Skip to main content

Full text of "The Bell System technical journal"

See other formats


? 



; 



■I 



1 > 

11 



■ 




reference 
collection 
book 



KC 



Kansas city 
public library 
kansas city, 
missouri 



iP^ 




^U' 



Ns^ 



Sii/ 



From the collection of the 



n 



R 



m 



o Jrre linger 

V * library 

p 



San Francisco, California 
2008 



f • « fl 



• • • • • 
« • ■ • 



HE BEL Li:iii8L^^S:'^T-,,E\M 

p.', j'i'— ' , 





meat journal 

l^^r/ A IN 



^OTED TO THE SCIENTIFIC ^^r>^ AND ENGINEERING 
»ECTS OF ELECTRICAL COMMUNICATION 



U M E XXXV JANUARY 1956 tf k k--- • ' t. N U M B E R-lv 

DiflPused Emitter and Base Silicon Transistors J ^' ^ '^ ^ ^^^° 

M. TANENBAUM AND D. E. THOMAS 1 

A High-Frequency Diffused Base Germanium Transistor c. a. lee 23 

Waveguide Investigations with Millimicrosecond Pulses 

a. c. beck 35 

Experiments on the Regeneration of Binary Microwave Pulses 

o. B. delange 67 

Crossbar Tandem as a Long Distance Switching System 

a. O. ADAM 91 

Growing Waves Due to Transverse Velocities 

J. R. pierce and l. r. walker 109 

Coupled Helices j. s. cook, r. kompfner and c. f. quatb 127 

Statistical Techniques for Reducing the Experiment Time in Re- 
liability Studies MILTON sobel 179 

A Class of Binary Signaling Alphabets david slepian 203 



Bell System Technical Papers Not Published in This Journal 235 

Recent Bell System Monographs 242 

Contributors to This Issue 244 



COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY 






; , * -^ -^ f - -.r » ' J " -' • 



THE BELL SYSTEM TECHNICAL JOURNAL 



ADVISORY BOARD 

F. E. K A P P E L, President, Western Electric Company 

M. J. KELLY, President, Bell Telephone Laboratories 

E. J. McNEELY, Executive Vice President, American 
Telephone and Telegraph Company 

EDITORIAL COMMITTEE 

B. MCMILLAN, Chairman H. R. HUNTLEY 

A. J. BUSCH F. R. LACK 

A. C. DICKIESON J. R. PIERCE 

R. L. DIETZOLD H. V. SCHMIDT 

K. E. GOULD C. E. SCHOOLEY 

E. L GREEN G. N. THAYER 



EDITORIAL STAFF 

J. D. TEBO, Editor 

M. E. s T R I E B Y, Managing Editor 

R. L. SHEPHERD, Production Editor 



THE" BELL SYSTEM TECHNICAL JOURNAL is pubUshed six times 
a year by the American Telephone and Telegraph Company, 195 Broadway, 
New York 7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Secretary; 
John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. 
Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 
cents per copy. Printed in U. S. A. 



THE BELL SYSTEM 

TECHNICAL JOURNAL 

VOLUME XXXV JANUARY 1956 number 1 

Copyright 1956, American Telephone and Telegraph Company 

Diffused Emitter and Base Silicon 

Transistors* 

By M. TANENBAUM and D. E. THOMAS 

(Manuscript received October 21, 1955) 

Silicon n-p-n transistors have been made in which the base and emitter 
regions were produced by diffusing impurities from the vapor phase. Tran- 
sistors with base layers 3.8 X 10~ -cm thick have been made. The diffusion 
techniques and the processes for making electrical contact to the structures 
are described. 

The electrical characteristics of a transistor with a maximum alpha of 
0.97 and an alpha-cutoff of 120 mc/sec are presented. The manner in which 
some of the electrical parameters are determined by the distribution of the 
doping impurities is discussed. Design data for the diffused emitter, dif- 
fused base structure is calcidated and compared with the rneasured char- 
acteristics. 

INTRODUCTION 

The necessity of thin base layers for high-frequency operation of tran- 
sistors has long been apparent. One of the most appealing techniques for 
controlling the distribution of impurities in a semiconductor is the dif- 
fusion of the impurity into the solid semiconductor. The diffusion co- 
efficients of Group III acceptors and Group V donors into germanium 
and silicon are sufficiently low at judiciously selected temperatures so 

* A portion of the material of this paper was presented at the Semiconductor 
Device Conference of the Institute of Radio Engineers, Philadelphia, Pa., June, 
1955. 



2 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

that it is possible to envision transistors with base layer thicknesses of a 
micron and frequency response of several thousand megacycles per 
second. 

A major deterent to the application of diffusion to silicon transistor 
fabrication in the past was the drastic decrease in lifetime which generally 
occurs when silicon is heated to the high temperatures required for dif- 
fusion. There was also insufficient knowledge of the diffusion parameters 
to permit the preparation of structures with controlled layer thicknesses 
and desired dopings. Recently the investigations of C. S. Fuller and co- 
workers have produced detailed information concerning the diffusion of 
Group III and Group V elements in silicon. This information has made 
possible the controlled fabrication of transistors with base layers suffi- 
ciently thin that high alphas are obtained even though the lifetime has 
been reduced to a fraction of a microsecond. In a cooperative program 
with Fuller, diffusion structures were produced which have permitted 
the fabrication of transistors whose electrical behavior closely approxi- 
mates the behavior anticipated from the design. This paper describes 
these techniques which have resulted in high alpha silicon transistors 
with alpha-cutoff of over 100 mc/sec. 

1.0 FABRICATION OF THE TRANSISTORS 

Fuller's work has shown that in silicon the diffusion coefficient of a 
Group III acceptor is usually 10 to 100 times larger than that of the 
Group V donor in the same row in the periodic table at the same tem- 
peratures. These experiments were performed in evacuated silica tubes 
using the Group III and Group V elements as the source of diffusant. 
Under these conditions a particular steady state surface concentration 
of the diffusant is produced and the depth of diffusion is sensitive to 
this concentration as well as to the diffusion coefficient. The experiments 
show that the effective steady state surface concentration of the donor 
impurities produced under these conditions is ten to one hundred times 
greater than that of the acceptor impurities. Thus, by the simultaneous 
diffusion of selected donor and acceptor impurities into n-type silicon 
an n-p-n structure will result. The first n-la,yer forms because the surface 
concentration of the donor is greater than that of the acceptor. The 
p-laycr is protluced because the acceptor diffuses faster than the donor 
and gets ahead of it. The final n-region is simply the original background 
doping of the n-type silicon sample. It has been possible to produce n-p-n 
structures by the simultaneous diffusion of several combinations of 
donors and acceptors. Often, however, the diffusion coefficients and 
surface concentrations of the donors and acceptors are such that opti- 

1 C. S. Fuller, private communication. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 3 

mum layer thicknesses (see Sections 3 and 4) are not produced by simul- 
taneous diffusion. In this case, one of the impurities is started ahead of 
the other in a prior diffusion, and then the other impurity is diffused 
in a second operation. 

With the proper choice of diffusion temperatures and times it has been 
possible to make n-p-n structures with base layer thicknesses of 2 X 10~* 
cm. The uniformity of the layers in a given specimen is better than ten 
per cent of the layer thickness. Fig. 1 illustrates the uniformity of the 
layers. This figure is an enlarged photograph of a view perpendicular 
to the surface of the specimen. A bevel which makes an angle of five 
degrees with the original surface has been polished on the specimen. This 
angle magnifies the layer thickness by 11.5. The layer is defined by an 
etchant which preferentially stains p-type silicon^ and the width of the 
layer is measured with a calibrated microscope. 

After diffusion the entire surface of the silicon wafer is covered with 
the diffused n- and p-type layers, see Fig. 2(a). Electrical contact must 
now be made to the three regions of the device. The base contact can 
be made by polishing a bevel on the specimen to expose and magnify 
the base layer and then alloying a lead to this region by the same tech- 




f.^ *f^'- *; 




'>i 



i * /i 



n-TfPE DIFFUSED LAV^ER 
fo-t^^E*OiFFUSED LAYER 





i»# 



OF^GIt^L n-TYPE 
CRYSTAl. 



I 1 EQUIVALENT TO 2 X lO"'* CM 

LAYER THICKNESS 

Fig- 1 — Angle section of a double diffused silicon wafer. The p-type center 
ayer is approximately 2 X 10-< cm thick. 



4 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

niques employed in the fabrication of grown junction transistors. Fig. 
2(b). However, a much simpler technique has been evolved. If the sur- 
face concentration of the donor diffusant is maintained below a certain 
critical value, it is possible to alloy an aluminum wire directly through 
the diffused n-type layer and thus make effective contact to the base 
layer, Fig. 2(c). Since the resistivity of the original silicon wafer is one 
to five ohm-cm, the aluminum will be rectifying to this region. It has 
been experimentally shown that if the surface concentration of the 
donor diffusant is less than the critical value mentioned above, the 
aluminum will also be rectifying to the diffused n-type region and the 
contact becomes merely an extension of the base layer. The n-layers 
produced by diffusing from elemental antimony are below the critical 
concentration and the direct aluminum alloying technique is feasible. 



-n + TYPE DIFFUSED LAYER 




-p-TYPE DIFFUSED LAYER 



n + 



n+ 




-ALUMINUM WIRE 

p + ALUMINUM DOPED 
REGROWTH LAYER 



n-TYPE 



(b) 




,^- ALUMINUM WIRE 

P + ALUMINUM DOPED 
, REGROWTH LAYER 



^M'nY ^-i-r 



n-TYPE 



(c) 



Fig. 2 — ■ Schematic illustralioii of (a) double diffused n-p-n wafer, (b) angle 
section method of making base contact, and (c) direct alloying method of making 
base contact. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 



AU-Sb PLATED 
POINT 



VAPORIZED Al 

LINE 
0.005 CM WIDE 




t MM 




Fig. 3 — Mounted double diffused transistor. 

Contact to the emitter layer is achieved by alloying a film of gold 
containing a small amount of antimony. Since this alloy will produce 
an n-type regrowth layer, it is only necessary to insure that the gold- 
antimony film does not alloy through the p-type base layer, thus shorting 
the emitter to the collector. This is controlled by limiting the amount of 
gold-antimony alloy which is available by using a thin evaporated film 
or by electroplating a thin film of gold-antimony alloy on an inert metal 
point and alloying this structure to the emitter layer. 

Ohmic, contact to the collector is produced by alloying the silicon 
wafer to an inert metal tab plated with a gold-antimony alloy. 



6 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

The transistors whose characteristics are reported in this paper were 
prepared from 3 ohm-cm n-type siHcon using antimony and ahmiinum 
as the diffusants. The base contact was produced by evaporating alumi- 
num through a mask so that a hne approximately 0.005 X 0.015 cm in 

o 

lateral dimensions and 100,000 A thick was formed on the surface. This 
aluminum line was alloyed through the emitter layer in a subsequent 
operation. The wafer was then alloyed onto the plated kovar tab. A 
small area approximately 0.015 cm in diameter was masked around the 
line and the wafer was etched to remove the unwanted layers. The unit 
was then mounted in a header. Electrical contact to the collector was 
made by soldering to the kovar tab. Contact to the base was made with 
a tungsten point pressure contact to the alloyed aluminum. Contact 
to the emitter was made by bringing a gold-antimony plated tungsten 
point into pressure contact with the emitter layer. The gold-antimony 
plate was then alloyed by passing a controlled electrical pulse between 
the plated point and the transistor collector lead. Fig. 3 is a photograph 
of a mounted unit. 

2.0 ELECTRICAL CHARACTERISTICS 

The frequency cutoffs of experimental double diffused silicon tran- 
sistors fabricated as described above are an order of magnitude higher 
than the known cutoff frequencies of earlier silicon transistors. This is 
shown in Fig. 4 which gives the measured common base and common 
emitter current gains for one of these units as a function of frequency. 
The common base short-circuit current gain is seen to have a cutoff fre- 
quency of about 120 mc/sec. The common emitter short-circuit current 
gain is shown on the same figure. The low-freciuency current gain is 
better than thirty decibels and the cutoff frequency which is indicated 
by the freciuency at which the gain is 3 db below its low-frequency 
value is 3 mc/sec. This is an exceptionally large common emitter band- 
width for a thirty db common emitter current gain and is of the same 
order of magnitude as that obtained with the highest frequency ger- 
manium transistors (e.q., p-n-i-p or tetrode) which had been made 
prior to the diffused base germanium transistor. 



^ Tlio iiicroasp in (•oiiiinon haso current gain ahovc unity (indicated by current 
gain in decibels being positive) in the vicinity of 50 mc/sec is caused by a reactance 
gain error in the common base measurement. This error is caused by a combination 
of the emitter to ground parasitic capacitance and the i)ositive reactance com- 
ponent of the transistor input impedance resulting from phase shift in the ali)ha 
current gain. 

' C. A. Lee, A High-Frequency Diffused Base Germanium Transistor, see 
page 23. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 



z 
< 

o 

I- 

z 

LJ 

a. 
cr 

D 
O 



40 



30 



20 



(0 



-\0 



-20 



-30 

















Ie = 


3 MA 


Vc 


= 10 VOLTS 






COMMON^ 
EMITTER 


N 




'OCCB — ^ ^^ 
OCq = 0.9716 

['=^"=106MC 

l-Ofg 

\ facb = i20MC 














\ 






COMMON 


BASE 
































\ 
























\ 


\ 





























0.1 0.2 0.5 1.0 2 6 10 20 50 100 200 

FREQUENCY IN MEGACYCLES PER SECOND 



500 1000 



Fig. 4 — ■ Short-circuit current gain of a double diffused silicon n-p-n transistor 
as a function of frequency in the common emitter and common base connections. 



Fig. 5 shows a high-freciueiicy lumped constant equivalent circuit 
for the double diffused silicon transistor whose current gain cutoff char- 
acteristic is shown in Fig. 4. External parasitic capacitances have been 
omitted from the circuit. The configuration is the conventional one for 
junction transistors with two exceptions. A series resistance rj has been 
added in the emitter circuit to account for contact resistance resulting 
from the fact that the present emitter point contacts are not perfectly 
ohmic. A second resistance r/ has been added in the collector circuit to 
account for the ohmic resistance of the n-type silicon between the col- 
lector terminal and the effective collector junction. This resistance exists 
in all junction transistors but in larger area low frequency junction 
transistors its effect on alpha-cutoff is sufficiently small so that it has 
been ignored in equivalent circuits of these devices. The collector RC 



Ce = TmmF 



Pq -]AU) 




Cc = 0.52//^F r ' _ ,50 co 



Tg = 150; 



a 



J^C( 



•Le 



'%=QOCO 



COMMON BASE CURRENT 
GAIN CUT-OFF FREQUENCY 



■ 120 MC 



Ic = 3 MA 
Vc = 10 VOLTS 



Fig. 5 ~ High-frequency lumped constant equivalent circuit for a double 
diffused silicon n-p-n transistor. 



8 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



cutoff caused by the collector capacitance and the combined collector 
body resistance and base resistance is an order of magnitude higher 
than the measured alpha cutoff frequency and therefore is not too serious 
in impairing the very high-frecjuency performance of the transistor. 
This is due to the low capacitance of the collector junction which is 
seen to be approximately 0.5 mmf at 10 volts collector voltage. The 
base resistance of this transistor is less than 100 ohms which is quite low 
and compares very favorably with the best low frequency transistors 
reported previously. 

The low-frequency characteristics of the double diffused silicon tran- 
sistor are very similar to those of other junction transistors. This is il- 
lustrated in Fig. 6 where the static collector characteristics of one of 
these transistors are given. At zero emitter current the collector current 
is too small to be seen on the scale of this figure. The collector current 



45 



40 



35 



30 



25 



20 



15 



10 



-5 























































































le=0 


2 


4 6 


8 


10 


12 












] 


J 


14/ 






^ 


J^ 


^ 


y^ 


^ 

















2 4 6 8 10 12 14 

CURRENT, If, IN MILUAMPERES 



Fig. 6 — Collector characteristics of a double diffused silicon n-p-n tran- 
sistor. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 



9 



0.98 



0.94 



0.90 



0.86 



a 



0.82 



0.78 



0.74 



0.70 





T=150°C, 




^ 


^ 






^ 


^ 


^ 




7 


<^ 


y 
^ 


^ 


^ 










\ 


/ 


9/ 


y 
















24, 5M 
65-W 


/> 


7 

/24.5 


















t35^y\ 


7 




















15ol 


/ 






















/ 


1 


1 


1 


_L. 




1 


1 


1 


,1 





0.1 0.2 0.4 0.6 1 2 4 6 8 10 20 

CURRENT, Ig, IN MILLIAMPERES 

Fig. 7 — Alpha as a function of emitter current and temperature for a double 
diffused silicon n-p-n transistor. 



under this condition does not truly saturate but collector junction re- 
sistance is very high. Collector junction resistances of 50 megohms at 
reverse biases of 50 volts are common. 

The continuous power dissipation permissible with these units is also 
shown in Fig. 6. The figure shows dissipation of 200 milliwatts and the 
units have been operated at 400 milliwatts without damage. As illus- 
trated in Fig. 3 no special provision has been made for power dissipation 
and it would appear from the performance obtained to date that powers 
of a few watts could be handled by these iniits with relatively minor 
provisions for heat dissipation. However, it can also be seen from Fig. 6 
that at low collector voltages alpha decreases rapidly as the emitter 
current is increased. The transistor is, therefore, non-linear in this 
range of emitter currents and collector voltages. In many applications, 
this non-linearity may limit the operating range of the device to values 
below those which would be permissible from the point of view of con- 
tinuous power dissipation. 

Fig. 7 gives the magnitude of alpha as a function of emitter current 
for a fixed collector voltage of 10 volts and a number of ambient tem- 
peratures. These curves are presented to illustrate the stability of the 
parameters of the double diffused silicon transistor at increased ambient 
temperatures. Over the range from 1 to 15 milliamperes emitter current 
and 25°C to 150°C ambient temperature, alpha is seen to change only 



10 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

by approximately 2 per cent. This amounts to only 150 parts per million 
change in alpha per degree centigrade change in ambient temperature. 
The decrease in alpha at low emitter currents shown in Fig. 7 has been 
observed in every double diffused silicon transistor which has been made 
to date. Although this effect is not completely understood at present it 
could be caused by recombination centers in the base layer that can 
be saturated at high injection levels. Such saturation would result in an 
increase in effective lifetime and a corresponding increase in alpha. The 
large increase in alpha with temperature at low emitter currents is con- 
sistent with this proposal. It has also been observed that shining a strong 
light on the transistor will produce an appreciable increase in alpha at 
low emitter currents but has little effect at high emitter currents. A 
strong light would also be expected to saturate recombination centers 
which are active at low emitter currents and this behavior is also con- 
sistent with the above proposal. 

3.0 DISCUSSION OF THE TRANSISTOR STRUCTURE 

Although the low frequency electrical characteristics of the double 
diffused silicon transistor which are presented in Section 2 are quite 
similar to those usually obtained in junction transistors, the structure 
of the double diffused transistor is sufficiently different from that of the 
grown junction or alloy transistor that a discussion of some design 
principles is warranted. This section is devoted to a general discussion 
of the factors which determine the electrical characteristics of the tran- 
sistors. In Section 4 the general ideas of Section 3 are applied in a more 
specialized fashion to the double diffused structure and a detailed cal- 
culation of electrical parameters is presented. 

One essential difference between the double diffused transistor and 
grown junction or alloy transistors arises from the manner in which the 
impurities are distributed in the three active regions. In the ideal case 
of a double-doped grown junction transistor or an alloy transistor the 
concentration of impurities in a given region is essentially uniform and 
the transition from one conductivity type to another at the emitter and 
collector junctions is abrupt giving rise to step junctions. On the other 
hand in the double diffused structure the distribution of impurities is 
more closely described by the error function complement and the emitter 
and collector junctions are graded. Tlu\se differences can have an appre- 
ciable influence on the electrical beha\'ior of the transistors. 

Fig. 8(a) shows the probable distribution of donor impurities, No , 
and acceptor impurities, A''^ , in a double diffused n-p-n. Fig. 8(b) is a 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 



11 




DONORS 

ACCEPTORS 



DISTANCE 

(a) 




DISTANCE *• 

(b) 

Fig. 8 — Diagrammatic representation of (a) donor and acceptor distributions 
and (b) uncompensated impuritj- distribution in a double diffused n-p-n tran- 
sistor. 



plot of Nd — Na which would result from the distribution in Fig. 8(a). 
Kromer has shown that a nonuniform distribution of impurities in a 
semiconductor will produce electric fields which can influence the flow 
of electrons and holes. For example, in the base region the fields between 
the emitter junction, Xe , and the minimum in the Nd — Na curve, x', 
will retard the flow of electrons toward the collector while the fields 
between this minimum and the collector jvmction, Xc , will accelerate the 
flow of electrons toward the collector. These base laj^er fields will affect 
the transit time of minority carriers across the base and thus contribute 

* H. Kromer, On Diffusion and Drift Transistor Theory I, II, III, Archiv. der 
Electr. Ubertragung, 8, pp. 223-228, pp. 363-369, pp. 499-504, 1954. 



12 THE BELL SYSTEM TECHNICAL JOUENAL, JANUARY 1956 

to the fre(iuency response of the transistor. In addition the base re- 
sistance will be dependent on the distribution of both diffusants. These 
three factors are discussed in detail below. 

Moll and Ross have determined that the minority current, /,„ , that 
will flow into the base region of a transistor if the base is doped in a non- 
uniform manner is given by 



f N(x) dx 



where rii is the carrier concentration in intrinsic material, q is the elec- 
tronic charge, V is the applied voltage, Dm is the diffusion coefficient of 
the minority carriers, and the integral represents the total number of 
uncompensated impurities in the base. The primary assumptions in this 
derivation are (1) planar junctions, (2) no recombination in the base 
region, and (3) a boundary condition at the collector junction that the 
minority carrier density at this point equals zero. It is also assumed that 
the minority carrier concentration in the base region just adjacent to the 
emitter junction is equal to the equilibrium minority carrier density at 
this point multiplied by the Boltzman factor exp (qV/kT). It is of special 
interest to note that Im depends only on the total number of uncom- 
pensated impurities in the base and not on the manner in which they 
are distributed. 

In the double diffused transistor, it has been convenient from the 
point of ease of fabrication to make the emitter layer approximately the 
same thickness as the base layer. It has been observed that heating sili- 
con to high temperatures degrades the lifetime of n- and p-type silicon 
in a similar manner. Both base and emitter layers have experienced the 
same heat treatment and to a first approximation it can be assumed that 
the lifetime in the two regions will be essentially the same. Thus as- 
sumptions (1) and (2) should also apply to current flow from base to 
emitter. If we assume that the surface recombination \'elocity at the 
free surface of the emitter is infinite, then this imposes a boundary 
condition at this side of the emitter which under conditions of forward 
bias on the emitter is equivalent to assumption (3). Thus an equation 
of the form of (3.1) should also give the minority current flow from base 
to emitter. Since the emitter efficiency, y, is given by 



^ J. Tj. Moll and I. M. Ross, The J)opendencc of Transistor Paramotors on tlie 
Distribution of Base Layer liesistivity, Proc. I.R.E. in press. 
8 G. Bemski, private comnmnication. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 13 

/m (emitter to base) 

-y = . . . 

/^(emitter to base) + /„j(base to emitter) 

proper substitution of (3.1) will give the emitter efficiency of the double 
diffused n-p-n transistor, 

1 



7 = 



J-'n 



Z).^''^-^"^ 



dx 



p .6 (3.2) 



\ (No - iVj dx 



In (3.2), Dp is the diffusion coefficient of holes in the emitter, /)„ is the 
diffusion coefficient of electrons in the base and the ratio of integrals is 
the ratio of total uncompensated doping in the base to that in the 
emitter. 

A calculation of transit time is more difficult. Kromer has studied 
the case of an aiding field which reduces transit time of minority carriers 
across the base region and thus increases frequency response. In the 
double diffused transistor the situation is more complex. Near the 
emitter side of the base region the field is retarding (Region R, see Fig. 8) 
and becomes aiding (Region A) only after the base region doping reaches 
a maximum. The case of retarding fields has been studied by Lee and 
by MoU.^ At present, the case for a base region containing both types of 
fields has not been solved. However, at the present state of knowledge 
some speculations about transit time can be made. 

The two factors of primary importance are the magnitude of the 
built-in fields and the distance over which they extend. In the double 
diffused transistor, the widths of regions R and A are determined by the 
surface concentrations and diffusion coefficients of the diffusants. It 
Can be shown by numerical computation that if region R constitutes no 
more than 30-40 per cent of the entire base layer width, then the overall 
effect of the built-in fields will be to aid the transport of minority car- 
riers and to produce a reduction in transit time. In addition the absolute 
magnitude of region R is important. If the point x' should occur within 
an effective Debye length from the emitter junction, i.e., if x' is located 
in the space charge region associated with the emitter junction, then the 
retarding fields can be neglected. 

The base resistance can also be calculated from surface concentrations 
and diffusion coefficients of the impurities. This is done by considering 
the base layer as a conducting sheet and determining the sheet con- 

' J. L. Moll, private communication. 



14 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

ductivity from the total number of uncompensated impurities per square 
centimeter of sheet and the approjiriate moliility weighted to account 
for impurity scattering. 

4.0 CALCULATION OF DESIGN PARAMETERS 

To calculate the parameters which determine emitter efficiency, transit 
time, and base resistance it is assumed that the distribution of uncom- 
pensated impurities is given by 

N(x) = Nicrfc f - N-2erJc^ + Nz (4.1) 

where A^i and A^2 are the surface concentrations of the emitter and base 
impurity diffusants respectively, Li and L^ are their respective diffusion 
lengths, and Nz is the original doping of the semiconductor into which 
the impurities are diffused. The impurity diffusion lengths are defined as 

Li = 2 V/M and L2 = 2 ^Ddo (4.2) 

where the D's are the respective diffusion coefficients and the f's are the 
diffusion times. 

Equation (4.1) can be reduced to 



r(^) = Ti erfc I - Ta erfc X^ + 1 (4.3) 



where 



For cases of interest here, r(^) will be zero at two points, a and 13, 
and will have one minimum at ^'. In the transistor structure the emitter 
junction occurs at ^ = ^v and the collector junction occurs at ^ = (3. 
Thus the base width is determined by 13 — a. The extent of aiding and 
retarding fields in the base is determined by ^'. The integral of (4.3) 
from to a, I\ , and from o to ^, I2 , are the integrals of interest in (3.2) 
and thus determine emitter efficiency. In addition I2 is the integral from 
which base resistance can be calculated. 

The calculations which follow apply only for values of ri/r2 and To 
greater than ten. Some of the simplifying assumptions which are made 
will not apply at lower values of these parameters where the distribution 
of both diffusants as well as the background doping affect the structure 
in all three regions of the device. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 



15 



4.1 Base Width 

From Fig. 8 and (4.3) it can be seen that for r2 ^ 10, a is essentially 
independent of r2 and is primarily a function of T1/T2 and X. Fig. 9 is a 
plot of a versus ri/r2 with X as the parameter. The particular plot is for 
r2 = 10 . Although as stated a is essentially independent of r2 , at lower 
values of r2, a may not exist for the larger values of X, i.e., the p-layer 
does not form. 

In the same manner, it can be seen that ^ is essentially independent of 
T]/T2 and is a function only of r2 and X. Fig. 10 is a plot of /3 versus F^ 
with X as a parameter. This plot is for Ti/Fo = 10 and at larger Fi/Fo , 
/3 may not exist at large X. 



10" 






\0' 



10 



r2=)o'' 








/// 


// 


/ 






^ 


::i 


ll 


r / 


/ 








m 


0/ / 


' 








> 




/os/ 


1 








i 


1 

/// 


'o.e/ 


/ 












f 0.7/ 


/ 








/// 




/ 












/ 




<.e 




I 


w. 


W 


/ 
/ 


/ 







1.0 



1.4 



1.8 



2.2 2.6 

a 



3.0 



3.4 



3.8 



Fig. 9 — Emitter layer thickness (in reduced units) as a function of the ratio 
of the surface concentrations of the diffusing impurities (ri/r2) and the ratio of 
their diffusion lengths (X). 



16 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



The base width 



W = ^ — a 



can be obtained from Figs. 9 and 10. a, 13 and iv can be converted to 
centimeters by nuiltiplying by the appropriate value of Li . 

4.2 Emitter Efficiency 

With the hmits a and /3 determined above, the integrals h and 1 2 can 
be calculated. Examination of the integrals shows that h is closely pro- 
portional to ri/r2 and also to r2 . On the other hand I2 is closely propor- 
tional to r2 and essentially independent of ri/r2 . Thus, the ratio of 
/2//1 which determines 7 depends primarily on ri/r2 . Fig. 11 is a plot 
of the constant /2//1 contours in the ri/T2 — X plane for lo/h ii^ the 
range from — 1.0 to —0.01. The graph is for r2 = 10 . Since from (3.2) 



7 = 



1 



1 _ ^h 

Dnh 



(4.4) 



for an n-p-n transistor, and assuming Dp/Dn = /^ for silicon, then 



to' 



(0- 



10' 



10 





1' 


1 












\= 


..J\ 














0.6- 
0.5- 


::ffl 


M 
















\u 


|6 In 
1 1° 










1 






\\\ 


( 


0.2 


0.1 












'/// 


















/// 










0.01/ 








ill 


7 


/ 






/ 








/// 


/ 


/ 






/ 







10 



20 50 



100 200 



500 1000 



Fig. 10 — (Collector junction dopth (in rodurod units) as a function of the sur- 
face concuMit.ration (in reduced units) of llie dilfusaiit wliicli determines the con- 
ductivity type of the l)ase layer (I'.') and liie ratio of tlie dilTusioii lengths (X) of 
the tAvo diffusing inii)urifies. 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 

10" 



17 



10 



H 

Ta 



10 



10 















r2 = io'* 


2 
















w 
















\v 


V 












2 
? 


1 


^ 












1 


\\ 


\ 












t 


^. 


\ 


\I2/I 


1 






2 


V 




,\ 


N-0 

VO.05 


02 






-i.o\ 


-0.3S^ 


32X^0 


'\ 









0.1 



0.2 



0.3 



0.4 



0.5 



0.6 



0.7 



Fig. 11 — ^Dependence of emitter efficiency upon diffusant surface concentra- 
tions and diffusion lengths. The lines of constant /2//1 are essentially lines of 
constant emitter efficiency. The ordinate is the ratio of surface concentrations of 
the two diffusants and the abscissa is the ratio of their diffusion lengths. 

/2//1 = — 1.0 corresponds to a 7 of 0.75 and /2//1 = —0.01 corresponds 
to a 7 of 0.997. 



4.. 3 Base Resistance 

It was indicated above that I2 depends principally on r2 and X. Fig. 12 
is a plot of the constant I2 contours in the r2 — X plane for I2 in the range 
from —10^ to —10. The graph is for Ti/To = 10. The base layer sheet 
conductivity, cjb , can be calculated from these data as 



Qb = —qtihTjiNz 



(4.5) 



where q, L\ and A^3 are as defined above and /I is a mobility properly 
weighted to account for impurity scattering in the non-uniformly doped 
base region. The units of gb are mhos per square. 



18 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



10- 


1 2= -10,00^ 


/ 


/ 




7/ 


1 






/ 


-5000/ 


r 


/ / 


// 


/ 






/ 


/ -1000/ 


// 


// 


/ / 1 


2 


1 


/ 




/-5oa 


/ / 


/ / 


/ 1 


^/^^ 


/ 


/ 


/ 


/I 


^/ 


/ / 


1 1 


10 


/ 


/ 


// 


v. 


/-ioy 


V 


11 




/ / 


/ 


/, 


// 


/-/ , 


(I 


5 


// 


/, 


-^ 


/J 


/ 


V/ 


/ 


2 


^ 


^ 


^ 


f^ 




u 


10 


102 


r 


/ / 


^ 


/ 
/ 


^ 






5 


— 1 





^/ 


V 


r 






10 




/ 


/ 











0.1 



0.2 



0.3 



0.4 



0.5 



0.6 



0.7 



Fig. 12 — Dependence of base layer sheet condiictivitj^ on diffusant surface 
concentrations and diffusion lengths. The lines of constant Ii are essentiallj' lines 
of constant base sheet conductivity. The ordinate is the surface concentration 
(in reduced units) of the diffusant which determines the conductivity type of the 
base layer and the abscissa is the ratio of the diffusion lengths of the two difi'using 
impurities. 

4.4 Transit Time 

With a knowledge of where the minimum value, ^', of (4.3) occurs, 
it is possible to calculate over what fraction of the base width the fields 
are retarding. The interesting quantity here is 

13 - a 

^ is a function of ri/r2 and X and varies only very slowly with ri/r2 . 
a is also a function of ri/r2 and X and varies only slowly with ri/r2 . 
The most rapidly changing part of bJi is l^ which depends primarily on 
r2 as noted above. Fig. 13 is a plot of the constant LR contours in the 
r2 — X plane for values of A/2 in the range 0.1 to 0.3. This graph is 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 



19 



lor data with ri/r2 = 10. As ri/r2 increases at constant r2 and X, AR 
decreases slightly. At ri/r2 = 10\ the average change in AR is a decrease 
of about 25 per cent for constant r2 and X when AR ^ 0.3. The error is 
larger for values of AR greater than 0.3. It was noted above that when 
AR becomes greater than 0.3, the retarding fields become dominant. 
Therefore, this region is of slight interest in the design of a high frequency 
transistor. 

4.5 A Sample Design 

By superimposing Figs. 11, 12 and 13 the ranges of r2 , ri/r2 and X 
which are consistent with desired values of y, gt and AR can be deter- 




0.7 



Fig. 1.3 — Dependence of the built-in field distribution on concentrations and 
diffusion lengths. The lines of constant aR indicate the fraction of the base layer 
thickness over which built-in fields are retarding. The ordinate is the surface 
concentration (in reduced units) of the diffusant which determines the conductiv- 
ity type of the base layer and the abscissa is the ratio of the diffusion lengths of 
the two diffusing impurities. 



20 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

mined by the area enclosed by the specified contour lines. It is also 
possible to compare the measured parameters of a specific device and 
observe how closely they agree with what is predicted from the estimated 
concentrations and diffusion coefficients. This is done below for the 
transistor described in Sections 1 and 2. 

The comparison is complicated by the fact that the exact values of the 
surface concentrations and diffusion coefficients are not known {Precisely 
enough at present to permit an accurate evaluation of the design theory. 
However, the following values of concentrations and diffusion coefficients 
are thought to be realistic for this transistor. 

iVi = 5 X 10^' /)i = 3 X 10"'' /i = 5.7 X lO' 

iV2 = 4 X 10'' Di = 2.5 X 10"" t^= 1.2 X lO' 

Nz = 10'' 

From these values it is seen that 

Ti/ra = 12.5; r, = 400; X = 0.6 

From Fig. 9, a = 1.9 and from Fig. 10, /3 = 3.6 and therefore w = 1.7. 
Measurement of the emitter and base layer dimensions showed that these 
layers were approximately the same thickness which was 3.8 X 10" cm. 
Thus the ifieasured ratio of emitter width to base width of unity is in 
good agreement with the ^'alue of 1.1 predicted from the assumed con- 
centrations and diffusion coefficients. 

From Fig. 11, lo/h ~ —0.01. If this value is substituted into (4.4), 
7 = 0.997. This compares with a measured maximum alpha of 0.972. 

From Fig. 12, lo = —15. Assuming an average hole mobility of 350 
cm' /volt. sec. and evaluating Li from the measured emitter thickness 
and the calculated a, (4.5) gives a value of gb = 1.7 X 10^ mhos per 
square. The geometry of the emitter and base contacts as shown in Fig. 
3 makes it difficult to calculate the effective base resistance from the 
sheet conductivity even at very small emitter currents. In addition at 
the very high inje{;tion levels at which these transistors are operated the 
calculation of effective base resistance becomes very difficult. However, 
from the geometr}^ it would be expected that the effective base re- 
sistance would l)c no greater than 0.1 of the sheet resistivity or 600 ohms. 
This is about seven times larger than the measured \'alue of 80 ohms 
reported in Section 2. 

From Fig. b3, A/^ is approximately 0.20. Thus there should be an over- 
all aiding elfect of the built-in fields. In addition the impurity gradient 
at the emitter junction is believed to be approximately lO'Vcm and the 



DIFFUSED EMITTER AND BASE SILICON TRANSISTORS 21 

space charge associated with this gradient will extend approximately 
2 X 10 ■' cm into the base region. The base thickness over which re- 
tarding fields extend is AR times the base width or 7.6 X 10~^ cm. Thus 
the first quarter of region R will be space charge and can be neglected. 
The frequency cutoff from pure diffusion transit is given by 

2A3D ,. , 

where W is the measured base layer thickness. Assuming D — 25 cmVsec 
for electrons in the base region, ,/'„ = (w mc/sec. Since the measured 
cutoff was 120 mc/sec, the predicted aiding effect of the built-in field 
is evidently present. 

These computations illustrate how the measured electrical parameters 
can be used to check the values of the surface concentrations and dif- 
fusion coefficients. Conversely knowledge of the concentrations and 
diffusion coefficients aid in the design of devices which will have pre- 
scribed electrical parameters. The agreement in the case of the transistor 
described above is not perfect and indicates errors in the proposed values 
of the concentrations and diffusion coefficients. However, it is sufficiently 
close to be encouraging and indicate the value of the calculations. 

The discussion of design has been limited to a very few of the important 
parameters. Junction capacitances, emitter and collector resistances are 
among the other important characteristics which have been omitted 
here. Presumably all of these quantities can be calculated if the detailed 
structure of the device is known and the structure should be susceptible 
to the type of analysis used above. Another fact, which has been ignored, 
is that these transistors were operated at high injection levels and a low 
level analysis of electrical parameters was used. All of these other factors 
must be considered for a detailed understanding of the device. The object 
of this last section has been to indicate one path which the more detailed 
analysis might take. 

5.0 CONCLUSIONS 

By means of multiple diffusion, it has been possible to produce silicon 
transistors with alpha-cutoff above 100 mc/sec. Refinements of the 
described technicjues offer the possibility of even higher frequency per- 
formance. These transistors show the other advantages expected from 
silicon such as low saturation currents and satisfactory operation at 
high temperatures. 

The structure of the double diffused transistor is susceptible to design 



22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

analysis in a fashion similar to that which has been applied to other junc- 
tion transistors. The non-uniform distribution of impurities produces 
significant electrical effects which can be controlled to enhance appre- 
cial)ly the high-frequency behavior of the devices. 

The extreme control inherent in the use of diffusion to distribute im- 
purities in a semiconductor structure suggests that this technique will 
become one of the most valuable in the fabrication of semiconductor 
devices. 

ACKNOWLEDGEMENT 

The authors are indebted to several people who contributed to the 
work described in this paper. In particular, the double diffused silicon 
from which the transistors were prepared was supplied by C. S. Fuller 
and J. A. Ditzenberger. The data on diffusion coefficients and concen- 
trations were also obtained by them. 

P. W. Foy and G. Kaminsky assisted in the fabrication and mounting 
of the transistors and J. M. Klein aided in the electrical characterization. 
The computations of the various solutions of the diffusion equation, (4.3), 
were performed by Francis Maier. In addition many valuable discussions 
with C. A. Lee, G. Weinreich, J. L. Moll, and G. C. Dacey helped formu- 
late many of the ideas presented herein. 



A High-Frequency Diffused Base 
Gernianiuni Transistor 

By CHARLES A. LEE 

(Manuscript received November 15, 1955) 

Techniques of impurity diffusion and alloying have been developed which 
make possible the construction of p-n-p junction transistors utilizing a 
diffused surface layer as a base region. An important Jeature is the high 
degree of dimensional control obtainable. Diffusion has the advantages of 
being able to produce uniform large area junctions which may be utilized in 
high power devices, and very thin surface layers which may be utilized in 
high-frequency devices. 

Transistors have been made in germanium which typically have alphas 
of 0.98 and alpha-cutoff frequencies of 500 mcls. The fabrication, electrical 
characterization, and design considerations of these transistors are dis- 
cussed. 

INTRODUCTION 

Recent work ■ concerning diffusion of impurities into germanium 
and silicon prompted the suggestion that the dimensional control in- 
herent in these processes be utilized to make high-frecjuency transistors. 

One of the critical dimensions of junction transistors, which in many 
cases seriously restricts their upper freciuency limit of operation, is the 
thickness of the base region. A considerable advance in transistor proper- 
ties can be accomplished if it is possible to reduce this dimension one or 
two orders of magnitude. The diffusion constants of ordinary donors 
and acceptors in germanium are such that, with'n realizable tempera- 
tures and times, the depth of diffused surface layers may be as small as 
10" cm. Already in the present works layers slightly less than 1 micron 
(10~ cm) thick have been made and utilized in transistors. Moreover, 
the times and temperatures required to produce 1 micron surface laj^ers 
permit good control of the depth of penetration and the concentration 
of the diffusant in the surface layer with techniciues described below. 

If one considers making a transistor whose base region consists of such 

23 



24 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

a diffused surface layer, several problems become immediately apparent : 

(1) Control of body resistivity and lifetime during the diffusion heat- 
ing cycle. 

(2) Control of the surface concentration of the diffusant. 

(3) INIaking an emitter on the surface of a thin diffused layer and 
controlling the depth of penetration. 

(4) Making an ohmic base contact to the diffused surface layer. 
One approach to the solution of these problems in germanium which has 
enabled us to make transistors with alpha-cutoff frequencies in excess 
of 500 mc/sec is described in the main body of the paper. 

An important characteristic feature of the diffusion technique is that 
it produces an impurity gradient in the base region of the transistor. 
This impurity gradiant produces a "built-in" electric field in such a 
direction as to aid the transport of minority carriers from emitter to 
collector. Such a drift field may considerably enhance the frequency 
response of a transistor for given physical dimensions. 

The capabilities of these new techniques are only partially realized 
by their application to the making of high frequency transistors, and 
even in this field their potential has not been completely explored. For 
example, with these techniques applied to making a p-n-i-p structure 
the possibility of constructing transistor amplifiers with usable gain at 
frequencies in excess of 1,000 mc/sec now seems feasible. 

DESCRIPTION OF TRANSISTOR FABRICATION AND PHYSICAL CHARACTERIS- 
TICS 

As starting material for a p-n-p structure, p-type germanium of 0.8 
ohm-cm resistivity was used. From the single crystal ingot rectangular 
bars were cut and then lapped and polished to the approximate dimen- 
sions: 200 X 60 X 15 mils. After a slight etch, the bars were washed in 
deionized water and placed in a vacuum oven for the diffusion of an 
n-type impurity into the surface. The vacuum oven consisted of a small 
molybdenum capsule heated by radiation from a tungsten coil and sur- 
rounded by suitable radiation shields made also of molybdenum. The 
capsule could be baked out at about 1,900°C in order that impurities 
detrimental to the electrical characteristics of the germaniinn be evapo- 
rated to sufficiently low levels. 

As a source of n-type impurity to be placed with the p-type bars in 
the molybdenum oven, arsenic doped germanium was used. The rela- 
tively high vapor pressure of the arsenic was reduced to a desirable range 
(about lO"* nun of Ilg) by diluting it in germanium. The use of ger- 
manium eliminated any additional problems of contamination by the 



A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 25 

dilutant, and provided a convenient means of determining the degree of 
dilution by a measurement of the conductivity. The arsenic concentra- 
tions used in the source crystal were typically of the order of 10 '-10^^/cc. 
These concentrations were rather high compared to the concentrations 
desired in the diffused surface layers since compensation had to be made 
for losses of arsenic due to the imperfect fit of the cover on the capsule 
and due to some chemical reaction and adsorption which occurred on the 
internal surfaces of the capsule. 

The layers obtained after diffusion were then evaluated for sheet con- 
ductivity and thickness. To measure the sheet conductivity a four-point 
probe method^ was used. An island of the surface layer was formed by 
masking and etching to reveal the junction between the surface layer 
and the p-type body. The island was then biased in the reverse direction 
with respect to the body thus effectively isolating it electrically during 
the measurement of its sheet conductivity. The thickness of the surface 
layer was obtained by first lapping at a small angle to the original surface 
(3^-2°~l°) and locating the junction on the beveled surface with a thermal 
probe; then multiplying the tangent of the angle between the two sur- 
faces by the distance from the edge of the bevel to the junction gives the 
desired thickness. Another particularly convenient method of measuring 
the thickness' is to place a half silvered mirror parallel to the original sur- 
face and count fringes, of the sodium D-Yme for example, from the edge 
of the bevel to the junction. Typically the transistors described here 
were prepared from diffused layers with a sheet conductivity of about 
200 ohms/square, and a layer thickness of (1.5 ± 0.3) X 10~ cm. 

When the surface layer had been evaluated, the emitter and base con- 
tacts were made using techniques of vacuum evaporation and alloying. 

o 

For the emitter, a film of aluminum approximately 1,000 A thick was 
evaporated onto the surface through a mask which defined an emitter 
area of 1 X 2 mils. The bar with the evaporated aluminum was then 
placed on a strip heater in a hydrogen atmosphere and momentarily 
brought up to a temperature sufficient to alloy the alimiinum. The 
emitter having been thus formed, the bar was again placed in the masking 
jig and a film of gold-antimony alloy from 3,000 to 4,000 A thick was 
evaporated onto the surface. This film was identical in area to the 
emitter, and was placed parallel to and 0.5 to 1 mil away from the 
emitter. The bar was again placed on the heater strip and heated to the 
gold-germanium eutectic temperature, thus forming the ohmic base 
contact. The masking jig was constructed to permit the simultaneous 
evaporation of eight pairs of contacts on each bar. Thus, using a 3-mil 
diamond saw, a bar could be cut into eight units. 



20 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Each unit, with an alloyed emitter and base contact, was then soldered 
to a platinum tab with indium, a sufficient quantity of indium being- 
used to alloy through the n-type surface layer on the back of the unit. 
One of the last steps was to mask the emitter and base contacts with a 
6- to 8-mil diameter dot of wax and form a small area collector junction 
by etching the unit attached to the platinum tab, in CP4. After washing 
in solvents to remove the wax, the unit was mounted in a header designed 
to allow electrolytically pointed wire contacts to be made to the base and 
emitter areas of the transistor. These spring contacts were made of 1-mil 
phosphor bronze wire. 

ELECTRICAL CHARACTERIZATION 

Of the parameters that characterize the performance of a transistor, 
one of the most important is the short circuit current gain (alpha) ver- 
sus frequency. The measured variation of a and q:/(1 — a) (short-circuit 
current gain in the grounded emitter circuit) as a function of frequency 
for a typical unit is shown in Fig. 1 . For comparison the same parameters 
for an exceptionally good unit are shown in Fig. 2. 

In order that the alpha-cutoff frequency be a measure of the transit 
time of minority carriers through the active regions of the transistor, any 
resistance-capacity cutoffs, of the emitter and collector circuits, must lie 
considerably higher than the measured /„ . In the emitter circuit, an 
external contact resistance to the aluminum emitter of the order of 10 



U1 

_J 

LU 

eg 

o 

lij 

Q 



•4U 

( 
30 

20 










>-( 


— , 


4.3 


MC 
















UNIT 0-3 p- 


n-p 


Ge 










Ie = 2 MA 
Vc =-10 VOLTS 
ao= 0.982 


















' 1 


s 


S. 1-a 




























6 DB 
OCT/> 


PER ^' 

VE 


■> 


^s 














1 



-10 






















l«l 






w 


> 


\ 


46 


3 M( 

1 


; 




































^ 


* 





0.1 0.2 0.4 0,6 1 2 4 6 8 10 20 40 60 100 200 400 1000 

FREQUENCY IN MEGACYCLES PER SECOND 



Fig. 1 — The grounded emitter and grounded base response versus frequency 
for a typical unit. 



A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 27 



40 



30 



10 

_l 

LU 

5 20 
o 

LJJ 

Q 



10 







o- 


^ 


« 








3.4 M 


C 










UNIT M-2 p-r 

Ie = 2MA 


1-p 


Ge 






















N 


-— oc 
1 \-oc 






Vc=-10 VOLTS 
OCo- 0.980 
























6Db\ 
PER A 
OCTAVE 


^N 


^'s 






























oc 








i-C 




v^ 540 MC 






























^\ 


\ 





-10 

0.1 0.2 0.4 0.6 1 2 4 6 e 10 20 40 60 100 200 400 1000 

FREQUENCY IN MEGACYCLES PER SECOND 

Fig. 2 — The grounded emitter and grounded base response versus frequency 
for an exceptionally good unit. 

to 20 ohms and a junction transition capacity of 1 fx^xid were measured. 
The displacement current which flows through this transition capacity 
reduces the emitter efficiency and must be kept small relative to the 
injected hole current. With 1 milliampere of ciu"rent flowing through the 
emitter junction, and conseciuently an emitter resistance of 26 ohms, 
I the emitter cutoff for this transistor was above 6,000 mc/sec. One can 
now see that the emitter area must be small and the current density 
high to attain a high emitter cutoff freciuency. The fact that a low base 
resistance requires a high level of doping in the base region, and thus a 
high emitter transition capacity, restricts one to small areas and high 
current densities. 

In the collector circuit capacities of 0.5 to 0.8 ^l^xid at a collector volt- 
age of — 10 volts were measured. There was a spreading resistance in the 
collector body of about 100 ohms which was the result of the small 
emitter area. The base resistance was approximately 100 ohms. If the 
phase shift and attenuation due to the transport of minority carriers 
through the base region w^ere small at the collector cutoff frequency, the 
(effective base resistance would be decreased by the factor (1 —a). The 
collector cutoff frequency is then given by 






where Cc = collector transition capacity 

and Re = collector body spreading resistance. 



28 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

However, in the transistors described here the base region produces the 
major contribution to the observed alpha-cutoff frequency and it is more 
appropriate to use the expression 



2irCcin + Re) 



where n = base resistance. This cutoff frequency could be raised by in- 
creasing the collector voltage, but the allowable power dissipation in the 
mounting determines an upper limit for this voltage. It should b noted 
that an increase in the doping of the collector material would raise the 
cutoff since the spreading resistance is inversely proportional to Na , 
while the junction capacity for constant collector voltage is only pro- 
portional to Na . 

The low-frequency alpha of the transistor ranged from 0.95 to 0.99 
with some exceptional units as high as 0.998. The factors to be con- 
sidered here are the emitter efficiency y and the transport factor (3. 
The transport factor is dependent upon the lifetime in the base region, 
the recombination velocity at the surface immediately surrounding the 
emitter, and the geometry. The geometrical factor of the ratio of the 
emitter dimensions to the base layer thickness is > 10, indicating that 
solutions for a planar geometry may be assumed.^ If a lifetime in the base 
region of 1 microsecond and a surface recombination velocity of 2,000 
cm/sec is assumed a perturbation calculation gives 

iS = 0.995 

The high value of ^ obtained with what is estimated to be a low base 
region lifetime and a high surface recombination velocity indicates that 
the observed low frecjuency alpha is most probably limited by the 
emitter injection efficiency. As for the emitter injection efficiency, within 
the accuracy to which the impurity concentrations in the emitter re- 
growth layer and the base region are known, together with the thick- 
nesses of these two regions, the calculated efficiency is consistent with 
the experimentally observed values. 

Considerations of Transit Time 

An examination of what agreement (^xists between the alpha-cutoff 
frequency and the physical measurements of the base region involves 
the me(;hanism of transport of minority carriers through the active 
regions of the transistor. The "active regions" include the space charge 



A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 29 

region of the collector junction. The transit time through this region 
is no longer a negligible factor. A short calculation will show that with 
— 10 volts on the collector junction, the space charger layer is about 
4 X 10"^ cm thick and that the frequency cutoff associated with trans- 
port through this region is approximately 3,000 mc/sec. 

The remaining problem is the transport of minority carriers through 
the base region. Depending upon the boundary conditions existing at the 
surface of the germanium during the diffusion process, considerable 
gradients of the impurity density in the surface layer are possible. How- 
ever, the problem of what boundary conditions existed during the diffu- 
sion process employed in the fabrication of these transistors w^ill not be 
discussed here because of the many uncertainties involved. Some quali- 
tative idea is necessary though of how electric fields arising from impurity 
gradients may affect the frequency behavior of a transistor in the limit 
of low injection. 

If one assumes a constant electric field as would result from an ex- 
ponential impurity gradient in the base region of a transistor, then the 
continuity eciuation may be solved for the distribution of minority 
carriers.* From the hole distribution one can obtain an expression for 
the transport factor j3 and it has the form 



/3 = e" 



r? sinh Z -{- Z cosh Z 



where 



1, Ne IqE 
^"2^^iV; = 2^^' 

z ^ [i^ + ,r' 

IV' 

Ne = donor density in base region at emitter junction 
Nc = donor density in base region at collector junction 

E = electric field strength 
Dp = diffusion constant for holes 

w = width of the base layer 



30 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



A plot of this function for various values of rj is shown in Fig. 3. For ?? = 0, 
the above expression reduces to the well known case of a uniformly doped 
base region. The important feature to be noted in Fig. 3 is that relatively 
small gradients of the impurity distribution in the base layer can produce 
a considerable enhancement of the frequency response. 

It is instructive to calculate what the alpha-cutoff f recjuency would be 
for a base region with a uniform distribution of impurity. The effective 
thickness of the base layer may be estimated by decreasing the measured 
thickness of the surface layer by the penetration of the space charge 
region of the collector and the depth of the alloyed emitter structure. 
Using a value for the diffusion constant of holes in the base region appro- 
priate to a donor density of about 10 Vcc, 

300 mc/s ^fa^ 800 mc/s 

This result implies that the frecjuency enhancement due to "built-in" 
fields is at most a factor of two. In addition it was observed that the 
alpha-cutoff frequency was a function of the emitter current as shown 
in Fig. 4. This variation indicates that at least intermediate injection 



<Si 



£L 




















'^ ^ 77 siNhZ +Z coshz 
Z=(L5z5+772)'/2 


0.8 
0.6 

0.4 




' > 






*~ 




:;^;~->i^ 




k.^ 


^ 


Nv 


^ 


N 






"\ 






V 


\ 


\ 




\ 


V 


^ 


\ 




\ 


>v, 










0.2 


A 


\ 


\ 


K 


\ 


\ 


\ 


i 


\ 


K 


\ \ 


\ 








0.08 


- 


^ 








\— 


^ 




A 




— \ — 


v\- 








0.06 
0.04 


- 


^^ 


\ 






^, 


^ 


\ 




^ 


1\ 


4i 


r 






0.02 






\ 


\ 




\ 


V 


\ 


\ 




V 










0.01 




1 


1 


1 


\ 




\ 

1 


1 


> 

1 


1 




1 




1 


_L 



0.1 



0.2 



0.4 0.6 0.8 1 



6 8 10 



20 



40 60 80 100 



w2 
<^-U} -g- , (RADIANS) 



Fig. .3 — The variation of | i3 | ver.sii.s frequency for various values of a uniform 
drift field in the base region. 



A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 31 



in 

_i 

LU 

m 
o 

LU 

a 

z 



b 






























n 






=7^" 


'^^-^^ 


S— 1' 




















i 


f 


\ 


^ ' 




' 


; Q ■ ■_;;;; -t 


Fv 


Rl 


k 








-5 




UNIT 0-3 p-n-p Ge 

o Ie = 2 MA 
A Ie=0.8MA 
D Ig=0.4MA 






\ 


k^ 


^ 
















\ 


\ 


\ 






10 




Vc 


= -K 


) VOLTS 

1 


1 








\ 


1 




1 


1 



10 



20 



30 40 50 60 80 100 200 300 400 

FREQUENCY IN MEGACYCLES PER SECOND 



600 800 1000 



Fig. 4 
current. 



The variation of the alpha-cutoff frequency as a function of emitter 



levels exist in the range of emitter current shown in Fig. 4. The conclu- 
sion to be drawn then is that electric fields produced by impurity 
gradients in the base region are not the dominant factor in the transport 
of minority carriers in these transistors. 

The emitter current for a low level of injection could not be deter- 
mined by measuring /„ versus /« because the high input impedance at 
very low levels was shorted by the input capacity of the header and 
socket. Thus at very small emitter currents the measured cutoff fre- 
quency was due to an emitter cutoff and was roughly proportional to 
the emitter current. At /e ^ 1 ma this effect is small, but here at least 
intermediate levels of injection already exist. 

A further attempt to measure the effect of any "built-in" fields by 
turning the transistor around and measuring the inverse alpha proved 
fruitless for two reasons. The unfavorable geometrical factor of a large 
collector area an a small emitter area as well as a poor injection effi- 
ciency gave an alpha of only 



a 



= 0.1 



Secondly, the injection efficiency turns out in this case to be proportional 
to oT^^'^ giving a cutoff freciuency of less than 1 mc/sec. The sciuare-root 
dependence of the injection efficiency on freciuency may be readily seen. 
The electron current injected into the collector body may be expressed as 



Je = qDnN 



1 -)- iu^Te 



1/2 



where q = electronic charge 



32 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Dn ^ diffusion constant of electrons 

Vi = voltage across collector junction 

Tic = density of electrons on the p-type side of the collector junction 

Te = lifetime of electrons in collector body 

Le = diffusion length of electrons in the collector body 

Since the inverse cutoff frequency is well below that associated with the 
base region, we may regard the injected hole current as independent of 
the frequency in this region. The injection efficiency is low so that 

7 ;^ ^ « 1 

J e 



Thus at a frequency where 



then 



cor, 



»1 



I 



-1/2 

An interesting feature of these transistors was the very high current 
densities at which the emitter could be operated without appreciable loss 
of injection efficiency. Fig. 5 shows the transmission of a 50 millimicro- 
second pulse up to currents of 18 milliamperes which corresponds to a 
current density of 1800 amperes/cm". The injection efficiency should 
remain high as long as the electron density at the emitter edge of the 
base region remains small compared to the acceptor density in the 
emitter regrowth layer. When high injection levels are reached the in- 
jected hole density at the emitter greatly exceeds the donor density in th(> 
base region. In order to preserve charge neutrality then 

p ^ n 

where p = hole density 

n = electron density 

As the inject(Hl hole density is raised still further the electron density 
will eventually become comparable to the acceptor density in the 
emitter regrowth layer. Tlie density of acceptors in the emitter regrowth 



A HIGH-FREQUENCY DIFFUSED BASE GERMANIUM TRANSISTOR 33 




30 46 60 75 90 

TIME IN MILLIMICROSECONDS 



>" 




9 
























"^ 


V 




4 


'^ 














\^ 




/ 












18 






V 




/ 













-15 



15 



30 45 60 75 90 

TIME IN MILLIMICROSECONDS 



105 



120 



136 



Fig. 5 — Transmission of a 50 millimicrosecond pulse at emitter currents up 
to 18 ma by a typical unit. (Courtesy of F. K. Bowers). 

region is of the order of 

and this is to be compared with injected hole density at the base region 
iside of the emitter junction. The relation between the injected hole 
density and the current density may be approximated by^ 



J. 



w 



where pi = hole density at emitter side of base region 

w = width of base region 

jA short calculation indicates that the emitter efficiency should remain 
'high at a current density of an order of magnitude higher than 1,800 
|amp/cm'. The measurements were not carried to higher current densities 
jbecause the voltage drop across the spreading resistance in the collector 
was producing saturation of the collector junction. 

CONCLUSIONS 

Impurity diffusion is an extremely powerful tool for the fabrication 
of high frequency transistors. Moreover, of the 50-odd transistors which 



34 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

were made in the laboratory, the characteristics were remarkably uni- 
form considering the ^•ariations usually encountered at such a stage of 
development. It appears that diffusion process is sufficiently controllable 
that the thickness of the base region can be reduced to half that of the 
units described here. Therefore, with no change in the other design 
parameters, outside of perhaps a different mounting, units with a 1000 
mc/s cutoff frequency should be possible. 

ACKNOWLEDGMENT 

The author wishes to acknowledge the help of P. W. Foy and W. Wieg- 
mann who aided in the construction of the transistors, D. E. Thomas who 
designed the electrical equipment needed to characterize these units, 
and J. Klein who helped with the electrical measurements. The numerical 
evaluation of alpha for drift fields was done by Lillian Lee whose as- 
sistance is gratefully acknowledged. 

REFERENCES 

1. C. S. Fuller, Phys. Rev., 86, pp. 136-137, 1952. 

2. J. Saby and W. C. Dunlap, Jr., Phys. Rev., 90, p. 630, 1953. 

3. W. Shocklej', private communication. 

4. H. Kromer, Archiv. der Elek. tlbertragung, 8, No. 5, pp. 223-228, 1954. 

5. R. A. Logan and M. Schwartz, Phys. Rev., 96, p. 46, 1954 

6. L. B. Valdes, Proc. I.R.E., 42, pp. 420-427, 1954. 

7. W. L. Bond and F. M. Smits, to be published. 

8. E. S. Rittner, Pnys. Rev., 94, p. 1161, 1954. 

9. W. M. Webster, Proc. I.R.E., 42, p. 914, 1954. 

10. J. M. Early, B.S.T.J., 33, pp. 517-533, 1954. 



Waveguide Investigations with 
Millimicrosecond Pulses 

By A. C. BECK 

(Manuscript received October 11, 1955) 

Pulse techniques have been used for many waveguide testing 'puryoses. 
The importance of increased resolution hy means of short pulses has led to 
the development of equipment to generate, receive and display pidses about 
5 or 6 millimicroseconds lo7ig. The equipment is briefly described and its 
resolution and measuring range are discussed. Domi7ia7it mode waveguide 
and antenna tests are described, and illustrated. Applications to midtimode 
waveguides are then considered. Mode separation, delay distortion and its 
equalization, and mode conversion are discussed, and examples are given. 
The resolution obtained with this equipment provides information that is 
difficult to get by any other means, and its use has proved to be very helpfid 
in ivaveguide investigations. 

CONTENTS 

1 . Introduction 35 

2. Pulse Generation 36 

3. Receiver and Indicator 41 

4. Resolution and Measuring Range 42 

5. Dominant Mode Waveguide Tests 43 

6. Testing Antenna Installations 45 

7. Separation of Modes on a Time Basis 48 

8. Delay Distortion 52 

9. Delay Distortion Ecjualization 54 

10. Measuring Mode Conversion from Isolated Sources 57 

11. Measuring Distril)uted Mode Conversion in 1 ong Waveguides 61 

12. Concluding Remarks 65 

1. INTRODUCTION 

Pulse testing techniques have been employed to advantage in wave- 
guide investigations in numerous ways. The importance of better resolu- 
tion through the use of short pulses has always been apparent and, from 
the first, eciuipment was employed which used as short a pulse as pos- 
sible. Radar-type apparatus using magnetrons and a pulse width of 
about one-tenth microsecond has seen considerable use in waveguide 
research, and many of the results have been published.' • - 

35 



36 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

To improve the resolution, work was initiated some time ago by S. E. 
Miller to obtain measuring equipment which would operate with much 
shorter pulses. As a result, pulses about 5 or 6 millimicroseconds long 
became available at a frequency of 9,000 mc. In a pulse of this length 
there are less than 100 cycles of radio frequency energy, and the signal 
occupies less than ten feet of path length in the transmission medium. 
The RF bandwidth required is about 500 mc. In order to obtain such 
bandwidths, traveling wave tubes were developed by J. R. Pierce and 
members of the Electronics Research Department of the Laboratories. 
The completed amplifiers were designed by W. W. Mumford. N. J. 
Pierce, R. W. Dawson and J. W. Bell assisted in the design and construc- 
tion phases, and G. D. Mandeville has been closely associated in all of 
this work. 

2. PULSE GENERATION 



These millimicrosecond pulses have been produced by two different 
types of generators. In the first equipment, a regenerative pulse gener- 
ator of the type suggested by C. C. Cutler of the Laboratories was used.^ 
This was a very useful device, although somewhat complicated and hard 
to keep in adjustment. A brief description of it will permit comparisons 
with a simpler generator which was developed a little later. 

A block diagram of the regenerative pulse generator is shown in Fig. 1. 
The fundamental part of the system is the feedback loop drawn with 
heavy lines in the lower central part of the figure. This includes a travel- 
ing wave amplifier, a waveguide delay line about sixty feet long, a crystal 
expander, a band-pass filter, and an attenuator. This combination forms 
an oscillator which produces very short pulses of microwave energy. 
Between pulses, the expander makes the feedback loop loss too high for 
oscillation. Each time the pulse circulates around the loop it tends to 
shorten, due to the greater amplification of its narrower upper part 
caused by the expander action, until it uses the entire available band 
width. A 500-mc gaussian band-pass filter is used in the feedback loop,^ 
of this generator to determine the final bandwidth. An automatic gain 
control operates with the expander to limit the pulse amplitude, thus 
preventing amplifier compression from reducing the available expansion. 

To get enough separation between outgoing pulses for reflected pulse 
measurements with waveguides, the repetition rate would need to be 
too low for a practical delay fine length in the loop. Therefore a r2.8-mc 
fundamental rate was chosen, and a gated traveling wave {\\\)v ampfifier 
was used to reduce it to a 100-kc rate at the output. This amplifier is 
kept in a cutoff condition for 127 pulses, and then a gate pulse restores 



I 



i 



t 



WAVEGUIDE TESTING WITH MILLIMIf'ROSECOND PULSES 



37 



it to the normal amplifying condition for fifty millimicroseconds, during 
which time the 128th pulse is passed on to the output of the generator 
as shown on Fig. 1. 

The synchronizing system is also shown on Fig. 1. A 100-kc quartz 
crystal controlled oscillator with three cathode follower outputs is the 
basis of the system. One output goes through a seven stage multiplier 
to get a 12.8-mc signal, which is used to control a pulser for synchroniz- 
ing the circulating loop. Another output controls the gate pulser for the 
output traveling wave amplifier. Accurate timing of the gate pulse is 
obtained by adding the 12.8-mc pulses through a buffer amplifier to the 
gate pulser. The third output synchronizes the indicator oscilloscope 
sweep to give a steady pattern on the screen. 

Although this equipment was fairly satisfactory and served for many 



OSCILLATOR 

AND CATHODE 

FOLLOWERS 

100 KC 



I 1 



MULTIPLIER 

100 KC TO 

12.8 MC 



SYNC 

PULSER 

0.02 A SEC 

12.8 MC 



500 MC 

BANDPASS 

FILTER 



GATE 

PULSER 

0.05 USEC 

100 KC 



A 



BUFFER 
AMPLIFIER 



"1 



CRYSTAL 
EXPANDER 



U 



AGC I 



WAVEGUIDE 

DELAY 

LINE 



TW TUBE 




■Y^ 



MILLI/iSEC/ 
9000 MC/' 
PULSES 
12.8 MC RATE 



MlLLIyUSEC 

9000 MC 

PULSES 

100 KC RATE 



GATED 
TW TUBE 



SYNC SIGNAL TO 
INDICATOR SCOPE 



Fig. 1 — Block diagram of the regenerative pulse generator. 



38 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

testing purposes, it was rather complex and there were some problems 
in its construction and use. It was difficult to obtain suitable microwave 
crystals to match the waveguide at low levels in the expander. Tliis 
would make it even more difficult to build this type of pulse generator 
for higher frequency ranges. Stability also proved to be a problem. The 
frequency multiplier had to be very well constructed to avoid phase 
shift due to drifting. The gate pulser also required care in design and 
construction in order to get a stable and flat output pulse. It was some- 
what troublesome to keep the gain adjusted for proper operation, and 
the gate pulse time adjustment required some attention. The pulse 
frequency could not be changed. For these reasons, and in order to get 
a smaller, lighter and less complicated pulse generator, work was carried 
out to produce pulses of about the same length by a simpler method. 

If the gated output amplifier of Fig. 1 were to have a CW instead of a 
pulsed input, a pulse of microwave energy would nevertheless appear at 
the output because of the presence of the gating pulse. This gating pulse 
is applied to the beam forming electrode of the tube to obtain the gating 
action. If the beam forming electrode could be pulsed from cutoff to its 
normal operating potential for a very short time, very short pulses of 
output energy could be obtained from a continuous input signal. How- 
ever, it is difficult to obtain millimicrosecond video gating pulses of suf- 
ficient amplitude for this purpose at a 100-kc repetition rate. 

A traveling-wave tube amplifies normally only when the helix is 
within a small voltage range around its rated dc operating value. For 
voltages either above or below this range, the tube is cut off. When the 
helix voltage is raised through this range into the cutoff region beyond 
it, and then brought back again, two pulses are obtained, one during a 
small part of the rise time and the other during a small part of the return 
time. If the rise and fall times are steep, very short pulses can be 
obtained. Fig. 2 shows the pulse envelopes photographed from the 
indicator scope screen when this is done. For the top trace, the helix was 
biased 300 volts negatively from its normal operating potential, then 
pulsed to its correct operating range for about 80 millimicroseconds, 
during which time normal amplification of the CW input signal was ob- 
tained. The effect of further increasing the helix video pulse amplitude 
in the positive direction is shown by the succeeding lower traces. The 
envelope dips in the middle, then two separated pulses remain — one 
during a part of the rise time and one during a part of the fall time of 
helix voltage. The pulses shown on the bottom trace have shortened to 
about six millimicroseconds in length. The helix pulse had a positive 
amplitude of about 500 volts for this trace. 



1 



WAVEGUIDE TESTIXG WITH MILUMICROSErOXD PULSES 



39 



Since only one of these pulses can be used to get the desired repetition 
rate, it is necessary to eliminate the other pulse. This is done in a simi- 
lar manner to that used for gating out the undesired pulses in the re- 
generative pulse generator. However, it is not necessary to use another 
amplifier, as was required there, since the same tube can be used for 
this purpose, as well as for producing the microwave pulses. Its beam 
forming electrode is biased negatively about 250 volts with respect to 
the cathode, and then is pulsed to the normal operating potential for 
about 50 millimicroseconds during the time of the first short pulse ob- 
tained by gating the helix. Thus, the beam forming electrode potential 
has been returned to the cutoff value during the second helix pulse, 
which is therefore eliminated. 
Il A block diagram of the resulting double-gated pulse generator is 
shown in Fig. 3. Comparison with Fig. 1 shows that it is simpler 
than the regenerative pulse generator, and it has also proved more 
satisfactory in operation. It can be used at any frequency where a sig- 
nal source and a traveling-wave amplifier are available, and the pulse 




Fig. 2 — Envelopes of microwave pulses at the output of a traveling wave am- 
lifier with continuous wave input and helix gating. The gating voltage is higher 
or the lower traces. 



40 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



frequency can be set anywhere within the bandwidth of the travehng- 
wave ampUfier by tuning the klystron oscillator. 

The pulse center frequency is shifted from that of the klystron os- 
cillator frequency by this helix gating process. An over-simphfied but 
helpful explanation of this effect can be obtained by considering that 
the microwave signal voltage on the helix causes a bunching of the elec- 
tron stream. This^ bunching has the same periodicity as the microwave 
signal voltage when the dc helix potential is held constant. However, 
since the helix voltage is continuously increased in the positive direction 
during the time of the first pulse, the average velocity of the last bunches 
of electrons becomes higher than that of the earlier bunches in the pulse, 
because the later electrons come along at the time of higher positive 
helix voltage. This tends to shorten the total length of the series of 
bunches, resulting in a shorter w^avelength at the output end of the 
helix and therefore a higher output microwave frequency. On the second 
pulse, obtained when the helix voltage returns toward zero, the process 
is reversed, the bunching is stretched out, and the frequency is de- 
creased. This second pulse is, however, gated out in this arrangement 
by the beam-forming electrode pulsing voltage. The result for this 
particular tube and pulse length is an effective output frequency ap- 
proximately 150 mc higher than the oscillator frequency, but this figure 
is not constant over the range of pulse frequencies available within the 
amplifier bandwidth. 



OSCILLATOR AND 

CATHODE FOLLOWERS 

100 KC 



KLYSTRON 

OSCILLATOR 

9000 MC 



BEAM FORMING 

ELECTRODE 

PULSER 



HELIX 
PULSER 



^ 



PULSED 
TW TUBE 



MILLI/aSEC 
9000 MC 
PULSES 



SYNC SIGNAL TO 
INDICATOR SCOPE 



Fig. 3 — Block diugram of the double-gated traveling wave tube millimicro- 
second pulse generator. 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



41 



3. RECEIVER AND INDICATOR 



The receiving equipment is shown in Fig. 4. It uses two traveUng- 
wave amplifiers in cascade. A wide band detector and a video amplifier 
then follow, and the signal envelope is displayed by connecting it to 
the vertical deflecting plates of a 5 XP type oscilloscope tube. The 
video amplifier now consists of two Hewlett Packard wide band dis- 
tributed amplifiers, having a baseband width of about 175 mc. The 
second one of these has been modified to give a higher output voltage. 
The sweep circuits for this oscilloscope have been built especially for 
this use, and produce a sweep speed in the order of 6 feet per micro- 
second. An intensity pulser is used to eliminate the return trace. These 
parts of the system are controlled by a synchronizing output from the 
pulse generator 100-kc oscillator. A precision phase shifter is used at 
the receiver for the same purpose that a range unit is employed in radar 
systems. This has a dial, calibrated in millimicroseconds, which moves 
the position of a pulse appearing on the scope and makes accurate 
measurement of pulse delay time possible. 

Fig. 4 also shows the appearance of the pulses obtained with this 
equipment. The pulse on the left-hand side of this trace came from the 



PULSE 

SIGNAL 

9000 MC 



SYNC 

SIGNAL 
100 KC 



TW TUBES 



VIDEO 
AMPLIFIER 




INTENSITY 

PULSER 

0.05/USEC 

100 KC 



PRECISION 

PHASE 

SHIFTER 



SWEEP 
GENERATOR 




DOUBLE-GATED 
PULSE 



REGENERATIVE 
PULSE 



Fig. 4 — Block diagram of millimicrosecond pulse receiver and indicator. The 
idicator trace photograph shows pulses from each type of generator. 



