-z) = 1 (53) Now the reciprocal of R(z)R( — z) has a power series expansion, in the region of interest. (Recall Section 8.) It follows that (53) may be multiplied by this quantity, without damaging the equality of power series coefficients. In other words (53) is equivalent to: Let Kk be the coefficient of z'^ in the polynomial expansion of the left hand side; and let Kk be the coefficient of z''' in the infinite series ex- pansion of the right hand side. Then, Kl n (1 - I) = Ko + K,z' + • . . Kr.z''' (55) R(z)R{-z) Substitution in (54) gives me K, + K,^ + • . . KJ- = E Kkz"' (56) In other w^ords, Kk = Kk , k ^ m (57) These relations are directly applicable to network synthesis, provided 640 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 the coefficients Kk can be calculated. Formulae for their calculation may be derived in the following way: If (55) is substituted in (50), the result is-.f a = T. C-2kT2k (58) Z Cuz'' = - log Z Kj' When 2 = 0, the second equation reduces to: Ko = e-^0 (59) If the functions of z are differentiated, a simple rearrangement gives: E kK,i'-' = [ - Z kC.J'-'] [Z Kj'] (60) The right hand side may be expanded as a single power series, and then like powers on the two sides may be equated separately. The result is: ivi = — C 2ivo 2Zo = -C-2K1 - 2CiKo (61) SKz = — C2K0 — 2CiKi — SC^Ko Synthesis calculations may now be carried out in the following stages. The assigned gain a. is expanded as a Tchebycheff polynomial series, to determine coefficients ^2^ , say through order k = n. The equations (61) are then used to calculate coefficients Kk , also through order n. Each successive coefficient is computed in terms of those previously deter- mined. Note that the Kk , k ^ n, are fixed by the same number of Cik — that is, orders k ^ n. Equation (57) is now applied to identify Kk with Kk , k ^ w. If all the network degrees of freedom are to be used to get Cok = Cok , index m = n, and (57) determines the polynomial in (55) completely. Other- wise, m < w, and coefficients K„,+i to Kn are to be adjusted in accord- ance with specifications of other kinds. When all the Kk have been determined, the singularities z„^ are found by root extraction methods, applied to the right hand side of the first equation of (55). The previous example might have been carried out in these terms, but happened to be simpler in the terms used. If (32) is regarded as a special case of (50), and if (32) is simplified (for purposes illustration) by using K, = 1, the corresponding R(z)R( — z) becomes simply f 1 - ^y Then Z Kkz'' is t We may think of these equations as defining an infinite network, with na- tural modes only, which would match the assigned gain a exactly. NKTWOKK SYXTIIKSIS ()41 2 ^'^'' = iT^iw = s (^ + « (ly (02) thus Kk and Kk become k + 1 Kk= .2k , k = 0 to 00 ~0 /v = 0 to n (63) If these Kk are used to evahiate the polynomial on the left hand side of (53), in accordance with (5o), and if the polynomial is then multiplied by the above special R(z)R( — z), the result is exactly (36). The error function H(z), of (30) and (42), may now be defined as follows : Kl n (l - I) Riz)R(-z) = 1 + Hiz) (64) The error a — a is again: a - a = - log I 1 + H(z) 1 (G5) If (64) is used to express (53) in terms of H{z), a "1" mny be sub- tracted from each side of the relation, to get me H(z) = 0 (66) ^^'hen tn = n, this requires an H(z) of the following form, in terms of the coefficients Kk derived from R{z)R{ — z): fr 2n+2 , ^ 2n+4 , H{z) = -^l^ "t^in, ^ '" (67) 17. CHARACTERISTICS OF Z„ As in the previous example, \z„\ > I when the approximation, a to a, is at all reasonable. The z^ are again zeros of 1 + H(z); but now H{z) is defined by (64). If | H(z) | < 1, when \z\ = 1, there will be the same number of zeros of 1 + H(z) as poles, in the region \ z \ < 1. Any poles would have to be poles of R{z)R{ — z). In Section 8, we noted that this function is regular in the region \z\ ^ 1. Hence, there will be no poles, and there wdll be no | 2, | < 1, luider ordinary accuracy con- ditions. As before, the 2<, can be chosen in accordance with the physical con- 642 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 ditions, provided none are pure imaginaries. Since an imaginary z^ is a negative real 2<, : There must he no negative real Zg . There will be no negative real zl if the polynomial ^ KkZ^ in (55) is non-zero at all negative real z". If the requirement is violated, initially, one or more Kk , of the highest orders, must be modified. Graphical methods are likely to be useful for this, combining plots of the original polynomial, and proposed changes. An approximation of the form C2k = Cik , k ^ m, will still be realized, but with m < n. The error function corresponding to m < n is as follows, in place of (67): H(z) = ^^^, (08) 18. ACCURACY The accuracy of match again may be estimated by the means of (41), using H{z) of (67) or (68). H{z), however, may not be so easily cal- culated as for the previous example. A simpler but less reliable estimate of accuracy is furnished by the error in the first unmatched coefficient in the Tchebycheff polynomial series. If Kk = Kk through k = m, the leading terms in various series are as follows. First, from (64) and (68), Kl n (l - I) Ri^)Ri-z) = 1 + ^-^'-^^-^' .-^^ . . . (69) using this in (52) gives : {Cu - C2k)z = -log 1 -f I ^ ) z Then, from the properties of logarithms, Zfn n ^,2fc Am+l — -Km+l 2m+2 /^iN (C2jfc — ^ik)^ = — i> z •" wi; -ft-o Consequently (also from (52)) : K^+l — Km+l , . a — a = ^ 1 27H-2 • • • K'^J Ao This is the same as the leading term of H{z), except that z'"''^^ is re- placed by — T2m+2 . If m = n, the same equation holds with /v,„+i = 0. (70) NETWORK SYXTIIESIS 643 The coefficient """^^ ^^ in (72) is a sort of average of the enve- lopes of llie ripples in a — a. The variability of the envelopes, across the useful interval, depends upon higher order coefficients, in compari- son with the leading term. Calculation of higher order coefficients is relatively complicated. 19. APPROXIMATION IN THE TCHEBYCHEFF SENSE The criteria (45) carry over to general assigned gains, as conditions on H(z) which, if realized, are usually sufficient to establish approxima- tion in the Tchebycheff sense. For this purpose we must use the H{z) of (04), rather than (G7) or (68) (which correspond explicitly to Cu = C2k , k ^ m). In terms of the polynomial and series representations of (55), the H(z) of (64) becomes: /C. + g.z' ■ ■ ■ K.z''- "^'^ = E^? " ^ ^'^^ The followang somewhat special problem is easily solved, in these terms, and has a direct bearing on various quite different synthesis techniques: A network is to be designed which combines the functions of an equalizer or simulator, with those of a filter, or selective network. In the useful interval, an assigned gain variation a is to be approximated in the Tchebycheff sense. At higher frequencies, there is to be a rapidly increasing loss, or "sharp filter cut-off." The number of natural modes, n, is to be more than sufficient to match a to the required accuracy, in the absence of a selectivity re(iuirement, the latitude being used to pro- duce the required sharp cut-off. In particular, n is to be large enough so that an n term match of Tchebycheff coefficients produces errors that are negligible compared with those accepted as a price of the sharp cut-off. On the above assumption of an ample n, the infinite series ^ K^z^ in (73) may be truncated after the term of order n, and the errors due to the truncation may he neglected in calculating the design error a — a.] Then (73) becomes „, , g. + g.z' ■ ■ ■ K^z" t The truncated series is merely the polynomial on the left side of (56) which tvould he obtained if the filter selectivity were ignored, and m were given the maximum value, n. 644 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 If a sharp cut-off were not required, this approximate H{z) could be made exactly zero, by using Kk — Kk for all coefficients. Then the actual design error would be determined by the approximation inherent in the use of (74) in place of (73). For high selectivity, however, Kn should be much larger than Kn , as large as possible within assigned limits on a — a in the useful range. (It is readily established that K„2" will deter- mine a at asymptotically high frequencies.) The other Kk are then to be adjusted so that a — a exhibits the desired "equal ripples." The following H{z) has the functional form (74), and also meets con- ditions (45): K\ Kn H{z) = Gz (75) Ko + Krz' • • • KnZ Multiplying G/" into the numerator gives a rational fraction which is obviously consistent with (74). The coefficients Kk of (74) which corre- spond to (75) are: Kk = Kk + GKn-k (76) In the useful interval [X) Kk/z^^\ is [^ KkZ^''\*. Hence the polynomials in z and \/z have identical magnitudes, in the useful interval; and, since also \z\ = 1, | H{z) \ = ] (7 | in (75). The phase variation, over the useful interval, is the same for H{z) as for z ", which yields the same number of ripples in a — a as an ordinary Tchebycheff filter of like degree, t The constant G is arbitrary, except that its sign must be properly chosen to avoid non-physical natural modes. Increasing G increases the filter selectivity, but also increases a — a in the useful interval. G and n are to be chosen together, to realize an assigned selectivity within an assigned limit on distortion. The above analysis may be related to the following filter problem: Required to design a filter which has flat gain, in the useful interval, but which has m assigned frequencies of infinite loss, in addition to n arbitrary natural modes (m ^ n). The n arbitrary natural modes may be regarded as compensating for gain variations due to the assigned frequencies of infinite loss, in the useful interval, while reinforcing their effects at other frequencies. Compensation of effects of the infinite loss points is the same as simulation of the effects of natural modes at the same (assigned) frequencies. The approximation in the useful interval t This assumes an a with the general characteristics described in Section 8, which are such that the numerator and denominator of the fraction in (75) will each have a net phase shift of zero, across the useful interval. NETWORK SYNTHESIS G45 is to be no better than necessary, so that there may be a maximum reinforcing of kjsses at other frequencies. In these terms, (58), and ^ Kkz' in (73), correspond to the assigned natural modes (at the same locations as the assigned frecjuencies of infinite loss). Then the ideal ^Z KkZ~'' is itself a polynomial, of degree yn ^ ?i, and (74) is e.xact, rather than an approximation to (73). Then (70) determines the n arbitrary modes in such a way that the net filter gain approximates zero in the Tchebycheff sense, over the useful interval. A different procedure for obtaining the same result is described in the author's paper "Synthesis of Reactance 4-Poles".'* The above analysis of the filter problem is of interest in relating the more general synthesis techniciues, in terms of Tchebycheff polynomial series, to previous filter theory. Similar filters have also been obtained by Matthaei , on a potential analogy l)asis. He includes, however, somewhat more general filter char- acteristics, for which he obtains only approximately equal ripple errors. Analysis of the sort described abo\-e may be used to clarify Matthaei's analysis of the conditions under which he obtains exactly equal-ripples. Equation (75) may be related to work of Bashkow . The (arbitrary) amplitude of the (equal) maxima of | a — « | , computed from H{z) of (75), depends only on \ G \ . The frequencies at which the maxima occur correspond to phase H{z) = st, which is independent of \ G \ . Thus, the locations of the maxima are invariant to the arbitrary amplitude, ivithin the range where (75) applies. (75) applies only when (74) may be used in place of (73). Generally, (74) only approximates (73), and the approximation introduces small variations in the maxima of | a — a | (when a corresponds to (7G)). If the maxima themselves are sufficiently small, the small variations will be large percentage variations; and the adjustments to compensate for the variations will yield significant shifts in the location of the maxima. In other words, the locations of the maxima of \ a — a\ , ret^uired for equal amplitudes, are largely invariant to the magnitude of the equal amplitudes, but only to an approximation which becomes worse as the amplitudes are decreased. Bashkow states the invariance of the freciuencies of maximum I a — a I , as a more or less empirical conclusion, based on a quite dif- ferent approach to the same synthesis problem. Equation (75) may be related also to work of Kuh. The natural modes z^ are zeros of 1 + H(z). In other words, H(za) = —1. Using the H{z) of (75) gives the following: o 0,, '>n I - -^1 Kn\ Ao + Kiz; + • • • Knz; = -Gz; n" . There- fore (85) requires zero coefficients in the expansion of the right hand side, from order n" + 1 to order n" + n'. Equating these coefficients to zero gives 7i' linear equations in the n' unknown Kk . Solving for the Kk determines polynomial D. The values calculated for the Kk may then be used in lower order coefficients of the expansion of the right hand side of (85), which are exactly the coefficients Kk of N. When n" — n' = 0 or 1, a continued fraction method is likely to be preferable. Various established techniques f may be used to convert the series ^ Kkz' into a continued fraction of the form: 22 KkZ^'' = ao + a 1 _. 1 z'^ . 1 (86) a2 + z- ai t See, for example, Fry's applications of continued fractions to network de- sign.' NETWORK SYNTHESIS G49 rf the continued fraction is truncated after the term of oi'der m, and is rearranged as a rational fraction N D, it will obey e(iuation (84). The degrees of A^ and D will be sucii that n" + n' = m, and n" — n' = 0 or 1. The contiinied fraction may be associated with the hypotlietical ladder network shown in Fig. 10, with variable impedance shunt branches proportional to z'. The impedance of the (tnuicated) ladder is N/D. 21. .VC'CUU.VCY The accuracy of match, a — a, may again be evaluated from the final network singularities; or by (41), with H{z) as in (30), before the singularities have been determined from roots of A'^ and D. A rougher estimate may again be obtained from the error in the first unmatched term in the Tchebycheff polynomial series. As before, (e([uation (72)), this is equal to the leading term in H{z), with z "'^' replaced by — T-2m+2 • The rational fraction in (30) is the same as our present N/D. In terms of N/D, (30) becomes: ^R{z)Ri-z) = 1-}- H(z) (87) If (8(i), or Fig. (10), is used to determine .V and D, the leading term in H(z) turns out to be: H{z) = ( ^ m+l -,2m+2 The corresponding mismatch in Tchebycheff polynomial terms is: (-)"' C-lmArl — C; m+2 (aia2 • • • amf'ttrti+xKo (88) (89) 22. ZEROS AND POLES When frequencies of infinite loss are to be chosen, as well as natural modes, the situation in regard to | 2<, I < 1 is somewhat less favorable. o V\Ar 82 a, A/A — r zf 33 Fig. 10 — A ladder network, representation of the continued fraction (86). 650 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 It is still true that 1 + H{z) will have the same number of zeros as poles in the region \z\ < 1, so long as a — 5 is reasonably small in the interval | 2 | = 1. In equation (87), however, the poles of 1 + H(z) include the zeros z„ of D (the arbitrary infinite loss points), as well as the poles of R{z)R{ — z). When the coefficients of D are to be chosen as in the previous section, the contour rule merely says that any z, and z^ in the region | 2 | < 1 will occur in like numbers. Fortunately, the frequent occurrence of | z„ | < 1 is softened by the following curious circumstance. Almost always, any | z, \ and ] z^ | < 1 are so nearly identical that factors {z — z„) and {z — z„) may be can- celled out without any important effect on H{z), or a — a. Cancellations of this sort were encountered a number of times before an explanation was discovered. Actually the explanation is quite simple. At any zero of 1 + H{z), H{z) = — 1. On the other hand, H{z) is small when \z\ = 1. Generally, it is much smaller in most of the inter- val \z\ < 1. For instance, when C-ik = Cok through k = m, H{z) is pro- portional to Z™'''^, in the neighborhood oi z = 0. As a result, | H{z) \ rarely becomes as large as 1, in the region | 2; 1 < 1, except in the very close proximity of a pole. In other words, in the region \z\ < 1, any zero Za is usually very close to a pole Za — usually so close that the corresponding factors z — z„ may be cancleled out without significant effect on a — a. The occurrence of non-physical natural modes {Rez^ = 0) is the same as before; but adjustments to correct for these, in an efficient manner, are much more complicated. In addition, there may be non-physical infinite loss points, z„ . To correct for non-physical singularities, the simplest procedure would be to change one or both of the highest order coefficients in N and D of (82), that is Kn" and Kn' . This would be inefficient, however, for it would spoil the match of Cn" to Cn" , or C„' to Cn' . The unmodified design, defined by (84), can match terms through order n" + n', and it is desirable to change only the highest order terms in adjusting the design. More efficient adjustments are in fact feasible. They sometimes re- quire an increased element of art; but the art may be based on specific principles. Some particularly useful principles are described in the next section. These apply to various other corrections besides correction of non-physical zeros and poles. Examples are reduction in phase to make two-terminal realization possible, and increase in shunt capacity in two-terminal designs. In general, they offer a means of making m < n' -{■ n" in (84), and using the remaining degrees of freedom to meet other conditions. NETWORK SYNTHESIS G51 23. MODIFICATION OF N/D Suppose Ni/Di and N2/D2 are two rational fractions, representing special choices of .¥ and D in (82), such that: AT *"!* Then suppose F is a function of z' such that F = 0 (91) The following combination represents an approximation to ^ Kkz'', of the order indicated :t m = rni , or Wo + ^i, whichever is the smaller. The left hand side of (92) may be used as N/D in (84), to match Cik to Cik through A: = m. Adjustment of the function F may be used to satisfy other requirements, in addition to accuracy specifications. Frequently, Ni/Di may be the rational fraction corresponding to ti-uncation of the continued fraction, (86), after the term in a„"+n' . Then N2/D2 is likely to be a truncation of order m < n" -\- n'. The corresponding F is likely to be proportional to /'', or at most a simple polynomial in /. When i^ is a constant, and m = 7i" -{- n' — I, the use of these particular rational functions in (92), to determine N/D, corresponds to matching C2k to C2/C through k = n" -\- n' — I, but leaving C2{n"+n') subject to adjustment. Specificially, C2{n"+n') depends on the choice of F, which may hinge upon such special conditions as the elimination of non-physical singularities. Problems which call for more complicated combinations are by no means uncommon. Skill may be needed in the choice of specific com- binations which will solve specific problems. Computations may be t The relationship is easily established by noting that: Ni - N2 - — - 2 Kkz'" ^ - 2 Kkz^ '^1±I^-^K^^^ = ^ ^F^ (93) D. + FD. D, A 652 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 systematized, to a considerable extent, by using the error formula (88), and other relations between the coefficients of the continued fraction, and the rational fraction truncations of various orders, f 24. APPROXIMATION TO BOTH GAIN AND PHASE The applications described in previous sections relate to approxima- tions to prescribed gain, a, without regard to the associated phase. Quite similar methods apply, however, to the simultaneous approxima- tion of gain and phase. The starting point is equation (20). Replacing products of factors by polynomials gives, in place of (81): a -{- i^ = ^ CkTk, k even and odd ZiCkz' = -log'^ The polynomials are now as follows, in place of (82) : N = lu + K['z ■ ■ • K"»2"" D = 1 4- Kiz • • ■ Kn'Z (If only natural modes are to be used, the suitable replacement for the first equation or (55) is here obtauied merely by using D = \, and K^ , z, z„ , in place of their squares.) A comparable expression is needed for the assigned gain and phase a. + Zj8. In place of (50), we now repeat (22), and redefine the coeffi- cients Kk in accordance with a + ZjS = ^ CfcTfc , fc even and odd E iC,/- - log R{z) = -log E Kkz' The definition of Kk has been changed in such a way that it is now related to Ck/2 exactly as it was previously (in 58) related to C^k . Equations (59), (61) may be applied to calculating the Kk by merely substituting therein a Ck/2 for every C2k ■ t For example, a simple recursion formula may be used to assemble the polj^- nomials A'^ and D which correspond to truncation of the continued fraction (86) at a number of different points. Specifically, P„ = P„_i -\ Pn-2 , where P a„a„_i is either N or D and P„ corresponds to truncation of the continued fraction after the term in a„ . The formula holds for n ^ 2. XKTNNOUK SV.XrilKSlS (),j3 E(iuatioii.s (84), (85), (8()) may now be modified, for the new A^, D and Kk , by merely using z in place of z' wherever it occurs (including z'' in place of 2"^).t The modifications of equations (84), (85), and the truncation of (86) after a„, now It^ad to (\- = Ck , A" ^ m, instead of the previous C-jt = C-ik . This means that //; must be twice as great to match coefficients out to the same actual ordiMs. This is to be expected since now one half of our design parameters are used to appioximate phase ^, leaving only half for approximating gain a. Eciuation (89) must be changed not only in n^gai-d to the orders of Ck , Ck , but also in regard to the factors | in (94), (9()). This gives 2 (aitto • ■ ■ am)" a,n+iKo The most important change is in regard to the zeros and poles z^ . The polynomials N and D now determine z^ and z„ directly, instead of their squares. There is no opportunity to adjust the sign of Re z„ by choosing the correct sign of y/zT^. When non-physical singularities ap- pear, adjustments of high order coefficients may be tried. Section 23 applies provided z is replaced by z. If the specification of the problem permits added delay, linear phase may be added to a + ?^ to increase the probability of phj^sical singularities!. (Addition of linear phase changes only Ci , in ^ CkTk . A negative change in Ci increases the delay.) 25. APPROXIMATION' TO AX ASSIGXED PHASE ^§ Sometimes it is recjuired to approximate an assigned phase, without regard to gain. More commonly, it is required to approximate an as- signed phase, using an "all-pass" network, which has a theoretically zero gain. These two problems, however, are very nearly identical, due to circumstances explained at the end of this section. For approximation to phase only, we go back to the /3 equation in (21). As before, products of factors {z — z^) are replaced by polynomial t and = in place of = . X The well known relation between the gain and j)huse of any j)hysical network (See for instance Bode'") may give some information regarding the reasonable- ness of 3. It must be remembered, however, that departures of network gain a, from the assigned gain a, outside the useful interval, may affect the permissible phase /3, within the interval. § Up to the present, apjjlications to phase problems have not been developed to the same extent as for gain. Techni k odd (98) Using n to represent n" + n', the total number of network singularities, we may write N and D as follows, in place of (82) or (95) : N = 1 -\- Kiz + Koz^ + K^z • • • + K„z" (99) D = I - Kyz + K22' - Kzz' •■■ i-TKnz'' Notice that A'' and D are related by D{z) = N{-z), (100) which is required by the form of the /3 equation in (21). To arrive at a design procedure most easily (but not the simplest design procedure), one may express the assigned gain jS in the follo^\dng w^ay (comparable to (58) and (96)) : i)8 = E CuTk , k odd J^Ckz" = -log^iCA./ Coefficients Kk may again be calculated by a modification of (61). This time C-2k is replaced by Ck , wherever it appears in (61), and then all even ordered Ck are made zero (since only odd terms appears in ^Z ^i^-^'' of (101)). Note that even ordered Kk are not usually zero, even though even ordered Ck are. The degrees of N and D, in (99), are such that we can make Ck = Ck through terms of order k = 2n. This requires merely: AT ^" ^ = E ^^' (102) As stated, the condition applies to Ck of both even and odd orders. Since even ordered Ck are zero, it means that at least n even ordered Ck will be zero, in addition to the match between n odd orders. (102) is sufficient to determine an N and a D without reference to (100). If the (equal) degrees w of (99) are assumed, however, the N and D deter- mined by (102) will be found to obey (100) automatically (pro\-ided E Kkz'' corresponds to an odd series ^ Ckz", as here assumed).! A simpler method for computing the same N and D takes advantage t This was discovered bj' Mrs. M. D. Stoughton. NETWORK SYNTHESIS 655 of the known relation (100), connecting N and D. Let E and 0 be re- spectively the sums of even and odd terms in N. Then A'' is E + 0 and (100) re(|uires D to be E - 0. The ratio 0/E may be related to X) ^kz' of (98) as follows : I = -tanh I E C,z' (103) Now let two convergent series, respectively even and odd, be such that: - = -tanhi^ZC,^' (104) Let coefficients Kk be defined by: J^ + 0 = E ^i^^' " (105) Then E and 0 are respectively the sums of the even and odd terms. The complete series may now be related to the (odd) series ^ Ckz'^ as follows : S hCz = -log E K'uz' (106) This fixes the Kk oi E -{- 0 in terms of the Ck . To make Ck = Ck through m odd orders, (102) is now replaced by 0 """ 0 , , E = B (10^) mo The symbol = designates equality of power series through m odd orders. {All even terms are now zero on both sides.) The right hand side may be expressed as a continued fraction of the following form, comparable to (86): 0 1 E tti 1 Z 02 1 Z (108) Truncation after only the m^^ term gives the 0/E of (107). The coefficients of 0 and E may be calculated by an appropriate modification of (61). (Calculate like Kk of (101), after dividing all Ck by 2). After 0 and E have been evaluated, by truncating the continued fraction (108), their sum gives polynomial A'' of (99). 656 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 The natural modes and frequencies of infinite loss are determined from the zeros of the polynomial A^. By (21), each zero is either a {z- plane) natural mode, z^ , or the negativ^e of an infinite loss point —z„ . If gain variations are inconsequential, there is likely to be some latitude in designating each zero as a 2,, , or as a —z^. A zero of N with a positive real part would make a non-physical natural mode, and hence it must be a —Za, corresponding to an infinite loss point. A zero of N with a negative real part can be a natural mode z^ , but this may not be required. It may be either a z„ or a, —z„ , provided conjugate zeros are assigned in the same way, and provided the total number of — z^ does not exceed the total number of z, . The latter con- dition requires: At least half the zeros of N must have negative real parts. The continued fraction (108) shows how many zeros will have nega- tive real parts, before any zeros have been calculated. The following theorem makes this easy: The number of zeros of N which have negative real parts is equal to the number of positive coefficients in the trun- cation of the continued fraction (108) ivhich determined N. When gain is not to be disregarded, but is to be exactly zero, the synthesis technique needs few changes. The phase of an "all-pass" net- work is related to the natural modes z^ as follows: il3 = E C,T, , k odd E ^ = -log ) ^ (109) 2 n(i + 4, This may be regarded as a special case of (20) for a = 0 (which makes Ck = 0 for k even, and also happens to require z^ = —z^). In functional form however, it is more like il3 of (21). It differs in only two regards. In the power series in z, each C^ is divided by two. In the rational frac- tion in z, all the zeros correspond to natural modes, and the poles cor- respond to frequencies of infinite loss; but the poles are also exactly the negatives of the zeros, as in the il3 eciuations of (21). Accordingly, the phase synthesis technique which ignores gain varia- tions may be applied to the zero gain form of the problem by cutting NETWORK SYNTHESIS 657 every Ck in two. All zeros of N = E -\- 0 must be construed as natural modes z, . Finally, the network must have as many infinite loss j)()ints as natural modes, such that z, = —z„ . (Integer n is now the number of" natural modes, rather than the total numl)er of singularities.) For physical networks, all the first /(. terms of the continued fraction (108) must now be positi\'c. To meet this condition it may be necessary to add linear phase to the assigned phase (by adding a negative cor- rection to Ci). It appears that sufficient linear phase will always lead to a physical design, provided the numlx'r of modes n is increased to retain a reasonable accuracy. 26. LINEAR PHASE When the assigned phase ^ is lineai', the calculations are relatively simple. If a delay I) is to be appi'oximated ov(n- a fre(iuency inter\-al extending to CO = Wc , i~^ = -DcocTi (110) If delay D is to be realized without regard to gain variations, the ap- propriate 0/E is g = tanh^ (111) A known continued fraction expansion of tanh A' may be applied to (111), to obtain the coefficients of (108) without bothering with (105).t The result may be arranged as follows: (112) Do3cZ Truncation of the continvied fraction gives 0/E, and then 0 -{- E. The zeros Za- turn out to be proportional to - , and therefore root extraction techniciues are required only for one D, for each n. The zeros are tabu- lated for sample n's, in Table II. t For tho expansion of tanh A', reference may t)e made to a text on continued fractions by Wall", page 349, equation 91.6. 0 1 E ^ 1 ^ Doicz 3-2 1 Du:cZ 5-2 658 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Table II — Z-Plane Natural Modes for Linear Phase n DwcZf 1 -2 2 -3 ± z Vs" 3 -4.64438 -3.67782 ± i 3.50876 4 -5.79242 ± i 1.73446 -4.20758 ± i 5.31484 5 -7.29348 -6.70392 ± i 3.48532 -4.64934 zh i 7.14204 6 -8.49668 ± i 1.73510 -7.47142 ± i 5.25256 -5.03190 ± i 8.98532 The error in the first mismatched Tchebycheff coefficient is a rough measure of accuracy. It may be shown to be (-)"(i)co.)=^"+^ 4"[l-3-5 ••• (2n - l)f(2n + 1) Cin+l — Cin+l — ,„r^ r. i- TTTT ^^^,/r.,_ 1 TV (113) This measure of accuracy is plotted in Fig. 11, for various numbers of natural modes n. A sample detailed curve of |8 — ^ is shown in Fig. 12, with dotted Unes corresponding to the estimated error (113). If delay D is such that the error is reasonable, all the zeros may be natural modes. If these are combined with a Hke number of infinite loss points, such that Za = —z^, an all-pass network will be obtained, instead of one which approximates D without regard to gain. The all pass network \\dll produce twice the delay, and twice the nonlinearity of phase. In other words, for an all pass network, both coordinates in Fig. 11 must be doubled. 27. SIMPLIFICATION OF SINGULARITY ARRAYS In complex communication systems, a single equalizer may be re- quired to correct for a number of effects. In a coaxial cable system, for instance, a single network in the standard repeater may be required to compensate for the following: Cable attenuation, characteristics of input and output networks, effects of interstages (significant because the feed- back is limited), and distortion due to variable controls at mean settings. Tchebycheff polynomial methods may not be efficient when applied NETWORK SYNTHESIS 659 / 1 0.8 / / 1 / / 0.6 0.5 0.4 0.3 0.2 0.1 1 / / / / / / / / / n = i / 3/ 1 'i / 0.08 / / 0.06 0.05 0.04 0.03 0.02 O.OI / / / / 1 1 10 20 30 40 50 60 80 100 200 300 400 600 1000 PHASE ANGLE IN DEGREES AT TOP USEFUL FREQUENCY (multiply by 2 FOR ALL-PASS NETWORKS) Fig. 11 — Estimated error for n natural modes approximating linear phase. 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CO/COq Fig. 12 — Error when six natural modes approximate a linear phase, with a slope giving 402° at top useful frequency. 660 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 directly to all these effects. They may still be useful, however, when applied in the following way: Separate arrays of singularities are determined, which match the separate effects to required accuracy, using any convenient methods. Minimum network complexity is not required at this point. Combining all the singularities in a single array gives an initial design which is sufficiently accurate, but may use many more singularities than are actually necessary. Tchebycheff polynomial metliods are now used to obtain a simpler set of singularities, which approximates the initial set to sufficient accuracy. This has been designated "boiling down" the original set. In a problem of this sort the assigned characteristic has the network form, as well as the network characteristic which is to approximate it. (The example discussed in Sections 10 to 14 is also a problem of this sort.) As a result, equations (20) and (21) apply to a and ^, as well as to a and /5. This makes it possible to replace ^ KkZ *" and ^ Kkz'', of (55), (56), (96) etc., by a finite rational fraction N/D. If both a and ^ are to be approximated, the following is derived from (20). n (i - P) N E ife' = log K. ) -H- = -log ^ (114) The singularities Za , z^ correspond, of course, to the network singu- larities which are to be boiled down. If only a, or only ^, is to be ap- proximated, suitable modifications are readily derived from (21). The boiling down is accomplished by requiring N ^ N ^ ^ where N/D is of lower total degree than N/D. If m = n" + n', and n" — n' = 0 or 1, the continued fraction method can again be used. This requires expansion of N/D in continued fraction form, instead of An example of a boiled down set of singularities is illustrated in Fig. 13. 28. GENERALIZATION OF THE USEFUL INTERVAL All the previous analysis applies to a useful frequency interval —o)c< CO < 4-Wc . Its important characteristics are as follows: It is a single continuous interval, with co = 0 at its center. Useful intervals with other NKTWOHR SVXTIIKSIS 661 other characteristics may be obtained, within limits, by chaiikT'>k , which represent a least squares approximation to a. When (117) relates p to z, 2;-plane singularities z„ may be defined by: a -\- h Iz^ — vl = ■ )- ^2 (118) 1 + c {■-3' I 2. 1 > 1, Re Za to have same sign as Re p^ An additional singularity, Zo , is also needed, corresponding to the finite poles of (117). It may be defined as follows: 1 + c L - -Y = 0 \ Zo/ (119) Uo 1 > 1 When Pa — p", in a of (2), is expressed in terms of z and z, , (117) introduces denominator factors (1 — z /zo) and (1 — I/zqz). As a re- sult, a of (21) must be changed to the following, for band-pass intervals: OC = z2 C2kT2k 11 { 1 — 72 I / Jl\n"-nl (120) When definite values have been chosen for a, h, c of (117) (in order that the Ck may be calculated), (1 — z'/zo) in (120) is not subject to arbitrary adjustment. This situation can be handled by defining N/D as the rational fraction in the a ofiuatioiis of (21), as before, but re- t For general discussions of orthogonal functions and least squares approxi- mations, see Courant and Hilhert^, and also a short text by Jackson.'^ 664 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 placing (84) by D me / _2\ n' ' — n> (121) Fig. 14 illustrates an application of the technique to the simulation of a coaxial cable attenuation (which is nearly proportional to -^/w)- 29. RECAPITULATION Tchebycheff polynomial series may be applied advantageously to a very ^dde range of network synthesis applications. The scope of their usefulness may depend upon the skill of the designer, as with any synthesis tools, but the underlying principles are reasonably simple. The most important principles are perhaps the following: A Tchebycheff polynomial series in frequency may be related to a power series in a new variable z. When the Tchebycheff polynomial series corresponds to a finite network gain or phase, the power series corresponds to an analytic function of 2, quite similar in form to the network function of p, with singularities at 2;-plane mappings of the 0 0.005 ■ 0.010 ■ 0.015 / \ A / \ A s/- -^ / \ / \ i \ V / / \ J V __^ 1 1 1 1 1 1 0.3 0.4 0.5 0.6 0.8 1.0 2 3 FREQUFNCY IN MEGACYCLES PER SECOND Fig. 14 — Simulation of a coa.xial cable attenuation — Attenuation at top useful frequency = 46 db; Network = four constant-resistance sections. NETWORK SYNTHESIS GG5 network singularities. This makes it possible to apply power series ap- proximation methods, in terms of z, to obtain approximations based on TchebychelT poljaiomial series, in terms of frequency. "Maximally flat" approximations in terms of z may be used to match tlie first m terms in the Tchebycheff polynomial series representing net- work gain or phase to the corresponding terms in the series representing assigned gain or phase. In this way, a Tchebycheff polynomial type of least squares approximation to the network function is made identical to the corresponding least stjuares approximation to the ideal function. The overall error, network function minus ideal function, is then the difference between the two least squares errors. The s'-plane analysis may also be manipulated, in a quite different way, to approach an equal ripple type of approximation (which usually represents approximation in the Tchebycheff sense). The complications are such that applications have been limited to problems of certain quite special types. On the other hand, analysis of this sort has been found useful in clarifying various other ways of seeking equal ripple approximations . REFERENCES 1. S. Darlington, "The Potential Analogue Method of Network Synthesis," Bell System Tech. J., 30, pp. 315-365, Apr. 1951. 2. G. L, Matthaei, "A General Method for Synthesis of Filter Transfer Func- tions as Applied to L-C and R-C Filter Examples," Stanford University Electronics Laboratory Technical Report No. 39, Aug. 31, 1951 (for Office of Naval Research, NR-078-360). 3. T. R. Bashkow, "A Contribution to Network Synthesis by Potential Anal- ogy," Stanford University Electronics Laboratory Technical Report No. 25, June 30, 1950 (for Office of Naval Research, NR-078-360). 4. E. S. Kuh, "A Study of the Network-Synthesis Approximation Problem for Arbitrarj' Loss Functions," Stanford University Electronics Laboratory Tech- nical Report No. U, Feb. 14, 1952 (for Office of Naval Research, NR-078-360). 5. R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. I, Chap. 2, Julius Springer, Berlin, 1931. 6. C. Lanczos, "Trigonometric Interpolation of Empirical and Analytic Func- tions,'' J. of Math, and Phys., 17, pp. 12.3-199, 1938-.39. 7. H. A. Wheeler, "Potential Analog for Frequency Selectors with Oscillating Peaks," Wheeler Monograph No. 15, Wheeler Laboratories, Great Neck, N. Y., 1951. 8. S. Darlington, "Synthesis of Reactance 4-Poles," J . of Math, and Phys., 18, pp. 257-353, Sept., 1939. 9. T. C. Fry, "Use of Continued Fractions in Design of Electrical Networks," American Math. Soc. Bulletin, 35, pp. 463-498, July-Aug., 1929. 10. H. W. Bode, Network Analysis and Feedback Amplifier Design, D. Van Nos- trand Co., New York, 1945. 11. H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co., New York, 1948. 12. Dunham Jackson, Fourier Series and Orthogonal Polynomials, The Mathe- matical Association of America, Oherlin, Ohio, 1941. A Carrier Telegraph System for Short-Haul Applications By J. L. HYSKO, W. T. REA and L. C. ROBERTS (Manuscript received May 14, 1952) A compact frequency-shift carrier telegraph system is described which pro- vides channels in the voice range and above the voice. The channel ter- minal unit incorporates arrangements for handling TWX supervisory sig- nals and employs no electro-magnetic relays. INTRODUCTION Most short Bel] System telegraph circuits, particularly those in the less-densely populated areas of the country, have customarily been operated over direct-current facilities obtained by compositing or sim- plexing physical telephone circuits. Many of these extend from a tele- graph repeater in a central office to another arranged as a subscriber set and mounted in the knee-well of the customer's teletypewriter table. Thus, for example, circuits are extended to Teletypewriter Exchange Service (TWX) subscribers located far from the switchboard. The TWX facilities are arranged to handle supervision as well as transmission. The form of supervision is identical to that obtained when local facihties are employed and hence uniform operating procedures are obtained at TWX switchboards for all subscriber stations without regard to their geographical location. During and immediately following World War II, the growth of the Bell System's telegraph business resulted in some shortage of dc facilities. It was foreseen that this shortage would be rapidly intensi- fied by the use of new short-haul carrier telephone systems, such as type Nl,^ in providing telephone circuits without adding physical con- ductors. Moreover, many of the existing direct-current facilities would be absorbed to meet signaling needs for the rapid expansion of tele- phone toll dialing. It therefore became evident that carrier telegraph methods must be adopted for relatively shoi't hauls in fringe areas. The existing 40C1 voice-frequency carrier telegraph system ' was SHORT-HAUL CARRIER TELEGRAPH 667 designed for application in large groujjs at telegraph central offices and for trunk-ser\'icc operation o\cr toll telephone circuits emplojdng stand- ard levels. It has proved very economical in this field. However, the very features which make for economy in large installations (such as amplitude modulation, common carrier supply and testing equipment, and standardized operating conditions) cause this equipment to be costly when it is applied a few channels at a time in outlying offices; these may not be ecjuipped with either telegraph battery supplies or telegraph boards. Moreover, the -40C1 equipment, being a carrier-on- for-mark and carrier-off-for-space system, does not lend itself to the pro\'ision of TWX toll subscriber line supervision identical to that of local stations without the addition of rather complex and expensive superNisory applicjue circuits. Where TWX supervision is involv^ed these super^^sory circuits are required to generate and recognize supervisory signal patterns capable of being distinguished from transmission space signals and communication breaks, which are long space signals. Consequently, it was decided to develop a new carrier telegraph system especially aimed at the needs of fringe areas. One of the problems to which much thought was given concerned the choice between ampli- tude-modulation and frequency-shift operation. A frequency-shift sys- tem provides some reduction in the effect of noise and other interference on transmission and it is also less affected by rapid level changes. Al- though these advantages were attractive, it was not clear that they were sufficient to justify the added complexity and cost entailed by the adoption of this type of transmission, in view of the quiet and stable cir- cuits encountered in the Bell System plant. What finally s^vlnlg the bal- ance to a frequency-shift system was its advantage in handling TWX supervisory signals. With transmission accomplished by shifting the car- rier frequency, supervisory signals could be sent by turning the carrier on and off. A cheap and simple circuit might then be used to distinguish between transmission and supervision. From the foregoing discussion it will be evident that during the twelve years since the 40C1 system was developed the needs of the Bell System have changed. Fortunately, the designer's art has concurrently made great strides in making available new miniature apparatus and elec- tronic techniques such as have been exploited so successfully in the 143A type electronic telegraph regenerative repeater, the V3 telephone repeater and the N-1 carrier telephone system. As a result, the channel terminal of the new 43A1 carrier telegraph system, being small, inexpensive, self-contained and all-electronic with no electro-mechanical 668 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 relays, is almost ideally suited to the needs of the smaller central offices and TWX stations. FREQUENCY ALLOCATIONS The 43 A 1 system provides two groups of ehannel-fre([uency alloca- tions, as follows: a — A three-channel high-frequency allocation, using frequencies be- tween the upper edge of the voice-frequency band and the lower edge of the type-C carrier telephone band. This allocation is primarily for opera- tion on open-wire lines but can also be operated on cable circuits where the loading provides a suitably high cut-off. b — ^A voice-frequency allocation capable of providing six channels on two-wire circuits or twelve channels on four-wire circuits. The chan- nels of this allocation are for operation over telephone speech channels on any of the standard facilities, including broad-band carrier and cable or open- wire physical circuits. The present frequency allocations are shown in Fig. 1. The voice- frequency s.ystem is based on twelve nominal midband frequencies spaced 170 cycles apart from 595 cj^cles to 2635 cycles, omitting 1615 cycles. The carrier frequency is shifted ±35 cycles about midband, and either the higher or the lower frequency may be used for marking sig- 4-WIRE VOICE FREQUENCY ± 35'^SHIFT CHANNEL NUMBER p e e e SMSMSMSMSMSM 2 3 4 5 6 7 P P 3MSMSMSMSMSM 9 10 II 12 13 14 2-WIRE VOICE FREQUENCY ±35'^ SHIFT CHANNEL NUMBER c' e " SMSMSMSMSMSM 2 3 4 5 6 7 P P P ^END \ / _^ ^ \ s \ V / / ^ y^ N N \ 1 I SIG REQU NAL ENCIE sh / y y \ \ 1 Y y /^ 1 1 1 -200 -175 -150 -125 -100 -75 -50 -25 0 25 50 75 100 125 FREQUENCY IN CYCLES PER SECOND FROM MID -BAND 150 175 200 Fig. 3 — Send and receive filter characteristics, VF allocation. 672 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 is provided between the first and second stages. With the rec gain control at maximum the third stage is driven to full output when the input to the receiving filter is greater than about —50 dbm*. Where it is not necessary to detect the presence or absence of carrier for super- vision as described below, the control is generally set for maximum gain. The limiter output is passed through a frequency discriminator con- sisting of two anti-resonant circuits in series, tuned so that one has a parallel resonance at the low frequency edge and the other at the high frequency edge of the channel band. The voltages appearing across the anti-resonant circuits are rectified separately by germanium varistor diodes and the resultant d-c output voltages are added algebraically, filtered and applied to the control grids of the output tubes. Since at normal receiving levels the limiter removes all magnitude variations, the output from the discriminator detector circuit is depend- ent in magnitude and sign only on the signaling frequency. A negative voltage from the detector causes cut-off of the amplifier tubes and a positive voltage causes plate current to flow. A switch between the discriminator and the detector provides means for reversing the out- put connections of the discriminator so that a positive voltage from the detector can be obtained with either the higher or the lower sig- naling frequency. Fig. 4 shows the dc voltage output versus frequency characteristic obtained with a typical discriminator. D O 0-40 / . J \ 1 f 1 J \ V y / ^ y 1 \* — 1 -^ "~fre' JIGNA QUEN CIES ( -240 -200 -160 -120 -80 -40 0 40 80 120 160 200 240 FREQUENCY IN CYCLES PER SECOND FROM MID -BAND Fig. -1 — Frequency characteristic of typical discriminator. 'dbm" is an abbreviation for "decibels with respect to one milliwatt." SIIORT-IIAri, CARRIKR TKLKGRAPII ()73 The low pass filter between the detector and last stage serves to remove carrier ripple and to decrease the effects of noise and other inter- ference having demodulated frequency components greater than about 40 cps, which is slightly higher than the "dotting" frequency of 100- word per minute signals. In order to prevent a change in the tuning of (he discriminator when the reversing switch is operated, a balanced low-pass filter structure without mutual inductance is employed. This presents high and nearly equal impedances to ground from the positive and negative sides of the detector circuit. Nearly all the \'oltage gain of the receiver appears ahead of the tletector. Since the detector output voltage applied to the grids of the beam power tetrodes is high enough to give an approximately square signal wave shape in the loop, no intermediate stage of dc amplification is needed following the detector. For unbiased signal reception, the de- modulated signals should be centered on the grid characteristic of the receiving tubes; that is, the marking and spacing voltages applied to the grid circuit should be s^Tnmetrical about a potential a few volts negative with respect to the receiving tube cathodes. To obvdate the need for a voltage source negative with respect to the cathodes, the signals are prebiased by unbalancing the detector so that the mean of the mark and space output voltages from the low pass filter is about —5 volts. Further adjustment of the mean signal value may be made by means of the REC BIAS potentiometer to compensate for bias of signals received from the line due to deviations in the mark and space frequencies from their theoretical values or to other causes originating at the sending terminal of the telegraph circuit as well as for bias due to discrepancies in the discriminator network or to differences between mark and space levels. These arrangements permit great freedom in the assignment of loop battery voltages. The cathodes of the final stage may be fixed at — 130- volt, — 48-volt or ground potential and the plates operated from ground, -|-48-volt or +130-volt potential. The remainder of the circuit may be powered by +130- volt battery for the plates and — 24-volt battery for the heaters of the tubes, regardless of the loop conditions. By means of the reversing switch mentioned above, current maj^ be caused to flow in the loop during the reception of the higher or the lower frequency. Thus not only can various frequency allocations be accom- modated, but the local circuit may be operated neutral (current for mark) or inverse neutral (no current for mark). One tube is used in the final stage for 20 ma or 30 ma loop current, and two for 60 ma loop current. 674 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Supervisory Circuit When the channel is used in TWX service as a toll subscriber line, the subscriber calls the operator to initiate a call by closing the power switch on his teletypewriter. This connects power to the teletypewriter motor, closes the transmission circuit to the teletypewriter and applies plate battery voltage to the transmitting oscillator in the channel ter- minal, resulting in the transmission of carrier current over the line. At the distant (switchboard) terminal the receipt of carrier current energizes a supervisory signal receiving circuit which is responsive to carrier-on and carrier-off conditions in the receive band. In this circuit, carrier voltage appearing at the plate of the limiter tube is rectified and applied to the grid of the supervisory triode. The operation of a relay in the plate circuit of this tube causes a line lamp at the switch- board to light. A disconnect signal is sent by the subscriber at the end of a call by opening the teletypewriter power switch. This removes the oscillator plate voltage. At the central office, the receipt of the resulting no-carrier signal de-energizes the supervisory recei\ang circuit and causes the super- visory lamp in the operator's cord circuit to light steadily. To recall the operator during a call the subscriber opens and recloses his power switch. This causes the cord lamp at the smtchboard to flash. An RC circuit slows the rise of current in the supervisory receiving tube to guard against false operation of the switchboard line lamp due to noise impulses during the carrier-off, that is, the idle condition. DC Circuits On the dc side of the channel terminal, provision is made for optional wiring arrangements to connect to the circuits of the various telegraph test boards, service boards and TWX switchboards, as well as to local teletypewriter loops, using telegraph voltages of either 130 or 48 volts. In offices where a negative 130-volt battery is not provided, operation with a single positive 130-volt battery is possible. The loop connections are made to an electronic circuit in the channel terminal which is similar to that employed in a recently-developed electronic loop repeater used in telegraph offices and which possesses several interesting features. Fig. 5 compares the action of this circuit, in transmitting toward the subscriber station, with that of more con- ventional arrangements : (a) shows a conventional open-and-close circuit and the wave shapes which it produces at the central office end and at the far end of a capaci- SHORT-HAUL CARRIER TELEGRAPH G75 tative loop. As is well known, the asymmetrical wave shape causes positive signal bias. (b) shows an "effective polar" circuit along with the wave shapes it delivers. This is the circuit con\'cntionally used to drive subscriber loops. It presents a constant low iniiKHJauce to the loop and might therefore be considered a "close-and-close" circuit. (c) shows the electronic loop circuit. The driving tetrodes are operated in their high-impedance region, above the knee of the plate-current pUite-voltage curve. They deliver a highly symmetrical rectangular wave to the loop and little or no bias results. This circuit presents a nearly constant high impedance to the loop and might be considered an "open-and-open" circuit. Although the rectangular wave is inferior to a peaked wave in that less a^'erage power is delivered for the same values of steady-state cur- rent and voltage, it provides entirely acceptable transmission for 19- gauge cable loops up to about 20 to 25 miles in length. Inasmuch as 80 volts potential is absorbed in the electron tube plate circuits, this is almost the maximum length over which 62.5 ma can be supplied when loop battery of 260 volts is iLsed. M_t_ (1) S* 1 _L IC (2)n ;-; CURRENT AT (1) CURRENT AT (2) (a) "OPEN AND CLOSE" CIRCUIT T (b) "close and close" CIRCUIT "T" T (c) "open and open" circuit Fig. 5 — Explanation of electronic loop circuit. 676 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 During open-and-close transmission by the subscriber, the high im- pedance termination of the loop at the tetrode plate circuits causes the current at the central office end of the loop to change very slowly — too slowly for good transmission at teletypewriter speeds. However, the voltage wave is very well shaped, and this is what is used to drive the grid of the transmitting tube. One noteworthy fact is that the bias of the signals repeated from loop to line is nearly independent of loop length; consequently no inductive wave shaping is required at the subscriber station, even in the longest operable loops. Because of the high impedance termination, loop current is insensitive to circuit resistance. The loop padding rheostat is, therefore, adjusted to build out the loop resistance to a standard value and the amount of loop current required for proper operation of the station teletypewriter is obtained by varying the screen grid potential of the tetrode tubes. Duplex Feature In half -duplex operation, one dc loop at each channel terminal serves for both sending and receiving. The central office end of this loop is connected to the grid of the sending triode and to the plates of the re- ceiving tetrodes. If a marking signal is being received from the carrier line while the teletypewriter contacts in the loop are closed, the receiving tubes conduct, current flows in the loop and the teletypewriter in the loop receives a marking signal. Under this condition the office end of the loop is positive with respect to the cathode of the sending triode; hence tliis tube passes a marking signal toward the carrier line. When a spacing signal is received from the carrier line the tetrodes are cut off, the loop current is reduced practically to zero, the teletypewriter re- ceives a spacing signal and the voltage at the office end of the loop be- comes more positive. Hence a marking signal continues to be transmitted to the line during the receipt of either mark or space signals from the line. When the subscriber opens the loop to send a spacing signal to the distant terminal, the potential of the sending triode grid becomes nega- tive with respect to its cathode, the tube cuts off and hence, as described previously, a spacing frequency is passed to the ac line. In full-duplex operation, two loops are provided at each channel terminal to permit sending and receiving simultaneously. The grid of the sending tube is disconnected from the plates of the tetrodes and transferred to a resistive connection which terminates the full-duplex sending loop. The loop circuits operate in the same way as described for half -duplex operation except that no break action is provided. SHORT-HAUL CARRIER TKLlOnRARH 677 Break Feature When the subscriber opens the loop at the teletypewriter to break transmission coming from the distant tei'minal, a clean-cut space should be transmitted to the line regardless of imy incoming signals. The re- sistor shunted between the plates and cathodes of the receiving tub(\s causes the central office end of the open loop to assume the same potential as the tetrode cathodes. This insures that a steady spacing potential will be applied to the send tube even though the teti'odes are cut off by an incoming space. This provides a rapid, clean break. However, if a large leakage exists across the loop conductors, the resistor will not be able to keep the sending tube in a cut-off condition and a ])reak by the subscriber will result in the incoming transmission being reflected in an inverted condition to the distant carrier terminal. In such a case the distant sending suliscriber would be broken by a "bust-up" of local copy or by operation of the keyboard break lock. This would normally be caused only by a trouble condition in cable loops. If a break signal is received over the line from the distant end while the near end subscriber is sending, his loop current is reduced to prac- tically zero. This operates the keyboard break lock thus breaking the subscriber. This circuit differs from the conventional loop circuit in that the receipt of a break signal does not stop the outgoing signals except via the break lock. TELEGRAPH DISTORTION On quiet circuits, total distortion per section averages 1 to 2 per cent at 60 words per minute and about 5 per cent at 100 words per minute. Plots of received signal distortion versus level of received carrier are shown on Fig. 6 for both signaling speeds. EQUIPMENT FEATURES The channel terminal employs a formed sheet-metal framework and occupies a space 10| inches high, 5^ inches wide and 7f inches deep over- all. Fig. 7 shows a 43 A 1 channel terminal. It is plug-terminated, and hence removable for maintenance or repair at a bench. The basic portion of the channel terminal is common to all frequency allocations. The oscillator network and send filter, which constitute the elements determining the transmitted freciuency, form a plug-termi- nated sub-assembly 7f inches high, 5| inches wide, and 1| inches deep. The receive filter and discriminator, w^hich select the received frequency, form a plug-terminated sub-assembly of the same size. 678 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 §30 25 O 20 I- a. O I- •n 15 t; 10 60 WPM V \ \ \ ^ 0 WPM \ --^ 1 1 -65 -60 -40 -35 -30 -25 -20 RECEIVING LEVEL IN DBM Fig. 6 — Telegraph distortion vs receiving level. A rear view of the channel terminal with the send frequency unit removed is shown in Fig. 8. With both frequency units in place, the rear of the channel terminal is almost completely enclosed. When they are removed, the wiring and apparatus terminals of the basic channel terminal are readily accessible for test and repair. Tube sockets, potentiometers, test points, switches and the inductor of the low-pass filter are mounted on the front panel. Small resistors, capacitors, and germanium diodes are assembled on a plastic "ladder" which is mounted vertically in the space between the frequency units. As shown in Fig. 9, three channel terminals may be mounted abreast on a welded metal frame which is fastened to any of the standard bay frameworks designed to accommodate 19-inch mounting plates. The unit mounting frame carries the multicontact receptacles into which the channel terminals are plugged. Twenty-four channel terminals may be mounted on an ll^-foot relay rack, with line coils and certain auxiliary equipment. Where arrangements for s\vitching between half and full-duplex opera- tion are required, duplex switches for a number of channel terminals are mounted on a narrow plate between the channel terminal mounting frameworks. Loop rheostats, when required, may be mounted adjacent to the channel terminals or in a loop pad bay along with other loop rheostats that may be associated with electronic loop repeaters. The latter arrange- SHORT-HAUL CARRIER TELEGHAPII 679 ment concentrates the heat dissipated by these rheostats at a place where it will not l)e harmful. Subscriber Set A channel terminal may also be mounted in a station set box appear- ing in the knee-well of a subscriber's teletypewriter table. This, called a 130B1 teletypewriter sul)scriber set, is illustrated in Fig. 10. It con- tains a line or hybrid coil and balancing network, as well as local circuit resistors and other miscellaneous apparatus. When so moimted, the Fig. 7 — Channel terminal, front view. 680 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 I IS- 8 — Channel terminal, rear view, sending network removed. mm^m^. Fig. 9 — Three channel terminals mounted on relay rack. .♦..Jij SIIOllT-IIAUL CAUKIEU TELEGRAPH 681 Fig. 10 — loitl')! til(t\ pi'w liter subscriber set including channel terminal. channel terminal is powered by the teletypewriter rectifier, which fur- nishes 130- volt dc and 20- volt ac power. The 130B1 set may be employed in private-line or TWX service. In the latter, the application and removal of oscillator plate battery is controlled as described above by the teletypeAwiter power switch, so that the equivalent of telephone "switch-hook" super\ision is attained. Supervision and transmission are largely independent. The telegraph receiving circuit at the central office terminal remains marking during recall and disconnect signals; hence these supervisory signals do not pass through the cord-circuit repeater to the TWX toll line. Since there is no frequency discrimination in the supervisory receiving circuit, either marking or spacing carrier from the station energizes the super- visory circuit. Hence a communication break (spacing) signal from the subscriber station is transmitted through the operator's cord without any effect on the supervision. On a TWX call to the subscriber station, a series of alternate marks and spaces, generated by applying 20-cycle ringing voltage to the grid 682 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 of the sending tube at the central office terminal, actuates the station ringer, which is connected to the local loop whenever the teletypewriter power s\vitch is in the OFF position. The circuit which terminates the TWX toll subscriber line at the switchboard office is operable with all existing types of TWX cord cir- cuit repeater. All the features of TWX service, including unattended service, are therefore available, POWER DRAINS A channel terminal dissipates about 25 watts. Tube heaters consume about half an ampere at 24 volts and the remainder of the channel term- inal, exclusive of its loop-terminating portion, consumes 50 ma at 130 volts. The loop terminating portion dissipates 20, 30 or 62.5 ma at 80 volts, depending upon the type of local circuit employed. LINE LEVELS The 43A1 system is capable of working with a great variety of line levels. The send level may be adjusted for any value from -|-6 dbm downward. The receiving equipment operates satisfactorily with —45 dbm or even —50 dbm. But the levels actually used are controlled by crosstalk and noise conditions in the line. Receiving levels are normally limited by lightning interference on open wire and by noise on cable circuits. The minimum tolerable levels are about —40 dbm on open wire, —45 dbm on four-wire cable circuits and —35 dbm on two- wire cable. In Fig. 11, a comparison is made of the effects of static on the 43 Al system and on the 40C amplitude-modulation system. It gives the results of simultaneous tests on the 2465-cycle bands of the two systems, using the static from a record made at Madison, Florida. The 43A1 channel could tolerate about 4 db stronger static than the 40C. SYSTEM LAYOUTS A typical circuit layout of the 43 A 1 system working in the frequency band between the voice and type-C carrier on an open-wire line is shown in Fig. 12. The telegraph circuits extend from 43A1 channel terminals located in a central office, at the left, to 130B1 sets in subscriber stations, at the right. In the central office, the send and receive paths of the chan- nel terminal are combined in a hybrid coil. With the moderate degree of balance provided by the network of this hybrid coil, the allowable dif- ference between send and receive levels of the middle channel may be SHORT-HAUL CARRIER TELEGRAPH 683 100 80 ^20 a. 6 - ^ \ - '\ N ^ \. - s \ ^ \ - \ \ \j 40C1 43 a\ \ \ - \ \ - > V \ - THE INTERFERENCE IS THE VALUE WHICH IS EXCEEDED 1% OF THE TIME, AS MEASURED IN A 10 KC BAND BY A METER HAVING A 10 MILLISECOND INTEGRATION TIME CIRCUIT. \ \ \ \ \ \ -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 SIGNAL-TO-INTFRFERENCE* RATIO IN DECIBELS Fig. 11 — Comparison of amplitude and frequency shift modulation with static interference. 35 db or more. The telegraph channels are next combined with the voice frequency circuit by means of a 150A filter, and are connected to the composite set and line through the low-pass section of the 121 A (type -C) carrier line filter. As a result of the cut-offs of the 150A high-pass and 121A low-pass filter components, the pass band of the telegraph is about 3.7 to 5.4kc. At the outlying terminal of the open-wire line, the telegraph is separated from the voice and type-C carrier circuits by similar filters and connected to the individual subscriber stations by a branching network and branch lines. The typical arrangement on a two-wire circuit in the voice frequency range is sho^\'n in Fig. 13. Six channels are available, using six of the twelve frequency bands for transmission east to west and the other six bands west to east. As in the high frequency case, a branching network and branch lines at the outlying end connect the circuits to the sub- scribers. Fig. 14 shows a layout in which branch lines are connected at intermediate points in the telephone circuit. At these intermediate points the impedance of the branching network is made high, in order to keep the balance at the telephone repeater from being harmed excessively. Though the network attenuates greatly the signals through it, the tele- graph level is usually sufficiently high so that this loss can be tolerated. 684 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 The branching network at the outlying terminal has low impedance. Taps on the transformers in the network permit the impedance ratios to be adjusted to suit the line impedances between which the network operates. Since several circuits may pass through this network, a short- circuit on one branch should not be capable of degrading transmission in the other branches. To prevent this, resistances are inserted in series with each branch of such a value that a short circuit will not cause more than 3 db excess loss in other channels. It has been sho\m by tests that, TYPE C REPEATER OR TERMINAL 121 A FILTER VOICE FREQUENCY REPEATER OR TERMINAL 43A1 CHANNEL TERMINALS 150A FILTER -1 HP J ' T SET LP -■ ' 1 1 r DOO T 000 ^ , 1 1 ' t J 43A1 43A1 43A1 CHI CH2 CH3 SUBSCRIBER SETS Fig. 12 — System layout, above the voice. BRANCHING NETWORK 130B1 SUBSCRIBER SETS TELEPHONE REPEATER ^-r-\ Fig. 13 — System laj-out, in voice band. e- t BRANCH LINES SHORT-HAUL fAHRIKU TELEGRAPH 685 in this frequency-modulation system, a sudden loss of 3 db causes little distortion. The series resistances may serve another purpose besides protecting against short circuits. If one branch has a much greater at- tenuation than the others, the resistance values in series with the shorter branches may be increased so that more energy is directed into the longer branch. Emergency Circuits If a circuit containing no intermediate branches fails, a regular mes- sage circuit can be patched in to replace it until the trouble is cleared. Fig. 15 shows trunks to be used for making this patch in the case of two- wire circuits. They contain 3 db pads which reduce the signal level to compensate for the change from 0 db transmission level on the regular line to +3 db level on the message circuit. The 43A1 system may operate also over a four- wire circuit, which accommodates twelve telegraph channels. A patch to an emergency mes- sage circuit would then be made at the four-wire patch bay. Since the circuit used for telegraph would usually be similar to those used for tele- phone message service, no pads to adjust levels would be required for this "W^ BRANCHING NETWORKS BRANCH — LINES--' nm^ 130 B1 SUBSCRIBER SETS Fig. 14 — Intermediate branch lines. BRANCHING NETWORK BRA^4CH LINES AND SUB- SCRIBER SETS -9DBM MESSAGE CIRCUIT Fig. 15 — Emergency circuit. 686 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 patch. Obviously echo suppressors must be disabled when a message circuit is used for telegraph. When the telegraph circuit contains one or more branches at inter- mediate points, it would be difficult and often impractical to use an ordinary message circuit to replace the telegraph stem in emergencies. The branching location frequently will not be manned and so no one will be available to patch the branch line to the message circuit. In such cases each intermediate branch circuit may be made good over a separate message circuit which is individual to it. Fig. 16 shows this arrangement. A patch trunk is provided between the 43 A 1 channel terminal at the central office and the telephone switchboard. At the switchboard which is nearest to the intermediate branch subscriber, the branch line is 43A1 CHANNEL TERMINALS SWITCHBOARD SWITCHBOARD Fig. 16 — Emergency circuit for intermediate branch. VAr DIFFERENTIAL LOOP REPEATER Fig. 17-Connection to other telegraph repeaters. SHOKT-IIAUL CARRIER TELEGRAPH 687 carried throusli a cut-off jack. Tho toll operators then can patch the circuit to the subscriber, thus by-j)assiiig the main line when it is in trouble. Since the telegraph eiieigy from only one channel is impicssed on tlie emergency circuit, no adjustment of levels is required. The channel terminals which are at the central office, shown at the left of Figs. 12 to 16, may be connected to subscribers over do loops or they may be connected to other types of telegraph repeaters. Fig. 17 indicates the latter connection schematically. Since the loop circuit must supply positive potential to the 43A1 channel terminal, the connecting repeater must be arranged to supply positive battery for marking signals. FUTURE EXTENSIONS It is expected that the field of application of the 43 A 1 system will be broadened by further development over the next few years. More fre- quencies will be pro\'ided, ])oth in and al)ove the voice band. Means will be designed for passmg TWX super\isory signals over a direct-current loop from a subscriber station to a channel terminal installed in a nearby central ofhce. The built-in supervisory arrangements of the 43 A 1 ec^uip- ment will be exploited to obtain inexpensive straightforward trunks for use both between TWX switchboards and from switchboards to Line Concentrating Units. The supervisory feature will also be employed in pri\-ate line ser\dce to provide an open circuit alarm. REFERENCES 1. R S. Caruthers, H, R. Huntley, W. E. Kahl, L. Pedersen, "A New Telephone Carrier System for Medium-Haul Circuits," Elec. Eng., 70, pp. 692-697, Aug. 1951. 2. B. P. Hamilton, "Carrier Telegraphy in the Bell System," Bell Labs. Record, 26, pp. 58-62, Feb. 1948. 3. J. A. Duncan, R. D. Parker and R. E. Pierce, "Telegraphy in the Bell System," A.I.E.E. Transactions, 63, pp. 1032-1044, 1944. 4. B. Ostendorf, Jr., "New Electronic Telegraph Regenerative Repeater," Elec. Eng., 69, pp. 237-240, March, 1950. 5. R. L. Case, "Transmission Features of V3 Repeaters," Bell Labs. Record, 27, pp. 94-95, March, 1949. The Type-0 Carrier System BY PAUL G. EDWARDS AND L. R. MONTFORT (Manuscript received June 11, 1952) INTRODUCTION While the sight of an open-wire toll line is a rarity in many parts of the East, considerable use is made of open- wire facilities in other sections of the country to provide toll and exchange service. At the present time there are about 170,000-route-miles of open-wire in the Bell System which carry some 1,400,000 pair-miles of wire used for toll service. It is estimated that about 60 per cent of this pair-mileage is used for carrier, although only about 10 per cent carries the full fifteen carrier channels, which is possible by employing type-C and type- J carrier systems. It is obvious that some of the remaining line pairs are available for addi- tional carrier growth, provided, of course, the demand for additional circuits exists, and there are carrier systems which can meet these de- mands economically. Type O is a multi-channel, open-wire carrier sys- tem which has been designed to pro\'ide, economically, additional circuits in the range from a minimum of about 15 up to a maximum of 150 miles, or more. The type-0 system is the open-wire counterpart of the type N short-haul cable system. Present open-wire toll lines vary from a single-arm line, with one or two pairs of wires, up to lines with five or six arms carrying thirty pairs. These lines may carry long-haul toll circuits up to about 1000 miles in length, short-haul toll circuits up to 150 miles, as well as tributary trunks and exchange circuits. Growth in the past of toll and tributary circuits on these lines has been provided by the addition of single-channel D or H systems, three-channel C systems, twelve-channel J systems or by other similar carrier systems. The full development of a line for open-wire carrier has, in the past, required expensive line rearrangements. For instance, most lines reach terminal and repeater offices over entrance cables which may be several miles in length. Impedance matching at the junction of the open-wire and cable is reciuired, and is provided by loading the cable at both voice and carrier frequencies, by employing junction line filters using non- TYFIO-O CAHIIIKH SYSTEM 089 loaded carrier pairs, or by the use of autotrausformers. In addition, transposition schemes are needed to reduce the crosstalk coupling be- tween open-wire pairs to tolerable amounts, and longitudinal and metallic filtering is necessary at repeater points to control interaction crosstalk, 'i'he C and J carriers have been designed essentially for long-haul use, and when the line transposition costs are added, these systems are likely to be more expensi\-e than adding wire for providing relief for the shorter circuits, which are recjuired in increasing number as the length decreases. In contrast to the heavy back-bone toll lead carrying both long and short haul circuits is the one or two arm line which may be a secondary route, or a line which terminates in a small town. The demands along this line are for short-haul toll service, trunks between tributary offices and their toll centers, and for exchange service. Because growth has been relatively slow, carrier has been employed only to a limited extent. Single-channel D or H systems ma}^ be found on these lines, or an adapta- tion of the M system for toll service, and possibly other miscellaneous systems. Only minor rearrangements of the line and entrance cables has been necessary because of the small percentage of circuits equipped with carrier facilities. A typical need for expansion on this type of line occurs when a manual tiibutary office is cut over to dial operation, and the operators are moved to the toll center. Additional circuits are immediately needed from the toll center to the tributary office because of certain factors introduced when the operators are moved some miles away. Experience has shown the desirability of being able to reach an operator a fairly large percent- age of the time because of special services, such as directory information, reports on the a^'ailability of toll circuits, service complaints, etc. This requires a substantial increase in the number of circuits to the tributary office in such instances. Development of this line, then, proceeds by adding single-channel carrier systems, until a point is reached where it is necessary to string more copper wire, which is costly and maj^ be in short supply, or to add multi-channel carrier systems. The situation in Iowa is typical of many areas in the Southern and Western parts of the country. Fig. 1 shows the principal open-wire and cable routes in Iowa used by the Bell System for toll business. The type-K transcontinental cable crosses the state, passing through Daven- port and Des Moines on its way to Omaha. Small branch cables serve Muscatine, Cedar Rapids and Atlantic, where the circuits are extended by open-wire facilities. In general, the transcontinental TD-2 radio relay system follows the K carrier route. A coaxial cable route extends north from Des Moines to Minneapohs, connecting at Iowa Falls with short 690 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 TYPE-O CARRIER SYSTEM 691 K cables to Fort Dodge and Waterloo. Eventually a second cable route will extend across the state through Waterloo and Fort Dodge, as shown by the dashed lines. A second coaxial route cuts across the southwestern corner on its waj^ to Kansas City and a third coaxial route coiniects Sioux City with Omaha. The rest of the state is served by open-wire lines. Fig. 2 shows the distribution with length of the Bell System open-wire short-haul toll and tributar}' circuits in Iowa in 1950, including both voice-freciuency aiul carrier facilities. It will ])e noted that 95 per cent of the circuits are less than 100 miles in length, while 90 per cent are less than 70 miles, the point where type C systems just become economi- cal. For tributary circuits 98 per cent are less than 30 miles in length. There arc a total of some 2700 toll and tributary circuits in Iowa. In addition, there is also a sizable connecting company development. Fig. 3 is a distribution of the number of circuits per group, where a group is composed of the circuits used for via or terminating business between two towns. There are about four circuits per group for short-haul toll and two circuits per group for tributary service in the median case. As the dial conversion program proceeds the average number of circuits per tributary group is expected to increase. Because of the short distances involved, and the small number of circuits per group, carrier development in Iowa has been restricted, to a large extent, to single-channel systems, and type M. Only four or five M channels can be operated on a given open-wire line, and while these systems have some transmission disadvantages, they have been employed to a large extent. However, further M development is blocked because of the expense of isolating M systems from adjacent lines. There are a few C systems on such routes as Sioux City-Spencer-Mason City, Waterloo- Dubuque, Muscatine-Keokuk, and Atlantic-Spencer. The type-0 system, therefore, is being made available to provide short-haul toll and tributary circuits on open- wire lines in the range from 15 to 150 miles. WTien completely developed it will provide four four-channel systems in the frequency range from 2 to 156 kc, as shown on Fig. 4. The use for separate channels of both sidebands of a single carrier, called a "twin-channel," results in economical use of the fre- ciuency space. The four channel systems are designated OA, OB, OC, and OD respectively, and cover substantially the same frequency range as the C and J systems. Considerable attention has been given to keeping the line rearrange- ments as simple as possible. The use of non-loaded entrance cable is proposed, and simple arrangements are available for adding 0 groups 692 THE BELL SYSTEMj[tECHNICAL JOURNAL, JULY 1952 100 90 80 70 60 50 40 ^ ^ -— ' ~~" ^ / /^ / / NORTHWESTERN BELL TELEPHONE CIRCUITS / TOTAL NUMBER OF CIRCUITS = 2720 TOTAL NUMBER OF CIRCUIT MILES = 86,400 1 ( END OF 1949 \ / ^ f 80 100 120 140 OPEN-WIRE LENGTH IN MILES Fig. 2 — Distribution of circuit lengths in Iowa in 1950. 5< 'iV, 70 °a: 30 TRIBUTARY^ - — — — / / ^' ,^^^ ^ / / TOLL / / ; / 1 i NORTHWESTERN BELL TELEPHONE CIRCUITS END OF 1950 / / / / / J \ / 1 / 1 1 / / > / 12 13 1 23456789 10 11 NUMBER OF CIRCUITS PER GROUP Fig. 3— Distribution of circuits per group in Iowa in 1950. OA -OB- 4_ -H^^H-^H^^ LOW HIGH LOW HIGH LOW HIGH LOW HIGH 24 32 40 50 60 70 80 90 100 110 120 130 140 150 160 FREQUENCY IN KILOCYCLES PER SECOND Fig. 4 — 0 carrier telephone — frequency allocation. TYPE-O CARHIKH SYSTKM ()93 aboYG exi.stiiig carrier systems, such as (■, wliich has a top frcfiuency of 30 kc. With the aid of the compandor, it is possible to apply the OB system to practically an.y open-wire pair transposed for C carrier opera- tion, thus nearly doubling the number of circuits without additional line rearrangements. Transposition arrangements which are expected to be less expensive to apply are being made available where the higher- frequency type-0 groups are invohed. Line losses of the order of 35 to 40 db can be spanned under normal wet weather conditions, and 50 db loss under sleet conditions with some transmission impairment, since sleet is relative^ infrequent in occurrence. This will result in repeater spacings of the order of 50 miles in sleet areas for the OB group, and 100 miles in other areas. For the higher frequency groups (OC and OD) those spacings will be approximately halved. CHOICE OF DEVELOPMENT APPROACH The type-0 carrier system followed the type-N system closely in time, and, in effect, covers the same range of circuit lengths for open wire lines that N provides for cables. It was both natural and expedient that many of the N features were carried over directly into the O design. It was necessar}^, however, to make important distinctions as well. These similarities and differences will be discussed m some detail. The transmitting and receiving voice frequency subassemblies are reused with substantially no modification. This provides the O system with the same compandor and the same 3700 cycle signaling system as used in X. An important difference between the two systems is concerned with the use of single sideband in the 0 rather than double sideband as in the N. This choice is an economic one. The double sideband system is relatively easier to design and less expensive than the single sideband arrangement. The use of double sideband in cables is practicable in many cables because of the relative abundance of conductors as com- pared with open wire pairs. In some cases the use of single sideband in cables may be attractive as compared with the cost of new outside plant for certain length ranges. Another distinction between the two systems is the provision of cir- cuits in smaller groups in 0. In N the basic group is 12, although systems may in some instances be partiallj^ equipped. In O, the desire to furnish smaller circuits groups resulted in the choice of a basic four-channel group. The full complement per pair for O, including a channel replacing the voice circuit, is sixteen channels. The regulation problem is more severe in the O design. It is necessary 694 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 to provide sufficient regulator range to accommodate line variations due to wet or icy lines. The repeater and receiving group regulator range common to four channels is in the order of 40 to 50 db, or approximately four times the regulating range of an N repeater. The range of the twin-channel regulator is comparable to the N individual channel regu- lator, but the O regulator is shared by two channels forming adjacent sidebands of the same carrier. The use of a single sideband imposes more severe requirements on channel band filters. The use of a material called ferrite, in combination with a crystal, affords an efficient channel band filter in a small space when compared to previous single sideband channel filters employing air-core coils. Ferrite coils are employed in a coil-condenser type of filter to provide separation for the various four-channel groups. While the N system employs only receiving channel band filters, O has filters in both the transmitting and recei\dng terminals. The O system employs the double modulation principle for all groups. This arrangement permits the use of only four channel band filter de- signs for all of the 32 channel frequency allocations. The frequency range for these basic channel bands has been selected to provide the most economical overall filter design. The use of die-castings has been extended in a number of ways. No- table among these is a die-cast framework, used in both the terminal and the repeater. The plug-in technique has been expanded to provide plug-in filters for channel and group band filters. DESCRIPTION OF THE SYSTEM The system will be described first by block schematics, second, by transmission characteristics, and finally by photographs. This descrip- tion will show representative features rather than describe the system completely. While the description wdll cover the complete O plan, it should be pointed out that the OB system is the first to be made available. It will be followed by the other 0 systems. In the schematic description, where a unit is common to all systems the designation is "Type 0." Where the arrangement is different for the several systems, a particular designation is used, such as "Type OB," etc. Schematics The O modulation plan is shown on Fig. 5. The single-sideband chan- nel filters for all groups are in the frequency range from 180 to 196 kc. TYPE-0 CARRIER SYSTEM 695 By the use of different group carrier frequencies the several four-channel groups are placed in their various locations. As indicated by Figs. 4 and 5, high and low group assignments are used for the two directions of each four-channel system. A repeater is provided for each four-channel system and, except in the case of OA, the high and low frequency groups are "frogged" at each repeater, as in the N system. Figure (5 shows a block diagram of a complete 0 system. On this figure, and in general on other figures showing filters, a letter code is assigned to designate the kind of filter, with a subscript letter to indicate the particular system in which the filter is used. The several filter designa- tions and number codes are collected for reference in Table I. Much of the apparent complexity of the system, particularly as regards filters, arises out of the use of a single pair for both directions of transmission. Another complexity is occasioned by the requirement that a complete complement of O systems will not always be provided. For example, OB, OC, and OD systems may be used above an existing C system, or similar systems. O 100 - "■-TERMINAL Fig. 5 — ^Type-0 modulation plan. 696 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 lotj -Jx: LU^ '^IS? 1/1 -I 2 UJ 0 52 ■7 1- :£. "> Q- , a. UJ 0 3 t OZ 1- 0 < u tr Ct rr f> u n H H < 1 _1 ^ < * " -) -J -) U 1 ) f^ 0 U1 0 i 0 5 < 5 u 10 -I < Q. §? (J)-' ^ ^_^ a UJ 1- > H TYPE-O CARRIER SYSTEM GUU transmitting and receiving group units to translate the sidebands be- tween the frequencj'^ range of the channel band filters and the correct line frequency allocation. The group receiving unit contains the directional filter for separating the four-channel transmitting and receiving groups at line frequencies. INIultiple points are indicated for the connection of other O carrier systems on the same pair, Channel Unit A block diagram of the channel unit is shown on Fig. 8. As indicated by the dashed line, the channel unit is comprised of four parts which are interconnected by plugs and jacks. These are: 1. The Compressor Sub-Assembly. This unit contains the compressor and a tei'minating arrangement to permit the system to be used for foin*-wire termination, or for two-wire termination at non-gain locations, i.e., those mthout SAvitching pad control. 2. The Expandor Sub- Assembly. This unit contains the expandor as well as the signal transmitting and receiving equipment. 3. The Carrier Frequency Sub- Assembly . This unit contains the trans- mitting and recei\'ing modulators, and is arranged to receive the plug-in transmitting and receiving channel band filters. 4. The Transmitting and Receiving Band Filters. These are combined in a single plug-in unit. Items 1 and 2 are practially identical to the corresponding sub-assem- blies for N carrier. Each channel receives its carrier supply for the trans- mitting side from an oscillator in the twin channel unit. On the receiving side the carrier is obtained by selecting and amplifying the transmitted carrier. The frequencies indicated on Fig. 8 are the same for all 0 systems, and apply to two of the four channels in the group. Figs. 4 and 5 show go and return channels in high and low frequency assignments in the same 0 system. However, as sho^vn on Fig. 9 covering the OB system, the fre- quency assignments applying to the channel band filters are above any of the 0 line frequencies and are in the frequency range from 180 to 196 kc for l)oth transmitting and receiving channel band filters. Transmitting and receiving channel band filters are paired in a single plug-in unit. In order to reduce the number of kinds of paired channel band filters (transmitting and receiving in the same plug-in luiit) the pairing has been done in a special way. If, for example, transmitting assignment 180-184 were paired with receiving assignment 180-184, etc., four different kinds of paired filters would result. Instead, assign- ment 180-184 is paired with assignment 192-196, and by making the 700 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 - LU iz CC LU nice 5 oQ- TYPE-O CARRIER SYSTEM roi ])lu^-iii filter iv\'er.sible in il.s .socket, this groupiiiji; cuu be made to serve two channels as follows: 180-184 Transmitting 192 IIH) Keceivinj-- A similar i)aire(l liltei" serx'es 184 188 Transmitting 188 192 Keeeivin"- nd and 192 19(1 Transmit tiiio- 180 181 deceiving 188 192 ^^ransmitting 184-188 lUH-eivina- Thus only two basic kinds of paired chainiel band filters are rcfiuired, I'ather than four kinds. In these filters, as well as the reversible giou}) filters, the filter d(\signations are so arranged that when the filter is in l)laoe the propel- filter designation is in view. Twin Channd Unit The twin chaimel unit is shown in somewhat more detail in Fig. 10. There are two kinds of twin channel units to serve the foiu* channel assign- ments, and the freciuencies shown on Fig. 10 correspond to those shown on Fig. 8. (and Fig. 9). A transmitting carrier adjustment permits the transmitted carrier level to be set properly in relation to the sideband levels. In order that the group regulators may function primarily on the carrier, and thus be substantially independent of voice or signaling sidebands, the carrier is transmitted approximately 6 db above the side- band level. On the receiving side of the twin channel unit a regulating amplifier controls the received le^'el of both sidebands. It does this from the carrier LINE FREQUENCIES IN KILOCYCLES FOR EACH CHANNEL 4 CHANNEL TERMINAL GROUP FREQUENCIES IN KILOCYCLES FOR EACH CHANNEL LOW GROUP 184 192 180 188 196 u. .-^U _J 12 3 4 236 GROUP CARRIER HIGH GROUP 60 I Ll. 256 GROUP CARRIER 4 3 2 1 Fig. 9 — Type-OB system frequencies. 702. THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 picked off by a narrow band crystal filter. This same carrier is supplied to the receiving side of the channel units for demodulating the associated sidebands. Group Transmitting Unit The OB group transmitting unit is shown on Figure 11. It receives the four sidebands and two transmitted carriers and places them in the proper high or low line frequency assignment. The transmitting group unit, depending on the optional connection to the group oscillator (Fig, 11), can be either a high group transmitting unit or a low group trans- mitting unit. For convenience the noise generator is contained in the group trans- mitting unit. On very quiet circuits this noise source provides a means of masking crosstalk. In ordinary usage the noise thus provided is not noticeable on the circuit, but is sufficient to reduce the chance of hearing intelligible crosstalk to a small value. Group Receiving Unit The OB group recei\dng unit is shown on Fig. 12. It is comprised of an amplifier and a regulator-modulator arrangement equipped with plug- in filters. The same basic arrangement is used for all receiving group units, as well as for all repeaters. Only the plug-in filters, and the fre- quencies from the associated oscillators are different. The directional CARRIER SUPPLY TO TRANSMITTING- MODULATORS TO RECEIVING BAND FILTERS OSCILLATOR 184 KG (or 192 KC) -Wv AA^ PICKOFF FILTER 192 KC (or 184 KC) CONTROL AMPLIFIER CARRIER SUPPLY TO RECEIVING MODULATORS TRANSMITTED CARRIER ADJUSTMENT REGULATING AMPLIFIER TRANSMITTED CARRIER TO OTHER TWIN-_ CHANNEL UNIT Fig. 10 — T}^pe-0 twin channel unit. ■AAV' — »- — 'WV-i 180-196 KC TYPE-0 CARRIER SYSTEM 703 filter is reversible as well as plug-in and thus serves either high or low groups. The receiving group band filter and its associated auxiliary filter have the same physical arrangement as in the repeater but they are never reversed. A dc feedback type regulator controls the gain of the regulating ampli- fier, and operates principally on the two received carriers, although the sidebands are fed back also. Group Oscillator The group oscillator contains two oscillators for supplying the group transmitting and group receiving units. These oscillators are interchange- able (by strapping) and permit the group units to operate in either the high or low groups. For conveneince the 3700 cycle signaling oscillator, common to all four channels, is contained in the group oscillator. Repeater As indicated on Figs. 5 and 6, a repeater is pro\dded for each four- channel system. An amplifier, regulator and modulator arrangement serves each direction. The directional bands are routed through the repeater by directional and auxiliary band pass filters as indicated on Figs. 13 and 14. At each repeater (except OA) the high- and low-fre- quency groups are "frogged" to improve transmission, particularly as FROM CHANNEL <^ UNITS FROM TWIN- CHANNEL UNITS COMB MULT ■ \Vv \AyV MODULATOR TRANS GROUP LPF E NOISE GENERATOR TO GROUP I REC UNIT 0 3700 CYCLES TO GROUP REC UNIT 1 THROUGH DIR FILTER IN GROUP REC UNIT 40-56KC (OR 60-76 KC) TO KEYING CIRCUITS Fig. 11 — Tjpe-OB group transmitting unit and group oscillator unit. 704 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 TYPE-0 CARRIER SYSTEM 705 z2 o5) uj :7 t— i/ia < =/ >- >- -, o UJ 2 <0 cc 2§S| Q 1/1 Z I- < Z o o -z m 5 Os ^a Q D UJ o => ID tn Q- < LU £D CC o a >> 706 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 regards automatic equalization of attenuation slope with frequency, and to obviate the necessity for additional line treatment. Some repeaters receive low group frequencies and transmit high group frequencies for both directions. Other repeaters receive high group fre- quencies and transmit low group frequencies. In N two kinds of repeaters were required. In O the reversal of the filters in their sockets provides both kinds of repeaters, and presents the proper designation to view. A regulator is provided in each direction of the same kind as in the group receiving unit. It should be noted at this point that the filters internal to the repeater (as opposed to directional filters) differ from the filters used in the re- ceiving group units. This is because the repeater always accommodates line frequencies on both sides of the amplifier, while the receiving group unit accommodates line frequencies on one side (which correspond to the repeater line frequencies) but always must supply channel band filter frequencies at the group amplifier output. Fig. 14 shows in somewhat greater detail than Fig. 13 the filter arrange- ment for an OB repeater. TRANSMISSION CHARACTERISTICS The overall channel band width is illustrated in Fig. 15. The approxi- mate frequency cutoffs are similar to N but for various reasons the several channels may show somewhat greater differences. The O sys- tem, being a single sideband system, has a filter cutoff at low (voice) frequencies, which the N does not have. Differences may exist between \r \r \f ^_ Ab + Bf HIGH PASS BAND PASS BAND PASS p}^4i§f "![>]- HD- BAND PASS \f TO OA CARRIER SYSTEM OR TO C CARRIER AND VOICE SYSTEMS Fig. 14 — Type-OB repeater filter arrangement. \f 60 76 I BAND PASS BAND PASS TO LINE FILTER \r TYI'K-O CATUUKIi SVSTKM 70/ 0 4 06 1.2 FREQUENCY Fig. 15 — Net loss frequency charactieristic. 1,6 2.0 2.4 2.8 IN KILOCYCLES PER SECOND upper and lower sideband filters. In addition, O has transmitting band filters while N does not. A situation is of interest which applies to both N and O, as well as to any other system employing the type of compandor controlled from the voice energy. Different frequency characteristics will be obtained with the compandor operating and with the compandor controls locked. Neither of these necessarily corresponds to the operating condition with speech or music. With the compandor controls free and using single frequency test tone, the characteristic obtained is a combination of the freciuency characteristics of the line and control circuits. With the con- trols locked, the characteristic is that of the line only. If the control circuit is substantially flat, there will be little distinction between the measurements. The curve of Fig. 15 is of the type obtained with free controls and with a substantially flat control circuit. A typical overall channel load characteristic is shown on Fig. 16. This characteristic includes not only the load curve of various amplifiers, modulators, etc., but shows also the order of match of the compressor and expandor load characteristics. This is a match of curves ha\'ing 2:1 slopes on a db basis over a v\ide range of volumes. A typical overall net loss variation for a non-repeatered circuit is shoA\Ti on Fig. 17. Principally because of the line regulator in the group receiving units (Fig. 18) a wide range of line loss is covered. A similar regulator is included in each repeater, and the extension of a system 708 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 5 -30 v"' // J / y^ ./ / .,/ > ^ — IDEAL / ^ /' / / -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 COMPRESSOR 1000-CYCLE INPUT IN DBM (aT ZERO LEVEl) Fig. 16 — Typical over-all channel load characteristic. with repeaters will not result in a substantial change of the net loss variation, assuming the repeater section losses do not exceed the range of the regulators. The line regulator is assisted by the twin channel regulator, for which a characteristic is shown on Fig. 19. This regulator is similar to the individual channel regulator of N, and serves two channels having a common carrier. This fact alone does not materially change the effective- ness of regulation since the carrier is adjacent to the sideband which 15 20 25 30 35 40 45 LINE LOSS IN DECIBELS Fig. 17 — Typical over-all channel net loss variation, nonrepeatered. TYPE-O CARRIER SYSTEM 709 i/l 0 _l OJ m — 1 ■ — — a (- Q. -3 t- D O u ? -I < u; U 2 < -a / / /I r / / 1 -30 -20 -10 INPUT IN DECIBELS Fig. 18 — Typical group receiving and repeater regulator characteristic. it controls in any case. There are other important differences, however, between X and 0 channel regulators. In N the channel regulator follows the channel band filter and thus tends to compensate for its flat trans- mission variations. Also the N regulator is controlled from the demodu- lator dc output and thus compensates in some degree for demodulator variations. In O, the twin-channel control is ahead of both the chaimel band filter and the channel demodulator, and therefore does not make up their variations. A statement might be intei-polated at this point to emphasize that the relative advantages of single-sideband and double-sideband trans- mission are by no means easily listed and evaluated, since the differences are many and devious, some necessarily and some fortuitously. An ex- ample worthy of note is that in N it is necessary to be concerned about relati\'e phase shift of the sidebands and in the instances of longer cir- -9 -8 -7 -6 -5 -4 -3 -2 -1 CHANGE IN CARRIER INPUT IN DECIBELS Fig. 19— Typical twin-channel regulator characteristic. 710 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 cuits, perhaps to equalize this phase shift in order to prevent serious reduction of signal output, or variation in channel net loss with fre- quency. No such concern applies to O. In regard to filter characteristics, it seems obvious that complete coverage is not feasible in this description. Instead typical curves only will be shown. Fig. 20 shows the general characteristics of filters for separating the wanted sideband from the carrier and unwanted sideband. The trans- mitting and receiving filters have similar shapes. The carrier pick-off filter characteristic is shown in the same figure. Fig. 21 shows the filter characteristics for separating the voice and signaling (3700 cycle) func- tions. Another filter case of interest is the line filter for separating, for ex- ample, the OA system from the OB system, and from the OC and OD systems, as well, if they are employed. Fig. 22 shows the configuration and loss characteristics of the Gl (537A or 538A) filter. A Cb directional filter (530A) characteristic is shown on Figure 23. This filter assembly includes two filters to accommodate the OB high and low group assign- CHANNELS TRANS BAND- PASS WANTED SPEECH UNWANTED CROSS -TALK REC BAND- PASS FILTERS 5432 10 12345 FREQUENCY IN KILOCYCLES PER SECOND (FROM CARRIER! Fig. 20^T3^pical channel band and carrier pick-off filter characteristic. TYPE-O CARRIER SYSTEM 711 s ^> SIGNALING, . BAND-PASS \ ^^ N J \ '^ ' \ / w k L'\ \ y1 \ \ \ /I \ ^ > X 1 \ \ 'A ^ \ \ / / VOICE, \ \ / / LOW-PASS \ \ \ / / /: \ \ V H^ \ / i \ / ! 1 \ 1 \ 1 I 1 1 1 A 1 1 y \ 1 y \ 1 . .* . 2000 2400 4800 2800 3200 FREQUENCY Fig. 21 — Typical receiving low pass and signaling filter characteristics. 3600 4000 4400 CYCLES PER SECOND ments. Similar characteristics apply to the Ab + Bb auxiliary filter (53 lA). The A and B characteristics are used in the group receiving filters Ab + Db , (531B) and Bb + Db , (531C). The D filter is a band- pass filter with relatively gradual cutoff to pass the 180-196 band for the channel filters, and has peaks at the group carrier frequencies of 236 kc and 256 kc. These filter characteristics are not shown. PHYSICAL ARRANGEMENTS A four-channel carrier terminal is shown on Fig. 24. This terminal includes four channel units shown in detail on Figs. 25, 26, 27 and 28. In Fig. 28 the unit is separated into the three sub-assemblies, of which as noted above, the two voice frequency sub-assemblies are substantially the same as for N carrier. The carrier sub-assembly with its plug-in channel band filters is shown on Fig. 29. The interior arrangement of the plug-in unit containing the trans- mitting and receiving band filters is shown on Fig. 30. This assembly contains an adjustable ferrite inductance, a miniaturized transformer, 712 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 and a crystal with the necessary fixed and adjustable capacitors. The small size is made possible partly by the high Q ferrite coil, and partly by the circuit configuration employing it. As compared with filters em- ploying air-core coils and having comparable cutoffs, the reduction in size of these filters is very striking. The group receiving unit is shown with its plug-in filters on Fig. 31. The filters are held in place by stud screws and nuts. The arrangement shown on Fig. 31 is also used with different filters for the repeater ampli- r. \ J \ LOW -PASS \ BRANCH \ > V HIGH-PASS \ BRANCH r ) 1 1 I ^^"■^ \ J 1 ^..^ \ 1 i 1 0 10 20 30 40 50 60 70 80 90 1( FREQUENCY IN KILOCYCLES PER SECOND Fig. 22 — Tj-pical line filter characteristics (Gl, 537A or 538A). TYPE-O CARRIER SYSTEM 713 fiers. The filter construction is shown on Fig. 32. This filter employs ferrite coils and condensers, having no crystals because of percentage band width considerations. These filters employ a form of printed wiring for interconnection of c()mj)onents. The terminal framework is shown on Fig. 33. This framework em- ploys aluminum die-castings in contrast to the frabricated framework of N. Tliis method permits the inclusion of a slide arrangement which guides the units into place, and insures proper registration of the plugs and jacks. Some units are above the framework; others are suspended from it. The same die-casting, inverted, serves both upper and lower 135 n- in 40 O 20 / ^ v / \ / \ n / \ / \ \ / \ \ / \ \ < ^ */ V '\ '\ /' \ / 1 \ // \ \ // \ \ \ // \ 1 i' /'> \ // \ / / N k \i \ / ,' \ 1 \ i V \ \ N 11 Ji \^ J V \ 1 ' / ' 1 L ^-N 1 / 1 1 1 \y 1 1 \ 1 \ ^ / 1 _/ 1 1 _LJ. 40 50 FREQUENCY 60 70 80 90 KILOCYCLES PER SECOND 100 200 400 Fig. 23^T\pical directional filter characteristics (Cb , 538A). 714 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Fig. 24 — Four-channel O terminal. plug-in units. Since there is no interference between plug-in units in inserting and removing them, can covers have been eliminated. This fact, and the somewhat wider distribution of units having a high con- centration of vacuum tubes, result in a relatively low temperatvn-e rise for O as compared with N. Blower facilities are not provided in the terminal. Typical of the units suspended from the framework is the twin chan- TYPE-0 CARRIER SYSTEM 71i 716 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Fig. 2',) Carrier suhasseinhly and channel band filter. Fig. 30 — Channel band filter-internal arrangement. 717 718 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 nel unit, shown on Fig. 34. The same basic die-casting is employed, \vith minor rearrangement of the die, for the two t\vin channel units, the group transmitting unit, and the group oscillator. The part of the slide arrangement associated with a plug-in unit is shown at the top of the twin channel unit. All plug-in units are held in by a common cover (Fig. 24) which en- closes the handles of the units. For additional support a rapid action fasterner holds the tops of the channel and receiving group units. Repeaters may be either pole mounted or placed in a central office. A group of two repeaters is shown on Fig. 35. This assembly employs a framework, shown on Fig. 36, which includes the same slide die-casting as the terminal. A central unit (also plug-in) accommodates the two Fig. 31 — Group receiving or repeater unit. TYPE-0 CARRIER SYSTEM 719 Fig. 32 — Typical directional or auxiliary band filter — internal view. I -I Fig. 33 — Terminal framework. 720 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Fig. 34 — Twin-channel unit. plug-in group oscillators, which are also shown in the photograph, to- gether with fuses, alarm lamps, etc. Pole mounted repeaters are housed in a cabinet, similar to that used for N. Such a cabinet, equipped with four repeaters is shown on Fig. 37. Since a maximum of four repeaters would have to be supphed by one pair of wires, it is not feasible to transmit power for the repeaters over line pairs. Instead the cabinet contains rectifiers and a line voltage regulator for obtaining 130 volts dc from commercial ac supply. For reserve power supply, a cabinet is available containing a 24-volt storage battery and a dynamotor to supply 130 volts dc to two repeaters (or two dynamotors to supply four repeaters) in case of power failure. ALARMS At terminals a common alarm, operating from carrier failure, performs the functions of: First, dropping all connected subscribers to prevent kt> J '^^\ ( '"^ w a Fig. 35 — Two repeater assembly. Fig, 36 — Repeater framework with oscillators. 721 722 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Fig. 37 — Typii Ml ;ii i.ingemeut of pole mounted repeaters. TYPE-0 CARRIER SYSTEM 723 Fig. 38— Test stand. their being held during the interval of failure; and second, to make all circuits busy at both terminals, to prevent false seizure by operators or automatic switching equipment. Since many 0 systems may be em- ployed in situations where one terminal is unattended, facilities are included whereby, after failure, the system can be tested from either end, through the use of one of the signaling channels. If it is indicated that the system is operable it can be placed in service again without the necessity of a trip to the unattended terminal. SPECIAL SIGNALING FEATURE Arrangements are provided by which two 0 circuits can have their E and M signaling control leads interconnected without the use of the signahng converter, which is otherwise required. This feature is employed when two circuits are connected together on a permanent or semi-per- manent basis to form a single trunk. TESTING To facilitate testing at terminal points a test stand (Fig. 38) has been provided which supports an 0, or N, channel unit during test and adjustment. By a patch cord, the channel unit can be connected to its original framework if desired. Built in pin jacks permit bridging meas- urements to be made at selected points in the transmission circuit. Efficient Coding By B. M. OLIVER (Manuscript received May 14, 1952) This paper reviews hrieflij a few of the simpler aspects of communication theory, especially those parts which relate to the information rate of and channel capacity required for sampled, quantized messages. Two methods are then discussed, whereby such messages can be converted to a "reduced" form in which the successive samples are more nearly independent and for which the simple amplitude distribution is more peaked than in the original message. This reduced signal can then be encoded into binary digits with good efficiency using a Shannon-Fano code on a symbol-by-symbol {or pair-by-pair) basis. The usual inefficiency which results from ignoring the correlation between message segments is lessened because this correlation is less in the reduced message. INTRODUCTION The term coding, as applied to electrical communication, has several meanings. It means the representation of letters as sequences of dots and dashes. It means the representation of signal sample amplitudes as groups of pulses ha^dng two or more possible amplitudes as in pulse code modulation. Lately, it has also come to be the generic term for any process by which a message or message wave is converted into a signal suitable for a given channel. In this usage single-sideband modula- tion, frequency modulation and pulse code modulation are examples of encoding procedures, while microphones, teletypewriters and television cameras are examples of encoding devices. This is a nice concept, but it is useful to distinguish between two classes of encoding processes and devices: those which make no use of the statistical properties of the signal, and those which do. In the first class, the encoding operation consists simply of a one-to-one conversion of the message into a new physical variable, as a microphone converts sound pressure into a proportional voltage or current, or of the one-to-one remapping of the message into a new representation without regard to probabilities, as by ordinary amplitude, frequency or pulse code modula- 724 EFFICIENT CODING 725 tion. In ordinary PCINI for example, the message samples are converted into groups of on-or-otf pulses. The particular combination of pulses in any group depends only upon the amplitude of the particular sample, not upon any other property of the message, and the same time is allotted to each group, regardless of the probability of that group or of the amplitude it represents. Almost all the processes and devices used in present day communication })el()ng to this first class. In the second class, the probabilities of the message are taken into account so that short representations are used for likely messages or likely subsequences, longer representations for less likely ones. Morse code, for example, uses short code groups for the common letters, longer code groups for the rare ones. Processes of the first class we may call non-statistical coding processes, or simply modulation or remapping processes. The time of transmission is the same for all messages of the same length, and all messages are handled by the sj^stem with eciual facility (or difficulty). These processes require no memory and have a small and constant delay They are inefficient in their use of channel capacity. Processes of the second class we may call statistical encoding proc- esses. These processes in general recjuire memory. The time of trans- mission of messages of the same length may be different so that if messages are to be accepted and delivered by the system at constant rates, variable delays may be necessary at the sending and receiving ends. They are more efficient in their use of channel capacity. It is with this second type of process that this paper is concerned, although pro- cesses of the first type may be used as component steps. Thus we con- sider systems of the type shown in Fig. 1, with the accent on the word "efficient". TRANSMISSION CIRCUITS AND THEIR VOCABULARIES C'ommvuiication circuits or channels can, of course, differ in many respects. Either the peak signal power or the average signal power may be limited. The transmission may be uniform over the band or vary with fre(iuency; it may be constant or subject to selective fading. The noise may be gaussian thermal or shot noise uniform across the band, or peaked at some frefjuency; or it may be largely impulse noise or erratic SIGNAL= EFFICIENT DESCRIPTION OF MESSAGE Fig. 1 — Reversible statistical encoding. 