42 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



newer double-gated pulse generator, while the pulse on the right was 
produced by the regenerative pulse generator. It can be seen that they 
appear to have about the same pulse width and shape. This is partly 
due to the fact that the video amplifier bandwidth is not c^uite adequate 
to show the actual shape, since in both cases the pulses are slightly 
shorter than can be correctly reproduced through this amplifier. The 
ripples on the base line following the pulses are also due to the video 
amplifier characteristics when used with such short pulses. 

4. RESOLUTION AND MEASURING RANGE 

Fig. 5 shows a piece of equipment which was placed between the pulse 
generator and the receiver to show the resolution which can be obtained. 
This waveguide hybrid junction has its branch marked 1 connected to 
the pulse generator and branch 3 connected to the receiver. If the two 
side branches marked 2 and 4 were terminated, substantially no energy 
would be transmitted from the pulser straight through to the receiver. 
However, a short circuit placed on either side branch will send energy 
through the system to the receiver. Two short circuits were so placed 
that the one on branch 4 was 4 feet farther away from the hybiid junc- 
tion than the one on branch 2. The pulse appearing first is produced l)y 
a signal traveling from the pulse generator to the short circuit on branch 
2 and then through to the receiver, as shown by the path drawn with 
short dashes. A second pulse is produced by the signal which travels 



BRANCH 
2 



SHORT 
CIRCUIT 



BRANCH 



FROM 
PULSER 



TO 
RECEIVER 




FIRST PULSE PATH 
SECOND PULSE PATH 



SHORT 

CIRCUIT 




DOUBLE-GATED PULSES 




REGENERATIVE PULSES 



Fig. 5 — W;iv(!guicle hyhriil ciicuil- uscxl to demonstrate resululion of milli- 
microsecond pulses. Trace photographs of pulses from each type of generator ;iie 
shown. 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



43 




TO RECEIVER 
\ 




TE° IN 3"DIAM copper GUIDE (ISO FT LONG) 



Fig. 6 — Waveguide arrangement and oscilloscope trace photos showing pres- 
ence and location of defective joint. The dominant mode (TEn) was used with its 
polarization changed 90 degrees for the two trace photos. 

from the pulse generator through branch 4 to the short circuit and then 
to the receiver as shown by the long dashed line. This pulse has traveled 
8 feet farther in the waveguide than the first pulse. This would be equiva- 
lent to seeing separate radar echoes from two targets about 4 feet apart. 
Resolution tests made in this way \vith the pulses from the regenerative 
pulse generator, and from the double-gated pulse generator, are shown 
on Fig. 5. With our video amplifier and viewing equipment, there is 
no appreciable difference in the resolution obtained using either type 
of pulse generator. 

The measuring range is determined by the power output of the gated 
amplifier at saturation and by the noise figure of the first tube in the 
receiver. In this equipment the saturation level is about 1 watt, and the 
noise figure of the first receiver tube is rather poor. As a result, received 
pulses about 70 db below the outgoing pulse can be observed, which is 
I enough range for many measurement purposes. 



5. DOMINANT MODE WAVEGUIDE TESTS 

Fig. 6 shows the use of this equipment to test 3'^ round waveguides 
such as those installed between radio repeater equipment and an an- 
tenna. This particular 150-foot line had very good soldered joints and was 
thought to be electrically very smooth. The signal is sent in through a 
transducer to produce the dominant TEn mode. The receiver is con- 
nected through a directional coupler on the sending end to look for any 



44 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Fig. 7 — Defective joint caused by imperfect soldering which gave the reflec- 
tion shown on Fig. 6. 

reflections from imperfections in the line. The overloaded signal at the 
left of the oscilloscope trace is produced by leakage directly through 
the directional coupler. The overloaded signal on the other end of this 
trace is produced by the reflection from the short circuit piston at the 
far end of the waveguide. The signal between these two, which is about 
45 db down from the input signal, is produced by an imperfect joint 
in the waveguide. The signal polarization was oriented so that a maxi- 
mum reflection was obtained in the case of the lower trace. In the 
other trace, the polarization was changed by 90°. It is seen that this 
particular joint produces a stronger reflection for one polarization than 
for the other. By use of the precision phase shifter in the receiver the 
exact location of this defect was found and the particular joint that was 
at fault was sawed out. Fig. 7 shows this joint after the pipe had been 
cut in half through the middle. The guide is quite smooth on the inside 
in spite of the discoloration of some solder that is shown here, but on 
the left-hand side of the illustration the open crack is seen where the 
solder did not run in properly. This causes the reflected pulse that shows 
on the trace. The fact that this crack is less than a semi-circumference 
in length causes the echo to be stronger for one polarization than for the 
other. 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



45 



Fig. 8 shows the same test for a 3" diameter ahiminum waveguide 
250 feet long. This line was mounted horizontally in the test building 
with compression couplings used at the joints. The line expanded on 
warm days hut the friction of the mounting supports was so great that 
it pulled open at some of the joints when the temperature returned to 
normal. These open joints produced reflected pulses from 40 to 50 db 
down, which are shown here. They come at intervals equal to the length 
of one section of pipe, about 12 feet. Some of these show polarization 
effects where the crack was more open on one side than on the other, 
but others are almost independent of polarization. These two photo- 
graphs of the trace were taken with the polarization changed 90°. 

Fig. 9 shows the same test for a 3" diameter galvanized iron wave- 
guide. This line had shown fairly high loss using CW for measure- 
ments. The existence of a great many echoes from random distances 
indicates a rough interior finish in the waveguide. Fig. 10 shows the 
kind of inperfections in the zinc coating used for galvanizing which 
caused these reflections. 



6. TESTING ANTENNA INSTALLATIONS 

The use of this equipment in testing waveguide and antenna installa- 
tions for microwave radio repeater systems is shown in Fig. 11. This 
particular work was done in cooperation wdth A. B. Crawford's antenna 
research group at Holmdel, who designed the antenna system. A direc- 
tional coupler was used to observe energy reflections from the system 
under test. In this installation a 3" diameter round guide carrying the 
TEu mode was used to feed the antenna. Two different waveguide 





TE,, IN 3"D1AM aluminum GUIDE (250 FT LONG) 



Fig. 8 — Reflections from several defective joints in a dominant (TEn) mode 
waveguide. The two trace photos are for polarizations differing by 90 degrees. 



46 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 





TO RECEIVER ^n 



^— a^IS^rv^ 



i— ^ 



TE 



■^-^Bi 



21 



I 



10 



TE,° IN 3" DIAM GALVANIZED IRON GUIDE (250 FT LONG) 



Fig. 9 ■ — Multiple reflections from a dominant (TEn) mode waveguide with a 
rough inside surface. The two trace photos are for polarizations differing by 90 I 
degrees. 

joints are shown here. In addition, a study was being made of the re- 
turn loss of the transition piece at the throat of the antenna which • 
connected the 3" waveguide to the square section of the horn. The I 
waveguide sections are about 10 feet long. The overloaded pulse at the 
left on the traces is the leakage through the directional coupler. The 




Fig. 10 — Rough inside surface of a galvanized iron waveguide which produced 
the reflections shown on Fig. 9. 



I 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



47 



other echoes are associated with the parts of the system from which 
they came by the dashed Hues and arrows on the figure. A clamped 
joint in the line gave the reflection shown next following the initial 
overloaded pulse. A well made threaded coupling in which the ends of 
the pipe butted squarel,y is seen to have a very much lower reflection, 
scarcely observable on this trace. Since there is ahvays reflection from 
the mouth and upper reflector parts of this kind of antenna, it is not 
possible to measure a throat transition piece alone by conventional CW 
methods, as the total reflected power from the system is measured. 
Here, use of the resolution of this short pulse equipment completely 
separated the reflection of the transition piece from all other reflections 
and made a measurement of its performance possible. In this particular 
case, the reflection from the transition is more than 50 db down from 
the incident signal which represents very good design. As can be seen, 



OPEN APERTURE 



FIBERGLASS COVER 
OVER APERTURE 

REFLECTION APPEARS 
-^TO COME FROM 16 FT 
N FRONT OF HORN MOUTH 




DIRECTIONAL TRANSDUCER CLAMPED THREADED ROUND-TO 
COUPLER JOINT COUPLING SQUARE 

TRANSITION 

Fig. 11 — Waveguide and antenna arrangement with trace photos showing re- 
flections from joints, transition section, and cover. 



48 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

the reflection from the parabohc reflector and mouth is also finite low, 
and this characterizes a good antenna installation. 

The extra reflected pulse on the right of the lower trace on Fig. 11 
appeared when a fiberglas weatherproof cover was installed over the 
open mouth of the horn. This cover by itself would normally produce a 
troublesome reflection. However, in this antenna, it is a continuation of 
one of the side walls of the horn. Consequently, outgoing signals strike 
it at an oblique angle. Reflected energy from it is not focused by the 
parabolic section back at the waveguide, so the overall reflected power 
in the waveguide was found to be rather low. However, measuring it 
with this equipment, we found that an extra reflection appeared to 
come from a point 16 feet out in front of the mouth of the horn when the 
cover was in place. This is accounted for by the fact that energy re- 
flected obliquely from this cover bounces back and forth inside the 
horn before getting back into the waveguide, thus traveling the extra 
distance that makes the measurement seem to show that it comes from 
16 feet out in front. 

7. SEPARATION OF MODES ON A TIME BASIS 

If a pulse of energy is introduced into a moderate length of round 
waveguide to excite a number of modes which travel with different 
group velocities, and then observed farther along the line, or reflected 
from a piston at the end and observed at the beginning, separate pulses 
will be seen corresponding to each mode that is sent. This is illustrated 




! t r 


t t 


TE„ TMo,TE2, 


TM„ TE3, 




(TEoi) 

^NOT EXCITED 


TO RECEIVER 






=^ 


^-^ 


=^^ 



t 


;ft 


t 




TMj, 


TE4I TE,2 


TM02 


TM3, AND 

TE5, TOO 

WEAK TO 

SHOW 



TE, 



•^^ " 



PROBE 3 DIAM ROUND GUIDE 

COUPLING (WILL SUPPORT 12 MODES) 



Fig. 12 — Arrangement for showing mode separation on a time basis in a multi- 
mode waveguide. The pulses in the trace ])]io(o have all traveled to the iiisloii and 
back. The earlier outgoing pulse due to direelional coupler unbalance is not shown. 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



49 



in Fig. 12. In this arrangement energy was sent into the round line from 
a probe inserted in the side of the guide. This couples to all of the 12 
modes which can be supported, with the exception of the TEoi circular 
electric mode. The sending end of the round guide was terminated. A 
directional coupler is connected to the sending probe so that the return 
from the piston at the far end can be observed on the receiver. Because 
of the different time that each mode takes to travel one round trip in 
this waveguide, which was 258 feet long, separate pulses are seen for 
each mode. The pulses in this figure have been marked to show which 
mode is being received. 

The time of each pulse referred to the outgoing pulse was measured 
and found to check very well with the calculated time. The formula for 
the time of transit in the waveguide for any mode is: 



T = 



0.98322V'1 - VnJ 



[where T = time in millimicroseconds 

L = length of pulse travel in feet 

Vnm ^^ A /Ac 

X = operating wavelength in air 

Ac = cutoff wavelength of guide for the mode involved. 

[ Table I — Calculated and Measured Value of Time for One 

Round Trip 










Time in Millimicroseconds 




Mode Designation 






Calculated 


Measured 


1 


TEn 


545 


545 


2 


TMoi 


561 


561 


3 


TE,i 


587 


587 


4 


TMn 


634 


634 


5 


TEoi 


634 


. 


6 


TE31 


665 


665 


7 


TM21 


795 


793 


8 


TE4: 


835 


838 


9 


TE12 


838 





10 


TM„2 


890 


890 


11 


TMn 


1461 


— 


12 


TE51 


1519 


— 



50 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

The calculated and measured \'alue of time for one round trip is given 
in Table I. 

In this experiment the operating wavelength was 3.35 centimeters 
This was obtained by measurements based on group velocit}' in a num- 
ber of guides as well as information about the pulse generator com- 
ponents. It represents an effective wa\'elength giving correct time of 
travel. The pulse occupies such a wide bandwidth that a measurement 
of its wavelength is difficult by the usual means. 

The dashes in the measured column indicate that the mode was not 
excited by the probe or was too weak to measure. These modes do not 
appear on the oscilloscope trace photograph. 

The relative pulse heights can be calculated from a knowledge of the 
probe coupling factors and the line loss. The probe coupling factors as 
given by M. Aronoff in unpublished work are expressed by the following 

For TE„„, modes: 

P = 2.390 r—^ 



i 



For TM„^ modes: 



TV- L a -. 

j\. nm ^ "flu 



X X 

P = 1.195€„ — - 



where 

P = ratio of probe coupling power in mode nm to that in mode TEn 

n = first index of mode being calculated 
Knm = Bessel function zero value for mode being calculated = Td/\c 

X = wavelength in air 

X(, = wavelength in the guide for the mode involved ' 

Xc = cutoff wavelength of guide for the mode involved 

€„ = 1 for w = 

€„ = 2 for n ?^ , 

d = waveguide diameter 

Formulas for guide loss as given by S. A. Schelkunoff on page 390 of 
his book Elect romagnelir Waves for this case where the resistivity of the 
aluminum guide is 4.14 X 10~^ ohms per cm cube are: 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 51 

For TE„,„ modes: 

a = 3.805 ! — 2 2 + V.an ) (1 " Vnm) 

\l\n,n — n / 

For TM„,„ modes: 

a = 3.805(1 - VnJy''' 
where: 

a = attemiation of this aluminum guide in db 

n — first index of mode being calculated 
Knm — Bessel function zero value for mode being calculated = TrtZ/Xc 

Vnm = A/Ac 

X = operating wavelength in air 

Xc = cutoff wavelength of guide for the mode involved 

d = waveguide diameter 

Table II gives the calculated probe coupling factor, line loss, and rela- 
tive pulse height for each mode. In the calculation of the latter, wave 
elUpticity and loss due to mode conversion were neglected, but the heat 
loss given by the preceding formulas has been increased 20 per cent for 
all modes, to take account of surface roughness. Relative pulse heights 
were obtained by subtracting the relative line loss from twice the rela- 
tive probe coupling factor. The relative line loss is the number in the 
itable minus 2.33 db, the loss for the TEn mode. 

The actual pulse heights on the photo of the trace on Fig. 12 are in 
fair agreement with these calculated values. Differences are probably 
due to polarization rotation in the guide (wave ellipticity) and conver- 
sion to other modes, effects which were neglected in the calculations, 
and which are different for different modes. 

Calculated pulse heights with this guide length, except for modes 
near cutoff, vary less than the probe coupling factors, because line loss 
is high when tight probe coupling exists. This is to be expected, since 
both are the result of high fields near the guide walls. 

The table of round trip travel time shows that the TE41 and TE12 
modes are separated by only three millimicroseconds after the round 
trip in this waveguide. They would not be resolved as separate pulses 
by this e(iuipment. However, the table of calculated pulse heights shows 
that the TE41 pulse should be about 22 db higher than the TE12 pulse. 



52 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Table II — Calculated Probe Coupling Factor, Line Loss and 
Pulse Height for Each Mode 



Mode 


Mode 


Relative Probe 


1.2 X Theoretical 


Calculated Relatix e 


Number 


Designation 


Coupling Factor, 
db 


Line Loss, db 


Pulse Heights, db 


1 


TEu 





2.33 





2 


TMoi 


+0.32 


4.88 


-1.91 


3 


TE2, 


+2.86 


4.85 


+3.20 


4 


TMu 


+2.80 


5.51 


+2.42 


5 


TEo, 


— 00 


1.73 


— 00 


6 


TE31 


+4.82 


8.21 


+3.76 


7 


TM2, 


+ 1.82 


6.92 


-0.95 


8 


TE41 


+6.80 


13.86 


+2.07 


9 


TE12 


-8.73 


4.70 


-19.83 


10 


TM02 


-1.68 


7.74 


-8.77 


11 


TMsi 


-0.82 


12.71 


-12.02 


12 


TE51 


+ 10.14 


32.09 


-9.48 



Since the TE12 pulse is so weak, it would not show on the trace even if 
it were resolved on a time basis. Coupling to the TM02 mode is rather 
weak, and the gain was increased somewhat at its position on the trace 
to show its time location. 

8. DELAY distortion 

Another effect of the wide bandwidth of the pulses used with this 
equipment can be observed in Fig. 12. The pulses that have traveled 
for a longer time in the guide are in the modes closer to cutoff, and are 
on the right-hand side of the oscilloscope trace. They are broadened 
and distorted compared with the ones on the left-hand side. This effect 
is due to delay distortion in the guide. This can be explained by refer- 
ence to Fig. 13. On this figure the ratio of group velocity to the velocity 
in an unbounded medium is shown plotted as a function of frequency 
for each of the modes that can be propagated. The bandwidth of the 
transmitted pulse is indicated by the vertical shaded area. It will he 
noticed that the spacing of the pulses on the oscilloscope trace on Fig. 
12 from left to right in time corresponds to the spacing of the group 
velocity curves in the bandwidth of the pulse from top to bottom. De- 
lay distortion on these curves is shown by the slope of the line across 
the pulse bandwidth. If the line were horizontal, showing the same group 
velocity at all points in the band, there would be no delay distortion. 
The greater the difference in group A-elocity at the two edges of the 
band, the greater the delay distortion. The curves of Fig. 13 indicate 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



53 



I that there should be increasing amounts of delay distortion reading 
ifrom top to bottom for the pulse bandwidth used in these experiments. 
;The effect of this delay distortion is to cause a broadening of the pulse. 
Examination of the pulse pattern of Fig. 12 shows that the later pulses 
corresponding in mode designation to the lower curves of Fig. 13 do in- 
deed show a broadening due to the increased delay distortion. One 
method of reducing the effect of delay distortion is to use frequency 
division multiplex so that each signal uses a smaller bandwidth. Another 
way, suggested by D. H. Ring, is to invert the band in a section of the 
waveguide between one pair of repeaters compared with that between 
an adjacent pair of repeaters so that the slope is, in effect, placed in the 
opposite direction, and delay distortion tends to cancel out, to a first 
order at least. 

The (luantitative magnitude of delay distortion has been expressed 
by S. Darlington in terms of the modulating base-band frequency 
needed to generate two side frequencies which suffer a relative phase 
error of 180° in traversing the line. This would cause cancellation of a 
single frequency AM signal, and severe distortion using any of the 



1.0 
















PULSE BANDWIDTH 


— >. 




<— 


















^^ 


^— 












UJ 

^0.9 

Q. 
</) 

OI 

mo.8 
u. 

z 

^0.7 

1- 

o 

O 

> 
o 

^ 0.5 

>- 

o 

3 0.4 

m 

> 

^0.3 

o 

(r 
o 

^0.2 

o 
io., 






■^^H;;^ 


. — 










/^ 


o^^ 




^ 




^ 




^ 


^^ 








/ 


/ 


y 




\a 


X 

y 


/ 




^ 


^ 








/ 


/ 


/ 








6 






^ 






/ 


7 


/ 


f 


// 


< 


f/> 


\ 


^-'^'fA 


y^. 








/ 


/ 


1 


/ 


1 


'L 


f4 




// 






1 


/ 


/ 


1 




1 


/A 




'/ 




















// 






// 


L 


^i 
















/ 1 










7 


\\ 


1 

































3 4 

FREQUENCY 



5 6 7 8 9 10 

IN KILOMEGACYCLES PER SECOND 



12 



1 Fig. 1.3 — Theoretical group velocity vs. frequency curves for the 3" diameter 
ivaveguide used for the tests shown on Fig. 12. The vertical shaded area gives the 
bandwidth for the millimicrosecond pulses employed in that arrangement. 



54 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

ordinary modulation methods. Darlington gives this formula: 

^) ^^^^ 

iLLi/ Vnm 

where : 

jB = base bandwidth for 180° out of phase sidebands 

/ = operating frequency (in same units as jB) 

X = wavelength in air 

L = waveguide length (in same units as X) 

Vnm = X/Xe 

Xc = cutoff wavelength for the mode involved 

With this equipment, the base bandwidth of the pulse is about 175 
mc, and when/5 from the formula above is about equal to or less than 
this, pulse distortion should be observed. The following Table III gives 
fB calculated from this formula for the arrangement shown on Fig. 12. 

It is interesting to note that pulses in the TMu and TE31 modes, for 
which jB is less than the 175-mc pulse bandwidth, are broadened, but 
not badly distorted. For the higher modes, where jB is much less than 
175 mc, broadening and severe distortion are evident. Another example 
is given in the next section. 

9. DELAY DISTORTION EQUALIZATION 

If the distance which a pulse travels in a waveguide is increased, its 
delay distortion also increases. Since the group velocity at one edge of 
the band is different than at the other edge of the band, the amount 
by which the two edges get out of phase with each other increases with 
the total length of travel, causing increased distortion and pulse broaden- 
ing. The Darlington formula in the previous section shows that jB 
varies inversely as the square root of the length of travel. This efTect 
is shown on Fig. 14. In this arrangement the transmitter was connected 
to the end of a 3" diameter round waveguide 107 feet long through a 
small hole in the end plate. A mode filter was used so that only the 
TEoi mode would be transmitted in this Avaveguide. Through another 
small hole in the end plate polarized 90° from the first one, and rotated 
90° around tlu^ plate, a directional coupler was connected as shown. 
The direct through guide of this directional coupler could be short cir- 
cuited with a waveguide shorting switch. Energy reflected from this 



fl 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 



55 



Table III - 


— Calculatee 


> Values of fB foe the Arrangement 


Shown in Fig. 1 


2 


Mode Number 


Mode Designation 


/B Megacycles 


Remarks 


1 


TEn 


324.0 




2 


TMoi 


237.7 




3 


TEn 


174.9 




4 


TMu 


124.1 




5 


TEoi 


124.1 


Not excited 


6 


TE31 


105.2 




7 


TMoi 


65.9 




8 


TE41 


59.1 




9 


TEi, 


58.6 


Veiy weakly excited 


10 


TMoo 


51.8 




11 


TM3: 


21.3 


Not observed 


12 


TE51 


20.0 


Not observed 


NUMBER OF 






R( 


3UND TRIPS 









TAPERED 

DELAY 

DISTORTION 

EQUALIZER 



WAVEGUIDE 

SHORTING 

SWITCH 



1/ 



'M 



^ 



te; 



>T0 RECEIVER 




NOT EQUALIZED 
(SWITCH CLOSED) 



EQUALIZED 
(SWITCH OPEN) 



TEqiIN 3 DIAM ROUND GUIDE 
(107 FT LONG) 



Fig. 14 — The left-hand series of pulses shows the build up of delay distortion 
with increasing number of round trips in a long waveguide. The right-hand series 
shows the im]irovement obtained with the tapered delay distortion equalizer 
shown at the right. 



56 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



switch was then taken through the directional coupler to the receiver 
as shown by the output arrow. The series of pulses at the left-hand 
photograph of the oscilloscope traces was taken with this waveguide i 
shorting switch closed. The top pulse shows the direct leakage across 
the inside of the end plate before it has traveled through the 3" round 
guide. The next pulse is marked one round trip, having gone therefore 
214 feet in the TEoi mode in the round waveguide. The successive pulses 
have traveled more round trips as shown by the number in the center 
between the two photographs. The effect of increased delay distortion 
broadening and distorting the pulse can be seen as the numbers increase. 
The values of fB from the Darlington formula in the previous section 
for these lengths are given in Table IV. 

It will be noticed that pulse broadening, and eventually severe dis- 
tortion, occurs as fB decreases much below the 175-mc pulse band- 
width. The effect is gradual, and not too bad a pulse shape is seen until 
fB is about half the pulse bandwidth, although broadening is very 
evident earlier. 

When the waveguide short-circuiting switch was opened so that the 
tapered delay distortion equalizer was used to reflect the energy, in- 
stead of the switch, the series of pulses at the right was observed on 
the indicator. It will be noted that there is much less distortion of these, 
pulses, particularly toward the bottom of the series. The ones at the top, 
have less distortion than would be expected, probably because of fre-, 
quency modulation of the injected pulse. The equalizer consists of a 
long gradually tapered section of waveguide which has its size reduced 
to a point beyond cutoff for the frequencies involved. Reflection takes 
place at the point of cutoff in this tapered guide. For the high frequency 
part of the pulse bandwidth, this point is farther away from the short- 
ing switch than for the low frequency part of the bandwidth. Conse- 
quently, the high frequency part of the pulse travels farther in one round 
trip into this tapered section and back than the low frequency part of 



Table IV — Values of fB from the Darlington Formula 
FOR the Arrangement Show^n in Fig. 14 



li 













Round Trip Number 


JB Megacycles 


Round Trip Number 


fB Megacycles 




1 
2 
3 
4 
5 


185.8 

131.4 

107.3 

92.9 

83.1 


6 

7 

8 

9 

10 


75.8 
70.2 
65.7 
61.9 

58.7 


















j 




















1 










1 





WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 57 

he pulse. This increased time of travel compensates for the shorter 
ime of travel of the high frequency edge of the band in the 3" round 
.vaveguide, so equalization takes place. Since this waveguide close to 
cutoff introduces considerable delay distortion by itself, the taper effect 
nust be made larger in order to secure the equalization. This can be 
ilone by making the taper sufficiently gradual. This type of equalizer 
ntroduces a rather high loss in the system. For this reason it might 
le used to predistort the signal at an early level in a repeater system, 
ilqualization by this method was suggested by J. R. Pierce. 

.0. MEASURING MODE CONVERSION FROM ISOLATED SOURCES 

I One of the important uses of this equipment has been for the meas- 
irement of mode conversion. W. D. Warters has cooperated in develop- 
ng techniques and carrying out such measurements. One of the prob- 
ems in the design of mode filters used for suppressing all modes except 
;he circular electric ones in round multimode guides is mode conversion. 
Since these mode filters have circular symmetry, conversion can take 
alace only to circular electric modes of order higher than the TEoi mode. 
This conversion is, however, a troublesome one, since these higher 
Drder circular modes cannot be suppressed by the usual type of filter. 

An arrangement for measuring mode conversion at such mode filters 
rom the TEoi to the TE02 mode is being used with the short pulse equip- 
:nent. This employs a 400-foot long section of the b" diameter line. Be- 
ause the coupled- line transducer available had too high a loss to TE02 , a 
3ombined TEoi — TE02 transducer was assembled. It uses one-half of 
:he round waveguide to couple to each mode. Fig. 15 shows this device. 

The use of this transducer and line is illustrated in Fig. 16. Pulses in 
:he TEoi mode are sent into the waveguide by the upper section of the 
transducer as shown. Some of the TEoi energy goes directly across to 
ohe TE02 transducer and appears as the outgoing pulse with a level 
down about 32 db. This is useful as a time reference in the system and 
s shown as the outgoing pulse in the photo of the oscilloscope trace 
ibove. The main energy in the TEoi mode propagates down the line as 
hown by dashed line 2, which is the path of this wave. Most of 
ohis energy goes all the way to the reflecting piston at the far end and 
ohen returns to the TE02 transducer where it gives a pulse which is 
narked TEoi round trip on the trace photograph above. Two thirds of 
;he way from the sending end to the piston, the mode filter being meas- 
ired is inserted in the line. When the TEoi mode energy comes to this 
node filter, a small amount of it is converted to the TE02 mode. This 



58 



THE BELL SYSTEM TECHNICAL JOURNAL 




Fig. 15 — A special experimental transducer for injecting the TEoi mode and' 
receiving the converted TE02 mode in a 5" diameter waveguide. 



continues to the piston by path 4 (with dashed Hnes and crosses) 
and then returns and is received by the TE02 part of the transducer. 
This appears on the trace photo as the TE02 first conversion. When the 
main TEoi energy reflected by the piston comes back to the mode filter, 
conversion again takes place to TE02 • This is shown by path 3 hav- 
ing dashed lines and circles. This returns to the TE02 part of the trans- 
ducer and appears on the trace photo as the TE02 second conversion. 
In addition, a small amount of energy in the TE02 mode is generated 
by the TEoi upper part of the transducer. It is shown by path 5, having' 



OUTGOING PULSE 



TEoi 

ROUND 

TRIP 



TE02 

SECOND 

CONVERSION 



TE02 

FIRST 

CONVERSION 



TE02 

ROUND 

TRIP 



' 




MODE FILTER 



Fig. 16 — Trace photos and waveguide paths traveled when measuring TEoi, 
to TE02 mode conversion at a mode filter with the transducer shown on Fig. 15 



All 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 59 

jihort dashes. This goes down through the waveguide to the far end 
Ijiston and back, and is received by the TE02 transducer and shown as 
[he pulse marked TE02 round trip. The pulse marked TEoi round trip 
las a time separation from the outgoing pulse which is determined by 
,he group velocity of TEoi waves going one round trip in the guide. The 
|rEo2 round trip pulse appears at a time corresponding to the group 
/elocity of the TE02 mode going one round trip in the guide. Spacing the 
node filter two-thirds of the way down produces the two conversion 
:)ulses equally spaced between these two as shown in Fig. 16. The first 
ponversion pulse appears at a time which is the sum of the time taken 
or the TEoi to go down to the filter and the TE02 to go from the filter 
uo the piston and back to the receiver. Because of the slower velocity 
bf the TE02 , this appears at the time shown, since it was in the TE02 
node for a longer time than it was in the TEoi mode. The second con- 
[/ersion, which happened when TEoi came back to the mode filter, comes 
jiarlier in time than the first conversion, since the path for this signal 
ivas in the TEoi mode longer than it was in the TE02 mode. This arrange- 
,nent gives very good time separation, and makes possible a measure- 
Inent of the amount of mode conversion taking place in the mode filters, 
viode conversion from TEoi to TE02 as low as 50 to 55 db down, can be 
neasured with this equipment. 

Randomly spaced single discontinuities in long waveguides can be 
ocated by this technique if they are separated far enough to give in- 
lividually resolved short pulses in the converted mode. Fig. 17 shows 




CONVERSION 

FIRST CONVERSION AT FAR END SECOND CONVERSION 

AT NEAR END SQUEEZED AT NEAR END 

SQUEEZED SECTION SECTION SQUEEZED SECTION 



TO RECEIVER 



TEJo — *- TEq, TE2, -• »- TE,o NEAR END 250 FT OF FAR END 

TRANSDUCER COUPLED LINE SQUEEZED 3"DIAM ROUND SQUEEZED 

TRANSDUCER SECTION GUIDE SECTION 

Fig. 17 — Arrangement used to explain the measurement and location of mode 
onversion from isolated sources. A deliberately squeezed section was placed 
t each end of the long waveguide, producing the pulses shown in the trace photo. 



60 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

an arrangement having oval sections deliberately placed in the wave- ' 
guide in order to explain the method. Pure TEoi excitation is vised, and 
the converted TE21 mode observed with a coupled line transducer giv- ; 
ing an output for that mode alone. ; 

Let us consider first what would happen with the far-end squeezed; 
section alone, omitting the near-end squeezed section from considera- • 
tion. The injected TEoi mode signal would then travel down the 250 , 
feet of 3" diameter round waveguide to the far end with substantially, 
no mode conversion at the level being measured. At this point it goes 
through the squeezed section. Conversion now takes place from the TEou, 
mode to the TE21 mode. Both these modes after reflection from the piston 
travel back up the waveguide to the sending end. The group velocity 
of the TE21 mode is higher than the group velocity of the TEoi mode, so 
energy in these two modes separates, and if a coupling system were 
used to receive energy in both modes, two pulses would appear, with at 
time separation between them. In this case, since the receiver is con- 
nected to the line through the coupled line transducer which is responsive 
only to the TE21 mode, only one pulse is seen, that due to this mode 
alone. This is the center pulse in the trace photograph at the top of 
Fig. 17. If only one mode conversion point at the far end of the guide 
exists, only this one pulse is seen at the receiver. It would be spaced a 
distance away from the injected outgoing pulse that corresponds m:^ 
time to one trip of the TEoi mode down to the far end and one trip of || 
the TE21 mode from the far end back to the receiver. 

Now let us consider what would happen if the near-end squeezed sec- 
tion alone were present. When the TEqi wave passes the oval section! 
just beyond the coupled line transducer, conversion takes place, andi 
the energy travels down the line in both the TEoi and the TE21 modes,:; 
at a higher group velocity in the TE21 mode. These two signals are re- 
flected by the piston at the far end and return to the sending end. The 
TE21 signal comes through the coupled line transducer and appears as 
the pulse at the left of the photo shown on Fig. 17. Now the TEoi energy 
has lagged behind the TE21 energy, and when it gets back to the near- 
end squeezed section, a second mode conversion takes place, and TE21 
mode energy is produced which comes through the coupled line trans-: 
ducer and appears at the receiver at the time of the right hand pulse. 
The spacing between these two pulses is equal to the difference in round 
trip times between the two modes. 

In general, for a single conversion source occurring at any point in 
the line, two pulses will appear on the scope. The spacing between these 
pulses corresponds to the difference in group velocity between the modes. 



WAVEGUIDE TESTING AVITH MILLIMICROSECOND PULSES 61 

{from the point of the discontiimity down to the piston at the far end, 

land then back to the discontinuity. If the discontinuity is at the far 

lend, this time difference becomes zero, and a single pulse is seen. By 

i [making a measurement of the pulse spacing, the location of a single 

i icon version point can be determined. 

[ In the arrangement illustrated in Fig. 17, two isolated sources of 

j conversion existed. They were spaced far enough apart so that they 

\ were resolved by this equipment, and all three pulses were observed. 

The two outside pulses were due to the first conversion point. The center 

pulse was caused by the other squeeze, which was right at the reflecting 

|:)iston. If this conversion point had been located back some distance 

rom the piston, it would have produced two conversion pulses whose 

'spacing could be used to determine the location of the conversion point. 

I The coupled-line transducers are calibrated for coupling loss by send- 

ng the pulse through a directional coupler into the branch normally 

ised for the output to the receiver. This gives a return loss from the 

lirectional coupler equal to twice the transducer loss plus the round 

rip line loss. 

1. MEASURING DISTRIBUTED MODE CONVERSION IN LONG WAVEGUIDES 

; Measurements of mode conversion from TEoi to a number of other 
nodes have been made with 5" diameter guides using this equipment, 
rhe arrangement of Fig. 18 was set up for this purpose. This is the same 
IS Fig. 17, except that a long taper was used at the input end of the 5" 
waveguide, and a movable piston installed at the remote end. 

One of the converted modes studied with this apparatus arrange- 
uent was the TMu mode, which is produced by bends in the guide, 
rhis mode has the same velocity in the waveguide as the TEoi mode. 
Therefore energy components converted at different points in the line 
tay in phase with the injected TEoi mode from which they are converted, 
rhere is never any time separation between these modes, and a single 

TO RECEIVER 



■ I ■■'■ '■■■ ^ 



^^i 



Si 




TEro— *TE^, COUPLED LINE -^^p^P, 5„ 0,^^^ MOVABLE 

TRANSDUCER TRANSDUCER HOLMDEL LINE PISTON 

FOR THE MODE 
BEING MEASURED 

Fig. 18 — Arrangement used for measuring mode conversion in the 5" diameter 
aveguides at Holmdel. 



62 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



narrow pulse like the transmitted one is all that appears on the indicator 
oscilloscope. It is not possible from this to get any information about 
the location or extent of the conversion points in the line. Moving the 
far end piston does not change the relative phases of the modes, so no 
changes are seen in indicator pattern or pulse level as the piston is 
moved. For the Holmdel waveguides, which are about 500 feet long, 
the total round trip T]\In mode converted level varies from 32 to 36 db 
below the input TEoi mode level over a frequency range from 8,800 to 
9,600 mc per second. 

All the other modes have velocities that are different than that of 
the TEoi mode. ^Vhen mode conversion takes place at many closely 
spaced points along the waveguide, the pulses from the various sources 
overlap, and phasing effects take place. In general, a filled-in pulse 
much longer than the injected one is observed. The maximum possible, 
but not necessary, pulse length is equal to the difference in time re- 
quired for the TEoi mode and the converted mode to travel the total 
waveguide length being observed. The phasing effects within the broad- 
ened pulse change its height and shape as a function of frequency and 
line length. 

Measurements of mode conversion from TEoi to TE31 in these wave- 
guides illustrate distributed sources and piston phasing effects. The 
TE3, mode has a group velocity 1.4 per cent slower than the TEoi mode. 
For a full round trip in the 500-foot lines, assuming conversion at the 
imput end, this causes a time separation of about two and one half 
pulse widths between these two modes. The received pulse is about two 
and a half times as long as the injected pulse, indicating rather closely 
spaced sources over the whole line length. For one far-end piston posi- 
tion, the received pattern is shown as the upper trace in Fig. 19. As 
the piston is moved, the center depressed part of the trace gradually 




ImK. 10 — Hocoivcd pulsr patterns willi llic .irraiijicnuMit of Fig. IS used for 
studying conversion to tlie Tlvn mode. 



WAVEGUIDE TESTING WITH MILLIMICROSECOND PULSES 63 

rises until the pattern shown in the lower trace is seen. As the piston 
is moved farther in the same direction the trace gradually changes to 
have the appearance of the upper photo again. Moving the far-end 
piston changes the phase of energy on the return trip, and thus it can 
be made to add to, or nearly cancel out, conversion components that 
originated ahead of the piston. When the time separation becomes 
great enough to prevent overlapping in the pulse ^^^dth, phasing effects 
cannot take place, therefore, the beginning and end of the spread-out 
received pulse are not affected by moving the piston. Energy converted 
at the sending end of the guide travels the full round trip to the piston 
and back in the slower TE31 mode, and thus appears at the latest time, 
which is at the right-hand end of the received pulse. Conversion at the 
piston end returns at the center of the pulse, and conversion on the 
return trip comes at earlier times, at the left-hand part of the pulse. 
The TEoi mode has less loss in the guide than the TE31 mode. Since the 
energy in the earlier part of the received pulse spent a greater part of 
the trip in the lower loss TEoi mode before conversion, the output is 
higher here, and slopes off toward the right, where the later returning 
energy has gone for a longer distance in the higher loss mode. The pulse 
height at the maximum shows the converted energy from that part of 
the line to be between 30 and 35 db below the incident TEoi energy 
level over the measured band\\ddth. 

Measurements of mode conversion from TEoi to TE21 in these wave- 
guides show these same effects, and also a phasing effect as a function 
of frequency. The TE21 mode has a group velocity 2.4 per cent faster 
than the TEoi mode. For a full round trip in the guides, this is a time 
separation of about four pulse mdths between the modes. At one fre- 
quency and one far-end piston position, the TE21 response shown as the 
top trace of Fig. 20 was obtained. Moving the far-end piston gradually 
changed this to the second trace from the top, and further piston mo- 
tion changed it back again. This is the same kind of piston phasing effect 
observed in the TE31 mode conversion studies. The irregular top of this 
broadened pulse indicates fewer conversion points than for the TE31 
mode, or phasing effects along the guide length. Since the TE21 mode 
has a higher group velocity' than the TEoi mode, energy converted at 
the beginning of the guide returns at the earlier or left-hand part of the 
pulse, and conversions on the return trip, having traveled longer in 
the slower TEoi mode, are on the right-hand side of the pulse. This is 
just the reverse of the situation for the TE31 mode. Since the loss in the 
TE21 mode is higher than in the TEoi mode, the right side of this broad- 
ened pulse is higher than the left side, as the energy in the left side has 



64 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



gone further in the higher loss TE21 mode. Conversions from the piston 
end of the guide return in the center of the pulse, and only in this re- 
gion do piston phasing effects appear. As the frequency is changed the ' 
pattern changes, until it reaches the extreme shape shown in the next- 
to-the-bottom trace, with this narrower pulse coming at a time corre- 
sponding to the center of the broadened pulse at the top. Further fre- 
quency change in the same direction returns the shape to that of the 
top traces. At the frequency giving the received pulse shown on the 
next-to-the-bottom trace, moving the far-end piston causes a gradual 
change to the shape shown on the lowest trace. This makes it appear 
as if the mode conversion were coming almost entirely from the part of 
the guide near the piston end at this frequency. The upper traces appear 
to show that more energy is converted at the transducer end of the 
waveguide at that frequency. It would seem that at certain frequencies 
some phase cancellation is taking place between conversion points 
spaced closely enough to overlap within the pulse width . At frequencies 
between the ones giving traces like this, the appearance is more like 
that shown for the TE31 mode on Fig. 19 except for the slope across the 
top of the pulse being reversed. The highest part of this TEoi pulse is 




Fiff. 20 — Received pulse patterns witli the urrangemeiit of Fig. 18 used for 
studying conversion to the TE21 mode. 



WAVEGUIDE TESTING WITH MILLIMICKOSECOND PULSES 65 

24 to 27 db below the injected TEoi pulse level for the 5" diameter 
Holmdel waveguides. 

12. CONCLUDING REMARKS 

The high resolution obtainable with this millimicrosecond pulse 
equipment provides information difficult to obtain by any other means. 
These examples of its use in waveguide investigations indicate the 
possibilities of the method in research, design and testing procedures. 
It is being used for many other similar purposes in addition to the illus- 
tratio)is given here, and no doubt many more uses will be found for 
such short pulses in the future. 

REFERENCES 

1. S. E. Miller and A. C. Beck, Low-loss Waveguide Transmission, Proc. I.R.E., 

41, pp. 348-358, March, 1953. 

2. S. E. Miller, Waveguide As a Communication Medium, B. S. T. J., 33, pp. 1209- 

1265, Nov., 1954. 

3. C. C. Cutler, The Regenerative Pulse Generator, Proc. I.R.E., 43, pp. 140- 

148, Feb., 1955. 

4. S. E. Miller, Coupled WaveTheory and Waveguide Applications, B. S. T. J., 33, 

pp. 661-719, May, 1954. 



Experiments on the Regeneration of 
Binary Microwave Pulses 

By O. E. DeLANGE 

(Manuscript received September 7, 1955) 

A sifnple device has been produced for regenerating binary pulses directly 
at microwave frequencies. To determine the capabilities of such devices one 
of them was included in a circidating test loop in which pidse groups were 
passed through the device a large number of titnes. Residts indicate that 
even in the presence of serious noise and bandwidth limitations pidses can 
be regenerated many times and still shotv no noticeable deterioration. Pic- 
tures of circulated pidses are included which illustrate performance of the 
regenerator. 

INTRODUCTION 

The chief advantage of a transmission system employing Ijinary pulses 
resides in the possibility of regenerating such pulses at intervals along 
the route of transmission to prevent the accumulation of distortion due 
to noise, bandwidth limitations and other effects. This makes it possible 
to take the total allowable deterioration of signal in each section of a 
long relay system rather than having to make each link sufficiently good 
to prevent total accumulated distortion from becoming excessive. This 
has been pointed out by a number of writers. i-- 

W. M. GoodalP has shown the feasibility of transmitting television 
signals in binary form. Such transmission reciuires a considerable amount 
of bandwidth; a seven digit system, for example, would require trans- 
mission of seventy million pulses per second. This need for wide bands 
makes the microwave range an attractive one in which to work. S. E. 
Miller* has pointed out that a binary system employing regeneration 
might prove to be especially advantageous in waveguide transmission. 

1 B. M. Oliver, J. R. Pierce and, C. E. Shannon, The Pliilosophv of PCM, Proc. 
I. R.E., Nov., 1948. 

'^ L. A. Meacham and Iv Peterson, An Experimental Multichannel Pulse Code 
Modulation System of Toll Quality, B. S. T. J., Jan. 1948. 

' W. M. Goodall, Television l)y Pulse Code Modulation, B. S. T. J., Jan., 1951. 

* S. E. Miller, Waveguide as a Communication Medium, B. S. T. J., Nov., 1954. 

67 



68 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



INPUT 



FILTER 



AUTOMATIC 

GAIN 

CONTROL 



REGENERATOR 



DETECTOR 



TIMING 

WAVE 

GENERATOR 



FILTER 



OUTPUT 




Fig. 1 — A typical regenerative repeater shown in block form. 



That the Bell System is interested in the long-distance transmission 
of television and other broad-band signals is evident from the number 
of miles of such broad-band circuits, both coaxial cable and microwave 
radio, ^ now in service. These circuits provide high-grade transmission 
because each repeater was designed to have a very fiat frequency charac- 
teristic and linear phase over a considerable bandwidth. Furthermore, 
these characteristics are very carefully maintained. For a binary pulse 
system employing regeneration the requirements on flatness of band and 
linearity of phase can be relaxed to a considerable degree. The compo- 
nents for such a system should, therefore, be simpler and less expensive 
to build and maintain. Reduced maintenance costs might well prove to 
be the chief virtue of the binary system. 

Since the chief advantage of a binary system lies in the possibility of 
regeneration it is obvious that a very important part of such a system is 
the regenerative repeater employed. Fig. 1 shows in block form a typical 
broad-band, microwave repeater. Here the input, which might come from 
either a radio antenna or from a waveguide, is first passed through a 
proper microwave filter then amplified, probably by a traveling-wave 
amplifier. The amplified pulses of energy are regenerated, filtered, am- 
plified and sent on to the next repeater. The experiment to be described 
here deals primarily with the block labeled "Regenerator" on Fig. 1. 

In these first experiments one of our main objectives was to keep the 
repeater as simple as possible. This suggests regeneration of pulses 
directly at microwave frequency, which for this experiment was chosen 
to be 4 kmc. It was suggested by J. R. Pierce and W. D. Lewis, both of 
Bell Telephone Laboratories, that further simplification might be made 
possible by accepting only partial instead of complete regeneration. 
This suggestion was adopted. 

For the case of complete regeneration each incoming pulse inaugurates 
a new pulse, perfect in shape and correctly timed to be sent on to the 

'A. A. Roetken, K. D. Smith and R. W. Friis, The TD-2 System, B. S. T. J., 
Oct., 1951, Part II. 



REGENERATION OF BINARY MICROWAVE PULSES 69 

next repeater. Thus noise and other disturbing effects are completely 
eliminated and the output of each repeater is identical to the original 
signal which entered the system. For the case of partial regeneration 
incoming pulses are retimed and reshaped only as well as is possible with 
simple equipment. Obviously the difference between complete and partial 
. regeneration is one of degree. 

One object of the experiment was to determine how well such a partial 
regenerator would function and what price must be paid for employing 
partial instead of complete regeneration. The regenerator developed 
consists simply of a waveguide hybrid junction with a silicon crystal 
diode in each side arm. It appears to meet the requirement of simplicity 
in that it combines the functions of amplitude slicing and pulse retiming 
in one unit. A detailed description of this unit will be given later. Al- 
though the purpose of this experiment was to determine what could be 
accomplished in a very simple repeater we must keep in mind that 
superior performance would be obtained from a regenerator which ap- 
proached more nearly the ideal. For some applications the better re- 
generator might result in a more economical system even though the 
regenerator itself might be more complicated and more expensive to 
produce. 

METHOD OF TESTING 

The regeneration of pulses consists of two functions. The first function 

is that of removing amplitude distortions, the second is that of restoring 

each pulse to its proper time. The retiming problem divides into two 

[parts the first of which is the actual retiming process and the second 

! that of obtaining the proper timing pulses with which to perform this 

lifunction. In a practical commercial system timing information at a 

[repeater would probably be derived from the incoming signal pulses. 

There are a number of problems involved in this recovery of timing 

pulses. These are being studied at the present time but were avoided in 

the experiment described here by deriving such information from the 

local synchronizing gear. 

Since the device we are dealing with only partially regenerates pulses 
it is not enough to study the performance of a single unit — we should 
•like to have a large number operating in tandem so that we can observe 
'what happens to pulses as they pass through one after another of these 
Tegenerators. To avoid the necessity of building a large number of units 
the pulse circulating technique of simulating a chain of repeaters was 
j employed. Fig. 2 shows this circulating loop in block form. 



70 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



HYBRID 

JUNCTION 

' NO. 3 




CW 

OSCILLATOR 

(4 KMC) 



TRAVELING WAVE 

AMPLIFIER 

(NOISE GENERATOR) 



Fig. 2 — The circulating loop. 



To provide RF test pulses for this loop the output of a 4 kmc, cw 
oscillator is gated by baseband pulse groups in a microwave gate or 
modulator. The resultant microwa\-e pulses are fed into the loop (heavy 
line) through hybrid junction No. 1. They are then amplified by a trav- 
eling-wave amplifier the output of which is coupled to the pulse regen- 
erator through another hybrid junction (No. 2). The purpose of this 
hybrid is to provide a position for monitoring the input to the regen- 
erator. A monitoring position at the output of the regenerator is pro- 
vided by a third hybrid, the main output of which feeds a considerable 
length of waveguide which provides the necessary loop delay. At the far 
end of the waveguide another hybrid (No. 4) makes it possible to feed 
noise, which is derived from a traveling-wave amplifier, into the loop. 
The combined output after passing through a band pass filter is ampli- 



REGENEKATION OF BINARY MICROWAVE PULSES 



71 



fied by another traveling-wave amplifier and fed back into the loop in- 
put thus completing the circuit. 

The synchronizing equipment starts out with an oscillator going at 
approximately 78 kc. A pulse generator is locked in step with this os- 
cillator. The output of the pulser is a negative 3 microsecond pulse as 
shown in Fig. 3A. After being amplified to a level of about 75 volts 
this pulse is applied to the helix of the first traveling-wave tube to re- 
I duce the gain of this tube during the 3-microsecond interval. Out of each 
12.8/xsec interval pulses are allowed to circulate for O.S/xsec but are blocked 
I for the remaining 3Msec thus allowing the loop to return to the quiescent 
i condition once during each period as shown on Figs. 3A and 3C. 

The S^sec pulse also synchronizes a short-pulse generator. This unit 
delivers pulses which are about 25 millimicroseconds long at the base 
and spaced by 12.8/isec, i.e., Avith a repetition frequency of 78 kc. See 
Fig. 3B. 

In order to simulate a PCM system it was decided to circulate pulse 



CIRCULATING INTERVAL 
9.8/ZS 



QUENCHING 
INTERVAL 

-3//S-*| 



(A) GATING CYCLE 




(B) SHORT SYNCHRONIZING PULSES 



--24 GROUPS OF PULSES 






(C) CIRCULATING PULSE GROUPS 



GROUP GROUP GROUP 
1 2 3 



lOOMyUS 



^ k ^^-o.4;uS-^^ I (D) PULSE GROUPS (EXPANDED) 

■ ' |300M/US| I I 



I 



(E) TIMING WAVE (40MC) EXPANDED 




TIME 



Fig. 3 — Timing events in the circulating loop. 



72 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

groups rather than individual pulses through the system. These were 
derived from the pulse group generator which is capable of delivering 
any number up to 5 pulses for each short input pulse. These pulses are 
about 15 milli-microseconds long at the base and spaced 25 milli-micro- 
seconds apart. The amplitude of each of these pulses can be adjusted 
independently to any value from zero to full amplitude making it pos- 
sible to set up any combination of the five pulses. These are the pulses 
which are used to gate, or modulate, the output of the 4-kmc oscillator. 

The total delay around the waveguide loop including TW tubes, etc.,' 
was 0.4)usec or 400 milli-microseconds. This was sufficient to allow time 
between pulse groups and yet short enough that groups could circulate 
24 times in the available 9.8jLtsec interval. This can be seen from Figs. 
3C and 3D. The latter figure shows an expanded view of circulating 
pulse groups. The pulses in Group 1 are inserted into the loop at the 
beginning of each gating cycle, the remaining groups result from circu- 
lation around the loop. 

When all five pulses are present in the pulse groups the pulse repeti- 
tion frequency is 40 mc. (Pulse interval 25 milli-microseconds). For this 
condition timing pulses should be supplied to the regenerator at the rate 
of 40 million per second. These pulses are supplied continuously and not 
in groups as is the case with the circulating pulses. See Fig. BE. In order 
to maintain time coincidence between the circulating pulses and the tim- 
ing pulses the delay around the loop must be adjusted to be an exact 
multiple of the pulse spacing. In this experiment the loop delay is equal 
to 16-pulse intervals. Since timing pulses are obtained by harmonic 
generation from the quenching frequency as will be discussed later this 
frequency must be an exact submultiple of pulse repetition frequency. 
In this experiment the ratio is 512 to 1. 

Although the above discussion is based on a five-pulse group and 
40-mc repetition frequency it turned out that for most of the experi- 
ments described here it was preferable to drop out every other pulse, 
leaving three to a group and resulting in a 20-mc repetition frequency. 
The one exception to this is the limited-band-width experiment which 
will be described later. - 

For all of the experiments described here timing pulses were derived 
from the 78-kc quenching frequency by harmonic generation. A pulse 
with a width of 25 milli-microseconds and with a 78-kc repetition fre- 
quency as shown in Fig. 3B supplied the input to the timing wave gen- 
erator. This generator consists of several stages of limiting amplifiers all 
tuned to 20 mc, followed by a locked-in 20-mc oscillator. The output of 
the amplifier consists of a train of 20-mc sine waves with constant ampli- 



til 



REGENERATION OF BINARY MICROWAVE PULSES 73 

tude for most of the 12.8Msec period but falling off somewhat at the end 
of the period. This-train locks in the oscillator which oscillates at a con- 
stant amplitude over the whole period and at a frequency of 20 mc. 
Timing pulses obtained from the cathode circuit of the oscillator tube 
pro^'ided the timing waves for most of the experiments. For the experi- 
ment where a 40-mc timing wave was required it was obtained from the, 
20 mc train by means of a frequency doubler. For this case it is necessary 
for the output of the timing wave generator to remain constant in ampli- 
tude and fixed in phase for the 512-pulse interval between synchronizing 
pulses. 

In spite of the stringent requirements placed upon the timing equip- 
ment it functioned well and maintained synchronism over adequately 
long periods of time without adjustment. 

PERFORMANCE OF REGENERATOR 

Performance of the regenerator under various conditions is recorded 
on the accompanying illustrations of recovered pulse envelopes. The 
first experiment was to determine the effects of disturbances which arise 
at only one point in a system. Such effects were simulated by adding 
disturbances along with the group of pulses as they were fed into the 
circulating loop from the modulator. This is equivalent to having them 
occur at only the first repeater of the chain. 

Some of the first experiments also involved the use of extraneous 
pulses to represent noise or distortion since these pulses could be syn- 
chronized and thus studied more readily than could random effects. In 
, Fig. 4A the first pulse at the left represents a desired digit pulse with 
' its amplitude increased by a burst of noise, the second pulse represents 
' a clean digit pulse, and the third pulse a burst of noise. This group is at 
1 the input to the regenerator. Fig. 4B shows the same group of pulses 
' after traversing the regenerator once. The pulses are seen to be shortened 
due to the gating, or retiming, action. There is also seen to be some ampli- 
tude correction, i.e. the two desired pulses are of more nearly the same 
j amplitude and the undesired pulse has been reduced in relative ampli- 
tude. After a few trips through the regenerator the pulse group was 
rendered practically perfect and remained so for the rest of the twenty- 
four trips around the loop. Fig. 4C shows the group after 24 trips. In 
'another experiment pulses were circulated for 100 trips without deteri- 
oration. Nothing was found to indicate that regeneration could not be 
repeated indefinitely. 
Figs. 5 A and 5B represent the same conditions as those of 4 A and 4B 



74 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 




Fig. 4 — Effect of regeneration on disturbances which occur at only one re- 
peater. A — Input to regenerator, original signal. B — Output of regenerator, 
first trip. C — Output of regenerator, 24th trip. 




Fig. 5 — l']ffect of regeneration on disturbances which occur at only one re- 
peater. A — Input to regenerator, first four groups. B — Output of regenerator, 
first four groups. C — Output of regenerator, increased input level. 



REGENERATION OF BINARY MICROWAVE PULSES 



75 




Fig. 6 — Effect of regeneration on disturbances which occur at only one re- 
peater. A — Input to regenerator, original signal. B — ^ Output of regenerator, 
first trip. C • — Oi^tput of regenerator, 24th trip. 



except that the oscilloscope sweep has been contracted in order to show 
the progressive effects produced by repeated passage of the signal through 
the regenerator. Fig. 5B shows that after the pulses have passed through 
the regenerator only twice all visible effects of the disturbances have 
been removed. Fig. 5C shows the effect of simply increasing the RF 
pulse input to the regenerator by approximately 4 db. The small "noise" 
pulse which in the previous case was quickly dropped out because of 
being below the slicing level has now come up above the slicing level 
and so builds up to full amplitude after only a few trips through the 
regenerator. Note that in the cases shown in Figs. 4 and 5 discrimination 
against unwanted pulses has been purely on an amplitude basis since 
the gate has been unblocked to pulses with amplitudes above the slicing 
level whenever one of these distiu'bing pulses was present. 

For Fig. 6A conditions are the same as for Fig. 4A except that an ad- 
ditional pulse has been added to simulate intersymbol noise or inter- 
ference. Fig. 6B indicates that after only one trip through the regenerator 
the effect of the added pulse is very small. After a few trips the effect 
is completely eliminated leaving a practically perfect group which con- 
tinues on for 24 trips as shown by Fig. 6C. For the intersymbol pulse, 
discrimination is on a time basis since this interference occurs at a time 



76 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 




Fig. 7 — Effect of regenerating in amplitude without retiming. A — Outputof 
regenerator, no timing, firt trip. B — Output of regenerator, no timing, 10th trip. 
Output of regenerator, no timing, 23rd trip. 

when no gating pulse is present and hence finds the gate blocked regard- 
less of amplitude. 

To show the need for retiming the pictures shown on Figs. 7 and 8 
were taken. These were taken with the amplitude slicer in operation but 
with the pulses not being retimed. Figs. 7A, 7B and 7C, respectively, 
show the output of the slicer for the first, tenth and twenty-third trips. 
After ten trips, there is noticeable time jitter caused by residual noise 
in the system; after 23 trips this jitter has become severe though pulses 
are still recognizable. It should be pointed out that for this experiment 
no noise was purposely added to the system and hence the signal-to- 
noise ratio was much better than that which would probably be encoun- 
tered in an operating system. For such a system we would expect time 
jitter effects to build up much more rapidly. For Fig. 8 conditions are 
the same as for Fig. 7 except that the pulse spacing is decreased by the 
addition of an extra pulse at the input. Now, after ten trips, time jitter 
is bad and after 23 trips the pulse group has become little more than a 
smear. This increased distortion is probably due to the fact that less 
jitter is now required to cause overlap of pulses. There may also be some 
effects due to change of duty cycle. For Fig. 9 there was neither slicing 
nor retiming of pulses. Here, pulse groups deteriorate very rapidly to 
nothing more than blobs of energy. Note that there is an increase of 



i 



REGENERATION OF BINARY MICROWAVE PULSES 



77 




Fig. 8 — ■ Effect of regenerating in amplitude without retiming. A — Output of 
regenerator, no timing, first trip. B — Output of regenerator, no timing, 10th 
trip. C — Output of regenerator, no timing, 23rd trip. 




Fig. 9 — Pulses circulating through the loop without regeneration. A — Origi- 
nal input. B — 4th trip without regeneration. C — 20th to 24th trip without re- 
generation. 



'8 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



iWWWMMMWWIWWMilJflM ^ I I . rlil l lT- i \m....: i n i T i inr- IH. 




Fig. 10 — The regeneration of band-limited pulses. A — Input to regenerator, 
first two groups. B — Output of regenerator, first two groups. C — Output of 
regenerator, 24th trip. 

amplitude with each trip around the loop indicating that loop gain was 
slightly greater than unity. Without the sheer it is difficult to set the 
gain to exactly unity and the amplitude tends to either increase or de- : 
crease depending upon whether the gain is greater or less than unity. 
Results indicated by the pictures of Fig 9 are possibly not typical of a 
properly functioning system but do show what happened in this par- . 
ticular sj^stem when regeneration was dispensed with. 

Another important function of regeneration is that of overcoming . 
band-limiting effects. Figs. 10 and 11 show what can be accomplished. . 
For this experiment the pulse groups inserted into the loop were as shown i| 
at the left in Fig. lOA. These pulses were 15 milli-microseconds wide at 
the base and spaced by 25 milli-microseconds which corresponds to a j 
repetition frequency of 40 mc. After passing through a band-pass filter 
these pulses were distorted to the extent shown at the right in Fig. lOA. 
From the characteristic of the filter, as shown on Fig. 12, it is seen that 
the bandwidth employed is not very different from the theoretical min- 
imum required for double sideband transmission. This minimum char- 
acteristic is shown by the dashed lines on Fig. 12. Fig. lOB shows that 
at the output of the regenerator the effects of band limiting have been 
removed. This is borne out by Fig. IOC which shows that after 24 trips 
the code group was still practically perfect. It should l)e pointed out 
that the pulses traversed the filter once for each trip around the loop, 



REGENERATION OF BINARY MICROWAVE PULSES 



79 




Fig. 11 — The regeneration of band-limited pulses. A — Input to regenerator, 
first two groups. B — Output of regenerator, first two groups. C — Output of re- 
generator, 24th trip. 



that is for each trip the input to the regenerator was as shown at the right 
of Fig. lOA and the output as shown by Fig. lOB. It is important to 
note that Fig. 12 represents the frequency characteristic of a single hnk 
of the simulated system. The pictures of Fig. 11 show the same experi- 
ment but this time with a different code group. Any code group which 
we could set up with our five digit pulses was transmitted equally well. 
In order to determine the breaking point of the experimental system, 
broad-band noise obtained from a traveling-wave amplifier was added 
into the system as shown on Fig. 2. The breaking point of the system is 
the noise level which is just sufficient to start producing errors at the 
output of the system.* The noise is seen to be band-limited in exactly 
the same way as the signal. With the system adjusted to operate properly 
the level of added noise was increased to the point where errors became 
barely discernible after 24 trips around the loop. Noise level was now 
reduced slightly (no errors discernible) and the ratio of rms signal to rms 
noise measured. Fig. 13A shows the input to the regenerator for the 23rd 
and 24th trips with this amount of noise added. Note that the noise has 

* The type of noise employed has a Gaussian amplitude distribution and there- 
fore there was actually no definite breaking point — the rate at which errors Oc- 
curred increased continuously as noise amplitude was increased. The breaking 
point was taken as the noise level at which errors became barely discernible on 
the viewing oscilloscope. More accurate measurements made in other experiments 
indicate that this is a fairly satisfactory criterion. 



80 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

28 



24 



IT) 

aJ2o 

03 



O 



16 



to 

If) 

g 12 
a. 

UJ 

5 8 



ll 










1 — 




1 




\ 












i 




A 


\ 


^ 








1 




1 

1 

1 
1 




/ 


< 


\ 






1 




1 

1 


/ 


/ 




\ 










1 


/ 






< 


V h* 


--20 M 


1 


--20MC *] 


1/ 








v. 




1 




/ 










1 


■~ TX^ 


^ 


■rrD*^ 


/J 







3950 3960 3970 3980 3990 4000 4010 4020 4030 4040 
FREQUENCY IN MEGACYCLES PER SECOND 

Fig. 12 — Characteristics of the band-pass microwave filter. 



m % 



I 




JYYYYYYTin 



Fig. 13. — The regeneration of pulses in the presence of broad-hand, random 
noise added at each repeater. A — Ini)ut to regenerator, 23rd and 24th trijis, 
broad-band noise added. B — Ini)ut to regenerator, 23rd and 24th trips, no added 
noise. C — 20-mc timing wave. 



\ 



KEGENERATION OF BINARY MICROWAVE PULSES 



81 




Fig. 14 — The regeneration of pulses in the presence of interference occurring 
at each repeater. A — Original signal with added moduhited carrier interference. 
B — Input to regenerator, 24th trip, niochilatod carrier interference. C — Output 
of regenerator, 24th trip, modulated carrier interference. 



produced a considerable broadening of the oscilloscope trace. Fig. 13B 
shows the same pulse groups with no added noise. These photographs are 
included to give some idea as to how bad the noise was at the l;)reaking 
point of the system. Of course maximum noise peaks occur rather infre- 
quently and do not show on the photograph. At the output of the re- 
generator effects due to noise were barely discernible. This output looked 
so much like that shown at Fig. 14C that no separate photograph is 
shown for it. 

Figs. 14A, 14B and 14C show the effects of a different type of inter- 
ference upon the system. This disturbance was produced by adding into 
the system a carrier of exactly the same frequency as the signal carrier 
(4 kmc) but modulated by a 14-mc wave, a frequency in the same order 
as the pulse rate. Here again the level of the interference was adjusted 
to be just below the l)reaking point of the system. A comparison between 
Figs. 14B and 14C gives convincing evidence that the regenerator has 
substantially restored the waveform. 

For the case of the interfering signal a ratio of signal to interference 
of 10 db on a peak-to-peak basis was measured when the interference 
was just below the breaking point of the system. This, of course, is 4 db 
above the theoretical value for a perfect regenerator. For the case of 



82 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

broad-band random noise an rms signal to noise ratio of 20 dl) was meas- 
ured.* This compares Avith a ratio of 18 db as measured by Messrs. 
Meacham and Peterson for a system employing complete regeneration 
and a single repeater, f 

Recently, A. F. Dietrich repeated the circulating loop experiment at 
a radio frequency of 11 kmc. His determinations of required signal-to- 
noise ratios are substantially the same as those reported here. From the 
various experiments we conclude that for a long chain of properly func- 
tioning regenerative repeaters of i-he type discussed here practically 
perfect transmission is obtained as long as the signal-to-noise ratio at 
the input to each repeater is 20 db or better on an rms basis. In an operat- 
ing system it might be desirable to increase this ratio to 23 db to take 
care of deficiencies in automatic gain controls, power changes, etc. 

From the experiments we also conclude that the price we pay for using 
partial instead of complete regeneration is about 3 to 4 db increase in 
the required signal-to-noise ratio. In a radio system which provides a 
fading margin this penalty would be less since the probability that two 
or more adjacent links will reach maximum fades simultaneously is very ' 
small. Under these conditions only one repeater at a time would be near 
the breaking point and the system would behave much as though the 
repeater provided complete regeneration. 

TIMING 

Although we have considered the problem of retiming of signal pulses 
up to now we have not discussed the problem of obtaining the necessary ' 
timing pulses to perform this function, but have simpl}^ assumed that a 
source of such pulses was available. As w^as mentioned earlier timing I 
pulses would probably be derived from the signal pulses in a practical »^ 
system. These pulses would be fed into some narrow band amplifier 
tuned to pulse repetition frequency. The output of this circuit could be 
made to be a sine wave at repetition frequency if gaps between the input 
pulses were not too great. Timing pulses could be derived from this sine 
wave. This timing equipment could be similar to that used in these ex- 
periments and described earlier. Further study of the problems of ob- 
taining timing information is being made. 

* For Gaussian noise it is not possible to specif.y a theoretical value of minimum 
S/N ratio without specifying the tolerable percentage of errors. For the number of 
errors detectable on the oscilloscope it seems rasonable to assume a 12 db peak 
factor for the noise. The peak factor for the signal is 3 db. The 6 db peak S/N 
which would be required for an ideal regenerator then becomes 15 db on an rms 
basis. 

t L. A. Meacham and E. Peterson, B. S. T. J., p. 43, Jan., 1948. 



" 



KEGENERATION OF BINARY MICROWAVE PULSES 



83 



' GATING 
PULSE 




INPUT 



OUTPUT 



Fig. 15A — Low-frequency equivalent of the partial regenerator. 



DESCRIPTION OF REGENERATOR 

This device regenerates pulses by performing on them the operations 
of ''slicing" and retiming. 
An ideal slicer is a device with an input-output characteristics such as 
shown by the dashed lines of Fig. 15C. It is seen that for all input levels 
below the so-called slicing level transmission through the device is zero 
but that for all amplitudes greater than this value the output level is 
finite and constant. Thus, all input voltages which are less than the slic- 
ing level have no effect upon the output whereas all input voltages 
greater than the slicing level produce the same amplitude of output. 
Normally conditions are adjusted so that the slicing level is at one-half 




INPUT LEVEL 



Fig. 15B — Characteristics of the separate branches with ditterential bias. 



84 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 




INPUT LEVEL 



Fig. 15C — Resultant output with differential bias. 




BRANCH 2 
BRANCH 1 



RESULTANT 



INPUT LEVEL 



Fig. 15D — Characteristics of the separate branches and resultant output with 
equal biases. 



of peak pulse amplitude — then at the output of the slicer there will be 
no effect whatsoever from disturbances unless these disturbances exceed 
half of the pulse amplitude. It is this slicing action which removes the 
amplitude effects of noise. Time jitter effects are removed by retiming, 
i.e., the device is made to have high loss regardless of input level except 
at those times when a gating pulse is present. 

Fig. 15A shows schematically a low-frequency equivalent of the re- 
generator used in these experiments. Here an input line divides into two 
identical branches isolated from each other and each with a diode shunted 
across it. The outputs of the two branches are recombined through neces- 
sary isolators to form a single output. The phase of one branch is re- 
versed before recombination, so that the final output is the difference 
between the two individual outputs. 

Fig. 15B shows the input-output characteristics of the two branches 
when the diodes are biased back to be non-conducting by means of bias 
voltages Vi and V2 respectively. For low levels the input-output char- 
acteristic of both branches will be linear and have a 45° slope. As soon 



REGENEKATION OF BINARY MICROWAVE PULSES 



85 



as the input voltage in a branch reaches a vakie equal to that of the back 
bias the diode will start to conduct, thus absorbing power and decrease 
the slope of the characteristic. The output of Branch 1 starts to flatten 
off when the input reaches the value Vi , while the output of Branch 2 
does not flatten until the input reaches the value V2 . The combined 
output, which is equal to the differences of the two branch outputs, is 
then that shown by the solid line of Fig. 15C and is seen to have a transi- 
tion region between a low output and a high output level. If the two 
branches are accurately balanced and if the signal voltage is large com- 
pared to the differential bias V2 — Vi the transition becomes sharp and 
the device is a good slicer. 

If the two diodes are equally biased as shown on Fig. 15D the outputs 
of the two branches should be nearly equal regardless of input and the 
total output, which is the difference between the two branch outputs, 
will always be small. 

Fig. 16 shows a microwave equivalent of the circuit of Fig. 15A. In 
the microwave structure lengths of wave-guide replace the wire lines and 
branching, recombining and isolation are accomplished by means of 
hybrid junctions. The hybrid shown here is of the type known as the lA 
junction. 

Fig. 17 shows another equivalent microwave structure employing only 
one hybrid. This is the type used in the experiments described here. The 
[output consists of the combined energies reflected from the two side 
jarms of the junction. With the junction connected as shown phase rela- 
Itionships are such that the output is the difference between the reflec- 



GATING 
PULSE 



^(— r-V\^^^ 



RF 
INPUT ARM 



PROBE 




TERMINATION 

I 



ARM 4 



I— vw-^ 



Fig. 16 — Microwave regenerator. 



86 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



tions from the two side arms so that when conditions in the two arms 
are identical there is no output. The crystal diodes coupled to the side 
arms are equivalent to those shunted across the two lines of Fig. 15A. 

Fig. 18, which is a plot of the measured input-output characteristic 
of the regenerator used in the loop test, shows how the device acts as a 
combined sheer and retimer. Curve A, ol)tained with equal biases on the 
two diodes, is the characteristic with no gating pulse applied i.e. the 
diodes are normally biased in this manner. It is seen that this condition 
produces the maximum of loss through the device. By shifting one diode 
bias so as to produce a differential of 0.5 volt the characteristic changes 
to that of Curve B. This differential bias can be supplied by the timing 
pulse in such a way that this pulse shifts the characteristic from that 
shown at A to that shown at B thus decreasing the loss through the de- 
vice by some 12 to 15 db during the time the pulse is present. In this way 
the regenerator is made to act as a gate — though not an ideal one. 

We see from curve B that with the differential bias the device has the 
characteristic of a slicer — though again not ideal. For lower levels of 
input there is a region over which the input-output characteristic is 
square law with a one db change of input producing a two db change of 
output. This region is followed by another in which limiting is fairly 
pronounced. At the 8-db input level, which is the point at which limiting 
sets in, the loss through the regenerator was measured to be approxi- 
mately 12 db. The characteristic shown was found to be reproducible 
both in these experiments at 4 kmc and in those bj'- A. F. Dietrich at 
11 kmc. 

For a perfect slicer only an infinitesimal change of input level is re- 



GATING 
PULSE 




■AAV-i_ 



ARM 2 



RF 

OUTPUT 



Fig, 17 — Microwave regenerator employing a single hybrid junction. 



REGENERATION OF BINARY MICROWAVE PULSES 



87 



ID 

m 
o 

LU 

a 



D 



3 

o 



-10 



-12 



-14 



-16 



-18 



-20 



-22 



-24 















V, = 0.5 
V2 = 


<-- 






12 DB 


LOSS 


i>— <! 


P'< 


JH " 


^ 










/I 


^ 


1 
1 

) 
( 
1 












( 


r 




1 

6DB 
1 
1 












r 





-6DB- 


1 
1 

.--J 










(B) 


/ 








V, = V2 


A 


f 




/ 










/ 


Y 






/ 








/ 


/ 






/I 


1 






(A)/ 


/ 








1/ 






/ 


^ 














y 


K 
















/ 















6 8 10 12 

INPUT LEVEL IN DECIBELS 



14 



16 



18 



Fig. 18 — Static characteristics of the regenerator employed in these experiments. 

f}uired to change the output from zero to maximum. The input level at 
which this transition takes place is the slicing level and has a very defi- 
nite value. For a characteristic such as that shown on Fig. 18 this point 
is not at all definite and the question arises as to how one determines the 
slicing level for such a device. Obviously this point should be somewhere 
on the portion of the characteristic where expansion takes place. In the 
case of the circulating loop the slicing level is the level for which total 
gain around the loop is exactly etiual to unity. Why this is so can be seen 
from Fig. 19 which is a plot of gain \'ersus input level for a repeater 
containing a sheer with a characteristic as shown by curve B of Fig. 18. 
Amplifiers are necessary in the loop to make up for loss through the re- 
generator and other components. For Fig. 11) we assume that these 
amplifiers have been adjusted so that gain around the loop is exactly 
unity for an input pulse having a peak amplitude corresponding to the 



88 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 




-3 



-2-1 1 2 3 4 5 

INPUT LEVEL IN DECIBELS ABOVE SLICING LEVEL 



Fig 19 — Gain characteristics of u repeater providing partial regeneration. 



point F' of Fig. 18. On Fig. 19 all other levels are shown in reference to 
this unity-gain value. 

From Fig. 19 it is obvious that a pulse which starts out in the loop 
with a peak amplitude exactly equal to the reference, or slicing level, 
will continue to circulate without change of amplitude since for this 
level there is unity gain around the loop. A pulse with amplitude greater 
than the slicing level will have its amplitude increased by each passage 
through a regenerator until it eventually reaches a value of +6 db. It 
will continue to circulate at this amplitude, for here also the gain around 
the loop isVmity.* Any pulse with peak amplitude less than the reference 
level will have its amplitude decreased by successive trips through the 
regenerator and eventually go to zero. We also see that the greater the 
departure of the amplitude of a pulse from the slicing level the more 
effect the regenerator has upon it. This means that the device acts much 
more powerfully on low level noise than on noise with pulse peaks near 
the slicing level. As examples consider first the case of noise peaks only 
1 db below slicing level at the input (peak S/N = 7 db). At this level 
there is a 1 db loss through the repeater so that at the output the noise 
peaks will be 2 db below reference to give a *S/A^ ratio of 8 db. Next 



* Note that llic ^-fi-dl) level is at a point of stable equilibrium whereas at the 
slicing level C(iuilil)rium is unstable. 



REGENERATION OF BINARY MICROWAVE PULSES 89 

consider noise with a peak level 5 db below slicing level (S/N =11 db) 
at the input. The loss at this level is 5 db resulting in a noise level 10 db 
below reference to give a S/N ratio of 16 db. We see that a 4 db improve- 
ment in S/N ratio at the input results in an 8 db improvement in this 
ratio at the output. 

Everything which was said above concerning the circulating loop ap- 
plies equally to a chain of identical repeaters. To set the effective slicing 
level at half amplitude at each repeater in a chain one would first find 
two points on the sheer characteristics such as P and P' of Fig. 18. The 
point P should be in the region of expansion and P' in the limiting region. 
Also the points should be so chosen that a 6 db increase of input from 
that at point P results in a 6 db increase in output at the point P'. If 
now at each repeater we adjust pulse peak amplitude at the sheer input 
to a value corresponding to that at point P' we will have unity gain 
from one repeater to the next at levels corresponding to pulse peaks. 
We will also have unity gain at levels corresponding to one half of pulse 
amplitude. The effective slicing level is thus set at half amplitude. Ob- 
viously the procedure for setting the slicing level at some value other 
than half amplitude would be practically the same. It should be pointed 
out that although half amplitude is the preferred slicing level for base- 
band pulses this is not the case for carrier pulses. W. R. Bennett of Bell 
Telephone Laboratories has shown that for carrier pulses the probability 
that noise of a given power will reduce signal pulses below half amplitude 
is less than the probability that this same noise will exceed half ampli- 
tude. This comes about from the fact that for effective cancellation there 
must be a 180° phase relationship between noise and pulse carrier. For 
this reason the slicing level should be set slightly above half amplitude 
for a carrier pulse system. 

The difference in performance between a perfect sheer and one with 
characteristics such as shown on Fig. 18 are as follows: For the perfect 
sheer no effects from noise or other disturbances are passed from one 
repeater to the next. For the case of the imperfect regenerator some ef- 
fects are passed on and so tend to accumulate in a chain of repeaters. 
To prevent this accumulated noise from building up to the breaking 
point of the system it is necessary to make the signal-to-noise ratio at 
each repeater somewhat better than that which would be required with 
the ideal sheer. For the case of random noise the required S/N ratio 
seems to be about 5 or 6 db above the theoretical value. This is due in 
part to sheer deficiency and in part to other system imperfections. 



90 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 19o() 

CONCLUSIONS 

It is possible to build a simple device for regenerating pulses directly 
at microwave frequencies. A long chain of repeaters employing this 
regenerator should perform satisfactorily as long as the rms signal-to- 
noise ratio at each repeater is maintained at a value of 20 db or greater. 
There are a number of remaining problems which must be solved before 
we have a complete regenerative repeater. Some of these problems are: 

(1) Recovery of information for retiming from the incoming pulse train; 

(2) Automatic gain or level control to set the slicing level at each re- 
peater; (3) Simple, reliable, economical, broad-band microwave ampli- 
fiers. (4) Proper filters — both for transmitting and receiving. Traveling- 
wave tube development should eventually result in amplifiers which 
will meet all of the requirements set forth in (3) above. Any improve- 
ments which can be made in the regenerator without adding undue 
complications would also be advantageous. 

ACKNOWLEDGMENTS 

A. F. Dietrich assisted in setting up the equipment described here and 
in many other ways. The experiment would not have been possible with- 
out traveling-wave tubes and amplifiers which were obtained through 
the cooperation of M. E. Hines, C. C. Cutler and their associates. I wish 
to thank W. M. Goodall, and J. R. Pierce for many valuable suggestions. 



Crossbar Tandem as a Long Distance 
Switching System 

By A. O. ADAM 

(Manuscript received March 4, 1955) 

Major toll switching features are being added to the crossbar tandem 
switching system for use at many of the important long distance switching 
centers of the nationwide network. These include automatic selection of one 
of several alternate routes to a 'particular destination, storing and sending 
forward digits as required, highly flexible code conversion for transmitting 
digits different from those received, and a translating arrangement to select 
the most direct route to a destination. The system is designed to serve both 
operator and customer dialed long distance traffic. 

INTRODUCTION 

The crossbar tandem switching system,^ originally designed for switch- 
ing between local dial offices, will now play an important role in nation- 
wide dialing. New features are now available or are being developed that 
will permit this system to switch all types of traffic. As a result, crossbar 
[ tandem offices will have widespread use at many of the important switch- 
ing centers of the nationwide switching network. 

This paper briefly reviews the crossbar tandem switching system and 
its application for local switching, followed by discussion of the general 
aspects of the nationwide switching plan and of the major new features 
required to adapt crossbar tandem to this plan. 

CROSSBAR TANDEM OFFICES USED FOR LOCAL SWITCHING 

Crossbar tandem offices are now used in many of the large metropolitan 
areas throughout the country for interconnecting all types of local dial 
offices. In these applications they perform three major functions. Basi- 
cally, they permit economies in trunking by combining small amounts of 

91 



02 THE BELL SYSTEM TECHXIf AL JOURNAL, JANUARY 1956 

traffic to and from the local offices into larger amounts for routing over 
common triuik groups to gain increased efficiency resulting in fewer over- 
all trunks. 

A second important function is to permit handling calls economically 
between different types of local offices which are not compatible from the 
standpoint of intercommunication by direct pulsing. Crossbar tandem 
offices serve to connect these offices and to supply the conversion from 
one type of pulsing to another where such incompatibilities exist. 

The third major function is that of centralization of equipment or 
services. For example, centralization of expensive charging equipment at 
a crossbar tandem office results in efficient use of such equipment and 
over-all lower cost as compared with furnishing this equipment at each 
local office requiring it. Examples of such equipment are remote control 
of zone registration and centralized automatic message accounting.^ Cen- 
tralization of other services such as weather bureau, time-of-day and 
similar services can be furnished. 

The first crossbar tandem offices were installed in 1941 in New York, 
Detroit and San Francisco. These offices were equipped to interconnect 
local panel and No. 1 crossbar central offices in the metropolitan areas, 
and to complete calls to manual central offices in the same areas. The war 
years slowed both development and production and it was not until the 
late 40's that many features now in use were placed in service. These 
later features enable customers in step-by-step local central offices on the 
fringes of the metropolitan areas to interconnect on a direct dialing basis 
with metropolitan area customers in panel, crossbar, manual and step- 
by-step central offices. This same development also permitted central 
offices in strictly step-by-step areas to be interconnected by a crossbar 
tandem office where direct interconnecting was not economical. Facilities 
were also made available in the crossbar tandem system for completing 
calls from switchboards where operators use dials or multifrequency key 
pulsing sets. 

Since a crossbar tandem office usually has access to all of the local 
offices in the area in which it is installed, it is attractive for handling 
short and long haul terminating traffic. The addition of toll terminal 
equipment at Gotham Tandem in New York City in 1947 permitted 
operators in New York State and northern New Jersey as well as distant 
operators to dial or key pulse directly into the tandem equipment for 
completion of calls to approximately 350 central offices in the New York 
metropolitan area. This method of completing these calls without the 
aid of the inward operators was a major advance in using tandem switch- 
ing ecjuipment for speeding completion of out-of-town calls. 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 



93 



CROSSBAR TANDEM SWITCHING ARRANGEMENT 

The connections in a crossbar tandem office are established through 
crossbar switches mounted on incoming trunk link and outgoing office 
link frames shown on Fig. 1. The connections set up through these 
switches are controlled by equipment common to the crossbar tandem 
office which is held only long enough to set up each individual connec- 
tion. Senders and markers are the major common control circuits. 

The sender's function is to register the digits of the called number, 
transmit the called office code to the marker and then, as subsequently 
directed by the marker, control the outpulsing to the next office. 

The marker's function is to receive the code digits from the sender 
for translation, return information to the sender concerning the de- 
tails of the call, select an idle outgoing trunk to the called destination 
and close the transmission path through the crossbar switches from the 
incoming to the outgoing trunk. 

GENERAL ASPECTS OF NATIONWIDE DIALING 

Operator distance dialing, now used extensively throughout the 
country, as well as customer direct distance dialing are based on the 
division of the United States and Canada into numbering plan areas, 
interconnected by a national network through some 225 Control Switch- 
ing Points (CSP's) equipped with automatic toll switching systems. 
^ An essential element of the nationwide dialing program is a universal 
numbering plan^ wherein each customer will have a distinctive number 
which does not conflict with the number of any other customer. The 
method employed is to divide the United States and Canada geographi- 



INCOMING 

TRUNK FROM 

ORIGINATING 

OFFICE 



TANDEM 

TRUNK 



TRUNK LINK FRAME 



9 ? 



TRUNK LINK 
CONNECTOR 



SENDER LINK 



SENDER LINK 
CONTROL CIRCUIT 



SENDER 



OFFICE LINK FRAME 



<? 9 



OFFICE LINK 
CONNECTOR 



J 4_ 

MARKER 



CONNECTOR 



OUTGOING 
TRUNK 



MARKER 



Fig. 1 — Crossbar tandem switching arrangement. 



94 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

cally into more than 100 numbering plan areas and to give each of these 
a distinctive three digit code with either a 1 or as the middle digit. 
Each numbering plan area will contain 500 or fewer local central offices 
each of which will be assigned a distinctive three-digit office code. 
Thus each of the telephones in the United States and Canada will have, 
for distance dialing purposes, a distinct identity consisting of a three 
digit area code, an office code of two letters and a numeral, and a sta- 
tion number of four digits. Under this plan, a customer will dial 7 digits 
to reach another customer in the same numbering area and 10 digits to 
reach a customer in a different numbering area. 

A further reciuirement for nationwide dialing of long distance calls is 
a fundamental plan"* for automatic toll switching. The plan provides a 
systematic method of interconnecting all the local central offices and 
toll switching centers in the United States and Canada. As shown on 
Fig. 2, several local central offices or "end offices" are served by a single 
toll center or toll point that has trunks to a "home" primary center 
which serves a group of toll centers. Each primary center, has trunks to 
a "home" sectional center which serves a larger area of the country. 
Similuj-ly, the entire toll dialing territory is divided into eleven very 
large areas called regions, each having a regional center to serve all the 
sectional centers in the region. One of the regional centers, probably 
St. Louis, Missouri, will be designated the national center. The homing 
arrangements are such that it is not necessary for end offices, toll centers, 
toll points and primary centers to home on the next higher ranking 
office since the complete final route chain is not necessary. For example, 
end offices may be served directly from any of the higher ranking switch- 
ing centers also shown in Fig. 2. 

Collectively, the national center, the regional centers, the sectional 
centers and the primary centers will constitute the control switching 
points for nationwide dialing. The basic switching centers and homing 
arrangements are illustrated in Fig. 3. 

TANDEM CROSSBAR FEATURES FOR NATIONWIDE DIALING 

The broad objective in developing new features for crossbar tandem 
is to provide a toll switching system that can be used in cities where 
the large capacity and the full versatilit}^ of the No. 4 toll crossbar 
switching system-'' may not be economical. 

The application of crossbar tandem two-wire switching systems at 
primary and sectional centers has been made possible by the extended 
use of high speed carrier systems. The echoes at the 2-wire crossbar 
tandem switching offices can be effectively reduced by providing a high 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 



95 



office balance and by the use of impedance compensators and fixed pads. 
A well balanced two-wire switching system, proper assignment of inter- 
toll trunk losses, and the use of carrier circuits with high speed of propa- 
gation will permit through switching Mdth little or no impairment from 
an echo standpoint. 

The new features for crossbar tandem will provide arrangements 
necessary for operation at control switching points (CSP's). These in- 
clude automatic alternate routing, the ability to store and send forward 




TP 



e 



I I NC = NATIONAL CENTER 
RC = REGIONAL CENTER 
/\ SC = SECTIONAL CENTER 
( J PC = PRIMARY CENTER 

Fig. 2 — Homing arrangement for local central offices and toll centers. 



TC = TOLL CENTER 
TP = TOLL POINT 
EG = END OFFICE 




96 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 



97 



digits as required, highly flexible code conversion (transmitting forward 

i different digits for the area or office code instead of the dialed digits), 

prefixing digits ahead of the called office code, and six-digit translation. 

ALTERNATE ROUTING 

The control switching points will be interconnected by a final or 
"backbone" network of intertoll trunks engineered so that very few 
calls will be delayed. In addition, direct circuits between individual 
switching offices of all classes will be provided as warranted by the 
traffic density. These are called "high-usage" groups and are not en- 
gineered to handle all the traffic offered to them during the busy hour. 
Traffic offered to a high-usage group which finds all trunks busy will be 
automatically rerouted to alternate routes®-^ consisting of other high- 
usage groups or to the final trunk group. The abi.ity of the crossbar 
tandem equipment at the control switching point to select one of several 
alternate routes automatically, when all choices in the first route are 
busy, contributes to the economy of the plant and provides additional 
protection against complete interruption of service when all circuits on 
a particular route are out of service. 

Fig. 4 shows a hypothetical example of alternate routing when a 
crossbar tandem office at South Bend, Indiana, receives a call destined 
for ^Youngstown, Ohio. To select an idle path, using this plan, the 
switching equipment at South Bend first tests the direct trunks to 
Youngstown. If these are all busy, it tests the direct trunks to Cleveland 
where the call would be completed over the final group to Youngstown. 
If the group to Cleveland is also busy, South Bend would test the group 



CHICAGO 



SOUTH BEND 

CROSSBAR 

TANDEM 



CLEVELAND 




-YOUNGSTOWN 



ITT5BURGH 



Fig. 4 — Toll network — alternate routing. 



98 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



in 

UJ 

I- 
o 
o 
a. 

m 
t- 
< 

2 

a. 
m 
I- 
_j 
< 



mm 

•"^^ 
1- I- 

5m< 

2cD2 
_JOO 

^^ 

< o 

O 1- 



gur-H 



I- < 

2 



^ b 



UJ 

z o 

UJ Q 

rO 



UJ 

z 



i 



Ul 



UJ 



m 



X 



ll 



C\j 



I 



UJ 



(To: 

1- 



X' 



Z 



o o o 









UJ 

<> 



m 



UJ 
(0 



Hi' 



Hi' 



X 



oo 



2-mmiiiii 



o 
(J 




X 



a 

UJ 
UJQ 

i£2 

U.UJ 

°5 
QO 

uicr 

-lU. 

_j 

<UJ 

UQ 
Ol 
Ul 



D 
CD 

Z 

^ 
a. 

H 
(- 

o 

UJ 



Hi 



2 in 



C 

3 

O 






o 

O 
+2 






Hi 



^ 



irt 

2 

O 

UJI- 

oo 

ZUJ 

aim 
I- to 

DO 

otr 

I 
I 
I 
I 



I I 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 99 

to Pittsburgh and on its last attempt it would test the final group to 
Indianapolis. If the call were routed to Pittsburgh or Indianapolis, the 
switching equipment at these points would attempt by first choice and 
alternate routes to reach Youngstown. The final choice backbone route 
would be via Indianapolis, Chicago, St. Louis, Pittsburgh, Cleveland to 
Youngstown. Should all the trunks in any of the final groups tested be 
busy no further attempt to complete the call is made. It is unlikely 
that so many alternate routes would be provided in actual practice 
since crossbar tandem can test only a maximum of 240 trunks on each 
call and, in the case illustrated, the final trunk group to Indianapolis 
may be quite large. 

The method employed by the crossbar tandem marker in selecting 
the direct route and subsequent alternate routes is shown in simplified 
form on Fig. 5. As a result of the translating operation, the marker 
selects the first choice route relay, corresponding to the called destina- 
tion. Each route relay has a number of contacts which are connected to 
supply all the information recjuired for proper routing of the call. Several 
of these contacts are used to indicate the equipment location of the 
trunks and the number of trunks to be tested. The marker tests all of 
the trunks in the direct route and if they are busy, the search for an 
idle trunk continues in the first alternate route which is brought into 
play from the "route advance" cross-connection shown on the sketch. 
As many as three alternate routes in addition to the first choice route 
can be tested in this manner. 

STORING AND SENDING FORWARD DIGITS AS REQUIRED 

The crossbar tandem equipment at control switching points must 
store all the digits received and send forward as many as are required to 
complete the call. 

The called number recorded at a switching point is in the form of 
ABX-XXXX if the call is to be completed in the same numbering 
plan area. If the called destination is in another area, the area code 
XOX or XIX precedes the 7 digit number. The area codes XOX or XIX 
and the local office code ABX are the digits used for routing purposes 
and are sufficient to complete the call regardless of the number of switch- 
ing points involved. Each control switching point is arranged to ad- 
vance the call towards its destination when these codes are received. 
If the next switching point is not in the numbering area of the called 
telephone, the complete ten-digit number is needed to advance the 
call toward its destination. If the next switching point is in the num- 



100 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

bering area of the called telephone the area code is not needed and seven 
digits will suffice for completing the call. 

For example, suppose a call is originated by a customer in South 
Bend, Indiana, destined for customer NAtional 4-1234 in Washington, 
D.C. If it is assumed that the route to Washington is via a switching 
center in Pittsburgh, then the crossbar tandem equipment at South 
Bend pulses forward to Pittsburgh 202-NA4-1234, 202 being the area 
code for the District of Columbia. Pittsburgh in turn will delete the 
area code and send NA4-1234 to the District of Columbia terminating 
area. 

As another example, suppose the crossbar tandem office at South 
Bend receives a call from some foreign area destined to a nearby step- 
by-step end office in Michigan. The crossbar tandem equipment re- 
ceives and stores a ten-digit number comprising the area code and the- 
seven digits for the office code and station number. Assuming that 
direct trunks to the step-by-step end office in Michigan are available, 
the area code and office code are deleted and the line number only is 
pulsed forward. To meet all conditions, the equipment is arranged to 
permit deletion of either the first three, four, five or six digits of a ten- 
digit number. 

CODE CONVERSION 

At the present time, some step-by-step primary centers reach other 
offices by the use of routing codes that are different from those assigned 
under the national numbering plan. This arrangement is used to obtain 
economies in switching equipment of the step-by-step plant and is 
accetpable with operator originated calls. However, with the intro- 
duction of customer direct distance dialing, it is essential that the codes 
used by customers be in accordance with the national numbering plan. 
The crossbar tandem control switching point must then automatically 
provide the routing codes needed by the intermediate step-by-step 
primary centers. This is accomplished by the code conversion feature 
which substitutes the arbitrary digits required to reach the called office 
through the step-by-step systems. Fig. 6 illustrates an application of 
this feature. It shows a crossbar tandem office arranged for completing 
calls through a step-by-step toll center to a local central office, GArden 
8, in an adjacent area. A call reaching the crossbar tandem office for a 
customer in this office arrives with the national number, 218-GA8-1234. 
To complete this call, the crossbar tandem equipment deletes the area 
code 218 and pulses forward the local office code and number. If the 



« 



CROSSBAK TANDEM AS A TOLL SWITCHING SYSTEM 



101 



call is switched to an alternate route via the step-by-step primary 
center, it will be necessary for the crossbar tandem equipment to delete 
the area code 218 and substitute the arbitrary digits 062 to direct the 
call through the switches at the primary center, since the toll center 
requires the full seven digit number for completing the call. 

PREFIXING DIGITS 

It may be necessary to route a call from one area to another and back 
to the original area for completion. Such a situation arises on a call 
from Amarillo to Lubbock, Texas, both in area 915 when the crossbar 
tandem switching equipment finds all of the direct paths from Amarillo 
to Lubbock busy as illustrated on Fig. 7. The call could be routed to 
Lubbock via Oklahoma City which is in area 405. A seven-digit number 
for example, MAin 2-1234, is received in the crossbar tandem office at 
Amarillo. Assuming that the call is to be switched out of the 915 area 
through the 405 area and back to the 915 area for completion, it is 
necessary for the crossbar tandem office in Amarillo to prefix 915 to the 
MAin 2-1234 number so that the switching equipment in Oklahoma 
City will know that the call is for the 915 area and not for the 405 area. 

Prefixing digits may also be needed at crossbar tandem offices to 
route calls through step-by-step primary centers. The crossbar tandem 
office in Fig. 8 receives the seven digit number MA2-1234 for a call to a 



701 

AREA 



218 

AREA 



NUMBER 

RECEIVED 

218-GA8-1234 



CROSSBAR 
TANDEM 



NUMBER 

OUTPULSED 

062-GA8-1234 



STEP-BY-STEP 
PRIMARY CENTER 



ALTERNATE 
ROUTE 

DIRECT 
ROUTE 



GA8-I234 



i^ 



GA8-t234 



SX S 

TOLL 
CENTER 



GA8-1234 



LOCAL 
CUSTOMER 





Fig. 6 — Code conversion. 



102 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



customer in the Madison office in the same area. However, since the 
toll center needs the full seven digit number for completing the call and 
since the step-by-step switches at the primary center "use up" two 
digits (04) for its switching, the crossbar tandem equipment must 
prefix 04 to the seven digit number. 



METHOD OF DETERMINING DIGITS TO BE TRANSMITTED 

The circuitry involved for transmitting digits as received, prefixing, 
code conversion and for deletion involves both marker and sender 
functions. The senders have ten registers (1 to 10) for storing incoming 
digits and three registers (A A, AB, AC) for storing the arbitrary digits 
that are used for prefixing and code conversion. 

On a ten-digit call into a crossbar tandem switchmg center the area 
code XOX, the office code ABX and the station number XXXX are 
stored in the inpulsing or receiving registers of the sender. The code 
digits XOX-ABX are sent to the marker which translates them to 
determine which of the digits received by the sender should be outpulsed. 
It also determines whether arbitrary digits should be transmitted ahead 
of the digits received and, if so, the value of the arbitrary digits to be 
stored in the sender registers AA, AB and AC. Case 1 of Fig. 9 assumes 
that a ten-digit number has been stored in the sender registers 1 to 10 



915 

AREA 



INCOMING 
TOLL CALL 



LOCAL 

OFFICE 




AMARILLO 
CROSSBAR 

TANDEM 
OFFICE 



NUMBER 

RECEIVED 

MA2-1234 




405 
AREA 



^< 


■^ 








.-^ 


^^o^^ 

.-^^^"J^^' 




OKLAHOMA CITY 
TOLL OFFICE 


LUBBOCK 

TOLL 

OFFICE 




MA 2 

LOCAL 

CO. 










CUSTOMER 





















MA 2-1234 



Fig. 7 — Prefixing. 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 



103 



and that the marker has mformed the sender the called number is to be 
sent as received. The outpulsing control circuit is connected to each 
register in turn through the steering circuit SI, S2, etc. and sends the 
digits stored. 

Case 2 illustrates a situation where the sender has stored ten digits 
in registers 1 to 10 and received information from the marker to delete 
the digits in registers 1 to 3 inclusive and to substitute the arbitrary 
digits stored in registers AA, AB and AC. The outpulsing circuit is 
first connected to register AA through steering circuit PSl, then to AB 
through PS2, continuing in a left to right sequence until all digits are 
outpulsed. 

Case 3 covers a condition where the sender has stored seven digits and 
has obtained information from the marker to prefix the two digits 
stored in registers AB and AC. Outpulsing begins at the AB register 
through steering circuit PS2 and then advances through steering circuit 
PS3 to the AC register, continuing in a left to right seciuence until all 
digits have been transmitted. 

These are only a few of the many combinations that are used to give 
the crossbar tandem control switching equipment complete pulsing 
flexibility. 



SIX-DIGIT TRANSLATION 

Six-digit translation will be another feature added to the crossbar 
tandem system. When only three digits are translated, it is necessary to 
direct all calls to a foreign area over a single route. The ability to trans- 
late six digits permits the establishment of two or more routes from the 
switching center to or towards the foreign area. This is shown in Fig. 



LOCAL 
OFFICE 




NUMBER OUTPULSED 
04-MA2-1234 



MADISON 
OFFICE 



MA2-I234 



CROSSBAR 
TANDEM 


t 












■ » 


' 


' — 1 n 






MA2- 
1 


1234 


EIVED 
4 






4 


1 
1 


TOLL 
CENTER 






— >- 


















MADISON 2- 

1234 



STEP-BY-STEP 
PRIMARY CENTER 



Fig. 8 — Prefixing. 



104 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



f/l 



10 with Madison and Milwaukee, Wisconsin, in area 414 and Belle 
Plaine Crossbar Tandem in Chicago, Illinois, in area 312. An economical 
trunking plan may provide for direct circuits from Chicago to each 
place. If only three-digit translation were provided in the Chicago 
switching equipment, the route to both places would be selected as a 
result of the translation of the 414 area code alone and, therefore, calls 
to central offices reached through Madison, would need to be routed 
via Milwaukee. This involves not only the extra trunk mileage, ])ut 
also the use of an extra switching point. With six-digit translation, both 
the area code and the central office code are analyzed, making it 
possible to select the direct route to either city. 

Six-digit translation in crossbar tandem will involve primarily the 
use of a foreign area translator and a marker. The translator will have 
a capacity for translation of five foreign areas and for 60 routes to each 
area. Since the translator holding time is very short, one translator is 
sufficient to handle all of the calls requiring six-digit translation, but 
two are always provided for hazard and maintenance reasons. 

On a call requiring six-digit translation the first three digits are 



CASE 1 ^ 

DIGITS RECEIVED 


t 




2 




3 




-IMPULSING 
4 5 


REGISTERS - 
6 7 


8 




9 




10 


\ 


X 









X 




A 




B 




X 




X 




X 




X 




X 












































. 






. 






; 






i. 




OUTPULSING 
CONTROL 




;Si 


: Sd 


S J 


Sa - 


S3 


So 


. S / 


- So 


b» * oiu 































































CASE 2 

DIGITS RECEIVED 



OUTPULSING 
CONTROL 







DIGITS CODE CONVERTED 
AA AB AC 



;: PS1 



X' 



PS2 ;:PS3 ;:S4 ):S6 ~:S6 ;;S7 ::S8 



10 



59 ;:S10 



CASE 3 

DIGITS RECEIVED 



DIGITS PREFIXED 
AB AC 



B' 



C 



OUTPULSING 
CONTROL 



i PS2 : : P 



B 



PS3 •:SI ':S2 ::S3 :;S4 : : S5 :'S6 ■;S7 



Fig. 9 — Method used for outpulsing digits. 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 



105 



translated in the marker and the second three digits in a foreign area 
translator which is associated with the marker. Fig. 11 shows, in simpli- 
fied form, how this translation is accomplished. 

The first three digits, corresponding to the area code, are received by 
a relay code tree in the marker which translates it into one of a thousand 
code points. This code point is cross-connected to the particular relay of 
the five area relays A(3-A4 which has been assigned to the called area. 
A foreign area translator is now connected to the marker and a corre- 
sponding area relay is operated in it. The translator also receives the 
called office code from the sender via the marker and by means of a 
relay code tree similar to that in the marker translates the office code 
to one of a thousand code points. This code point plus the area relay is 
sufficient to determine the actual route to be used. As shown on the 
sketch, wires from each of the code points are threaded through trans- 
formers, two for each area. When the marker is ready to receive the 
route information, a surge of current is sent through one of these threaded 
wires which produces a voltage in the output winding to ionize the 
T- and U- tubes. Only the tubes associated with the area involved in 
the translation pass current to operate one each of the eight T- and U- 
relays. This information is passed to the marker and registered on 
corresponding tens and units relays. These operate a route relay which 



WISCONSIN 




MICH. 



J 



ILLINOIS 



CHICAGO = 
' f BELLE \ 

1 AREA IplaINeJ 
\312 I 

^- — 1 



I N D. 



ROUTE WITHOUT 6 DIGIT TRANSLATION 



ROUTE WITH 6 DIGIT TRANSLATION 



Fig. 10 — Six-digit translation. 



106 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 




Fig. 11 — Method used for foreign area translation. 



CROSSBAR TANDEM AS A TOLL SWITCHING SYSTEM 107 

provides all the information necessary for routing the call to the central 
office involved. 

CUSTOMER DIRECT DISTANCE DIALING 

Crossbar tandem will provide arrangements permitting customers in 
step-by-step offices to dial their own calls anywhere in the country. 
Centralized automatic message accounting previously mentioned will 
be used for charging purposes. While the basic plan for direct distance 
dialing provides for the dialing of either seven or ten digits, it will be 
necessary for the customer in step-by-step areas to prefix a three-digit 
directing code, such as 112, to the called number. This directing code 
is required to direct the call through the step-by-step switches to the 
crossbar tandem office so that the seven or ten digit number can be 
registered in the crossbar tandem office. 

When a customer in a step-by-step office originates a call to a distant 
customer whose national number is 915-CH3-1234, he first dials the 
directing code 112 and then the ten-digit number. The dialing of 112 
causes the selectors in the step-by-step office to select an outgoing trunk 
to the crossbar tandem office. The incoming trunk in the crossbar tandem 
office has quick access to a three-digit register. The register must be 
connected during the interval between the last digit of the directing 
code and the first digit of the national number to insure registration of 
this number. This arrangement is used to permit the customer to dial 
all digits without delay and avoids the use of a second dial tone. If this 
arrangement were not used, the customer would be required to wait 
after dialing the 112 until the trunk in the tandem crossbar office could 
gain access to a sender through the sender link circuit which would 
then signal the customer to resume dialing by returning dial tone. 

After recording the 915 area code digits in the case assumed, the 
CH3-1234 portion of the number is registered directly in the tandem 
sender which has been connected to the trunk while the customer was 
dialing 915. When the sender is attached to the trunk, it signals the 
three-digit register to transfer the 915 area code digits to it via a con- 
nector circuit. Thus when dialing is complete, the entire number 915- 
CH3-1234 is registered in the sender. 

Crossbar tandem is being arranged to serve customers of panel and 
No. 1 crossbar offices for direct distance dialing. At the present time, 
ten digit direct distance dialing is not available to these customers 
because the digit storing equipments in these offices are limited to 
eight digits. Developments now under way, will provide arrangements 
for expanding the digit capacity in the local offices so that ultirnately 



108 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

calls from custoniers in panel and No. 1 crossbar offices may be routed 
through crossbar tandem cr other equivalent offices to telephones 
anywhere in the country. 

CONCLUSION 

The new features developed for crossbar tandem will adapt it to 
switching all types of traffic at many important switching centers of 
the nationwide toll network. Of the 225 important toll switching centers 
now contemplated, it is expected that about 80 of these will be ecjuipped 
with crossbar tandem. 

REFERENCES 

1. Collis, R. E., Crossbar Tandem System, A.I.E.E. Trans., 69, pp. 997-1004, 1950. 

2. King, G. v.. Centralized Automatic Message Accounting, B.S.T.J., 33, pp. 

1331-1342, 1952. 

3. Nunn, W. H., Nationwide Numbering Plan, B.S.T.J., 31, pp. 851-859, 1952. 

4. Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, B. S.T.J. , 31, pp. 

832-850, 1952. 

5. Shipley, F. F., Automatic Toll Switching Systems, B.S.T.J., 31, pp. 860-882, 

1952. 

6. Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- 

ments in Alternate Routing Trunk Networks, B.S.T.J., 33, pp. 277-302, 1954. 

7. Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Laboratories 

Record, 32, pp. 51-57, Feb. 1954. 



Growing Waves Due to Transverse 

Velocities 

By J. R. PIERCE and L. R. WALKER 

(Manuscript received March 30, 1955) 

This paper treats propagation of slow waves in two-dimensional neu- 
tralized electron floiv in which all electrons have the same velocity in the 
direction of propagation hut in which there are streams of two or more veloci- 
ties normal to the direction of propagation. In a finite beam in which 
' electrons are reflected elastically at the boundaries and in which equal dc 
currents are carried by electrons with transverse velocities -\-Ui and — Wi , 
there is an antisi/mmetrical growing ivave if 

Up ~ {rUi/Wf 

and a symmetrical growing wave if 



y- 



i{Tu,/wy 



Here cop is plasma frequency for the total charge density and W is beam 
width. 

INTKODUCTION 

i It is well-known that there can be growing waves in electron flow when 
the flow is composed of several streams of electrons having different 
velocities in the direction of propagation of the waves. ' While Birdsall 
considers the case of growing waves in electron flow consisting of streams 
which cross one another, the growing waves which he finds apparently 
occur when two streams have different components of velocity in the 
direction of propagation. 

This paper shows that there can be growing waves in electron flow 
consisting of two or more streams with the same component of velocity 
in the direction of wave propagation but with different components of 
velocity transverse to the direction of propagation. Such growing Avaves 
can exist when the electric field varies in strength across the flow. Such 
waves could result in the amplification of noise fluctuations in electron 

' flow. They could also be used to amplify signals. 

109 



110 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Actual electron flow as it occurs in practical tubes can exhibit trans- 
verse velocities. For instance, in Brillouin flow, ' • if we consider electron 
motion in a coordinate system rotating with the Larmor frequency we 
see that electrons with transverse velocities are free to cross the beam 
repeatedly, being reflected at the boundaries of the beam. The trans- 
verse \-elocities may be completely disorganized thermal velocities, or 
they may be larger and better-organized velocities due to aberrations at 
the edges of the cathode or at lenses or apertures. Two-dimensional 
Brillouin flow allows similar transverse motions. 

It would be difficult to treat the case of Brillouin or Brillouin-like flow 
with transverse velocities. Here, simpler cases with transverse velocities 
will be considered. The first case treated is that of infinite ion-neutra- 
lized two-dimensional flow with transverse velocities. The second case 
treated is that of two-dimensional flow in a beam of finite width in which 
the electrons are elastically reflected at the boundaries of the beam. 
Growing waves are found in both cases, and the rate of growth may be 
large. 

In the case of the finite beam both an antisymmetric mode and a 
symmetric mode are possible. Here, it appears, the current density 
required for a growing wave in the symmetric mode is about ^^ times 
as great as the current density required for a growing wa^•e in the anti- 
symmetric mode. Hence, as the current is increased, the first growing 
waves to arise might be antisymmetric modes, which could couple to a 
symmetrical resonator or helix only through a lack of symmetry or 
through high-level effects. 

1 . Infinite two-dimensional flow 

Consider a two-dimensional problem in which the potential varies 
sinusoidally in the y direction, as exp{—j^z) in the z direction and as exp 
(jut) with time. Let there be two electron streams, each of a negative 
charge po and each moving with the velocity ?/o in the z direction, but 
with velocities Wi and —ih respectively in the y direction. Let us denote 
ac quantities pertaining to the first stream by subscripts 1 and ac quan- 
tities pertaining to the second stream by subscripts 2. The ac charge 
density will be denoted by p, the ac velocity in the y direction by y, 
and the ac velocity in the z direction by i. We will use linearized or 
small-signal equations of motion.^ We will denote differentiation with 
respect to ?/ by the operator D. 

The equation of continuity gives 

jupi = -D(piUi + po?yi) + j|8(piWo + pnii) (1.1)1 

jcopo = -D{-p-iHi -\- pi)lj':d + il3(P2''o + Poi2) (1.2) 



t; 






GROWING WAVES DUE TO TRANSVERSE VELOCITIES 111 

Let US define 

dx = i(co - ^u,) + u,D (1.3) 

do = ./(w - i8wo) - uj) (1.4) 

We can then rewrite (1.1) and (1.2) as 

f/iPi = Poi-Diji + j(3zi) (1.5) 

dopi = Pi^{ — Dy2 + .7/3i2) (1.0) 

We will assume that we are dealing ^^•ith slow waves and can use a po- 
tential V to describe the field. We can thus write the linearized equations 
of motion in the form 

r/iii = -j-^F (1.7) 

m 

d2h = -j-^V (1.8) 

m 

drlji = - DV (1.9) 

m 

d,y, = 1 DV (1.10) 

w 

From (1.5) to (1.10) we obtain 

^m = ~ PoiD' - ^')V (1.11) 

m 

d'p2= --poiD'- ^')V (1.12) 

m 

Now, Poisson's equation is 

{D' - ^')V = _^L±£! (1.13) 

From (1.11) to (1.13) we obtain 

{D' - /3^)y = - Kco/ (^1 + ^^ (D' - /3^)7 (1.14) 



9 ^ 
— Z— po 

2 m 

Wp = 

e 

Here Wp is the plasma frequency for the charge of both beams. 



(1.15) 



112 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Either 



or else 



(2)' - /3')7 = 

— C0„" (c/l" + ^2") 



^ 2 di^ d.} 

We will consider this second case. 

W(< should note from (1.3) and (1.4) that 

d{ = u^-D^ - (co - /5(/„)" + 2yD(co - |8?/.o)«i 

^2^ = ?<i-D" - (co - ^ihf - 2jD{o^ - l3uo)ui 

di' + f/o' = 2{u{D' - (co - iSwo)'] 

rfiW = [uiD' + (co - /3;/„)T 

Thus, (1.17) becomes 



(1.16) 



(1.17) 



(1.18) 
(1.19) 
(1.20) 
(1.21) 

(1.22) 
WD"" + (co - j8mo)2]^ 

If the quantities involved vary sinusoidally with y as cos ru or sin yy, 



-co, 



\u{lf - (co - /3ao)'] 



then 



Our equation becomes 



D' 



-7 



(1.23) 



CO 



P L 



1 + 



CO — jS'Uo 



T^Wi^ 



_ / co - 13^0 Y" 
\ 7^1 / 



(1.24) 



What happens if we have many transverse velocities? If we refer back 
to (1.14) we see that we will have an equation of the form 



1 = E - 14 



2^pn 



2 I din + C?2n 



d^d ^ J ^^-^''^ 

"In (fin / 

Here cop„^ is a plasma frequency based on the density of electrons having 
transverse velocities ±Un . Equation (1.25) can be written 

(co - |(3//o)" "| 



i = E 



A^ 

'M„2 r _ (g, - /3uo)2 -['^ 

L 7-'"n^ J 



(1.2()) 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 



113 




(u;-/3Uo 



Fig. 1 

Suppose we plot the left-hand and the right-hand sides of (1.26) versus 
(co — ^Uo)- The general appearance of the left-hand and right-hand sides 
of (1.26) is indicated in Fig. 1 for the case of two velocities Un . There 
will always be two unattenuated waves at values of (w — /3wo) > y Ug 
where Ue is the extreme value of lu; these correspond to intersections 3 
and 3' in Fig. 2. The other waves, two per value of Un , may be unat- 
tenuated or a pair of increasing and decreasing waves, depending on the 
values of the parameters. If 



CO 



pn 



-yhir? 



> 1 



there will be at least one pair of increasing and decreasing waves. 

It is not clear what will happen for a Maxwellian distribution of veloci- 
ties. However, we must remember that various aberrations might give a 
very different, strongly peaked velocity distribution. 

Let us consider the amount of gain in the case of one pair of transverse 
velocities, ±i/i . The equation is now 



2 2 
7 Ui 

C0„2 



[ 



1 + 



CO — |3wo 



)•] 



[ ■ - (^OI 



(1.27) 



Let 



/5 = ^+i^ 

Wo Wo 



(1.28) 



114 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



1 .u 
0.9 
0.8 


\ 




















\ 








































0.7 
0.6 






\ 




















\ 


\, 




















\ 


^ 












0.5 
0.4 
0.3 










\^ 






















\ 


>s. 




















\ 
























V 




0.2 


















\ 




















> 


\ 


0.1 




















\ 




















\ 























\ 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 

v2 



m 

Fig. 2 
This relation defines e. Equation (1.27) becomes 



2 2 
0}J 



1 - e^ 



(1 + e^)^ ^'-''^ 

In Fig. 2, e is plotted versus the parameter y^Ui/oip^. We see that as the 
parameter falls below unity, e increases, at first rapidly, and then more 
slowly, reaching a value of ±1 as the parameter goes to zero (as cop' 
goes to infinity, for instance). 

It will be shown in Section 2 of this paper that these results for infinite 
flow are in some degree an approximation to the results for flow in narrow 
beams. It is therefore of interest to see what results they yield if applied 
to a beam of finite width. 

If the beam has a length L, the voltage gain is 



The gain G in db is 



G = 8.7 '^ € db 

Wo 



(1.30) 



(1.31) 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 115 

Let the width of the beam be W. We let 

Thus, for n = 1, there is a half -cycle variation across the beam. From 
(1.31) and (1.32) 

G = 27.s(^^^\ne db (1.33) 



Now L/uo is the time it takes the electrons to go from one end of the 
beam to the other, while W/ui is the time it takes the electrons to cross 
the beam. If the electrons cross the beam A'' times 

iV = ^4 (1-34) 

Thus, 

G = 27.SNnedb (1.35) 

While for a given value of e the gain is higher if we make the phase 
vary many times across the beam, i.e., if we make n large, we should 
note that to get any gain at all we must have 






2 . //iTTUlV 
0)r> > 



(1.36) 



W 



If we increase oop , which is proportional to current density, so that cop 
passes through this value, the gain will rise sharply just after cOp" passes 
through this value and will rise less rapidly thereafter. 

.?. A Two-Dimensional Beam of Finite Width. 

Let us assume a beam of finite width in the ^/-direction ; the boundaries 
lying a,t y = ±^o • It will be assumed also that electrons incident upon 
these boundaries are elastically reflected, so that electrons of the incident 
stream (1 or 2) are converted into those of the other stream (2 or 1). The 
condition of elastic reflection implies that 

yi = -h (2.1) 

Zi = 22 Sit y = ±2/0 (2.2) 

and, in addition, that 

Pi = p2 at y = ±?/o . (2.3) 

since there is no change in the number of electrons at the boundary. 



116 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

The equations of motion and of continuity (1.7-1.12) may be satisfied 
by introducing a single quantity, ^, such that 

V = dx dzV (2.4) 

ii = -J - /3 d, ^2^ (2.5) 

m 

zi = —j — di di\p (2.6) 

m 

yi=-d, d^Dyp (2.7) 

m 

112=- di d^Di^ (2.8) 

m 



Pi 



m 



poiD' - ^') dirl^ (2.9) 



P2 = -- Po(i)' - n di'rl^ (2.10) 

m 



Then, if we introduce the symbol, 12, for co — jSuo 

yi + y^ = 2j-d,d2D^yp (2.11) ' 

m 

h- Z2 = 2j - di diUiD^ (2.12) 

m 

PI - P2 = 2j- po{D' - l3')uiQDi^ (2.13) 

m 

It is clear that if 

Drjy = D^xl^ = y = ±yo (2.14) 

the conditions for elastic reflection will be satisfied. The equation satis- 
fied by rf/ may now be found from Poisson's equation, (1-13), and is 

{D' - /3^) dx' di^P = '-^{D'- fi'){d,' + di)^l. 

we 

or 

{D' - ^')[{u,'D' + ny + coJiu.'D' - n')] = (2.15) 

which is of the sixth degree in D. So far four boundary conditions have, 
been imposed. The remaining necessary pair arise from matching the 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 117 

internal fields to the external ones. For y > ijo 

V = Voe-'^'-e~^" (2.16) 



and 



Similarlv 



^ + i37 = at 2/ = 2/0 
dy 



dV 

— - ^V = at y = -7/0 (2.17) 

dy 

The most familiar procedure now would be to look for solutions of 
(2,15) of the form, e''^. This would give the sextic for c 

(c' - /3')[(WiV + nY + a;/(niV - n')] = (2.18) 

with the roots c = ±|8, ±ci , ±C2 , let us s^y. We could then express \p 
as a linear combination of these six solutions and adjust the coefficients 
to satisfy the six boundary equations. In this way a characteristic equa- 
tion for l3 would be obtained. From the S3anmetry of the problem this 
has the general form F(l3, Ci) = F(i3, C2), where Ci and Co are found from 
; (2.18). The discussion of the problem in these terms is rather laborious 
and, if we are concerned mainly with examining qualitatively the onset 
of increasing waves, another approach serves better. 

From the symmetry of the equations and of the boundary conditions 
we see that there are solutions for \p (and consequently for V and p) 
which are even in y and again some which are odd in y. Consider first the 
even solutions. We will assume that there is an even function, ^i(y), 
periodic in y with period 2yo , which coincides with \l/(y) in the open 
interval, —yo<y<yo and that \pi(:y) has a Fourier cosine series repre- 
sentation : 

hiy) = E c„ cos \ny X„ = — n = 0, 1, 2, • • • (2.19) 
1 yo 

yp inside the interval satisfies (2.15), so we assume that ypiiy) obeys 
(D^ - ^')[{u,'D' + ^'f + o.,\u,'D' - ^-)^, 



+00 



(2.20) 



= Z) 5(2/ - 2m + lyo) 



where 6 is the familiar 5-function. Since D\p and D^\p are required to vanish 
at the ends of the interval and \l/, D'^ and Z)V are even it follows that all 



118 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

of these functions are continuous. We assume that xpi = \l/, D\pi = D\l/, 
DVi = D~\p, D% = D^yp and D% = D*xl/ at the ends of the intervals. 
From (2.20), Wi'D^i ^ -H as y ^ ijo . 
Since 

2 8iy - 2m + lyo) = ^ + - £ (-1)" cos Ky (2.21) 

we obtain from (2.20) 
/ 1 



2?/oi/'i 



,/32ff(i22 - Wp2) 



+ 2i;(-l)" ^"^'"^ 



Since 

^ + ^F = (Z) + /3)(t.x^Z)^ + fi^)V, 

using (2.4), the condition for matching to the external field, 

dV 

^ + /37 = 0, 

dy 

yields, using D\p = DV = and Ui*D^\f/ = — i^, the relation 

(ui'D' + fi')Vi = 3^/3 at 2/ = 2/0 . 
Applying this to (2.22), we then obtain, finally, 
yo ^ 1 



+ 2Z 



r (^2 4- X„2)[(i22 - Ml2X„2)2 - cOp2(Q2 + ,,^2X„2)] 



(2.22) 



(2.23) 



For the odd solution we use a function, yp2(y), equal to ;/'(?/) in — //o < 
y < yo and representable by a sine series. To ensure the vanishing of D^p 
and 7)V at ?/ = ±?/o it is appropriate to use the functions, sin n„y, where 
Mn = (n -\- l'2)ir/yo . The period is now iyo and we define \p2(y) in /yo < 
y < 32/0 by the relation i;'2(2/) = ^{2yo — y) and in — 32/o < 2/ < — 2/o by 
^2(2/) = ^{ — '^Uo — y)- Thus, we write 

00 

1^2(2/) = 2 C?n sin UnV Hn = (w + 3^)^7/0 



^2(2/) ^^i" ho supposed to satisfy 



GROWING "WAVES DUE TO TRANSVERSE VELOCITIES 119 

+M (2.24) 

= 2 [^(y - 4m + lyo) - Ky - 4m - lyo)] 

m=— 00 

The extended definition of i/'2 (outside — /yo <y < ijo) is such that we may 

again take \pi = \p, , D% = DV at the ends of the interval. ?/i*DVi is 

still equal to — }4 at ij = ijo . Now 

+ 00 

£ [5(y - 4m + iW - ^(y — 4m - l^/o)] 

(2.25) 

= — 2 (—1)" sin /i„?/ 
2/0 

so from (2.24) we may find 

v^L = -T (-l)"sin/xnj/ , ^ 

Matching to the external field as before gives 
and applied to (2.26) we have 

00 /rfi 2 2\2 

_y^ = y (^ - uinn) , . 

The equations (2.23) and (2.27) for the even and odd modes may be 
rewritten using the following reduced variables. 

. = ^« 

IT 
1 _ Wj/0 _ Wo 

(2.23) becomes 

^' 4- 2 y ^ (n' - k^ _ _ . 

and (2.27) transforms to 



„^ 2^ + (n + 3^)2 [{n + 1^)2 - /c2]2 - s\{n + 3^)^ + k'] (2 99) 

= — tt;? 



120 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



We shall assume in considering (2.28) and (2.29) that the beam is 
sufficiently wide for the transit of an electron from one side to the other 
to take a few RF cycles. The number of cycles is in fact, coz/o/ttwi , and, 
hence, from the definition of z, we see that for values of A: less than 2, 
perhaps, z is certainly positive. 

Let us consider (2.29) first since it proves to be the simpler case. If we 
transfer the term ttz to the right hand side, it follo^^•s from the observa- 
tion that z is positive (for modest values of h), that it is necessary to 
make the sum negative. The sum may be studied qualitatively by sketch- 
ing in the k^ — d' plane the lines on which the individual terms go to 
infinity, given by 

[(n + 3^)^ - k'f 



8' = 



(n -f K)' + k' 



(2.30) 




3.5 



Fig. 3 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 



121 



77 



0.4 

0.3 



0.2 0.4 0.6 0.8 1.0 



1.2 1.4 

(X/TT 



1.6 



1.8 2.0 2.2 2.4 



Fig. 4 

Fig. 3 shows a few such curves (n = 0, 1, 2). To the right of such curves 
the individual term in question is negative, except on the Hne, k^ = 
{n + V^) , where it attains the value of zero. Approaching the curves 
from the right the terms go to — oo . On the left of the curves the func- 
tion is positive and goes to + oo as the curve is approached from the 



10 



... 




/ 


/ 








/ 


/ 




J, 


L 








/ 


/ 






/ 








/ 


/ 




/ 


L 


/ 






/ 


/ 




/ 




Y 






/ 


/=, 




/ 






A 


V 


/ 


/ 




/ 








' 


\ 


/ 




A 


-0 








1 


/ 


\ 


/ 












y 




/ 


\ 


^^ 










A 


>< 






■^ 




^^ 






>C 


^ 


"\ 


^ 






'^ 




^ 



3 4 5 6 7 8 9 

Fig. 5 



J 22 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G 

left. Clearly in the regions marked + which lie to the left of every curve 
given by (2.30), the sum is positive and we cannot have roots. Let us 
examine the sum in the region to the right of the n = curve and to the 
left of all others. On the line, A;^ = J4» the sum is positive, since the first 
term is zero. On any other line, k' = constant, the sum goes from + °° 
at the n = 1 curve monotonically to — oo at the n = curve, so that 
somewhere it must pass through 0. This enables us to draw the zero- 
sum contours qualitatively in this region and they are indicated in Fig. 3. 
We are now in a position to follow the variation in the sum as k varies 
at fixed 5 . It is readily seen that for 5 < 0.25, because —wz is negative 
in the region under consideration, there will be four real roots, tw^o for 
positive, two for negative k. For 5' slightly greater than 0.25, the sum has 




Fig. 6A 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 123 

a deep minimum for k = 0, so that there are still four real roots unless z 
is very large. For z fixed, as 5^ increases, the depth of the minimum de- 
creases and there will finally occur a 5" for which the minimum is so shal- 
low that two of the real roots disappear. Call z(0) the value of ziork = 0, 
write the sum as 2(5^ k^) and suppose that 2(5o^ 0) = —irziO), then for 
small k we have 

S(5^ e) = -«(0) + (6^ - 8o') §, + k'§,= -«(0) -"^ k 

do^ dk^ Ua 

as 

dB dk^ 




^ = ^± / ".^(^-^0^) + 



'^ a/ 
dk' y 

The roots become complex when 



aA-2 




S.2 J 2 (Ul/Uo) 

= do — 



52 as 

d8^ dB 



Since Ui/uq may be considered small (say 10 per cent) it is sufficient to 



look for the values of 5o^. 
When k = we have 



-TZ = 2X) 

2z 



z (n + y,y 



z^ + 52 

irz" 



z'-\-in-\- y^r (n -1- y^y- - s' 

' H ^ + i ^ 

\in + 3^)2 - 52 ^ (n + 1^)2 + zy 
(5 tan -Kb -\- z tanh irz) 



z" + 52 



Fig. 4 shows the solution of this equation for various 2(0) or oiyo/iruo . 
Clearly the threshold 5 is rather insensitive to variations in uyo/ir^io . 

Equation (2.28) may be examined by a similar method, but here some 
complications arise. Fig. 5 shows the infinity curves for n = 0, 1, 2, 3; 
the n = term being of the form k^/k^ — 8^. The lowest critical region 
in 5^ is the neighborhood of the point fc^ = 6^ = ]^i, which is the intersec- 
tion of the n = and n = 1 lines. To obtain an idea of the behavior of 



124 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G 

the left hand side (l.h.s.) of (2.28) in this area we first see how the point 
k^ = f = 1^ can be approached so that the l.h.s. remains finite. If we 
put k^ = H + £ and a' = ^ + ce and expand the first two dominant 
terms of (2.28), then adjust c to keep the result finite as f -^ we find 

= 1 3^' - 5 
^ ~ 4 32^ + 1 

c varies from — % to \i as z goes from to c» , changing sign at 2^ = %. 
Every curve for which the l.h.s. is constant makes quadratic contact with 
the Jine 5" — V3 = c(/v" — ]i) at Jc' = 5' = I/3. If we remember that 
the l.h.s. is positive for A;' = 0, < 5" < 1 and for A;^ = 1, < 5^ < 1, 



1 

2 


lik 




3 






w-oX 


k^ 




/ 




1 




y( I 




3 


SHADED AREAS // 
NEGATIVE yV 

X /' 
/ // 

/ /I 
/ / / 


X 


\ 








n = i^v 












3 



3 



Fig. 6B 



GROWING WAVES DUE TO TRANSVERSE VELOCITIES 125 

since there are no negative terms in the sum for these ranges and again 
that the l.h.s. must change sign between the n = and n — I Unes for 
any k^ in the range < k^ < 1 (since it varies from T oo to ±0°), this 
information may be combined with that about the immediate vicinity 
of 5 = k = V^ to enable us to draw a Hue on which the l.h.s. is zero. 
This is indicated in Figs. 6A and 6B for small z and large z respec- 
tively. It will be seen that the zero curve and, in fact, all curves on which 
the l.h.s. is equal to a negative constant are required to have a vertical 
tangent at some point. This point may be above or below /c^ = ^ (de- 
pending upon the sign of c or the size of z) but always at a 3^ > ^. For 
5 < H there are no regions where roots can arise as we can readily see 
by considering how the l.h.s. varies with k"^ at fixed 5^ For a fixed d^ > }/s 
we have, then, either for k^ > ]4 or k^ < V^, according to the size of z, 
a negative minimum which becomes indefinitely deep as 5^ -^ ^. Thus, 
since the negative terms on the right-hand side are not sensitive to small 
changes in 5^, we must expect to find, for a fixed value of the l.h.s., two 
real solutions of (2.28) for some values of 5^ and no real solutions for some 
larger value of 5 , since the negative minimum of the l.h.s. may be made 
as shallow as we like by increasing 6". By continuity then we expect to 
find pairs of complex roots in this region. Rather oddly these roots, which 
will exist certainly for 5' sufficiently close to V^ + 0, will disappear if 
5^ is sufficiently increased. 

REFERENCES 

1. L. S. Nergaard, Analysis of a Simple Model of a Two-Beam Growing-Wave 

Tube, RCA Review, 9, pp. 585-601, Dec, 1948. 

2. J. R. Pierce and W. B. Hebenstreit, A New Type of High-Frequency Amplifier, 

B. S. T. J., 28, pp. 23-51, Jan., 1949. 

3. A. V. Haeff, The Electron-Wave Tube — A Novel Method of Generation and 

Amplification of Microwave Energy, Proc. I.R.E., 37, pp. 4-10, Jan., 1949. 

4. G. G. Macfarlg,ne and H. G. Hay, Wave Propagation in a Slipping Stream of 

Electrons, Proc. Physical Society Sec. B, 63, pp. 409-427, June, 1950. 

5. P. Gurnard and H. Huber, Etude E.xp^rimentale de L'Interaction par Ondes 

de Chargd^d'Espace au Sein d'Un Faisceau Electronique se Deplagant dans 
Des Champs Electrique et Magn^tique Croisfe, Annales de Radio^lectricite, 
7, pp. 252-278, Oct., 1952. 

6. C. K. Birdsall, Double Stream Amplification Due to Interaction Between Two 

Oblique Electron Streams, Technical Report No. 24, Electronics Research 
Laboratory, Stanford University. 

7. L. Brillouin, A Theorem of Larmor and Its Importance for Electrons in Mag- 

netic Fields, Phys. Rev., 67, pp. 260-266, 1945. 

8. J. R. Pierce, Theory and Design of Electron Beams, 2nd Ed., Chapter 9, Van 

Nostrand, 1954. 

9. J. R. Pierce, Traveling-Wave Tubes, Van Nostrand, 1950. 



Coupled Helices 

By J. S. COOK, R. KOMPFNER and C. F. QUATE 

(Received September 21, 1955) 

An analysis of coupled helices is presented, using the transmission line 
approach and also the field approach, with the objective of providing the 
tube designer and the microwave circuit engineer with a basis for approxi- 
mate calcidations. Devices based on the presence of only one mode of propa- 
gation are briefly described; and methods for establishing such a mode are 
given. Devices depending on the simultaneous presence of both modes, that 
is, depending on the beat wave phenomenon, are described; some experi- 
mental results are cited in support of the view that a novel and useful class of 
coupling elements has been discovered. 

CONTENTS 

1. Introduction 129 

2. Theory of Coupled Helices 132 

2.1 Introduction 132 

2.2 Transmission Line Equations 133 

2.3 Solution for Synchronous Helices 135 

2.4 Non-Synchronous Helix Solutions 137 

2.5 A Look at the Fields 139 

2.6 A Simple Estimate of b and x 141 

2.7 Strength of Coupling versus Frequency 142 

2.8 Field Solutions 144 

. 2.9 Bifilar Helix 146 

2.10 Effect of Dielectric Material between Helices 148 

2.11 The Conditions for Maximum Power Transfer 151 

2.12 Mode Impedance 152 

3. Applications of Coupled Helices 154 

3.1 Excitation of Pure Modes 156 

3.1.1 Direct Excitation 156 

3.1.2 Tapered Coupler 157 

3.1.3 Stepped Coupler 158 

3.2 Low Noise Transverse Field Amplifier 159 

3.3 Dispersive Traveling Wave Tube 159 

3.4 Devices Using Both Modes 161 

3.4.1 Coupled Helix Transducer 161 

3.4.2 Coupled-Helix Attenuator 165 

4. Conclusion 167 

Appendix 

I Solution of Field Equations 168 

II Finding r I73 

III Complete Power Transfer 175 

127 



128 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

GLOSSARY OF SYMBOLS 

a Mean radius of inner helix 

h Mean radius of outer helix 

h Capacitive coupling coefficient 

Bio, 20 shunt susceptance of inner and outer helices, respectively 

Bi, 2 Shunt susceptance plus mutual susceptance of inner and outer 

helices, respectively, Bm + Bm , Boo + B^ 
Bm Mutual susceptance of two coupled helices 
c Velocity of light in free space 

d Radial separation between helices, h-a 

D Directivity of helix coupler 

E Electric field intensity 

F Maximum fraction of power transferable from one coupled helix 

to the other 
F(ya) Impedance parameter 

7i, 2 RF current in inner and outer helix, respectively 
K Impedance in terms of longitudinal electric field on helix axis 

and axial power flow 
L ]\Iinimum axial distance required for maximum energy transfer 

from one coupled helix to the other, X6/2 

Axial power flow along helix circuit 

Radial coordinate 

Radius where longitudinal component of electric field is zero for 

transverse mode (about midway between a and b) 

Return loss 

Radial separation betw^een helix and adjacent conducting shield 

Time 

RF potential of inner and outer helices, respectively • 

Inductive coupling coefficient 

Series reactance of inner and outer helices, respectively 

Series reactance plus mutual reactance of inner and outer helices, 

respectively, Xio + Xm , X20 + Xm 

Mutual reactance of two coupled helices 

Axial coordinate 

Impedance of inner and outer helix, respectively 

Attenuation constant of inner and outer helices, respectively 

General circuit phase constant; or mean circuit phase constant. 

Free space phase constant 

Axial phase constant of inner and outer helices in absence of 

coupling, V^ioXio , VBioXio 



p 


r 
f 


R 


s 
t 

F1.2 


X 

Xva, 20 
Xl, 2 


Xm 


Z 
Zil, 2 


Oil, 2 


^0 
^10. 20 



COUPLED HELICES 129 

181 , 2 May be considered as axial phase constant of inner and outer 
helices, respectively 

(Sft Beat phase constant 

jSc Coupling phase constant, (identical with ^b when /3i = JS2) 

I3ce Coupling phase constant when there is dielectric material be- 

tween the helices 

/3d Difference phase constant, [ /3i — /32 [ 

(8f Axial phase constant of single helix in presence of dielectric 

^t, ( Axial phase constant of transverse and longitudinal modes, re- 
spectively 

7 Radial phase constant 

jt, ( Radial phase constant of transverse and longitudinal modes, 
respectively 

r Axial propagation constant 

Tt. ( Axial propagation constant for transverse and longitudinal 
coupled-helix modes, respectively 

e Dielectric constant 

e' Relative dielectric constant, e/eq 

En Dielectric constant of free space 

X General circuit wavelength; or mean circuit wavelength, \/XiX2 

Xo Free space wavelength 

Xi, 2 Axial wavelength on inner and outer helix, respectively 

X6 Beat wavelength 

Xc Coupling wavelength (identical with Xb when (5i = /So) 

yj/ Helix pitch angle 

i/'i, 2 Pitch angle of inner and outer helix, respectively 

CO Angular frequency 

1. INTRODUCTION 

Since their first appearance, traveling-wave tubes have changed only 
very little. In particular, if we divide the tube, somewhat arbitrarily, 
into circuit and beam, the most widely used circuit is still the helix, and 
the most widely used transition from the circuits outside the tube to the 
circuit inside is from waveguide to a short stub or antenna which, in 
turn, is attached to the helix, either directly or through a few turns of 
increased pitch. Feedback of signal energy along the helix is prevented 
by means of loss, either distributed along the whole helix or localized 
somewhere near the middle. The helix is most often supported along its 
whole length by glass or ceramic rods, which also serve to carry a con- 
ducting coating ("aquadag"), acting as the localized loss. 

We therefore find the following circuit elements within the tube en- 
velope, fixed and inaccessible once and for all after it has been sealed off: 



130 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

1 . The helix itself, determining the beam voltage for optimum beam- 
circuit interaction ; 

2. The helix ends and matching stubs, etc., all of which have to be 
positioned very precisely with relation to the waveguide circuits in 
order to obtain a reproducible match ; 

3. The loss, in the form of "aquadag" on the support rods, which 
greatly influences the tube performance by its position and distril)ution. 

In spite of the enormous bandwidth over which the traveling-wave 
tube is potentially capable of operating — a feature new in the field of 
microwave amplifier tubes — it turns out that the positioning of the tube 
in the external circuits and the necessary matching adjustments are 
rather critical; moreover the overall bandwidths achieved are far short 
of the obtainable maximum. 

Another fact, experimentally observed and well-founded in theory, 
rounds off the situation: The electro-magnetic field surrounding a helix, 
i.e., the slow wave, under normal conditions, does not radiate, and is 
confined to the close vicinity of the helix, falling off in intensity nearly 
exponentially with distance from the helix. A typical traveling-wave 
tube, in which the helix is supported by ceramic rods, and the whole 
enclosed by the glass envelope, is thus practically inaccessible as far as 
RF fields are concerned, with the exception of the ends of the helix, 
where provision is made for matching to the outside circuits. Placing 
objects such as conductors, dielectrics or distributed loss close to the 
tube is, in general, observed to have no effect whatsoever. 

In the course of an experimental investigation into the propagation of 
space charge waves in electron beams it was desired to couple into a long 
helix at any point chosen along its length. Because of the feebleness of 
the RF fields outside the helix surrounded by the conventional sup- 
ports and the envelope, this seemed a rather difficult task. Nevertheless, 
if accomplished, such a coupling would have other and even more im- 
portant applications; and a good deal of thought was given to the 
problem. 

Coupled concentric helices were found to provide the solution to the 
problem of coupling into and out of a helix at any particular point, and to 
a number of other problems too. 

Concentric coupled helices have been considered by J. R. Pierce, 
who has ti'cated the problem mainly with transverse fields in mind. 
Such fields were thought to be useful in low-noise traveling-wave tube 
devices. Pierce's analysis treats the helices as transmission lines coupled 
uniformly over their length by means of nuitual distributed capacitance 
and inductance. Pierce also recognized that it is necessary to wind the 



COUPLED HELICES l,']! 

two helices in opposite directions in order to obtain well defined trans- 
verse and axial wave modes which are well separated in respect to their 
velocities of propagation. 

Pierce did not then give an estimate of the velocity separation which 
might be attainable with practical helices, nor did anybody (as far as we 
are aware) then know how strong a coupling one might obtain with such 
heUces. 

It was, therefore, a considerable (and gratifying) surprise^' ^ to find 
that concentric helices of practically realizable dimensions and separa- 
tions are, indeed, very strongly coupled when, and these are the im- 
portant points, 

(a) They have very nearly equal velocities of propagation when un- 
coupled, and when 

(b) They are wound in opposite senses. 

It was found that virtually complete power transfer from outer to 
inner helix (or vice versa) could be effected over a distance of the order 
of one helix wavelength (normally between i^fo and 3^^o of a free-space 
wavelength. 

It was also found that it was possible to make a transition from a co- 
axial transmission line to a short (outer) helix and thence through the 
glass surrounding an inner helix, which was fairly good over quite a con- 
siderable bandwidth. Such a transition also acted as a directional coupler, 
RF power coming from the coaxial line being transferred to the inner 
helix predominantly in one direction. 

Thus, one of the shortcomings of the "conventional" helix traveling- 
wave tube, namely the necessary built-in accuracy of the matching 
parameters, was overcome by means of the new type of coupler that 
might evolve around coupled helix-to-helix systems. 

Other constructional and functional possibilities appeared as the 
work progressed, such as coupled-helix attenuators, various tj^pes of 
broadband couplers, and schemes for exciting pure transverse (slow) or 
longitudinal (fast) waves on coupled helices. 

One central fact emerged from all these considerations: by placing 
part of the circuit outside the tube envelope with complete independence 
from the helix terminations inside the tube, coupled helices give back to 
the circuit designer a freedom comparable only with that obtained at 
much lower frequencies. For example, it now appears entirely possible 
to make one type of traveling wave tube to cover a variety of frequency 
bands, each band requiring merely different couplers or outside helices, 
the tube itself remaining unchanged. 

Moreover, one tube may now be made to fulfill a number of different 



132 THE BELL SYSTEM TECHNICAI- JOURNAL, JANUARY 1956 

functions; this is made possible by the freedom with which couplers 
and attenuators can be placed at any chosen point along the tube. 

Considerable work in this field has been done elsewhere. Reference 
will be made to it wherever possible. However, only that work with 
which the authors have been intimately connected will be fully reported 
here. In particular, the effect of the electron beam on the wave propaga- 
tion phenomena will not be considered. 

2. THEORY OF COUPLED HELICES 

2.1 Introduction 

In the past, considerable success has been attained in the under- 
standing of traveling wave tube behavior by means of the so-called 
"transmission-line" approach to the theory. In particular, J. R, Pierce 
used it in his initial analysis and was thus able to present the solution 
of the so-called traveling-wave tube equations in the form of 4 waves, 
one of which is an exponentially growing forward traveling wave basic 
to the operation of the tube as an amplifier. 

This transmission-line approach considers the helix — or any slow- 
wave circuit for that matter — as a transmission line with distributed 
capacitance and inductance with which an electron beam interacts. 
As the first approximation, the beam is assumed to be moving in an RF 
field of uniform intensity across the beam. 

In this way very simple expressions for the coupling parameter and 
gain, etc., are obtained, which give one a good appreciation of the 
physically relevant quantities. 

A number of factors, such as the effect of space charge, the non-uniform 
distribution of the electric field, the variation of circuit impedance with 
frequency, etc., can, in principle, be calculated and their effects can be 
superimposed, so to speak, on the relatively simple expressions deriving 
from the simple transmission line theory. This has, in fact, been done and 
is, from the design engineer's point of view, quite satisfactory. 

However, phj^sicists are bound to be unhappy over this state of 
affairs. In the beginning was Maxwell, and therefore the proper point to 
start from is Maxwell. 

So-called "Field" theories of traveling-w^ave tubes, based on Maxwell's 
equation, solved with the appropriate boundary conditions, have been 
worked out and their main importance is that they largely confirm the 
results obtained by the inexact transmission line theory. It is, however, 
in the nature of things that field theories cannot give answers in terms of 



COUPLED HELICES 133 

simple closed expressions of any generality. The best that can be done 
is in the form of curves, with step-wise increases of particular param- 
eters. These can be of considerable value in particular cases, and when 
exactness is essential. 

In this paper we shall proceed by giving the "transmission-line" type 
theory first, together with the elaborations that are necessary to arrive 
at an estimate of the strength of coupling possible with coaxial helices. 
The "field" type theory will be used whenever the other theory fails, or 
is inadequate. Considerable physical insight can be gotten with the use 
of the transmission-line theory; nevertheless recourse to field theory is 
necessary in a number of cases, as will be seen. 

It will be noted that in all the calculations to be presented the presence 
of an electron beam is left out of account. This is done for two reasons: 
Its inclusion would enormously complicate the theory, and, as will 
eventually be shown, it would modify our conclusions only very slightly. 
Moreover, in practically all cases which we shall consider, the helices are 
so tightly coupled that the velocities of the two normal modes of propaga- 
tion are very different, as will be shown. Thus, only when the beam 
velocity is very near to either one or the other wave velocity, will 
growing-wave interaction take place between the beam and the helices. 
In this case conventional traveling wave tube theory may be used. 

A theory of coupled helices in the presence of an electron beam has 
been presented by Wade and Rynn,^ who treated the case of weakly 
coupled helices and arrived at conclusions not at variance with our views. 

2.2 Transmission Line Equations 

Following Pierce we describe two lossless helices by their distributed 
series reactances Xio and A'20 and their distributed shunt susceptances 
Bio and ^20 . Thus their phase constants are 

/3io = V^ioA'io 

Let these helices be coupled by means of a mutual distributed reac- 
tance Xm and a mutual susceptance B^ , both of which are, in a way 
which will be described later, functions of the geometry. 

Let waves in the coupled system be described by the factor 

jut — Tj; 

e e 



\v 



here the F's are the propagation constants to be found. 



134 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



The transmission line equations may be written: 

r/i - jB,V, + jB„y2 = 
rFi - iXi/i + jXJo = 

r/o - JB0V2 + jB„yi = 

TV2 - jXJa + jXJ, - 

where 

B, - 5io + 5« 

Bo = B20 + Bm 

X2 = X20 -f" Xm 

1 1 and 1 2 are eliminated from the (2.2.1) and we find 

F2 ^ + (r- + XiBi + x^Bj 

Fi 

F2 



(2.2.1) 



X\Bm + B%Xm 

+ (r- + X2S2 + x^Bj 



XlBm + 5lX„ 



(2.2.2) 



(2.2.3) 



These two equations are then multipUed together and an expression for 
r of the 4th degree is obtained : 

r' + (XiBi + X2B2 + 2Z,„Bjr' 

+ (X1Z2 - Xj){B,B2 - Bj) = 
We now define a number of dimensionless quantities: 



(2.2.4) 



B, 



BiB. 



Xm 



= h' = (eapacitive coupling coefficient)' 



= X = (inductive coupling coefficient) 



XiXo 

B\Xi = ^1, B2X2 = (82' 
X1B1X2B2 = 13^ = (mean phase constant) 
With these substitutions we obtain the general equation for T~ 



T' = 13' 



2 \(3-r ^ I3{' ^ 



y 4v^2'^^/3i^ 



_ (2.2.5) 



+ 26.r - (1 - .r-)(l - U') 



COUPLED HELICES 135 



(2.2.6) 



If we make the same substitutions in (2.2.2) we find 

Fi T ZiL /3(/3i?> + /3o:r) . 
where the Z's are the impedances of the heUces, i.e., 

Z,. = VXJB, 

2.3 Solution for Synchronous Helices 

Let us consider the particular case where (Si = (S-z = |S. From (2.2.5) 
we obtain 

r' = -I3\l + xb db (x + b)] (2.3.1) 

Each of the above values of T" characterizes a normal mode of propaga- 
tion involving both helices. The two square roots of each T" represent 
waves going in the positive and negative directions. We shall consider 
only the positive roots of T , denoted Tt and Tt , which represent the 
forward traveling waves. 

Ttj = i/3Vl + xb ± {x + b) (2.3.2) 

If a: > and 6 > 

I r, I > |/3i, I r,| < 1^1 

Thus Vt represents a normal mode of propagation which is slower than 
the propagation velocity of either helix alone and can be called the 
"slow" wave. Similarly T( represents a "fast" wave. We shall find that, 
in fact, X and b are numerically equal in most cases of interest to us; we 
therefore write the expressions for the propagation constants 

r. = M^ + H(-^ + b)] 

(2.3.3) 

r. = Ml - Viix + b)] 

If we substitute (2.3.3) into (2.2.6) for the case where /3i = (82 = /3 and 
assume, for simplicity, that the helix self-impedances are equal, we find 
that for r = Tt 

Y% _ 



for r = T; 



F2 

-— = -f 1 

Yx ^ 



136 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Thus, the slow wave is characterized by equal voltages of unlike sign on 
the two helices, and the fast wave by equal voltages of like sign. It fol- 
lows that the electric field in the annular region between two such coupled 
concentric helices will be transverse for the slow wave and longitudinal 
for the fast. For this reason the slow and fast modes are often referred 
to as the transverse and longitudinal modes, respectively, as indi- 
cated by our subscripts. 