726 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 static discharges. The best type of signal for one channel may be very poor for another. In the following sections it is assumed that the channel transmission characteristic is flat in ampUtude and delay over a definite band and zero outside. It is also assumed that the channel has a definite peak signal power limitation, and that the noise is white gaussian noise. Such a channel is no mere academic ideal. It is in fact quite closely approached in practice by many circuits. Moreover, the conclusions based on these assumptions can usually be modified or extended to other actual cases, such as that of noise with non-uniform spectral distribution (as for example the coaxial cable). If the bandwidth of the channel is W, we can (using single sideband modulation, if necessary) transmit over it without distortion from fre- quency limitation signals containing frequencies from 0 to W (or —W to W in the Fourier sense). Such a wave can assume no more than 2W independent amphtudes per second. Any set of samples of the wave taken at regular intervals -^ serves to specify the wave completely. The wave may be thought of as a series of (sin x)/x pulses centered on the samples and of proportional height, and indeed the wave may be reconstructed from the samples in this fashion. This is the well-known sampling theorem . Thus a message source of bandwidth W can supply at most 2W independent symbols (samples) per second, and this same number can be transmitted as overlapping, but independently dis- tinguishable pulses by a circuit of bandwidth W. Since, as will appear later, channels which are to transmit signals resulting from efficient statistical encoding must be relatively invailner- able to noise, we shall assume that the pulses on the channel are quan- tized. This allows regenerative repeatering to be used to eliminate the accumulation of noise . If there are b quantizing levels, and if the levels are sufficiently separated so that the probability of noise causing in- correct readings is negligibly small, then the capacity of the channel in bits/sec is^ C = 2W log2 h. (1) Such a circuit talks in an alphabet of h "letters" and uses a language in which all combinations of these letters are allowed. There are no for- bidden or impossible "words". The circuit has a vocabulary of h one- letter words, h two-letter words, 6" n-letter words. The basic ineffi- ciency in present day electrical communication is that we build circuits with unrestricted vocabularies and then send signals over them which EFFICIENT CODING 727 use only a tiny fraction of this vocabulary. If all the letters of the written alphabet were used with ef(ual prol)ablUty and if all combinations of letters were allowed, then many words which are now long could l)e made shorter, and written text would be less than one third as long as English. Similarly, if we could arrange to let our circuits use their en- tire vocabulary with equal probability, they could describe our messages with much less time (or bandwidth) on the average. EXCHANGE OF BANDWIDTH AND SIGNAL TO NOISE RATIO It was the advent of wide band ¥M, and other modulation methods which exchange bandwidth for signal-to-noise ratio, which revealed the inadequacy of earlier concepts of information transmission and ulti- mately led to the development of modern communication theory, or information theory . One of tlie more familiar results of this theory is the expression for the maximum capacity of a channel (listurl)ed by white noise: C =W log, (^1 + ^^ (2) in which C is the capacity in bits/sec, W is the bandwidth and P/N the ratio of average signal power to average noise power. This capacity can only be approached, never exceeded, and is only reached when the sig- nal itself has the statistics of a white noise. The expression sets a limit for practical endeavor, and also gives the theoretical rate of exchange between W and P/N. A practical quantized channel, operated so that the loss of informa- tion due to incorrectly received levels is negligible requires about 20 db more peak signal power than the average signal power of the ideal channel to attain the same capacity \ However, bandwidth and signal- to-noise ratio are still exchanged on the same basis. For example, a satis- factory tele\Tlsion picture could be sent over a channel with, say, 100 levels. This would require a (peak) signal to rms noise ratio of some 40 + 20 = 60 db. The bandwidth could be halved by a sort of reverse PCM: by using one pulse- to represent two picture elements. But there are 10,000 combinations of two samples each of which can have any of 100 values. Hence the new combination pulse would need 10,000 distinguishable levels and this would require a signal to noise ratio of 80 + 20 = (2 X 40) + 20 = 100 db. It is evident that while bandwidth compression by non-statistical or straight signal remapping means is not an impossibility, it is neverthe- 728 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 less impractical when the signal to noise ratios are already high What we should really try to do is make our descriptions of our messages more efficient so that less channel capacity is required in the first place. The saving can then be taken either in bandwidth or in signal-to-noise ratio, whichever fits the requirements of our channels best. MESSAGES Messages can either be continuous waves like speech, music, or tele- vision; or they can consist of a succession of discrete characters each with a finite set of possible values, such as English text. Because a finite band- width and a small added noise are both permissible, continuous signals can be converted to discrete signals by the processes of sampling and quantizing . This permits us to talk about them as equivalent from the communication engineering viewpoint. Since many of the principles which follow are easier to think of with discrete messages and since quantization of the channel is assumed for reasons already stated, we shall think of our messages as always being available in discrete form. Let S = the symbol (or sample) rate of the message S Wo = K = the original bandwith of the continuous message ^ = number of quantizing levels. Then if all the message samples were independent and if all quantizing levels were equally likely, the information per sample would be Ho = log2 ( bits (3) the information rate would be Ho = S logo ( bits/sec (4) and the message would use the full capacitj^ of a channel with / quan- tizing levels, and bandwidth S/2. Or by remapping k message samples /log A (with the i possible levels) into I .-^—r I k samples, a channel mth b levels yog b/ /log A and bandwidth W = S/2 ( ,-^— I could be loaded to full capacity. yog b/ However, it is not true that the successive samples of typical messages are independent, nor is it true that the various sample amplitudes are in general equiprobable. If these things were tiTie, speech and music would sound like white noise, pictiu'es would look like the snowstorm EFFICIENT CODING 729 a TV set produces on an idle channel. Written text would look like WPEIPTNKUH WFIOZ — : a random secjuence of letters. The statistics of the message, in particular the correlations between the various sam- ples, greatly reduce the number of sequences of given length which are at all likely. As a result the information rate is less, and fewer bits per second are required to describe the average message. A sequence of M binary digits can describe any of 2 possible mes- sages. Conversely any of A^ messages can be described by logo A'^ binary digits. The information rate, H, of a message source is therefore given by // = hm bits/symbol n— >oo 11/ where N = number of message sequences of length n. If the successive symbols of the message are independent but not equiprobable, then a long sequence will contain Xi symbols of type 1, X2 of type 2, etc. The number of possible combinations of these symbols will be A^ = II ^i!' y so that log A^" = log n\ — ^ log Xj\ 7 For large enough n, all the Xj will be large also and we may write, by Stirling's approximation log A^ -> log \/27rn -\- nlogn — n — ^ [log ^/2TXj + J But since Zl ^j = ^) and since for large n, Xj — » p(j)n where p{j) is the probability of the j*'' symbol, we have log A^ -^ log V'27rn + n log n — n X log \/2TrXj — i n Z) vU) log p{j) - n log n + n H, = hm !^^ = -E VU) log p{j) (5) n-»oo n J which is the expression Shannon derives more rigorously . //i is a maxi- mum when all the p{j) are equal to \/ C. Then H\ = log-, f = Ho . The more uneciual the p(j), i.e., the more peaked the probability distribution, the smaller II y becomes. 730 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 If the successive samples are not independent, the message source will pass through a sequence of states which are determined by the past of the message*. In each state there will be a set of conditional probabilities describing the choice of the next symbol. If the state is i and the condi- tional probability (in this state) of the next symbol being thej^^ is pi(j), then the information produced by this selection is Hi = -Ep.(i)logp,(j). (6) The average rate of the source is then found by averaging (G) over all states with the proper weighting; thus H = J: p{i)Hi = -E p(i)Z P^U) log Piij). (7) » i i The greater the correlation between successive symbols or samples of a message, the more peaked the distributions Pi(j) become on the average, and this results in a lower value for H. As Shannon points out, the in- formation rate of a source, as given by (7), is simply the average un- certainty as to the next symbol when all the past is known. But in a properly operating communication channel the past of the message is available at both ends, so that it should be possible to signal over the channel at the rate H bits/message symbol, rather than Ho as we now do. In present day communication systems we ignore the past and pretend each sample is a complete surprise. By completely efficient statistical coding it should be possible to re- duce the required channel capacity by the factor H/Ho . Whether or not this improvement can be actually reached in practice depends upon the amount of past required to uniquely specify the state of the message source. If long range statistical influences exist, then long segments of the past must be remembered. If there are m symbols in the past which determine the present state and each symbol has ^ possible values, there will be /" states possible (although only 2""" of these are at all prob- able for large m) . If m is large the number of possible states becomes f an- * In a philosophical sense the state of a message source may be dependent on many other factors besides the past of the message. If the source is a human being, for example, the state will depend on a large number of intangibles. If these could really be taken into account the resulting H for the message might be quite low. If the universe is strictly deterministic one might say that H is "really" always zero. When we describe the drawing of balls from the urn in terms of probabilities, we admit our ignorance as to the exact detail of the mixing operation which has occurred in the urn. Likewise the information rate of a source is a measure of our ignorance of the exact state of the source. From a communication engineering standpoint, the knowledge of the state of the source is confined to that given by the past of the message. EFFICIENT CODING 731 tastically large and complete statistical encoding becomes an economic impossibility if not a technical one. Let B'I be a particular combination (the /") of k symbols in the i)ast of the message. Each of these combinations at least partially determines the state of the system. TIence we can write an approximation to (7): F,= -T. pili'i) Z pAj) log pu){j) (8) i J Fk -^ JJ, as k -^ 00 . If only m symbols in the past influence the present state, then k need only be as great as //(, in order that F„» = //. In any case the setiuence F\ , Fo , • • • I'\ is monotone decreasing. Naturally one should always pick the A' symbols in the past which exert the great- est effect upon the present state, i.e. which cause Pb\{J) to be as highly peaked as possible, on the average. In English these would be the im- mediately previous letters; in television, the picture elements in the im- mediate space-time vicinity of the present element. Suppose we break the message up into blocks of length k. Each of these l)locks may be considered to be a character in a new (and huge) alpha- l)ct. If we ignore any influences from previous blocks, i.e. if we consider the blocks to be independent, then the information per block will be simply -ZyXi^i)logp(B-). (9) i Since there are A- symbols per block, the information per symbol, Gk is Gk= - iZpiB'i) log p{B\). (10) /v i As A; — > 00 , Ga — > //, since the amount of statistical influence ignored (between blocks) becomes negligible compared with that taken into account. If d is the number of binary digits refiuired to specify a message n symbols long, then as n — ^ ^, d/nH — > 1. For large n there are thus 2"" messages which are at all likely out of 2""° = T possible sequences (in an f letter alphabet). The probability that a purely random source will produce a message (i.e., a sequence with all the proper statistics) is therefore p ^ 2""^"°""^ (11) for large ?i. Even ii Ho — H is small, p -^ 0 rapidly for large n. This is why white noise never produces anything resembling a picture on a television screen, for instance. For in television signals. Ho — H > 1 732 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 even for very complicated picture material, and n = 250,000 for a single frame. As given by (11), p, also represents the fraction of the possible signals on a channel of / levels which are likely ever to be used by messages of length n without statistical encoding. STATISTICALLY MATCHED CODES Since a sequence of binary digits can be remapped by a non-statistical process into a channel with b quantizing levels, or indeed into a wide variety of other signalling alphabets, it suffices to consider statistical coding processes and codes which reduce the message to a sequence of binary digits. An efficient code is then one for which the average number of binary digits. He , per message symbol lies between Ho and H. As the efficiency increases H/Hc — > 1 , so this ratio may be taken as an efficiency index. With highly efficient processes, the sequences of binary digits produced will have little residual correlation, i.e., they will be nearly random sequences. Since the encoding process must be reversible the receiver must be able to recognize the beginnings and ends of code groups. Since we have at our disposal only zeros and ones, the divisions between code groups must either be marked by a special code group reserved for this purpose, or else the code must have the property that no short code group is duplicated as the beginning of a longer group. A code which satisfies this latter requirement and which is capable of unity efficiency is the so-called Shannon-Fano code, developed inde- pendently by C. E. Shannon of Bell Telephone Laboratories and R. M. Fano of the Massachusetts Institute of Technology. This code is con- structed as follows: One writes down all the possible message sequences of length k in order of decreasing probability. This list is then divided into two groups of as nearly equal probability as possible. One then writes zero as the first digit of the code for all messages in the top half, one as the first digit for all messages in the bottom half. Each of these groups is again divided into two subsets of nearly equal probability and a zero is written as the second digit if the message is in the top subsets, a one if it is in the bottom. The process is continued until there is only one message in each subset. Fig. 2a shows the code which results when this process is applied to a particularly simple probability distribution piB)) = (l/2)\ Here each code group is a series of ones followed by a zero. The receiver knows a code group is finished as soon as a zero appears. Although the longer groups contain mostly ones, their probability is less and on the average as many zeros are sent as ones. EFFICIEXT CODING '83 MESSAGE CODE STEP NO. PROB. 1 •/2 0 (1) (2) (3) (4) (5) 2 '/4 1 0 3 Vs 1 1 0 4 Me 1110 5 '/32 1 1 1 1 0 6 '/64 111110 7 MESSAGE CODE STEP NO, PROB 1 % 0 0 (2) (1) (3) (2) (4) (3) 2 '/4 0 1 3 '/8 1 0 0 4 '4 1 0 1 5 '/16 110 0 6 Me 110 1 7 '/32 1110 0 MESSAGE CODE STEP NO. PROB. 1 '4 0 0 0 (3) (2) (3) (1) (3) (2) (3) 2 '/a 0 0 1 3 Va 0 1 0 4 '/« 0 1 1 5 '/a 1 0 0 e '/a 1 0 1 7 '/a 1 1 0 8 '/a 1 1 1 (a) (b) (c) Fig. 2 — Shaunon-Fano codes for three different distributions. The successive bisections are indicated b}- the dashed lines and the number gives tiie step at which that bisection took place. If the successive message segments are independent, the code will gen- erate a random sequence of zeros and ones. Fig. 2b shows the code which results with another distribution. Here the termination of each code group is more complicated but the non-duplicative property exists so the receiver can still identify the groups. Fig. 2c shows the code which results when all the p(5t) are equal. It is the ordinary binary code. The length of each code group is equal to log l/p{Bi), for the cases shown in the figures. This is true in general so long as it is possible to divide the list into subgroups which are of exactly equal probability. When this is not possible, some code groups may be one digit longer as Shannon shows. The average number of digits per message symbol using this code is therefore given by -l/k Z piBl) log piB\) 2 = 0 db _ 2 — 01 from which „ 0i(l - 02) "" 1-0! With these values of a and 6: P = 1 — 01 — (01 — 02)' 1 - 0? If 02 = 01 , then the expressions simplify to a = 01 , 6 = 0, p = 1 — 01 . As can easily be shown, if 0(a;) = e""'''' , then all the coefficients except a are zero, and a has the value e~". In other words, if the autocorrela- tion function is of exponential shape, the previous sample alone is needed 742 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 for linear prediction. Samples before this add no further information as to the location of the mean of the conditional distributions.* It happens that in typical television signals the autocorrelation for small displacements shows a very nearly exponential behavior. Thus linear prediction on the basis of the previous picture element alone is a natural method for television, particularly in view of the simplicity of apparatus required. Linear prediction is easily instrumented. Fig. 7 shows in block sche- matic form the essentials of a linear predictor. Samples of the message are applied to a delay line. Taps along this line separated by the inter- symbol time of the message, or multiples thereof, make the desired past symbols available. The signals from these taps are merely attenu- ated by amounts corresponding to the coefficients a,h, c • • • and added. A differential summing amplifier is shown to allow for negative coeffi- cients, and also to accomplish the subtraction of the predicted sample amplitude from the present sample amplitude. A complete linear predictor-subtractor is nothing but a transversal (time domain) filter whose impulse response is f(t) = 8(t) - a8(t - r) - 1)8(1 - 2r) • • • and whose equi\'alent frequency response is therefore 77!/ \ i — iiiiT 7 — 2ia)T F{co) = 1 — ae —he where r is the delay between taps. If, for example, simple previous value prediction is used (a = l;h,c--- =0) F{co) = 1 - e-'^' = 2i sin ^ e~^ . * From the preceding expression for p, we see that p = 0 (i.e., perfect predic- tion is possible) if: (l - <{>■{)' = (1 - lr l - 2 = ±(1 - 4>l) f02 = 1 [4>2 = 2l - 1. If 02 = 1, the message samples alternate between two independent but constant values. For this case a = 0, b = 1. If <^2 = 2(t>^ — 1 the autocorrelation is a cosine wave so the message consists of samples of a sinusoid. In this case a = 2i , b = — 1. If <^i is nearly unity, the sinusoid is of low frequency, and the prediction approaches "slope" prediction (i.e. extrapolation of a straight line through the last two samples). In any case where perfect prediction is possible the wave is periodic and there- fore ^ = 0. EFFICIENT CODING 743 It is often argued that linear predietion is therefore uothin"; more than pre-distortion (freciueiicj^-wisc). If the message is unquantized and un- sampled, and if the signal from the predictor is applied to the channel as straiglit amplitude or single side-band modulation, the allegation is certainly true. Pre-distortion is a perfectly valid way of improving the statistical match between message, and channel, and destination as the optimum filter theory of Weiner and Lee shows. On the other hand, when the message is sampled and quantized, and when the output of the linear predictor is further encoded into a sequence of binary digits, and these are possibly remapped onto a higher base for the channel, then the information is being handled digitally throughout, and the usual reasons for a certain type of predistortion no longer apply. The best linear predictor will usually be quite different for the two cases. Even though analogue operations (such as subtraction of amplitudes) are used for convenience, the quantization makes the operation discrete and hence equivalent to a digital process. At the beginning of this section, we were a little vague as to whether the prediction should shift the modes or the means of the conditional distributions to zero amplitude. If the object of the prediction-subtrac- tion operation is to minimize the power in the error signal, then certainly the means should be shifted to zero. The coefficients as determined from the autocorrelation function do this aside from quantizing granularity. They specify an optimum least-square predictor, i.e., one which tends to minimize e, = ^ifp(j)- TERMINATION ISOLATING AMPLIFIERS COEFFICIENT MAGNITUDES DIFFERENTIAL SUMMING AMPLIFIER IF PRESENT SAMPLE IS SUBTRACTED, OUTPUT WILL BE "ERROR" SIGNAL IF PRESENT SAMPLE IS OMITTED, OUTPUT WILL BE ''PREDICTION " Fig. 7 — A linear predictor. 744 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Power reduction is an index of merit when many reduced signals are to be sent by frequency division over one channel, as we have said. When the object is to reduce the channel capacity required for a single message source, then it is the upper bound entropy of the reduced signal which should be minimized, not the power. That is we want — zJi vij) log p(j) to be minimized. For certain types of signals this requires the modes to be shifted to zero, although this is by no means a general rule. Shifting the modes to zero may actually increase the entropy of the "reduced" signal over that of the original message, by adding too many new symbol levels, as the example in the last section shows. If the original message has / quantizing levels, the reduced message after predictive-subtractive coding will in general contain more than ^ levels since an error of more than " can be made in either direction. An n-gramming operation, on the other hand, never increases the al- phabet. 2 >- 4 PRESENT SYMBOL (j j 3 4 5 6 7 1 a"' a"2 a-3 i ^-^ a"^ a~^ a-' 1 a-^ a-2 | a-^ a-^ a-^ a-2 a-' 1 a-' | a-2 a-^ a"3 a~2 a"^ i | a"' a"^ 3-4 3-3 3-2 3-1 , 3-5 3-4 3-3 3-2 1 a-6 a-s 1 J_i 1 ! FIRST ORDER ROWS THEN ADD COLUMNS FOR DIGRAMMER DISTRIBUTION -^ ADD THIS WAY FOR SIMPLE DISTRIBUTION OF ORIGINAL MESSAGE - V / ^w ^^r / ^ Fig. 8 — Joint probability distribution (divide all coefficients by the sum over each array). EFFICIENT CODING 745 Other operations besides simple subtraction of the predicted symbol from the present sj^mbol are of course possible. However, in most cases it would seem that if a more complicated operation were indicated, /i-^ramniin<2; would have provided a better start. ILLUSTRATIVE EX.\MPLE Let us compare the operation of w-gramming and prediction-subtrac- tion techniques on a hypothetical message. We will assume the message has digram statistics, but that longer range statistical influences either do not exist or are ignored. The statistics are then specified entirely by the joint probability distribution p(i, j) of a pair of symbols. Let us assume that there are ^ quantizing levels, and that piij) = /va"''"'l where a is a constant > 1, and K is given by K is the factor which assures that ^ ij p(i, j) = L Thus the most likely level is that of the previous sample. A sample differing by one level is 1/a times as likely, one differing by m levels is aT"" times as likely. Figure 8 shows a plot of the relative values of p(i, j) (neglecting the factor K). For ^ = 4, the total array would be the 4x4 portion enclosed by the dashed line. This sort of distribution is rather similar to those of typical television signals, as shown by pre- liminary measurements, although typical values of a have yet to be determined. With no statistical coding, the required channel capacity is Ho = log2 i bits/sample. If the simple distribution of individual samples is taken into account, the required channel capacity is reduced to Hi = -J^pii) log p{i) where pii) = Z) p(h j) = Z) p(h j) = p(j) i i Hi may be computed from the array of relative coefficients by adding the rows to form the sums 746 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 In terms of these sums, we have Hi = \og^ -Kj2Si\ogSi. Since, with the assumed distribution, the Si are all nearly equal very little reduction in channel capacity is achieved by this step. With linear prediction, the modes of the distributions {i = j) could be centered at zero merely by sending the difference between the present and previous sample (previous value prediction). This would give a reduced signal whose distribution may be found by adding the array along the diagonals. The required channel capacity is then given by: H, = -Kf log Ki - 2i: ^:ii:^ log ^^^^ The distribution of the signal from a digrammer is found by rearrang- ing each row of the table in order of decreasing probability and then adding the resulting columns. Call these sums Sd . The digrammer out- put will thus require a channel capacity: Ho = -T.KSa log KSd 1 ' = log j^ - K^Sd log Sd K d=i Lastly, the true rate of the source is given by H = -Yl Pii) S ViU) log piij) = log i, - //i + 2/v i: ^^ k log a Values for the above quantities were computed for a = 2 and n = 2, 3, 4, 6, 8, 16, 32, 00 . For the case of a = 2, we find that K = [3f - 4(1 - 2"')]"' and that as ^ -^ 00^ Hl,Hd,H -^i ^ log2 3 = 2.918 bits. The results are shown in the Table I and also are plotted in Fig. 9. While Ho and Hi increase without limit as ( is increased, H^ , Hd, and H quickly approach a definite limit. This limit exists because we assumed that the decrease in joint probability as a function of number EFFICIENT rODIXG 747 Table I Number of levels ffo Hi Hl Hd H 1 0 0 0 0 0 2 1 1 1.252 0.918 0.918 3 1.585 1.583 1.777 1.437 1.422 4 2 1.995 2.074 1.764 1.750 6 2.583 2.575 2.381 2.157 2.131 8 3 2.988 2.552 2.370 2.343 16 4 3.99 2.768 2.678 2.641 32 5 5 - t 2.850 2.818 2.782 00 log =C log » 2.918 2.918 2.918 (These figures were computed by slide rule so the fourth figure is not very significant.) of levels off the diagonal was the same regardless of f. In typical signals this is not true. The decrease is more apt to depend on ainplitude differ- ence and the finer the quantum step, the more levels a given difference represents. As a result, the probability will fall off less per level off the diagonal, and doubling ( will in general add one bit to H. On the other hand, doubling the sampling rate will not in general double the required channel capacity, for the closer spaced samples will 6.0 / / / / / / / '». / Ho / / ■^ ^S=S3 ss= y / -^ -^ J Hl^ A / ^ / / 1 1 1 5 6 7 8 9 10 20 30 40 50 60 80 100 200 NUMBER OF LEVELS, I Fig. 9. 748 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 be more highly correlated. Thus in TV, doubling the horizontal resolu- tion would not double the bandwidth for the same picture material if use were made of the statistics. (Of course, increased resolution in TV might encourage the use of more detailed scenes and this would increase the required bandwidth.) It should also be noticed that for small (, linear prediction actually makes matters worse. The increase in the number of levels in the error signal more than offsets the peaking of the distribution. Since all the conditional distributions in this message are of similar shape (after ordering), Hd and H are almost the same, for all /. The difference between H^ and Hd is slight except for small ( because the distribution we assumed is unimodal throughout. Fig. 10 shows the simple probability distributions for (a) the original message, (b) the reduced signal from linear prediction, and (c) the re- duced signal from the digrammer. VARIABLE DELAY AND OTHER PROBLEMS We have seen in the last two sections how it is possible to convert a message for which i7 <{As/l) = Tis/l) T{s[l+ As/i), (2) averaged over as much area as practicable. This space-domain auto- correlation is much easier to measure than the time-domain autocorrela- tion. We need merely measure the relative optical transmission of two identical cascaded transparencies, shifted from register by a variable PARALLEL — LIGHT -OPAL GLASS ' CONDENSER COMPOSED OF TWO 5-INCH ASPHERIC LENSES LIGHT SOURCE GE-1493 Fig. 2 — Basic arrangement of picture autocorrelator. STATISTICS OF TELEVISION SIGNALS 755 Fig. 3 — Close-up view of slide holding assembly and shifting mechanism of picture autocorrelator. amount. The averaging process is inherent in such a measurement. The apparatus used to measure autocorrelation is sho^^^l in Figs. 1 and 2. The chamber at the bottom contains a Ught source of very constant intensity and a convex lens to collimate the light. The middle part, made of accurately machined aluminum, holds the two identical slides of the picture under test, and an aperture exposing a large circular area of the slides. The top chamber contains a collector lens and a photo- multiplier tube which (on a microammeter not shown) gives a sensitive indication of the total light transmitted through the slides. Fig. 3 shows a close-up \'iew of the slide-holding assembly. Two close-fitting graduated aluminum rings permit accurately determined rotation of both slides or one slide, and the micrometer drive permits translational displacements measurable to within one mil (moving the two slides by equal and opposite amounts); the separation between picture elements is approxiinately 7.5 mils horizontally and 5 mils vertically (for the 2Y by 3i" shde size used). The light transmission is always a maximum when the two slides are in precise register (As = 0). For large shifts the transmission fluctuates about a nonzero asymptote. The nonzero asymptote results from the fact that the average transmission is always positive, and the fluctuation from the fact that large displacements introduce substantial amounts of new picture material into the aperture. Since these components tend to 750 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 obscure the correlation effects, it is useful to make additional measure- ments which enable us to subtract them out completely. This leaves us with a 'pure' autocorrelation A(As/d_), which is then normalized so as to have a peak value of unity. It is given by A{As/0) T^{-'ill)-T.{fll)T.{-f/e) (3) T2(0) - Tim where T2 ( —pr— /^ ) is the transmission through the two cascaded slides As shifted by equal and opposite amounts — at an angle 6 with the hori- zontal, and 2'i ( — /^ ) is the transmission of a single slide with displace- As ment — at the same angle 6. SCENE C SCENE D Fig. 4 — Test pictures whose statistics are included in this article. STATISTICS OF TELEVISION SIGNALS 757 SHIFT, AS, NUMBER OF VERTICAL PICTURE ELEMENTS ■100 -80 -60 -40 -20 0 20 40 60 80 100 -i — r — I — I — I — I — I — I — \ — I — I — I I I I I T" SHIFT, AS, NUMBER OF HORIZONTAL PICTURE ELEMENTS -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 1 1 \ 1 1 1 1 1 1 I I I T 0.6 A (AS) 0.2 j^ -0.5 -0.2 -0.1 0 0.1 0.2 SHIFT, As, IN INCHES Fig. 5 — Plots of autocorrelation in horizontal and vertical directions for two pictures. Fig. 4 shows some pictures for which autocorrelation measurements have been made. The results can be presented in the various ways shown in Figs. 5, 6, and 7. Fig. 5 shows conventional plots of A versus As in the horizontal and vertical directions. Scene B is seen to have more correlation than Scene D, and curve shapes range from remarkably- linear to somewhat like exponential. Fig. 6, gi\ang contours of constant autocorrelation, brings out the variation with the angle 6. Scene A happens to have its greatest correlation in the vertical direction, but that was not found to be a general rule by any means; Scene B, for ex- ample, has its greatest correlation in the horizontal direction. No pre- ferred directions appear to exist in general. In Fig. 7 attention is focused on the more local correlation, for small values of As. The average cor- relation among horizontally adjoining picture elements, designated by Aio , is seen to be approximately 0.99 for Scene B and only 0.75 for Scene C. A20 denotes the correlation for a horizontal spacing of two picture elements while Aoi denotes the correlation among vertically adjoining picture elements. It should l)e pointed out that the pictures which gave the above results were not band-limited to the standard 4-mc resolution. However, before the results were used quantitatively, the proper band limitation was applied mathematically. This has the effect of rounding off the peaks of the curves, decreasing the autocorrelation drop within the first Ny- quist interval by up to approximately 24 per cent. 758 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 120° 90° 60° >r:^ '^''^^\>y/ /X^"^ "~~^>A^\ 150° / / //^>C / /Oc*^ 5m\ IS0° \ 1/ fA l^ ) 10 / 15 20 25 30 \ ^^^ J SHIFT IN MILS 210° \ ^^^ y/X^ 2 ^ yZ^ '^\ Fig. 6 — Contours of constant autocorrelation for Scene A. In general there are no preferred directions of correlation. 1.0 AS IN ELEMENTS 0 5 10 15 I I I I I I I II I I I I I I I q 5 10 I I I I I I I 1 1 1 T" AS IN ELEMENTS VERT. 0 5 10 15 I II I I I HOR. 0 VERT. I I I I I I I 1 I 5 10 HOR. A(S) HORIZONTAL VERTICAL^^^ (90°)-' SCENE B A,o = 0.991 A2o= 0.980 Aoi = 0.979 1 1 1 1 1 1 1 1 1 1 I \ \ SCEN JE C \ V A,o = 0.75 A2o=0.43 Aoi = 0.69 \ ^ ^VERTICAL (90°) HORIZON (0°) TAL^yS ^ 0 20 40 60 60 too 0 20 40 60 SO 100 SHIFT, As, IN MILS SHIFT, AS, IN MILS Fig. 7— Plots of autocorrelation for small shifts. Aio is the autocorrelation for a shift of one horizontal elemental distance, A20 for two horizontal elemental dis- tances, and Aoi for one vertical elemental distance. Alternatively Aio may be described as the average correlation between horizontally adjoining elements, etc. STATISTICS OF THLEVISION SIGNALS 759 PROBABILITY DISTRIBUTIONS A i)i-()ba!nlity (list ribut ion of amplitiulcs is generally shown as a plot of probability density' versus signal amplitude. Probability density, say, corresponding to amplitude Xi , is the probability of finding the signal amplitude between .ri and Xi + dx, divided by the differential amplitude increment dx. Conversely, the probability of finding the signal ampli- tude between .Ti and xy + dx is given by ])(xi)dx, p(x) being the pi'oba- bility density corresponding to amplitude x. If a cathode-ray spot is deflected, say horizontally, by the signal in (luestion, its average dwell time at any point is directly proportional to the corresponding probability density. In the optical system shown in Fig. 8, a cylindrical lens maps each point into a vertical line which is UNDEFLECTED IMAGE ""n DEFLECTED /IMAGE "^ LIGHT-PROOF ■"ENCLOSURE Fig. 8 — Basic arrangement of probabiloscope. \HIGH TRANSMISSION then tapered in intensity by an optical density wedge before reaching a high-contrast photographic film. Depending on the dwell time at any amplitude level, the corresponding tapered line has enough average intensity to blacken the film up to a certain level. This level is pro- portional to log p(x), since the density wedge is tapered exponentially so that the intensity of each tapered line of light reaching the film diminishes, say, by a factor of ten for each inch we travel up the line. The film in effect traces out a contour of constant exposure. Two or three iterated photographic printings increase the effective gamma sufficiently to yield a contour of ample sharpness. This contour is then changed to a sharp line by a simple dark room trick: while the film is in the development tray, already fully developed, it is momentarily exposed to light. The blackened portion of the film is unaffected, the clear portion is fully blackened, while the transition contour, being partly opaque, is not fully blackened. By printing from this film we then 7G0 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 obtain a well-defined black-on-white curve of p(x) versus x on a loga- rithmic probability scale. The logarithmic scale has the advantage of making the curve shape independent of exposure length and giving uni- form relative accuracy over the entire range. Fig. 9 shows some typical results obtained by means of the "proba- biloscope." The two small curves are distributions of two different still pictures. The left-hand end corresponds to black, the right-hand end to peak white; the blanking intervals (slightly blacker than black) cause the peaks at the extreme left. (The signals did not contain any synchro- nizing pulses.) The tall and slender curve at the right of Fig. 9 is the distribution of errors resulting from previous-value prediction of one of the pictures in Fig. 4. The peak corresponds to zero error which is seen to be most probable, as it should be if the prediction criterion is good. Increasingly larger errors are increasingly improbable or rare. The six decades of probability density spanned by the curve were ob- tained in three separate exposures and subsequently joined, since stray 1 W'- TT TT 1 SCENE A 1 s ■li -,--'' + . ::: ::- : ORIGINAL SIGNAL _ TT- - -T T T-- - .--.\l .jl t 10-1 m--\ F :- :■: 4- htt-::::i tT^ 5 fw% f :| P hi[r 'A ^. ^ ■---++ -- - -- T JL: ' + (r"'x in-2 *'' \ I ;Si::- + :± ih ... --: ::i:;-^s! 5 TFt uti it : .. '': :-- ::-.| '-\-- ? " " ' T - f- ffl 11: ^fl if i^ SCENE A PREVIOUS VALUE a-" M i 1 ■• ii t: 1 rj- :l|::::::::::::::::: 2 •• •■ ;;|:: ::::::: : :::j::::: 10-' 1 i = f = # ^fi^ii^y 2^|i 11:: E 1 — 5 §=■■- -r -It - lT:r"-f ,0-2... / i:::::::, ::::::: : %t: %-- ;:;;;;;;;i! 2.:-- -- - :x^::::- 1 4:5 "i'X .1; .... X jl . .. : :j :::.. . .. !;.:::: ^ 10-3 - r t T 5 |ttj * 'i p.:::.: :: H t gS :.. '-X :::::::::::: : ;-: : gJ :! 2 -■ .._ -- ::';: : ^, ::: 10-4::: ;:;:;;;.::::::;; , ,!:::::: 5 ::: *---::-:- ;■ ::::: : ::::: , :;' ^X 10-5 02468 10 02468 10 AMPLITUDE, X, IN ARBITRARY UNITS Fig. 9 — Typical probability distributions as obtained from the probabiloscope. Curves at left are for video signals; right-hand curve is for difference between video signal and delayed replica. STATISTICS OF TELEVISION SIGNALS 701 light limits the useful range of the probabiloscope to approximately two decades. In obtaining those sections of the curve corresponding to the few and far-between large errors, a long exposure was used and the cathode-ray beam was blanked whenever passing through the range of zero or small errors. The vertical scale on all ciu'ves is determined solely by the density taper of the optical density wedge. If this scale is to repre- sent true probability density, instead of a proportional quantity, it should be sliifted up or down so as to make the area under the curve equal to unity. APPLICATION OF RESULTS The statistics measured can be put to various uses, such as in the design of better predicting or coding schemes. The most interesting application is probably in estimating the reduction in channel capacity which the measured statistics show to be theoretically possible. In other words, the results can give us various lower bounds to the re- dundancy of television signals. For the sake of illustration, suppose that the signal is quantized into 64 amplitude levels. An ordinary television channel assumes all 64 levels to be equally likely, hence is prepared to accommodate log2 64 or 6 bits per sample. But the simple amplitude distribution of the signal is not flat, so that all 64 levels are 7iot equally likely. The maximum possible associated average information content per sample is given by 64 ^max = JL Pi log Pi , (4) t where pi is the simple probability of the signal's falling into the iih level. Since the 64 p,'s are unequal, //max is necessarily less than 6 bits. For all available data the average value of //max turns out to be ap- proximately 5 bits, indicating a one-bit redundancy. The latter figure is essentially independent of quantization. The prediction error signal still contains all the useful picture in- formation. The maximum possible information content per sample (max- imum in that all samples are assumed to be completely independent) is still given by (4) but in this case the 64 values of p.- are obtained from the peaked error distribution. The average* result from all available data turns out to be approximately 3.4 bits below the 6-bit ceihng, show- * This average was computed by averaging the various redundancy values ob- tained for the individual pictures, rather than averaging all statistical data and then finding one corresponding average redundancy. The average computed here is more favorable and can be realized onl}^ if optimum coding is performed on a short-term basis rather than on the basis of one set of long-term statistics. 702 THE BELL SYSTEM TECHNICAL JOURNAL, JILY 1952 ing that the original signal must have contained at least 3.4 hits of redundancy. The autocorrelation can also furnish a lower bound to the redundancy, as has been pointed out by P. Elias in his Letter to the Editor of the Proceedings of the I.R.E. for July, 1951. If, for example, the correlation Aio , between horizontally adjoining picture elements, is high, the cor- responding lower-bound redundancy is \'er3^ roughly eriual to R ~ —^ log., (1 — .4io) bits/sample. (5) Alternatively, taking the Fourier transform of the autocorrelation yields the power spectrum P(/), from which we can find the lower-bound re- dundancy through the relation R = ^ j logo P(/) df + i logo W -f log2 K bits/sample, (6) 1 r^ where W = bandwidth in cps, and — = / P(/) df. K Jo Using either method, one obtains approximately 2.4 bits for the average* of the a\'ailable data. This is an approximate bound, in that it applies strictly only to functions having gaussian amplitude distri- butions. Suppose, then, that we have exposed an average redundancy of at least 3 bits per sample. This means a potential 3-bit reduction in the chamiel capacity required for television transmission. In a 6-bit system (64 amphtude levels) this means a 50 per cent reduction, and hence a potential halving of the band^^^Ldth with the aid of an ideal coding scheme. It is true that the decorrelated signal is somewhat "frail," i.e., \Tilner- able to interference, so that it might be desirable to use a "rugged" system of the PCM variety for transmission. Thus, if a Shannon-Fano code were used, the 3-bit decorrelation should enable us to send tele- ^'ision by an average of 3 on-off pulses per picture sample rather than 6. This represents a two-to-one saving over the usual PCM bandwidth. More spectacular reductions are hkely to be achie^■able onlj^ by tap- ping the large-scale redimdancies mentioned earlier. FRAME-TO-FRAME CORRELATION There is, of course, a great deal of interest in the possibility of utilizing the similarity between successive frames. Accordingly, adjacent-frame * See previous footnote. STATISTICS OF T i;hl-;\ ISK ).\ SIGNALS 703 eon'olation was measured for two typical motion-picture films, by means of the apparatus descrilxMl in tiie section on autocorrelation.* The i-esults were 0.80 and 0.8(), after coi'i'ectioii for tlu> l-inc t)aiid\vi(lth limitation. This means that "[)re\ious-frame" pixMliction can ivmoxe only slifi;htly mor(> than one bit of redundancy per sample. More complicated schemes would pi'esumably be more successful in taking advantage of the large frame-to-frame redundancy which undoubtedly exists. ACKNOWLEDGMENT Many of the ideas expressed in this paper are due to B. M. Oliver, whose resourcefulness is hereby gratefully acknowledged. * The expression used in evaluating tlie correlation between frame 1 and frame 2 (an}^ two frames) is rp rp2 _ -^ 12 ~ -^ 1 (-7) where T12 is the optical transmission of frames 1 and 2 in cascade, Ti is the average of the individual transmission of frames 1 and 2, and Tn is the average of the transmissions of two cascaded slides of frame 1 and two cascaded slides of frame 2, respectively. In all cascade transmission measurements, the two frames must be in precise register. Experiments with Linear Prediction in Television By C. W. HARRISON (Manuscript received February 28, 1952) The correlation present in a signal makes possible the prediction of the future of the signal in terms of the past and present. If the method used for prediction makes full use of the entire pertinent past, then the error signal — the difference between the actual and the predicted signal — will be a com- pletely random wave of lower power than the original signal but contaiyiing all the information of the original. One method of prediction, which does not make full use of the past, but which is nevertheless remarkably effective with certain signals and also appealing because of its relative simplicity, is linear prediction. Here the prediction for the next signal sample is simply the sum of previous signal samples each midtiplied by an appropriate weighting factor. The best values for the weighting coefficients depend upon the statistics of the signal, but once they have been determined the prediction may be done with relatively simple apparatus. This paper describes the apparatus used for some experiments on linear prediction of television signals, and describes the residts obtained to date. INTRODUCTION Linear prediction is perhaps the most expedient elementary means of removing first order correlation in a television message. Before discussing the advantages and disadvantages of linear prediction, it might be well to consider what is generally meant by correlation in a tele\dsion pic- ture and why it should be removed. Almost every picture that has recognizable features contains both linear and non-linear correlation. Each type of correlation helps in identifying one picture from another; however, linear prediction is only effective in removing linear correlation, and for this reason, future ref- erences to correlation ^\'ill refer only to its linear properties. With tele- vision, a signal is obtained as the result of scanning; hence, the cor- 764 LINEAR PREDICTION IN TELEVISION 7fi5 relation is pvidout in both space and time. Briefly, correlation is that relation which the "next" elemental part of the signal has with its past. To lea\e correlation in a, message is to be I'edundant, and this effec- tively loads the transmission medinm with a lot of excess "words" not necessar}' to the description of the picture at the receiving end. It is then more "efficient" to send only the information necessary to identify the picture, and to restore the redundancy at the receiver. EFFICIENT TR.\NSMISSION The more efficient we are in sending pictures over a given transmission line, the more alarmed we become at the increasing amount of equipment that is required at the transmitting and receiving terminals. Cei'tainly the design wll be a compromise between the complexity of apparatus and the efficiency achieved. The ingenuity of engineers will be taxed along these lines for years to come; however basically, the general form of these systems will be similar to that shown in Fig. 1. Although not always separable, four essential operations are required-namely, decor- relating, encoding, decoding and correlating. The transmitting decor- relator and the encoder encompass the principal design problems, since the decoder and correlator at the receiving end perform the reverse operations which interpret the code and add in the redundancy that was removed. Decorrelation involves prediction, and as the predictors are more nearly made to predict the future of the signal, the more the output signal from the decorrelator resembles random noise. The essential picture information is still present, which means that our original picture signal can be obtained at the receiving end without theoretical degrada- tion. The basic job of the encoder is to match the picture information out of the decorrelator to the channel over which it is to be transmitted. There are several encoding operations. The first concerns the rate of information into the encoder, and that required out of it. In the case of television, there are fiat, highly correlated areas as well as areas containing more concentrated detail. This means that the information OUT TRANSMITTER RECEIVER DECORRE- LATOR — ENCODER DECODER — CORRE- LATOR ♦ TRANSMISSION MEDIUM Fig. 1— Block diagram of an efficient transmission system employing reversible decorrelating and encoding means. 766 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 rate varies when the picture is scanned at the conventional uniform scan- ning rate. The output of the encoder feeds a transmission hne that has a definite channel capacity, and if maximiun efficiency is to be obtained from this transmission mediiun, then the rate of information into it must be held relatively uniform at a value near the channel capacity. It is the job of the encoder to take the varying rate of information from the decorrelator and feed it to the channel at a constant rate. At the receiving end, the decoder must take the constant rate of information and dehver it to the correlator at the variable rate as originally fed into the transmitter's encoder. Thus, to perform this task, a variable or elastic delay to run ahead or behind, depending on the information con- tent of the picture being scanned, is an important part of the encoder. Over a long period of time, the variable delay would average out to some fixed value. Tliis variable delay must never run out, even when the detail is concentrated. There are instances when this condition could not be met, such as an extended reproduction of a snow storm; however, with good design the system should fail "safe" — a slight degradation of picture quality. This condition can be made infrequent enough to cause little concern. The encoder design must also account for noise as well as bandwidth of the channel and must consider the ultimate effect of an error that may be introduced by noise along the transmission line. As more re- dundancy is removed to get at the "essence" of the picture signal, the more important it is to guard this "essence," as mistakes presented to the receiver will propagate themselves longer in the absence of correla- tion. Errors can be minimized by rugged systems of modulation such as PCM, where the signal-to-noise ratio of the transmission line deter- mines the base of the PCM system selected. In any event, the encoder must send the information so that the effect of errors will not appreciably disturb the picture. DECORRELATION AND LINEAR PREDICTION Fig. 2 illustrates, in a general way, a means of decorrelating the sig- nal, Si(t). For purposes of explanation, the encoder and decoder have been omitted, and the transmission between the receiving and sending terminals, idealized. The predictors, P, are identical, and base their prediction, Sp(t), on the signal's past history. In this way, the output of the computer represents the discrepancy between the actual value of the signal sample and the predictor's prediction. By this means we are sending only our mistakes- — the amount by wliich the next pictm'e ele- ment surprises us. For example, if the computer is so designed that it LINEAR PKIODU'TION IN TELEVISION 7G7 COMPUTER s,(t) I — *^ — I- I " S,(t)-Sp(t) .- fe_ Sp(t)l PREDICTOR Sp(-t) s,(t) Fig. 2 — Decorrelator and correlator showing reversible nature of this method of removing redundancy. bases its prediction on the "previous frame," and we are transmitting a "still," there will be no surprises after the first frame and consequently no output signal. Certainly it is redundant to send the same picture more than once. linear prediction pro\ades an easily instrumented means of removing redundancy. With linear prediction the next signal sample is simply the sum of the previous signal samples, each multiplied by an appropriate weighting factor. The best values for these weighting coefficients de- pend on the statistics of the signal. Fig. 3 is a block diagram of a decorrelator employing linear predic- tion. The delayed versions of the input signal can be obtained from taps along the delay line. The weighting coefficients for each of the delayed signals are selected by loss in their respective paths as shown by the amplitude controls. The polarity of each signal can be determined by the switches. The output is simply the sum of these weighted signals. If we consider the signal on a continuous basis (not quantized or sampled), linear predictors can be characterized as ordinary linear filters used to predistort the frequency spectrum of the signal. As such, they can be designed in the frequency or time domain. However as will Fig. 3 — General block diagram of decorrelator employing linear prediction. Linear prediction bases its prediction on the weighted sum of previous signal samples. 7G8 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 be shown, it is much easier to recognize circuit configurations that reduce redundancy in the time domain. To this end, and for purposes of encoding, the signal is thought of as signal samples uniformly spaced at Nj^quist intervals. Thus, amplitude values obtained by sampHng a 4.0 mc picture signal at | microsecond intervals serve to specify the signal completely. Fig. 4 shows a small portion of a television raster where the signal is represented by signal values spaced at Nyquist intervals, T. The coordinates sho^vn are designated mth respect to the "present value" of the signal, So,o ■ The positive coordinate directions are shown by the arrows. The past is represented by positive coordinates — the future by negative coordinates. In this way, the previous value of the signal <-T- ->(<- T- ->j<- -r - ->j<- r — >4<- T — > 3,2 2,2 1,2 0,2 -1,2 -2,2 V, 3,1 2,1 1,1 0,1 -1,1 -2,1 / LINES OF •INTERLACED FIELD 3,0 2,0 1,0 Sx,y r = NYQUIST INTERVAL _ _1_ " 2W X Fig. 4 — A small portion of a television raster showing geometrical location of signal samples with relation to the "present value" of the signal, So,o than the closest horizontal sample, *Si,o . The error signal, e, is given by where T is a line time. The error output has a maximum peak amplitude of twice the input. The filter characteristic can be expressed as ■ \2 ~ "2~J F(co) = _2sm-J Fig. 6 — Example of "slope" prediction. Here the next signal value is assumed to lie on a straight line that intersects the two previous signal values. LINEAR I'UKDU'TION IN TELEVISION 771 "Planar" prediction, shown in Fi^. 8, is effectively tandem operation of "previous value" and "previous line" prediction. Planar prediction may also be thought of as the \alue represented by a plane above the present value of the signal when passed through three adjacent signal PREVIOUS LINE PREDICTION Fig. 7 — Example of "previous line" prediction. Here the error signal is the difference between the actual value of the signal and the value of the signal on the line directly above. PLANAR PREDICTION SIGNAL OF PREVIOUS LINE / / /^ \ / / / - e S,,o So,o Sp= 5, e = S, + 5, ' 0,0 ■-'1,0 -^0,1 ■'" -^1,1 Fig. 8 — An example of • planar" prediction. Here the prediction is represented by a plane that has been passed through three adjacent signal values. 772 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 values, namely aSi.o , aSo.i and >Si.i . The predicted signal is given by Sp = 01,0 4" So^i — *Si,i . The filter characteristic is given by F(w) = 4 sin — sin -— e ^ ^ ^ The peak error ampUtude for "Planar" prediction can be four times that of the input signal. "Planar" prediction has several good characteristics. For example, if S>\,\ and »Si,o were white and >So.i black, then *So,o would be predicted to be black. Thus a change horizontally from white to black would produce no errors. Similary, if *Si,i and O O ^\ 2,2 -1,2 ^"-y INTERLACED ^" / FIELD / / > O O ' 2,1 -',1 ^ * o X 1,0 0,0 Sp = S)^o "" ^2^1 + ^2,2 ~ S)^3 + So,3 ~ S-i^a"*" S-i,t Fig. 9 — Past signal samples required for "circular prediction" — a type of prediction which removes horizontal, vertical, and d=52° straight line picture contours. LINEAR PREDICTION IN TELEVISION 773 Therefore, indefinite extension of this straight Hne contour deleting philosophy is not a paying means of prediction, at least not at the present state of the art of wide band delay lines. Furthermore, the increasing diameter of the circle for extension of circular prediction would decrease its accuracy for finely concentrated detail. Fig. 10 shows the relative position of picture elements nearest >So.o if a mde band field delay were available. The methods of prediction dis- 0,1 ;- INTERLACED FIELD 1,0 .1. Fig. 10 — Small portion of television raster showing signal samples, including those of previous field, which would enable time extrapolation-space interpolation as a method of prediction. cussed have been essentially an extrapolation in space; however, with a field delay, interpolation in space, and extrapolation in time would also be possible. EXPERIMENTAL CIRCUITRY Experunentally, those types of predictors that involve only a few Nyquist intervals of delay are easiest to mechanize. Fig. 11 shows a simplified schematic of a decorrelator that enables an evaluation of linear prediction schemes having error signals given by e = ao,o«So.o ± ai,o V \ 3 , / / \ \ > / V k > / ^ \ / \ \ 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 ^1,0/^0,0 Fig. 13 — Power reduction for various weighting coefficients for the previous horizontal sample. areas as can be observed in the background of Scene A. Where the separation of a white to black area is made, the error signal is large. It is tliis large error signal that informs the receiver of this change in bright- ness, and until another change occurs, the error output is again zero. This type of performance produced the flat grey appearance of the back- ground. In this way, the picture represents only changes in brightness — a first difference type of picture. It may be noted that horizontal contour lines have vanished leaving only vertical contours which pertain to the brightness changes that have occurred. This effect is especially evident in Scene C. The power reductions given in the lower left hand corner of these pictures are consistent with their complexity. Fig. 16 shows the error signal appearance for "slope" prediction. When compared to the error signal for "previous value" prediction a finer vertical granularity is observed, and this is attributed to sudden SCENE A SCENE B ORIGINAL 'W 1 SCENE C Fig. 1-i — Three pictures as photographed from the face of a kinescope. Scene "A" is a picture of average complexitj*. Scene "B" is a simple, rather soft picture. Scene "C" is a complex, highly detailed picture. Roughly, these pictures represent the gamut of pictures normally expected to be transmitted. 778 SCENE A SCENE B PREVIOUS VALUE 17 DB PREVIOUS VALUE 22 DB , 4 SCENE ^ C i PREVIOUS VALUE 10.5 DB m 1 \ ■ ■ '<,■!■ r^>\ '■ , '• ^l^^^l^iyi Fig. 15 — Three pictures showing the appearance of tlie error signal when using "previous value" prediction. Note the absence of horizontal contours. 779 SCENE A SCENE B SLOPE 20 DB SCENE C SLOPE 8.5 DB Fig. 16 — Three pictures showing the appearance of the error signal when using 'slope" prediction. 780 SCENE A -^ PREVIOUS LINE 14.7 DB HP SCENE B SCENE C PREVIOUS LINE - 8.0 DB Fig. 17 — Three pictures showing the appearance of the error signal when using "previous line" prediction. Note the absence of vertical contours. 781 SCENE A '9^SBHi PLANAR 17.5 DB MAX 20.0 DB ^Mi SCENE B PLANAR 18.7 DB MAX 21.0 DB SCENE C PLANAR 10.3 DB MAX 12.5 DB Fig. 18 — Three pictures showing ihe appearance of the error signal when using "planar" prediction. Note the absence of horizontal and vertical contours. 782 LINKAK PHKDKTIOX IN TKLKVISION 7S'o,o. If the closest horizontal sample was taken at the same distance from the present value of the signal as the pre^ious line sample, then the power reduction using these signal \'alues individually for prediction would be essentially the same for most pictures. Fig. 18 shows the error signal appearance for "planar" pi-ediction. Here, vertical as well as horizontal contours are deleted. In Scene A the ti'ee trunk has almost completely vanished. In Scene B the picture has an extremely flat appearance. Scene C exhibits the lack of hori- zontal and \'ertical contours best, since only sloping contours are left. The power reduction figures at the lower left hand corner also show values foi' minimiun error power. For most pictures, the error power can be reduced by a factor of one-half again over the planar coefficients by modifying the weighting coefficients. The coefficients for this modified planar case are given by Sp = 3*^1,0 + 3*^n,l ~ 3*J1,1 These coefficients generally produce an error signal with less power than the coefficients used for "planar" prediction. While all pictures contain redundancy, the error signals from these simple hnear predictors shown in Figs. 15, 16, 17 and 18 can \'isually l)e noted still to contain large amounts of redundancy. The contours of the models and of the various objects are readily identifiable. Were all redundancy removed, the picture would be completely chaotic and would appear very much like random noise, although greater efficiency in transmission would be achieved. For lichei' rewards, more sophisticated methods of prediction will be reciuired. ACKNOWLEDGMENT The author wishes to acknowledge with grateful ajipreciation the in- valuable guidance of Dr. B. 'SI. ()li\(>r. It was j)rinci])ally through iiisef- orts that this study was made j)ossible. Generalized Telegraphist's Equations for Waveguides By S. A. SCHELKUNOFF (Manuscript received April 30, 1952) In this paper Maxwell's partial differential equations and the boundary conditions for waveguides filled with a heterogeneous and non-isotropic medium are converted into an infinite system of ordinary differential equa- tions. This system represents a generalization of ^'telegraphisVs equations" for a single mode transmission to the case of multiple mode transmission. A similar set of equations is obtained for spherical waves. Although such generalized telegraphist's equations are very complicated, it is very likely that useful results can be obtained by an appropriate modal analysis. From a purely mathematical point of view the problem of electro- magnetic wave propagation inside a metal waveguide reduces to obtain- ing that solution of Maxwell's eciuations which satisfies certain boundary conditions along the waveguide and certain terminal conditions at the ends of the waveguide. If the medium inside the wa\eguide is homo- geneous and isotropic and if the cross-section of the waveguide is either rectangular or circular or elliptic, the desired solution is obtained by the method of separating the variables. The method works for some other special cross-sections. It works also if the medium inside a rectangular waveguide consists of homogeneous, isotropic strata parallel to one of its faces. Similarly, it works if the medium inside a circular waveguide consists of coaxial, homogeneous, isotropic layers. But in general if the medium is either nonhomogeneous or non-isotropic or both, the method does not work. The mathematical reason for this is that the solution is of a more complicated form than a simple production of functions, each depending on a single coordinate. Any function that one usuall}^ en- counters in physical problems, and therefore a solution of Maxwell's equations, may be expanded in a series of orthogonal functions. Sets of such functions are pro\-ided by the solutions for waveguides filled ^\'ith homogeneous media. Such functions already satisfy the proper boundary conditions and the problem is to obtain series which also satisfy 784 GENERALIZED TELEGRAPHIST'S EQUATIONS 785 Maxwell's equations. From the physical point of view this method represents a conversion of Maxwell's equations into generalized "tele- graphist's equations." Thus it is already known that Maxwell's partial diflfcreutial eciuations and the boundary conditions alojig a waveguide are convertible into a set of independent ordinary differential equations, each resembling tele- graphist's equations for electric transmission lines. ^ Each ecjuation de- scribes a "mode of propagation" in terms of concepts well known in electric circuit theory. A waveguide can be considered as an infinite system of transmission lines. If the medium inside the waveguide is homogeneous and isotropic and if the surface impedance of the boundary is zero, the method of separating the variables enables us to obtain a set of "normal", that is, uncoupled modes of propagation. Any irregularity or "discontinuity" in the waveguide provides a coupling between some, or all, modes of propagation. The irregularity may be in a boundary of the waveguide or in the dielectric within it. A heterogeneous dielectric may be considered as a homogeneous dielectric with distributed irregu- larities. Similarly a heterogeneous non -isotropic dielectric may be con- sidered as a homogeneous isotropic dielectric with distributed irregu- larities. Such irregularities provide a distributed coupling between the ^'arious modes appropriate to homogeneous isotropic waveguides. Our problem is to calculate the coupling coefficients. The generalized tele- graphist's equations, obtained in this manner, are very complicated in that they represent an infinite number of coupled transmission modes. They are useful, however, in suggesting a physical picture of w^ave propagation under complicated conditions, and can be used in approxi- mate analysis when we can ignore all but the most tightly coupled modes. For example, this picture was successfully employed })y Alber- sheim in studying the effect of gentle bending of a waveguide on propa- gation of circular electric waves. In this case it was important to consider the coupling between only two modes, TEoi and TMn , which have the same cutoff frequency in a straight waveguide. More recently, Stevenson obtained exact equations for waves in horns of arbitrary shape.* His equations express the propagation of the axial components of E and //. The various modes are coupled through the boundary of the horn. In 1 S. A. Schelkunoff, "Transmission Theory of Plane Electromagnetic Waves," Proc. Inst. Radio Engrs., Nov. 1937, pp. 1457-1492. 2 S. A. Schelkunoff, "Electromagnetic Waves," D. van Nostrand Co., (1943), pp. 92-93. ^ W. J. Albersheim, "Propagation of TEoi Waves in Curved Waveguides," Bell System Tech. J., Jan. 1949, pp. 1-32. * A. F. Stevenson, "(general Theory of Electromagnetic Horns," J. Appl. Phys., Dec. 1951, pp. 1447-1460. 786 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 the present paper we shall consider waveguides of cotistant cross-section and conical horns of arbitrary shape filled with a heterogeneous and non-isotropic dielectric and derive the equations for propagation of the generalized voltages and currents representing the transverse field com- ponents. The various modes are coupled through the medium. It is very- likely that our ecjuations can ])e generalized to include the coupling through the boundary. To understand the mechanism of coupling between the various modes through the medium consider Maxwell's equations curl E = -jo,B, curl H = 'J -\- jc,D, (1) where "^J is the density of conduction current while the other letter symbols have the usual meanings. In the most general linear case the components of B and D are linear functions of the components of H and E respectively, with the coefficients depending on the coordinates. These equations can always be rewritten as follows curl E = ->m// - M, curl H - jcceE + ./, (2) ion where M and ./ are the densities of magnetic and electric polarizati currents. M = MB - uH), J = "J + MD - eE), (3) and n, e are constants (not necessarily those of vacuum). If M and / were given, they would act as sources exciting various modes of propa- gation in a homogeneous, isotropic waveguide. If M and J are functions of H and E, they can still be considered as the sources, acting on power borrowed from the wave, of the \'arious modes. Thus ^1/ and J will provide the coupling between the modes existing in a homogeneous, isotropic waveguide. Thus in order to derive the generalized telegraphist's equations we shall first consider the various modes of propagation in a homogeneous isotropic wave guide. Each mode is described by a trans^'erse field distri- bution pattern*^ T(u, v), where u and v are orthogonal coordinates of a point in a typical cross-section. This function is a solution of the follow- ing two-dimensional partial differential ecjiiation AT = — 6162 "a {^^ ^ ] -\- 1 (^2^ _du V Ci du / dv Ke-i dv = -X% (4) * See Reference 2. 6 S. A. Schelkunoff, "Electromagnetic Waves," D. van Xostrand Co. (1943), Chapter 10. GENKRALIZKI) TKLIXiHArillST S EQIATIONS 787 wiiere x is the separation constant and Ci , Co are the scale factors in the expression for the elementary distance ds' = e-i (Jii + e-j dv . (5) In the case of TM wa\'es the V'-fnnction must vanisii on the boundary of zero impedance. This boundary condition I'estricts x to a doubly in- finite set of ^•alues Xmn witii the con-espondinj^- functions 7'm„ . In the case of TE wa\'es the normal dei-i\'ati\'e of the 7'-fvuiction must \'anisli on the boundary of zero impedance. Since we have to consider both types of ^va^'es simultaneously, we shall distinguish between them by enclosing the subscripts in parentheses for TM waves and in brackets for TE wa\'es. The double subscript designation of various modes has been standardized only for rectangular and circular waveguides. For wa\'(^guides of other shapes the standard is to use a single subscript by ai'ranging the modes in the order of their cutoff frequencies. For con- venience, we shall use this convention in the general case and denote T]\I modes b.y T'(„)(;/, r), and TE modes by T[„]{u, v). The correspond- ing cutoff constants will be xm and X[«] • Ii^ what follows it is under- stood that whenever the T-functions should be designated by double subscripts, our single letter subscripts should be considered as symbols for ordered double subscripts. The transverse field components may be derived from the potential and stream functions, V and 11 for TM waves and U and ^ for TE waves. Thus Et = - grad T^ - flux ^, Hi = flux n - grad U, (G) where the components of the gradient and flux of a scalar fiuiction W are dW dW grad„ W = —— , grad, W = — — , ei du eo dv (7) dW dW flux„ W = ^^ , flux„ W = -^^ . €■> dv ei du The T-fiuictions corresponding to the various modes of the same va- riety are orthogonal ; that is, the followhig integrals over the cross-section vanish, \JT(n)T^m) dS = 0, JJTmTi,,,] dS = 0, if m ^ n. (8) It should be stressed that 7'(„) and !/'[,„] are not, in general, orthogonal. ^ See Reference 6. 788 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Similarly the gradients of the J'-functions of the same variety as well as the fluxes, are orthogonal, ||(grad T(„))-(grad T(„)) dS = //(flux T(„))-(flux T(„)) dS (9) = //(grad Tin]) • (grad Tim]) dS = //(flux T^) • (flux Ti^]) dS = 0, if m 5^ w. The following gradients and fluxes of the T-f unctions are ortho- gonal for all m and 7i, //(grad T(„)).(flux T^m]) dS = //(grad ri„j)-(flux T(„)) dS = //(grad T(„))-(flux T^m)) dS = 0. (10) On the other hand, grad T[m] and flux T[n] are not, in general, orthogonal. If all modes are present, the potential and stream functions are V = -VM(z)Tin){u,v), U = -IUz)Tm{u,v), ^ ^ (11) ^ = -Vln]{z)TMiu, V), U = -lM(z)Tin](u, v) , where the tensor summation convention is used: whenever the same letter subscript is used in a product, it should receive all values in a given set and the resultmg products should be added. The negative signs have been inserted m order to avoid a preponderance of negative signs in later equations. Substituting in (9), we have Et = Vm grad T^ + V^] flux T^^] , (12) Ht = — /(„) flux Tm + I[n] grad Tin] . The ^-functions for the various modes are determined by equation (4) and the boundary conditions except for arbitrary factors related to the power levels of the modes. If we choose these constants in such a way that //(grad T) • (grad T) dS ^ x' ffr' dS = 1, (13) then the complex power carried by the wave is gi\'en by an expression similar to that in an ordinary transmission line, P = WmII) + hVmiltn] . (14) GENKRALIZKD TELEGRAPHIST'S EQUATIONS 789 Hence, the F's and /'s correspond to the voltages and currents in (■()ui)lcd transmission lines. In an expanded form equations (12) are ^^(") 1 T/ ^^["1 ti V dT(n) ^r dr[n] — -r— + y [n] -- - , ii„ = V (n) — — Vln] " T" , ei du e-> dv €•> dv ei du Eu — V^n) 1— + V[n] --„-, Ev = V(n) T^ — V [n] (15) Ci du e-i dv Co dv C] du To these we add the following expansions for the longitudinal compo- nents of E and H E, = X(«)T%.(„)(2)7\„)0/, r), //, = X(«]A',[n](2)7^[.)(M, v). (lO) Equations of this form satisfy automatically the boundary conditions on E, and //, . The multipliers X;. have been inserted arbitrarily in order to make the phj^sical dimensions of the second factors to correspond to those of \'oltage and current. Let us now write Maxwell's equations in an expanded form dE. dE,, . „ dH, dH„ . ^ e-i dv dz e-i dv dz dE^ dE, . a//„ dH, . ^ , „. dz 61 du dz 6] du —^-— - ^„ = -J^eiCoB, , —- • - — = JUC1C2D, . du dv du dv Substituting from (15) and (16) hi the left column of (17), we find 62 dv dz e-i dv dz exdu „ dTf^n) . dYjn-) dT(n) . dV[„] dT[n] . „ , . ei du dz ei du dz e-idv > v / y d'T(n) _ y A /^!r ^Zinl\ _ T/ ^'^(n) _ x. d ( Cl dT [n]\ '""'dudv '"'auU'a^/ ^'"^d^Td^ ^"^a^U^'aT/ (20) = —jojeieiBg. In view of (4) the last equation reduces to X[n]V[n]Tln] = -jc^B^. (21) 790 THE BELL SYSTEM TECHNICAL J(3URNAL, JULY 1952 Multiplying (18) by [-a7\„)/e2 dv] dS, (19) by [a7\„o/ei du] dS, adding, and integrating over the cross-section, we obtain -X(.)T .,(») + -^ = jco jj ^B. ^-^ - B. ^ dS . (22) In the first term the summation con\'ention should be ignored. Multiply- ing (18) by [dT[,n]/ex du] dS, (19) by [dT[m]/e2 dv] dS, adding, and in- tegrating we find dV[m] . ff I n dT[m] , „ dT[„,] Jc Bu -^ + B.. "-^ dS. (23) dz JJ \ e\ du 62 dv / Multiplying (21) by T[n,] dS and integrating, we have ^[»] = -> f{B.T[„] dS. (24) Subjecting the right column of (17) to a similar treatment, we obtain three additional equations. Summarizing, we have ^ = j. ff (5„ ^ - B„ ^) dS + X(.)T^.(.) , (25) dz J J \ 62 dv 6i du J ^ = -2. ff (Z). ^ + D. ^) dS, (26) dz J J \ 6i du 60 dv / ^ = -j^ ff (b. ^ + 5. '^) dS, (27) dz J J \ 6i du 6-2 dv / '^^"' = Jo^ ff (-Du ^ + D. '-^^ dS + xt.]/-MH , (28) dz J J \ 6i dv Ci du Ff.] = -jo: fJBJ\,n^ dS, /(.) = -jo: \\D.T,m, dS. (29) In the last terms of equations (25) and (28) the summation convention should be ignored. If the components of B and D are linear functions of the components of H and E respectively, then with the aid of (15) and (16) they can be expressed as linear functions of F(„) , V[n] , /(„) , I[n] , Vz,(„) , L,[n] - Solving (29) for Fs,(„) and I,,[n] and making the appropriate substitu- tions in (25), (26), (27), (28), we obtain the generalized telegraphist's GENERALIZKD TKLKGItAl'llIS T S EQUATIONS 791 equations m the following;" form = —Z(m)in)I(n) — Z(.m)[n]T[n] — 7\„i) („) T' („) — ?'(»,)[«] l'^[„l , dv (m) dz = — F( „,)(,.) F(„) — Y{m)[n]V[„] — Tim){n)I{n) — 'L\m){n\I [n] , d/(„o dz (30) dV[m\ dz dl[m] dz — ~Z[m]l7i]I[n] — Z[m]{n)I{n) — T[,„][n]^ [n] — T [,n]{n)V (n) , l[m][nlT [«] — y[m]{n)V(„) — T[„,][n]T[n] — T [„,]{„) I (n) • Th(> ti'iuisfcr impedances Z, i\\v Iranster admittances F, the voltage transfer coefficients 7', and the current transfer coefficients 'T between \arious modes are in general functions of z. They are constants if the properties of the waveguide are independent of the distance along it; in this case the problem of solving the generalized telegraphist's equa- tions reduces to solving an infinite system of linear algebraic ecjuations and the corresponding characteristic equation. Similar ecjuations may be derived for spherical waves either in an un- limited medium or in a medium bounded by a perfectly conducting coni- cal surface of arbitrary cross-section. If the latter is circular and if the flare angle is 180°, we have a plane boundary. Hence, the case of spheri- cal waves in a non-homogeneous medium is included. In the spherical case we shall use the general orthogonal system of coordinates (r, u, v) where r is the distance from the center and {u, v) are orthogonal angular coordinates. In this system the elements of length ds and area dS are given bv 2/272. 2 J 2^ , , * 2 ds = dr" -\- r {e\ du + e^ dv"), dS = r dQ, c/O = CiC-^ du dv. {\^\) The trans\'erse field components may be expressed in a form similar to that foi- waveguides rEt = - grad V - flux n, rHi = flux n - grad U, (32) where grad and flux of a typical scalar function are defined l)y ecjuations (10). Instead of (11) we have V = -VUr)TUu, v), n = -/(„)(r)r(„)(u, v), (33) ^ = -F[„](r)r[„](7A,.), U = -I,„(r)T^.,(n,v), where the ^'-functions satisfj^ e(iuation (4) and appropriate boundary conditions. These functions, their gradients and fluxes are orthogonal. 792 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 They are assumed to be normalized as follows J|(grad T) • (grad T) dn = x' JJt' d^ = 1 , (34) where dQ is an elementary solid angle. Hence, equation (14) will again represent the complex power flow in the direction of propagation. The various field components may then be expressed as follows ^^(.n) . jT dT[n] j^ _ ^-r dT(„) ,, dT{„] ^ a ' ^["1 ^ ' rJ^i' — y (.n) — — V [n] - , Si du 62 dv 62 dv Ci du rM„ - 1 („) — -— + 1 [n] —-— , rHu - — ^ (n) — + I [n] —^- , Bi du 62 dv e-i dv ex du r Er = X(n) Vr,{n) T (n) , T Hr = X[n] /f . [n] T [n] . It should be noted that the physical dimensions of Vt.m and Ir,[n\ are not those of voltage and current. Substituting in Maxwell's equations and using transformations similar to those in the case of plane waves, we find '^ - - // (* dT{m) ^n dT(r, 62 dv ei du dJ^ dr dV[m] ">) 1 JO I .. ^-2 _rB,-^]dn-\- xwr n.c^), dr dllm] dr j. K (rD,. "^ + tD. ^') da, J J \ ei du e-i dv / f[(rBj-^^rB.^-^)dU, (36) J J \ Ci du e-z dv / = jc f( (- vD. ^ + rD. ^') dfi + X[.ir-^.fH , J J \ 62 dv 6i du/ = -jcu F[.] = -jo^ ff(r'Br)T[^^dn, Ion) = -jc^ fl{r'Dr)T^,n) dQ. Returning to the plane wave case and assuming the following general linear relations Bu = UuuHu + fJLitvHy + HuzHz , Du = e„„£^„ + euvEv -\- tuzEz , Bv = fJivuHu + Hvvtlv -\- UvzHz , Dy = truEu + e^Ev + t^zE , , (37) Bz = nziiHu + i^ivH„ + nzzHz , Dz = tzuEu + tzvEv + e33£'2 , GENERALIZED TELEGRAPHIST'S EQUATIONS 793 we find Hu = i(n) -Muu — ^ + Mu. —3- + i[n] Muu ^ + W„. ^ L ^2 ay Ci 6w J |_ ei du 62 dy J L eiov eiduj |_ eidu 62 ^y J [ n — T I ^^(n) , 53^1 r>z - i(„) -/i,^ — — + /i.„ — - 62 ov €1 du 3uJ [_ ei du 62 dv J + ^z.Mt^zzX[n]T[n] , (38) -/^u — K („) fu„ 1- €„„ — + I' j„ L ei du 62 5y J dT, , [n] _ ^ ^y'ln] 62 dv "" Ci du_ + 1^2,(n)e„jX(n)^(n) , L ei du Cidvj L Cidv eidtij + ^z.(n)€i,jX(n)^{n) , D. = T7 r "^ («) I = F(„) €.„ — ^ + e,„ ei du dT(n) e-i dv L 62 dy + Vz,{n)izzX(.n)T(n) 62 dv ^^ ei du J Substituting from equations (38) into equations (25) to (29) we ob- tain dV ^=-^-"^'"'// dT(n) dT(m) . dT{n) dT(m) e-2 dv 62 dv ei du 6] du \dS + J'«^-'[nl // Muu 7 r livv r r— {•iy) J J [_ e\ du e-2 dv 62 dv ei du dT(m) dT[n] dT(m) i r, — r fJ-vu r— r— ao 62 dv ei du ei du J _ ^^ (n) dT(m) _ dT(n) dT(m) 61 du 62 ay 62 ay ei du dT^rn) + Mut. dT^ dT 62 ^y 62 + J(^L.ln] // Muz ^ — Mi^ ^^ X[nl7'[„] dS + X(m)F^,(m) , J J [_ 62 dv ei du J 794 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 dz = -i"i^.. // + j^v,, // dT{n) dT(m) _|_ dT (n) dT(m) ei du ei du ei dv 62 dv ^^ dT(n) dT(m) j^ dT(n) dT (^m) ~r ^uv :r~ ' — :r~ — r ^mi 62 dv ei du ei du 6-2 dv _ dT[n] dT(m) . dT[n] dT (m) 62 dv ey du dT[„] dT(m) _ ei du 62 dv dT[n] dT (m) j^y^An) jj e„,^^ + e,. ei dii Ci du dT(^m) dV[m] ■ T ff -Mi^^jj h^uu ex du dT{n) dT[,n] 62 dv ei du 62 dv 62 dv 62 dv X(.n)T(^„) dS, dS (40) dS f^vv dT(^n) dT[m] 61 du 62 dv dT(^„) dT[m] , dT(^n) dTlm] 61 du e\ du dT[n] dT[m] dTln] dT[n,] f^„u — ;r~ + Ml'" 62 dv 62 dv g] du 6\ du 62 dv 62 dv dT[n] dTlm] I dT[n] dT[,n] dl[m] dz - j^Jz.ln] jj = icoF(„) // - j^Vln] jj P^uz Z 1- Ml' 62 dv 6\ du dT[m] Ci du 62 dv _ dS (41) dS e\ du 62 dv dT^n) dT[m] e\ du 62 dv + 6., X[n]T[n] dS, dT(n) dT[m] eo dv 6\ du _ dT i^n) dT[m] . dT(„) dl [m] 62 dv 62 dv dT[n] dT[m] . dT[n] dTlm] ei du 61 du 62 dv 62 dv ei du 61 du _ dT[n] dT[m] _ dTln] dT[,n] 61 du 62 dv 62 dv e\ du _ dS (42) dS + J '-^-"> // _^ dT[m] . ^ dT[m] 62 dv " Ci du . X(n)T^n) dS + X[m]/2,[m] , Iz.ln] I jcOtJLzzXl7i]T[n]T[m] dS = —V[m] J J \_ 62 dv e\du_ dT T ff A \ ^^[^] 1 ^^[J'] — i [p] / / JW M^« — T- + M^z. ex du 62 dv _ T[.] dS (43) T[m] dS, GEXi:UALlZi;i) rKl.KOKAlMllST S EQUATIONS 795 - Too + r,,//., j<^ 7'(„o r/N (44) ^ ^7'[p] a7'[„] t;H I Cj,. - — — ('2 5/' f^i f>?/. 7\„o r/N. If we solve the last two e(iiiatioiis for /,,[„] and !%,(«) and substitute in the preceding four ecjuatioiis, we shall obtain the telegraphist's ecjua- tions ill their final form (30). Tims, let 'Z['n][n] = jj ju}tX,^Xln]Tln]Tl„,]dS, (45) From these coefficients we obtain another set ""Z[„][„,] = normalized co-factor of ~Z[„,][n] , 'Z(„)(m) = normalized co-factor of 'Y(^m-)(^n-, . Then, (4(0 + 7'(p)'F[n][m] jj — I[p]'Yin][m] jj V2,(n) = —i(m) Z(„)(,„) + y[p]'Z^n)(.m) jj ji J^ yco (Pi _ e-2 dv '" ei du _ dT[p] dT[p] Ci du 62 dv d T(p) dT(^p) Cidu 62 dv 7V«] dS T,„, dS, 7^(m) dS (47) _ dTip] dTip] 62 dv Bi du (m) d.S. Before substituting in e(iuations (39) to (42), the summation index m in (47) should be changed to avoid conflict with m in the former equa- tions. It does not seem necessary to make these final substitutions in their most general form. The results are very complicated and in prac- tice the various coefficients are not independent. Some coefficients may 796 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 vanish; others may be small. In isotropic media, /i„„ = n^^ = i^zz = M, «uu = fill. = Uz = (■ and the mutual coefficients vanish. In gyromag- netic media subjected to a strong magnetic field in the 2-direction, the permeability coefficients of superposed ax: fields are^ iiuu l^vv = M, Mru = "Mur, Hzu = yiz = 0. (48) If the entire waveguide is filled with such a medium, assumed to be homogeneous, equations (43) and (44) become Iz.[n]j(^l^zzX[n] II T[„]Tlm]dS = — T Vz.(n)j^^X(n) 11 T (_n)T (_„,-, dS = — 7(, [rn] , (49) In view of the orthogonality of the ^-functions and the normalization conditions (13), we have z,[m] — X[m] V [m] V z, (m) X(m) J (m) , (50) where the summation convention is waived. In this case all the transfer coefficients in equations (30) vanish, 1 (.m)(.n) — J (m)[n] — i [m][»] — i [m](n) — i (m)(.n) — i (m)[nl \0i.) = T[„,](n) = T[m]ln] = 0. The transfer impedances and admittances are ^(m)(n) = 0, ii n 9^ m, 2 1 X(.m) -r = jo:tx + -r- , ii n = m; jcoe <(m)[n] — —Ji^f^uv II dT[n] dT(m) , dT d du d du e T[n-\ dT(m) ■2 dV 62 dv J 6162 du dv; Y(m){n) = 0, \in 9^ m, = jwe, if n = m; Y(m)[n] = 0, all m, n; dT[n] dT[m] _ dT[n] dT[m] du dv Z[m][n] = juHuv 11 dv du = join, if n = m; (52) du dv, \i n 9^ m, 8 C. L. Hogan, "The Ferromagnetic Faraday Effect at Microwave Frequencies and Its Applications — The Microwave Gyrator, Bell System Tech. J., Jan. 1952, p. 9. GENERALIZED TELEGRAPHIST'S EQUATIONS ^lml(n) = jW«. J J ^'[mllnl = 0, if 71 5^ 77/, 797 dT(n) dTlm] , d _€] du d du e-i 2 dv 62 dv J 6162 du dv; = /a)6 4- -P^— '- , if /i = 7W.; Y[m]{n) = 0, all m, n. We note that Z(;„)[„', = — Z[n](m) ; -^[m][n] = — -^[n][m] , {n 9^m). In rectangular waveguides we choose cartesian coordinates; then 61= 62 = 1, H = X, V = y and _i . .-1/2 . pirx . Qir// ^(P9) = lp«X(P«)(a^) s"i ^— - sill V^ , a 0 rr 1 -1 / 7\-i/2 sirx tiry Tisi] = IstXistMb) cos — cos -T^ , a 0 22 2 "^ 2 _2 _P^,?'r"^2 X(pg) ~ X[P9] ~ ^ r —TT — Xp3 > where Ipq = 2 if neither p nor g is equal to zero and log = Ipo = \/2. Hence 1 1 ^ (53) K^ ff [ ru/ \ ■ ^^^ p^^ ^'^y ■ Q'^y X // {h/a)sp sin — cos - — cos —^ sin 2— x Jj [_ a a 0 b f / /i.\, «^^ • P'TX- . tTry qwy ■+■ {a/b)tq cos — sm ^— sin -i- cos ^-r- a a b b = J<^tixu i,,hm[is/ay + (t/by][i - i-y^'m - i-y^] Xp.Xst(s' - p'){q' - f-) a s 9^ p, q 9^ t, = 0, if s = p or g = ^; ip,i..(pV - gV)[i - {-y-"'][i - (-)"+'] dx dy (54) '[pr()iiuij2;iu'ticnuMliunic)ii theTE|ioi aiidTKioi] modes may be understood by taking into account their mutual coupling hut ignoring thcii- coupling to other modes. The e(iuations of propaga- tion become dVm dz dI[io] —jwnf [10] — Ja;/Xx.v(8/7r")/[ui] = - {J^e+ .^— - V [10] , ds \ jionzzd^ = i'^Mi.v(8/7r")/[io] — iw/i7[oi] , d\ [01] ((il) dz = - ( jwe + ) F[oi] dz \ J^iJ-: For exponentially propagated waves we have l'^[io] = F[io]e ", F[oi] = F[oi]e , J [10] — i[10]C , -t [01] — i[01]C (02) When the mutual permeability is zero, we have two independent modes whose phase constants are (2\l/2 / 2\l/2 wVe - 5 ) , /3oi = ( wVe — — r? ) • (03) The phase constants of the pertiu'bed modes may be expressed in terms of the unpertiu'bed constants and the coefficient of coupling. When the losses are neglected, the mutual permeability is a pure imaginary. In this case it is con\-enient to define a real coupling coefficient k = ^^ . (64) Substituting from (62) in (61) and using (64), we find /3F[10] = WM^IlO] — ,/COyu/>"/[01] , (8/[10] = ( COe — )F[10] |8F[oi] = jcciJ.J:I[io] + w,u/[oi] , j8/[oii = ( we — r- i v [oi] (jiyizzO-/ (65) (66) 800 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 Eliminating Fdo] and V'[oi] , we have (/3^ — /3Io)2(io] = —jJ0loim] , (/3^ - ^oi)/[oi] = jk^lJ[io] . Multiplying term by term, we obtain the characteristic equation ^' - {0lo + ^l)^' + (1 - k')^lo^l = 0. (67) Solving, we have /3' = K^io + /3oi) ± mlo -/3oi)' + 4:k'l3lo^lf". (68) The effect of coupling is to increase the larger phase constant and de- crease the smaller one ; that is, to make the slower wave slower, and the faster wave faster. Let us assume a > b; then /3io > /3oi . Taking the upper sign in (68) and substituting in the second equation of the set (66), we have (69) /[Oil M^oi/lSio) ) 1 //3io ^ 2 V/3o: /Soi 2[io] P + iv' + k')'" /3io From (65) and (69) we find V[Q\] _ /3io /[oi] _ jk(0io/^oi) F[10] Pll /[lO] V + {f + A;2)V2 • (70) Hence, the ratio of the power carried in the TE[oi] mode to that in the TE[io] mode is (71) Po\ t^[01]/(011 ^ This ratio increases as k increases and p decreases. If the phase constants of the vmcoupled modes are equal, then p = 0 and Poi = ^10 for all values of the coupling coefficient. In this case (68) becomes iS' = ^Io(l ±k) or /3 = Ao(l ± kf'\ (72) In terms of the original constants. /5 = _(m ± ^ jVx.) (co^e - -^)_ ni/2 (73) The cutoff frequencies of both normal modes are seen to be independent of either the transverse permeability or the mutual permeability. Since GENERALIZKI) TKLKGRAPHIST's EQUATIONS 801 Hxv is a pure imaginary, it effectively increases the transverse permeabil- ity for one mode and dec^reases it for the other. To avaluate the effect of higher order TE and TM modes on wave pro[)agation we may substitute from (68) in all terms of the character- istic tMiuation for telegraphist's ecjuations except the first two diagonal terms and recalculate the jS's. Alternatively we may replace TE[io] and TE[on modes by the normal modes just obtained, recalculate the cou- pling coefficients, and evahuite the effect of the mode with the greatest coupling to the modes under consideration. Photoelectric Properties of lonically Bombarded Silicon By EDWIN F. KINGSBURY and RUSSELL S. OHL (Manuscript received March 25, 1952) In the course of investigation of the rectifying 'properties of silicon very interesting photoelectric properties were found. The first photo-cells were cut from bulk silicon in which a natural potential barrier tvas found. A typical spectral characteristic of such a cell is shown. This early work was followed by the discovery of the ionic bombardment method of producing photo active silicon surfaces. The effects of the temperature of the target and of the energy of the bombarding particles in the photoelectric properties is illustrated by characteristic curves. Relative equi-energy spectral response characteristics as a function of wavelength are illustrated. The photon efficiency as a function of wavelength of a typical cell is shown. INTEODUCTION Because of the importance that barriers have come to assume in the general field of semiconductors the authors have been urged to publish results of their early experiments in this field. These experiments were undertaken in the course of a search for semiconductive material suitable for use as point contact rectifiers. Before March 1941 one of the writers discovered a well-defined bar- rier having a high degree of photovoltaic response. The barrier was found only in melts of some lots of commercially available high-purity silicon. This barrier showed a high photovoltaic sensitivity to radiation from incandescent lamps. The existence of this natural barrier was first observed in rods cut from melts for resistivity measurements. These rods showed a high de- gree of photovoltaic response, were found to have a high thermoelectric coefficient, and had good rectifying properties. The fact that one end of the rod developed a negative potential when ilhmiinated or heated and that when supplied with a negative potential showed low resistance to current flow across the barrier led to the terminology of n-type 802 PHOTOELECTRK' I'ROPKRTIKS OF lONICALLY HOMBAKDKD SIIJCOX 0.2 5 < 3 1 1 1 1 ^,^'' \ o"^ ''' \ \ \ ■^<^ .t> .'' \ < \ \ -' '-y' \ X \ • ^ • ^ v< 3LTAGE ^ N >= • • / /^ /' • "^ ^-- . , f?F S^STA 1 / NCE — — 1 / 1 / (b) 20 10 15 HI a. \u Q. 10 2 < 2 0 I- z (t D O -10 -15 \ -of^ !> ^/ / , A^ J<^ ^ ^ 6;*^ -6 -5 -4 -3 -2 -1 VOLTS CC1 Fig. 2a — Spectral response of internal barrier in silicon. Fig. 2b — Voltage and current photosensitivity of internal barrier in silicon. Fig. 2c — Rectification characteristic of internal barrier, dark and illuminated. PHOTOELECTRIC PROPERTIES OF lONICALLY BOMBARDED SILICON 805 A typical spectral response curve of such a barrier is shown in Fig. 2a while Fig. 2b gives its open circuit voltage, short-circuit current and resistance when illuminated by a tvinslcii light of 2848°K color tempera- ture. This cell resistance was taken as e(iual to that of an added series resistance which reduced the short-circuit photocurrent to one-half. The value so obtained is somewhat higher than the corresponding ratio of the voltage and current given in the figure. Fig. 2c giv(\s the })ehavior as a rectifier in the dark and with a stated light on the barrier. Cells whose barrier was near the surface were made by cutting close to the natural one as shown in Fig. lb. This cut exposed large photoactive areas. Surface barrier activity was occasionally found on the top surface of some melts. These surface type cells showed a wider spectral response toward the visible than the internal barrier type. This was found to be due to the spectral absorption characteristics of the bulk sihcon. A further discussion of this appears later in the paper. These early barrier cells showed remarkable stability under severe temperature conditions. For instance, they could be heated to redness in air and quenched in water with no serious change in their character- istics. They were tested in hquid nitrogen, under water and in oil without injury. They could be illuminated with direct sunlight with no injury to their response characteristics other than the temporary effect of the increased temperature. Several of these internal barrier cells have been in use in test circuits for more than ten years with no serious change in their photoresponse properties. These observations seemed to indicate clearly that a very high degree of stability could be expected from sihcon photocells. However, there were serious practical disadvantages to the early cells. Those sho\vn in Fig. la were active near the exposed barrier itself which was usually a strip along the surface about ^ mm wide. On the other hand, the surface types as shown in Fig. lb showed irregular re- sponsiveness over the surface area. From these early studies it was clear that if a good method could be found to activate large areas of silicon surfaces uniformly, cells could be made which might compete with other kinds of surface barrier type cells already available. The search for such a process resulted in the ionic bombardment method of activating silicon surfaces. Such surfaces also have desirable rectifying properties.^ METHOD OF PREPARATION Hyper-purity sihcon was used for bombardment type cells to avoid the formation of natural barriers due to minute impurities and to give 806 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 better control of the sensitivity. After being cast in a fused silica crucible, the roughly cyhndrical piece was ground to a cylinder about 1|" di- ameter, a process which removed crucible contamination and gave a convenient size for shcing into wafers about 0.025" tliick. The two faces were then made approximately flat and parallel after which one was left rough and the other ground and polished down to a good optical surface. In most cases the discs were then cleaned by soaking for approximately fifteen minutes in a solution of hydrofluoric acid, rinsed in distilled water and dried. The activation consisted of exposing the polished face to a uniform beam of positive ions of helium at a pressure of 10~^ to 10~^ mm of ff^ Fig. 3 — Intermediate and large size photocells made by ion bombardment. Back of the intermediate also shown. mercury. The energy of the particles used in different units ranged from 100 to 30,000 electron volts. During this bombardment the silicon sm-face was kept at a favorable temperature, about 395°C. After activation, collector electrodes of evaporated rhodium were ap- plied. Cells of three sizes have been constructed, two of which are sho^^^l in Fig. 3, the intermediate and the large one, of exposed active areas about 0.40 and 8.0 sq. cm. respectively. A small one had an area around 0.005 sq. cm. INIost of the measurements reported in this paper have been made with the intermediate size. EFFECT OF ION VELOCITY That ion velocit}^ has a profound effect on the voltage current char- acteristic of bombarded siu'faces is shown in Fig. 5. These characteristics were obtained by placing a tungsten point contact under 10 gm of force, PHOTOELECTRIC PROPERTIES OF lONICALLY BOMBARDED SILICON 807 ■ y / y / / 7' / / ^7 \ A, V / / .-^ //o-^ c^ ^' / ^ ^"^ / '^ f .^ / r^ /^ .^ A /"" / ^ /r / / / A LOG AMPERES Fig. 4 — ^Rectification characteristic of the large photocell. J L J L Fig, cells. -200 -150 -100 -50 O 50 -?00 -150 -100 -50 0 50 -POO -150 -100 -50 0 50 VOLTAGE 5 — Effect of bomliardment voltage on the rectification of Ihe intermediate 808 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 2.4 2.0 2.4 2.8 LOG - BOMBARDING Fig. 6— Photocurrent at a constant illuminance versus the bombarding voltage. on the photo active surfaces of the medium size cells whose spectral response is given in Fig. 8. However, in order to show the rectifying prop- erty of the barrier without the complication of a point contact, a disc of hyper-purity silicon I5" in diameter and about 0.025" thick was given an optical polish on both faces. One face was bombarded with 30-kv ions. 10 :: i 8 2 ^ o 260 280 300 320 340 360 380 400 420 440 460 480 500 TEMPERATURE IN DEGREES CENTIGRADE Fig. 7 — Photovoltage at a constant illumination versus temperature of the bombarded silicon surface. PHOTOELECTRIC PROPERTIES OF lONICALLY BOMBARDED SILICON 809 cr 0.2 uj 0 z UJ 1.0 / \ 100-226 V / \ \ \ \ \ V / r-\ V 570-696 V \ / \ \ V / '"^ V 970-1096V / \ / \ 1 \ V / r^ 3KV / / / \ V / r \ 10 KV / \ / \ \ V / /^ \ 30 KV / \ / \ I / \ V 1.2 0.4 06 0.8 WAVELENGTH IN 1.0 1.2 0.4 MICRONS Fig. 8 — Spectral response of the intermediate size cells at various bombarding voltages. Electrodes 1|" in diameter of evaporated rhodium metal were applied in like manner to each surface. Contact was made to the collector electrodes by means of tin discs. Fig. 4 gives the forward and backward log voltage- log current relation of this large cell. Without bombardment such an arrangement shows ohmic conductivity so it is evident that the treat- ment is responsible for the development of a potential barrier beneath the surface. It is beheved from the high dark resistivity of the bom- barded layer that the intrinsic properties of the silicon are developed therein. Thus an intrinsic -p type potential barrier is produced similar to a degree to the n-p junction. One would expect the depth of this bar- rier to be related to the velocity of the ions. Consequently a study has been made of the effect of ion velocity on the photoelectric properties. The photoelectric current at constant illuminance for a series of cells prepared by bombardment with ions of different energies is shown in Fig. 6. It is remarkable how quickly and completely the current sensi- tivity saturates at approximately 500 volts. 810 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 EFFECT OF SURFACE TEMPERATURE DURING BOMBARDMENT In the preparation of photocells it was found that the surface tem- perature during bombardment had a pronounced effect on the efficiency. In order to study this effect it was necessary to determine the surface temperature of the silicon itself. Since it was impractical to measure the silicon temperature during bombardment, a calibration was made of the surface temperature in terms of the temperature of the graphite heating block. This calibration was carried out by tw^o platinum/platinum rhodiimi thermocouples made of 5-mil wires. The fused thermojunction beads were held in contact with the surfaces by miniature tungsten springs. Temperature measurements ^^dth the thermojunction in contact with the silicon surface were subject to error from the slightest con- tamination at the point of contact. Perhaps the most difficulty was due to a reaction between the platinum and silicon at temperatures above 400°C. The effect of surface temperature on the photoresponse is shown in Fig. 7. It is apparent that maximum sensitivity results when the target is kept at about 395°C. Perhaps by coincidence this is also the temper- ature at which no Hall Effect is observable in this hyper-pure material. Cells prepared at temperatures above the critical value show lower back resistances than those prepared at the critical temperature and conversely those at temperatures below the critical value have higher back resistances but a much reduced photoresponse. EFFECT OF TOTAL BOMBARDING CHARGE The 4)hotoresponsiveness improves as the total bombarding charge is increased until it has reached about 600 microcoulombs per sq. cm. Further bombardment produces no appreciable improvement. In certain exploratory tests a total charge of about 9000 microcoulombs at 30 kv has been applied. Under these severe conditions the surface may show small areas having a slightly etched appearance. No extensive tests have been made to determine the effect of the rate of application of the bombarding charge. The apparatus was designed for use at a rate of about 5 microamperes per sq. cm. It is known how- ever, that between the limits of about 2.5 and 10 microamperes per sq. cm. the effects are subject only to the total charge or the total number of ions which strike the silicon surface. EFFECT OF BOMBARDMENT VOLTAGE IN SPECTRAL RESPONSE Six spectral curves are shown in Fig. 8 which illustrate the result ob- tained with the intermediate size cells over the bombardment voltage PHOTOKLKl "rUIC I'UOl'KUriKS OF lOXICAhLV B( )MHA KDIOD SILICON 811 range previously mentioned. The peak of the lowest voltage cell is definitely toward the blue compared with the other five whose maximum is constant at about 0.725 /x- One objective in this studj^ was to obtain evidence relating to the depth of the barrier below the silicon surface as a function of the energy of the bombarding particles. The higher the velocity of the particles the further beneath the surface one would expect the barrier to be located and as a r(\sult there might be a shift in the spectral characteristic toward the red with increasing depth of the barrier due to the relatively greater absorption at the blue end. There is however, a selective or secondary maximum at the jjeak which sharpens it and imllifies the effect of the warping of the entire curve. The blue to red shift can be shown as in Fig. 9 by plotting the ratio of the responses in Fig. 8 at 0.50 n and 1.0 /x. Thus at low voltage the blue to red ratio is high and decreases as the l)ombarding potential is raised. In the spectral curves it will be noted that there are a number of secondary humps located near the top of the curves and extending down on the blue side. There is a strong tendency for them to occur at definite wavelengths and to be evenly spaced regardless of the bombarding voltage. SPECTRAL MEASUREMENTS ON THE LARGE CELLS AND TH^ EFFECT OF MATERIAL COMPOSITION For the large cells, two grades of silicon were used both prepared by pyrolytic reduction of SiCU and called "hyper-pure". These will be 2.8 0.8 \ \ ) \ \ \ \ s^ o \ \ o s \, o^ ^ "v, ^ 0.8 1.6 4.0 4.8 2.4 3.2 LOGio -VOLTS Fig. 9 — Ratio of blue to near infra-red response versus bombarding voltage. 812 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 designated B and C, the former being from the same soiu'ce as "sihcon B" referred to in the paper by Scaff et al.^ A typical analysis is given therein. The C silicon was from another source and a spectroscopic analy- sis indicated it was somewhat more impure than B thus agreeing with observed differences in its electrical and optical characteristics. An opti- cal variation of considerable interest is shown in Fig. 10 where the spectral transmittance of the two grades of silicon is compared in the infra-red for polished plates each 0.0195'' thick. The transmittance of B decreases a little with increasing wavelength but C goes down much more. Both however, start to get transparent at about the same point, 1.1 ^ and also show corresponding absorption bands superimposed on the main curve. Briggs* has compared the trans- 0.2 (.■ ~- X \ ^ B^ -A / \y -^ f V "~c" \ > \. 1 -^^ '^— •« / 6 16 18 20 « 10 12 14 WAVELENGTH IN MICRONS Fig. 10 — Spectral transmittance of B and C grades of silicon, polished 0.0195" thick. mittance from 2 ^ to 12 /x of the A and B silicons in Scaff 's paper where the former was much more impure than the C grade. The absorption of the A sihcon increased so rapidly out in the infra-red that a much thinner sample was used for the measurements than for the B material. If this difference in thickness is allowed for, the effect of impurity is very striking. The spectral response of large area cells made of the B and C materials and bombarded with 1000- volt helium positive ions is shown in Fig. 11. The two curves are similar in shape except the one for C silicon is some- what narrower and in addition is shifted toward the blue. Both have some of the secondary humps noted previously. All the cells shown in this paper have indicated a long wave limit of about 1.2 jLt. Actually some response can usually be detected out to about 1.3 \i. Measurements made some years ago on the internal barrier units also gave a limit around 1.3 /x but relatively more response at 1.2 /i with peaks close to 1.10 /i- This difference is reasonable because light was PHOTOELECTRIC PROPERTIES OF lONICALLY BOMBARDED SILICON 813 projected along the barrier plane and not normal to it as in the. latest units, SO that with the rapidly increasing transparency in this region, less infra-red radiation was lost. However, the blue was rapidly at- tenuated. When illuminated by tungsten hght of 2848°K color temperature, the large B cells gave 2160 microamps per lumen and the C unit 638. Cor- recting for a surface reflectance of 0.385, the net sensitivities would be 3510 and 1040. These measurements were made with between 4- and 5-footcandles illuminance on the cells, a region in which the response is proportional to the intensity. At much higher values of illuminance there was some falling off of response so that the effective sensitivity was a little lower. The above measurements were made on a ten ohm microam- meter which is too low a resistance to affect the linearity. The inter- mediate cells ran approximately 3000 microamps per lumen in the most sensitive region of bombardment without correction for surface reflec- tion and at 10- to 20-footcandles for the same tungsten lamp using a meter of 76 ohms. 0.6 0.5 1/ it II 7^ \ \ \ \ i" \ \ \ 1 1 1 / \ \ \ / \ \ 1 \ \ / \ \ / 1 \ \ \ k 1 \ \ \ \ ; / \ \ ' / \ \ / / — \ — \ — - // \ \ // \ \ ' / \ \ / \ \ // \ \ 1 / \ \ 1 V 1 \ / \ / \ \ \ \ \ \ \ ^ \ \ \ \ \ \ V \ \ s \ ^ ^ ^v^ "■^ 0.6 0.7 O.S 0.9 1.0 WAVELENGTH IN MICRONS Fig. 11 — Spectral response of large size photocells of B and C grades of silicon. 814 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 PHOTON EFFICIENCY It is of interest to examine the spectral photon efficiency of a cell made by bombardment. As an example, there may be taken the 3-kv cell whose spectral response is shown in Fig. 8. When illuminated by a tung- sten light of 2848°K color temperature at 10-foot candles, a sensitivity of 3090 microamps per lumen was secured. Allowing for a surface re- flectance loss of 0.385, this value becomes 5020 for the radiation actually absorbed. From these data the sensitivity in microamps per microwatt at the peak 0.725 u, calculates to be 0.388 and the photon efficiency, i.e., the electrons per photon, 0.66. Fig. 12 gives the efficiency through the spectrum. Note that the efficiency rises some on the short wave side shifting the peak of the equi-energy curve (Fig. 8) over to 0.625 n. This increase is evident from the fact that if the equi-energy curve decreased linearly from the peak at 0.725 ju to zero at 0 /x, the photon efficiency would remain constant and ec^ual to that at 0.725 m- For the purpose of the above calculation, the curve in Fig. 8 has been taken as going to zero at about 0.40 iu, a fact experimentally checked. If unity is considered to be the maximum possible efficiency at any wavelength, 72 per cent of it is attained at 0.625 m and nearly half of the spectral range is 50 per cent or higher. 0.7 / / -\ 0.6 / \ / \ >- u 5 0.5 1 \ / \ (J li- \ V / \ Z o O 0.3 I Q. 0.2 0.1 \ / N k \ \ 0 ^ 0.6 0.7 0.8 0.9 1.0 WAVELENGTH IN MICRONS Fig. 12 — Spectral photon efficiency of the 3-kv cell of Fig. 8. PHOTOELECTRIC PROPERTIES OF lONR'ALLY HOMHAliDKl) SILICON 815 CONCLUDING REMARKS Those oxiK'iimonts have served not only to introduce us to some of the phenomena inxolved in .semiconchictor harriers hut have also yielded photo cells having desirable properties. These cells ha^ e a high degree of stability and will stand treatment ruinous to most other cells. 'I'hey have a very high current sensitivity to tungsten light and daj'lighl. They re- (|uire no associated battery and can be mad(^ in large areas. I'lilikc \hc material used in many types of photo cells, silicon does not ha\e the disadvantage of scarcit^y. All tests to date indicate that an indefinitely long life may be expected cNen inider extreme illumination. Fig. 11 sug- gests that it may be possible to control to some extent the spectral re- sponse in the region from the deep blue into the infra-red. The long wave limit is set by the edge of the al)sorption characteristic. REFERENCES 1. U. S. Patent No. 2,402,839, Filed Mar. 27, 1941. U. S. Patent No. 2,402,662, Filed May 27, 1941. U. S. Patent Xo. 2,443,542, Filed May 27, 1941. 2. J. H. Scatf, H. C. Theuerer and E. E. Schumacher; also W. G. Pfaiin and J. H. Scaff, rrans. A. I. M. E., 185, pp. 383-392, 1949. 3. P. S. Ohl, Bell System Tech. J., Jan., 1952. Also see this paper for more details regarding the method of preparing silicon. 4. H. B. Briggs, Phys. Rec, 77, pp. 727-728, Mar. 1, 1950. Abstracts of Bell System Technical Papers* Not Published in This Journal Mechanical Properties of Discrete Polymer Molecules. W. 0. Baker\ W. P. Mason^ and J. H. Heiss\ J. Polymer Sci, 8, pp. 129-155, Feb., 1952. (Monograph 1937). Post-War Achievements of Bell Laboratories: II. O. E. Buckley\ Bell Tel. Mag., 30, No. 4, pp. 224-237, 1951-1952 A Portable, Direct-Reading Microwave Noise Generator. E. L. Chin- nock . Proc. Inst. Radio Engrs., 40, pp. 160-164, Feb., 1952. (Mono- graph 1939). This paper discusses the factors which influenced the design of a directly calibrated portable microwave noise source, utilizing a fluorescent lamp. The variation of the noise power output and the impedance match as a function of the operating temperature are considered, and the portable unit is described. The Quantum Theory. K. K. Darrow\ Sci. Am., 186, pp. 47-54, Mar., 1952. (Monograph 1940). Concerning the early years of this fundamental concept of modern physics — how Max Planck formulated it at the turn of the century and how others en- larged it up to 1923. Performance of Ultrasonic Vitreous Silica Delay Lines. M. D. Fagan\ Tele-Tech, 11, pp. 43-45, 138+, Mar., 1952. (Monograph 1951). Results of tests at 10 and 60 mc with resistive terminations of 75 to 1000 ohms. Low terminating impedance values yield wide bands but involve higher insertion losses. Phase Transition of NDiDiPO^. B. T. Matthias\ Phys. Rev., v. 85, p. 141, Jan. 1, 1952. Engineering Local Television Facilities and Their Operation. B. D. * Certain of these papers are available as Bell System Monographs and may be obtained on request to the Publication Department, Bell Telephone Labora- tories, Inc., 463 West Street, New York 14, N. Y. For papers available in this form, the monograph number is given in parentheses following the date of pub- lication, and this number should be given in all requests. 1 Bell Telephone Laboratories. 816 ABSTRACTS OF TECHNICAL ARTICLES 817 WicKLiNE* and J. E. Farley*. Elec. Eng., 71, pp. 252-257, Mar,, 1952. All the means of electrical communication are called into play when a city- wide coverage of an event is to be televised. How telephone and television facil- ities were utilized on the day that Chicago welcomed General MacArthur is explained in this article. Echo Distortion in the FM Transmission of Freouency-Division Mul- tiplex. W J. Albersheim^ and J, P. Schafer\ Proc. Inst. Radio Engrs., 40, pp. 316-328, March, 1952. The composite multiplex signals generated by frequency-division methods long standard in telephone communication can be transmitted by the new trans- continental broad-band FM radio relays. Signal intermodulation by echoes must be minimized. Such intermodulation is investigated in this paper experimentally and analj^tically. Two types of echoes are considered: (1) weak echoes with de- lay's exceeding 0.1 microseconds, caused mainlj' by mismatched long lines; and (2) powerful echoes with delays shorter than 0.01 microseconds, caused by multi- path transmission, and leading to selective fading. By use of random noise sig- nals, the distortion is evaluated as a function of various parameters of the echo, the base-band, and the rf modulation. Motion of a Ferromagnetic Domain Wall in FczOa. J. K. Galt , Phys. Rev., 85, pp. 664-669, Feb. 15, 1952. Experiments have been made on a sample of FesOi cut from a single crystal in such a way that its ferromagnetic domain pattern includes an individual domain wall whose motion can be studied. This sample has a permeability which is high (about 5000) at low frequencies and drops off rapidly above 1000 cycles. A hysteresis loop and data on wall velocity vs applied field were also taken. The data are discussed in terms of recent developments in the theory of the ferromagnetic domain wall. It appears that this theory explains our data satis- factorily, and that in using it to explain our data we determine some of the fundamental magnetic constants of Fe304. We are also able to gain some insight into domain wall motion in ferrites generally in this way. The Drift Mobility of Electrons in Silicon. J. R. Haynes^ and W. C- Westphal\ Phijs Rev., 85, p. 680, Feb. 15, 1952. Formulas for the Group Sequential Sampling of Attributes. H. L. Jones*. Ann. Math. Statistics, 23, pp. 72-87, March, 1952. Soine Fundamental Properties of Transmission Systems. F. B. Llewellyn^ Proc. Inst. Radio Engrs., 40, pp. 271-283, March, 1952. The problem of the minimum loss in relation to the singing point is investi- gated for generalized transmission systems that must be stable for any combina- ' Bell Telephone Laboratories. * Illinois Bell Telephone Company. 