It should be noted here that we arbitrarily chose h and x positive. A 
different choice of signs cannot alter the fact that the transverse mode is 
the slower and the longitudinal mode is the faster of the two. 

Apart from the interest in the separate existence of the fast and slow 
waves as such, another object of interest is the phenomenon of the simul- 
taneous existence of both waves and the interference, or spatial beating, 
between them. 

Let V2 denote the voltage on the outer hehx; and let Vi , the voltage 
on the inner halix, be zero at z = 0. Then we have, omitting the common 
factor e'" , 

(2.3.4) 

Since at 2 = 0, Fi = 0, Vn = — V(^ . For the case we have considered we 
have found Fa = — V^ and Vn = V^ . We can write (2.3.4) as 



Fi = I {e~'^' - e-^n 



V, = ^ {e''^' + e-'n 



(2.3.5) 



F2 can be written 



= Ye-"'''''^''^''' cos [-jj^(r, - Vi)z\ 
In the case when x = 6, and /Si = /32 = /8 

F2 = Ye"'^' cos Wiix + h)^z\ (2.3.6) 

Correspondingly, it can be shown that the voltage on the inner helix is 

y, = j\Tfr^^' sin Wiix + h)^z\ (2.3.7) 

The last tAvo equations exhibit clearly what we have called the spatial 
beat phenomonou, a wave-like transfer of power from one helix to thc^ 



COUPLED HELICES . 137 

other and back. We started, arbitrarily, with all the voltage on the outer 
helix at 2 = 0, and none on the inner; after a distance, z', which makes 
the argument of the cosine x/2, there is no voltage on the outer helix 
and all is on the inner. 

To conform with published material let us define what we shall call 
the "coupling phase-constant" as 

^, = ^{h + x) (2.3.8) 

From (2.3.3) we find that for (Si = ^2 = |S, and x = h, 

Tt - Ti = jl3c 

2.4 Non-Synchronous Helix Solutions 

Let us now go back to the more general case where the propagation 
velocities of the (uncoupled) helices are not equal. Eciuation (2.2.5) can 
be written: 



Further, let us define 



(2.4.1) 



r- = -^- [1 + (1/2)A + xb ± 

V(l + xb)A + (1/4)A2 + (6 + xy] 
where 

L /3 _ 
In the case where x = h, (2.4.1) has an exact root. 

r,, , = j^ [Vl + A/4 ± 1/2 Va + (a; + by] (2.4.2) 

We shall be interested in the difference between Tt and Tt, 

Tt-Tf = j^ Va + (x + by- (2.4.3) 

Now we substitute for A and find 

Tt- Tc = j V(^i - ^2y + ^M& + 4' (2.4.4) 

Let us define the "beat phase-constant" as: 

Pb = V(/3i - /32)2 + nb + xy 

so that 

r, - r, = jA (2.4.5) 



(3a = \ i5i - iSo 



138 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

and call this the "difference phase-constant," i.e., the hase constant cor- 
responding to two uncoupled waves of the same frequency but differing 
phase velocities. We can thus state the relation between these phase 
constants : 

^b' = &I + ^c (2.4.6) 

This relation is identical (except for notation) with expression (33) in 
S. E. Miller's paper. ^ In this paper Miller also gives expressions for the 
voltage amplitudes in two coupled transmission systems in the case of 
unequal phase velocities. It turns out that in such a case the power trans- 
fer from one system to the other is necessarily incomplete. This is of 
particular interest to us, in connection with a number of practical 
schemes. In our notation it is relatively simple, and we can state it by 
saying that the maximum fraction of power transferred is 

(2.4.7) 




or, in more detail, 



iS/ + iSc- (^1 - iS2)- + ^Kh + xY 

This relationship can be shown to be a good approximation from (2.2.6), 
(2.3.4), (2.4.2), on the assumption that h ^ x and Zx 'PH Z2 , and the 
further assumption that the system is lossless; that is, 

I 72 I ^ + I Fi I ^ = constant (2.4.8) 

We note that the phase velocity difference gives rise to two phenomena : 
It reduces the coupling w^avelength and it reduces the amount of power 
that can be transferred from one helix to the other. 

Something should be said about the case where the two helix imped- 
ances are not equal, since this, indeed, is usually the case with coupled 
concentric helices. Equation (2.4.8) becomes: 

I F2 1 _^ \Vx\_ ^ (3Qj^g^^j^^ (2.4.9) 



Z2 Z\ 

Using this relation it is found from (2.3.4) that 



F2 , /Zi 

FiT z, 




(1 ± Vl - /^) (2.4.10) 



When Ihis is combined with (2.2.6) it is found that the impedances droj) 
out with the voltages, and that "F" is a function of the |S's only. In other 



COUPLED HELICES 139 

words, complete power transfer occurs when ,81 = /So regardless of the 
relative impedances of the helices. 

The reader will remember that (3io and (820 , not jSi and ^o , were defined 
as the phase constants of the helices in the absence of each other. If the 
assumption that h ^ x is maintained, it will be found that all of the de- 
rived relationships hold true when (Sno is substituted for /3„ . In other 
words, throughout the paper, /3i and /So may be treated as the phase con- 
stants of the inner and outer helices, respectively. In particular it should 
be noted that if these ciuantities are to be measured experimentally each 
helix must be kept in the same environment as if the helices were coupled ; 
onl}^ the other helix may be removed. That is, if there is dielectric in the 
annular region between the coupled helices, /Si and ^2 must each be 
measured in the presence of that dielectric. 

Miller also has treated the case of lossy coupled transmission systems. 
The expressions are lengthy and complicated and we believe that no 
substantial error is made in simply applying his conclusions to our case. 

If the attenuation constants ai and ao of the two transmission systems 
(helices) are equal, no change is required in our expressions; when they 
are unequal the total available power (in both helices) is most effectively 
reduced when 

^4^'^l (2.4.11) 

Pc 

This fact may be made use of in designing coupled helix attenuators. 

2.5 A Look at the Fields 

It may be advantageous to consider sketches of typical field distribu- 
tions in coupled helices, as in Fig. 2.1, before we go on to derive a quanti- 
tative estimate of the coupling factors actually obtainable in practice. 

Fig. 2.1(a) shows, diagrammatically, electric field lines when the 
coupled helices are excited in the fast or "longitudinal" mode. To set up 
this mode only, one has to supply voltages of like sign and equal ampli- 
tudes to both helices. For this reason, this mode is also sometimes called 
the "(+-f) mode." 

Fig. 2.1(b) shows the electric field lines when the helices are excited in 
the slow or "transverse" mode. This is the kind of field required in the 
transverse interaction type of traveling wave tube. In order to excite 
this mode it is necessary to supply voltages of equal amplitude and 
opposite signs to the helices and for this reason it is sometimes called the 
"(-| — ) mode." One way of exciting this mode consists in connecting one 



140 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

helix to one of the two conductors of a balanced transmission line 
("Lecher"-line) and the other hehx to the other. 

Fig. 2.1(c) shows the electric field configuration when fast and slow 
modes are both present and equally strongly excited. We can imagine 
the two helices being excited by a voltage source connected to the outer 




(a) FAST WAVE (longitudinal) 




(b) SLOW WAVE (transverse) 




(C) fast and slow waves combined SHOWING SPATIAL "BEAT" PHENOMENON 



Fig. 2.1 — Typical electric field distributions in coupled coaxial helices when 
thej^ are excited in: (a) the in-phase or lonfritudinal mode, (b) the out-of-phase or 
transverse mode, and (c) both modes equally. 



COUPLED HELICES 141 

helix only at the far left side of the sketch. One, perfectly legitimate, 
view of the situation is that the RF power, initially all on the outer helix, 
leaks into the inner helix because of the coupling between them, and then 
leaks back to the outer helix, and so forth. 

Apart from noting the appearance of the stationary spatial beat (or 
interference) phenomenon these additional facts are of interest: 

1) It is a simple matter to excite such a beat- wave, for instance, by 
connecting a lead to either one or the other of the helices, and 

2) It should be possible to discontinue either one of the helices, at 
points where there is no current (voltage) on it, without causing reflec- 
tions. 

2.6 A Simple Estimate of h and x 

How strong a coupling can one expect from concentric helices in prac- 
tice? Quantitatively, this is expressed by the values of the coupling fac- 
tors X and h, which we shall now proceed to estimate. 

A first crude estimate is based on the fact that slow-wave fields are 
known to fall off in intensity somewhat as c where (3 is the phase con- 
stant of the wave and r the distance from the surface guiding the slow 
wave. Thus a unit charge placed, say, on the inner helix, will induce a 
charge of opposite sign and of magnitude 

-Pib-a) 

on the outer helix. Here h = mean radius of the outer helix and a = 
mean radius of the inner. We note that the shunt mutual admittance 
coupling factor is negative, irrespective of the directions in which the 
helices are wound. Because of the similarity of the magnetic and electric 
field distributions a current flowing on the inner helix will induce a simi- 
larly attenuated current, of amplitude 

on the outer helix. The direction of the induced current will depend on 
whether the helices are woimd in the same sense or not, and it turns out 
(as one can verify by reference to the low-freciuency case of coaxial 
coupled coils) that the series mutual impedance coupling factor is nega- 
tive when the helices are oppositely wound. 

In order to obtain the greatest possible coupling between concentric 
helices, both coupling factors should have the same sign. This then re- 
fiuires that the helices should be wound in opposite directions, as has 
been pointed out by Pierce. 

When the distance between the two helices goes to zero, that is to say, 



142 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

.if they lie in the same surface, it is clear that both coupling factors h and x 
will go to unity. 

As pointed out earlier in Section 2.3, the choice of sign for h is arbi- 
trary. However, once a sign for h has been chosen, the sign of x is neces- 
sarily the opposite when the helices are wound in the same direction, and 
vice versa. We shall choose, therefore, 

the sign of the latter depending on whether the helices are wound in the 
same direction or not. 

In the case of unequal velocities, (5, the propagation constant, would 
be given by 

1^ = VM~2 (2.6.2) 

2.7 Strength of Coupling versus Frequency 

The exponential variation of coupling factors with respect to frequency 
(since /3 = co/y) has an important consequence. Consider the expression 
for the coupling phase constant 

/3. = I3{b + x) (2.3.8) 

or 

l/3e| = 2/3^"^^'""^ (2.7.1) 

The coupling wavelength, which is defined as 



Ac 
is, therefore, 



27r 



(2.7.2) 



or 



Xc- -e 



X, = ;^ g(2./x)u.-«) (2.7.3) 

where X is the (slowed-down) RF wavelength on either helix. It is con- 
venient to multiply both sides of (2.7.1) with a, the inner helix radius, 
in order to obtain a dimensionless relation between /3c and /3: 

^,a = 2/3ac~^''"''°^"" (2.7.4) 

This relalion is j)l()Ued on Fig. 2.2 for several values of b/a. 



COUPLED HELICES 



143 



3.00 



2.75 



2.50 



2.25 



2.00 



/3ca 



1.75 



1.50 



1.25 



i.OO 



0.75 



0.50 



0.25 













^^ 








— - 


/ 

/ 
/ 












/ 


/ 


/ 


/ 
/ 
/ 










l-y 


/ 




/ 
/ 

/ 
/ 
J 












/ 


/ 




/ 
/(/Jc3)max 

/ 










/ 






/ 
/ 

/ 












1 


/ 




/ 
/ 
/ 

/ 
















^ 


/ 






b 


= 1.5 








/ 


/ 


/ 
/ 

f 










^ 




, 


V 


/ 


^^ 










'" 


\ 


1 






-\ 




"^^^ — 1 


75 








L 










2.0 






■\ 




/ 




"^ 


■-^ 


3.0 








— - 






0.5 



1.5 



2.0 



2.5 

/3a 



3.0 



3.5 4.0 



4.5 



5.0 



Fig. 2.2 — Coupling pliase-constant plotted as a function of the single helix 
phase-constant for synchronous helices for several values of b/a. These curves 
are based on simple estimates made in Section 2.7. 



There are two opposing tendencies determining the actual physical 
length of a coupling beat-wavelength: 

1) It tends to grow with the RF wavelength, being proportional to it 
in the first instance; 

2) Because of the tighter coupling possible as the RF wavelength 
increases in relation to the heli.x-to-helix distance, the coupling beat- 
wavelength tends to shrink. 

Therefore, there is a region where these tendencies cancel each other, 
and where one would expect to find little change of the coupling beat- 
wavelength for a considerable change of RF freciuency. In other words, 
the "bandwidth" over which the beat-wavelength stays nearly constant 
can be large. 

This is a situation naturally very desirable and favorable for any 
device in which we rely on power transfer from one helix to the other by 



144 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

means of a length of overlap between them an integral number of half 
beat-wavelengths long. Ob^'iously, one will design the helices in such a 
way as to take advantage of this situation. 

Optimiun conditions are easily obtained by dijfferentiating ^c with 
respect to (3 and setting d^c/d^ equal to zero. This gives for the optimum 
conditions 



^opt — 



1 



b — a 



(2.7.5) 



or 



Pc opt 



2e 



h — a 



= 2e ')8opt 



(2.7.6) 



Equation (2.7.5), then, determines the ratio of the helix radii if it is re- 
quired that deviations from a chosen operating frequency shall have 
least effect. 

2.8 Field Solutions 

In treating the problem of coaxial coupled helices from the transmis- 
sion line point of view one important fact has not been considered, 
namely, the dispersive character of the phase constants of the separate 
helices, /3i and fS-i . By dispersion we mean change of phase velocity with 
frequency. If the dispersion of the inner and outer helices were the same 
it would be of little consequence. It is well known, however, that the 
dispersion of a helical transmission line is a function of the ratio of helix 
radius to wavelength, and thus becomes a parameter to be considered. 
When the theory of wave propagation on a helix was solved by means of 
Maxwell's equations subject to the boundary condition of a helically 
conducting cylindrical sheath, the phenomenon of dispersion first made 
its appearance. It is clear, therefore, that a more complete theory of 




/i 






'V^ 'TV 







Fig. 2.3 — ShoMtli liolix arrangement on which the field equations are based. 



COUPLED HELICES 145 

coupled helices will require similar treatment, namely, Maxwell's equa- 
tions solved now with the boundary conditions of two cylindrical heli- 
cally conducting sheaths. As shown on Fig. 2.3, the inner helix is specified 
by its radius a and the angle 1^1 made by the direction of conductivity 
with a plane perpendicular to the axis; and the outer helix by its radius 
h (not to be confused with the mutual coupling coefficient 5) and its 
corresponding pitch angle i/'-j . We note here that oppositely wound helices 
require opposite signs for the angles \f/i and i/'o ; and, further, that helices 
with equal phase velocities will ha\'e pitch angles of about the same 
absolute magnitude. 

The method of solving Maxwell's equations subject to the above men- 
tioned boundary conditions is given in Appendix I. We restrict our- 
selves here to giving some of the results in graphical form. 

The most universally used parameter in traveling-wave tube design is 
a combination of parameters: 

/3oa cot \pi 

where (So = 27r/Xo , Xo being the free-space wavelength, a the radius of 
the inner helix, and xpi the pitch angle of the inner helix. The inner helix 
is chosen here in preference to the outer helix because, in practice, it will 
be part of a traveling-wave tube, that is to say, inside the tube envelope. 
Thus, it is not only less accessible and changeable, but determines the 
important aspects of a traveling-wave tube, such as gain, power output, 
and efficiency. 

The theory gives solutions in terms of radial propagation constants 
which we shall denote jt and yt (bj^ analogy with the transverse and 
longitudinal modes of the transmission line theory). These propagation 
constants are related to the axial propagation constants ^t and j3( by 

Of course, in transmission line theory there is no such thing as a radial 
propagation constant. The propagation constant derived there and de- 
noted r corresponds here to the axial propagation constant j^. By 
analogy with (2.4.5) the beat phase constant should be written 

How^ever, in practice ^0 is usually much smaller than j3 and Ave can there- 
fore write with little error 

iSfc = 7e — li 
for the beat phase constant. For practical purposes it is convenient to 



146 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



J.OU 
































_^^ 









1 




3i.Z0 






COT ^2 _ „„„ 












;:^ 


^ 














COTV'i 


-0.90.. 


\ 




!^ 




























-0.82,^ 




^ 


:J^ 
















2.80 
















>^ 


<r 


















Q 2.40 
^2.00 
















^ 
































/. 




































^ 


































// 


/ 




































^ 




















|=,.25 






1. 60 


































1.20 






i 




































/// 
































0.80 







































0.5 



1.0 



2.0 2.5 

/io a COT ^, 



3.0 



3.5 



4.0 



4.5 



Fig. 2.4.1 — Beat phase-constant plotted as a function of /3oa cot i^i • These 
curves result from the solution of the field equations given in the appendix. For 
hi a = 1.25. 

normalize in terms of the inner helix radius, a: 



jSbO 



7<a — 7/a 



This has been plotted as a function of /5o a cot i/'i in Fig. 2.4, which 
should be compared with Fig. 2.2. It will be seen that there is considerable 
agreement between the results of the two methods, 

2.9 Bifilar Helix 

The failure of the transmission line theory to take into account dis- 
persion is well illustrated in the case of the bifilar helix. Here we have 
two identical helices wound in the same sense, and at the same radius. 
If the two wires are fed in phase we have the normal mode characterized 
by the sheath helix model whose propagation constant is the familiar 
Curve A of Fig. 2.5. If the two wires of the helix are fed out of phase we 
have the bifilar mode; and, since that is a two wure transmission system, 
we shall have a TEM mode which, in the absence of dielectric, propa- 
gates along the wire with the velocity of light. Hence, the propagation 
constant for this mode is simplj' /3oa cot \p and gives rise to the horizontal 



COUPLED HELICES 



147 



1.80 



1.60 



(0 

n 1.40 
<5. 



I 
to 

t.OO 



0.80 



0.60 



b. 
















^ 




^>. 




"^ 












"a"'" 










A 








s 


^ 


N. 


\. 


\^ 


















i 


& 












\ 


\ 


^ 


■^ 


0.82 














w 










^ = -0.98 

COT^, 


^ 


0.90 


^V 


^ 










// 


/ 










\ 


\ 


v 










J, 


/ 






















\ 


\ 


\, 








t 


























\ 










f 




























\ 








f 











































































































































0.5 



1.0 



1.5 



2.0 2.5 

/3oaCOTi^, 



3.0 



3.5 



4.0 



4.5 



Fig. 2.4.2 — Beat phase-constant plotted as a function of /3oa cot ^i-i . These 
curves result from the solution of the field equations given in the appendix. For 
hia = 1.5. 

line of Curve B in Fig. 2.5. Again the coupling phase constant j3c is given 
by the difference of the individual phase constants: 



^cO- — /3oa cot \f/ — ya 



(2.9.1) 



which is plotted in Fig. 2.6. Now note that when /So <3C 7 this equation is 
accurate, for it represents a solution of the field equations for the helix. 

From the simple unsophisticated transmission line point of view no 
coupling between the two helices would, of course, have been expected, 
since the two helices are identical in every way and their mutual capacity 
and inductance should then be equal and opposite. 

Experiments confirm the essential correctness of (2.9.1). In one experi- 
ment, which was performed to measure the coupling wavelength for the 
l)ifilar helices, we used helices with a cot 1/' = 3.49 and a radius of 0.036 
cm which gave a value, at 3,000 mc, of ^oa cot i^ = 0.51 . In these experi- 
ments the coupling length, L, defined by 

(/3oa cot xp — 7a) — = TT 
a 

was measured to be 15.7o as compared to a value of 13.5a from Fig. 2.6. 
At 4,000 mc the measured coupling length was 14.6a as compared to 



148 THE BELL vSYSTEM TECHNICAL JOURNAL, JANUARY 1956 



1.20 


b 
a 


1.76 








^ 


^^ 


^ 


^ 


X, 
























/ 


y 






^ 


^ 
S. 


X 
















1.00 








/ 


V 










\ 


^. 




-\ 










/ 










/ 














\ 


P> 










•^.82 






0.80 








r — 
















\ 




^ 










<5. 




















COT^ 
COT^ 


N 

^ = -0.9 
1 


k^ 


"^^^ 


0.90 






(0 0.60 




















a >s^ 


X 








<0 

'^ 0.40 
































^ 




. 












































































0.20 







































< 






































D 





5 


1 





1 


5 


2 


.0 


2 


.5 
^1 


3.0 


3.5 


4.0 


4 



Fig. 2.4.3 — Beat phase-constant plotted as a function of ^^a cot -^x . These 
curves result from the solution of the field equations given in the appendix. For 
hi a = 1.75. 

12.6a computed from Fig. 2.6, thus confirming the theoretical prediction 
rather well. The slight increase in coupling length is attributable to the 
dielectric loading of the helices which were supported in quartz tubing. 
The dielectric tends to decrease the dispersion and hence reduce /3,. . This 
is discussed further in the next section. 



2.10 Effect of Dielectric Material hetween Helices 

In many cases which are of interest in practice there is dielectric ma- 
terial between the helices. In particular when coupled helices are used 
with traveling-wave tubes, the tube envelope, which may be of glass, 
quartz, or ceramic, all but fills the space between the two helices. 

It is therefore of interest to know whether such dielectric makes any 
difference to the estimates at which we arrived earlier. We should not be 
surprised to find the coupling strengthened by the presence of the di- 
electric, because it is known that dielectrics tend to rob RF fields from 
the surrounding space, leading to an increase in the energy flow through 
the dielectric. On the other hand, tlio dielectric tends to bind the fields 
closer to the conducting medium. To find a qualitative answer to this 
question we have calculated the relative coupling phase constants for 
two sheath helices of infinite radius separated by a distance "d" for 1) 



COUPLED HELICES 



149 



1.00 


b 






































-a-^.u 






^ 


^^ 


^ 
































^ 


j^ 






^ 








COT Tp2 






^ 


^ 


C 0.60 

)^ 

1 0.40 
m 








y 










^ 






COT }^, 

^ 1 




























> 


-^ 


S^ 




































^ 




. , 


— 


-0. 


90 


=- 


-- 
























V 


^, 


i 


























^ 





^0^ 


98 




0.20 


1 







































1 




































( 


3 





5 


1 





1 


5 


2 




>oac 


2.5 


3 


.0 


3 


.5 


4 





4. 



Fig. 2.4.4 — Beat phase-constant plotted as a function of /3oa cot ^i . These 
curves result from the solution of the field equations given in the appendix. For 
b/a = 2.0. 

the case with dielectric between them having a relative dielectric con- 
stant e' = 4, and 2) the case of no dielectric. The pitch angles of the two 
helices were \p and —xp, respectively; i.e., the helices were assumed to be 
synchronous, and wound in the opposite sense. 
■ Fig. 2.7 shows a plot of the ratio of /3,,.//3, to ^d^ versus /3o (f//2) cotiA, 



1.00 



0.80 



to 
n 

«5. 0.60 

II 
m 

i 0.40 



0.20 



b 




































a-o.u 


































































^y 


^ 






























>< 


^ 


y 






















COT ^2 




^ 


\>^ 












^-- 














COT 5^, 


-^ 


r 












/, 


^ 
^ 


==^ 


"^^ 


N^ 








^^ 


y^ 










^ 


^ 


^ 




f/ 






^ 


^ 


N. 






^ 






-c 


).90 


-^ 










r 










\ 
















































-^ 


_ 


-o.s 


?8 


, 


^ 



0.5 



1.0 



f.5 



2.0 2.5 

/JoacoT;^, 



3.0 



3.5 



4.0 



4.5 



Fig. 2.4.5 — ■ Beat phase-constant plotted as a function of (3o« cot ^\ . These 
curves result from the solution of the field equations given in the appendix. For 
Va = 3.0. 



150 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



2.4 




0.5 ).0 1.5 2.0 2. 



5 3.0 3.5 
/3oaCOT^ 



4.0 4.5 



Fig. 2.5 — Propagation constants for a bifilar helix plotted as a function of 
/3oa cot i/-! . The curves illustrate, (A) the dispersive character of the in-phase 
mode and, (B) the non-dispersive character of the out-of -phase mode. 

where ^^ is the coupling phase-constant in the presence of dielectric, 
/3j is the phase-constant of each helix alone in the presence of the same 
dielectric, ^c is the coupling phase-constant with no dielectric, and (3 is 
the phase constant of each helix in free space. In many cases of interest 
/3o(d/2) cot lA is greater than 1.2. Then 






3£ + 1 " 
_2£' + 2_ 



g—(v'2« '+2-2)^0 (dl2) cot \l/ 



(2.10.1) 



Appearing in the same figure is a similar plot for the case when there is a 
conducting shield inside the inner helix and outside the outer, and 
separated a distance, "s," from the helices. Note that 

c? = 6 — a. 



It appears from these calculations that the effect of the presence of 
dielectric between the helices depends largely on the parameter /So (d/2) 
cot \{/. For values of this parameter larger than 0.3 the coupling wave- 
length tends to increase in terms of circuit wavelength. For values smaller 
than 0.3 the opposite tends to happen. Note that the curve representing 
(2.10.1) is a fair approximation down to /3o(c?/2) cot i/' = 0.6 to the curve 
representing the exact solution of the field equations. J. W. Sullivan, in 
unpublished work, has drawn similar conclusions. 



COUPLED HELICES 



151 



2.11 The Conditions for Maximum Power Transfer 

The transmission line theory has led us to expect that the most efficient 
power transfer will take place if the phase velocities on the two helices, 
prior to coupling, are the same. Again, this would be true were it not for 
the dispersion of the helices. To evaluate this effect we have used the 
field equation to determine the parameter of the coupled helices which 
gives maximum power transfer. To do this we searched for combinations 
of parameters which give an equal current flow in the helix sheath for 
either the longitudinal mode or the transverse mode. This was suggested 
by L. Stark, who reasoned that if the currents were equal for the indi- 
vidual modes the beat phenomenon would give points of zero RF current 
on the helix. 

The values of cot T/'2/cot 4/i which are required to produce this condi- 
tion are plotted in Fig. 2.8 for various values of b/a. Also there are shown 
values of cot ^2/cot \{/i required to give equal axial velocities for the helices 
before they are coupled. It can be seen that the uncoupled velocity of the 
inner helix must be slightly slower than that of the outer. 

A word of caution is* necessary for these curves have been plotted 
without considering the effects of dielectric loading, and this can have a 
rather marked effect on the parameters which we have been discussing. 
The significant point brought out by this calculation is that the optimum 



u.^o 




r 


N 
















0.24 
0.20 


/ 




\ 


\ 














/ 






N 














<D 


/ 








\, 














/ 








N 












/ 










N^ 












^ 0.12 












\ 


\^ 








~j 












■^v 








CD 
















^■^^^ 








0.08 


f- 
















^- 


-^ 


0.04 




































04 0.8 1.2 1.6 2.0 2.4 

/3oaCOT J^, 



2.8 



3.2 



3.6 



4.0 



Fig. 2.6 — The coupling phase-constant which results from the two possible 
modes of propagation on a bifilar helix shown as a function of jSoo cot i/-! . 



152 THE BELL SYSTEM TECHXICAL JOURNAL, JANUARY 1956 

2.6 

2.4 

2.2 
2.0 

i.8 

1.6 






u 



1.4 



i.2 



1.0 



0.8 



0.6 



0.4 



0.2 



















PROPAGATION 














DIRECTION 




\ 










\ 


^, 










\ 


L 






VA 








\ 


s 


PLANE SHEATH -^^^^"'^ XdiELECTRIC, 
HELICES \^^r e' 
CONDUCTING 
SHIELD 




\ 






\ 












\ 






















s=oo 




\ 


\, 














APPROXIMATION 


^^ 


^, 


s 


\ 




















"N 


'^ 




\ 


^ 
























"^^^ 


■^^ 



























o.t 



0.2 0.3 



0.4 0.5 



0.6 



0.7 



0.8 



0.9 



1.0 



1.1 



1.2 



/iofcOT^^ 



Fig. 2.7 — ^ The effect of dielectric material between coupled infinite radius 
sheath helices on their relative coupling phase-constant shown as a function of 
fiod/2 cot \pi . The effect of shielding on this relation is also indicated. 

condition for coupling is not necessarily associated with equal \'elocities 
on the uncoupled helices. 



2.12 Mode Impedance 

Before leaving the general theor_y of coupled helices something should 
be said regarding the impedance their modes present to an electron beam 
traveling either along their axis or through the annular space between 
them. The field solutions for cross woimd, coaxiall}^ coupled helices, 
which are given in Appendix I, have been used to compute the imped- 
ances of the transverse and longitudinal modes. The impedance, /v, is 
defined, as usual, in terms of the longitudinal field on the axis and the 
power flow along the system. 



COUPLED HELICES 



153 



K = 







F{ya) 



In Fig. 2.9, Fiya), for various I'atios of inner to outer radius, is plotted 
for both the transverse and longitudinal modes together with the value 
of F{ya) for the single helix {b/a = co). We see that the longitudinal 
mode has a higher impedance with cross wound coupled helices than 
does a single helix. We call attention here to the fact that this is the 
same phenomenon which is encountered in the contrawound helix^, where 
the structure consists of two oppositely wound helices of the same radius. 
As defined here, the transverse mode has a lower impedance than the 
single helix. This, however, is not the most significant comparison; for 
it is the transverse field midway between helices which is of interest in 
the transverse mode. The factor relating the impedance in terms of the 
transverse field between helices to the longitudinal field cni the axis is 
Er (f)/Ei(0), where f is the radius at which the longitudinal component 
of the electric field E^ , is zero for the transverse mode. This factor, 
plotted in Fig. 2.10 as a function of /3oa cot \l/r , shows that the impedance 
in. terms of the transverse field at f is interestingly high. 



1.00 




0.72 



1.6 2.0 2.4 

/3o a COT Ifi 



4.0 



Fig. 2.8 — The values of cot ^^./cot \pi required for complete power transfer 
plotted as a function of /3tia cot \pi for several values of b/a. For comparison, the 
value of cot ^2/cot \//i required for equal velocities on inner and outer helices is also 
shown. 



154 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



F(ra) 



7.0 
6.5 
6.0 
5.5 
5.0 
4.5 
4.0 

3.5 

• 
3.0 

2.5 

2.0 

1.5 

1.0 

0.5 



0.5 (.0 (.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 

/SoaCOTii', 

Fig. 2.9 — ■ Impedance parameter, F(ya), associated with both transverse and 
longitudinal modes shown for several values of b/a. Also shown is F{ya) for a 
single helix. 

It is also of interest to consider the impedance of the longitudinal 
mode in terms of the longitudinal field between the two helices. The 
factor, ^/(f)/£'/(0), relating this to the axial impedance is plotted in 
Fig. 2.11. We see that rather high impedances can also be obtained with 
the longitudinal field midway between helices. This, in conjunction with 
a hollow electron beam, should provide efficient amplification. 













LONGITUDINAL WAVE 


V 










COT U/2 


\ 


\^. V 






\ 


\ 


\ "^ 






^=-0.90 




k \ 


\ 




COT U/^ 




V \ 


\ 


\ 






b.ooV ^ 


\ 


\ 
\ 














a 


\ 


\ 




^ 














\ 


\ 


\ 














\-o\ 


\ 
\ 

\ 


\ 
\ 














\ 


\ 


> 














\ 


\ 




\ 












\ 


\ 
\ 


\ 




\ 

\ 














L \ 




i 


\ 














\ 




\ 


\ 
















\ 
\ 

\ 


\ 
\ 


V 














\ 
\ 


,25\ 












yp 


\ 


\ 




\ 












XT' 


\ 


\ 




\ 














\ 




\ 


^ 










\ ' 








\ 
















\ 
\ 




\ 

\ 
\ 












\ ' 


1 


\ 




\ 












\ 


\ 




\ 












\ 


\ 


\ 






k 




\ 






\ 


\ 
\ 


\ 






\ 




\ 






\ 




\ 






\ 






\ 




\ 


\ 

\ 




\ 




\ 
\ 




\ 


\ 






\ \ 




\ 




% 






\ 




N, 


\ 


\ 


\ 
\ 
s 

> 






\ 

\ 

\ 




\, 




V, 


^.0 


\\ 




\ 








N 




s^5 


\ 


\ 


1 


\ 

\ 


























1=1.2^ 




^^^ 


^ 


^^ 


^ 
















'""'^-^ 


^^ 


"^^ 


>., 


'- 




















^^ 












^ 


■ 


^^ 


^==^ 





3. APPLICATION OF COUPLED HELICES 



When we come to describe devices which make use of coupled helices 
we find that they fall, quite naturally, into two separate classes. One 



COUPLED HELICES 



155 



class contains those devices which depend on the presence of only one of 
the two normal modes of propagation. The other class of devices depends 
on the simultaneous presence, in roughly equal amounts, of both normal 
modes of propagation, and is, in general, characterized by the words 
"spatial beating." Since spatial beating implies energy surging to and 
fro between inner and outer helix, there is no special problem in exciting 
both modes simultaneously. Power fed exclusively to one or the other 




/bo a COT jfi, 



Fiji;. 2.10 — The relation l)et\veen the impedance in terms of the transverse 
field between conpled helices excited in the out-of -phase mode, and the impedance 
in terms of the longitudinal field on the axis shown as a function of /3oa cot tpi . 



156 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



helix will inevitably excite both modes equally. When it is desired to 
excite one mode exclusively a more difficult problem has to be solved. 
Therefore, in section 3.1 we shall first discuss methods of exciting one 
mode only before going on to discuss in sections 3.2 and 3.3 devices 
using one mode only. 

In section 3.4 we shall discuss devices depending on the simultaneous 
presence of both modes. 

3.1 Excitation of Pure Modes 

3.1.1 Direct Excitation 

In order to set up one or the other normal mode on coupled helices, 
voltages with specific phase and amplitudes (or corresponding currents) 



E|(f) 
E|(o) 



10^ 
5 

10^ 



10^ 



10' 



10 



10" 



COT 


ip? 












■ — = -0.90 

COT i^, 




1 














/ 














/ 












1 


' 










l-.o/ 


1 












L 










































L 




























1 


/ 












J 


l\.2b 












/ 










J 


/ J 


/ 










^ 


'^ 







































3 A 

/ho a COT 1fi^ 



Fig. 2.11 — -The relation Ijetween the impedance in terms of the longitudinal 
field between couj)led helices excited in the in-phase mode, and the impedance in 
terms of the longitudinal field on the axis shown as a function of /3offl cot \pi . 



COUPLED HELICES 157 

have to be supplied to each helix at the input end. A natural way of doing 
this might be by means of a two-conductor balanced transmission line 
(Lecher-line), one conductor being connected to the inner helix, the other 
to the outer helix. Such an arrangement would cause something like the 
transverse (-| — ) mode to be set up on the helices. If the two con- 
ductors and the balanced line can be shielded from each other starting 
some distance from the helices then it is, in principle, possible to intro- 
duce arbitrary amounts of extra delay into one of the conductors. A delay 
of one half period would then cause the longitudinal ( + + ) mode to be 
set up in the helices. Clearly such a coupling scheme would not be 
broad-band since a frequency-independent delay of one half period is not 
realizable. 

Other objections to both of these schemes are: Balanced lines are not 
generally used at microwave frequencies; it is difficult to bring leads 
through the envelope of a TWT without causing reflection of RF energy 
and without unduly encumbering the mechanical design of the tube plus 
circuits; both schemes are necessarily inexact because helices having 
different radii will, in general, require different voltages at either input 
in order to be excited in a pure mode. 

Thus the practicability, and success, of any general scheme based on 
the existence of a pure transverse or a pure longitudinal mode on coupled 
helices will depend to a large extent on whether elegant coupling means 
are available. Such means are indeed in existence as will be shown in the 
next sections. 

3.1.2 Tapered Coupler 

A less direct but more elegant means of coupling an external circuit 
to either normal mode of a double helix arrangement is by the use of the 
so-called "tapered" coupler.^' ^' ^^ By appropriately tapering the relative 
propagation velocities of the inner and outer helices, outside the inter- 
action region, one can excite either normal mode by coupling to one 
helix only. 

The principle of this coupler is based on the fact that any two coupled 
transmission lines support two, and only two, normal modes, regardless 
of their relative phase velocities. These normal modes are characterized 
by unequal wave amplitudes on the two lines if the phase velocities are 
not equal. Indeed the greater the phase velocity difference and /or 
the smaller the coupling coefficient between the lines, the more their 
wave amplitudes diverge. Furthermore, the wave amplitude on the line 
with the slower phase velocity is greater for the out-of-phase or trans- 
verse normal mode, and the wave amplitude on the faster line is greater 



158 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G 

for the longitudinal normal mode. As the ratio of phase constant to 
coupling constant approaches infinity, the ratio of the wave amplitudes 
on the two lines does also. Finally, if the phase velocities of, or coupling 
between, two coupled helices are changed gradually along their length 
the normal modes existing on the pair roughly maintain their identity 
evan though they change their character. Thus, by properly tapering the 
phase velocities and coupling strength of any two coupled helices one 
can cause the two normal modes to become two separate waves, one 
existing on each helix. 

For instance, if one desires to extract a signal propagating in the in- 
phase, or longitudinal, normal mode from two concentric helices of equal 
phase velocity, one might gradually increase the pitch of the outer helix 
and decrease that of the inner, and at the same time increase the diameter 
of the outer helix to decrease the coupling, until the longitudinal mode 
exists as a wave on the outer helix only. At such a point the outer helix 
may be connected to a coaxial line and the signal brought out. 

This kind of coupler has the advantage of being frequency insensitive ; 
and, perhaps, operable over bandwidths upwards of two octaves. It 
has the disadvantage of being electrically, and sometimes physically, 
quite long. 

3.1.3 Stepped Coupler 

There is yet a third way to excite only one normal mode on a double 
helix. This scheme consists of a short length at each end of the outer helix, 
for instance, which has a pitch slightly different from the rest. This 
has been called a "stepped" coupler. 

The principle of the stepped coupler is this: If two coupled transmis- 
sion lines have unlike phase velocities then a wave initiated in one line 
can never be completely transferred to the other, as has been shown in 
Section 2.4. The greater the velocity difference the less will be the maxi- 
mum transfer. One can choose a velocity difference such that the maxi- 
mum power transfer is just one half the initial power. It is a characteristic 
of incomplete power transfer that at the point where the maximum trans- 
fer occurs the waves on the two lines are exactly either in-phase or out-of- 
phase, depending on which helix was initially excited. Thus, the condi- 
tions for a normal mode on two equal-velocity helices can be produced 
at the maximum transfer point of two unlike velocity helices by initiating 
a wave on only one of them. If at that point the helix pitches are changed 
to give equal phase velocities in both helices, with equal current or volt- 
age amplitude on both helices, either one or the other of the two normal 
modes will be propagated on the two helices from there on. Although the 



COUPLED HELICES 159 

pitch and length of such a stepped coupler are rather critical, the re- 
quirements are indicated in the equations in Section 2.4. 

The useful bandwidth of the stepped coupler is not as great as that 
of the tapered variety, but may be as much as an octave. It has however 
the advantage of being very much shorter and simpler than the tapered 
coupler. 

3.2 Low-Noise Transverse-Field Amplifier 

r One application of coupled helices which has been suggested from the 
very beginning is for a transverse field amplifier with low noise factor. 
In such an amplifier the EF structure is required to produce a field which 
is purely transverse at the position of the beam. For the transverse mode 
there is always such a cylindrical surface where the longitudinal field is 
zero and this can be obtained from the field equation of Appendix II. 
In Fig. 3.1 we have plotted the value of the radius f at which the longi- 
tudinal field is zero for various parameters. The significant feature of 
this plot is that the radius which specifies zero longitudinal field is not 
constant with frequency. At frequencies away from the design frequency 
the electron beam will be in a position where interaction with longitudinal 
components might become important and thus shotnoise power will be 
introduced into the circuit. Thus the bandwidth of the amplifier over 
which it has a good noise factor would tend to be limited. However, this 
effect can be reduced by using the smallest practicable value of b/a. 

Section 2.12 indicates that the impedance of the transverse mode is 
very high, and thus this structure should be well suited for transverse 
field amplifiers. 

3.3 Dispersive Traveling-Wave Tube 

Large bandwidth is not always essential in microwave amplifiers. In 
particular, the enormous bandwidth over which the traveling-wave tube 
is potentially capable of amplifying has so far found little application, 
while relatively narrow bandwidths (although quite wide by previous 
standards) are of immediate interest. Such a relatively narrow band, if 
it is an inherent electronic property of the tube, makes matching the 
tube to the external circuits easier. It may permit, for instance, the use 
of non-reciprocal attenuation by means of ferrites in the ferromagnetic 
resonance region. It obviates filters designed to deliberately reduce the 
band in certain applications. Last, but not least, it offers the possibility 
of trading bandwidth for gain and efficiency. 

A very simple method of making a traveling-wave tube narrow-band 



160 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



0.5 



1.W 






































1.8 






























































^ 


^ 










1.7 












COT \p. 


_^ 


^ 


^ 






<^ 


















T^ = -0. 

COT ^, 


82^ 


^ 




^ 


^ 


^ 






l=- 


1.6 












^ 


^ 


^ 


-0.90 


^ 
























^ 


^ 


^ 
















* 




1.5 


— - 





















-0.9 


8 


' 


^-' ' 










^ 




^ 
































1.4 


-^ 




















































































COT ^i'2 

^-^ = -0.82 

COT UJ^ , 


-0.9j 


, 














^- 


1.3 





















. 


— 


"ZH 


' 1 
-0.98 














































1.2 




























_ 


"71 














| = ,.25 




— 


^= 








CO' 


r 1//. 






— T-" 


H 
























COT 5^, 


= -0.82 - 
-0.90 ■ 


' / 
^ 


1 
/ 

/ 
















i.n 
















-0 


.98 


~ 



















1.0 



1.5 



2.0 



2.5 3.0 

/3o a COT j^. 



3.5 



4.0 



4.5 



5.0 



Fig. 3.1 — The radius r at which the longitudinal field is zero for transversely 
excited coupled coaxial helices. 



is by using a dispersive circuit, (i.e. one in which the phase velocity varies 
significantly with frequency). Thus, we obtain an amplifier that can be 
limed by varying the beam voltage; being dispersive we should also 
expect a low group velocity and therefore higher circuit impedance. 

Calculations of the phase velocities of the normal modes of coupled 
concentric helices presented in the appendix show that the fast, longitu- 
dinal or (+ + ) mode is highly dispersive. Given the geometry of two 
such coupled helices and the relevant data on an electron beam, namely 
current, voltage and beam radius, it is possible to arrive at an estimate 
of the dependence of gain on frecjuency. 

Experiments with such a tube showed a Ijandwidth 3.8 times larger 
than the simple estimate would show. This we ascribe to the presence 



COUPLED HELICES 161 

of the dielectric between the helices in the actual tube, and to the neglect 
of power propagated in the form of spatial harmonics. 

Nevertheless, the tube operated satisfactorily with distributed non- 
reciprocal ferrite attenuation along the whole helix and gave, at the 
center frequency of 4,500 mc/s more than 40 db stable gain. 

The gain fell to zero at 3,950 mc/s at one end of the band and at 
4,980 mc/s at the other. The forward loss was 12 db. The backward 
loss was of the order of 50 db at the maximum gain frequency. 

3.4 Devices Using Both Modes 

In this section we shall discuss applications of the coupled-helix princi- 
ple which depend for their function on the simultaneous presence of both 
the transverse and the longitudinal modes. When present in substantially 
equal magnitude a spatial beat-phenomenon takes place, that is, RF 
power transfers back and forth between inner and outer helix. 

Thus, there are points, periodic with distance along each helix, where 
there is substantially no current or voltage; at these points a helix can be 
terminated, cut-off, or connected to external circuits without detriment. 

The main object, then, of all devices discussed in this section is power 
transfer from one helix to the other; and, as will be seen, this can be ac- 
complished in a remarkably efficient, elegant, and broad-band manner. 

3.4.1 Coupled-Helix Transducer 

It is, by now, a well known fact that a good match can be obtained 
between a coaxial line and a helix of proportions such as used in TWT's. A 
wire helix in free space has an effective impedance of the order of 100 
ohms. A conducting shield near the helix, however, tends to reduce the 
helix impedance, and a value of 70 or even 50 ohms is easily attained. 
Pro\'ided that the transition region between the coaxial line and the 
helix does not present too abrupt a change in geometry or impedance, 
relatively good transitions, operable over bandwidths of several octaves, 
can l)e made, and are used in practice to feed into and out of tubes em- 
ploying helices such as TWT's and backward-wave oscillators. 

One particularly awkward point remains, namely, the necessity to lead 
the coaxial line through the tube envelope. This is a complication in 
manufacture and reciuires careful positioning and dimensioning of the 
helix and other tube parts. 

Coupled helices offer an opportunity to overcome this difficulty in the 
form of the so-called coupled-helix transducer, a sketch of which is 
shown in Fig. 3.2. As has been shown in Section 2.3, with helices having 



162 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



the same velocity an overlap of one half of a beat wavelength will result 
in a 100 per cent power transfer from one helix to the other. A signal in- 
troduced into the outer helix at point A by means of the coaxial line will 
be all on the inner helix at point B, nothing remaining on the outer helix. 
At that point the outer helix can be discontinued, or cut off; since there 
is no power there, the seemingly violent discontinuity represented by the 
'open" end of the helix will cause no reflection of power. In practice, un- 
fortunately, there are always imperfections to consider, and there will 
often be some power left at the end of the coupler helix. Thus, it is de- 
sirable to terminate the outer helix at this point non-reflectively, as, for 
instance, by a resistive element of the right value, or by connecting to it 
another matched coaxial line which in turn is then non-reflectively ter- 
minated. 

It will be seen, therefore, that the coupled-helix transducer can, in 
principle, be made into an efficient device for coupling RF energy from 
a coaxial line to a helix contained in a dielectric envelope such as a glass 
tube. The inner helix will be energized predominantly in one direction, 
namely, the one away from the input connection. Conversely, energy 
traveling initially in the inner helix will be transferred to the outer, and 
made available as output in the respective coaxial line. Such a coupled- 
helix transducer can be moved along the tube, if required. As long as the 
outer helix completely overlaps the inner, operation as described above 
should be assured. By this means a new flexibility in design, operation 
and adjustment of traveling-wave tubes is obtained which could not be 
achieved by any other known form of traveling-wave tube transducer. 
Naturally, the applications of the coupled-helix transducer are not 
restricted to TWT's only, nor to 100 per cent power transfer. To obtain 




Fig. 3.2 — A simple coupled helix transducer. 



COUPLED HELICES 1G3 

power transfer of proportions other than 100 per cent two possibilities 
are open: either one can reduce the length of the synchronous coupling 
helix appropriately, or one can deliberately make the helices non-syn- 
chronous. In the latter case, a considerable measure of broad-banding 
can be obtained by making the length of overlap again equal to one half 
of a beat-wavelength, while the fraction of power transferred is deter- 
mined by the difference of the helix velocities according to 2.4.7. An 
application of the principle of the coupled-helix transducer to a variable 
delay line has been described by L. Stark in an unpublished memo- 
randum. 

Turning again to the complete power transfer case, we may ask: 
How broad is such a coupler? 

In Section 2.7 we have discussed how the radial falling-off of the RF 
energy near a helix can be used to broad-band coupled-helix devices 
which depend on relative constancy of beat-wavelength as frequency 
is varied. On the assumption that there exists a perfect broad-band match 
between a coaxial line and a helix, one can calculate the performance of 
a coupled-helix transducer of the type shown in Fig. 3.2. 

Let us define a center frequency co, at which the outer helix is exactly 
one half beat-wavelength, \b , long. If oj is the frequency of minimum 
beat wavelength then at frequencies coi and co2 , larger and smaller, 
respectively, than co, the outer helix will be a fraction 5 shorter than 
}i\b , (Section 2.7). Let a voltage amplitude, Vo , exist at the point where 
the outer helix is joined to the coaxial line. Then the magnitude of the 
voltage at the other end of the outer helix will be | F2 • sin (x5/2) | which 
means that the power has not been completely transferred to the inner 
helix. Let us assume complete reflection at this end of the outer helix. 
Then all but a fraction of the reflected power will be transferred to the 
inner helix in a reverse direction. Thus, we have a first estimate for the 
"directivity" defined as the ratio of forward to backward power (in db) 
introduced into the inner helix: 



D = 



10 log sin" 




(3.4.1.1) 



We have assumed a perfect match between coaxial line and outer helix; 
thus the power reflected back into the coaxial line is proportional to 
sin^(x5/2). Thus the reflectivity defined as the ratio of reflected to 
incident power is given in db by 



i^ = 10 log sin' ^ (3.4.1.2) 




164 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

For the sake of definiteness, let us choose actual figures: let /3a = 2.0. 
and hi a = 1.5. And let us, arbitrarily, demand that R always be less than 
-20 db. 

This gives sin (7r5/2) < 0.316 and 7r5/2 < 18.42° or 0.294 radians, 
8 < 0.205. With the optimum value of (Sea = 1.47, this gives the mini- 
mum permissible value of I3ca of 1.47/(1 + 0.205) = 1.22. From the 
graph on Fig. 2.2 this corresponds to values of jSa of 1.00 and 3.50. 
Therefore, the reflected power is down 20 db over a frequency range of 
aj2/aji = 3,5 to one. Over the same range, the directivity is better than 
10 to one. Suppose a directivity of better than 20 db were required. 
This requires sin (7r5/2) = 0.10, 8 = 0.0638 and is obtained over a fre- 
quency range of approximately two to one. Over the same range, the 
reflected power would be down by 40 db. 

In the above example the full bandwidth possibilities have not been 
used since the coupler has been assumed to have optimum length when 
jSctt is maximum. If the coupler is made longer so that when I3ca is maxi- 
mum it is electrically short of optimum to the extent permissible by 
the quality requirements, then the minimum allowable (S^a becomes even 
smaller. Thus, for h/a =1.5 and directivity 20 db or greater the rea- 
lizable bandwidth is nearly three to one. 

When the coupling helix is non-reflectively terminated at both ends, 
either by means of two coaxial lines or a coaxial line at one end and a 
resistive element at the other, the directivity is, ideally, infinite, irrespec- 
tive of frequency; and, similarly, there will be no reflections. The power 
transfer to the inner helix is simply proportional to cos (t8/2). Thus, 
under the conditions chosen for the example given above, the coupled- 
helix transducer can approach the ideal transducer over a considerable 
range of frequencies. 

So far, we have inspected the performance and bandwith of the 
coupled-helix transducer from the most optimistic theoretical point of 
view. Although a more realistic approach does not change the essence 
of our conclusions, it does modify them. For instance, we have neglected 
dispersion on the helices. Dispersion tends to reduce the maximum at- 
tainable bandwidth as can be seen if Fig. 2.4.2 rather than Fig. 2.2 is 
used in the example cited above. The dielectric that exists in the annular 
region between coupled concentric helices in most practical couplers 
may also affect the bandwidth. 

In practice, the performance^ of coupled-hc^lix transducers has been 
short of the ideal. In the first place, the match from a coaxial line to a 
helix is not perfect. Secondly, a not inappreciable fraction of the RF 
power on a real wire helix is propagated in the form of spatial harmonic 



COUPLED HELICES 



165 



28 



26 



24 



22 



20 



18 



)6 

in 

_i 

LU 

m (4 
u 



12 



10 













r\ 


















\ 








\ 
\ 

\ 
















' * / 
' * / 
1 t / 




[\ 


n 


[ 1 




I 


















1 1 




\j 


^ 
















\ 






Wf 


\ 




1 


\ 


\ 












I / 
I / 
\ / 




\1^ 


U~ 


/ 

/ 
/ 




\ 




.' 
1 










A 




\J 










\- 












/ \ 














\ 


A 








/ 
















Vi 
































\ 
\ 

\ 


1 

1 

/ 
1 






p 


OUPLER DIRECTIVITY 
ETURN LOSS 








\ 


\ 


1 


J 
























\ 


A 


























V 


I 


/ 
























l 





1.5 



2.5 3 4 

FREQUENCY IN KILOMEGACYCLES 



Fig. 3.3 — • The return loss and directivity of an experimental 100 per cent 
coupled-helix transducer. 

wave components which have variations with angle around the helix- 
axis, and coupling between such components on two helices wound in 
opposite directions must be small. Finally, there are the inevitable me- 
chanical inaccuracies and misalignments. 

Fig. 3.3 shows the results of measurements on a coupled-helix trans- 
ducer with no termination at the far end. 



3.4.2 Coupled-Helix Attenuator 

In most TWT's the need arises for a region of heavy attenuation 
somewhere between input and output; this serves to isolate input and 
output, and prevents oscillations due to feedback along the circuit. Be- 
cause of the large bandwidth over which most TWT's are inherently 
capable of amplifying, substantial attenuation, say at least 60 db, is 



166 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

required over a bandwidth of maybe 2 octaves, or even more. Further- 
more, such attenuation should present a very good match to a wave on 
the heHx, particularly to a wave traveling backwards from the output 
of the tube since such a wave will be amplified by the output section of 
the tube. 

Another requirement is that the attenuator should be physically as 
short as possible so as not to increase the length of the tube unneces- 
sarily. 

Finally, such attenuation might, with advantage, be made movable 
during the operation of the tube in order to obtain optimum performance, 
perhaps in respect of power output, or linearity, or some other aspect. 

Coupled-helix attenuators promise to perform these functions satis- 
factorily. 

A length of outer helix (synchronous with the inner helix) one half of a 
beat wavelength long, terminated at either end non-reflectively, forms a 
very simple, short, and elegant solution of the coupled-helix attenuator 
problem. A notable weakness of this form of attenuator is its relatively 
narrow bandwidth. Proceeding, as before, on the assumption that the 
attenuator is a fraction 8 larger or smaller than half a beat wavelength 
at frequencies coi and W2 on either side of the center frequency co, we find 
that the fraction of power transferred from the inner helix to the attenu- 
ator is then given by (1 — sin" (ir8/2)). The attenuation is thus simply 

A = sin^ (I) 

For helices of the same proportions as used before in Section 3.4.1, we 
find that this will give an attenuation of at least 20 db over a frequency 
band of two to one. At the center frequency, coo , the attenuation is in- 
finite; — in theory. 

Thus to get higher attenuation, it would be necessary to arrange for a 
sufficient number of such attenuators in tandem along the TWT. More- 
over, by properly staggering their lengths within certain ranges a wdder 
attenuation band may be achieved. The success of such a scheme largely 
depends on the ability to terminate the helix ends non-reflectively. Con- 
siderable work has been done in this direction, but complete success is 
not yet in sight. 

Another basically different scheme for a coupled-helix attenuator rests 
on the use of distributed attenuation along the coupling helix. The diffi- 
culty with any such scheme lies in the fact that unequal attenuation in 
the two coupled helices reduces the coupling between them and the moi'c 
they differ in respect to attenuation, the less the coupling. Naturally, one 



COUPLED HELICES 167 

would wish to have as Httle attenuation as practicable associated with 
the inner helix (inside the TWT). This requires the attenuating element 
to be associated with the outer helix. Miller has shown that the maxi- 
mum total power reduction in coupled transmission systems is obtained 
when 

ai — 0:2 



where ai and 012 are the attenuation constants in the respective systems, 
and ^b the beat phase constant. If the inner helix is assumed to be loss- 
less, the attenuation constant of the outer helix has to be effectively equal 
to the beat wave phase constant. It turns out that 60 db of attenuation 
requires about 3 beat wavelengths (in practice 10 to 20 helix wave- 
lengths). The total length of a typical TWT is only 3 or 4 times that, 
and it will be seen, therefore, that this scheme may not be practical as 
the only means of providing loss. 

Experiments carried out Avith outer helices of various resistivities and 
thicknesses by K. M. Poole (then at the Clarendon Laboratory, Oxford, 
England) tend to confirm this conclusion. P. D. Lacy" has described a 
coupled helix attenuator which uses a multifilar helix of resistance 
material together with a resistive sheath between the helices. 

Experiments were performed at Bell Telephone Laboratories with a 
TWT using a resistive sheath (graphite on paper) placed between the 
outer helix and the quartz tube enclosing the inner helix. The attenua- 
tions were found to be somewhat less than estimated theoretically. The 
attenuator helix was movable in the axial direction and it w^as instructive 
to observe the influence of attenuator position on the power output from 
the tube, particularly at the highest attainable power level. As one might 
expect, as the power level is raised, the attenuator has to be moA-ed nearer 
to the input end of the tube in order to obtain maximum gain and power 
output. In the limit, the attenuator helix has to be placed right close to 
the input end, a position which does not coincide with that for maximum 
low-level signal gain. Thus, the potential usefulness of the feature of 
mobility of coupled-helix elements has been demonstrated. 

4. CONCLUSION 

In this paper we have made an attempt to develop and collect together 
a considerable body of information, partly in the form of equations, 
partl}^ in the form of graphs, which should be of some help to workers 
in the field of microwave tubes and devices. Because of the crudity of the 
assumptions, precise agreement between theory and experiment has not 



168 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

been att-aiiu>(l iiur can it l)c expected. Nevertheless, the kind of physical 
phenomena occurring with coupled helices are, at least, qualitatively 
described here and should permit one to develop and construct various 
types of (lexices with fair chance of success. 

ACKNOWLEDGEMENTS 

As a final note the authors wish to express their appreciation for the 
patient work of Mrs. C. A. Lambert in computing the curves, and to 
G. E. Korb for taking the experimental data. 

Appendix i 
i. solution of field equations 

In this section there is presented the field equations for a transmission 
system consisting of two helices aligned with a common axis. The propa- 
gation properties and impedance of such a transmission system are dis- 
cussed for various ratios of the outer helix radius to the inner helix radius. 
This system is capable of propagating two modes and as previously 
pointed out one mode is characterized by a longitudinal field midway 
between the two helices and the other is characterized by a transverse 
field midway between the tw^o helices. 

The model which is to be treated and shown in Fig. 2.3 consists of an 
inner helix of radius a and pitch angle \pi which is coaxial with the outer 
helix of radius 6 and pitch angle \j/2 . The sheath helix model will be 
treated, wherein it is assumed that helices consist of infinitely thin sheaths 
which allow for ciuTent flow- only in the direction of the pitch angle \p. 

The components of the field in the region inside the inner helix, be- 
tween the two helices and outside the outer helix can be written as 
follows — inside the inner helix 

H,, = BrIoM (1) 

E., = B^hM (2) 

H,, = j - BMyr) (3) 

7 

Hr, = ^^ BMyr) (4) 

7 

E,, = -j "^ BMyr) (5) 

7 

Er, = -^ BJ,(rr) (()) 

7 



COUPLED HELICES 169 

and between the two helices 

H,, = BMrr) + BJuirr) ' (7) 

E., = BJoiyr) + B^oiyr) (8) 

H,, = ^~ [B,h(yr) - B^^(yr)] (9) 

7 

Hr, = -^ [53/1(7/0 - BJuiyr)] (10) 

7 

E,, = - J ^ [B^hiyr) - BJuiyr)] (11) 

7 

Er, = -^ [BMyr) - BJv,{yr)\ (12) 

7 

and outside the outer hehx 

H.^ = B,Ko(yr) (13) 

E,, = 58/vo(7r) (14) 

^.s = -J- BsK,{yr) (15) 

7 

Hr, = ^^ 5,Ki(7r) (16) 

7 

^,, = i — BJuiyr) (17) 

7 

^r« = ^^ 58Ki(7r) (18) 

7 

With the sheath helix model of current flow only in the direction of wires 
we can specify the usual boundary conditions that at the inner and outer 
helix radius the tangential electric field must be continuous and per- 
pendicular to the wires, whereas the tangential component of magnetic 
field parallel to the current flow must be continuous. These can be written 
as 

E, sin t/' + E^ cos ^ = (19) 

' E, , E^ and (H, sin \f/ -f H^ cos \p) be equal on either side of the helix. 
By applying these conditions to the two helices the following equations 
are obtained for the various coefficients. 



170 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



First, we will define a more simple set of parameters. We will denote 

Io(ya) by /oi and h{yh) by /02 , etc. 

Further let us use the notation introduced by Humphrey, Kite and 
James" in his treatment of coaxial helices. 



Poi ^ laiKoi P02 = ToiKa2 Rq = I01K02 

Pn = InKn P12 = InKu Ri = /iii^i2 

and define a common factor (C.F.) by the equation 

r(/3oa cot hY p p (/3oa cot ^pif cot i/'z „ r, 

\_ (yay {jay cot t^i 

+ Ro' — PoiP 



(20) 



.,] 



(21) 



With all of this we can now write for the coefficients of equations 1 
through 18: 






y ju j8oa cot \pi 1 02 

U iQoa cot 1^1 7oi/vi2 RiSoa cot i^i) 

y M ""to C.F. L 

^4 _ _ • / £_ /3oa cot 1^1 /pi/ii r( 

B^~ -^ T M 7^ C.F. L" (7a)^ 



5 
5 






(7a)'^ 
(/3oa cot 1^2)^ 



cot 1A2 p 
cot ;^i J 

P12 — jPo2 



■] 



B5 

B, 
Bt 



Ro 
C.F. 



Ro — 



((Soa cot xl/iY cot 1/' 



(7a^) 
(/3oa cot 1^2) 



cot l/' 



;«'] 



(7a)^ 



12 — -P02 



B7 _ • . /£ i3oa cot lAi 1 /oi r 
5; ~ "^ y M 7a C.F. K12 L 



Bs _ (|8oa cot i/'i)" cot 1/^2 /pi "" 
B2 {yay coT^i C.F.Po 



P02R1 — 
P02R1 - 



cot l/'2 
cot i/'i 

cot l// 
cot \l/ 



2R0 
- P12R0 



(22) 
(23) 
(24) 
(25) 
(26) 
(27) 
(28) 



The last equation necessary for the solution of our field problem is the 
transcendental equation for the propagation constant, 7, which can be 



COUPLED HELICES 



171 



written 



Ro — 



(i8o a cot \J/iY cot ^2 „ 
(yaY cot 4/1 



[ 



= P02 - 



(jSo a cot \p2) D 

? Vi -^ 12 



Poi - 



(/3oa cot ^0" 
(yay 



_ (29) 



11 



The solutions of this equation are plotted in Fig. 4.1. 

There it is seen that there are two values of 7, one, yt , denoting the 
slow mode with transverse fields between helices and the other, yt , 
denoting the fast mode with longitudinal fields midway between the two 
helices. 



5.0 



4.S 



4.0 



3.5 



3.0 



ra 



2.6 



2.0 



1.5 



1.0 



0.5 







4 = 1.25 








// 




/ 






COT 5^2 
COT ^1 

0.82 

0.90 

0.98 




^ 


#■ 










/t 


// 
















A 




na 
















/ 




















f 










A 


y 

f 




/ 


r 














/ 


I 


/ 

if 

< 










A 








/ 








-<^^ 


^ 


•y 








L 




-- 


•=**^ 















0.5 1.0 1.6 



2.0 2.5 3.0 

/3o a COT yj 



3.5 4.0 4.5 5.0 



Fig. 4.1.1 —-The radial propagation constants associated with the transverse 
and longitudinal modes on coupled coaxial sheath helices given as a function of 
|3oa cot ^i-i for several values of hja = 1.25. 



172 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



These equations can now be used to compute the power flow as defined 
by 

P = }4 Re j E XH' 
which can be written in the form 



dA 



(30) 



r^;^(o)T 
L ^'p J 



fo © ^^-' ''' 



(31) 



where 



[F{ya, yb)] = 



(( 



W + 



(i8oa cot i/' 
{yar 



^ /n^) 



(In' - /oi/2i)(C.F.)- 



- A'02' + 



240 (C.F.)' 

(i8oa cot 1/^1)' 
(t«)'^ 



/or/n- r 



(80a cot ;^iY 



' 



/Vl2" i^O - 



ya 

((Soft cot i^i)' cot \p2 
(ya)'' cot i/'i 



Rx 



- ) (/02/22 — /12') 4" (/ii — /01/21) 



, /p (/3oa cot 1A1)- cot \i/2 p Wp (^0^ ^'0^' "^2)'' p 



(ya)'^ cot i/'i 



(7a)^ 



( - ) i'lInKu + /02/V22 + /22X02) — (2/iiKii + /01K21 + /21/voi) 

ot ^2)'^ p T 



(32) 



2 , (^ofl cot l/'i) J ■> 
•'01 i- 7 r;; ^11 



(l3oa cot 



- I (K02K22 — K12 ) — (K01K21 — Kn) 



.a, 



+ 



(/3oa cot i^i)" A^ 



■ 2 , (/Soa cot i/'2)" J 2 J. 2 



(7a)'^ 



cot 1/^2 p J. 
I 02itl — -— r- i 12A0 

cot 1^1 



[/Vo2A'22 — /V12"] 



In (32) we find the power in the transverse mode by using values of 



COUPLED HELICES 



173 



5.0 




0.5 



2.0 2.5 3.0 

/3o a COT y/ 



5.0 



Fig. 4.1.2 — The radial propagation constants associated with the transverse 
and longitudinal modes on coupled coaxial sheath helices given as a function of 
^ofl cot \}/i when h/a — 1.50. 

yt obtained from (29) and similarly the power in the longittidinal mode is 
found by using values of yi . 



II. FINDING r 

When coaxial helices are used in a transverse field amplifier, only the 
transverse field mode is of interest and it is important that the helix 
parameters be adjusted such that there is no longitudinal field at some 
radius, f, where the cylindrical electron beam will be located. This condi- 
tion can be expressed by equating Ez to zero at r = f and from (8) 



BMyr) + B^,{yf) = 



(33) 



174 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

which can be written with (25) and (26) as 
(jSott cot ipiY cot \f/2 



K(i2 Ri 



[ 



02 ilO 



(7a)- cot \{/i 
= /oi 



Ri 



loM 



(/3oa cot \l/2)- 

■I 02 — 7 rr, rn 



(34) 



Koiyf) 



This equation together with (29) enables one to evaluate f/a versus j8oa 
cot \l/i for various ratios of b/a and cot i^2/cot xpi . The results of these 
calculations are shown in Fig. 3.1. 



5.0 



4.5 



4.0 



3.5 



3.0 



7a 



2.5 




2,0 



0.5 



Fig. 4.1. .3 — The radial propagation constants associated with the transverse 
and longitudinal modes on coupled coaxial sheath helices given as a finiclion of 
0oa cot \{/i when b/a = 1.75. 



i 



COUPLED HELICES 



175 



5.0 



7a 




2.0 2.5 3.0 

/Oo <3 COT ^, 



3.5 



4.0 



4.5 



5.0 



Fig. 4.1.4 — The radial propagation constants associated with the transverse 
and longitudinal modes on coupled coaxial sheath helices given as a function of 
/3oa cot yp\ when 6/a = 2.0. 



III. COMPLETE POWER TRANSFER 

For coupled heli.x applications we require the coupled helix parame- 
ters to be adjusted so that RF power fed into one helix alone will set up 
the transverse and longitudinal modes equal in amplitude. For this 
condition the power from the outer helix will transfer completely to the 
inner helix. The total current density can be written as the sum of the 
current in the longitudinal mode and the transverse mode. Thus for the 
inner helix we have 



-i&li 



J a = Jate-'''' + Jate 



.-J^<2 



(35) 



17G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



7?, 2.5 




Fig. 4.1.5 — The radial propagation constants associated with the transverse 
and longitudinal modes on coupled coa.xial sheath helices given as a function of 

/3oa cot i/-! when hi a = 3.0. 



and for the outer helix 
For complete power transfer we ask that 

•J hi — J hi 

when Jo is zero at the input {z = 0) 
or 

Jbt _ Jbt 

J at J at 



\ 



(36) \ 



(37) 



COUPLED HELICES 177 

Now J at is equal to the discontinuity in the tangential component of 
magnetic field which can be written at r = a 

J at = {H,z cos ^i — //^5 sin \pi) — (H,i cos i/'i - H^o sin \f/i) 

\^'hich can be written as 

Ja( = - (H,i - H,3)a((cot i/'i + tau xj/i) slu \Pi (38) 

and similarily at r = h 

Jb( = — (H^7 — H,s)b({cot \p2 + tan 4^2) sin i/'2 (39) 

Equations (38) and (39) can be combined with (37) to give as the condi- 
tion for complete power transfer 

At = -At (40) 

where 

^ = V (yay / ni) 

(T J^ _i- r V \( T? (/3oa cot <Ai)'^ cot 1^2 „ \ 
\ {yo,y cot i/'i / 

In (40) At is obtained by substituting jt into (41) and At is obtained by 
substituting 7 < into (41). 

The value of cot i/'o/cot i/'i necessary to satisfy (40) is plotted in Fig. 
2.8. 

In addition to cot i/'o/cot i/'i it is necessary to determine the interference 
wavelength on the helices and this can be readily evaluated by consider- 
ing (36) which can now be written 

or 

/, = /,,.-«^'+^''-''^> cos ^ilJZ^ , (48) 

and 

J, = J.ce-'''^'^'^''"' cos M/3i^ (49) 

where we have defined 

iSfcO = {yta — jta) (50) 

This value of /S^ is plotted versus /3oa cot i/'i in Fig. 2.4. 



178 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

BIBLIOGRAPHY 

1. J. R. Pierce, Traveling Wave Tubes, p. 44, Van Nostrand, 1950. 

2. R. Kompfner, Experiments on Coupled Helices, A. E. R. E. Report No. 

G/M98, Sept., 1951. 

3. R. Kompfner, Coupled Helices, paper presented at I. R. E. Electron Tube 

Conference, 1953, Stanford, Cal. 

4. G. Wade and N. Rynn, Coupled Helices for Use in Traveling-Wave Tubes, 

I.R.E. Trans, on Electron Devices, Vol. ED-2, p. 15, July, 1955. 

5. S. E. Miller, Coupled Wave Theory and Waveguide Applications, B.S.T.J., 

33, pp. 677-693, 1954. 

6. M. Chodorow and E. L. Chu, The Propagation Properties of Cross-Wound 

Twin Helices Suitable for Traveling-Wave Tubes, paper presented at the 
Electron Tube Res. Conf., Stanford Univ., June, 1953. 

7. G. M. Branch, A New Slow Wave Structure for Traveling-Wave Tubes, paper 

presented at the Electron Tube Res. Conf., Stanford Univ., June, 1953. 
G. M. Branch, E.xperimental Observation of the Properties of Double Helix 
Traveling-Wave Tubes, paper presented at the Electron Tube Res. Conf., 
Univ. of Maine, June, 1954. 

8. J. S. Cook, Tapered Velocity Couplers, B.S.T.J. 34, p. 807, 1955. 

9. A. G. Fox, Wave Coupling by Warped Normal Modes, B.S.T.J., 34, p. 823, 

1955. 

10. W. H. Louisell, Analysis of the Single Tapered Mode Coupler, B.S.T.J., 34, 

p. 853. 

11. B. L. Humphrey's, L. V. Kite, E. G. James, The Phase Velocity of Waves in a 

Double Helix, Report No. 9507, Research Lab. of G.E.C., England, Sept., 
1948. 

12. L. Stark, A Helical-Line Phase Shifter for Ultra-High Frequencies, Technical 

Report No. 59, Lincoln Laboratory, M.LT., Feb., 1954. 

13. P. D. Lacy, Helix Coupled Traveling-Wave Tube, Electronics, 27, No. 11, 

Nov.. 1954. 



Statistical Techniques for Reducing the 
Experiment Time in Reliability Studies 

By MILTON SOBEL 

(Manuscript received September 19, 1955) 

Given two or more processes, the units from which fail in accordance with 
an exponential or delayed exponential law, the problem is to select the partic- 
ular process with the smallest failure rate. It is assumed that there is a com- 
mon guarantee period of zero or positive duration during which no failures 
occur. This guarantee period may be known or unknown. It is desired to 
accomplish the above goal in as short a time as possible without invalidating 
certain predetermined probability specifications. Three statistical techniques 
are considered for reducing the average experiment time needed to reach a 
decision. 

1 . One technique is to increase the initial number of units put on test. 
This technique will substantially shorten the average experiment time. Its 
effect on the probability of a correct selection is generally negligible and in 
some cases there is no effect. 

2. Another technique is to replace each failure immediately by a new 
unit from the same process. This replacement technique adds to the book- 
keeping of the test, but if any of the population variances is large (say in 
comparison with the guarantee period) then this technique will result in a 
substantial saving in the average experiment time. 

3. A third technique is to use an appropriate sequential procedure. In 
many problems the sequential procedure results in a smaller average experi- 
ment time than the best non-sequential procedure regardless of the true 
failure rates. The amount of saving depends principally on the ^'distance'" 
between the smallest and second smallest failure rates. 

For the special case of two processes, tables are given to show the proba- 
bility of a correct selection and the average experiment time for each of three 
types of procedures. 

Numerical estimates of the relative efficiency of the procedures are given 
by computing the ratio of the average experiment time for two procedures of 
different type with the same initial sample size and satisfying the same 
probability specification. 

179 



180 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

INTRODUCTION 

This paper is concerned with a study of the advantages and disad- 
vantages of three statistical techniques for reducing the average dura- 
tion of hfe tests. These techniques are: 

1. Increasing the initial number of units on test. 

2. Using a replacement technique. 

3. Using a sequential procedure. 

To show the advantages of each of these techniques, we shall consider 
the problem of deciding which of two processes has the smaller failure 
rate. Three different types of procedures for making this decision will 
be considered. They are: 

Ri , A nonsequential, nonreplacement type of procedure 
E,2 , A nonsequential, replacement type of procedure 
Rs , A sequential, replacement type of procedure 
Within each type wq will consider different values of n, the initial 
number of units on test for each process. The effect of replacement is 
shown by comparing the average experiment time for procedures of 
type 1 and 2 with the same value of n and comparable probabilities of a 
correct selection. The effect of using a sequential rule is shown by com- 
paring the average experiment time for procedures of type 2 and 3 with 
the same value of n and comparable probabilities of a correct selection. 

ASSUMPTIONS 

1. It is assumed that failure is clearly defined and that failures are 
recognized without any chance of error. 

2. The lifetime of individual units from either population is assumed 
to follow an exponential density of the form 

f{x; e,g) =\ e-^^-")/" iov x -^ g 

f(x; e,g) = iorx<g 

where the location parameter g ^ represents the common guarantee 
period and the scale parameter 6 > represents the unknown parameter 
which distinguishes the two different processes. Let Ox ^ do denote the 
ordered values of the unknown parameter 6 for the two processes; then 
the ordered failure rates are given by 

Xi = 1/(01 + {/) ^ Xo = 1/(02 -f g) (2) 

3. It is not known which process has the parameter di and which has 
the parameter dt . 



REDUCING TIME IN RELIABILITY STUDIES 181 

4. The parameter g is assumed to be the same for both processes. It 
may be known or unknown. 

5. The initial number n of units put on test is the same for both pro- 
cesses. 

6. All units have independent lifetimes, i.e., the test environment is 
not such that the failure of one unit results in the failure of other units 
on test. 

7. Replacements used in the test are assumed to come from the same 
population as the units they replace. If the replacement units have to 
sit on a shelf before being used then it is assumed that the replacements 
are not affected by shelf-aging. 

CONCLUSIONS 

1. Increasing the initial sample size n has at most a negligible effect 
on the probability of a correct selection. It has a substantial effect on the 
average experiment time for all three types of procedures. If the value of 
n is doubled, then the average time is reduced to a value less than or 
equal to half of its original value. 

2. The technique of replacement always reduces the average experi- 
ment time. This reduction is substantial when ^ = or when the popu- 
lation variance of either process is large compared to the value of g. 
This decrease in average experiment time must always be weighed against 
the disadvantage of an increase in bookkeeping and the necessity of 
having the replacement units available for use. 

3. The sequential procedure enables the experimenter to make rational 
decisions as the evidence builds up without waiting for a predetermined 
number of failures. It has a shorter average experiment time than non- 
sequential procedures satisfying the same specification. This reduction 
brought about by the sequential procedure increases as the ratio a of 
the two failure rates increases. In addition the sequential procedure 
always terminates with a decision that is clfearly convincing on the basis 
of the observed results, i.e., the a posteriori probability of a correct 
selection is always large at the termination of the experiment. 

SPECIFICATION OF THE TEST 

Each of the three types of procedures is set up so as to satisfy the 
same specification described below. Let a denote the true value of the 
ratio 61/62 which by definition must be greater than, or equal to, one. 
It turns out that in each type of procedure the probability of a correct 
selection depends on 6i and 62 only through their ratio a. 



182 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1950 

1. The experimenter is asked to specify the smallest value of a (say 
it is a* > I) that is worth detecting. Then the interval (1, a*) represents 
a zone of indifference such that if the true ratio a lies therein then we 
would still like to make a correct selection, but the loss due to a wrong 
selection in this case is negligible. 

2. The experimenter is also asked to specify the minimum value P* > 
\'2 that he desires for the probability of a correct selection whenever 
a ^ a*. In each type of procedure the rules are set up so that the proba- 
bility of a correct selection for a = a* is as close to P* as possible without 
being less than P*. 

The two constants a* > 1 and \2 < P* < 1 are the only quantities 
specified by the experimenter. Together they make up the specification 
of the test procedure. 

EFFICIENCY 

If two procedures of different type have the same value of n and satisfy 
the same specification then we shall regard them as comparable and 
their relative efficiency will be measured by the ratio of their average 
experiment times. This ratio is a function of the true a but we shall 
consider it only for selected values of a, namely, a = 1, a = a* and 
a = CO . 

PROCEDURES OF TYPE Ri — • NONSEQUENTIAL, NONREPLACEMENT 

"The same number n of units are put on test for each of the two pro- 
cesses. Experimentation is continued until either one of the two samples 
produces a predetermined number r (r ^ n) of failures. Experimenta- 
tion is then stopped and the process with fewer than r failures is chosen 
to be the better one." 



Table I — Probability of a Correct Selection — Procedure 

Type Ri 
(a = 2, any g '^ 0, to be used to obtain r for a* = 2) 



n 


r = 1 


r = 2 


r = 3 


r = i 


1 


0.667 








. — . 




2 


0.667 


0.733 


— 


— 




3 


0.667 


0.738 


0.774 


— 




4 


0.667 


0.739 


0.784 


0.802 




10 


0.667 


0.741 


0.78!) 


0.825 




20 


0.667 


0.741 


. 790 


0.826 




00 


0.667 


0.741 


0.790 


0.827 





Note: The value for ?• = is obviously 0.500 for any n. 



REDUCING TIME IN RELIABILITY STUDIES 183 

We shall assume that the number n of units put on test is determined 
by non -statistical considerations such as the availability of units, the 
availability of sockets, etc. Then the only unspecified number in the 
above procedure is the integer r. This can be determined from a table 
of probabilities of a correct selection to satisfy any given specification 
(a*, P*). If, for example, a* = 2 then we can enter Table I. If n is 
given to be 4 and we wish to meet the specification a* = 2, P* = 0.800 
then we would enter Table I with n — 4 and select r = 4, it being the 
smallest value for which P ^ P*. 

The table above shows that for the given specification we would also 
have selected r = 4 for any value of n. In fact, we note that the proba- 
bility of a correct selection depends only slightly on n. The given value 
of n and the selected value of r then determine a particular procedure 
of type Ri , say, Ri(n, r). 

The average experiment time for each of several procedures R\{n, r) 
is given in Table II for the three critical values of the true ratio a, 
namely, a = \, a = a* and a = oo . Each of the entries has to be multi- 
plied by 6-1 , the smaller of the two d values, and added to the common 
guarantee period g. For n = oo the entry should be zero (-\-g) but it 
was found convenient to put in place of zero the leading term in the 
asymptotic expansion of the expectation in powers of I/71. Hence the 
entry for n = 00 can be used for any large n, say, n ^ 25 when r ^ 4. 

We note in Table II the undesirable feature that for each procedure 
the average experiment time increases with a for fixed 62 . For the se- 
quential procedure we shall see later that the average experiment time 
is greater at a = a* than at either a = 1 or a = 00 . This is intuitively 
more desirable since it means that the procedure spends more time when 
the choice is more difficult to make and less time when we are indifferent 
or when the choice is easy to make. 

PROCEDURES OF TYPE R2 — NONSEQUENTIAL, REPLACEMENT 

"Such procedures are carried out exactly as for procedures oiRi except 
that failures are immediately replaced by new units from the same 
population." 

To determine the appropriate value of r for the specification a* = 2, 
P* = 0.800 when g = we use the last row of Table I, i.e., the row 
marked n = ^ , and select r = 4. The probability of a correct selection 
for procedures of type Ro is exactly the same for all values of n and de- 
pends only on r. Furthermore, it agrees wdth the probability for pro- 
cedures of type Ri with n = co so that it is not necessary to prepare a 
separate table. 






PL, 



II 



H 

PM 

H 

K 
P 

o 

.^ 
Pi d 

1 '^ 
I «3 



'a 



2 S 









CC Cl t-- o 
00 r^ r-i o 

O '^ (N o 



^ Ci o o 
»o CO o CO 

00 ^ (M CD 

.-I o deo 



Oi r^ r^ CD 

^ ^ lO o 

■* CO i-i o 

rH 00(N 



CO CO CD CO o 
CO 00 CO »o o 

00 o CO ^ o 
1— I .— I o oco 



(M ^ t^ C5 (M 

t^ ■* O CO "tH 
lO O C^ T— I CO 

T-H O O O <M 



r^ »o r— I C2 CO 

1—1 CO CO O CD 
(M t^ <M T-H o 

^ O O O (M 



O CO CO 1— I CO o 

o CO 00 T-H o o 

lO 00 lO <M i— I O 

1-1 o doo(M 



O lO-* (M ■* O 

o t^ t^ t^ 00 CO 

C^ CD ^ >— I O CD 
1— I O O O O '-H 



t^ t^ CO iM -^ O 
t— I 1— t CD CO CD lO 

a; kO CO r-H o (M 

O O O O O T-H 



O O CO o o o o 
O O CO lO O lO o 
O lO CO (M T-H O O 

^ o o o o o ^ 



t- CO (M t^ t^ CO t^ 

CD CO (M CD CD CO CD 
CD CO C^ >— I O O CD 

d> d> d CD d> d> d> 





s 


II 

a 


o o r^ vo o lO o 

O >0 CD (M lO (M O 
lO (M »-< 1-^ O O lO 




ooooooo 



--H iM CO "* O O 



184 



REDUCING TIME IN RELIABILITY STUDIES 



185 



Table III — Value of r Required to Meet the Specification 
(a*, P*) FOR Procedures of Type R2 (g = 0) 















a* 
















p* 






























1.05 


1.10 


1.15 


1.20 


1.25 


1.30 


1.35 


1.40 




1.45 



1.50 


2.00 


2.50 



3.00 


0.50 
































0.55 


14 


4 


2 


2 


1 


1 


1 


1 


1 


1 


1 


1 


1 


0.60 


55 


15 


7 


5 


3 


3 


2 


2 


2 


1 


1 


1 


1 


0.65 


126 


33 


16 


10 


7 


5 


4 


3 


3 


3 


1 


1 


1 


0.70 


232 


61 


29 


17 


12 


9 


7 


6 


5 


4 


2 


1 


1 


0.75 


383 


101 


47 


28 


19 


14 


11 


9 


7 


6 


3 


2 


1 


0.80 


596 


157 


73 


43 


29 


21 


17 


13 


11 


9 


4 


2 


2 


0.85 


903 


238 


111 


65 


44 


32 


25 


20 


16 


14 


5 


3 


3 


0.90 


1381 


363 


169 


100 


67 


49 


37 


30 


25 


21 


8 


5 


4 


0.95 


2274 


597 


278 


164 


110 


80 


61 


49 


40 


34 


12 


7 


5 


0.99 


4549 


1193 


556 


327 


219 


160 


122 


98 


80 


68 


24 


14 


10 



It i.s also unnecessary to prepare a separate table for the average ex- 
periment time for procedures of type R2 since for g = the exact values 
can be obtained by substituting the appropriate value of n in the ex- 
pressions appearing in Table II in the row marked n = oo . For example, 
for /( = 2, /• = 1 and a = 1 the exact value for ^ = is 0.500 62/2 = 
0.250 62 , and for n = 3, r = 4, a = 00 the exact value for g = is 
4.000 62/3 = 1.333 62 . It should be noted that for procedures of type R2 
we need not restrict our attention to the cases r ^ n but can also con- 
sider r > //. 

Table III shows the value of r recjuired to meet the specilication 
(a*, F*) with a procedure of type R2 for various selected values of a* 
and P*. 



procedures of type R3 — sequential, replacement 

Let D{t) denote the absolute difference between the number of fail- 
ures produced by the two processes at any time t. The sequential pro- 
cedure is as follows: 

"Stop the test as soon as the inequality 



Dit) ^ 



In [P*/{1 - P*)] 



In 



a 



(3) 



is satisfied. Then select the population with the smaller number of fail- 
ures as the better one." 

To get the best results we will choose (a*, P*) so that the right hand 
member of the inequality (3) is an integer. Otherwise we would be operat- 
ing with a higher value of P* (or a smaller value of a*) than was specified. 



186 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Table IV — Average Experiment Time and Probability of a 

Correct Selection — Procedure Type R3 

(a* = 2, P* = 0.800, ^ = 0) 

(Multiply each average time entry by d^) 



n 


a = 1 


a = 2 


a = 00 


1 


2.000 


2.400 


2.000 


2 


1.000 


1.200 


1.000 


3 


0.667 


0.800 


0.667 


4 


0.500 


0.600 


0.500 


10 


0.200 


0.240 


0.200 


20 


0.100 


0.120 


0.100 


oc 


2.000/w 


2.400/n 


2.000/n 


Probability 


0.500 


0.800 


1.000 







For example, we might choose a* = 2 and P* = 0.800. For procedures 
of type R3 the probability of a correct selection is again completely in- 
dependent of n; here it depends only on the true value of the ratio a. 
The average experiment time depends strongly on n and only to a limited 
extent on the true value of the ratio a. Table IV gives these quantities 
for a = 1, a = 2, and a = 00 for the particular specification a* = 2, 
p* = 0.800 and for the particular value ^ = 0. 

efficiency 

We are now in a position to compare the efficiency of two different 
types of procedures using the same value of n. The efficiency of Ri rela- 
tive to R2 is the reciprocal of the ratio of their average experiment time. 
This is given in Table V for a* = 2, P* = 0.800, r = 4 and n = 4, 10, 20 
and 00 . By Table I the value P* = 0.800 is not attained for n < 4. 

In comparing the sequential and the nonsequential procedures it was 
found that the slight excesses in the last column of Table I over 0.800 

Table V — Efficiency of Type Ri Relative to 

Type R2 
{a* = 2, P* = 0.800, r = 4:,g = 0) 



{ 



n 


a = 1 


a = 2 


a = 00 


4 
10 
20 

00 


0.501 
0.837 
0.925 
1.000 


0.495 
0.836 
0.917 
1.000 


0.480 
0.835 
0.922 
1.000 



I 



REDUCING TIME IN RELIABILITY STUDIES 



187 



Table VI 


— Efficiency of 
(«* = 2, P* 


Adjusted Ri Relative To R^ 
= 0.800, ^ = 0) 


n 


a = 1 


a = 2 


a = 00 


4 
10 
20 

00 


0.615 
0.754 
0.818 
0.873 


0.575 
0.708 
0.768 
0.822 


0.419 
0.528 
0.573 
0.612 



had an effect on the efficiency. To make the procedures more comparable 
the values for r = 3 and r = 4 in Table I were averaged with values p 
and 1 — p computed so as to give a probability of exactly 0.800 at a = a*. 
The corresponding values for the average experiment time were then 
averaged with the same values p and 1 — p. The nonsequential pro- 
cedures so altered will be called "adjusted procedures." The efficiency 
of the adjusted Ri relative to Rz is given in Table VI. 

In Table VI the last row gives the efficiency of the adjusted procedure 
7^2 relative to Rz . Thus we can separate out the advantage due to 
the replacement feature and the advantage due to the sequential fea- 
ture. Table VII gives these results in terms of percentage reduction of 
average experiment time. 

We note that the reduction due to the replacement feature alone is 
greatest for small n and essentially constant with a while the reduction 



Table VII — Per Cent Reduction in Average Experiment Time 
DUE TO Statistical Techniques 

(a* = 2,P* = 0.800, ^ = 0) 



a 


K 


Reduction due to 

Replacement 

Feature Alone 


Reduction due to 

Sequential 
Feature Alone 


Reduction 

due to both 

Replacement 

and Sequential 

Features 


1 


4 
10 
20 

00 


29.5 

13.7 

6.3 

0.0 


12.7 
12.7 
12.7 
12.7 


38.5 
24.6 
18.2 
12.7 


2 


4 
10 
20 

00 


30.1 

13.9 

6.6 

0.0 


17.8 
17.8 

17.8 
17.8 


42.5 
29.2 
23.2 
17.8 


cc 


4 
10 
20 

00 


31.5 

13.6 

6.3 

0.0 


38.8 
38.8 

38.8 
38.8 


58.1 
47.2 
42.7 
38.8 



188 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

due to the sequential feature alone is greatest for large a and is inde- 
pendent of n. Hence if the initial sample size per process n is large we 
can disregard the replacement techniciue. On the other hand the true 
value of a is not known and hence the advantage of sequential experi- 
mentation should not be disregarded. 

The formulas used to compute the accompanying tables are given in 
Addendum 2. 

ACKNOWLEDGEMENT 

The author wishes to thank Miss Marilyn J. Huyett for considerable 
help in computing the tables in this paper. Thanks are also due to 
J. W. Tukey and other staff members for constructive criticism and 
numerical errors they have pointed out. 

Addendum 1 

In this addendum we shall consider the more general problem of select- 
ing the best of k exponential populations treated on a higher mathemati- 
cal level. For k = 2 this reduces to the problem discussed above. 

DEFINITIONS AND ASSUMPTIONS 

There are given k populations H, (^ = 1, 2, • • • , k) such that the life- 
times of units taken from any of these populations are independent 
chance variables with the exponential density (1) with a common (known 
or unknown) location parameter g ^ 0. The distributions for the k popu- 
lations are identical except for the unknown scale parameter 6 > which 
may be different for the k different populations. We shall consider three 
different cases with regard to g. 

Case 1 : The parameter g has the value zero (g = 0). 

Case 2: The parameter g has a positive, known value (g > 0). 

Case 3: The parameter g is unknown (g ^ 0). 
Let the ordered values of the k scale parameters be denoted by 

di^ e.-^ ■■■ ^ dk (4) 

where equal values may be regarded as ordered in any arbitrary manner. 
At any time / each population has a certain number of failures associated 
with it. Let the ordered values of these integers be denoted by ri = ri{t) 
so that 



I 



ri g r2 ^ • • • ^ r-fc (5) ^ 



i 



REDUCING TIME IN RELIABILITY STUDIES 189 

For each unit the life beyond its guarantee period will be referred to 
as its Poisson life. Let Li{t) denote the total amount of Poisson life 
observed up to time t in the population with Vi failures (z = 1, 2, • • • , fc). 
If two or more of the r^ are equal, say Vi = rj+i = • • • = r^+y , then we 
shall assign r, and L; to the population with the largest Poisson life, 
ri+i and L^+i to the population with the next largest, • • • , ri+_, and Lj+,- 
to the population with the smallest Poisson life. If there are two or more 
equal pairs (ri , Li) then these should be ordered by a random device 
giving equal probability to each ordering. Then the subscripts in (5) as 
well as those in (4) are in one-to-one correspondence with the k given 
populations. It should be noted that Li(t) ^ for all i and any time 
t ^ 0. The complete set of quantities Li{t) {i = 1, 2, • • • , k) need not 
be ordered. Let a = 61/62 so that, since the 6i are ordered, a ^ 1. 

We shall further assume that : 

1 . The initial number n of units put on test is the same and the start- 
ing time is the same for each of the k populations. 

2. Each replacement is assumed to be a new unit from the same popu- 
lation as the failure that it replaces. 

3. Failures are assumed to be clearly recognizable without any chance 
of error. 

SPECIFICATIONS FOR CASE 1 : gf = 

Before experimentation starts the experimenter is asked to specify two 
constants a* and P* such that a* > 1 and l'^ < P* < 1. The procedure 
Ri = Rsin), which is defined in terms of the specified a* and P*, has 
the property that it will correctly select the population with the largest 
scale parameter with probability at least P* whenever a ^ a*. The initial 
number n of units put on test may either be fixed by nonstatistical con- 
siderations or may be determined by placing some restriction on the 
average experiment time function. 

Rule Rs : 

"Continue experimentation with replacement until the inequality 

k 

^ ^*-(^.-a) ^ (1 _ p*)/p* (6) 

i=2 

is satisfied. Then stop and select the population with the smallest num- 
ber of failures as the one having the largest scale parameter." 



190 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Remarks 

1. Since P* > Y2 then (1 — P*)/P* < 1 and hence no two popula- 
tions can have the same vahie ri at stopping time. 

2. For A: = 2 the inequality (6) reduces to the inequalitj^ (3). 

3. The procedure 7^3 terminates onl}^ at a failure time, never between 
failures, since the left member of (G) depends on t only through the 
quantities 7-i{t). 

4. After experimentation is completed one can make, at the lOOP per 
cent confidence level, the confidence statement 



ds ^ di S a* 9, (or di/a"" 



^ ds S e,) 



(7) 



where 6s is the scale parameter of the selected population. 



Numerical Illustrations 



»l/4 



Suppose the preassigned constants are P* = 0.95 and a* = 19' 
2.088 so that (1 - P*)/P* = ^9- Then for A; = 2 the procedure is to 
stop when r-i — ri ^ 4. For A; = 3 it is easy to check that the procedure 
reduces to the simple form: "Stop when ?'2 — ri ^ 5". For A; > 3 either 
calculations can be carried out as experimentation progresses or a table 
of stopping values can be constructed before experimentation starts. 
For A: = 4 and A; = 5 see Table VIII. 

In the above form the proposed rule is to stop Avhen, for at least one 



Table VIII — Sequential Rule for P* = 0.95, a* = 19 
A: = 4 fc = 5 



1/4 



r2 — ri 


rs — ri 


n — ri 


5 


5 


9 


5 


6 


6 


6 


6 


6 



ri — ri 


ra — ri 


n — ci 


Ti — n 


5 


5 


9 


10 


5 


5 


10 


10 


5 


6 


6 


8 


5 


6 


7 


7 


5 


7 


7 


7 


6 


6 


6 


6 



* Starred rows can be omitted without affecting the test since every integer in 
these rows is at least as great as the corresponding integer in the previous row. 
They are shown here to ilhistrate a systematic method which insures that all the 
necessary rows are included. 



REDUCING TIME IN RELIABILITY STUDIES 191 

row (say row j) in the table, the observed row vector (r^ — Vi , 
Ts — Ti , ■ ■ ■ , Vk — z'l) is such that each comyonent is at least as large as 
the corresponding component of row j. 

Properties of Rs for k = 2 and g = 

For A- = 2 and ^ = the procedure Rs is an example of a Sequential 
Probability Ratio test as defined by A. Wald in his book.^ The Average 
Sample Number (ASN) function and the Operating Characteristics (OC) 
function for Rs can be obtained from the general formulae given by 
Wald. Both of these functions depend on di and 0-2 only through their 
ratio a. In our problem there is no excess over the boundary and hence 
Wald's approximation formulas are exact. When our problem is put into 
the Wald framework, the symmetry of our problem implies equal proba- 
bilities of type 1 and type 2 errors. The OC function takes on comple- 
mentary values for any point a = 61/62 and its reciprocal 62/61 . We shall 
therefore compute it only for a ^ 1 and denote it by P{a). For a > 1 
the quantity P(a) denotes the probability of a correct selection for the 
true ratio a. 

The equation determining Wald's h function is 



1 + a 1 + a 
for which the non-zero solution in h is easily computed to be 

h{a) = }^ (9) 



In 



a 



Hence we obtain from Wald's formula (3:43) in Reference 5 



s 

a 



Pia) = -^^ (10) 

where s is the smallest integer greater than or equal to 

S = In [PV(1 - P*)]/ln a* (11) 

In particular, for a = 1"^, a* and 00 we have 

Pi^^) = 1/2, ^(«*) ^ P*, P(^) = 1 (12) 

^\'e have written P(l"^) above for lim P{x) as x -^ 1 from the right. The 
procedure becomes more efficient if we choose P and a* so that *S' is an 
integer. Then s ^ S and P(a*) = P*. 

Letting F denote the total number of observed failures required to 



192 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

terminate the experiment we obtain for the ASN function 

and, in particular, for a = 1, oo 

E(F; 1) = s- and E{F; oo) = s (14) 

It is interesting to note that for s = 1 we obtain 

E{F; a) = 1 for all a ^ 1 (15) 

and that this result is exact since for s = 1 the right-hand member S \ 

of (3) is at most one and hence the procedure terminates with certainty ' 

immediately after the first failure. ' 

As a result of the exponential assumption, the assumption of replace- ; 

ment and the assumption that ^ = it follows that the intervals between \ 

failures are independently and identically distributed. For a single popu- ' 

lation the time interval between failures is an exponential chance vari- ; 

able. Hence, for two populations, the time interval is the minimum of j 

two exponentials which is again exponential. Letting r denote the i 

(chance) duration of a typical interval and letting T denote the (chance) j 
total time needed to terminate the procedure, Ave have 

E{T; a, 62) = E{F; a)E(r; a, d^) = E{F; a) (^^^ (f^) (16) 

I 

Hence Ave obtain from (13) and (14) 

E{T; a, 02) = - -^ ^^^ for a > 1 (17) 

n a — 1 a* + 1 

E{T; 1, d,) = ^ and E{T; <^, 0,) = ^ (18> 

For the numerical illustration treated above Avith k = 2 we have 

na) = ^-^ (19) : 

P(l+) = ^; P(2.088) = 0.95; P(oo) = 1 (20) 

EiF-a) = 4^^4^ = 4 ^--+ Vy + '^ (21) 

a— la*-f-l a*-t-l 

E{F; 1) = 16.0; /iXF; 2.088) = 10.2; E{F; 00) = 4 (22), 



REDUCING TIME IN RELIABILITY STUDIES 193 

E(T; 1, ^2) = — ; E{T; 2.088, 6^ = — ; 

n n (23) 

n 

For /.• > 2 the proposed procedure is an application of a general se- 
quential rule for selecting the best of A- populations which is treated in 
[1]. Proof that the probability specification is met and bounds on the 
probability of a correct decision can be found there. 

CASE 2: COMMON KNOWN ^ > 

In order to obtain the properties of the sec^uential procedure R:>. for 
this case it will be convenient to consider other sequential procedures. 
Let (S = 1/6-2 — 1/^1 so that, since the di are ordered, jS ^ 0. Let us 
assume that the experimenter can specify three constants a*, /3* and 
P* such that a* > 1, /3* > and ^ 2 < -P* < 1 ai^d a procedure is de- 
sired which will select the population with the largest scale parameter 
with probability at least P* whenever we have both 

a ^ a* and i3 ^ /3* 

The following procedure meets this specification. 

Rule Rs': 

"Continue experimentation with replacement until the inec^uality 

fi «*-(^i-'-i>e-^*(^i-^i)^ (l_p*)/p* (24) 

1=2 

is satisfied. Then stop and select the population with the smallest nimiber 
of failures as the one having the largest scale parameter. If, at stopping 
time, two or more populations have the same value ri then select that 
particular one of these with the largest Poisson life Li ." 

Remarks 

1 . For k = 2 the inequality reduces to 

(r, - n) In a* + (Li - L2) 13* ^ In [P*/a - P*)] (25) 

If <7 = then Li = Li for all t and the procedure R/ reduces to R3 . 

2. The procedure R/ may terminate not only at failures but also be- 
tween failures. 



194 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

3. The same inequality (24) can also be used if experimentation is 
carried on without replacement, one advantage of the latter being that 
there is less bookkeeping involved. In this case there is a possibility 
that the units will all fail before the inequality is satisfied so that the 
procedure is not yet completely defined for this case. One possibility 
in such a situation is to continue experimentation with new units from 
each population until the inequality is satisfied. Such a procedure will 
terminate in a finite time with probability one, i.e., Prob{ T > To} -^0 
as To — > 00, and the probability specification will be satisfied. 

4. A procedure R3 (ni , n-z , ■ • • , rik , ti , t2 , • • • , tk) using the same 
inequality (24) but based on dilTerent initial sample sizes and/or on 
different starting times for the initial samples also satisfies the above 
probability specification. In the case of different starting times it is 
required that the experimenter wait at least g units of time after the last 
initial sample is put on test before reaching any decision. 

0. One disadvantage of R3 is that there is some (however remote) 
possibility of terminating while ri = r2 . This can be avoided by adding 
the condition r^ > n to (24) but, of course, the average experiment time 
is increased. Another way of avoiding this is to use the procedure R3 
which depends only on the number of failures; the effect of using R3 
when g > will be considered below. 

6. The terms of the sum in (24) represent likelihood ratios. If at any 
time each term is less than unity then we shall regard the decision to 
select the population with n failures and Li units of Poisson life as opti- 
mal. Since (1 — P*)/P* < 1 then each term must be less than unity at 
termination. 

Properties of Procedure Rz for k = 2 p 

The OC and ASN functions for Rs will be approximated by comparing 
R3' with another procedure R/ defined below. We shall assume that P* 
is close to unity and that g is small enough (compared to d^) so that the 
probability of obtaining two failures within g imits of time is small 
enough to be negligible. Then we can write approximately at termination 

Li^nT - r,g {i = 1, 2, • • • , A:) (26) 

and 

Li - Li ^ (r, - r,)g (i = 2, 3, • • • , A:) (27) 

Substituting this in (24) and letting 

5* = a* c^*" (28) 

suggests a new rule, say R/' , which we now define. 



REDUCING TIME IN RELIABILITY STUDIES 195 

h'ule R/ 

"Continue experimentation with replacement until the inequality 

k 

X 6*-(^i-'-i) ^ (1 - P*)/P* (29) 



is satisfied. Then stop and select the population with n failures as the 
one with the largest scale parameter." 

For rule Rz" the experimenter need only specify P* and the smallest 
value 5* of the single parameter 

8 = ^' e''''"''-''"''' = ae'^ (30) 

62 

that he desires to detect with probability at least P*. 

We shall approximate the OC and ASX function of R/' for k = 2 
by computing them under the assumption that (27) holds at termina- 
tion. The results will be considered as an approximation for the OC and 
ASN functions respectively of R/ for /,■ = 2. The similarity of (29) 
and (6) immediately suggests that we might replace a* by 5* and a by 
5 in the formulae for (6). To use the resulting expressions for R^ we 
would compute 5* as a function of a* and /3* by (28) and 5 as a function 
of a and /3 by (30). 

The similarity of (29) and (6) shows that Z„ (defined in Reference 5, 
page 170) under (27) with gr > is the same function of 5* and 5 as it 
is of a* and a when g = 0. To complete the justification of the above 
result it is sufficient to show that the individual increment ^ of Z„ is the 
same function of 5* and 8 under (27) with ^ > as it is of a* and a 
when ^ = 0. To keep the increments independent it is necessary to as- 
sociate each failure with the Poisson life that follows rather than with 
the Poisson life that precedes the failure. Neglecting the probability 
that any two failures occur ^^•ithin g units of time we have two values for 
z, namely 

^ -(.nt-g)/ei -ntl$2 

z = log^^^ = -log 5 (31) 

and, interchanging 61 and ^2 , gives z — log 5. Moreover 



196 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

r r - e-(— «)/«v"^^^^ dx dy 

Jg Jg 6-1 



Prob \z = -logSj 



^2 -0[92(n-l)+9l"l/9lfl2 _i_ ^1 -H9in+Bi(n-l)]l9ie2,) 

- e + - e /o9\ 



1 + 5 



Thus the OC and ASN functions under (27) with g > bear the same 
relation to 5* and 5 as they do to a* and a when ^ = 0. Hence, letting 
w denote the smallest integer greater than or equal to 

^ In [P*/(l - P*)] ^ \n[P*/{l-P*)] 

In 8* gl3* + In a* ^' ' 

we can write (omitting P* in the rule description) | 

7^15; /?/ («*, /5*){ ^ P{5; /^.^"(S*)! ^ ^-^^^ (34) 

<w. I ■ — - tor 5 > 1 {So) 

^ \8 - l/\5"' +1/ 



w~ for 5=1 

W'e can approximate the average time between failures by 



I 



and the average experiment time by « 

E{T; /?/(«*, ^*)} ^ E{F; R,'(a*, 0*)\ [^.^ f^'^ _^ ^'^, (37) 

n{Oi -T 02 -f- zg) 

Since 5 ^ 1 then 5"(1 + 5") is an increasing function of w and by 
(33) it is a non-increashig function of 5*. By (28) 5* ^ a* and hence, 
if we disregard the approximation (34), 

P{8; AV(«*)1 - ^!^{py^/_p.^y..n ^* ^ P{S;R/m} (38) 

Clearly the rules Ri{a*, P*) and R/ {a*, P*) are equivalent so that 
for g > we haA-e 

P{8;R-s{a*)} ^ P{8;R/ia*)] (39) 



REDUCING TIME IN RELIABILITY STUDIES 197 

and hence, in particular, letting 8 = 8* in (38) we have 

P{8*;R,(a*)} ^ P{8*;R,"(8*)] ^ P* (40) 

since the right member of (34) reduces to P* when W is an integer and 

5 = 5*. The error in the approximations above can be disregarded when 
g is small compared to 02 . Thus we have shown that for small values of 
g/d2 the probability specification based on (a*, ^*, P*) is satisfied in the 
sense of (40) if we use the procedure Rsia*, P*), i.e., if we proceed as if 

It would be desirable to show that w^e can proceed as if g = for all 
values of g and P*. It can be shown that for swfficiently large n the rule 
Ri{a*, P*) meets it specification for all g. One effect of increasing n 
is to decrease the average time E{t) between failures and to approach 
the corresponding problem without replaceme^it since g/E{T) becomes 
large. Hence we need only show that Ri{a*, P*) meets its specification 
for the corresponding problem without replacement. If we disregard the 
information furnished by Poisson life and rely solely on the counting of 
failures then the problem reduces to testing in a single binomial whether 

6 = di for population IIi and 6 = do for population 112 or vice versa. Let- 
ting p denote the probability that the next failure arises from 111 then 
we have formally 

tia'-V = -. — ; — versus Hi-.p = 



1 + a ^ 1 + a 

For preassigned constants a* > I and P* (V2 < P* < 1) the appropri- 
ate sequential likelihood test to meet the specification: 

"Probability of a Correct Selection ^ P* whenever a ^ a*" (41) 
then turns out to be precisely the procedure Rsia*, P*). Hence we may 
proceed as if gr = when n is sufficiently large. 

The specifications of the problem may be given in a different form. 
Suppose 01* > 02* are specified and it is desired to haxe a probability of a 
correct selection of at least P* whenever ^1 ^ 0i* > 02* ^ 02 . Then we 
can form the following sequential likelihood procedure R3* which is 
more efficient than Rsia*, P*). 

Rule /?3*.- 

"Continue experimentation without replacement until a time t is 
reached at which the inequality 



198 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

is satisfied. Then stop and select the population with ri failures as the 
population with d = di". 

It can be easily shown that the greatest lower bound of the bracketed 
quantity in (42) is 0i*/^2*. Hence for di*/d2* = a* and P* > i 2 the time 
required by Rz*{6i*, 62*, P*) ivill always be less than the time required 
by R,(a*,P*). 

Another type of problem is one in which we are given that 6 = di* 
for one population and d = 62* for the A; — 1 others where 6]* > 62* are 
specified. The problem is to select the population with 6 = di*. Then 
(42) can again be used. In this case the parameter space is discrete with 
k points only one of which is correct. If Rule R3* is used then the 
probability of selecting the correct point is at least P*. 

Equilibrium Approach When Failures Are Replaced 

9 

Consider first the case in which all items on test are from the same 

exponential population with parameters (6, g). Let Tnj denote the length 
of the time interval between the j^^ and the j + 1^* failures, (j = 0, 
1, • • • ), where n is the number of items on test and the 0*'' failure de- 
notes the starting time. As time increases to infinity the expected number 
of failures per unit time clearly approaches n/(0 + g) which is called the 
equilibrium failure rate. The inverse of this is the expected time between 
failures at equilibrium, say E{Tn^). The question as to how the quanti- 
ties E{Tnj) approach E(Tn^) is of considerable interest in its own right. 
The following results hold for any fixed integer 71 ^ 1 unless explicitly 
stated otherwise. It is easy to see that 

^^(^i) ^ E{TnJ ^ E(T„o) (43) 

since the exact values are respectively 

e /, e-^-^'^/^X ^ g+d ^ , d 



< 



^ 9+ - (44) 



n — 1 \ n / n n 

In fact, since all units are new at starting time and since at the time of 
the first failure all units (except the replacement) have passed their 
guarantee period with probability one then 

^(^i) ^ E(Tnj) S E{Tn,) (j ^ 0) (45) 

If we compare the case g > with the special case g = we obtain 

E{2\j) ^ - (y= 1,2, •••) (46) 

n 



REDUCING TIME IN RELIABILITY STUDIES 199 

and if we compare it with the non-replacement case {g/Q is large) we 
obtain 

^(n,) ^ -^. (i = 1, 2, • . • , n - 1). (47) 

These comparisons show that the difference in (46) is small when g/0 is 
small and for j < n the difference in (47) is small when g/d is large. 

It is possible to compute E{Tnj) exactly for g ^ but the computa- 
tion is extremely tedious for j ^ 2. The results for j = 1 and are given 
in (44). Fori = 2 



E(Tn2) = 



n 



(n + 2)(/i - 1) -(n-2)gie 



1 - ' ' ': -e 

n 



+ Vl^iI g-(«-i)p/^ ri-2_ -un-i),ie I {n>2) 



n — \ v?{n — 1) 

and 



2{n-l)glB 



(48) 



E{T,.^ = ^ - ^ [1 - ^e-'" + e-'"'\ (49) 

For the case of two populations with a common guarantee period g 
we can write similar inequalities. We shall use different symbols a, h for 
the initial sample size from the populations with scale parameters Oi , O2 
respectively even though our principal interest is in the case a = b = n 
say. Let Ta,b.j denote the interval between the j^^ and j -f P* fail- 
ures in this case and let X, = l/di (i = 1,2). We then have for all values 
of a and b 

[aXi + b\o]-' ^ E(TaXj) ^ E(Taxo) 

= g + [aXi + b\,]-' (j = 0,1,2, ■■■, ^) (50) 

J?(T ^ (gl + g){e2 + g) .riN 

a{92 -h 9) + b{di + g) 

The result for E(Ta,b.i) corresponding to that in (43) does not hold if 
the ratio di/62 is too large; in particular it can be shown that 

-0[(a-l)Xi+6X2l-l 



E{T.,b..) = ^ "^^ ^' ^ 



aXi + 6X2/ \(a — l)Xi 4- 6X2 



_ Xie 



aXi + 6X2 



+ / ^X2 Y 1 \r x^e-''^'^^''-''''-'- 



(52) 



,aXi -\- bX2/\aXi + (& — 1)^2 L 0X1 + ^^2 

is larger than E{Ta,h.J for a = 6 = 1 when ^/^i = 0.01 and g/di = 0.10 



200 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

SO that QilQi = 10. The expression (52) reduces to that in (44) if we set 
di = 02 = 6 and replace a and h by n/2 in the resulting expression. 

Corresponding exact expressions for E(Ta.b,j) for j > 1 are extremely 
tedious to derive and unwieldy although the integrations involved are 
elementary. If we let g —^ oo then we obtain expressions for the non- 
replacement case which are relatively simple. They are best expressed 
as a recursion formula. 



E(.Ta,bj) = — , ,. ETa-\,b,}-l 



+ m^ ^"— ^^ = '^ 



(53) 



EiT.,b.d = "^^ ^ 



aXi + 6X2 (a — l)Xi + 6X2 

I 0X2 1 ( h > ^^ 

"^ aXi + 6X2 aXi + (6 - 1)X2 ' = 



(54) 



E(Tafij) ^ g + di/a fori ^ a and j = (55) 

E{Ta,oJ = dr/(a -j) for 1 ^ i ^ a - 1 (56) 

Results similar to (55) and (56) hold for the case a = 0. The above 
results for gr = 00 provide useful approximations for E{Ta,b,j) when g 
is large. Upper bounds are given by M 

E{Ta,bj) ^ [aXi + (6 - i)X2r (i = 1, 2, • • • , h) (57) 

E(Ta.bj+b) ^ [(a - j)Xr' (i = 1, 2, • . • , a - 1). (58) 

Duration of the Experiment 

For the sequential rule R^' with k = 2 we can now write down approxi- 
mations as well as upper and lower bounds to the expected duration 
E{T) of the experiment. From (50) 



I 



g + ..5^;^.\ s E(T) = E /?(r.,,) 



c-l 

n(Xi -f X2) ^ '''^ ' ~ § '^^^ "'"'^^ (59) 

+ \FA¥; 5) - c]i!;(T„,„,.) 



where c is the largest integer less than or equal to E{F\ 5). The right ex- 
pression of (59) can be approximated by (53) and (54) if g is large. If 
c < 2n then the upper bounds are given by (57) and (58). A simpler 



j 



REDUCING TIME IN RELIABILITY STUDIES 201 

upper bound, which holds for all \'aliies of c is given by 

E{T) ^ E{F- b)E{Tn,n..) = E{F; 8) (g + ^^ (60) 

CASE 3: COMMON UNKNOWN LOCATION PARAMETER ^ ^ 

In this case the more conservative procedure is to proceed under the 
assumption that </ = 0. By the discussion above the probability require- 
ment will in most problems be satisfied for all ^ ^ 0. The OC and ASN 
functions, which are now functions of the true value of g, were already 
obtained above. Of course, we need not consider values of g greater than 
the smallest observed lifetime of all units tested to failure. 

Addendum 2 

For completeness it would be appropriate to state explicitly some of 
the formulas used in computing the tables in the early part of the paper. 
For the nonsequential, nonreplacement rule Ri with /c = 2 the proba- 
bility of a correct selection is 

P(a; R,) = [ [ Mu, OAfrix, 6,) dy dx (61) 

where 

fXx, e) = '- C(l - e^'"y-' e-^^"-^+^"^ (r ^ n) (62) 

and C" is the usual combinatorial symbol. This can also be expressed in 
the form 

P{a; R,) = 1 - (rC:r Z ^~^^"' 

;=i n - r -\-j (63) 

C'-l{B[r, n-r+l+a(n-r+ j)]}-' 

where B[x, y] is the complete Beta function. Eciuation (66) holds for 
any g ^ 0. 

For the rule Ri the expected duration of the experiment for k = 2 
is given by 

E{T) = r x{fr(x, d,)[l - Frix, 62)] + frix, d,)[l - Fr(x, ^i)] } dx (64) 

•'0 

where frix, 6) is the density in (62) and Fr{x, B) is its c.d.f. This can 



202 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

also be expressed in the form 

^iKC^ZZt (-1) c.-. c, . 

plus another similar expression in which 6i , a are replaced by 62 , a~^ 
respectively. For ^ > we need only add g to this result. This result 
was used to compute E(T) in table lA f or a = 1 and a = 2. For a = oo 
the expression simplifies to 

E{T) = e^rC: ± erl ^~^^'^\ (66) 

which can be shoAvn to be equivalent to 

E{T) = e,f: ^— (67) 

REFERENCES 

1. Bechhofer, R. E., Kiefer, J. and Sobel, M., On a Type of Sequential Multiple 

Decision Procedures for Certain Ranking and Identification Problems with 
k Populations. To be published. 

2. Birnbaum, A., Statistical methods for Poisson processes and exponential 

populations, J. Am. Stat. Assoc, 49, pp. 254-266, 1954. 

3. Birnbaum, A., Some procedures for comparing Poisson processes or popula- 

tions, Biometrika, 40, pp. 447-49, 1953. 

4. Girshick, M. A., Contributions to the theory of sequential analj'sis I, Annals 

Math. Stat., 17, pp. 123-43, 1946. 

5. Wald, A., Sequential Analysis, John Wiley and Sons, New York, 1947. 



I 



A Class of Binary Signaling Alphabets 

By DAVID SLEPIAN 

(Manuscript received September 27, 1955) 

A class of binary signaling alphabets called "group alphabets" is de- 
scribed. The alphabets are generalizations of Hamming^ s error correcting 
codes and possess the following special features: {1) all letters are treated 
alike in transmission; {2) the encoding is simple to instrument; (3) maxi- 
mum likelihood detection is relatively simple to instrument; and (4) in 
certain practical cases there exist no better alphabets. A compilation is given 
of group alphabets of length equal to or less than 10 binary digits. 

INTRODUCTION 

This paper is concerned with a class of signahng alphabets, called 
"group alphabets," for use on the symmetric binary channel. The class 
in question is sufficiently broad to include the error correcting codes of 
Hamming,^ the Reed-Muller codes," and all "systematic codes''.^ On 
the other hand, because they constitute a rather small subclass of the 
class of all binary alphabets, group alphabets possess many important 
special features of practical interest. 

In particular, (1) all letters of the alphabets are treated alike under 
transmission; (2) the encoding scheme is particularly simple to instru- 
ment; (3) the decoder — a maximum likelihood detector — is the best 
I possible theoretically and is relatively easy to instrument; and (4) in 
certain cases of practical interest the alphabets are the best possible 
theoretically. 

It has very recently been proved by Peter Elias^ that there exist group 
alphabets which signal at a rate arbitarily close to the capacity, C, of 
the symmetric binary channel with an arbitrarily small probability of 
error. Elias' demonstration is an existence proof in that it does not 
show explicitly how to construct a group alphabet signaling at a rate 
greater than C — e with a probability of error less than 5 for arbitrary 
positive 5 and e. Unfortunately, in this respect and in many others, our 
understanding of group alphabets is still fragmentary. 

In Part I, group alphabets are defined along with some related con- 

203 



204 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

cepts necessary for their understanding. The main results obtained up 
to the present time are stated without proof. Examples of these concepts 
are given and a compilation of the best group alphabets of small size 
is presented and explained. This section is intended for the casual reader. 

In Part II, proofs of the statements of Part I are given along with 
such theory as is needed for these proofs. 

The reader is assumed to be familiar with the paper of Hamming, 
the basic papers of Shannon* and the most elementary notions of the 
theory of finite groups. 

Part I — Group Alphabets and Their Properties 

1.1 INTRODUCTION 

We shall be concerned in all that follows with communication over the 
symmetric binary channel shown on Fig. 1. The channel can accept 
either of the two symbols or 1 . A transmitted is received as a with 
probability q and is received as a 1 w'ith probability p — 1 — g : a trans- 
mitted 1 is received as a 1 with probability q and is received as a with 
probability p. We assume ^ p ^ ^^. The "noise" on the channel 
operates independently on each symbol presented for transmission. The 
capacity of this channel is 

C = 1 + P log2P + q log29 bits/symbol (1) 

By a K-leUer, n-place binary signaling alphabet we shall mean a collec- 
tion of K distinct sequences of n binary digits. An individual sequence 
of the collection will be referred to as a letter of the alphabet. The integer 
K is called the size of the alphabet. A letter is transmitted over the 
channel by presenting in order to the channel input the sequence of n 
zeros and ones that comprise the letter. A detection scheme or detector for 




INPUT X OUTPUT 



Fig. 1 — The symmetric binary channel. 



A CLASS OF BINARY SIGNALING ALPHABETS 



205 



a given /v-letter, n-place alphabet is a procedure for producing a sequence 
of letters of the alphabet from the channel output. 

Throughout this paper we shall assume that signaling is accomplished 
with a given /i-letter, n-place alphabet by choosing the letters of the 
alphabet for transmission independently with equal probability l/K. 

Shannon^ has shown that for sufficiently large n, there exist K-letter, 
n-place alphabets and detection schemes that signal over the symmetric 
binary chaimel at a rate R > C — e for arbitrary £ > and such that 
the probability of error in the letters of the detector output is less than 
any 5 > 0. Here C is given by (1) and is shown as a function of p in 
Fig. 2. No algorithm is known (other than exhaustvie procedures) for 
the construction of A'-letter, /i-place alphabets satisfying the above 
inequalities for arbitrary positive 8 and e except in the trivial cases C — 
and C = 1. 

1.2 THE GROUP -S„ 

There are a totality of 2" different w-place binary sequences. It is fre- 
quently convenient to consider these sequences as the vertices of a cube 
of unit edge in a Euclidean space of n-dimensions. For example the 5- 
place sequence 0, 1, 0, 0, 1 is associated with the point in 5-space whose 



o.e 



0.6 




0.4 



0.2 



Fig. 2 — The capacity of the symmetric binary channel. 
C = 1 + p log2 p + {I - p) log2 (1 - p) 



206 



THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



coordinates are (0, 1, 0, 0, 1). For convenience of notation we shall gen- 
erally omit commas in writing a sequence. The above 5-place sequence 
will be written, for example, 01001. 

We define the product of two n-ylace hinarij sequences, aicii • • • a„ and 
^1^2 • ■ • bn as the n-place binary sequence 

fli + hi , a-i ■]- h-i , ■ ■ • , ttn + hn 

Here the a's and 6's are zero or one and the + sign means addition 
modulo 2. (That is + 0=1 + 1 = 0, 0+1 = 1+0=1) 
For example, (01101) (00111) = 01010. With this rule of multiplication 
the 2" w-place binary sequences form an Abelian group of order 2". 
The elements of the group, denoted by Ti , T'2 , • • • , Tin, say, are the 
n-place binary sequences ; the identity element I is the sequence 000 • • • 
and 

IT, = Til = T. ■ T,Tj = TjTr, TiiTjT,) = iTiTj)Tk ; 

the product of any number of elements is again an element; every ele- 
ment is its own reciprocal, Ti = Tf^, TI = /. We denote this group 
by Bn . 

All subgroups of Bn are of order 2 where k is an integer from the set 
0, 1, 2, • • • , n. There are exactly 



N{n, k) = 



(2" - 2") (2" - 2') (2" - 2') • • • (2" - 2'-') 



(2^ - 2»)(2'^ - 20(2* - 22) 
= N(n, n — k) 



{2" - 2'-') 



(2) 



distinct subgroups of Bn of order 2 . Some values of N(n, k) are given in 
Table I. 



Table I — Some Values of A^(n, k), the Number of Subgroups 
OF Bn OF Order 2''. N(n, k) = N{n, n — k) 



n\k 





1 


2 


3 


4 


5 


2 




3 


1 








3 




7 


7 


1 






4 




15 


35 


15 


1 




5 




31 


155 


155 


31 


1 


6 




63 


651 


1395 


651 


63 


7 




127 


2667 


IISU 


11811 


2667 


8 




255 


10795 


97155 


200787 


97155 


9 




511 


43435 


788035 


3309747 


3309747 


10 




1023 


174251 


6347715 


53743987 


109221651 



000 


000 


000 


000 


000 


000 


000 


100 


100 


100 


010 


010 


001 


no 


010 


001 


oil 


001 


101 


no 


on 


110 


101 


111 


on 


111 


111 


101 



A CLASS OF BINARY SIGNALING ALPHABETS 207 

1.3 GROUP ALPHABETS 

An ?i-place group alphabet is a 7v-letter, n-place binary signaling alpha- 
bet whose letters form a subgroup of Bn . Of necessity the size of an 
n-place group alphabet is /v = 2 where k is an integer satisfying ^ 
k ^ n. By an (n, k)-alphahet we shall mean an n-place group alphabet of 
size 2^. Example: the N{3, 2) = 7 distinct (3, 2)-alphabets are given by 
the seven columns 

(i) (ii) (iii) (iv) (v) (vi) (vii) 



(3) 



1.4 STANDARD ARRAYS 

Let the letters of a specific (n, /i:)-alphabet be Ai = / = 00 • • • 0, 
Ao , As , • ■ ■ , A^ , where ju = 2 . The group Bn can be developed accord- 
ing to this subgroup and its cosets: 

/, A2, A3, ■■• ,A^ 

S2 , S2A2 , S2A3 , • • • , S2A^ 
Sz , S3A2 , S3A3 , • • • , SsA^ 

Bn = ; (4) 

Sr f SyA2 , SpAz , • ' • , SfAfi 

In this array every element of Bn appears once and only once. The col- 
lection of elements in any row of this array is called a coset of the (n, k)- 
alphabet. Here *S2 is any element of B„ not in the first row of the array, 
S3 is any element of Bn not in the first two rows of the array, etc. The 
elements S2 , S3 , • • • , Sy appearing under I in such an array will be 
called the coset leaders. 

If a coset leader is replaced by any element in the coset, the same coset 
will result. That is to say the two collections of elements 

Si , ^1^2 , SiSz ; ■ • ■ , SiA^ 

and 

SiA,, , (SiAu)A2 , (SiAMs ,■■■ {SiAk)A, 

are the same. 



208 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 195G 

We define the weight Wi = w{Ti) of an element, Ti , of Bn to be the 
number of ones in the n-place binary sequence T,- . 

Henceforth, unless otherwise stated, we agree in dealing with an ar- 
ray such as (4) to adopt the following convention: 

the leader of each coset shall be taken to be an . . 

element of minimal weight in that coset. 

Such a table will be called a standard array. 

Example: Bi can be developed according to the (4, 2)-alphabet 0000, 
1100, 0011, nil as follows 



(6) 



0000 


1100 


0011 


nil 


1010 


Olio 


1001 


0101 


1110 


0010 


1101 


0001 


1000 


0100 


1011 


0111 


)W"ever, 


^^-e should 


write. 


for exan 


0000 


1100 


0011 


nil 


1010 


0110 


1001 


0101 


0010 


1110 


0001 


1101 


1000 


0100 


1011 


0111 



(7) 



The coset leader of the second coset of (6) can be taken as any element 
of that row since all are of weight 2. The leader of the third coset, how- 
ever, should be either 0010 or 0001 since these are of weight one. The 
leader of the fourth coset should be either 1000 or 0100. 

1.5 THE DETECTION SCHEME 

Consider now communicating with an (n, fc) -alphabet over the sym- 
metric binary channel. When any letter, say A,, of the alphabet is 
transmitted, the received sequence can be of any element of B„ . We 
agree to use the following detector: 

if the received element of Bn lies in column i of the array (4), the 

detector prints the letter Ai ,i = 1,2, • • • , ju. The array (4) is to (8) 

be constructed according to the convention (5). 

The following propositions and theorems can be proved concerning 
signaling with an (n, /c)-alphabet and the detection scheme given by (8). 

1.6 BEST DETECTOR AND SYMMETRIC SIGNALING 

Define the probability /,• = ((Ti) of an element Ti of Bn to be A = 
^wi^n-uf ^yYiere p and q are as in (1) and Wi is the weight of Ti . Let 



A CLASS OF BINARY SIGNALING ALPHABETS 209 

Qi , i = 1 , 2, • • • , jLi be the sum of the probabilities of the elements in 
the iih. column of the standard array (4). 

Proposition 1. The probability that any transmitted letter of the 
(n, A;) -alphabet be produced correctly by the detector is Qi . 

Proposition 2. The equivocation^ per symbol is 



1 ** 
Hy{x) = — S Qi log2 Qi 



n i=i 

Theorem 1 . The detector (8) is a maximum likelihood detector. That 
is, for the given alphabet no other detection scheme has a greater average 
probability that a transmitted letter be produced correctly by the de- 
tector. 

Let us return to the geometrical picture of w-place binary sequences 
as vertices of a unit cube in n-space. The choice of a i^-letter, n-place 
alphabet corresponds to designating K particular vertices as letters. 
Since the binary sequence corresponding to any vertex can be produced 
by the channel output, any detector must consist of a set of rules that 
associates various vertices of the cube with the vertices designated as 
letters of the alphabet. We assume that every vertex is associated with 
some letter. The vertices of the cube are divided then into disjoint sets, 
Wi , Wi , • • • , Wk where Wi is the set of vertices associated with tth 
letter of the signaling alphabet. A maximum likelihood detector is char- 
acterized by the fact that every vertex in Wi is as close to or closer to 
the iih. letter than to any other letter, i = 1,2, • • • , K. For group alpha- 
bets and the detector (8), this means that no element in the iih. column 
of array (4) is closer to any other A than it is to ^i , z = 1, 2, • • • , ;u. 

Theorem 2. Associated with each {n, /(;)-alphabet considered as a point 
configuration in Euclidean n-space, there is a group of n X n orthogonal 
matrices which is transitive on the letters of the alphabet and which 
leaves the unit cube invariant. The maximum likelihood sets 1^1 , 
W2 , • • • Wn are all geometrically similar. 

Stated in loose terms, this theorem asserts that in an (n, A;)-alphabet 
every letter is treated the same. Every two letters have the same number 
of nearest neighbors associated with them, the same number of next 
nearest neighbors, etc. The disposition of points in any two W regions 
is the same. 

1.7 GROUP ALPHABETS AND PARITY CHECKS 

Theorem 3. Every group alphabet is a systematic^ code: every syste- 
matic code is a group alphabet. 



210 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

We prefer to use the word "alphabet" in place of "code" since the 
latter has many meanings. In a systematic alphabet, the places in any 
letter can be divided into two classes : the information places — A; in 
number for an (n, /c)-alphabet — and the check positions. All letters 
have the same information places and the same check places. If there 
are k information places, these may be occupied by any of the 2 /v-place 
binary sequences. The entries in the n — k check positions are fixed 
linear (mod 2) combinations of the entries in the information positions. 
The rules by which the entries in the check places are determined are 
called parity checks. Examples: for the (4, 2)-alphabet of (6), namely 
0000, 1100, 0011, nil, positions 2 and 3 can be regarded as the informa- 
tion positions. If a letter of the alphabet is the sequence aia^a^ai , then 
ai = a2 , tti = az are the parity checks determining the check places 1 
and 4. For the (5, 3)-alphabet 00000, 10001, 01011, 00111, 11010, 10110, 
01100, 11101 places 1, 2, and 3 (numbered from the left) can be taken 
as the information places. If a general letter of the alphabet is aiazazaiai , 
then a4 = a2 -j- as , Ob = ai -j- a2 -|- ^3 . 

Two group alphabets are called equivalent if one can be obtained from 
the other by a permutation of places. Example: the 7 distinct (3, 2)- 
alphabets given in (3) separate into three equivalence classes. Alpha- 
bets (i), (ii), and (iv) are equivalent; alphabets (iii), (v), (vi), are equiva- 
lent; (vii) is in a class by itself. 

Proposition S. Equivalent (n, fc) -alphabets have the same probability 
Qi of correct transmission for each letter. 

Proposition 4- Every (n, /c) -alphabet is equivalent to an (n, k)- 
alphabet whose first k places are information places and whose last n — k 
places are determined by parity checks over the first k places. 

Henceforth we shall be concerned only with (n. A;) -alphabets w^hose 
first k places are information places. The parity check rules can then 
be written 

k 
ai = S Tij-ay , t = /b -j- 1, • • • , n (9) 

where the sums are of course mod 2. Here, as before, a typical letter of 
the alphabet is the sequence aia^ • ■ - ttn . The jn are k(n — k) quantities, 
zero or one, that serve to define the particular (n, A;)-alphabet in question. 

1.8 MAXIMUM LIKELIHOOD DETECTION BY PARITY CHECKS 

For any element, J\ of Bn we can form the sum given on the right of 
(9). This sum maj^ or may not agree with the symbol in the ?'th place of 



A CLASS OF BINARY SIGNALING ALPHABETS 211 

T. If it does, we say T satisfies the tth-place parity check; otherwise T 
fails the zth-place parity check. When a set of parity check rules (9) is 
giN'cii, we can associate an (n — /i^-place binary sequence, R{T), with 
each element T of 5„. We examine each check place of T in order starting 
with the (k -\- 1 )-st place of T. We write a zero if a place of T satisfies 
the parity check; we write a one if a place fails the parity check. The re- 
sultant sequence of zeros and ones, written from left to right is R(T). 
We call R(T) the parity check sequence of T. Example: with the parity 
rules 04 = 02 -j- 03 , 05 = Oi -j- 02 -j- c^s used to define the (5, 3)-alphabet 
in the examples of Theorem 3, we find i?(11000) = 10 since the sum of 
the entries in the second and third places of 11001 is not the entry of 
the fourth place and since the sum of Oi = 1, 02 = 1, and 03 = is 
= 05 . 

Theorem 4- Let I, A2 , • • • ^^^ be an {n, /c)-alphabet. Let R{T) be the 
parity check sequence of an element T of B„ formed in accordance with 
the parity check rules of the (n, /c) -alphabet. Then R(Ti) = R(T2) if 
and only if Ti and T2 lie in the same row of array (4). The coset leaders 
can be ordered so that R{Si) is the binary symbol for the integer i — 1. 

As an example of Theorem 4 consider the (4, 2)-alphabet shown with 
its cosets below 



0000 


1011 


0101 


1110 


0100 


nil 


0001 


1010 


0010 


1001 


0111 


1100 


1000 


0011 


1101 


0110 



The parity check rules for this alphabet are 03 = oi , 04 = Oi -j- ^2 • 
Every element of the second row of this array satisfies the parity check 
in the third place and fails the parity check in the 4th place. The parity 
check sequence for the second row is 01. The parity check for the third 
row is 10, and for the fourth row 11. Since every letter of the alphabet 
satisfies the parity checks, the parity check sequence for the first row is 
00. We therefore make the following association between parity check 
sequences and coset leaders 

00 -^ 0000 = Si 

01 -^ 0100 = S2 

10 -^ 0010 = S, 

11 -^ 1000 = ^4 

1.9 INSTRUMENTING A GROUP ALPHABET 

Proposition 4 attests to the ease of the encoding operation involved 



212 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

with the use of an (n, fc) -alphabet. If the original message is presented as 
a long sequence of zeros and ones, the sequence is broken into blocks of 
length k places. Each block is used as the first k places of a letter of 
the signaling alphabet. The last n-k places of the letter are determined 
by fixed parity checks over the first k places. 

Theorem 4 demonstrates the relative ease of instrumenting the maxi- 
mum hkelihood detector (8) for use with an (n. A:) -alphabet. When an 
element T of Bn is received at the channel output, it is subjected to the 
n-k parity checks of the alphabet being used. This results in a parity 
check sequence R{T). R(T) serves to identify a unique coset leader, say 
Si . The product SiT is then formed and produced as the detector out- 
put. The probability that this be the correct letter of the alphabet is Qi . 

1.10 BEST GROUP ALPHABETS 

Two important questions regarding (n, fc)-alphabets naturally arise. 
What is the maximum value of Qi possible for a given n and k and which 
of the N(n, k) different subgroups give rise to this maximum Qi? The 
answers to these questions for general n and k are not known. For many 
special values of n and k the answers are known. They are presented in 
Tables II, III and IV, which are explained below. 

The probability Qi that a transmitted letter be produced correctly by 
the detector is the sum, Qi = ^i f{Si) of the probabilities of the coset 
leaders. This sum can be rewritten as Qi = 2Zi=o «« P^Q^~^ where a, is 
the number of coset leaders of weight i. One has, of course, ^a, = v = 

/ y) \ T? ' 

2^"'' for an (n, /(;)-alphabet. Also «> ^ ( . ) = -7-7 — '■ — n- ! since this is the 

\t / tlin — t) 

number of elements of Bn of weight i. 

The (Xi have a special physical significance. Due to the noise on the 
channel, a transmitted letter, A, , of an (n, /c)-alphabet will in general be 
received at the channel output as some element T of Bn different from 
Ai .li T differs from Ai in s places, i.e., if w{AiT) = s, we say that an 
s-tuple error has occurred. For a given (n, fc)-alphabet, ai is the number 
of i-tuple errors which can be corrected by the alphabet in question, 
i = 0, 1,2, ■ • • , n. 

Table II gives the a{ corresponding to the largest possible value of Qi 
for a given k and ?i for k = 2,3, •••w— l,n = 4--- ,10 along with a 
few other scattered values of n and k. For reference the binomial coeffi- 
cients ( . ) are also listed. For example, we find from Table II that the 
best group alphabet with 2 =16 letters that uses n = 10 places has a 



A CLASS OF BINARY SIGNALING ALPHABETS 213 

1 A Q C 'J ** Q 

probability of correct transmission Qi = q + lOg p + 39g p" + l-Ag'p . 
The alphabet corrects all 10 possible single errors. It corrects 39 of the 

possible f .^ j = 45 double errors (second column of Table II) and in 

addition corrects 14 of the 120 possible triple errors. By adding an addi- 
tional place to the alphabet one obtains with the best (11, 4)-alphabet 
an alphabet with 16 letters that corrects all 11 possible single errors and 
all 55 possible double errors as well as 61 triple errors. Such an alphabet 
might be useful in a computer representing decimal numbers in binary 
form. 

For each set of a's listed in Table II, there is in Table III a set of 
parity check rules which determines an {n, A)-alphabet having the given 
a's. The notation used in Table III is best explained by an example. A 
(10, 4)-alphabet which realizes the a's discussed in the preceding para- 
graph can be obtained as follows. Places 1, 2, 3, 4 carrj- the information. 
Place 5 is determined to make the mod 2 sum of the entries in places 
3, 4, and 5 ecjual to zero. Place 6 is determined by a similar parity check 
on places 1, 2, 3, and 6; place 7 by a check on places 1, 2, 4, and 7, etc. 

It is a surprising fact that for all cases investigated thus far an {n, k)- 
alphabet best for a given value of p is uniformly best for all values of 
p, ^ p ^ 1 2. It is of course conjectured that this is true for all n and /,-. 

It is a further (perhaps) surprising fact that the best {n, fc) -alphabets 
are not necessarily those with greatest nearest neighbor distance be- 
tween letters when the alphabets are regarded as point configurations on 
the n-cube. For example, in the best (7, 3)-alphabet as listed in Table 
III, each letter has two nearest neighbors distant 3 edges away. On the 
other hand, in the (7, 3)-alphabet given by the parity check rules 413, 
512, 623, 7123 each letter has its nearest neighbors 4 edges away. This 
latter alphabet does not have as large a value of Qi , however, as does 
the (7, 3)-alphabet listed on Table III. 

The cases /.; = 0, 1, /? — 1, n have not been listed in Tables II and III. 
The cases k = and k = n are completely trivial. For k = 1, all n > 1 
the best alphabet is obtained using the parity rule a> = 03= • • • = 
a„ = oi . If n = '2j, 

If n = 2j + 1, Qi = i: (^') pY-\ 

For k = n — 1, /; > 1. the maximum Qi is Qi = g"~ and a parity rule 
for an alphabet realizing this Qi is o„ = oi . 

If the a's of an (/<, A)-alphabet are of the form a, = ( . j , i = 0, 1, 




214 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



Table II — Probability of No Error with Best 
Alphabets, Qi = 2Z «»P*2"~' 







(?) 


k = 2 


k = 3 


k = 4 


k = 5 


k = 6 


k = 7 


k = 8 


k = 9 


* = 10 




i 



ai 


(li 


a 


a, 


ai 


Oi 


fli 


ai 


a; 


n = 4 


1 


1 




















1 


4 


3 























1 


1 


1 
















n = 5 


1 

2 



5 
10 

1 


5 
2 

1 


3 

1 


1 














71 = 6 


1 
2 


6 
15 


6 
9 


6 
1 


3 



















1 


1 


1 


1 


1 












n = 7 


1 
2 
3 


7 
21 
25 


7 

18 

6 


7 
8 


7 


3 

















1 


1 


1 


1 


1 


1 










n = 8 


1 
2 
3 


8 
28 
56 


8 
28 
27 


8 

20 

3 


8 

7 


7 


3 















1 


1 


1 


1 


1 


1 


1 










1 


9 


9 


9 


9 


9 


7 


3 








n = 9 


2 
3 

4 


36 

84 

126 


36 
64 

18 


33 
21 


22 


6 

















1 


1 


1 


1 


1 


1 


1 


1 








1 


10 


10 


10 


10 


10 


10 


7 


3 






n = 10 


2 
3 

4 


45 
120 
210 


45 

110 

90 


45 
64 

8 


39 
14 


21 


5 















1 


1 


1 


1 




1 


1 


1 


1 






1 


11 


11 


11 


11 




11 


11 


7 


3 




n = 11 


2 
3 
4 
5 


55 
165 
330 

462 


55 
165 
226 

54 


55 

126 

63 


55 

61 




20 


4 













1 


1 


1 








1 


1 


1 


1 




1 


12 


12 


12 








12 


12 


7 


3 


n = 12 


2 
3 
4 
5 


66 
220 
495 
792 


66 
220 
425 
300 


66 
200 
233 








19 


3 








A CLASS OF BINARY SIGNALIHG ALPHABETS 215 

2, • • • , j, «j+i = f some integer, aj+o = ay+s = • • • = «„ = 0, then 
there does not exist a 2 -letter, w-place alphabet of any sort better than 
the given (n, A)-alphabet. It will be observed that many of the a's of 
Table II are of this form. It can be shown that 

Proposition 5 ii n -\- I „ /"t"! q 1^2"^* — 1 there exists 

no 2'''-letter, n-place alphabet better than the best (n, /c) -alphabet. 
When the inequality of proposition 5 holds the a's are either «o = 1, 

""'' - 1, all other « = 0; or ao = 1, «i = (Vj , «2 = 2"~' - 1 - 

, all other a = 0; or the trivial ao = 1 all other a = which holds 

uhen k = n. The region of the n — k plane for which it is known that 
(n, A-)-alphabets cannot be excelled by any other is shown in Table IV. 

1.11 A DETAILED EXAMPLE 

As an example of the use of {n, A") -alphabets consider the not un- 
realistic case of a channel with -p = 0.001, i.e., on the average one binary 
digit per thousand is received incorrectly. Suppose we wish to transmit 
messages using 32 different letters. If we encode the letters into the 32 
5-place binary sequences and transmit these sequences without further 
encoding, the probability that a received letter be in error is 1 — 
(1 _ pf = 0.00449. If the best (10, 5)-alphabet as shown in Tables II 
and III is used, the probability that a letter be wrong is 1 — Qi = 
1 - r/" - lOgV - 21gy - 24/)' - 72p' + • • • = 0.000024. Thus 
by reducing the signaling rate by ^^, a more than one hundredfold re- 
duction in probability of error is accomplished. 

A (10, 5)-alphabet to achieve these results is given in Table III. Let 
a typical letter of the alphabet be the 10-place sequence of binary digits 
aia2 ■ • • agttio . The symbols aia^Ozaia^ carry the information and can be 
any of 32 different arrangements of zeros and ones. The remaining places 
are determined by 

06 = ai -j- a-i -j- a4 -j- ^5 

a? = tti -j- oo -f a4 -j- as 

as = ai -j- a2 + a.3 + Os 

ag = Oi + 02 4- Qi -j- 0,4 

Oio = Oi + a-i -j- 03 4- 04 4- «5 

To design the detector for this alphabet, it is first necessary to deter- 
mine the coset leaders for a standard array (4) formed for this alphabet. 



•Jl 

t-l 

a 
pa 
< 
M 

Ph 
< 

cc 

o 

H 



O 

H 

ti; 
O 

H 

I— I 

-< 
Ph 



P3 



t^ 00 



-f ^ cc 

CC C^) !M 



O t^ X 



lO a; t^ oc 



00 C2 



^ ^ CC 
CC (N C^l 



t- GC 



lO ic lO -r 

-f -^ CC CT 
CC C^ CM C^I 

;C 1^ X c: 







^ 




cc -+ 


-f -^ cc 


-f -^ cc cc 


-r -^ cc re cc 


(M <N 


CC C^ CM 


CC CM CM CM 


re T-l CM CM CM 



ic :c I- y: — 



i 







re cc 


CO 


ce 


C^l cc 


CM cc re 


re C^J CM CM 


CM re re c^i 


CM re re CM 


^— .-H 


.-H -— C^l 


_ ,_ — ,-H 


r— ^- T-H (M ,— . 


T-^ CM ^ ^ CM -^ 


'^^ lo 


•^ lO « 


-* iC <£) t^ 


•^ lO CO t^ oc 


"* >OCD t^OO C5 



C^l 



ex 

re 



C^l CM C^l 

re-rocot^ ce-^iocot^oc 



C^l C^l 



' >o 



re f lO CO 



re T lO CO t^ oC' 



iCi 



CO 



oc 



210 



















1—1 1—1 


















^CM 


















1-H 1-H 


















cO'f -* 


















CM CM CO 










1-1 1-l 








T— 1 1-H 1-H 










Ot-h 








Oi-HCM 










1—1 1—1 








I-H 1-H 1-H 


















00 


















^cot-oo 


















^^iCiO 










134 

124 

1 123 








12351 

123 

1 124 
2134 


Ol ^ 








01 .—1 1—1 








"^ 1-H 1-H 1-H 










r^ 








t- t- t^ 










coco 








CO CO lO 










•^iO-* 








^^10^-^ 


CO 








"^^00 CO 








"''^ CO CO CO 


-* 't< jvj 








^'*(N^ 








'^'* CM CM CM 


CO(M^ 








«^^^ 








COCM^^^ 


^-^o 








-^-^O-H 








^'~' 0--HCM 


GOO-J T-1 








00 01 1-1 1—1 








00 02 1-H 1-H 1-H 


CO 








CO iC 










^-t 








OCOCO^^ 










iOiO>Oj^ 








'^'^'^coco 










-^-^^0^ 








'^'^^cacM 










CO!M C^ ^ 








CO <M CM ^ ^ 










T— I 1-H T-H _^ 








1— 1 1 — 1 r— * -^ 










o 








C "—1 










t^ 00 o i-H 








t^ 00 C5 1— 1 >— 1 










iCi 


















'I* 


















lOiOiO'*^ j^ 


















•r-^ c^cc ^ 


















CO CM iM O) ^ 


















CO t^ 00 02 ^ 




































•* 








^ 










CO 






^co 








CO^ ^ -* 


CM 






'^'cOCM 








'f CM (M CO CO 








CO ^ "* coco j^ 


1— H 








CO T-H 1— 1 ^H C^l 









1— 1 CM r-( CM .-< ^ 


1— 1 








10 CO t^ 00 cr. 






10 CO t^ 00 rH 


1-H 












CO 






coco 




CO CO 






CM 






(N (M 




CO CM CM 


CI CO CO 


r— 1 




coco CM 


T—l 


T~i 




o*o*c<i^^^ 


1— 1 CM CO r- 1 .— 1 


CM 




1— ( 




CO CO CM 1-1 1—1 1—1 





1-H 




^ CM CO -1 1-H -H ^ ^ ^ 


^lOfOi^ccoi 




•^ lO CO t^ 00 Ci 


I— 1 


1— 1 




rflOCOt^OOOli-Hi-Hi-H 








CM 






C^ C^l 


CM CM CM 




CM (M 






!M 














T-H 






1—1 


1— t 


t— 1 1 — I 


1— 1 ^H I— 1 CM CM 


1-H 


T— t 





r-i .-< i-H CM CM CM 


1— t 





1-H 


^ ^ ^ ^ CM CM CM ^ ^ ^ 


CO 'tl lO CO t^ 00 05 


>— 1 


CO ■* 10 CO t> 00 Oi 


i-H 


r— ( 


CO-*lOCOt^00C2l-Hr-l^ 











I-l 








CM 


T— 1 








1— ( 








1— t 


II 








II 








II 


e 








e 








e 



217 



218 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Table IV — Region of the n-k Plane for Which it is Known 

THAT [n, fc)-ALPHABETS CaNNOT Be EXCELLED 

k 

30 

29 

28 • • • 

27 .... 