818 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 tion of passive terminating impedances. It is concluded that the loss maj' ap- proach zero db only m those cases where the image impedances seen at the ends of the system are purely resistive. Moreover, in such cases, the method of over- coming the transmission loss, whether by conventional repeaters or by series and shunt negative impedance loading, or otherwise, is quite immaterial to the ex- ternal behavior of the system as long as the image impedances are not changed. The use of impedance-correcting networks provides one means of insuring that the phase of the image impedance of the over-all s^'stem approaches zero. Gen- eral relations are derived which connect the image impedance and the image gain of an active system with its over-all performance properties. The Arithmetic of Menage Numbers. J. Riordan\ Duke Math. JL, 19, pp. 27-30, March, 1952. ^4 Recurrence Relation for Three-Line Latin Rectangles. J. Riordan\ Am. Math. Monthly, 59, pp. 159-162, March, 1952. Capacitors arid Comynunications. Inductive Coordination of Lines. A. R. Waehner^ and W. E. BloeckerI Elec. Light and Power, 30, pp. 105- 108, 114, March, 1952. Although the use of capacitors on power lines has been expanding, their use has caused relatively few cases of noise on communication lines and these have been satisfactorilj' corrected. The causes of trouble and remedial measures were the subject of a recent, joint E.E.I.-Bell System study described here. Book Reviews Antennas: Theory and Practice. By Sergei A. Schelkunoff and Harald T, Friis, 639 + xxii pages, John Wiley and Sons, Inc., New York (1952). Price: $10.00. This is a recent addition to Wiley's Applied Mathematics Series edited by I. S. Sokolnikoff. It contains a thorough and balanced treatment of electro- magnetic radiation and electrical properties of various tj^^es of antennas. In these days of rapid expansion of microwave engineering it would have been easy to neglect the older and less glamorous long-wave and short-wave antennas. The authors are to be congratulated on their impartiality. The exposition is lucid. While the entire quantitative theory of antennas is based on Maxwell's equations, unnecessary mathematics is conspicuous b}' its absence, and physical explanations are abundant. The book begins with a long chapter on Physical Principles of Radiation. This chapter is almost a book within the book. It touches upon the most impor- tant ideas and problems of antenna analysis and contains a number of simple but useful formulas. Circuit and field concepts are compared, and the similari- ties as well as the differences between them are exliibited. Maxwell's equations are stated in a form which is particularly easy to understand. In this form, one ^ Bell Telephone Laboratories. 2 American Telephone and Telegraph Company. ^ Line Material Company. ABSTRACTS OF TKCFINICAL AiniCLKS 819 eqiuitiuu expresses a relation between tlie u\-eragc electric intensity tangential to a given curve and the time rate of change of the average magnetic intensity normal to a surface bounded b}' this curve. The other equation expresses a com- plementary relation. The reatler will l)e impressed by a simple phj-sical picture from which the authors are able to deri\'e the expression for the radiation field of a short antenna. In this chapter they discuss the effect of heat loss and imped- ance mismatch on the elficiency of antennas. Among other topics will be found di- r(M'ti\'e radiation and reception, large antenna arraj's, horns, leflectors, and lenses. After this extended general introduction a more detailed analysis of \'arious problems begins. Chapter 2 is devoted to Maxwell's equations and Chapter 3 to plane waves on conductors and in free space. The main topic in Chapter 4 is the derivation of the expressions for the complete field surrounding a short antenna from Maxwell's equations. The authors have made a si)ecial effort to show the connection between this field and the oscillating charge in the antenna. Applications of this l>asic theory begin with (-hapter 5 devoted to directive radiation. This chapter is concerned with radiation ]mtterns of vaiious arrays ami with calculation of radiated i)ower. A novel method, the method of momonts (pp. 162-195), is likel}' to prove valuable when spatial distrilnitions of antenna current are complicated (as in the case of shunt-fed antennas). Chapter 6 ex- plains methods for calculating directivites and effective areas of antennas. Some ground effects are considered briefly in Chapter 7. In Chapter 8, the dis- cussion of current distributions in antennas made up of tliin wires is particularly thorough. First, simjjle approximations are developed; then the effects of var- ious factors are carefuUj' examined. Various reciprocity and cii'cuit equivalence theorems, so useful in antenna analysis, are collected in Chapter 9. Beginning with Chapter 10 the general theory is applied to specific antenna tj'pes. Thus, small antennas are treated in Chapter 10; quarter-wave, half-wave and full-wave antennas in Chapter 11; general dipole antennas in Chapters 12 and 13; rhombic antennas in Chapter 14; miscellaneous types of w4re antennas in Chapter 15; horn antennas in Chapter 16; slot antennas in Chapter 17; re- flectors in Chapter 18; and lenses m Chapter 19. Practical engmeers will be delighted with the appendices which contain in a compact form some of the most useful information about transmission lines, (.lipole antennas, antenna array's, optimum horns, and lenses. Teachers will welcome the numerous problems scattered throughout the book. Advanced Antenna Theory. By Sergei A. Schelkunoff, 216 + xii pages, John Wiley and Sons, Inc., New York (1952). Price: $6.50. This book is a recent addition to Wiley's Applied Mathematics Series edited by I. S. Sokolnikoff. It is concerned with recent advances in antenna theory and is divided into six chapters. General expressions in spherical coordinates are derived for electromagnetic fields in free space and in the presence of conducting cones and thin wires diverging froni a common point. In Chapter 2 these expres- sions are applied to dipole antennas, vee antennas, end-fed antennas, etc. Chap- ter 3 gives an account of Stratton and Chu's theory of spheroidal antennas. Integral equations in antenna theory and Hallen's method of their solution are tieated in Chapters 4 and 5. The book is concluded with a chapter on natural oscillations in antennas. A substantial number of problems and several interest- ing appendices will be found at the end. Contributors to this Issue Sidney Darlington, B.S., Harvard University, 1928; B.S. in E.E., Massachusetts Institute of Technology, 1929; Ph.D., Columbia Uni- versity, 1940. Bell Telephone Laboratories, 1929-. Dr. Darlington has been engaged in research in applied mathematics with emphasis on network theory. Paul G. Edwards, B.E.E., Oliio State University, 1924; E.E., Ohio State University, 1929. Western Union Telegraph Company, 1919-22; American Telephone and Telegraph Company, 1922-34; Bell Telephone Laboratories, 1934-. His main concern in the Laboratories has been with toll transmission problems, including voice frequency and carrier systems. Member of the I.R.E., A.I.E.E., Sigma Xi, Tau Beta Pi, and Eta Kappa Nu. C. W. Harrison, B.S. in E.E., Purdue University, 1938; M.S., Lehigh University, 1940. Bamberger Broadacasting Company, 1939-41. Bell Telephone Laboratories, 1941-. Mr. Harrison is a member or the tele- vison research group. He formerly designed radio receivers and, later, microwave relay repeaters. Member of the I.R.E. John L. Hysko, B.S. in E.E., Cooper Union, 1921. U. S. Army, 1918-19. Bell Telephone Laboratories, 1919-. Mr, Hysko's principal activities in the Laboratories have been in the development of amphtude- modulation and frequency-shift carrier telegraph systems for land line, radio teletypewriter and submarine cable applications. Edwin F. Kingsbury, B.S., Colgate University, 1910. United Gas Improvement Company, 1910-18. U. S. Army, 1918-19. Eastman Kodak Company, 1919-20. Bell Telephone Laboratories, 1920-51. Mr. Kings- bury retired in 1951 after a career which was primarily concerned with television research and development, especially that part dealing with photoelectric and electrooptical problems. Member of the Frankhn In- stitute, the Optical Society of America, and Phi Beta Kappa; Fellow of the American Physical Society and the American Association for the Advancement of Science. 820 CONTRIBUTORS TO THIS ISSUE 821 Erniost R. Kretzmer, B.S., Wor(;ostcr Polytecluiic Institute, 1944; M.S., Massachusetts Institute of Technology, 1946; Sc.D., Massachu- setts Institute of Technology, 1949. As a member of the Electrical En- gineering Department at Massachusetts Institute of Technology, Dr. Kretzmer taught from 1944-46 and conducted research there from 1946-49. Bell Telephone Lal)oratories, 1949-. He works in the tele\'ison research group, where he has been principally concerned with decor- relation of television signals. Member of I.R.E. and Sigma Xi. L. R. Montfort, E.E., University of Mrginia, 1926; American Tele- l^hone and Telegraph Company 1926-34; Bell Telephone Laboratories, 1934-. Mr. Montfort has been concerned with the engineering of carrier systems. This has included field work with new systems and field tests prior to the design of new systems. During the end of World War II and for a shoi't time thereafter, he assisted in the engineering and test- ing of microwave radio systems. Member of A.I.E.E., Tau Beta Pi, Theta Tau, and Sigma Phi Epsilon. Russell S. Ohl, B.S. in Electro-Chemical Engineering, Pennsylvania State College, 1918; U. S. Army, 1918 (2nd Lieutenant, Signal Corps); Vacuum tube development, Westinghouse Lamp Company, 1919-21; Instructor in Physics, University of Colorado, 1921-1922. Department of Development and Research, American Telephone and Telegraph Com- pany, 1922-27; Bell Telephone Laboratories, 1927-. Mr. Ohl has been engaged in various exploratory phases of radio research, the results of wliich have led to numerous patents. For the past ten or more years he has l)een working on some of the problems encountered in the use of millimeter radio waves. Member of American Physical Society and Alpha Chi Sigma and Senior Member of the I.R.E. B. ]\I. Oliver, B.A., Stanford University, 1935; M.S., California Institute of Technology, 1936; Ph.D., California Institute of Technology, 1939. Bell Telephone Laboratories, 1939-52. During World War II, Dr. Ohver was engaged in radar research and the rest of his employ- ment before leaving the laboratories was in the television research group. Member of I.R.E. and Phi Beta Kappa. Wilton T. Rea, B.S., Princeton University, 1926; American Tele- phone and Telegraph Company, 1926-34; Bell Telephone Laboratories, 1934-. Except for the years 1941-45, when he worked on military pro- jects. Mr. Rea has been principally concerned ^^^th telegraphy. As Tele- 822 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952 graph Development Engineer, he is in charge of the development of tele- graph and telephotograph systems. Senior member of I.R.E. and member of A.I.E.E. and Phi Beta Kappa. L. C. Roberts, A.B., Harvard University, 1916; B.S. in E.E., Harvard University, 1919; B.S. in E.E., Massachusetts Institute of Technology, 1918; American Telephone and Telegraph Company, 1917-34; Bell Telephone Laboratories, 1934-. Mr. Roberts has been primarily concerned with the development of dc and carrier telegraph except during World War II when he worked on multichannel and single-channel radio tele- typewriter developments. Member of A.I.E.E. S. A. ScHELKUNOFF, B.A., M.A. in Mathematics, The State College of Washington, 1923; Ph.D. in Mathematics, Columbia University, 1928. Engineering Department, Western Electric Company, 1923-25; Bell Telephone Laboratories, 1925-26. Department of Mathematics, State College of Washington, 1926-29. Bell Telephone Laboratories, 1929-. Dr. Schelkunoff has been engaged in mathematical research, especially in the field of electromagnetic theory. PHE BELL SYSTEM Jechnical ournai »E VOTED TO THE SC I E N T I F IC^^^ AND ENGINEERING SPECTS OF ELECTRICAL COMMUNICATION OLUME XXXI SEPTEMBER 1952 NUMBERS ^&- ' Auotmatic Switching for Nationwide Telephone Service A. B. CLARK AND H. S. OSBORNE 823 Fundamental Plans for Toll Telephone Plant J. J. pilliod 832 Nationwide Numbering Plan w. h. nunn 851 Automatic Toll Switching Systems f. f. shipley 860 Mathematical Theory of Laminated Transmission Lines — ^Part I SAMUEL p. morgan, JR. 883 Electrical Noise in Semiconductors h. c. Montgomery 950 important Design Factors Influencing ReUabihty of Relays J. R. FRY 976 Lnpedance Bridges for the Megacycle Range n. t. wilhelm 999 Abstracts of Bell System Papers Not Published in this Journal 1013 Contributors to this Issue 1020 COPYRIGHT 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD S. BRACKEN, President, Western Electric Company F. K. KAPPEL, Vice President, American Telephone and Telegraph Company M. J. KELLY, President, Bell Telephone Laboratories EDITORIAL COMMITTEE A. J. B U S C H F. R. L A C K W. H. DOHERTY J. W. MCRAE G. D. EDWARDS W. H. N U N N J. B. FISK H. I. ROMNES E. I. GREEN H. V. SCHMIDT R. K. H O N A M A N EDITORIAL STAFF PHILIP C.JONES, Editor M. E. STRIFE 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. Cleo F. Craig, President; Carroll 0. Bickelhaupt, Secretary; Donald R. Belcher, 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 VOL u ME XXXI SEPTEMBER 1952 number 5 Copyright, 1952, American Telephone and Telegraph Company Automatic Switching for Nationwide Telephone Service By A. B. CLARK and H. S. OSBORNE (Manuscript received May 15, 1952) .4 plan for automatic long distance switching, which will ultimately cm- brace the entire area of the United States and extend into Canada and per- haps Mexico, has been formulated and important steps have been taken toward its realization. The plan contemplates that when a telephone cus- tomer places a call with a long distance operator, this operator will be able to establish a connection to any desired telephone simply by playing a 10 or 11 digit code into an automatic mechanism. She will receive distinctive signals when the called telephone answers or when the telephone or the toll circuits are busy. She will completely control the establishment of the connection and will have available to her the information necessary for proper billing of the call. The plan also contemplates that telephone cus- tomers will ultimately be able to dial long distance calls themselves, wherever may be the locations of the calling and called telephones. INTRODUCTION Ever since the invention of the telephone 76 years ago, development work has been pressing forward both in telephone transmission and in switching. These two fields have been closely interrelated in the develop- ment of telephone service on a nationwide basis, and neither could have progressed as it has without corresponding progress in the other. The first development of equipment for the mechanical switching of telephone lines was the local dial system to enable one customer to be 823 824 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 connected with another in the same town. It was a natural step to de- velop the equipment so that operators in nearby towns could complete toll calls through this local dial equipment. This was done first by using the local equipment and then with progressive modifications making it more and more suitable for toll. By these means through the decades of the 20's and 30's regional networks were developed for operator toll dialing, using step-by-step types of equipment, particularly in Southern California, Connecticut and Ohio. Also many short haul toll calls in metropolitan areas were handled in connection with the panel type dial equipment which was developed for automatic switching in these areas. Also during this period the I'ange of customer dialing in large metro- politan areas was extended, where local service is measured by message registers, through arrangements for the multiple registration of calls for which the charge was more than one local unit. An important feature of switching development in this period was the perfecting of "common control" switching systems for large metro- politan areas endowed with a high degree of intelligence and great reliability. As will be shown, still more extensive and complicated func- tions must be performed by the common control systems of a nationwide automatic switching system. Also throughout this period great advance was made in the quality and stability of long distance circuits. Telephone connections, some with as many as five circuits in tandem, were being regularly established by telephone operators with satisfactory overall transmission. The limita- tion was in the speed and accuracy with which multiple switches could be made by operators rather than in the overall transmission charac- teristics. Several factors have worked together to bring about a big expansion of long distance telephone service. These include the great growth in the numbers of telephones in service, improvements in long distance trans- mission, in switching, and in methods of traffic operation. Since auto- matic switching becomes increasingly attractive as the traffic density increases, this large growth pointed toward the desirability of further mechanizing the switching operations. In 1943 there was cut into service in Philadelphia the first installation of the No. 4 toll crossbar system. This system was designed to enable general automatic switching of toll connections in and out of large metro- politan areas and had many of the capabilities necessary for nation- wide switching. The various considerations already mentioned, coupled with the sue- NATION W I I)K Al TOM A'l'IC SWITCHING 825 cess of the No. 4 installation at Philadelphia, led to studies of the service and operating results which might l)e expected from a nationwide extension of automatic switching. The conclusion was reached that this would be a desirable obj(M-ti\'e of the Hell System companies and would r(>sult in a \-erv substantial fui'thei- improx'ement in the spewed and ac- curacy- of handling of long distance messages. Accordingly, during the next few years, a national plan was pi'epar(>d and was adopted by the telephone companies. OKXKHAL PLAN FOK NATIOXW IDE AU'roMATIC SWITCHING The features of this nationwide plan and the present status of its application form the subject of the three technical papers which accom- pany this introductory paper. ' ' " The basic requirements to be met in the development of this plan included the following: 1. It should be suitable for the nationwide extension of automatic switching both by originating toll operators and by the customers direct. When this work was commenced it was clear that a program leading toward general nationwide operator dialing was desirable. Subsequent de\elopments have confirmed the wisdom of making the basic plan consistent with general nationwide customer dialing as well since it now appears that a very wide extension of this form of service wall become desirable. 2. The plan must provide for satisfactory overall service betw^een any two telephones in this country and Canada. I'^nder manual operation satisfactory overall service was provided for by the general toll switching plan in use since about 1930. This plan is modified to recognize the far greater speed and accuracy of automatic switching compared with manual swit(;hing. This involves also modifi- cations of transmission design standards so that the overall connections will continue to be satisfactory. 3. The system must be designed for instantaneous service, so that delays due to lack of circuits or equipment would be very infrequent. This is necessary, both from the standpoints of service and the avoidance of tieups, particularly of the automatic switching machinery. A trunking system must therefore be devised which will most economi- cally meet this I'equirement, considering overall costs of lines, switch- ing equipment and operation. 4. Machines must be designed for use at strategic points in the net- work, called "control switching points", to perform automatically the various tasks required to make the overall plan operative and economical.. 826 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 5. The entire plan must be such as to provide satisfactorily for growth, for flexibility to meet changing conditions and for minimum overall costs of operation. FUNDAMENTAL PLANS FOR TOLL PLANT Mr. Pilliod's paper, pages 832 to 850, discusses the fundamental lay- out of plant for nationwide operator toll dialing. This is subject to changes from time to time with further specific studies, as is the case with all far-reaching fundamental plans of this type. The additional requirements imposed by nationwide customer dialing are still under study as will be discussed a little later. The national toll switching plan is modified so that there may be a maximum of eight toll circuits switched together to connect any two telephones compared with the previous limit of five. In order to handle the entire traffic of the country, approximately 100 control switching points are necessary at which highly intelligent common control switch- ing systems of the No. 4 crossbar type will be placed. A very important feature of the layout is a trunking plan providing for a high degree of use of alternate routes. To design all of the toll cir- cuit groups of the country for a no-delay service would be very expen- sive. However, taking advantage of the extreme rapidity of automatic switching and the ability to build into the machine capacity for using a large number of .alternate routes, a trunking system has been devised in which only about one-sixth of the toll circuit groups of the country need be engineered on a very liberal basis. These are called final groups and are the groups to which the machine ultimately appeals if all of the more direct circuit groups are busy. These more direct circuit groups can then be engineered on a basis providing for high usage of the cir- cuits, recognizing that when one group is busy the machine appeals to another and so on until as a last resort the final group is used. In determining means for handling all of the toll messages with a relatively small number of control switching points, tremendous ad- vantage was derived from modern transmission developments, par- ticularly carrier systems which give a great economy from the concen- tration on a long distance route of large numbers of telephone circuits - numbers often running into the thousands. As a result, a considerable degree of circuitous routing and back hauling of circuits is economical if by these means the circuits can be concentrated on heavy routes. This in turn lends itself to a plan using a minimiun of control switching points. NATIOXW IDK AITOMATIC SWircillNci 827 NATIONWIDE XU.MBKHING I'LAX In the previous use of automatic switchiiif!; by loll operators, the operators wei'c furnished with codes by means of which could be selected the various circuits necessary to reac^h the destination. These codes Avere dialed, followed by the local number of the called party. With this sys- tem, toll operators calling a given telephone from diffei'ent remote cities would, in general, use different codes coi-respoiiding to the different circuit groups which the}' must select. For nationwide toll dialing even by operators this system would luivc impossible complications, and for nationwide customer dialing it is clear that the code to be dialed must uniquely represent the office which serves the called telephone and that office only and not be dependent upon the route to be followed to reach it. In other words, it in\{)l\es the development of what is called a destination type code. Anothtn- descrip- tion of this code plan is to say that for toll dialing purposes each tele- phone in the country (and Canada) must have a distinctive telephone number different from that of every other telephone. It is also clear that as a practical matter this number should be based upon the local telephone number of the customer prefixed b}- a minimum number of digits, following easily understood rules. To bring this about has involved a very high order of planning. Such a plan has been perfected and forms currently the basis for the deter- mination of the coding of all new telephone offices and for changes in office codes when these are necessary. The development of this is the subject of Mr. Nunn's paper. CUSTOMER TOLL DL\L1NG When the customer is to dial long distance calls directly without as- sistance from any operator, two additional requirements are imposed beyond those necessary for nationwide operator dialing. 1. The customer normally is connected to a local central office but for the purpose of nationwide toll dialing he must be connected to the nationwide toll network. At present he does this by dialing a code such as '211" which comiects him with the long distance operator. This pro- cedure could be continued. However, since the customer must in any event dial 10 digits for the longest hauls to designate the called telephone, it is desirable if possible to cut out this preliminary step. That would mean modifying the local central office equipment so that it would receive the 10 digit numbers and transmit them on to the toll equipment. This is a simple undertaking for local central offices using the latest 828 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 19o2 type of local central office equipment, called No. 5 crossbar, which was designed with this in view. For older types of equipment, the job is more difficult. 2. The switching equipment must be provided with automatic means for recording all of the information necessary for charging the call. In the case of operator dialing this is now done manually by the operator. Great advances have been made in recent years in the development of automatic message recording equipment. In 1944 there was placed in service in California the first installation in this country of automatic ticketing equipment. This equipment is associated with step-by-step local switching equipment and automatically prints for each call a ticket similar to that prepared by the operator with manual operation. In 1948 there was installed in Media, near Philadelphia, a greatly im- proved type of message recording equipment in which the information appears in the form of punched holes in a tape. This equipment is much more economical than the earlier system and also lends itself to the automatic preparation of toll statements or bills. The present forms of equipment have been designed to be associated with local central offices. A careful study has been made of their field of application and of the basic plan necessary to provide for a general nationwide extension of customer dialing. This indicates that there will be a large field for automatic message accounting ecjuipment associated wdth the toll network and arranged to receive orders for toll messages from a number of local dial offices. This centralized AMA equipment, as it is called, is under development and an initial installation will be made next year in Washington, D. C. In this installation the range of customer dialing will be limited and certain service features will be lacking, which it is planned to add later. The nationwide extension of customer toll dialing involves many op- erating problems in addition to those relating to the design of the plant. These problems involve the extent to which customers wish to dial long distance calls, requiring 10 pulls of the dial, the accuracy of dialing, the treatment of WTong numbers, provision for giving subscribers information regarding telephone numbers in distant cities, information on charges and many other questions. Recognizing that the best way to develop these questions is a trial, arrangements were made to open such a trial last fall at Englewood, N. J. This office is equipped with a No. 5 crossbar system so that arrange- ments for such a trial could readily be made there. The Englewood customers are able to dial directly any of about eleven million telephones in ten metropolitan areas scattered throughout the countiy, including NATION WIDK AUTOMATIC SWITCHING 829 Boston, New York, Pittsburgh, Cleveland, Chicago and San Francisco and the Bay area. The results of this trial have been \'ery encouraging. Subscribers are continuing to dial over 95 per cent of all the culls which can be dialed. Errors due to wrong ninnbers are at a minimiun and other difficulties are relatively low. In so far as this trial can answer the questions, the results are all in favor of the nationwide extension of customer dialing as the development and installation of facilities suitable for this purpose make it possible to do so. In \iew of the prospect of nationwide customer dialing, fundamental plan studies are now being made by the Telephone Companies through- out the country of the whole layout of plant including the distribution of centralized automatic message accounting equipments with the future general application of this method of operation. The present indication is that the number of points at which toll operating centers will be re- (juired will be greatly reduced. This will react in important ways on the design of telephone buildings, telephone equipment installations and toll circuit routes. AUTOMATIC TOLL SWITCHING AND ACCOUNTING EQUIPMENT All of these plans depend upon the successful development of striking innovations in toll switching and automatic message accounting equip- ments. The plans in turn react upon the features to be incorporated in such equipments and upon the schedule of their development. Mr. Shipley's paper, pages 860 to 882, tells about the more important fea- tures of these equipments and the problems which are involved in their development. CONCLUSIONS Experience with operator toll dialing shows clearly that it provides a marked improvement in toll service. This improvement will increase as progress is made toward the full application of the nationwide automatic switching plan. The development of long distance dialing by customers is at an early stage. The results of recent trials, however, indicate that nationwide customer dialing has service advantages and will generally be received with enthusiasm by telephone users. It is anticipated, therefore, that customer dialing will rapidly expand both on a regional and on a nation- wide basis. The service advantages of nationwide automatic switching are not 830 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 measured entirely by the increased speed and improved accuracy of coiuiections. An important factor is the continued abihty of the tele- phone system to meet the rapidly increasing demand for telephone service without making excessive demands on the available supply of labor. The development of local dial operation was absolutely necessary to handle the great growth of local telephoning which has taken place. Today, in many places, requirements for people for toll operations are very heavy and an increased amount of automatic toll switching is becoming more and more necessary to make possible handling the rapidly increasing number of long distance telephone messages. With this development there has been a marked increase of employ- ment. The Bell Companies today employ 244,000 operators compared with 131,000 in 1941. They have also employed many people to build and install about 300-million dollars worth of toll dialing ecjuipment, to constinict places to house it, maintain it and carry out operating rearrangements. With respect to the future, even with the nationwide automatic switch- ing plan in full operation and the local central offices arranged to permit customer dialing, there will still be a large amount of work for operators. They will be required to handle information and assistance traffic, person-to-person calls, collect calls and other classes of calls which do not lend themselves to customer handling, as well as any individual calls which the customers may not wish to dial themselves. The Bell Companies have necessarily taken the lead in planning and applying these new developments. The plans, however, are all laid in such a way as to include telephone users in Independent Telephone Company offices. The Independent Companies are being kept fully in- formed of these plans as they develop and are participating, as the development of their own plant makes it practicable and desirable, in extending the benefits of the new forms of operation to their own customers. This long-term development has required the very close cooperation of all parts of the Bell System - American Telephone and Telegraph Company General Department, Bell Telephone Laboratories, Western Electric Company, Long Lines and all of the Bell Operating Companies. Each installation of equipment and circuits and each operation is a part of a nationwide system and must be closely coordinated. The close interrelation and working together of the various parts of the Bell Tele- phone System, research and development, manufacturing, engineering and operating are necessary for the effective planning and execution of this tremendous project. NATIONWIDE AUTOMATIC SWITCHING 831 BIBLIOGRAPHY 1. F. J. Sc'iuldcM- and .J. X. Rcvnolds, "Cr().ssl)iir Dial Telephone Switching Sys- tem," A.I.E.E. Transactions, 58, pp. ITil 1!»2, I'CW. 2. L. G. Ahraliani, A. J. Busch and K. F. Shiplrv, "Crossbar Toll Switching Sys- tem," A.I.E.E. Transactions, 63, pp. 'M)2 m), \U\\. 3. J. J. Pilliod, "Fundamental Plans for Toll Teiephonc' Plant." Page 832 of this issue. 4. W. H. Xuiui, "Xationwidc Xumlieiinfr Plan." Pafi;e S51 of this issue. 5. F. F. Shipley, "Automatic Toll Switching Systems." Page .S60 of this issue. 6. H. S. Osborne, "A ( ieneral Switching Plan for Telephone Toll Service," A .I.E.E. Transactions, 49, pp. 154<) 1557, 1930. 7. F. A. Korn and J. (1. Ferguson, "Xo. 5 Crossbar Dial Telephone Switching System," A.I.E.E. Transactions, 69, Part 1, pp. 244-254, 1950. 8. O. A. Friend, "Automatic Ticketing of Telephone Calls," A.I.E.E. Transac- tions, 63, pp. 81-88, 1944. 9. John Meszar, "Fundamentals of the Automatic Telephone Message Account- ing System," A.I.E.E. Transactions, 69, I'art 1, pp. 255-269, 1950. Fundamental Plans for Toll Telephone Plant By J. J. PILLIOD (Manuscript received Maj' 15, 1952) This paper covers the general switching plan and fundamental plant layout proposed for handling telephone toll messages throughout the United States and Canada using aidomatic toll switching. There has been rapid growth in the number of telephones and in the volume of toll traffic, particularly long haul. Toll facilities are provided under fundamental plans, an essential part of which is a toll switching plan for setting up connections quickly between any two telephones. The introduction of mechanical operation and the general improvement in the transmission performance of the communication plant over a period of years make the introduction of certain modifications in the fundamental plans possible and advantageous at this time. The important new features and the service improvements which are provided by the proposed plans are outlined in this paper. The principal types and characteristics of circuit facilities available for use in the intertoll network are also described. GENERAL ASPECTS OF TOLL SWITCHIXG PROBLEMS Switching plans providing for the systematic routing of toll telephone traffic have been employed b}- the communication industiy for many 3^ears. These plans have contributed directly to the high c^ualit}' of long distance telephone service enjoyed by the public in the United States and Canada. This generally excellent service is the result of the coopera- tive work of many organizations including the Bell Operating Companies, many independent connecting Companies and others in the United States as well as in adjoining countries. The techniques employed today reflect a great amount of research and engineering and improvements in manufacturing skill and in construction, maintenance and operating methods developed over a period of many years. Throughout the United States and Canada there are approximate^ 20,000 different places - cities, towns, and villages - that serve as toll 832 FUNDAMENTAL PLANS FOR TOLL TELEPHONE PLANT 833 couiiocting points. The telephone offices in each of these places have access through the toll network to practically all of the 50,000,000 tele- phones in the United States and Canada and also to most of the tele- phones in the rest of the world. Currently the Bell Operating Companies are handling toll calls at an a\'erage rate of over 7,000,000 during a business day. The many millions of different connection possibilities which this number of calls involves require a definite and comprehensive switching plan. Whenever practicable and economical direct circuits are used to handle toll message traffic between two given points. Much of the traffic in the coinitr^^ is handled this way. However, a substantial volume of business, about 20 per cent, is handled as a matter of economy, by switch- ing toll circuits together. Althougli the volume of traffic between different points may ^'ary o^'er a wide range, it is nevertheless important that adequate service be provided for all possible connections. For example, there are about 110 circuits from Chicago terminating in the toll office serving Minneapolis and St. Paul. These handle about 5500 calls per day. On the other hand, only a few calls a year may be involved between some point in Western Minnesota and a point in Florida. The switching plan described in this paper is devised for the purpose of efficiently and effec- tively establishing connections between any two points regardless of their separation and regardless of whether traffic volume be a few calls per year or many calls per hour. ELEMENTS OF THE PROBLEM In order to illustrate the problem a specific example may be useful. Fig. 1 is a map of Wisconsin and ]\Iinnesota on which nearly 1200 circles indicate points at which exchange facilities may be connected to the toll network. The extent of the coverage in this area is typical of that found throughout the country. The 150 odd larger circles represent existing offices known as "toll centers" - that is, places where operators record toll calls and perform other operations necessary to establish toll connections. These places have switching arrangements of various types depending on how they fit into the switching plan. Some may operate as control switching points in the nationwide plan as described later. IMore than 1,000 smaller circles on the map represent "tributaries" - that is, towns where little or no toll operating is done. Toll connections to and from these points are completed at the toll centers which in gen- eral do the toll operating required. 834 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 In the United States and Canada as a whole, there are approximately 2,600 toll centers. The remainder of the toll connecting points — about 17,500 — are tributaries. Fig. 2 gives an idea of the variety and complexity of the network of circuit groups required to interconnect the toll centers in one area. Here each line represents a group of circuits, known as "intertoll trunks," between two toll centers. Each group may contain anywhere from one to several dozen trunks. The location of the lines on the map is unrelated to the geographical routing of the trunks, and only a part of the circuit groups are shown. To get a complete picture one should visualize that a cluster of relatively short circuit groups radiates from each toll center to its tributaries, of which there may be up to 15 or more. Physically, the plant consists of a network of open wire lines, cables and radio systems. On these, voice frequency or carrier operation is employed in each section as required to provide the necessary intertoll trunks. The routes of the lines in Minnesota and Wisconsin are shown by Fig. 3. In this area there are no radio routes carrying telephone cir- cuits, but a radio system between Chicago and Minneapolis is in the planning stage. Areas like Wisconsin and Minnesota must, of course, be connected to- gether, and Fig. 4 shows the major Bell System toll routes that accom- plish this. On a map of this kind it is not possible to include anything like the detail shown in Fig. 3. One must visualize, therefore, that each state contains a network of routes generally comparable to those shown for Wisconsin and Minnesota. This then represents the interconnection problem to be met by an orderly switching plan that will provide efficient, reliable and fast toll telephone service between any two points. EARLIER TOLL SWITCHING PLANS Very early in the telephone industry it became evident that: (1) There must be a plan for connecting circuits together. (2) Switching centers with suitable equipment must be established in accordance with this plan. (3) Trunks must be provided in adequate numbers to connect every place to one or more switching centers and to interconnect the switching centers. (4) All this must be done in a way that makes it possible to provide good service at reasonable cost. As time went on, early plans crystallized into what became known as the General Toll Switching Plan. A paper presented at the summer con- vention of the A.I.E.E. in Toronto in 1930 by Dr. H. S. Osborne outlined FUNDAMEXTAL PLAN'S FOR TOLL TELEPHONE PLANT 835 the piiiiciples of this comproheiisivo plan for han(llinROUTING SEQUENCES I MONTREAL J ALBANY r' SYRACUSE T 1 _^-\- eUFFAUOl I HAVEN^ ^^ ^--tfCRANTON,. ' ^Ac? \ I $ I NEWARirrNEW 1 SACRAMENTO 1 U" (^ \ ^0.. AND \^ \ / ) UeES.o^^^. \/ I lincolnT"'"""*""""^' Champaign i \iNDiANAPOLl5_c:oLUMeus/' '^'' ,'"""-^^M \ \ S" A -^ /-' ^9 WASh^lNGTON ,' \SPRlNGF=IELO ]*TERRE I j. *CINCINNATr'' ^^ • ^T^--^ CITY ^"''"^ ^) \ ''r ^,'^ 1 /RtCHMONOlfc : ; CENTRALIA*' / VlOUISVILLE NCHARLESTON ^ ML ] *-~ ^""" I /'"EVANSVILLE I .-J ± .LOS ANGELES •ALBUQUERQUE '"?' B NATIONAL CENTER ■ REGIONAL CENTER A SECTIONAL CENTER REGIONAL SWITCHING AREA BOUNDARIES Fig. 7— Tentative locations of i-ontrol sivileliing points in Unitod States and Canada, FUNDAMENTAL PLANS FOR TOLL TELEPHONE PLANT 839 tically all overflows from the hi^h usage jijroup.s during the heavy traflic periods. The "high usage" atid "finar' gi'oups which could he used for routing calls l)et\vc(Mi Ilihhing, Minn(>s(>(a and DaAciiporl , Iowa ai'e shown by Fig. 5. Generalization of the Toll Switching Plan. The generalization of the arrangements discussed for the Chicago region is illustrated in Fig. G. This shows diagi'ammatically all types of switching points in two regions and also indicates the relative position occupied by the National Center in the switching plan. On this chart, the solid lines represent the "final groups" of trunks, and the dotted lines represent "high usage" trunks. Examination of this chart will indicate that the mechanical switching system need perform only rela- tively simple toll switching operations at the toll centers. At other points the system must attempt to complete the call over the most favorable routes, in planned sequence, until the "final" route is selected. For example, from a given primary outlet such as POl on a call des- tined for a toll center in the other region such as TC2, the switching equipment would attempt to complete the call, in sequence over the routes marked 1 to 6. Should Route 6, which is the "final" route, be selected because all of the trirnks in the "high usage" groups marked 1 to 5 were busy at the time, the switching equipment at the SC would in turn try routes marked A, B, C, etc., in attempting to complete the call. A fairly complete pattern of circuit groups is indicated in this illustration. Depending on the relative locations of the points concerned and the traffic load re- quirements, certain of the "high usage" groups shown may not exist. It is expected, however, that most TC's will have high usage groups to points other than their "home" PO's. Also each PO can be expected to have high usage groups to sectional centers other than its "home" SC. All regional centers will be interconnected with direct trunks, regardless of geographical location. Control Switching Points Because of rapid and complex switching operations required by the automatic equipment at PO's and higher order switching points, (SC's, RC's and the NC) these switching centers are called Control Switching Points (CSP's). As covered by a companion paper, the switching equipment required at the CSP's is quite complex. This equipment must have a high degree note: the short line from each PO and sc points toward its home csp >RIES FUNDAMENTAL PLANS FOR TOLL TELEPHONE PLANT 839 tically all overflows from the hijj;h usage {groups during the heavy IrafFic l)eriods. The "high usage" and "final" groups which could be used for routing calls Ix'lwccn Ilihhiiig, Minnc^sota and Daxcnpoi'l, Iowa are shown by Fig. ,'). Generalization of the Toll Switching Plan The generalization of the arrangements discussed for the Chicago i-egion is illustrated in Fig. G. This shows diagrammatically all types of switching points in tw'o regions and also indicates the relative position occupied by the National Center in the switching plan. On this chart, the solid lines represent the "final groups" of trunks, and the dotted lines represent "high usage" trunks. Examination of this chart will indicate that the mechanical switching system need perform only rela- tively simple toll switching operations at the toll centers. At other points the system must attempt to complete the call over the most favorable routes, in planned sequence, until the "final" route is selected. For example, from a given primary outlet such as POl on a call des- tined for a toll center in the other region such as TC2, the switching equipment would attempt to complete the call, in sequence over the routes marked 1 to 6. Should Route 6, which is the "final" route, be selected because all of the tiTuiks in the "high usage" groups marked 1 to 5 were busy at the time, the switching equipment at the SC would in turn try routes marked A, B, C, etc., in attempting to complete the call. A fairly complete pattern of circuit groups is indicated in this illustration. Depending on the relative locations of the points concerned and the traffic load re- quirements, certain of the "high usage" groups shown may not exist. It is expected, however, that most TC's will have high usage groups to points other than their "home" PO's. Also each PO can be expected to have high usage groups to sectional centers other than its "home" SC. All regional centers will be interconnected with direct trunks, regardless of geographical location. Control Switching Points Because of rapid and complex switching operations required by the automatic equipment at PO's and higher order switching points, (SC's, RC's and the NC) these switching centers are called Control Switching Points (CSP's). As covered by a companion paper, the switching equipment required at the CSP's is quite complex. This equipment must have a high degree 840 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 of built-in capability to perform quickly the circuit .selection work as- sociated with the alternate routing features of the switching plan. In adflition, to help provide the ti'aiismission margins needed for satisfac- tory operation of the plan as contemplated, it must be arranged to con- nect circuits on a four-wire basis rather than on a two-wire basis, the latter being the an-angement used at most toll centers. The switching ecjuipment at a CSP must not only pro\'ide for connecting one toll cir- cuit to another; it must also perform the very important function of tying the toll networks which sei've limited local areas together so that collectively they work as a smoothly functioning nationwide system. This becomes practicable when there is coordination between the design of the individual limited networks and the design of the overall system. The location of control switching points indicated by the nationwide plan is shown in Fig. 7. This also indicates the home switching center of higher order associated with each switching point. As the number of CSP's increases, the cost of the toll circuit plant decreases because each CSP can then be located closer to the cluster of ordinary toll centers which it ser\^es. How^ever, because of the cost of the CSP eciuipment, it is necessary to weigh the cost of circuit facilities with the ecjuipment costs in a way that will result in the minimum overall cost. Certain of the smaller Primary Outlets are being studied with the view of reclassifying them as Tandem Outlets (TO's). A Tandem Outlet occupies the same relative position in the switching plan as a Primary Outlet but is not a control switching point. The switching equipment employed is less complex than that used at control switching points and therefore pro- vides for only limited alternate routing and does not have all the ad- vantages of four-wire transmission. Effects of Customer and Operator Toll Dialing Customer dialing of short-haul toll calls has been in use, particularly in metropolitan areas, for some years. A trial of long-haul customer dialing over the intertoll trunk network and through the switching equip- ment provided for operator toll dialing was instituted at Englewood, New Jersey, in the Fall of 1951. The local ecjuipment includes automatic message accounting and permits Englewood customers to dial directly to about eleven million telephones in ten metropolitan areas across the country. A trial installation of customer toll dialing, utilizing automatic message accounting eriuipment on a centralized basis rather than at each local office, is planned for Washington, D. C, in the Fall of 1953. Ini- tially customers will dial toll calls within the Washington metropolitan FUNDAMENTAL PLANS FOR TOLL TELEPHONE PLANT 841 area and to such points as Baltimore and Ainiapolis. The favorable results and general acceptance of the trial at T^nglewood indicate ex- tensive application of customer dialing of toll calls as conditions warrant. The general introduction of customer toll dialing as this becomes desirable will affect the lunnber and location of ordinary toll centers since calls handled by operators may be limited to assistance calls and to person-to-person, collect and others which cannot be customer dialed. Indications are that toll operation for a number of smaller centers can be combined as the local service is converted to dial operation with operator toll dialing. Studies now in progress indicate that the lumiber of toll centers may be reduced by one half or more over a period of years in many areas. Reactions on Toll Plant Layout The expanded general toll switching plan for nationwide dialing con- templates a degree of alternate routing far in excess of that used with the former switching plan designed for manual operation. This change along with the reduction in toll centers will have a marked effect on the normal flow of many traffic items through the intertoll network. As a result the arrangement of the present intertoll trunks will be significantly modified both in number, routing and terminating points. It is necessary to take these facts into account in engineering toll plant additions so that they will lead toward an advantageous layout for future nationwide dialing as well as meet the needs of the more immediate future. Fortun- ately, the effect is in the direction of greater concentration of circuits in main routes so that with the new cable and radio facilities available, o\er-aIl economy and better service should result. TYPES OF TRANSMISSION FACILITIES USED AND INCLUDED IN SWITCHING PLAN The domestic toll network is an outgrowth of the demands of the business and the advance in communication technique over many years. At present, about 100,000 intertoll trunks over twenty-five miles in length and many thousand shorter toll trunks are in service throughout the countrj'. They are provided generally by voice frequency or carrier frequency facilities. The choice of transmission facility on a given route is dependent on a number of factors, such as cost, length of haul, luimber of trunks in the cross-section, numbers of trunks to be terminated at intermediate points, the types of terrain to be transversed, storm and 842 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 I Fig. S Terminal cquiinnciit (if tyi)e-Nl cable carrier system. Provides twelve message channels with self contained signaling equipment over two pairs of cable conductors in same sheath. FUNDAMENTAL PLANS FOR TOLL TELEPHONE PLANT 843 Fig. 9 — Coaxial Cable. Cross section of cable containing four pairs of coaxials. Each pair can accommodate one two-way coaxial carrier system. other conditions affecting service continuity and the transmission re- quirements of the circuits to be provided. Voice frequency facihties equipped with repeaters as required are used on both open wire lines and cables. At voice frequencies it is customary to derive three trunks known as a phantom group, from two pairs of open wires or from one "quad" (two pairs) of loaded cable conductors. In general the use of voice frequency facilities is now limited to shorter circuits. Considerations of economy and service improvement led to the intro- duction of carrier operation into all types of toll plant as rapidly as the state of the art permitted. This directly affects the toll switching plan from the standpoint of routing and location of switching centers. At present, carrier systems use four broad categories of facilities : open wire, conventional paired or quadded cables, coaxial cable and radio. Several types of open wire carrier systems permitting from one to fifteen telephone channels above the frequency band of the voice channel are now in use. In general these systems are used when> trunk cross-sec- tions are relatively small and where the terrain and weather conditions make open wire lines economical. Cable carrier systems at present permit the operation of up to twelve telephone channels on two pairs of cable conductors. These conductors may be in one cable or divided between two separate cables, depending 84-4 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Fig. 10 — Microwave radio relay tower at Cotoctin Mountain, Marj-land, on a New York-Washington radio route. There are 300 message circuits in service with more planned. on the type of carrier system (Fig. 8) . Coaxial cable transmission systems currently provide up to 600 telephone channels per pair of coaxials (Fig. 9). A new coaxial system, under development, is expected to pro- duce about 1,800 telephone channels per pair of coaxials. Most of the applications of radio for toll telephone service now contem- plated, involve the use of pomt-to-pomt microwave systems. By employ- FUNDAMENTAL I'LAXS FOK TOLL TKLKIMIONE PLANT 845 iiig rhaniioliiifz; (equipment at the tormiiiuls of these systems similar to that used for the present coaxial system, eaeh pair of radio channels may provide up to (lOO telephone channels. Several pairs of such radio channels may be operated throuf>h the same antennas (Fi^- 10). Radio systems are also u.seful in some cases where the number of toil triniks re(iuire(l is moderate, where diversity is desired or where water or other natural l)arri(M's make tiie proxision of wire circuits difficult oi' impracticable. The type of facility to be used on a particular route is sometimes affected by re(iuirements for other services such as teletypewriter, tele- vision network facilities, program facilities, private lines and other factors. Trend to Carrier Type Facilities and Advantages to Toll Sivi telling Plan About 70 per cent of the long haul toll message mileage in Bell Operat- ing Companies is provided on carrier type facilities as contrasted with 7 per cent in 1930 (Fig. 11). From the transmission standpoint, carrier facilities offer marked ad- Fig. 11 — Growth in Bell Sy.stem iiitertoll trunk mileage showing trend toward more extensive use of carrier type facilities. 846 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Fig. 12 — Toll switchboard position with kej- set used lur lull dialing. vantages. The}^ are iiiherentl}^ of the "four-wire" type which minimizes the number of possible singing and echo paths on a circuit. Also, the speeds of propagation over carrier systems are generally higher than o\'er voice frequency systems thereby further minimizing the echo problem. These features are of great advantage in reducing limitations on circuit design and layouts of the general toll switching plan. Signaling Systems In addition to the ability to carry messages, intertoll trunks must be provided with suitable signaling facilities. ' These must provide a means of: first, attracting the attention of the distant point, either an operator or automatic equipment, to the fact that a connection is to be established ; and second, in the case of dial operation, transmitting coded information in the form of pulses for establishing the connection; and third, trans- mitting a general class of super\'isory signals including connect and disconnect signals, on and ofT switch hook signals, recall signals and FUNDAMENTAL PLANS FOR TOLL TELEPHONE PLANT 847 hutsy sifijnals which arc (>sscntial to the efficient operation of the switchiuf^ phuit. 'fhc circuit (lesi<>;ii conleinphited in the o\'erall plan must tafvc into accouni this rc(iuircnicnt lor tiansniit tina; sii>;nafs as well as speecli, to ol)tain a< 'curacy and speed in set t int; up and takinu; (h)wn connections. TKANSMISSIOX DESIGN ASPECTS OF CIUCUITS FOR NATIONWIDE TOLL DLVLING '\l\o more extensiyp use of alternate I'outins to<2;ether with the increase in maximum possible number of trunks in tandem associated with nation- wide toll dialing, tends to increase the problems of assuring adequate transmission of speech and signals on all possi])le connections. On the other hand, the use of four- wire switching at important points and the definiteness of the routing patterns permit more efTectiye use of the ayailable facilities and thus tend to simplify the problem. Extensive studies indicate that on the whole, the new toll switching plan will make feasible still further improvements in transmission. This is, of course, a desirable objective. Tranf^mission Design of Tnmks With dial operation, the number of trunks in tandem in a given toll connection may vary on successive calls. To avoid undesirable trans- mission contrasts and other adverse effects, it is important that every trunk be designed to operate as closely as possible to the theoretically correct transmission loss. The problem is complicated by the fact that the extent to which the echo, noise and crosstalk will limit the perform- ance of an individual link is not directly proportional to the length of the circuit. In fact, the minimum loss at which a particular circuit used singly or in various built-up combinations can theoretically be operated depends on the number, length and characteristics of the other circuits connected in tandem with it. Arrangements for precisely adjusting the loss in the indi\'idual trunks for each call would be complicated. Adequate performance can be achieved however by compromise methods Avhich provide for automatic adjustments in the loss of each trunk in accordance with the following: 1. When a trunk is switched to other intertoll trunks at both ends it is operated at the minimum loss practicable. This loss is known as "via net loss." (VXL) 2. When the trunk is switched to another intertoll trunk at one end only, the loss is increased two db. 3. When the trunk is not switched to another intertoll trunk at either 848 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 end a further loss of two db is added. This loss which is four db greater than the via net loss is known as "terminal net loss." (TXL) The data and methods used in the derivation of the via net loss are rather complex and not within the scope of this paper. Assignment of Facilities Among Trunks The definite routing patterns established for the toll machine switching operation impose more severe transmission conditions on certain classes of circuits than on others. For example, a trunk in a ''final" group be- tween a TC and a PO can become involved in an eight-link connection, whereas a trunk in a "high usage" group, say, between a PO and another PO will not be involved in more than a three-link connection. This creates a need and provides an opportunity for allocation of the available facilities among the various tnmk groups in a way that will provide the best overall service. For example, to the extent practicable it is desirable to assign carrier grade facilities to trunks in "final" groups that may be involved in connections with the maximum number of links. Facilities with less favorable transmission characteristics ma}" then be reserved for trunks in groups that are used for connections involving fewer links. TRANSMLSSION PERFORMANCE Table I shows the approximate range of transmission losses between toll centers under the manual plan compared to ranges that appear practicable under the proposed fundamental plan, which, of course, per- mits more links in tandem. Trunk Transmission Stability It is as important that the transmission loss of a trunk used in the contemplated toll dialing network be maintained at or close to its assigned value at all times as that the assigned value be right. On multi- switched connections even a relati^'ely small consistent excess or defi- ciency in the loss in the indi\idual trunks can accumulate to overall excesses or deficiencies in loss large enough to cause difl^culty - by making it hard for people to hear if the attenuation becomes too great or by creating excessive echo, crosstalk or noise if the loss becomes appreciably less than normal. This subject has been extensiveh' studied for the past several years and it appears that some changes in practices and the introduction of FUNDAMENTAL PLANS FOR TOLL TKLKl'IIOXK PLANT 849 Table I— Approximate Range of Losses I^etwkkx '1'oll Centers in db No. of Links in IntertoU Connection Manual Plan Proposed Plan 1 2 5 8 4-12 8-14 9-20 4-8 5-12 6-13 7-13 new metliods of m(>asiiriiig rosults will lead to marked improvements. It is of some interest that one of the major factors in securing improvement appears to be the application of a statistical method of evaluating per- formance along somewhat the same lines as the "(juality control" methods used in other fields of industry. Since, with operator toll dialing only one operator is involved in many connections and with customer toll dialing there is no operator on the connection it is extremely important that everything be right. This is typical of the reciuirements of any large scale "push button" operation (Fig. 12). conclusion The fundamental plans proposed for Telephone Toll Switching provide a basis for the progressive mechanization of toll service. The installation of suitable switching mechanisms at Control Switching Points and the provision of toll trunks utilizing the new instrumentalities will implement the toll switching plan. The plan is sufficiently flexible to adjust for changes in the telephone art as they develop. Also, the plan can fit in with the requirements of those Companies whose plants connect with the Bell operating network should they desire to arrange for operator or customer toll dialing. Average speed of service will be improved. The flexibility in plant design inherent in the new toll switching plan will increase service security and improve the utilization of the entire toll plant. In addition, adequate provision is made for the progressive introduction of customer toll dialing as this becomes practicable and desirable. bibliography 1. H. S. Osborne, "The General Switching Plan for Telephone Toll Service," A.I.E.E. Transaclions, 49, pp. 1549 1557, 19,30. 2. C. M. Alapes, "Carrier is King," Bell Tel. Mnq., 28, pp. 191 203, Winter 1949-50. 850 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 3. J. J. Pilliod and 11. L. Rvan, "Operator Toll Dialing," BcJl Td. Mag., 24, pp. 101-115, SumnuM- 1945. 4. E. W. Baker, "Toll Dialing is Expanding Throughout the Nation," Bell Tel. Mag., 30, pp. 253-264, Winter 1951-52. 5. F. F. Shipley, "Automatic Toll Switching Systems." Page 860 of this issue. 6. C. A. Dahlhom, A. W. Horton, Jr. and D. "L. Moody, "Application of iMulti- frequencv Pulsing in Switching," A.I.E.E. Transactions, 68, pp. 392-396, 1949. 7. N. A. Newell and A. Weaver, "Single-frequency Signaling System for Super- vision and Dialing over Long Distance Telephone Trunks," A.I.E.E. Trans- actions, 70, (7 pages), 1951. 8. Articles prepared by American Telephone and Telegraph Company for infor- mation of Dial Intere.xchange Committee of the United States Indejjendent Telephone Association. Published in Telephony on dates indicated. a. Nationwide Operator Toll Dialing, January 12, 1946. b. New Toll Switching Plan for Nationwide Dialing, May 10 and 17, 1947. c. Nationwide Toll Dialing— Use of Tandem CDO's, July 3, 1948. Nationwide Numbering Plan By W. 11. NUNN (Manuscript received May 15, 1952) In telephone language a numbering plan gives each telephone in a city, a town, or a geographical area an identify or designation different from that given any other telephone in the same area. There is a wide variation in the types of numbering arrangements in use today in the Bell System, and this paper gives the reasons for this diversity, and examples of the various numbering plans now in use. With the introduction of modern toll switching facilities and the extension of toll dialing to nationwide scope, it ivas realized that an improvement in the method of dialing toll calls to distant cities, was essential in order to realize the inaximum speed and accuracy inherent in toll dialing. A nationwide numbering plan covering the United States and Canada has been designed. Each of the more than 20,000 central offices in the two countries are to be given a distinctive designation which identifies that particular office. This designation is to consist of a regional or area code and a central office code The new switching equipment for the key points in the toll network is being designed so that any toll opera- tor, wherever located, will use the same designation or code for reaching a given office. The combination involved in laying out these areas and the composition of the area codes are presented. A total of 152 codes are available of which approximately 90 are assigned to the present numbering plan areas. Ulfimafrly each central office will be given a type of number consisting of an office name and five numerical digits, such as LOcust 4-6678, in which the first two letters of the office name become the two letters of the central office code. The entire program will take a considerable number of years to realize, but is one which must be accomplished in order to achieve the best residts in operator toll dialing and the ultimate goal of nationwide customer toll dialing. In telephone langnage a numbering plan is exactly what the name im- plies, a plan or system of giving each telephone in a city, a town or any geographical area an identity or designation which is different from that given every other telephone in this same area. This designation is the 851 852 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 telephone number; it appears in the directory and in most cities on the telephone instrument itself. It is the address of the telephone in the telephone network. Just as it is essential for efficient postal and delivery service to have streets and house numbers clearly marked, it is important for good telephone service that the telephone numbering plan be such that it will be used with convenience and accuracy by the telephone customer. A telephone number is comprised of two elements, a designation for the central office to which the telephone is connected and a number within the central office which identifies one particular telephone from all others served by that office. If there is only one central office in the city or town, the office designation is frequently omitted. A dial office is designed to serve up to 10,000 numbers with a limitation of four digits. Typical numbers are therefore MAin 2-1234, ADams-2345, 5-6789 and 3456, the office designations being MAin 2, ADams and 5 with the last four digits in all cases representing the number within the central office. There is a wide variation in the types of numbering arrangements in use today in the Bell System. This diversity arises from the fact that telephone communities vary greatly with respect to the number of telephones served, ranging all the way from New York City with its more than three million telephones and three hundred central offices to small villages and rural communities with perhaps a few score or a few hundred telephones. In the 1920's when the Bell System embarked upon its program of converting local offices to dial operation each exchange or city was in general an entity unto itself. Customers dialed local calls within their own city but all calls involving a toll or multi-unit charge required handling by operators for timing and ticketing. There was no advantage, therefore, in making a numbering plan for a given city more compre- hensive than required to serve the telephones and central offices in that city with a suitable allowance for the expected growth. Thus there were formed a multitude of local dial communities, large and small, within which customers could dial their own calls and connections between these telephone communities were established by operators. Over the years these basic numbering plans which were originally established for local dialing have in many of the cities proved inadequate to furnish as many office codes as later events have shown are required. This is due to a variety of causes. The station growth in many places has outstripped all expectations and the number of central offices required to serve this unprecedented demand for service consume many more office codes than the original plans provided for. XATIONWIDE XUMHERIXG PLAN 853 In many places local service areas were changed so that customers could call into contiguous exchanges at local rates. To enable customers to dial into these neary-by places the original numbering plans required expansion to include this increased number of offices. In addition, with the athance in the telephone art many cities introduced e(iuipment for automatic charging on multi-unit and short haul toll calls so that cus- tomers could dial such calls directly instead of placing them with an operator for completion. In order to enable customers to dial these calls, it was necessary to expand the original city numbering plans to encom- pass wider and wider geographical areas. In expanding the various types of numbering plans to serve a larger nimiber of central offices than were originally anticipated, various ex- pedients were resorted to. In the largest cities having three-letter office codes a numeral was substituted for the third letter thus \ery materially increasing the code capacity from about 325 to about 500 and making it possible to form a number of codes using the same office name. The name CAXal for example, instead of serving but one office may ser^'e a number of offices, CAnal 2, CAnal 3, CAnal 4, etc. In the medium size cities ha\'ing two-letter codes, expansion meant adding a digit to the code to all or in some cases to only a part of the offices in the city. The fi\'e-digit places Avere usually expanded by adding a digit to some of the numbers so that some of the telephones had five digits and others six digits in their numbers. As a result of choosing originally a numbering plan which at the time seemed adequate and most suitable for the city involved and in many cases being forced to expand to meet changing needs, we now have in the Bell System a considerable variety of different numbering plans. These are given in Table I. The numbering plans given are all adequate to serve the present local dialing needs for the cities in which they appear. Having reviewed the numbering plan situation as it exists today in the \-arious cities and towns, let us turn to the problem of handling toll calls. Under ringdown operation there is an operator at the outward toll center where the call originates and another operator at the terminating or inward toll center. On built-up toll connections there are additional operators at each intermediate toll switching point. The inward toll operators, who are familiar with the numbering plans in the offices served by their particular toll center, can be relied upon to connect to the de- sired station even though there is uncertainty on the part of the calling customer or the outward toll operator regarding the precise pronuncia- tion or spelling of the name of the called office or the particular form of numbering system used at the called city. 854 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Under operator toll dialing the inward operator is replaced by dial switching equipment under the control of the outward operator; hence the outward operator has no one to rely upon but herself in completing a toll connection to a distant city. With the present method the operator dials a code for each circuit group in the connection followed by the number of the called party which may consist of any number of digits from three to seven. The operator must refer to her position bulletin or to a routing operator for the correct circuit group codes unless she hap- pens to remember them. Where the office to be reached has central office names, the operator must rely on routing information to determine how many letters of the name are to be dialed. The great variation in the number of digits to be dialed on different calls is a source of some dif- ficulty and confusion to the operators. The present system of operator toll dialing by which operators use codes depending upon the routes to reach a desired destination, is a great improvement over the old manual handling methods. However, with the introduction of more modern toll switching facilities and the nationwide extension of toll dialing, it was realized that an improvement in the methods for dialing toll calls to distant cities was essential in order to realize the maximum speed and accuracy inherent in toll dialing. These handicaps in the present toll dialing methods are to be overcome by establishing a nationwide numbering plan covering the United States and Canada by which each of the more than 20,000 central offices in the two countries is to be given a distinctive designation which identifies that particular office and that office only. This designation is to consist of Table I — Different Types of Numbering Plans Place Directory Listing Customer Dials Ordinarily Referred to as Philadelphia, Pa. LOcust 4-5678 LO 4-5678 Two-five Los Angeles, Cal. PArkway 2345 and PA 2345 and Combined two-four •* REpublic 2-3456 RE 2-3456 and two-five Indianapolis, Ind. MArket 6789 MA 6789 Two-four El Paso, Texas PRospect 2-3456 PR 2-3456 and Combined two-five and 5-5678 5-5678 and five digit San Diego, Cal. Franklin 9-2345 F 9-2345 One letter, four and Franklin 6789 F6789 five digit Des Moines, Iowa 4-1234 and 62-2345 4-1234 and 62- Combined five and 2345 six digit Binghamton, N. Y. 2-5678 2-5678 Five digit Manchester, Conn. 5678 and 2-2345 5678 and 2-2345 Combined four and five digit Winchester, Va. 3456 3456 Four digit Ayer, Mass. 629 and 2345 629 and 2345 Combined three and four digit Jamesport, N. Y. 325 325 Three digit NATIONWIDE NUMBERING PLAN 855 two elements, a regional or area code and a central office code. Any outward toll operator, wherever located, will use that same designation in r(>aching that office through the dial toll switching network. In a sense, all of the thousands of offices involved are to be treated as though the}' were contained in one huge multi-office cit3^ Toll opera- tors will use the area code and the office code in reaching an office situated outside her own numbering plan area, while on calls to points within her own numbei-ing plan area she will dial only the number as listed for toll in the directory. In principle the method employed is to divide the two countries geographically into numbering plan areas and to give each of these areas a distinctive code. Refer to Fig. 1. Within each of these numbering plan areas each office will have a code unlike that of any other office in the same numbering plan area and also unlike any area code. Hence for toll dialing pui'poses each office will have an area code and central office code which will form a combination unlike that of any other central office in the two countries. In this geographical division into numbering plan areas, border lines between states and between Canadian provinces have generally been used as mmibering area boundaries. Since about 500 central offices are the maximum number which can be served in a numbering plan area, it is necessar>^ to divide the larger and more populous states and provinces into two or more areas making, of course, due allowance for growth. Xew York state with the largest number of central offices is divided into six numbering plan areas; Pennsylvania, Illinois, Texas and Cali- fornia have four areas each. Other divided states have three or two areas depending upon the number of offices to be served. Approximately 90 areas are being provided, with 14 states and two provinces served by two or more numbering plan areas, the remaining states and provinces by one area each. In fixing the intrastate numbering plan area boundaries of subdivided states, among other considerations effort was made to avoid cutting across heavy toll traffic routes in order to have as much of the toll traffic as possible terminating in the area in which it originated. The advantage of arranging the numbering plan areas in this manner is readily apparent since on this traffic which does not pass an area boundary the area code is not required. Let us now consider the composition of the area codes. As indicated previously they must be of a type which will enable the switching equip- ment to distinguish them from the codes of central offices. On the telephone dial plate letters are assigned only to the dial posi- tions 2 to 9, inclusive (on some dial plates a Z appears on the 0 position 856 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 but the Z is never used in a central office code), hence any office code will always avoid a 1 or a 0 in the first two places. The digits 1 and 0 can therefore be used in area codes to distinguish these from office codes. It is not practical to use them as initial digits of area codes since custo- mers dial 0 to reach operators and the local dial equipment is arranged to ignore an initial 1 for technical reasons. A 1 or 0 in the second place, however, can be employed in an area code without conflicting with an}^ central office codes or interfering with any existing practices. Accord- ingly the area codes will consist of three digits with either a 1 or a 0 as the middle digit, 516, 201, etc. A few codes of this type are now in use, leaving a practical total of 152 of these area codes available as compared to approximately 90 assigned to our present numbering plan areas. This will provide a comfortable spare for additional future numbering plan areas or possibly for reaching overseas points which may later be in- corporated into the toll dialing network. As shown in Fig. 1, states and provinces such as Montana or Alberta which are contained in a single numbering plan area will have area codes with a 0 as the middle digit to distinguish them from areas in divided states such as Texas where the middle digit will be a 1. This is to enable toll operators to differentiate between the two classifications of areas. On calls to single area states the operators will always know that every call to the state in question uses the one area code, whereas on calls to subdivided states additional information will be rec^uired to de- termine which of the several area codes should be employed to reach the particular destination. It is proposed to show on the operator position bulletin the codes of all single area states and the codes of all frequently called cities in multi-area states. The area codes of the less frequently called places in the multi-area will be obtained from a routing operator. Within each numbering plan area each of the 500 or fewer offices are to be given a three-digit office code which will be different from that of any other office code in that same area. Ultimately each central office will be given a 2-5 type of number consisting of an office name and five numerical digits, such as LOcust 4-5678, illustrated for Philadelphia. In the larger cities customers will dial seven digits, LO 4-5678, on local calls to numbers in the same exchange. In many of the smaller places the customers on local calls will dial only the numerical digits, the office name being employed for toll dialing purposes only. Considering the thousands of central offices which now have numbers other than the 2-5 type and the fact that to change existing numbering systems is a difficult and often costly procedure, it will be a number of years before this ultimate objective is realized. As a practical measure, NATION' WIDE NUMHi;i{IN(; I'I,\N 857 tluM'pfoio, it will bo iipocssary during this interim jxMiod, Ix'loi'o the centi'al office names witli the 2 ") typ(> of luimlxM' ai'e estal)lishe(l vvory- whei'e, to employ foi' oixM'atoi' loll (lialiiiu,' office codes which in many eases may not be deri\-ed tVom the customers' t(>lephone nuiubei'. Ill dialinjj; to a combined 2 4 and 2") city, f'oi- example ]>os Anfi;eles, the three-diji;it office code f'oi' the Park\va\' office which has six difiits in the local numl)er, will be PAR, wheivas to reach the Republic 2 office havinji seven digits in the local number, the office code will be RE2. To call a telephone in Winchester, \'a., with oni\- foui' digits in the local numl)er, the operator will us(> a code consisting of numerical digits oidy, such as 294 which, of course, must be different fi-om e\'eiy other office code in this numlxM'ing plan area. To secure the particular office code to be used in reaching an office where the called mmiber does not fiu'iiish complete information, the toll operator must I'efer to a position bulletin or the route operator. This reference work, of course, takes time and therefore imposes a dela}' in completing the call. In addition to giving a distinctive three-digit code to eaeh office within each numbering plan area, each toll center will also be given a three-digit code to enable outward operators to reach inward informa- tion, and delayed call operators at toll centers in distant cities. Calls to these operators will be routed in the same manner as calls to customers except that the operator codes will be used instead of a station mmiber and a toll center code in place of a central office code. The central office names now in use in the various cities in the System were chosen, generally speaking, on the basis of their suitability for customer dialing within the cit}^ itself. Many of these names are un- familiar words to operators and customers in distant cities and the use of these names contributes materially to the operator dialing errors. This situation is gradually being corrected by using for new offices, names from a System approved list and replacing existing names which experience has shown to be particularly troulilesome by names from this list. While numbering plans are important in operator toll dialing, they play an even more essential part in the dialing of toll calls by customers. ( )i)erators can be trained to adapt their dialing procedures to the type of local numbering system encountered in the called city even though more time is consumed and more errors result than would be the case if all telephone numbers were of a uniform type. Customers, however, could not be expected to follow any plan which reciuires a \'ariety of different procedures to be used in reaching different cities. Only a num- bering system which is readily understandable and w^hich customers find 858 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 convenient to use and one which they can use with a very high degree of accuracy will suffice. The need for accuracy is readily apparent since with the customer's telephone being given access to the intertoU network without the intervention of an operator, a call which is misdialed can be routed to a telephone thousands of miles from the desired destination. At present customer dialing of toll and multi-unit calls is for the most part confined to situations where the call can be completed by the use of the number as listed in the directory without any additional digits being dialed. In a few cases as from Camden, N. J. to Philadelphia and certain offices in Northern New Jersey to New York City, the code 11 is prefixed to the listed number. In the case of the current trial of cus- tomer toll dialing at Englewood, N. J., the customers are using area codes such as 415 for Oakland, Cahfornia, 312 for Chicago, etc., diaUng only into those cities which now have the 2-5 type of numbering. From the Englewood experience it can be confidently predicted that this form of dialing, i.e., an area code followed by a telephone number consisting of a uniform number of digits, is one that customers will use with a reasonable degree of convenience and accuracy. The problem therefore to meet the requirements for nationwide customer toll dialing, is to establish universally for all central offices regardless of size and loca- tion a uniform pattern of numbering for toll purposes. The only form of number completely filling the needs is the 2-5 system, which is that used in the largest cities today. Accordingly, in order to implement the program for customer dialing of toll calls on a nationwide basis, it will be necessary to place all tele- phone numbers on a 2-5 basis with the code of each office different from that of every other office in the same numbering plan area. Thus each of the 50,000,000 telephones in the United States and Canada will have, for toll dialing purposes, a distinct identity consisting of ten digits; a three- digit area code, an office code of two letters of an office name and a numeral, and four digits of the station number within the office. Typical numbers for toll dialing would therefore be 601-CA3-4567 or 317-MA7- 6789. As with operator toll dialing, on a toll call which terminates in the same numbering plan area in which it originates, the area code will be omitted and the office code and station number^ — a, total of seven digits will be used. With this universal 2-5 type of number, local calls in and about the larger and medium sized exchanges will be completed by dialing the entire seven-digit number. For many of the smaller places in the more isolated sections, 5-digit or 4-digit dialing will frequently be employed where this number of digits will be adequate for all of the telephones Fig. J— Nationwide loll ili;iling luciis in tho UTiito niarkei'. The woi'k time of the decoder has becMi in {\w order of a half second. The markei- delei-mines tlie identity of the frames on which the in- comiiifi; and outjj;()injj; circuits are located, finds an idle path between the two circuits and sets up the connection. After checking the path through th(> switches to be sure that there are no troubles it Jiotifies the sender that its task has been completed and then leaves the connection. Its work time has also been in the order of a half second. In th<> meantime other digits have been coming in to the sender but it does not wait foi' all of them to arrive before advancing the call. When the marker selected the circuit to New York a signal was immediately sent foiward to summon a sender in the New York switching system. The process of attaching the sender in New York was carried on con- currently with the establishment of the coimection through th(> switches in Atlanta. When the New York sender is attached a signal is sent to the Atlanta sender to advise it that pulsing may proceed. It immediately sends the area code 207 to New York by MF pulsing and follows it with the remaining digits of the called number, AC4-2345, as they are received from the operator, ending with a start pulse, and then leaves the connec- tion. All common control equipment in Atlanta is now free. In New York, as soon as the Maine area code is received it is submitted to the decoder. Upon examination of the code the decoder finds that it is insufficient for routing purposes. New York has a direct circuit group to Portland over which traffic to some offices in Maine is routed, but other offices are reached through Bangor by Avay of Boston. In order to determine which route to take the decoder must know what office is desired. It, therefore, gives the sender a signal saying "come again when you have six digits" and leaves the connection. When the sixth digit arrives the sender again calls for a decoder and gives it the complete code 207-AC4. The decoder again translates the area code, which now directs it to the foreign area translator which serves the Maine area, and submits the complete code to that translator. From the ensuing translation it learns that the route is by way of Boston and that all digits should be sent forward by MF. It then calls for a marker and releases the foreign area translator. Subseciuent operation is the same as previously described for Atlanta and the complete ten-digit number now arrives at Boston. At that point 868 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 both codes are again translated since Boston also has a choice of routes to Maine, and the route to Bangor is selected. The translating equipment in Boston knows that Bangor is in the Maine area and that the area code will, therefore, not be needed. However, since Bangor is a TO having no senders, the Boston sender must pulse forward all of the digits needed to complete the call through switches in Bangor, Houlton and Monticello. It is assumed that Houlton is arranged to route the call to Monticello on receipt of the digits AC4. Numerical digits 2345 will route the call through the Monticello switches to the called customer's line. These digits are all registered in the Boston sender but the digits required to switch the call through Bangor are not and must be supplied. An arbitrary set of digits beginning with "1" can be used for this purpose since no office code begins with "1" and there will, therefore, be no conflict. The decoder in Boston, therefore, gives the sender the proper set of arbitrary digits, say 16, to be placed ahead of the office code AC4. The sender sends forward by the DP method 16-AC4-2345 driving switches in Bangor, Houlton and JMonticello to the called subscriber's line, and ringing starts automatically. The talking connection is now established and the common control equipment at all intermediate points is free. When the called subscriber answers, the Atlanta operator's cord lamp is extinguished. When he hangs up the lamp lights to denote end of conversation. The removal of the operator's cord automatically releases the entire connection, the release of each link causing the next in line to release. In setting up this call all of the characteristic CSP features were em- ployed, automatic alternate routing in Atlanta, six-digit translation in New York and Boston, digit storing and variable spilling at all CSP's with substitution of arbitrary digits for the area code at Boston. TRANSMISSION All talking connections through the CSP system are made on a four- wire basis, that is, separate pairs of conductors are provided for trans- mission in the two directions. This is done in order to simplify the problem of maintaining satisfactory balance so that the loss introduced by extra links in a connection can be held to a minimum value. The importance of this feature is emphasized hy the fact that the switching plan permits as many as eight intertoll trunks to be connected in tandem for the completion of a call. The advantages of four-wire switching were fully explained in the paper on the toll crossbar system now in service. AUTOMATIC TOLL SWITCIIIXr, SYi^TEMR (SCO SIGNALING III following the progress of the call from Atlanta to Monticello, Maine, it was observed that besides the transmission of information in the form of digits it was necessary to pass a number of control and supervisory signals o\'er the toll lines. These included seiziu'e and disconnect signals in the forward direction and switchhook supervisory signals and sender attached signals in the reverse direction. On some calls it is also necessary to send flashing signals to indicate busy lines or trunks and ringing signals in either direction when operators are called in at intermediate or termi- nating points to assist in establishing connections. For the early toll dialing installations the signaling method most widely used was the composite method whereby signaling channels for the three circuits of a phantom group are derived from three of the conductors with the fourth being used for earth potential compensation. Hirect current is used for signaling. This is a simple, reliable and eco- nomical method of signaling and will continue to be used on circuits where it can be applied. Where circuits are obtained from carrier systems, however, conductors are not available in sufficient numbers for signaling channels and other methods must be employed. Since carrier is used almost exclusively on the long haul circuits it was necessary to provide a signaling system to accompany it before toll dialing could be expanded beyond networks of limited range. To meet this situation a system using a frequency of 1600 cycles was developed and has been in service since 1948. Signaling is done by application and removal of the 1600-cycIe signaling current. The system is used in the same manner as the composite signaling system, to carry dial pulses as well as supervisory signals when used on circuits that require it. The set of leads brought out of the signaling unit are identical in function to those brought out of the composite signaling unit so that toll line relay circuits will operate in the same manner with either type of signaling. Since 1600 cycles is in the voice range the signaling current can be carried over the same channel that carries the speech current but the signaling circuits must, of course, be protected against false operation due to speech and precautions must likewise be taken to insure that the signaling tone does not interfere with speech. Protection against inter- ference between signaling and speech is more difficult at 1600 cycles than at higher frequencies because there is more energy in voice ciu'rents at the lower range. That value was chosen, ne\'ertheless, so that it would be possible to operate over the narrow band circuits that were estab- lished to relieve shortages occasioned by the war. 870 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 A new 2600-cycle system to be used only on the broader band circuits has since been developed. It is simpler and more economical than the 1600-cycle system. The older carrier systems, having been designed when practically all toll operation was by the ringdown method, made no provision for signaling since that was all done by short applications of the 1000 cycles when there was no speech on the line. Some of the new carrier systems for short haul applications are designed to provide their own signaling channel for each voice channel. PRINCIPAL ELEMENTS OF THE CSP SYSTEM 1. Crossbar Switch Frames Crossbar switches are used for incoming and outgoing link frames on which the trunks (both toll lines and trunks to and from local offices and switchboards) are terminated, and for sender link frames used ^o give trunks access to senders. These frames are similar to those in the toll crossbar systems now in service. Since they have been described in a previous paper they will be passed over with only a mention of their capacity. Each incoming or outgoing link frame normally has terminals for 300 trunks. As many frames are provided as required for the size of the office. In the smaller offices one train of switches with complete interconnection of incoming and outgoing frames is pro\nded. In the larger offices two trains each with its own set of markers are provided. When this is done the incoming trunks are multipled to both trains and an extra build out bay is provided on the incoming frame to provide 400 terminals per frame. Since each train has a theoretical limit of 40 incoming and 40 outgoing frames the maximum size of an office is theoretically 80 of each. Practical considerations, however, such as the number of markers that can be efficiently operated in a group and the maximum size office it is feasible to operate as a single administrative unit will limit an installa- tion to about 60 incoming and 60 outgoing frames. The sender link frame gives 100 trunks access to 40 senders. 2. Senders Two separate groups of incoming senders are provided, one to receive DP and the other MF pulsing. Whether the system is installed in a step-by-step or a panel-crossbar area both groups of senders will always be needed. MF will be received from senders in other CSP's and from switchboard positions. DP will be received from switchl>oard positions AUTOMATIC TOLL SWITCIIIXr, SYSTEMS S7 1 at TCs not 0(iuippe(l to send MF and in sonic cases from dialinj; A boards in the local area of the C'SP itseli". Aside from the type of pulses received the functions of the two sendeis are identical. They have a capacity for receiving and sending eleven digits. They must register arbitrary digits given them by the decoder and send them out as directed. They will send out digits by cither the DP or the MF method as reciuired to control switches in distant offices, and in some installations will also send digits to an outgoing sender in the same office by the dr key pulsing method, which employs direct current in xarious coml)ina1ions of \alue atid j)oIarity through a pair of conductors. When the CSP is in a panel-crossbar area a group of outgoing senders is pi'ovided to transmit either the type of pulses re(iuired by the eciuip- ment in local panel offices or the type used to reach manual offices. 3. MarLrrs The marker has been stripped of its usual translating functions and performs most of its duties on instructions from the decoder. It is told what leads to test for idle circuits and where they are to be found in the trunk block connector, but having found an idle circuit it carries on the process of setting up the coiniection independently of the decoder. Having contact with both the incoming and outgoing trunks through connecting circuits, it determines what frames they are located on, con- nects itself to those frames, selects a path through them and sets up the connection. In a single-train office one group of markers common to the office is provided. In a two-train office there is a group of markers associated with each train of switches. A single group of decoders serxcs the entire office whether one or two trains of switches are provided. An impoi'tant element of the decoder is the translator which will be discussed separately. The decoder contains several hundr(>(l r(>lays. A large gi'oup is used for registering the information furnished by the translator. Others use this infoi'mation to control the action of the markers and senders. One group of decoder relays which is of particular interest is the array used for automatic selection of alternate routes. It is composed chiefly of one relay for each C'SP to which the office has a direct grouj) of toll lines. The relays are arranged in an orderly pattern sinudating the 872 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 pattern of the CSP network for the country as seen from the CSP concerned and are interconnected in a pattern of progression correspond- ing to the fixed order of alternate route selection. Group busy leads from the toll line groups are connected to the contacts of the relays in such a manner that if a group is busy the relay corresponding to the next choice route in the chain will be operated. In this way the lowest choice route having an idle circuit will be speedily selected without testing individual trunks of separate groups. The decoder learns from the translator which relay in the array to operate first and the choice of the best route a\'ail- able follows automatically. The principle will be readily understood by reference to the simplified sketch in Fig. 3. Contacts not shown on the relaj^s cause the translator to select the route corresponding to the last relay operated in the chain. 5. Translators The magnitude of the translating job for nationwide dialing led to the decision to develop a new translator operating on a principle radically different from that employed in other crossbar systems. In previous systems translation is done by relays. The code digits - never more than three - operate a group of relays which cause a single terminal corre- sponding to the code to be selected. A cross-connection is made between HIGH-USAGE ROUTES FINAL ROUTES ALL TRUNKS BUSY INDICATION I -INDICATION FROM TRANSLATION OF THE FIRST ALTERNATE ROUTE '-INDICATION THAT ALL TRUNKS IN THE GROUP ARE BUSY Fig. 3 — Alternate route array for the decoder at a sectional center. AUTOMATIC TOLL SWITCHING SYSTEMS 878 Fij£. 4 — Cai'd t fuiishitor- the code point and a route relay associated with the trunk group to be selected. The route relay has a number of contacts which are cross- connected to supply the information required for proper routing of the call. When changes in routing or equipment location of trunks within the office are made it is necessary to change cross-connections. With the nationwide dialing plan in operation routing changes or opening of new offices in one part of the country will necessitate trans- lator changes in many offices, some of them far removed from the scene of the event that forces them to be made. The changes in any one CSP will, therefore, be frequent and to make them by running cross-connec- tions would be cumbersome and expensive. The new translator uses punched cards instead of relays, making it possible to effect changes by the simple process of removing old cards and inserting new ones in a machine. This can be done in a very short time and not only saves labor but requires less out-of-service time for the equipment. Fig. 4 is a photo- graph of the machine. A metal card about 5 by lOf inches is provided for each area code and also one for each office code that must be translated in a particu- 874 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 lar C\SP, the cards representing destinations. The capacity of a single machine is about 1000 cards. The cards are lined up in a box as in a liUng drawer, with tabs along the bottom of the cartl resting on select bars which run the length of the box. One-hinidred and eighteen holes are punched out in all cards in fixed positions so that in the normal condition 118 tunnels are formed from one end to the other. A light source at one end of the box shines through the tuiniels upon phototransistors (Fig. 5) at the other end but the phototransistors are disabled until, concurrently with the dropping of a card, voltage is applied to them. All tabs along the bottom of the card are cut off except those which serve to identify the particular card. When a code is presented to the machine a combination of select bars corresponding to the code is Fig. 5 — Transistor. AUTOMATIC TOLL SWITCIIIXG SYSTEMS 875 lowcfcd. 'The caid h;i\in (h'oppinj^ot' the card would cnt off all lifi;ht channels hut on each card some holes luv. eniard and throufjh tliese holes the lifiht continu(>s to shine, enerf>;izinfj; the corresponding pliototi'ansistors. The ('()ml)ination of enlarged holes furnishes all of the information needed for I'outing the call to the destination represented by the card. Fig. (3 shows the functions of th(> \-ariou.s groups of tabs and holes. The designations will not appear on the actual card. Fig. 7 is a pholo- gi'aph of an actual card prepared foi' us(v a. Seh'ctiiuj Tabs - Input Information. The sole use of the information presented to the card translator is to enable it to select the proper card. The information presented is in the form of code digits with accompan}^- ing indications of their nature. The information is recognized by cutting off tabs along the bottom of the card in proper combinations. The groups of tabs labeled A, B, C, D, E and F are for the six code digits. For each digit used two tabs are left since the digits are registered in the sender on a two-out-of-five basis and the leads from the sender will operate the select bars directly. If the card represents an ordinaiy three-digit code all tabs will be cut off except two each of the A, B and C tabs, two of the four CG tabs and perhaps either the VO or NVO tab. The CG (card group) tabs are used in combination to indicate three- digit, six-digit and alternate route card groups. The VO and NVO (via only and not via only) tabs are used when the group of toll lines over which the call will be routed is divided into one subgroup of a trans- mission gi-ade suitable for only terminal traffic and another subgroup for either terminal or switched traffic. If the card represents an ordinary six-digit code two tabs will be left in each of the digit posiTions, and a different pair in the CG group. h. Punch Holes -Output Information. The output information from the card translator is recognized in the decoder and marker by relays operated in the combinations set up by enlargement of associated holes in the card. The output from the phototransistors is amplified by other transistors to fire cold cathode tubes which in turn operate the relays. The pretranslation group on the top line of Fig. 7 indicates how many digits the sender must supply for a complete translation. The term ''pretranslation" implies that fvu'ther translation is reciuired. This is not always true. In many ca.ses only the first three digits need to be translated and all information needed for routing the call is supplied by this card. In many cases the six digits of the area and office code are needed and the routing information will be on another card to be selected 876 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 in ID §D iD 'D §D ,§D ID iD ■^D "D iD D ID D D D D D D ID ID 0 ID I ID: iD' ID ID„ m iDi «i D D D D iD iD^ iD iD iD ^D §D §D iD sD iD ■sD iD lU iD ID iD HD ?D ^D ^D ,?D ID iD ID ^D ID §D §D ID sD sD ID^ ^D D^ iD iD ID ID ID ID iD sD Fig. 6 — Card layout. OAN OA Al'TOMA'PIC TOT.L SWITCH! \(; SYSTEMS 877 Fig. Punched card. later. For certain calls such as calls to operators only four or five digits are needed. These are treated as six-digit calls by having the sender suppl}' the extra one or two digits to fill the complement. The NCA hole enlarged means "no come again", that is, three digits are sufficient, and translation will proceed. The other holes enlarged mean respectively "come again when 3'ou have four, five or six digits", and no fiu'ther translation is done until the sender comes back a second time, prob- ably to a different decoder, with an indication that it has the required number of digits. The OGT appearance holes are used in a two-train office to tell which train the outgoing tmnk appears on and enable the decoder to select a marker in the proper group. The remaining holes on the top lines are for controlling operation of traffic meters. The translator box number holes in the second line arc punched on the area code cards to indicate which machine contains the indi^•idual cards for the called area when six-digit translation is required. The INDl hole in the second line and the IND2 hole in the fourth line are index holes and are never enlarged. Any card that drops will always cut off the light through those channels. This serves as an indica- tion that a card has actually dropped and that the phototransistors a.ssociated with the other holes should be prepared for action. The index holes also aid in trouble detection and in proper disposition of calls where cards are deliberately omitted or wliere operators have dialed a blank code in error. 878 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 The class holes indicate such things as type of pulsing and nature of the signaling channel used on the trunk group out of the office. The area code control holes in the third line are to tell the decoder what to do al)out dropping or spilling forward an area code registered in the sender or supplying an area code when none is registered. This information is needed primarily in corniection with alternate routing. The alternate route pattern number holes tell the decoder at what point to enter its chain of alternate route relays for the first choice alternate. Provision is made for a maximum of 100 entry points. The holes on the fourth line are for making proper disposition of calls when no circuits are available on any routes, telling how many digits to expect on certain calls and other items of a detailed technical nature. The code conversion holes on the fifth line supply the arbitrary digits to replace code digits on calls routed through step-by-step TO's or TC's. Provision is made for one, two or three digits as required. The variable spill control holes in the sixth line tell whether to spill all digits received, skip the first three code digits or skip six code digits. The remaining holes define the location on the equipment of the test leads for the trunk group o\'er which the call will be routed. The notches around the edges are used for proper positioning and re- moval of cards. An individual card is removed from the stack by first keying the code to cause it to drop so that it may be identified. Since a card can easily be located in this manner it is unnecessary to keep cards in any ordered position in the box. At least one translator is provided in every decoder. It contains the cards for all offices in the home numbering area of the CSP, for certain operator codes, the single three-digit card for each toll numbering area and a card for each toll line group out of the office that can be used as an alternate route. If there are other areas to which the volume of traffic is very high and for which six-digit translation is required the cards for those areas are put in a second machine in each decoder. Cards for other areas are put in foreign area translators common to the office and acces- sible to all decoders on a one-at-a-time basis. An emergency translator is provided to permit removal of all cards to it from any translator which may require prolonged repair work. 6. Traffic Control Panel The traffic control panel is located in the operating room. The ecjuip- ment in it consists of a key for each group used as an alternate route. When a particular key is operated no alternate routed traffic will be AUTOMATIC TOLL SWlTt'Hl.NG SYSTEMS 879 offered to the group represented by it nor to any group ul)()ve it in the fixed alternate I'oute pattern. This is done to relieve offices which are overloaded by either untoi'eseen or predicted traffic peaks. MAINTENANCE The maintenance facilities for the new C'SP system are basically similar to those of the older toll crossbar system with the lUM-essary ad- dition of e(iuipment to test the new features introduced. The send(M' test framc^ is, of course, ol)liged to test the ('8P ieatures added to the sender and \\\v trouble indicatoi' frame is changed to operate with the new decoders, translators and markers. In place of the lamp trouble indicator the new trouble leccjrdei' in- troduced with the latest local crossbar system is used. Whene\'er trouble is encountered it punches on a card a record of the circuits involved and of the important events that had occurred in the progress of the call, as an aid to the maintenance man in locating the trouble. A sample ti'ouble recorder card is shown in Fig. 8. Automatic eciuipment for testing the operation and transmission fea- tures of intertoll trunks has also been designed both for the older sys- tems and for the new CSP system. SWITCHING ASPECTS OF CUSTOMER TOLL DIALING In the course of developing the switching system for CSP's the re- quirement for handling long distance traffic dialed by customers as well as that dialed bj^ operators was kept in mind. The trial of long dis- tance customer dialing now in progress in Englewood, N. J., confirms the soundness of the basic plan and exemplifies the principles involved in full realization of the plan. With a toll network laid out to accept a distinctive ten-digit number for any telephone in the country and route it to the proper destination, the remaining tasks to be performed are to provide for delivery of the number to the toll netw^ork from the cus- tomer's dial instead of from an operator and to provide an automatic record of the call for charging purposes. In Englewood both tasks were quite easily performed. The Englewood local office eciuipment is of the most modern type and includes AMA* facilities. When it was in the development stage the ultimate reciuirement for nationwide customer dialing was foreseen and provision was made for expanding the digit capacity of the switching equipment at small expense. Also the designs of the accoiuiting center were such that corre- sponding changes could readily be made. In the new local office switching 880 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 system, arrangements were included for sending forward the complete number, as received, to the toll office by MF pulsing. The system was also designed to be capable of automatic alternate routing and this feature is used in the trial. Expansion of the program will, of course, demand that similar ar- rangements be provided for the older types of local switching systems already in service. More extensive modification will be required to make them capable of giving the customer the same service. For them, as for the most modern system, however, AMA equipment is admirable for recording the information necessary for charging for the calls. The requirement for customer toll dialing that senders (or directors) and recording equipment be provided has a bearing on the type of equipment used at TC's and TO's. For calls handled by operators and for calls received by the customers through such offices the only disad- vantage of step-by-step ec^uipment without senders at those points is that the CSP equipment at other points must be somewhat more com- plicated and expensive than it would otherwise need to be. But with customer dialing, if senders and recording equipment are not provided either in the local office or in the TC or TO, the calls must be routed by the most direct means possible to a CSP where such equipment is pro- vided. Thus some advantages that might be gained from having them at the TC or TO would be lost: 1. In some cases an indirect route to the CSP would need to be taken for the sole purpose of recording the call. For example, a call which might normally be switched from a TC through a TO to another TC would need to be connected to the CSP for making the record. ^■^^^^^^■1 ■^^H 0 5 10 IS JO ?5 11 ▼ I 0 M c Dr°Ml'l°v°ci 1 Rl"lR2 «.«,. ■■■- 1 fIF MfT CFROSTIMSTt HTRF T5T 5DT OCT 0 1 2 3^4' "5 '"s" 7 B sL 1 2 ("""V'a 7 A 1 DO P«0 1 0 1 3 3 4 i « 7 8 9 D , 2 rrri , . Jo-rMo""," M EU TO 11 UO 1 2 3 ■,"°5 6 7 6 U9 TOT, T2UD , —rrr-T , J.. , 1 AO 1 ? 4 «7 eo . 2 4 B7 CO 1 2 4 C7 DO 1 2 4 D7 ED r 2 4 B7 CO 1 2 4"c"*Do' 1 2 4 D? EO 1 2 JD 60 60« VO NVO "bo° °ck" 'c FMPF T5A TSB ISC SBdI lV'i" "l "Va 1 IvONVOCCo" ° '"2°"'cM Icsi . «» R«, RAJBMCSOGSr CSJG53C S4 CSS CO CI C2 C3 CB RtS «9 RO ROIT ROTC 1 ""r5°"rs'3 'of"ubr 1 NCA CAA CAbCAsi IT TC lic| 0 ' 2|tpc|o 7 2|h to 'tV"uo'"V'°2 4 tItO 1 UO 1 2 4 COLcInAC AC AHA ATaI TO I ? ' 4 T7 UO 1 2 4 U7 1 1 0 °i" '"2 '" 1 CO.. co...«.o~ 1 INSK S«5 5«6| MN TN UN HO 1 2 4 H7 TO 1 2 4 T7 UO 1 7 A 1 2 4 u;|o 1 2 4 7|tO Tl U0"l' 2 4 U7 1 TO Tl lo" 1 '"2 4 cc CR r°r"o°f"f;°b''f'mfs"t"°" '""PCRRPCR Inn TN UN HO ;"'-" •'-'^;'''^°-^""^" """'"uo 1 MB RO RRO FOF FUB f'r'o'f" h[o' ooc co..i.i.(.» U DC MF 5>D IFD >0D ISO OLC " !°c""fd'Vc' CKCHTRSMISMC C«!CCK CB«I1CT DHL RLT IchkI T TC RCRRUE RCD RCA DCBDCB2AT8CPLARS ICDT.SRDRl B !G n'KK« OFK IFK 'iV'liRito™ UI 8 OSCMRL «L |cL« ClB CIC COa""a m7 h"b'"t ""',a"\i. UB TSA OGA I^^HI^^H ^^B Fig. 8 — Card for the new trouble reco AUTOMATIC TOLL SWITCHING SYSTEMS 881 2. There is no operator at the TC or TO to select an alternate route and with the equipment there incapable of automatic alternate routing the economies and service protection inherent in the alternate routing procedure would be lost. If step-by-step toll switching equipment is already provided at a TC or TO, senders (or directors) could be added, making it in effect a common control switching system. This measure would permit auto- matic alternate routing and the further addition of recording equipment would eliminate the indirect routings for recording purposes. A further benefit from having common control equipment in TC's or TO's can be realized in some instances. When a customer is served by a local office that has no senders he must dial one or more directing digits (probably three digits) ahead of the seven or ten-digit number in order to get to an office where senders are provided. It is, of course, desirable to avoid this extra burden on the customer. Where the equipment in a TC or TO can be used in common for switching local and toll traffic the customers whose lines are terminated in that office will be dialing directlj' into senders, if the equipment uses common control, and will, therefore, benefit in that they will not have to dial directing digits. CONCLUSION The new system was designed to implement the nationwide switching plan, which integrates the switching network of the entire nation into a single unit. This switching job, requiring a high order of mechanical intelligence, is the most comprehensive ever performed by any system. The skillful manipulation of code digits enables the provision of a ;•"_:■ c». 601 i«B cf r«L Wi Bit 1 «i fto uo iMl i« lui im; t«j . 15 IPi TB 51 US « 0 I JP1LS01.5JS SM SMI SMO TL nC«TKS IR IB,. 5I» U«L HF lOf 10 1 T^l lo 1 ^ < I 0 1 2 J 4 S 6 J « s 10 " IJ 1! 1« IS 16 1' It i« '""'-"•"' '■°"' 1 « alo l" "T's 0 1 ; 3 4 5 6 I !? 10 II 12 ""'a' """'l 1? H 19 20 2. 22 2 J 24 25 26 27 28 11 » 11 !2 !5 J4 )S )6 57 3« )9 1 E o\t 0 | t" o" °C '"o"" T T" CNOCNt ICBOCBIte C»B «TA BTBDIC RJO 0 12 3 4 5 6 7°""«°'"V"To' '""'2 11 14 IS 16 17 It l? 0 1 2 3°''4"'5'"6 7 1 9 0 1 2 5 4 S 6 I t 9 10 11 12 IJ 14 IS 16 17 u 19 10 1 2 3 4 5 6 7 8 19 »0 1 2 3 4 5 6 7 t 119 «»C01 llo 1 ?345lo 1 234slo 1 24 /lo 1 2471 0 12 3 4 5 17 8 9 012)«56»B9 « S C 0 ( 0 1 2 14 log OL ri f» C» CIO tLl OT PC CiIC^OJC P« PB 5A SB SC HO HU Ici PX°5I M.lcil < B"c°t« A« Nf I« 107 oduced with the latest local crossbar system. 882 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 numbering plan covering the entire country with a minimum number of digits to give each customer a distinctive number. It also obviates the need for extra expense to make step-by-step toll offices satisfactory operating elements of the plan in those locations where CSP features are not essential. The automatic and almost instantaneous selection of alternate routes makes it possible to give ^'irtual no-delay service without greatly in- creasing the cost of outside plant and to make multi-switch connections at a speed comparable to that for local service. The translating equipment simplifies administration of the plan which demands coordination of activities on a nationwide basis. The numbering plan, the switching plan and the CSP equipment which implements them make it feasible to offer nationwide dialing service to customers without the aid of operators when automatic charging facilities and local office switching arrangements for handling the three extra digits of the national nimiber are provided. It will be readily ap- preciated that so far as the CSP switching equipment is concerned it is immaterial whether the digits it receives come from an operator or from a customer. The call will be routed to its destination and .supervision for charging purposes will be furnishes in the same manner in either event. The new system represents an important step in the process of con- tinually improving the long distance switching methods of the Bell System with consequent improvement of the service to all telephone customers in the United States and Canada. REFERENCES 1. J. J. Pilliod, "Fundamental Plans for Toll Telephone Plant," pp. 832 of this issue. 2. L. G. Abraham, A. J. Busch and F. F. Shipley, "Crossbar Toll Switching System," A.I.E.E. Transactions, 63, June Section, pp. 302-309, 1944. 3. C. A. Dahlbom, A. W. Horton, Jr. and D. L. Moody, "Application of Multi- frequency Pulsing in Switching," A.I.E.E. Transactions, 68, June Section, pp. 505-510, June 1949. 4. W. H. Nunn, "Nationwide Numbering Plan," pp. 851 of this issue. 5. F. J. Scudder and J. N. Reynolds, "Crossbar Dial Telephone Switching Sys- tem," A.I.E.E. Transactions, 58, May Section, pp. 179-192, 1939. 6. N. A. Newell and A. Weaver, "Single Frequency Signaling for Telephone Trunks," Presented at Winter General Meeting of A.I.E.E., Jan. 31, 1951. 7. F. A. Korn and J. G. Ferguson, "The Number 5 Crossbar Dial Telephone Switching System," A.I.E.E. Tranasctions, 69, First Section, pp. 233-254, 1950. 8. J. Meszar, "Fundamentals of the Automatic Telephone Message Accounting System," Presented at the Winter General Meeting of A.I.E.E., Jan. 31, 1951. Mathematical Theory of Laminated Transmission Lines — Part I By SAMUKL P. MOlU; AN, JR. .1 inatlu'niadcdl (inulysis is given of the low-losa, broad-bdnd, UuninaUd transmission lines 'proposed by A. M. Clogston, including bulk idealized parallel-plane lines and coaxial cables. Part I deals with ''Clogston i" lines, which have laminated conductors with a dielectric, chosen to provide the proper phase velocity for waves on the line, jilUng the space between the conductors. Part II will treat lines having an arbitrary fraction of their total volume filled with laminations and the rest with dielectric, and will be concerned in particular with '"Clogston ^" lines, in which the entire propaga- tion space is occupied by laminated material. The electromagnetic problem is first formulated in general terms, and then specialized to yield detailed results. The major theoretical questions treated include the determination of the propagation constants and the fields of the principal mode and the higher modes in laminated transmission lines, the choice of optimum proportions for these lines, the calculation of the fre- quency dependence of attenuation due to the finite thickness of the laminae, the increase in loss caused by improper phase velocity (dielectric mismatch) in Clogston 1 lines and by nonuniformity of the laminated material in Clogston 2 lines, and the effects of dielectric and magnetic dissipation. TABLE OF CONTENTS I. Introduction 884 II. Wave Propagation Between Plane and Cylindrical Impedance Sheets. 887 III. Surface Impedance of a Laminated Boundary 89G IV. Principal Mode in Clogston 1 Lines with Infinitesimally Thin Laminae 908 V. Effect of Finite Lamina Thickness. Frequency Dependence of Attenu- ation in Clogston 1 Lines 921 VL Effect of Dielectric Mismatch 931 VII. Dielectric and Magnetic Losses in Clogston 1 Lines 940 Appendix I : Bessel Function Expansions 944 Table of Symbols 946 883 884 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 I. INTRODUCTION A recent theoretical i)aper' by A. M. Clogston presents the very interesting disc(j\ery that luider certain conditions skin effect losses in the conductors of a transmission line at elevated frequencies can be much reduced by laminating the conducting surfaces, parallel to the direction of current flow, with alternate thin layers of conducting and insulating material. The requirements are that the thickness of each conducting layer must be considerably smaller than the skin depth in the conductor, and the phase ^•elocity of waves on the transmission line must be held very close to a certain critical value, which depends on the relative thicknesses and the electrical properties of the conducting and insulating layers. Under these conditions the "effective skin depth" of the laminated surface is greatly increased; in other words, the eddy cur- rents induced by a high-frequency alternating field will penetrate much farther into such a laminated structure than into a solid conductor, with consequent marked reduction of ohmic losses in the metal. The metal losses can also be made to vary much less with fretiuency, over a fixed band, than the ordinary skin effect losses, which are known to be very nearly proportional to the square root of frequency. Clogston goes on to show that a laminated material composed of alternate thin conducting and insulating layers may itself be regarded as a transmission medium. For example, if the space in a coaxial cable which is ordinarily occupied by air or other dielectric be filled with a large number of coaxial cylindrical tubes which are alternately conduct- ing and insulating, the cable will propagate various transmission modes, and under the proper circumstances some of these modes will exhibit lower attenuation constants than the transmission mode in a conven- tional coaxial cable of the same size at the same frequency. Experimental verification of Clogston 's theory of laminated conductors has been obtained- at the Bell Telephone Laboratories, and the trans- mission properties of a line filled with laminated material have also been measured at these Laboratories and found in reasonable agreement with theory. However experiments with structures as complex as those pro- posed by Clogston are by no means simple, and the experimental work on laminated conductors is still in an early, exploratory stage. Inasmuch as the experiments are necessarily time-consuming, it has been thought 1 A. M. Clogston, Proc. Inst. Radio Enqrs., 39, 767 (1951), and Bell System Tech. J., 30, 491 (1951). References will be to the Bell System Technical Journal article, although except for equation numbers the two papers are identical. 2H. S. Black, C. O. Mallinckrodt, and S. P. Morgan, Jr., Proc. Inst. Radio Engrs., 40, p. 902 (1952). LAMINATED TRANSMISSION LINES. I 885 desirable to carry out simultaneously as complete a theoretical treat- ment of Clogston-type transmission lines as possible. Clogston's original paper brought out the fundamental ideas by analysis of idealized trans- mission lines bounded by infinite parallel planes. The ])resent paper con- siderably e.xtentls the theoretical analysis of pai'allel-plane systems, and also treats laminated transmission lines l)ounded by coaxial circular cylinders, which are of course the structures of practical engineering interest. Part I of this paper deals with both plane and coaxial lines having laminated conductors and ha\'ing the space between the conductoi'S filled with a suitable main dielectric, which may so far as the theory is con- cerned also be a nonconducting magnetic material. Structures of this type are called "Clogston 1" transmission lines. Although in principle the total space may be divided between the main dielectric and the laminated stacks in any desired ratio, we suppose in Part I that the width of the main dielectric is several times the total thickness of the laminations. When this is true, the principal mode fields in the main dielectric are almost identical to the fields of the transverse electro- magnetic (TEM) mode between pei'fectly conducting planes or cylinders. The phase velocity is controlled by the properties of the main dielectric, while the attenuation constant is determined by the surface impedances of the laminated boundaries (and the dissipation, if any, in the main dielectric). The calculation of the surface impedance of a laminated plane or cylindrical stack is reduced, using the generalized impedance concept de\'eloped by Schelkunoff, to the calculation of the input impedance of a chain of transducers with known impedance elements, the chain also being terminated in a kno^\^l impedance. We are thus able to employ the language and the I'esults of one-dimensional transmission theory to solve our three-dimensional field problem. In the remaining sections of Part I we introduce various simplifying appi'oximations and special assumptions into the general equations in order to obtain simple and explicit results. We first calculate the propa- gation constant and the field components of the principal mode under the assumption that the individual conducting laminae are extremely thin compared to the skin depth at the operating frequency, and show that the attenuation constant is substantially independent of frequency so long as this assumption is valid. We then give formulas for the reduc- tion of the effective skin dejith in the stacks and the consec^uent increase of attenuation with frequency when the laminae are of finite thickness. Xext we investigate the efTect of varying the phase velocity of the line away from the optimum value given by Clogston ; and in the last section 886 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 we discuss losses due to imperfect dielectrics and lossy magnetic ma- terials. Part II will be largely devoted to transmission lines of the so-called "Clogston 2" type, in which the entire propagation space is filled with the laminated medium, though to a lesser extent we shall also consider transmission lines having an arbitrary fraction of their total volume filled with laminations and the I'est with dielectric. We shall first derive expressions for the propagation constant and the fields of the lowest Clogston 2 mode assuming infinitesimally thin laminae, so that the attenuation constant is essentially independent of freciuency, and then go on to investigate the transition of the lowest Clogston 1 mode into the lowest Clogston 2 mode as the space occupied by the main dielectric is gradually filled with laminations. We shall also discuss the higher modes which can exist in Clogston 1 and Clogston 2 lines with infinitesi- mally thin laminae. Next the effect of finite lamdna thickness on the variation of attenuation with frecjuency in a Clogston 2 will be investi- gated, and then the important cjuestion of the influence of nonuni- formity of the laminated medium on the transmission properties of the line. We shall conclude with a short section on dielectric and magnetic losses. Insofar as possible, plane and coaxial lines will be treated together throughout the paper. Since however Bessel functions are not so easy to manipulate as hyperbolic functions, there will be a few cases where explicit formulas are not yet available for the cylindrical geometry. In these cases the formulas derived for the parallel-plane geometry usually provide reasonably good approximations, or if greater accuracy is desired specific examples may be worked out numerically from the fundamental equations in cylindrical coordinates. The purpose of the present paper is to set up a general mathematical framework for the analysis of laminated transmission lines, and to treat the major theoretical questions which arise in connection with these lines. In view of the length of the mathematical analysis, we have not devoted much space to numerical examples, although a large number of specific formulas are given which may be used to calculate the theoretical performance of almost any Clogston-type line that happens to be of interest. A considerable part of our work is directed toward evaluating the effects of deviations from the ideal Clogston structure. Both theoreti- cal and experimental results suggest that the limitations on the ultimate applications of the Qogston cable ai-e likely to be imposed by practical problems of manufacture. These limitations, however, depend upon engineering (questions which we shall not consider here. LAMIXATIOD TRANSMISSION' LINKS. I 887 II. WAVE PROPAGATION BETWEEN PLANE AND CYLINDRICAL IMPEDANCE SHEETS We shall consider waves in a homogeneous, isotropic medium of dielectric constant e, permeability n, and conductivity g (rationajized MKS units). When convenient we shall also describe the medium in terms of the secondary electromagnetic constants a and tj, defined by (T — ■\/i(jijx{g + ^coe) , 7? = \/i(,}n/(g + iue) . (1) The ([uantity a is called the intrinsic propagation constant and t? (lui intrinsic impedance of the medium. We begin by considering structures bounded by infinite planes parallel to the x-z coordinate plane, and we confine our attention to transverse magnetic waves propagating in the ^-direction. We assume that the only non-vanishing component of magnetic field is Hj , and that all the fields are independent of x. Then the non-zero field components, written to indicate their dependence on the spatial coordinates, are Hx(y, z), Ey{y, z) and Ez(y, z), the time dependence e*"' being understood through- out. The field components are shown in Fig. 1. The field vectors are connected by Maxwell's two curl ec^uations, which reduce in the present case to dHJdz = {g + ioit)E,j , (2) dHx/diJ =—({/ + io:e)E, , and dEy/dz - dEJdy = 2co/x/^x . (3) If we eliminate Ey and E^ we get d'HJdy' + d'H./dz' = a'H, , (4) where a is the intrinsic propagation constant defined above. It is easy to sec that (4) is satisfied by a wave function of exponential form, say //. = e~'" ~'\ (5) pr()\-ided that the constants k and 7 are such that k' + 7" = 0-". (0) We may regard k and 7 as the (possibly complex) propagation constants in the y- and ^-directions respectively. Either may be chosen at will and the other is then determined by the condition (6). The electric field com- THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ponents corresponding to any particular H^ are easily obtained from equations (2). A concept important in what follows is that of wave impedances^ at a point. For a wave whose field components are Hx , Ey , E^ , the wave impedances looking in the positive and negative y- and ^-directions at a typical point are defined to be, respectively, Zy = EJHx , Zt = — Ey/Hx , (7) Z^ = —EJHx , Z^ = Ey/Hx . For waves of the type that we consider, Zy and Z~ are functions of y only, so that if two media ha\'ing different electrical properties are separated by the plane y = yo , the continuity of the tangential compo- z {^) .^ , f^y ^0'7c -^E-, W//:\/.////^/y//////////////////^^^^ Fig. 1 — Transmission line bounded by parallel impedance sheets. nents of E and H across the boundary can be assured by merely re- quiring the continuity of Z^ (say) at t/ = yo . This is equivalent to the requirement that the sum of the impedances Z^ and Z^ looking into the media on opposite sides of the boundary be zero. A similar condition holds for the impedances Zt and Z7 at a boundary z — z^ . As an example of the use of the wave impedance concept, we shall consider the propagation of a transverse magnetic wave between parallel impedance sheets which are separated by a distance 6. For the moment nothing is specified about the structure of the sheets except that the normal surface impedance looking into each is ^(7), for a wave whose propagation constant in the ^-direction is 7. The fact that in general Z will depend upon 7 should be noted, since in some cases this dependence ^ S. A. Schelkunoff, Electroniagnetic Waves, D. van Nostrand Co., Inc., New York, 1943, pp. 249-251. Since in our problem three field components vanish identi- cally, we need only two of the six impedances which are defined in the general case. ^ Reference 3, pp. 484-489. LAMINATED TRANSMISSION LINES. I 889 is quite important. The slieets are located at y = ±^^, as sliowii in Fi}z;. 1, and the spaee between them is fill(Ml with a, mecHum whos(> elect I'ical constants are eo , md ? (/» (<>i' ^n > 'Jo , il w(^ wish to use the deiiNcd constants). From the symmetry of the l)oun(lary conditions it is evident that for any particular mode If^r must be (Mther an e\'en function or an odd func- tion of // about the plane // = 0. Takiiia; the ex'en case first, we have Hx = ch Koy e~^% (8) E, = {/o 7 ch '^01/ c -IX E. = — — KO sh '^■o// e -yz go -\- iojeo where kq + t" = o'o . (9) If we replace f/n + /toeo by a^^/rjo and ku by (o-q — 7")', the boundary con- dition at y = \b, namely Zi = Z(7), (10) 1 )ecome!s \{al - Y)'b tanh i( pi , as shown in Fig. 2. We suppose that the radial impedances looking from the main dielectric into the inner and outer cylinders are, respectively, Z7|p=p, = Zi{y), ^J|p=po = Zi{y). (35) LAMINA TKI) TRANSMISSION LINKS. I Si)3 TluMi from (33) and (84) the bouiulary coiiditioiis are AIo{koPi) — BKo{koPi) _ ry ^ s .4/i(koP2) + BKiiKopi) m\ Avhero K'O — (ffo 7 '/(tp Ko go + ?"coe(i r7o(l — y'/(Tu)\ (37. If e(iuatious (36) are to he satisfied by values of A and B which are not both zero, it is easily shown tluit a necessary and sufficient condition is r]i)f,Ki,{Kupi) + Zi{y)Ki{Kopi) 77op/vo(^oP2) — ^2(7) A'l(^■oP2) VOpIo(Kopi) — Zi('y)Ii(koPi) rjOp/o(KoP2J + Z2(y)Il{KQP2) (38) and (38) is a transcendental e(iuation for the determination of the propa- gation constants of all the circular magnetic modes in the coaxial line. As ill the discussion of the parallel-plane line, we shall confine our attention to the principal mode and shall assume forthwith that the wall losses are small." Since for the principal mode we expect that 7 will be nearly equal to ctq , we may write 70 for o-q and eyaluate Zi and Z2 at 70 ; and we may replace the modified Bessel functions in (38) by their ap- Fig. 2 — Transmission line bounded l)y coaxial inii)edance cylinders. ^ J. A. Stratton, Electrumniinctic Theori/, McGraw-Hill, New York, 1941, pp. 551-554, gives a similar treatment of the jMiiicipal mode in an ordinary coaxial cable with solid metal walls. 894 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 proximate values for small argument. From the series given in Dwight 813.1, 813.2, 815.1, and 815.2, we have loix) ^ 1, h{x) ^ ir, Koix) ^ -(0.5772 + log ir) = -log 0.8905a;, (39) A'i(.r) ^^- -\- ^x log 0.8905a,-, X for I X I <3C 1, where log represents the natural logarithm. If we put these approximations into (38) and if we suppose that the wall im- pedances are so small that I cropiZi(7o)/ 2r?o | « 1, | (Top2Z2{'Yo)/'2rio | « 1, (40) we obtain, after a little algebra, 2 2 2 (To[Zl{yo) / pi + Z2(.yo)/p2] /.,x Ko = (To — y = — T J — ^-^ . 141 j rjn log (po/pi) Now further assuming that 1 Zi(yo)/pi + Z2{yo)/p2 ^ ^ /^2) 8 (Tor/o log (p2/pi) we get by the binomial theorem I Zi{yo)/pi + Z2{yQ)/p2 /.Q^ 7 = o'o + ^ — -. . , . • 143; 2i7o log (p2/pi) If we formally set ^o = 0, A\e find that the attenuation and phase con- stants of the principal mode in a coaxial line with low-loss walls and no dissipation in the main dielectric are Zi(7o)/pl + Z2{yo)/p2 /. .X a = lie 7 = Ke ;r — z -. — -— , 144; 2rjo log (p2/pi) o T / I T^ Ziiyo)/pl + Z2(yo)/p2 /,-N jS = Im 7 = ojVMofo + Im — , — ^-r . i4o; 2770 log (P2/Pl) As before, these approximations for a and 13 will ultimately break down as the frequency approaches zero, but they will certainly be valid over the frequency range in which we are interested in the present paper. 8 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Revised Edi- tion, Macmillan, New York, 1947. We shall refer to Dwight for a number of stand- ard series expansions. LAMINATED TRANSMISSION LINES. I 895 The magnetic field lines of the principal mode will of course be circles and the electric held will he largely radial, l)ut with a small longitudinal compoiuMil unless the wall impedances are rigorously zero. The general exj)r(>ssi()ns (33) for the fields may be reduced to simple approximate fornuilas if we use the fact that kq is given \)y (41) and kop is small com- pared to unity. The ratio A H may be obtained from either of equations (3()). Introducing the approximations (39) for the Bessel functions and carrying out a little algebra, we get the following approximate expres- sions for the fields: H, E. E. I - 2irp vol ^. 2wp -yz / (46) '?i(2^1og^+?!^hog^ .pi P P2 P J 27r log (po/pi) where the amplitude factor / is equal to the total current flowing in the inner cylinder. Incidentally we note that the above results might have been tlerived from more elementary arguments if we had started with the fields in a coaxial line with perfectly conducting walls and treated the effect of finite wall impedance as a small perturbation. If we consider an ordinary coaxial cable with solid metal walls at a frequency high enough so that there is a well-developed skin effect on both conductors, then to a good approximation Zi(7o) = ^2(70) = (1 + i)/giBr , (47) where r/i and 5i are the conductivity and the skin thickness of the metal; and the attenuation and phase constants are given by the well-known expressions 1/pi + 1/P2 (3 = u 2vogi8i log (P2/P1) ' 1/pi + 1/P2 (48) (49) 2r]ogi8i log (P2/P1) If necessary we may take account of dissipation in the main dielectric of either a plane or a coaxial transmission line by assigning complex values to eo and /xo , say ' See, for example, C. G. Montgomery, Principles of Microwave. Circuits, M. I. T. Rad. Lab. Series, 8, McGraw-Hill, New York, 1948, pp. 365-369 and 382-385. 896 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 €o = 60 — ?eo = eo(l — i tan €)E,, dE./dy = -[K'/'{g -Vio:e)]H,, where k is dehned by equation (G). Xow if we formallj^ identify Hx with "current" and E^. with "voltage", equations (53) are just the equations of a uniform one-dimen- sional transmission line extending in the ?/-direction, with series im- pedance K /{g -\- icof) per unit length and shunt admittance {g + z'we) per unit length; in other words a transmission line w^hose propagation constant is k and whose characteristic impedance is -qy , where K = a{l - y/(T-y, vv = K/'ig + iue) = 77(1 - '//) + Via + ^y - ui 26 (03) /vi is the impedance seen when we look into a semi-infinite stack of double layers if the first layer is of type 1, while K2 is the impedance seen if the first layer is of type 2. In calculations relating to Clogston 1 lines with dissipative walls, the real parts of Ki and K2 will both be positive. By a straightforward procedure we may express the matrix elements a, (B, 6, 3D in terms of Ki , K2 , T, and M, and then transform e( [nation 'See, for example, E. A. Guillemiii, Comtnunication Networks, 2, Wiley, New York. 1935, pp. 161-166. 'OF. Abeles. Comptes Rendus, 226, 1872 (1948). This result was called to the author's attention by Mr. J. G. Kreer. 900 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 (60) into Finally we obtain from (59) and (64) an expression for the impedance Zo looking into a plane stack of ?i double layers when the nth layer is backed by a surface whose impedance is Zn , namely Eo ^ ^ZnJKie"^ + gge"""^) + K1K2 sh nV Ho ~ Zn sh nV + K^ie""'' + /^2e"'^) ^0 = TT = — ~ — ; — ir^ — ■ y^^) For the cylindrical geometry, matters are a good deal more compli- cated. If we consider waves having field components H^ , Ep , Ez in a homogeneous, isotropic shell bounded bj'' coaxial cylindrical surfaces, and assume a propagation factor 6"*^^ Maxwell's equations (27) and (28) may be written Ep = [y/{g + i^e)]H, , (66) and d{-pH^)/dp = -(fir + iwi)pEz , (67) dE,/dp = -[K/{g + ^■coe)p](-pi7^). If desired, we might identify E^ with "voltage" and —pH^ with "current" and regard equations (67) as describing a nonuniform radial 'transmis- sion line, having series impedance K/{g + icoe)p per unit length and shunt admittance {g + io>t)p per unit length. Since, how^ever, in equations (34) we have already defined the radial wave impedance to be a field ratio without the extra factor of p, we shall carry out the analysis of the present paper directly in terms of the field components Ez and — H^ . From the general expressions (33) for the fields in cylindrical co- ordinates, we can show that the matrix relation between the tangential field components E^ , —H^ at two radii pi and p2 is given by EipiY -H{p,) UpiiKoiTu + Knloi) Vpi^Pii^oiIoi — Ko2loi)\ I Eipi) — (A^i/i2 - /Vi2/n) Kp-2iKnh2 + KoJn) }\-H{p2) Vp , (68) LAMINATED TRANSMISSION LINES. I 901 wlicro = (cr' - y'V, Vp = ^(1 - y~/o-')\ m aiul we have iisetl tlie ahhreN'iatioiis Irs = Ir(lips), Krs = /^^(/vp,). 70) It may be verified that the dehM'niinaut .1/ of the sciuare matrix a{)- l)eariiig iti ((18) is simply M = p./ Pi/ pi (71) III i)riii('iple equation ((38) permits us to determine by matrix multi- phcation the relation between the tangential fields at the inner and outer surfaces of a coaxial double layer, or of a laminated stack of any number of double layers, such as is shown in Fig. 4. The difficulty is that the elements of the matrix of a single layer are not functions only of the electrical properties of the layer and its thickness, but depend in a more complicated way on the inner and outer radii separately. Whereas in the plane case we had merely to take the ?ith powder of a single matrix, we are now faced with the problem of multiplying together n matrices, each of which (Uffers more or less from all the others. An exact expres- sion for the result is practically out of the ciuestion; but we can make some reasonable approximations if we assume that each individual layer is thin compared to its mean radius, so that the matrix elements do not change much from one layer to the next. If the thickness t {= p-i — pi) of a single layer is small compared to pi , then the Bessel function combinations appearing in (68) may be ex- panded in series, as shown in Appendix I, and the circuit parameter matrix takes the following approximate form, ' + 2^J ch Kt — - — sh d rjp 2k pi .' + 2., sh kI Vp L '^^u sh Kt ' + 2.J , (72) ch Kt + - — sh Kti 2/vpi wliere terms of the order of t/pi represent the first-order curvature cor- rections. If we use the same value of pi , say p, for both parts of a double layer, then up to first order the elements of the matrix of the double laver become 902 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 a 1 + 1 + e = 1 + 2-p J u + w 25 U + t-i 2p . Ch Kill Ch K2k + — Sh Kltl Sh K2^2 L2«i ?72p ch Ki^i sh K'd-i + i?ip sh /v'l^i ch ^2^2 sh Ki^i ch K'lt-y -j- —7 ch K'l/i sh K2I2 K\p Zk2P ]. + _2k2p 2kip_ sh Kid sh /C2/2 , ■ 1 1 — sh Kltl ch Kd-^ -f- — ch Kill sh /co/o .^p 7j2p (73) + |_2t72pKiP 277lpK2/C sh Kltl sh K2^2 , 3D = 1 + tl + t. 2p ^72^ — sh Ki^i sh /C2/2 + ch Kltl ch k2^2 + ■ 1 1 ^ ■ r— sh Ki/a ch /C2^2 + ^^~Z ch m^i sh Kitz ZKip ZK2P As in the analogous equations (58) for a plane double layer, the sub- scripts 1 and 2 refer to the first and second layers respectively. If we have a stack of double layers in which all the layers of the same kind have the same thickness and same electrical constants, then the only term in (73) which varies from one double layer to the next is the mean radius p. Depending on the circumstances, we may wish to use a single value of p for the whole stack, or a few different values, or even, if high-speed computing machinery is available to carry out the matrix multiplications, a different value of p for each double layer. The matrix of the whole stack then becomes a product of powers of as many different matrices as we have chosen values of p. Obviously this method is better adapted to the numerical analysis of special cases than to the general theoretical treatment of a stack whose ratio of outer radius to inner radius is unspecified. In principle we are now able to compute the normal surface impedance of any laminated plane or coaxial stack at a given frequency provided that we know the electrical constants and the thickness of each layer, the number of layers, the propagation constant 7 in the z-direction, and the normal impedance Z„ of the material behind the last layer. Since the general formulas even for plane stacks are quite complicated, however, we shall introduce at this point some very good approximations which will be valid for all of the following work. LAMINATED TRANSMISSION LINES. I 903 Henceforth we shall take the layers of thickness ti to be such good conductors that the ratio co€i/(/i of displacement current to conduction current is negligible in comparison with unity. For metals like copper this is an excellent appro.ximation at even the highest engineering fre- quencies. Then on introducing the characteristic skin thickness 5i , we have for the conducting layers, (Ti = y/ioJuiQi = (1 + i)/5i , (74) Vi = Vioip-i/gi = (1 + ^OM^i J where 5i = WoifiiQi . (75) For pure copper the permeability and conductivity are Ml = 1.257 X 10~^ henrys-meter~\ gi = 5.800 X 10 mhos -meter \ from which we obtain the numerical values cri = 1.513 X 10* (1 + i)v5^ meters"', (77) 7,1 = 2.609 X 10 * (1 + ^0 V/mc ohms, and 6.609 X 10-5 2.602 ., 5i = 7^= meters = ~~Jj= mils, (78) V jMo VjMc where /mc is the frequency in Mc-sec~\ Referring to equations (56) and (69) and bearing in mind the above numerical values, we see that for the conducting layers we have Ki ;^ ai = (1 + i)/8i , (79) Viy = Vip ^ Tji = (1 4- r)/giSi , to a very good approximation, since in our applications the quantity y will always be of the order of 2iri/Xv , where the vacuum wavelength X„ is at least a few meters, while the skin thickness 5i will be at most a small fraction of a centimeter. For the insulating layers of thickness ^2 we shall set the conductivity fif2 equal to zero, so that 0-2 = i(^\/niti , V2 = y/ml^ ' (80) We denote the relative dielectric constant and permeability by eor and jU2r respectively; dissipation in the insulating layers may be included 904 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 if necessary by making tir and/or \iir complex. In MKS units we have t2 = e2rCD , M2 = M2rM« > (81) where the electrical constants of vacuum are 12 e„ = 8.854 X 10"'' farads •meter~\ /x„ = 1.257 X 10~ henrys • meter" . (82) It follows that •" = <" V«,.. = ^;— = 299.8 ""'^'' ' (83, V2 = Vv^/n2r/e2r = 376.7\/M2r/e2r ohmS, where as usual the subscript v refers to vacuum. It is clear that unless we deal with ferromagnetics, the quantities ao and rjo will be of roughly the same order of magnitude as a-„ and tjv ■ From (56) and (69) we have K2 = 0-2(1 — 7 /<^2y, , o . (84) /I 2/ 2x| ^ ^ T?2j/ == r?2p = 7?2(1 - 7 /0-2) , where since 0-2 and 7 are both of the same order of magnitude as 2-Ki/\v , in general no further approximations can be made. In all of what follows we shall assume that the thickness to of each insulating layer is very small compared to the vacuum wavelength at the highest operating frequency ; in practice h will be at most a few mils and Xi, at least a few meters. Then the quantity | ^2^2 |, which is of the order of 2irt2/'Kv , will be so small that to an excellent approximation we may set sh K2/2 = '<2^2 and ch ^2^2 = 1- Using this simplification, together with the fact that 771^ « r]2y for all frequencies which may concei^'ably be of interest, it is not difficult to show from (58) that the matrix ele- ments of the plane double layer reduce to a = ch Kiti , (S> = V2yK2t2 ch Klf\ + Viy sll K'l^i , e = — sh Klh , (^^) 3D = '?^%h .1^ + ch .1/1 . The determinant of the matrix is unity, and from (61) the propagation constant per section is defined by LAMINATED TRANSMISSION LINES. I 905 ch r = sli Kiti + ch Kid (86) while from (03) the iteiatue impedances are A'l = — hrjiyK-J-i + \/ ihmvKiii)' + VlymyKiti COth Kid + vlv ) A'o = +I7?2J/^•2/•2 + \/{hV2vK2t2)- + T?]yr72j,K2^2 COth Kj^i + r}ly . If we make the same simplifications in the approximate expressions (73) for the matrix elements of a coaxial double layer, we obtain (87) a = (B = 1 + 1 -pj ch Kifi — ~ ; sh Kiti , Zkip 1 + tl + /2' 2-p . V-loK-'lti C'h K\ti + [l + .^ + (2 - 'i=^) ^" e = SD = 1 + 2pJ 'yip sh Klti , 77ip sh Kid , (88) ^1 + 1-2 _ Vl ^P 2772p/ClK2^2pJ Vlp "n-ipf^it sh m/i + 1 + d + 2^; 2p ch Kid ■ In the preceding equations no restrictions have been laid on the thicknesses d and h except the trivial requirement that d shall be small compared to a wavelength. We shall now consider the limiting case in which both ^1 and ti are infinitesimally small. When we make this last and most drastic approximation we do not expect that the idealized structure thus obtained will show all of the features which are of interest in a physical transmission line with finite layers; but the results of the simplified analj'sis will be useful in some cases nevertheless. It need scarcely be pointed out that we are dealing here only with a mathematical limiting process, in which we assume that each layer, no matter how thin, always exhibits the same electrical properties as the bulk material. If this assumption be regarded as mu'ealistic, it may be observed that the (juantity which we actually allow to tend to zero is the ratio of layer thickness to skin depth. The skin depth may be made as large as desired by lowering the frequency, so that the formulas which we derive by 906 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 letting ti and ti approach zero at a finite frequency will also hold for finite thicknesses if the frequency is sufficiently low. We shall let 6 denote the fraction of the stack which is occupied by conducting material, so that d = h/{k + ^2), (89) w^here at present t\ and h are both infinitesimal. Then the stack may be regarded as a homogeneous, anisotropic medium, characterized by an average dielectric constant e perpendicular to the layers, an average permeability /i parallel to the layers, and an average conductivity g parallel to the layers. Sakurai has treated such an artificial anisotropic medium, and from his formulas we find that when the layers are al- ternately conductors and insulators, the average electrical constants are, to a very good approximation, 6 = 62/(1 - 6), M = 0M1 + (1 - ^)M2 , (90) g = Ggi- Sakurai has also shown that the average values of the electrical con- stants may be used in Maxwell's equations for the average (macroscopic) fields, due regard being paid to the orientations of the field vectors with respect to the laminae. For the plane stack, these equations read dHJdz = iwlEy , dHJdy = -gE, , (91) dEy/dz - dEz/dy = tco/i^x , where the bars denote average values. By analysis exactly similar to that carried out at the beginning of this section for a homogeneous, isotropic medium, we may find the relation between the tangential field compo- nents E: , Hx at the two surfaces of a stack of infinitesimally thin layers. (The bars representing average values may be omitted, since the tan- gential components of E and H are continuous across the boundaries of the layers.) We obtain a matrix relation analogous to (55), namely EiO) \ /ch T(S K sh T(s\ /E{s) ' 1 M ■' ^^^^ H(0) I \-- sh Tts ch T(S / \h(s) " T. Sakurai, J. Phys. Soc. Japan, 5, 394 (1950), especially Section 3. LAMINATKI) TRANSMISSION LINES. I 907 where s is the thickness of the stack. Tlie propagation constant T( per unit distance normal to the stack and the characteristic impedance K of the stack are given by r, = W / 2- I 2x ^ (o) Me + 7 ) K = Tr/g = —^ (a;>e + 7") -\i (93) (9^) 1\ and K niixy also be derived from equations (8G) and (87) by limiting processes; we have r, = lim T/(k + ^2), K = lim Ki = lim K2 (95) (96) It should perhaps be noted that terms of the order of coti/^i and we^/^i compared to unity were omitted in the expressions (90) for e and g, and in the derivations of r< and K. Since, however, under all practical circumstances the omitted terms appear to be insignificant, we shall not take space to write out the formally more complicated results which woukl be obtained by keeping them. In a cylindrical stack of infinitesimal layers, the average fields satisfy d(pH^)/dp = gpE, , ^E^/^p - dE,/dz = iwilH^ . (97) The relation between the tangential field components E, , —H^ at two radii po and p„ is expressed by a matrix equation analogous to (08), nameh^ E{p,) -Hipo) TtPniKooIln + Ki,Jm)) KT(Pn{KooI(,n — Konh (98) E(p.y ^ (A^o/U - /Vl„/l0) TfPn(K,oIon + Kojio) /\ -^^(pj/ ' '2 In Reference 1, equations (11-17) through (11-26) give examples of ecjuations in which these smaU terms have been retained. 908 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 where /.. = Ir{l\Ps), Kr. = Kr(V,p,), (99) and r^ and K are given, as in the plane case, by (93) and (94). IV. PRINCIPAL MODE IN CLOGSTON 1 LINES WITH INFINITESIMALLY THIN LAMINAE An idealized parallel-plane Clogston 1 transmission line is shown schematically in Fig. 5. It consists of a slab of dielectric of thickness b, with electrical constants mo , «o , bounded above and below by laminated stacks each of thickness s. Outside each stack there may be an insulating or a conducting sheath, of which nothing more will be assumed at present than that its normal surface impedance Zn(y) is known. The total dis- tance between the sheaths will be denoted by a, where a = b -\- 2s. The corresponding Clogston 1 coaxial line is shown in Fig. 6. We de- note the thickness of the inner and outer stacks by Si and So respectively, while a is the radius of the inner core (if any), and b is the inner radius of the sheath around the outer stack. The inner and outer radii of the main dielectric are pi = a + Si and p2 = b — So , respectively. In practice the core may be a dielectric rod and the sheath may be a conducting shield, but in the present theoretical analysis we shall merely assume that the radial impedances Za(y) and Zb(y) looking into the core and the sheath are known. In Part I of this paper we shall deal with "extreme" Clogston 1 lines, in which the space occupied by the stacks is small compared to the space occupied by the main dielectric. We may then regard the laminated boundaries as impedance sheets guiding waves whose phase velocity is Zn(r) Fig. 5 — Parallel-plane Clogston 1 transmission line. LAMINATED TRANSMISSION LINES. I 909 (letermiiKMl l)>- tlic i)i()i)(Mties of tlu> mniii dicloclric, as discussed in Sec- tion II, and \v(> nuiy use tiie intrinsic propaj^ation constant of the main dielectric in calculating the svu'face impedance of tli(> boundaries. This aj^proximation simplifies the analysis of C'logston 1 lines a {^reat deal. We siiall treat the general case, in which an arbitrary fraction of the total space is filled with laminations, in Section IX of Part II, as a part of our study of Clogston 2 lines. In this section we shall assume that the laminae are infinitesimally thin, so that the stacks may be completely characterized by their average l)roperties e, fi, and g. The case of finite laminae will be taken up in the next section. We shall also assume throughout that dielectric and mag- netic dissipation may be neglected except, as in Section VII, where the contrary is explicitly stated. In general the current density and the other held (|uantities in a plane stack of infinitesimally thin layers will be linear combinations of the functions sh Tfij and ch Tdj, where y is distance measured into the stack, and T( is the propagation constant per unit distance, as given by (93). The qualitative behavior of the fields in a cylindrical stack will be similar. In particular, if the stack is thick enough the current density and the fields will fall off as e"^'^, and we can define an "efTective skin depth" A by A = l/(Re r,). (100) Cloiiston's fundamental observation was that in order to minimize the Fig. 6 — Coaxial Clogston 1 transmission line. 910 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ohmic losses in a stack carrying a fixed total current the current density should be uniform across the stack, and that we can achieve uniform cur- rent density by adjusting the mo«o product of the main dielectric so as to make Tf equal to zero. If in equation (93) we set 7 = To = iu-\/tioeo , (101) then Ti will be zero if M0€0 = fii = [dn, + (1 - e)M2][62/(l - d)]. (102) Equation (102) will be referred to henceforth as Clogston's condition' If the permeabilities of the various materials are all equal, the condition reduces to eo = 6 = 62/(1 - e) , (103) which is the form employed by Clogston in Reference 1. When Clogston's condition is satisfied, r^ = 0 and the effective skin depth of the stack is infinite ;^^ that is, the current density is uniform in any stack of finite total thickness. The quantities Tt and K vanish simultaneously, but the limiting value of their ratio is finite; and the matrix of the plane stack, as given by (92), takes the form (104) Accordingly we obtain, for the surface impedance Zo(7o) of the stack, which is, as might have been expected, just the impedance between opposite edges of a unit square of material of conductivity g and thick- ness s through which the current density is uniform, in parallel with the sheath impedance Zniyo). It follows from equations (20) and (21) of Section II that the attenuation and phase constants of the principal mode in a plane Clogston 1 line with infinitesimally thin laminae, Clogston's condition being satisfied exactly, are " This statement is certainly accurate enough for all practical purposes, al- though an exact calculation which takes into account the small terms that were neglected in the approximate formula (93) for Tf shows that the effective skin depth is \J'2-Kd, where Xo is the length of a free wave in the main dielectric. The exact result is derived by Clogston in Reference 1, equation (11-26). In practice, finite lamina thickness will restrict us to effective skin depths much smaller than this theoretical limit. LAMINATED TRANSMISSION LINES. I \1\\ 1 , , a = Re -77: , . /ry f M , (!()()) TjoOlgrs + l/Z„(7o)J In general the sheath impedance Zn{yu) will b(» large compared to the impedance \/gs of the stack, since even if the sheath is an electrically I hick metal plate of the same material as the conducting layers, its impedance is Z„(7„) = (1 + i'l'gA, (LOS) whereas ds will usually be several times the skin thickness 8y in the fre- quency range of interest. If the sheath is free space, its impedance is a fortiori much greater than 1/gs, since then it may be shown that Z„(7o) = -iVvinoreor - l)^ (109) where ??„ = 376.7 ohms is the intrinsic impedance of free space, and /ior and €or are the relative permeability and relative dielectric constant of the main dielectric. Under most circumstances, therefore, we may neglect l/Z„(7o) in comparison with gs, and obtain the very simple results, a = l/r],bgs, (110) 13 = coVmI^o . (Ill) To this approximation the line exhibits neither amplitude nor phase distortion. For a coaxial stack of infinitesimally thin layers with Clogston's con- dition satisfied, the stack matrix given in (98) reduces to 1 (112) 9' / 2 2>, -— (.p„ - pdj where po and p„ denote the inner and outer radii of the stack. It follows from (112) that Zrjyo) ^ 1 ^ 1 pi Igipl - «^) + a/Za{yo) gsiia + |si) + a/Za{yo) ' ( 1 10) Z,(yo) 1 1 P2 \g(b' - pi) + 6/25(70) gS2{b - W + b/Z,{yo) ' 912 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 where ^1(71)) and Z-^iyo) are the radial impedances looking into the stacks at pi and p2 respectively, and Z„(7()) and Zhiji)) are the radial impedances looking into the core and the outer sheath. From equations (44) and (45) of Section II, the atteiuiation and phase constants of a coaxial C'log- ston 1 cable with infinitesimally thin layers, Clogston's condition being- satisfied exactly, are Zi(yo)/pi + Z2(yo)/p2 ... ,. oi = Re — — -— , (114) 2r?o log (po/pi) /3 / I T^ Zl(yo)/Pl + ^2(to)/P2 /,,.x /3 = coVMotu + Im — — ^- , (llo) 2r/o log (po/pi) where Zi{yo)/pi and Z2(yo)/p2 are given by (113). The impedances Za{yo) and Zb(yn) may be computed if we know the structure of the core and the sheath. For a solid, homogeneous core and a homogeneous sheath of effectively infinite thickness, we have . . r]K /o(Ka) 7 t \ '"^' ^^o('^^^) mnX Zaiyo) = — rT~\ ' Zbiyo) = — , (116) a i]{Ka) V^o, (119) and again to this approximation there is neither amplitude nor phase distortion. The formulas which have just been derived on the assumption of LAMINATIOD TRANSMISSION LINES. I 913 infiiiitosimally thin laminae approacli validity for laminae of finite thickness as the frequency is reduced, provided of course that we do not go to such extremely low frequencies that the attenuation per wave- lonoth becomes large. We shall show in the next section tiiat the effect of finite lamina thickness is to introduce a frequency dependence into the attenuation and phase constants, in addition to the variations (if any) which av\sv from the fr(M|uencv dependence of the coi-e and sheath impedances. We next writc^ down approximate expr(\ssions for Wiv field components in a })lan(^ Clogston 1 line with infinitesimally thin laminae. In the main dielectric we have, from eciuations (22) of Section II, (120) ^ 2Zoiyo)Hoy -y, 111, 1^^ 6 J for —\b^ u ^ \h, where i/o is an arbitrary amplitude factor and Z(i(7o) is given by (105). In the stacks the fields are Ey y^^ Holl + gZo{yo)ihb T ijW, (121) E, ^ ±Zo(To)i^oe~"^ for 56 ^ I 7/ I ^ ha, where in cases of ambiguous sign the upper sign refers to the upper stack (y > 0) and the lower sign to the low^er stack {y < 0). It should be noted that whereas the tangential field components Hj: and E, are continuous through the stack, the normal field component Ey is discontinuous at layer boundaries. From equation (52) we have, in th(> conducting layers, Ey = -(y/g,)H,, (122) while in the insulating layers, Ey = -(7Aco62)7/.. (123) To our approximation, therefore, the only contributions to the average field Ey come from the insulating layers. The average current density .A in either stack is iniiform, being 914 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 given by J. = gE. = ±gZo{yo)Hoe-'' . (124) The total current per unit width carried by the stack is just J^s, where s is the thickness of the stack; there will also be small currents in the sheaths unless we assume the sheath impedance to be infinite. The potential difference between any two points i/i and 2/2 in the same trans- verse plane may easily be found from V{y2) rV2 F(2/i) = - ''Vl Eydy. (125) For a Clogston 1 line of the proportions which we have been considering, the potential difference across the stacks will be small compared to the potential difference across the main dielectric. In a coaxial Clogston 1 with infinitesimally thin laminae, the fields in the main dielectric are given to a good approximation by equations (46) of Section II, namely Ha E. E. 2irp €0 2irp / 2t log (p2/pi) (126) ?i^log^^ + ?^logei L Pi P P2 P J where / is an arbitrary amplitude factor and ^1(70) and ^2(70) are ex- pressed by (113). In the inner stack we have Hd E. Ziiyo)! Vgip' - a~) 2irpi _ e 2xpi + 2p ' pZa(yo)_ gip' - a) 2p + P^a (70) _ (127) E. Zi{yo)I 2irpi while in the outer stack, LAMINATIOD TRAXSMLSSIOX LINES. I 915 //. E. Z 2(70) I 'gib' - p') + 27rp2 L 2p pZb{'Yo)_ M Ziiyo)! gib' - pO + e 27rp2 L ^P pZbi'y(i)_ (128) ^ _Z2(7o)/ -1 -C/z ' — ' -^ e 27rp2 The average current densitj^ in either stack is uniform and is given by 7. = gE, , (129) tliough in general the current density will not be the same in the two stacks because of the difference in cross-sectional areas. The potential difference between the surface of the inner core and any other point in the same transverse plane is Vip) - Via) = -fE, dp. (130) If the stacks are thin compared to the thickness of the main dielectric, as we are assuming throughout Part I, then the potential difference across the stacks ^\\\\ be small compared to the potential difference across the main dielectric, and the characteristic impedance Zk of the Clogston 1 cable will be approximately the same as the characteristic impedance of an ideal coaxial cable with perfect conductors of radii Pi and po and the same main dielectric, namely Z, = 6O4/S log ^' ohms. (131) r ^Or Pi We shall defer making any field plots for Clogston-type transmission lines until Section IX of Part II, when we shall discuss the transition from Clogston 1 to Clogston 2 as the space originally occupied by the main dielectric is gradually filled with laminations. Our present results will then appear as the limiting case in which the thickness of the stacks is small compared to the thickness of the main dielectric. In conclusion we shall mention briefly the question of how to dispose a given amount of laminated material in a Clogston 1 coaxial cable so as to achieve the minimum attenuation constant. The whole problem of optimum proportions for Clogston cables is a complicated one of which an adequate treatment would require a separate paper in itself, with the results depending to a great extent on engineering considerations which limit the ranges of the parameters that we can vary in any practical case. Here we shall discuss only the following rather highly idealized problem : 916 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Given a coaxial Clogston 1 with infiuitesimally thin laminae, having a high-impedance core and a high-impedance sheath of fixed radius b, and in which the total thickness Si -+- S2 of both stacks is a fixed constant 2s. Assuming that 2s is small compared to b, what should be the radius a of the core, and how should the total stack thickness be divided between the outer and inner stacks so as to minimize the attenuation constant of the line? Finalh^, what should be the fraction 6 of conducting material in the stacks to minimize the attenuation constant, if the electrical constants of the conducting and insulating layers are fixed, but the properties of the main dielectric are at our disposal? If the two inequalities si « a, S2 « 6, (132) are satisfied (these restrictions will be removed in Section IX, when we discuss Clogston cables having an arbitrary fraction of their total volume filled with laminations), then equation (118) for the attenuation constant of a Clogston 1 with infiuitesimally thin laminae and high-impedance boundaries becomes, approximately. 1 asi 0S2J (133) 2r]og log (b/a) If we write S2 = 2s - Si , (134) and vary Si and S2 in accordance with this relation while holding a and b constant, it is easy to show that the expression on the right side of (133) is a minimum when 2s\/b 2s-\/a r^o-\ Si = —^ ^, S2 = —^ ^. (13o) V a + V f> Va + V& These equations tell us the most efficient way to divide the stacks in a Clogston 1 when the radii of the core and the outer sheath are a and b re- spectively, still assuming of course that the thickness of each stack is small compared to its mean radius. If we introduce the optimum values of Si and So into (133), we get 1 1 ■\/a Vb_ (136) 2770^ (si + S2) log (b/a) If b is fixed, the last expression is a minimum, considered as a function of a, when log (b/a) = 1 -\- \^ctjh. (137) LAMINATED TRANSMISSION LINES. I U17 The root of tliis transcendental equation is b/a = 4.3827, a = 0.228176. (138) Substitutiiiii' this xahio of h a into (135), we find si = 1.3535s, S2 = 0.r)465s, (139) si/so = 2.0935; wiiile from (130) and (138) the minimum \-aluo of the attenuation constant is - 3.238 ^^^^^ Vog{si + S2)h To find the ()i)tinium value of 9, we observe that e(iuat ion (118) forthe attenuation constant of a Clogston 1 cable with infinitesimaily thin laminae and high-impedance boundaries may be written in the form (eo/Mo)' t-r 1 \ r-i ^^\ « = —^ .f(a, 6, Si , .So), (141) where the first factor depends on the electrical constants of the com- ponents of the cable, while /(a, 6, Si , S2) is a function only of the geometry. By (110) the attenuation constant of a plane Clogston 1 has the same form, only with a different dependence on the geometrical factors. Now assume that the geometrical proportions of the line are fixed, and that the electrical constants ijli , gi , ^2 , and €2 of the conducting and insula- ting layers are given, but that the constants /xo , to of the main di- electric and the fraction of space 9 occupied by conducting layers in the stacks are at our disposal. The M(tfn product of the main dielectric is to be codetermined with 9 so that Clogston's condition (102) is always satisfied. Solving (102) for 9 gives - M060 - M2e2 (^^2) jUoeo + I'm! — ^2)^2 Hence the first factor in the expression (141) for a may be written (co/jLto)' _ coUoeo + (mi ~ M2)g2] ^S'l gifjL~n\n(>€a — M2€2] (143) If we minimize the right side of (143) with respect to to , all other quanti- ties being held constant, by equating to zero the derivative with respect 918 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 to €o and then solving for eo , we get Moeo = M(mi + 2m2) + (mi + S^x^^i2)% . (144) From (142) the value of 6 is 3mi + (mi + 8M1M2) and the corresponding attenuation constant is proportional to (co/mo)' (co/mo}' 3mi + (mi + 8/X1/X2)'' %i 9i Ml + (mi + 8miM2)^ (146) It will be observed that so far we have determined only the optimum value of the product moco , and so we are still free to alter the ratio of Mo to Co while holding the product of these two quantities constant. For given values of m and m2 , we obtain the lowest attenuation constant by making eo as small as possible and mo as large as possible, subject of course to the practical restriction that €0 cannot be lower than the dielectric constant of free space. However if we permit m2 and mo to be simul- taneously increased, as by magnetic loading of both the insulating layers and the main dielectric, we find from (146) that on paper it is possible to decrease the attenuation constant without any definite limit. This observation is in accord wdth the fact that the attenuation constant of an ordinary coaxial cable may be decreased indefinitely, with a corre- sponding decrease in the velocity of propagation along the cable, if we are willing to assume an unlimited amount of lossless magnetic loading. If Ml = M2 , (144) and (145) take the form Moeo = 3m262 , d = 2/3, (147) from which we have the result given by Clogston: If the conducting and insulating layers are infinitesimally thin and have equal permea- bilities, then minimum attenuation is achieved when the thickness of the conducting layers is twice the thickness of the insulating layers. In this case, from (146) and (147) the attenuation is proportional to (^o/mo)' ^ 3(eo/Mo)' Q^gs When Mo = M2 , corresponding to no magnetic loading, we must take €0 = 3€2 , and (148) reduces to 1* Reference 1, pp. 513-514. LAMINATKD TK.WSMISSIOX LINES. I 1)19 while it we lixul the iiiaiii dielectric so that ^x^, = 3/i2 iind we can take fo = €2 , we have (cq/mo) ^ V3 (e2//X2) Qf^QN 6gi '2gi ' which is just one-third of the value with no magnetic loading. As Clogston has pointed out, if the limitation is on the total thickness of conducting material in the stacks rather than on the stack thicknesses themselves, we shall find it advantageous to use a small value of 6 (a high "dilution" of conducting material) so as to make the average dielectric constant 62/(1 — 6) of the stacks, which has to be matched by the main dielectric, as small as possible. We shall see later that the effect of hnite lamina thickness is in fact to limit the total thickness of conduct- hig material which it is useful to employ in a single stack at high frequen- cies, so that for physical stacks of non-magnetic layers at high frequencies the optimum \'alue of d is less than 2/3. Quantitative results w^hich take into account the finite thickness of the layers will be obtained in Section XI. To illustrate the use of some of the equations derived above by means of a numerical example, w'e shall compare the attenuation constant of a conventional coaxial cable with that of a Clogston 1 cable of the same size. If a and b denote the radii of the inner and outer conductors of a conventional coaxial cable, and we take b/a = 3.5911 to minimize the attenuation constant, then we have from equation (48) of Section II, on setting pi = a and p2 = 6, ^-^1^^, (151) TjogiSib where 170 is the intrinsic impedance of the main dielectric, which may be air. For a Clogston 1 coaxial cable with infinitesimally thin laminae, no magnetic material in the stacks (mi = M2 = Mt), and the optimum propor- tions given by (139) and (147), w^e have - 4.857 (J.2) 7?0^l(Sl + S2)b' where b is the outside radius of the outer stack and rjo is the intrinsic impedance of the main dielectric, which cannot be air in a Clogston cable. The ratio of the attenuation constant «<; of this Clogston cable to the 920 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 attenuation constant as of an air-filled standard coaxial of the same size, made of the same conducting material, is ac ^ 2.704 8i . . where /xor and eor refer to the main dielectric of the Clogston cable. Since the attenuation constant of a standard coaxial cable is propor- tional to the sciuare root of freciuency in the range we are considering, while the attenuation constant of the ideal Clogston cable is independent of frequency in this range, there will be a crossover frequency above which the Clogston cable has a lower attenuation constant than a con- ventional coaxial cable of the same size. If we are dealing with copper conductors and if frequencies are measured in Mc-sec~ and linear dimensions in mils, then from equations (78) and (153) we find that the crossover frequency is given approximately by ^ 49.50 M^r) (Sl + S2)mils For example, let us take an ideal Clogston 1 cable of outer diameter 0.375 inches, excluding the sheath, with no magnetic loading, and assume the following values: a = 42.8 mils 6 = 2/3 h = 187.5 mils e^r = 2.26 (polyethylene) sx = 12.69 mils €o. = 3e2r = 6.78 (155) 52 = 6.06 mils Hor — Mir = M2r = 1 Si + So = 18.75 mils This cable has a lower attenuation constant than a standard air-filled coaxial of the same size at frequencies above about 1 Mc-sec~ , the ap- proximate formula (154) yielding 0.955 Mc-sec~^ for the crossover frequency and the exact ecjuation (118), taken in conjunction with (151), yielding 1.251 Mc-sec~ . The reader is cautioned that the comparison given by (153) between Clogston and conventional cables is based upon certain highly idealized assumptions. In the first place we have neglected the finite thickness of the laminae, which will in fact cause the attenuation constant of a physical Clogston cable to increase with increasing frequency, and ultimately to cross over again and become higher than the attenuation constant of a conventional air-filled coaxial. We have also neglected dielectric and magnetic losses, which are likely to be directly propor- tional to frequency and by no means negligible at the upper end of the LAMINATED TRANSMISSION LINES. I 1)21 froquoncy Inuul. In practice, too, the noeo product of the main dioloctric must be held very close to the Clojiston value or the benefit of the lai-jic effective skin depth is lost; and the stacks must be extremely uniform oi- ajiain the depth of penetration is fj;reatly reduced. We shall take up all these matters in later sections, and shall see that while the n^sulls just given represent ultimate limits of performance, the practical impioxe- ments which can be achie\-ed o\'er conv(Mitional cables depend upon the degree to which one can solve the manufacturing ])roblems that tend to make every actual Clogston cable differ morc^ or less from the ideal striic- tuvc considered above. V. EFFECT OF FINITE LAMINA THICKNESS. FREQUENCY DEPENDENCE OF ATTENU.\TION IN CLOGSTON 1 LINES The principal effect of finite lamina thickness in a Clogston cable is to introduce a frequency dependence into the propagation constant, and in particulai" to cause the attenuation constant to increase, with increas- ing frequency, above the value which we have found for infinitesimally thin laminae (or for finite laminae at low frequencies). The increased losses are associated with the fact that the penetration depth in a lami- nated stack decreases with increasing freciuency, even when Clogston's condition is exactly satisfied, if the laminae are of finite thickness. We shall now obtain expressions for the surface impedance of a plane lami- nated stack of n double layers, such as is shown in Fig. 3, when Clogston's condition is satisfied but the individual layers are of finite thickness. We first observe that Clogston's condition (102) implies . r. 0M1 + (1 - ^)m2' = tCOUo i — ~ — L (1 - ^)M2 . = —io^md = —vicTih (1 - d)h 6 (15G) where in the last step we have used the fact that in the conducting layers rjiy is equal to 771 and ki is equal to ai to a very good approximation. We now introduce the dimensionless parameter 0 = a,h = (1 + i)h/8i ^ K,U , (157) which may be regarded as a measure of the electrical thickness of the individual conducting layers. From (86) and (156) we have, for the prop- agation constant per double layer, 922 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 [+!© + a®' - ecoth0 + I)*], [- ^0 + (\e' - 0 coth 0 + 1)*], (158) (159) ch r = ch 0 - i0 sh 0, and from (87), for the iterative impedances, _©_ @_ since rjiy = ki/qi = S/gik . If the thickness ^i of each conducting layer is moderately small com- pared to the skin depth 5i at the highest frequency of interest, the quanti- ties r, Ki , and K2 may conveniently be expanded in powers of 0. The identity 2 ch a:; — 1 = 2 sh la; enables us to transform (158) into sh' ir = i(ch 0 - 1) - i0 sh 0 0^ 48 0 0^ 1 + 15+560 + (160) (161) after we expand sh 0 and ch 0 by Dwight 657.1 and 657.2 and collect terms. Taking the square root by the binomial theorem gives sh hV = - 4V3 0* 170' 30 50400 + (162) the negative sign being introduced because from (157) 0 is a positive imaginary number and we want Re F > 0. Then 2sh" L WS \ 30 ^ 50400 + ■©_' 0* 0^ V3 L 2 60 + 525 (163) provided that we expand the sh ^ function by Dwight 706. From (159) we get Ki = Ko = 1 J_ ■(3 - iVs) &' + i\/S r\i ^'V3 45 0* - 1575 0'-f (3 -f V3) 0' + iV3 45 0^ ^VS .2.6 (164) 1575 ^ © + LAMINATED TRANSMISSION LINES. I 923 where we have expanded coth 0 by D wight 657.5 and chosen the sign of the square root to make Re Ki and Re Ko both positive. Our first observation is that when the lamina thickness is finite the effective skin depth of the stack is also finite. We have, from (157) and (163), ,4 r = -L r^ + i^ V3 L^i 155 J 8^1 5255? (165) niid (he average propagation constant pvv unit distance into tlic stack is r 1 i\ = ^tl ih + ^>) Vsdi + /o) + ill M 1561 5256"i (166) If as usual we define the effecti\'e skin depth A to be the distance, meas- ured into an infinitely deep stack, at which the current density has fallen to 1/e of its value at the surface, then keeping only the first term in (166) we have . 1 Vsih + t2)8l V3(^i + t2) .._. A = :=; = 2 = 72 — J ^^"^^ Re r^ ^1 TTiJugifti a result also given by Clogston.^^ The number A'' of double layers in one effective skin depth is A V35? VS N = T^Mifk ih + ^2) tl while the total thickness Ta of conducting material in these layers is (168) Ta = Nh = ^1 TruiQift] (169) T^ is essentially the thickness of conducting material in each stack which is effectively carrying current; it is evident that for small values of ^1/61 this efTective thickness is inversely proportional to the frequency / and to the thickness h of the individual conducting layers, but independent of the thckness /2 of the insulating layers, provided that <2 is very small compared to the length of a free wave in the insulating material. In the general case, still assuming of course that Clogston's condition is satisfied, the surface impedance Zoijo) of a plane Clogston stack is given by equation (65) of Section III, namely Zoijo) = |Zn(7o)(A>"" + AV"') + K,K, sh nr Z„(7o) sh nr + UKif"^ + ^26""^) (170) '^Reference 1, equation (III-44). 924 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 where Zniyo) is the impedance of the surface behind the stack. If 0 = 0, (170) reduces to (105) of Section IV, that is, -^0(70) = ::: , — . ,ry , — r = —ff^ — ; — TTfTl — \ > (171) grs + l/Z„(7o) gxTx + l/Zn(yo) where Ti is the total thickness of conducting material in the stack. If Zniyo) is infinite, then for all values of 0 and 71. we have Zo(7o) = -^ [|© + a©' - © coth 0 + 1)^ coth nT]; (172) and if Re nT is large, corresponding to a stack many effective skin depths thick, then for any Znijo) we have ^1(70) = K, . (173) Once Zo(yo) has been computed for a particular frequency, the at- tenuation and phase constants of the plane Clogston 1 line at that fre- quency are given, as in Section II, by a = Re Zo(yo)/r,ob, (174) 0 = ojaZ/xoTo -f- Im Zo(yo)/rioh. (175) Explicit expressions for the surface impedance of a coaxial stack of finite layers have not been derived. However, if in a coaxial Clogston 1 the thickness of each stack is small compared to its mean radius, or if the depth of penetration given by (167) is small compared to the radius of the surface near which the currents flow, then the parallel-plane formula (170) may be used for the stack impedances Zi(yo) and ^2(70) which are to be substituted into the equations of Section II for the attenuation and phase constants, namely ^ Zi(7o)/pi + Zi{y^lpi , . a = Re ^ — , (176) 27J0 log (P2/P1) o / , T^ Zl(7o)/pi + Z2(70)/P2 /.-„x jS = coVmo€o + Im — — J— . (177) 2770 log (P2/P1) If the plane approximations are regarded as insufficiently accurate, one can compute the surface impedance of a cylindrical stack by repeated multiplication of matrices similar to the one given by equations (88) of Section III. This procedure would obviously involve considerable numeri- cal computation, but we can hardly expect that it would reveal anything qualitatively new for Clogston cables of the proportions considered in Part I. LAMINATED TRANSMISSION LINES. I 925 It will bo instructive to comi)aro the impedanee of a laminated plane stack with the impedance of a solid metal ])lat(^ over the full fre(iuency ran^e from zero to ver>- lii,u;li ficMiueiicies."' If the stack contains /( con- ducting laycM's, each of thickness d , and the metal plate is of thickness J\ = nti , the impedances of the plate and of the stack will be e(|ual at zero frequency, and also at \-ery high fre(iuencies where the fii'st layer of the stack is alnnidy many skin depths thick. For simplicity we assume that l)()th the plate and the stack are backed by infinite-impedance surfaces at all fretiuencies. To orient ourselves we shall (l(>fine tln-ee critical frequencies, for which n^spectix'cly the thickness of the solid plate is equal to one skin depth in the metal, th(^ thickness of the stack isecjual to one "effective skin depth", and the thickness of a single conducting layer is eciual to \/3 skin depths in the metal. These frequencies are /2 = VS/(jrfiigihT^ = VSnfi {T, = Ta), (178) fz = 3/(TrnigA) = Sn'fy {h - V35,). The approximate forms of the surface impedance functions of the plate and the stack in the various freciuency ranges are then quite simple. In the range 0 ^ / ^ /i , the surface impedance of the solid plate is approximately constant and given by Z„(Tn) ^ l/giT, , (179) while in the I'ange.f ^ A we see approximately the surface impedance of an infinite plate, Zo(to) ;^ (1 + i)/gA = (1 + iW^i^^^J/gi , (180) which is proportional to \/f. The surface impedance of the stack is approximately constant in the range 0 ^ / ^ /2 , where Zo(Tn) ^ l/r/iT^i , (181) while in the range /2 ^ / ^ /s it is approximately eciual to the impedance A'l of an infinitely deep stack of moderately thin layers as given by the first of equations (164), namely Zo(7o) ^ (l/VS + i)TrnAf, (182) '* In this connection see also ■Reference 1, Fig. 2, ]). 494. Clogston compares a laminated stack with a solid i)late of the same total thickness as the stack, hence a plate which contains more conducting material than the stack. 92G THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 which is directly proportional to frequency (and independent of con- ductivity). For/ ^ /a the stack acts much like an infinitely thick solid plate, for which Zo(7o) ^ (1 + i)/gA = (1 + i) VWT^ , (183) an impedance again proportional to -\/f. The real parts of the approximate expressions for surface impedance may be plotted on log-log paper, where power-law relationships are represented by straight lines, to give quite a good idea of the way in which the stack resistance varies over the entire frequency range. To show how the exact resistance departs from the approximate formulas in the transition regions, we have calculated the resistance of a particular stack over the full frequency range from equation (172), and also the resistance of the corresponding solid plate from the formula Zoiyo) = (1 + OVmT^ coth [(1 -f i)V^mgJTr], (184) and plotted the results, together with those for an infinite plate and an infinite stack, in Fig. 7. The actual numerical values were chosen solely for ease in plotting, and are of no particular significance. It should be noted that the exact curves oscillate slightly around the asymptotic lines in the transition regions. For example, the resistance of the laminated stack is actually higher than the resistance of the solid plate at certain frequencies slightly above /s . These oscillations appear clearly in the numerical results, but are scarcely visible on the plots because of the logarithmic compression of the upper ends of the frequency and re- sistance scales. We shall next obtain an expression for the rate at which the surface impedance of a laminated stack begins to depart from its dc value as the frequency is increased. For this purpose we must expand the various factors appearing in equation (170) for Zo(7o) in powers of ©. Using the expansions (163) and (164) which have already been derived for r, A^i , and K2 , it is a matter of straightforward if tedious algebra to show that: e =1--^U- ^ U-f---, (I80) 6 360 in sh nF = — ; ■2V3 ■ . e^ _ (175n' - 48) e ^ 30 12600 , (187) LAMINATIOD TRANSMISSION LINES. I 927 J- — n r 1 7 •■ n r Aie + A 26 A'iA'2 sh nV = (15n - 4) _ (175n^ - 70n - 16) "^ 30 4200 ^ + 1 (ir)/i + 4) ., (l7o7i- + 707i - 16) 30 0 4200 0' + 1 in (cjihY qVs 0' + (188) (189) (190) By siibstitiitiiig the above series into equation (170), we can obtain the variation of the stack impedance with frequency so long as /i/5i is sufficiently small. Although in principle there would be no difficulty in taking into account an arbitrary sheath impedance Z„(yo), for brevity we shall restrict ourselves here to the case in which the sheath impedance is so high that at all frequencies of interest the current in the sheath may be neglected. Then we have equation (191) (see next page). < 5^2 : - ^y ^ _,^ i^\ ^ E INFINIT PLATE E ^ y - ^' / / ,^ ^ " / E ^ ^ / ^INITE TACK - FINITE ,PLATF ,^ ^ y^ FINITE STACK /INF i^ ^ ^ l^ ?1 / - / / 1 1 / 1 1 1 z / 1 1 ~ i^l / / Ifs H 1 1 /in 1 1 \ 111 1 .1. Lllll 1 1 1 111 \ 1 111! 1 1 1 III 2 5 12 5 FREQUENCY f Fig. 7 — Surface resistance H of solid i)lates and laminated stacks versus frequency/ on log-log scale. 928 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 which can be reduced to ^0(70) = ^ (3?i - 1) q2 _ (on'' - 1) ^4 , ngi^i L 6 180 .2,2 1 gVT^iL^ ' h\ ' 9^1 (192) the last expression being valid if the number of double layers is not too small {n ^ 5, say). To this approximation the fractional changes in the resistance and reactance of the stack are (193) ^ =T\A ^ T\t\.Mg\f Ro 981 9 AX T:ti ^ = -TT = TAttm^U (194) where i?o = l/giTx (195) is the dc resistance. From the exact calculations described above it appears that (193) and (194) are valid up to the neighborhood of the critical frequency h = V3/(TMihTi), (196) at which frequency the approximate formulas yield AR/Ro = 1/3, AX/Ro = V3. (197) For/ > /2 , however, these approximations rapidly break down. We may now answer the question: What must be the thickness ti of the individual conducting layers in a plane stack which contains a given total thickness Ti of conducting material, if at a specified top frequency fm the resistance of the stack is not to have increased by more than a specified small fraction of its dc value? We find that the permissible value of ^1 is TTHigiliJm y lio and we note that this value of ^i is inversely proportional both to fm and to Ti . If we measure h and Ti in mils and/„ in Mc -sec^^ then on putting LAMINATED TRANSMISSION LINES. I 929 ill the numerical values of m and g^ for copper, we have Kjmhlc\i Umils f ^0 For a plane Clogston 1 with stacks of equal thickness, the attenuation constant is given by (174), and the fractional change in attenuation with frequency is equal to the fractional change in resistance of either stack, as calculated from (193). For a coaxial Clogston 1 with stacks thin enough so that the plane approximation is valid we may also use (193), but the fractional changes in resistance will be different for the two stacks if these are of different thicknesses, and the fractional change in the attenuation constant must be calculated from equation (17G). If Rw and R20 are the dc resistances "per square" of the two stacks, and ARi and ARo their increments as obtained from (193), then the fractional increase in attenuation is given approximately by Aa ^ ARi/pi + AR2/P2 (200) oco Rio/ Pi + R20/P2 For either plane or cylindrical geometry we find that if we scale up a particular Clogston line by multiplying the thicknesses of the stacks and the main dielectric by the same factor, then the low-frequency at- tenuation constant will be divided by the square of the scale factor. However, the permissible thickness of the individual conducting layers, if we are to have the attenuation flat to a specified degree up to a fixed frequency, is inversely proportional to the scale factor. Thus if we double the overall dimensions of the line and double the amount of conducting material in the stacks, we shall divide the low-frequency attenuation constant by four, but we shall have to make the individual layers half as thick in order to maintain the same relative increase in attenuation constant at the same top frequency fm • In addition it is clear that if we double the top frequency while maintaining the same requirement on Aa/aa for a line of giv^en dimensions, we shall also have to cut the thick- ness of the individual layers in half. As a numerical example, let us return to the cable whose specifications were given })y (155) at the end of Section IV. For this cable we have: Pi = 55.49 mils dsi = 8.4(3 mils p.> =181.44 mils ds. = 4.04 mils (201) P2/P1 = 3.270 R20/R10 ^ S1/S2 = 2.094 930 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 If the conducting layers are copper, we find that equation (200) for the fractional increase in attenuation becomes, numerically, Aa/ao ^ 0.121(^i)Li8/L . (202) If for example the copper layers are 0.1 mil thick and the polyethylene layers 0.05 mil thick, since we are assuming 6 ^ 2/3, then the attenuation constant has increased by 10 per cent of its "flat" value at a frequency of about 9.1 Mc-sec~ . We may also ask for the upper crossover frequency, above which the Clogston cable will have a higher attenuation constant than a standard air-filled coaxial of the same size. Such a crossover frequency must exist because the dielectric loading of the Clogston cable (in our case eor = 6.78) introduces a factor -s/eor ii^to the asymptotic expression for the attenuation constant at extremeh^ high frequencies when the stacks look like solid metal walls; in addition there will be slight differences due to the fact that the geometric proportions of the conventional and Clogston cables are not exactlj^ the same. We assume, subject to a posteriori verification, that the upper cross- over frequency lies between the critical frequencies f-i and /s , defined by (178), for each stack. Then we have in effect infinitelj^ deep stacks of moderately thin laminae, whose surface resistances are equal and are given by (182) to be Ri = R.2 ^ 7rM//V3 = 5.79 X 10"'(;i)„u,s/mc ohms. (203) The attenuation constants of the conventional and Clogston cables are obtained from (151) and (176) respectively, where for the conventional coaxial we set 770 = rj^ . After a little arithmetic we find for the upper crossover frequency in this particular case, /mc ^ 2.79/(«i)Li8 . (204) Thus if the copper layers are 0.1 mil thick, the upper crossover frequency is about 280 Mc-sec~ , which turns out to lie well inside the interval between the critical frequencies /o and /a for both stacks. Comparing this result with the result at the end of Section IX, we see that a 0.375-inch Clogston 1 cable with 0.1 -mil copper conductors and the other specifications given by (155) is nominally better than a con- ventional air-filled coaxial cable of the same size in the frequency range from about 1 Mc -sec" to 280 Mc -sec" . We are still neglecting the effect of failure to .satisfy Clogston's condition exactly, the effect of stack non- uniformity, and dielectric losses. All of these factors will be present to a greater or less degree in any physical embodiment of a Clogston cable, LAMINATED TRANSMISSION LINES. I 981 and will i('(lu('(\ or in cxtrcMiu^ cast's wow eliminate, tli(> freiiueney ranji;e over which the C'lo^ston cable exhibits lowei' loss than a conx-ent ional coaxial cable. VI. EFFECT OF DIELECTRIC MISMATCH We may think of Clogston's relation (102) as a condition imposed on the phase N-elocity in a laminatctl transmission line to maximize the depth of eddy current i)enetration into the stacks. If this condition is not exactly satisfied, that is, if the mo^o protluct of the main dielectric is not eciual to the fie product of the stacks, then the effective skin depth of the stacks is finite at finite fre(|uencies and decreases with increasing freciuency e\'en in the ideal case of infinitesimally thin layers, while if the layers are of finite thickness the effective skin depth is even less than it would be with a perfectly matched main dielectric. The losses in the stacks at moderate frequencies where Clogston's penetration effect is of importance are correspondingly increased by the presence of dielectric mismatch. For a quantitative discussion we define the amount of dielectric mismatch A(;Uo€o) by A(/i(ieo) = MoCu — Me, (205) and also the dielectric mismatch parameter /,■ by k = ^(^"^"^ = (1 - S) A(/ioeo) .,-,Q^.x In terms of k, the general expressions for T, Ki , and Ko in a plane stack of finite layers take a relatively simple foi'm. We have = iM'iC- — Miifnl (207) = -/coMid + l:)ti = -(1 + Idvi'Tifi ^^ —{i -\- k)r]iyKiti , after a little rearrangement, where the only approximation that has been made so far is to set rjiy ^ r?, and ki '^ ai . Substituting (207) into (86) and (87) gives ch r = ch t) - 1(1 + /Ot) sh t), (208) and 932 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ^1 = -r [Kl + k)S + VKl + ky& - (1 + k)@ coth e + l], (209) K2 = -^ [-UI + A-)© + Vi(l + AO^e^ - (1 + k)@ coth 0 + l], where as usual 0 = a,h = (1 + t)/i/5i ;^ Kih . (210) If /c = 0, equations (208) and (209) evidently reduce to (158) and (159) of the preceding section. For a stack of infinitesimally thin layers, the constants T( and K are given by equations (93) and (94) of Section III, namely (w lie — co>oeo) (-2ik)"d/di, (211) K = Tr/g = {-2ik)^/g,h. (212) Up to this point we have set no restrictions on the magnitude of A", and we have not even assumed that k is necessarily real. Throughout the rest of this section, however, we shall assume that A; is a positive or negative real number, as it must be if there is no dielectric or magnetic dissipation. In practice both the lamina thickness and the amount of dielectric mismatch will be as small as it is feasible to make them. It will be useful, therefore, to obtain approximate expressions for r, Ki , and Ko under the assumptions I 0 I « 1, I A; 1 « 1. (213) Then equation (208) yields sh' ir = i(ch 0 - 1) - id + A;)0 sh 0 ^ _fc 2 _ (1 + 2k) 4 _ (214) 4 48 If I A: I « 1 we can neglect 2A; compared to unity in the coefficient of 0 , but since we have made no assumptions as to the relative magni- tudes of I 0 I and I A- I, we cannot drop either the term in kQ" or the term in 0*. If we replace sh |r by §r in (214), we get LAMINATED TRANSMISSION LINES. I 933 T ^l-kB' - (-)Vl2]* ^ (215) -iisgn k)lVWJWT^' - l^iA)?}, wliere \\v liaxc (akcn (li(> s(iuar(» root of the coiiipk^x (luaiiiily by Dwight 58.2, aiul + 1 if /v > 0, sgn k = (216) - 1 if /v < 0. Similarly, from (209), ^1^1 L ■(1 + k) ^2 e^+0^_,_(Lzi^^^e^_...] ^ [h@' + ev-fc - 0V12] iti 1 j^^ ^^ ^2 - iXsgn fc)[\/i(^iA)^ + 9k' - Wi/SiTf}, K2 ^ :;y^^ {[VHh/W+~9k' + K^i/5i)']' - Vs/i/^i - «"(sgn /c)[\/i(■)« coth B + 1]^ coth nV], which simpHfios, foi- iiifiiiitcsimally thin hiyors, to Z„(7n) = K cotli V,s. (228) It" th(> stuck is ninny (effective skin (lei)ths thick, we luix'C Z„(7.i) = A'l , (229) while if tlic in(h\i(hial hiycrs arc inhnite.simally thin, Zo(7o) = K , (230) where A'l and K are given by (209) and (211), respectively. When Zn(7o) is known, the attenuation and phase constants of the j)arallel-phine Clogston 1 are given as usual by a = Ke Zii(7o)/77o^, 0 = coVMoeo + Im Z„(7i,)/r7,|6. For the coaxial cable we use 2^i(7o)/pi + Z2(7o)/p2 = Re (8 = coVmoco + Im 27^0 log (p2/pi) ' 2r/o log (p2/pi) (231) (232) (233) (234) but the impedances of the cylindrical stacks are easy to compute only if we can employ the parallel-plane approximation for each stack. To take Ao I.U o.a 06 0.4 02 0 1 \ \J \ y / \ V — — J - — Fig. 8 — Relative skin (lf'|)th \^\i, in a stack of finite layers versus dielectric mismateti jiarameter A-, measured in units of {[\lh\Y. 936 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 curvature effects into account would require a considerable amount of numerical calculation. Equation (98) of Section III provides an explicit expression for the surface impedance of a cylindrical stack of infini- tesimally thin layers in the presence of dielectric mismatch, in terms of Bessel functions of complex argument; but if the layers are of finite thickness we can at present do nothing better than multiply out the matrices of the individual layers step by step. The variation of the surface impedance of a laminated stack with frequency over the full frequency range is not quite so simple in the presence of dielectric mismatch as when Clogston's condition is exactly satisfied, but a somewhat analogous discussion may be given. As in the preceding section, we consider a plane stack of n conducting layers each of thickness h , where n(i = Ti , and backed by an infinite-imped- ance surface. When the mismatch parameter is k, the three critical frequencies are: /2 = V3/(7rMi6fi/iTi\/rT3^^0 , (235) = A/3n/i/VrT3^- (2^1 = Ta), /s = 3/(7rMig/i) = 3n'/i (^1 = V35i). In the range 0 ^ / ^ /2 , the surface impedance of the stack is ap- proximately constant, being given by Zo(7o) ^ l/giTi . (236) In the range f-> ^ f ^ fz , we have Zo(7o) ^ K, , (237) where Ki is given by (217) provided that k is small compared to unity. For infinitesimally thin layers the upper critical frequency fs is infinite, and we have for / ^ /2 , Zo(yo) ^ I k p(l - i sgn k)/gA . (238) = (1 - t sgn fc)\/7rMi I k I f/gi , which is proportional to s/f. If the layers are of finite thickness but k = 0, we have the result obtained in the preceding section, Zo(7o) ^ (l/\/3 + i)TiXitJ, (239) which is proportional to / up to the critical frequency fo . If neither the mismatch parameter k nor the layer thickness h is zero, then the surface LAMINATED TKAxNSMISSION^LINES. I 937 impedance ^0(70) cannot be represented bj'' a simple power of / in the range jt ^ / ^ /s . At frequencies above /» , if the hiyer thickness is finite, the impedance is approximately that of a solid conductor, namely ZoCto) ^^ (1 + 0/{/i5i = (1 + i)V7rfIJ7i, , (240) whicii is proportional to \/J. Since in general the surface resistance depends upon the two param- eters ti/8i and k, it is not possible to plot a single curve which shows the variation of resistance with frequency under all possible conditions of dielectric mismatch. However if we compare a matched stack of finite layers with a similar mismatched stack, we see that the asymptotic behavior of Zoiyo) is the same for both stacks at very low and very high frequencies. A numerical study of the exact equation for Zu(yo) shows that in the neighborhood of the critical frequency /2 , the resist- ance of the mismatched stack is higher than the resistance of the matched stack. (The critical frequency /2 as defined in (235) is a function of the mismatch parameter k, but will be of the same order of magnitude for a slightly mismatched stack as for a perfectly matched stack.) The resistance of the mismatched stack exhibits relatively small fluctuations above and below the resistance of the matched stack in the neighborhood of the upper critical frequency /s , but this region is not of as much prac- tical interest as the region near /2 , where the stack resistance is defi- nitely increased by the effect of dielectric mismatch. An explicit expression for the rate at which the surface impedance of a mismatched stack begins to depart from its dc value as the frequency is increased has been worked out only for the ideal case of infinitesimally thin layers. For a plane stack of infinitesimal layers backed by an in- finite-impedance surface, equation (228) gives, at moderately low fre- quencies. K r. . (r^sY (TrsY 2ikTl 4k^T{ 381 4551 from which the fractional changes in resistance and reactance are AR 4k'n ^kW^glTlf (241) Ro 4551 45 ' AX ^ j2kTl ^ J2kiniigyTlf Ro 35i 3 (242) (243) 938 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 The admissible value of | A; |, if the fractional change in resistance is not to exceed a specified value AR/Ro at a given top frequency /„» , is k\ = 3\/55l /aR ^ 3V5 /aR . . 2T\ y Ro 27rM JmTl y Ro ' ^^ ^ which is inversely proportional both to fm and to the square of the total thickness of conducting material in the stack. If we express Ti in mils, fm in Mc-sec~ , and assume the conducting layers to be copper, we get 22.71 /AE (/mJMcl-' l)mi)s f Ri /■ (245) lo The variation with frequency of the surface impedance of a matched stack of finite layers at moderate frequencies (say / ^ /2) is given by equation (192) of Section V; but no simple formula has yet been de- rived for the surface impedance of a mismatched stack of finite layers in this frequency range. The derivation of such a formula would appear to involve nothing more than some rather formidable algebra, the diffi- culties centering around the fact that in the general case we can make no a priori assumptions as to the relative magnitudes of k and (ti/8i) . It is reasonable to suppose, however, that if both dielectric mismatch and finite lamina thickness contribute appreciably to AR/Ro , the per- missible values of I /c I and ^i individually will be less, if we are to achieve a given flatness of the attenuation versus frequency curve, then the permissible value of either if the other factor were unimportant. To exhibit the effect of dielectric mismatch from a slightly different point of view, we may plot the surface resistance of an infinitely deep plane stack of moderately thin layers (a finite stack several effective skin depths thick would show essentially the same behavior) at a fixed frequency, as a function of the mismatch parameter k. The surface re- sistance is just Re Ki , which may be obtained from (217) if k and ti/Si are assumed small compared to unity. Fig. 9 shows the dimension- less quantity Re gidiK, = ;^ [Vl(ti/8^y + 9^:^ + Wi/^iYf, (246) for the three values h/Si = 0, /i/6i = 0.1, and ti/8i = 0.2. For an elec- trically thick solid conductor we have simply RegAK, = 1; (247) hence to get any benefit from the laminated stack we must have Re gidiKi smaller than unity. Actually, if we meet Clogston's condition by laminatp:d traxsmissiox lines. I 989 ^ .^ N N V t, V y \ ^. A "% ^ \, / / ^ ^ y// \ II \ 1 -0.08 -0.06 -0.04 -0.02 0.08 0.10 Fig. 9 — Xormalized stack resistance Re g\h\K\ versus dielectric inisniatch l)arameter fc, for different values of i\lh\. raising the dielectric constant and thus lowei'ing the impedance of the main dielectric, then since the attenuation constant of the line is pro- portional to the ratio of stack resistance to dielectric impedance, we must have Re ^i5i/vi considerably smaller than unity to obtain a lower attenuation with the Clogston line than with an ordinary air-filled line having solid metal walls. For a plane Clogston 1 line with stacks of ecjual thickness, the frac- tional change in the atteimation constant with freciuency is equal to the fractional change in the resistance of either stack, whether this change ari.ses from the effects of finite lamina thickness or from di- electric mismatch or both. The fractional change in the attenuation con- stant of a coaxial Clogston 1 depends not only on the change in resist- ance of each stack, but also on the geometric proportions of the cable, in the manner expressed by e(iuation (200) of Section V. The effect of dielectric mismatch on the overall attenuation versus frecjuency characteristic of a Clogston cal)le is in general to reduce the total frequency range (in Mc-sec~') over which the Clogston cable has a smaller attenuation constant than a conventional air-filled coaxial cable of the same size. To calculate the lower crossover frequency we may ordinarily neglect finite lamina thickness effects and use equation (241) for the stack impedances, while at the upper crossover freciuency the stack impedances are very nearly e(|ual to A'l , as given l)y (217). 940 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 It should be remembered that mismatch of the mo^o product of the main dielectric will usually be accompanied by a change in the dielectric impedance \/)Lto/co • Thus under certain conditions the lower crossover frequency may even be reduced by choosing eo slightly below the Clog- ston value, inasmuch as the increase in dielectric impedance may more than compensate for the increase in stack resistance at low frequencies; but it appears that this will be paid for in a steeper slope of the attenu- ation versus frequency curve and a consequent greater reduction of the upper crossover frequency. It would be very useful to make a numerical study of the effects of dielectric mismatch in Clogston cables having a variety of different pro- portions; but in the present paper space limitations restrict us to a few observations concerning orders of magnitude. For the cable which we considered at the end of the preceding section, it turns out than an increase or decrease of 1 per cent in the value of eo makes a change of at most a very few per cent in either crossover frequency ; with a matched dielectric, we recall, these crossover frequencies were about 1 Mc-sec~ and about 280 Mc • sec" respectively. However if we had designed a laminated cable with thicker stacks or thinner laminae or both, so as to increase the theoretical factor of improvement over a conventional cable in the working frequency range, we should have found that the tolerable deviation of co from Clogston's value, instead of being of the order of 1 per cent, was more nearly of the order of 0.1 per cent or even smaller; and the greater the improvement striven for, the more stringent the re- quirement of accurate dielectric match. VII. DIELECTRIC AND MAGNETIC LOSSES IN CLOGSTON 1 LINES Dielectric and magnetic dissipation in the main dielectric and in the stacks can be taken into account by introducing complex dielectric con- stants and permeabilities for the lossy materials. Thus we may write (248) w^here in the most general case the loss tangents may all be different, though it will be assumed that they are all small compared to unity, so that the problem may be treated by first-order perturbation methods. Co = eo — Uo = eo (1 — i tan <^o), / C2 = «2 - leo = 62 (1 — i tan (f)2), 1 Mo = Mo . // / , — ?/io = Mo (1 - i tan ^j). Ml = Ml — iHi = Hi (1 — i tan fi), M2 = M2 - iHo = H2 (1 — i tan f 2) , LAMINATED TRANSMISSION LINES. I 941 The avoraso rule of enorj>;y dissipation por unit volume in a lossy di- cloctric hy a harmonically xaryinji; elect lic field of maximum amplitudes A' is just \u)t"l'j~, since the imaginary part e" of the complex dielectric constant coti-esponds to a conductivity ij = we". Similarly the average rate of energy dissipation per vuiit volume in a lossy majj;netic mateiial by a harmonically varying magnetic field of maximum amplitude // is ^o:n"H'. The part of the attenuation constant which arises from di- electric and magnetic dissipation is one-half the ratio of power dissipated per- unit IcMigth of line to total transmitted power, provided of course that the attenuation per wavelength is small. Since the loss tangents of the various materials are assumed small, we can use the fields found for the lossless case to calculate the transmitted and dissipated power. If the volume occupied by currents in the stacks is small compared to the volume of the main dielectric, so that we can neglect the power flow in the stacks in the direction of wave propagation compared to the power flow in the main dielectric, then the part of the attenuation constant which is due to dielectric and magnetic dissipation is given by equation (51) of Section II, namely ad = lo^VM^Ctan 00 + tan fo) = "" V"'"'"' ^^an ')(tiin 02 + tan {') -.coVm,,.,, 1 + (2V36)V^^4^ ' wliich ivdiiees to (249) if \V(> ne<>;l('ct the terms in s b. The total attenua- tion is tlie sum of the metal losses, 2 + tan f), llT Opi AP2 "^ ^co/i' — — -— (tan 4>2 + tan f). zr op2 (266) The part of the attenuation constant which is due to dielectric and magnetic dissipation is therefore ^ APo + APi + AP2 "' 2(Po + Pi + P2) P2 1*"/^' / S\ S>\ log — (tan 00 + tan fo) + ^yi I ^ + ~ ) (tan ^2 + tan t) _ 1 /-n Pi 3 jUQ \Pi P2/ log '-^ + i 4/?^? (^ + 'A Pi 3 K ^^e' \pi P2/ We need scarcely point out that if the loss tangents are not small compared to unity, it may be impossible to satisfy Clogston's condition (102) very closely with a real value of 6, and the resulting mismatch may reduce the depth of penetration and increase the metal losses in the stacks. In practice, however, the loss tangents will be of the order of 0.001 or even 0.0001, and matching the imaginary parts of nata and Jit will be much less of a practical problem than matching the real parts. Appendix I BESSEL FUNCTION EXPANSIONS Let Pi and po be the inner and outer radii of a cylindrical shell and let the thickness t, given by t = p2 — pi , (Al) be less than pi . Then, following Schelkunoff, we may replace the Bessel functions appearing in equation (68) of Section III by their Taylor ex- pansions, namely 1 S. A. Schelkunoff, Bell System Tech. J., 13, pp. 561-562 (1934). LAMINATED TRANSMISSION LINES, I 945 n-o n! KoUpi) = KoUpi + kO = 2-/ — 7- -^o" (kpi), ri=o n! /i(kP2) = hixp-i) = X^ — - /o" ^ (kpi), n=o n! (A2) /m(k-p,) = -Koixp,) = -E ^ Al"+^\.pO. „=o n! It follows that where «> (A" Ko{Kpi)IliKp-2) + /Vi(kP2)/o(/vPi) = — 2J r ^n+l(«Pl)) °° f -/")" 7\o(k-Pi)/o('vP2) — Ku{KP'>)Iit{Kpl) = —22 r~ '^"('^Pl)) n=o n! /Vi(/CPi)/i(kP2) — A'i(kPo)/i(k'Pi) = 23 r ^n+l('^Pl)) 00 /An /Vi(kpi)/(i(kp2) + Iu{kp2)Ii(kpi) = 2^ — r- ^nC^'Pl), A„(.r) = Io(x)K',"\r) - Ko(x)n''\x), B,,(x) = /o(.r)/vr'(.r) - A'o(^-)/o"^(:r). (A3) (A4) The quantities .4„(.r) and B„(.r) turn out to be finite polynomials in 1, .r, the general expressions for the coefficients having been derived in a rather inaccessible monograph by Pleijel. When x is large, however, the leading terms are quite simple. From Pleijel's analysis, or directly by substituting the asymptotic series for Io(x) and Ko{x) into (A4), we find AUx) = 1/x + 0(l/x'), A2m+i(x) = -mix + 0{\lx), 2 4 (A5) Bim{x) = vi/x + 0{\/x ), where m is a positive integer or zero. If we substitute these approximations into the first of equations (A3), we obtain 2 H. Pleijel, Berdkning af Motstand och Sjdlfinduktion, K. L. Beckmans Bok- tryckeri, Stockholm, 1906. 946 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Ko(kPi)Ii(kP2) + KiiKpo)Io(Kpi) J_ y (Kt)"" 1 ^ (m + DUO""""' Kpi 1 m=o i2m)l (kpiY m=o (2w+ 1)! = — ch K^ liP2 1 (xpi)- _ 1 rf 2/cpij ch /v'/ — ^ -J- {x sh x) 2 a.r 1 (A6) 2(kpi)- sh d. The other three equations may he treated similarly. Doing so, and remembering that P2/P1 = 1 + //pi , (A7) we obtain the results which were quoted in Section III, namely Kp'iiKoiIn + Ki2l(n) 1 + TT- I ch d — - — sh d, Kp2{KoiIo2 — K02I01) KpiiKnIvi — Knhi) zpi ' + 2k] ''' ^■^' 1 + -^1 sh d, -plj 2kpi (A8) / 2pi_ ch d + - — sh d, 2kpi Kp2{KnL)2 + Knoln) up to first order in t/pi . Table of Symbols Note: Rationalized MKS units are employed throughout. The sub- scripts 0, 1,2 applied to symbols representing material constants, such as €, /i, g, (T, and r], have the significance that 0 refers to the main dielec- tric in a Clogston line, while 1 refers to the conducting layers and 2 re- fers to the insulating layers in the stacks. Bars denote a^'erage ^'alues. Subscripts not included in the present table are explained in the context where they are used, a, (B, e, 3D: Elements of the general circuit parameter matrix (Section III). a: Distance between outer sheaths of plane Clogston line. Radius of inner core of coaxial Clogston line. b: Thickness of main dielectric in plane Clogston line. Inner radius of outer sheath of coaxial Clogston line. LAMIXATKD TRANSMISSION LINES. I 947 C: A parameter related to the degree of iioimiiiforniity in a laminated medium (Section XII). E: Electric field intensity; coordinate subscripts indicate components. /: Frequency. g: Electrical conductivity. g: 6gi ; average conductivity parallel to laminated stack. //: Magnetic field intensity; coordinate sul)scripts indicate components. /;: —iKo', a transverse separation constant (Section X). / : Electric current. J: Electric current density; coordinate subscripts indicate components. K: Characteristic impedance of stack of infinitesimally thin laminae. A'l , K> : Characteristic or iterati\'e impedances of laminated stack (introduced in Section III). /,•: A parameter related to dielectric mismatch in a Clogston 1 line (Section VI). M: The general circuit parameter matrix (aCBCSD-matrix). m: A mode numbei'. n: Number of double layers in a laminated stack. p: A mode number. q: A parameter related to the propagation constant in a Clogston 2 line (Section XI). R: A-c resistance of a laminated stack. r: Ratio of attenuation constants of Clogston and conven- tional lines (Section XII). s: Thickness of a laminated stack. Si , So : Thicknesses of inner and outer stacks in a coaxial Clog- ston 1. T: Total thickness of conducting material in a laminated stack (subscripts explained in context). Ta : Total thickness of conducting material in one effective skin depth. t: Thickness of an electrically homogeneous layer. Time. ^1 : Thickness of a single conducting layer. ^2 : Thickness of a single insulating layer. V: Electric potential. w: An abbreviation for II y in Section XII. 948 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 X: AC reactance of a laminated stack. x: Rectangular coordinate in the direction of magnetic field in a plane Clogston line. y: Rectangular coordinate in the direction normal to the stacks in a plane Clogston line. Z: Surface impedance of a plane or cylindrical boundary; ratio of tangential components of the electric and mag- netic fields (subscripts explained in context). Zk : Characteristic impedance of a transmission line. z: Rectangular coordinate in the direction of wave propa- gation. a: Re 7; attenuation constant. /3: Im 7; phase constant. T: Propagation constant per double layer normal to lam- inated stack. r^ : r/(fi + ^2); average propagation constant per unit dis- tance normal to laminated stack. 7: Propagation constant in longitudinal direction. A: Effective skin depth; the depth at which the current den- sity in an infinite plane stack has fallen to 1/e of its value at the surface. A small change in a quantity. 8: \/2/o3iJLg; skin thickness in a solid conductor. e: Dielectric constant (capacitivity or permittivity). e: €2/(1 — 6); average dielectric constant measured normal to laminated stack. €, : e/€i, ; relative dielectric constant. €v : Dielectric constant of vacuum; 8.854 X 10~ farads -me- ter" . e', e'': Real and (negative) imaginary parts of complex dielectric constant. f: taiC^(n''/fi'); phase angle of complex permeability. ■q: \/iwm/ (g + ^'we) ; intrinsic impedance of medium. 77„ : Intrinsic impedance of vacuum; 376.7 ohms. rjy , r]p : 7j(l — y'/aY; characteristic impedance looking in the y- or p-direction in a homogeneous medium. 0: (1 + i)ii/5i ; a parameter related to the electrical thick- ness of a conducting layer. d: h/(ti -\- ^2); fraction of stack volume filled by conducting layers. (c: (o- — 7 )'; transverse propagation constant in the y- or p-direction in a homogeneous medium. LAMINATED TRANSMISSION LINES. I 940 A: A parameter related to the propagation constant in a Clogston 2 (Section XII). X: Wavelength. Xi, : Free-space wavelength. H'. Permeability. p.: dm -{- (\ — 6)pL2 ; average permeability measured parallel to laminated stack. Hr : m/mi> ; relative permeability. /i^ : Permeability of vacuum; 47r X 10^ hcnrys • meter^ . n', n" : Real and (negative) imaginary parts of complex per- meability. ^: y/a -\- \; normalized coordinate transverse to a plane Clogston 2 line (Section XII). p: Radial coordinate in cylindrical system. pi , p2 : Inner and outer I'adii of main dielectric in coaxial Clogston line. : Angular coordinate in cylindrical system. Phase angle, tan~^(e'V«')) of complex dielectric constant. x: —iV( ; a transverse separation constant. CO : Angular frequency in radians • second" . FUNCTION SYMBOLS Real part. Imaginary part. Natural logarithm. Hyperbolic sine. Hyperbolic cosine. Bessel functions of the first kind. Bessel (Neumann) functions of the second kind. Modified Bessel functions of the first kind. Modified Bessel functions of the second kind. Re: Im; log; sh: ch: J f) , Ji No ,N, h, /i: Ko ,Ki Electrical Noise In Semiconductors By H. C. MONTGOMERY (Manuscript received June 3, 1952) Transistors, diodes, and single crystal filaments of germanium have com- mon noise properties: a spectrum varying inversely with frequency, and strong dependence on the biasing current. Theoretical attempts to explain this noise are reviewed briefly. Experiments with single crystal filaments indicate that the noise resides in the behavior of the minority carrier. In one type of experiment, the correlation of noise voltages in adjacent por- tions of a filament is quantitatively related to the lifetime and transit time of minority carrier. In another, the effect of a magnetic field on the noise is found in accord with calculated changes in lifetime of the minority carrier. In the development of the transistor it was recognized quite early that electrical noise in the device was considerably in excess of Johnson noise, particularly at low frequencies. Noise having a similar spectrum had been observed many years earlier in microphonic carbon contacts carrying a current, and in copper oxide rectifiers, composition resistors, and crystal diodes. Flicker noise in vacuum tubes appears to be a re- lated phenomenon. A number of attempts have been made to deter- mine the mechanism of production of noise of this sort, but none have been particularly successful. In this paper we will first surve}^ the more important characteristics of noise in germanium diodes and transistors. This will be followed by a partial hypothesis as to the nature of the noise mechanism. We will then discuss experimental work on noise in filaments of single crystal germanium carrjnng a dc current. These experiments strongly support the hypothesis, and in fact led to its formulation in the first place. I. NOISE IN DIODES AND TRANSISTORS There are many similarities in the noise phenomena found in diodes and transistors of both the point contact and junction type. It seems likely that the noise mechanism is similar in all these devices. One of the most characteristic features of the noise in such structures 950 ELECTRICAL NOISE IN SEMICONDUCTORS 951 is t ho spectrum. Tho spectral dousity fpowcM- per unit baiulwidlli) \-:irics in\-orsch' as the tVcciucMicy, according lo llic relation (IW f "(If wiiei'e the exponent lies between L and 1.") with an average about 1.2. This type of spectrum will \)c referred to as a 1/f spectrum. Measure- mcMits of the spectra of silicon point contact diodes have been reported by P. 11. Miller^ for the freciuency range 20 cycles to 300 kilocy(des. Spectra of point contact transistors measured by the author have been reported elsewhere"' "* for the range 20 to 15,000 cycles. Typical spectra foi- p-n junction type diodes and transistors are shown in Fig. 1. Almost witiiout (\\c(>ption, oiu' measurements and those reported in the litera- ture have shown the 1/f spectrum over most of the frequency range covered. There is some evidence from the related fields of flicker noise and carbon microphone noise that the 1/f spectrum may extend to fre- quencies well below 0.1 cycle per second. Some departures from this type of spectrum have been noted in the neighborhood of 100 kc, as shown in the curves. 1000 800 ai 80 O 60 1 2 4 6 8 10 20 40 60 100 200 400 FREQUENCY IN KILOCYCLES PER SECOND Fig. 1— The spectrum of noise in ii-p-n transistors varies inversely with fre- quency. 952 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 / 1 / ^/ / \ _ / y^ 1 1 1 // r J / ^y / I c/ / ^ . / «/ y 1 1 1 1 A '/c / r / ---/:" / ' \ L / /'" f / \ / r \ -'A i / / / 1 / IJ / A- PRODUCT p-n JUNCTION A / E / B - c - EXPERIME^ POINT CON JTAL p-n JUNCTION TACT UNIT FORWARD BIAS REVERSE BIAS / A y i y 0 y / / 2 5 -4 2 5-3 DC BIAS CURRENT IN AMPERES Fig. 2 — The short-circuit noise current from a point contact or junction diode generally increases with dc bias current. A second characteristic feature of noise in all semiconductor devices is that it is current dependent. In the absence of biasing current only- Johnson noise is observed. When biasing current is present the noise power may be as much as three or four orders of magnitude above Johnson noise. As a general thing the noise increases as the bias is in- creased, although some minor exceptions to this rule are noted, usually at bias values where the slope of the current-voltage curve is changing rapidly. To illustrate the bias-dependent behavior, the noise properties of some germanium diodes of various types are shown in Fig. 2. The short circuit noise current in a 1-cycle band at 1000 cycles is plotted as a func- tion of dc bias current, some of the data being for forward bias, but most for reverse bias. Several curves are shown for each type of unit, and ELECTRICAL NOISE IN SEMICONDUCTORS 953 these are typical of the variations encountered. There is a general ten- dency for noise current to increase in proportion to bias current, l)ut in limited regions the individual units may have slopes considerably dif- ferent from unity. It would perhaps be more logical to plot cui'rent densities rather than total cm-rents, but because of the general form of the relations this makes little difference in the overall picture, und there is some difficulty in estimating the appropriate area for the point contact units. There is an almost unlimited number of different ways of representing noise data. For example, noise current, current density, voltage, or available power may be expressed as a function of various bias parameters. Of a good many combinations tried, none gave an outstandingly simple picture of noise behavior, and the representation used in Fig. 2 is probably as good as any for an overall picture of diode noise. The noise behavior of transistors depends on two bias parameters. Selection of the emitter cm-rent and collector voltage for the parameters usually leads to a rather simple representation. It often turns out that the noise behavior as an amplifier over the commonly used range of bias ^•alues depends largely on the collector voltage and is relatively inde- jjcndent of the emitter bias. Data of this sort were shown for point contact transistors in a previous reference, and have been given for an n-p-n transistor by Wallace and Pietenpol. A somewhat more com- plete family of curves is shown in Fig. 3 for a recent n-p-n transistor. A few attempts have been made to determine the effect of tempera - a 20 ^-o - -:rA r" ,— ^ "^ Fl S2 r^ EMITTER BIAS - IN MILLIAMPERE5 1.0 ^.^_ ^^'y' ^ y l^.'^n L^ ^ ^^' —i 0.1 _ o" ^J v^ K>. r" zzi b' 1— — -o — • -p- r"o5^ 1 1 1 1 1 1 0.2 20 40 60 0.6 0.8 1.0 2 4 6 8 10 COLLECTOR BIAS IN VOLTS Fig. 3 — The noise figure of an 7i-p-n transistor depends in a fairly simple way on emitter current and collector voltage. 