26 

25 

24 

23 

22 

21 

20 

19 

18 

17 

16 

15 

14 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 



\ 



1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 22 24 26 28 30 n 

This can be done by a \'ariety of special methods which considerably 
reduce the obvious labor of making such an array. A set of best »S's along 
with their parity check symbols is given in Table V. 

A maximum likelihood detector for the (10, 5)-alphabet in question 
forms from each received sequence 6162 • • • &10 the parity check symbol 
C1C2C3C4C5 where 

Ci = h 4- ^h 4- ^3 + Ih + ^5 

C2 = 67 -(- 6i -]- h-i + hi \- Ih 

Cs = &8 + ^^1 + h -j- Ih + ^5 

Ci = bg + hi 4- h-i -i- h-.i -(- hi 

C5 = />in + hi + />, + h, 4- hi 4- 65 

According to Table V, if CiC-jCiAf'b contains less than three ones, the de- 
tector should brint hih^kihih^ . The detector should piint (/m 4- 1)^2^3^4'':. 
if the parity check sequence C1C2C3C4C5 is either 11111 oi- 11110; the dv- 



A CLASS OF BINARY SIGNALING ALPHABETS 



219 



Table V — Coset Leaders and Parity Check Sequences 

FOR (10, 5) -Alphabet 



ClCiCsCiCb 


^ s 


CIC2C3C4C6 


5 


00000 


0000000000 


11100 


0000100001 


10000 


0000010000 


11010 


0001000001 


01000 


0000001000 


11001 


0001000010 


00100 


0000000100 


10110 


0010000001 


00010 


0000000010 


10101 


0010000010 


00001 


0000000001 


10011 


OOIOOOOIOO 


1 1000 


0000011000 


OHIO 


0100000001 


10100 


0000010100 


01101 


0100000010 


10010 


0000010010 


01011 


0100000100 


10001 


0000010001 


00111 


0100001000 


01100 


0000001100 


11110 


1000000001 


01010 


0000001010 


11101 


OOOOIOOOOO 


01001 


0000001001 


11011 


OOOIOOOOOO 


00110 


0000000110 


10111 


0010000000 


00101 


0000000101 


01111 


0100000000 


00011 


0000000011 


mil 


1000000000 



tector should print 61(62 -j- l)b3lhh^ if the parity check sequence is 01111, 
00111, 01011, 01101, or OHIO; the detector should print hMb-i + 1)6465 
if the parity check sequence is 10111, 10011, 10101, or 10110; the de- 
tector should print 616263(64 -j- 1)65 if the parity check sequence is 11011, 
11001, 11010; and finally the detector should print 61626364(65 -j- 1) if the 
parity check sequence is 11101 or 11100. 

Simpler rules of operation for the detector may possibly be obtained 
by choice of a different set of S's in Table V. These quantities in general 
are not unique. Also there may exist non-equivalent alphabets with 
simpler detector rules that achieve the same probability of error as the 
alphabet in question. 



I'vrt II — Additional Theory and Proofs of Theorems of Part I 

' 2.1 the abstract group Cn 

It will be helpful here to say a few more words about Br, , the group 

of n-place binary sequences under the operation of addition mod 2. This 

j group is simply isomorphic with the abstract group Cn generated by n 

\ commuting elements of order two, say ai, a-2 , ■ ■ ■ , a„ . Here a,:ay = 

<i,ai and a/ = /, i, j = 1, 2, • • • , n, where / is the identity for the 

group. The eight distinct elements of C3 are, for example, /, o-i , a-y , 

(h , (iici-, , aio-.i , a-itti , aia-ittz . The group C„ is easily seen to be isomorphic 

I with the Ai-fold direct product of the group Ci with itself. 



220 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

It is a considerable saving in notation in dealing with C„ to omit the 
symbol "a" and write only the subscripts. In this notation for example, 
the elements of d are 7, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 
234, 1234. The product of two or more elements of C„ can readily be 
written down. Its symbol consists of those numerals that occur an odd 
number of times in the collection of numerals that comprise the sym- 
bols of the factors. Thus, (12)(234)(123) = 24. 

The isomorphism between Cn and Bn can be established in many ways. 
The most convenient way, perhaps, is to associate with the element 
iii-2H ■ ■ ■ ik of Cn the element of Bn that has ones in places ii ,1-2, • • • , ik 
and zeros in the remaining n — k places. For example, one can associate 
124 of C4 with 1101 of Bi ; 14 with 1001, etc. In fact, the numeral no- 
tation afforded by this isomorphism is a much neater notation for Bn 
than is afforded by the awkward strings of zeros and ones. There are, 
of course, other ways in which elements of C„ can be paired with elements 
of Bn so that group multiplication is preserved. The collection of all such 
"pairings" makes up the group of automorphisms of C„ . This group of 
automorphisms of Cn is isomorphic with the group of non-singular linear 
homogenous transformations in a field of characteristic 2. 

An element T of C„ is said to be dependent upon the set of elements 
Ti , T2 , • • ■ , Tj oi Cn if T can be expressed as a product of some ele- 
ments of the set Ti , T2 , • • • , Tj ; otherwise, T is said to be independent 
of the set. A set of elements is said to be independent if no member can 
be expressed solely in terms of the other members of the set. For example, 
in Cs , 1, 2, 3, 4 form a set of independent elements as do likewise 2357, 
12357, 14. However, 135 depends upon 145, 3457, 57 since 135 = 
(145) (3457) (57). Clearly any set of n independent elements of Cn can 
be taken as generators for the group. For example, all possible products 
formed of 12, 123, and 23 yield the elements of C3 . 

Any k independent elements of C„ serve as generators for a subgroup 
of order 2*". The subgroup so generated is clearly isomorphic with Ck ■ 
All subgroups of C„ of order 2'' can be obtained in this way. 

The number of ways in which k independent elements can be chosen 
from the 2" elements of C„ is 

F{n, k) - (2" - 2'')(2" - 2')(2" - 2') • • • (2" - 2'-') 

For, the first element can be chosen in 2" — 1 ways (the identity cannot 
be included in a non-trivial set of independent elements) and the second 
element can be chosen in 2" — 2 ways. These two elements determine a 
subgroup of order 2\ The third element can be chosen as any element of 
the remaining 2" — 2" elements. The 3 elements chosen determine a 



I 



A CLASS OF BINARY SIGNALING ALPHABETS 221 

subgroup of order 2l A fourth independent element can be chosen as 
any of the remaining 2" — 2 elements, etc. 

Each set of k independent elements serves to generate a subgroup of 
order 2''. The quantity F{n, k) is not, however, the number of distinct 
subgroups of C„ of this order, for, a given subgroup can be obtained 
from many different sets of generators. Indeed, the number of different 
sets of generators that can generate a given subgroup of order 2^ of C„ 
is just F{k, k) since any such subgroup is isomorphic with Ck . Therefore 
the number of subgroups of Cn of order 2'' is N{n, k) = F(n, k)/F(k, k) 
which is (2). A simple calculation gives N(n, k) = N(n, n — k). 

2.2 PROOF OF PROPOSITIONS 1 AND 2 

After an element A of 5„ has been presented for transmission over 
a noisy binary channel, an element T of 5„ is produced at the channel 
output. The element U = AT oi Bn serves as a record of the noise 
during the transmission. U is an n-place binary sequence with a one at 
each place altered in A by the noise. The channel output, T, is obtained 
from the input A by multiplication by U: T = UA. For channels of the 
sort under consideration here, the probability that U be any particular 
element of Bn of w^eight w is p^'g"""'. 

Consider now signaling with a particular (n, /b) -alphabet and consider 
the standard array (4) of the alphabet. If the detection scheme (8) is 
used, a transmitted letter A i will be produced without error if and only 
if the received symbol is of the form SjAi . That is, there will be no 
error only if the noise in the channel during the transmission of Ai is 
represented by one of the coset leaders. (This applies (or i = 1,2, • • • , 
fi = 2 ). The probability of this event is Qi (Proposition 1, Section 1.6). 
The convention (5) makes Qi as large as is possible for the given alpha- 
bet. 

Let X refer to transmitted letters and let Y refer to letters produced 
by the detector. We use a vertical bar to denote conditions when writing 
probabilities. The quantity to the right of the bar is the condition. We 
suppose the letters of the alphabet to be chosen independently with 
ec^ual probability 2" . 

The equivocation h{X \ Y) obtained when using an (n, fc)-alphabet 
with the detector (8) can most easily be computed from the formula 

h(X I F) = h{X) - h(Y) + h(Y I X) (10) 

The entropy of the source is /i(X) = k/n bits per symbol. The probability 
that the detector produce Aj when Ai was sent is the probability that 
the noise be represented by AiAjSt , ^ = 1,2, • • • , v. In symbols, 



222 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Pr{Y -. Ai I X -^ Ad = Z Pr{N -^ AiA.Sc) = QiA^A,) 

where Q{Ai) is the sum of the prol)abiUties of the elements that are in 
the same column as Ai in the standard array. Therefore 

Pr{Y -> .4,) = E Pr{Y -> A, \ X -^ AdPr{X -^ A^ = ^ E QU,A,) 

= 4, since E Q^A.A^ = E QUi) = 1. 
This last follows from the group property of the alphabet. Therefore 

/i(lO = -- E P>iy -^ A,) log Pr{Y -^ A,) = - bits/symbol. 
n n 

It follows then from (10) that 

h{X I Y) = h(Y I X) 

The computation of h(Y \ X) follows readily from its definition 

h{Y I X) = E Prix -^ AdhiY \ X -^ Ai) 

i 

= -E Prix -> AdPriY -^ Aj \ X -> Ai) 

log PHY -^Aj I X-^Ai) 
= -^,1211 PriN ->AiScAj) log E PriN -> AiS„,Aj) 



I 



= -^,ZQiAiAj)'}ogQiAiAj) 

Zi ij 

= - EQU,)logQ(A,) 

i 

Each letter is n binary places. Proposition 2, then follows. 

2.3 DISTANCE AND THE PROOF OF THEOREM 1 

Let A and B be two elements of Bn ■ We define the distance, diA, B), 
between A and B to be the weight of their product, 

d{A, B) = w(AB) (11) 

The distance between .4 and B is the number of places in which A and 
B difTer and is jnsl the "Hamming distance." ^ In terms of the n-cube, 
diA, B) is Ihe minimum mmiber of edges that must be traversed to go 



A CLASS OF BINARY SIGNALING ALPHABETS 223 

from vertex ^4 to vertex B. The distance so defined is a monotone fnne- 
tion of the Euchdean distance between vertices. 

It follows from (11) that if C is any element of B„ then 

d{A,B) = cJ(A(\BC) (12) 

This fact shows the detection scheme (8) to be a maximum likelihood 
detector. By definition of a standard array, one has 

d(Si , I) ^ d(S,Aj , I) for all i and j 

The coset leaders were chosen to make this true. From (12), 

d(S, , I) = d(SiA,„S,- , / .4„.^S,) = d(SiA,n , A,„) 

d(SAj , /) - diS^AjSiAm , I SiAJ = diAjA,n , SiAr.) 

= d{SiAm , A() 

where Af = AjA^ . Substituting these expressions in the inecjuality 
above yields 

d(SiAm , A„,) ^ d(SiAm , At) for all i, m, I 

This equation says that an arbitrary element in the array (4) is at least 
as close to the element at the top of its column as it is to any other letter 
of the alphabet. This is the maximum likelihood property. 

2.4 PROOF OF THEOREM 2 

Again consider an (n, /c) -alphabet as a set of vertices of the unit n-cube. 
Consider also n mutually perpendicular hyperplanes through the cen- 
troid of the cube parallel to the coordinate planes. We call these planes 
"symmetr}^ planes of the cube" and suppose the planes numbered in 
accordance with the corresponding parallel coordinate planes. 

The reflection of the vertex with coordinates (ai , a^ , • • • , a^ , • • • , a,j) 
in symmetry plane i yields the vertex of the cube whose coordinates 
are (ai , oo , ■ • • , a, -j- 1, • • • , 0,0 . More generally, reflecting a given 
vertex successively in symmetry planes i, j, k, ■ • ■ yields a new vertex 
whose coordinates differ from the original vertex precisely in places 
i, j, k ■ ■ ■ . Successive reflections in hyperplanes constitute a transfor- 
mation that leaves distances between points unaltered and is therefore 
a "rotation." The rotation obtained by reflecting successively in sym- 
metry planes ?', j, k, etc. can be represented by an ?i-place symbol having 
a one in places ?', j, k, etc. and a zero elsewhere. 

We now regard a given {n, /j)-alphabet as generated by operating on 
the vertex (0, 0, • • ■ , 0) of the cube with a certain collection of 2 ro- 



224 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



tation operators. The symbols for these operators are identical with the 
sequences of zeros and ones that form the coordinates of the 2 points. 
It is readily seen that these rotation operators form a group which is 
transitive on the letters of the alphabet and which leave the unit cube 
invariant. Theorem 2 then follows. 

Theorem 2 also follows readily from consideration of the array (4). 
For example, the maximum likelihood region associated with / is the 
set of points I, So , S3 , • • • , Sy . The maximum likelihood region asso- 
ciated with A; is the set of points Ai , AiS^ , AiSs , ■ • ■ , AiSy . The 
rotation (successive reflections in symmetry planes of the cube) whose 
symbol is the same as the coordinate sequence of Ai sends the maximum 
likelihood region of / into the maximum likelihood region oi Ai , i = 
1, 2, • • • , M. 

2.5 PROOF OF THEOREM 3 

That every systematic alphabet is a group alphabet follows trivially 
from the fact that the sum mod 2 of two letters satisfying parity checks 
is again a letter satisfying the parity checks. The totality of letters satis- 
fying given parity checks thus constitutes a finite group. 

To prove that every group alphabet is a systematic code, consider 
the letters of a given (w, /c) -alphabet listed in a column. One obtains in 
this way a matrix with 2 rows and n columns whose entries are zeros 
and ones. Because the rows are distinct and form a group isomorphic to 
Ck , there are k linearly independent rows (mod 2) and no set of more 
than h independent rows. The rank of the matrix is therefore h. The 
matrix therefore possesses k linearly independent (mod 2) columns and 
the remaining n — k columns are linear combinations of these A;. Main- 
taining only these k linearly independent columns, we obtain a matrix of 
k columns and 2*' rows with rank k. This matrix must, therefore, have k 
linearly independent rows. The rows, however, form a group under mod 
2 addition and hence, since k are linearly independent, all 2" rows must 
be distinct. The matrix contains only zeros and ones as entries; it has 2 
distinct rows of k entries each. The matrix must be a listing of the num- 
bers from to 2^^ — 1 in binary notation. The other n — k columns of 
the original matrix considered are linear combinations of the columns of 
this matrix. This completes the proof of Theorem 3 and Proposition 4. 

2.6 PROOF OF THEOREM 4 

To prove Theorem 4 we first note that the parity check sequence of 
the product of two elements of Bn is the mod 2 sum of their separate 





A CLASS OF BINARY SIGNALING ALPHABETS 225 

parity check sequences. It follows then that all elements in a given coset 
have the same parity check sequence. For, let the coset be Si , SiA2 , 
SiAz , ■ ■ • SiA^ . Since the elements I, A^ , A3, • • • , A^ all have parity 
check sequence 00 • • • 0, all elements of the coset have parity check 
R(Si). 

In the array (4) there are 2" cosets. We observe that there are 2"~* 
elements of Bn that have zeros in their first k places. These elements 
have parity check symbols identical with the last n — k places of their 
symbols. These elements therefore give rise to 2"~ different parity check 
symbols. The elements must be distributed one per coset. This proves 
Theorem 4. 

2.7 PROOF OF PROPOSITION 5 

If 

n ^ 2"-' - 

we can explicity exhibit group alphabets having the property mentioned 
in the paragraph preceding Proposition 5. The notation of the demon- 
stration is cumbersome, but the idea is relatively simple. 

We shall use the notation of paragraph 2.1 for elements of Bn , i.e., 
an element of Bn will be given by a list of integers that specify what 
places of the sequence for the element contain ones. It will be convenient 
furthermore to designate the first k places of a sequence by the integers 
1, 2, 3, • • • , k and the remaining n — k places by the "integers" 1', 2', 
3', • • • , r, where ( = n — k. For example, if n = 8, /c = 5, we have 

10111010^ 13452' 
10000100^ 11' 
00000101 ^ 1'3' 

Consider the group generated by the elements 1', 2', 3', • • • , (' , i.e. 
the 2' elements /, 1', 2', ■■■,(', 1'2', 1'3', • • • , 1'2'3' ■■■('. Suppose 
these elements listed according to decreasing weight (say in decreasing 
order when regarded as numbers in the decimal system) and numbered 
consecutively. Let Bt be the zth element in the list. Example: if ( ^ 3, 
Ih = 1'2'3', B2 = 2'3', B, = 1'3', B, = 1'2', B, - 3', B, = 2', B, - 1'. 

Consider now the (n, /^-alphabet whose generators are 

ISi , 2B, ,W,, ■■• , kBk 
We assert that if 



22G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

>r% n—k 
.. — 2 - 







this alphabet is as good as any other alphabet of 2 letters and n places. 

In the first place, we observe that every letter of this (n, A-)-alphabet 
(except /) has unprimed numbers in its symbols. It follows that each of 
the 2' letters /, 1', 2', • ■ • , (', V2', ■■■ , V2' ■■■ (' occurs in a different 
coset of the given (n, A-)-alphabet. For, if two of these letters appeared 
in the same coset, their product (which contains only primed numbers) 
would have to be a letter of the (n, k) alphabet. This is impossible since 
every letter of the (/i, A) alphabet has unprimed numbers in its symbol. 
Since there are precisely 2 cosets we can designate a coset by the single 
element of the list Bi , Bi , ■ • ■ , B-ii = I which appears in the coset. 

We next observe that the condition 

71 ^ 2 — 

guarantees that J5a+i is of weight 3 or less. For, the given condition is 
equivalent to 

'-■-©-o-o-e 

We treat several cases depending on the weight of Bu+i . 

If Bk+\ is of weight 3, we note that for i = 1,2, • • • , A-, the coset con- 
taining Bi also contains an element of weight one, namely the element 
i obtained as the product of Bi with the letter iBi of the given (n, A;)- 
alphabet. Of the remaining (2 — A') 5's, one is of weight zero, C are of 

weight one, f j are of weight 2 and the remaining are of weight 3. We 

have, then an = 1, ai = f + A- = n. Now every B of weight 4 occurs in' 
the list of generators \Bi , 2B-2 , • • • , kBk . It follows that on multi- 
plying this list of generators by any B of weight 3, at least one element 
of weight two will result. (E.g., (l'2'3')(il'2'3'40 = j4') Thus every 
coset with a B of weight 2 or 3 contains an element of weight 2 and 
a2 = 2 — ao — cn] . 

The argument in case Bk+i is of weight two or one is similar. 

2.8 MODULAR REPRESENTATIONS OF C„ 

In order to explain one of the methods used to obtain the best (//, A)- 
alphabets listed in Tal)les II and III, it is necessary to digress here lo 
present additional theory. 



I 



A CLASS OF BINARY SINGALING ALPHABETS 



227 



It has been remarked that every (n, /v)-alphabet is isomorphic with 
Ck . Let us suppose the elements of Ci, hsted in a column starting with / 
and proceeding in order /, 1, 2, 3, • • • , /.', 12, 13, ■••,(/.•— 1)/,-, 123, 

, 123 • • • k. The elements of a given (n, A-)-alphabet can be 

paired off with these abstract elements so as to preserve group multipli- 
cation. This can be done in many different ways. The result is a matrix 
with elements zero and one with 7i columns and 2 rows, these latter 
being labelled by the symbols /, 1,2, • • • etc. What can be said about 
the columns of this matrix? How many different columns are possible 
when all (n, A)-alphabets and all methods of establishing isomorphism 
with Ck are considered? 

In a given column, once the entries in rows 1,2, • • • , /,• are known, the 
entire column is determined by the group property. There are therefore 
only 2 possible different columns for such a matrix. A table showing 
these 2 possible columns of zeros and ones will be called a modular repre- 
senfafion table for Ck ■ An example of such a table is shown for /,• = 4 in 
Table VI. 

It is clear that the colunuis of a modular representation table can also 
be labelled by the elements of Ck , and that group multiplication of these 
column labels is isomorphic with mod 2 addition of the columns. The 
table is a symmetric matrix. The element with row label A and column 
label B is one if the symbols A and B have an odd number of different 
numerals in common and is zero otherwise. 

Every (n, /c)-alphabet can be made from a modular representation 
table by choosing w columns of the table (with possible repetitions) at 
least k of which form an independent set. 



Table VI — Modular Representation Table for Group C4 

I 12 3 4 12 13 14 23 24 34 123 124 134 234 1234 

I 

1 

2 

3 

4 

12 

13 

14 

23 

24 

34 

123 

124 

134 

234 

1234 















































n 





1 











1 




1 










1 






1 








1 








1 















1 






1 











1 




























1 














1 







1 










1 






1 





1 


1 













1 























1 





1 





1 




1 










1 












1 








1 


1 































1 


1 





1 















1 















1 





1 


1 




1 





























1 


1 







1 










1 












1 


1 


1 


() 







1 

















1 





1 


1 





1 


















1 






1 





1 





1 


1 


1 






















1 








1 


1 


1 


1 




1 

















1 





1 


1 


1 


1 




















1 






u 



228 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

We henceforth exclude consideration of the column / of a modular 
representation table. Its inckision in an (n, /v)-alphabet is clearly a waste 
of 1 binary digit. 

It is easy to show that every column of a modular representation table 
for Ch contains exactly 2 " ones. Since an (n, /v)-alphabet is made from 
n such columns the alphabet contains a total of n2 '~ ones and we have 

Proposition 6. The weights of an (n, /c)-alphabet form a partition of 
n2''~^ into 2* — 1 non-zero parts, each part being an integer from the set 
1,2, ■■■ ,n. 
The identity element always has weight zero, of course. 

It is readily established that the product of two elements of even 
weight is again an element of even weight as is the product of two ele- 
ments of odd weight. The product of an element of even weight with an 
element of odd weight yields an element of odd weight. 

The elements of even weight of an (n, A;) -alphabet form a subgroup 
and the preceding argument shows that this subgroup must be of order 
2*" or 2*""^ If the group of even elements is of order 2''~\ then the collec- 
tion of even elements is a possible (n, k — l)-alphabet. This (n, k — 1) 
alphabet may, however, contain the column / of the modular represen- 
tation table of Ck-i ■ We therefore have 

Proposition 7. The partition of Proposition 6 must be either into 
2^ — 1 even parts or else into 2 " odd parts and 2^—1 even parts. 
In the latter case, the even parts form a partition of a2 "" where a is 
some integer of the set k — I, k, ■ • • , n and each of the parts is an in- 
teger from the set 1, 2, • • • , n. 

2.9 THE CHARACTERS OF Ck 

Let us replace the elements of Bn (each of which is a sequence of zeros 
and ones) by sequences of 4-1 's and — I's by means of the following 
substitution 

The multiplicative properties of elements of Bn can be preserved iti this 
new notation if we define the product of two 4-1,-1 symbols to be the 
symbol whose tth component is the ordinary product of the ?'th compo- 
nents of the two factors. For example, 1011 and 01 10 become respectively 
-11 -1 -1 and 1 -1 -11. We have 

(-11 -1 -1)(1 -1 -11) = (-1 -11 -1) 



1 














1 














-1 














-1 



A CLASS OF BINARY SIGNALING ALPHABETS 229 

corresponding to the fact that 

(1011) (0110) = (1101) 

If the +1,-1 symbols are regarded as shorthand for diagonal matrices, 
so that for example 



-11 -1 -1 



then group multiplication corresponds to matrix multiplication. 

(While much of what follows here can be established in an elementary 
way for the simple group at hand, it is convenient to fall back upon the 
established general theory of group representations for several proposi- 
tions. 

The substitution (13) converts a modular representation table (col- 
umn / included) into a square array of +l's and — I's. Each column (or 
row) of this array is clearly an irreducible representation of Ck ■ Since Ck 
is Abelian it has precisely 2 irreducible representations each of degree 
one. These are furnished by the converted modular table. This table also 
furnishes then the characters of the irreducible representations of Ck 
and we refer to it henceforth as a character table. 

Let x"(^) be the entry of the character table in the row labelled A and 
column labelled a. The orthogonality relationship for characters gives 



E x'{A)/{A) = 2'8., 



ACCk 



Z x%A)x"(B) = 2'b 

<xCCk 



AB 



where 8 is the usual Kronecker symbol. In particular 

E xiA)x\A) = Z AA) = 0, ^^I 

ACCk ACCk 

Since each x (A) is +1 or — 1, these must occur in eciual numbers in any 
column ^ 9^ I. This implies that each column except / of the modular 
representation table contains 2 ~ ones, a fact used earlier. 

Every matrix representation of Ck can be reduced to its irreducible 
components. If the trace of the matrix representing the element A in an 
arbitrary matrix representation of Ck is x{A), then this representation 
contains the irreducible representation having label ^ in the character 
table dp times where 



230 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



(h = ^. E x{A)AA) (14) 



2^- 



A C Ck 



Every (n, A)-alphabet furnishes iis with a matrix representation of Ck 
by means of (13) and the procedure outUned below (13). The trace xi^.) 
of the matrix representing the element A of C\ is related to the weight 
of the letter by 

x(A) = n - 2w(A) (15) 

Equations (14) and (15) permit us to compute from the weights of an 
(u, /,)-alphabet what irreducible representations are present in the alpha- 
bet and how many times each is contained. It is assumed here that the 
given alphabet has been made isomorphic to Ck and that the weights are 
labelled by elements of Ck ■ 

Consider the converse problem. Given a set of mmibers ivi , Wn , • ■ ■ , 
W'lk that satisfy Propositions 6 and 7. From these we can compute 
cjuantities %/ = n — 2wi as in (15). It is clear that the given ty's will 
constitute the weights of an (/t, A)-alphabet if and only if the 2^ x» can 
be labelled with elements of (\ so that the 2 sums (14) {fi ranges over 
all elements of Ck) are non-negative integers. The integers d^ tell what 
representations to choose to construct an in, A)-alphabet with the given 
weights Wi . 

2.10 CONSTRUCTION OF BEST ALPHABETS 

A great many different techniques were used to construct the group 
alphabets listed in Tables II and III and to show that for each n and k 
there are no group alphabets with smaller probability of error. Space 
prohibits the exhibition of proofs for all the alphabets listed. We content 
ourseh'es here with a sample argument and treat the case n = 10, k = 
4 in detail. 

According to (2) there are A^(10, 4) = 53,743,987 different (10, 4)- 
alphabets. We now show that none is better than the one given in Table 
III. The letters of this alphabet and weights of the letters are 

1 

167 8 10 5 

2 6 7 9 10 5 

3 5 6 8 9 10 6 

4 5 7 8 9 10 6 
1289 4 
13579 5 



A CLASS OF BINARY SIGNALING ALPHABETS 



231 



14569 
23578 
24568 
3 4 6 7 
12 3 5 7 9 
12 4 5 7 10 

1 3 4 8 10 

2 3 4 9 10 

12 3 4 6 7 8 9 



5 
5 
5 
4 
6 
6 
5 
5 
8 



The notation is that of Section 2.1. By actually forming the standard 
array of this alphabet, it is verified that 



ao =1, Oil = 10, 



«2 



39, 



a:i 



14. 



Table II shows ( .-> ) = ^5, whereas a-z = 39, so the given alphabet 

does not correct all possible double errors. In the standard array for the 
alphabet, 39 coset leaders are of weight 2. Of these 39 cosets, 33 have 
only one element of weight 2; the remaining 6 cosets each contain two 
elements of weight 2. This is due to the two elements of weight 4 in the 
given group, namely 1289 and 3467. A portion of the standard array 
that demonstrates these points is 



1289 



3467 



12 


89 


• 


18 


29 


• 


19 


28 


. 


34 




67 


36 




47 


37 




46 


] 




• 



In order to have a smaller probability of error than the exhibited 
alphabet, it is necessary that a (10, 4)-alphabet have an a^ > 39. We 
proceed to show that this is impossible by consideration of the weights 
of the letters of possible (10, 4)-alphabets. 

We first show that every (10, 4)-alphabet must have at least one ele- 
ment (other than the identity, /) of weight less than 5. By Propositions 
• ') and 7, Section 2.8, the weights must form a partition of 10-8 = 80 into 
1 5 positive parts. If the weights are all even, at least two must be less 
than 6 since 14-6 = 84 > 80. If eight of the weights are odd, we see from 
8-5 + 7-() = 82 > 80 that at least one weight must be less than 5. 



232 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

An alphabet with one or more elements of weight 1 must have an 
«2 ^ 36, for there are nine elements of weight 2 which cannot possibly 
be coset leaders. To see this, suppose (without loss of generality) that 
the alphabet contains the letter 1. The elements 12, 13, 14, • • • 1 10 can- 
not possibly be coset leaders since the product of any one of them with 
the letter 1 yields an element of weight 1 . 

An alphabet with one or more elements of weight 2 must have an 
ai S 37. Suppose for example, the alphabet contained the letter 12. 
Then 13 and 23 must be in the same coset, 14 and 24 must be in the 
same coset, ■ • • , 1 10 and 2 10 must be in the same coset. There are at 
least eight elements of weight two which are not coset leaders. 

Each element of weight 3 in the alphabet prevents three elements of 
weight 2 from being coset leaders. For example, if the alphabet contains 
123, then 12, 13, and 23 cannot be coset leaders. We say that the three 
elements of weight 2 are "blocked" by the letter of weight 3. Suppose an 
alphabet contains at least three letters of weight three. There are several 
cases: (A) if three letters have no numerals in common, e.g., 123, 456, 
789, then nine distinct elements of weight 2 are blocked and a-2 S 36; 
(B) if no two of the letters have more than a single numeral in common, 
e.g., 123, 345, 789, then again nine elements of weight 2 are blocked and 
a-2 ^ 36; and (C) if two of the letters of weight 3 have two numerals in 
common, e.g., 123, 234, then their product is a letter of weight 2 and l)y 
the preceding paragraph ao ^ 37. If an alphabet contains exactly two 
elements of weight 3 and no elements of weight 2, the elements of weight 

3 block six elements of weight 2 and 0:2 ^ 39. 
The preceding argument shows that to be better than the exhibited 

alphabet a (10, 4)-alphabet with letters of weight 3 must have just one 
such letter. A similar argument (omitted here) shows that to be better 
than the exhibited alphabet, a (10, 4)-alphabet cannot contain more 
than one element of weight 4. Furthermore, it is easily seen that an 
alphabet containing one element of weight 3 and one element of weight 

4 must have an ao ^ 39. 
The only new contenders for best (10, 4)-alphabet are, therefore, 

alphabets with a single letter other than / of weight less than 5, and this 
letter must have weight 3 or 4. Application of Propositions 6 and 7 show 
that the only possible weights for alphabets of this sort are: 35 6 and 

5 46' where 5' means seven letters of weight 5, etc. We next show that 
there do not exist (10, 4)-alphabets having these weights. 

Consider first the suggested alphabet with weights 35 6'. As explained 
in Section 2.9, from such an alphabet we can construct a matrix repre- 
sentation of ('4 having the character x(/) = 10, one matrix of trace 4, 



A CLASS OF BINARY SIGNALING ALPHABETS 233 

seven of trace and seven of trace —2. The latter seven matrices cor- 
respond to elements of even weight and together with / must represent 
a subgroup of order 8. We associate them with the subgroup generated 
by the elements 2, 3, and 4. We have therefore 

x(/) = 10, x(2) = x(3) = x(4) = x(23) 

= x(24) = x(34) = x(234) = -2. 

Examination of the symmetries involved shows that it doesn't matter 
how the remaining Xi ai"e associated with the remaining group elements. 
We take, for example 

x(l) = 4, x(12) = x(13) = x(14) = x(123) 

= x(124) = x(134) = x(1234) = 0. 

Now form the sum shown in equation (14) with /3 = 1234 (i.e., with the 
character x^" obtained from column 1234 of the Table VI by means 
of substitution (13). There results c?i234 = V-i which is impossible. There- 
fore there does not exist a (10, 4) -alphabet with weights 35 6 . 

The weights 5 46 correspond to a representation of d with character 
x(/) = 10, 0^, 2, ( — 2)^ We take the subgroup of elements of even weight 
to be generated by 2, 3, and 4. Except for the identity, it is clearly im- 
material to w^hich of these elements we assign the character 2. We make 
the following assignment: x(/) = 10, x(2) = 2, x(3) = x(4) = x(23) = 
x(24) = x(34) = x(234) = -2, x(l) = x(12) = x(13) = x(14) = 
x(123) = x(124) = x(134) = x(1234) = 0. The use of equation (14) 
shows that ^2 = \'2 which is impossible. 

It follows that of the 53,743,987 (10, 4)-alphabets, none is better than 
the one listed on Table III. 

Not all the entries of Table III were established in the manner just 
demonstrated for the (10, 4)-alphabet. In many cases the search for a 
l)est alphabet was narrowed down to a few alphabets by simple argu- 
ments. The standard arrays for the alphabets were constructed and the 
best alphabet chosen. For large n the labor in making such a table can 
be considerable and the operations involved are highly liable to error 
when performed by hand. 

I am deeply indebted to V. M. Wolontis who programmed the IBM 
CPC computer to determine the a's of a given alphabet and who pa- 
tiently ran off many such alphabets in course of the construction of 
Tables II and III. I am also indebted to Mrs. D. R. Fursdon who eval- 
uated many of the smaller alphabets by hand. 



234 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 



REFERENCES 

1. R. W. Hamming, B.S.T.J., 29, i)p. 147-160, 1950. 

2. I. S. Reed, Transactions of tlie Piofossional (iroup on Information Tlieorv, 
^ PGIT-4, PI). 3S-49, 1954. 

3. See section 7 of R . W. Hamniinji's paper, loc. cit. 

4. I.R.E. Convention Record, I'art 4, pp. 37-45, 1955 National Convention, 

March, 1955. 

5. C. E. Shannon, B.S.T.J., 27, pp. 379-423 and pp. 623-656, 1948. 

6. Birkhoff and MacLane, A Snrvey of Modern Algebra, Macmillan Co., New 

York, 1941 . Van der Waerden, Alodern Algebra, Ungar Co., New York, 1953. 
Miller, Bliclifeldt, and Dickson, Finite Groups, Stechert, New York, 1938. 

7. This theorem has been previously noted in the literature by Kiyasu-Zen'iti, 

Research and Development Data No. 4, Ele. Comm. Lai)., Nippon Tele. 
Corp. Tokyo, Aug., 1953. 

8. F. D. Murnaghan, Theory of Group Representations, Johns Hopkins Press, 

Baltimore, 1938. E. Wigner, Gruppentheorie, Edwards Brothers, Ann Arbor, 
Michigan, 1944. 



I 



Bell System Technical Papers Not 
Published in This Journal 

Allen, L. J., see Fewer, D. R. 

Alllson, H. W., see Moore, G. E. 

Baker, W. 0., see Winslow, F. H. 

Barstow, J. M.^ 

Color TV How it Works, I.R.E. Student Quarterly, 2, pp. 11-16, 
Sept., 1955. 

Basseches, H.^ and ^McLean, D. A. 

Gassing of Liquid Dielectrics Under Electrical Stress, Ind. c^- Engg. 
Chem., 47, pp. 1782-1794, Sept., 1955. 

Beck, A. C} 

Measurement Techniques for Multimode Waveguides, Proc. I.R.E., 
MRI, 4, pp. 325-6, Oct. 1, 1955. 

Becker, J. A.^ 

The Life History of Adsorbed Atoms, Ions, and Molecules, N. Y. 
Acad. Sci. Ann., 58, pp. 723-740, Sept. 15, 1955. 

Hlackwell, J. H., see Fewer, D. R. 
BooRSE, H. A., see Smith, B. 

HozoRTii, R. M.,' Getlin, B. B.,' Galt, J. K.,' Merritt, F. R.,' and- 

^'ager, W. a.' 
Frequency Dependence of Magnetocrystalline Anisotropy, Letter to 
the Editor, Phys. Rev., 99, p. 1898, Sept. 15, 1955. 



1. Bell Telephone Laboratories, Inc. 

235 



236 THE BELL SYSTEM TECHXICAL JOURNAL, JANUARY 1956 

BozoRTH. R. M.\ TiLDEX, E. F..' and Williams, A. j/ 
Anisotropy and Magnetostriction of Some Ferrites, Phys. Rev., 99, 
pp. 17S8-1798, Sept. 15, 1955. 

Bridgers, H. E.,^ and Kolb, E. D.^ 

Rate-Grown Germanium Crystals for High-Frequency Transistors, 
Letter to the Editor, J. Appl. Phys., 26, pp. 1188-1189, Sept., 1955. j 

BULLIXGTOX, K.^ 

Characteristics of Beyond-the-Horizon Radio Transmission, Pioc. 
I.R.E., 43, pp. 1175-1180, Oct., 1955. 

BULLIXGTOX, K.^ IXKSTER, W. J.,^ and DVRKEE, A. L.^ 

Results of Propagation Tests at 505 Mc and 4,090 Mc on Beyond- 
Horizon Paths, Proc. I.R.E., 43, pp. 1306-1316, Oct., 1955. 

Calbick, C. J.' 

Surface Studies with the Electron Microscope, X. Y. Acad. Sci. Ann., 
58, pp. 873-892, Sept. 15, 1955. 

Cass, R. S., see Fewer, D. R. 

DuRKEE, A. L., see Bullington, K. 

Fewer, D. R..' Blackwell. J. H..' Allex. L. J..^ and Cass, R. S." 
Audio-Frequency Circuit Model of the 1-Dimensional Schroedinger 
Equation and Its Sources of Error, Canadian J. of Pins., 33, pp. 483- 
491, Aug., 1955. 

Francois, E. E., see Law, J. T. 

Davis, J. L., see Suhl, H. 

Galt, J. K., see Bozorth, R. "SI., and Yager, W. A. 

Garn, p. D.,' and Hallixe, Mrs. E. W.' 

Polarographic Determination of Phthalic and Anhydride Alkyd Res- 
ins, Anal Cliem., 27, pp. 15()3-15G5, Oct., 1955. 

1. Bell Telephone Laboratories, Inc. 

4. University of Western Ontario, London, Canada 

5. Bell Telephone Company of Canada, Montreal 



TECHNICAL PAPERS 237 

Getlin, B. B., see Bozorth, R. M. 

GlANOLA, V. F} 

Application of the Wiedemann Effect to the Magnetostrictive Coupling 
of Crossed Coils, J. Appl. Phys., 26, pp. 1152-1157, Sept., 1955. 

Goss, A. J., see Hassion, F. X. 

Green, E. I.^ 
The Story of 0, American Scientist, 43: pp. 584-594, Oct., 1955. 

Halline, Mrs. E. W., see Garn, P. D. 

Harrower, G. A.^ 
Measurement of Electron Energies by Deflection in a Uniform Electric 
Field, Rev. Sci. Instr., 26, pp. 850-854, Sept., 1955. 

Hassion, F. X.,^ Goss, A. .1.,^ and Trumbore, F. A.^ 
The Germanium-Silicon Phase Diagram, J. Phys. Chem., 59, p. 1118, 
Oct., 1955. 

Hassion, F. X.,^ Thurmond, C. D.,^ and Trumbore, F. A.^ 

On the Melting Point of Germanium, J. Phys. Chem., 59, p. 1076, 
Oct., 1955. 

Hines, I\I. E.,' Hoffman, G. W.,' and Saloom, J. A.^ 

Positive-Ion Drainage in Magnetically Focused Electron Beams, J. 
Appl. Phys., 26, pp. 1157-1162, Sept., 1955. 

Hoffman, G. W., see Hines, M. E. 

Inkster, W. J., see Bullington, K. 

Kelly, M. J.' 
Training Programs of Industry for Graduate Engineers, Elec. Engg., 
74, pp. 866-869, Oct., 1955. 

KoLB, E. D., see Bridgers, H. E. 
1. Bell Telephone Laboratories, Inc. 



1 



238 THE BELL SYSTEM TECHXICAL JOURXAL, JANUARY 1 9 of) 

Law, J. T./ and Francois, E. E.' 

Adsorption of Gasses and Vapors on Germanium, X. Y. Acad. Sci. 
Ann., 58, pp. 925-936, Sept. 15, 1955. 

LovELL, Miss L. C, see Pfann, W. G. 

Matreyek, W., see Winslow, F. H. 

McLean, D. A., see Basseches, H. 

Merritt, F. R., see Bozorth, R. M., and Yager, W. A. 

Meyer, F. T.' 

An Improved Detached-Contact Type of Schematic Circuit Drawing, 
A.LE.E. Commun. ct Electronics, 20, pp. 505-513, Sept., 1955. 

Miller, B. T.' 

Telephone Merchandising, Telephony, 149, pp. 116-117, Oct. 22, 
1955. 

Miller, S. L.^ 

Avalanche Breakdown in Germanium, Phys. Rev., 99, pp. 1234-1241, 
Aug. 15, 1955. 

Moore, G. E.,^ and Allison, H. W.^ 
Adsorption of Strontium and of Barium on Tungsten, J. Chem. 
Phys., 23, pp. 1609-1621, Sept., 1955. 

Neisser, W. R.,^ 

Liquid Nitrogen Coal Traps, Rev. Sci. Instr., 26, p. 305, Mar., 1955. 

Ostergren, C. N." 

Some Observations on Liberahzed Tax Depreciation, Telephony, 149, 
pp. 16-23-37, Oct. 1, 1955. 

Ostergren, G. N. 

Depreciation and the New Law, Telephony, 149, pp. 96-100-104-108, ; 
Oct. 22, 1955. I 

Rape, N. R., see Winslow, F. H. 

1. Bell Telephone Laboratories, Inc. 

2. American Telephone and Telegraph Co. 



\\ 



technical papers 239 

Pedekskn, L. 
Aluminum Die Castings for Carrier Telephone Systems, A.I.E.E. 
Commun. & Electronics, 20, pp. 434-439, Sept., 1955. 

Peters, H.^ 
Hard Rubber, Tnd. and Engg. Chem., Part II, pp. 2220-2222, Sept. 
20, 1955. 

Pfann, w. c;.' 

Temperature-Gradient Zone-Melting, J. Metals, 7, p. 961, Sept., 1955. 

Pfann, W. G.,' and Lovell, Miss L. C.^ 
Dislocation Densities in Intersecting Lineage Boundaries in Ger- 
manium, Letter to the Editor, Acta. Met., 3, pp. 512-513, Sept., 1955. 

Pierce, J. P.' 
Orbital Radio Relays, Jet Propulsion, 25, pp. 153-157, Apr., 1955. 

Poole, K. M.' 
Emission from Hollow Cathodes, J. Appl. Phys., 26, pp. 1176-1179, 
Sept., 1955. 

Saloom, J. A., see Hines, M. E. 

Slighter, W. P.^ 
Proton Magnetic Resonance in Polyamides, J. Appl. Phys., 26, pp., 
1099-1103, Sept., 1955. 

Smith, B./ and Boorse, H. A. 
Helium II Film Transport. II. The Role of Surface Finish, Phys. Rev. 
99, pp. 346-357, July 15, 1955. 

Smith, B.,^ and Boorse, H. A. 
Helium II Film Transport. IV. The Role of Temperature, Phys. Rev., 
99, pp. 367-370, July lo, 1955. 

SuHL, H.,^ Van Uitert, L. G.,^ and Davis, J. L.^ 
Ferromagnetic Resonance in Magnesium-Manganese Aluminum Fer- 
rite Between 160 and 1900 Mc, Letter to the Editor, J. Appl. Phys., 
26, pp. 1181-1182, Sept., 1955. 

1. Bell Telephone Laboratories, Inc. 

6. Columbia University, New York City 



240 THE EELL SYSTEM TECHNICAL JOURNAL, JANUARY 1956 

Thurmond, C. D., see Hassion, F. X. 

TiDD, W. H/ I 

Demonstration of Bandwidth Capabilities of Beyond -Horizon Tropo- 
spheric Radio Propagation, Proc. I.R.E., 43, pp. 1297-1299, Oct., 1955. 

Tien, P. K.,' and Walker, L. R.' 
Large Signal Theory of Traveling -Wave Amplifiers, Proc. I.R.E., 43, 
p. 1007, Aug., 1955. 

TiLDEN, E. F., see Bozorth, R. M. 

Trumbore, F. a., see Hassion, F. X. 

IThlir, a., Jr.^ 

Micromachining with Virtual Electrodes, Rev. Sci., Instr., 26, pp. 
965-968, Oct., 1955. 

Ulrich, W., see Yokelson, B. J. 

Van Uitert, L. G., see Siihl, H. 

Walker, L. R., see Tien, P. K. 

Weibel, E. S.' 
Vowel Synthesis by Means of Resonant Circuits, J. Acous. Soc, 27, 
pp. 858-865, Sept., 1955. 

Williams, A. J., see Bozorth, R. M. 

WiNSLow, F. H.,' Baker, W. O.,^ and Yager, W. A.^ 

Odd Electrons in Polymer Molecules, Am. Chem. Soc, 77, pp. 4751- 
4756, Sept. 20, 1955. 

WiNSLow, F. II.,' Baker, W. O.,' Rape, N. R.' and Matreyek, W.' 
Formation and Properties of Polymer Carbon, J. Polymer Science, 16, 
p. 101, Apr., 1955. 

Yager, W. A., sec Bozorth, R. M. 
1. Bell Tc;l(;i)li()ne liaboratorics, Inc. 



TECHNICAL PAPERS 241 

Yagkr, W. a./ Galt, J. K/ and Merritt, F. R.' 
Ferromagnetic Resonance in Two-Nickel-Iron Ferrites, Phys. Rev., 
99, pp. 1203-1209, Aug. 15, 1955. 

YoKELSON, B. J.,^ and Ulrich, W.^ 

Engineering Multistage Diode Logic Circuits, A.I.E.E. Commun. & 
Electronics, 20, pp. -466-475, Sept., 1955. 

1. Bell Telephone Laboratories, Inc. 



Recent Monographs of Bell System Technical 
Papers Not Published in This Journal* 

Arnold, W. O., and Hoefle, R. R. 

A System Plan for Air Traffic Control, ]\Ionograph 2483. 

Beck, A. C. 

Measurement Techniques for Multimode Waveguides, ]\Ioiiograph 
2421. 

Becker, J. A., and Brandes, R. G. 

Adsorption of Oxygen on Tungsten as Revealed in Field Emission 
Microscope, Alonogiaph 24U3. 

Boyle, W. S., see Germer, L. H. 

Brandes, R. G., see Becker, J. A. 

Brattain, W. H., see Garrett, C. G. B. 

Garrett, C. G. B., and Brattain, W. H. 

Physical Theory of Semiconductor Surfaces, Monograph 2453. 

Gerner, L. H., Boyle, W. S., and Kisliuk, P. 

Discharges at Electrical Contacts — II, Monograph 2499. 

Hoefle, R. R., see Arnold, W. 0. 

KisLiuK, P., see Germer, L. H. 

Linvill, J. G. 

Nonsaturating Pulse Circuits Using Two Junction Transistors, Mono- 
graph 2-17."). I 

* Copies of these monographs may 1)0 ()l)l;tin(Ml on request to the Pul)licat ion 
Department, Hell Telephone Laboratories, Iiie., 463 West Street, New York 14, 
N. Y. The numbers of the monographs should be given in all requests. 

242 



MONOGRAPHS 243 

Mason, W. P. 

Relaxations in the Attenuation of Single Crystal Lead, Monograph 
2454. 

Mkykr, F. T. 
An Improved Detached-Contact-Type of Schematic Circuit Drawing, 
Monograph 2456. 

VoGEL, F. L., Jr. 

Dislocations in Low-Angle Boundaries in Germanium, Monograph 
2455. 

Walker, T.. R. 

Generalizations of Brillouin Flow, Monograph 2432. 

Warner, A. W. 

Frequency Aging of High -Frequency Plated Crystal Units, Monograph 
2474. 

Weibel, E. S. 

On Webster's Horn Equation, Monograph 2450. 



Contributors to This Issue 

A. C. Beck, E.E., Rensselaer Polytechnic Institute, 1927; Instructor, 
Rensselaer Polytechnic Institute, 1927-1928; Bell Telephone Labora- 
tories, 1928 -. With the Radio Research Department he was engaged 
in the development and design of short-wave and microwave antennas. 
During World War II he was chiefly concerned with radar antennas and 
associated waveguide structures and components. For several years 
after the war he worked on development of microwave radio repeater 
systems. Later he worked on microwave transmission developments 
for broadband communication. Recently he has concentrated on further 
developments in the field of broadband communication using circular 
waveguides and associated test equipment. 

J. S. Cook, B.E.E., and M.S., Ohio State University, 1952; Bell 
Telephone Laboratories, 1952 -. Mr. Cook is a member of the Research 
in High-Frequency and Electronics Department at Murray Hill and 
has been engaged principally in research on the traveling- wave tube. 
Mr. Cook is a member of the Institute of Radio Engineers and belongs 
to the Professional Group on Electron Devices. 

0. E. DeLange, B.S. University of Utah, 1930; M.A. Columbia Uni- 
versity, 1937; Bell Telephone Laboratories, 1930 — . His early work was 
principally on the development of high-frequency transmitters and re- 
ceivers. Later he worked on frequency modulation and during World 
War II was concerned with the development of radar. Since that time 
he has been involved in research using broadband systems including 
microwa^'e and baseband. Mr. DeLange is a member of the Institute 
of Radio Engineers. 

R. KoMPFNER, Engineering Degree, Technische Hochschule, Vienna, 
1933; Ph.D., Oxford, 1951; Bell Telephone Laboratories, 1951 -. Be- 
tween 1941-1950 he did work for the British Admiralty at Birmingham 
University and Oxford University in the Royal Naval Scientific Service. 
He invented the traveling-wave tube and for this achievement Dr. 
Kompfner i-eceived the 1955 Duddcll Medal, bestowed by the Physical 
Society of England. In the Laboratoi'ies' Research in High Frequency 

244 



CONTRIBUTORS TO THIS ISSUE 245 

and Electronics Department, he has continued his research on vacuum 
tubes, particularly those used in the microwave region. He is a Fellow 
of the Institute of Radio Engineers and of the Physical Society in 
London. 

Charles A. Lee, B.E.E., Rensselaer Polytechnic Institute, 1943; 
Ph.D., Columbia University, 1953; Bell Telephone Laboratories, 1953-. 
When Mr. Lee joined the Laboratories he became engaged in research 
concerning solid state devices. In particular he has been developing 
techniques to extend the frequency of operation of transistors into the 
microwave range, including work on the diffused base transistor. During 
World War II, as a member of the United States Signal Corps, he was 
concerned with the determination and detection of enemy counter- 
measures in connection with the use of proximity fuses by the Allies. 
He is a member of the American Physical Society and the American 
Institute of Physics. He is also a member of Sigma Xi, Tau Beta Pi 
and Eta Kappa Nu. 

John R. Pierce, B.S., M.S. and Ph.D., California Institute of Tech- 
nology 1933, 1934 and 1936; Bell Telephone Laboratories, 1936-. Ap- 
pointed Director of Research — Electrical Communications in August, 
1955. Dr. Pierce has specialized in Development of Electron Tubes and 
Microwave Research since joining the Laboratories. During World War 
li II he concentrated on the development of electronic devices for the 
[I Armed Forces. Since the war he has done research leading to the develop- 
;j ment of the beam traveling- wave tube for which he was awarded the 
h 1947 Morris Liebmann Memorial Prize of the Institute of Radio Engi- 
[li neers. Dr. Pierce is author of two books: Theory and Design of Electron 
Ij Beams, published in second edition last year, and Traveling Wave Tubes 
il (1950). He was voted the ''Outstanding Young Electrical Engineer of 
[| 1942" by Eta Kappa Nu. Fellow of the American Physical Society and 
J the I.R.E. Member of the National Academy of Sciences, the A.I.E.E., 
I Tau Beta Pi, Sigma Xi, Eta Kappa Nu, the British Interplanetary So- 
il ciety, and the Newcomen Society of North America. 

C. F. QuATE, B.S., University of Utah 1944; Ph.D., Stanford Uni- 
i versity 1950; Bell Laboratories 1950-. Dr. Quate has been engaged in 
rj research on electron dynamics — the study of vacuum tubes in the 
;| microwave frequency range. He is a member of I.R.E. 

I David Slepian, University of Michigan, 1941-1943; M.A. and Ph.D., 
li Harvard LTniversity, 1946-1949; Bell Telephone Laboratories, 1950-. Dr. 



24G THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1950 

Slepian has been engaged in mathematical research in communication 
theory, switching theory and theory of noise. Parker Fellow in physics. 
Harvard University 1949-50. Member of I.R.E,, American Mathemati- 
cal Society, the American Association for the Advancement of Science 
and Sigma Xi. 

Milton Sobel, B.S., City College of New York, 1940; M.A., 1946 and 
Ph.D., 1951, Columbia University; U. S. Census Bureau, Statistician, 
1940-41; U. S. Army War College, Statistician, 1942-44; Cohunbia Uni- 
versity, Department of Mathematics, Assistant, 1946-48 and Research 
Associate 1948-50; Wayne University, Assistant Professor of Mathe- 
matics, 1950-52; Columbia University, Department of Mathematical 
Statistics, Visiting Lecturer, 1952; Cornell University, fundamental re- 
search in mathematical statistics, 1952-54; Bell Telephone Laboratories, 
1954-. Dr. Sobel is engaged in fundamental research on life testing 
reliability problems with special application to transistors and is a con- 
sultant on many Laboratories projects. Member of Institute of Mathe- 
matical Statistics, American Statistical Association and Sigma Xi. 

Morris Tanenbaum, A.B., Johns Hopkins University, 1949; M.A., 
Princeton University, 1950; Ph.D. Princeton University, 1952; Bell 
Telephone Laboratories, 1952-, Dr. Tanenbaum has been concerned 
with the chemistry and semiconducting properties of intermetallic com- 
pounds. At present he is exploring the semiconducting properties of 
silicon and the feasibility of silicon semiconductor devices. Dr. Tanen- 
baum is a member of the American Chemical Society and American 
Physical Society. He is also a member of Phi Lambda LTpsilon, Phi Beta 
Kappa and Sigma Xi. 

Donald E. Thomas, B.S. in E.E., Pennsylvania State College, 1929; 
M.A., Columbia University, 1932; Bell Telephone Laboratories, 1929- 
1942, 1946-. His first assignment at the Laboratories was in submarine 
cable development. Just prior to World War II he became engaged in 
the development of sea and airborne radar and continued in this work I 
until he left for military duty in 1942. During World War II he was made ' 
a member of the Joint and Combined Chiefs of Staff Committees on 
Radio C-ountermeasures. Later he was a civilian memlior of the Depart-' 
ment of Defense's Research and Development Board Panel on Electronic 
Countermeasures. Upon rejoining the Laboratories in 1946, Mr. Thomas 
was active in the development and installation of the first deep sea re- 
peatered submarine telephone cable, hctwcen Key West and Havana,' 



COXTIUBUTOKS TO THIS ISSUE 247 

which went into service in 1950. Later he was engaged in the develop- 
ment of transistor devices and circuits for special applications. At the 
present time he is working on the evaluation and feasibility studies of 
new types of semiconductors devices. He is a senior member of the I.R.E. 
and a member of Tau Beta Pi and Phi Kappa Phi. 

Laurence R. Walker, B.Sc. and Ph.D., McGill University, 1935 
and 1939; LTniversity of California 1939-41; Radiation Laboratory, 
Massachusetts Institute of Technology, 1941-45; Bell Telephone La- 
boratories, 1945-. Dr. Walker has been primarily engaged in the develop- 
ment of microwave oscillators and amplifiers. At present he is a member 
of a physical research group concerned with the applied physics of solids. 
Fellow of the American Physical Society. 



IHE BELL SYSTEM 

Jechnical journal 

VOTED TO THE SC I E N T I FIC^^^ AND ENGINEERING 
PECTS OF ELECTRICAL COMMUNICATION 



LUME XXXV MARCH 1956 NUMBER 2 






An Experimental Remote Controlled Line Concentrator \.f^ y 

A^E. JOEL, JR. 249 

Transistor Circuits for Analog and Digital Systems 

F. H. BLECHER 295 

Electrolytic Shaping of Germanium and Silicon a. uhlir, jr. 333 

A Large Signal Theory of Traveling-Wave Amplifiers p. k. tibn 349 

A Detailed Analysis of Beam Formation with Electron Guns of the 
Pierce Type w. e. danielson, j. l. rosenfeld and j. a. saloom 375 

Theories for Toll Traffic Engineering in the U.S.A. r. i, Wilkinson 421 

Crosstalk on Open -Wire Lines 

W, C, BABCOCK, ESTHER RENTROP AND C. S. THAELER 515 



Bell System Technical Papers Not Published in This Journal 519 

Recent Bell System Monographs 527 

Contributors to This Issue 531 



COPYRIGHT 1956 AMERICAN TELEPHONE AND TELEGRAPH COMPANY 



THE BELL SYSTEM TECHNICAL JOURNAL 



ADVISORY BOARD 

F. K. K A P P E L, President Western Electric Company 

M. J. KELLY, President, Bell Telephone Laboratories 

E. J. McNEELY, Executive Vice President, American 
Telephone and Telegraph Company 

EDITORIAL COMMITTEE 

B. MCMILLAN, Chairman 

A. J. BUSCH H. R. HUNTLEY 

A. C. DICKIBSON F. R. LACK 

R. L. DIETZOLD J. R. PIERCE 

K. E. GOULD H. V. SCHMIDT 

E. I. GREEN C. ESCHOOLEY 

R. K. HON AM AN G. N. THAYER 

ED ITORI AL STAFF 

J. D. TEBO, Editor 

M. E. s T R I E B Y, Managing Editor 

R. L. SHEPHERD, Production Editor 



THE BELL SYSTEM TECHNICAL JOURNAL is published six times 
a year by the American Telephone and Telegraph Company, 195 Broadway, 
New York 7, N. Y. Qeo F. Craig, President; S. Whitney Landon, Secretary; 
John J. Scanlon, Treasurer. Subscriptions are accepted at $3.00 per year. 
Single copies are 75 cents each. The foreign postage is 65 cents per year or 11 
cents per copy. Printed in U. S. A. 



THE BELL SYSTEM 

TECHNICAL JOURNAL 

VOLUME XXXV MARCH 1956 number 2 

Copyright 1958, American Telephone and Telegraph Company 

An Experimental Remote Controlled 
Line Concentrator 

By. A. E. JOEL, JR. 

(Manuscript received June 30, 1955) 

Concentration, which is the process of connecting a number of telephone 
lines to a smaller number of switching paths, has always been a funda?nental 
function in switching systems. By performing this function remotely from 
the central office, a new balance between outside plant and switching costs 
may be obtained which shows promise of providing service more economi- 
cally in some situations. 

The broad concept of remote line concentrators is not new. However, its 
solution with the new devices and techniques now available has made the 
possibilities of decentralization of the means for switching telephone con- 
nections very promising. 

Three models of an experimental equipment have been designed and con- 
structed for service. The models have included equipment to enable the evalua- 
tion of new procedures required by the introduction of remote line concentra- 
tors into the telephone plant. The paper discusses the philosophy, devices, 
and techniques. 

CONTENTS 

1 . Introduction 250 

2. Objectives 251 

3. New Devices Emploj^ed 252 

4. New Techniques Emploved 254 

5. Switching Plan ". 257 

249 



250 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

6. Basic Circuits 261 

a. Diode Gates 261 

b. Transistor Bistable Circuit 262 

c. Transistor Pulse Amplifier 263 

d. Transistor Ring Counter 264 

e. Crosspoint Operating Circuit 266 

f . Crosspoint Relay Circuit 267 

g. Pulse Signalling Circuit 268 

h. Power Supply 269 

7. Concentrator Operation 270 

a.Line Scanning 270 

b. Line Selection 272 

c. Crosspoint Operation and Check 273 

8. Central Office Circuits 274 

a. Scanner Pulse Generator 279 

b. Originating Call Detection and Line Number Registration 280 

c. Line Selection 282 

d. Trunk Selection and Identification 284 

9. Field Trials 286 

10. Miscellaneous Features of Trial Equipment 287 

a. Traffic Recorder, b. Line Condition Tester 288 

c. Simulator, d. Service Observing 290 

e. Service Denial, f . Pulse Display Circuit 291 

1. INTRODUCTION 

The equipment which provides for the switching of telephone connec- 
tions has ahvays been located in what have been commonly called "cen- 
tral offices". These offices provide a means for the accumulation of all 
switching equipment required to handle the telephone needs of a com- 
munity or a section of the community. The telephone building in which 
one or more central offices are located is sometimes referred to as the 
"wire center" because, like the spokes of a wheel, the wires which serve 
local telephones radiate in all directions to the telephones of the 
community. 

A new development, made possible largely by the application of de- 
vices and techniques new to the telephone switching field, has recently 
been tried out in the telephone plant and promises to change much of . 
the present conception of "central" offices and "wire" centers. It is 
known as a "line concentrator" and provides a means for reducing the 
amount of outside plant cables, poles, etc., serving a telephone central 
office by dispersing the switching equipment in the outside plant. It is 
not a new concept to reduce outside plant by bringing the switching 
equipment closer to the telephone customer but the technical difficulties 
of maintaining complex switching equipment and the cost of controlling" 
such equipment at a distance have in the past been formidable obstacles 
to the development of line concentrators. With the invention of low 
power, small-sized, long-life devices such as transistors, gas tubes, and 
sealed relays, and their application to line concentrators, and with the 
development of new local switching systems with greater flcxibilit}', it 
has been possible to make the progress described herein. 



REMOTE CONTROLLED LINE CONCENTRATOR 251 

2. OBJECTIVES 

Within the telephone offices the first switching equipment through 

which dial lines originate calls concentrates the traffic to the remaining 

equipment which is engineered to handle the peak busy hour load with 

the appropriate grade of service.^ This concentration stage is different for 

different switching systems. In the step-by-step system^ it is the line 

' finder, and in the crossbar systems it is the primary line switch.^ Pro- 

1 posals for the application of remote line concentrators in the step-by- 

i step system date back over 50 years/ Continuing studies over the years 

have not indicated that any appreciable savings could be realized when 

such equipment is used within the local area served by a switching center. 

When telephone customers move from one location to another within 

a local service area, it is desirable to retain the same telephone numbers. 

The step-by-step switching system in general is a unilateral arrangement 

where each line has two appearances in the switching equipment, one 

for originating call concentration (the line finder) and one for selection 

of the line on terminating calls (the connector) . The connector fixes the 

line number and telephone numbers cannot be readily reassigned when 

moving these switching stages to out-of-office locations. 

Common-control systems^ have been designed with flexibility so that 
the line number assignments on the switching equipment are independ- 
ent of the telephone numbers. Furthermore, the first switching stage 
in the office is bilateral, handling both originating and terminating calls 
through the same facilities. The most recent common-control switching 
system in use in the Bell System, the No. 5 crossbar,^ has the further 
advantage of universal control circuitry for handling originating and 
terminating calls through the line switches. For these reasons, the No. 
5 crossbar system was chosen for the first attempt to employ new tech- 
niques of achieving an economical remote line concentrator. 

A number of assumptions were made in setting the design require- 
ments. Some of these are influenced by the characteristics of the No. 5 
crossbar system. These assumptions are as follows: 

1. No change in customer station apparatus. Standard dial telephones 
to be used with present impedance levels, transmission characteristics, 
dial pulsing, party identification, superimposed ac-dc ringing,^ and sig- 
naling and talking ranges. 

2. Individual and two-party (full or semi-selective ringing) stations 
to be served but not coin or PBX lines. 

3. Low cost could best be obtained by minimizing the per line 
equipment in the central office. AMA^ charging facilities could be used 
but to avoid per station equipment in the central office no message reg- 
ister operation would be provided. 



252 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

4. Each concentrator would serve up to 50 lines with the central office 
control circuits common to a number of concentrators. (Experimental 
equipment described herein was designed for 60 lines to provide addi- 
tional facilities for field trial purposes.) No extensive change would be 
made in central office equipment not associated with the line switches 
nor should concentrator design decrease call carrying capacities of exist- 
ing central office equipment. 

5. To provide data to evaluate service performance, automatic traffic 
recording facilities to be integrated with the design. 

6. Remote equipment designed for pole or wall mounting as an addi- 
tion to existing outside plant. Therefore, terminal distribution facilities 
would not be provided in the same cabinet. 

7. Power to be supplied from the central office to insure continuity 
of telephone service in the event of a local power failure. 

8. Concentrators to operate over existing types of exchange area fa- 
cilities without change and with no decrease in station to central office 
service range. 

9. Maintenance effort to be facilitated by plug-in unit design using 
the most reliable devices obtainable. 

3. NEW DEVICES EMPLOYED 



»! 



I 



Numerous products of research and development were available for 
this new approach. Only those chosen will be described. 

For the switching or "crosspoint" element itself, the sealed reed switch 
was chosen, primarily because of its imperviousness to dirt.* A short coil 
magnet with magnetic shield for increasing sensitivity of the reed 
switches were used to form a relay per crosspoint (see Fig. 1). 

A number of switching applications^ '^^ for crosspoint control using 
small gas diodes have been proposed by E. Bruce of our Switching Re- 
search Department. They are particularly advantageous when used in 
an "end marking" arrangement with reed relay crosspoints. Also, these 
diodes have long life and are low in cost. One gas diode is employed for 
operating each crosspoint (see Fig. 6). Its breakdown voltage is 125v ± 
lOv, A different tube is used in the concentrator for detecting marking 
potentials when termination occurs. Its breakdown potential is lOOv ± 
lOv. One of these tubes is used on each connection. 

Signaling between the remote concentrator and the central office con- 
trol circuits is performed on a sequential basis with pulses indicative of 
the various line conditions being transmitted at a 500 cycle rate. This 
frequency encounters relatively low attenuation on existing exchange 
area wire facilities and j^et is high enough to transmit and receive in- 
formation at a rate which will not decrease call carrjdng capacitj^ of the 



REMOTE CONTROLLED LINE CONCENTRATOR 



253 






Fig. 1 — Reed switch relay. 

central office equipment. To accomplish this signaling and to process the 
information economically transistors appear most promising. 

Germanium alloy junction transistors were chosen because of their 
; improved characteristics, reliability, low power requirements, and mar- 
gins, particularly when used to operate with relays.^^ Both N-P-N and 
P-N-P transistors are used. High temperature characteristics are par- 
ticularly important because of the ambient conditions which obtain on 
pole mounted equipment. As the trials of this equipment have progressed, 



254 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Table I— Transistor Characteristics 



Code No. 


Type and Filling 


Alpha 


Max. Ico at 28V 
and 65°C 


Emitter Zener 
Voltage at 20^=1 


M1868 
M1887 


p-n-p Oxygen 
n-p-n Vacuum 


0.9-1.0 
0.5- .75 


150 Ma 
100 Ma 


>735 
>735 



considerable progress has been made in improving transistors of thi.s 
type. Table I summarizes the characteristics of these transistors. 

For directing and analyzing the pulses, the control employs semicon- 
ductor diode gate circuits." The semiconductor diodes used in these 
circuits are of the silicon alloy junction type,^^ Except for a few diode.s 
operating in the gas tube circuits most diodes have a breakdown voltage 
requirement of 27v, a minimum forward current of 15 ma at 2v and a 
maximum reverse current at 22v of 2 X 10^^ amp. 

4. new techniques employed 

The concentrator represents the first field application in Bell System 
telephone switching systems which departs from current practices and 
techniques. These include: 




Fig. 2 — Transistor packages, (a) Diode unit, (b) Transistor counter, (c) 
Transistor amplifiers and bi-stable circuits, (d) Five trunk unit. 



REMOTE CONTROLLED LINE CONCENTRATOR 255 

1. High speed pulsing (500 pulses per second) of information between 
switching units. 

2. The use of plug-in packages employing printed wiring and encap- 
sulation. (Fig. 2 shows a representative group of these units.) 

3. Line scanning for supervision with a passive line circuit. In present 
systems each line is equipped with a relay circuit for detecting call orig- 
inations (service requests) and another relay (or switch magnet) for 
indicating the busy or idle condition of the line, as shown in Fig. 3(a). 
The line concentrator utilizes a circuit consisting of resistors and semi- 
conductor diodes in pulse gates to provide these same indications. This 
circuit is shown in Fig. 3(b). Its operation is described later. The pulses 
for each line appear at a different time with respect to one another. 
These pulses are said to represent "time slots." Thus a different line is 
examined each .002 second for a total cycle time (for 60 lines) of .120 
second. This process is known as "line scanning" and the portion of the 
circuit which produces these pulses is known as the scanner. Each of the 
circuits perform the same functions, viz., to indicate to the central office 
equipment when the customer originates a call and for terminating calls 
to indicate if the line is busy. 

4. The lines are divided for control and identification purposes into 
twelve groups of five lines each. Each group of five lines has a different 
pattern of access to the trunks which connect to the central office. The 
ten trunks to the central office are divided into two groups as shown in 
Fig. 4. One trunk group, called the random access group, is arranged in 
a random multiple fashion, so that each of these trunks is available to 
approximately one-half of the lines. The other group, consisting of two 
trunks, is available to all lines and is therefore called the full access 
group. The control circuitry is arranged to first select a trunk of the 
random access group which is idle and available to the particular line to 
which a connection is to be made. If all of the trunks of this random ac- 
cess group are busy to a line to which a connection is desired, an attempt 
is then made to select a trunk of the full access group. The preference 
order for selecting cross-points in the random access group is different 
for each line group, as shown in the table on Fig. 4. By this means, each 
trunk serves a number of lines on a different priority basis. Random ac- 
cess is used to reduce by 40 per cent the number of individual reed relay 
crosspoints which would otherwise be needed to maintain the quality 
of service desired, as indicated by a theory presented some years ago.^^ 

5. Built-in magnetic tape means for recording usage data and making 
call delay measurements. The gathering of this data is greatly facilitated 
by the line scanning technique. 



256 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



CROSSBAR CROSSPOINT 

OR 

SWITCH CONTACTS 



-^ 



TO LINE 



-^ 



TO OTHER 

CENTRAL OFFICE 

EQUIPMENT 



9 9 



r 



^ 



LR 






■^f- 



CO 



c 



HI 



"H 



1_ 



(a) 



■:l 



LINE BUSY 



SERVICE 
REQUEST 



I + 5V 



CROSSPOINT 



■^ 



TO LINE 




-^ 



TO 

CENTRAL 

OFFICE 



^4- 



-X 



-16V 



-16 VOLTS -NORMAL 
(RECEIVER ON HOOK) 

-3 VOLTS -AWAITING SERVICE 
(RECEIVER OFF HOOK) 

-16 VOLTS -CROSSPOINT CLOSED 
(RECEIVER OFF HOOK) 



S\. 



-¥^ 




-16 V 



^ 



LINE BUSY 



+ 15 VOLT 
TIME SLOT PULSE 
FROM SCANNER 



GATE 



SERVICE 
REQUEST 



Fig. 3 — (a) Relay line circuit, (b) Passive line circuit. 



REMOTE CONTROLLED LINE CONCENTRATOR 



257 



5. SWITCHING PLAN 

The plan for serving lines directly terminating in a No. 5 Crossbar office 
is shown in Fig. 5(a). Each line has access through a primary line switch 
to 10 line links. The line links couple the primary and secondary switches 
together so that each line has access to all of the 100 junctors to the trunk 
link switching stage. Each primary line switch group accommodates 
from 19 to 59 lines (one line terminal being reserved for no-test calls). 
A line link frame contains 10 groups of primary line switches.^* 
. The remote concentrator plan merely extends these line links as trunks 
to the remote location. However, an extra crossbar switching stage is 
introduced in the central office to connect the links to the secondary line 
switches with the concentrator trunks as shown in Fig. 5(b). Since each 
line does not have full access to the trunks, the path chosen by the marker 
to complete calls through the trunk link frame may then be independent 
of the selection of a concentrator trunk with access to the line. This 
arrangement minimizes call blocking, simplifies the selection of a matched 
path by the marker, and the additional crossbar switch hold magnet 
serves also as a supervisory relay to initiate the transmission of disconnect 
signals over the trunk. 

In addition to the 10 concentrator trunks used for talking paths, 2 
additional cable pairs are provided from each concentrator to the central 
office for signaling and power supply purposes. The use of these two pairs 
of control conductors is described in detail in Section 6g. 

The concentrator acts as a slave unit under complete control of the 
central office. The line busy and service request signals originate at the 



LINE 



60 LINES 

I 



o.-»-o 























5 


9 




























7 '^ 


/ ^ 








/ ■v 














/ 


p, ■^ 


i' 


\. 






^ 


V 


































































•y 






\ 


/ s 


f 


\ 


^ 






\ 


,• \ 




'^ 
























^ 


1 > 


f 


































































< > 











1 2 3 



5 6 



8 9 10 11 



Fig 4. — Concentrator trunk 
to line crosspoint pattern and 
preference order 



CONCENTRATOR 
TRUNKS 



9 


9 


9 


9 


9 


9 


9 


9 


9 


9 


9 


9 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


8 


6 





5 


4 


7 


5 


3 


1 


4 


7 


2 


1 


7 


3 


1 


5 


2 





6 


4 


6 


5 





3 


1 


7 


2 


3 


6 


2 


4 








6 


3 


5 





4 


6 


2 


3 


7 


1 


6 


2 


4 


1 


7 



1 



5 6 



8 



VERTICAL GROUPS OF FIVE LINES EACH " 

ORDER OF PREFERENCE 



GAS TUBE REED -RELAY 
CROSS POINTS 



10 11 



258 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



49 
LINES 



TEN GROUPS 
OF LINES 



49 

LINES I 



CENTRAL OFFICE 
LINE LINK FRAME 



LINE SW 







1 


CON- 
NECTOR 


1 
1 


1 


— 


-- 1 

1 




CON- 
NECTOR 



I 

TO MARKER 




TRUNK 
LINK 
FRAME 



Fig. 5(a) — No. 5 crossbar system subscriber lines connected to line link frame. 



60 

LINES 



60 
LINES 



60 
<? 



10 



CONTROL 



CE^4TRAL OFFICE 



TEN CONCENTRATOR 
, TRUNKS 



I 

JL 



TWO CONTROL PAIRS 



60 





10 



TEN POLE- 
MOUNTED 
^CONCENTRATOR 
UNITS AT 
DIFFERENT 
LOCATIONS 



CONTROL 



TEN CONCENTRATOR 
TRUNKS 



TWO CONTROL PAIRS 



CONCENTRATOR 

TRUNK SW JUNCTOR 

SW 



10 
9 C> 




TRUNK 

LINK 
FRAME 



TO MARKER 



CONCENTRATOR LINE LINK 
FRAME 



Fig. 5(b) — No. 5 crossbar system subscriber lines connected to remote line 
concentrators. 



REMOTE CONTROLLED LINE CONCENTRATOR 



259 




Fig. 6 — Line unit construction. 



concentrator only in response to a pulse in the associated time slot or 
when a crosspoint operates (a line busy pulse is generated under this 
condition as a crosspoint closure check). The control circuit in the 
central office is designed to serve 10 remote line concentrators connected 
to a single line link frame. In this way the marker deals with a concen- 
trator line link frame as it would with a regular line link frame and the 
marker modifications are minimized. 

The traffic loading of the concentrator is accomplished by fixing the 



260 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




Fig. 7(a) — Line unit. 



number of trunks at 10 and equipping or reassigning lines as needed to 
obtain the trunk loading for the desired grade of service. The six cross- 
points, the passive line circuit and scanner gates individual to each line 
are packaged in one plug-in unit to facilitate administration. The cross- 
points are placed on a printed wiring board together with a comb of plug 
contacts as shown in Fig. 6. The entire unit is then dipped in rubber and 
encapsulated in epoxy resin, as shown in Fig. 7(a). 

This portion of the unit is extremely reliable and therefore it may be 
considered as expendable, should a rare case of trouble occur. The passive 
line circuit and scanner gate circuit elements are mounted on a smaller 
second printed wiring plate (known as the "line scanner" plate, see Fig. 
7(b) which fits into a recess in the top of the encapsulated line unit. Cir- 










Fig. 7(b) — Scanner plate of the line unit shown in Fig. 7 (a). 



REMOTE CONTROLLED LINE CONCENTRATOR 



261 



cuit connection between printed wiring plates is through pins which ap- 
pear in the recess and to which the smaller plate is soldered. 



6. BASIC CIRCUITS 



a. Diode Gates 



All high speed signaling is on a pulse basis. Each pulse is positive and 
approximately 15 volts in amplitude. There is one basic type of diode 
gate circuit used in this equipment. By using the two resistors, one con- 
denser and one silicon alloy junction diode in the gate configuration 
shown in Fig. 8, the equivalents of opened or closed contacts in relay 
circuits are obtained. These configurations are known respectively as 
enabling and inhibiting gates and are shown with their relay equivalents 
ill Figs. 8(a) and 8(b). 

In the enabling gate the diode is normally back biased by more than 
the pulse voltage. Therefore pulses are not transmitted. To enable or 



INPUT 



ENABLING GATE CIRCUIT 
CI 



OUTPUT 




(a) 



ENABLING GATE SYMBOL 



INPUT 



OUTPUT 



CONTROL 

EQUIVALENT RELAY CIRCUIT 
OUTP UT 
INPUT f 



CONTROL 



CHli^ 



INPUT 



INHIBITING GATE CIRCUIT 
Cl 



OUTPUT 



INHIBITING GATE SYMBOL 



INPUT 




OUTPUT 



CONTROL 

EQUIVALENT RELAY CIRCUIT 
OUTPUT 



DhUHl 



Fig. 8 — Gates and relay equivalents. 



262 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

open the gate the back bias is reduced to a small reverse voltage which is 
more than overcome by the signal pulse amplitude of the pulse. The 
pulse thus forward biases the diode and is transmitted to the output. 

The inhibiting gate has its diode normally in the conducting state so 
that a pulse is readily transmitted from input to output. When the bias 
is changed the diode is heavily back biased so that the pulse amplitude 
is insufficient to overcome this bias. 

The elements of 12 gates are mounted on a single printed wiring board 
w4th plug-in terminals and a metal enclosure as shown in Fig. 2(a). All 
elements are mounted in one side of the board so that the opposite side 
may be solder dipped. After soldering the entire unit (except the plug) 
is dipped in a silicone varnish for moisture protection. 

b. Transistor Bistable Circuit 

Transistors are inherently well adapted to switching circuits using but 
two states, on (saturated) or off.^^ In these circuits with a current gain 
greater than unity a negative resistance collector characteristic can be 
obtained which will enable the transistor to remain locked in its conduct- 
ing state (high collector current flowing) until turned off (no collector 
current) by an unlocking pulse. At the time the concentrator develop- 
ment started only point contact transistors were available in quantity. 
Point contact transistors have inherently high current gains (>1) but 
the collector current flowing when in the normal or unlocked condition 
(Ico) was so great that at high ambient temperatures a relay once op- 
erated in the collector circuit would not release. 

Junction transistors are capable of a much greater ratio of on to off 
current in the collector circuit. Furthermore their characteristics are 
amenable to theoretical design consideration.^^ However, the alpha of a 
simple junction transitor is less than unity. To utilize them as one would | 
a point contact transitor in a negative resistance switching circuit, a 
combination of n-p-n and p-n-p junction transistors may be employed, i 
see Fig. 9(b). Two transistors combined in this manner constitute a ' 
"hooked junction conjugate pairs." This form of bi-stable circuit was j 
used because it requires fewer components and uses less power than an 
Eccles-Jordan bistable circuit arrangement. It has the disadvantage of a 
single output but this was not found to be a shortcoming in the design 
of circuits employing pulse gates of the type described. In what follows 
the electrodes of the transistor will be considered as their equivalents 
shown in Fig. 9(b). 

The basic bi-stable circuit employed is shown in Fig. 10. The set 



REMOTE CONTROLLED LINE CONCENTRATOR 



263 



EMITTER 



COLLECTOR 




EMITTER 




n-p-n 



COLLECTOR 



BASE 

fa) 

POINT CONTACT 
TRANSISTOR 

Ic 



BASE 



(b) 



CONJUGATE PAIR 

ALLOY JUNCTION 

TRANSISTORS 



C _ 



0C> 1 



Fig. 9 — Point contact versus hooked conjugate pair. 

pulse is fed into the emitter (of the pair) causing the emitter diode to 
conduct. The base potential is increased thus increasing the current 
flowing in the collector circuit. When the input pulse is turned off the 
base is left at about —2 volts thus maintaining the emitter diode con- 
( lucting and continuing the increased current flow in the collector circuit. 
The diode in the collector circuit prevents the collector from going 
positive and thereby limits the current in the collector circuit. To reset, 
a positive pulse is fed into the base through a pulse gate. The driving of 
tlie base positive returns the transistor pair to the off condition. 

c. Transistor Pulse Amplifier 

This circuit (Fig. 11) is formed by making a bi-stable self resetting 
circuit. It is used to produce a pulse of fixed duration in response to a 



TRANSISTORS 

p-n-p 



SET 



RESET 




I-5V 



-I6V 



F/F 



Fig. 10 — Transistor bi-stable circuit. 



264 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



pulse of variable width (within limits) on the input. Normally the emitter 
is held slightly negative with respect to the base. The potential difference 
determines the sensitivity of the amplifier. When a positive input pulse 
is received, the emitter diode conducts causing an increase in collector 
current. The change in bias of the diode in the emitter circuit permits 
it to conduct and charge the condenser. With the removal of the input 
pulse the discharge of the condenser holds the transistor pair on. The 
time constant of the circuit determines the on time. When the emitter 
potential falls below the base potential, the transistor pair is turned off. 

The amplifiers and bi-stable circuits or flip-flops, >as they are called 
more frequently, are mounted together in plug-in packages. Each pack- 
age contains 8 basic circuits divided 7-1, 6-2, or 2-6, between amplifiers 
and fhp-flops. Fig. 2(c) shows one of these packages. They are smaller 
than the gate or line unit packages, having only 28 terminals instead of 
42. 

The transistors for the field trial model w^ere plugged into small hear- 
ing aid sockets mounted on the printed wiring boards. For a production 
model it w^ould be expected that the transistors w^ould be soldered in. 

d. Transistor Ring Counter 

By combining bi-stable transistor and diode pulse gate circuits to- 
gether in the manner shown in Fig. 12 a ring counter may be made, with 



INPUT 



p-n-p 



^w 



^vW-" 



I 

+ 5V 



OUTPUT 




-16 V 



INPUT 



OUTPUT 



Fig. 11 — Transistor pulse amplifier. 



REMOTE CONTROLLED LINE CONCENTRATOR 



265 



COUNT 
INPUT 

lie 



STAGE NUMBER 
3 




NOTE: 

LEADS A-0 TO A-4 
ARE OUTPUT LEADS 
OF RESPECTIVE STAGES 



1 I I \ r 

s 's 's 's 's 



Fig. 12 — Ring counter schematic. 



a bi-stable circuit per stage. The enabling gate for a stage is controlled 
by the preceding stage allowing it to be set by an input advance pulse. 
The output signal from a stage is fed back to the preceding stage to turn 
it off. An additional diode is connected to the base of each stage for re- 
setting when returning the counter to a fixed reference stage. 

A basic package of 5 ring counter stages is made up in the same frame- 
work and with the same size plug as the flip-flop and amplifier packages, 
see Fig. 2(b). A four stage ring counter is also used and is the same 
package with the components for one stage omitted. The input and out- 
put terminals of all stages are available on the plug terminals so that 
the stages may be connected in any combination and form rings of more 
than 5 stages. The reset lead is connected to all but the one stage which 
is considered the first or normal stage. 

Other transistor circuits such as binary counters and square wave 
generators are used in small quantity in the central office equipment. 
They will not be described. 



266 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



CONCENTRATOR 

LINE BUSY 



CENTRAL OFFICE 



TO ALL CROSSPOINTS 
/ SERVED BY TRUNK 



+ 130 V 




VG 
VF 



I 



L..1.. 



/ 



TO ALL CROSSPOINTS 
FOR SAME LINE 



SELECTION 

FROM 

" CENTRAL 

OFFICE 



i-65V 




I + 100V 



Fig. 13 — Crosspoint operating circuit. 



e. Crosspoint Operating Circuit 

The crosspoint consists of a reed relay with 4 reed switches and a gas 
diode (Fig. 1). The selection of a crosspoint is accomplished by marking 
with a negative potential ( — 65 volts) all crosspoints associated with a 
line, and marking with a positive potential ( + 100 volts) all crosspoints 
associated with a trunk (Fig. 13). The line is marked through a relay 
circuit set by signals sent over the control pair from the central office. 
The trunk is marked b}^ a simplex circuit connected through the break 
contacts of the hold magnet of the crossbar switch associated with the 
trunk in the central office. Only one crosspoint at a time is exposed to 
165 volts which is necessary and sufficient to break down the gas diode 
to its conducting state. The reed relay operates in series with the gas 
diode. A contact on the relay shunts out the gas diode. When the marking- 
potentials are removed the relay remains energized in a local 30-voll 
circuit at the concentrator. The holding current is approximately 2.5 ma. 

This circuit is designed so that ringing signals in the presence or ab- 
sence of lino marks will not falsely fire a crosspoint diode. Furthonnoi'o, 



REMOTE CONTROLLED LINE CONCENTRATOR 



267 



a line or trunk mark alone should not be able to fire a crosspoint diode 
on a busy line or trunk. 

When the crosspoint operates, a gate which has been inhibiting pulses 
is forward biased by the —65 volt signal through the crosspoint relay 
winding. The pulse which initiates the mark operations at the concentra- 
tor then passes through the gate to return a line busy signal to the central 
office over this control pairs which is interpreted as a crosspoint closure 
check signal. 

f. Crosspoint Release Circuit 

The hold magnet of the central office crossbar switch operates, remov- 
ing the +100- volt operate mark signal after the crosspoint check signal 
is received. A slow release relay per trunk is operated directly by the 
hold magnet. When the central office connection in the No. 5 crossbar 
system releases, the hold magnet is released. As shown in Fig. 14, with the 
hold magnet released and the slow release relay still operated, a — 130- 
volt signal is applied in a simplex circuit to the trunk to break down a 
gas tube provided in the trunk circuit at the concentrator. This tube in 



CONCENTRATOR 



CENTRAL OFFICE 



TO ALL CROSSPOINTS 
SERVED BY SAME TRUNK 




130V I 



Fig. 14 — Crosspoint release circuit. 



268 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



breaking down shunts the local holding circuit of the crosspoint causing 
it to release. The — 130-volt disconnect signal is applied during the 
release time of the slow release relay which is long enough to insure the 
release of the crosspoint relay at the concentrator. 

The release circuit is individual to the trunk and independent of the 
signal sent over the control pairs. 

g. Pulse Signalling Circuits 

To control the concentrator four distinct pulse signals are transmitted 
from the central office. Two of these at times must be transmitted 
simultaneously, bvit these and the other two are transmitted mutually 
exclusively. In addition, service request and line busy signals are trans- 
mitted from the concentrator to the central office. The two way trans- 
mission of information is accomplished on each pair by sending signals in 
each direction at different times and inhibiting the receipt of signals 
when others are being transmitted. 

To transmit four signals over two such pairs, both positive and nega- 




CONTROL 
PAIR NO. 1 



VF 



M 



LB 




D 



SR 



-16V 




VG 



CONTROL 
PAIR NO. 2 





16 V 



M 



CONCENTRATOR 
AMPLIFIERS 



I 



CENTRAL OFFICE 

AMPLIFIERS 

PER CONCENTRATOR 



Fig. 15. — Signal transmission circuit. 



REMOTE CONTROLLED LINE CONCENTRATOR 269 

tive pulses are employed. Diodes are placed in the legs of a center tapped 
transformer, as shown in Fig. 15, to select the polarity of the trans- 
mitted pulses. At the receiving end the desired polarity is detected by 
taking the signal as a positive pulse from a properly poled winding of a 
transformer. The amplifier, as described in Section 6c responds only to 
positive pulses. If pulses of the same polarity are transmitted in the 
other direction over the same pair, as for control pair No. 1, the outputs 
of the receiving amplifier for the same polarity pulse are inhibited 
whenever a pulse is transmitted. 

As shown in Fig. 15, the service request and line busy signals are 
transmitted from the concentrator to the central office over one pair of 
conductors as positive and negative pulses respectivel3^ The trans- 
mission of these pulses gates the outputs of two of the receiving ampli- 
fiers at the concentrator to permit the receipt of the polarized signals 
from the central office. This prevents the pulses from being used at the 
sending end. A similar gating arrangement is used with respect to the 
signals when sent over this control pair from the central office. The pulses 
designated VG or RS never occur when a pulse designated SR or LB 
is sent in the opposite direction. The transmission of the VF pulse over 
control pair No. 2 is processed by the concentrator circuit and becomes 
the SR or LB pulses. Li section 7 the purpose of these pulses is described. 

The signaling range objective is 1,200 ohms over regular exchange 
area cable including loaded facilities from sfation to central office. 

h. Power Supply 

Alternating current is supplied to the concentrator from a continuous 
service bus in the central office. The power supply path is a phantom 
circuit on the two control pairs as shown in Fig. 16. The power trans- 
former has four secondary windings used for deriving from bridge 
rectifiers four basic dc voltages. These voltages and their uses are as 
fofiows: —16 volts (regulated) for transistor collector circuits and gate 
biases, -|-5 volts (regulated) for transistor base biases, -|-30 volts (regu- 

, lated) for crosspoints holding circuits and — 65 volts for the marking and 
operating of the line crosspoints. For this latter function a reference to 
the central office applied -flOO volt trunk mark is necessary. The refer- 
ence ground for the concentrator is derived from ground applied to a 
simplex circuit on the power supply phantom circuit. Series transistors 
and shunt silicon diodes with fixed reference breakdown voltages are 

I used to regulate dc voltages. 



270 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Total power consumption of the concentrator is between 5 and 8 watts 
depending upon the number of connections being held. 

7. CONCENTRATOR OPERATION 

a. Line Scanning 

The sixty lines are divided into 12 groups of 5 lines each. These group- 
ings are designated VG and VF respectively corresponding to the 
vertical group and file designations used in the No. 5 crossbar system. 
Each concentrator corresponds to a horizontal group in that system. 

To scan the lines two transistor ring counters, one of 12 stages and 
one of 5 stages, are employed as shown in Fig. 17. These counters are 
driven from pulses supplied from the central office control circuits and 
only one stage in each is on at any one time. The steps and combinations 
of these counters correspond to the group and file designation of a par- 
ticular line. Each 0.002 second the five stage counter (VF) takes a 
step and between the fifth and sixth pulse the r2-stage counter (VG) 
is stepped. Thus the 5-stage counter receives 60 pulses or re-cycles 12 
times in 120 milliseconds while the 12-stage counter cycles but once. 

Each line is provided with a scanner gate. The collector output of each 
each stage of the VG counter biases this gate to enable pulses which 
are generated by the collector circuit of the 5-stage counter to pass on 



-65V 



+ 30V 



+ SV 



-I6V 




115 V 
AC 



MOTOR 
GENERATOR 




TO 

COMMERCIAL 

AC 



REGULATORS 



Fig. 16 — Power supply transmission circuit. 



REMOTE CONTROLLED LINE CONCENTRATOR 



271 



to the gate of the passive line circuit, Fig. 3(b). If the line is idle the 
pulses are inhibited. If the receiver is off-hook requesting service (no 
(•rosspoint closed) then the gate is enabled, the pulse passes to the service 
request amplifier and back to the central office in the same time slot 
as the pulse which stepped the VF counter. If the line has a receiver 
off-hook and is connected to a trunk the pulse passes through a contact 
of the crosspoint relay to the line busy amplifier and then to the central 
office in the same time slot. 

At the end of each complete cycle a reset pulse is sent from the central 
office. This pulse instead of the VG pulse places the 12-stage counter in 
its first position. It also repulses the 5 stage VF counter to its fifth stage 
so that the next VF pulse will turn on its first stage to start the next 

j cycle. The reset pulse insures that, in event of a lost pulse or defect in 
a counter stage, the concentrator will attempt to give continuous ser- 
\'ice without dependence on maintaining synchronism with the central 

I office scanner pulse generator. Fig. 18(a) shows the normal sequence of 

I line scanning pulses. 

, When a service request pulse is generated, the central office circuits 



t] 




04 



r 



VF 5- STAGE 
COUNTER 



03 



TO 10 

INTERMEDIATE 

GATES EACH 

V 




02 



I 




01 



00 



TO 5 GATES EACH 
I 



1 23456 789 10 
I I I I I I I I I I 

I I I I I I I I I I 



VG 12-STAGE COUNTER 




59\ 



58 



57 



56 



55 



GATE PER LINE 

■ FEEDS PASSIVE 

LINE CIRCUITS 



/ 



VG 



RES ET 
VF 



FROM 

CENTRAL 

OFFICE 



Fig. 17 — Diode matrix for scanning lines. 



272 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

common to 10 concentrators interrupt the further transmission of the 
vertical group pulse so that the line scanning is confined to the 5 lines 
in the vertical group in which the call originated. In this way the cen- 
tral office will receive a service request pulse at least every 0.010 sec as 
a check that the call has not been abandoned while awaiting service. 
Fig. 18(b) shows the detection of a call origination and the several 
short scan cycles for abandoned call detection. 

b. Line Selection 

When the central office is ready to establish a connection at the con- 
centrator a reset pulse is sent to return the counters to normal. In gen- 
eral, the vertical group and vertical file pulses are sent simultaneously 
to reduce holding time of the central office equipment and to minimize 
marker delays caused by this operation. For this reason the VG and VF 
pulses are each transmitted over different control pairs from the central 
office. The same polarity is used. 

On originating calls it is desirable to make one last check that the 
call has not been abandoned, while on terminating calls it is necessary 

L* 120MS >| 

■M k-2MS I 

PULSE ' 

012340123401234 0123401234 



VF - 

VG 
LB 
RS 



J__l I I I I I I I I I I I I I I 1 I I I I 1 I L 

1^ \1 ^ 



(a) REGULAR LINE SCANNING 



VF 



123401 23401 23401 2340123401 
_l_l 1 I I I I I I I I I I I I I I I I I L_l I I I I 



,5 ,6 

VG 1 1 — 



LB- 

RS- 

SR- 

M- 



(b) CALL ORIGINATION SERVICE REQUEST FROM LINE 6/3 

12 3 1 
I I I I I 



VG- 

LB 

RS — 1>- 

SR h 



1° 1' 1^ |3 1^ 1^ 1^ 



jLi. 



M - 



H^-l^-- 



"7 

RESULTS FROM CONC CONTROL RECEIVED 'OPERATE ""CROSSPOINT 'NORMAL SCANNING 

CKT AT CENTRAL OFFICE ONLY IF CROSSPOINT CLOSURE IS RESUMED 

RECEIVING FROM MKR VG , LINE 6/3 HAS INDICATION 

VF, HG INFORMATION BECOME BUSY 



(C) LINE SELECTION FOR LINE 6/3 



Fig. 18 — Pulse sequences, (a) Regular, (b) Call origination, (c) Line selection. 



REMOTE CONTROLLED LINE CONCENTRATOR 273 

to determine if the line is busy or idle. These conditions are determined 
in the same manner as described for line scanning since a service re- 
quest condition would still prevail on the line if the call was not aban- 
doned. If the line was busy, a line busy condition would be detected. 
However to detect these conditions a VF pulse must be the last pulse 
transmitted since the stepping of the VF counter generates the pulse 
which is transmitted through an enabled line selection and passive 
line circuit gates. Fig. 18(c) shows a typical line selection where the num- 
ber of VF pulses is equal to or less than the number of VG pulses. In 
all other cases there is no conflict and the sending of the last VF pulse 
need not be delayed. On terminating calls, the line busy indication is 
returned to the central office within 0.002 sec after the selection is com- 
plete. During selections the central office circuits are gated to ignore 
any extraneous service request or line busy pulses produced as a result 
of steps of the VF counter prior to its last step. 

c. Crosspoint Operation and Check 

Associated with each concentrator transistor counter stage is a reed 
relay. These relays are connected to the transistor collector circuits 
through diodes of the counter stages when relay M operates. The con- 
tacts of these reed relays are arranged in a selection circuit as shown 
in Fig. 19 and apply the —65 volt mark potential to the crosspoint 
relays of the selected line. 

After a selection is made as described above a "mark" pulse is sent 
from the central office. This pulse is transmitted as a pulse of a different 
polarity over the same control pair as the VF pulses. The received 
pulse after amplification actuates a transistor bistable circuit w^hich has 
the M reed relay permanently connected in its collector circuit. The 
bi-stable circuit holds the M relay operated during the crosspoint opera- 
tion to maintain one VF and one VG relay operated, thereby applying 
— 65 volts to mark and operate one of the 6 crosspoint relays of the 
selected line as described in section 6e, and shown on Fig. 13. 

The operation and locking of the crosspoint relay with the marking 
potentials still applied enables a pulse gate associated with the holding 
circuit of the crosspoint relays in each trunk circuit. The mark pulses 
are sent out continuously. This does not affect the bi-stable transistor 
circuit once it has triggered but the mark pulse is transmitted through 
the enabled crosspoint closure check gate shown in Fig. 20 and back 
to the central office as a line busy signal. 

With the receipt of the crosspoint closure check signal the sending 



274 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

of the mark pulses is stopped and a reset pulse is sent to the concentra- 
tor to return the mark bi-stable circuit, counters and all operated selec- 
tor relays to normal. The concentrator remains in this condition until 
it is resynchronized with the regular line scanning cycle. 

A complete functional schematic of the concentrator integrating the 
circuits described above is shown in Fig. 21. Fig. 22(a) and (b) show an 
experimental concentrator built for field tests. 

8. CENTRAL OFFICE CIRCUITS 

The central office circuits for controling one or more concentrators 
are composed of wire spring relays as well as transistors, diode and reed 



VG 



RS 



VF 



M 



-20V 



-20 V 
o- 



VF-5 STAGE 
COUNTER 






r 



-65V 



n 



04 



03 



02 



00 



6 RELAY I W-, wo w<- ,^, -o p 

PACKAGE J '-|„p_„p_^_p-_-p-^Ui 



TO CONTACTS OF 4 
INTERMEDIATE RELAYS 



6 RELAY 
PACKAGE 



TO CONTACTS OF 4 . 
INTERMEDIATE RELAYS 



n 




L 



59 



n 



58 



i^ 



34 33 32 31 30 , 



29 28 27 26 25 I 5^ 



57 56 55 



TO 4 

INTERMEDIATE 

RELAYS 



' 

-/ *■ 



I 



I I 2 



b^'" 



/ 7_ 
8_ 
9 

10 






cr 

LU 

h- 

z 

o 
o 



< 

I- 

OJ 

I 

(J 
> 



I 



TO 4 
INTERMEDIATE il 
RELAYS 



P^ig. 19 — Line selection and marking. 



I 



REMOTE CONTROLLED LINE CONCENTRATOR 



275 



relay packages similar to those used in the concentrator. The reed 
relays are energized by transistor bi-stable circuits in the same manner 
as described in Section 7c. The reed relay contacts in turn operate wire 
spring relays or send the dc signals directly to the regular No. 5 crossbar 
marker and line link marker connector circuits. 

Fig. 23 shows a block diagram of the central office circuits. A small 
amount of circuitry is provided for each concentrator. It consists of the 
following: 

1. The trunk connecting crossbar switch and associated slow relays 
for disconnect control. 

2. The concentrator control triuik circuits and associated pulse ampli- 
fiers. 

3. An originating call detector to identify which concentrator among 
the ten served by the frame is calling. 

4. A multicontact relay to connect the circuits individual to each 
concentrator with the common control circuits associated with the line 
link frame and markers. 

The circuits associated with more than one concentrator are blocked 
out in the lower portion of Fig. 23. Much of this circuitry is similar to 
the relay circuits now provided on regular line link frames in the No. 5 
crossbar system.^ Only those portions of these blocks which employ the 
new techniques will be covered in more detail. These portions consist 
of the following: 

1. The scanner pulse generator. 

2. The originating line number register. 



T 




TO ALL TRUNK LINES 
+ 30V 



A/vV 



U 



j^Wv- 




-65V 



T 

I 
I 
I 
I 
I 

CONCENTRATOR 

TRUNK 
I 
I 
I 
I 
I 

i 



Fig. 20 — Crosspoint closure check. 



aoidzio nvbiNBo oi 
iinoaio ONnvNois via 




276 




Fig. 22(a) — Complete line concentrator unit. 

r 5 -STAGE COUNTER 

12 -STAGE COUNTER 




-fO TRUNK CiftCUITS 

AMPLIFIERS 
RECTIFIERS 



Fig. 22(b) — Identification of units within the line concentrator. 

277 



a. 

Q-tU 
O 

z 
o 
o 

1 



2 



o< 

z 

o 



CO 

UJ O 

<:^ 

o: u 
tu z 

D. o 

(J 



UJ£t Q. 

10 



I 



^-; 



C-: 



n 



W 



liJO 

IOq 
Q 



I 
I 

I 
J 



o 



o 

LJJ 



o 
u 

a. 
o 

^ 
cc 

I- 
z 

UJ 

o 

z 
o 
u 



Z 111 
Di- 
ce I- 
l-< 



>^av/^l 



Asng 



i3S3a 



9A 



dA 



VAVVV 



tr 

zy= 
Di- 

CEZ 

I-LIJ 



Nl 



9H 



lAIS 



"11 



Z 

CEUJ 

I 1 

UJ 
CO 



S3 



(O 



u 



90 



ui- 

UJ z 
-lO 

ujo 
(/I 



Hj 



UJ(- 

ZU 

-iUJ 



9A 



HH 



dA 



d: p 

uJuJt; 

z -id: 

< DUJ 

en "J 



; < UJ 



CC 
O 



O UJ 

zmCE 

^UjS 

cc 2 cc 
On 



IdA 



cc 
o 

I- 

O 
UJ 

z 
z 

O 
O 



< 

CC 
U. 

z 
_J 



UJ 




^^ 


1- 
O 

u 
o 

_l 


UJUJ 


^tr 


oruj 


<u. 


^Si 


Q. 





tn 

IT 
UJ 

:£. 

.CC 

< 

2 

o 

I- 



3A 



IS 



i9H 



F-^: 



I I 



a0J.VdlN33N00 Oi 



§(0 

u 



cc 
o 

I- 

?i 

UJ O 

zO 
-Jq: 

CC 

< 

2 



278 



REMOTE CONTROLLED LINE CONCENTRATOR 



279 



3. The line selection circuit. 

4. The trunk identifier and selection relay circuits. 

(For an understanding of how these frame circuits work through the line 
(link marker connector and markers in the No. 5 system, the reader 
should consult the references.) 

The common central office circuits will be described first. 



a. Scanner Pulse Generator 

The scanner pulse generator, shown in Fig. 24, produces continuously 
the combination of VG, VF and RS or reset pulses, described in connec- 
tion with Fig. 18(a), required to drive the scanners for a number of 
concentrators. The primary pulse source is a 1,000-cycle transistor 
oscillator. This oscillator drives a transistor bi-stable circuit arranged 
as a binary counter such that on each cycle of the oscillator output it 
alternately assumes one of its states. Pulses produced by one state drive 
a 5-stage counter. Pulses produced by the other state through gates 
drive a 12-stage counter. 

The pulses which drive the 5-stage counter are the same pulses which 
are used for the VF pulses to drive scanners. Each time the first stage 
of the 5-stage counter is on, a gate is opened to allow a pulse to drive 
the 12-stage counter. The pulses which drive the 12-stage counter are 
also the pulses used as the VG pulses for driving the scanners. They 
are out of phase with the VF pulses. 

When the last stage of the 12-stage counter is on, the gate which 



r VFC 




Fig. 24 — Scanner pulse generator. 



280 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195(5 



transmits pulses to the 12-stage counter is closed and another gate is 
opened which produces the reset pulse. The reset pulse is thereby trans- 
mitted to the scanners in place of the first vertical group pulse. At the 
same time the 5 and 12-stage counters in the scanner pulse generator 
are reset to enable the starting of a new cycle. 

In the central office control circuits, out of phase pulses on lead TP 
similar to those which drive the VG counters at the concentrator are 
used for various gating operations. 

b. The Originating Call Detection and Line Number Registration 

The originating call detector (Fig. 25) and the originating line num- 
ber register (Fig. 26) together receive the information from the line 
concentrator used to identify the number of the line making a service i 
request. The receipt of the service request pulse from a concentrator i 
in a particular time slot will set a transistor bi-stable circuit HGT of { 
Fig. 25 associated with that concentrator if no other originating call is 
being served by the frame circuits at* this time. 

The originating line number register consists of a 5 and 12-stage 
counter. These counters are normally driven through gates in syn- 
chronism with the scanning counters at concentrators with pulses sup- 
plied from the scanner pulse generator. When a service request pulse 
is received from any of the concentrators served by a line link frame, a 
pulse is sent to the originating line number register which operates a 
bi-stable circuit over a lead RH in Fig. 26. This bi-stable circuit then 
closes the gates through which the 5- and 12-stage counters are being 
driven, and also closes a gate which prevents them from being reset. 



TO TRAFFIC I 
RECORDER I 




TO ORIGINATING 
CALL REGISTER 



I TO CONCENTRATOR 
I CONTROL TRUNK 



Fig. 25 — Originating call detector. 



EEMOTE CONTROLLED LINE CONCENTRATOR 



281 



In this way, the number of the line which originated a service request is 
locked into these counters until the bi-stable circuit is restored to nor- 
mal. 

The HGT bi-stable circuit of Fig. 25 indicates which particular con- 
centrator has originated a service request. A relay in the collector cir- 
cuit has contacts which pass this information on to the other central 
office control circuits to indicate the number of the concentrator on the 
frame which is requesting service. This is the same as a horizontal group 
on a regular line link frame and hence the horizontal group designation 
is used to identify a concentrator. 

With the operation of this relay, relays associated with the counters 
of the originating line number register are operated. These relays indicate 
to the other central office circuits the vertical file and vertical group 
identification of the calling line. Contacts on the vertical group relays 
are used to set a bi-stable circuit associated with lead RL of Fig. 25 each 
time the scanner pulse generator generates a pulse corresponding to the 
vertical file of the calling line number registered. 

The operation of the HGT bi-stable circuit inhibits in the concentra- 
tor control trunk circuit (Fig. 27) the transmission of further VG and 

SRS 



FROM 
CONCENTRATOR 

CONTROL 
TRUNK CIRCUIT 



RB 
RH 



RH 




FROM 
SCANNER 

PULSE 
GENERATOR 



VF 



VFO-4 
VG 



RS 



1 



*" 5-STAGE COUNTER ^ 



12-STAGE COUNTER 



^ ^ 



Fig. 26 — Originating line number register. 



282 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

reset pulses to the concentrator so that, as described in Section 7a, 
only the VF counter continues to step once each 0.010 sec. So long as 
the line continues to request service this service request pulse is gated 
to reset the RL bi-stable circuit within the same time slot that it was 
set. If, however, a request for service is abandoned the RL bi-stable cir- 
cuit of Fig. 26 will remain on and permit a TP pulse from the scanner 
pulse generator to reset the HGT bi-stable circuit which initiated the 
service request action. 

Whenever the RH bi-stable circuit of Fig. 26 is energized it closes a 
gate over lead SRS for each concentrator to prevent any further service 
request pulses from being recognized until the originating call which 
has been registered is served. The resetting of the RH bi-stable circuit 
occurs once the call has been served. When more than one line concen- 
trator is being served it is possible that the HGT bi-stable circuit of 
more than one concentrator will be set simultaneously as a result of 
coincidence in service requests from correspondingly numbered lines in 
these concentrators. The decision as to which concentrator is to be 
served is left to the marker, as it would normally decide which horizontal 
group to serve. 

c. Line Selection 

On all calls, originating and terminating, the marker transmits to the 
frame circuits the complete identity of the line which it will serve. In 
the case of originating calls it has received this information in the manner 
described in Section 8b. In either case, it operates wire spring relays 
VGO-U and VFO-4, which enable gates so that the information may be 
stored in the 5- and 12-stage counters of the line selection circuit shown " 
in Fig. 28. 

The process of reading into the line selection counters starts when 
selection information has been received by the actuation of the HGS 
bi-stable circuit in the concentrator control trunk circuit of Fig. 27. 
This action stops the regular transmission of scanner pulses if they 
have not been stopped as a result of a call origination. At the same time 
it enables gates for transmission of information from the line selection 
circuit. Fig. 28. 

The ST bi-stable circuit of the line selection circuit is also enabled 
to start the process of setting the line selection counters. The next TP 
pulse sets the Rl bi-stable circuit. This bi-stable circuit enables a gate 
which permits the next TP pulse to set the counters and transmit a re- 
set pulse to the concentrator through pulse amplifier RIA. At the same 
time bi-stable circuit ST is reset to prevent the further read-in cr reset 



\ 



REMOTE CONTROLLED LINE CONCENTRATOR 



283 



pulses and to permit pulses through amplifier OPA to start the out- 
pulsing of line selections. These pulses pass to the VGP and VFP leads 
as long as the VG and VF line selection counters have not reached 
their first and last stages respectively. The output pulses to the con- 
centrator are also fed into the drive leads of these counters so that, as 
the counters in the concentrator are stepped up, the counters in the 
central office line selection circuit are stepped down. When the first 
stage of the VF counter goes on, the VF pulses are no longer transmitted 
until the first stage of the VG counter goes on. This insures that a VF 
pulse is the last to be transmitted. Also this pulse is not transmitted 
until the other frame circuits have successfully completed selections of 
an idle concentrator trunk. Then bi-stable circuit VFLD is energized, 



TO ORIGINATING 

CALL DETECTOR 

I 




VF- 

FROM 
VG U SCANNER 
PULSE 
GENERATOR 



FROM LINE 
[-SELECTION 
CIRCUIT 



Fig. 27 — Concentrator control trunk circuit. 



284 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



producing, during its transition, the last VF pulse for transmission to 
the concentrator. 



d. Trunk Selection and. Identification 

The process of selecting an idle concentrator trunk to which the line 
has access utilizes familar relay circuit techniques.^^ This circuit, in 
Fig. 29, will not be described in detail. One trunk selection relay, TS, is 
operated indicating the preferred idle trunk serving a line in the particu- 
lar vertical group being selected as indicated by the VG relay which 
has been operated by the marker. 

The TS4 and TS5 relays select trunks 8 and 9 which are available to 
each line while the 4 trunks available to only half of the lines are selected 
by relays TS0-TS3. The busy or idle condition of each trunk is indicated 
by a contact on the hold magnet associated with each trunk through 



TRUNK 
SELECTION 
COMPLETE T 



VFLD 



_l 
O 

cr 

I- 
z 
Oh 

^5 
tr u 
O cr 



Z : 

LU : 

^! 

o 
o 

o 

t- 



VFP 



VGP_ 
RS 



VFLI 



VFL 2 



0-1 



5-STAGE COUNTER 



VF4X -2V 



VGO 



y^ 




12-STAGE COUNTER 



0-0 



ST 



OPA 



R1A 



ST 




FROM SCANNER 
PULSE GENERATOR I 



TP 



Fig. 28 — Line selection circuit. 



REMOTE CONTROLLED LINE CONCENTRATOR 



285 



relay HG which operates on all originating and terminating calls to the 
particular concentrator served by these trunks. The end chain relay 
TC of the lockout trunk selection circuit^^ connects battery from the 
SR relay windings of idle trunks to the windings of the TS relays to 
permit one of the latter relays to operate and to steer circuits, not shown 
on Fig. 29, to the hold magnet of the trunk and to the tip-and-ring con- 
ductors of the trunk to apply the selection voltages shown on Figs. 13 
and 14. 

The path for operating the hold magnet originates in the marker. 
The path looks like that which the marker uses on the line hold mag- 
net when setting up a call on a regular line link frame. For this reason 
and other similar reasons this concentrator line link frame concept has 
been nicknamed the "fool-the-marker" scheme. 

Should a hold magnet release while a new call is being served the 
ground from the TC relaj^ normal or the TS relay winding holds relay 



CONCENTRATOR TRUNK 
SWITCH CROSSPOINTS 



SR [ 




LINE LINK 
NUMBER FROM 
MARKER I 



-48 V 



Fig. 29 — Trunk selection and identification. 






286 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



SR operated through its own contact until the new call has been set up. 
This prevents interference of disconnect pulses applied to the trunk 
when a selection is being made and insures that a disconnect pulse is 
transmitted before the trunk is reused. 

A characteristic of the No. 5 crossbar system is that the originating 
connection to a call register including the line hold magnet is released 
and a new connection, known as the "call back connection", is estab- 
lished to connect the line to a trunk circuit after dialing is completed. 

With concentrator operation the concentrator trunk switch connection 
is released but the disconnect signal is not sent to the concentrator as 
a result of holding the SR relay as described above. However, the marker 
does not know to which trunk the call back connection is to be estab- 
lished. For this reason the frame circuits include an identification proc- 
ess for determining the number of the concentrator trunk to be used 
on call back prior to the release of the originating register connection, i 

Identification is accomplished by the marker transmitting to the 
frame circuits the number of the link being used on the call. This in- 
formation is already available in the No. 5 system. The link being used 
is marked with —48 volts by a relay selecting tree^" to operate the TS 
relay associated with the trunk to which the call back connection is to 
be established. Relay CB (Fig. 29) is operated on this type of call in- 
stead of relay HG. The circuits for reoperating the proper hold magnet 
are already available on the TS relay which was operated, thereby rc- 
selecting the trunk to which the customer is connected. The concen- 
trator connection is not released when the hold magnet releases and 
again the marker operates as it would on a regular line link frame call. 

9. FIELD TRIALS 

Three sets of the experimental equipment described here have been 
constructed and placed in service in various locations. The equipment 
for these trials is the forerunner of a design for production which will 
incorporate device, circuit and equipment design changes based on the 
trial experiences. Fig. 30 shows the cabinet mounted central office trial 
equipment with the designation of appropriate parts. 

For the field trials described, the line links on a particular horizontal 
level of existing line link frames were extended to a separate cross-bar 
switch provided for this purpose in the trial equipment. The regular line 
link connector circuits were modified to work with the trial control 
circuits whenever a call was originated or terminated on this level. N(i 
lines were terminated in the regular primary line switches for this level. 



REMOTE CONTROLLED LINE CONCENTRATOR 



287 



10. MISCELLANEOUS FEATURES OF TRIAL EQUIPMENT 

There are a number of auxiliary circuits provided with the trial equip- 
ment to aid in the solutions of problems brought about by the concepts 
of concentrator service. One of the purposes of the trials was to deter- 
mine the way in which the various traffic, plant and commercial ad- 



CONCENTRATOR 
TRUNK SWITCH 



SERVICE OBSERVING 

TEST CONTROL-] 

SIMULATOR 

TRUNK DISCONNECT 
RELAYS 



CONCENTRATOR 
CONNECTOR RELAYS 



FRAME RELAY 
CIRCUITS 



SERVICE DENIAL 



FRAME ELECTRONIC 
CIRCUITS 



POWER SUPPLY 



LINE CONDITION 
TESTER 




Fig. 30 — Trial central office equipment. 



288 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

ministrative functions could be economically performed when concen- 
trators become common telephone plant facilities. The more important 
of these miscellaneous features are discussed under the following head- 
ings : 

a. Traffic Recording 

J 
To measure the amount and characteristics of the traffic handled by 

the concentrator a magnetic tape recorder, Fig. 31, was provided for 

each trial. The number of the lines and trunks in use each 15 seconds 

during programmed periods of each day were recorded in coded form 

with polarized pulses on the 3-track magnetic tape moving at a speed of 

1}/2" per second. Combinations of these pulses designate trunks busy on 

intra-concentrator connections and reverting calls. 

The line busy indications were derived directly from the line busy 
information received during regular scanning at the concentrator. Dur- 
ing one cycle in each 15 seconds new service requests were delayed to 
insure that a complete scan cycle would be recorded. Terminating calls 
were not delayed since marker holding time is involved. Trunk condi- 
tions are derived for a trunk scanner provided in the recorder. 

In addition to recording the line and trunk usage, recordings were 
made on the tape for each service request detected during a programmed 
period to measure the speed with which each call received dial tone 
and the manner in which the call was served. In this type of operation 
the length of the recording for each request made at a tape speed of 
only \i!' per second is a measure of service delay time. 

As may be observed from Fig. 31 the traffic recorder equipment was 
built with vacuum tubes and hence required a rather large power supply. 
It is expected that a transistorized version of this traffic recorder serv- 
ing all concentrators in a central office will be included in the standard 
model of the line concentrator equipment. With this equipment, traffic 
engineers will know more precisely the degree to which each concentra-- 
tor may be loaded and hence insure maximum utilization of the concen- 
trator equipment. 

b. Line Condition Tester 

It has been a practice in more modern central office equipment to 
include automatic line testing equipment.^^ An attempt has been made 
to include similar features with the concentrator trial equipment. The 
line condition tester (see Fig. 30) provides a means for automatically 
connecting a test circuit to each line in turn once a test cycle has been 



I 



REMOTE CONTROLLED LINE CONCENTRATOR 



289 



! , P ] 'f 



^ u 




POWER SUPPLIES 

AND 

PROGRAMMER 



I 



Fig. 31 — Traffic recorder. 



290 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

manually initiated. This test is set up on the basis of the known concen- 
trator passive line circuit capabilities. Should a line fail to pass this 
test, the test circuit stops its progress and brings in an alarm to summon 
central office maintenance personnel. The facilities of the line tester are 
also used to establish, under manual control, calls to individual lines as 
required to carry out routine tests. 

c. Simulator 

As the central office sends out scanner control pulses either no signal, 
a line busy or service request pulse is returned to the central office in 
each time slot. The simulator test equipment, shown in Fig. 30, was 
designed to place pulses in a specific time slot to simulate a line under 
test at the concentrator. 

In addition to transmitting the equivalent of concentrator output 
pulses the simulator can receive the regular line selection pulses trans- 
mitted to the concentrator for purposes of checking central office opera- 
tions. It is possible by combined use of the line tester and simulator to 
observe the operation of the concentrator and to determine the probable 
cause when a fault occurs. 

d. Service Observing 

The removal of the line terminals from the central office poses a num- 
ber of problems in conjunction with the administration of central office 
equipment. One of these is service observing. 

To maintain a check on the quality of service being rendered by the 
telephone system, service observing taps are made periodically on tele- 
phone lines. This is normally done by placing special connector shoes 
on line terminations in the central office. 

To place such shoes at the remote concentrator point would lead to 
administrative difficulties and added expense. Therefore, a method was 
devised to permit service observing equipment to be connected to con- 
centrator trunks on calls from specific lines which were to be observed. 
This mcithod consisted of manual switches on which were set the number 
of the line to be observed in terms of vertical group and vertical file. 
Whenever this line originated a call and the call could be placed over the 
first preferred trunk, automatic connection was made to the service ob- 
serving desk in the same manner as would occur for a line terminated 
directly in the central office. 

In addition, facilities were provided for trying a new service observ- 
ing technique where calls originating over a particular concentrator 



REMOTE CONTROLLED LINE CONCENTRATOR 291 

trunk would be observed without knowledge of the originating line num- 
ber. For this purpose a regular line observing shoe was connected to 
one of the ten concentrator trunk switch verticals in the trial equipment 
and from here connected to the service observing desk in the usual 
manner. 

The basic service observing requirements in connection with line 
concentrator operation have not as yet been fully determined. How- 
ever, it appears at this time that the trunk observing arrangement may 
be preferable. 

e. Service Denial 

In most systems denial of originating service for non-payment of 
telephone service charges, for trouble interception and for permanent 
signals caused by cable failures or prolonged receiver-off-hook conditions 
may be treated by the plant forces at the line terminals or by blocking 
the line relay. To avoid concentrator visits and to enable the prompt 
clearing of trouble conditions which tie up concentrator trunks, a ser- 
vice denial feature has been included in the design of the central office 
circuits. 

This feature consists of a patch-panel with special gate cords which 
respond to particular time slots and inhibit service request signals pro- 
duced by a concentrator during this period. In this way service requests 
can be ignored and prevent originating call service on particular lines 
until a trouble locating or other administrative procedure has been 
invoked. 

f. Display Circuit 

A special electronic switch was developed for an oscilloscope. This 
arrangement permited the positioning of line busy and service request 
pulses in fixed positions representing each of the 60 lines served. Line 
busy pulses were shown as positive and service request pulses as negative. 
This plug connected portable aid, see Fig. 32, was useful in tracing calls 
and identifying lines to which service may be denied, due to the existence 
of permanent signals. 

Other circuits and features, too detailed to be covered in this paper, 
have been designed and used in the field trials of remote line concen- 
trators. Much has been learned from the construction and use of this 
equipment which will aid in making the production design smaller, 
lighter, economical, serviceable and reliable. 

Results from the field trials have encouraged the prompt undertaking 



292 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




Fig. 32 — Pulse display oscilloscope. 



REMOTE CONTROLLED LINE CONCENTRATOR 293 

of development of a remote line concentrator for quantity production. 
The cost of remote line concentrator equipment will determine the ul- 
timate demand. In the meantime, an effort is being made to take advan- 
tage of the field trial experiences to reduce costs commensurate with 
insuring reliable service. 

The author wishes to express his appreciation to his many colleagues 
at Bell Telephone Laboratories whose patience and hard work have 
been responsible for this new adventure in exploratory switching de- 
velopment. An article on line concentrators would not be complete 
without mention of C. E. Brooks who has encouraged this development 
and under whose direction the engineering studies were made. 

BIBLIOGRAPHY 

1. E. C. Molina, The Theory of Probabilities Applied to Telephone Trunking 

Problems, B.S.T.J., 1, pp. 69-81, Nov., 1922. 

2. Strowger Step-bv-Step System, Chapter 3, Vol. 3, Telephone Theory and 

Practice by K.B. Miller. McGraw-Hill 1933. 

3. F. A. Korn and J. G. Ferguson, Number 5 Crossbar Dial Telephone Switching 

System, Elec. Engg., 69, pp. 679-684, Aug., 1950. 

4. U.S. Patent 1,125,965. 

5. O. Myers, Common Control Telephone Switching Systems, B.S.T.J., 31, pp. 

1086-1120, Nov., 1952. 

6. L. J. Stacy, Calling Subscribers to the Telephone, Bell Labs. Record, 8, pp. 

113-119, Nov., 1929. 

7. J. Meszar, Fundamentals of the Automatic Telephone Message Accounting 

System, A. I. E. E. Trans., 69, pp. 255-268, (Part 1), 1950. 

8. O. M. Hovgaard and G. E. Perreault, Development of Reed Switches and 

Relays, B.S.T.J., 34, pp. 309-332, Mar., 1955. 

9. W. A. Malthaner and H. E. Vaughan, Experimental Electronically Controlled 

Automatic Switching System, B. S.T.J., 31, pp. 443-468, May, 1952. 

10. S. T. Brewer and G. Hecht, A Telephone Switching Network and its Electronic 

Controls, B.S.T.J., 34, pp. 361-402, Mar., 1955. 

11. L. W. Hussey, Semiconductor Diode Gates, B.S.T.J., 32, pp. 1137-54, Sept., 

1953. 

12. U. S. Patent 1,528,982. 

13. J. J. EbersandS. L. Miller, Design of Alloyed Junction Germanium Transis- 

tor for High-Speed Switching, B. S.T.J. , 34, pp. 761-781, July, 1955. 

14. W. B. Graupner, Trunking Plan for No. 5 Crossbar System, Bell Labs. Record, 

27, pp. 360 365, Oct., 1949. 

15. G. L. Pearson and B. Sawyer, Silicon p-n Junction Alloy Diodes, I.R.E. Proc, 

42, pp. 1348-1351, Nov." 1952. 

16. A. E. Anderson, Transistors in Switching Circuits, B.S.T.J., 31, pp. 1207- 

1249, Nov., 1952. 

17. J. J. Ebers and J. L. Moll, Large-Signal Behavior of Junction Transistors, 

I. R. E. Proc, 42, pp. 1761-1784, Dec, 1954. 

18. J. J. Ebers, Four-Terminal p-n-p-n Transistors, I. R. E. Proc, 42, pp. 1361- 

1364, Nov., 1952. 

19. A. E. Joel, Relay Preference Lockout Circuits in Telephone Switching, Trans. 

A. L E. E., 67, pp. 720-725, 1948. 

20. S. H. Washburn, Relay "Trees" and Symmetric Circuits, Trans. A. I. E. E., 

68, pp. 571-597, 1949. 

21. J. W. Dehn and R. W. Burns, Automatic Line Insulation Testing Equipment 

for Local Crossbar Systems, B.S.T.J., 32, pp. 627-646, 1953. 



Transistor Circuits for Analog and 
Digital Systems* 

By FRANKLIN H. BLECHER 

(Manuscript received November 17, 1955) 

This paper describes the application of junction transistors to precision 
circuits for use in analog computers and the input and output circuits of 
digital systems. The three basic circuits are a summing amplifier, an inte- 
grator, and a voltage comparator. The transistor circuits are combined into 
a voltage encoder for translating analog voltages into equivalent time inter- 
vals. 

1.0. INTRODUCTION 

Transistors, because of their reliability, small power consumption, 
and small size find a natural field of application in electronic computers 
and data transmission systems. These advantages have already been 
realized by using point contact transistors in high speed digital com- 
puters. This paper describes the application of junction transistors to 
precision circuits which are used in dc analog computers and in the 
input and output circuits of digital systems. The three basic circuits 
which are used in these applications are a summing amplifier, an inte- 
grator, and a voltage comparator. A general procedure for designing 
these transistor circuits is given with particular emphasis placed on new 
design methods that are necessitated by the properties of junction 
transistors. The design principles are illustrated by specific circuits. 
The fundamental considerations in the design of transistor operational 
amplifiers are discussed in Section 2.0. In Section 3.0 an illustrative 
summing amplifier is described, which has a dc accuracy of better than 
one part in 5,000 throughout an operating temperature range of to 
50°C. The feedback in this amplifier is maintained over a broad enough 
frequency band so that full accuracy is attained in about 100 micro- 
seconds. 

The design of a specific transistor integrator is presented in Section 

* Submitted in partial fulfillment of the requirements for the degree of Doctor 
of Electrical Engineering at the Polytechnic Institute of Brooklyn. 

295 



296 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



I 



4.0. The integrator can be used to generate a voltage ramp which is 
linear to within one part in 8,000. By means of an automatic zero set 
(AZS) circuit which uses a magnetic detector, the slope of the voltage 
ramp is maintained constant to within one part in 8,000 throughout a 
temperature range of 20°C to 40°C. 

The voltage comparator, described in Section 5.0, is an electrical de- 
vice which indicates the instant of time an input voltage waveform 
passes through a predetermined reference level. By taking advantage 
of the properties of semiconductor devices, the comparator can be de- 
signed to have an accuracy of ±5 millivolts throughout a temperature 
range of 20°C to 40°C. 

In Section 6.0, the system application of the transistor circuits is 
demonstrated by assembling the summing amplifier; the integrator, and 
the voltage comparator into a voltage encoder. The encoder can be used J 
to translate an analog input voltage into an equivalent time interval 
with an accuracy of one part in 4,000. This accuracy is realized through- 
out a temperature range of 20°C to 40°C for the particular circuits 
described. 

2.0. FUNDAMENTAL CONSIDERATIONS IN THE DESIGN OF OPERATIONAL 
AMPLIFIERS 

The basic active circuit used in dc analog computers is a direct coupled 
negative feedback amplifier. With appropriate input and feedback net- 
works, the amplifier can be used for multiplication by a constant coef- 
ficient, addition, integration, or differentiation as shown in Figure 1 
The accuracy of an operational amplifier depends only on the passive 
components used in the input and feedback circuits provided that there 
is sufficient negative feedback (usually greater than 60 db). The time 
that is required for the amplifier to perform a calculation is an inverse 
f miction of the bandwidth over which the feedback is maintained. 
Thus a fundamental problem in the design of an operational amplifier 
is the development of sufficient negative feedback over a reasonably 
broad frequency range. The associated problem is the realization of 
satisfactory stability margins. Finally there is the problem of reducing 
the drift which is inherent in direct coupled amplifiers and particularly 
troublesome for transistors because of the variation in their character- 
istics with temperature. 

The first step in the design is the blocking out of the configuration 
for the forward gain circuit (designated A in Fig. 1). Three primary re- 
quirements must be satisfied: 

(1) Stages must be direct coupled. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



297 



(2) Amplifier must provide one net phase reversal. 

(3) Amplifier must have enough current gain to meet accuracy re- 
quirements. 

Three possible transistor connections are available: (a) the common 
base connection which may be considered analogous to the common 
grid vacuum tube connection; (b) the common emitter connection 
which is analogous to the common cathode connection; and (c) the 
common collector connection which is analogous to the cathode follower 
connection. These three configurations together with their approximate 
equivalent circuits are shown in Fig. 2. It has been shown^ that for 
most junction transistors the circuit element a is given by the expression 



a = sech 



W 



(1 + PTrn) 



1/2 



(1)^ 



where W is the thickness of the transistor base region, Lm is the diffusion 
length and t„, the lifetime of minority charge carriers in the base region, 



Rk 

I — ^AV 



E- "J 

-^ Wvr 



Eo 



Rk A/3EL Rk ^ 

•=0" Rj (i-A/i)"^ Rj ^l 

(a) MULTIPLICATION BY A 
CONSTANT COEFFICIENT 



E, 



R. 



E ^2 

E3 ^^ 



Rk 

I — vv\- 



Eo = E 



Rk A/bEj 



p, Rj (i-A/3) 

(b) ADDITION 



N r- . 

•RKEf: 



c 



§i — vw- 



£[Eo] 



A/3 £[el] sl[eQ 



Eo 



^N^^^^?|^-PH«[EJ 



(d) DIFFERENTIATION 



(l-A/3) pRC ~ pRC 
(C) INTEGRATION 

note; £[Eo] = LAPLACE TRANSFORM OF OUTPUT VOLTAGE 
£[Ei1 = LAPLACE TRANSFORM OF INPUT VOLTAGE 

p = jco 
Fig. 1 — Summary of operational amplifiers. 



* This expression assumes that the injection factor y and the collector efficiency 
at are both unity. This is a good approximation for all alloy junction transistors 
and most grown junction transistors. 



298 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

and 7? — ju. At frequencies less than Ua/^ir, (1) can be approximated by 



a = 



1 + ^ 



(2)' 



COa 



where ao is the low frequency value of 



a ^ 1 



l/TT 
2U. 



and 



2.4Z). 



CCa = 



w 



(Dm is the diffusion constant for the minority charge carriers in the base 
region). A readily measured parameter called alpha (a), the short 
circuit current gain of a junction transistor in the common base connec- 



SCHEMATIC 



Zc = 



EQUIVALENT CIRCUIT 



s ^ — 

*■ e/ 

(V 






b 



=^ V\V 



aZcLe 



Tb 



(a) COMMON BASE 
Lb 



: 




rb 



Zed -a) 



aZc'Lb 

'X, 



■re 



(b) COMMON EMITTER 

i^b rb 




aZc 



re 



aZcLb 



Zed -a) 



(C) COMMON COLLECTOR 



re 



1 + prcCc 



a 

P 


— 


ao 


i-hP 



re = COLLECTOR RESISTANCE 
Cc = COLLECTOR CAPACITANCE 



ZTT 



ALPHA-CUTOFF FREQUENCY 



Fig. 2 — Basic transistor connections. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 299 

tion, is related to a by the equation 

aZe -{■ n /^x 

Ze + n 

For most junction transistors the base resistance, n , is much smaller 
than the collector impedance | Zc |, at frequencies less than Wa/27r. There- 
fore, a ^ a and Ua/^ir is very nearly equal to the alpha-cutoff frequency, 
the frequency at which | a | is down by 3 db. 

The transistor parameters r^ and n are actually frequency sensitive 
and should be represented as impedances. However, good agreement 
between theory and experiment is obtained at frequencies less than 
Wa/27r with re and n assumed constant. 

The choice of an appropriate transistor connection for a direct coupled, 
negative feedback amplifier, is based on the following reasoning. The 
common base connection may be ruled out immediately because this 
connection does not provide current gain unless a transformer interstage 
is used. The common emitter connection provides short circuit current 
gain and a phase reversal for each stage. Thus if the amplifier is com- 
posed of an odd number of common emitter stages, all three requirements 
previously listed, are satisfied. A common emitter cascade has the addi- 
tional practical advantage, that by alternating n-p-n and p-n-p types of 
transistors, the stages can be direct coupled with practically zero inter- 
stage loss. 

The common collector connection provides short circuit current gain 
but no phase reversal. Consequently, the dc amplifier cannot consist 
entirely of common collector stages and operate as a negative feedback 
amplifier. This paper will consider only the common emitter connection 
since, in general, for the same number of transistor stages, the common 
emitter cascade provides more current gain than a cascade composed of 
both common collector and common emitter stages. 

2.1 Evaluation of External Voltage Gain 

Since the equivalent circuit of the junction transistor is current acti- 
vated, it is convenient to treat feedback in a single loop transistor ampli- 
fier as a loop current transmission (refer to Appendix I) instead of as a 
loop voltage transmission which is commonly used for single loop vacuum 
tube amplifiers.^ Fig. 3 shows a single loop feedback amplifier in which 
a fraction of the output current is fed back to the input. A is defined as 
the short circuit current gain of the amplifier without feedback, and jS is 
defined as the fraction of the short circuit output current (or Norton 



300 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



equivalent circuit current) fed back to the input summing node. With 
these definitions, 

he = A/in' (4) 



la - 131. 



sc 



where /sc is the Norton equivalent short circuit current. 
From Kirchhoff's first law 

/in = /in + Iff 
Combining relations (4) to (6) yields 



'sc 



A 



(5) 



(6) 



(7) 



/in 1 - A^ 
Expression (7) provides a convenient method for evaluating the external 









^ 








[|N 












I IN 


















> 




^ 



























Fig. 3 — Single loop feedback amplifier. 

voltage gain of an operational amplifier. Fig. 4 shows a generalized op- 
erational amplifier with N inputs. With this configuration, 



IN 



j=i L 



TTT he r, I 



Zj 



(S) 



where Ej , j = 1,2, • ■ • , N, are the N input voltages referred to the 
ground node. 
Zj,j— 1,2, • ■ ■ , N, are the A^ input impedances 
ZiN is the input impedance of the amplifier measured at the 
summing node with the feedback loop opened. 



Eo 



//i 



sc 



UT 



'IN 



la = 



A 



Eovr = 



Zk 

/sc ~ / 



(3 



Rl Zovt 



(5>) 



(10) 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 301 



where Zovr is the output impedance of the amplifier measured with the 
feedback loop opened. The expression for the output voltage is obtained 
by combining (7), (8), (9), and (10). 



E, 



OUT 



N y 

= zL ^i 7" 

;=1 ^i 



A^ + 



3 = 1 ^1 _ 



(iir 



where 



A^ = A 



1 - 



'IN 



\Ri 



+ 



/OUT 



1 _^ ^ + ^^^ 

Rl Zovr 



IA/3 is equal to the current returned to the summing node when a unit 



Ei 



Z, 






MN 



1/3 Zk 



I IN 



Zn 



Zls 



J7 



1/5 Zk 



Equt 




NORTON EQUIVALENT CIRCUIT 



Fig. 4 — Generalized operational amplifier. 

icurrent is placed into the base of the first transistor stage (/in = 1). 
If I A^ 1 is much greater than ] Zj^'/Zr \ and 



1 + L 



'IN 



then 



N 



Eqvt — ~ 2^ J^j nT 



(12) 



y=i 



The accuracy of the operational amplifier depends on the magnitude of 
AjS and the precision of the components used in the input and feedback 
networks as can be seen from (11). There is negligible interaction between 
the input voltages because the input impedance at the summing node is 
equal to Zin' divided by (1 — A^)? This impedance is usually negligibly 
tsmall compared to the impedances used in the input circuit. 

* In general, E,- and Eout are the Laplace transforms of the input and output 
fvoltages, respectively. 



302 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



2.2. Methods Used to Shape the Loop Current Transmission 

An essential consideration in the design of a feedback amplifier is the 
provision of adequate margins against instability. In order to accomplish 
this objective, it is necessary to choose a criterion of stability. In Ap- 
pendix I it is shown that it is convenient and valid to base the stability 
of single loop transistor feedback amplifiers on the loop current trans- 
mission. In order to calculate the loop current transmission of the dc 
amplifier, the feedback loop is opened at a convenient point in the cir- 
cuit, usually at the base of one of the transistors, and a unit current is 
injected into the base (refer to Fig. 24). The other side of the opened 
loop is connected to ground through a resistance (r^ -j- r^) and voltage 
Veli • In many instances, the voltage re/4 can be neglected. If | Zj? | and 



3=1 Zj 



I 



are much greater than | Z 



IN 



then A/3 is very nearly equal to the loop 
ciu'rent transmission. For absolute stability^ the amplitude of the loop 
current transmission must be less than unity before the phase shift 
(from the low frequency value) exceeds 180°. Consequently, this charac- 
teristic must be controlled or properly shaped over a wide frequency 



10 

_J 

LU 

O 
LU 
Q 



< 

o 

\- 
z 

LU 

a. 
o 



40 














U), 




a;,' 








Wa 


^{\-\-S)u)^ 




\ 


ao 


^" 


■\ 


\, 




\J 


i 












"^ 


\ 






30 


1- 


ao+cT 


t- — 


. ao 






\ 

7~' 


\ 




\ 


^ 


AM PL 


ITUC 


E 








\ 


\ 


20 
10 


i-ao 


+ 7_ 






AMPLITUDE' 

(WITH LOCAL 

FEEDBACK) 




\ 


\ 




\ 
\ 


S 


< 


PHASE (WITH 
LOCAL FEEDBACK) 


















phase\ 


\ 




y 

\ 
> 


\ 


\ 




ao 











^'^ 

f' 


\+S 1 
















-270° 




X 


— . 


s 
\ 




\ 


N 


d 


«/c 


-10 
20 
30 
40 








"^ 




























N 


\ 


^ 


^ 




































\ 


^ 


\ 


































\ 





•180 



-200 



-220 



•240 



•260 uj 

_l 

2 
< 



-280 



•300 



LU 

< I 

I 

Q- 



-320 



■340 



,02 2 5 ,q3 2 ^ ,o4 •=; S ,q5 t S ,q6 



5 ■/^4 2 5 ,„s 2 5 ,„6 2 5 ^q7 2 5 jq8 
FREQUENCY IN CYCLES PER SECOND 



Fig. 5 — Current transmission of a common emitter stage. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 303 

band. In addition, it is desirable that the feedback fall off at a rate equal 
to or less than 9 db per octave in order to insure that the dc aniplifier 
has a satisfactory transient response. 

Three methods of shaping are described in this paper; local feedback 
shaping, interstage network shaping, and (3 circuit shaping. Local feed- 
back shaping will be described first. The analysis starts by considering 
the current transmission of a common emitter stage, ecjuivalent circuit 
shown in Fig. 2(b). If the stage operates into a load resistance Rl , then 
to a good approximation the current transmission is given by 



where 



Gr = r" = ^ ~ ^° +/ (13)^ 

^^ 1 + ^ + '^ 

wi a)aCOc(l — ao -\- 8) 

RL+Te 



8 = 



COl = 



(1 - ao + 8) 
1 + 5,1 



-^ ^ alpha-cutoff frequency 
Ztt 



1 



Uc 



(7?x, + re)Cc 



It is apparent from expression (13) that if (1 — Oo + 8) is less than 0.1, 
then the current gain of the common emitter stage falls off at a rate of 
6 db per octave with a corner frequency at wi .f A second 6 db per octave 
cutoff with a corner frequency at [co^ + (1 + 5)aJc] is introduced by the 
p" term in the denominator of (13). A typical transmission characteristic 
is shown in Fig. .5. The current gain of the common emitter stage is unity 
at a frequency equal to 

ao 



1 +5 I 1 



* Expressions (13) and (14) are poor approximations at frequencies above 

' coo/27r. 

' t Strictly speaking the corner frequency is equal to 01/2 tt. However, for sim- 
plicity, corner frequencies will be expressed as radian frequencies. 



304 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Since the phase crossover of A|S* is usually placed below this frequency, 
the principal effect of the second cutoff is to introduce excess phase. This 
excess phase can be minimized by operating the stage into the smallest 
load resistance possible, thus maximizing Wc . j 

An undesirable property of the common emitter transmission charac- 
teristic is that the corner frequency coi occurs at a relatively low fre- 
quency. However, the corner frequency can be increased by using local 
feedback as shown in Fig. 6(a). Shunt feedback is used in order to pro- 
vide a low input impedance for the preceding stage to operate into. The 
amplitude and phase of the current transmission is controlled prin- 
cipally by the impedances Z\ and Z2 . If | A& \ is much greater than one, 
and if /3 ;^ ^1/^2 , then from (7) the current transmission of the stage is 
approximately equal to — Z2/Z1 . Because of the relatively small size of 
A^ for a single stage, this approximation is only valid for a very limited 
range of values of Zi and Z2 . If Zi and Zi are represented as resistances 
R\ and Ri , then the current transmission of the circuit is given to a good 
approximation by 



tto 



h. _ R2 1 — gp + 7 

^^ = /i= ~{R2 + n)r_^p_^ v' 



where 



7 = 



coi = 



COc = 



Co/ COaCOcCl — Oo + 7), 

R\ + Te _, Rl + Te 

R2 + ^6 



I 

(14) 



(/?2 + 


rb)rc 


i22 + n 

(1 +ao 


+ ro 
+ 7) 


1 + 7 

1 





{R, -f re)Cc i 



By comparing (14) with (13), it is evident that the negative feedback 
has reduced the low-frequency current gain from ao/(l — ao) (5 may 
usually be neglected) to 



( 



R2 \ I «0 \ _ , ^2 



R2 + rj \1 - ao + 7/ ^1 + re 



(if 7 > 1 - ao) 



.-•! 



* The phase crossover of A/3 is equal to the frequency at which the phase shift 
of A/3 from its low-frequency value is 180°. 



I 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 305 

The half power frequency, however, has been increased from 

1— Oo , 1— Oo + 7 

t:^ 1 + 7 , 1 

as shown by the dashed curves in Fig. 5.* 

The bandwidth of the common emitter stage can be increased without 
reducing the current gain at dc and low-frequencies by representing Zi 
by a resistance Ri , and Z2 by a resistance R2 in series with a condenser 
C2 . If I/R2C2 is much smaller than co/, then the current transmission of 
the stage is given by (14) multiplied by the factor 



P 



1 + 



C04 
P 



(15) 



where 



602 



Wi 



H^^i 



1 - cro + 



Ri + re 



C2(/?2 + r6)(l - ao + 7) 



The current transmission for this case is plotted in Fig. 6(b). The con- 
denser d introduces a rising 6 db per octave asymptote with a corner 
frequency at wi . At dc the current gain is equal to 



ao 



1 — ao + 5 



A second method of shaping the loop current transmission char- 
acteristic of a feedback amplifier is by means of interstage networks. 
These networks are usually used for reducing the loop current gain at 
relatively low frequencies while introducing negligible phase lag near 
the gainf and phase crossover frequencies. Interstage networks should 
be designed to take advantage of the variable transistor input impedance. 
The input impedance of a transistor in the common emitter connection 



* In Figs. 5 and 6(b), the factor R^/iRi + n) is assumed equal to unity. This is 
' a good approximation since in practice R2 is equal to several thousand ohms while 
rt is equal to about 100 ohms. 

t The gain crossover frequency is equal to the frequency at which the magni- 
tude of Al3 is unity. 



306 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



is given by the expression 



'INP 



UT = ?'6 + ^e(l — Gi) 



(16)1 



where Gj is the current transmission given by (13). If Gi at dc is much' 
greater than 1, then the input impedance and the current transmission 
of the common emitter stage fall off at about the same rate and with 
approximately the same corner frequency (wi). The input impedance 
finally reaches a limiting value equal to r^ + Vb . 

A particularly useful interstage network is shown in Fig. 7(a). This 
network is analyzed in Appendix II and Fig. 7(b) shoAvs a plot of the 




60 



50 



40 



30 



20 



Z 
< 



z 

UJ 



10 



tr 
cr 

D 
U 

-10 



-20 



-30 




(a: 



EQUIVALENT CIRCUIT 









\ 


\ 






(b) 






























\ 








AMPLITUDE 




an 














\ 






(WITHOUT LOCAL 






















\ 






FEEDBACK) 




1-ao+d" 


^ - 












^" 


\ 






■*•> 


1 


^ 




























s 


.AMPLITUDE 


























^4 




^>CiL 




^'' 














,^_ 




•— ^ ■ 


■~^ 


r"**^ 




cvz 








r^ 






i 
















/ 




"^v 








^ 






' 1 










>^ 


/ 




V 


\ 
\ 
\ 










X 


V 


^ ao 




i-ao+ 


7 - 






/ 


A 


/ ^ 


\ 


s. 






\ 




k 


\ 




\ 


\ 








/ 


/ 






PH/ 


>s> 




/ 


\ 


\ 
\ 

\ 
V 




\ 


\ 
























PHASE N 






\ 
























(WITHOUT LOCAL 


\ 




s 
























FEEDBACK) 


s 


w 




k. 






- 


























^-. 


■"••^^, 











120 



140 



-160 10 

UJ 

m 
cr 

-180 liJ 

Q 
Z 

-200 ^ 

z 
< 



-220 , 






10 



2 5p2 5,2 5.2 5,2 5 

-!•= in^ m^ in5 



10'= 



lO-^* 10^ 10= 

FREQUENCY IN CYCLES PER SECOND 



lO'' 



-240 ' 



260 



- -280 



10' 



Fig. 6 — Negative feedback applied to a common emitter stage. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



307 



resulting current transmission. The amplitude of the transmission falls 
off at a rate of 6 db per octave with the corner frequency C05 determined 
by C'3 and the low frequency value of the transistor input impedance. 
The inductance L3 introduces a 12 db per octave rising asymptote with 
a corner frequency at C03 = WLsCs . The corner frequencies C03 and C05 
are selected in order to obtain a desirable loop current transmission 
characteristic (specific transmission characteristics are presented in Sec- 
tions 3.0 and 4.0). The half power frequency of the current transmission 
of the transistor, wi , does not. appear directly in the transmission char- 
acteristic of the circuit because of the variation in the transistor input 
impedance with frequency. 
The overall (3 circuit of the feedback amplifier can also be used for 




i-ao+(J 



I ^ 
s 

LU 
Q 

z 
I - 



<l< 

z 
< 

15 



Z 
UI 

cc 

D 

u 

Q 
UJ 

y 

< 

2 

a 
o 

z 



40 



20 



-20 



-40 



-60 



-80 

















(b) 












/ 


/ 




CU5 




















u 


■^3/ 


y' 




* 


^^^ 
























A^ 


^ 






^s^ 




















1 




\. 






^N, 




















/ 




^s^ 








S^ 


















/ 






\ 






\ 


AMPLITUDE 












/ 
/ 






\ 








\^ 
















/ 






\ 


\ 
\ 
\ 

\ 






> 


X 


X 


V 








1 

1 
/ 
/ 
/ 




1 
1 
1 
1 

1 


_ 




\ 

\ 


\ 










\ 


\ 


s,. 


/ 


cu,(rb+ 


' Te 


\ 


\-do+l 


W 






\ 


.PHASE 










^ 




Tb+le-l-Ra-K^iLa 








^^ 












/ 


\. 


















*^.., 





— 




^** 






X 


— - 


N 


- 



-135 



10 



LU 
UJ 

isog 



z 
< 

-225 1}^ 



< 

I 
a. 