954 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ture on noise behavior. Such experiments have been rather unsatisfac- tory because the changes in impedance and gain characteristics as a function of frequency are of the same order as the changes in noise prop- erties. This makes the interpretation ambiguous. B}^ and large, such experiments suggest that changes in noise with temperature are rather small, perhaps of the order of the change in absolute temperature, and not at all like the exponential changes associated with a diffusion process. This observation does not necessarily rule out a diffusion-like noise process; it might indicate merely that we are not looking at the right part of the spectrum to observe exponential changes with temperature. II. A HYPOTHESIS REGARDING THE NOISE MECHANISM Considerable work has been done on the theory of current-dependent noise having a 1/f spectrum. Among the earliest was that of Schottky^ in connection with flicker noise in vacuum tubes. He considered the arrival of foreign atoms on the emitting surface of the cathode as a random series of events governed by a diffusion law with a charac- teristic time constant, and arrived at a 1/f rather than a 1/f spectrum, and a highly temperature sensitive process. Surdin pointed out that by postulating a series of decay processes with suitably distributed time constants a 1/f spectrum could be achieved. From physical arguments regarding the emission process from cathodes, INIacfarlane' obtained a range of relaxation times and a 1/f spectrum, in a process which was highly temperature dependent. Richardson gave a very general anal- ysis of the noise properties of sj^stems in which the conductivity was governed by a diffusion process. One conclusion was that a geometrically simple diffusion process in one, two, or three dimensions could not lead to 1/f spectrum, although by some highly specialized assumptions about the geometry of a contact surface he was able to obtain such a spectrum. DuPre, in considering a hj^pothesis somewhat resembling that of Sur- din, showed that the required range of activation energies was phys- ically reasonable, and that the assumptions could be set up in such a way as to make the process relatively temperature independent. Several of the abo^•e authors and Van der Ziel discuss the physical basis for applying flicker noise theory to the noise in semiconductors. Although this theoretical work has contributed a great deal to distinguishing be- tween suitable and unsuitable mechanisms, there is still no specific physical theory of noise in semiconductors which can be tied in a (juan- titative manner to experimental results. The experimental work described in the remainder of this paper has ELECTRICAL NOISE I\ SKMICONDUOTORS 0")") led to a hypothesis regarcHiis the noise meclianism, which is by no means a compk^tc^ exphination, hut which nuiy h(^ a useful step in that dii-ec- tioii. This hypothesis resulted largely from tlie experimental woik, hut it seems worth while to desci'ihe it first to h(>lp appreciate the siji;nifi- cance of some of the experimental results. It has been observed that in many semiconductor structures the noise \'oltag"e is approximately proportional to the dc bias current. This relation sufi;jj;ests that tiie noise is tiu^ result of fluctuations of the conduct i\ity of tlie materiah wiiich modulate the bias current and j)ro- duce a tiuctuatino; voltage across tlie specimen. Such fluctuations in conducti\-ity could result from variations in concenti'ation of the mi- nority carrier (holes in /;-type material, electrons in p-type). The mag- nitude of the ol^served noise and the type of spectrum seem to demand that the fluctuation be coarse-grained in time to a much greater extent than could i)e accounted for by random statistical fluctuations of carrier density. Experiments of Haynes^ on lifetime and transit of injected car- riers in rods of germanium have occasionally indicated finite sources of minority carriers in the material. Our hypothesis is that such sources of carriers are rather generally distributed over the material (although mostly too small to be noticed in experiments of the Haynes type), and that their acti^•ity is being modified at a slow rate by some unspecified local influence in a suitable way to agree with the observed noise spec- trum. The experiments described below involving noise correlation phenom- ena and the effect of a magnetic field on noise point strongly to an im- portant role for the minority carrier in the noise mechanism, and hence strongly suggest some such hypothesis as that just described. III. NOISE IN SINGLE CRYSTAL FILAMENTS It was found se^'eral years ago that a filament cut from single crystal germanium of high purity exhibits noise well above Johnson noise when a dc current is flowing in it. It is not clear whether this noise arises in the body of the material or on the surface, but to date no method of preparing the sample has eliminated this noise, and it is a prominent feature even at bias fields as low as 10 volts per centimeter. This noise seems to have most of the characteristics of the noise in diodes and transistors: it has the l/f spectiiim, is current dependent, and is quite stable with time. It has been the subject of considerable study in the hope that a better understanding of it would illuminate the whole field of semiconductor noise. 956 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Samples, referred to as "bridges", have been cut from thin slabs of single crystal germanium, by a technique devised by W. L. Bond, often of a form shown in Fig. 4. Side arms for both the current and the noise measuring electrodes have been found necessary to avoid spurious noise at the electrodes. A large inductance in the bias circuit greatly reduces the effect of any noise voltage generated at the bias electrodes. The spurious noise power from this source is seldom more than a few per cent of that being measured. It should be noted that the contact area for the noise measuring electrodes should not be on a por- tion of the specimen carrying bias current, otherwise spurious noise may be generated at these electrodes. Typical dimensions for the straight central filmanent of the bridge are 0.05 x 0.05 x 0.7 cm. The side arms have sandblasted surfaces to suppress holes or electrons injected at the electrodes. The central portion may be etched, sandblasted, or other- wise treated at will. The enlarged circular areas are rhodium plated to provide good contacts to each side arm. Measurements of the noise spectrum in such bridges with several dif- ferent etching treatments and with sandblasted surfaces are charac- terized by the 1/f spectrum over a wide frequency range.* Fairly ex- tensive measurements have been made in the audio frequency range, and a few covering the range from 20 cycles to 1 megacycle. A typical spectrum is shown in Fig. 5. The current dependence of the noise is shown in Fig. 6 for a number of samples, mostly n-type, one p-type, and with various resistivities. The outstanding feature is that noise voltage always increases with dc bias voltage. In many cases there is direct proportionality at the lower bias values, increasing to a square law at higher biases. There are some axj LENGTH .a BOUT 7 MM TO NOISE MEASURING AMPLIFIER Fig. 4 — Filament with side arms cut out of a single crystal of germanium. * Departures from the 1/f spectrum at frequencies of the order of 100 kilo- cycles and above were first discovered by G. B. Herzog and A. Van der Ziel. See Reference 13. ELECTRICAL NOISE IN SEMICONDUCTORS 957 to'* 102 1 2 5 ,2 5 -.2 5 A Z 5 ^2 5 n 10 10^ I0-* 10^ 10^ 10^ FREQUENCY IN CYCLES Fig. 5 — Typical spectra of noise in single crystal filaments carrying a dc cur- rent. exceptions to this trend. Also, there are large variations in the magni- tude of the noise. An average unit shows a noise voltage about three times Johnson noise at a bias of 10 volts per centimeter. The noise behavior at reduced temperatures has been investigated. Results on three different bridges are shown in Fig. 7. The open circuit noise voltage is shown as a function of temperature for constant bias voltage. Although the curves show rather large irregularities, there seems to be no general trend for noise to decrease with decreasing tem- perature over the range covered, from — 200°C to room temperature. The surface treatment applied to a bridge may affect the noise very substantially. A sandblasted surface usually gives the lowest noise. Etching the surface may raise the noise voltage by a factor of ten or more, though the iV' resistance changes only a few per cent. The tech- nique of washing and drying the surface may have an important effect 958 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 too 80 d 50 ^40 o o o - 30 to 20 O o to - / - / 9 TYPE ^ t f p-TYPE / I / V / 1 5 / / 1 / 3 1 =!ESI ■^ O 5TI\ HM- MTY -cm: / / f' / 1 It "/ „.d /A J I - / f ' M // / i - / / /// fJ j 1 / // / ^ 7 / V/j ^/ r J / J 'a ^ / ^/. / t / / / ^^ V / 1 1 /^ 1 1 I 2 3 4 5 6 8 10 20 30 40 50 60 80 100 BIAS FIELD IN VOLTS PER CENTIMETER Fig. 6 — Variation of noise with dc bias in single crystal filaments. on the noise. Some of these processes also affect the hfetime of carriers in the bridge to a large extent. However, there seems to be no direct and simple relation between the two effects, since treatments have been foinid which change the noise by a large factor with almost no effect on lifetime, and vice versa. Fig. 8 shows measurements of noise voltage on several dozen bridges at a uniform bias of 10 volts per centimeter, all having sandblasted siu^- faces, mostly of n-type but a few of p-type germanium, and with widely different values of resistivity, produced by varying impurity concentra- tions. There is considerable scatter in the results, but there is a fairly obvious tendency for noise voltage to increase in proportion to resis- tivity. Since Johnson noise also increases in proportion to resistivity in a structure of fixed dimensions, the conclusion is that with constant bias voltage the ratio of current induced noise to Johnson noise tends to be ELECTRICAL NOISE IN SEMICONDUCTORS !).")'.) -200 -180 -160 -140 -120 -100 -60 -60 -40 -20 0 20 40 TEMPERATURE IN DEGREES CENTIGRADE Fig. 7 — \'ariati()ii of noise with temperature in .single crystal filanuuits. independent of the resistivity of the material. From the data it also appears that there is no consistent difference between n- and p-type material. Noise does not appear to depend on orientation of the filament with respect to the crystal axes. Filaments orientated along the 100, 110, and 1 1 1 directions and I'otated in several wavs about these directions showed 200 UJ ^100 > 80 o 60 o o ^40 <^ LU O ? 20 _i O > UJ in 10 i 8 - 6 1- 5 4 UJ 2 1 - • n-TYPE ▲ p-TYPE • • • • • • • - • - > • • - • • • - A A • • • • • • .-" ^^ ^' ▲ • • • ,,'-' ,-'' - .- L.o' >■ X - ^e • • ^'' .-' ,^ ,.-' 1 1 1 1 1 1 .1. 1 1 1. 0.1 0.2 40 60 100 0.4 0.6 1.0 2 4 6 8 10 20 RESISTIVITY IN OHM-CENTIMETERS Fig. 8 — Variation of noise with resi.stivit_y in single crystal filaments. 960 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 no significant differences in noise behavior. It should be noted, however, that small variations might be hidden in the large scatter in the data from undetermined causes. IV. NOISE AND MAGNETIC FIELDS An important role for the minority carrier in the noise mechanism was first clearly indicated in experiments on the effect of a magnetic field on noise in germanium filaments. It has been found experimentally that the noise in a single crystal filament may change by a substantial factor when the filament is subjected to a steady transverse magnetic field. The following discussion will show that this behavior is in har- mony with the hypothesis of noisy injection of minority carriers, as set forth in a preceding section.* The physical picture on which this treatment is based involves the random injection of holes into an n-type filament by hole sources which may be either in the interior or on the surface of the filament. f It is assumed that the spectrum of the noise arises from the fluctuating na- ture of the noise source. The effect which any source has will depend upon the lifetime of the holes which it emits. If these holes remain in the filament for a long time, they will produce more noise than if they remain in the filament for a short time. We shall be concerned with the effect of magnetic fields upon these lengths of time and shall not deal in this paper with the fluctuations of the noise sources themselves. If a transverse magnetic field is applied to an n-type germanium fila- ment, a Hall effect voltage is set up and the holes will be deflected to- wards one surface of the filament. Since recombination takes place prin- cipally at the surfaces, this may cause a substantial change in the lifetime of the holes. In order to determine the effect of the magnetic field on the noise we proceed along the following lines. (a) We assume that the observed noise is due to fluctuations in the conductivity of the filament produced by fluctuations in the hole con- centration. Since these fluctuations are small, we may take the change in conducitivity to be proportional to the change in average hole den- * The following semi-quantitative theory of the dependence of noise on mag- netic field is taken with some modification from unpublished work of W. Shock - ley and H. Suhl, on the basis of which the calculations leading to the curves of Figs. 10 and 11 were carried out. It is hoped that this work may be published in the near future. t To simplify the terminology, the discussion is based on n-type material with holes as minority carrier. An exactly similar argument could be made for p-type material with electrons as the minority carrier. There is some experimental evi- dence of the similarity of behavior of n- and p-type germanium, though most of the experimental work has been done with n-type. ELECTRICAL NOISE IN SEMICONDUCTORS 9i)l sity. (b) We restrict the noise measurements to frequencies low enougli so that the period is long compared to the lifetime of a hole. It is tlien evident that the contribution of a hole source to the noise is proportional to the fiuctuatin<>; hole current jicnerated ])y the source and to the aver- age lifetime of the holes. This lifetime depends on the position of the source in the filament, the absorption properties of the surfaces and the electric and magnetic fields.* (c) We assume that the generation prop- erties of the soiu'ces are unaffected by the magnetic field, hence, the calculation of the effect of the field on the noise reduces to a problem of calculating the change in lifetime produced by the field, (d) We neglect body recombination in comparison with surface recombination. In ger- manium filaments of the size usually dealt with, this approximation causes only a small error in the lifetime, (e) Individual soui'(;es (or at any rate groups of sources over regions small compared to the dimen- sions of the filament) will be considered to be statistically independent; therefore, the total effect on the noise can be determined by summing the squares of the contributions from individual sources. Hence we wish to evaluate the following expression: Change in noise power at field H = {t (H))/{t (0)) (1) where the symbol () indicates an average over all the noise sources. The statments (a) to (e) represent the principal assumptions in de- veloping the theory. In order to calculate t as a function of the magnetic field, H, we con- sider a steady state case in which a current of holes Jo is injected into a region in which the average lifetime is t. If the density of holes in the region is p(x, y, z), the total number is P = I p{x, y, z) dx dy dz. However, P = Jor/q, where q is the charge carried by a hole. Therefore, T = "T / vi^, y, 2) d^ dy dz (2) Jo J This is the method of evaluating r which is used in the qualitative dis- cussion which follows, and also in the calculation of the curves of Figs. 10 and 11. * It should be pointed out that a consetiuence of the hole injection theory of noise in a filament is that marked frequency dispersion should occur when the frequency- being studied is high enough so that a i)eriod is short compared to the lifetime of holes in the filament. However, we shall neglect this important and interesting aspect of the problem. 962 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 Three cases will be treated. In all of these it will be supposed that the width of the filament parallel to the magnetic field is relatively large, so that effects from the edges can be neglected. Also, we are concerned only with average effects over the long dimension of the filament. This permits us to deal with a one-dimensional problem. We shall consider first the case in which holes are supposed to be injected from the sur- faces, and the two surfaces have equal and rather large recombina- tion rates. In Fig. 9, part (a) shows how holes injected from each surface are distributed across the thickness in the absence of a mag- netic field, and part (b) shows the distribution with a moderate field. The form of these distributions may be determined from the following arguments. H=0 H>0 THICKNESS ' THICKNESS' (a) (b) Fig. 9 — Excess hole density across the thickness dimension, (a) with no magnetic field, (b) with moderate magnetic field. If we suppose a steady hole current Jo emitted from the left-hand surface of the filament, then a relatively high concentration pi of holes will appear directly in front of the surface. Some of these holes will re- combine upon the surface, the rate Ji being given by Ji = PiSq where S is the recombination constant for the surface. The balance of the holes will diffuse through the filament to recombine upon the right surface at a rate J2 = p^Sq and we note that Ji -\- J2 = J. Because of the high recombination rate, P2 w'ill be very small; hence, J2 will be much smaller than Ji . In the ab- sence of a magnetic field the gradient is uniform, and the concentrations will be linear, as shown in part (a) of the figure. An identical argument ELECTKU'AL XOISK l.\ SKMIt'OMDUCTORS 91)3 applies to pr , the coiu'eutratioii of holes emitted from ihc li^ht-haiid side. UndcM- the iiilliuMice of a magnetic field pusluii rifi;ht, the eoneeiit rations will ('hanj«;<' to those shown in part (b) of the fignre. The magnetic field will {)nll holes through the filament and tend to pre- voni diffusion from right to left. For some moderate value of field, the value of ./■> is not increased enough to change ./i appreciably, so the value of pi is nearly the same as with no field. At tlu^ same time the efTects of diffusion are suppressed by the field so that the concentration pi extends nearly to the right side of the figure. By the same action, (he concent i-ation of holes emitted from the right surface drops to zero very quickly. From the curves of Fig. 2 and relation (2), we see that the ai'ea under the density curve, and hence the lifetime of holes injected at the left is at most doubled by the magnetic field, while the lifetime of those in- jected at the right is reduced nearly to zero. Recalling that the noise is proportional to a summation of the scjuare of the lifetimes, we see that the noise power is at most doubled at a suitable value of magnetic field. Higher \'alues of field will sweep so many holes to the right-hand surface as to substantially reduce pi , so at very high fields the noise decrea.ses monotonically to zero. Thus it is seen that the noise behavior is the result of competing ten- dencies. On the one hand, the magnetic field helps holes escape from the surface at which they are emitted, but on the other hand it tends to push these holes against the opposite surface and thereby reduce their lifetime. The relative importance of those two tendencies depends on the surface recombination properties and the strength of the magnetic field. Calculation of the lifetime along the lines just discussed involves solu- tion of the continuity eciuation ax- ax with suitable boundary conditions. The results of such a calculation carried out by Shockley and Suhl in tlie work already referred to are plotted in Fig. 10. , In order to make the results independent of sample dimensions, the following parameters are used. The first i)arameter is proportional to the applied magnetic field, and is defined as the effecti\e transverse potential in units of kT/q: 964 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ^ = fIt- = l72tEH X 10"' (3) /vi/g where t = thickness of the filament (cm) H = magnetic field (oersteds) E = applied electric field (volts per cm) Eh = effective transverse component due to Hall effect (volts per cm) q = unit electronic charge kT = Boltzman's constant X absolute temperature. The constant may be derived by noting that kT/q is 1/40 volt at room temperature, and that the effective transverse field. Eh , may be ex- pressed as follows. (See Reference 14, Section 8.8.) Eh = eE = (On + dp)E = (m„ + Hp)HE X 10"' = 4.3 X 10~' HE where 9 = Hall angle ju„ = Hall mobility for electrons (2800 cmV volt-sec) fXp = Hall mobility for holes (1500 cm /volt-sec). The other dimensionless parameter is proportional to the rate of surface recombination, and is defined as the ratio of the surface recombination velocity to the diffusion velocity from the center: t/' = st/2D = st/8Q where s = recombination velocity characteristic of the surface (cm/ sec) D = diffusion constant (cm /sec). The numerical constant is given for holes at room temperature. The noise changes are expressed in decibles, that is, ten times the common logarithm of the ratio of noise powers with and without the magnetic field. A second case is that in which generation and recombination are on the surfaces, but the two surfaces have unequal absorption properties. It might be expected that rather large increases in noise would result when the magnetic field was poled to pull holes away from the surface with high absorption properties, and this turns out to be the case when the calculations are carried out. The results are shown in Fig. 11 for a ELECTRICAL NOISE IN SEMICONDUCTORS 9G5 1 ^ - VOLUME GENERATION SURFACE RECOMBINATION CONSTANT yj-. ■ 100 vr«--^ "■■ 1 V- ■-^ 10 ^ s ^ ^ N, " ■^ \ \ •v 00 s \ V 1.0 '*. -^^ •>v 5^ = 0.1 ^^^ 4 6 8 10 12 14 16 18 20 22 MAGNETIC FIELD PARAMETER, § Fig. 10 — Calculated magnetic effect for similar surfaces. 24 26 28 /^ ^ *^ A ^, = 0.1 5^2 ='0 B ^, = 0.5 ^2=S0 / A-2 ^ ^ / 1 f ^^ _— - -— _ >•._ ■"^ B-2 "■■ n / \, f \ s. / 1 \ \, 1 11 N \ ^1 / \ V \ V t / / h n \ \ \j V / / •} \ U'l A t / ^N B-1 .'" ^' / f N ^^' / / V ■"N ^1. -16 -14 -12 -10 -8-6-4-2024 6 8 10 12 14 16 18 20 MAGNETIC FIELD PARAMETER, ^ Fig. 11 — Calculated magnetic effect for dissimilar surfaces. Curve 1 of each pair is for the contribution from the surface having the lower recombination constant. 960 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 14 12 10 8 6 4 2 0 -2 -4 -6 -8 -10 n ^1 ---Lif^ ^ ~'5 / / , / n.^i^ ^lI^—— / / 1 JK^ /^ ^^ ^?^, "^^C];^ ■*"^i^ ^^ ^^^-''' / / ** X^ ^-' •12 -8-4 0 4 8 12 16 20 MAGNETIC FIELD PARAMETER, $ Fig. 12 — Experimental magnetic effect for dissimilar surfaces. case where the two recombination parameters are 0.1 and 10, and for a second case where the parameters are 0.5 and 50. In this figure, sepa- rate curves have been shown for noise due to holes generated on each of the two surfaces. The total noise would be gotten by adding the noise powers represented by the two curves after appropriate weighting for the contributions of the two surfaces. At present we do not see any way of determining the weighting factor. A third case is that in which it is assumed that the noisy generation of holes is uniform throughout the body of the filament, but that re- combination takes place on the surfaces only. These assumptions seem at first sight to be in contradiction to the statistical mechanical prin- ciple of detailed balancing, which states that under equilibrium condi- tions all processes occur with equal frequency in the forward and reverse directions. Thus it would seem that if holes are generated in the interior, we must consider recombination in the interior also. Actually this is not necessary under the non-equilibrium conditions which prevail during noise measurements. There is no necessity for the noise generated by a source and a sink for holes to be simply related to the strength of this source. Thus we may suppose there are relatively weak sources and sinks for holes in the interior, but that the hole absorption and genera- tion of the sources is very noisy compared to the recombination and generation processes on the surfaces. If this is the state of affairs, most ELECTRICAL NOISE IN SEMICONDUCTORS 9()7 of the noise will bo <>;eiienited in tlu> inlei'ior, hut a hole }»;enerate(l in the interior will be much more likely to recombine on the sui'tace. The (lotted (•ui'\'e of Fiii;. 10 has been calculated assumiii<>; a uniform dis- tribution of noise sources throujihout the interior of the filament and ('([ual and very larj>;e i'ecoml)ination constants for the two surfaces. It is S(HMi that for this rase the reduction of lifetime i)i'(Ml()miiiates, ;ind there is a monotonic decrease in noise witii increasint>; magnetic held. Experimental work has given ivsults which in most cases are in fair (lualitative agreement with the calculated relations. Measurements for three filaments, each of which had one high recombination and one low recombination surface, are shown in Fig. 12. The recombination param- eters, as shown on tlu^ curves, were of the order of ^ = 10 for one sur- face, and \f/ = 0.5 for the other. The general shape of the curves is cjuite similar to the calculated curves of Fig. 11. The maxima are of the right order of magnitude, and occur at reasonable values of the field param- eter $. The lack of detailed agreement between the measured and cal- culated curves is not surprising, because the experimental conditions did not fulfill the assumptions made for the calculations in several re- spects. The filaments were neither wide enough nor long enough so that edge and end effects could be overlooked. The recombination properties of the surfaces could not be measured directly, but had to be estimated from other filaments which had been similarly treated. One experi- mental ciu've shows a secondary maximum on the opposite side of the origin. This might indicate a defective portion of one surface having an anomalous recombination constant. Experimental results are shown in Fig. 13 for four filaments, each of which had nominally equal recombination (constants for the two sur- faces. These may be compared with the calculated curves of Fig. 10. It will be noted that the experimental curves are not symmetrical about $ = 0. This lack of symmetry is probably due to dissymmetry in the zy ^^.^=1^ ■=^-=^: =-. — .^^ — ''' _ •— • ^--^ ^■^' — ■ ^•5^-' , " '"' ■^7.6 -12 12 -8-4 0 4 8 MAGNETIC FIELD PARAMETER, <|) Fig. 13 — Experimental magnetic etTect for similar surfaces. 968 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 samples, particularly the fact that one surface of each filament was cemented to a support, which would probably change the surface recom- bination properties somewhat. Aside from the lack of symmetry, the behavior of the two filaments with the higher recombination constants is in reasonable agreement with the calculated curves. The filaments with the lower recombination constants are in poor agreement with calculated values, in that the noise does not fall off with increasing field nearly as fast as calculated. The cause of this behavior is not under- stood. These experimental curves may also be compared with the dotted curve of Fig. 10, calculated on the assumption of volume generation and surface recombination. The similarity is quite poor in all cases. The somewhat better agreement with the surface generation calculations than with the volume generation calculations is not the basis for anything more than a very tentative feeling that the experimental results sup- port the surface generation viewpoint. While there are many discrepancies in detail between the experi- mental and calculated relations between noise and magnetic field, these are at least partially understandable in terms of the differences between the experimental setup and the theoretical model. The high degree of qualitative agreement considerably strengthens the hypothesis of noisy injection of minority carriers as an important element in the noise process. V. NOISE CORRELATION PHENOMENA The noisy hole injection hypothesis leads one to expect certain corre- lation phenomena in the noise voltage observed in neighboring portions of a filament. Consider first noise measurements at a frequency so low that the transit time of a hole* along the filament is negligibly small. This might be a frequency of one kilocycle in a typical experiment. The holes have an average lifetime, from which can be determined an average life path, which is defined as the product of the lifetime by the drift velocity under the existing electric field. Noise voltage measure- ments across segments of the filament much shorter than a life path should be highly correlated, since nearly all the holes which make a transit of one segment will make an almost simultaneous transit of the other segment. On the other hand, noise voltages across segments much longer than a life path should show little correlation, because most of the holes appearing in the two segments are from different sources, and the sources have been assumed to be statistically independent. * As before, the concepts apply equally well to electrons in p-type material. ELECTRICAL NOISE IN SEMICONDUCTORS 909 A second situation arises when noise is measiu'ed at frequencies high enough so that the transit time of holes between segments is not neg- ligible. In this case we should expect the correlation between the noise voltages to be improved by incorporating in one channel of the meas- uring circuit a delay equal to the transit time between segments. In order to calculate the correlation resulting from the first situalion, we set up a theoretical model based on a few simplifying assumptions: (a) The noise process may be represented by an array of nois}' hole cur- rent generators which are statistically independent; (b) These generators are uniformly distributed along the filament over the segments where the noise is to be observed, and for a suHicient distance on either side to produce uniform conditions over the segments; (c) The hole currents from the generators decay exponentially with a decay constant deter- minal)le from the lifetime; (d) Measurements are made at low enough frequencies so that time of transit of holes may be neglected. We will consitler later an alternative to the second assumption. The correlation coefficient between two voltages of instantaneous values Vi and Vo may be defined as Pi2 = ViV2/{vl X vD 1/2 where the bars represent time averages. To evaluate this expression, the contribution of a single generator to the noise voltage in each segment is determined by integrating over the appropriate portion of the decay curve. The total contribution from all generators to the mean voltage product and the mean squared voltages is then determined by inte- grating the product or square over all the generators. The details are carried out in the appendix, and lead to the solid curve of Figs. 14-16, in which the ordinates are the correlation between noise voltages in two segments of a filament and the abscissae are the ratio of life path of a hole to the segment length. In an experiment the lifetime r of holes remains fixed, determined chiefly by the recombination properties of the surface. Consequently the life path / is proportional to the hole velocity, which is determined by the electric field, according to the relation / = tijlE where E is the applied field in volts per centimeter and n is the drift mobility of holes. Hence, by \-ai\ving the biasing voltage a large range of life path values can be obtained. The correlation is measui'ed by carrying the noise voltages through separate amplifying channels having identical pass bands extending 970 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 from 800 to 1300 cycles per second. A switching ai'rangement makes it possible to apply either of the output voltages or their sum or difference to a rectifier-meter combination. From the readings of the meter the correlation can be computed according to the relation Pl2 = (>S' - If)/4:VJ^2 (6) Vi and V2 are rms values of the individual noise voltages, and S and D are the rms values of their sum and difference. The equivalence to expression (5) can be seen by noting that S' D' = (vi + ^2)^ — (^1 — V2)' = -4^11^2 Results of correlation measurements on three bridges are shown in Figs. 14-16. In each case the calculated curve is shown for reference. The values of / were calculated from decay measurements on optically injected holes, as described by J. R. Haynes," using a value for mobility of 1700 cm /volt-sec. In Fig. 14 the agreement with the theoretical model is very good. The scatter in the points is due to fluctuations in the noise, which are quite large in the band used for these measurements. In Fig. 15 the agreement could be made quite good with a lateral shift - 0.6 2 0.5 o ^•^ •* - • .-< • • • • ^ • < • • •/ • / ;/ • • / /' > • / • • ^ / 1 1 1 0.1 0.2 03 0.4 0.6 0.8 1.0 2 3 4 6 8 10 2C RATIO OF LIFE PATH TO SEGMENT LENGTH Fig. 14 — Noise correlation. The solid curve is calculated, the points experi- mental. ELECTRICAL XOISE IN SE.MICO.NDUCTOHS 971 0.9 O 0.7 0.6 Z 0.5 O ^ ^ -^ ^ • • • y /•-^ '< • / 4 • t / / , / / / / / • / / / r /I / / A / / / ^ y / / • " ' • \ 1 0.1 0.2 0.3 0.4 0.6 0.8 1.0 2 3 4 6 8 10 20 RATIO OF LIFE PATH TO SEGMENT LENGTH Fig. 15 — Noise correlation. The dotted curve includes allowance for losses at the side arms. by a factor of two. In Fig. 16 the form of the experimental curve seems different from that calculated. In particular, the slope is steeper, and the curve tends to level off at a correlation of about 0.8. It seems pos- sible to explain the discrepancies between the experimental data and the calculations on the basis of two considerations which were not in- cluded in the model, (a) The pair of side arms separating the two seg- ments of the filament serve to drain off some holes which would other- wise contribute to the correlation. The dashed curve in Fig. 15 shows the calculated effect, on the assumption that the absorption in the side arms is equivalent to an extra section of filament equal in length to half a segment. The actual distance across the side arms is only 20 per cent of a segment, but it is not hard to believe that the decay rate in this region might increase by a factor of two or three due to the reduced elec- tric field and loss of holes down the side arms, (b) The model assumed a unifoim distribution of noise sources along the filament. There is ex- perimental evidence that the distribution may be quite spotty. This can have a substantial effect on the form of the correlation curve. For example, the dashed curve in Fig. 16 shows the curve calculated for noise sources lumped at the mid-point of each segment. Other assumed positions might shift the curve considerably along the horizontal axis. 972 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 0.9 ulO.S LU _l u 5 0.7 O o 2 0.6 < 2 0.5 O 1- jO.4 LU Ct a 8 0.3 OJ O0.2 0.1 0 .^ ^ ^•'•.:4^ pr ;-^ •^••» <•*•> • V/'' • / / / / / t • / / / / 4 • • / / / y / • 1 • • k • : • t S • ^ • 1 • • • 1 1 1 0 1 0 2 0 3 0 4 0 6 0 8 1. 0 2 3 4 e e 10 RATIO OF LIFE PATH TO SEGMENT LENGTH Fig. 16 — Noise correlation. The dotted curve is calculated for lumped noise sources. In view of these considerations there seems to be very satisfactory agree- ment between the experimental results and the model. Another type of experiment involves noise measurements at frequen- cies high enough so that the transit time of a hole across a segment is an appreciable fraction of a cycle. In this case the correlation between noise voltages from adjacent segments can be improved by putting a time delay in one channel of the measuring circuit. Measurements were made by taking the noise voltages from the two segments through sepa- rate amplifying channels having identical pass bands extending from 17 to 24 kilocyles. The outputs of the two channels were put on the vertical and horizontal plates of a cathode ray oscilloscope, forming a sort of Lissajous pattern. The patterns differ from those obtained with sinus- oidal voltages in that the elliptical figures are filled in solid, due to the continual variation in amplitude of the noise. A phase shifting device is included in one channel, and as the phase is shifted to give optimum correlation, the elliptical pattern narrows down and approaches a line inclined at 45°. For a quadrature phase shift, the pattern becomes cir- cular, and in practice this setting can be determined more precisely than the in-phase setting, largely because the backgroinid noise in the circuit is less troublesome. With the phase shift for optimum correlation determined, the delay at the center of the pass band is easily calculated, ELECTRICAL NOISE IN SEMICONDUCTORS 973 and since the band is not very wide, the variation in dehiy over the hand is not important. Frt)m the drift mobiht,y of holes we may estimate the transit time between segments, according to the relation t = L/fiE whei(^ t = transit time, seconds L = distance between segment mid-points, cm A' = applied field, volts/cm M = mobility of holes, cm"/volt-sec. Data for a bridge of /(-type germanium of resistivity about 20 ohm- cm are gi\-en in Table I. The transit distance, L, after a small correction Table I E volt/cm Delay micro sec. Bridge Temp. "K. Mobility cm^/volt-sec. Transit Time micro sec. 10 21.1 298 1700 18.0 14 15.7 299 1690 12.9 20 11.8 301 1670 9.1 30 9.2 305 1640 6.2 40 9.3 313 1580 4.8 for reduced field across the side arm, was taken as 0.305 cm. As noted in the table, the bridge temperature rose somewhat at the higher bias values, and the assumed values of mobility have been modified accord- ing to the inverse three-halves power of the absolute temperature. The delay required for optimum correlation is shown in the second column of the table, and the calculated transit time between segments in the last column. It is seen that the two are in reasonably good agreement, especially at low fields. When the direction of the field is reversed, an ecjual delay is required, but in the opposite channel of the measuring circuit, as would be expected. Here, again, we have experimental evi- dence supporting the noisy hole injection hypothesis. The cause of the discrepancy shown in the table at higher fields is not understood. It is possible that trapping phenomena increase the transit time over that calculated from the mobility. There is some evidence for this sort of behavior in lifetime experiments, but to date there does not seem to be enough information for any estimate of magnitude of such an effect. VI. GENERAL COMMENTS These studies of electrical noise in semiconductors leave little doubt that the noise is closely related to the behavior of the minority carriers. 974 THE BELL SYSTEM TECHNICAL JOURXAL, SEPTEMBER 1952 It is not yet clear whether the noise is a surface or a volume property of the material, but it is well established that the surface properties have an important connection with the magnitude of the noise. From some of the experimental work it seems likely that the generation and recombination processes are separate and have different noise prop- erties. Because of the nonequilibrium situation, this does not violate the principle of detailed balancing. It seems probable that a more complete understanding of the generation and recombination processes and a clearer picture of the origin of noise in semiconductors may be expected to develop together. VII. ACKNOWLEDGEMENT The analysis leading to the theoretical relations between noise and magnetic field is the work of W. Shockley and H. Suhl, under w^hose di- rection the calculations leading to the curves of Figs. 10-11 were carried out. The continued interest of Dr. Shockley in the experimental work has been invaluable. The author is indebted to many associates for help- ful discussion of certain problems, and also for the construction of many of the devices and materials which entered into the experimental work. Appendix Suppose that a source of holes located at a point Xo in a filament pro- duces a fluctuating current of holes of rms value Ji in a specified fre- cjuency band. The hole current is swept down the filament by a field E and is assumed to decay exponentially according to the relation J = ./ie~^^-'°^/' (1) where the life path f may be expressed in terms of drift velocity v, hole mobility ju, and lifetime r C = Vt = ijlEt. Assuming that the frequency of measurement is low enough to justify neglecting the hole transit time, the noise voltage due to holes from a single source is proportional to the number of holes present in the seg- ment. This is obtained by integrating (1) over an appropriate range dv = Ji f e-(^-'«)/' dx _ {K.e^'^'^e-" - e-"'] x, < a (2) ~ |ki[1 - e-^^'°^^'] a < .To < h where Kx is an omnibus constant \\hich will cancel out in the final result. ELECTRICAL N'OISE IN SEMICONDUCTORS 07") Under the assumption that {]\o sonrces are statistically indcpciKlcnl , the total voltafio scinarcd is ol)tain('d l)y into,<>;ratinfj; the s(|naic of (2) oxer all t he sources. /*n 2 2 T I "ixoKr -all -(o+L)/k1^3d® oK UKI-AVS !)(S3 serious conditions. A sroup of coils aw su])jo('Iays in soi'Nicc in the tclophoiic plant on \\\v basis of nnmbci's of foniid oixmi tronl)l('s on both l>'pcs of contacts. As might bo o.xpoctcd the results Nai'iod widely, with the twin contact being superior ])y a factor of anywhere from 3 to 100 with pci'liaps 10 as a rc^isonablc lignre. MAGNETIC STABILITY Magnetic niateiials in rela>'s liaxc been found to change^ in tlieii' magnetic characteristics with time and temperatures to which they are subjected in their normal usage. This effect is known as aging. The dii'oc- tion of th(> change is such as to decrease the permeability and increase the coercive force of the material. The dogi'ce of change in certain ap- plications, such as i-elays in mai'ginal and time delay circuits, may be so large as to be of serious concern. A high grade of magnetic iron which has been extensively used in the telephone system has been found to age considerably under conditions simulating operation in the plant. Aging of iron is attributed to the pre- cipitation of impurities such as carbon, nitrogen, and oxygen. The solu- bility of these elements decreases with decreasing temperature. When iron is cooled from a high temperature, impurities, such as carbides and nitrides, do not have sufficient time to precipitate completely, so a super- saturated solid solution is produced. C'onsociuently the impurities tend to continue to precipitate slow^ly at low temperatures \vhere the diffusion rate is extremely sIoav, and internal strains are ])rodnced which affect the magnetic properties.'^ It has been found that if these parts are annealed in atmospheres of dry hydrogen instead of the ordinary "pot" anneal, this aging effect is greatly reduced. Not only is the aging effect reduced to where it is of no great engineering importance, but the magnetic properties of the material are improved. The maximum permeability is increased and the coercive force is decreased both by a factor of about two. The use of relays in critical applications is thus greatly enhanced. The degree by which magnetic materials change by aging may be determined readily by laboratory tests. Long time aging effects can be simulated by baking ring samples of the material or the relays at 100°C for several hundred hours and measuring the magnetic jn-opei'tios of the ring specimens oi' the operating charactei'istics of the relays befoi-(> and after aging. Foi' "pot" annealed magnetic iron the effect of such aging is to decrease the maximum permeability by about 50 per cent and to approximately double the coercive force. When the iron is hyth-ogen 992 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 annealed, the corresponding changes caused by aging will be about a 15 per cent decrease in maximum permeability and 15 to 20 per cent increase in coercive force. The improvement in aging effect on relay performance obtained by the hj^drogen treatment is illustrated in Fig. 7.' This was obtained on a design of relay having a closely coupled magnetic circuit for use in time delay circuits. The ordinates show the change in residual grams; hours of aging are plotted as abscissae. Residual grams represent the force with which the armature is held attracted to the core by the residual flux remaining in the magnetic circuit after the electrical circuit through its winding is opened. This force is a measure of the coercive force of the magnetic material. As was noted, the effect of aging is to increase the coercive force and hence the residual grams. For this relay, a change in residual grams will cause a change in the delay time of the relay under a given adjustment and is therefore important. How hydrogen annealing improves the pull characteristics of a relay is shown in Fig. 8. This was taken on a relay designed for sensitive and marginal circuit applications. The curves show the grams pull, plotted as ordinates, produced on the relay armature at a given air gap by various values of ampere turns on the relay plotted as abscissae. The abilit}^ of the hydrogen treated relay to operate given loads on considerably smaller currents is obvious. This improvement is due to the higher permeabilities obtained by the hydrogen anneal. There are other magnetic materials available for use in relays and in which the aging effect is practically non-existent or is considerably smaller than that just described. Several kinds of nickel-iron alloys known as permalloy are widely used in the telephone sj^stem where their W 120 2 O 100 < O 80 < I O 20 STANDARD "pot" ANNEAL^,^.-^''^ ^ ^ / /" A r / HYDROGEr Nl ANN EAL _ — /, --'' 0 100 200 300 400 500 600 700 800 900 1000 HOURS AGING Fig. 7 — Improvement in aging effect by hydrogen anneal. DESIGN FACTORS INFLUENCING RELIABILITY OF RELAYS 993 excellent magnetic properties are needed in difficult applications. They are substantially non-aging. Low silicon-iron alloys are being more widely emploj^cd. They have good magnetic properties and the aging effect is small. Where the intrinsically inferior magnetic properties of low carbon steel alloys, such as SAE 1010, can be tolerated they are used. While initially they have poorer magnetic characteristics than magnetic iron, their aging effect is considerably smaller. cti\'e is for a 40-year life. 'Die effects of weai- on peitoiinaiice to a great (^xtent can ofttimes be counteracted by ingenious design. Fig. 10 is an illusti'ation of such a case.'' Thv diagram on the left shows a moving system of a relay in which the contact springs ai-e stud ac- tuated. The moving spi-ings are tensioned toward the ai'mature and exert a force tending to open the contacts. When the armature operates, the stud presses the moving springs into engagement with the stationary springs. There is no contact force when engagement is first made and further flexing of the spring is necessary to build up the contact force to the desired \'alue when the armature reaches its fully operated posi- tion. As the contacts and studs Avear, it is apparent that the contact force and conseciuently the load on the armature decreases rapidly. The stud wear becomes cumulative in its effect on tlie outside pair of springs as more springs are added to the pile-up. The diagram to the right shows a moving system of a relay using what is called "lift-off" card actuation. The moving springs are ten- . STUD ACTUATION CARD LIFT-OFF ACTUATION I UNOPERATED STUD RING STAKED HERE tk: H =^ OPERATED r 5 B Fig. 10 — Two moving sj^steins of relays in relation to i\w. effeets of wear on their performance. 996 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 sioned, before assembly, toward the stationary spring by an amount necessary to give the desired contact force. Two supplementary springs are provided to support the card and tensioned to restore the armature and contact springs to their unoperated position. Upon operation, the motion of the card permits the contacts to close, and when engagement of the contacts occurs, the contact force reaches its predetermined value very rapidly. Further motion of the card, provided for by the width of the slot in the card, allows for wear of the contact and card without appreciably affecting the contact force or the load on the armature. The effect of this wear on the contact force is shown in Fig. 11 for both types of actuation. For the stud actuated relay, as the contact and stud wear continues, the contact force decreases very rapidly. After 0.010 inch wear only about 6 grams remains out of an original 26 grams. This is accounted for by the fact that the combined stiffness of the moving spring in engagement with the stationary spring is 2 grams per 0.001 inch deflection. This recjuires 0.013 inch contact follow to establish a contact force of 26 grams when the relay is adjusted initially. For the card "lift-off" actuated relay where the moving spring had been pre- tensioned to give a contact force of 25 grams initially, after 0.010 inch wear of the contacts, the contact force will have decreased about 1 gram. This is because the stiffness of the moving spring is about 0.1 gram per 0.001 inch deflection. Card wear does not affect the contact force so long as it is provided for by the width of the slot in the card. 24 ^ CARD •-■r-l OFF ACIUATION ~ , in 22 O - \ I Z 18 - N. \ o 16 a. £ 14 - \ 1 §10 - STUD \ ACTUATION \ 1 : < cc 6 ^ 4 - ^ n1 ; 2 - In. 0 L 1 1 1 1 1 1 .. J.-. I ll ,\ 0.010 ^^ DO li-Z i-z 0.008 ^- hZ ZO 0.006 ^5 o 0.004 QjE;^ UJx o> 0.002 '-' 0 0.002 0.004 0.006 0.008 0.010 0.012 STUD AND CONTACT WEAR IN INCHES Fig. 11 — Comparison of effects of wear on contact pressure of a relay. DESIGN FACTORS INFLUENCING RELIABILITY OF HKLAY.S 997 Fig. 12 — Magnetic circuit of a relay having embossed pole faces. Another instance where the effects of mechanical variations upon its performance have been largely nullified by design, is in the design of slow release copper sleeve relays. To make most effective use of the copper sleeve, which causes the delayed action, it is desirable to provide as low a reluctance as possible of the magnetic circuit when the relay is in the operated position. Instead of providing small non-magnetic separators in the air-gap between armature and the core as is usually y / y / EMBOSSED ;7^ /^ ^ -^ y /^ >^LAT / / 0.5 1.0 1.5> 2.0 DEVIATION IN DEGREES BETWEEN CORE AND ARMATURE POLE SURFACES Fig. 13 — Comparison of flat and embossed pole surfaces and their magnetic closed circuit reluctance with misalignment. 998 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 done -with the ordinary (luick-to-release relays, for slow release relays the armature is allowed to contact the core, finish to finish. When plane flat pole face surfaces are provided, it is e\j)eiisi\e and diflicult to insure in commercial practice that pi-ecise and uniform alignment of the pole face surfaces will ol)tain. A'ariations in the aligmnent of these two sur- faces will cause variations in the closed magnetic circuit reluctance and conseciuently on the release time of the rela3^ In Fig. 12 is shown a design where the necessity for holding the align- ment of core and armature so precisely is not so great.* A spherical surface of rather large radius is embossed on the front end of the armature, so that with commercial variations in alignment, the armature always pre- sents a point on the surface of a sphere for contacting the flat surface of the core. Similarly, the legs of the armature where the}^ pi\'ot on the front ends of the hinge bracket are likewise embossed. The results of the effects of these structural differences on the closed circuit re- luctance are shown in Fig. 13 for a design with flat surfaces and one with embossed surfaces. While it is true that with perfect alignment the relay with flat surfaces will give longer release times, it is apparent that as variations in alignment occur from time to time and from relay to relay, it will have larger variations in performance than the relay with the embossed surfaces. This is a feature which has proven of great value in the manufacture of slow release relays of reasonable time precision. REFERENCES 1. C. Schneider, "Cellulose Acetate Filled Coils," Bell Labs. Record, 29, p. 514, Nov., 1951. 2. W. C. Slauson, "Improved U, UA and Y Tvpe Relaj's," Bell Labs. Record, 29, p. 466, Oct., 1951. 3. B. F. Runvon, "Contacts for Crossbar Apparatus," Bell Labs. Record, 18, p. 278, Alav, 1940. 4. P. W.' Swenson, "Contacts," Bell Labs. Record, 27, p. 50, Feb., 1949. 5. H. M. Knapp, "The UB Relay," Bell Labs. Record, 27, p. 355, Oct., 1949. 6. L. H. Germer and F. E. Haworth, J. Appl. Phijs., 20, p. 1085, 1949. 7. L. H. Germer, J. Appl. Phijs., 22, p. 955, 1951. 8. R. M. Bozorth, Ferromagnetism, D. Van Xostrand Co., Inc., 1951. 9. F. A. Zupa, "The Y-Type Relay," Bell Labs. Record, 16, p. 310, May, 1938. Impedance Bridges for the Megacycle Range By H. T. WIT-1IK1.M (Maiui.scrii)f received August 19, 1952) This paper reviews ac bridges developed for use in the Bell System for the measurement of impedance parameters, particularly at frequencies in the megacycle range. Three recent bridges designed for measuring networks and components for coaxial systems are described. INTRODUCTION The need during recent 3'ears for increased accuracy of impedance measurement in the megacycle range has led to advances in the art of l)ridge measurement. A particular stimulus has been the development of a new coaxial system, designated L-3, for transmitting over distances up to several thousand miles a continuous frequency band extending roughly from 0.3 to 8 megacycles per second. Such a system will be capable of providing on a single coaxial unit the combination of a single television chainiel and as many as GOO one-way telephone channels. The large loss inherent in transmitting this wide frequency band o\'ei' the cable makes it necessary to provide an amplifier about every four miles, and these amplifiers and associated networks have created diffi- cult measurement problems. MEASUREMENT PROBLEMS The measurement problems arise partly from the wide frequency l)and, approximately thirty times the minimum frequency. This makes equalization of the system for satisfactory transmission very difficult, particularly in transmitting a television signal which covers a frecjuency Inuid equivalent to about a thousand telephone channels and which must be equalized for phase as well as loss. Even more important, however, ai'c the problems arising from the close spacing of the amjjlifiers, with the i-esult that a transcontinental circuit re(iuires up to a thousand amplifiers in its path. Departures in 999 1000 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 individual transmission characteristics will produce cumulative errors, making it necessary to maintain close control over the manufacture and adjustment of all of these amplifiers and associated networks. This calls for networks of highly refined design and requires ancillary meas- urement facilities of greater precision than heretofore available at these higher frequencies. The design of transmission networks to meet exacting requirements is a subtle art, embracing on the one hand the use of complex mathe- matical manipulation to produce theoretical networks having the de- sired loss and phase characteristics, and requiring, on the other hand, a down-to-earth knowledge of the properties of the actual components used including parasitic effects and interaction of the various elements when assembled into a network. To furnish this knowledge, to measure the component resistors, capacitors, inductors and transformers which are the building blocks of the networks, to evaluate the ever-present parasitic effects, to determine simplified circuit equivalents of the more complex components such as transformers, and to answer other ques- tions too numerous to mention, measurements of impedance parameters - precise measurements - are required. EXISTING BRIDGE TECHNIQUE For measuring impedance and admittance parameters, that is R,. L, C and G, suitable ac bridges, ordinarily simply designated as impedance bridges, have long held a high place in the Bell System because of their inherent reliability and precision, and their ability to cover a wide range of A^alues. The development of many of the original bridges^' ^' ^' ^ for frequencies above the audio range stemmed from the needs of the earlier carrier systems. With this development came also analysis of shielding technique, standardization of capacitance,^' ^ and a syste- matic classification of bridge methods^ by J. G. Ferguson in 1933, in which bridges were grouped into two major types designated as ratio- arm and product-arm, respectively. Following this classification, com- bined impedance and admittance bridges were developed,^' ^° utiliz- ing a single set of bridge standards for both kinds of parameters by changing the configuration of the bridge network. There have also been special purpose bridges^ , 12. i , h ^^^^ ^^^ ^^ audio and the lower carrier frequencies. More recently, coaxial impedance standards^^ having values calculable from physical dimensions have been developed. Bridges for frequencies above one-half megacycle were used in the Bell System as early as 1919,^ but relatively few bridges were built until the mid 1930's when new carrier systems required bridges in the IMPEDANCE BRIDGES FOR MEGACYCLE RANGE 1001 megacycle range. A ratio-arm bridge^ using external standards was developed for precise measurements up to three megacycles. Intercon- nection of bridge and standards using coaxial cords provided flexibility of configuration resulting in an admittance bridge for high impedances and a series-reactance bridge for medium impedances. These two bridge circuits are shown schematically in (a) and (b) of Fig. 1. A separate, self-contained Maxwell product-arm inductance bridge, shown sche- matically in Fig. Ic and illustrated in Fig. 2, was designed primarily for measuring low-impedance parameters up to one megacycle/sec. Inductance was measured using calibrated air capacitors, and resistance was measured by means of conductance decades employing wire-wound resistors. The bridge included a double-shielded coupling transformer and complete shielding not shown in the simplified schematic. To show clearly the scope and inter-relation of these three bridge methods, it is helpful to plot their ranges on a Slonczewski reactance/frequency chart ' shown in Fig. 3. In this chart, the top frequency shown for the ratio-arm bridge is three megacycles, and for the Maxwell bridge is one megacycle, as these are considered boundaries (a) ADMITTANCE BRIDGE (b) SERIES REACTANCE BRIDGE (c) MAXWELL INDUCTANCE BRIDGE Fig. 1 — Simplified schematics showing the basic circuits of three existing bridges for use at frequencies up to about three megacycles. 1002 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 for their best performance, even though both bridges are useable at higher fi'equencies. It will be observed that while there is some over- lapping of the three ranges, all three methods are necessary to obtain the impedance coverage shown. It should be emphasized that all the ranges shown cover both capacitive and inductive reactances. In the case of the admittance and series-reactance bridges, inductive imped- ances are measured by using a resonating capacitor, in parallel or series, respectively, with the apparatus being measured. In the Maxwell inductance bridge, capacitive impedances are measured by using a fixed resonating inductor in series with the impedance under test. A complete accuracy statement for these bridges is necessarily complex, but in general accuracies of ±0.25 per cent for the major component Fig. 2 — One-megacycle Maxwell inductance bridge, shown schematically in Fig. Ic, designed for relay-rack mounting. IMI'EDANCK HlUnCJKS F()l{ M K(i A( 'Y< LK KAN'GE 1003 was o])taincd over most of the ranjie plottinl on the I'eactaiico chart. These bridges have been very successful foi- the pui-pose foi- wiiicli they were designed, l)ut they ai-e not useable up to th(> eight megacycles oi- highei' i-e(|uii-(Ml l)v the \/A system. REQUIREMENTS OF BRIDGES FOR l3 SYSTEM When the L3 system was contemplated, it was evident that new ))ii(lges would l)e needed. It was recjuired to be able to measure virtually any impedance value at frecjuencies up to and beyond the second harmonic of the 8.4 megacycle upper limit of the system. Accoi'diiigly, a top 0.01 0.05 0.1 1 3 10 FREQUENCY IN MEGACYCLES PER SECOND Fig. 3 — Reactance/freciueiicy chart showing tlio mfa.surcmciil ratifjc of tlie bridges shown in Fig. 1. 1004 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 frequency of twenty megacj^les was decided upon as a design objective with a basic accuracy of ±0.5 per cent for the major component. The immediate need was for a general-purpose bridge, but it was expected that special-purpose bridges having better accuracy would be required later. GENERAL PURPOSE 20-MEGACYCLE BRIDGE It was decided first to develop a single bridge unit which would em- brace both admittance and series impedance methods, and thereby cover a reactance range from a few ohms up to nearly a megohm, as sho^^^l in Fig. 4. Such a bridge would combine the features of (a) and (b) of Fig. 1. There were numerous departures from the earlier designs, however, in- cluding the use of a series range capacitor to reduce the size of the series capacitance standard, the use of deposited carbon resistors, the form and construction of both conductance and resistance standards, and especially the use of transformer-coupled inductive ratio arms. O 10^ ['admittance ^ k<> METHOD CsJ .^^ 0.1 0.5 1 10 20 100 FREQUENCY IN MEGACYCLES PER SECOND Fig. 4 — Reactance/frequency chart applj'ing to the general-purpose bridge shown in Fig. 5. IMPEDANCE BRIDGES FOR MEGACYCLE RANGE 1005 The successful use of a center-lapped 1 raiisfoinier for latio anns in a 465-KC direct capacitance hiidge ' iiulicaied that the icsistanct! i-alio arms r1, r2 of Fig. I might be omitted if a .suital)l(> transformer could i)c developed for higher freciuencie.s. The traiisfoiiner group of the T>a])<)ra- tories succeeded in producing a transformer with a deviation from unity ratio of less than 0.1 per cent over a fr(H|uency range from 0.5 to 20 megacycles. This was made possilile l)y precise location of the windings in tine milled grooves in the form of re\(>rsed helices, cut on a longitudin- ally-split brass cylinder for the inner winding, and on a surrounding phenol OSCILLATOR Fig. 5 — Schematic of the 20-megacycle general-purpose bridge showing both the series (impedance) and parallel (admittance) bridge circuits combined in a single unit. fibre cylinder for the bifilar outer winding which serves as the bridge ratio arms. Electrostatic shielding limits the direct capacitance be- tween primary and secondary to less than 0.01 /i/x/. The core material is compressed powdered molybdenum permalloy. This transformer was the nucleus around which the general purpose bridge was built, and the resulting bridge is shown schematically in Fig. 5. In Fig. 5, the letters a, b, c and d designate the four bridge corners, and T is the ratio-arm transformer already described. Apparatus to be measured by the admittance method is connected to terminals c and d, and is balanced by the calibrated capacitor cp and conductance standard GP. To use the series reactance method, cp and gp are set at minimum settings, apparatus to be measured is connected to terminals xl and x2, 1000 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 and is balanced by the calibrated capacitor cs and resistance standard rs. The ranf2;e capacitor cr consists of several mica capacitors for extending the range of cs, as will be described below ; and zs is merely a compensat- ing impedance, essentially an inductive two-ohm resistor. The circuit is thus l)asically ([uite simple and avoids the use of switches or other complications which would impair performance at these high freciuencies. Capacitors cp and cs are worm-driven air capacitors with a range of about 220 mm/, and were specially designed for this bridge. In the case of cs, any direct conductance between rotor and stator would result in an effective series resistance which would vary both with freciuency and capacitor setting, and therefore re(|uire laborious correction. This was avoided by arranging the construction so that the rotor and stator are mounted on independent insulating supports to the ground panel, thereby completely eliminating direct conductance from rotor to stator. While this results in some conductance from test terminal xl to ground, the amount is small and its effect is negligible because of the relatively low impedance values measured. In the case of cp, on the other hand, it is important to minimize series resistance and inductance to avoid con- ductance and capacitance corrections which would change both with frecjuency and capacitor setting. This was accomplished by careful design of the rotor brush using silver contact surfaces and center-fed connections to both rotor and stator. The conductance standard, gp, and resistance standard, rs, were de- signed to emphasize high-freciuency performance. Deposited carbon resis- tors^* on ceramic rods |" in diameter and |" long mounted on small decade rotors were used, so arranged that only one resistor on a rotor is in the circuit at any time, and that adjacent resistors are short-circuited by means of auxiliary shorting brushes to eliminate shunting admittance which might vary with frecjuency. For gp the resistance values are such that the two lower decades and the slide-wire rheostat each have a residual conductance of 333 micromhos, thereby avoiding the use of resistors exceeding 3,000 ohms in value which would be more likely to vary with frequency. The structure is designed to minimize series in- ductance and to maintain constant capacitance for all settings. For rs, on the other hand, it is necessary to maintain constant inductance for all settings. This was accomplished by adding small wire-loop compen- sating inductors in series with individual resistors in the 10-ohm and 100-ohm decades when necessary. To minimize the over-all inductance, the resistor rotors are placed very close together and are dri^Tn l\v gear- ing from the corresponding dials. The range capacitor, cr, has already been mentioned. It consists of a IMl'KDANCK BHlUCiKS VOH M lU; ACM LI", ll\\(;K 1007 i-()t()i' switch oil which are mounted (ix'c uncalibi'titcd mica capacitors which ciiahic cs to moasui'c liotli positive and ncji;ati\(' reactance \ahies up to 10,000 njjif without adcHtioiial switching!;. 'The 20 ju/a/ cii capacitor covers capacitance measurements u]) to 00 ^fj.f\ the 40 /x/x/' capacitoi- covers up to 150 nfif; the 80 ^jlhJ up to 000 mm'': '!»' '40 mm/ up to 10,000 MM./; iiiid th(> 200 ^ijjif capacitor covers all the posit i\-e series reactance measurements. Since the cr capacitor permits the l)ri(l}2;e to be balanced with the test leads short-circuited, the value of the effective resistance under test is simply etjual to the difference between RS readings for the measui'cmcnt balance and the short-circuit balance, and the reactance under test is determined from a computation of the two r(>adinji;s of cs. A front N'iew of the geneiiil purpose bridge is shown in Fig. (i. 'Die four lower dials are for gp; a})ove them ai'e the four us dials; and aboxc them is the CR dial. The capacitors cs and cp are located adjacent to the test terminals, but are operated remotely by the dial knobs at the extreme right end of the bridge. This was done to remove the operatoi''s hands as far as possible from the test terminals. Near the test terminals is a coaxial connector engraved a. This allows plug-in capacitors (cu in Fig. 5) to be added in parallel with cp for extending the capacitance range. Com- pact silvered mica capacitors in steps of 200 ^ifxf are used. Fig. 7 shows the interior of the same bridge with cp and cs in the lower foreground, GP at the left and rs in the upper right. Fig. 6 — Front view of the general -i)uii)()sc bridge shown in Fig. 5. The t)ri(lgo is appro.ximately 10| inches high and 19 inches wide. 1008 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ""mini!!! I ■IP' Fig. 7 — Interior view of the general -purpose bridge. The panel edge shown in the foreground is the left edge of the bridge shown in Fig. 6. FIVE-MEGACYCLE MAXWELL INDUCTANCE BRIDGE To facilitate the measurement of low-valued inductors, there was need for a direct-reading inductance bridge inasmuch as such measurements entail considerable computation effort when using the general-purpose bridge. Accordingly, it was decided to build a five-megacycle Maxwell inductance bridge to cover a range from 0.001 microhenry up to 10 micro- henries, and effective resistance values up to 11 ohms. The basic circuit is the same as the Maxwell bridge in Fig. 1, but the design embraces such refinements as glass-sealed deposited-carbon resistors for the con- ductance standard, and a worm-driven center-fed variable air capacitor. Special woven-wire resistors on spools of Teflon are used for the tw^o fixed arms, and are compensated to give a constant product of practically zero IMPEDANCE BRIDGES FOR MEGACYCLE RANGE 1009 Fig. 8 — The tivo-inegarj-clc Maxwell inductance bridge is approximatelj^ 12j inches high and 19 inches wide. Test terminals are at upper left, and the three knobs at lower right are zero-balance adjusters. Fig. 9 — Interior of bridge of Fig. 8 showing the shielding for test terminal XI in foreground; at the left is the calibrated air capacitor; at the right are the con- ductance decades using glass-sealed dejiosited-carbon resistors. 1010 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 phase angle over the entire frequency range. The result is a direct-reading bridge sho\m in Figs. 8 and 9 which has greatly facilitated the develop- ment of inductors in the megacycle range. The accuracy for major component varies from ±0.25 per cent at one megacycle up to ±1 per cent at five megacycles. TEN-MEGACYCLE ADMITTANCE BRIDGE De\^elopment of capacitoi's for the L3 coaxial system has recjuired a new ten-megacycle admittance bridge. Intended especially for determin- ing temperature coefficient and fre([uenc,y characteristics of small capa- citors, the bridge is capable of measuring capacitance values up to 200 MM/^vith a precision of drO.Ol mm/, and a wide range of conductance values. Unlike the other two bridges described which make grounded measure- ments only, this bridge is arranged for direct and balanced-to-ground measurements as well. This is accomplished by using the ratio-arm trans- former already described in combination with a simple grounding circuit using a three-position key, as shown in the bridge schematic of Fig. 10. ® © "D" @ @ ^_ Fig. 10 — Ten-megacycle admittance l)ridge with three-position key for shift- ing ground to B for measuring direct admittance, to junction of CI and C2 for balanced admittance, and to D for grounded admittance. Unknowns may be connected from A to D or C to D. IMPEDAXCK RRIDGES FOR MrXiACYCLF, UANT.F. 1011 The (liroct-capacitanco measiiromeiits are useful iu the (lovelopmoui of low \alue(l capacitors, aud the balanced-to-jiiround measurements are helpful in evaluating low-admittance off-ground networks. CONCLUSION' Bridges have been developed for the measurement of impedance and admittance parameters at megacycle frequencies with accuracies hei-eto- fore i)ossible only at much lower fre(iuencies. Se\'eral of the twenty- megacycle general-purpose biidges ha\e been built and are furnishing useful measurements of net woiks and components. Experience with these l)rid^(>s has indicated ranges for which supplementary si)ecial-purpose bridges would l)e desirable, and two such bridges have been built: a Maxwell ])ridge for low-valued inductors, and an admittance bridge for low-\alued capacitors. One feature of all of these bridges not generally available in commercial measuring instruments for megacycle freciuen- cies is the provision of standards having a range of several decades. These allow balances to be made with greater precision over a wider I'ange of phase angles in the apparatus under test, and assure that the absolute accuracy will not be limited by readability. This added precision is \'eiy useful in comparing similar components or in measuring characteristics such as temperature coefficient. ACKNOWLEDGMENTS The de\elopment of impedance bridges during the past thirty years has been under the direction of J. G. Ferguson, and the work described in this article has been under the supervision of S. J. Zammataro. Their assistance in the preparation of this paper has been ^'ery helpful and is hereby acknowledged. It is a pleasure also to acknowledge the contribu- tions of a number of the author's colleagues particularly J. E. Xielsen who was largeh^ responsible for the twenty-megacycle general-purpose bridge, and L. E. Herborn for the fi\e megacycle Maxwell bridge. REFERENCES 1. W. J. Shackelton, "A Shielded Bridge for Inductive Impedance .Measure- ments," Bell System Tech. ./., 6, pp. 142-171, Jan., 1927. 2. W. J. Shackelton and J. G. Ferguson, "Electrical Measurement of Communi- cation .\pparatus," Bell System Tech. ./., 7, pp. 70-89, Jan., 1928. 3. J. G. Ferguson, "Measurement of Inductance bv the Shielded Owen Bridge," Bell System Tech. ./., 6, pp. 375 386, July, 1927. 4. S. J. Zammataro, "Im])edance Bridges," Bell Labs. Record, 8, p]i. 1(17 170, Dec, 1929. 5. J. G. Ferguson, "Shielding in High-Freciuency Measurement," Bell Si/strm Tech. J., 8, pp. 560-575, Aug., 1929. 1012 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 6. J. G. Ferguson and B. W. Bartlett, "The Measurement of Capacitance in Terms of Resistance and Frequency," Bell System Tech J., 7, pp. 420-437, July, 1928. 7. W. D. Voelker, "An Improved Capacitance Bridge for Precision Measure- ments," Bell Labs. Record, 20, pp. 133-137, Jan., 1942. 8. J. G. Ferguson, "Classification of Bridge Methods for Measuring Imped- ances," Bell System Tech. J., 12, pp. 452-468, Oct., 1933. 9. S. J. Zammataro, "An Inductance and Capacitance Bridge," Bell Labs. Record, 16, pp. 341-346, June, 1938. 10. H. T. Wilhelm, "Impedance Bridge with a Billion-to-One Range," Bell Labs. Record, 23, pp. 89-92, Mar., 1945. 11. H. T. Wilhelm, "Measuring Inductance of Coils with Superimposed Direct Current," Bell Labs. Record, 14, pp. 131-135, Dec. 1935. 12. H. T. Wilhelm, "A Bridge for Measuring Core Loss," Bell Labs. Record, 19, 92-96, Nov. 1940. 13. C. H. Young, "Measuring Inter-Electrode Capacitances," Bell Labs. Record, 24, pp. 433-438, Dec, 1946. 14. H. T. Wilhelm, "Maxwell Bridge for Measuring Loading Coils," Bell Labs. Record, 28, pp. 453-457, Oct., 1950. 15. Carl Englund, "Note on Radio Frequency Measurements," Proc. Inst. Radio Engrs., 8, pp. 326-333, Aug., 1920. 16. C. H. Young, "A 5-Megacycle Impedance Bridge," Bell Labs. Record, 15, pp. 261-265, Apr., 1937. 17. T. Slonczewski, "A Versatile Nomagram for Circuit Problems," Bell Labs. Record, 10, pp. 71-73, Nov., 1930. 18. R. O. Grisdale, A. C. Pfister, W. van Roosbroeck, Pyrolitic Film Resistors — Carbon and Borocarbon," Bell System Tech. J., 30, pp. 271-314, Apr., 1951. 19. R. A. Kempf, "Coa.xial Impedance Standards," Bell System Tech. J., 30, pp. 689-705, July, 1951. Abstracts of Bell System Technical Papers* Not Published in This Journal A Full Automatic Private-Line Teletypewriter Switeliitu/ System. W. M. Bacon' and G. A. Locke'. Trans. A.I.E.E., 70, Part 1, pp. 473-480, 1951. (Monograph 1837). This paper describes a full automatic teletj-pewriter message switching s3-stem for use in private-line networks involving one or more switc^hing centers and a multipHcity of local or long-distance lines, each of which may lia\-e one or more stations. This system provides fast telet^'pewriter communication from any station to any other station or gi-oup of stations in the network. At its point of origin a message first is perforated in tape accompanied l)y suitable directing and end-of-message characters, thereafter it is transmitted automatically, stored temporaril}' in perforated tape at a switching office, and then i-outed at high speed to its point or points of destination. Important features are the arrange- ments pro\dded to permit efficient use of long full duplex transmission lines, the full automatic handling of multiple-address messages with only a single originat- ing transmission, and the various guards and alarms which are provided to protect against loss of messages in case of trouble. Operational Study of a Highway Mobile Telephone System. L. A. Dorff\ Trans. A.I.E.E., 70, Part 1, pp. 31-37, 1951. (Monograph 1838). The Dynamics of the Middle Ear and Its Relation to the Acuity of Hear- ing. H. Fletcher\ ./. Acoust. Soc. Am., 24, pp. 129-131, March, 1952. The transformer action of the middle ear as measured by Bek6sy is shown to be the principal cause for the low acuity of hearing for low fre(}uencies. Because of the \'ery low mechanical imi)edance across the basilar membrane at low fre- quencies, large acoustical pressures in fi-ont of the ear drum produce appieciable acoustical pressures across the basilar memi)rane. For example, at 100 ci)s this pressure is thirty times and at 6000 cps it is one-tenth that created acn-oss the basilar membrane. Diffusion of Donor and Acceptor Eleynents Into Germanium. C. S. Fuller . Phys. Rev., 86, pp. 130-137, April 1, 1952. * Certain of these papers are available as Bell System Monographs and may be obtained on request to the Publication Def^artment, Bell Telephone Lal)oratories, Inc., 463 West Street, New York 14, X. Y . For pa])ers available in this form, the monograph number is given in parentheses following the date of puijlication, and this number should be given in all requests. ' Bell Telephone Laboratories 1013 1014 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 A Submarine Telephone Cable with Submerged Repeaters. J. J. Gilbert . Trans. A.I.E.E., 70, Part 1, pp. 564-572, 1951. (Monograph 1815). Physical Structure and Magnetic Anisotropy of Alnico 5. Part /. R. D. Heidenreich^ and E. A. Nesbitt\ Jl. Appl. Phys., 23, pp. 352-371, March, 1952. (Monograph 1970). It is concluded from electron metallograpliic results that the high coercive force and anisotropy of Alnico 5 are caused by a very finel}^ divided precipitate produced by the permanent magnet heat treatment. This precipitate is a transi- tion structure rich in cobalt and is face-centered cubic with ao = lOA and ap- pears as rods growing along the [100] diiections of the matrix crystal when no magnetic field is applied during heat treatment. The size of the i)recipitate rods at oi^timum properties is approximately 75-lOOA l)y 400A long. The spacing between rows of rods is about 200A. The rods are not distinctly resolved in the electron images unless they are grown by aging at 800°C. Their orientation and structure is clearlj' evident in the electron diffraction patterns at all stages of growth. The precipitate responds to a magnetic field applied during heat-treat- ment both bj' suppression of nuclei making an angle greater than about 70° with the field and by the forcing of the rods off the [100] direction into that of the field. The precipitate rods tend to scatter in direction about the field vectoi- when the field is off the [100] but are aligned accurately when the field is along [100]. Energy of a Bloch Wall on the Band Picture. I. Spiral Approach. C. Herring'. Phys. Rev., 85, pp. 1003-1011, March 15, 1952. It is shown that the band or itinerant electron model of a solid is capable of accounting for the "exchange stiffness" which determines the properties of the transition region, known as the Bloch wall, which separates adjacent ferromag- netic domains with different directions of magnetization. In this treatment the constant spin function usually assigned to each running electron wave is replaced by a variable spin function. At each point of space the spin of a moving electron is inclined at a small velocit.y-dependent angle to the mean spin direction of the other electrons, and this gives rise to an exchange torque which makes the spin direction of the given electron precess as it moves through the transition region, the precession rate being just sufficient to keep it in approximate alignment with the macroscopic magnetization. Physical insight into the mechanisms involved is i)rovided by a rigorous solution of the wall i^roblem for a ferromagnetic free electron gas in the Slater-Fock approximation, although it is known that the free electron gas is not likely to be fei-romagnetic in higher approximations. Rough upper limits to the exchange stiffness constants for actual ferromagnetic metals can be calculated without using an}" empirical constants other than the saturation moment and the lattice constant. The results are only a few times larger than the observed values. Elastic and Plastic Properties of Very Small Metal Specimens. C. Herring' and J. K. Galt . Phys. Rev., 85, pp. 1060-1061, March 15, 1952. (Monograph 1977). '■ Bell Telephone Laboratories ABSTUACTS OF TEC'IIXKAL AUTKLES 1015 .1 Scanner for Rapid MeasKrement of Kuvrlopr Dclai/ Ih'slorlion. 1.. K. Hint' and \\'. .). Alhkhshkim'. Proc. I.R.K., 40. pp. l")l !")!», Ai)ril, l<).VJ. (.M()ii()^rai)li l«.)(;7). A inoasuiin^ (Icxicc is (l('scril)e(l which instaiitaiicously (hsplays tlie envelope (lolay-i'io(iu(Micy charactpristic on a cathode-ray screen. Loop and one-way lueasnienients ol' long-distance racho networks can he carried out. Tlie freciuencv raiii^e extends from (il) to SO nief^acycles; the hndts of accuracy are 1 niiUinncro- secoiid or 2 ])ei' cent of the nieasui'cd delay ranf:;e. Comparison of two charac- teristics can he cairied out hv superposition of alternate scaiming ti'aces. The (l('\ice has h(H'n found us(>ful in nieasurinjj; the delay distortion of the TD-2 radio-iclay system and in desii>;ninii; and adjusting the delay ecjualizers needed to correct it. Numerical Integration Xcar a Sinytdarilij. E. I.. Kaplan . ./. Math. rhi/s., 31. pp. 1-28, April, 1952. (Monograph 1980). Measurement of Diffusion in Semiconductors by a Capacitance Method. K. B. AIcAfee\ W. Shockley^ and M. Sparks'. Phys. Rev., 86, i)p. 137-138, April, 1952. Probing the Space Charge Layer in a p-n Junction. G. L. Pearson , \V. T. Kead' and W. Shockley'. F'hys. Rer., 85, pp. 1055-1057, March 15, 1952. Control Methods Used in a Stiid.y of the ]'owels. G. E. Peterson and 11. L. Barney^. J. Acoust. Soc. Am., 24, pp. 175-184, March, 1952. (Monograph 1982) Relationship.s between a listener's identification of a spoken vowel and its properties as revealed from acoustic measurement of its sound wave have been a subject of study b}- many investigators. Roth the utterance and the identifica- tion of a vowel depend upon the language and dialectal backgrounds and the \ocal and auditory characteristics of the individuals concerned. The ijurpose of this pai)er is to discuss some of the control methods that have been used in the evaluation of these effects in a vowel study program at Bell Telephone Labora- tories. The plan of the study, calibration of recording and measuring ecjuipment, and methods for checking the performance of both speakers and listeners are described. The methods are illustrated from results of tests involving some 76 speakers and 70 listeners. Current Mtdtiplication in the Type-A Transistor. W. E. Sittner . Proe. I.R.E., 40, pp. 448-454, April, 1952. (Monograph 1969). One of the basic phenomena exhibited by transistors is current multii)lication. In transistors of the point-contact ty\K^. (one of these has been called the Ty])e-A), the mechanism giving rise to this effect has been somewhat uncertain. Four ' Bell Telephone Laboratories 1016 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 possible mechanisms of the current multipUcation process in the Tj'pe-A transistor are discussed. One of the mechanisms is based on trapping holes in the collector bai-rier of the semiconductor. B3' means of this trapping model, the effect of emitter curi'ont and temperature cm the current multiplication is predicted. It is shown that these predictions are in reasonable accord with experiment. Furthermore, assuming this model to hold, the trap density and activation energy (produced b}' forming) may be evaluated. Faraday Rotation of Guided Waves. H. Suhl^ and L. R. Walker . Phys. Rev., 86, pp. 122-123, April 1, 1952. Transistor Forming Effects in n-Type Germanium. L. B. Valdes , Proc. I.R.E., 40, pp. 445-448, April, 1952. (Monograph 1969). Some of the effects of electrical forming of the collector of an n-type germanium transistor are discussed. Evidence is presented for the existence of a region of p-type germanium underneath the formed electrode, together with some indica- tion of the size of the formed region. These experiments lend support to the p-n hook mechanism in that they explain the observed high values of alpha in tran- sistors. This relation is discussed. Domain Structure of Perminvar Having a Rectangular Hysteresis Loop. H. J. Williams' and M. Goertz\ Jl. Appl. Phys.,-2Z, pp. 316-323, March, 1952. (Monograph 1985). An investigation has been made of the magnetic domain structure of Permin- var (43 per cent Ni, 34 per cent Fe, 23 per cent Co) ring specimens having rec- tangular hysteresis loops after heat-treatment in a magnetic field. Domain patterns obtained with colloidal magnetite showed curved domain boundaries extending completely around the rings, forming circles concentric with them. Changes in magnetization occur when an apphed field causes the circular bound- aries either to expand or contract so that there is a change in the relative values of clockwise and counter-clockwise flux. A nucleus of reversed magnetization was formed by making a small notch in a specimen, and this decreased the co- ercive force and hysteresis loss by a factor of two. It was found that in a 180° domain boundary it was possible to make the change in spin orientations, which occurs in going from one side of the boundary to the other, have either a right- or left-hand screw relation, by the application of a field of appropriate sign per- pendicular to the surface. The effect of superposing an applied alternating field was also investigated, and an effective permeabihty of 4,000,000 was obtained. Measuring Techniques for Broad-Band. Long-Distance Radio Relay Systems. W. J. Albersheim'. Proc. I.R.E., 40, pp. 548-551, May, 1952. (Monograph 1971). Line-up and maintenance of radio relay systems require sensitive yet rapid measurements. These are obtained by scanning the systems response as func- tions of time, frequency, and amplitude. Parameters thus scanned include the 1 Bell Telephone Laboratories ABSTRACTS OF TECHNICAL ARTICLES 1017 ti'ansient response to step functions; frequency chai'actoi-istics of gain, pliase, impedance and their frequency derivatives; and amplitude characteristics of outi)ut nonlinoarity and of intermodulation products. Aluminum Die Castings — The Effect of Process Variables on Their Properties. W. Babington^ and D. TT. KleppingerI Proc. A.S.T.M., 51, pp. 169-197, 1951. Diffusion in Alloys and the Kirkendall Effect. J. Bardeen^ and C. Herring^ pp. 261-288 of Imperfections in Nearly Perfect Crystals, Wiley X. Y., 1952, 490 p. Edited by W. Shockley, J. H., HoUomon, R. Maurer and F. Seitz. Symposium held at Pocono Manor, Oct. 12-14, 1950, by Committee on Solids, National Research Council. Lighlning Protection for Fixed Radio Stations. D. W. Bodle . Tele- Terh, 11, pp. 58-60, 126+ , June, 1952. Common fi;rounds, parallel conducting paths, and discharge gai)s provide tlii-ee important means for avoiding equipment damage from high current surges. Protection of connecting facilities must also be considered to preserve service. Compression Tests on Lead Alloys at Extrusion Temperatures. G. M. Bouton' and G. S. Phipps'. Proc. A.S.T.M., v. 51, pp. 761-770, 1951. Load-deflection measurements made during compression tests on lead and lead-alloy cylinders at various temperatures show the effects of alloying in- gredients on the force required to produce deformation. The curves also furnish clues as to changes taking place in the materials during the course of the test. The load, P, to produce definite small deformation in pure lead at various tem- peratures, T, are shown to follow the relationship P = Ae"^"^, where A and B are constants for the material. This is the same relationship found by others in extrusion studies. The elements added to lead were those most commonly used in the manufacture of cable sheath, namely, antimony, arsenic, bismuth, silver, tellurium, and tin. The results show that the stronger alloys now used for cable sheathing deform less readily at extrusion temperatures than pure lead or the weaker alloys. RF Phase Control in Pulsed Ma(jnetro7is. E. E. David, Jr'. Proc. I.R.E., 40, pp. 669-685, June, 1952. This pajDer describes the behavior of a magnetron oscillator started in the l)resence of an externally applied rf exciting signal whose frequency is not greatly different from the unperturbed steady-state frequency of the magnetron. Effect of Prior Strain at Low Temperatures on the Properties of Some Close-Packed Metals at Room Temperature. W. C. Ellis^ and E. S. Greiner . J. Metals, 4, pp. 648-651, June, 1952. (Monograph 1966). ' Bell Telephone Laboratories ^ Frankford Arsenal, Philadelphia, Pa. 1018 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 The Fatigue Test as Applied to Lead Cable Sheath. G. R. Gohn^ and W. C. Ellis'. Proc. A.S.TJI., 51, pp. 721-740, 1951. This puper discusses the more important factors affecting the design of labora- tory test metliods suitable for ol)taining significant fatigue data from reversed bending tests on cantilever-beam specimens of lead cable sheathing alloys. Data are presented to show the effect of cycling rate, temperature, shape of specimen, alloy additions, and aging on fatigue life. The close correlation be- tween bending fatigue tests on sti'ip specimens and full size sections of cable is demonstratefl. The fatigue data are analyzed in terms of (1) cycle life versus deflection, (2) cycle life versus strain, and (3) cycle life \'ersus stress. Photo- micrographs illustrating representative laboratory and field failures are included. Thermal Conductivity of Germanium. A. Grieco' and H. C. Mont- gomery\ Phys. Rev., 86, p. 570, May 15, 1952. Bell System Cable Sheath Problems and Designs. F. W. Horn' and R. B. Ramsey'. Trans. A.I.E.E., 70, Part 2, pp. 1811-1816, 1951. (Monograph 1917). Powdered Standards for Spectrochemical Analysis. E. K. Jaycox'. Applied Spectroscopy, 6, pp. 17-19, May, 1952. (Monograph 1978). Engineering for Low Product Cost and High Product Quality at the Western Electric Company. A. C. Jones^ Ind. Quality Control, 8, pp. 53-59, May, 1952. The Approximation unth Rational Functions of Prescribed Magnitude and Phase Characteristics. J. G. Linvill'. References. Proc. I.R.E., 40, pp. 711-721, June, 1952. A successive-approximations method is applied to the selection of network functions having desired magnitude and phase variation with frequency. The first ajjproximation, the first set of pole and zero locations, can be selected on the basis of known solutions to similar problems or through use of a set of curves. In succeeding approximations the i)ole and zero locations are adjusted to decrease the deviation of the earlier approximations from the desired characteristics. The process adjusts the magnitude and phase characteristics simultaneously. Its flexibility permits accommodation of practical constraints not possible with other methods. The Magnetic Structure of Alnico 5.E. A. Nesbitt' and R. D. Heiden- reich'. Elec. Eng., 71, pp. 530-534, June, 1952. (Monograph 1981). In the investigation of Alnico 5, two problems arose. What is the mechanism which enables the alloy to respond to heat treatment in a magnetic field? What causes the alloy to have a high coercive force of 600 oersteds? The first prol:)]em has been solved and progress has been made toward solving the second. 1 Bell Telephone Laboratories 2 Western Electric Company ARSTHACTS OF TKCFIXICAL A irri('I,l':S 1019 Sim/Je-Frequency Signaling System for Supervision, and Dialing Over Long-Dislance Telephone Trunks. N. A. Xewell' and A. Weaver'. Trans. A.I.K.E., 70, Part 1, pp. 489-494, 1951. (Monograph 1841). The .single-freeiueiu'v sifinalinj;' system for lonjf-distanre teleplione tiuiiks frees dial calls from the range ami other limitations imposed by dc signaling methods. It uses alternating currents in the voice range as the signaling medium and so can he used with any trunk of any length or type of line facility which meets voice-tiansmission lecjuirements. The signaling recjuirements, design problems, main features of the circuit and eciuipment arrangements, and the operation of this .system are outlined in this i)aper. The .system described is the hrst practical arrangement of its type satisfactorily to meet all the conditions (jf telephone service in the Bell Telephone System. Experimental Information on Slip Lines. W. T. Read, Ju . pp. 129- lol of Imperfections in Nearly Perfect Crystals, Wiley, N. Y., 1952, 490 p. Edited by W. Shockley, J. H. Hollomon, R. Maiirer and F. Seitz. Symposium held at Pocono Manor, Oct. 12-14, 1950, by Committee on Solids, National Research Council. On the (k'ometry of Dislocations. W. T. Read, Jr.' and W\ Shockley^ pp. 77-94 of Imperfections in Nearly Perfect Crystals, Wiley, N. Y., 1952, 490 p. Edited by W. Shockley, J. H. Hollomon, R. Maurer and F. Seitz. Symposium held at Pocono Manor, Oct. 12-14, 1950, by Com- mittee on Solids, National Research Council. A Servo System for Heterodyne Oscillators. T. Slonczewski\ Trans. A.I.E.E., 70, Part 1, pp. 1070-1072, 1951. (Monograph 1883). A constant rate of progression of frecjuency of a motor-driven heterodyne oscillator is obtained by comparing its output with a frequency standard. The result is fed into a servo loop which drives the motor at the proper speed. When used in connection with a level recorder a linear frequency scale is obtained which is more accurate than the static calibration of the oscillator. Metallic Rectifiers in Telephone Power Plants. D. E. Trucksess\ Trans. A.I.E.E., 70, Part 2, pp 1464-1467, 1951. (Monograph 1987). Metallic rectifiers are a comparatively new means of converting power from alternating current to direct current. Most of the component ai)paratus used in the Telephone Systems operates with direct current while the normal power source is alternating current. Therefore a static device without expendable parts which is obtainat)le in small and laige current capacity lends itself as a means for power conversion in telephone power plants. ' Bell Telephone Laboratories Contributors to this Issue A. B. Clark, B.E.E., University of Michigan, 1911. A. T. & T. Co., 1911-34; Bell Telephone Laboratories, 1934-. Toll Transmission De- velopment Engineer, 1929; Toll Transmission Development Director, 1934; Director of Transmission Development, 1935; Director of Systems Development, 1940; Vice President, 1944. Bell System Chairman of Joint Subcommittee on Development and Research of the Edison Elec- tric Institute and Bell System since 1938. Since June, 1951, Mr. Clark has been in charge of coordinating all Bell System programs at the Laboratories. During World War II he served both as a consultant to and a member of various divisions of the Office of Scientific Research and Development. In 1944 he was appointed Consultant to the Secretary of War, and in connection with this work made trips to the European and Mediterranean theaters of operation. Member of I.R.E., Tau Beta Pi, Sigma Xi, and A.A.A.S. and Fellow of A.I.E.E. and the Acoustical Society of America. J. R. Fry, M.E., Cornell University, 1915. Western Electric Company, 1915-25. Bell Telephone Laboratories 1925-. Mr. Fry has been Assistant Switching Apparatus Engineer in the Switching Apparatus Development Department since 1946. Except for the years 1941-45, when he worked on military projects, most of Mr. Fry's Bell System service has been devoted to the design and development of electromagnetically operated switching apparatus such as relays, switches, registers, and selectors. Member of Eta Kappa Nu. H. C. Montgomery, A.B., University of Southern Cahfornia, 1929; M.A., Columbia University, 1933. Bell Telephone Laboratories, 1929- Prior to the war, Mr. Montgomery was engaged in studies of hearing acuity and the analysis of speech sounds. His recent work in the transis- tor physics group has been concerned with fluctuation phenomena in semiconductors. Samuel P. Morgan, Jr., B.S., California Institute of Technology, 1943; M.S., California Institute of Technology, 1944; Ph.D., California Institute of Technology, 1947. Bell Telephone Laboratories, 1947-. A 1020 CONTRIBUTORS TO THIS ISSUE 1021 research mathematician, Dr. Morgan specializes in electromagnetic theory, lie has been particularly concerned with problems of wave guide and coaxial cable transmission. Member of the American Physi- cal Society, Tau Beta Pi, and an associate member of Sigma Xi. W. H. NuNN joined the Home Telephone and Telegraph Company of T>os Angeles in 1*)]"). lie l^ecame Plant Staff iMigineer in 1927; Traflic l']ngine(M- in WVIS; (ieneral Ti-aftic iMigineer, Oregon, 1935; (ieneral Trattic iMigineer, Northern California and Nevada, 1940; Traffic Opera- tions Engineer, Pacific Telej)h()ne and Telegra[)h Company, 1942; and General Ti'affic Manager, Xorthei-n California and Nevada, 1947. In July of 1919 he transferred to the American Telephone and Telegraph Company as Traffic Facilities Engineer, and since March of this year has been Assistant Chief Engineer. H. S. Osborne, B.S., Mass. Inst, of Technology, 1908; Eng. D., Mass. Inst, of Technology, 1910; A. T. & T., Co., 1910-. Since joining the American Telephone and Telegraph Company in 1910, Mr. Osborne has been with the company continuously: as engineer in the Transmission and Protection Department until 1914; assistant to Transmission and Protection Engineer, 1914-1920; Transmission Engineer, 1920-1939; Operating Pesults Engineer, 1939-1940; Plant Engineer, 1940-1942; Assistant Chief I<]ngineer, 1942-1943; and Chief Engineer from 1943 until his retirement in August of this year. During the war Mr. Osborne was Special Consultant in the office of the Secretary of War and a mem- ber of the Telegraph Committee, War Communications Board. In addi- tion, he is a member of the Industry Advisory Council, Federal Specifica- tions Board ; of the Industry Advisory Committee for Supply Cataloging, Munitions Board; and of the Domestic Communications Industry Ad- visory Committee to N.P.A. For many years he has been active in the work of the A.I.E.E.: Chairman, Standards Committee, 1923-1926; member. Committee on Communications, 1931-1934; member, Edison Medal Committee, 1936-1943 and 1947-1952; Chairman, Committee on Award of Institute Prizes, 1936-1939; Chairman, Technical Program Committee, 1936-1939; member. Publication Committee, 1936 1939; Chairman, Special Committee on Institute Activities, 1936-1937; mem- ber. Committee on Planning and Coordination, 1936-1942, 1945-1946, and 1947-1949; member, Alfred Noble Prize Committee, 1937-1942; Chairman, Finance Committee, 1939-1942; President, 1942-1943; Chair- man, Execnitive Committee, 1942-1943; member. Board of Directors and Executive Committee, 1942-1945; member, John Fritz Medal lioard of Award, 1942-1946; Chairman, Board of Trustees, A.I.E.E. Retire- 1022 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 ment System, 1944-1945; member, Hoover Medal Board of Award, 1945-1951; and Chairman, Board of Trustees of Volta Memorial Fund, 1949-. He also has been active in the American Standards Association. He was long a member of the Board of Directors, and was Chairman of the Standards Council from 1942-1945 and Vice President 1948-1951. Since 1949 he has been President of the U. S. National Committee of the International Electrotechnical Commission. He is a member of the Joint Conference Committee on Standards of the Department of Commerce and ASA, and Chairman of the U. S. N. C. Executive Council Subcom- mittee. He is Fellow of the American Institute of Electrical Engineers, Acoustical Society of America, American Physical Society, American Association for the Advancement of Science, and of the Institute of Radio Engineers; and is a member of the American Society for Engineer- ing Education and of Tau Beta Pi. J. J. PiLLioD, E.E. 1908, D.E. (Hon.) 1939, Ohio Northern University; A. T. & T. Co., 1908-. From 1910 until 1943 Mr. Pilliod was associated with the Long Lines Department and the General Engineering Depart- ment of the American Telephone and Telegraph Company. From 1914 to 1918 he was Division Plant Engineer in Chicago; 1918-1920, Engineer of Transmission, New York City; 1920-1941, Engineer in charge of Long Lines Engineering Department; and 1941-1943, General Manager of the Long Lines Department. In 1943 he assumed his present position as Assistant Chief Engineer of the American Telephone and Telegraph Company. From October 1942 to April 1943 he was Chief of Signal Section, Production Division, Army Service Force. He is a Fellow of the A. I. E.E. and is a Trustee of Ohio Northern University and of Vassar College. F. F. Shipley, B.S. in E.E., Purdue University, 1925. A. T. & T. Co., 1925-34; Bell Telephone Laboratories, 1934-. Since 1948, Mr. Shipley has been switching engineer in charge of planning large automatic switching systems, both local and toll. This includes panel, crossbar, and large step-by-step systems. Member of the A.I.E.E., Tau Beta Pi, and Eta Kappa Nu. H. T. Wilhelm, B.S. in E.E., Cooper Union, 1927; E.E., Cooper LTnion, 1936. Western Electric Company, 1922-24; Bell Telephone Lab- oratories, 1925-. Since joining the Laboratories Mr. Wilhelm's work has been with the Transmission Apparatus Development Department, where he has designed electrical measurement apparatus and de\'eloped test methods. Member of A.I. E.E. and Tau Beta Pi. THE BELL SYSTEM meal lournal DEVOTED TO THE SCIENTIFIC ^W^ AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION i^OLUME XXXI NOVEMBER 1952 NUMBER 6 -^5> A New General Purpose Relay for Telephone Switching Systems AETHTJK C. KELLER 1023 Comparison of Mobile Radio Transmission at 150, 450, 900 and 3700 Mc W. RE A YOUNG, JR. 1068 Common Control Telephone Switching Systems oscar myers 1086 Mathematical Theory of Laminated Transmission Lines — Part II SAMUEL p. MORGAN, JR. 1121 Transistors in Switching Circuits a. eugene anderson 1207 Abstracts of Bell System Papers Not Published in this Journal 1250 Contributors to this Issue 1256 COPYRIGHT 1952 AMERICAN TELEPHONE AND TELEGRAPH COMPANY THE BELL SYSTEM TECHNICAL JOURNAL ADVISORY BOARD 8. BRACKEN, President, Western Electric Company F. R. KAPPEL, Vice President, American Telephone and Telegraph Company M. J. KELLY, President, Bell Telephone Laboratories EDITORIAL COMMITTEE E. I. GREEN, Chairman A. J. B U S C H F. R. L A C K W. H. DOHERTY J. W. McRAE G. D. E D W A RDS W. H. NUNN J. B. FISK H. I.ROM NES R. K. HONAMAN H.V.SCHMIDT EDITORIAL STAFF PHILIP C.JONES, 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. Cleo F. Craig, President; Carroll O. Bickelhaupt, Secretary; Donald R. Belcher, 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 XXXI NOVEMBER 1952 number6 Copyright, 1952, American Tele-phone and Telegraph Company A New General Purpose Relay for Telephone Switching Systems By ARTHUR C. KELLER (Manuscript received July 14, 1952) This paper describes a new general purpose electromagnetic relay for use in telephone switching systems. It is a wire spring relay known as the AF type relay and, with variations which provide slow release or marginal char- acteristics, it is known as the AG and AJ relay, respectively. Fig. 1 shows a typical AF type relay. Fig. 2 shows all of the parts of the relay and Fig. 3 is a drawing showing the relay assembly. 1. BACKGROUND The general purpose relay is one of the most important components of telephone switching systems.^ These relays constitute the most repetitive buiUUng block in switching equipment. Since several million are produced annually, low manufacturing cost is extremely desirable. Also of prime importance are low operating and maintenance costs. General purpose relays are, therefore, under constant observation and study by the tele- phone operating companies as the users, by the Western Electric Com- pany as the manufacturer, and by Bell Telephone Laboratories as the designer. The AF wire spring relay and its variations are the result of such studies. A general purpose relay for telephone switching sj^stems must meet a large number of diverse requirements. It must be capable of being assembled with any one of a variety of magnet coils having a wide range 1 S. P. Shackleton and H. W. Purcell, "Relays in the Bell System", Bell System Tech. J., Jan., 1924, p. 1. 1023 1024 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 Fig. 1 — AF type relay, with contact cover detached. of resistance values and to operate contacts which vary from one pair to as many as fourteen or more. The basic relay design must also be capable of pro\dding such features as fast operation and release, slow release, high sensiti^dty, heavy duty and marginal operation. These functions are performed satisfactorily in present crossbar switching sys- tems by U, UA, UB and Y type relays.^- ^- '^- ^ However, ■\^^th an objective of a fort3^-year life for new s^^^tching sj'stems and a trend toward unat- tended operation of switching offices, it is important to attain the best in the performance and reliability of relays. The general purpose relay must be designed to produce the best eco- nomic balance, when used in telephone switching S3'stems, so that the annual charges are minimized. The major ingredients of these annual charges are manufacturing expense, operating electrical power, speed of operation and release, space recjuired and maintenance costs which in- clude reUability and life. 2. REQUIREMENTS AND OBJECTIVES The requirements for a new general purpose relaj^ were initialh' broadly stated to be performance and maintenance at least equal to the 2 H. N. Wagar, "The U-Tvpe Relay", Bell Lab. Record, May, 1938, p. 300. 3 H. M. Knapp, "The UB Relay", Bell Lab. Record, Oct., 1949, p. 355. ^ F. A. Zupa, "The Y-Type Relay", Bell Lab. Record, May, 1938, p. 310. *W. C. Slauson, "Improved U, UA and Y Tj^pe Relays", Bell Lab. Record, Oct., 1951, p. 466. NEW GENERAL PURPOSE RELAY 1025 ARMATURE AND HINGE SPRING CONTACT H COVER i> CLAMP ■■■■■■■ ■ 11 ti s MOUNTING NUT RESTORING . SPRING CARD CO.^ ,-^:.:, CORE ASSEMBLY ill ! u'. TWIN WIRE MAKE CONTACTS »l^hl iifm STATIONARY ,•.,-, ,v >- in WIRE BREAK CONTACTS ASSEMBLIES Fig. 2 — Parts of the wire spring relay. 1026 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 U and Y type relays but with substantially lower manufacturing costs. As the development of the relay proceeded, it became possible to expand the requirements without appreciably altering the expected relay cost. In particular, it became possible to design the new relay to operate and release faster or to use less electrical power, to operate more often before appreciable wear occurred, etc. The improved performance character- istics of the AF Avdre spring relay, as described later, are of equal eco- nomic importance to those associated with lower manufacturing cost. The broad requirements were reduced to the following design objec- tives: 1. Lower cost — 50 per cent of U type relaJ^ 2. Reduced operating electrical power. 3. Faster operate and release times. 4. Long life — one billion operations. 5. Improved contact performance. These broad design objectives do not specifically state a large number of other characteristics which must be at least as favorable as those of the U and Y relay family. This refers to such items as: space rec^uired, magnetic interference, wiring costs, contact combinations, field servicing and repairs. 3. DESIGN POSSIBILITIES The studies of new relay design possibilities started with a careful review of the U type relay experience. In fact, much of the early thinking- considered various modifications of U type and other existing relays. In general, these studies indicated that about half of the manufacturing cost of U type relays came from assembly and adjusting operations. Accordingly, these operations required major revision for a substantial CONTACT COVER STATIONARY WIRE ASSEMBLY TWIN WIRE ASSEMBLY ^FOR MAKES / /FOR BREAKS — MOUNTING NUT CORE ARMATURE AND HINGE SPRING MOUNTING BRACKET ^COIL CLAMP Fig. 3 — Top view of the relay, showing location of parts NEW GENER.\L PURPOSE RELAY 1027 cost reduction of the relay. It became evident that the development of new manufacturing methods as well as new tlesigns were essential in leaching the ambitious objectives. For these reasons, the manufacturing engineers of the Western Electric Company were actix'e pailicipants in the development of the new relay from the beginning. Many new forms of relay designs were considered and studied includ- ing such tj'pes as miniature, magnetic contact, piezoelectric, etc. As a result, one general form, first proposed by H. C. Harrison, gave the most promise of meeting the manifold requirements. This is the wire spring type characterized by the wire spring subassemblies with code card oper- ation of pretensioned, low stiffness springs. Actually, the general form of the wire spring relaj' proposed by IMr. Harrison constitutes an entire new class of relaj-s with many possible variations. These include various t^'pes of code card operation and various forms of contact operation, operated by any of a number of magnet structiu'es. The new class of relays has the following important advantages: 1 . Pretensioned, low stiffness ^vire springs make possible (a) assembly to give close control of contact force without individual spring adjust- ment; and (b) essentially constant contact force thi'oughout the life of the relay and its contacts. 2. Wire spring subassemblies make possible (a) favorable manufacture of a multiplicity of contact springs by molding; (b) lower assembly costs because fewer piece parts are needed; and (c) simple code card operation. 3. Code card operation makes possible (a) standardized and simple assembly; (b) accurate control of contact position; (c) essential elimina- tion of locked contacts ; (d) complete independence of twin contacts ; and (e) simple means for pro\'iding a large number of contact combinations. A continuous and comprehensive study was necessary of the char- acteristics and probable manufacturing costs of many forms of the wire spring relay famil3^ As a result, after passing through se\'eral major designs, the basic design of the present relay was adopted. H. M. Knapp and C. F. Spahn proposed important features of this design. This form represented advantages over other types in 1. reducing the number and amount of dimensional variations con- trolling the contact gaps. In turn, this made possible smaller armature movement, shorter operating and release times and less chatter of the contacts ; 2. reducing the number of code cards required to provide the large number of contact combinations needed in switching systems; 3. reducing the manufacturing and wiring costs; 4. increasing the mechanical life. 1028 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 TENSION I CARD HOLDS CONTACTS APART TWIN CONTACTS--- CARD PERMITS CONTACTS LOSE ICAKD PI CONT> TO CL UNOPERATED OPERATED (a) MAKE CONTACT STATIONARY CONTACT :k. CARD DOES NOT TOUCH TWIN WIRES TENSION CARD FORCES CONTACTS APART UNOPERATED OPERATED (b) BREAK CONTACT -RESTORING SPRING TWIN CONTACTS (MAKE) COMMON STATIONARY CONTACT- TWIN CONTACTS (BREAK) BACKSTOP- UNOPERATED (C) TRANSFER CONTACT Fig. 4 — Principle of contact operation. NEW GENERAL PURPOSE RELAY 1029 4. PRINCIPLE OF CONTACT OPERATION OF THE AF RELAY The AF relay uses what has been called the "single card system" for actuating the contacts. This is in contrast to other code card systems wliich require two, three or four coded cards in each relay. The method for obtaining individual make and break contacts with this system is shown in Figs. 4a and 4b, and a means for obtaining transfer contacts, in which both make and break twin contacts are associated with a common stationary contact, is shown in Fig. 4c. As indicated on the figures, the following principles are incorporated in this method of actuation: 1. In general, three basic wire spring assemblies are required. Two of these carry movable twin wires for make and break contacts and ai'e identical except for some details in forming at the terminal ends for convenience in wiring. The twin wire assemblies are mounted on either NORMAL /POSITION AFTER ASSEMBLY CONTACT MOTION JTION f(| I — CDC I-ST^ "-''''''^ ■^nVree" POSITION OF TWIN r,^^, cr "nM f?"-^-?''' WIRES BEFORE ASSEMBLY DEFLECTION (r ---STATIONARY i ^t^ CONTACT Fig. 5 — Contact forces are controlled by relatively large predeflcctions of the twin wires. side of the stationary wire assembly, which consists of a group of rela- tively heavy wires molded into plastic sections, one a short distance be- hind the contacts and one near the rear of the relay. These sections are rigidly supported in the relay structure. 2. Moving twin contacts on separate twin wires are used ^^'ith every stationary contact. This arrangement assures good reliability and greater freedom from open contacts in the presence of dust and dirt. In addition, contact chatter is reduced as both contacts must be open simultaneously in order to interrupt the circuit. 3. As shown in Fig. 5, each group of twin wires is tensioned toward the stationary wires by means of large predeflections before assembly, so that the contact forces are determined by this predeflection. Good con- trol of the contact force is assured without need for hand adjustment because small variations in deflection of the low stiffness springs do not result in appreciable changes in force. For this reason, the force is stable and is not appreciably affected b}' wear of the contacts. 4. The twin wires are actuated by a single pimched fiber card. Since the tension in the twin wires is always in a direction to hold the contacts closed, the card serves to hold the make contacts open when the relay is unoperated and the break contacts open when the relay is energized. 1030 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 5. The card is supported by the armature on one side and a restoring spring on the other. The restoring spring supphes the force to hold the armature against the backstop and to hold make contacts open when the relay is unoperated, while the armature supplies the force to hold the break contacts open when the relay is operated. However, since the armature must also overcome the tension in the restoring spring, the entire spring load must of course be overcome by the pull of the arma- ture. 6. The twin contacts are held in good registration with their asso- ciated stationary contacts by means of molded guide slots in the station- ary plastic member just behind the card. These guide slots are slightly wider than the diameter of the twin wires so that these wires are free to move in the direction of the armature movement, but are restrained against lateral motion. 7. The close proximity of the card to the contacts is important in minimizing contact chatter and in substantiallv eliminating locked con- tacts, i.e., contacts which fail to open because of interlocking of rough- ened surfaces. The close spacing results in a rigid coupling between the card and contacts, so that the static and dynamic forces associated with the armature and card are available to break loose any incipient lock which might develop. As the armature moves toward the core, the particular point in its travel at which make contacts close and break contacts open depends upon the dimensions of the card between the surface which bears against the armature and the surfaces which engage the twin wires. By proper selection of these dimensions, any contact can be controlled to operate early or late in the travel as desired. By this means, several sequential contact arrangements may be obtained. For example, if the break con- tact in Fig. 4c is controlled by the card dimensions to open earlier in the travel than its associated make contact closes, the resulting arrangement is called an "early break-make" transfer. Similarly, an "early make- break" transfer, often called a "continuity" may be obtained by selection of card dimensions which will assure that the make contact closes before the break contact opens. If both contacts operate simultaneously, the result is a "non-seciuence" transfer. From the above it is evident that the card surfaces which engage the twin wires must be in different positions for early contacts as compared with late contacts. This is illustrated in Fig. 6 which shows an early break-make, an early make-break and a non-sequence transfer side by side. Of the contact pairs shown, only two operate early, and this is accomplished by means of steps in the actuating surfaces of the card. NEW GENERAL PURPOSE RELAY 1031 Tluis, if no se([ueiu'OS were re(|uire(l, the card would luive a single straight surface for makes and another for breaks, and only one card variety would be needed for all ('oml)inations of makes, breaks, and non-sequence transfers. Where seciuences are neetled, howex-er, additional card varie- ties are reciuired with steps in the actuating surfaces for the early contacts. In order to obtain a wider variety of the contact combinations in- cluding various numbers of make contacts, break contacts, sequence transfers and non-sequence transfers on the same relay, it is necessary to provide a variety of different coded stationary and twin wire as- semblies, as well as a variety of cards, some of which are illustrated in Fig. 7. The twin wire assemblies differ as to the number of twin wires provided and in the position of these wires across the width of the molded section. The stationary ^^^re assemblies are always provided with a full comple- ment of twelve wires in order to support the front molded section, which is held in place by spring tension in these wires. However, only certain of the wires may have contacts at the ends. These stationary contacts consist of base metal blocks with 0.010 inch thick precious metal sur- faces on either or both sides as needed for makes, breaks or transfers, and any of the three varieties may be welded to any wire. Thus precious metal is pro\'ided only where needed for the particular contact arrange- ments desired. SHOULDER FOR EARLY MAKES QO£j CONTACTS STATIONARY CONTACTS SHOULDER FOR EARLY BREAKS UNOPERATED Fig. 6 — Early break-make, early make-break and non-sequence transfer con- tacts, showing how early contacts are obtained by means of shoulders on the actuating card. 1032 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 tsd hsd tsd tsJ ^^^^■■■^^^^ ^^^^li^Bi^^^^ ^^^■■■■11^^^^ ^^^■■i^^^^ Htttninii nHiuniM •H!tH|l)!!l litliHinm iiiiHittinitiiniiiiii ^ ilDIIUIilll UllllilllMI^^^BllllillltlHI IIIUIHIill ■i'i u * ^ ^i IRE ASSEMBLIES ^^^ ^^IV ^^^ ^^V Fig. 7 — A few varieties of the coded parts used to obtain various contact combinations. NEAV GENER.\L PURPOSE RELAY 1033 By using dilTerent coinhiiuit ions of stal ionai'v and 1 win w iic assemblies with each card variety, a large number of different contact combinations may be obtained. While most of these needed for telephone switching S3^stems use either no sequences at all or a single stage of sequence, a ivw combinations are provided with "preliminary" contacts. These com- binations include two stages of sequence, in which some contacts operate at each of three different points in the armature travel. The preliminary contacts operate earliest in the travel. These are followed by the early contacts of sequence transfers and finally by the late contacts, including ordinary makes and breaks. To be sure the desired sequences will be maintained during the life of the relays, it is necessary to provide margins in the form of armature travel allowances at each stage. Combinations with sequences will there- fore require total armature travels which are longer than those with no sequences, and two stages of sequence will require more travel than a single stage. According^, the AF relay is provided with a choice of three armature travels to correspond with the number of sequences needed. At the card, these travels are 0.02G inch (short) for no sequences, 0.044 inch (intermediate) for one stage and O.OGO inch (long) for two stages. Thus, combinations including ordinary makes, breaks and non-se- quence transfers use short traA'el. Where sequence transfers are also needed, intermediate travel is used and the early contacts of the se- fiuence transfers operate first. Long travel is used only where prelim- inary contacts followed by seciuence transfers are needed. 5. ARMATURE SYSTEM AND MAGNETIC CIRCUIT The armature system and the associated magnetic circuits constitute the basic motor element of an electromagnetic relay. The size of the motor element is determined, in part, by the work it must do and here a basic factor is the contact force. On the basis of analytical as well as experimental studies, it was decided to use a contact force of about six grams per single contact, i.e., about twelve grams for the combined force of the twin contacts. Other important factors which react on the design of the magnet are the speed reciuii-ed, winding space, heating,^ sensitiv- ity, etc. The detailed analysis of the magnetic S3^stem and the asso- ciated measurements will be coxcred in separate papers. The magnetic structure chosen is shown in Fig. 8. The armature is a flat member of I' shape whicli piox'ides desiiably large jiole face areas. •5 R. L. Pock, Jr., "Internal Temperatures of l{ela\' Wiiiding.s", Hell Sijtitcin Tech. J., Jan., 1951, p. 141. 1034 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 CORE *^ ARMATURE Fig. 8 — Magnetic structure of the AF relay. The core is a simple one-piece E-shape section of sufficient thickness of 1 per cent sihcon u-on to produce the magnetic flux needed to meet the force and speed requirements and to provide the main member to which all other parts are assembled. The silicon iron has appreciably higher electrical resistance than ordinary magnetic iron and this, together with the rectangular cross-sections of the legs, reduces eddy currents as needed for high speed operation and release. The one-piece construction avoids welded or butt joints common to many magnets. These joints are re- sponsible for added reluctance and hence decrease the magnet sensitivity and require added electrical power to operate a given load. The relatively wide spacing of the legs increases leakage reluctance and, in turn, in- creases the useful magnetic flux. After a cellulose acetate filled coiF has been assembled to the middle leg of the core, a core plate, shown in Fig. 9, is forced over the ends of the E-shaped core to hold the three legs in good alignment for proper mating with the armature. The core plate also provides the backstop for the armature and serves as a means of gang adjustment of the con- tacts covered more completely under the Relay Adjustment section of this paper. The armature is spring supported in a very definite manner to produce ^ C. Schneider, "Cellulose Acetate Filled Coils", Bell Lab. Record, Nov., 1951, p. 514. NEW GENERAL PURPOSE RELAY 1035 CORE PLATE Fig. 9 — Legs of the core are held in alignment by the core plute, which is forced over the ends after the coil is assembled. a minimum of rebound when it is released from its operated position. The conditions for reducing armature rebound were described previously* and make it necessary to proportion the forces at the front and rear of the armatiu'e properly. The magnitude and the ratio of these forces are a function of the mass distribution of the armature. The magnet design must not only meet such functional recjuirements as speed, sensitivity, etc., but it must meet these for several values of armature travel as needed by the variety of contact combinations pro- vided. Another requirement is that the relay be designed to fit on a 2-inch mounting plate and this, in turn, restricts the width of the E-shaped magnet core to slightly less than two inches. The relay is normally mounted with the 2-inch dimension in the vertical direction to allow the contact surfaces to be in vertical planes. The corresponding hori- zontal dimension in which the relay can be mounted is 13^ inches except for a few special cases. As described in more detail under the section on Relay Performance, the improved magnet design has resulted in a reduc- tion of the magnetic interference between mounted relays to values which are negligible for most practical purposes. For comparison with the U type relay, the following typical constants of the magnet are of interest (see Table I). The closed gap reluctance, (Ro , is the reluctance of the magnetic circuit, excluding leakage paths, with the armature operated and with the iron near maximum permeabil- ity. The coil constant, G, is the ratio of the square of the number of turns to the resistance for a full sized coil. The sensitivity, S, is a measure * E. E. Sumner, "Relay Armature Rebound Analysis", Bell Sijsteni Tech. J, Jan., 1952, p. 172. 1036 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 of the ultimate work capacity of the magnet as related to the power input and has been defined as S = oirG/Sio ergs per watt. The favorable low value of closed gap reluctance for the new relay- results from adequate cross-sections of magnetic material, the absence of joints, proper mating of the armature and core, and large pole face areas. A low value of reluctance also insures less magnetic interference to other relays and from other relays. Table I Closed Gap Reluctance (Ro, cm Coil Constant G, kilomhos Sensitivity S, ergs per watt. . . . AF Relay 0.028 160 90,000 U Relay 0.065 160 39,000 Although the coil constants are the same for the two relays, as can be seen from Table I, the sensitivity of the new relay is more than double that of the U relay, because of the lower closed gap reluctance. 6. MOLDED WIRE SPRING SUBASSEMBLIES One of the major features of the new relay is the use of molded wire spring subassemblies. Fig. 10 shows a wire spring relay with twelve make contacts, and Fig. 11 shows a comparison of the wire spring assemblies used in this relay and the corresponding parts of the U relay. From this it is clear that the number of parts handled in the assembly of the con- tact spring members is greatly reduced in the new relay. Not all relays will have twelve contacts and in those cases where fewer contact springs are needed the comparison will not be so unfavorable to the U relay. For six contacts, about one-half of the parts shown will be needed for the U relay, whereas the new relay will again require two \vire spring combs. In the new relay three wire spring combs are needed for any contact combination which includes both make and break contacts up to twelve makes and twelve breaks. Four wire spring combs are needed for a relay having twenty-four make contacts. Two problems had to be solved in providing molded wire spring combs, namely, wire straightening and molding of a multiplicity of wires. Both of these were studied cooperatively at Bell Telephone Laboratories and the Western Electric Company. Wire is straightened by rotating cam and die members around the unstraightened wire which causes alternating flexing of the wire. For best results, it was found important to shape the cams properly and to NEW GENERAL 1>1 Hl'OSK UKLAY 1037 Fig. 10 — AF relay with twelve make contacts. push, rather than to pull, the \\'ire through the rotating cams. By this means it is expected to get straight wire without producing an appreci- able tuist in it. The Western Electric Company has developed a multiple head wire straightening machine which can be directly associated with the molding press. A multiplicit}^ of straightened wires is fed into a molding press where plastic molding is used to hold them in proper location. Molding of wire required that the plastic, fed into the die, avoid any appreciable dis- tortion of the -^ares between unsupported sections. A considerable amount of development work, chiefly by the Western Electric Company engineers, was required to achieve this result. Transfer molding of a thermosetting phenolic plastic has been chosen as the most suitable for producing stable wire spring subassemblies. This is based on the need for stability of the wire positions and because of the ability of the ma- terial to withstand the effects of heat. Fig. 12 shows continuous ladders of molded wire spring sections before cutting to length. The molded sections have a number of features of design importance be^'ond liolding the wires in place. These added features are provided by shaping molded sections to make the remainder of the relay simpler. In particular, these features provide registration pins and holes, guides for 1038 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 Fig. 11 — A comparison of the molded wire assemblies for twelve make contacts with the corresponding U relay parts. NEW GENERAL PURPOSE RELAY 1039 STATIONARY WIRE ASSEMBLIF TWIN WIRE ASSEMBLIES Fig. 12 — Molded stationary and twin wire a>ssenil)li(!s, before cutting to length. the ends of the twin wires, cov^er anchorage, damping matei'ial sup- port, etc. 7. CONTACTS AND CONTACT WELDING Since the primary piupose of the relay is to open and close electrical circuits through the contacts, there has been a special effort to make these contacts as reliable as possible. Accordingly, palladiimi is used for all contact surfaces. This use of precious metal substantially eliminates opens due to corrosion. Palladium not only gives outstanding reliability but studies indicate that its use results in the best economic balance be- tween manufacturing cost and service because of the reduced main- tenance expense. Open circuits due to particles of dirt between the contact surfaces are largely eliminated by the use of a contact cover, complete independence of the twin contacts described in the section Relay Performance, and the dynamic characteristics of the wire springs. However, to further reduce the incidence of dirt troubles, the surfaces of the twin contacts are coined to a cylindrical shape. This greatly reduces the effective bear- ing area between the twin contacts and the flat surfaces of the single contacts. Thus, even if an occasional dirt particle should come to rest on one of the contact surfaces, there is small likelihood that it would be in the contact area. Since welding contacts to wires instead of flat springs is relatively new, considerable attention was given to the dcxclopment of suitable 1040 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 techniques. The basic requirements for satisfactory welds are: 1. Sufficient strength to withstand the forces encountered during manufacture and service; 2. Accurate positioning of the contacts on the wires, and 3. Low cost. These requirements apply to both stationary and twin contacts. How- ever, because of differences in geometry, entirely different methods have been developed for welding the two types of contacts. The twin contacts are produced by spot welding precious metal con- tact tapes to the tips of the twin wires. The diagram of the welding circuit is shown in Fig. 13. The condenser c is charged by a power supply to a predetermined voltage. The condenser is then discharged through the primary of the welding transformer t giving rise to the low voltage r,^,.,^r, ELECTRONIC POWER SWITCH SUPPLY TWIN WIRE 'ASSEMBLY Fig. 13 — Diagram showing the essential elements of the spot welding process used for the twin contacts. liigh current surge which produces the weld. The contacts are then sheared to length and the surfaces are coined to a C3dindrical shape. The spot wielding process did not appear best for welding the station- ary contacts to the ends of the wires because of the need to grip the wires with heavy welding electrodes in the limited space directly behind the contacts. Accordingly, a type of welding known as "percussive welding" was developed, which permits one of the electrodes to be placed near the wiring end of the wire springs without developing excessive heat in the wires and which also permits the accurate positioning needed for the contacts in order to control the point of contact closure on the as- sembled relay. The welding circuit is shown in Fig. 14. The condenser c is charged by means of a direct current power supply, and the condenser voltage also appears on the stationary wire. As the contact to be welded is moved toward the end of the wire, the condenser discharges forming an arc which melts the abutting surfaces of the contact and wire. The constants of the electrical circuit and mechanical system were chosen to assure melting a proper amount of metal at a controlled rate to assure high weld strength. The parts are held together during the very brief NEW GEXKHAL ITRPOSK UKLAY 1011 cooling period as the wekl is completed. A small resistance u is adde(l ill series with the discharge circuit to limit the curiciit and control the arcing period. That high weld strengths are obtained by this process is indicated in Fig. 15 which shows typical distributions of Aveld strength for both the percussive-welded contacts and the spot-welded twin contacts. The plots show the percent of contat'ts with weld strengths e(iual to or less than any prescribed value Axitliin the range of the chart. As shown, the per- cussive welds are generall}^ stronger than the sj)ot welds which is, in part, due to larger welded areas. Although percussive welding is more suitable for the stationary con- tacts welded in the factory, it is planned that occasional replacement of both stationary and twin contacts will be made in the field by spot POWER SUPPLY -Wv — I AW R CONTACT 'Jlr^ MOTION I I H= \ STATIONARY WIRE ASSEMBLY Fig. 14 — Diagram showing the essential elements of the percussive welding process used for the stationary contacts. welding. Tliis will be done with the Bell System field welding eciuip- ment^ provided with suitable electrodes. In this case, however, more expensive all-palladium stationary contacts of special shape would be used to facilitate the spot welding and individual hand adjustment for final position of the contacts will be necessary. 8. STANDARDIZED ASSEMBLY OF CODED PARTS Since assembly was one of the most promising fields for reducing costs in a new relay design, special effort was made to reduce the assembly cost of the AF relay. Some of the major design features which contribute to low cost assembly are: 1. The continuous molding and fabricating processes for the wire spring subassemblies, Avhich avoid all individual handling of wires and contacts. 9W. T. Pritchard, "Relav Contact Welder", Bell Lab. Record, April, 1044, p. 374. 1042 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 50 E 15 ^^ ^ ■^ ^ ^ PERCUSSIVE-WELDED STATIONARY CONTACTS '^ ^^ SPOT-WELDE TWIN CONTAC" D rs ^ ^ ^ 0.1 1 10 20 30 40 50 60 70 80 90 99 99.9 PER CENT OF CONTACTS WITH WELD STRENGTHS ^ VALUE SHOWN Fig. 15 — Tj'pical weld strength distributions for the stationary and twin con- tacts. The horizontal scale is graduated so that normal distributions will plot as straight lines. 2. Clamping the relay pile-ups by means of a simple spring clamp instead of the more conventional method using screws. 3. A single, easily mounted, operating card. Less obvious, but equally important is the basic philosophy whereby a large variety of different relay codes are obtained by assembling parts which for assembly purposes are essentially identical for each code. As previously described, the spring combination for each relay is controlled by selection of the proper code card, twin T\qre assemblies with wires in the proper positions for that combination, and a stationary wire assembly with the right kind of contacts welded to the proper Avires. At the present time six different card varieties, fifty twin wire assemblies and seventy- five stationary wire assemblies have been standardized. The twin uire assemblies are provided with any number from one to twelve pairs of wires in various positions while the stationary wire assemblies have from one to twelve contacts in matching positions, with the added variable that each contact may have precious metal on either or both sides as needed. With these it is possible to obtain more than 300 differ- ent contact combinations, although only about 100 of these are now needed. Yet, with a few exceptions, each relay code is assembled from NEW GENERAL PURPOSE RELAY 1043 the same number of ))arts ])ut together in the same manner. Hy usiiifz; a Iditional \'arieti(\s of cards and wire spi-iiij); assemblies tlie total iininher of eontaet combinations which are i)ossible with the basic design is many times larger than the 300 indicated aboN'e. Other examples of coded parts which are assembled in a standardized manner are the coils, core plates and restoring sjjrings. Although coils vary greatly as to turns, resistance, etc., all are a.ssembled to the coi'es t)y the sam{> procedure, using itlentical spoolheads. The three \alues of armature trax'el are controlled by selection of core plates with the proi)er size of openings, but all core plates are assembled alike. Similarly, the restoring springs are pro\'id{Hl in se\'en \'arieties including six different thicknesses and seven predefiections to give the desired restoring force, but these \'ariations do not affect the assembly operations. Standardized assembly of coded parts is of value, not only in reducing the cost of hand assembly operations, but also in providing a more uni- form product and as a principle which may make machine assembly practicable. 9. RELAY ADJUSTMENT Since adjustment expense accounts for a considerable part of the manufacturing costs of older type relays, special efforts were made in the design of the AF relay to reduce the need for adjustment. As a result several types of adjustment used with other relays have been eliminated completely and the remaining adjustments have been simplified. All individual contact adjustment has been eliminated and only two types of factory adjustments are made with the AF relay. These include adjust- ment of the restoring spring to control armature back tension and a gang adjustment of the stationary contacts to control the points in the armature travel at which the contacts operate. Even these adjustments are needed for only a fraction of the relays as close control of the tol- erances in manufacture often causes the back tension and contact operate points to fall within acceptable limits as the relays are assembled. The gang adjustment of the stationary contacts is made by bending the arms of the core plate, thereby changing the position of the front molded section of the stationary wire assembly which rests on the ends of the arms. Each arm may be bent by means of a screwdriver inserted in the slot as shown in Fig. 16. Rotation of the screwdriver in a counter- clockwise direction causes the upper end of the core plate arm to move to the left, carrying with it the upper end of the stationary wire assembly, including the stationary contacts. This reduces the gap between these 1044 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 Fig. 16 — Contacts maj' be gang adjusted to operate at the proper points in the armature travel by bending the arms of the core plate with a screwdriver. stationary contacts and the make twin contacts, thereby causing these contacts to operate earher in the armature travel. Since the break twin contacts rest against the stationary contacts, these are also moved to the left, reducing the space between the break twin wires and the actuating surface of the card. Thus, bending the core plate arms to the left causes both make and break contacts to operate earlier in the armature travel, while bending the arms to the right causes these contacts to operate later. By bending both arms in the same direction, the operate points of all contacts may be shifted in the same direction. On the other hand, separate arms permit adjustment of the upper relay contacts independ- ently of the lower contacts, thereby increasing the latitude of adjustment. The parts of the relay are dimensioned so that no adjustment of the core plate arms is rec^uired, except to compensate for variations in manu- facture of the relay parts. Hence relays assembled from parts made with sufficient accuracy do not generally require adjustment. The restoring springs ma}' be adjusted for the proper armatiu'c back tension by the use of a simple spring bending tool applied to the side arms. However, springs are pro^'ided with various predeflections and thicknesses to correspond with various numbers of make twin contacts which must be held open in the unoperated position. Again, no adjust- ment for back tension is necessarj' except to compensate for variations NEW GENERAL PURPOSE RELAY 1045 in manufacture, as the restoring spring tension is normally just sufficient to overcome the tension of the make twin wires and hold tlu; ;unuiture against the backstop within acceptable force limits. Close control of the tension bends in the wires and restoring springs reduces the fi'e(|uenc.y with which adjustments are needed and a large portion of the relays do not require this adjustment. Types of factor)^ adjustment which are common on other relays but which have been eliminated entirely on the AF relay include adjustments for contact force, individual adjustment of contacts for contact operate point, and adjustment for armature travel. Contact force is controlled by means of the large predeflections of the twin wires as mentioned previously. Individual contact adjustment is eliminated by close control of tolerances combined with the single card method of actuation, and by the simpler gang adjustment used when necessary. Adjustment for armature travel is eliminated by the use of close tolerances on the con- trolling dimensions of the parts. Adjustments of worn relays in the field may be limited to gang adjust- ment of the contacts and back tension adjustment as described above. Other adjustments may include burnishing the contacts to remove sur- face irregularities, replacement of contacts and individual contact adjust- ment as mentioned previously, and replacement of the card if it should become badly worn or damaged. If card replacement is necessary this may be done without dismounting the relay from the mounting plate and without disconnecting the associated wiring. 10. RELAY PERFORMANCE As previously stated, the broad objective in the design of the AF relay has been to reduce the annual charges for the use of this relay in the telephone system. Part of this reduction comes from lower manufacturing costs; the remainder comes from savings associated with the improved performance characteristics, such as long life with relatively low main- tenance expense, reduced power consumption, and increased speed which reduces the number of units of certain types of equipment, such as markers, needed for telephone central offices. A brief description of some of the principal characteristics of the new relay follows. Load and Pull Characteristics Typical load and pull curves for a wire spring relay with twelve early break-make transfer contacts are shown in Fig. 17. The abscissa shows the motion of the armature as it travels from the luioperated position 1046 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 to the operated position, and back again. This is measured at the center- h'ne of the card and hence is also the card motion. In the unoperated position the armature rests against a backstop, which is part of the core plate. In the operated position it rests against 0.006-inch thick non- magnetic separators which prevent the armature from touching the core. The ordinate shows the spring load, which opposes the armature motion 0.01 0.02 0.03 0.04 | 0.05 armature travel in inches released position (against backstop) Fig. 17 — Typical load and pull characteristics of a wire spring relay with twelve early break-make transfer contacts. NEW GENERAL PURPOSE RELAY 1047 towuid the core, and the inagiK'tic pull acting on the armature for \-arious numbers of ampere turns in the winding. These pull and load curves are also measured at the card. Examination of the load curxes shows several features of the I'elay. 'i'he armature hack tension, or force, holding the armature against the backstop is al)out Go grams in this case. As the ai'mature moves toward the core, the spring load increases along the upper of the two nearly- jiarallel load curx'cs until it reaches a final value of about 440 grams in the operated position. As the armature is allowed to return to its original position, a second cur\-e, just below the original curve, is obtained. The area between these two cur\es is a measure of the energy loss due to mechanical hysteresis, or friction, in the relay. As can be seen from the curves, the friction in the new relay is very low and is a small fi'action of the spring load at all \alues of armature travel. The shape of the load curves is characteristic of AF relays with inter- mediate travel (0.044 inch). The load increases rapidly in two regions, corresponding to the intervals in which the early and late contacts operate. The rapid increases are caused by the armature and card picking up the additional load of the twdn wire springs. Each of the 48 twin wires is picked up almost abruptly at various points and the summation of these additions to the load gives the irregular appearance shown. The pull curves of Fig. 17 are for essentially static conditions since the armature was restrained to move slowly through its travel while the curves were automatically recorded. These curves are of interest because they show the ampere turns necessary to assure operation of the relay and also values which will assure the armature will not leave the back- stop. For example, the "critical load point," or point on the load curve which requires the greatest number of ampere turns, is seen to occur at 0.025-inch travel and 250 grams, Avhich under static conditions would require at least 160 ampere turns in the winding to assure complete operation. On the other hand, as little as 94 ampere turns could cause the armature to leave the backstop and might cause operation of one or two contacts. Hence, a lower value must be maintained to assure that the armature will remain at rest against the backstop. This information is important for relays having non-operate requirements. Similar informa- tion may be obtained for limiting ampere turn values which will assure that the armature will remain in the operated position (hold require- ments) and, again, w'hich will assure complete release to the backstop position (release requirements). 1048 THE BELL SYSTEM TECHNICAL JOURN'AL, NOVEMBER 1952 0.016 0.014 Q 8z 0.012 ujO WW .-,UJ -^ 0.010 UJ_I (=<-> UJ5 0.008 ii oi^ 0.006 oo \\ ^ \, \\ \ ^^,- U RELAY \ N k ^j i AF TYPE -4 Fig. 22 — Independent action of the twin contacts is limited on U and UB relays because both contacts are mounted on a single spring which is notched. The AF relay achieves complete independence by mounting the contacts on separate wires. 1054 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 welded to the armature. The special features consist largely of variations in finish and material which do not greatly affect the manufacturing processes. The only added parts are the damping members. These are molded from soft but stable polyisobutylene with grooves to receive the twin wires. One damper is attached to each side of the shelf provided on the stationary wire assembly. The twin wires pass through the grooves and are cemented in place. As shown in Fig. 25, these dampers reduce the vibration of the twin wires between the card and the molded section at the rear, thereby reducing the slide between the wires and the card. Early designs of relays indicated that wear between the twin wires and the card was excessive and that changes in materials would not produce the improvement needed for very long life, particularly with high-speed relays. A fundamental study^" of the conditions which cause wear was made and it was found that reduction of the sliding motion between the wires and card to 0.001 inch or less was necessary to sub- stantially eliminate such wear. The AF relay meets this requirement. The necessity for such a requirement will be better understood when it is uj 90 80 \ AF RELAY \ \ \ \ U RELAY \ V \ \ \ V \ > \ y 20 0 0.004 0.008 0.012 0.016 WEAR OF THE CONTACTS AND SOME OTHER RFi AY PARTS IN INCHES Fig. 23 — Contact forces on the AF relay remain almost constant with wear, while U relay contacts lose force rapidly. low. P. Mason and S. D. White, "New Techniques for Measuring Forces and Wear in Telephone Switching Apparatus", Bell System Tech. J., May, 1952, p. 469. NEW GENERAL PURPOSE RELAY 1055 DAMPERS Fig. 24 — Where very Ions life is needed, polyisol)Utyleiie dampers are mounted between the twin wires and a molded shelf on the stationary wire assembly. noted that, for one billion operations, the total slide corresponds to a distance of about thirty-two miles. Stability The AF relay is a distinct improvement in stability compared with earlier designs when subjected to shock or temperature and humidity changes. Under severe and repeated variations in temperature and hu- midity, the largest changes in contact separation are not more than 0.002 Wi — \ r- ■ ■ ■ I 1 1 1 1 1 VWl/WVWWAMA/VvAMMAAAAA^ V V V V ^ UNDAMPED TWIN WIRES (Q = 80) 1 DAMPED TWIN WIRES (Q=12) 1 1 1 1 \ \ 1 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1C TIME AFTER RELAY OPERATES IN SECONDS Fig. 25 — Oscillograms showing the effectiveness of the polyisobutylene dampers in reducing the vibration of twin wires following operation of the relay. The vibration is measured in the horizontal plane, about midway along the length of the wires. 1056 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 to 0.003 inch. Tlie improved stability is expected to permit final adjust- ment and inspection of the relay at the time it is assembled without need for readjustment after it is wired into equipment and installed into service after shipment. Magnetic Interference In the past it has often been necessary to maintain large spacing be- tween relays where critical values of current to operate or release the relays must be maintained. In some cases special iron shields were used for further magnetic isolation. Without these precautions, the leakage flux from adjacent relays entered the magnetic circuit of the critical relays and the operate or release currents varied according to whether the adjacent relays were energized. ]\Iagnetic interference between AF relays is substantially eliminated as shown in Fig. 26. This is largely because of the low reluctance of the magnetic circuit resulting from the one-piece core and the large pole face areas between the core and armature. As shown in the figure, the I = INTERFERING RELAYS C = CONTROL RELAY & & uj.^q: 20 O^'O \. Ure LAY ■ " AF REL AY ^ r^ 25 50 75 100 125 150 175 200 225 250 275 AMPERE TURNS TO JUST OPERATE CONTROL RELAY Fig. 26 — Typical magnetic interference between AF relays and between U relays, with the rela.ys mounted in the pattern shown. NEW GKXKUAL PURl'OSK UKl.AY 105 AG ARMATURE AND HINGE SPRING BUFFER SPRING AJ ARMATURE AND HINGE SPRING Fig. 27 — Additional parts for AG and AJ relays. ntcasmeiiients were made by svii'roiindiiig a control relay with eight adjacent closely-spaced interfering rela^ys. The ampere tiu'ns to just operate the control relay were \'aried b}^ changing the mechanical load on the relay, and for each \-aliie the change caused by simultaneously energizing the adjacent relays was observed. The impro\'ement of AF relays with respect to the U type is seen to be of the order of ten times for most of the range, with the effects of the adjacent relays being well tnider 10 per cent up to 300 ampere turns. This is small enough so that no shields or precautions with respect to spacing are required. 11. AG AND A J TYPE RELAYS The AG and AJ type relaj's include modifications of the basic AF design to provide slow release, sensitive, marginal and other additional characteristics. For the most part these modifications are not extensive and the assembled relays closely resemble the AF design. The additional parts most often used in the AG and AJ relays are shown in Fig. 27. Both relays use thicker armatures with longer side legs than the AF relay, and the armature of the AG relay has a spherical (Miibossing instead of noiuuagnetic separators. This I'educes the magnetic ciicuit reluctance of the AG relay when it is in the opei-ated position. In addition, for longer release times, a metal sleeve is assembled over the middle leg of the core, inside the coil. In>. ANTENNA - -—J — ^. -^^^^ ^ - X .^ " mobile transmitter (assuming six land receivers with DIPOLE antennas) 1 1 1 - - 10,000 5000 2000 1000 500 200 CO 100 H H50 20 10 5 100 150 200 300 400 600 800 1000 1500 2000 3000 5000 FREQUENCY IN MEGACYCLES PER SECOND Fig. 3 — Transmitter power at antenna input required for urban and suburban coverage. (Mobile antennas are assumed to be quarter-wave ships.) A word of explanation is needed at this point about the gain antennas which were assumed in one of the curves of Fig. 3. These are antennas which tend to concentrate radiation toward the horizon in all directions. Limits for the amount of gain were based upon the considerations (1) that a set of radiating elements greater than about 50 feet in extent would be impractical to build for this service, and (2) that the vertical width of the beam should not be less than about 2 degrees in order that valleys and hilltops will be covered. The amounts of gain possible within these limits are as follows: Frequency mc. Gain-db 150 450 900 3700 8 13 15 15 The mobile antennas were assumed to be quarter-wave whips or the equivalent. MOBILE RADIO TUANSMISSION 1073 Use of gain antennas for the land I'cccivers would result in still further lowering the required mobile transmitter poAver. This is not shown on Fig. 3 because the amount of reduction cannot l)e accurately stated on the basis of present knowledge. It appears certain that tiie reihution Avill be at least equal to the antenna gain, and may be appreciably' more than this, as indicated later. The system modulation and ])ass-band weic assumed in I lie al)oye discussion to be the same at all frequencies. This would not. be ix^alistic if the tolerance allowed for frequency instability were a iixed jx'icentage of operating frequenc3^ It may be justified, how-eyer, because^ the neces- sity for frequency economy and for best transmission })erfoi"mance demands better percentage stability at higher fre(iuen(!ies. A spot check of transmission, ob.serying cii'cuit merits by listening, has been made to determine the \'alidity of the al)oye results in a very general way. T^and transmitter powers were adjusted so that the equiva- lent dipole po\\er at 450 mc was 3 db less than at 150, and power at 900 mc was 1 db less than at 150 mc. This approximates the powers shown on the "dipole" curve of Fig. 3. The map of Fig. 4 shows the results of this test. While the comparison of circuit merits generally shows a preferred frequency at any given location, the performance appears to be about equal w^hen all locations are considered. TEST EQUIPMENT ARRANGEMENTS Tests of transmission out\vard from the land transmitting station were made on signals radiated from antennas on the roof of the Long Lines Building, 32 Avenue of the Americas, New York City. These antennas wxre 450 feet above ground. One of the existing Mobile Service trans- mitters served for the 150-mc tests. Special experimental transmitters were set up for the 450, 900, and 3700-mc tests. All were capable of frequency modulation. The mobile unit was a station wagon equipped to receive and measur- signals at the various frequencies. The receiving equipment was ar- ranged for rapid conversion from 150 to 450 to 900 mc. The bandwidth (about 50 kc) and system modulation (±10 kc) were identi(;al at all three freciuencies (equal to the existing standards at 150 mc). The 3700- mc tests were handled separately. It was not possible to employ the same bandwidth and deviation, but this does not invalidate the com- parison of signal propagation at the A'arious frequencies. A most useful tool in making these measurements was a device known as a "Level Distribution Recorder", or simply "LDR". This was built miomo CD lO a, ■ 1074 MOBILE RADIO TRAN'SMISSION 1075 especially for these tests and is similar to its foi'eninners which have been used in the past for measuring- atmospheric static noise. Tlie LDR, in combination with a caliliiated radio r(M'eiver, is capable of taking as many as twenty instantaneous sami)les of radio sij>;nal str(Mif>;th per sec- ond, sorting the samples by amplitude, and rendering infoiination on a "batch" of samples from which a statistical distribution (•ur\"e can be plotted. The LDR was also us(xl for measui'ing the statistical distribution of audio noise in the output of the radio receiver. The LDR was, in this case, associated with a special con\-erter whose characteristics resemble those of a 2B noise measuring set. No arrangements were made for measuring radio proi)agation from mobile unit to a land receiver. It was felt that the comparison by fre- quencies woidd be substantially the same as in the outward direction of transmission. It does not follow, however, that the background electri- cal noise, against which an r-f signal must compete, will be the same at mobile and land receivers. Strength of r-f signal required at land receiv- ers for satisfactory transmission was measured at several typical locations. RECEIVED R-F SIGNAL STRENGTHS AND PATH LOSSES The first factor in evaluating mobile radio transmission is the strength of the r-f signal which is received. This is inversely related to the loss in the r-f path. The mobile units of a mobile system are either moving around or, if stationary, are located at random. Since the effects of the many geographical features, buildings, and the like, which influence propagation can combine differently for different locations of a car, even where the locations are only a fraction of a wavelength apart, the only meaningful measure of signal strength is a statistical one. Such statistical answers were obtained by making and recording many instantaneous samples of field strength with the aid of the LDR, mentioned above. It is of interest to note that whenever the sample measurements were confined to a relatively small area, say 500 to 1000 feet or less in extent, the amplitude distribution of these samples tended strongly to follow along the particular curve known as a Rayleigh distribution. Such a curve and a typical set of experimental points are shown in Fig. 5. The same distribution was obtained at all of the frequencies tested, including 3700 mc. The rapidity of signal fluctuation, as the car moved, was pro- portional to frequency, but this does not affect amplitude distribution. Such a distribution could have been predicted if it had been postulated that the transmitted signal reached the car antenna by many paths having a random loss and phase relationship. It is thus inferred that in general the signal reaches a car by many simultaneous paths. 1076 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 ^ ^^ ^ — ^Cf -" •^ J**" y 0 MEASURED DATA TAKEN ON BETHUNE STREET, N.Y.C. RAYLEIGH DISTRIBUTION y ^^ / 1 1 1 0.01 0.1 0.5 1 5 10 20 30 40 50 60 70 80 90 95 99 99.9 99.99 PER CENT OF SAMPLES WHICH ARE SMALLER THAN THE ORDINATE Fig. 5 — Tj^pical distribution of test samples of r-f signal strength taken over a small area. With the shape of the distribution known, only one other vakie need be given in order to specify the propagation to such a small area. This might be the median, the average, the rms, or any single point on the curve. The one used most often here is the median, that is, the value which is larger than 50 per cent of the samples and smaller than the other 50 per cent. Measurement of the median value by this statistical method was found to be accurately reproducible, and therefore is presumed to be reliable. Successive batches of 200 samples each, all covering the same test area, yielded median values which differed not more than 0.5 db when none of the conditions changed; i.e., transmitter power, antenna gain, and receiver calibration remained the same. This accuracy may seem surprising when it is realized that individual samples differ fre- quently b}^ 10 db, and often as much as 30 to 40 db. It was presumed at the outset of the tests that the different frequencies would exhibit different propagation trends with distance. For this reason the samples have been grouped by distance. In presenting these results, it was convenient to express the measurements of received RF signal in terms of path losses. By this it is meant the loss between the input to a dipole antenna at the transmitter and the output of a whip antenna on the test car. These path losses will have, of course, the same distribution as the received r-f signal. The results of the path loss measurements are given in Figs. 6, 7, and 8 for 150, 450, and 900 mc respectively. These values represent the loss between the input to a half-wave dipole antenna at one end of the path and the output of a quarter-wave whip at the other end. They are shown here as a function of distance from the land station. For distances under MOBILE RADIO TRANSMISSION 107' tea miles the data are the result of tests in Manhattan and the Bronx. For each distance a test course was laid out approxiniatoly following a circle with that distance as a radius. The data for ten miles atid greater distances were obtained on two series of tests along ladials from the land transmitter, one of which followed Route 1 through New Rochelle, N. Y., and the other followed Route 10 toward Dover, N. J. For refer- ence, a cur\'e has been gi\'en on each of these figures which shows the computed loss based upon the assumption of smooth earth. A curve labeled "1 per cent" means that in one per cent of the sample measurements the loss was less than that indicated on the ordinate. The meaning of the labels on the other curves is similar. The curve labeled "50 per cent" is, of course, the median. It will be apparent that the assumption of smooth earth is not applica- ble to the area tested. The data for median losses are in the order of 30 db greater than the ^'alue computed over smooth earth. This additional loss may be thought of as a "shadow" loss arising from the presence of manj^ buildings and structures. The disti-ibution of losses given in these three curves is wider than the Ravleigh distribution of Fig. 5. This is because the data for each 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 DISTANCE FROM TRANSMITTER IN MILES Fig. 6 — Measured path loss at 150 mc in Manhattan and the Bronx and suburbs. (Note: Data for 10 miles and greater were taken on Route 1 toward New Rochelle and on Route 10 toward Dover.) 1078 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 J 130 < c r K X ^. X 'V. ^ < \ \ *"*T k ^'^^ ( ^ N^ ■*«> ^^^/; 1.'" ^ --^ ""'"'O— K ^x \ \ \ \ N I ? — '— ■ ■~-o~ — — < k X h^ -i- \ \ 1.^ N kN ^'"7 A\ *■--- '^> ^JO \ .»-< 1 1 1 1 V 0.1 0.2 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 8 10 20 30 DISTANCE FROM TRANSMITTER IN MILES Fig. 7 — Measured path loss at 450 mc in ]\Ianhattan and the Bronx and suburbs. (Note: Data for 10 miles and greater were taken on Route 1 toward New Rochelle and on Route 10 toward Dover.) distance are a summation over many different locations rather than a set of samples covering one location. The data for ten miles and further from the transmitter were taken on routes through suburban areas. The losses at twelve miles appear to be less than the average trend indicated by the curves. This is because data taken at the top of the First Orange Mountain weigh heavily at this distance. It is of interest to note that the losses at distances of ten miles and over are 6 to 10 db less than might have been predicted from the trend at smaller distances, where the measurements were made in city areas. This probably reflects the fact that there is a considerable difference in the character of the surroundings, such as height and num- ber of buildings in the suburban territory as compared with the city itself. The median curves of loss have been replotted for three frequencies on Fig. 9. This permits a better comparison uith frequency. Except verj'- close to the transmitter, the performance at the various frequencies seems to differ by an essentially constant number of db, while exhibiting the same trend with distance. The similarity between frequencies is appar- MOBILK n\mo TRANSMISSION 107!) (Mitly much }i;reiiler than tlic similarity bctweon tlic median Naliic and the \alu(' computed over smootli earth for any j>;iven i'red by transmitter power and receiver sensitivity. Only those locations for which path loss was relati\-ely low could be tested. .\ comparison of results at these locations is given in Figure 10. The cui v(\s labeled "1 mi.", "2 mi.", and "4 mi." for Manhattan are the median N-alues obtained along test routes wiiich followed circles of 1, 2 and -i miles radius from the transmitters. The other curves refer to selected small areas at greater distances on the Hutchison River Parkway and New Jersey Route 10, as indicated. Although the data at 3700 mc not extensive, the trend with frequency' seems clear. Alore specific data for path losses measured along the routes toward l)o\er and New Rochellc are given in Fig. 11. Each value plotted here is the median of about 200 samples taken in a smaU area at the distance indicated. The strong effect of the First and Second Orange mountains at fourteen and sixteen miles on the Dover route is of interest. The coverage desired in these mobile telephone systems extends into suburban locations. It follows that a comparison of coverage by the 0.1 0.2 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 8 10 DISTANCE FROM TRANSMITTER IN MILES Fig. 8 — Measured path loss at 900 mc in Manhattan and the Bron.x and suburbs. (Note: Data for 10 miles and greater were taken on Route 1 toward New Kocliclle and on Route 10 toward Dover.) 1080 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 0.3 0.4 0.6 0.8 1.0 2 3 4 5 6 8 10 DISTANCE FROM TRANSMITTER IN MILES Fig. 9 — Median values of measured path losses. (Note: Data for 10 miles and greater were taken on Route 1 toward New Rochelle and on Route 10 toward Dover.) various frequencies should be based upon measurements taken in the suburbs. The data from the New Rochelle and Dover series have been used as a basis for the points and the curve given in Figure 1. Each of the circle points shows the path loss at a given frequency relative to that at 150 mc for a particular location. Their spread indicates that the MANHATTAN --9.0U ■'■^ NSMIT 7 ^ NEW JERSEY- ROUTE 10 A, y^ ..., -— .'- -' -" *•** ^ P' ( >- tr' — ^y- , ^ r .'>'^'' o ,^-- — D_ ^ ** ,^ ) 3 -^ y- — -' 300 400 500 600 800 1000 1500 2000 FREQUENCY IN MEGACYCLES PER SECOND 3000 4000 Fig. 10 — RF path losses at locations for which 3700 mc measurements were made. MOBILE RADIO TRANSMISSION 1081 ON A RADIAL THROUGH DOVER / / \ V 1/ A h 'A N/A 1 / t ■ U \ / N^ V v' V i /I ii / 'k / (50 MC //C \ y 5 /' ! / 1 / .450 / ^;^_ IJ \ \ \ \ -^ _J00 \ 14 16 18 20 22 24 26 DISTANCE FROM TRANSMITTER IN MILES Fig. 11 — Median r-f path losses along selected routes, (a) On a radial through Dover, (b) On a radial through New Rochelle. comparison of frequencies is different at different locations. The "crosses" are the median values of these points, so placed that there are as many points above as below. The points for 3700 mc are taken from the data of Fig. 10. The crosses of Fig. 1 are considered to be the most reliable all-aroiuid comparison of propagation at the different frequencies. RELATION OF SPEECH-NOISE RATIOS TO R-F SIGNAL POWER Speech-to-noise ratios were measured at all of the test locations by the use of the level distribution recorder as described earlier. During the course of any given test the audio noise from the receiver varied con- siderably and these variations were recorded on the LDll. It was found by correlation between subjective observations of circuit merit and the 1082 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 median value of noise that the latter is equi\'alent in noise effect to a steady random noise of the same \^aliie. In the FAI receiver, the level of speech is essentially not affected by the strength of RF signal and so a measurement of the output noise is directly related to the speech-to- noise ratio. The speech-to-noise ratios given here are computed from noise measurements by assuming that speech of —14 vu level is applied to the system at a point where one milliwatt of 1,000 cycles tone would produce a 10-kc frequency deviation. The strength of the speech signal at the receiver output is expressed in the same units as are used for the noise. As might have been expected the median speech-to-noise ratios cor- relate strongly with the amounts of r-f signal received at the various locations. This correlation has been e^■aluated in order that the most likely relationship between speech-to-noise ratio and received r-f signal may be known for the different frequencies. These are shown in Fig. 12, where each circle represents the median speech-to-noise value measured at one test location plotted against the median r-f signal received at that location. The solid lines have been drawn in to show the trend. The bending at the top of the curve is inconsequential. It only represents the limit imposed in the test setup by tube microphonic noise, vibrator noise, etc. The curves show, for example, that in order to produce a commercial grade of transmission, which requires a 12 db speech-to-noise ratio, the median r-f signal must be 122.5 db below one watt at 150 mc. The data given in Fig. 12 pertain only to the suburban locations. Measurements in Manhattan have not been included, even though they indicate that larger signals are required, because the limit of system coverage is to be found in the suburbs. The data on the solid curves of Fig. 12 have been used to derive the curve of Fig. 2 which plots the value of r-f signal required at the mobile receiver for a commercial grade of transmission. The dotted curve of Fig. 2, which shows the median signal required in locations where noise picked up by the antenna is less than set noise, is based on the assumption of an 8 db noise figure for a practical 150-mc receiver, 11 db at 450 mc, and 12 db at 900 mc and higher. Measurements have been made of the effect of noise picked up by the antenna at land receiver stations. These are expressed here in terms of the carrier strength required for just-commercial grade of transmission (12 db speech-to-noise ratio) as compared with the value required when there is no antenna noise and only receiver noise is present. These com- parison measurements were made by injecting a steady carrier into the receiver with an antenna connected normally, and again with a dumjny antenna connected. Although these tests were made mth a steady rather ^^ ^.^•' -- 150 MC • _/] '" .A • / '/ / • o / ' • • • J* •• at o Q 35 O 30 Z 20 • •• :M'> ,,- -•- 450 MC • > .'< • • • • / / • / /: • / c ?* / • • -130 -120 -110 -100 MEDIAN RF SIGNAL RECEIVED IN DBW Fig. 12 — Correlation between median values of speech-to-noise ratio and r-f signal strength in suburban locations. (Note: Each point represents the speech- to-noise ratio and the r-f signal received at one location, (a) 150 mc. (b) 450 mc. (c) 900 mc. 1083 1084 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 than randomly varying signal, it is felt that the comparative results will apply to the random signal case as well. Tests were made at 150, 450 and 900 mc, at four locations of interest, and with dipole and 7 db gain anteinias. Not all combinations were tested, but enough to permit some interesting comparisons. The locations tested were as follows: A: On the Long Lines building, a 27-st()i'v building in downtown Manhattan. B: On the Graybar- Varick building, a lO-story building in downtown Manhattan. C: On the telephone building which houses the Melrose exchange, a 7-story building in the center of the Bronx. D: On the 3-story telephone exchange building in Lynbrook, Long Island. Table I describes the generally prevailing noise situation at these locations. Higher noise was encountered occasionally at some of the sites, due in at least one case to operation of elevators in the building. However, these occasions were so brief and infrequent that the general i)ackground of noise is considered to be a better value to use in estimating systems performance. The trend toward lower site noise at higher frequencies, already noted for mobile installations, is seen to apply to land receivers as well. Table I — R-F Signal Input to Receiver for 12 db Speech-to-Noise Ratio (Given in db Above That Needed to Override Set Noise*) Frequency Antenna Dipole 7-db Gain A B C D 150 mc 450 900 150 450 900 150 450 900 150 450 900 10 1 12 4 0 11 1 1 5 0 0 0.5 1.5 2.5 1 4 0 * Noise figures in the test receivers were 9, 12 and 12 db for 150, 450, and 900 mc, respectively. MOBILE RADIO TRANSMISSION 1085 These data bring out another interesting and significant fact. Where noise collected i)y a dipole anteiuia is discernible over set noise, the noise collected by the 7 db gain antenna at the same site is, surprisingly, less. This means that the gain antenna picks up less noise power than a dipole. Since it picks up 7 db more signal from a distant car, a gain antenna thus provides a double impro\'ement in liaiismission at those sites for which ambient noise is controlling. An explanation of this behavior maj' l)e surmised if it is assumed that the sources of noise are numerous and are scattered around at street le\'el (motor vehicles, mostly). The overall noise received is a sum of contri- Initions from all sources, weighted for distance and the receiving antenna pattern. A gain antenna of the type considered here tends to ignore the strong nearb}^ noise sources because they are below the antenna beam. The sources, which are nearly enough in the beam to count, are also further away and are attenuated by distance. The amount of data given in Table I does not seem sufficient to war- rant stating a firm figure as to the amount of improvement obtainable from a gain antenna. However, substantial improvement at 150 mc is indicated, and this might have the effect of bringing the value of mobile transmitter power required at 150 mc down to the value required at 450 mc, assuming gain antennas in both cases. ACKNOWLEDGMENTS A number of people participated at one time or another in setting up and carrying through these tests. It is not possible to name them all, but the principal participants were R. L. Robbins, R. C. Shaw, W. Strack, D. K. White, and F. J. Henneberg. The program was supervised b}^ D. jMitchell. The special radio equipment required was designed and furnished by W. E. Reichle and his group. Common Control Telephone Switching Systems By OSCAR MYERS (Manuscript received August 1, 1952) In the development of dial telephone switching systems two fundamentally different arrangements have been devised for controlling the operations of the switches. In one arrangement the switch at each successive stage is directly responsive to the digit that is being dialed. Systems using this method of operation are called direct dial control systems, an example being the step- by-siep system a^ commonly used in the Bell System. In the other arrange- ment the dialed information is stored for a short time by centralized control equipment before being used in controlling the switching operations. Systems using the second arrangement are known as common control systems, ex- amples of which are rotary, panel and crossbar. These two arrangements have different economic fields of use, the direct dial control being better suited for the smaller telephone exchanges and the common controls for the larger exchanges, especially those in metropolitan areas. A history of the evolution of these types of switching systems is presented, followed by a discussion of their comparative merits for various fields of use. HISTORY Invention of machines for switching telephone connections started shortly after the invention of the telephone. A forerunner of the step-by- step system, the Connolly and McTighe "girlless" telephone sj^stem,* was patented in 1879 and the first patent on the Strowger step-by-step systemf was issued in 1891. The first commercial installation of auto- matic switching equipment was made at La Porte, Indiana, in 1892. This installation used step-by-step mechanisms. In the early 1900's many telephone engineers regarded full automatic switching as uneconomical but technically feasible if restricted to single office exchanges with individual flat rate lines. They were, however, un- * U. S. Patent 222,458— 1879— Connolly and McTighe. t U. S. Patent 447,918— 1891— Almon B. Strowger. 1086 COMMON' CONTROL SWITCHING SYSTEMS 1087 portaiii about the future of this method of operation. It appeared to them that the greatest promise in tlie use of automatic api)aratus was ill distributing calls to manual "A" operators and in the elimination of tiie "B" operators. Consideration was being given to systems capable of operating on either a semi-mechanical or a full mechanical basis depend- ing on whether the dial was located at the "A" board or at the sub- scriber's station. Development was also under way to provide arrange- ments for trunking calls between dial offices and to overcome the numerous weaknesses and deficiencies of existing dial systems. The Strowger Company, the Bell System, and several other companies were plaiming or developing automatic and semiautomatic systems at that time. These included the full automatic, the network automatic, the automatic operator, and the semiautomatic. Short descriptions of some of them follow. EARLY FULL AUTOMATIC SYSTEMS The full automatic systems were mostly direct dial control. They included the Strowger, the Western Electric 100-line and 20-line, the Clark, the Faller* and the Lorimer systems. The Strowger system of the middle 1890's provided 100-point two- digit selectors, one for each line. For each group of 100 lines the 100 outlets of each selector were multipled to the corresponding outlets of the other selectors serving the group. Each outlet of the group ran to a two-digit connector, each connector having access to 100 lines. Thus every group of 100 lines had 100 selectors and a maximum of 100 con- nectors and could reach 10,000 lines in a full office. Each group of con- nectors, up to the maximum of 100 connectors per group, had a multiple of 100 terminating lines. This was therefore a 4-digit single-office system theoretically of 10,000 lines capacity, requiring 1 selector and 1 connector per line. Subscribers in a given originating group of 100 lines had only one path to a particular terminating group of 100 lines. Since a selector was pro\'ided for each line, no dial tone was necessary. The switches used the familiar up and around motion. The exchanges of this type that were installed were small, the largest being in the order of 1000-line capacity. This type was followed by a new arrangement when automatic trinik selection was introduced. This provided multiple paths to each terminating group of 100 lines; the selector at this stage became a single- digit switch. The Western Electric 100-line system could actually serve onl}- 99 *U. S. Patent 686,892— Ernest A. Faller— Nov. 19, 1901. 1088 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 lines. (The record does not disclose why one of the terminals of the system could not be assigned.) It used a rotary selector per line directly driven by a single train of pulses generated by a lever operated dial at the sta- tion. The selector had 100 points and the number of pulses sent corre- sponded to the number of the called line. The 20-line system was similar to the 100-line system. The Clark system was a single motion rotary step-by-step system using 75-point switches which accommodated a maximum of 74 lines. (Here again there is no record as to why one terminal was not used for a line.) It did not provide a busy test. There were no relays in this system. "automatic operator" SYSTEMS The Faller and the Lorimer systems were called "automatic operator" systems but they were actually versions of direct dial. The Faller system was apparently never used commercially, but the Lorimer system was. The inventors of the Lorimer system had several objectives. One was to produce a system which could be installed in 100-line building blocks, called sections. As little as one section could be installed and operated alone. Additional sections in increments of 100-line capacity could be added as required up to the limit ot 10,000 lines. Another object was to get good contacts and they therefore employed switches with heavy contacts like those used in power switches. The power needed to drive switches with such contacts led to the adoption of a common power drive for a number of switches instead of electromagnets individual to the switches. Still another aim was to provide a minimum of equipment on a per line basis and to provide equipment only to the extent required by traffic. Line relays were therefore omitted in early offices and the 100-line sections were divided into divisions, maximum 10 divisions per section, with arrangements for omitting divisions if not required by traffic. The Lorimer system was a direct dial system operated from a pre-set calling device. It had a line finder stage, a selector stage and a con- nector stage. The calling device, wound up by a crank, had four settable levers, one for each digit, each of which grounded one terminal in its own set of ten terminals corresponding to the digit set up. The levers also operated a visual indicator. In the calling device there was also a switch driven over its terminals by a magnet-controlled escapement. Pulses were sent from the central office to control the escapement and the central office equipment was driven in synchronism with the station COMMON CONTROL SWITCHING SYSTEMS 1089 switch until :i grouiuled .station teriniiiul was found. The central office equipment was then stopped but the station switch continued stepping until the starting point for the next digit was reached. When the central office equipment was ready for the next digit the process was repeated until the called line Avas reached. The Lorimer system has now disai)pearetl from the scene in spite of a number of attractive features. The reasons for this disappearance are not clear from available records, but some reasonable conjectures can be made. For one thing, the pre-set calling device must have been expensive both in first cost and to maintain; it was also designed for a maximum of four digits and a re-design for more than four digits w^ould have en- tailed .substantial effort for developing both the calling device and the central office equipment. There is also some evidence to indicate that the system cost more than either step-by -step or panel. THE NETWORK AUTOMATIC SYSTEM The network automatic was a proposed form of semiautomatic in which the subscribers retained their manual instruments and were served by small unattended branch offices, each of which had a single group of trunks to a central operator office. On originating calls the branch offices acted as concentrators, automatically connecting calling lines to trunks to the central office where the operators -were located and who asked for the called number as in straight manual practice. Called Unes were reached through the branch offices by the operators at the central office who were pro\'ided with keysets to control the branch office equipment. SEMI-AUTOMATIC SYSTEMS There were several plans for other types of semi-automatic systems. Most of them contemplated replacing the "B" operator by a machine under control of the "A" operator. The plan of using machines under (;ontrol of the "A" operators to replace the "B" operators w^as operated successfully in Saginaw, Mich, with Strowger apparatus. A similar plan was in operation in Los Angeles, and several groups of engineers studied improvements and variations. STATUS IN 1905 The status of automatic switching by 1905 w'as this: there w'ere several single office cities which had commei'cial installations of Strowger stej)- hy-stcp eciuipmcnt with severe limitations vvvw for this field of u.se; a 1090 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 number of Western Electric Company 100-line and 20-line automatics were in commercial service ; a small amount of semi-automatic equipment was also in operation with the equipment under direct control of the "A" operator's dial; and planning and development work were under way to remove some of the limitations and extend the field of use of the auto- matic and semi-automatic systems. The rotary dial was developed in 1896. However, many of the early systems did not use this type of dial. Various calling devices were used for a number of years. Among these were lever operated pre-set devices, keysets of several types, and dials with holes (in one case as many as 100) in which a peg could be inserted to act as a stop for an arm which was pulled around and allowed to restore. In all the early systems, regardless of the device used, the signals generated at the calling station directly controlled the selections. RECOGNITION OF NEED FOR ACCESS TO LARGER TRUNK GROUPS While mechanisms and circuits were being developed for direct dial control switching, work of a theoretical nature was going on which was to have an important effect on future designs. This work consisted of traffic probability studies and observations the outcome of which was the development of formulae and curves on the efficiency of trunk groups which influenced strongly the views of engineers as to the eco- nomical sizes of switches. G. T. Blood of the American Telephone and Telegraph Company in 1898 found that the binomial distribution closely fitted the observed data on the distribution of calls. The first compre- hensive paper on the matter was one by M. C. Rorty in 1903, Application of the Theory of Probability to Traffic Problems. Curves accompanying his paper indicated that trunking efficiency improved with group size. Subse- quent work by E. C. Mohna in postulating that the grade of service experienced by a particular call applied to every call in the office and in developing the Poisson approximation to the binomial expansion formed the basis for trunking theory as used in the Bell System. Fig. 1 is a reproduction of three curves produced by Molina on July 6, 1908, showing the average load carried by various numbers of trunks for three probability conditions namely P.Ol, P.OOl and P.OOOl corresponding to an all trunks busy condition encountered by calls once in a hundred, once in a thousand, and one in ten thousand times respectively. From these curves it can be seen, for example, that ten trunks can carry a load averaging shghtly over four calls with a probability of loss of P.Ol. Twenty trunks can carry an average of over eleven simultaneous calls COMMON CONTROL SWITCHING SYSTEMS 1091 with the same P. 01 loss but witli an increase of efficiencj' of 15 per cent. The efficiency' rises from 41 to 56 per cent. EVOLITTION OF PRINCIPLE OF TRANSLATION These studies had considerable effect dh the trend of sj^stem design. For example, it appeared that grouping subscriber lines on the con- nectors in groups of more than 100 might result in some economy and that other economies were possible if the limitations imposed by decimal selections were avoided. However, a new invention, namely translation, was reciuired before sy.stems could operate with large access switches and non-decimal selec- tions. Translation is a mechanical rearrangement which permits con- version of the decimal information received from the dial to non-decimal forms for switch control and other purposes. When translation is made changeable b}' some means such as cross-connections, it is the basis ot much of the flexibility of common-control systems. Translation was first proposed by E. C. Molina late in 1905. A patent application* for a Translating and Selecting Syston was filed on .\pril 20, 19()(). 6 8 10 12 14 AVERAGE = (LOAD CARRIED) Fig. 2— Bypath system. Patent No. 1,083,456 issued to E. C. Molina, Jan. 6, 1916. 1092 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 A necessary feature of systems employing translation of a series of digits such as an office code is digit storage. It was only a small step from the concepts of translation and digit storage to arrangements which provided these features in common circuits. Common controls with translation were first employed in the rotary system. THE ROTARY AND PANEL SYSTEM DEVELOPMENTS The rotary system was a full-fledged common-control system using register-senders to store the dialed information, to translate it to control the two-hundred point ten-level power-driven switches in selecting out- going trunks from the originating office and in making line selections in the terminating office. The translation of the digits used for selecting trunks was changeable, but the translation of the numerical digits was fixed in permanent wiring of the register-senders. In a search for less expensive cabling arrangements than those required by the rotary system, the panel bank employing punched metallic strips was developed. Each bank in the selectors of this system can accommo- date 100 outlets with three wires per outlet, and five banks are stacked into a frame over which 60 power-driven selectors can hunt. For several years, starting in 1907, parallel development of the rotary and panel systems was carried on and desirable features of one were incorporated in the other. The panel system also has register-senders with changeable translation for selecting trunks and fixed translation for controlling selections in the terminating equipment. The major differences in the early designs of rotary and panel were due to the different access of the two systems and to differencesin the methods of controlling the selectors. Both panel and rotary use revertive pulsing to control the selections. With revertive pulsing as the selectors progress they send back pulses which the sender counts. When a selector reaches the desired position, the sender stops it by opening the pulsing circuit. Both panel and rotary, like the Lorimer system, use a continuously operated power dri^'e com- mon to a number of switches because the increased size of switch which the greater access of these systems required, made a separate power drive economical. The panel and rotary systems were originally designed for semi- mechanical operation with automatic distribution of calls to operators as a possible adjunct and with provision for full automatic operation if it proved desirable, by locating the dial or some other calling device at the subscriber's station rather than at the operator's position. This was a reasonable plan when development of these systems was started. Studies indicated that semi-mechanical systems could reduce the number of COMMON CONTUOI. .SWlTL'illNG SYSTEMS UMi opiMators i(Hiuiro{l ])y an amount iaiifi;iiig- iVoin 30 to 50 per cent l)y eliminating the "B" operators and increasing the efiiciency of tlie "A" operators. At that time, full automatic systems were subject to a nunilxi of shortcomings such as (he comj)lications and uiu-eHabiHty of liie i)ulsing de\-ice at tlie subscriber's station, tlie need for a local battery at the station, and the lack of arrangements for party line and message latc service. Furthermore, there was considerable doubt as to the ability of the .subscriber to dial with acceptable accuracy the si.\ or seven mmierical digits required in some of the multi-offiee exchanges. There was an acute need for relief from the difficulties of manual operation after the start of World War I. Telephone growth was so rapid that it appeared for a time that the demand for new operators, ])articu- larly in the large cities, might outstrip the availal)le supply. Comjx'tition fi'om other industry for female help was also increasing. As mor(> offices were added, the situation was further aggravated by the increasing com- plexity of operation. On account of the increasing number of trnnkcd calls, the growing number of central offices, and the increasing amount of manual tandem operation, the ciuality of service was being degraded. DEVELOPMENT OF A LARGE CITY NUMBERING PLAN By 191G, the full automatic system (Strowger) had established a competitive position with manual for single-office cities, and both manual and full automatic offices were considered to be more economical than semi-mechanical for such cities. Because the number of dial pulls for a single office was four or less, little concern was felt about dialing accuracy. For the multi-office cities it appeared that full mechanical operation would improve service and be more economical than either the semi- mechanical sj'stem or manual and would reduce the pressing need for operators. However, in spite of these factors urging the adoption of a dial system and even though automatic equipment was actually used in Los Angeles and Chicago in the first decade of the century, there w^as a reluctance to adopt full automatic operation in the very large multi-office cities because of the lack of a suitable numbering plan. A cumbersome plan was under consideration for handling dial traffic in these cities. This required the use of seven-digit numbei's with the dial customers being called on to use arbitrary three-digit numerical codes for the office names. At the same time, the existing office names would be retained for use by the manual customers. Adoption of this dual tirrangement would have required the provision of a cumbersome directoiy, but worse than that, it was felt that dialing seven numerical digits would be too 109-4 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 confusing to customers and that consequently there would be an exces- sive number of dialing errors. It was therefore planned to use semi- mechanical operation for cities like New York, retaining an operator between the customer and the machine. While this scheme did not save as many operators as the full mechanical method, it was believed neces- sary to have trained operators so that the customers would not be sub- jected to the complications of dialing. Under the proposed arrangement, the customer would pass the office name and number orally, and the operator would substitute the dial code for the office name and key or dial the code and number into the machine. Trial installations of the semi-mechanical panel system placed in service in the Waverly and Mulberry offices, Newark, N. J., in 1915 demonstrated that this method could provide reliable and improved telephone service under severe conditions. However, in 1917 W. G. Blauvelt of the American Telephone and Tele- graph Company proposed a numbering plan which would permit the cus- tomer to dial up to seven digits with acceptable accuracy and which would also be satisfactory for manual operation. This arrangement con- sisted of the use of one to three letters and four numbers. The first one, two or three letters of the office name were printed in bold type in the directory as an indication to dial customers that these were to be dialed ahead of the four numbers. Manual customers used the office name as be- fore. Letters as well as numbers were placed on the dial plate in line with the finger holes of the dial. This proposal was immediately adopted and further Bell System development proceeded along the lines of full auto- matic operation. The Bell System planned to use the panel system in large cities not only because of the trunk efficiency which was possible with the use of the large panel switch, but also because trunking, being no longer under direct control of the dial in this system, was divorced from num- bering. The panel system was also attractive because it had flexibility for growth and for contingencies such as the introduction of new types of service. These advantages would be provided by the common senders and translators of that system. EARLY INSTALLATIONS OF COMMON CONTROL SYSTEMS Early in 1918 tentative schedules were set up for 6-digit panel offices for Kansas City and Omaha and late that year a 7-digit office was recom- mended for the Pennsylvania office in New York City. When the Atlantic office in Omaha w^as placed in service on Dec. 10, 1921, it became the first commercial installation of a full automatic panel system. Commercial installations of rotary equipment preceded the first com- COMMON CONTROL SWITCHING SYSTEMS 1095 mercial panel offices. A semi-mechanical rotary system was installed in Landskrona, Sweden, in 1915 but remained in service for only a short time. A similar system was installed later in 1915 in Angiers, France. The first full mechanical rotary installation was at Darlington, England, in 1914. This sj'stem is still in service. A common control system using Strowger switches, the director sys- tem, was developed in 1922. This development was prompted by the desire to provide automatic equipment in the London, England, multi- office exchange where the layout of the outside plant required consider- able tandem trunking if a reasonably economical trunk network was to be achieved. All of the outside plant in London for the manual system was iniderground and it was required that this arrangement be retained when dial equipment was installed. This tended to fix the routes of tele- phone cables and to make it expensive to open new direct routes as new offices were opened. The trunking economies of tandems were extremely desirable under tliis condition and common controls with translation were necessary for a practical scheme capable of operating with the tandems. The director scheme, which in principle parallels the sender- translator scheme of the panel system, was designed to meet this situ- ation. The director system was first placed in operation in Havana, Cuba, in 1924 and later in London in 1927. EVOLUTION OF THE MARKER PRINCIPLE In retrospect, it is obvious that the development thinking up to the early 1920's was limited by the belief that it was necessary to have the selectors do the testing for idle trunks even with common controls. This arrangement had been successfully used in the step-by-step system and it was natural to follow the same plan in the panel, rotary and director systems. Subsequent development of the common-control idea, starting with an experimental "coordinate" system in 1924, has resulted in marker systems in which the trunk testing is done by the markers. The coordinate system derived its name from the method of operation of its switch, the process resembling the method of marking a point by the use of coordinates. The switch was essentially a large version of the crossbar switch and selected and held a set of crosspoints by the opera- tion of horizontal and vertical members. Translation of the called office code, selection of a trunk, and operation of the switches to connect a transmission circuit to the trunk were functions of a new circuit, the marker, which the sender called into use for a fraction of a second after it had received the office code digits. When the marker does the testing for idle trunks the trunk access from 1096 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 a particular switch is no longer a limiting factor in the size of the trunk group. Once markers were invented it became possible to design systems using markers to do the trunk testing and any type of switch to do the connecting. When a trunk has been selected by the marker, the appro- priate smtches can be operated to connect to the marked terminal. The maximum size of trunk group need not be limited by the number of terminals on one switch. With a primary-secondary switch array groups much larger than those accessible on a single switch can be handled. The coordinate system was not developed for commercial use. The first commercial marker system was PResident 2, a No. 1 crossbar office cut into service in Brooklyn, New York, in February, 1938. Improved crossbar systems have been developed since then including No. 5 cross- bar and several types of toll crossbar systems There is an interesting sidelight on the development of crossbar sys- tems. The crossbar switch was invented by J. N. Reynolds of the Western Electric Company in 1913.* At that time proposed plans for using this switch assumed that it would be used as a line switch. The arrangements did not appear attractive and no serious attempt was made to develop a commercial system using the switch either as a line switch or as a selector. A number of years later an improved version of the crossbar switch was developed by the Swedish telephone administration. Their plans con- templated the use of the switch as a selector in a direct dial control system. In 1930 W. H. Matthies of Bell Telephone Laboratories visited Sweden and, impressed with the possibilities of the switch, ordered samples from Sweden after his return to the United States. Work was started to improve the switch and to develop a modern system around it. The crossbar switch, as previously mentioned, was a small version of the coordinate switch and the development of No. 1 crossbar was therefore started on a plan which was based on principles used in the coordinate system some of which had been successfully applied to the panel system with the adoption of the decoder in 1927. TYPES OF COMMON CONTROL SYSTEMS Four basic variations have been used in systems with common con- trols. These are (1) digit storage in common circuits on a decimal basis and control of switches by the stored digits without translation ; (2) digit storage in the common circuits on a decimal basis, fixed translation and control of switches in a fixed pattern by the translated information; (3) a modification of the preceding plan in which the translation can readily * U. S. Patent No. 1,131,734— J. N. Reynolds— issued March 16, 1915 and re- issued December 26, 1916. COMMON CONTROL SWITCHING SYSTEMS 1097 CALLING TELEPHONE TRANSMISSION PATH PRE- SELECTOR BACKWARD SELECTOR LINE FINDER CALLED TELEPHONE Fig. 1 — Curves developed by E. C. Molina for trunk engineering. be cluiugccl for any item of traffic; and (4) a still furtiier variation where the function of hiuiting for an idle path is removed from the selectors and placed in new circuits called markers. Each variation resulted in improvements over preceding methods of operation. The first plan is the simplest but also the least flexible. An advantage of this arrangement as well as of the other plans which also store the digits over step-by-step is that the interdigital time does not control the group size. By-path sA^stems are examples of this method of operation. A system of this type is shown in Fig. 2. By-path systems use an auxiliary switch train that is under direct control of the dialed pulses to set up a connection. The talking circuit is then established over a parallel system of switches. The auxihary train releases after the talking connection is set up and is available for use in setting up other connections. The Lori- mer system avoided the penalties resulting from hunting during the interdigital interval by storing the digits at the station. A further step in the direction of flexibility, but with added compli- cation, can be taken by a fixed translation from a decimal to a non- decimal basis, i.e., a form of translation wherein a given decimal digit or a set of decimal digits is always changed into the same predetermined non-decimal equivalent. This permits the use of switches with less than ten groups of outlets thereby providing economies by permitting larger groups of outlets with a given size of switch. A third variation with still greater flexibility than the first two, but also with greater complication, is a system with changeable translation. Changeable translation is achieved by providing some means such as cross-connections for readily changing the output pattern of the trans- lators generally for sets of digits as, for example, for the called office 1098 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 1 it 01 0) 01 01 o o o (0 o> 01 01 o 8 ID 01 8 o Oi 0> 01 m o 8 01 Ol 01 8 o o z2 HjtlL M-^ „ (A) .(rt fc U.U Q < H ^ U- OJ Z CK UJ 0-J< iJJ COMMON CONTROL SWITCHING SYSTEMS 1099 codes. Changeable translation of office codes removes tiie limitation that the trunks for a given office designation must be located in a definite position on the switches which is the necessary result of fixed translation. Increased flexibility of numbering is now possible because office designa- tion changes no longer require rearrangements of switch multiple. More economical arrays of switches are also possible because the switching plan can conform to traffic requirements without regard to numbering. Other ad\'antages of translation— and as a practical matter, flexible transla- tion— -include the ability to operate with tandems, to operate with more than one type of outpulsing, and to operate with varying numbers of digits. The originating equipment of the panel system is an example of a system using changeable translation. This type of translation is also used for called line numbers as well as office codes in No. 1 and No. 5 crossbar thereby permitting these systems to shift lines for load balancing pur- poses without requiring numbering changes. Finally, there is the most flexible but also the most complicated plan of all in which the selection of paths and trunks or lines is divorced from the selectors and placed in markers. In this plan the size of group is not limited by the number of terminals that a switch can hunt over in one sweep. No. 1 crossbar is an example of a system using the marker method of operation. In this system a switch generally has access to only ten trunks but on any one call a marker can test 160 trunks distributed over a number of switches. Typical common control arrangements for systems using translation are shown in Fig. 3 for the panel system and in Fig. 4 for No. 1 crossbar. The advantages noted are, in each case, the fundamental ones. Many others are inherent in common control and some will be brought out in further discussion. A number of common control systems embodying the principles dis- cussed have been designed. Rotary, panel and coordinate have been previously mentioned. Although the coordinate system never reached the commercial stage as a complete system, some of its features were adopted in the panel system starting in 1927 with the introduction of the decoder to replace the original three digit panel translator which used special panel selectors and pulse generating drums to do the translating job. This translator was limited in the digit combinations and number of three digit codes it could handle and also demanded a great deal of atten- tion by the maintenance force. In place of the panel translators a small group of all-relay decoders, ranging from three to six, depending on traffic, was provided for each office. Senders were connected to decoders for about one-third of a second per call to obtain the information derived 1100 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMRER 1952 from translation of the three office code digits. The connector for making the momentary connection of the large number of leads reciuired between the senders and decoders presented new problems which were solved by the development of new relay preference and lockout circuits to permit as many simultaneous connections between senders and decoders as there were decoders and to permit an even distribution of calls to decoders. Decoder circuits were completely self-checking for trouble, provided for second trial in another decoder when trouble was discovered, and re- corded troubles on a lamp bank trouble indicator. In the early 1930's, encouraged by the success of decoders, the Bell System started development of the No. 1 crossbar system with markers in both originating and terminating equipments and with improved features over the coordinate system which it resembled in many respects. Self-checking circuits, second trials and trouble indicators which had proven highly successful in the decoder type panel system were important features of No. 1 crossbar. Automatic alternate routing and the ability to operate with non-consecutive PBX assignments were major new features introduced in this system for the first time. The subsequently developed No. 5 crossbar system included a number of improvements, the chief of which from a common-control standpoint was the use of common markers for originating and terminating business and the use of the call back feature in setting up the connection. In this system the common equipment records the calling line identification as well as the called number, and after setting up to the called line or outgoing trunk, breaks down the connection to the common equipment ORIGINATING EQUIPMENT I TERMINATING EQUIPMENT CALLED TELEPHONE DISTRICT JUNCTOR rv PRI SEC LINE LINK FRAME (used FOR BOTH originating and terminating calls) LINK / / OUTGOING^/ TRUNK ORIGINATING SENDER sender LINK ORIGINATING MARKER Fig. 4 — No. 1 crossbar. INCOMING TRUNK TERMINATING SENDER SENDER LINK TERMIN- ATING MARKER COMMON CONTROL SWITCHING SYSTEMS ilUl from the callinji; line and llicii ic-cstablislics a coiincclion l)ack to the calHnjj; line. Coniinoii coutiols ha\(' Ix'cii cinploycd by llic I^cll Syslcin in a nuinhci' of systems in adchtion to those ali'eady mentioned. These inchuh' ])anel sender tandem, ci'osshar tandem, and Xo. 4, A4A and 4 A toll erossbur. COMPARISOX OF COMMON' COXTKOl, SVSTlsMS AND DlltKCT DIAL CONTROL SYSTEMS Botii direct dial contrt)l and conunon conliol systems liave been de- A'eloped to meet a wide range of situations for both large and small exchanges but, as previously noted, direct dial control systems ha\-e found their greatest field of use in the smaller exchanges and common control systems in the larger ones. The reasons for this can be brought out by a discussion of some of the features which have an important bearing on costs. These include the features affecting numbering plans, tiunking arrangements, flexibility, quality of service, maintenance and engineering. A discussion of all the factors affecting costs Avill not be attemptetl. However, some of the more important ones will be covered. RELATION BETWEEN TY'PE OF SY'STEM AND NUMBERING PLANS The requirements of a good numbering plan are well known. A good plan must be universal, i.e., must use the same number for reaching a called line regardless of the point of origin of the call in the area covered by the numbering plan, must permit dialing with acceptable accuracy, must permit directory listings that are readily understood by both dial and manual customers, and should use a minimum number of digits to reduce the labor of diahng. In small networks a satisfactory plan can be set up with almost an}^ kind of system. However, especially in large networks, modern common control systems have outstanding advantages with respect to numbering. These advantages of common controls are derived from the mow, flexible method of operation. Direct dial control systems use up tlu^ digits in the various stages of the switching operations whereas common control systems momentarily store them and can retransmit them. The result is that where direct dial control systems are used the numbering plan and the switching and trunking plans must conform whereas with conunon controls numbering, switching and trunking are not directly dependent on each other because the digits can be stored and translated. The effects of these differences on permissible latitude in numbering arrangements can be brought out by some examples. 1102 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 Direct dial control systems cannot operate economically with a uni- versal numbering plan in a network requiring any given call to have the possibility of completion over a variable number of links. The need for operating in this fashion arises Avhen calls may be completed directly to the called office or via one or more tandem or toll systems. Numbering difficulties of a plan which attempts to use tandems with direct dial control systems can be illustrated by reference to Fig. 5. Assume that A, B, C represent three direct dial control type offices in a 6-digit number- ing plan area and that these are connected b}'' direct trunks between offices. Office B is designated ACademy (22 on the dial) and office C is designated BLue Hills (25 on the dial). Analysis of the trunk layout in this network indicates, let us say, that trunking economies can be made by establishing a tandem and that the direct route from A to C is no longer economical as compared to the route via the proposed tandem. The digits 25 must now select a route via tandem. However, if we use both digits for selecting the route to tandem we have none left for select- ing the route to office C at the tandem office. Since this plan mil not work, let us see what results if we assume that the tandem trunks are selected by means of the first digit. Now all calls starting with the code digit 2 at ofiice A must be routed via tandem and even though economies call for a direct route to the ACademy office from A we are forced to use the uneconomical route through tandem for this office. Actually we must consider the economy of routing the traffic for all offices whose codes begin with a given digit via tandem, or routing it over direct trunks, or we must change the designation of one of the offices. We could, of course, adopt the undesirable expedient of using non-universal numbering, i.e., numbering that varied by points of origin, as, for example, by introduc- ing extra digits on calls through tandem from A to C and omitting them on calls from B to C, y' >, LOCAL OFFICE ^^ ° /ACADEMY (22) LOCAL OFFICE ^ A LOCAL OFFICE ^ J BLUE HILLS (25) Fig. 5 — Trunking scheme with a tandem office. COMMON" CONTROL SWITCHING SYSTEMS 1103 It is a situation such as has been described which has led to the prac- tice, in some cases, of putting offices whose designations begin with the same first digit in the same buihhiig in step-b3^-step areas. This, of course, leads to restrictions. Another alternative is to use selector repeaters. With these devices a "mitlaufer" action takes place in the local and tandem office selectors, i.e., both the local office selectors and the tandem office selectors follow the dial pulses until sufficient information is received to determine the route, whereupon the unneeded etjuipment is released. This equipment makes possible both the direct route to office B and the route via tandem to office C without an office designation change. However, selector re- peaters are expensive and the cost of introducing them may be con- siderable. They also waste some trunk and equipment capacity because selector repeaters operate by seizing both local selectors and tandem trunks on every call. More often than not, perhaps, it would be cheaper to forego the trunk economy than to introduce the selector repeaters. Now take the same network and assume common control equipment at all points. Prior to the introduction of the tandem the local offices translate the first two digits into information for selecting an outgoing trunk and then outpulse only the last four numerical digits directly to the called office. When the tandem is introduced, the translation at office A is changed to select a trunk to tandem on calls to BLue Hills and to tell the sender at A to spill ahead the code digits or equivalent information as well as the line number for these calls. For calls to ACad- emy the existing arrangement is retained. There is no special problem at tandem since the code for the called office, BLue Hills, is made available there. The translator at the tandem office tells the tandem sender to omit the office code digits in outpulsing to BLue Hills. There is an essential difference in the coding between direct dial con- trol and common control which is obscured by the use of the same codes in the examples. In the direct dial control case the codes are route codes (sometimes called group codes) ; that is, the digits directly correspond to the route through the switches and are expended in the switching oper- ations. In the common control case they are destination codes and it is not necessary to have them conform to the route nor are they used up in the switching process. Only common control systems can operate with destination codes. Therefore common control systems are required where it is necessary to route calls to some offices by direct trunks and calls to other offices via tandems without numbering restrictions. Another example of a numbering difficulty with direct dial control systems tracing back to the use of route codes, is illustrated by an 1104 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 extreme example iu Fig. G. This iiguie shows a multi-switch route through four automatic intertoll switching systems, A, B, C, D, to a customer whose listed number is 2345 in the centi-al office, MAin 2, MAin 2 is in numbering plan area 217, a different area from that of the calling office. Typical digit combinations are shown at each place for reaching the next place with direct dial control systems. On a call from the A toll center area to the nimrber MA 2-2345, the originating toll op- erator must dial 16 digits, such as 059 076 097 157 2345. Calls starting at intermediate points or in other networks use different numbers depending on the route. (Note that the route codes start with 0 or 1 to distinguish them from local codes.) It is rather obvious that dialing such combina- tions is cumbersome and requires elaborate routing information at each toll center. Intertoll calls through direct dial control systems are there- fore generally limited to being smtched at one place along the route, with infreciuent use of two switching points. However, with common control systems the situation is quite different. The originating point need dial only the ten digits of the destination 217 MA 2-2345. At each point except the one preceding the called area the full complement of digits is sent ahead. At that point the area code is dropped. At the last point, D, which is assumed to have direct circuits to the called office, MA 2 is skipped and 2345 is sent ahead. If calling and called points had been in the same numbering plan area, only seven digits would have been required. Note that since destination codes are used all points outside the numbering plan area dial the same 10 digits to reach a given line and all points within dial the same seven digits. While only a small proportion of toll calls require multi-switch con- nections of the type just described, connections such as these are never- theless required for an economically feasible nationwide network in which all calls are dialed to completion, and this objective cannot be attained practically "wdthout systems operating with destination codes. (059) (076) (097) CALLED NUMBER 2345 IN MAIN 2 CENTRAL OFFICE TYPICAL DIGITS \ \ \ \ \ \ DIRECT DIAL 0590760971572345 0760971572345 0971572345 1572345 T 23.45 . COMMON CONTROL 217 Ma'z 2345 217 MA2 2345 217 MA2 2345 MA2 2345 |; 2345 : Fig. 6 — Numbering with direct dial control and common control systems. COMMON' COVrUOL SW 1 !'( 11 1 .\<} SYSTEMS ] 10") Alst), as brought out later, destiuatiou codes arc re(iuire(l iu order to realize the importaut tiunkiug economies of automatic alternate routing. CODE CONVERSION In passing, another feature of some eoniniou control S3'slems, namely code conversion, can be brought out here because the illustration, I'ig. (>, fits. (\dls originating in a common control system can use office name codes (such as MA 2 for calls to the ]\IAin 2 oflice) to n^ach destinations \ia step-by-step switching equipment where route codes (such as 157) ai-e widely used. The translating e(|uii)meiit at tiie common control ofhce can be arranged to substitute arbitrary digits for the office name code digits or in some cases to prefix arbitrarj^ digits ah(\i(l of the called number. The arbitrary digits substituted or prefixed conform to the re(iuirements of the office using route codes. In Fig. 6, office C when equipped with common controls could be arranged to convert MA 2 to 157, and therefore codes conforming to the nationwide numbering plan could be used for area 217 even though the calls were routed through step-l)y-step equipment. RKLATION BETWEEN TYPE OF SYSTEM AND TRUNKING ECONOMIES The pro^'ision of a system w^hich makes the most economical use of the trunk plant is important in any network but it is not as important in a small network as in a large one. Small networks can derive only small economies from arrangements whicfi permit saving trunks. P^or example, in a single office network the trunks consist of wires ruiniing from originating to terminating equipment in the same building plus relativel}^ cheap associated relay circuits. However, in a large toll net- work the trunks may include expensive repeaters, signaling equipments, carrier equipment and perhaps echo suppressors, as well as transmission channels running up to hundreds of miles in length and expensive toll relay circuits. For the larger networks there is therefore considerable urge to save as many trunks as possible. It is important therefore to operate these networks with switching plant that makes the most effi- cient use of the trunk plant by providing full access to groups, and to use an arrangement that permits the trunking economies of I'outes via tandems and of automatic alternate routing. These are features provided by common control systems and hel|) explain why these systems are more attractive in the larger networks, l)oth toll and local. The cost of rearrangements for growth, new routes, load l)alancing and for restoring ser\ic(> under cincrgency conditions \-ai-y with the type 1106 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 of system. Because of the flexibility of common controls such rearrange- ments are easier to make and usually cost less than in direct dial control systems. Also the frequency of rearrangements is greater in the larger places. Therefore this is another factor in favor of using common controls for those places. SUPERIORITY OF COMMON CONTROL SYSTEMS WITH RESPECT TO SWITCH ACCESS It has already been mentioned that the efficiency of trunks increases as the size of the group in which they are selected increases. Recognition of this fact early in the development of machine switching (about 1905) led to the invention of common controls. An ordinary step-by-step selec- tor has access to only ten outlets on a level. Access to more than ten outlets can be obtained by providing graded multiple or by the use of rotary out-trunk switches,* or by combinations of these. Whenever it is necessary to employ graded multiple or rotary out-trunk switches, there is still some slight loss of efficiency as compared to full access. In a system such as the panel system in which trunk hunting is a function of the selectors, the maximum number of trunks accessible to a call at any stage of selection is limited by the number of outlets accessible to the switch at that stage. A panel district or office selector, for example, can test a maximum of 90 trunks in a single group, 90 being the maximum number of terminals to which trunks can be assigned on a single panel bank, the remaining ten of the 100 terminals on a bank being reserved for overflow purposes. In the step-by-step system a corresponding limi- tation is avoided by a combination of graded multiple and rotary out- trunk switches with the penalty of a slight loss of efficiency. Marker systems avoid this limitation, also, by having the markers select trunks before they select the paths to the trunks. Crossbar systems with markers can readily test several hundred trunks for a given call. In some crossbar systems — No. 1, for example — trunks are tested in sub-groups of forty, therefore marker holding time is increased when there is more than one sub-group to be tested. This increase in marker holding time is largely avoided in systems like the toll crossbar systems by providing special testing arrangements in which a single indication per sub-group tells the marker which sub-group has one or more available trunks, whereupon the marker only tests the individual trunks of a sub-group in which it is assured that it can find an available trunk. * A rotary out-trunk switch is arranged to hunt over a single group of outgoing trunks and to connect to an idle one. It is arranged fnr preselection and switches not in use will advance from busy trunks. COMMON CONTROL SWITCHING SYSTEMS 1107 The maximum access of ten terminals on a level in ordinary step-by- step is not inherent in the system and might be overcome by a difTerent switch design. A review of how a direct dial control system operates will help to clarify this point. At each switching stage, two actions take place. First, the switch follows the dial pulses until it reaches a group of outlets corresponding to the dialed digit. Then in the interval following this digit and before the pulses of the ne.xt digit arrive the switch hunts over the outlets for an idle path to reach the next stage. The number of paths from a switch level is therefore limited by the number of terminals the switch can hunt over in the interdigital interval. Assuming, for example, an interdigital interval of six-tenths of a second and a hunting speed of 100 terminals per second, GO outlets could be provided. However, if such a high speed of hunting could be attained, and the 60 outlets were pro- vided, 60 terminals would be required per group even for small ones Avhich are in the majority. Hence such a switch would be wasteful of terminals. Direct dial control systems have generally employed switches with ten outlets per level although special arrangements such as twin levels have been employed to increase the number of outlets. A twin level switch provides terminals for two trunks at each rotary step and thus twenty trunks per level can be reached. TRUNK ECONOMIES FROM TANDEM OPERATION WITH COMMON CONTROL SYSTEMS An important factor in trunk economies is the ability to use tandems. The numbering difficulties that direct dial control systems have with tandems have already been discussed. Tandems permit major tiTink economies on two scores. First, tandem routings take advantage of the efficiency which results from concentrating the smaller items of traffic and handling them over common trunk groups. Fig. 7 shows how this economy is attained. Ten offices completely interconnected by one-way trunks require 90 interoffice trunk groups. Ten offices interconnected only by way of tandem require only 20 groups. The groups by Avay of tandem are larger in size than the individual direct groups they replace and because of increased efficiency with group size fewer trunks are required. There is a second possibility for an increase of efficiency, an example of which occurs when part of the offices are in business districts and part in residential districts. The peaks of trunked traffic from these different types of offices frequently occur at different hours, hence the trunks through tandem can be provided more economically for a given grade of 1108 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 service than by an arrangement which must care for the peaks of each office separately. The non-coincidence of peaks of traffic of different types of offices permits economies both on trunks to tandem and trunks from tandem. For example, assume that a given office completes calls via tandem to some offices which have a morning busy hour and to others which have an evening busy hour. Then the group to tandem must provide capacity to handle the traffic for the busier hour of the two, but this capacity need care only for the peak traffic to part of the destinations. If individual direct groups had been provided instead of a common group to tandem, each group would have required capacity for its own peak, regardless of when it occurred. The common group to tandem therefore benefits by the noncoincidence of the peaks. A corre- sponding situation also occurs on trunks from tandem. Each group com- pletes calls to a given destination from a number of originating offices whose peak hours may not coincide, and hence groups from tandem derive economies similar to those of the incoming groups to tandem. Tandems are also required for alternate routing. Alternate routing is an arrangement to provide trunking economies by using a limited num- ber of direct trunks for the traffic between two offices, and permitting the calls which do not find an available direct trunk to overflow to one or more tandems in succession. Because of the ability to load the direct circuits very heavily and yet provide good service by taking the overflow from and to a number of offices through a common tandem point, sub- stantial economies are possible. Automatic alternate routing is practical only with common control systems. Common controls are needed to DIRECT TRUNKS ONLY TANDEM TRUNKS ONLY (90 groups) (20 groups) Fig. 7 — Reduction of tlu! nuinljeiol' trunk groups by tlie use of a tandem office. COMMON CONTROL SWITCHING SYSTEMS 1109 provide the digit storage and digit spilling features in the oflice that does the alternate routing so that it can spill forward to the alternate route point the digits the latter requires. Common controls have other ad\-antages with respect to trunking which have already been covered in part. They also simplify the problems of assignment and load balancing as groups change in size or as new groups are added. An example of the difference in the methods of han- dling trunk groAvth in step-by step and crossbar is of interest. In step- bv-step when groups grow beyond 10 trunks a grade must be introduced in the switch wiring, or trunks must be sub-grouped or rotary out trunk switches used. If further growth occurs, regrades must be made or re- arrangements may be required in the sub-grouping or in the rotary out- trunk switches. In a crossbar system, howe\xr, in most cases added trunks are merely assigned to spare switch terminals which are left \-acant for this purpose. ROUTINGS FOR IRREGULAR CONDITIONS Common controls are adapted to the efficient recognition and handling of irregular conditions such as permanent signals, vacant codes, and dis- continued or temporarily intercepted lines. Registers or senders detect line troubles which cause permanent signals or receivers off the hook by a timing circuit which waits for a short time for dialing to start. If the dialing does not start ^^dthin the interval al- lowed the line is directed to a common group of permanent signal trunks which may appear before operators or at a test board. In No. 5 crossbar a trouble recorder card can be produced on which the location of the line in trouble is indicated. The step-by-step system indicates permanent signals by alarms to the maintenance force on a line group basis, and lines in trouble must be traced. Vacant codes are detected by the translators, decoders and markers of common control systems and the calls are routed to a common tiunk group which appears before operators or which returns "no such numl^er tone." The corresponding arrangement in step-by-step requires con- nections from the switch multiple to operator or tone trunks. In systems like No. 1 crossbar and No. 5 crossbar which have common controls in the terminating equipment, lines on which service has been discontiinied or temporarily intercepted can be recognized by the mark- ers and the calls rerouted to a common group of intercepting trunks. For example, temporary discontinuation of service is indicated by lifting a single cross-connection at the numboi- group frame. In the step-by-step 1110 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 system, however, one intercept trunk is commonly provided per 100 numbers and lines whose service is to be intercepted must be cross- connected to these trunks. FURTHER ADVANTAGES OF COMMON CONTROL SYSTEMS ACCRUING FROM THEIR ABILITY TO OPERATE WITH TANDEMS Some of the economies permitted by common control systems operat- ing with tandems have been previously mentioned. Tandems are also useful because they provide centralized points at which special features can be concentrated with considerable saving. For example, tandems are used for pulse conversion and for concen- tration of message charging equipment. Pulse conversion is needed when it becomes necessary to change from one type of pulsing to another, as, for example, on calls from a panel office to a step-by-step office. Panel can send out only revertive and panel call indicator pulses and step-by- step can receive only dial pulses. The two systems are therefore incom- patible without special arrangements. The following are some of the plans which might be used for handling calls to step-by-step. First, all the panel senders could be modified to send out dial pulses. Second, spill senders could be provided at the outgoing trunks in the panel office or at the incoming trunks in the step-by-step office to receive, say, revertive pulses and convert them to dial pulses. Finally, if there is a tandem in the area, the tandem senders could be arranged (as they actually are) to accept revertive or panel call indicator pulses and send out dial pulses. The first two arrangements are usually more expensive than the last. Therefore, when pulse conversion is required it is generally done by routing calls via tandem. To complete calls in the reverse direction, that is from step-by-step to panel, there is a requirement that is due to the use of the step-by-step system, namely that in cases where second dial tone is not employed the equipment at the called office or at an intervening tandem must be ready to accept the step-by-step pulses which are being dialed by the customer within a short time after the incoming trunk is seized. To meet this requirement, special high speed and costly link mechanisms are required to attach senders to incoming trunks or the incoming trunks must be arranged to record and store one or two digits. When calls are made be- tween two systems both using senders, however, cheaper and slower link mechanisms can be employed because the calling senders are arranged to wait for a sender attached signal from the called office. COMMON CONTROL SWITCHING SYSTEMS I I I I ADVANTAGES OF COMMON CONTROLS FOR AUTOMATIC UECOUDING OF IN- FORMATION FOR CHARGING The crossbar tandem system offers an economical method tor makiiiji; a record for charging i)uri)osos on mnlti-unit bnlk billed calls called re- mote control of zone registration. At present this is limited to use with oiiginating panel offices. The tandem is arranged to send back signals to the originating office for operating the customer's message register u]) to six times for the initial period on one call and also to operate it on over- time. Thus the application of extended customer dialing can be eco- nomically increased b}^ apphnng this arrangement in places which cannot justify the registration ari'angements available in the panel system itself which are economical only for a relatively heavy volume of this business. Local crossbar sj^stems provide these features economically enough to obviate the need for tandem control of message registers for calls orig- inating in the crossbar offices. When tandem offices are required to control the equipment which lecords customers' charging data, they must be equipped with common controls if the arrangement is to be economical. The data includes the origin of the call — the particular trunk group incoming to tandem over which the call arrives — and the destination — the called office code. These elements must be analyzed and combined to determine the basis for the amount charged. Since elaborate equipment is required for these func- tions, economy must be attained by providing a minimum amount of equipment to do the job. This objective is accomplished by providing the re(iuired features in the common controls. In tandems arranged for re- mote control of zone registration, for example, the number of times the customer's message register is operated is determined partly by the choice of trimk group at the originating office and partly by the tandem markers. In addition to remote control of zone registration, there are several other methods of determining and recording charging data which also re([uire the use of common control equipment. These are automatic ticketing, automatic message accounting and coin zone dialing. In automatic ticketing, which is used with step-by-step systems, calls which are to be ticketed are directed to outgoing trunks which select senders and other common equipment which determine the calling line number, reconstruct the called office code and store and outpulse the digits required for selections beyond the local office. The calling line number and the called office code are transmitted by the common equip- ment to the outgoing trunk which is equipped with a ticket printing 1112 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 device which prints this information and other data required for charging. The tickets can be used for bulk bills as well as detail records since they can be summarized at the accounting center by manual methods for calls on wliich detail information is not required. Automatic message accounting is used with crossbar systems both for bulk billing and detailed call records. With this system the data required for charging is perforated on j^aper tape by common central office equip- ment. The arrangement has been described in the technical literature* and will not be further described here. Both the ticketing method and automatic message accounting require the collection of a large amount of data and the ability to do a compli- cated job in handling and recording this data. Tliis demands elaborate and expensive equipment which is practical only when provided on a common basis so that it can be called into service for a short time and then restored to the common pool for other calls. Direct dial control systems without common controls can only have message registers on the line and therefore can handle notliing but bulk billed calls. Furthermore because of the expense of arrangements for determining multiple unit charge data and for operating the message register more than once on a call, multiple operation of message registers on individual calls is not practical. From coin stations in direct dial control systems the customer may dial calls only to offices within the local charge zone. However, in panel and crossbar areas the "coin zone dialing" arrangement is available to permit coin customers to dial beyond the local zone. With this plan calls are routed to a tandem office where completion is delayed until an oper- ator can plug into the trunk to tandem and supervise the collection of the required coins. The amount to be collected is indicated by trunk lamps which appear in a switchboard multiple. Common controls enter into this scheme at the originating office to route the call to tandem and to determine the charge, and at the tandem office so that the digits can be stored while the call is held up prior to collection of the coins. TYPES OF PULSING Direct dial control systems are restricted to operation with dial pulses and are usually limited to pulsing speeds of about 10 pulses per second and about one digit per second. Dial pulsing has range limitations which can be overcome by the addition of pulse repeaters at appropriate points. Common control systems store the digits in senders which can regen- * A.I.E.E. Transactions, 69, Part 1, pp. 255 to 268, 1950. COMMOX CONTROL SWITCHING SYSTEMS 1113 ciatc Ihciu ill \;iri()us lypc^s and combinalioiis of tyjx's ol' jjiilsing. Types of outpulsing found today in \aiious systems include revertive, panel call indicator, dial pulsing, dc key pulsing, and multi-froquoncy pulsing. Panel sender tandem and No. 4 toll can also sentl digital information ahead to operators by the call announcer method which uses voice an- nouncements derived from recordings on film. Provision for receiving and sending several types of pulsing in one system makes it more flexible since it can then comiect to a variety of cciuipments. Regenerating the jiulses adds to the range without the need of adding pulse repeaters. Some of the advantages which common control systems dei'ivc from the al)ility to operate with a modern type of pulsing can be brought out by a ])rief description of multi-frequency pulsing which is a relatively recent development. Digital information is transmitted over any facility capable of handling voice by sending spurts of alternating current which consist of pairs of frequencies in the voice range selected out of five freciuencies. There are ten such pairs. At the receiving end a check is ma(l(^ to insure that exactly two frequencies are received for each digit. When onh' one or more than two frequencies per digit are detected the call is not set up but a reorder signal is returned to the originating end. In addition to the advantages of being capable of transmission over voice facilities, including repeaters and carrier systems, and of providing checks for accuracy, this type of pulsing can be transmitted at the rate of seven digits per second at present. Operators can be provided with keysets capable of sending IMF pulses into either local or distant SAAdtching equip- ment with improA'ed operating resulting from the higher speed and otlier advantages of MF pulsing. It is quite feasible to add new types of pulsing to common control systems. Multi-freciuency pulsing has only recently been added to cross- })ar tandem, for example, although it has been in use with other crossbar systems for some time. In this case it required the development of new senders capable of receiving and sending the MF pulses. The addition of these senders, even in existing offices, is not a difficult job. IMPROVED STATION APPAK\TUS The stations in most exchanges are pro\'idcd with dials whic^h operate at approximately 10 pulses per second. In stcp-liy-stej) exchanges this pulsing speed is the maximum permitted by the capabilities of the .switches. In panel and crossbar areas the common equipment is capable of operating with higher speed dial pulsing, and PBX and central office operators in these areas are usually given dials that operate at about 18 pulses per second. 1114 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 Even fast dials are inefficient as compared to the push button keysets used bj^ operators for key pulsing and it is obvious that subscriber sets with push buttons would be faster and more convenient than dials. Such sets were used at Media, Pa., on an experimental basis and have func- tioned in a highly satisfactory manner. Their introduction merely re- quired the design and installation of registers to receive the pulses they generate. This was done with little difficulty or expense at the central office end. However, with ordinary step-by-step systems such devices are impractical because of the short interdigital interval they allow and be- cause of the cost of adding the pulse receiving equipment in every selector and of providing translation to change the key pulses into a form to drive the switch. CLASSES OF SERVICE Differences in the handling of calls from non-coin, coin and PBX lines and differences in rate treatments require the recognition of classes of customers at the central office. In step-by-step separate groups of line finders are provided to permit segregation in classes and where routings for different classes vary, separate selector multiples are required for these routings. Class distinctions within a line finder group can be made by normal post springs and by marking a fourth conductor in the line circuit. Common control systems permit the economical handling of many classes of service. The No. 5 crossbar, for example, is most flexible in this respect. As many as thirty classes of service can be handled in a single line link frame, including coin and non-coin. Special handling, reroutes and restrictions are mostly functions of the common controls and inefficiencies due to segregation of traffic in small groups of switching equipment are largely avoided. DOUBLE CONNECTIONS In systems such as panel and step-by-step in which selectors do the hunting, several selectors may be hunting over the same terminals simul- taneously, and since there is an unguarded interval just after an idle terminal has been found before it is made busy by the release of the busy testing relay, double connections occur. Considerable effort and expense have been expended to reduce the probability of double connections in these systems. In systems which employ markers, on the other hand, the trunk testing schemes do not normally permit double connections to COMMON CONTROL SWITCHING SYSTEMS 111') occur. In most marker systems a lockout arrangement permits onl}' one marker at a time to test trunks in a given group. There are cases where trunks arc common to two offices and two markers are allowed to test trunks simultaneously. In these cases special circuit arrangements are provided at nominal expense to avoid double connections. Modern com- mon control S3^stems with markers are, therefore, free of double con- nections resulting from weaknesses of the sj^stcm and they can occur only as a consequence of defects in circuits or apparatus. THEORETICAL OFFICES It is sometimes desirable to assign more than one office designation to customers in a single central office unit. A new unit may be planned for sometime in the future and if growth on the existing unit can be taken with a new office designation, then when this new office is placed in service it can be done without directory changes by transferring a block of lines from the old unit. Another occasion for assigning more than one designation to a single unit arises when customers served by the unit are in two rate zones, and service to lines in one of the rate zones must be restricted or extra charges collected. The lines served by an additional designation are called a theoretical office. Common control systems handle theoretical offices with little difficulty. In the first case mentioned the translating equipment in the originating offices recognizes that the physical office and theoretical office designations require identical treat- ment until the new unit is cut into service at which time translator cross- connection changes take care of the new routings. AVhere different rate treatments are involved, records for billing purposes depending on both the origin and destination of the call can be made by methods previously mentioned. In some cases where the billing data is determined at a tandem office and different treatments for the same destinations must be given to customers calling from one office, split trunk groups must be provided to tandem, one for each treatment. In the step-by-step system, theoretical offices can be opened up by multipling two selector levels together. For example, if the physical office is designated 25 and it is desired to open a theoretical office, say 26, the 5 and G levels on the proper second selectors in the network can be strapped until the 26 office is changed to a physical office. At that time the levels are split and trunks to the new office are connected to the 6 levels of the second selectors. Restrictions in reaching blocks of numbers can be applied by spHtting selector multiples and intercepting calls to restricted blocks from one of the splits. 1116 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 ADAPTABILITY TO NEW SERVICE FEATURES One of the major advantages of common controls, which has been covered in part but which deserves further emphasis, is their adaptability to new service features. Key sets and new dialing devices can be intro- duced at customers' stations and operator positions by readily feasible modifications of registers and senders. New pulsing schemes can also be introduced as they are developed as evidenced by the introduction of multi-frequency pulsing over the past few years. Nationwide customer dialing, now under development, can be readily introduced in existing common control systems by economical modifications without the use of either directing codes or second dial tone. Step-by-step systems re- quire at least partial senderization to provide equivalent service. In short, the flexibility of common controls and the concentration of the control elements in a relatively few circuits makes the addition of new service features easier and more economical than in direct dial systems. MAINTENANCE ASPECTS Experience has shown that switches with a large amount of motion, especially those with brushes which wipe over bank terminals, tend to wear excessively and require considerable maintenance effort and even replacement, at times. On the other hand, s^\itches with short motions and relay-like action require little maintenance and tend to have long life. Furthermore, the switches which employ wiping brushes mostly use base metal contacts, whereas relay-like switches can readily be equipped with precious metal contacts — and in most cases are so equipped — ^with the elimination of the transmission noise to which base metal contacts are subject. The crossbar switch is a relay type of switch Avith precious metal contacts and considerations such as those mentioned influenced its adoption. The advantages of relay type switches are not necessarily limited to common control systems since such switches have been used in direct dial control systems. The first use of the crossbar switch in Sweden was in a step-by-step system, for example. However, economical arrangements for using such switches in large systems require markers. This is because economy must be achieved by having more than one call ()(!cupv a switch at a time and marker control is necessary to attain this Important maintenance advantages have been introduced in systems using decoders and markers. In this category are the self-checking fea- tures, second trials with changed order of preference, and trouble report- ing features. In No. 5 crossbar the al)ility to report the location of a line COMMON CONTROL SWITCIIIXG SYSTEMS 1117 with a ixM'maneiit signal by perforating a liouhlc ircoidci- card has cliniinated the need for tracing permanents. A number of schemes are employed to detect tr()ul)les in markers and (l(>co(lers and in circuits which connect to them. These include detectoi's for wrong sequences of operations, wrong combinations of i'e]a3\s, exces- si\-e current, false potential and lack of contiiuiity. These are generally iiiti-odu(;ed at small cost since the circuits to which they are applied ai'c small multipliers. Howe\'er, some of them do a majoi- job of testing since they reach out and test the numerous elements of the switcliing system to which markers have access. In this category are the tests of the cross- bar linkages for opens, false grounds and double connections, tests of the switch crosspoints for continuity, tests of lines for false grounds, and for receivers off the hook on coin first coin lines. To obtain clear trouble records, markers are designed with interlocked progress signals. This has made trouble analysis easier and has tended to improve design by eliminating relay races. Starting with the panel system tests have also been intioduced in senders for detecting open and reversed trunks. These tests have been of considerable help in maintaining outside plant and in detecting condi- tions that could lead to false charges. DIJ^ADVANTAGES OF COMMON CONTROLS Up to this point the stress has been mainly on the ad\'antages of com- mon controls. There are also some disadvantages. One of the major ones is the substantial getting started cost due to the necessity of providing a minimvmi amount of common equipment. This minimum is provided to maintain operation in case of trouble and during intervals when, for example, cross-connections require change because of changed or added routes. The minimum requirements establish economic })arriers which tend to prohibit the economical use of common controls for small iso- lated systems. Another disadvantage is the performance of common control systems under severe and protracted overloads. Experience with these systems indicates that although they compare quite favorably to direct dial con- trol systems with respect to capability of handling moderate overloads, they are not able to handle severe overloads as well. In part this is a consequence of the fact that elements in common control systems are used at high efficiency and hence there is relatively less free equipment at full load for soaking up an overload than there is in systems that operate with smaller and less efficient groupings. AMienever the number 1118 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 of calls presented to the system exceeds the capacity of the common control elements provided, the excess calls are delayed. The things which customers, operators and connecting switching machines do when they encounter delays tend to aggravate the overload. The reactions of oper- ators and customers to delays can be illustrated by two examples. The first is taken from the operation of a network of No. 4 toll crossbar systems when one of the No. 4's is heavily overloaded. Operators placing calls through the overloaded system encounter, let us say, an abnormal number of "no circuit" conditions in the outgoing trunks. This causes them to make additional attempts to get circuits. These additional at- tempts plus the excessive number of fii'st attempts overload the markers. Sender holding time is then increased because of delays in connecting to the markers and this, added to the abnormal number of sender usages, results in a further shortage of senders. Operators trying to place calls through the system are therefore slowed down because of slow "sender attached" signals. (These are the signals which tell the operators that they can start keying or dialing.) Senders in connecting systems are also delayed w^aiting for senders to become idle in the overloaded office. The overload therefore tends to spread to all connecting systems. However, it is possible to provide remedies which limit the reaction to the overloaded system. These remedies are arrangements to rapidly clear out senders waiting for senders ahead. Automatic alternate routing is also useful in routing traffic around overloaded systems. The second example is taken from local systems. Here the reaction of customers to delays compounds the overload. A severe overload results in a shortage of senders, much as described above. A shortage of senders in a local system causes dial tone delays. There are always some custom- ers who either do not listen for dial tone or who will not wait very long for it, and who start to dial before senders are attached to their lines. The result of such dialing is either a partial digits condition under which the sender waits for a considerable interval for a full complement of digits, or a wrong number when the first digit is clipped. The delays reduce sender capacitj^ still further and the wrong numbers further in- crease the attempts. The load "snowballs" and the ability of the system to handle calls degenerates. Here again arrangements are available to control the overload. These include features for blocking calls before they reach the senders and markers, and for returning paths busy signals with a minimum of com- mon circuit holding time. While there is, then, a somewhat greater capacity for overloads in step-by-step because of less efficient use of equipment, common control COMMON' CONTROL SWITCHING SYSTEMS 1119 systems do a good job of haiidliiiji; moderate overloads and, by provision of load control features, viui operate satisfactorily even with severe o\'erloads. From a maintenance standpoint, a disadvantage of common controls is the relative complexity of the circuits. While this has introduced a training problem, maintenance forces have had no difficulty in acquiring the knowledge^ nec(h'd to do a conipe^tcnt niaint(>nanc(^ jol). CONCLUSION The full fledged common control systems exemplified by the crossbar local and toll s^'stems have a number of important advantages over systems where the switches are dii\(Mi directly by the customer's dial. The advantages arise largely from the ability to store digits, to translate them, use them fiexibl}' for switching within the office, and transmit as many of them as desired to distant points for subsequent switching operations. The digits can be converted to others of different value whenever it is adA'antageous to do so. The inherent flexibility of common control equipment makes it possible to adopt any kind of numbering plan for a local area or a nationwide network that is best suited for the purpose without regard to the manner in which calls will be trunked fi-om one point to another. Codes can be assigned at will to represent destina- tions and the best route for the call can always be taken. The best route may in some cases involve tandem operation or even a half-dozen switches in tandem. It may be the I'oute selected as an alternate after previous trial of one or more other routes. A connection may be set up between offices of different types and over trunk groups requiring differ- ent forms of pulsing. These conditions may be met b}" common control equipment and the ability to meet such conditions makes it possible to provide cheap step-by-step eciuipment in places for which it is best suited, compensating for some of its deficiencies with common control eciuipment in other places. With marker type common controls, ti-unk groups out of an office can be of any desired size regardless of the switch design. The individual crossbar switch, for example, gives access to only ten or twenty outlets as normally wired but full access single trunk groups of hundreds of trunks can be employed in some crossbar systems. Schemes for recording billing data, aside from the relatively simple ones where metering equipment is associated with the customer's line and operated once per call, make use of common control equipment. This seems to be necessary where detail records must be made on indi- vidual culls for charging purposes. 1120 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 As improvements in the art are made they can more readily be in- corporated in common control systems than in step-by-step systems. For example, new subsets which may employ keys or other sending devices different from the dial can be accommodated by provision of proper facilities in senders and registers. Also, improved high speed pulsing arrangements can be easily incorporated in systems which do not require the switches themselves to be directly driven by pulses from the calling device. BIBLIOGRAPHY 1. Bailey, W. J., "Lorimer automatic exchange at Hereford," P. 0. Elect. Engrs. J., 6, Part 2, pp. 97-155, July, 1913. 2. Aitken, William, Automatic telephone systems, D. Van Nostrand Co., N. Y., 1924. 3. Miller, K. B., Telephone theory and practice, McGraw-Hill, N. Y., 1933. 4. Rorty, M. C, "How the theory of probability may be applied to telephone traffic," Western Electrician, 36, p. 356, May 6, 1905, Abstract. 5. Craft, E. B., and others, "Machine switching telephone system for large metro- politan areas," Bell System Tech. J., 9, pp. 266-274, Jan., 1934. 6. Bronson, F. M., "Tandem operation in the Bell System," Bell System Tech. J., 15, pp. 380-404, 1936. 7. Scudder, F. J., and Reynolds, J. H., "Crossbar dial telephone switching system," Bell System Tech. J., 18, pp. 76-118, Jan., 1939. 8. Abraham, L. G., and others, "Crossbar toll switching system," A.I.E.E. Trans., 63, pp. 302-309, 1944. 9. Friend, O. A. ."Automatic ticketing of telephone calls," A.I.E.E. Trans., 63, ^ pp. 81-88, 1944. 10. Smith, A. B., "The 'director' for automatic telephone switching systems," A.I.E.E. Trans., 67, pp. 611-619, 1948. 11. Meszar, J., "Fundamentals of the AMA system, ".4 ./.E'.iS'. Trans., 67, Part 1, pp. 255-269, 1950. Mathematical Theory of Laminated Transmission Lines — Part II By SAMUEL P. MORGAN, JR. lliis part of the paper continues the analysis of the low-loss, broad-band, laminated transniission lines proposed by A. M. Clogston, and deals particularly with ''Clogston 2" lines, in which the entire propagation space is filled with laminated material. TABLE OF CONTENTS Page VIII. Principal Mode in Clogston 2 Lines with Infinitesimally Thin Laminae 1121 IX. Partially Filled Clogston Lines. Optimum Proportions for Principal Mode _ 1133 X. Higher Modes in Clogston Lines 1150 XI. Effect of Finite Lamina Thickness. Frequency Dependence of Atten- uation in Clogston 2 Lines 1163 XII. Effect of Nonuniformity of Laminated Medium 1181 XIII. Dielectric and Magnetic Losses in Clogston 2 Lines 1201 Appendix II: Optimum Proportions for Heavily Loaded Clogston Cables 1203 Appendix III: Power Dissipation in a Hollow Conducting Cylinder. 1204 VIII. PRINCIPAL MODE IN CLOGSTON 2 LINES WITH INFINITESIMALLY THIN LAMINAE In Part I* of this paper we have set up a general mathematical frame- work for the analysis of Clogston-type laminated transmission lines and have applied it to Clogston 1 lines having laminated conductors, but with the total thickness of the laminations small compared to the overall dimensions of the line, so that most of the forward power flow takes place in the main dielectric. In Part II we shall consider Clogston 2 lines, which instead of containing a main dielectric have the propaga- tion space entirely filled with laminations; and we shall also derive results, in Sections IX and X, for the general laminated transmission * S. P. Morgan, Jr., Bell System Tech. J., 31, 883 (1952). Since the two parts of the paper are very closely related, the sections, equations, figures, and footnotes have been numbered consecutively throughout the whole paper. A table of symbols appears at the end of Part I. 1121 1122 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 line in which the relative fractions of space occupied l)y the main dielec- tric and the laminations are arbitrary. A parallel-plane Clogston 2 line is shown schematically in Fig. 10. It consists of a stack of alternate layers of conducting and insulating material, whose total thickness is a. As before, the electrical constants of the conducting and insulating layers are denoted by ^i , c/i and eo , M2 respectively; and the fraction of conducting material in the stack is called e. The stack is bounded at ^ = ±|a by sheaths whose normal surface impedance is Zn(y), where 7 is the longitudinal pi-opagation constant of the mode under consideration. Zn(7) Fig. 10 — Parallel-plane Clogston 2 transmission line. Zb(/) Fig. 11 — Coaxial Clogston 2 transmission line. LAMINATED TRANSMISSION LINES. II 1123 The cross section of a coaxial Clof>stoii 2 cable is shown schematically in Fig. 11. It consists of a laminated coaxial stack bounded internally by a cylindrical core of radius a, which may be equal to zero so far as the theoretical analysis is concerned, and externally by a cylindrical sheath of radius b. We denote the radial impedance looking into the core at p = a by Za(y), and the radial impedance looking into the sheath at p = h by Zb(y). In this section we shall assume the laminae to be infinitesimally thin, so that the stack may be regarded as a homogeneous, anisotropic medium, completely characterized by its average electrical constants. The case of finite lamina thickness will be treated in Section XI. We shall neglect dielectric and magnetic dissipation throughout, except in Section XIII. For modes of the type which we consider, whose only field components are Hj^ , Ey , E^ in the plane line or //^ , E^ , E, in the coaxial line, the average electrical constants of the stack are given by equations (90) of Section III, namely, € = €2/(1 - d), n = en, + {I - e)n, , (268) g = 6gi ■ As observed in Section III, Maxwell's equations for the average fields in such an artificial anisotropic medium take the form, for a plane stack, dHJdz = iwiEy , dHJdij = -gE,, (269) dEy/dz - dEJdy = iwflH^ ; while for a cylindrical stack, dH^/dz = —iweEp , d(pH^)/dp = gpE. , (270) dE,/dp - dEp/dz = luifiH^ . We wish to determine the modes which can propagate in the laminated medium when guided by plane or cylindrical impedance sheets. This problem was solved for a homogeneous, isotropic dielectric in Section II of Part I ; and the method of solution is so similar for the anisotropic 1124 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 medium that we shall omit details of the analysis and pass at once to the results. In the parallel-plane line, the modes for which /7x is an even function of y about the center plane y = 0 have field components given by Hx = ch T(y e 7 E. = ch Tty e E^ = -KshTfye"'', (271) up to an arbitrary amplitude factor, where r^ and K are defined, as in Section III, by T/ = — (.w Aie + 7 ) we K = T,/g = J_ /■ 2_. > 2n ._ (w/xe + 7 j [_o:eg (272) (273) Matching impedances at the boundaries y = ±^a leads to the condition K tanh h^ia = -Zn(y). (274) In the odd case, the fields are given by Hg, = sh Tty e~''% Ey = -^ sh Tty e""', (275) zoje E,= -KchTtye"", up to an arbitrar}^ amplitude factor, and the boundary condition be- comes K coth iTta = -Zn(y). (276) General expressions for the field components in the coaxial line are H^ = U7i(r,p) + BKriTtpW, E, = — . [AhiTtp) + BKiiTtpW", iwe (277) E. = K[AUTtp) - BlUTtpW^ LAMINATED TRANSMISSION LINES. II 1125 where .1 and 11 arc arbitrary constants and Ff and A' ai'c defined as boforo. The lioiiiidary conditions at p = a and p ^ h take the foi-m AIo{l\a) - BIu{Y,a) ^ y . . "^ Ahir^a) + Blu{l\a) ''"^^^' ^^^^^^ AUTtb) - BKojTcb) _ ^ AI,{r,h) + BK,{Tcb) ''^^^' and these e([nati()ns can be satisfied by \alues of ,1 and />' that are not botii zero if and only if /avo(r,a) +Za{y)IuiTca) ^ KKoJT^b) - ZMK,{Tth) , KhiTca) - ZMhiV^a) KhiVcb) + Z,(y)I,{T,b) ' ^*^^^^ Now K is given in terms of T( by equation (273), while frf)m equa- tion (272) we have y- - -co'/Ze - {io)e/g)T]. (280) Hence if the dependence of the boundaiy impedances on y is known, equations (274) and (276) for the plane line and equation (279) for the coaxial line are transcendental relations from which in principle we may determine T( , and therefore y, for each mode of the type that we are considering. If the value of T( for a particular mode satisfies the in- equality r^, osfig « 1 , (281) then on taking the square root of the right side of equation (280) by the binomial theorem, we find that the attenuation and phase constants of the given mode are approximately a = Re 7 = -Re ^' , (282) /3 = Im 7 = CO V^ - Im — ^= . (283) 2^V)"/e Throughout the rest of this section we shall consider only the lowest or principal mode. In a parallel-plane line the principal mode corre- sponds to the lowest root in Tt (that is, the root having the smallest modulus) of equation (274), which may be written in the form ^TcataiDh^Tta = -^gaZniy). (284) 1120 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 We may express 7 in terms of 1\ by equation (280), and so if Zn{y) varies with 7 in any reasonably simple way, or better yet if Zn(y) is essentially independent of 7 in the range of interest, equation (284) may be solved numerically for F^ by successive approximations. A numerical solution of equation (284) is, however, rarely necessary, since the right-hand side of the equation is just the ratio of the sheath impedance Zn{y) to the resistance "per square", namely l/(^ga), of all the conducting layers in a stack of thickness |a in parallel, and this ratio will almost always be large compared to imity. This is another way of saying that the total one-way conduction current in the stack is large compared to the sum of the conduction and displacement currents in either sheath. Even if the sheaths are infinitely thick metal plates of conductivity gi , we have from equation (79) of Section III, since 7 W ioiy/jie , hgaZniy) = idgxam = (1 + i)ea/28i , (285) and for most frequencies of interest the thickness ^da of conducting ma- terial in half the stack will be several times the skin thickness 81 in the metal. If the medium outside the stack is free space, then Zn(y) will be a few hundred ohms and a fortiori the right side of (284) will be large compared to unity. So long as the inequality 1^ 2 gaZ„(y) I » 1 (286) is satisfied, the lowest root of (284) will be approximately r, = iw/a; (287) and so from (282) and (283) the attenuation and phase constants of a plane Clogston 2 line with infinitesimally thin laminae and high-im- pedance walls are . (288) 2\^fx/e ga ' ^ = wV^e. (289) To this approximation, there is neither amplitude nor phase distortion. The principal mode in a coaxial Clogston 2 corresponds to the lowest root in T( of equation (279). To solve this equation numerically with finite boundary impedances Za(y) and Zb(y), w^hile possible in principle, would evidently be a major undertaking. We shall therefore assume throughout the present paper that the total conduction and displace- ment currents flowing in the core and the sheath are negligible compared (292) LAMI.VATKD TRANSMISSION' LINES. IT 1127 to the conduction currents in the laminated medium. This is equivalent to assuming that the bounchuy impedances Za(y) and Zi,(y) arc effec- tively infinite, so that equation (279) reduces to the simple form Equation (290) may l)e converted to one involving ordinary Bessel and Xeumaini functions by the substitution r' = -x', r, - ix. (291) Tiicn since K„(r,p) = wr'"^'' [./„(xp) - iXnixp)], the e(iuation may easily l)e transformed into Ji(xa)Xi(xh) - ./i(x6).Vi(xa) - 0. (293) For any given value of the ratio a/b, equation (293) has an infinite number of real roots in x- The lowest root xi has been tabulated^* as a function of h/a, and may be written in the form X, = #^, (294) where fi{a/b) is a monotone decreasing function of a/b which is equal to 1.2197 when a/b = 0 and to 1 when a/b = 1. Hence the attenuation and phase constants of the principal mode in a coaxial Clogston 2 with infinitesimally thin laminae and high-impedance walls are rflia/b) , . a = j= , (295) 2VM/e g(b - ay /3 = coV^e , (296) and again to this approximation tiicre is neither amplitude nor phase distortion. Comparing equations (288) and (295), we see that the attenuation constant of the principal mode in a coaxial Clogston 2 with infinitesimally thin laminae (that is, the low-frequency attenuation constant if the laminae are of finite thickness) is equal to the attenuation constant of ** E. Jahrike and F. Emde, Tables of Functions, fourth ed., Dover, New York, 1945, pp. 204-207. What we call irji{a/h) is tabulated by Jahnke and Emde, p. 205, as {k — l)x/^', where k = b/n, while our/i(o/6) is plotted as 1 -(- a on p. 207. 1128 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 the principal mode in a plane Clogston 2 times the factor fi{a/b), provided that the thickness of the plane stack is equal to the thickness 6 — a of the coaxial stack. The functions /i (a/6) and/i(a/6)/(l — a/b) are plotted against a/b in Fig. 12. From the plots it is apparent that fi(a/b) decreases steadily from a value of 1.488 at a/b = 0 to 1 at a/b = 1, while fi(a/b)/(l —a/b) increases steadily from 1.488 at a/b = 0 to infinity at a/b = 1. Therefore if the stack thickness b — a is fixed, the attenuation constant will be smaller the greater is the mean radius of the stack; while if the outer radius b is fixed, the at- tenuation constant will be reduced by reducing the radius a of the inner core, and the lowest attenuation will be achieved when a = 0. It should be noted that our expressions for the attenuation and phase constants of Clogston 2 lines cannot be valid down to the mathe- matical limit of zero frequency, since the inequality (281), on which we based the approximations (282) and (283) for a and j8, \\\\\ ultimately break down as the frequency approaches zero. A similar failure of the approximate expressions which we used for the attenuation and phase constants of Clogston 1 lines was pointed out in Section II of Part I. Here, as before, we shall limit the use of the term "low frequency" to frequencies still high enough so that the attenuation per radian is small and the approximate formulas (282) and (283) for a and jS are valid. l.i) \ / i \ / Nj \f,' (a/b) > /f,^(a/b)/[i-(a/b)]^ \ / ^ >^ ^ 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 a/b Fig. 12 — Curves related to the function f\{a/h) = {b- aYxl/^', where /i(xia)A^i(xi&) - /,(xib)iVi(xio) = 0. LAMINATED TRANSMISSION LINES. II 1129 Usually we shall be able to apply these formulas down to frequencies of a few kc-sec~ . The field components of the principal mode in a plane Clogston 2 with infinitesimally thin laminae and high-impedance boundaries at ?/ = ±^a are given by equations (271), on substituting for r^ from (287). We have, approximately, Hx = Ho cos — e ■yz Ev= - A/-. Ho cos ^ e-'' , (297) ye a J, TT jj . Try -y^ hz = — Ho sm -^ e , ga a where Ho is an arbitrary amplitude factor, and in the coefficient of the expression for Ey we have replaced y by its approximate value iw\/fle. The bars have been omitted from i/^ and E, since these field components are continuous at the boundaries of the laminae. The potential difference between any two points in the same trans\^erse plane is the integral of —Ey between the points. In particular, the total potential difi'erence between the upper and lower sheaths is V= -T Ey dy = ^4/^ Hoe--". (298) The average value of the conduction current density J^ is Jz = gEz =- Ho sin ^ e"'', (299) a a and the current per unit width flowing in the positive ^-direction in the upper half of the stack is Jo J, dy = Hoe"''', (300) so that the ratio of voltage between the sheaths to total one-way cur- rent per unit width is The fields of the principal mode in a coaxial Clogston 2 with infinites- mally thin laminae and high-impedance boundaries are given by equa- 1130 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 tions (277), which simplify somewhat if we write ix\ for T( , replace the modified Bessel functions with ordinary Bessel functions according to (292), and remember that H^, must vanish at the high-impedance boundaries. We then get, approximately, H, = H,[ i(xi6)/i(xip) - ./l(xl&)A^l(xlP)]e""^ VI E,= j^'^. HolN.ixmiixw) - Ji(xih)N^{xip)]e~'% (302) E. = ^ Ho[Ni(xib)MxiP) - Jiixib)No{xip)]e~'% 9 where Ho is an arbitrary amplitude factor. The potential difference between any two points in the same transverse plane is the integral of —Ep between the points. Thus the total potential difference between the core and the outer sheath is V = - [ Ep dp J a -Vl ^' [iVi(xi&)/o(x.p) - J.i.xih)No{xip\y" (303) c Xi 'p. 2Ho Mxib) 1 e TTXi Lxia-^i(xia) Xi^J after some transformations using equation (293) and the well-known identity No(x)Jr(x) - Ni(x)Joix) = 2/t,x. (304) The average value of the conduction current density J z is 7. = gEz = Hoxi[Ni(xih)Jo(xip) - Ji(xib)No(xipW''. (305) The current reverses at p = c, where a < c < b and c satisfies Ni(xib)Mxic) - Ji{xib)No(xic) = 0; (306) hence the value of c may be found with the aid of a table of Bessel func- tions or from plotted curves. The total one-wa}' current in the outer part of the stack is I = 2w Jzp dp = 2TcHo[Ji{xib)N^(xic) - Ni(xib)J,ixicW (307) ^HoMxib) xiJoixic) e 18 Reference 18, p. 208. LAMINATED TRANSMISSION LINES. II 1131 whoi'c ill the last step \v(> iia\'o made use of (804) and (306). The ratio of voltafie aci'oss the stack to total one-way current is, from (303) and (307), V I M ./o(xic) € 2ir 1 1 Xi'>-/i(xi'>) Xifl^/i(xi«) (308) If there is no inner core, so tiiat a = 0, the e.xpressions which we have just deri\ed become indeterminate forms, and it is simplest to make an independent calculation of the fields for this special case. The Neumann functions are now e.xcluded because of tlieir singularity at p = 0, and the condition (293) is replaced b.y ^i(x^) = 0, from winch XI = 3.8317/6. The expi-essions for the fields are (309) (310) (311) Xi E. = ^ HoMxip)e'"'% Q where //o is now a different arbitrary amplitude factor. The total potential difference across the stack becomes, aftei- putting in numerical values, y = -I E,dp = 0.3661 Vm/c Hobe~ Jo The conduction current density is Jz = gE, = xJhJo{x\p)e"^'; and Jz changes sign at Jo(xi^) = 0, c = 2.4048/xi = 0.6276/>. The total one-way current is / = 27rf//„./i(xic)e~"-' = 2.047//obe"'', and the ratio of total \oltage to t(jtal current is V/I = 0.1788Va7^ • (312) (313) (314) (315) (316) 1132 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 The fields of the principal mode in both plane and coaxial Clogston 2 lines will be plotted in the next section, when we shall also be able to show the fields in various transition structures between the extreme Clogston 1 and the complete Clogston 2. As a numerical example, let us compare the attenuation constant of a conventional coaxial cable with that of a completely filled Clogston 2 cable of the same size. If a and h denote the radii of the inner and outer conductors of a conventional coaxial cable of optimum proportions (5/a = 3.5911), then at frequencies high enough to give a well-developed skin effect on both conductors, the attenuation constant is given by equation (151) of Section IV, namely 1.796 , . oc = —r , (317) where rjo is the intrinsic impedance of the main dielectric, which may be air. On the other hand, the attenuation constant of a Clogston 2 cable of outer radius h, with infinitesimally thin laminae and no inner core, is, from equations (282), (291), and (310), 7.341 ^ ^ « = /^-,, . (318) It will be shown in the next section that for infinitesimally thin laminae whose permeabilities are all equal, the optimum value of 6 is 2/3. As- suming no magnetic materials and setting 6 = 2/3, we find that the ratio of the attenuation constant ac of the Clogston cable to the at- tenuation constant «« of an air-filled standard coaxial cable of the same size, made of the same conducting material, is a./«3 = 10.62 V^ 8i/b. (319) For copper conductors, 5i is given by equation (78) of Section III, and the crossover frequency above which the Clogston cable has a lower attenuation constant than the standard coaxial cable turns out to be /mo = 763.5€2r/&mil8 , (320) where frequency is measured in Mc-sec~ and the radius of the cable in mils. We also note that at the crossover frequency the electrical radius of the inner conductor of the standard coaxial is 2.96 v'€2r ^i » so that the use of equation (317) for a, appears to be (barely) justified. Applying equation (320) to an ideal Clogston 2 cable of outer diameter 0.375 inches, excluding the sheath, with copper conductors, polyethylene LAMINATED TILVNSMISSION LINES. II 1133 insulation, and no inner core, we have h = 187.5 mils, eor = 2.26, (321) and the crossover frequency is about 50 kc-sec~ . It must be emphasized that several factors which have not yet been taken into account will conspire to reduce the practical improvement in transmission that can be obtained with a Clogston 2 cable. As we have already seen for Clogston 1 lines in Part I, the effect of finite lamina thickness in a Clogston 2 ^^^ll be to cause the attenuation constant to increase with increasing frequency, and ultimately to become higher than the attenuation constant of a conventional coaxial cable. Dissipa- tion in the insulating layers may also contribute appreciably to the total loss at the upper end of the frequency band. Perhaps most important of all, the a\'erage electrical properties of the laminated mediimi must be held extremely uniform across the stack, or the field pattern of the principal mode Anil be distorted and its attenuation constant corre- spondingly increased. In later sections we shall discuss these effects, in order to estimate the stringency of the requirements on a physical Clogston cable if its factor of improvement over a conventional cable is to approximate closely to the theoretical limit given, for example, by equation (319). IX. PARTIALLY FILLED CLOGSTON LINES. OPTIMUM PROPORTIONS FOR PRINCIPAL MODE The distinction which has heretofore been made between Clogston 1 and Clogston 2 lines is rather artificial, inasmuch as both structures are limiting cases of the general Clogston-type line in which an arbitrary fraction of the total space is occupied by laminated material and the rest by an isotropic main dielectric. We shall now consider the modes which can propagate in a general partially filled line, restricting ourselves for simplicity to stacks of infinitesimally thin layers backed by high- impedance walls. Under these assumptions we first set up equations which must be satisfied by the propagation constants and the fields of all modes having only H^ , Ey , Ez or H^ , Ep , Ez field components in a partially filled Clogston line, and then proceed to a study of the lowest or principal mode. We exhibit field plots for this mode at various stages of the transition between the extreme Clogston 1 and the complete Clogston 2 geometry, and investigate the conditions under which the attenuation constant passes through a minimum as the space occupied by the stacks is increased. This leads to the determination of certain opti- mum proportions for a line intended to transmit the principal mode. 1134 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 111 Section X we shall give a similar but briefer treatment of the various higher modes which can exist in partially or completely filled Clogston lines. The notation for the parallel-plane line is established in Fig. 5 of Part I. The stacks are bounded externally by high-impedance sheaths at 2/ = ±l«, while the main dielectric is bounded by the planes y = ±^5 = ±(^a — s). No restrictions are placed on the relative thick- nesses h and s of the main dielectric and the stacks. The average electri- cal constants of the stacks are e, Jx, and g, while the electrical constants €o and Mo of the main dielectric are assumed to satisfy Clogston's con- dition (102) but are otherwise arbitrary. As in Section II, the modes may be divided into two classes, according to whether Hx is an even function or an odd function about the center plane y ■= 0. The normal surface impedance Z{y) looking into either stack may be obtained from equation (92) of Section III; if the imped- ance of the outer sheath is effectively infinite we have Z{y) = K coth ViS = (Tt/g) coth Tts, (322) where T( = coe (w>e -f 7") (323) Substituting for Z(y) into equations (11) and (13) of Section II, we find that the impedance-matching conditions become tanh iKoh tanh r,s = - — - , (324) g Kg ioieo r g for the even and odd modes respecti\-ely, where From (323) Ave have coth i/cofc tanh Vts = _ !^« L' ^ (325) g Ko lio = (o-o — 7")' = ( — co'moCo — 7 )'• (326) 7" = -coVe - (iw€/g)T] , (327) and so from (326), 4 = -a)"(Mo€o - /ie) + (io}e/g)r] . (328) If Clogston's condition is satisfied, namely fjLaeo = p.e, (329) LAMINATED TRANSMISSION LINES. II 1135 then Ko = (/coe'g)r', Kn = Vi^'e/g T/ , (330) and the equations tor th(^ ovon and odd modes beeome, respectively, hmhWi^gr^bUmhl\s = - '-^i/'y' = -^aM, (331) ^ V 9 Mo K 9* il 1 /• - - T^ 7 4. 1 T1 ^" Aw€ M A cotli ^v^w«/^ r<6 tanh l\.s = a/ — = —~a/- ^yg Mo K '^ . (332) g For reference we shall now write down the field components of llif various modes. The fields in the main dielectric are given by e(iuations (8) and (12) of Section II, while the fields in the stacks may be ol)tained without difficulty if we recall that the tangential field components must be continuous at the inner boundary of each stack and that the tan- gential magnetic field must vanish at the high-impedance surface which forms the outer boundary of the stack. Taking the even modes first, we have for the fields in the main di- electric, Hx = Ho ch ^•()?/ e"^^ E„ = -Ho^ ch K„y c~^^ (333) icoeo Ez = — II 0 -. — sh KoU 0"^% for — 16 ^ ?/ ^ \h, where //o is an arbitrary amplitude factor, 7 and Ko are given in terms of V( by (327) and (330), and V( satisfies (331). The fields in the stacks are i/x = //c%i?^sh^,(|aT2/)c-^^ sh V(S E, = -Ih — . ^^4?^ sh Tciha T y) e~'\ (334) E, = ±Ho ^ %|?^ ch r,(ia T y) e-'% g sh r^s for ^h ^ \ y \ ^ |a, Avhere in case of ambiguous signs the upper sign is to be associated with the upper stack (y > 0) and the lower sign with the lower stack {y < 0). The continuity of E^ at ?/ = ±^6 is a conse- quence of equation (324) or (331). 1136 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 For the odd modes, the fields in the main dielectric are Hx = Hq sh Kay e"^', Ey= -Ho^sh Koij 6""% (335) Ez = —Ho -^ ch Koy e"'% for —\h ^ y ^ ^b, where Ho is again an arbitrary amplitude factor and 7 and kq are defined as before in terms of F^ , which is now a root of (332). The fields in the stacks are Hx = ±Ho %i^ sh rX^a T y) e-'\ sh T(S Ey = ^Ho -^ '^|?5 sh r,(ia -F y) e''^ (336) ?coe sh 1 (S E. = +Ho ^ '-^ ch T^{ha T y) 6''% g sh TfS for |6 ^ \ y \ ^ ^a, where again the upper signs refer to the upper stack and the lower signs to the lower stack. The continuity of E^ at y = ±|6 is now a consequence of equation (325) or (332). The notation for the partially filled coaxial cable is shown in Fig. 6 of Part I, where as before we assume that the laminae are infinitesimally thin, the boundary impedances are effectively infinite, and the main dielectric satisfies Clogston's condition. The radius of the inner core is a and that of the outer sheath is b, while the stack thicknesses are Si and S2 respectively; but no restrictions, other than obvious geometrical limita- tions, are placed on the relative values of a, b, Si , and S2 . The inner and outer radii of the main dielectric are denoted by pi (= a + Si) and po (= b — S2) respectively. The boundary conditions at the surfaces of the main dielectric will be satisfied, as in Section II, by matching radial impedances at the stack-dielectric interfaces. If the impedance Za looking into the core at p = a is effectively infinite, then the impedance looking into the inner stack at pi is given by equation (98) of Section III to be ^^ ^ Tj Ko(r,pi)/i(r,a) -f Kr{T(a)Io(Tcpi) ,33^. g Ki(r,a)7i(r,pi) - Ki(Tfp,)h{Tfa) ' Similarly, if the sheath impedance Zi is infinite, then looking into the LAMINATED TRANSMISSION LINES. II 1137 ()ii((M- stack at pi \vc have z ='^ ^^'o(r^P2)/i(r,6) + /vi(r,6)/o(r,p2) .^^^. ^' g /vi(r,p2)/i(r,6) - /vi(r,/>)/i(r,p.>) " 'i'he coiKlitiou that the radial impedances shall be matched at the sur- faces of the main dielectric is given by equation (38) of Section II, which lakes the form KoKo(koPi) + iweoZiKiJKoPl) _ KoKo{koP2) — io}€oZ2Kl{KoP2) CiOn) KoIo{koPi) — iCiieoZiIi(KoPl) KohiKopo) + ZCO€oZ2/i(koP2) where kq is related to T( by equation (330). If we substitute the expres- sions (337) and (338) for Zi and Zo into (339), we have a single equation whose roots in T( correspond to all the circular transverse magnetic modes on the coaxial Clogston line. The propagation constant y of each mode is given in terms of Tf by equation (327). Once the boundary conditions have been satisfied for a particular mode by a suitable determination of F^ , it is a routine matter to obtain the field components for this mode. In the main dielectric the fields are of the form given by equations (33) of Section II. Hence for pi ^ p ^ P2 we have H^ = U/i(kop) + BKriKopW, E, = -X- [A/i(kop) + BK.iK.pW', (340) E.= -!^ U/o(kop) - BIU{K,p)]e-'% where one of the constants A and B is arbitrary, but the ratio A/B must be taken equal to either side of equation (339). The fields in the stacks are of the form of equations (277) of Section VIII, where the constants are to be determined so that H^ = 0 at p = a and p = h, and so that the tangential field components are continuous at pi and p2 . Imposing these conditions, we find that in the inner stack, for a ^ p ^ Pi , H^ = C[Ki(r,a)7i(r,p) - /i(^,a)/^l(^,p)]e-^^ E, = ^, C[Ki(r,a)/x(r,p) - /i(r,a)/^i(r,p)]e-^ (341) icoe E. = ^ C[K,{Tca)UTtp) + /l(^,a)/Co(^,p)]e-^^ g 1138 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 where ^ ^ ^/i('<:opi) + BKiJKopi) , 7^i(r,a)/:(r,pO - /i(r,a)/Ci(r,pO • ^'^^^^ 111 the outer stack, for p- ^ p ^ b, E, = — . Z)[A'i(r,6)A(r,p) - h{Vth)K,{T(p)]e"'% (343) ?coe E. = ^4 D[lUVth)UVcp) + h{Vch)K,{Tcp)]e-'\ 9 where _ ^/l('^0P2) + BKi(koP2) i^i(r,6)/i(r,p2) - /i(r,6)K:(r,p2) ' ^'^"^"^^ For the remainder of the present section we shall confine our attention to the principal mode. In the parallel-plane line this mode corresponds to the lowest root in r^ of equation (331), that is, tanh I V^W^ Tfh tanh T(S = -^a/^-^ . (345) Mo K Q We note that the right side of the equation is very small compared to unity, being of the order of the square root of the ratio of displacement current density in the insulators to conduction current density in the conductors, and also that the coefficient of Vi in the first factor on the left will under all ordinary conditions be much smaller than the coefficient of r^ in the second factor. Hence in seeking the lowest root we are justi- fied in replacing the first hyperbolic tangent on the left side of (345) by its argument, so that the equation becomes (346) (347) (348) Since the right side of (348) is a positive real constant, the equation has Tis tanh V(S = -^~ . Mo 0 If we now let r' = -x', r, = ix, we obtain xs tan xs = — T" . Mo 0 LAMINATKD TUANSMISSION' LINES. II 1139 exactly oiu^ I'oot in the iiit(M\al 0 < x-"^' = 2^. wliicli may most easily 1)(^ found iVom a taldo" of the fmictioii .r tan .r. If we call this root xi> e(iiiatioii {'.V27) for the pi'opaiiat ioii constant 7 IxM-omes 7" = -co"Me + (iwe/g)xl ; (349) and on takina,' the sciuarc root l)y the l)inoniial Ihcorcm we lind for the at teinial ion and phase constants of the principal mode, 2 ^' (350) (3 = wV^i- (351) It is easy to \erify that (350) reduces to the expressions previously obtained for the attenuation constants of Clogston 1 and Clogston 2 lines in the limiting cases s <^ ^a and s = ^a respectivel3^ If s for pi ^ p ^ p_' , d{-pH,)/dp = 0, d{gE,) /dp = (piOC^/pp) ( - p//^) ; and for P2 ^ p ^ h, d(-pH,)/dp = -p(gE.), d(gE.)/dp = (x'/p)(-pH,). (3691) (369ii) (369iii) The ([uantities —pH^ and gEz must be continuous at pi amd po ; and the two-point boundary condition at the infinite-impedance surfaces p ^ a and p — b, namely -aH^ia) '-= -bH^ib) - 0, (370) determines a sequence of eigen^^alues xi , xi , x^ , " " • , of which the low- est corresponds to the principal mode. It is a routine matter to integrate eciuations (369) in terms of Bessel fmictions and logarithms, and to show that the continuity and boundary conditions lead exactly to equation (364). If equations (369) are set up on a differential analyzer with adjustable values of x , cl/^, Pi/b, pz/b, and mo/m, it is a simple procedure to make a few runs with different choices of x", and so to locate the appi'oximate 1144 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 2 value of xi which satisfies the boundary conditions for the given values of the other parameters. If additional accuracy is wanted, it is then not too difficult to refine this approximate value by desk computation. The results of quite a number of exploratory calculations which were made on the Laboratories' general purpose analog computer will be shown later in this section. The fields of the principal mode in the main dielectric of a Clogston cable are given approximately by equations (46) of Section II, namely ^P^V^/-^-^ (371) E. Zl . P2 . Z2 , Pi — log - + — log 2ir log {pi/pi) \_pi p Pi p J ' for pi ^ p ^ P2 , where / is an amplitude factor equal to the total current flowing in the positive 2-direction in the inner stack, and Zi and Z2 are given by writing xi for x in (362) and (363) respectively. The fields in the inner stack are H, I Ni(xia)Ji{xip) - JiixicQNiixip) -1 27rpi Ni(xia)Jiixipi) - Ji(xia)Ni{xiPi) (372) E r^ a/E -1^ Nl{xia)Jl{Xlp) - Jl(xiCi)Nl(xip) -7Z " '^ y e 2xpi iVi(xia)^i(xiPi) - /i(xia)A''i(xiPi) ' ^ ^ XI _^ NiixicQJoixip) - '/i(xia)iVo(xip) g-7. ' ^ g 2tpi iVi(xia)J'i(xipi) - /i(xia)iVi(xiPi) for a ^ p ^ pi ; while in the outer stack we have H^^-l- ^^(xib)J,(xip) - Ji{xih)N,(xip) ^-y. '^ 2-KP2 N i{xih)J i{xip2) — Jiixib)Ni{xip2) "'^ y -e 2-Kp, N^(xih)Ji{xiP2) - Mxib)Nr{xiP2) ' ^ ^ ^ ^ XI _i_ Ni(xib)Jo{xip) - Ji{xib)No(xip) g-T. ' '^ g 27rp2 Ni(xib)Ji{xip2) - Ji{xib)NiixiP2) for P2 ^ p S b. The potential and current distributions may be calcu- lated in the usual way from the fields. As numerical examples we have plotted in Fig. 14 the fields of the principal mode in two Clogston coaxial cables. Fig. 14(a) shows a LAMINATED TUAXSMISSION LINES. II 1145 cable ill wiiich mo = m, eo = e, and with the dimensions a = 0.0846, Pi = 0.4156, and pi = 0.8316, these proportions iiaving been found optimum, as discussed below, for a cable without magnetic loading in wiiich the total thickness of both stacks is arbitrarily chosen equal to \h. Fig. 14(b) shows the helds of a complete Clogston 2 with no inner core, the scale being chosen so that the total one-way current is the same in both cases. The attenuation constant of the first cable is 1.234 times that of the second one. Fig. 14 may be compared with Fig. 13 for the plane geometry, whence it should be possible to visualize ap- proximately the fields of the principal mode in other coaxial structures representing various stages of the transition between the extreme Clog- ston 1 and the complete Clogston 2 cable. Now let us consider a Clogston line with infinitesimally thin laminae, having fixed external dimensions and containing only materials with given electrical constants. We may pose two questions: (1) Supposing that for some practical I'eason the total available thickness of laminated material is also fixed, how should this material be divided between the two stacks to minimize the attenuation constant of the line? (2) Assum- ing that the total thickness of laminations in the line is at our disposal, what is the optimum amount of laminated material from the standpoint of minimizing the attenuation, and how should this material be distrib- uted in the optimum case? For plane transmission lines the first question is trivial; the stacks should always be of equal thickness. In a coaxial cable, if the filling ratio (si + S2)/6 is gi\'en, the proportions of the cable are completely determined when we specify, for example, the relative radius a/6 of the core and the relative thickness Si/(si + S2) of the inner stack. The opti- , f F Hc^ = H OR E H OR E Fig. 14 — Fields of principal mode in partially and completely filled coaxial Clogston lines with mo = M, eo = «. 1146 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 0.25 I , 1 ^ , ^ 1 , 1 1 1 0.75 0.20 0 3.0 \ \ S,/(S -.-c ^ ^ '■■^2; X ~~~" ^^ (a) ^ >^ v.,,.^^ ^_ 0.70 0.65 ^ 0.55 \ \ \ \,^ ^N (b) 0.5 0 0.1 0,2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (S, + S2)/b Fig. 15 — Relative proportions and relative attenuation constants of optimum Clogston cables with different filling ratios and mo = m, «o = «■ mum values of these two ratios in the extreme Clogston 1 case, where (si + So) « b, have already been given in equations (138) and (139) of Section lY, while in a complete Clogston 2 with Si + So = h, the stacks should be' divided at the radius 0.62766, where according to equation (314) the current density is zero. For intermediate filling ratios, with any fixed magnetic loading ratio mo/m, the optimum distribution of laminated material can most easily be found numerically by calculating Xi or (xiby for a number of different choices of the ratios a/b and Si/(si + S2), and then locating the minimum by double interpolation. The results of applying this numerical procediu-e to Clogston cables with various filling ratios and no magnetic loading are plotted in Fig. 15, the necessary values of xib having been found on the analog com- puter and then refined by desk computation. Fig. 15(a) shows the op- timum values of a/b and Si/(si + S2) as functions of the filling ratio (si + S2)/b, while Fig. 15(b) shows the corresponding value of {xib/S.8S)', LAMINATED TRANSMISSION LINES. II 1147 which by equation (365) is proportional to the attenuation constant. We note that the Clogston 2 Hne with fiUing ratio unity lias the low- est attenuation constant of any cable of the same size without mag- netic loading, but that the attenuation constant increases only slowly as the filling ratio decreases, so long as the ratio is greater than about one-half. It also appears that the minimum in xi^, considered as a func- tion of ajh and Si/(si + S2) for a fixed filling ratio, is quite broad, which means that in practice one can attain very nearly optimum perform- ance even while deviating somewhat from the optimum proportions. If the filling ratio is at our disposal, then the solution of the optimum problem is as follows: When there is no magnetic loading of the main dielectric relative to the stacks, that is, when mo ^ m, then minimum attenuation is obtained with a complete Clogston 2. If on the other hand there is magnetic loading of the main dielectric, so that juo > m, then minimum attenuation is obtained with a filling ratio less than unity, whose value is a function of the ratio mo/ju. According to equation (350), the attenuation constant of a plane Clogston line is 2 Xi (374) 2Vm/ c g where xi is given by equation (348), 2m M XI tan xis = — = —77- -. . (375) To find the minimum value of xi when a and mo/m are given, we differ- entiate (375) with respect to s and set dx\/ds equal to zero. This gives Xi sec^ xi-s = ~7T~ T2 , (375.5) which when solved simultaneously with equation (375) leads to sin xis = Vm/mo , XI = - sin~^ Vm/mo • (376) Substituting this value of xi hito (375) and solving for s in terms of a, we get ^ _ 1^ MO sin~^ V^o /o77>v Mo sin 'Vti/iJ.Q + V m(mo — m) and from (374) the corresponding attenuation constant is 2 « = /^T- - 2 [sii"!"^ Vm/mo + \/m(mo - m)/mo]"- (378) VM/e ga 1148 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 As mo/m increases from unity to very large values, the optimum value of s decreases from ^a toward \a, so that the filling ratio decreases from unity toward one-half. If mo/m < 1, equation (376) does not yield a real solution, but the complete Clogston 2 is still the physical structure hav- ing the lowest attenuation. For a coaxial Clogston line without magnetic loading the optimum filling ratio is unity, as we have seen above, while in the presence of magnetic loading a smaller filling ratio is optimum. This filling ratio and the optimum distribution of the laminated material in the cable can be determined by numerical analysis for any given value of juo/m- It is reasonably evident on physical grounds, and can be proved mathe- matically by a variational argument applied to the lowest eigenvalue of equations (369), that whatever may be the radii pi and p2 of the main dielectric, the lowest attenuation constant is achieved when a = 0, that is, when there is no core inside the inner stack. (This is only a mathematical limit; from a practical standpoint, the use of a small core in the manufacturing process is not likely to make any significant in- crease in the attenuation of the cable.) For each value of the loading ratio no/fi, therefore, we have merely to minimize the value of (xi&)' as a function of the two ratios pi/6 and pz/b, which can be done by the double interpolation procedure mentioned earlier. We find that as mo/m in- creases from unity to very large values, the optimum value of pi de- creases from 0.62766 toward 0.39306, while p2 increases from 0.62766 to- ward 0.82266, so that the filling ratio decreases from unity toward 0.5704. The limiting values of pi and p2 when mo/m ^ 1 are derived from equa- tion (364) by the method shown in Appendix II. As a numerical example we have considered a Clogston cable with Ho = 3/i. The optimum proportions of this cable and the corresponding value of xi are approximately Pi = 0.4266, p2 = 0.7966, xi = 2.720/6; (379) and the minimum attenuation constant is 3.699 , , (380) Vm/c gb~ ' The attenuation constant of a complete Clogston 2 with the same stack parameters jl and e is, from (318), 7.341 , , (381) V m/€ gb" ' so that the attenuation constant of the optimum loaded cable is only LAMINATED TRANSMISSION LINES. II 1149 about 0.504 times that of the optimum unloaded one. In this example, of course, we have said nothing about the effects of magnetic dissipa- tion. In the above Avork we have assumed that the electrical constants M, e, g of the stacks and /zo , fo of the main dielectric were all fixed quanti- ties. We now consider the case in which the electrical constants of the conducting and insulating layers are given, but the fraction 6 of conduct- ing material in the stacks may be varied. We also suppose that the con- stants of the main dielectric are at our disposal, so that Clogston's condition may always be satisfied. When then is the optimum value of 67 If we express e, m, and g in terms of the constants of the individual layers by equations (268) of Section VIII, we find that the expression for the attenuation constant of the principal mode in a Clogston line becomes €2X1 efW, -f (1 - e)y^,fgx ' (382) where xi is the lowest root of equation (348) for a plane line or equation (364) for a coaxial cable. We wish to minimize a as a function of 6. If the conducting and insulating la.yers have different permeabilities (mi 7^ M2), then in the general partially filled line xi depends on 6, through the factor ^ in equation (348) or (364), as well as on the geometric pro- portions of the line. In the limiting case of an extreme Clogston 1 line we found in Section IV, equation (145), that the optimum value of d is a Ml + (mi + 8MliU2)'' 3/il + (mi + S/XliUo)' (383) while in a Clogston 2 with no main dielectric, it turns out from (348) or (364) that xi is independent of 6, and an elementary calculation shows that the value of 6 which minimizes a is Q ^ 3 (mi — 2/X2) + (9mi — 4/X1JU2 + 4m2)' /gg.N 8 (mi — M2) For the general partially filled line, however, there seems to be no simple expression for the optimum value of 6. If the conducting and insulating layers have equal permeabilities, then the average permeability /I (= mi = M2) is independent of 6, and matters are much simpler. Since xi is also independent of 6, the minimum value of a in equation (382) is achieved when 6 = 2/3, (385) 1150 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 that is, when the thickness of the conducting layers is twice the thickness of the insulating layers. Thus the result obtained in Section IV for extreme Clogston 1 lines is shown to hold for Clogston lines with an arbitrary degree of filling, provided only that the permeabilities of the conducting and insulating layers are equal. We emphasize that the preceding results apply only when the layers are infinitesimally thin. If the layers are of finite thickness, then the optimum value of 6 will be less than that calculated for infinitesimally thin layers. The case of finite layers will be discussed in Section XI. X. HIGHER MODES IN CLOGSTON LINES We shall now investigate certain of the higher modes which are possible in Clogston-type transmission lines. As elsewhere in this paper, we shall restrict ourselves to modes having H^ , Ey , Ez or H^ , Ep , Ez field components only, and for simplicity we shall assume stacks of infinitesi- mally thin laminae backed by high -impedance boundaries; but we shall place no restrictions on the relative thicknesses of the stacks and the main dielectric. We shall suppose, however, that the main dielectric always satisfies Clogston's condition. From physical considerations we anticipate the existence of higher modes of two types: (1) In a partially filled Clogston line containing a finite thickness of main dielectric, there will be a group of modes very similar to the modes which can propagate between perfect conductors when the frequency is high enough to allow one or more field reversals in the space between the conductors. In a Clogston line these modes will have most of their field energy in the main dielectric, and for lack of a better term may be called "dielectric modes". They will all be cut off at sufficiently low frequencies, and for this reason are not likely to be of much engineering importance. The cutoff frequency of any particular dielectric mode is approximately inversely proportional to the thickness of the main di- electric, so that these modes cannot exist in a completely filled Clogston 2. (2) There will also be a group of modes which are closely bound up with the laminated stacks, and which correspond to one or more current reversals in the stacks themselves; we shall call these the "stack modes".^^ The stack modes will propagate do^^^l to zero frequency on either a partially or a completely filled Clogston line. They mil have higher attenuation constants than the principal mode, but occasions may arise in which they are of considerable practical importance. We shall therefore consider these modes in some detail in what follows. 2^ The stack modes in plane lines were discussed bv Clogston in Reference 1, Sections IV-VI. LAMINATED TRANSMISSION LINES. II 1151 As we have seen in the preeetUnp; section, the even and odd modes in a plane Clogston hne witli infinitesimally tiiin laminae and iiigh-imped- ance boundaries correspond respectively to the roots of equations (331) and (332), namely tanh hV^^ T(h tanh T(S = -- A/ — , (386) Mo K g coth hVi^ r,?> tanh T(S = --A/ — . (387) Mo K 9 In either case the propagation constant y is related to T( by y' = -co'/xe - (icc-e/g)T-t , (388) and the field components are given by (333) and (334) for the even modes, or by (335) and (336) for the odd modes. Our first observation relative to equations (386) and (387) is that the right-hand sides of these equations are extremely small compared to unity. Since the right-hand members are of the order of magnitude of (coe/g) , at least one of the two factors on the left side of each equation must be of the order of (oje/g)*, which is still small compared to unity. If we consider the factors separately, there "U'ill be an infinite number of values of Tt for which each vanishes, since the hyperbolic tangent \'anishes whenever its argument is equal to rmri, where m is any integer, and the h^^perbolic cotangent vanishes whenever its argument equals (m + ^)iri. Since the coefficients of r^ in the two factors on the left side of either equation have different phase angles, we see that both factors cannot vanish simultaneously for any non-zero value of T^ . However as we have noted earlier the coefficient of Tt in the first factor is very much smaller than the coefficient of T( in the second factor, and so in equation (386) both hjnperbolic tangents may be small in the neighborhood of the first few non-zero roots of the second one. On the other hand the second hyperbolic tangent will not be small in the neigh- borhood of the non-zero roots of the first one; and in equation (387) the hyperbolic tangent and cotangent will never be small simultaneously. With these remarks in mind we shall proceed to a more detailed study of the various higher modes. One group of modes is given to a good approximation by the condition that the first factor on the left side of equation (386) or (387) vanishes, that is, \/ioil/g Vth ^ rrnri, (389) where ??i = 1, 2, 3, • • • , and the even values of m correspond to the even 1152 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 modes while the odd values correspond to the odd modes. In this section we shall exclude the case m = 0, which corresponds to the principal mode discussed in the preceding section. From (389) and equation (330) of Section IX, we get KO T (391) The fields of the ?nth mode are given by substituting these quantities into (333) and (334) if m is even, or (335) and (336) if m is odd. From equation (388), making use of Clogston's condition, the propaga- tion constant of the wth mode of this family is given by 7" ;^ — co>oeo + m'-K /b' = — 47rVXo + nfT /b , (392) where Xo is the wavelength of a free wave in the main dielectric at the operating frequency. To this approximation the values of y are the same as the propagation constants of the family of modes (with H^ , Ey , and E^ field components only) which are possible in a dielectric slab of thickness b between perfectly conducting planes. The cutoff wavelength of the mth mode is \c = 2b/7n, (393) the propagation constant being real, to the present approximation, if Xo > X,. and pure imaginary if Xo < Xc . We see that the cutoff fre- quency is inversely proportional to the width of the main dielectric, so that this family of modes is not possible in a completel}^ filled Clog- ston 2. It is worth noting that the effective skin depth of the stacks for the mth mode is, from (390), A = Rer, ITT y g Xo K ^1^9 If the mode is just above cutoff, then A is of the order of magnitude of 5i (= \/2/w/ii6ri), but as co increases indefinite!}^ A also increases in- definitely, for the ideal stack of infinitesimally thin laminae. The physi- cal explanation is simple: When the mode is near cutoff the phase velocity is very high, but as the frequency is increased the phase velocity approaches the velocity of a free wave in the main dielectric, for which the effective skin depth of the stacks was designed by Clogston's condi- LAMINATED TRANSMISSION LINES. II 1153 tion to be infinite. By increasing the /xoeo product of tlie main dielectric, it would be possible to make the effective skin liiickness of a stack of infinitesimally thin layers infinite for any given mode at any single specified frequency, but at the moment this possibility appears of scarcely more than academic interest. Of course the practical limitation on effective skin depth at high frequencies is the finite thickness of the layers, a consideration which we do not take into account in the present section. The attenuation constants of the dielectric modes, when these modes are above cutoff, may be calculated either by obtaining the small cor- rections to the values of T( due to the fact that the right side of equation (386) or (387) is not rigorously zero, or by taking one-half the ratio of dissipated power per unit length to transmitted power. Either method gives for the mth mode, assuming the stack thickness s to be large compared to A, r.= '^'^ V27^ (395) b- MO Vl - (Xo/X.)2 Ecjuation (395) assumes conchicting laj'ers very thin compared to the skin depth, a situation which may be difficult to achieve at frequencies high enough to permit the modes of this family to propagate. Another family of modes which can exist on a parallel-plane Clogston line is given by the condition that the second factor on the left side of equation (386) or (387) shall be nearly equal to zero. As pointed out above the even modes present a slight complication; since the coefficient of T( in the first hyperbolic tangent on the left side of (386) is very small, this factor may be comparable to or smaller than the term on the right side in the neighborhood of the first few roots of the equation, in which case the second hyperbolic tangent will not be small compared to unity at these roots. For all the modes in which we can conceivably be inter- ested, however, | T(b \ will be a small fraction of the very large nimiber 2'\/g/cce, and we may therefore replace the first hyperbolic tangent on the left side of (386) by its argument. Thus on making the usual sub- stitution, r'. = -X, r, = ix, (396) we get for the even modes, xs tan xs = — -y , (397) Mo 0 which is the same as equation (348) of the preceding section. On the 1154 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 other hand, tlie odd modes of this family are all given approximately by tan xs = 0, (398) since the hji^erbolic cotangent on the left side of (387) will never be small for the same value of F^ (or x) as the hj^jerbolic tangent. Equation (397) has an infinite number of positive real roots, which may be denoted by Xi , X3 , X5 , • • • , X2p+i , • • • , (399) and it is clear that pr/s < X2P+1 ^ (p + Dtt/s. (400) If the thickness h of the main dielectric is not zero, so that the right side of equation (397) is finite, the higher roots X2p+i approach nearer and nearer to pir/s as p increases; but if 5 = 0, then X2P+1 = (p + ^)7r/.s = (2p + l)T/a (401) for all p. The positive roots of equation (398) may be called X2 , X4 , X6 , • • • , X2p , • • • , (402) where X2p = pyr/s (403) for all p; and both sets of roots may be combined in the single sequence Xi , X2 , X3 , • • • , Xp , • • • . (404) The advantages of designating the principal mode as the first rather than the zero-th mode seem to outweigh the minor disadvantage that in the sequence (404) the odd subscripts correspond to what we have been calling the even modes, and vice versa. The attenuation and phase constants of the pth. mode are obtained in terms of Xp from (388) and (396). Under the usual assumption that the attenuation per radian is small, we have 2Wig (405) jS = us/fil, (406) which become, for the completely filled Clogston 2, pV2 (407) «-2VA7e^a^' /3 = co-v/yue. (408) LAMINATED TRANSMISSION LINES. II 1155 From (330) and (396) we have for the pth mode, r^ = ixv, (409) K, = (-1 + 0V«e/2^ Xp . (410) The fields maj' be obtahied by substituting T( and ko into equations (333) and (334) when p is odd, or (335) and (330) when p is even. For tlie modes in which we are interested, that is, for sufficiently small values of p, we may replace sh kqij by k^ij and ch kqij bj^ unity when \y\ ^2^. Then for the modes with odd subscripts 2p + 1 we have, in the main dielectric, Ey ^-A/f^Hoe-"'"-'', (411) for —^6 ^ y ^ \h, while in the stacks, H, ^ Ho ^"^ X2p+xiha ^ y) g-T2p+i^ "" ^ ° sin X2p+is ' Ey^-iA H. '^ ^^--^^^'^ ^ ^^ .-— , (412) K e sm X2p+iS E ^ ^X2p+i ^ cos X2p+i(|a ^ y) g-72p+i^ ^ "^ g sin X2p+is for ^& ^ I y I ^ |a, where the upper signs refer to the upper stack and the lower signs to the lower stack. Of course the arbitrary amplitude factor Ho need not be the same for different values of p. Similarly for the modes with even subscripts 2p the fields in the main dielectric are H.^0, F Ci-" 0 gs for —^b ^ y ^ hh, while in the stacks, E, ^ i-y f^ Hoe-''-', gs (414) 1156 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 s Ey^^^Ho sin P-I^tS^ e-^^\ gs s for |6 ^ I 2/ I = 2«, and again the upper signs refer to the upper stack and the lower signs to the lower stack. In a complete Clogston 2 the expressions for the fields simplify a good deal. For the modes with odd subscripts 2p -{- 1 the fields are, for -^a ^ ij ^ ^a, i/,;^^,C0S^-?^-+il^6-— , a Ey^-ji/l i/o COS (?^±i)iL^ e-^-- (415) ye a ga a while for the modes with even subscripts 2p, H, ^ Ho sin ?^^ 6-^^"^ a Ey^ - \/\ Fo sin ?^ e-^^-^ (416) € a ^.;^-?^i/ocos?^.-^^^ ga a The fields of the higher modes in a plane Clogston 2 are simply related to the fields of the principal mode shown in Fig. 13(d). The fields of the pth mode may be obtained conceptually by stacking up p "layers", each of thickness a/p, the fields in each layer being identical with the fields of the principal mode except for the scale reduction and a phase difference of 180° between adjacent layers. Equation (407) shows that the attenuation constant of the pth mode in a plane Clogston 2 with infinitesimally thin laminae and high-impedance walls is just p times the attenuation constant of the principal mode. It may be observed that if we are considering a partially filled plane Clogston line with 6 > 0, then the propagation constants of the 2pth LAMINATED TRANSMISSION LINES. II 1157 and tlie (2p + l)st stack modes will be nearly the same for sufficiently large values of p (how large depends on the ratio of stack thickness to main dielectric thickness). Except for differences in sign, the fields in the stacks will also be the same up to second order differences which our approximations do not show. The physical interpretation is that for a thick enough main dielectric and/or sufficiently large values of p, the fields are confined almost entirely to the two stacks, being rela- tively small in the main dielectric, while the stacks act like a pair of almost independent Clogston 2 lines each of thickness s and carrying a particular Clogston 2 higher mode. Figs. 16 and 17 show field plots for the second and third stack modes (i.e., the first and second higher modes) in the same four plane Clogston lines that were used to exhibit the behavior of the principal mode in Fig. 13. Xote however that these plots are not normalized and that the horizontal scales on the figures are not all the same. The following table gives X2S and xss as functions of the fraction s/^a of the total space filled by the stacks, and also the quantities (x^a/Tr) and (x3 right side in powers of 0 by Dwight 657.2 and 657.8. If we replace sin pw/^n by p7r/4/i and neglect the square of this quantity in comparison with unitj^, (453) becomes 4n2 e' 12 (454) Iiitrodncing this expression for q into equation (443) and substituting for 0 from (452), we get for the propagation constant, 1 n V4wWifi'i^? 12" (455) As the frequency approaches zero in a Clogston 2 line of finite thick- ness, the term in l/w dominates the other terms in square brackets in eciuation (455). Thus at veiy low frequencies the attenuation and phase constants of the pth mode are given approximately by _ pird /we _ pt " ~ 27\ y 2^ ~ "a /3 pird we 97i (456) (457) Trif 9 ■zg a y g where 2Ti (= 2nti) is the total thickness of conducting material in the stack of thickness a. To this approximation the attenuation and phase constants are equal, and are proportional to the square root of frequency We note that at very low frequencies, 7^ 2ip~T^d^(j:e 1 dp-ir-e/fii 2 0-1 ^Tlg tcofjLigi 2-2 ag (458) which will be very small compared to unity for lines of all reasonable dimensions, in agreement with our assumption (ii) above. 1168 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 As the frequency is increased there will he a range in which the terms in parentheses in equation (455) are small compared to unity, so that the square root may be expanded by the binomial theorem. This gives a = ^ ^ + -^^^ , (459) /3 = icoV^e . (4()p) If the line is of finite total thickness a and the frequency is so low or the laminae are so thin that the first term on the right side of (459) is large compared to the second, we have approximately P IT 2\/iJL/e ga' (4G1) This is the frequency-independent attenuation constant that we found in Section X, equation (407), for the pth mode in a plane Clogston 2 with infinitesimally thin laminae. We shall call the range over which the attenuation is essentially flat the "low-frequency" range. On the other hand, if the laminae are of finite thickness the second term on the right side of (459) ultimately becomes dominant, and the attenuation constant is then given approximately by o^'ulgll _ Tr'nlgtlf 24:\/fI/e ()\/M/e (462) This is also the attenuation constant of a plane wave in an unbounded laminated medium (except at very high frequencies), as may be seen by letting the stack thickness a tend to infinity in equation (459). By "high frequencies" we shall mean the frequency range in which the attenuation constant is approximately proportional to / . Finally at very high frequencies when | 0 | » 1, w^e have from (451), q^2/@, (463) and so from (443), 7 = ?'co\//i2e2 1 + 2dn, (1 - 0)m20J (464) Expanding by the binomial theorem and substituting for 0 from (452), we get after a little rearrangement, LAMINATED Tli-VNSMIttSlOX LINES. II IKi'J ^ [I'l/ei hgih 7=-7 J^^ . (4<'.5) /5 / — i (466) Comparing tliese expressions with (Hjuations (25; and (20) of Section II, we see that they correspond to waAcs in parallel-phme transmission lines of width t^ , bounded by electrically thick solid conductors. Wc shall call this range, in which a is proportional to the square root of tr(M|U(nicy, the "ver}^ high frequency" range. At these frequencies the propagation constant is the same in a Clogston 2 lino of finite thickness as in an infinite laminated medium. In order to describe the various frequency ranges more precisely, we shall define the three critical frequencies /i , /2 , and /s to be the fre- (luencies at Avhich the approximate expressions for the attenuation constants in two adjacent frequency ranges are equal. These frequencies are closely related to the critical frequencies which we defined in equa- tions (178) of Section V, when w^e were discussing the surface impedance of a plane stack of finite layers. For a stack containing a total thick- ness Ti of conducting material, we recall that the critical frequencies were /i , where Ti = 8i ; f2 , w^here Ti = Ta ; and /s , where h — \/3 5i . For the pth. mode in a Clogston 2 containing a total thickness 27*1 of conducting material, the frequencies turn out to be /; = /; = / = pV^M p~Trd mgj'l 16/1 rzr- Ji , 2fMigitiTi 36m El 2 (467) _(1 — 6)n-i_\ TTfj'igitl 4^6 h, where of course p = I for the principal mode. Thus the attenuation (■(justant is given- approximately by (456) for 0 ^ / ^ /i , by (461) for h ^ / ^ /2 , by (462) for /.f ^ f ^ f, , and by (465) for / ^ /a . If we plot the foregoing approximate expressions for the atteiniation constant against frequency on log-log paper, we can get a good idea of the \'ariation of the attenuation of a Clogston 2 over the entire frequency range. Both the approximate expressions and the exact results for a particular numerical case are plotted in Fig. 20, for a Clogston 2 of linite thickness and also for an infinite laminated medium. The actual 1170 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 5 2 E - ^^ V 5 - 'A '/ " ^ XACT APPROXIMATE i 2 / 5 2 E / f - / } 5 2 E y " P/ INFINITE 'MEDIUM CLOGSTON 2.y / = i coml)ination of Bessel functions CnixpP) = N,(xJ>).f„(xpP) - Ji(xph)yn(xpP); (474) and X;- •■'^ tli(' /^th positive root of Ci(xa) = Ni(xh)J^(xa) - ./i(xM.V,(xa) = 0. (475) According to ecjuation (434) of Section X, we may wi'ite 2)Trfpia/b) , . Xp = —, • , (4/()) b — a where the functions fp{a/h) are of the order of unity. The total current /(p) flowing in the positive ^-direction between the inner core and a cylinder of arbitrary radius p is just Up) = 2ir f pJ, dp = 27r//opCi(xpp). (477) •'a The thickness of the jth conducting layer in a stack of finite layers may be written t, = e(k + ^2) = d(pj - p;_i) = dip, , (478) where Apy represents the thickness h + 1-2 of the jth. double layer. Hence approximately II j = 2Trpj^J^Apj = 2TrHoXppj-iCi)(xppj~i)'^pj , (479) it being remembered that the conduction current in the conducting layer is essentially equal to the total current in the double layer, since the displacement currents are negligil)le. The cuircnt flowing inside the radius py-i is, from (477), 7y_i = 2irHnPj-.iCi(xpPj~i), (480) and so the {)ower dissip;itc(j pci' unit Icuglli in tlic./lli c(»iiductor is 1174 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 given by (472) as AP, = TtHoHoPj^I dYi X'pCoixppj-i) + 0T4 C'i(XpPj-i) Apj. (481) The total power AP dissipated per unit length in the whole stack is obtained by summing APj o\'er j. Approximately the sum by an in- tegral, we have AP = ttHoHo 9 irHoHoXp j pixlClixpp) + ~ Ci(xpp)] dp 2g 1 + 3xp5i_ (482) [b^Cl(xph) - a'Clixpa)]. The average transmitted power P when the laminae are infinitesimally thin is Re i [ f E,Hlpdpd Jo Ja = tta/'^ HoHt I pCl(xpp) dp = h^AAHoHtlb'Clixph) - a'Clixpo)]. (483) If we assume the same value for P when the laminae are of finite thick- ness, then from (482) and (483) the attenuation constant of the line is AP 2 Xp 2P 2 Vm/^ g L 1 + 3xl5t_ (484) The similarity of equation (484) to equation (468) for the parallel- plane line becomes obvious if we write Xp in the form (476) and denote the total thickness ^(6 — a) of conducting material in the coaxial stack by 22^1 . We then have pV^/p(a/6) 1 + u\t\ 2\/M/e g{h - of 1 + 3pV^/p(a/6)6t_ 4.2rp2 2 2r2-| (485) and as the ratio a/b approaches unity the function fp(a/b) approaches unity and (485) becomes identical with (468). We recall that fi{a/b) was plotted against a/b in Fig. 12. For the principal mode in a cable LAMINATED TRANSMISSION LINES. II 117") with no inner core (a = 0), c(iuation (485) takes the lorni « = /— . -,2 [1 + O.S9mt\T\n\g\f-]. (480) V M/ e go It should be emphasized that whereas equation (4G8) was obtained fiom a I'igorous solution of the boundary-value problem for the plane line, eciuation (48.")) for the coaxial cahle has been derixcnl on the l)asis of certain physical assumptions and appi'oximations whose effect on the accuracy of the final result is not very eas.y to estimate. Presumably one might check the acciu-acy of (485) for a particular Clogstou cable l)y setting up the matrix relation between the known surface impedances of the core and the outer sheath and solving numerically for the propa- gation constant. It should not be too difficult to solve the matrix equa- tion by cut-and-try methods for a cable having, say, two hundred double layers, if the matrix of each double layer were assumed to be given by ecjuations (88) of Section III, and high-speed computing machinery were used to perfoi-m the matrix multiplications. In the absence of any such lunnerical results, however, we shall merely assume that equation (485) furnishes a reasonable approximation to the attenuation constant of a coaxial Clogston 2 in the frequency range /i ^ / ^ fs , where /i and /s are the critical frequencies defined by (467). The first conclusion which we can draw from (485) is that the maxi- mum permissible thickness of the conducting layers in a coaxial Clog- ston 2 with high-impedance boundaries, if the attenuation constant of the pth mode is not to exceed its "flat" value ao b}' more than a speci- fied small fraction ^a/ao at a top frequency fm , is , VS pfp{a/b) /a^ . ..^^. h = —^r^ J— A/ — , l-ioO or, putting in numerical values for copper, _ 36.84p/,(a/6) /K^ K'^-l lJmi\s\JmjMc Y Q!o For the principal mode in a Clogston cable with no inner core, this becomes 44.93 /a^ (^i)mii3 = /oT ^ /'/ ^ 4/ — • (489) {^ilJmiliKjmjMc y ao As a second application of equation (485), we shall determine the upper crossover frequency at which the attenuation constant of a Clogston 2 is equal to the attenuation constant of a conventional coaxial 1170 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 cable of the same size. Since the lower crossover frequency was found at the end of Section VIII, we shall then know the theoretical limits of the frequency range over which a gi\'en Clogston cable can have lower loss than the corresponding standard coaxial. According to eciuation (317) of Section VIII, a conventional coaxial cable of radius h and optimum proportions has an attenuation constant 1.796 _ 1.796 VTTMigi/ VAto.^eo fi'i^i^ Vmo/co git> (490) We shall assume that the upper crosso^'er occurs in the high-frequency langc where the attenuation constant of a Clogston 2 is approximately pi-oportional to / . Then for the pth mode in a cable with no inner core (a = 0), equation (485) gives 2Th\TlfAglf TT'tifilgf (491) A little algebra shows that the two attenuation constants are equal ^vhen / = 1 10.7" TT^igi L2Tit'i (492) If the conventional cable is air-filled, then assuming copper conductors and no magnetic materials, Ave find that equation (492) becomes, nu- merically, 1 h 33.02 (2Ti),nils(/l)mils 1 - (493) If we consider a 3/8-inch Clogston cable AAith 0.1-mil copper conduc- tors, 0.05-mil polyethylene insulators, and no inner core, then b = 187.5 mils 2Ti = 125 mils h = 0.1 mils e = 2/3 62. = 2.26 (494) We found in Section VIII that the lower crossover frequency for this cable is about 50 kc-sec~\ while from equation (493) the upper cross- over frequency turns out to be 15 Mc-sec~ . We next discuss the problem of maximizing the frequency band over which the attenuation constant of a Clogston cable of given diameter LAMIXATKD TKAXSMISSION LINES. II 11// does not exceed a specified value." We suppose tliat the tliickuess /i ot" tlie conductors is fixed, but that the proportion ot" conductinj^' ma- teiial in the cable may be adjusted by changing the thickness of the insulators. Let am be the value of the attenuation constant which is not t o be exceeded in the operating frequency range, and let Jm be the f re- (luency at which this maximum attenuation is reached. What should l)e the fraction 6 of conducting material in the cable in order to maxi- mize Jm ? It is tacitly assumed that «,„ is at least slightly greater than the minimum "flat" attenuation constant which is possible with a cable of the given diameter, since obviously we do not wish to work ontii-ely ill the very-low-frequency range. In the frequency range of interest the attenuation constant of the 79th motle is given bj' equation (484), which may be written 2 rfifi 2 2 2 J.2 2\/m/€ g Gx/m/c g where Xp i^^ ti root of (475) and indepcnident of 0. Sohing (495) for the frequency fm at which a. is equal to am , and su])stituting for e, jl, and g from (268), we obtain J m — V3 r2[^Mi + (1 - 0)m2]^[(1 - d)/e2fgiam xl"" iriJLigiti (4%) A routine calculation shows tliat fm is a maximum, considered as a function of 6, when 6 satisfies amgieid^i -f 2(1 - 0)m-->] _ .^ 2 (,,.^. [0M1 + (1 - ^Wd - ^)-4 ■^'''' ^ ^^ Equation (497) is easih' reduced to a quartic equation in 6, which may be soh-ed without difficulty when the other parameters are gi\'en. The maximum value of /„ is then obtained by substituting 6 back into (49G). We shall now investigate in more detail the case in which Ml = M2 , (498) that is, the permeabilities of the conducting and insulating layers arc equal. In this case the low-frequency attenuation constant ao , which is just tlio first term on t)ie right side of eciuation (495), is given by «„ = ^"^ .^^- , (499) 20\/l - d VfJ^-i/e-i gx ' 2" A similar problem was first investigated in an uniJiihlished memnr.indiim l)y H. S. Black. 1178 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 and ao has a minimum when e = 2/3. (500) The minimum value of the low-frequency attenuation constant, which we may call aoo , is just «oo 3V3 Writing 3V3 we find that equation (49(5) takes the form J m V3 Xp "3V3 (1 - ef ar. 26 aoc (501) (502) (503) for any value of 6. From equation (497), fm is a maximum when 0 satis- fies 6(2 - 6) 8 aoo (1 - 6)^ 3 V3 «„. ' which is equivalent to the quartic equation 64aoo - W + W -f 27c (0 - 1) = 0. (504) (505) If dm is the root of (505) which lies between zero and one, then the corresponding value of fm is Jm — V3 Xp '2 - 30. irHiQitidm [_2 — dm _ (506) We observe from either (503) or (506) that fm is inversely proportional to ti . The values of dm and C[(2 - 3dm)/{2 - dm)f are plotted in Fig. 21 against am/aoo , which is just the ratio of the maximum attenuation constant to the minimum low-frequency attenuation constant which can be achieved with a Clogston cable of the same diameter. When am/ocoo is unity, then dm = 2/3 and fm is zero to the present approximation (a better estimate of fm would be the critical frequency fi defined by equation (467)). For values of am/aoo greater than about five, dm is LAMINATED TRANSMISSION LINES. II 1179 gi\x'u to a good approximation b}' 4aoo e. 0.77 0:00 SVSa. while /. 2.20Xp Oim Trmgiti aoo (507) (508) The low-frequency attenuation constant ao of a Clogston cable with 6 = Bm will of course be greater than aoo if Om is not ecjual to 2/3. This is not really a disadvantage, however, since by assumption we only wish to insure that a ^ a,„ over the operating band, and the nearer a ap- proaches to am oxev the whole band the less serious will be the equaliza- tion problem. It may be showTi that the ratio ao/am decreases from unity toward one-half as am/ am is increased indefinitely. Physically this means that the low-frequency attenuation constant of an optimum Clogston cable is always at least half as great as the attenuation constant at the upper end of the band, and the cable never contains more conducting material than would correspond to a total stack thickness of about two effective skin depths at the highest operating frequency. We conclude with a few numerical formulas relating to the principal mode in a completely filled Clogston cable with copper conductors and no inner core. The low-frequency attenuation constant ao of such a \ \ y^ y Vm ,/ y \ \ / y .2-em. '/z y -^ y — - / E 4 ;lected the effects of (helecl ric loss and of stack uoiuuiiformity. Neither of these effects can l)e completely eliminated in a physical Clogston cable, and both will exert iiuncvisinKly adver.sc influences on the attenua- tion constant as the frequency is raised. Xn. EFFECT OF NONUNIFORMITY OF LAMINATED MEDIUM In the previous analysis of laminated transmission lines we have treated only perfectly unifoi'in structures, in which every conducting hiyer is identical to every other conducting layer in thickness and in electrical properties, and all the insulating layers are similarly identical to each other. In practice, however, it will not be possible to lay down large numbers of absolutely identical thin layers, and we therefore need to know the effect on transmission of slight nonuniformities in the himinated stacks. Some indication that stack uniformity will be a very ci'itical problem in laminated cables which are expected to give large improvements in attenuation over conventional coaxial cables of the same size may be obtained from the results of Section VI, which showed that in a Clogston 1 line, where the phase velocity is determined by the Me procluct of the main dielectric, this product must be controlled very accurately to maintain the desired deep penetration of current into the laminated stacks. In a Clogston 2, where the main dielectric has been replaced by extensions of the stacks, one might expect similarly stringent requirements on the uniformity of the laminated material if the desired current distribution is to be maintained. In this section we estimate the effects of stack nonuniformity by studying some particular idealized cases of nonuniformity in a parallel- plane Clogston 2 with infinitesimally thin layers, in which the average electrical properties of the stack vary only in the direction perpendicular to the layers. The principal conclusion is that if one attempts to realize with a Clogston line an attenuation constant Avhich is a small fraction, say of the order of one-tenth, of the attenuation constant of a conven- tional line of the same dimensions at the same fi-equency, then hmg- range \'ariations in the properties of the stack (as distinguished from short-range random fluctuations) must be controlled to within a few parts in 10,000. The price is less steep if the overall improvement sought 1182 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 is less, but in all practical cases it appears that the average properties of the stack must be held constant against slow variations to a fraction of a per cent. The requirement of extraordinarily high precision is in addition to the requirement that the individual layers must be extremely thin if a Clogston cable is to improve on a conventional coaxial cable at all in the megacycle frequency range. For purposes of analysis, we consider a parallel-plane Clogston 2 transmission line bounded by infinite-impedance sheaths at y = ± |a, as shown schematically in Fig. 10. The individual layers are supposed to be infinitesimally thin, so that near any given point the average elec- trical constants of the stack are 6 = 62/(1 - d), M = ^Mi + (1 - e)^l2 , (518) g = 0gi ■ The quantities e, jl, and g may vary, continuously or mth a finite num- ber of finite discontinuities, as functions of the transverse cooi'dinate y, owing to variations in any or all of mi , S^i , M2 , e2 , and 6; but they are not supposed to vary with x or z. We shall be concerned with modes in which the fields are independent of X, and in which the only field components are Hx , Ey , and Ez . Then Maxwell's equations are given by (269) of Section VIII, and reduce, if we write the field components in the form H^{y)e"'% Ey(y)e~'^\ and Ez(y)e~^', to —jHx = iwiEy , dHx/dy = -gE,, (519) —yEy — dEz/dy = iujlHx . If we eliminate Ey and Ez from these equations we obtain j2i (Tlh _ldgdlh _ .^_. ^ _^ 7 dy^ g dy dy WjJit H, = 0, (520) where Hx and Ez must be continuous at any points of discontinuity of e, p., or g. The tangential magnetic field must vanish on the infinite- impedance surfaces at ?/ = ±^a; hence we have the boundary con- ditions Hx{-ha) = Hxiia) = 0. (521) These boundary conditions, taken in conjunction with the differential LAMINATED TRANSMISSION LINES. II 1183 e(|iuiti()u (520), define the values of 7 wliicli are the propagation constants of t!i(^ \-ariou.s modes of the Hue. While it is possible to find special foruis of th(> fuiu^tious €('//), jl(ij), a:ui (jiu) such that (520) can be solved exactly in terms of known func- tions, it is easier to make certain approximations in the beginning which retain only the important terms. For this purpose we shall write e = Co + At, M = juo + Afi, g = g^ + Ag, (522) where eo , mo , and yo are constants representing the average values of e, jl, and g across the stack, so that the average values of At, Ajl, and A^ across the stack are zero.* Furthermore the fractional variations in the stack parameters will be assumed small compared to unity; in practical cases they will never be larger than a few per cent and will usually be only a fraction of one per cent. Referring now to equation (520), we see that the coefficient of i/x contains the large factor co/xg, which is of the order of l/5i, as compared with the term d''HJdy~, which is presumably of the order of (l/a)Hx . Hence small changes in e and p. will make relatively large changes in the coefficient of H^ , since 7 is a constant. On the other hand, the coefficient of dHx/dy will be small for any reasonable variations in the small quantity Ag/g^ . Hence we shall neglect this term entirely and deal with the approximate equation d-H. dy- ■loijlg Hx = 0. (523) If we substitute (522) into (523) and drop second order terms in Ae eii , Afl/fio , and Ag/go , we find that the coefficient of Hx becomes ig r 2_. , 2n ■ — [w ^e + 7 J coe tgo 1 + ^ - ^^ ^0 Co , and if 7 + w"/xoeo + w fioeo r^ = ^- lo) Moeo + 7 J, Am Ae Mo €o/_ (524) (525) * The j)reseut use of zero subscripts on *o , Mo , and go has of course nothing to do with the earlier convention that associated zero subscripts with the main dielectric in Clogston 1 lines. 1184 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 SO that 7 = — a)/Zo€o — {io:h/gQ)Vt, (526) then (524) becomes, approximately, - 1 1 + ^» - ^ ^0 Co. ,2 , ,. _ , / A/I , Ae .Mo Co L \J"o eo/_ (527) In all cases of interest we shall find that (A/x/mo + Ae/eo) is smaller than or at most of the same order of magnitude as Tf/io)jJogo . Hence the differential equation (523) takes the approximate form dif 2 I . _ - / Am , Ae Mo Co H, = 0, (528) where v] is determined by the two-point boundary conditions (521). The variations of the stack parameters appear in (528) only in the term (A/z//Zo + Ae/eo), which is some as yet unspecified function of ij. For convenience we shall write this term in the form ^ + ^' = -^*>(i/), (529) Mo Co oinogoa- where C is a dimensionless parameter and | the former mode has the smaller attenuation constant. It is not difficuh, although the details will be omitted here, to investi- gate the behavior of the e'gen values for small T and to show that no matter whether $o < | or ^o > h, the eigenvalue which starts from tt at C = 0 tends to the asymptotic value which has the smaller real part, so that this eigenvalue, whether its asymptotic form be given by (543) or (544), may be called Ai . It appears that if ^o < 1) then Im Ai is positive for positive C , while if |o > I; then Im Ai is negative for posi- tive C. An interesting mathematical phenomenon appears when ^o = hj so that the discontinuity in /(|) is exactly at the center of the stack. In this case, when C is small Ai and Ao are both real, Ai being somewhat greater than tt" and A2 somewhat less than 47r". For a certain value of (' the two eigenvalues coincide; this value is approximately C = 17.9, Ai = A2 = 25.6. (545) For larger \alues of (', Ai and A2 are complex conjugates (it seems to be immaterial which is which) whose asymptotic forms are gi\'en by (543) and (544) with ^0 = i Approximate values of Ai and A2 were found for the symmetric case, ^0 = 0.5, and for one unsymmetric case, ^0 = 0.6, on the Laboratories' general piu'pose analog computer for 0 ^ C ^ 100, and were refined afterward by desk computation, using a method of successive approxi- mations to solve ecjuation (542). The real and imaginary parts of Ai/tt' and Ao/x' are plotted in Fig. 22 for the symmetric case, where it should be noted that different vertical scales are used for Re A/tt and Im A/V". The corresponding eigenf unctions lOi(^) and wt{^) are shown in Fig. 23 for C = 0, C = 17.9, which corresponds to etiual eigenvalues, and C = 100. It will be recalled that w{^) is equal to Hx{y), and the other field components can be derived from H^ by equations (519) if desired. Fig. 24 shows plots of Ai/x^ and A2/7r^ for the unsymmetric case ^0 = 0.6. 1188 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 f(o 5 +1 0 1 / 1/2 1 ^ IM s^eA z/TT^ < ,.-•" 0) cr 2 ) -ReA ,/7r^ = ^qA /tt^ '<;;^^ |1T v^^ n 0 / 1 .y 50 C Fig. 22 — Real and imaginary parts of Ai/tt^ and ^iJ-k'- for a nonuniform stack whose average properties are constant except at a single, symmetric discon- tinuity. iii) f(^) a symmetric rectangular step. Let m = 'Wo' 2(1 - ^o) ' "21;' Mo < ^ < 1 - 1^0 1 - 1^0 < ^ ^ 1, (546) where ^o is some fixed number between 0 and 1 but not, in the cases of interest, extremely close to either 0 or 1. Inasmuch as /(^) has even sym- metry about ^ = ^, every mode will preserve the (even or odd) sym- metry about 1=2 which it has when C = 0. We shall consider the lowest even mode,* which has the eigenfunction sin x^ when C — 0. Solutions of (532) having even symmetry about ^ = \ (we need consider only the region 0 ^ ^ ^ | on account of the symmetry) and satisfying the boundary conditions (533) are given by * For large C the lowest even mode will be confined essentially to 2sO ^ s ^ 1 2SU ) while the lowest odd mode will be confined to the two regions 0 g ? < U^ :ind 1 - Uo < $ ^ 1. If ^0 > 2/3, the latter mode will ultimately have a lower attenuation constant than the former; but we shall not take space to investigate it here. LAMINATED TRANSMISSION LINES. II 1 1 SO ^'- 0 ^ t < i^„ , w(^) = A sin [A + iCmon, w(^) = B cos [A - iC/2(l - $n)]-(| - ^), ko < ^ ^ h (547) Tlic ro(|iiiroments that w and dw/d^ must i)o contiiiiious at ^ = |^o lead t(» the (Miuations A sin i[A + /CV2^o]-st,) = B cos i[A - /6V2(1 - ^o)]-{l - ^o) , .1[A + iCmo\- cos i[A + /C/2^o]-s^o = B[\ - /C/2(l - ^o)]- sin i[A - 7-C/2(l - ^0)]^! - t„), (548) which will be consistent if tlie t'oHowiug characteristic ecination is satisfied: tan |[A + iC/2^,]ko ^ cot |[A - iC /2(l - ^o)]^(l - ^0) ,^^g^ [A + iC/2^o\^ [A - tC/2(l - ^0)]^ The roots in A of equation (549) are the eigenvahies corresiioncHnj;- to the even modes of the symmetrical structure. When C = 0, the roots of (549) are A = tt', 9x", • • • .It appears tliat for C > 0 we have Re Ai > tt^ and Im Ai > 0. For large C the asymi)- totio expression for Ai turns out to be C = 100 C = 100 (a) (b) Fig. 23 — Real and iniaginary parts of the first two eigenfunctions, wi = wi +ivi and wi = U2 + ivi , for the nonuniform stack of Fig. 22. 1190 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 < (r), (555) where h\ and hi are the pair of independent solutions of Stokes' equation which have been tabulated for complex arguments by the Computation Lal)oratoiy of Harvard University. ^^ (The solution may also be ex- C =100 Fig. 26 — Real and imaginary parts of the first eigenfunction, Wi = u\ -\- ivi for the nonuniform stack of PMg. 25. " Tables of the Modified Hankel Functions of Order One-Third and of Their Deriv- atives, Harvard University Press, Cambridge, Mass., 1945. 1192 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 pressed, though less conveniently, m terms of Bessel functions of order one-third.) It is easy to show that the boundary conditions (533) at ^ = 0 and ^ = 1 require AhM + Bhin) = 0, Ahiir,) + Bh.2(T2) = 0, (556) where Tl (A + iC) (2cy ' Equations (55G) will be consistent if (A - iC) (2cy ■ /li(Ti)/l.2(r.>) - /ii(r2)/i2(ri) = 0; (557) (558) and this is the relation which must be satisfied by the eigenvalues Ai , A2 , A3 , • • • , for any given value of C. Approximate values of Ai and A2 have been found using the analog computer for the range 0 ^ C ^ 100, with spot checks by numerical solution of equation (558) ; and Ai/tt^ and A2/7r" are plotted in Fig. 27. The eigenf unctions are qualitativelv similar to those shown in Fig. 23 for the stack with a symmetric discontinuity. As in the symmetric ex- ample in case (i) above, we find that for small positive C, Ai is real and greater than x-, while A2 is real and less than At-. The two eigenvalues coincide at C ^A9. Ai = A2 :^ 29. (559) For larger values of C, Ai and A2 are complex conjugates. Their asymp- 12 10 $3 cr 2 f +1 0 -1 iO ^ "^ 1 /. i/w2 - ^^1/2 1 ^ ReAs/TT^ ^ f^e^^ /^ Aa/S ) Re A ^ M tro n/^! ^-^n^^ ./e. 1 ^^^"^ 6< E 50 c Fig. 27 — Real and imaginary parts of Aj/x- and Aj/'x- for a nonuniform stack whose average properties vary linearly across the stack. LAMINATKD TIIANSMISSIOX LINES. II 1193 totic foi'ins as r ^ » may be (locluccd by coiisidiM-iui;- the bchaA'ior of //i(>) and Ii-At) i\)V large arguments, and arc Ai = A* :^ 1.169(2C)'^ + i\C - 2.025(20^1. (oCO) The magnitudes of both the real and iniaginaiy pai'ls of Ai and Ao thus increase indefinitely with C. ('i'^ f(^) ^ sinv!ioi(Jal function. Let f(t) = -cos2j.7r^, 0 ^ ^ ^ 1, (5G1) where j/ = ^, 1, 2, 3, 4, • • • , so that/(^) goes through v complete cycles in 0 ^ ^ ^ 1. Then equation (532) reads (fw/d^~ + [A + iC cos 2j'7r^]w(^) = 0. (562) If we make the transformations W(t) = ui^), X = A/v IT , ^ = -iC/2vV-, \\e get cf]V/dT- + [X - 2?? cos 2t]W(t) = 0, (564) and the boundary conditions (533) become Tr(0) = W(u7r) - 0. (565) Equation (564) is one of the standard forms of Alathieu's equation. We are interested in solutions which are periodic with period 2 in ^, and w^hich approach the form sin niT^ when C ^ 0. In terms of t and ?>, the function corresponding to the ?nth mode in the Clogston line must reduce to the form W(t) — sin - T. (566) .5—0 V For any value of ?^, this function may be denoted l)y' W(t) = sCmAr, t?). (567) (563) 2' See N. W. McLachlan, Theory and Application of Mathieu Functions, O.xford, 1947, pp. 10-25, especially p. 13 and p. 10. In this roforence a or 6 corresponds to our X, q to our t?, and v to our m/i>. 1194 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 In our problem ?? is (negative) imaginary and m/v may be an integer or a rational fraction. For any given ?? and mjv the conditions (565) to- gether with the limiting form (566) determine an eigenvalue X, and hence by (563) determine A; but only a small amount of work has been published on the eigenvalues of Mathieu functions with imaginary parameter or of fractional order. We shall look at some special cases. V = \. The function /(^) is one-half cycle of a cosine curve which varies from — 1 to + 1 ; we expect results similar to those found for the symmetric discontinuity of case (i) and the linear variation of case (iii). The eigenfunctions of the first two modes (w = 1 and m = 2) are se2(T, ^) and se4(r, d-). The eigenvalues of these two functions for purely imaginary t? have been computed by Mulholland and Goldstein out to a point which corresponds to C = 8x , and an asymptotic formula is given for larger values of C. The values of Ai/tt" and A^/tt are plotted forO -^ C S. 100 in Fig. 28; the corresponding eigenfunctions resemble those shown in Fig. 23 for the stack with a symmetric discontinuity. Again we find that Ai and A2 are real for small positive C, equal for a particular value of C, and conjugate complex for larger C. The leading terms of the asymptotic formula are, in our notation, Ai A* A2 [4.7124C"^ - 3.0842 - 1.0901C"* - • • • ] + i{C - ^:jY1^& - 1.0901C"- - (568) J/ = 1. Here/(^) is one full cycle of a cosine function, varying from — 1 f +1 0 -1 .4) ^ ^ y^ ' / ^ y\/z 1^ ReAa 7^ \ "9^^ 7^ p,eN2/ /TT^ r ;^' ^ ReA, /tt^ '{oTn^ V*' \^N^ / 1 x-** 12 20 30 50 C Fig. 28 — Real and imaginarj^ parts of Aj/tt^ and Aa/Tr^ for a nonuniform stack whose average properties vary as one-half cycle of a cosine function across the stack. 29 H. P. Mulholland and S. Goldstein, Phil. Mag. (7), 8, 834 (1929). In this reference \a or 4/5 corresponds to our X and 89 to our »?. LAMINATED TUANSMISSIOX LINES. II 1195 Fig. 29 — Real and imaginary parts of A,/7r- for a nt)nuiiiform stack whose avorago propcM-ties vary as one cycle of a cosine function across the stack. to -(-1 and back to —1. The eigenf unction of the lowest mode (m = 1) is sei(r, t>), and the values of Ai may be obtained from Reference 29 for ten equally spaced values of C out to C = 327r". Since our ^ is negative imaginary, in the notation of this reference we have Ai = 4:t^^i . Ap- proximate values of Ai/x obtained on the analog computer for C at smaller intervals in the range 0 ^ C ^ 100 are plotted in Fig. 29; and the eigenfunctions are similar to those shown in Fig. 26 for the symmetric rectangular step. The leading terms of the asymptotic formula for Ai when C is large are as follows : Ai ^ [3.1416(7^ - 2.4674 - 0.9689C~' - • • • ] (569) + i[C - 3.1416C' - 0.9689(r^ - • • • ]. V = 3. Now /(^) is a three-cycle cosine function and the lowest mode corresponds to se'^ir, t?). Approximate values of Ai/tf" for 0 ^ C ^ 100 were obtained on the analog computer and are plotted in Fig. 30; the eigenfunctions are shown in Fig. 31 for C = 0 and C = 100. v » 1. For a i/-cycle cosine variation, the lowest eigenfunction is sei/^(r, t?), and for the lowest eigenvalue there is an approximate formula given by McLachlan.^" Incidentally this formula predicts no imaginary part for Ai if t? is purely imaginary and p> 1, which agrees approximately with the results of our analog computations for i/ = 3; we found the imaginary part of Ai to be only about 1 per cent of the real part even for C = 100. If C is fixed, one expects that asv ^ oo the effects of the I'apid fluctuations in/(^) will average out, so that Ai will ultimately approach '" Reference 28, p. 20, equation (6), where a corresponds to our \i , q to our I?, and V to our l/v. McLachlan's formula was ostensibly derived for real q, but the derivation appears equally valid for complex q. 1196 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 3 a. 1 f( + 1 0 -1 [AAA.. r^ \ji\j r ^^ ^ Re A Jjr^ -^ "^ ImA, /TT^ E 20 30 50 C Fig. 30 — Real and imaginary parts of Aj/tt^ for a nonuniform stack whose average properties vary as three cycles of a cosine function across the stack. the value -k appropriate to a uniform stack. McLachlan's formula shows that this is indeed the case; in our notation, the leading terms give ■w + e %vh 1 + e 8v2 (570) assuming of course that the second term is reasonably small compared to the first. This concludes our discussion of special types of nonuniformity. We shall now attempt to get an idea of what the numerical results mean in terms of the practical requirements on stack uniformity in a laminated transmission line which is expected to show a specified reduction in at- tenuation constant below a conventional line of the same dimensions. For this purpose we shall compare a plane Clogston 2 line having in- finitesimally thin layers with a plane air-filled line of the same 'width a, bounded by electrically thick solid conductors. At frequencies for which the conductor thickness of the "standard" __a. > a. o ^^^y^^\ .X^^^^\ d ~" ~ C = 100 Fig. 31 — Real and imaginary parts of the first eigenfunction, w\ = Wi + iv\ , for the nonuniform stack of Fig. 30. LAMINATED TRANSMISSION LINES. II 1197 air-filled line is great compared to the skin depth 8i , its atteinialioii constant a, is given by equation (25), namely as = 1/T]vgi8ia, (571) where 77^ is the intrinsic impedance of free space. By o(|iiati()u (535), the attenuation constant ac of the lowest mode in a plane Clogston 2 with infinitesimally thin layers is Ai ac = Re ^ /^-jr . 2 . (572) If we assume noimiagnetic materials and put in the optimum value of d, namely d = 2/3, we obtain for a uniform stack with Ai = tt^, 12.82 \/ev /,-»«\ «co = V^ , (573) VvQia- where €2r is the relative dielectric constant of the insulating layers. The attenuation constant of the conventional line is proportional to the square root of frequency, whereas the attenuation constant of the uniform Clogston 2 is independent of frequency up to some frequency at which the effect of finite lamina thickness begins to be appreciable. If we confine ourselves to the low'-frequency, flat attenuation region, and denote the ratio of attenuation constants by r, then from (571) and (573), r = ttco/as = 12.82 V€^ 5i/a, (574) and the crossover frequency above which the uniform Clogston line is better than the conventional line occurs when a/81 = 12.82V^ . (575) In the following numerical example we shall assume polyethylene in- sulating layers, wdth so that (574) becomes €2r = 2.26, (576) r = 19.275i/fl. (577) If the stack in a Clogston line is not uniform, then regardless of the thinness of the layers the attenuation constant will no longer be inde- pendent of frequency, but will increase with frequency at a rate depend- ing on the nature and the magnitude of the nonuniformity. Since from equation (572) the attenuation constant is proportional to Re Ai , 1198 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 while from equation (529) or (530), C is proportional to frequency for a given stack, we see that our plots of Re Ai/tt'' versus C need only the in- troduction of appropriate scale factors to read directly the variation of attenuation with frequency due to nonuniformity in the stack. Although a nonuniform Clogston line may still be better under some conditions than a conventional line of the same size, the crossover frequency will be higher and the improvement at any given frequency will be less than if the stack were uniform. Among the various interpretations which may be given to our nu- merical results, we shall consider here only the following: Suppose we have a plane Clogston 2 line which, if it were perfectly uniform, would have an attenuation constant smaller, at a certain frequency, than the attenuation constant of the corresponding conventional line by a given factor, say one-half, one-fifth, or one-tenth. For these particular attenua- tion reduction factors the ratio of a to bi may be calculated from (574), or from (577) if the msulation is polyethylene. The question is: What variation in e across the stack is permissible if we are willing to have the actual attenuation constant of the Clogston line be double its ideal value; in other words, if we will settle for attenuation reduction factors of unity (no improvement), two-fifths, or one-fifth instead of the ideal values one- half, one-fifth, or one-tenth? To answer this question for any particular type of nonuniformity, we have only to find, from the plot of Re Ai/tt^ versus C, the value of C for which Re Ai/tt = 2. Then the fractional difference between the maximmn and minimum values of e corresponding to this value of C is given by equa- tions (530) and (531) to be €inax Cmin oLiO\ ,. f \ fK^Q\ Umax Jmin^ , K'^^o) where we have taken ^o ^ 2/3, and/^ax and /,„!„ are the extreme values of the function /(^) which describes the type of nonuniformity bemg con- sidered. The special types of nonuniformity which have been studied above fall roughly into three different classes. In four of the cases, namely the sym- metric and unsymmetric single discontinuities, the linear variation, and the half-cycle cosine variation, the function /(^) varies monotonically from one side of the stack to the other. In the symmetric rectangular step and the one-cycle cosine variation, /(^) oscillates from one extreme value to the other and back again, while in the three-cycle cosine variation, f(^) ex- hibits three complete oscillations across the stack. The following table LAMINATED TRANSMISSION LINES. II 1199 shows the pormissiblc total variation in e for each of thoso typos of lum- uuiformitv. Case Symmetric discontinuity. . Unsymmetric discontinuity Linear Half-cycle cosine Rectangular step One-cycle cosine Three-cycle cosine 16.5 28.0 42.6 29.5 53.0 59.8 78.9 Cmax — Cm in r = Vi 0.0167 0.0295 0.0430 0.0298 0.0535 0.0604 0.0797 = H 0.0027 0.0047 0.0069 0.0048 0.0086 0.0097 0.0128 Ho 0.0007 0.0012 0.0017 0.0012 0.0021 0.0024 0.0032 It would be easy to construct a similar table for any other values of the attenuation ratio r, and for any specified degradation due to nonuniformity. It is, howe\'er, already ob^^ous that the greater the improvement for ^^'hich one strives, that is, the smaller the ratio r, the more stringent will be the requirement on (emax — emin)/€o ; in fact, the permissible value of this quan- tity is proportional to r". In an}^ practical case the value of e will have to be controlled against long-range variations within a fraction of a per cent, and if attenuation reduction factors of the order of one-fifth or one- tenth are contemplated, the variations probably cannot exceed a few hun- dredths of a per cent. It also appears that a steadj^ increase or decrease in the value of I across the stack will be the most serious type of nonuni- formity, since the effects of very rapid fluctuations A\ill tend to average out. Clearly the nonuniform laminated transmission lines which we have been considering in this section are \'ery highly idealized, even if we disregard the geometrical differences between plane and coaxial stiiictures. Any real Clogston cable ^^■ill be built up of layers of finite thickness with unavoid- able random fluctuations from layer to layer, superimposed on slower variations in the average properties of the layers from one side of the stack to the other. The thickness of an individual layer will also vary more or less in both directions parallel to the layer, so that the properties of the stack will be functions of the coordinates <^ and z as well as of p. A few qualitative remarks are in order concerning these neglected effects. The effect of finite lamina thickness in a nonuniform stack can be cal- culated, by the method employed in Section XI for a uniform coaxial stack, if we make the plausible assumption that the macroscopic current distribution remains the same as for infinitesimally thin layers. The results 1200 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 will certainly be qualitatively the same for uniform and slightly nonuniform stacks, so long as the nonuniformity does not seriously distort the field pat- tern of the operating mode. Some idea of how the effects of rapid random fluctuations in the average properties of the stack may be expected to average out is given by equa- tion (570), which assumes for the function f(^) a i^-cycle cosine ^^ariation across the stack. As a numerical example, suppose that with this variation of e We have '"^^ ~ '"'° = 0.01 (579) in a line designed to give an attenuation reduction ratio of r = Ko- (580) Assuming polyethylene insulating layers, we have for this line 8i/a = 1/192.7 = 0.00519, (581) and from (578) the corresponding value of C is C = 247.6. (582) The value of v for which the relative increase in Re Ai due to the fluctua- tions is, say, one-quarter is given by C' 1 C 4' V2 = 17.7. (583) Thus a 1 per cent fluctuation in e, repeated at intervals of about one- eighteenth of the stack width, will cause only a 25 per cent increase in attenuation, even for a Clogston line which is designed to have only one- tenth of the attenuation constant of a conventional line of the same size. Finally there is the question of the effects of variations in the average properties of the stack in both directions parallel to the layers. Mathemati- cal analysis of even a simple case of longitudinal variation would be much more difficult than what has been done here; yet on physical grounds it seems very likely that such variations A\ill add an appreciable amount to the total attenuation of the line. If we consider two cross sections of a laminated cable separated by a certain distance and having different trans- verse nonuniformities, the field pattern of the lowest mode vnW be differ- ent at the two cross sections, and so in traversing the intervening distance the power will be partly reflected and partly converted to higher modes with higher attenuation constants. The reflected or mode converted power Avill be at least partly lost, with a consequent increase in the overall at- LAMINATED TRANSMISSION LINES. II 1201 t{3tuuition of tlie cable. Hence the estimate of the increase in uLlenuation which one gets from the present analysis, considering only the variations transverse to the layers at an average cross section, is certain to be opti- mistic in that it neglects completely the effects of variations in other directions. XIII. DIELECTRIC AND MAGNETIC LOSSES IN CLOGSTON 2 LINES To discuss dielectric and magnetic losses in Clogston 2 lines we may take the electrical constants of the conducting and insulating layers to be complex; thus Ml = Ml — ^Mi = Mi(l — i tan fi), M2 = M2 — i/2' = M2(l — i tan ^2), (584) €2 = €2 — ie-i = €2(1 — '' tan ;ati()ii factor e~^^ and make the usual approximations gi/o)ei » 1, Ki --^ 0-1 , Ki/gi ^ 771 , (A21) for a good conductor. Tlic constants A and B are determined by the boundary conditions H^(pr) = /i/27rpi , H^(p.2) = /2/27rp2 , (A22) \\hicli follow directly from Ampere's circuital law. We find without diffi- culty ^ ^ {h/2Trp2)Ki{aipi) — {Ii/2irpi)Ki(aip2) Kl{(Tipi)Ii{(TiP2) — Ki{cip2)h{(TlPi) ^ _ (/i/27rpi)/i((rip2) — (/2/27rp2)/i(q-ipi) -K^i(o-ipi)/i(crip2) — Ki{crip2)Ii{(Tipi) (A23) The average power dissipated in the conducting cylinder is equal to one- half the real part of the inward normal flux of the complex Poynting vector E X H*. For the average power P dissipated per unit length we have P = Re h[2rp2EMHt(p2) - 2ir p^EMKip,)] = Re \[EMlt - EMI*x] = Re ^''^ (i^n/12 - i^i2/n) I2irp (Kn/02 + Ko2/n) (A24) (/1/2 + /r/2) , IJl 2ir(X\p\p2 27rpj + ^-^ (Koi/12 + A'12/ 01 where /„ = /r(o-lp.s), Krs = Kr{(J\P,). (A25) The combinations of Bessel functions appearing in (A24) are just those for which we gave approximate expressions in etiuations (A8) of AppcMidix I, assuming the thickness h (= p2 — p\) of the conducting cylindei' to be small compared to pi . Substituting these approximations into (A24) and 1206 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 rearranging, we get 1 I hit pi P = Re 47r^i^i (Tih coth ait\ + 2pi_ {hit + Ith) pi + 1 - hh a - csch (Jiti (A26) aih coth aih — -^z— Pi L ^Pi_ up to first order in ij p\ . If we set 0-1 = (1 + i)lh , (A27) then on expanding the right side of (A26) in powers of ti/8i up to the fourth, we obtain P = h - h 47r£?iii I V, PlP2 + .4 r T T* 4/1 hh , 7(hi; + /r/2) , hh 8\ L45p, + + 45piJ (A28) 360\/pip2 where we have approximated pi(l + /i/2pi) by \/pip2 in the interest of symmetry. Now writing APj for P, Alj for h — h , Pj-i for pi , and /y_i for h , and neglecting curvature corrections of the order of 4/pi entirely, we have, on setting h and h both equal to Ij-i in the coefficient of (h/di) , the ap- proximate relation ^Pj = F^T^ [l ^^i I' + ^ I ^i-^ I'l ' (A29) 4irgriiipy_i L 35i J which is just equation (472) of Section XI. Transistors in Switching Circuits By A. EUGENE ANDERSON (Manuscript received August 1, 1952) The general transistor properties of small size and weight, low power and voltage, and potential long life suggest extensive application of transistors to pulse or switching type systems of computer or computer-like nature. It is possible to devise simple regenerative circuits which perform the nor- mally employed functions of waveform generation, level restoration, delay, storage {registry or memory), and counting. The discussion is limited to point contact type transistors in which the alpha or current gain is in excess of unity and to a particular feedback configuration. Such circuits, ichich are of the so-called trigger type, are postulated to involve negative resistance. On this basis an analysis, which approximates the negative resistance characteristic by three intersecting broken lines, is developed. Conclusions which are useful to circuit and device design are reached. The analysis is deemed sufficiently accurate for the first order equilibrium calculations. Transistors having properties specifically intended for pulse service in the circuits described have been developed. Their properties, and limitations, and parameter characterizations are discussed at some length. INTRODUCTION It is proposed to discuss some of the properties of transistors which are appUcable to switching or pulse-type circuits, to develop elementary analysis methods and to describe a few circuits. The bounds or limits of the field of switching are diflficult to define. The common thread usually involves definite states of being as "open or closed", "off or on", "0 or 1", and so on, rather than a continuum of conditions. Even when consideration is given to more than two states, the thought involves distinct recognition of each state. The field is termed to be non-linear in distinction to linear manipulation of informa- tion. Any number of anomalies in definition may be raised. Without attempting either to define or to limit the field, some of the functions which are often employed are: wave form generation, as rectangular pulses, sawtooth waves, etc.; memory or storage which may 1207 1208 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 be for short, intermediate or long periods and inv^olves the retention of information for subsequent use; operations involving addition, sub- traction, multiplication and division; translation of information from one form or code to another; gating, involving the routing of signals according to a predetermined pattern or set of conditions; regeneration of signals in amplitude and wave form; delay, which may be thought of as a form of storage; and timing. Some of these functions are simple; others result from fairly complex structures of simpler functions. Present trends in electronic switching systems are toward compli- cated automata as exemplified by digital computers.^ The reliability, power consumption and physical size of the electron devices employed largely determines the degree of realizability of such systems. It is believed that the transistor will find a significant application in this field. The transistor can reduce power consumption by the elimination of heater or filament power. In addition, particularly in broadband ap- plications as in high speed pulse systems, the "B" power may be reduced by the order of one or two decades if not more. Transistor circuits with 0.02 /iS rise time have been made to operate Avith an input power of 20 milliwatts which compares with approximately 2.5 watts (1-watt heater, 1.5-watt plate) for an equivalent tube circuit. Transistors have operated with less than one microwatt input power.- Such power reductions result from the low operating voltages, low internal resistances and low capacitances of transistors. Low internal impedances greatly reduce the importance of stray wiring capacitances thereby making mechanical design much simpler and often eliminating the need for isolating or buffer amplifiers. The transistor can contribute definite reduction in size directly. Fig. 1 shows a "bead" transistor which has a volume of approximatelv 1/1000 of a cubic inch and a weight of 5/1000 ounce. Indirectly the transistor can contribute to size reduction through the use of smaller, lower voltage, lower dissipation components. The reduction of power supply require- ments in terms of size, regulation and capacity is also quite appreciable. Transistors have been subjected to shocks in excess of 20,000 G with- out change in characteristics. Vibration tests have shown no resonances in the transistor shown in Fig. 1 to several thousand cycles. Harmonic accelerations of 100 G at 1000 cycles have produced no detectable cur- rent modulation. ' L. N. Ridenour, "High Speed Digital Computers", J . Appl. Ph7js., 21, pp. 263-270, April, 1950. ^ R. L. Wallace, Jr. and W. J. Pietenpol, "Some Circuit Properties and Ap- plications of n-p-n Transistors", Bell System Tech. J., 30, pp. 530-563, Jul}', 1951. TRANSISTORS IV S\\ ri'( 11 1 N'(; ('[KCriTS 1 209 Life reliability and expectancy :ii'c diniciili lo dclci-iniiic due to the relative infancy of transistors, the dcliiiitc rmitciicss of time, tiic many variables involved and the I'ate of development ))r()j>;ress. Average life is presently estimated to be in excess of 70, 01)0 hours. Ijfe is a function of the operating conditions and may l)e materially reduced accordingly. Transistors also have limitations. Xoise at present is high for point- contact types as compared to electi-on tubes; input impedances are low, which may be either advantageous or disadvantageous; power output may be limited; frequency response is relatively low; circuit instability may cause design difficulties; and the devices are sensitive to tempera- ture changes. There is also an absence of a long practical experience with a consequent art background in both devices and circuits. A comprehensive review of transistor properties is given in the paper l)y J. A. Morton.3 While it is difficult to define the switching field, it is no less difficult to discuss circuit and device properties on a general basis. This is related to the non-linear nature of the circuits and devices in distinction to the virtually classical linear small-signal field. The lack of a classical method of analysis is a serious handicap in the synthesis of contemporary circuits and de'^dces. When new devices, as the transistor, are to be considered, Fig. 1 — A miuKUunj ;\\ i^pe transistor (M1689). ' J. A. Morton, "Present Status of Transistor Development", Bell System Tech. J., 31, pp. 411-442, May, 1952. 1210 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 the problem is multiplied due to the lack of a long background of experi- ence. It has been assumed that negative resistance is a common thread among "trigger circuits" and oscillators regardless of the device em- ployed — electron tube, gas tube, transistor, mechanical structures, etc. This is not a new or novel idea and there is no intent to present it as such.* Rather, it is used as a pattern upon which a certain degree of transistor analysis may be based, leading to simple understanding. The analysis assumes that the negative resistance characteristic can be broken into three regions; each region is then considered on a linear basis. Section I will deal with simple circuit properties; Section II with analysis and Section III with device properties. I — Simple Circuit Properties The common property ascribable to switching functions is that of definiteness of state. The condition of the function is either "off" or "on". Switches are either open or closed; relays are operated or not; tubes are in cutoff or overload; doors are open or closed and so on. This is common regardless of the phenomena being exploited. There is an intermediate region between these two conditions usually characterized by a time which is related to how fast the function may go from one state to another. Functionally the times of closing and opening are taken to be zero; practically, they are of determining im- portance. Relays replace hand-operated switches and electronic devices replace relays as speed becomes important. Obviously, no function or system can be faster than its state-devices. All such state-devices will have separate attendant properties such as the degree of reverse coupling between the controlling signal and the controlled signal. Separated into families, however, there are those which are passive and those which are active. The latter are threshold devices in which a small amount of signal or control energy causes the translation of a relatively larger amount of stored energy into dynamic energy which consummates the change in state. As long as the control * See for example "Negative Resistances, Their Characteristics and Effects. Sinusoids, Relaxation Oscillations and Relaxation Discontinuities", Walter Reichardt, Elektrische Nachtrichen-Technik, 20, pp. 76-87, March, 1943; "Uniform Relationship Between Sinusoids, Relaxation Vibrations and Discontinuities", Walter Reichardt; Elektrische Nachtrichten-Technik, 20, pp. 213-225, Sept., 1943. For transistors: "Counter Circuits Using Transistors", E. Eberhard, R. O. Endres and R. P. Moore, RCA Review, pp. 459-476, Dec. 1949; "A Transistor Trigger Circuit", H. J. Reich and Ungvary, Rev. Sci. Instr., 20, p. 8, p. 586, Aug., 1949; and "Some Transistor Trigger Circuits", Proc. Inst. Radio Engrs., 39, pp. 627-632, June, 1951, P. M. Schultheiss and H. J. Reich. TRANSISTORS IN' SWITCHING CIRCUITS 1211 signal is below (he initial (hi-eshoUl there is no response and any change is directly related to the passive transmission of the control signal alone. When the signal exceeds the threshold the second state is assnmed. Watch escapements, thyratrons, antl the whole family of oscillators fall into this category. When the simplest cases of such functions are analyzed, they are found to involve in one way or another two stable states separated by a region in which there is positi\'e feedback and gain in excess of unity with a resultant equivalent negative resistance. The proposition that a negative resistance characteristic is common to trigger or threshokl switching circuits is tacitly assumed. The next step is to examine tlu^ transistor for such behavior and to classify the proper- ties. NEGATIVE RESISTANCE IN THE TRANSISTOR That the transistor* can exhibit negative resistance has been demon- strated analytically^ and experimentally. The resistances seen looking into the emitter and collector of the transistor with grounded base are shown in Fig. 2. In the equations and discussion to follow, the symbol conventions are as follows: External circuit elements are capitalized as R^ , Rb , and Re . The symbols Rn , R12 , Rn and R21 define the open-circuit transistor Vm , and Vb define the equivalent circuit resistances; the svmbols r^ , Vc R|N [^ RiN — R? f^iz '^21 Tc+rb- rb(''b+ I'm) R'„ ■'■ ■" Te+rb+Rf Fig. 2 — Emitter and collector driving point resistances. * Discussion is limited primarily to point contact transistors with a's or cur- rent gains greater than unity. * R. M. Ryder and R. J. Kircher, "Some Circuit Aspects of the Transistor", Bell System tech. J., 28, pp. 367-400, July, 1949. 1212 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 element values. Network resistances which contain both device and external elements are primed. For example, 7^22 = Rii -\- Re -\- Rh , where R22 = Tc -\- r^ . See also references 3 and 5. Taking the collector or output resistance, Fig. 2, for example, ^6(^6 + r™) Rin = {Tc + Vb) — (1) Ri rt + n-\- Re can be negative or positive depending upon the relative magnitudes of the two terms. Actually, of course, r„ has a phase factor and so is frequency dependent. Frequencies wherein r^ is essentially resistive will be assumed. For negative resistance, r„, must be large, R^ small and Th not too small or else augmented by external resistance. Negative resistance is thus predicted on a small-signal linear basis. The large- signal behavior may be studied experimentally by adding sufficient resistance as Re to the first or positive term to insure stability. This is shown in Fig. 3 with the resultant characteristic. External base re- sistance Rb has been added and R^ is zero. Fig. 3 illustrates the pattern of a three- valued characteristic : Regions I and III are portions with positive slope, indicating stable operating / / / / n \ / \ I -20 -3 -2 -1 0 Fig. 3 — Collector large-signal negative resistance characteristic. TRANSISTORS liN SWncillXG CIRCUITS J213 regions, separated by Region II, a region oi negative slope, indicating the possibility of instability. In this particular case, Region I has high lesistance and Region III very low resistance. An evaluation of the emitter or input characteristic leads to similar lesults, using the circuit of Fig. 4. Rh has been added here also and Re taken as 2ero. The general pattern is again present. Region I has high, positive resistance; Region Tl, negative resistance; and Region III very low, positive resistance. BIASES AXD LOAD LINES — BISTABLE OPERATION The negative resistances of Figs. 3 and 4 are both of the so-called open-circuit stable type. If loads are applied to the circuit terminals of Fig. 2 which are larger in magnitude than the negative resistances, the circuits will be stable; that is, there will be single operating points. This is shown in Fig. 4 by the dashed load lines marked, Rt , Rt , Rt . A load resistance smaller in magnitude than the negative resistance may intersect the characteristic in three positions as shown by the load line Re . The load line R^ can be made to have single or multiple intei'sections l)y biasing properly as shown in Fig. 5, where the three possibilities are shown as Re , Re , R'i ■ Single or multiple intersections result in ac- cordance with the choice of emitter bias, F„ , as shown. It can be shown Fig. 4 — Idealized emitter large-signal negative resistance characteristic. 1214 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 that the intersection of load hne Re with the characteristic at b in Fig. 5 is unstable whereas those at a and c are stable. Experiment in the multiple intersection case shows also that as F„ is slowly increased (decreased in absolute magnitude) the load line moves upward and that the assumed operating point, a, moves up along the Region I portion of the charac- teristic. At the turning point shown on the current axis, the operating point suddenly flips to the high current region, returning along the curve to c as F„ is returned to the original value. A decrease in V^e toward F^^ moves the operating point at c down- ward along the characteristic until it "escapes" past the lower turning point and flips to the Region I portion, returning to a as V^t is returned to the original value. This then is an elementary switching circuit, a bistable trigger circuit or "flip-flop". A positive emitter pulse will cause the circuit to flip to high current, a negative pulse to low current. The triggers may be applied to emitter, base or in combination with proper attention to polarity. Trigger sensitivities are shown in Fig. 6. Such a circuit is often used for register or storage purposes. It can store one bit of information for a potentially infinite period, be sampled for the presence of such information, and be cleared or restored to the original condition for reuse when the stored information is no longer useful. Fig. 5 — Emitter negative resistance characteristic showing possible multiple operating points. TRANSISTORS IX SAVITCHIXG CIRCUITS 1215 With the addition of suitable steering circuits it can be made to count b}^ a scale of two. MONOSTABLE AND ASTABLE CIRCUITS The addition of a capacitor to the circuit as in Fig. 7(a) leads to either nionostable or astable operation. In Fig. 7(b) the normal operating point is stable at a as discussed previously b}^ virtue of the l)ias \'\( . As Vft is increased, as b}" a trigger, the load line is mo\'ed up and over tiie turning point. AVithout capacitor C in the circuit, the operating point would move to b with the resultant rapid change in voltage and current. However, a capacitor has in effect voltage inertia; this is equi\-alent to saying that a capacitor is a short-circuit to a voltage change. Both the capacitance and the rate of change of voltage are assumed high. Thus at the turning point the capacitor effecti\'ely short-circuits the emitter and the operating point snaps along dotted line (1) to intersect the characteristic. This point is ciuasi-stable and tiie capacitor is discharged along line (2) to the second turning point where the emitter is again effectively short-circuited and the operating point snaps along (3) to intersect the Region I portion of the character- istic. This point is also quasi-stable and the operating point moves slowly up to the initial or dc stable operating point. A single trigger thus causes a complete cycle of operation. The emitter current shifts Fig. 6 — Bistable circuit showing trigger sensitivity, A. 1216 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 rapidly to a high value of current, falls relatively slowly to an inter- mediate value, then shifts rapidly to a small negative value and finally returns slowly to the original value. The emitter current and voltage are sketched in Fig. 8. It is a so-called "single-shot" circuit. Alternately the rest or do stable point can be chosen to be in Region III, at high current, by choice of positive instead of negative bias F„ . Practical considerations as ease in triggering and average power consumption usually indicate a preference for the Region I dc stable point. O) -h 0 +Vf *le 1 1 1 . v.. T , J^''^ MONOSTABLE CHARACTERISTIC +Vf \ J 1 \ Vff \ "If >. 0 \R. +If fj 1 ^^^ 4;7 \ ^.,<^ il 3 ^'^M; ^y<^ (C) ' 1 ASTABLE 1 1 CHARACTERISTIC -Vf Fig. 7 — Monostable and astable characteristics. TRANSISTORS IN SWITCHING CIRCUITS 1217 ? +1/ 5 -If TRIGGER Tt TRIGGER Fig. 8 — Idealized monostable relaxation oscillator waveforms. When the load line and bias are chosen to result in intersection in the negative resistance portion, astable operation or continuous oscillation results. This mode is illustrated in Fig. 7(c). Proper bias and Rt > \ —Rin \, Region II, are required. The operating point formed by the intersection of the load line on the negative resistance portion of the characteristic would normally be stable. However, the capacitor provides an ac short- circuit in parallel with R^ causing the path (1), (2), (3), (4) to be followed continuously. Another form of physical explanation of this relaxation oscillation, usually applied to gas tubes, is that the capacitor C is charged slowly throuch Rt to a critical or breakdown value whereupon the tube or device rapidly discharges the capacitor. When the capacitor charge is dissipated, the device discharge can no longer be maintained due to the IR drop in Rt and the tube or device open-circuits and the capacitor is recharged. The above suggests a strong simularity to gas tube behavior and this is indeed so. In fact, the modes described above are common to all open-circuit stable negative resistance devices; only the parameters and device phenomena are different. The primitive circuits of Fig. 7 have pi'operties basic to several switching functions. These may be deduced from the waveforms of Fig. 8 which are essentially identical to both the mono.stable and astable cases. The emitter current has a rectangular Avaveform which suggests the generation of rectangular pulses; and, for the astable case, regenera- tive amplification for both amplitude and wave shape, pulse rate or 1218 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 frequency division and delay. As shown the current waveform is not particularly good, having neither a flat top nor a fiat base line. Practically, the waveform may be derived from the collector by means of a small load resistor to obtain a flat base line. When the emitter current is negative there is sensibly no transfer action, hence, the collector current will be constant during the re-charge portion of the cycle instead of exponential as shown. The slope of the top is inherent and may be removed by clipping. Pulse rise time, the time required for transition from low current to high current, of 0.1 us is quite easily obtainable; 0.02 us with average input powers of 20 7nw have been obtained. Fall time is usually longer than the rise time by factors of 3 or 4. It is to be noted in Fig. 8 that there has been shown a delay between the trigger application and the current transition. Such delay is not peculiar to transistors, but is common to all trigger type devices and circuits. The delay is shown here exaggerated in order to establish its existance and is associated with the static charging of the circuit and the d3aiamic delay of the de\dce concerned. The trigger-transition delay with tran- sistors is usually less than 0.1 us. The voltage waveform of Fig. 8 has a sawtooth form and may thus be employed to generate linear time bases or sweeps. The normal methods for linearization such as a high charging voltage F„ and a high charging resistance Re or other constant current means are appli- cable here as in other device circuitry. Free-running and driven sweeps may be obtained with the astable and monostable circuits respectively. Since the collector characteristic sho^^^l in Fig. 3 is also open-circuit stable, the same sort of circuits can be constructed using the output characteristic. Bistable, monostable and astable circuits are shown in Fig. 9. The resistances seen looking into the base are given in Fig. 10. These circuits are short-circuit stable. That is, high values of Rb result in instability. Bistable, monostable and astable circuits can be constructed also, but use is made of an inductor instead of a capacitor. The reactance of the inductor affords a quasi-open-circuit in the same manner as the capacitor afforded a cjuasi-short-circuit in the previous cases. Circuit examples are shown in Fig. 11. SUMMARY These simple circuits by no means exhaust the s\vitching circuit possibilities of the transistor; rather, they are the simplest. The simple circuit is often satisfactory and may sometimes be employed with little more understanding than that given. More often, however, problems TRANSISTORS IN SWITCHING CIRCUITS 1219 relating to the sensitivity, constancy of s(-nsili\ity, opei'ating currents and \-oltages, interchaugeability and the like re(iiiire a much more quantitative understanding in order to create circuit designs liaving specific properties.* An equal need also exists in transistor design for analytic circuit relationships. Such information is useful first, in the (a; COLLECTOR BISTABLE CHARACTERISTIC (b) COLLECTOR MONOSTABLE CHARACTERISTIC -Ic Rc\ -V, (c) COLLECTOR ASTABLE CHARACTERISTIC Fig. 9 — Collector connection switching circuits. * See, for example, J. R. Harris, "A Transistor Shift Register and Serial Adder", Proc. IRE, Nov., 1952; R. L. Trent, "A Transistor Reversible Binary Counter", Proc. Inst. Radio Engr., Nov., 1952; H. G. Follingstad, J. N. Shive, R. E. Yaeger, "An Optical Position Encoder and Data Transmitter", Proc. Inst. Radio Etujr., Nov., 1952. 1220 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 creation of optimized designs and, second, in the maintainance of proper parameter controls in manufacture. Finally, the more detailed the understanding, the more likely will be the creation of new circuits and new devices. A complete analytical treatment will not be attempted here; con- sideration will be limited to the equilibrium case and in particular to the simple circuits described. II — Analysis In order to deal analytically with circuits and devices it is necessary to have analytic expressions for the device characteristics. For small signal analysis this is relatively easy. In large signal applications, as in switching, the situation is not so simple. The problems arise because of the high degree of nonlinearity wherein the simplifying assumptions employed in small signal analysis are by no means valid. Further, it is desirable to retain dc terms in many cases. The method to be employed here is the so called broken-line method which involve^ approximating the negative resistance characteristic by three intersecting straight lines. The assumption is made that there are three distinct regions of operation in each of which the device is sepa- rately linear, but involving different parameter values for each region. The approximation is shown in Fig. 12. The assumption that the negative resistance characteristic can be simulated by three straight lines is reasonably valid for gross considerations; for fine detail near the <:^ R,N R,N = rb+r- + '€ I 'm If ; Rc+Pf^rc-rr, Tb VW RiN = rb+rc-Tm + Tc (fm-r-c) Rf+re+ Tc-rpn Fig. 10 — -Base driving point resistances. TRANSISTORS IN SWITCIUXG (IIUCUITS 1221 T_ "-Vbb (a) BASE BISTABLE CHARACTERISTIC Oi. 'L :Rb rVbb ( b) BASE MONOSTABLE CHARACTERISTIC r :Rb pVbb BASE ASTABLE (C) CHARACTERISTIC Fig. 11 — Base connection switching circuits. 1222 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 turning points the approximation is by no means accurate although affording zero order information. Preparatory to the analysis of the negative resistance characteristics, it is necessary to obtain analytic expressions for the transistor currents and voltages. This in turn involves the following steps: 1. Identification of the three regions in terms of the device character- istics, 2. Idealization of the device characteristics to obtains simple, linear relations, and 3. E\'aluation of the device parameters in each of the three regions. Fig. 13 is a family of open circuit characteristics for a typical switching type transistor. Specifically, in small signal terms, Table I Parameter Rn = Rl2 = Rn = R22 == die dVc dL Also die dL R21 R22 Equivalent Tee Rn = r, + Ti Ru = n R21 = Tm + n R22 = Tc + n To + n The above set, normally employed for small signal analj^sis, will be assumed to be constant within a given region, but changing in value from region to region. IDENTIFICATION OF THE THREE REGIONS It may be recalled with the aid of Fig. 12 that the negative resistance characteristic consists of a negative resistance region bounded on each side by a region of positive resistance. Thus the device is first passive in nature with little or no gain, then very abruptly exhibits considerable gain with the resultant negative resistance, and finally becomes very abruptly passive again with little or no gain. It would seem quite clear that abrupt changes in the transmission properties of a device should be associated with equally abrupt changes TRANSISTORS IN SWITCHING CIRCUITS 1223 in the forward transfer characteristic. In the case of tlie transistor, the behavior of the forward transfer properties is given by the forward transfer impedance, Rn . Examining the Rn family in Fig. 13, it is seen that in the normal, positi\'e emitter current region the slope, R^ , is high indicating the possibility of high forward gain. When /, is negative, however, the slope is zero or nearlj^ so, changing very abruptly at /« = 0. Further, it is to be noted that as I^ is made negative, the collector voltage is unaffected, remaining constant for further change in /« . Thus it may be said that the collector voltage is saturated.* REGION I REGION n Fig. 12 — Broken-line idealization of negative resistance characteristic- sion into regions. -divi- If, on the other hand, the emitter current is increased, at constant collector current, it is found that at a critical emitter current the slope again becomes zero or nearly so. There are also two further observa- tions. First, the collector ^'oltage is reduced to a verj- small value and second, that the critical emitter current is related to the collector current. From the small-signal relation, Vc = R21L + RoJc (2) or Vc = R220CU + R22IC , (3) * It is tacitly assumed that in the relation y = f(x) that there are extremes at which rj becomes essentially constant and independent of further change in the independent variable x. The point farthest removed from the origin at which the dependent variable becomes constant is termed saturation. The point closest to the origin at which the dependent variable becomes constant is termed cutoff. 1224 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 11 w V A V \ w \ i\ i\ \ N \ 1 W \\ A s \, i\i A \ \ ^ ^ w V \ V \ \ \\ \ \ ^ d\ \ \ \ \ \ \ V II \ <*> >— 1 o ^^ r- o t ^ ■^ 1 — o s -^ """^ .,,__l q ^'«~. ^ o k ^ -^ '■ d 1 II d -1 q -T q rvi 1 II o - . 30VnOA a3iillM3 33Vi"ioA aoioanoD TRANSISTORS I.V SWITCHIXG CIRCUITS 1225 the critical emitter current for collector \'oltaj2;e cut oft' may be obtained by setting Vc = 0, as, /. = -- (4) a This relationship is dual to th(> gild \()ltage-plate voltage relation in tubes for plate current cutoff as, 1'^ = —(Vp/^l). The criteria, for de- fining the three regions are thus established as: Region I (Collector \'oltage Saturation): /, < 0 (;")) Region II (Active): 0 < I, < -l^ (0) a Region III (Collector Voltage Cutoff): 7^ > -- (7) a The identification of de\'ice parameters will be made for the several regions by a single prime for Region I as i\ , none for Region 11 as 7\ , and three primes for Region III as i\ . LIXEARIZATIOX OF THE CHARACTERISTICS AND APPROXIMATIONS The next step is to linearize the characteristics and to make suitable approximations in order to obtain simple linear equations of the ter- minal currents and ^'oltages. The relations which require linearization are the device parameters Rn , Rr2 , R21 and R22 which are in general functions of the currents as Rij = /(/i , I2). LINEARIZATION OF Rn AND 7?io In terms of the equivalent tee circuit, which has been and will be employed, Rn is given as Rn = ?% + I'b . Also, Rio = Vb . It is convenient to separate r^ and n and discuss each separately since Vb is fairly constant and ft will have widely different regional values. In the R12 family of Fig. 13, it may be seen that R12 or n is fairly constant in all three regions and will be so taken here. Further, in the simple circuits under consideration, external base resistance Rb has been inserted so that minor variations in n in the total of n + Rb are inconsequential since usually Rh y> rb. The approximation that n is constant is subject to review where finer detail is necessary, particularly at low emitter currents where the rate of change of n is at a maximum. The emitter resistance r^ is approximately the resistance of a diode in the forward direction. As such, r^ is high when the emitter current is 1226 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 negative and low when the emitter current is positive. Experimentally, it is found convenient to give three values to r^ and hence to Rn , one for each region as shown in Fig. 14. This recognizes the non-linearity with It in the forward direction and assumes that a single value in the reverse direction is sufficient. As the circuitry becomes more sophisti- cated a more precise approximation will undoubtedly be required, particularly near I^ = 0. It may be noted that in the functional relation Rn = /(/« , Ic) that Rn is taken to be a function of It only. The contribution of Ic is to shift the characteristic in voltage by VbAIc increments. Thus the relation- ship of Ve = f(Ie , Ic) can be written very simply as 7. = Rnle + Rnic (8) Since the problem has been linearized to first order terms only, the currents and voltages are total instantaneous or dc values as indicated by the capital letters. IDEALIZATION OF Roi As indicated previously, Rn will be small in Regions I and III and large in Region II. Since Ru = Vm -\- n , Rn can be no less than rb ; the defining approximations will be applied to r^ . In Region I when the emitter current is negative, Vm is taken to be zero and reflects the device approximation that the emitter current under this condition is entirely electron current. This is not always a true approximation, particularly near 7^ = 0, and limiting tests are employed in transistor testing. REGION in Fig. 14 — Idealization and regional division of input characteristic (Rn). TRANSISTORS IN SWITCHING CIRCUITS 1227 Til Rof2;ioii IT, /•,„ has, of course, high vahies and in general r^ » /"/. . Pending further investigation, r,„ will be assumed finite, but very small in Region III. IDEALIZATION OF R^i In the output family of Fig. 13 it may be noted that 7^22 has two values, a high value for h > —aU and a low value for Ic < —al, . The high value corresponds to Regions I and II and the low value to Region III. To a first order the two values are separately constant which was not true of earlier transistors in which 7^22 underwent ex- tensi\'e degradation in magnitude as Ic and /« increased. The lower limit to which 7^22 can fall in Region III is n , since 7^22 = Tc + /■(, , implying that Tc is zero in Region III. This is approximately, but not acciu'ately true. As a/, approaches —Ic in magnitude, the \-oltage across the collector barrier becomes nearly zero so that r^ has a low, but finite value. Under this condition, the hole current is very high and heavy conductivity modulation of the collector barrier re- sistance occurs. Thus the collector resistance in Region III is indeed quite low and may be neglected for many circuit computations. In the functional relation R21 = /(/« , /c) it has been assumed that 7?22 is a function of Ic alone. Further, the approximation involves first order terms only and hence the functional relation Vc = /(/e , /c) may be written as: Vc = RnL + 7^22/0 (9) Here again, as in the input case, the currents and voltages are total instantaneous or dc values as indicated by the capital letters. It is believed desirable, however, to give one more consideration to the output relations. When 7^ = 0, the collector characteristic is ap- proximately that of a diode in the reverse direction. A diode has low reverse resistance until the voltage across the barrier exceeds a few tenths of a volt and then has quite high resistance, approaching infinite slope in the case of junction diodes. ' This effect is shown exaggerated in the idealized output family of Fig. 15. The current and voltage at the break in the /« = 0 curve have been termed /co and Vgo respectively. I tin and Fco are quite evident in junction devices; in point contact devices they are not nearly so evident due to the lower value of R22 and the higher voltages and currents normally employed. Where currents and * See Reference 2. ' Holes and Electrons, W. Shockley, Van Nostrand, p. 91, 1950. 1228 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 I voltages are of the order of several milliamperes and a few volts, Ico and Fco naay normally be neglected, /co and V^ do have considerable interest to the device designer, however. The net circuit interpretation of /ro and Fco is to effectively transfer the current-voltage axis from 0, 0 to lea , Vco . Therefore, Vc - Vco = R2lh + (/c - /co)/?22 (10) or Vc - FcO = (jm + n)h + He - Ico) (re + V,) (H) Making the approximation that Vco = /co/?22 and rearranging, equa- tion (10) becomes, or Vc — IcoRii + IdaR-22 = Riil( + IcRi Ve + /co(i?22 - R22') = R21L 4- TcR22 (12) (13) which is of the usual form except that a small dc generator of magnitude Ico{R22 — R22 ) has been added in series with the collector. Since R22 = Tc + n and R22 = r'o' + n , Ico{R22 - R'22) = Icoirc - r'/') (14) The output family equation with equivalent circuit parameters is region ie (collector CO *- VOLTAGE CUTOFF) COLLECTOR CURRENT, I,; 2.A. / / / / / t Vco .... :>^ r 1 1 LU a. 0 1- 0 tu _j _i 0 0 I- REGION n J ACTIVE n \ M I (collector VOLTAGE SATURATION) Fig. 15 — Idealization and regional divi.sion of output characteristic (Rn)- TR.\NSISTORS IN SWITCHING CIRCUITS 1229 then : Vc + lA^c - r'/') = (r,„ + n)!, + (r. + rb)Ic (15) SUMMARY OP IDEALIZATION OF CHARACTERISTICS The resuhs of the ideaHzatioii of Ihe device charactcfistics nw, ,sum- nuuized in Fig. 16. Here are given analytic expressions for the input and ()uti)ut voltages in terms of the ini)ut and output currents; the r(>gi()ns are defined symbolically and by tj^pical values; and an e(iuivalcnt circuit is given. It may be noted that the equivalent circuit is identical to the small-signal equivalent tee, excepting the small dc generator /co(/'c — fc ) which usually may be neglected when dealing with con- temporary point contact transistors. To obtain any of the negative resistance characteristics it is only necessary first to solve the two equations simultaneously for the ap- propriate voltage in terms of the appropriate current, and then second to insert into the resultant equation the proper parameter values, region by region, to obtain three equations. These equations, when plotted, result in an idealized characteristic similar in form to that of Fig. 12. A detailed example plus synopsis of the properties of the several connections will be given in the following sub-section. i_ 'e -VW a- Ic ■^llH T" Vf = (rf + rt,)lf+rblc PARAMETER REGION ^f r~b ^C I'm SYMBOL TYPICAL SYMBOL TYPICAL SYMBOL TYPICAL SYMBOL TYPICAL I ^e 100 K ^b 160 ^C 20 K r~m 0 n r-f 100 ^D 160 rc 20 K r^m 50 K M Tf'" 25 Tb 50 Pc'" 70 fm'" 30 Ico '''s -50/iA Fig. 16 — Broken-liiK; transistor cMHiat ions, rcf^ional parameter values and equivalent tee circuit. 1230 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 THE ALPPL\ OR CURRENT GAIN FACTOR* The derivation just given has been in terms of the equivalent circuit parameters, r, , rh , r^ , and r^ . Another circuit factor, alpha or the short-circuit current gain, is also quite useful. Alpha has been defined in Table 1 as the negative ratio of the incremental change in output current to the incremental change in input current under the condition of short-circuit output terminals. Thus alpha is restricted in interpretation to a specific device termina- tion and care should be taken in the employment of alpha when other terminations are involved. For example, the circuit current gain under general conditions is given by Rix/R^i . The ratio R21/R22 has been sometimes termed ac . Thus, ^21 rb -{■ Rb + Tm (.^. Uc = —f = (lb; R22 n -\- Rb + r, -\- Re Depending upon the magnitudes of Rb and Re , the two current gain ratios may be markedly different. In Region II where Vm and re are very large the effects of Rb and Re in equation (16) often may be neglected. The circuit current gain, ac , may then be taken as the device alpha. In Region I, Vm has been taken as zero; hence the current gain will be somewhat less than unity, given by (?'{, + Rb)/(rb + -Rs + ^c + Re), and is definitely not zero. Eciually, in Region III, the circuit current gain is not zero but rather approaches the ratio, (rj, + Rb)/{rb -\- Rb + Re)- If Rb^ Re , the ratio is nearly unity, ANALYSIS OF NEGATIVE RESISTANCE CHARACTERISTIC The objectives of the circuit analysis, as stated previously, are: 1. To determine operating conditions such as proper values of loads, biases, trigger sensitivities and operating currents and voltages, 2. To determine the relationships of the device parameters to the circuit behavior in order that these parameters may be optimized, properly characterized and controlled for required circuit performance. For example, the trigger sensitivity may be given by the voltage difference between the load line intersection with the Region I portion of the characteristic and the upper peak or turning point of the charac- teristic as shown in Figs. 6, 7 and 9. The sensitivity A is thus determined by the nearness of the bias point to the peak of the characteristic. Since the bias is normally fixed, variations in the sensitivity will arise * This section is inserted parenthetically as clarifj'ing material due to the use of the a-factor in subsequent discussion. TR.\NSISTORS I.V SWITCHING CIRCUITS 1231 from variations in the peak point. Thus it is necessary to know the relationsliips which (let(M-mine the currents and voltao;es of the peak and N'alh'N' points in ()i(h'r hrst to achieve a design and second, toestab- Ush controls on tlie proper (le\'ice parameters. In this example the emitter nei>;ative i-esistance characteristic will be solved and analyzed. The solutions for the other characteristics follow in the same manner and will be summarized. EV.VLUATION' OF P:M1TTEH CH.VR.VCTERISTIC AS AN EX.\MPLE To obtain the emitter charactei'istic, it is necessary to solve the two equations of Fig. 16 for T"« in terms of /« . The equations as given are for the short-circuit case. Since the general case will involve external parameters as R^ and Re , the equations will be modified to include these parameters. The effects of external parameters may be applied very easily since, V\ = F« - IMe (17) and Vc = Vcc - IcRc (18) where T^„ and T^^ are supply voltages; V, and Vc are measured from the appropriate terminal to the far end of the external base resistor. External Rh adds directly to rh . Thus the two equations become: F« = (r, + ff. + n + Rb)Ie + (n + Rb)Ic (19) Vcc + (re - rc')Ico = (rm + n-\- Rb)I, + (n + /?6 + r, + Re)h (20) In manipulation of equations (19) and (20) it is often more easy to do so in the functional form, Fi = Rnh + RvJ2 (21) Vo = Rnh + R22h (22) with substitution at the evaluation stage. The R"s here include both device and circuit parameters.* Solving equations (19) and (20) simultaneously, the following rela- * Here the primes indicate generalized open circuit driving point resistance rather than reference to Region I. The duplication of symbols is regretted. 1232 THE BELL SYSTEM TECFINTIfAL JOURNAL, NOVEMBER 1952 tionsliip between ]\ and /« is obtained: F, = r, + tie + Tb + Kb — j — 5 — j j — 5 Tb -\- Kb + re -\- Re (23) Equation (23) is general for the given circuit; it suffers, however, in difficulty in interpretation due to the numerous terms. In the regional evaluation to follow, approximations will be made which bring out the significant factors although decreasing the accuracy somewhat. The (''c — I'c )Ico terms will be neglected. It is assumed also that large ex- ternal base resistance Rb is employed. EVALUATION IN REGION I In Region I, from Fig. 10, r,„ is zero and r, is large so that r,' » (rb + Rb)- Also, by assumption, rb « Rb ■ Applying these approxima- tions, equation (23) becomes, ^-■■^-:^. + ^,;y;^. (24) Equation (24) is the equation of a straight line, having slope r^ and an intercept on the voltage axis at (VccRb)/(Rb + fc + Re)- The small- signal input impedance is just the slope value or r^ . The short-circuit case where Re is zero is the most adverse device condition in the sense that the dc term will th6n be most dependent upon device parameters. When Re = 0, equation (24) becomes F.I ^ r'je -f -J^ (25) Kb + Te EVALUATION IN REGION II In Region II all parameters are finite and the only approximations which may be made are vt « Rh and r^ « Rb . Thus, F. p _ RbjRb + Tm) Rb -\- Vm '' + ,, + t+ R. ^''^ If Rb is not too large, it may be approximated that (Rb + rm)/ (Rb + re) = a. Taking Re = 0, thus, Fai ^ Rb(l - a) + -i^ (27) Kb -f- re TliANSlSTOUS I\ SW ITCHl.NCi CIKCUITS 1233 Equation (27) is also the equation of a .strai{»ht line lia\-inf;- the xoltaftc axis interc'ei)t of {\'cRb)/{Ih + /',) the same \'alue as in Kegion 1. The slope, Rb(l — a), is negative provided a > 1. KVALUATIO.V IX REGION III In Region III it may be assumed that rb « lib , Vc « Rb and /'l" « Rb . Other suitable approximations will depend largely upon the magnitude of Re . From equation (23) F.I ' "\ -i /// , ,, Rb{Rb + /'m ) ''e -r tib — Rb + !•:" + RJ Rb + R V R,. L + ^' ' (28) If Re is large, that is, large compared to /% , but small compared to Region II Tc , then (28) becomes, T' r>^ tibiae J 1^ V cc^b (i^(\\ * '"' - Ih+^e ^' + WVR. ^^^^ Under these conditions, the circuit is essentially independent of de^'ice parameters. This is useful where a high independence of device parameters is required, but does not focus the attention upon the device parameters as does the Re ^ ^ case. This is the condition under which the transistor might be operated when it is desired to obtain the maximum ON current, or conversely the minimum internal switch re- sistance. Where Re = 0, ecjuation (28) becomes, T^in = [/•:" + r'e" - r::']L + F. (30) Since /% and (r<; — r^ ) are quite small the short-circuit currents may be very high. Where the transistor is considered as a switch between emitter and cbllectbr circuits, the "switch" voltage drop, as V^c , is given by the first term of equation (30). KVALIATIOX OF REGION J-REGION II TRANSITION Earlier, trigger sensitivities were mentioned as being the small voltage and current differences between the turning points of the negative resistance characteristic and the stable operating points. The determina- tion of the turning points and their stability is of great importance since it is usually desired to obtain maximum stable sensitivity. The voltage and current at the two turning points* have been given the subscript p and v for the low and high current conditions respectively as shown in the synopses of Fig. 17, 18 and 19. V^p and I^p in theshort- * Sometimes termed "peak" and "valley". 1234 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 I^ .^f V,:^ V^ .1 >Rb -V, GENERALIZED EMITTER CIRCUIT Fig. 17 — Sj-nopsis of emitter ne circuit or R^ = Re ^ 0 case for example may be obtained by a simul- taneous solution of equation (24) for Region I and equation (27) for Region II. Thus VMb F. and Rb + Tc Lr, = 0. (31) (32) That the low current turning point falls on the emitter current axis, i.e., I(p = 0, is a consequence of the original assumption that Tm, = 0 TRANSISTORS IN SWITCHING CIRCriT.-* 1235 &] I, (r. + n + Rb +R.) Synopsis in + Rb){n + Rb+ rj '■b + Rb + i-c + Re (Vcc + /c.(r, - r'e")) rb + Rb + re + Re (rb + Rb) xirnatc Short Circuit Case rb «: /?b ; r„ r'/' «: 74 ; Rb < r„, /■„.; /?< = /?. = 0; TcXrc - rn <^ Vc 1 I a II a III F. = 7.r/ + VcRb Rb + re F, = LRoir - a) + VcRb Rb + re T% = IM" + r'e" - r;'/) + Ve /.p - 0; F.p = VeRb Rb+ re V,„ = V, 1 + i^bCl ^— 'a F„ i2b + re Vtp Rb nice characteristic and properties. for /, < 0 and r„ > 0 for I, > 0. This is not a precise assumption and the turmng point will usually lie slightly in the positive emitter current region. For very small triggers or more accurate calculations, considera- tion must be given to closer approximations of r^ = /i(/«) and Rn = hih). The consequences of equation (31) can be quite serious. Vc and ^6 are of course fixed, but Tc is variable from unit to luiit, with temperature and perhaps with life. The variability of V,p can result in failure to trigger, self-triggering or lock-up at high current. L T I 'f fc _?^ Ico(rc-rc"') _t t. GENERALIZED COLLECTOR CIRCUIT I V\A- ±.. SIMPLIFIED COLLECTOR CIRCUIT Synopsis General Vcc + I,o{re — Tc'") = Ic Te + Re + n + Ri - in + RbKn + Rb + rj' r, + R, + rh+ Rb V,.in + R,> + rJ r, + R, + n +Ri, Approximate Short Circuit Case where R, = Rc = 0; n^Rb] Icoiu - n'") < F„ , i?6 ; point for the .short-circuit case is deter- mined from a simultaneous sohition of the pertinent equations for Regions II and UK ('((nations (27) and (30). Thus, VcTe V ?^ V I 1 1 1 III ' ' l\ (34) (re + /?6)/?6(l - a)J Where it ma>' l)e appi-oximated that /v » /f^ , as has ah'eady been done in bringinsi; in a, (Hjuations (33) and (34) become, V '-^ V r, + Vc - r (36) Rb (1 - a) In this order of approximation, T^„ is nearly equal to T^ . Any \-aria- tion in the lower trigger point is primarily with /e, , due chiefly to any change in a. It is interesting to note that the trigger point will mo\^e along the Region III curve given by (30). The ratio of F^t, to V,p is often of interest to estimate voltage swings or perhaps as a design objective in some switching circuits. Thus from (36) and (31), F.„ [R,(l - «) + r'/' + ;•:" - r'^'](R, + r,) V^v Rla - a) — a) then (37 V,v Rb + Tc If r" + Tc — Vm « Rb{l — a) then (37) becomes: (37) (38) Vtp Rb If Rb is verj' large, the ratio approaches unity with the implication of the existence of only two regions. This is eciuivalent to saying that the negati\-e resistance becomes infinite over an infinitely short range. The proper choice of Rb in terms of (38) may well be a design problem where it is desired to ha\e a high i-atio of l\,, to T'^,. , as in lockout circuits. SYNOPSIS OF NEGATIVE RESISTANCE CHARACTERISTICS Synopsis for the three negative resistance characteristics are gi\-en in Figs. 17, 18 and 19. The solution and analysis pi'ocedures are the same as outlined for the emitter charactei-istic. It should be not(xl that the base characteristic is short-circuit stable in distinction to the emitter 1238 thp: bell system techxical journal, November 1952 rmU rb r, _''J^^ ho(rc-r, "T^AA 1 \W imlA +Vb l/ Icp/ Ibv -lb 0 /vbp +Ib 1 N. Vbv \ .Vc_=Vc_c_^ -Vb _,-.-— -"''n'^ .-^ Fig. 19 — Sj'nopsis of base n( and collector characteristics which are open-circuit stable. It would have been more appropriate to solve the base circuit in terms of con- ductances rather than resistances. The magnitudes of negati\-e resistance obtained in this connection are quite low which may be misleading; the negative conductance is quite high, however, which is desired in short- circuit stable negative resistance circuits. Care should be taken in the literal employment of the approximate regional relationships in Figs. 17, 18 and 19. They are very definitely approximate and are intended to illustrate behavior and the limiting condition only to bring out the relative importance of device param- eters. It is suggested that calculations be started with the general case and approximations be made as are valid. For example, the con- TRANSISTORS IN SWITCHING CIRCUITS 289 (Synopsis •al Vb = hin + Rb+ >\ + R,) + I dr. + R,) ] Vcc + IcXrc - r'c") = hir, + R, - r„) + 7,(r. + R. + R, - /•„.) (r. + RMr. + R. - rj^ rb + Rb + A'. + r. - r. + R. + rc + Re- r„. [\\c + Loir, - r'e")]{r, + R,) + U + R,-\- rc+ Re- r„ ).\iin;itc Short Circuit Case A', = R^ ^ (^ ■ I^Xtc - K") «C T'. ; r. «: red - «) n I n II n III Tc + r'J r; + Te n = /.i':^'U "•'■• \ — a I /•«(! — a) /bp ~ Vbp = 0 1 - ,,. = r.(,-^^i^) ince characteristic and properties. elusion is reached in the collector characteristic that the negative re- sistance (Region II) is independent of the base resistance or fee(ll)ack. This is true for only the limited range where r^ « Rb « Tc . EX.\MPLE OF CALCULATED AND EXPERIMENTAL CHARACTERISTICS An example to illustrate the analysis is shown in Fig. 20 where both experimental and calculated characteristics for the emitter circuit are given. In this example there is appreciable load resistance; hence Vc , r't" and r^' are of no consequence since they will all be very small com- pared to the Re of 2.2K ohms. Also, Rb = Q.SK ohms is much greater than n ; hence n can be neglected. Since Vc is —45 volts, the Ico term may also be neglected. 1240 THE BELL SYSTEM TECHNICAL JOURNxVL, NOVEMBER 1952 Computing V^p first, The calculated value of — 10.9 \'olts compares quite fa^'orably to the measured — U.O volts. Region II is given in this case, approximately by. F. . ' "^ 7?6 f re + Rc^ V Rk /?6 + r-c + /?c F. ^ (^6.8K + e.8A- + 19i^ + 2.2k) ^' " '^"^ ^^'^ V, ^ (-8.9/0/. - 10.9 (42) The first term is of course the slope in Region II and is the magnitude of the negative resistance. The calculated value is —8900 ohms whereas the measured value was approximately —9200 ohms. The Region III approximation, derived also from the general rela- tionship is, Rb -{-Re Rb -\- Re ^ (mKK2.2K)) ^ 45(6.8A0 . . (Q.SK + 2.2K) ' 6.8/C + 2.2K ^ ^ or F.iii = (1785)/. - 34 (45) The relation for Region III agrees quite well in slope but not in dc value as may be seen in Fig. 20. Since in this example the Region III behavior is determined essentially by the circuit parameters, it is surmised that the nominal 45-volt battery employed in taking the data was actually 47 volts. The Region I check is essentially perfect since the approximation given in Fig. 17 is quite good. Note the error at the intersection of the Regions II and III. The broken-line method predicts a sharp transition whereas the actual case is gradual. The deviation is due to the gradual changes in r^ and re as the collector voltage approaches cutoff and is the largest gross error in the approximation. It is believed that analysis of this sort will reasonably predict circuit behavior and lead to device requirements. There must be a thorough understanding of the approximations involved and the accuracy will be directly related to the degree to which the original idealized character- istics are approximated. Extended, by means of more than three broken TRAX.S1S1\)KS I\ SWITCIII.VG CIUCUITS 1241 EMITTER CURRENT, If, IN MILLIAMPERES 0 1 2 3 4 5 >-15 5-; I^--^^^^ A 1 1 ^ , / V y >2.2 K I ^6.8 K -^45V Tf = 120/1 r^j =^ i6on Tc = 19,000 /I Tm = 58,000 n Tf ' = 500,000 A 1 \ \ > \ \ ^ \ \ ^^-' ^^ V ^- .--•' ^ --" > v< ,^'- '^ .-^ "^ / ^^-' -"'' .''■ 1 Fig. 20 — Experimental aud calculated emitter negative resistance characteristic. lines, the method will yield fine detail to the degree to which device parameters are known and patience will permit. Transient behavior and analysis have not been discussed and are needed for a more complete understanding, particularly where transitional speeds are of concern.* Ill — SwiTCHixG Type Transistor Properties An examination of the circuit approximations given in Figs. 17, 18, and 19 will reveal that the transistor and circuit designers will want to know nearly all there is to know about the device characteristics. This is not particularly surprising since the device is used over its entire range rather than over a limited portion as in the case of small-signal applica- tions. The same examination of the circuit relations will also show tjiat * A treatment of the transient behavior between regions is given in B. O. Farley, "Dynamics of Transistor Negative Resistance", Proc. Inst. Radio Erigr., Nov., 1952. Analysis and the solution for the periods of the monostable and astable cases, a.ssuming infinite region to region transition speed, are given in G. E. Mc- Dufhe, Jr., "Pulse Duration and Repetition Rate of a Transistor Multivibrator", Proc. Inst. Radio Engr., Nov., 1952. 1242 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 virtually all of the device parameters should be constant from unit to unit and with ambient conditions. It can be shown that for small-signals a device may be uniquely characterized by five measurements. In terms of the parameters used here these might be ^n , Rn , R22 , R21 and the dc bias point or equally, ''e , n , re , Vm and the bias point. Since the problem was linearized in the approximation, it follows that 15 such measurements, five in each region, are necessary for proper switching device characterization. The indicated extensive testing required may be reduced somewhat by suitable approximations. It is clear that the switching device designer and producer must reconcile themselves to making more tests for ac- curate characterization than when small-signal devices are concerned. What will be given here is a description of typical developmental switching transistors in terms of the parameters which have evolved as a result of practical approximations. The method will be to discuss device properties and measurements region by region; then to discuss the properties at the transition points. Temperature, frequency and life behavior will be taken up separately. REGION I PROPERTIES In Region I, the emitter current is negative. Hence the emitter resistance r^ is large and is essentially that of a diode in the reverse direction. At present r^ is measured by a simple dc test of the current which flows at a nominal —10 volts. Both r^ and n will be discussed further under the Region I-Region II transition properties. The Region I collector resistance is one of the most important param- eters in switching. This is because of its determining nature in the turning point voltages in Figs. 17, 18, and 19. Actually, what is of concern is not the small-signal slope shown as r^* in Fig. 21, but rather the dc current and voltage relationship shown as Tco . For example in Fig. 17, it may be seen that V^p is given by the voltage drop determined by the product of Rb and the dc collector current. Fig. 21 is an idealization of the R22 characteristic and has been de- signed to bring out the diode nature of the collector by emphasizing the saturation current and voltage, Ico and Vco . In junction devices the break in the /^ = 0 characteristic at Ico is quite evident whereas in present point contact devices the transition is smooth due to the much lower values of re . The device significance is the same, however; Ico varies rapidly with temperature whereas Tc varies at a considerably lower rate. * The actual parameter is of course ^22 , but since Ri2 = re -\- ri, and n <^ re R22 is taken as re . TRANSISTORS I.V SWITCHING CIRCUITS 1243 In junction devices the proper measurements would he of /^o and fc . Since /dt is difficult to define in point contact devices, Tco has been measured as an ap{)i-o.\imation. In the idealization, r^ and r^i arc re- lated as, . Uc - Ico) >M = = 1\ im The measiu'ement of 7-^0 is made at a collector voltage which is typical of the applications in the range of perhaps —10 to —45 volts. A constant dissipation line has been drawn on Fig. 21, which reveals the desirability of having Vco very large in order to operate at higher voltages and to secure high efficiency through lower dissipation in the OFF or rest condition. UEGIOX II PROPERTIES The Region II low frequency properties are essentially identical to those of transistors intended for small-signal applications. A possible exception is the somewhat less attention paid to the base resistance, n , which is critical to small-signal applications. The characterization con- sists of a normal small-signal set plus dc bias values. UEGIOX III PROPERTIES The Region III properties have been defined largely by a figure of Ir_= -5.5 MA COLLECTOR CURRENT r^= Vc CONSTANT CONSTANT DISSIPATION Fig. 21 — Idealized output characteristic illu.strating parameters. 1244 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 merit measurement shown as Fc(3, —5.5) in Fig. 21. This measurement is the voltage from collector to base under the condition that I^ > — {Ic/a). In this instance /j has been chosen to be 3 mA and Ic to be — 5.5 mA. The collector current value is chosen on the basis of the smallest tolerable value of alpha expected so as to place the point of measurement near the 7^22 knee, but in Region III or overload. The T^c(3, —5.5) measurement is a good measurement for defining the general behavior. T"c(3, —5.5) taken with the r^o measurement constitute a very good defining set for checking the transistor as in re-measuring. For design purposes, the Fc(3, —5.5) measurement is not sufficient. It provides an approximate value for Tc , but does not define r'/ and r„ . A second dc measurement, the collector to emitter voltage drop, V,c , has been employed experimentally also. An improved char- acterization will undoubtedly involve separate measurements of r^ , III , /// Tm and Tc . REGION-TO-REGION TRANSITION PROPERTIES The transition between Regions I and II is accompanied by abrupt changes in r, and r^ . The theory assumes that both of these parameters change at an infinite rate at a fixed emitter current, taken as h = 0. Unfortunately neither of these assumptions is strictly true, r, undergoes a gradual change from high to low values which is only approximated by the three assigned values. In particular the behavior near /, = 0 is of con- cern when dealing with small triggers. The forward transfer impedance changes at a finite rate also. Further, the emitter current at which the maximum rate of change occurs will vary from unit to unit. Present practice also has been to measure a rather than r„i . The rational for doing so is not too good since r„ is quite likely the better parameter to characterize. Alpha has a strong phys- ical appeal, fits well into the circuit problems and is easy to measure. Since a = (r^ + r,„)/(r6 + Vc) it is necessary to assume that n and fc are constant near /, = 0, an only fair approximation. Having made the approximation, the typical a behavior shown in Fig. 22 may be taken as a measure of /•,„ . Three values are measured, the first of which, ai , in Region II, is redundant to the Region II small-signal measure- ments. The two limits, 0:2 and 0:3 , serve to place lower and upper limits on the absolute values of a at the Regions I-II transition. These limits in turn place a lower value on the rate of change in a within the I, ± A range shown. It may be noted that a in Region I is finite. There is a lower limit TRANSISTORS IN" SWITCHING CIROUITS 1245 1^^ \ X A) 1 1 1 1 1 \ L- 1 1 1 1 : -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 EMITTER CURRENT, If , IN MILLIAMPERES Fig. 22 — Alpha characteristic. even though /•„, is zero since ay -^ {n/n + Tc). The vahies normally encountered at /« = 0 — A are usually in excess of this lower limit. The feedback resistance r^ tends to rise as /« — > 0 which may be important to some trigger circuits. As the circuitry becomes more sophisticated, it is expected that more attention will need to be paid to the behavior of r^ , r^ and n at and near /, = 0. The transition from Region II to Region III is determined from the relation /« = —(Ic/a). The problem is quite similar to the control of the M factor in tubes where plate current cut-off is given by Vg = — (Vp/n). Present practice has been to depend upon the ai values and upon the lower limit placed on alpha in the Vc(S, —5.5) measurement. Further effort is needed here also. TYPICAL PARAMETER VALUES AND DISTRIBUTIONS Integrated distribution curves for the parameters of a typical develop- mental switching transistor are shown in Fig. 23. The unit-to-unit variations are deemed to compare favorably with those of commercial electron tubes. The parameter of most serious vai'iability is r^o which is unfortunate since Vco is so important to trigger sensitivity stability. TEMPERATURE, FREQUENCY AND SHOCK PROPERTIES Transistor parameters are reasonably constant with temperatures below room temperature. Above room temperatures some of the param- eters are variable, r^ and n are fairly constant, changing very little to 70°C. Vc and r^ decrease fairly rapidly, maintaining a ratio such that alpha rises slightly, r^ and Tco change most I'apidly and, while both of 1246 THE BELL SYSTEM TECHNICAL JOTTRNAL, NOVEMBER 1952 100 75 «-> 50 N \ k A N \ y,2 \ \ \ \ V" t" \ ^; V \, ^ V \ ^^ 100 200 300 400 500 0 20 40 60 RESISTANCE IN OHMS 100 120 XlO^ 20 \ ' 0 iVc(3,5.5) \^ \, V ^ S^^ 60 SO OHMS 120 140 0 XIO^ 2 3 VOLTS 80 \ \ \ \ \ ^ «^r" --N OCz \oC3 \ \ K 1 \i V \\ X 0.3 0.4 I 2 3 4 5 6 7 8 0 0.1 0.2 oc oc Fig. 23 — Variation in parameters of developmental switching type transistor (M1698). these parameters are of little consequence in small-signal applications, they are quite important in switching, particularly Tco . Early transistors might exhibit a change in Tco at 60°C of 3 to 1 or more from room temperature values. The transistors of which the data in Figs. 13 and 23 are typical have an Vco temperature coefficient of about —0.75 per cent/°C. That is, the room temperature value of r^o might be reduced by 30 per cent at 70°C. The improved temperature behavior implies a corresponding reduction in variation in trigger sensitivity. Parameter values, large-signal and small-signal, are sho^\Ti in Fig. 24 as a function of temperature. TRANSISTORS IN SWITCHING CIRCUITS 1247 Variation iu characteristics will arise from self-enfj;en(lere(l heat, that is, dissipation. Transistors may be thermally unstable under constant Noltage conditions. Since the switching properties are e.xhibited under short-circuit or constant voltage terminations, thermal propei'ties are of concern. The limitations involved are similar to those of any positive feedback circuit. If the thermal loss through radiation and conduction exceeds the heat input, the system Avill be stable. The practical sig- nihcance is to place limitations on dissipation and to employ designs which result in rapid heat loss. Other design criteria such as miniaturiza- tion may limit the latter. If perfect switching characteristics were obtainable, dissipation would be of little consequence in switching. This is akin to saying that neither a short-circuit nor an open-circuit dissipates any energy. Further, the perfect device has zero transition time and therefore involves no loss. The transistor has finite resistance both open and closed and a finite although rapid transition time. There is some advantage however. A constant dissipation cur\'e sho\Mi as a dotted line has been included in Fig. 21. Small-signal operation at mid-range currents and voltages results in fairly low limitations on both cvuTent and voltage. The inter- section with the i?22 voltage saturation line (/« = 0) is at fairly high voltage. Similarly, the intersection with the collector voltage cut off line is at high current. For constant dissipation, approximately, Voltage saturation: Voltage cutoff: V Ills,* Depending upon the circuit the assumed dissipation limit may or may not be exceeded during the transitions. Should the limit be exceeded, «, Tb ■^" 'JIZS" ^e ^ 60 i ^° ? 40 u Z 30 < K (/) 7> 20 to Jm_ ^ — . -~~ .^co Tc ""s. 20 30 40 50 60 70 80 20 30 40 50 60 70 80 TEMPERATURE IN DEGREES CENTIGRADE Fig. 24 — Temperature behavior of characteristics of developmental switching- type transistor (M1689). * This includes both emitter and collector dissipation. See equation (30). 1248 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 and substantially so, there are normally no serious consequences due to the very rapid transitions and consequent low thermal energy generated. Transistors may not be able to tolerate excess dissipation on this basis if the circuits are slow, that is with transition times in excess of perhaps a few tenths of a microsecond. Such conditions may arise, for example, if loads are inductive. In many such cases, shunting capacitor networks will often permit a rapid transition with consequent transfer of current to the inductive load. The frequency response of point contact transistors can be sufficiently good to insure switching type operation with rise times of the order of 0.1 to 0.01 us. Fall times may be somewhat longer due to the hole storage effect. In regenerative circuits, operating speeds are faster than might be imagined from the small-signal frequency cutoff. Reliable operation with rise times of 0.1 ixs is obtained with only nominal attention to frequency cutoff. Speeds of the order of 0.02 jus require a 10 mc. lower limit. Present junction transistors are substantially slower. Accurate life estimates are difficult to make due to the rapid rate of development, the relative age of the transistor and the number of parameters involved. A given device is quite likely to be obsolete and forced to give way to an improved version before sufficient models can be obtained for life tests. A small quantity of transistors having proper- ties similar to those of Fig. 20 and 21 have been operated for over 6,000 hours with an indicated life of 30,000 hours. Other similar transistors with longer life histories have indicated lives of better than 70,000 hours. The pattern appears to be similar to that of electron tubes — an early failure and change rate followed by a very slow exponential rate. It is believed that life is extended by low power operation and is de- creased by high temperature operation. The relatively high noise level of transistors does not appear to be a significant problem at present when considered in terms of automata. Systems employing switching type circuits in pulse communication will of course be concerned. It is suggested that the non-concern for noise in non-transmission type systems is largely a reflection of the ease with which high magnitudes of state changes are obtained. With design trends toward low power and low operating levels, noise will undoubtedly set a lower limit of level operation in such systems also. The extreme resistance of the transistor to shock and vibration with a consequent absence of microphonism may in some applications result in effective lower noise. Shocks in excess of 20,000G have resulted in no damage. No evidences of current modulation in excess of noise have been detected with vibrational forces of the order of lOOG at frequencies TRANSISTORS IN SWITCHING CIRCUITS 12-19 as high as 1000 cj^cles in tests on the transistor of Fig. 1 . Transistors have been inchided in plastic embedded ciicuils without cliangc of fliaracteristics. SUMMARY -TRAN'SISTOR l'R< )l'KKr[ KS Transistors have been designed witli properties expressly intended for switching applications.* The characteristics are acceptable for con- temporary switching type eii'cuits and sufficiently reproducible to per- mit interchangeability of devices in circuits of normal reciuirements. The characterization has been sufficiently unique to permit the calcula- tion of first order circuit performance. The characterization is not suffi- ciently complete to permit determination of the complete transient behavior. In terms of the circuits described, the major parameter limitation is concerned with the ^■ariability of the d-c collector resistance among units and with temperature. It is expected that future circuit develop- ment will place additional rerjuirements on the transistor, particularly as related to the transitions between regions. It is also to be expected that future circuit designs may establish new or modify present device requirements. A major consideration for computer or computer-like systems, re- liabilitj^, particularly with respect to time and temperature, has not been established, but appears to be favorable. ACKNOWLEDGMENT It is impossible to properly acknowledge credit to all of those who contributed to the concepts, data and results of this paper. Particular acknowledgment is due to J. A. Morton who provided first the method of attack for the analysis and second, continued stimulation. Acknowl- edgment is also given to A. J. Rack who first classified and explained the several simple circuits of the first section. J. J. Kleimack provided transistor data and R. L. Trent, circuit data. * See Reference 3 also. Abstracts of Bell System Technical Papers* Not Published in This Journal An Approximate Quantum Theory of the Antiferromagnetic Ground State. P. W. Anderson'. Phijs. Rev., 86, pp. 694-701, June 1, 1952. (Monograph 1995). A careful treatment of the zero-point energy of the spin-waves in the Kramers- Heller semiclassical theory of ferromagnetics leads to surprisingly exact results for the properties of the ground state, as shown by Klein and Smith. An analogous treatment of the antiferromagnetic ground state, whose properties were un- known, is here carried out and justified. The results are expected to be valid to order 1 /S or better, where S is the spin quantum number of the separate atoms. The energy of the ground state is computed and found to he within hmits found elsewhere on rigorous grounds. For the hnear chain, there is no long-range order in the ground state ; for the simple cubic and plane square lattices, a finite long-range order in the ground state is found. The fact that this order can be observed experimentalh^, somewhat puzzling since one knows the ground state to be a singlet, is explained. Method of Synthesis of the Statistical and Impact Theories of Pressure Broadening. P. W. Anderson^ Letter to the Editor. Phys. Rev., 86, p. 809, June 1, 1952. Arcing at Electrical Contacts on Closure. Part III. Development of an Arc. L. H. Germer' and J. L. Smith\ Jl. Applied Phys., 23, pp. 553-562, May, 1952. (Monograph 2002). A description is given of a s^-stem made up of experimental electrodes and an oscilloscope by means of which the potential across the electrodes can be recorded with a time resolution of about 10~' sec. and a potential sensiti\dty of 1-trace width per volt. The closure of the electrodes to produce a short arc is sjTichro- nized with the oscilloscope sweep so that the beginning of the arc is photographed. As an arc starts the potential across the electrodes decreases more or less gradually from the applied voltage to a steady value characteristic of the metal of the electrodes. The course of this change is extremely variable as is also the time over which the change is spread. The average value of the time appears to * Certain of these papers are available as Bell System Monographs and may be obtained on request to the Publication Department, Bell Telephone Labora- tories, Inc., 463 West Street, New York 14, N. Y. For papers available in this form, the monograph number is given in parentheses following the date of pub- lication, and this number should be given in all requests. ' Bell Telephone Laboratories 1250 ABSTRACTS OF TECHNICAL ARTICLES 1251 vary with ciicuit iiuluctance and with the natuie of tlic electrode surfaces. For inactive silver surfaces and an inductance of 0.10 /xlt the averaj^e value of the time is about 0.007 n sec, and foi' active surfaces and the .same inductance O.OI 1 n sec. P\)r active surfaces and an in;n j^roup, from 1929 until 1918. iSince 1948 he has Ikumi ii nieniher of tile Switching L^ngineering Department, and has contributed to the de- sign or development of practically all of the switching developments of the Laboratories, particularly in the field of common controls. His work has covei'ed panel, crossbar, automatic message accounting, toll, cross- bai tandem, and other systems. Member of^\.. L E. E. W. Rae Young, Jr., B.S., in E.E., University of Michigan, 1937; Bell Telephone Laboratories, 1937-. Mr. Young is in the Systems Studies De- partment, where he is giving consideration to new system possibilities for meeting future communication needs. During World War II, he worked in radar development and, later, on systems problems in radio communi- cations. From 1945-50 ]\Ir. Young helped set up Bell System perform- ance reciuirements for mobile radio telephone eciuipment. Member of I. R. E. and Sigma Xi. Index to Volume XXXI A Adniillanco Inipochince Bridj^cs for the Megacycle Range, //. T. Wilhelm, jiagcs OW 1012. Alphahrts A Comparison of Signalling Alphal)els, K. X. Gilbert, ])ages 504-522. Anderson, A. E., Transistors in Switching Circuits, pages 1207-1249. Application of Boolean Algebra to Switching Circuit Design, N. E. Slaehler, pages 280-305. Armatures, Relay Relay Armature Rebound Analysis, E. E. Sunnier, ])ages 172-200. Automatic Switchmg for Nationwide Telephone Service, .1. B. Clnrk and II. S. Osborne, pages 823-831. Automatic Toll Switching S}stems, /'". /■". Shipley, pages 860-882. B Baker, W . 0. and Heiss, J. //., Interaction of Polymers and Mechanical Waves, pages 306-356. Barium Titanate New Techniques for Measuring P'orces and Wear in Telephone Switching Apparatus, W. P. Mason and S. D. While, pages 469-503. Bridges, Impedance Impedance Bridges for the Megacycle Range, H. T. Wilhelm, pages 999-1012. C Cables. Sheaths and Sheath Losses Principal Strains in Cable Sheaths and Other Buckled Surfaces, /. L. Hopkins, pages 523-529. Carrier Telegraph System for Short-Haul Applications, J. L. Hysko, W . T. Rea and L. C. Roberts, pages 666-687. Circuits, Switching An Application of Boolean Algebra to Switching Circuit Design, R. E. Slaehler, i)ages 280-305. Circuits, Traveling-Wave-Tube Pro]:)agation Studies at Microwave Frequencies by Means of Very Short Pulses, 0. E. DeLange, pages 91-103. Circuits, Trigger-Type Transistors in Switching Circuits, .4. E. Anderson, pages 1207-1249. Clark, A. B. and Osborne, H. S., Automatic Switching for Nationwide Telejjhone Service, pages 823-831. Clos, Charles and Wilkinson, R. I., Dialing Habits of Telci)hone Customers, pages 32-67. Coding, Communication Efficient Coding, B. M. Oliver, pages 724-750. Coils, Relay Important Design Factors Influencing Reliability of Relays, J. R. Fry, ])ages 976-998. Common Control Telephone Switching Systems, Oscar Myers, pages 1086-1120. Communication Theory Efficient Coding, B. M. Oliver, pages 724-750. Comparison of Signalling Alphabets, E. N. Gilbert, pages 504-522. Computers Selective Fading of Microwaves, A. B. Crawford and W. C. Jakes, Jr., pages 68-90. Contacts A New General Purpose Relav for Telephone Switching Systems, .1 . C. Keller, jiagcs 1023-1067. Important Design Factors Inlluencing Reliability of Relays, /. R. Fry, pages 976-998. iv THE BELL SYSTEM TECHNICAL JOURNAL, 1952 Craivford, A. B. and Jakes, W . C, Jr., Selective Fading of Microwaves, pages 68-90. Crystals, Germanium Electrical Noise in Semiconductors, H. C. Montgomery, pages 950-975. D Darlington, Sidney, Network Synthesis Using Tchebycheff Polynomial Series, pages 613- 665. DeLange, 0. E., Propagation Studies at Microwave Frequencies by Means of Very Short Pulses, pages 91-103. Design Factors Influencing Reliability of Relays, Important, /. R. Fry, pages 976-998. Dialing Habits of Telephone Customers, Charles Clos and R. I. Wilkinson, pages 32-67. Dialing, Nationwide Automatic Switching for Nationwide Telephone Service, .4. B. Clark and H. S. Os- borne, pages 823-831. Nationwide Numbering Plan, IF. H. Xiinn, pages 851-859. Dielectric Mismatch Mathematical Theory of Laminated Transmission Lines, S. P. Morgan, Jr., pages 883-949; 1121-1206. Direct Dial Control Systems in Switching Common Control Telephone Switching Systems, Oscar Myers, pages 1086-1120. Dynamics Relay Armature Rebound x\nalysis, E. E. Sumner, pages 172-200. E Edwards, P. G. and Montfort, L. R., The Type-0 Carrier System, pages 688-723. Efficient Coding, B. M. Oliver, pages 724-750. Elasticity Interaction of Polymers and Mechanical Waves. W . O. Baker and J. H. Heiss, pages 306-356. Elasticity Mechanical Properties of Polymers at Ultrasonic Frequencies, W. P. Mason and H. J. McSkimin, pages 122-171. Electric Circuit Theory Network Representation of Transcendental Impedance Functions, .If. A". Zinn, pages 378-404. Electric Networks Introduction to Formal Realizability Theory, Brockway McMillan, pages 217-279; 541-600. Network Synthesis Using Tchebvcheff Polynomial Series, Sidnev Darlington, pages 613-665. Electrical Noise in Semiconductors, H. C. Montgo^nery, pages 950-975. Electromagnetism Mathematical Theory of Laminated Transmission Lines, S. P. Morgan, Jr., pages 883-949; 1121-1206. Electronically Controlled Automatic Switching System, An Experimental, IF. .4. Mal- Ihaner and H. E. Vaughan, pages 443-468. Experiments with Linear Prediction in Television, C. W. Harrison, pages 764-783. F Faraday Effect The Ferromagnetic Faraday Effect at Microwave Frequencies and Its Applications — The Microwave Gyrator, C. L. Hogan, pages 1-31. Ferromagnetic Faraday Effect at Microwave Frequencies and Its Applications — The Microwave Gyrator, C. L. Hogan, pages 1-31. Formal Realizability Theory, Introduction to, Brockway McMillan, pages 217-279; 541- 600. Fr\<, J . R., Important Design Factors Influencing Reliability of Relays, pages 976-998. Fundamental Plans for Toll Telephone Plant, /. /. Pilliod, pages 832-850. G General Purpose Relay for Telephone Switching Systems. A New, A. C. Keller, pages 1023-1067. INDEX V Gilbert, E. N.. A Comparison of Signalling Alphabets, pages 504-522. Graphs A Comparison of Signalling Alphabets, E. N. Gilbert, pages 504 522. G3Tator The Ferromagnetic Faraday Effect at Microwave Frequencies antl Its Applications — The Microwave Gyrator, C. L. Hogan, pages 1-31. H Harrison, C. IT., Experiments with I>incar Prediction in Television, pages 764-783. Hayward, W. S., Jr., The Reliability of Telephone Traffic Load Measurements by Switch Counts, pages 357-377. Ileiss, J. n. and Baker, W. 0., Interaction of Polymers and Mechanical Waves, pages 306-356. Hogan, C. L., The Ferromagnetic Faraday Effect at Microwave Frequencies and Its Applications — The Microwave Gyrator, pages 1-31. Hopkins, I. L., Principal Strains in Cable Sheaths and Other Buckled Surfaces, pages 523-529. Hysko, J. L., Rea, W. T. and Roberts, L. C, A Carrier Telegrai)h System for Short-Ilaul Applications, pages 666-687. I Impedance Bridges for the Megacycle Range, H. T. Willielm, pages 999-1012. Impedance Functions Network Representation of Transcendental Impedance Functions, .1/. A'. Zinn, pages 378-404. Impedance Aleasurements Mechanical Properties of Polymers at Ultrasonic Frequencies, IF. P. Mason and H . J . }fcSkimin, pages 122-171. Interaction of Polymers and Mechanical Waves, IF. 0. Baker and J. H. Heiss, pages 306-356. Ionization Properties of Ionic Bombarded Silicon, R. S. Old, pages 104-121. Iron Oxide A New Recording Medium for Transcribed Message Services, J. Z. Menard, pages 530-540. J Jakes, W. C, Jr. and Crau'ford, A. B., Selective Fading of Microwaves, pages 68-90. K Keller, A. C, A New General Purpose Relay for Telephone Switching Systems, pages 1023-1067. Kingsbury, E. F. and Old, R. S., Photoelectric Properties of lonically Bombarded Silicon, pages 802-815. Kretzmer, E. R., Statistics of Television Signals, pages 751-763. L Laminated Transmission Lines, Mathematical Theory of, S. P. Morgan, Jr., pages 883- 949. Lee de Forest and William Shockley Discuss Electronics, page 612. Level Distribution Recorder Comparison of Mobile Radio Transmission at 150, 450, 900, 3700 Mc, IF. R. Young, Jr., pages 1068-1085. Linear Prediction Experiments with Linear Prediction in Television, C. W. Harrison, pages 764-783. M Magnetic Materials Important Design Factors Influencing Reliability of Relays, J. R. Fry, pages 976-998. Magnetic Recording A New Recording Medium for Transcribed Message Services, J. Z. ^fenard, pages 530-540. Vi THE BELL SYSTEM TECHNICAL JOURNAL, 1952 Malthaner, W . A. and Vanghan, H. E., An Experimental Electronically Controlled Auto- malic Switching System, pages 443-468. Manufacturing Processes Important Design Factors Influencing Reliability of Relays, /. R. Fry, pages 976-998. Mason, W . P. and McSkimin, H. J., Mechanical Properties of Polymers at Ultrasonic Frequencies, pages 122-171. Mason, W . P. and White, S. D., New Techniques for Measuring Forces and Wear in Tele- phone Switching Apparatus, pages 469-503. McMillan, Brockivav, Introduction to Formal Realizability Theory, pages 217-279; 541- 600. McSkimin, H. J . and Mason, W . P., Mechanical Properties of Polymers at Ultrasonic Frequencies, pages 122-171. Measuring Forces and Wear in Telephone Switching Apparatus, New Techniques for, IF. P. Mason and S. D. While, pages 469-503. Mechanical Properties of Polymers at Ultrasonic Frequencies, W . P. Mason and II. J. McSkimin, pages 122-171. Menard, J. Z., A New Recording Medium for Transcribetl Message Services, pages 530- 540. Message Services, Transcribed A New Recording Medium for Transcribed Message Services, J. Z. Menard, pages 530-540. Microwaves Propagation Studies at Microwave Freciuencies liy Means of Very Short Pulses, L. E. DeLange, pages 91-103. Selective Fading of Microwaves, ,1. B. Crawford and W. C. Jakes, Jr., pages 68-90. Miniaturization Present Status of Transistor Development, /. .1. Morion, pages 411-442. Mittag-Leffler's Theorem Network Representation of Transcendental Impedance Functions, M . K. Zinn, pages 378-404. Mobile Radio Transmission at 150, 450, 900, and 3700 Mc, W. R. Young. Jr., pages 1068-1085. Monlforl, L. R. and Edwards, P. G., The Type-0 Carrier System, pages 688-723. Montgomery, H. C, Electrical Noise in Semiconductors, pages 950-975. Morgan, S. P., Jr., Mathematical Theory of Laminated Transmission Lines, pages 883- 949; 1121-1206. Morton, J. A., Present Status of Transistor Development, pages 411-442. Myers, Oscar, Common Control Telephone Switching Systems, pages 1086-1120. N Nationwide Numbering Plan, W. H. Nunn, pages 851-859. Network Rejiresentation of Transcendental Impedance Functions, ,1/. K. Zinn, pages 378-404. Network Synthesis Using Tchebycheff Polynomial Series, Sidney Darlington, pages 613- 665. Noise Theory Electrical Noise in Semiconductors, H. C. Montgomery, pages 950-975. Numbering Plan, Telephone Nationwide Numbering Plan, W. H. Nunn, pages 851-859. Nunn, IF. //., Nationwide Numbering Plan, pages 851-859. O Ohl, R. S. and Kingsbury, E. F., Photoelectric Properties of lonically Bombarded Silicon, pages 802-815. Ohl, R. S., Properties of Ionic Bombarded Silicon, pages 104-121. Oliver, B. M ., Efficient Coding, pages 724-750. Osborne, H. S. and Clark, A. B., x\utomatic Switching for Nationwide Telephone Service, pages 823-831. P Photoelectric Properties of lonically Bombarded Silicon, E. F, Kingsbury and R. S. Ohl, pages 802-815. iNDKX vn PilUod, J. ./., Fundanu-ntal Plans for Toll Telc-ijlionc Plant., images 832-850. Plastics Interaction of Polymers and Mechanical Waves. IP. O. Baker aiid J. If. Heiss, pages 306-356. Polymers Interaction of Polvmers and >[echanical Waves, li'. 0. Baker and J . II. Hcits, pages 306-356. Mechanical Properties of Polymers at I'ltrasonic Fref|uencies, II'. /'. Mason and II. J. McSkimin, pages 122-171. Present Status of Transistor Development, J. A. Morton, pages 411-442. Propagation Studies at Microwave Frec|uencies by Means of Very Short Pulses, 0. E. DeLange, pages 91-103. Properties of Ionic Bombarded Silicon, K. S. Old. pages 104-121. Pulse Measurements Pro])agation Studies at Microwave Frequencies by Means of Very Short Pulses, O. E. DeLangc, pages 91-103. Radio. Fading Selective Fading of Microwaves, .1. B. Crawford and W. C. Jakes, Jr., pages 68-90. Radio Transmission Propagation Studies at Microwave Frequencies by Means of Very Short Pulses, DeLange, pages 91-103. Selective Fading of ^Microwaves, .1. B. Cranford and IF. C. Jakes, Jr., pages 68-90. Radiotelephone Service, Mobile Comparison of Mobile Radio Transmission at 150, 450, 900, and 3700 Mc, W . R. Young, Jr., pages 1068-1085. Rea, W. T., Hysko, J. L. and Roberts, L. C, \ Carrier Telegraph System for Short-Haul AppHcations, pages 666-687. Realizability Theory Introduction to Formal Realizability Theory, Brockway McMillan, pages 217-279. 541-600. Rebound of Relay .\rmature Relay Armature Rebound Analysis, E. E. Sumner, pages 172-200. Recording Medium for Transcribed ^Message Services. A New, /. Z. Menard, pages 530- 540. Relaxation Mechanical Properties of Polymers at Ultrasonic Frequencies, IF. F. Mason and II. J. McSkimin, pages 122-171. Relay Armature Rebound Analysis, E. E. Sumner, pages 172-200. Relays An Application of Boolean Algebra to Switching Circuit Design. R. E. Slaelder, pages 280-305. Important Design Factors Influencing Reliability of Relays, /. R. Fry, pages 976-998. Relays, Electromagnetic — Types AF, AG and AJ .V Xew General Purpose Relays for Telephone Switching Systems, .1 . C. Keller, pages 1023-1067. Reliability of Telephone Traffic Load Measurements by Switch Counts, IF. S. Flayward, Jr., pages 357-377. Roberts, L. C, Hysko, J. L. and Rea, W . F ., .\ Carrier Telegraph System for Short-Haul Applications, pages 666-687. Sampling The Reliability of Telephone TrafTic Load Measurements by Switch Counts, II'. S. Haynard, Jr., pages 357-377. Schelkunojj, S. A., Generalized Telegraphist's Equations for Wave-guides, pages 784-801. Selective Fading of Microwaves, .1. B. Crawford and W . C. Jakes, Jr., pages 68-90. Semiconductors Electrical Xoise in Semiconductors, //. C. Montgomery, pages 950-975. Shannon-Fano Code Efficient Coding, B. M . Oliver, pages 724-750. viii THE BELL SHSTEM TECHNICAL JOURNAL, 1952 Shipley, F. F., Automatic Toll Switching Systems, pages 860-882. Signals, Dialing An Experimental Electronically Controlled Automatic Switching System, W. A. Mal- thaner and H. E. Vaughan, pages 443-468. Silicon, Electrical Properties Properties of Ionic Bombarded Silicon, R. S. Old, pages 104-121. Silicon, Photoelectric Properties Photoelectric Properties of lonically Bombarded Silicon, E. F. Kingsbury and R. S. Ohl, pages 802-815. Solids Mechanical Properties of Polymers at Ultrasonic Frequencies, W. P. Mason and H. J. McSkimin, pages 122-171. Solutions Interaction of Polymers and Mechanical Waves, W. 0. Baker and J. H. Heiss, pages 306-356. Mechanical Properties of Polymers at Ultrasonic Frequencies, W . P. Mason and H. J. McSkimin, pages 122-171. Staehler, R. E., An Application of Boolean Algebra to Switching Circuit Design, pages 280-305. Statistics of Television Signals, E. R. Kretzmer, pages 751-763. Strains in Cable Sheaths and Other Buckled Surfaces, Principal, /. L. Hopkins, pages 523-529. Sumner, E. E., Relay Armature Rebound Analysis, pages 172-200. Switch Counts, Traffic The Reliability of Telephone Traffic Load Measurements by Switch Counts, W . S. Hayivard, Jr., pages 357-377. Switches, Crossbar Automatic Toll Switching Systems, F. F. Shipley, pages 860-882. Switching A New General Purpose Relay for Telephone Switching Systems, A . C. Keller, pages 1023-1067. An Experimental Electronically Controlled Automatic Switching System, W. A. Mal- thaner and H. E. Vaughan, pages 443-468. Automatic Switching for Nationwide Telephone Service, A. B. Clark and H. S. Os- borne, pages 823-831. Automatic Toll Switching Systems, F. F. Shipley, pages 860-882. Common Control Telephone Switching Systems, Oscar Myers, pages 1086-1120. Fundamental Plans for Toll Telephone Plant, /. /. Pilliod, pages 832-850. Nationwide Numbering Plan, W . H. Nunn, pages 851-859. Transistors in Switching Circuits, A. E. Anderson, pages 1207-1249. Switching Circuit Design An Application of Boolean Algebra to Switching Circuit Design, R. E. Staehler, pages 280-305. Switching Equipment New Techniques for Measuring Forces and Wear in Telephone Switching Apparatus, W. P. Mason and S. D. While, pages 469-503. T Tchebycheff Polynomial Series Network Synthesis Using Tchebycheff Polynomial Series, Sidney Darlington, pages 613-665. Telegraph Systems, Carrier — 40C1 A Carrier Telegraph System for Short-Haul Applications, /. L. Hysko, W. T. Rea and L. C. Roberts, pages 666-687. Telegraphist's Equations for Waveguides, Generalized, S. A. Schelkunojf, pages 784-801. Telephone Apparatus New Techniques for Measuring Forces and Wear in Telephone Switching Apparatus, W. P. Mason and S. D. While, pages 469-503. Telephone Calls Automatic Switching for Nationwide Telephone Service, A. B. Clark and H. S. Os- borne, pages 823-831. INDEX IX Dialing Habits of Telephone Customers, Charles Clos arid R. I . Wilkinson, ])ages 32-67. Telephone Circuits An Experimental Electronically Controlled Automatic Switching System, II'. .1. Mal- thaner and H. E. Vaughan, pages 443-468. Telephone Codes Nationwide Numbering Plan, W . H. Nunn, pages 851-859. Telephone Service. Testing Dialing Hal)its of Telephone Customers, Charles Clos and R. I. Wilkinson, pages 32-67. Telephone Systems, Carrier — Type O The Type-0 Carrier System, P. G. Edivards and L. R. Montfort, pages 688-723. Telephone Systems, Dial Common Control Telephone Switching Systems, Oscar Myers, pages 1086-1120. Telephone Systems, Toll Fundamental Plans for Toll Telephone Plant, J. J. Pilliod, pages 832-850. Telephone Traffic The Reliability of Telephone Traffic Load Measurements by Switch Counts, II . S. Hayicard, Jr., pages 357-377. Telephone Transmission Fundamental Plans for Toll Telephone Plant, ./. /. Pilliod, pages 832-850. Teletypewriters A Carrier Telegraph System for Short-Haul Api)lications, /. L. Hysko, W. T. Rea and L. C. Roberts, pages 666-687. Television Signals Experiments with Linear Prediction in Television, C. IF. Harrison, pages 764-783. Statistics of Television Signals, E. R. Kretzmer, pages 751-763. Temperature Measurements Properties of Ionic Bombarded Silicon, R. S. Old, pages 104-121. Thirtieth Anniversary (of The Bell System Technical Journal), page 611. Toll Traffic Automatic Toll Switching Systems, F. F. Shipley, pages 860-882. Fundamental Plans for Toll Telephone Plant, /. /. Pilliod, pages 832-850. Transistors Electrical Noise in Semiconductors, H. C. Montgomery, pages 950-975. Transistors, Types M1689, M1698, M1729, M1734'^, M1752 Present Status of Transistor Development, /. ^4. Morion, pages 411-442. Transistors in Switching Circuits, A. E. Anderson, pages 1207-1249. Translators Automatic Toll Switching Systems, F. F. Shipley, pages 860-882. Transmission Lines Network Representation of Transcendental Impedance Functions, M . K. Zinn, pages 378-404. Transmission Lines, Clogston 1 and 2 Mathematical Theory of Laminated Transmission Lines, S. P. Morgan, Jr., pages 883-949; 1121-1206. Transmission Measurements Comparison of Mobile Radio Transmission at 150, 450, 900, and 3700 Mc, W. R. Young, Jr., pages 1068-1085. Trunking Dialing Habits of Telephone Customers, Charles Clos and R. J. Wilkinson, pages 32-67. Type-0 Carrier System, P. G. Edwards and L. R. .Montfort, pages 688-723. Vaughan, H. E. and Malthaner, W. A., An Experimental Electronically Controlled Auto" matic Switching System, pages 443-468. Viscosity Interaction of Polymers and Mechanical Waves, W . O. Baker and J . H. Ileiss, pages 306-356. Mechanical Properties of Polymers at Ultrasonic Frequencies, W . P. .Mason and H. J . McSkimin, pages 122-171. X THE BELL SYSTEM TECHNICAL JOURNAL, 1952 w Waveguides Generalized Telegraphist's Equations for Waveguides, 5. A. Schelkunof, pages 784- 801. Wear Studies of Telephone Apparatus New Techniques for Measuring Forces and Wear in Telephone Switching Apparatus, W. P. Mason and S. D. White, pages 469-503. White, S. D. and Mason, W. P., New Techniques for Measuring Forces and Wear in Tele- phone Switching Apparatus, pages 469-503. Wilhelm, H. T., Impedance Bridges for the Megacycle Range, pages 999-1012. Wilkinson, R. I. and Clos, Charles, Dialing Habits of Telephone Customers, pages 32-67. Wiring and Wrapping Tools A New General Purpose Relay for Telephone Switching Systems, A . C. Keller, pages 1023-1067. Young, W. R., Jr., Comparison of Mobile Radio Transmission at 150, 450, 900, and 3700 Mc, pages 1068-1085. Zinn, M. K., Network Representation of Transcendental Impedance Functions, pages 378-404. Printed in U.S.A. A'