-270 



102 " "^ \0^ " = 10^ -^ = 105 

FREQUENCY IN CYCLES PER SECOND 



Fig. 7 — Interstage shaping network. 



lO'' 



308 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

shaping the loop current transmission. If the feedback impedance Zk 
(Fig. 4) consists of a resistance Rk and condenser Ck in parallel, then 
the loop current transmission is modified by the factor 



1 + 



CO; 






1 + ^ 



COS 



(17) 



where 



C07 = 



C08 = 



RkCk 

(Rl_±_Rk) 

RlRkC K 



Since Zk affects the external voltage gain of the operational amplifier, 
(11), the corner frequency C07 must be located outside of the useful fre- 
quency band. Usually it is placed near the gain crossover frequency in 
order to improve the phase margin and the transient response of the 
amplifier. 

In Sections 3.0 and 4.0, the above shaping techniques are used in the 
design of specific operational amplifiers. 

3.0. THE SUMMING AMPLIFIER 

3.1. Circuit Arrangement 

The schematic diagram of a dc summing amplifier is shown in Fig. 8. 
From the discussion in Section 2.0 it is apparent that each common 
emitter stage will contribute more than 90 degrees of high-frequency 
phase lag. Consequently, while the magnitude of the low-frequency : 
feedback increases with the number of stages, this is at the expense of , 
the bandwidth over which the negative feedback can be maintained. 
It is possible to develop 80 db of negative feedback at dc with three 
common emitter stages. This corresponds to a dc accuracy of one part 
in 10,000. In addition, the feedback can be maintained over a broad 
enough band in order to permit full accuracy to be attained in about 
100 microseconds. Thus it is evident that the choice of three stages repre- 
sents a satisfactory compromise between accuracy and bandwidth ob- 
jectives. 

The output stage of the amplifier is designed for a maximum power 
dissipation of 75 milliwats and maximum voltage swing of ±25 volts 



I 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



309 



when operating into an external load resistance equal to or greater than 
50,000 ohms. A p-n-p transistor is used in the second stage and n-p-n 
transistors are used in the first and third stages. This circuit arrangement 
makes it possible to connect the collector of one transistor directly to 
the base of the following transistor without introducing appreciable 
interstage loss. ''Shot" noise" and dc drift are minimized by operating 
the first stage at the relatively low collector current of 0.25 milliamperes. 
The 110,000-ohm resistor provides the collector current for the first 
stage, and the 4,700-ohm resistor provides 3.8 milliamperes of collector 
current for the second stage. The series 6,800-ohm resistor between the 
xcond and third stages, reduces the collector to emitter potential of the 
second stage to about 4.5 volts. 

The loop current transmission is shaped by use of local feedback ap- 
plied to the second stage, by an interstage network connected between 
the second and third stages, and by the overall (3 circuit. The 200-ohm 
resistor in the collector circuit of the second stage is, with reference to 
Fig. 6(a), Zi . The impedance of the interstage network can be neglected 
since it is small compared to 200 ohms at all frequencies for which the 
local feedback is effective. The interstage network is connected between 
the second and third stages in order to minimize the output noise voltage. 
^^'ith this circuit arrangement, practically all of the output noise voltage 



iE 



250 K 



IN 



+ 33V 




5MUf 

Hf- 



20on 



n-p-n 



250 K 
2.4 K 200 n 



0.01/U.F 




p-n-p 



■llOK 



100 K POT. 

MANUAL 

ZERO SET 



I 

+ 33V 



I 

+ 4.5V 




OUT 



5>UH 



-45V -27V +33V 



Fig. 8 — ■ DC summing amplifier. 



310 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



120 



100 



UJ 

U 80 

LU 

o 

z 
" 60 

< 



< 

IS 



LU 

a: 
tr 

D 
U 

Q. 

o 
o 

_) 



40 



20 



■20 



-40 



















,-' 




../>' 

--T 


.-' 


— 


364 




LOCAL _ 
FEEDBACK"^- 


^-' 


.-1 

41,000 






-^ 


^ 




d 6 


630 




\, 


. 12 

N 
\ 


?,000 


--.., 












s^-> 






^-. 

'S 


'-.. 


\ 

\ 

V 
\ 
\ 
V 




2ND •^^ 

STAGE ^ 












s 


\ 




-.. 


"-. 
^-. 




1ST & 3RD 
STAGES 
















N 


\ 


0.5/ZF 


^■-S;:-.-, 
























^ 


\, 


\ 
\ 


























> 


\ 





10' 



10-^ 



10- 



10' 



10' 






FRFOUENCY IN CYCLES PER SECOND 

Fig. 9 — Gain-frequency asymptotes for summing amplifier. 

is generated in the first transistor stage. If the transistor in the first 
stage has a noise figure less than 10 db at 1,000 cycles per second, then 
the RMS output noise voltage is less than 0.5 millivolts. 

Fig. 9 shows a plot of the gain-frequency asymptotes for the sum- 
ming amplifier determined from (13), (14), (15), (17), and (A6) under 
the assumption that the alphas and alpha-cutoff frequencies of the tran- 
sistors are 0.985 and 3 mc, respectively. The corner frequencies intro-' 
duced by the 0.5 microfarad condenser in the interstage network, thel 
local feedback circuit, and the cutoff of the first and third stages are so 
located that the current transmission falls off at an initial rate of about' 
9 db per octave. This slope is joined to the final asymptote of the loop 
transmission by means of a step-type of transition.^ The transition is 
provided by 3 rising asymptotes due to the interstage shaping network, 
and the overall /S circuit. An especially large phase margin is used in order 
to insure a good transient performance. 

Fig. 10 shows the amplitude and phase of the loop current trans- 
mission. When the amplitude of the transmission is db, the phase angle 
is -292°, and when the phase angle is —360°, the amplitude is 27.5 db 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



311 



100 



LU 

m 
u 

LU 
Q 



<^ 

z 
< 

H 
Z 
UJ 

a. 
a. 

D 

o 

Q. 
O " 

o 

_l 



80 



60 



40 



20 



20 



-40 



— 


■~-^ 


^•"v 






































> 


\ 


^ 


AM 


PLITL 


DE 




























\ 

1 


\ 

\ 




\ 


s 






























> 


\. 


._ 


s 


>^ 


^r 


•—'' 


y^ "N phase 

'PHASE 'nCROSSOVER 






















s 


\ 






\ 


V 




























/ GAIN^-N^ 
CROSSOVER 


N 


-27.5 DB 
95 = -360° 


































sv 







■160 



-200 



to 

LU 
-240 ^ 

O 

LU 

Q 

-280 7 



•320 



-360 



-400 



■440 



10= 
FREQUENCY IN CYCLES PER SECOND 



10^ 



10' 



Fig. 10 — Loop current transmission of the summing amplifier. 

below db. The amplifier has a 68° phase margin and 27.5 db gain margin. 
In order to insure sufficient feedback at dc and adequate margins against 
instability, the transistors used in the amplifier should have alphas in 
the range 0.98 to 0.99 and alpha-cutoff frequencies equal to or greater 
than 2.5 mc. 



3.2. Automatic Zero Set of the dc Summing Amplifier 

The application of germanium junction transistors to dc amplifiers 
does not eliminate the problem of drift normally encountered in vacuum 
tube circuits. In fact, drift is more severe due principally to the varia- 
tion of the transistor parameters alpha and saturation current with 
temperature variation. Even though the amplifier has 80 db of negative 
feedback at dc, this feedback does not eliminate the drift introduced by 
[the first transistor stage. Because of the large amount of dc feedback, 
the collector current of the first stage is maintained relatively constant. 
The collector current of the transistor is related to the base current by 
the equation 



Ic = 



/c 



+ 



a 



I — a 1 — a 



(18) 



[The saturation current, Ico , of a germanium junction transistor doubles 
(approximately for every 11°C increase in temperature. The factor 
a/(l — a) increases by as much as 6 db for a 25°C increase in tempera- 



312 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

ture. Consequently, the base current of the first stage, Ih , and the output 
voltage of the amplifier must change with temperature in order to main- ' 
tain Ic constant. The drift due to the temperature variation in a can be 
reduced by operating the first stage at a low value of collector current. 
With a germanium junction transistor in the first stage operating at a 
collector current of 0.25 milliamperes, the output voltage of the amplifier 
drifts about ±1.5 volts over a temperature range of 0°C to 50°C. It is 
possible to reduce the dc drift by using temperature sensitive elements 
in the amplifier. • In general, temperature compensation of a transistor 
dc amplifier requires careful selection of transistors and critical adjust- 
ment of the dc biases. However, even with the best adjustments, tem- 
perature compensation cannot reduce the drift in the amplifier to within 
typical limits such as ±5 millivolts throughout a temperature range of i 
to 50°C. In order to obtain the desired accuracy it is necessary to use 
an automatic zero set (AZS) circuit. t 

Fig. 11 shows a dc summing amplifier and a circuit arrangement fori 
reducing any dc drift that may appear at the output of the amplifier. 
The output voltage is equal to the negative of the sum of the input volt- 
ages, where each input voltage is multiplied by the ratio of the feedback 
resistor to its input resistor. In addition, an undesirable dc drift voltage ^ 
is also present in the ovitput voltage. The total output voltage is 

^o.t = -i:^y|^ + Adrift (1!))^ 

In order to isolate the drift voltage, the A^ input voltages and the output 
voltage are applied to a resistance summing network composed of re- 
sistors Ro , Ri , R2 , • • • , Rn ■ The voltage across Rs is equal to 

Es=^ Adrift (20) 

if 

R,«Ro,R/; j = 1,2, ■■' ,N 
and 

RoRj = RkR,'; j = 1,2, ■■■ ,N 

The voltage E, is amplified in a relatively drift-free narrow band dc 
amplifier and is returned as a drift correcting voltage to the input of the 
dc summing amplifier. If the gain of the AZS circuit is large, the drift 
voltage at the output of the summing amplifier can be made very small. 
Fig. 12 shows the circuit diagram of a summing amplifier which uses 
a mechanical chopper in the AZS circuit.^^ The AZS circuit consists of a 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 313 

resistance summing network, a 400-cycle synchronous chopper, and a 
tuned 400-cycle amplifier. Any drift in the summing amplifier will pro- 
duce a dc voltage Es at the output of the summing network. The chopper 
converts the dc voltage into a 400 cycles per second waveform. The 
fundamental frequency in the waveform is amplified by a factor of about 
400,000 by the tuned amplifier. The synchronous chopper rectifies the 
sinusoidal output voltage and preserves the original dc polarity of Eg . 
The rectified voltage is filtered and fed back to the summing amplifier 
as an additional input current. The loop voltage gain of the AZS circuit 
at dc is about 54 db. Any dc or low-frequency drift in the summing 
amplifier is reduced by a factor of about 500 by the AZS circuit. The 
drift throughout a temperature range of to 50°C is reduced to ±3 
millivolts. 

Since the drift in the summing amplifier changes at a relatively slow 
rate, the loop voltage gain of the AZS circuit can be cutoff at a relatively 
low frequency. In this particular case the loop voltage gain is zero db at 
about 10 cycles per second. 



4.0. THE INTEGRATOR 

4.1. Basic Design Considerations 

The design principles previously discussed are illustrated in this sec- 
tion by the design of a transistor integrator for application in a voltage 



VvV 




-OUT 



Fig. 11 — DC summing amplifier with automatic zero set. 



314 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 315 



encoder. The integrator is required to generate a 15-volt ramp which is 
linear and has a constant slope to within one part in 8,000. This ramp is 
to have a slope of 5 millivolts per microsecond for an interval of 3,000 
microseconds. 

The first step in the design is to determine the bandwidth over which 
the negative feedback must be maintained in order to realize the desired 
output voltage linearity. The relationship between the output and input 
voltage of the integrator can be obtained from expression (11) by sub- 
stituting (1/pc) for Zk and R for Zj (refer to Fig. 1). 



£l-C'outJ — 



pRC 



A/3 + Zr^'pC 



1 - AjS + 



-nN_ 
R 



(21) 



where ce[£'ouT] and JSiii'iN] are the Laplace transforms of the output and 
input voltages, respectively. In order to generate the voltage ramp, a 
step voltage of amplitude E is applied to the input of the integrator. The 
term Zy^ jR is negligible compared to unity at all frequencies. Therefore, 



£L-£'outJ — 



E \ A& 



+ 



EZ 



IN 



1 



'^-RC Ll - A&\ pR \\ - A^_ 
It will be assumed that A/3 is given by the expression 

-K 



(22) 



A^ = 



V 



)0 + ^T 



(i + -M(i + ^ 



(23) 



Expression (23) implies that A/3 falls off at a rate of 6 db per octave at 
low frequencies and 12 db per octave at high frequencies. The output 
\ voltage of the integrator, as a function of time, is readily evaluated by 
substituting (23) into (22) and taking the inverse Laplace transform of 
the results. A good approximation for the output voltage is 



^OUT — 



E 



RC 



+ 



2K 



^-[(2w2+«l)(/2] ^;„ -x/W 



sm 



Vk> 



OJo 



■iC02M 



ER 



(24)^ 



IN 



R 



[1 _ e-(-i'W _!_ g-[(2<-2+.i)t/2i ^Qg ^Tkc.,!] 



The linear voltage ramp is expressed by the term — (Et/RC) . The 
additional terms introduce nonlinearities. The voltage ramp has a slope 
of 5 millivolts per microsecond for E = —21 volts, R = 42,000 ohms, 

* In evaluating jE'out it was assumed that Zm' was equal to a fixed resistance 
Rin' , the low frequency input resistance to the first common emitter stage. A 
complete analysis indicates that this assumption makes the design conservative. 



31G 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



and C = 0.1 microfarads. For these circuit values, and K = 10,000 
(corresponding to 80 db of feedback) the nonhnear terms are less than 
1/8,000 of the linear term (evaluated when / = 4 X 10"^ seconds) if 
/i ^ 30 cycles per second, J2 ^ 800 cycles per second, and if the first 
1000 microseconds of the voltage ramp are not used. Consequently, 80 
db of negative feedback must be maintained over a band extending from 
30 to 800 cycles per second in order to realize the desired output voltage 
linearity. 

4.2. Detailed Circuit Arrangement 

Fig. 13 shows the circuit diagram of the integrator. The method of 
biasing is the same as is used in the summing amplifier. The 200,000-ohm 
resistor provides approximately 0.5 milliamperes of collector current for 
the first stage. The 40,000-ohm resistor provides approximately 0.9 
milliamperes of collector current for the second stage. The output stage 
is designed for a maximum power dissipation of 120 milliwatts and for 
an output voltage swing between —5 and +24 volts when operating 
into a load resistance equal to or greater than 40,000 ohms. 



J+'08V 



• + 108V 



42 K 



D2 

44- 



C 



0.01>(/F o.l/iF 

2.4K 



270 K 



I 

+ I08V 




1MEG 



200n 

\ — vw 

2>U.F 



200 K 



rVWA/^An 

j 100 K [ 



POT. I 




I 



OUT 



-10.5V 



+ 108V 



+ 4.5V 



•45V -10.5V 



Fig. 13 — Integrator. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 317 



1 !!3 

{ LU 

I 5 
u 
ai 
a 



1 z 
< 



140 
120 


N 








































































100 


.^^ 






\ 




AMPLITUDE 

v. 






















80 


^""^ 






\ 


\ 

\ 


\ 


N 


\, 






















60 










\ 
\ 

> 


\ 




\ 


\ 


s 


















40 
20 












\ 




'^— -- 


,'' 


S 


"-s 


S, 


PHASE 
































^ 


\ 


\ 


■\ 


PHASE 
. CROSSOVER 



-20 
-40 
























GAIN-" 
CROSSOVER 


\ 


\ 


— ?n HR 


































95=- 


360° 



■80 



-120 



160 



■200 



■240 



•280 



UJ 

_J 

z 
< 

LU 
lO 
-320 < 
I 
Q. 



-360 



-400 



■440 



10 



2 S .- 2 5 .^3 2 5 ,^^ 2 ^ 105 2 ^ ,0« ' ' 10^ 



lO'^ 



w 



FREQUENCY IN CYCLES PER SECOND 

Fig. 14 — - Loop current transmission of the integrator. 

The negative feedback in the integrator has been shaped by means of 
local feedback and interstage networks as described in Section 2.2. The 
loop current transmission has been calculated from (13), (14), (15), and 
(A6) and is plotted in Fig. 14. The transmission is determined under the 
assumption that the alphas of the transistors are 0.985 and the alpha- 
cutoff frequencies are three megacycles. Since the feedback above 800 
cycles per second falls off at a rate of 9 db per octave, the analysis in 
Section 4.1 using (23), is conservative. The integrator has a 44° phase 
margin and a 20 db gain margin. In order to insure sufficient feedback 
between 30 and 800 cycles per second and adequate margins against 
instability, the transistors used in the integrator should have alphas in 
the range 0.98 to 0.99 and alpha-cutoff frequencies equal to or greater 
than 2.5 megacycles. 

The silicon diodes Di and D2 are rec^uired in order to prevent the 
integrator from overloading. For output voltages between —4.0 and 21 
volts the diodes are reverse biased and represent very high resistances, of 
the order of 10,000 megohms. If the output voltage does not lie in this 
range, then one of the diodes is forward biased and has a low resistance, 
of the order of 100 ohms. The integrator is then effectively a dc amplifier 
with a voltage gain of approximately 0.1. The silicon diodes affect the 



318 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



linearity of the voltage ramp slightly due to their finite reverse resistances 
and variable shunt capacities. If the diodes have reverse resistances 
greater than 1000 megohms, and if the maximum shunt capacity of each 
diode is less than 10 micromicrofarads (capacity with minimum reverse 
voltage), then the diodes introduce negligible error. 

As stated earlier, the integrator generates a voltage ramp in response 
to a voltage step. This step is applied through a transistor switch which 
is actuated by a square wave generator capable of driving the transistor 
well into current saturation. Such a switch is required because the 
equivalent generator impedance of the applied step voltage must be very 
small. A suitable circuit arrangement is shown in Fig. 15. For the par- 
ticular application under discussion the switch *S is closed for 5,000 
microseconds. During this time, the voltage E = —217 appears at the 
input of the integrator. At the end of this time interval, the transistor 
switch is opened and a reverse current is applied to the feedback con- 
denser C, returning the output voltage to —4.0 volts in about 2500 micro- 
seconds. An alternate way of specifying a low impedance switch is to say 
that the voltage across it be close to zero. For the transistor switch, con- 
nected as shown in Fig. 15, this means that its collector voltage be within 




FIRST STAGE 

OF DC 

AMPLIFIER 



10.5V 



50 K 150 K 

' — WV-HVW 



RESIDUAL 
VOLTAGE BALANCE 



(TO AZS) 



Fig. 15 — Input circuit arrangement of the integrator. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 319 

one millivolt of ground potential during the time the transistor is in 
saturation. Xow, it has been shown that when a junction transistor in 
the common emitter connection is driven into current saturation, the 
minimum voltage between collector and emitter is theoretically equal to 

— in - (25) 

q oci 

where k is the Boltzmann constant, T is the absolute temperature, q is 
the charge of an electron ((kT/q) = 26 millivolts at room temperature), 
and ai is the inverse alpha of the transistor, i.e., the alpha with the 
emitter and collector interchanged. There is an additional voltage drop 
across the transistor due to the bulk resistance of the collector and 
emitter regions (including the ohmic contacts). A symmetrical alloy 
junction transistor with an alpha close to unity is an excellent switch 
because both the collector to emitter voltage and the collector and emit- 
ter resistances are very small. 

At the present time, a reasonable value for the residual voltage* be- 
tween the collector and emitter is 5 to 10 millivolts. This voltage can be 
eliminated by returning the emitter of the transistor switch to a small 
negative potential. This method of balancing is practical because the 
voltage between the collector and emitter of the transistor does not 
change by more than 1.0 millivolt over a temperature range of 0°C to 
50°C. 

4.3. Automatic Zero Set of the Integrator 

A serious problem associated with the transistor integrator is drift. 
The drift is introduced by two sources; variations in the base current of 
the first transistor stage and variations in the base to emitter potential 
of the first stage wdth temperature. In order to reduce the drift, the 
input resistor R and the feedback condenser C must be dissociated from 
the base current and base to emitter potential of the first transistor stage. 
This is accomplished by placing a blocking condenser Cb between point 
T and the base of the first transistor as shown in Fig. 15. An automatic 
zero set circuit is required to maintain the voltage at point T equal to 
zero volts. This AZS circuit uses a magnetic modulator known as a 
"magnettor."^^ 

A block diagram of the AZS circuit is shown in Fig. 16. The dc drift 
current at the input of the amplifier is applied to the magnettor. The 
carrier current required by the magnettor is supplied by a local transistor 

* The inverse alphas of the transistors used in this application were greater 
than 0.95. 



320 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



oscillator. The useful output of the magnettor is the second harmonic of 
the carrier frequency. The amplitude of the second harmonic signal is 
proportional to the magnitude of the dc input current and the phase of 
the second harmonic signal is determined by the polarity of the dc input 
current. The output voltage of the magnettor is applied to an active 
filter which is tuned to the second harmonic frequency. The signal is 
then amplified in a tuned amplifier and applied to a diode gating circuit. 
Depending on the polarity of the dc input current, the gating circuit 
passes either the positive or negative half cycle of the second harmonic 
signal. In order to accomplish this, a square wave at a repetition rate 
equal to that of the second harmonic signal is derived from the carrier 
oscillator and actuates the gating circuit. 

A circuit diagram of the AZS circuit is shown in Figs. 17(a) and 17(b). 
The various sections of the circuit are identified with the blocks shown 
in Fig. 16. The active filter is adjusted for a Q of about 300, and the gain 
of the active filter and tuned amplifier is approximately 1000. The AZS 
circuit provides ±1.0 volt of dc output voltage for ±0.05 microamperes 
of dc input current. The maximum sensitivity of the circuit is limited 
to ±0.005 microamperes because of residual second harmonic generation 
in the magnettor with zero input current. 

When the transistor integrator is used together with the magnettor 
AZS circuit, the slope of the voltage ramp is maintained constant to 
within one part in 8,000 over a temperature range of 20°C to 40°C. 

5.0. The Voltage Comparator 

The voltage comparator is one of the most important circuits used in 
analog to digital converters. The comparator indicates the exact time 
that an input waveform passes through a predetermined reference level. 
It has been common practice to use a vacuum tube blocking oscillator 
as a voltage comparator. ^^ Due to variations in the contact potential, 
heater voltage, and transconductance of the vacuum tube, the maximum 



DC 
INPUT 











AC 








MAGNETTOR 




ACTIVE 
FILTER 




\ 


GATING 
CIRCUIT 






^ 
















A 








■~ 










OSCILLATOR 




GATING 
PULSE 



































DC 

OUTPUT 



Fig. 16 — Block diagram of AZS circuit. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 321 

accuracy of the circuit is limited to about ±100 millivolts. By taking 
advantage of the properties of semiconductor devices, the transistor 
blocking oscillator comparator can be designed to have an accuracy of 
±5 millivolts throughout a temperature range of 20°C to 40°C. 

5.1. General Descri'ption of the Voltage Comparator 

Fig. 18 shows a simplified circuit diagram of the voltage comparator. 
Except for the silicon junction diode D\ , this circuit is essentially a 
transistor blocking oscillator. For the purpose of analysis, assume that 
the reference voltage Vee is set equal to zero. When the input voltage V, 
is large and negative, the silicon diode Di is an open circuit and the jiuic- 
tion transistor has a collector current determined by Rb and Ebb [Expres- 
sion (18)]. The base of the transistor resides at approximately —0.2 
volts. As the input voltage Vi approaches zero, the reverse bias across 
the diode Di decreases. At a critical value of Vi (a small positive poten- 
tial), the dynamic resistance of the diode is small enough to permit the 
circuit to become unstable. The positive feedback provided by trans- 
1 former Ti forces the transistor to turn off rapidly, generating a sharp 
I output pulse across the secondary of transformer T-z . When Vi is large 
and positive, the diode Di is a low impedance and the transistor is main- 
tained cutoff. In order to prevent the comparator from generating more 
than one output pulse during the time that the circuit is unstable, the 
natural period of the circuit as a blocking oscillator must be properly 
chosen. Depending on this period, the input voltage waveform must 
have a certain minimum slope when passing through the reference level 
in order to prevent the circuit from misfiring. 

I The comparator has a high input impedance except during the switch- 
1 ing interval.* When Vi is negative with respect to the reference level, the 
\ input impedance is equal to the impedance of the reverse biased silicon 
i diode. When Vi is positive with respect to the reference level, the input 
I impedance is equal to the impedance of the reverse biased emitter and 
! collector junctions in parallel. This impedance is large if an alloy 
; junction transistor is used. During the switching interval the input im- 
■ pedance is equal to the impedance of a forward biased silicon diode in 
series with the input impedance of a common emitter stage (approxi- 
mately 1,000 ohms). This loading effect is not too serious since for the 
circuit described, the switching interval is less than 0.5 microseconds. 

The voltage comparator shown in Fig. 18 operates accurately on 
voltage waveforms with positive slopes. The voltage comparator will 
operate accurately on waveforms with negative slopes if the diode and 

* The switching interval is the time required for the transistor to turn off. 



322 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




note: all capacitors and inductors 
IN tuned circuits have a 

tolerance of ±0.1% 



Fig. 17(a) — AZS circuit. 

battery potentials are reversed and if an n-p-n junction transistor is 
used. 



5.2. Factors Determining the Accuracy of the Voltage Comparator 

Fig. 19 shows the ac equivalent circuit of the voltage comparator. In 
the equivalent circuit Ri is the dynamic resistance of the diode Di , Rg 
is the source resistance of the input voltage, and R2 is the impedance of 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



323 



the load R^ as it appears at the primary of the transformer T2 . Ri is a 
function of the dc voltage across the diode Z)i . At a prescribed value of 
Ri , the comparator circuit becomes unstable and switches. The relation- 
ship between this critical value of Ri and the transistor and circuit 
parameters is obtained by evaluating the characteristic equation for the 
circuit and by determining the relationship which the coefficients of the 
equation must satisfy in order to have a root of the equation lie in the 
right hand half of the complex frequency plane. To a good approxima- 
tion, the critical value of Ri is given by the expression 



R, -\-R„ + n = 



Mao 



RiCc -\- 






(26) 



N'^Rr 



where M is the mutual inductance of transformer Ti and R2 — ly h^l 
Since the transistor parameters which appear in expression (26) have only 
a small variation with temperature, the critical value of Ri is independent 
of temperature (to a first approximation). 

It will now be shown that the comparator can be designed for an ac- 
curacy of ±5 millivolts throughout a temperature range of 20°C to 40°C. 
In order to establish this accuracy it will be assumed that the critical 
value of 7^1 is equal to 30,000 ohms. This assumption is based on the 




30/iF 



TO LC FILTER 

IN MAGNETTOR 

NPUT CIRCUIT 



4/iF 



+33V 



I+33V 

Fig. 17(b), 900-cycle carrier oscillator. 



324 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



data displayed in Fig. 20 which gives the volt-ampere characteristics of a 
silicon diode measured at 20°C and 40°C. Throughout this temperature 
range, the diode voltage corresponding to the critical resistance of 
30,000 ohms changes by about 30 millivolts. Fortunately, part of this 
voltage variation with temperature is compensated for by the variation 
in voltage Vb-e between the base and emitter of the junction transistor. 
From Fig. 18, 



V, = Vo - Vb-e + Ve 



(27) 



For perfect compensation (Vi independent of temperature), Vb-e should 
have the same temperature variation as the diode voltage Vd . Experi- 




REFERENCE 
I LEVEL 

-I ADJUSTMENT i+ 






Fig. 18 — Simplified circuit diagram of voltage comparator. 




Fig. 19 — Equivalent circuit of voltage comparator. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



325 



0.7 



_) 
O 
> 



0.6 



R, = 30,000 OHMS 



20°C 



> 



u:o.5 

< 

I- 
_l 

o 
> 

o 



0.3 




2 3 4 5 6 

DIODE CURRENT, Ip, IN MICROAMPERES 

Fig. 20 — Volt-ampere characteristic of a silicon junction diode. 

mentally it is found that Yh-e for germanium junction transistors varies 
by about 20 millivolts throughout the temperature range of 20°C to 
40°C. Consequently, the variation in Yi at which the circuit switches is 
±5 millivolts. 

It is apparent from Fig. 20 that the accuracy of the comparator in- 
creases slightly for critical values of R\ greater than 30,000 ohms, but 
decreases for smaller values. For example, the accuracy of the comparator 
is ±10 millivolts for a critical value of U\ equal to 5,000 ohms. In gen- 
eral, the critical value of R\ should be chosen between 5,000 and 100,000 
ohms. 



5.3. A Practical Yoltage Comparator 

Fig. 21 shows the complete circuit diagram of a voltage comparator. 
The circuit is designed to generate a sharp output pulse* when the input 
voltage waveform passes through the reference level (set by Yee) with a 
positive slope. The pulse is generated by the transistor switching from 
the "on" state to the "off" state. To a first approximation the amplitude 
of the output pulse is proportional to the transistor collector current 
during the "on" state. When the input voltage waveform passes through 
the reference level with a negative slope an undesirable negative pulse is 
generated. This pulse is eliminated by the point contact diode D2 . 

The voltage comparator is an unstable circuit and has the properties 

* For the circuit values shown in Fig. 21, the output pulse has a peak amplitude 
of about 6 volts, a rise time of 0.5 microseconds, and a pulse width of about 2.0 
microseconds. 



32G 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



of a free running blocking oscillator after the input voltage Vi passes 
through the reference level. After a period of time the transistor will 
return to the "on" state unless the voltage Vi is sufficiently large at this 
time to prevent switching. In order to minimize the required slope of the 
hiput waveform the time interval between the instant Vi passes through 
the reference level and the instant the transistor would naturally switch 
to the "on" state must be maximized. This time intei-val can be con- 
trolled by connecting a diode D3 across the secondary winding of trans- 
former Ti . When the transistor turns off, the current which was flowing 
through the secondary of transformer Ti(Ic) continues to flow through 
the diode D3 so that L2 and D3 form an inductive discharge circuit. The 
point contact diode D3 has a forward dynamic resistance of less than 10 
ohms and a forward voltage drop of 0.3 volt. If the small forward re- 
sistance of the diode is neglected, the time required for the current in the 
circuit to fall to zero is 



T = 



0.3 



(28) 



During the inductive transient, 0.3 volt is induced into the primary of 
transformer Ti (since N = 1) maintaining the transistor cutoff. The 
duration of the inductive transient can be made as long as desired by 
increasing L2 . However, there is the practical limitation that increasing 
L2 also increases the leakage inductance of transformer Ti , and in turn, 



I 



I 



-4.5V 



5.1K 



250A 




:iD2 



>3K 



A-l- 



OUTPUT 
PULSE 



V- 



INPUT 
WAVEFORM 



PULSE 
AMPLITUDE^, 
ADJUSTMENT^ 



•^ 



2.5 MEG POT. 
I- 



jr 



ee' 



Ij, = 4 MILS 

L, = L2= 5 MILLIHENRIES 

L', = L2= 5 MILLIHENRIES 

COEFFICIENT OF 

COUPLING = 0.99 



REFERENCE 
g^ LEVEL 

''adjustment 

MA 1 



I 
I 

-46V 



I 
I 

-t-1.5V 



100 OHM 
POT. 



I 
I 

-1.5V 



Fig. 21 — Voltage comparator. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 



327 



increases the switching time. The circuit shown in Figure 21 does not 
misfire when used with voltage waveforms having slopes as small as 25 
millivolts per microsecond, at the reference level. 



6.0. A TRANSISTOR VOLTAGE ENCODER 



6.1. Circuit Arrangement 



The transistor circuits previously described can be assembled into a 
voltage encoder for translating analog voltages into equivalent time 
intervals. This encoder is especially useful for converting analog informa- 
, tion (in the form of a dc potential) into the digital code for processing 
in a digital system. Fig. 22 shows a simplified block diagram of the 
encoder. The voltage I'amp generated by the integrator is applied to 
amplitude selector number one and to one input of a summing amplifier. 
The amplitude selector is a dc amplifier which amplifies the voltage ramp 
in the vicinity of zero volts. Voltage comparator number one, which 
follows the amplitude selector, generates a sharp output pulse at the 
exact instant of time that the voltage ramp passes through zero volts. 

The analog input voltage, which has a value between and —15 
volts,* is applied to the second input of the summing amplifier. The 
output voltage of the summing amplifier is zero whenever the ramp 



INTEGRATOR 



N0.1 



N0.1 




3000^65 



SUMMING 
AMPLIFIER 




AMPLITUDE 
SELECTORS 



VOLTAGE 
COMPARATORS 



ANALOG 

INPUT VOLTAGE 

0-^-16V 



N0.2 



N0.2 



Fig. 22 — • Simplified block diagram of voltage encoder. 



* If the analog input voltage does not lie in this range, then the voltage gain 
of the summing amplifier must be set so that the analog voltage at the output of 
the summing amplifier lies in the voltage range between and +15 volts. 



328 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



voltage is equal to the negative of the input analog voltage. At this 
instant of time the second voltage comparator generates a sharp output 
pulse. The time interval between the two output pulses is proportional 
to the analog input voltage if the voltage ramp is linear and has a con- 
stant slope at all times. 

6.2. The Amplitude Selector i 

The amplitude selector increases the slope of the input voltage wave- 
form (in the vicinity of zero volts) sufficiently for proper operation of the 
voltage comparator. The amplitude selector consists of a limiter and a 
dc feedback amplifier as shown in Fig. 23. The two oppositely poled 
silicon diodes Di and D2 , limit the input voltage of the dc amplifier to 
about ±0.65 volts. The dc amplifier has a voltage gain of thirty, and so 
the maximum output voltage of the amplitude selector is limited to 
about ±19.5 volts. The net voltage gain between the input and output 
of the amplitude selector is ten. 

The principal requirement placed on the dc amplifier is that the input 
current and the output voltage be zero when the input voltage is zero. 
This is accomplished by placing a blocking condenser Cb between point 
T and the base of the first transistor stage, and by using an AZS circuit 
to maintain point T at zero volts. The dc and AZS amplifiers are identical 
in configuration to the amplifiers shown in Fig. 12. The dc amplifier is 



50 K 

-VvV 



50 K 



:|N 



D 



1:: 



SILICON 
DIODES 



Dp 



1.5 MEG 



Cb 

250 /ZF 



500 K 




I 



OUT 



I V^^ »— AAA^ 

50 K 1.5 MEG 



-1 



Fig. 23 — Block diagram of the amplitude selector. 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 329 

designed to have about 15.6 db less feedback than that shown in Fig. 10 
since this amount is adequate for the present purpose. 

The bandwidth of the dc ampHfier is only of secondary importance 
because the phase shifts introduced by the two amplitude selectors in 
the voltage encoder tend to compensate each other. 

6.3. Experimental Results 

The accuracy of the voltage encoder is determined by applying a 
precisely measured voltage to the input of the summing amplifier and by 
measuring the time interval between the two output pulses. The maxi- 
mum error due to nonlinearities in the summing amplifier and the voltage 
ramp is less than ±0.5 microseconds for a maximum encoding time of 
3,000 microseconds. An additional error is introduced by the noise voltage 
generated in the first transistor stage of the summing amplifier. The 

! RMS noise voltage at the output of the summing amplifier is less than 
0.5 millivolts. This noise voltage produces an RMS jitter of 0.25 micro- 

I seconds in the position of the second voltage comparator output pulse. 

; The over-all accuracy of the voltage encoder is one part in 4,000 through- 

' out a temperature range of 20°C to 40°C. 

1 
I 

i ACKNOWLEDGEMENTS 

! 

I The author wishes to express his appreciation to T. R. Finch for the 
^ advice and encouragement received in the course of this work. D. W. 
! Grant and W. B. Harris designed and constructed the magnettor used 
' in the AZS circuit of the integrator. 

I Appendix I 

I RELATIONSHIP BETWEEN RETURN DIFFERENCE AND LOOP CURRENT 
i TRANSMISSION 

} In order to place the stability analysis of the transistor feedback ampli- 
fier on a sound basis, it is desirable to use the concept of return differ- 
ence. It will be shown that a measurable quantity, called the loop current 
transmission, can be related to the return difference of aZc with reference 
Ve .*• t In Fig. 24, N represents the complete transistor network exclusive 
of the transistor under consideration. The feedback loop is broken at 
the input to the transistor by connecting all of the feedback paths to 

* In this appendix it is assumed that the transistor under consideration is in 
the common emitter connection. The discussion can be readily extended to the 
other transistor connections. 

t This fact was pointed out by F. H. Tendick, Jr. 



330 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Te+rb 



^6^4 ( 'V 



-aZcLb 




N 

COMPLETE 
AMPLIFIER 
EXCLUSIVE 

OF THE 
TRANSISTOR 
IN QUESTION 



Fig. 24 — Measurement of loop current transmission. 



ground through a resistance (/•<; + n) and a voltage r J4 • Using the 
nomenclature given in Reference 8, the input of the complete circuit is 
designated as the first mesh and the output of the complete circuit is 
designated as the second mesh. The input and output meshes of the 
transistor under consideration are designated 3 and 4, respectively. The 
loop current transmission is equal to I3', the total returned current when 
a unit input current is applied to the base of the transistor. 

The return difference for reference Ve is equal to the algebraic differ- 
ence* between the unit input current and the returned current h'. 1 3 is 
evaluated by multiplying the open circuit voltage in mesh 4 (produced 
by the unit base current) by the backward transmission from mesh 4 to 
mesh 3 with zero forward transmission through the transistor under 
consideration. The open circuit voltage in mesh 4 is equal to (re — aZc). 
The backward transmission is determined with the element aZc , in the 
fourth row, third column of the circuit determinant, set equal to Ve . 
Hence, the return difference is expressed as 

A43 



Fr' = 1 + {aZc - re) 



(Al)t 



Fr' = 



A''* + {aZc - r.)A 



43 



ir', 



(A2) 



Fr'.= 



A^'' 



= 1+ Tr' 



(A3) 



The relative return ratio Tr', is equal to the negative of the loop current 
transmission and can be measured as shown in Fig. 24. The voltage reh 
takes into account the fact that the junction transistor is not perfectly 



* The positive direction for the returned current is chosen so that if the original 
circuit is restored, the returned current flows in the same direction as the input 
current. 

t A''« is the network determinant with the element aZc in the fourth row, third 
column of the circuit determinant set equal to r, . 



TRANSISTOR CIRCUITS FOR ANALOG AND DIGITAL SYSTEMS 331 



unilateral. Fortunately, in many applications, this voltage can be neg- 
lected even at the gain and phase crossover frequencies. 

In the case of single loop feedback amplifiers. A""* will not have any 
zeros in the right hand half of the complex frequency plane. A study of 
the stability of the amplifier can then be based on F^-, or T^-, . 

Appendix II 

INTERSTAGE NETWORK SHAPING 

This appendix presents the analysis of the circuit shown in Fig. 7(a). 
The input impedance of the common emitter connected junction tran- 
sistor is given by the expression 

^iNPUT = n-\- re(l - Gl) (A4) 

where Gi is the current transmission of the common emitter stage, ex- 
pression (13). The current transmission A of the complete circuit is equal 
to 

A = ^ = ^ 

I\ Zz -\- ^ IN PUT 



G, 



(A5) 



where Z3 = i?3 + V^ + (l/p<^3). Combining (13), (A4), and (A5) yields 



ao 



A = 



1 — ao + 5 



1 + 



C03 



+ V 






\ W5/ I, Wl 



(A6) 



+ p^ 



W5 , CsOO^in + Te -\- R3) 



_C0iC03- 



+ 



PCO5 



where 



WaWc(l — tto -}- 6) J ' CO3^C0aC0c(l — ^Q "j- 6) j 
^ ^ Rl + Te 

COl = 
Wc = 



CO3 



OJs = 



(1 - ao + 5) 




1 + 6 _^ 1 

1 




(R^ + r,)Co 
1 




1 




~. . ^« 


C 


(1 - ao 


+ 5)J^ 



332 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Expression (A6) is valid if l/ws ^ 1/coi + RzCz . The denominator of the 
expression indicates a falUng 6 db per octave asymptote with a corner, 
frequency at ws . The second factor in the denominator can be approxi- 
mated bj^ a falHng 6 db per octave asymptote with a corner frequency at 



COl 



1 ^^ 

n + 



(1 - ao + 5) 



] 



n -\- Te -^ Rz -\- W1L3 

pkis additional phase and amplitude contributions at higher f recjuencies 
due to the y and p terms. If 



COzCzRz 



then the circuit has a rising 12 db per octave asymptote with a corner 
frequency at C03 . Fig. 7(b) shows the amplitude and phase of the current 
transmission. 



REFERENCES 

1. Felker, J. H., Regenerative Amplifier for Digital Computer Applications, 

Proc. I.R.E., pp. 1584-1596, Nov., 1952. 

2. Korn, G. A., and Korn, T. M., Electronic Analog Computers, McGraw-Hil 

Book Company, pp. 9-19. 

3. Wallace, R. L. and Pietenpol, W. J., Some Circuit Properties and Applications 

of n-p-n Transistors, B. S.T.J. , 30, pp. 530-563, July, 1951. 

4. Shockley, W., Sparks, M. and Teal, G. K., The p-n Junction Transistor, 

Physical Review, 83, pp. 151-162, July, 1951. 

5. Pritchard, R. L., Frequenc}' Variation of Current-Amplification for Junction 

Transistors, Proc. I.R.E., pp. 1476-1481, Nov., 1952. 

6. Early, J. M., Design Theory of Junction Transistors, B.S.T.J., 32, pp. 1271- 

1312, Nov., 1953. 

7. Sziklai, G. C, Symmetrical Properties of Transistors and Their Applications, 

Proc. I.R.E., pp. 717-724, June, 1953. 

8. Bode, H. W., Network Analysis and Feedback Amplifier Design, Van Nos- 

trand Co., Inc., Chapter IV. 

9. Bode, H. W., Op. Cit., pp. 66-69. 

10. Bode, H. W., Op. Cit., pp. 162-164. 

11. Bargellini, P. M. and Herscher, M. B., Investigation of Noise in Audio Fre- 

quency Amplifiers Using Junction Transistors, Proc. I.R.E., pp. 217-226,' 
Feb., 1955. 

12. Bode, H. W., Op. Cit., pp. 464-468, and pp. 471-473. 

13. Keonjian, E., Temperature Compensated DC Transistor Amplifier, Proc: 

I.R.E., pp. 661-671, April, 1954. 

14. Kretzmer, E. R., An Amplitude Stabilized Transistor Oscillator, Proc. I.R.E.,« 

pp. 391-401, Feb., 1954. i 

15. Goldberg, E. A., Stabilization of Wide-Band Direct-Current Amplifiers for 

Zero and Gain, R.C.A. Review, June, 1950. 

16. Ebers, J. J. and Moll, J. L., Large Signal Behavior of Junction Transistors. 

Proc. I.R.E., pp. 1761-1772, Dec, 1954. 

17. Manlej', J. M., Some General Properties of Magnetic Amplifiers, Proc. I.R.K. 

March, 1951. 

18. M.I.T., Waveforms, Volume 19 of the Radiation Laboratories Series. McGraw 

Hill Book Company, pp. 342-344. 



Electrolytic Shaping of Germanium 
, and Silicon 

^ By A. UHLIR, JR. 

i (Manuscript received November 9, 1955) 

Properties of electrolyte-semiconductor barriers are described, with em- 
phasis on germanium. The use of these barriers in localizing electrolytic 
! etching is discussed. Other localization techniques are mentioned. Electro- 
lytes for etching germanium and silicon are given. 

I 

INTRODUCTION 

I 

I Mechanical shaping techniques, such as abrasive cutting, leave the 
surface of a semiconductor in a damaged condition which adversely 
affects the electrical properties of p-n junctions in or near the damaged 
j material. Such damaged material may be removed by electrolytic etch- 
ing. Alternatively, all of the shaping may be done electrolytically, so 
that no damaged material is produced. Electrolytic shaping is particu- 
[ larly well suited to making devices with small dimensions. 
I A discussion of electrolytic etching can conveniently be divided into 
[■ two topics — the choice of electrolyte and the method of localizing the 
ji etching action to produce a desired shape. It is usually possible to find 
1 an electrolyte in which the rate at which material is removed is accurately 
proportional to the current. For semiconductors, just as for metals, the 
I choice of electrolyte is a specific problem for each material ; satisfactory 
j electrolytes for germanium and silicon will be described. 

The principles of localization are the same, whatever the electrolyte 

used. Electrolytic etching takes place where current flows from the 

semiconductor to the electrolyte. Current flow may be concentrated at 

I certain areas of the semiconductor-electrolyte interface by controlling 

the flow of current in the electrolyte or in the semiconductor. 

LOCALIZATION IN ELECTROLYTE 

Localization techniques involving the electrolytic current are appli- 
cable to both metals and semiconductors. In some of these techniques, 

333 






334 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

the localization is so effective that the barrier effects found with n-type 
semiconductors can be ignored; if not, the barrier can be overcome by 
light or heat, as will be described below. 

If part of the work is coated with an insulating varnish, electrolytic 
etching will take place only on the uncoated surfaces. This technique, 
often called "masking," has the limitation that the etching undercuts 
the masking if any considerable amount of material is removed. The i 
same limitation applies to photoengraving, in which the insulating coat- 
ing is formed by the action of light. 

The cathode of the electrolytic cell may be limited in size and placed 
close to the work (which is the anode). Then the etching rate will be 
greatest at parts of the work that are nearest the cathode. Various 
shapes can be produced by moving the cathode with respect to the I 
work, or by using a shaped cathode. For example, a cathode in the form | 
of a wire has been used to slice germanium. 

Instead of a true metallic cathode, a "virtual cathode" may be used 
to localize electrolysis.^ In this technique, the anode and true cathode 
are separated from each other by a nonconducting partition, except for 
a small opening in the partition. As far as localization of current to the 
anode is concerned, the small opening acts like a cathode of equal size 
and so is called a virtual cathode. The nonconducting partition may 
include a glass tube drawn down to a tip as small as one micron diameter 
but nevertheless open to the flow of electrolytic current. With such a 
tip as a virtual cathode, micromachining can be conducted on a scale 
comparable to the wavelength of visible light. A general advantage of 
the virtual cathode technique is that the cathode reaction (usually 
hydrogen evolution) does not interfere with the localizing action nor 
with observation of the process. :| 

In the jet-etching technique, a jet of electrolyte impinges on the 
work.^'* The free streamlines that bound the flowing electrolyte are 
governed primarily by momentum and energy considerations. In turn, 
the shape of the electrolyte stream determines the localization of etch- 
ing. A stream of electrolyte guided by wires has been used to etch semi- 
conductor devices.^ Surface tension has an important influence on the 
free streamlines in this case, 

PROPERTIES OF ELECTROLYTE-SEMICONDUCTOR BARRIERS 

The most distinctive feature of electrolytic etching of semiconductors 
is the occurrence of rectifying barriers. Barrier effects for germanium 
will be described; those for silicon are qualitatively similar. 

The voltage-current curves for anodic n-type and p-type germanium 



ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 



335 



[in 10 per cent KOH are shown in Fig. 1. Tlie concentration of KOH 

[is not critical and other electrolytes give similar results. The voltage 

'drop for the p-type specimen is small. For anodic n-type germanium, 

! however, the barrier is in the reverse or blocking direction as evidenced 

by a large voltage drop. The fact that n-type germanium differs from 

p-type germanium only by very small amounts of impurities suggests 

that the barrier is a semiconductor phenomenon and not an electro- 

i chemical one. This is confirmed by the light sensitivity of the n-type 

1 voltage-current characteristic. Fig. 2 is a schematic diagram of the 

! arrangement for obtaining voltage-current curves. A mercury-mercuric 

loxide-10 per cent KOH reference electrode was used at first, but a gold 

(wire was found equally satisfactory. At zero current, a voltage Vo exists 

j between the germanium and the reference electrode ; this voltage is not 

[included in Fig. 1. 

I The saturation current Is , measured for the n-type barrier at a 
\moderate reverse voltage (see Fig. 1), is plotted as a function of tempera- 
Iture in Fig. 3. The saturation current increases about 9 per cent per 
[degree, just as for a germanium p-n junction, which indicates that the 

I 



40 



35 



30 



^25 

Lil 

O 20 



15 



10 





1 






12 OHM-CM 
n-TYPE 




/ 












DAR\<. 








1 
/ 


/ 










1 
1 
1 
1 

1 


/ 










1 
1 
1 

1 
1 










WITH ; 
LIGHT ^' 
1 


1 










I 
1 

1 












1 




n 






i 


1 

1 
/ 
/ P- 


FYPE 



10 20 30 40 50 60 

CURRENT FLOW IN MILLIAMPERES PER CM^ 



Fig. 1 — Anodic voltage-current characteristics of germanium. 



336 



THE BELL SYSTEM TECHXICAL JOURNAL, MARCH 1956 



current is proportional to the equilibrium density of minority carriers 
(holes). The same conclusion may be drawn from Fig. 4, which shows 
that the saturation current is higher, the higher the resistivity of the 
n-type germanium. But the breakdown voltages are variable and usu- 
ally much lower than one would expect for planar p-n junctions made, 
for example, by alloying indium into the same n-type germanium. 

Breakdown in bulk junctions is attributed to an avalanche multipli- 
cation of carriers in high fields.^ The same mechanism may be responsible 
for breakdown of the germanium-electrolyte barrier; low and variable 
breakdown voltages may be caused by the pits described below. 

The electrolyte-germanium barrier exhibits a kind of current multi- 
plication that differs from high-field multiplication in two respects: it 
occurs at much lower reverse voltages and does not vary much with 
voltage.^ This effect can be demonstrated very simply by comparison 
with a metal-germanium barrier, on the assumption that the latter has 
a current multiplication factor of unity. This assumption is supported 
by experiments which indicate that current flows almost entirely by 
hole flow, for good metal-germanium barriers. 

The experimental arrangement is indicated in Fig. 5(a) and (b). The 
voltage-current curves for an electrolyte barrier and a plated barrier on 
the same slice of germanium are shown in Fig. 5(c).* The curves for the 



REFERENCE 
ELECTRODE 



CATHODE 




LIGHT 



Fig. 2 — Arrangement for obtaining voltage current characteristics. 



* In Fig. 5 the dark current for the phited barrier is much hirger than can be 
exphained on the basis of hole current; it is even higher than the dark current for 
the electrolyte barrier, which should be at least 1.4 times the hole current. This 
excess dark current is believed to be leakage at the edges of the plated area and 
probably does not affect the intrinsic current multiplication of the plated barrier 
as a whole. 



ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 



337 



10 



2 
o 

a. 

01 

a. 
to 

Ui 

oc 

LU 
Q. 

5 
< 

_) 



m 
cc 
tr 

3 
U 

z 
o 

cc 

3 
(0 



•> I 



10" 













/ 


{ 


- 










/ 




- 








1 


/ 












7 














/ 














/ 
























/ 










- 




/% 










- 




/ 












/ 














/ 














/ 












n/ 


^ 












y 















i<:i 



10 20 30 40 50 60 

TEMPERATURE IN DEGREES CENTIGRADE 

Fig. 3 — Temperature variation of the saturation current of a barrier between 
5.5 ohm-cm n-type germanium and 10 per cent KOH solution. 



illuminated condition were obtained by shining light on a dry face of a 
slice while the barriers were on the other face. The difference between 
the light and dark currents is larger for the electrolyte-germanium bar- 
rier than for the metal-germanium barrier, by a factor of about 1.4. 

The transport of holes through the slice is probably not very different 
for the two barriers. Therefore, a current multiplication of 1.4 is indi- 
cated for the electrolyte barrier. About the same value was found for 
temperatures from 15°C to 60°C, KOH concentrations from 0.01 per 
cent to 10 per cent, n-type resistivities of 0.2 ohm-cm to 6 ohm-cm, 
light currents of 0.1 to 1.0 ma/cm^, and for O.IN indium sulfate. 

Evidently the flow of holes to the electrolyte barrier is accompanied 
by a proportionate return flow of electrons, which constitutes an addi- 
tional electric current. Possible mechanisms for the creation of the 
electrons will be discussed in a forthcoming article. 



338 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

7 



> 4 



LU 

o 
> 




I 



0,5 1.0 1 

CURRENT F 



5 2.0 2.5 3.0 3.5 4.0 

LOW IN MILLIAMPERES PER CM^ 



4.5 



Fig. 4. — Anodic voltage -current curves for various resistivities of germanium. 



SCRATCHES AND PITTING 

The voltage- current curve of an electrolyte-germanium barrier is 
very sensitive to scratches. The curves given in the illustrations were : 
obtained on material previously etched smooth in CP-4, a chemical I 
etch.* '' 

If, instead, one starts with a lapped piece of n-type germanium, the 
electrolyte-germanium barrier is essentially "ohmic;" that is, the voltage 
drop is small and proportional to the current. A considerable reverse 
voltage can be attained if lapped n-type germanium is electrolytically 
etched long enough to remove most of the damaged germanium. How- 
ever, a pitted surface results and the breakdown voltage achieved is 
not as high as for a smooth chemically-etched surface. 

The depth of damage introduced by typical abrasive sawing and 
lapping was investigated by noting the voltage-current curve of the 



Br2 



Five parts HNO3 , 3 parts 48 per cent HF, 3 parts glacial acetic acid, ^0 P^-^t 



ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 



339 



electrolyte-germanium barrier after various amounts of material had 
been removed by chemical etching. After 20 to 50 microns had been re- 
moved, further chemical etching produced no change in the barrier 
characteristic. This amount of material had to be removed even if the 
lapping was followed by polishing to a mirror finish. The voltage-current 
curve of the electrolyte-germanium barrier will reveal localized damage. 
On the other hand, the photomagnetoelectric (PME) measurement of 



I 

-< — 








REFERENCE 
ELECTRODE 


CATHODE- 


-- 


-^ 












■< 


■y 


GLASS TUBING 

CEMENTED 

TO Ge 


E 


LECTROLYT 


z i 




N-Ge 






■^ 










1 

1 

1 
1 





<rri> 



(a) 




ELECTROPLATED 
INDIUM 





METAL TO N-Ge 
CONTACT 

ELECTROLYTE TO 
N-Ge BARRIER 



(c) 



2 4 6 

CURRENT, I, IN MILLIAMPERES 

PER CM 2 



Fig. 5 — Determination of the current multiplication of the barrier between 
6 ohm-cm n-type germanium and an electrolyte. 



340 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




Fig. 6 — Electrolytic etch pits on two sides of 0.02-inch slice of n-type germa- 
nium. Half of the slice was in contact with the electrolyte. 

surface recombination velocity gives an evaluation of the average con- 
dition of the surface. A variation of the PME method has been used 
to study the depth of abrasion damage; the damage revealed by this 
method extends only to a depth comparable to the abrasive size. 

A scratch is sufficient to start a pit that increases in size without limit 
if anodic etching is prolonged. However, a scratch is not necessary. Pits 
are formed even when one starts with a smooth surface produced by 
chemical etching. A drop in the breakdown voltage of the barrier is 
noticed when one or more pits form. The breakdown voltage can be 
restored by masking the pits with polystyrene cement. 

Evidence that the spontaneous pits are caused by some features of 
the crystal, itself, was obtained from an experiment on single-crystal 
n-type germanium made by an early version of the zone-leveling process. 
A slice of this material was electrolytically etched on both sides, after 
preliminary chemical etching. Photographs of the two sides of the slice 
are shown in Fig. 6. Only half of the slice was immersed in the electro- 
lyte. The electrolytic etch pits are concentrated in certain regions of 
the slice — the same general regions on both sides of the slice. It is 
interesting that radioautographs and resistivity measurements indicate 
high donor concentrations in these regions. Improvements, including 
more intensive stirring, were made in the zone-leveling process, and the 
electrolytic etch pit distribution and the donor radioautographs have 
been much more uniform for subsequent material. 

Several pits on a (100) face are shown in Fig. 7. The pits grow most 
rapidly in (100) directions and give the spiked effect seen in the illustra- 
tion. Toiler prolonged etching, the spikes and their branches form a com- 
plex network of caverns beneath the surface of the germanium. 

High-field carrier generation may be responsible for pitting. A locally 



ELECTROLYTIC SHAPING OF GERMAXIUM AND SILICON 



341 




Fig. 7 — Electrolytic etch pits on n-type germanium. 

high donor concentration would favor breakdown, as would any con- 
cavity of the germanium surface (which would cause a higher field for 
a given voltage) . Very high fields must occur at the points of spikes such 

jas those shown in Fig. 7. The continued growth of the spikes is thus 
favored by their geometry. 

Microscopic etch pits arising from chemical etching have been corre- 

;lated with the edge dislocations of small-angle grain boundaries. A 

I specimen of n-type germanium with chemical etch pits was photomicro- 
graphed and then etched electrolytically. The etch pits produced elec- 
trolytically could not be correlated with the chemical etch pits, most 
of which were still visible and essentially unchanged in appearance. 
Also, no correlation could be found between either kind of etch pit and 
the locations at which copper crystallites formed upon immersion in a 
copper sulfate solution. Microscopic electrolytic etch pits at dislocations 

j in p-type germanium have been reported in a recent paper that also 
I mentions the deep pits produced on n-type germanium.^* 
y Electrolytic etch pits are observed on n-type and high-resistivity 
silicon. These etch pits are more nearly round than those produced in 
germanium. 

In spite of the pitting phenomenon, electrolytic etching is success- 



342 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



I 

fully used in the fabrication of devices involving n-type semiconductors. 
Pitting can be reduced relative to "normal" uniform etching by any 
agency that increases the concentration of holes in the semiconductor. 
Thus, elevated temperatures, flooding with light, and injection of holes 
by an emitter all favor smooth etching. 



SHAPING BY MEANS OF INJECTED CARRIERS 



I 



Hole-electron pairs are produced when light is absorbed by semi- 
conductors. Light of short wavelength is absorbed in a short distance, 
while long wavelength light causes generation at considerable depths. 
The holes created by the light move by diffusion and drift and increase 
the current flow through an anodic electrolyte-germanium barrier at 
whatever point they happen to encounter the barrier. In general, more 
holes will diffuse to a barrier, the nearer the barrier is to the point at 
which the holes are created. For n-type semiconductors, the current 
due to the light can be orders of magnitude greater than the dark cur- 
rent, so that the shape resulting from etching is almost entirely deter- 
mined by the light. As shown in Fig. 3, the dark current can be made 
very small by lowering the temperature. 

An example of the shaping that can be done with light is shown in 
Fig. 8. A spot of light impinges on one side of a wafer of n-type germanium 
or silicon. The semiconductor is made anodic with respect to an etching 
electrolyte. Accurately concentric dimples are produced on both sides of 
the wafer. Two mechanisms operate to transmit the effect to the oppo- 
site side. One is that some of the light may penetrate deeply before 
generating a hole-electron pair. The other is that a fraction of the car- 
riers generated near the first surface will diffuse to the opposite side. 
By varying the spectral content of the light and the depth within the \ 



\ 




-n-TYPE SEMICONDUCTOR 



LIGHT 



I I 



Fig. 8 — Double dimpling with light. 



ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 



343 



wafer at which the light is focused, one can produce dimples with a vari- 
,'ety of shapes and relative sizes. 

I It is obvious that the double-dimpled wafer of Fig. 8 is desirable for 
{the production of p-n-p alloy transistors. For such use, one of the most 
[important dimensions is the thickness remaining between the bottoms 
of the two dimples. As has been mentioned in connection with the jet- 
I etching process, a convenient way of monitoring this thickness to de- 
Itermine the endpoint of etching is to note the transmission of light of 
[suitable wavelength.^ There is, however, a control method that is itself 
[automatic. It is based on the fact that at a reverse-biased p-n junction 
[Or electrolyte-semiconductor barrier there is a space-charge region that 
is practically free of carriers. When the specimen thickness is reduced 
so that space-charge regions extend clear through it, current ceases to 
flow and etching stops in the thin regions, as long as thermally or op- 
tically generated carriers can be neglected. However, more pitting is to 
be expected in this method than when etching is conducted in the pres- 
ence of an excess of injected carriers. 

A p-n junction is a means of injecting holes into n-type semiconduc- 
tors and is the basis of another method of dimpling, shown in Fig. 9. 
The p-n junction can be made by an alloying process such as bonding 
an acceptor-doped gold wire to germanium. The ohmic contact can be 
made by bonding a donor-doped gold wire and permits the injection of 
a greater excess of holes than would be possible if the current through 
the p-n junction were exactly equal to the etching current. Dimpling 
without the ohmic contact has been reported.^ 



14 



OHMIC CONTACT 



p-n JUNCTION 




Fig. 9 — Dimpling with carriers injected by a p-n junction. 



344 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



CONTROL BY OHMIC CONDUCTION 

The carrier-injection shaping techniques work very well for n-typei 
material. It is also possible to inject a significant number of holes intos 
rather high resistivity p-type material. But what can be done about: 
p-type material in general, short of developing cathodic etches? ] 

The ohmic resistivity of p-type material can be used as shown in Fig.!^ 
10. More etching currect flows through surfaces near the small contact 
than through more remote surfaces. A substantial dimpling effect is 
observed when the semiconductor resistivity is equal to the electrolyte 
resistivity, but improved dimpling is obtained on higher resistivity 
semiconductor. This result is just what one might expect. But the math- 
ematical solution for ohmic flow from a point source some distance from 
a planar boundary between semi-infinite materials of different conduc- 
tivities shows that the current density distribution does not depend on 
the conductivities. An important factor omitted in the mathematical 
solution is the small but significant barrier voltage, consisting largely of 
electrochemical polarization in the electrolyte. The barrier voltage is; 
approximately proportional to the logarithm of the current density; 
while the ohmic voltage drops are proportional to current density. Thus,- 
high current favors localization. 

ELECTROLYTES FOR ETCHING GERMANIUM AND SILICON » 

The electrolyte usually has two functions in the electrolytic etching 
of an oxidizable substance. First, it must conduct the current necessary 
for the oxidation. Second, it must somehow effect removal of the oxida- 
tion product from the surface of the material being etched. 

The usefulness of an electrolytic etch depends upon one or both of: 



ANY CONTACT, 
PREFERABLY OHMIC 




^//yyyy//y/y/y////////y////y///// yyyyyyyyyyy7^ 



Fig. 10 — Dimpling by ohmic conduction. 



ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 345 

the following situations — the electrolytic process accomplishes a reac- 
tion that cannot be achieved as conveniently in any other way or it 
permits greater control to be exercised over the reaction. Accordingly, 
chemical attack by the chosen electrolyte must be slight relative to the 
electrochemical etching. 

A smooth surface is probably desirable in the neighborhood of a p-n 
junction, to avoid field concentrations and lowering of breakdown 
voltage. Therefore, a tentative requirement for an electrolyte is the 
production of a smooth, shiny surface on the p-type semiconductor. Such 

\ an electrolyte will give a shiny but possibly pitted surface on n-type 

j specimens of the same semiconductor. 

The effective valence of a material being electrolytically etched is 

; defined as the number of electrons that traverse the circuit divided by 
the number of atoms of material removed. (The amount of material 

! removed was determined by weighing in the experiments to be described.) 
If the effective valence turns out to be less than the valence one might 
predict from the chemistry of stable compounds, the etching is sometimes 
said to be "more than 100 per cent efficient." Since the anode reactions 
in electrolytic etching may involve unstable intermediate compounds 
and competing reactions, one need not be surprised at low or fractional 
effective valences. 

Germanium can be etched in many aqueous electrolytes. A valence of 
almost exactly 4 is found. That is, 4 electrons flow through the circuit 
for each atom of germanium removed. For accurate valence measure- 
ments, it is advisable to exclude oxygen by using a nitrogen atmosphere. 
Potassium hydroxide, indium sulfate, and sodium chloride solutions are 
among those that have been used. Sulfuric acid solutions are prone to 

) yield an orange-red deposit which may be a suboxide of germanium/* 

I Similar orange deposits are infrequently encountered with potassium 

I hydroxide. 

Hydrochloric acid solutions are satisfactoiy electrolytes. The reaction 

I product is removed in an unusual manner when the electrolyte is about 
2N hydrochloric acid. Small droplets of a clear liquid fall from the etched 
regions. These droplets may be germanium tetrachloride, which is denser 
than the electrolyte. They turn brown after a few seconds, perhaps be- 
cause of hydrolysis of the tetrachloride. 

Etching of germanium in sixteen different aqueous electroplating 
electrolytes has been mentioned. Germanium can also be etched in the 
partly organic electrolytes described below for silicon. 

One would expect that silicon could be etched by making it the anode 
in a cell with an aqueous hydrofluoric acid electrolyte. The seemingly 



346 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 | 

) 

likely oxidation product, silicon dioxide, should react with the hydro-! 
fluoric acid to give silicon tetrafluoride, which could escape as a gas. In 
fact, a gas is formed at the anode and the silicon loses weight. But the 
gas is hydrogen and an effective valence of 2.0 ± 0.2 (individual deter- 
minations ranged from 1.3 to 2.7) was found instead of the value 4 that i 
might have been expected. The quantity of hydrogen evolved is con- 
sistent with the formal reaction 



Si —> Si"*"'" + me (electrochemical oxidation) 

Si+™ + (4-to)H+ -^ Si+' + Vz (4-m)H2 (chemical oxidation) 






where m is about two. The experiments were done in 24 per cent to 48 
per cent aqueous solutions of HF at current densities up to 0.5 amp/cm^. 

The suggestion that the electrochemical oxidation precedes the chemi- 
cal oxidation is supported by the appearance and behavior of the etched 
surfaces. Instead of being shiny, the surfaces have a matte black, brown, 
or red deposit. 

At 40 X magnification, the deposit appears to consist of flakes of a; 
resinous material, tentatively supposed to be a silicon suboxide. A re- 
markable reaction can be demonstrated if the silicon is rinsed briefly in 
water and alcohol after the electrolytic etch, dried, and stored in air for 
as long as a year. Upon reimmersing this silicon in water, one can observe 
the liberation of gas bubbles at its surface. This gas is presumed to be 
hydrogen. To initiate the reaction it is sometimes necessary to dip the 
specimen first in alcohol, as water may otherwise not wet it. The speci- 
mens also liberate hydrogen from alcohol and even from toluene. 

Thus, chemical oxidation can follow electrolytic oxidation. But 
chemical oxidation does not proceed at a significant rate before thei 
current is turned on. 

Smooth, shiny electrolytic etching of p-type silicon has been obtained; 
with mixtures of hydrofluoric acid and organic hydroxyl compounds,; 
such as alcohols, glycols, and glycerine. These mixtures may be an- 
hydrous or may contain as much as 90 per cent water. The organic 
additives tend to minimize the chemical oxidation of the silicon. They; 
also permit etching at temperatures below the freezing point of aqueous 
solutions. They lower the conductivity of the electrolyte. 

For a given electrolyte composition, there is a threshold current 
density, usually between 0.01 and 0.1 amps/cm , for smooth etching.; 
Lower current densities give black or red surfaces with the same hy- 
drogen-liberating capabilities as those obtained in aqueous hydrofluoric 
acid. 



ELECTROLYTIC SHAPING OF GERMANIUM AND SILICON 347 

In general, smooth etching of siHcon seems to result when the effective 
valence is nearly 4 and there is little anodic evolution of gas. The elec- 
I trical properties of the smooth surface appear to be equivalent to those 
! of smooth silicon surfaces produced by chemical etching in mixtures of 
i nitric and hydrofluoric acids. On the other hand, the reactive surface 
[produced at a valence of about 2, with anodic hydrogen evolution, is 
I capable of practically shorting-out a silicon p-n junction. The electrical 
j properties of this surface tend to change upon standing in air. 

ACKNOWLEDGEMENTS 

Most of the experiments mentioned in this paper were carried out by 
my wife, Ingeborg. An exception is the double-dimpling of germanium 
by light, which was done by T. C. Hall. The dimpling procedures of 
Figs. 9 and 10 are based on suggestions by J. M. Early. The effect of 
light upon electrolytic etching was called to my attention by 0. Loosme. 
W. G. Pfann provided the germanium crystals grown with different 
degrees of stirring. 

REFERENCES 

1. J. F. Barry, I.R.E.-A.I.E.E. Semiconductor Device Research Conference, 

Philadelphia, June, 1955. 

2. A. Uhlir, Jr., Rev. Sci. Inst., 26, pp. 965-968, 1955. 

3. W. E. Bailey, U. S. Patent No. 1,416, 929, May 23, 1922. 

4. Bradley, et al. Proc. I.R.E., 24, pp. 1702-1720, 1953. 

5. M. V. Sullivan and J. H. Eigler, to be published. 

6. S. L. Miller, Phys. Rev. 99, p. 1234, 1955. 

7. W. H. Brattain and C. G. B. Garrett, B.S.T.J., 34, pp. 129-176, 1955. 

8. E. H. Borneman, R. F. Schwarz, and J. J. Stickler, J. Appl. Phvs., 26, pp. 

1021-1029, 1955. 

9. D. R. Turner, to be submitted to the Journal of the Electrochemical Society. 

10. R. D. Heidenreich, U. S. Patent No. 2,619,414, Nov. 25, 1952. 

11. T. S. Moss, L. Pincherle, A. M. Woodward, Proc. Phys. Soc. London, 66B, 

p. 743, 1953. 

12. T. M. Buck and F. S. McKim, Cincinnati Meeting of the Electrochemical 

Society, Mav, 1955. 

13. F. L. Vogel, W. G. Pfann, H. E. Corey, and E. E. Thomas, Phys. Rev., 90, 

p. 489, 1953. 

14. S. G. Ellis, Phys. Rev., 100, pp. 1140-1141, 1955. 

15. Electronics, 27, No. 5, p. 194, May, 1954. 

16. F. Jirsa, Z. f. Anorg. u. AUgemeine Chem., Bd. 268, p. 84, 1952. 



\ 



A Large Signal Theory of Traveling 
Wave Amplifiers 

Including the Effects of Space Charge and Finite 
Coupling Between the Beam and the Circuit 

By PING KING TIEN 

Manuscript received October 11, 1955) 

The non-linear behavior of the traveling-wave amplifier is calculated in 
this paper by numericalhj integrating the motion of the electrons in the 
presence of the circuit and the space charge fields. The calculation extends 
the earlier work by Nordsieck and the srnall-C theory by Tien, Walker and 
Wolontis, to include the space charge repulsion between the electrons and 
the effect of a finite coupling between the circuit and the electron beam. It 
however differs from Poulter's and Rowers works in the methods of calcu- 
lating the space charge and the effect of the backward wave. 

The numerical work was done using 701 -type I.B.M. equipment. Re- 
sults of calcidation covering QC from 0.1 to 0.4, b from 0.46 to 2.56 and k 
from 1.25 to 2.50, indicate that the saturation efficiency varies between 
23 per cent and 37 per cent for C equal to 0.1 and between 33 per cent and 
Jf.0 per cent for C equal to 0.15. The voltage and the phase of the circuit wave, 
the velocity spread of the electrons and the fundamental component of the 
charge-density modidation are either tabulated or presented in curves. A 
method of calculating the backward wave is provided and its effect fully 
discussed. 

1. INTRODUCTION 

Theoretical evaluation of the maximum efficiency attainable in a 
traveling-wave amplifier requires an understanding of the non-linear 
behavior of the device at various working conditions. The problem has 
been approached in many ways. Pierce/ and later Hess,^ and Birdsalf 
and Caldwell investigated the efficiency or the output power, using cer- 
tain specific assumptions about the highly bunched electron beam. They 
either assume a beam in the form of short pulses of electrons, or, specify 

349 



350 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

an optimum ratio of the fundamental component of convection current 
to the average or d-c current. The method, although an abstract one, 
generally gives the right order of the magnitude. When the usual wave 
concept fails for a beam in which overtaking of the electrons arises, we 
may either overlook effects from overtaking, or, using the Boltzman's 
transport equation search for solutions in series form. This attack has 
been pursued by Parzen and Kiel, although their work is far from com- 
plete. The most satisfying approach to date is Nordsieck's analysis.' 
Nordsieck followed a typical set of "electrons" and calculated their 
velocities and positions by numerically integrating a set of equations of 
motion. Poulter has extended Nordsieck equations to include space 
charge, finite C and circuit loss, although he has not perfectly taken into 
account the space charge and the backward wave. Recently Tien, 
Walker, and Wolontis have published a small C theory in which "elec- 
trons" are considered in the form of uniformly charged discs and the 
space charge field is calculated by computing the force exerted on one 
disc by the others. Results extended to finite C, have been reported by 
Rowe,^*^ and also by Tien and Walker.^^ Rowe, using a space charge 
expression similar to Poulter's, computed the space charge field based on 
the electron distribution in time instead of the distribution in space. This 
may lead to appreciable error in his space charge term, although its 
influence on the final results cannot be easily evaluated. 

In the present analysis, we shall adopt the model described by Tien, 
Walker and Wolontis, but wish to add to it the effect of a finite beam to 
circuit coupling. A space charge expression is derived taking into account 
the fact that the a-c velocities of the electrons are no longer small com- 
pared with the average velocity. Equations are rewritten to retain terms 
involving C. As the backward wave becomes appreciable when C in- 
creases, a method of calculating the backward wave is provided and the 
effect of the backward wave is studied. Finally, results of the calculation 
covering useful ranges of design and operating parameters are presented 
and analyzed. 

2. ASSUMPTIONS 

To recapitulate, the major assumptions which we have made are: 

1. The problem is considered to be one dimensional, in the sense that 
the transverse motions of the electrons are prohibited, and the current, 
velocity, and fields, are functions only of the distance along the tube and 
of the time. 

2. Only the fundamental component of the current excites waves on 
the circuit. 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 351 

3. The space charge field is computed from a model in which the 
helix is replaced by a conducting cylinder, and electrons are uniformly 
charged discs. The discs are infinitely thin, concentric with the helix and 
have a radius equal to the beam radius. 

4. The circuit is lossfree. 

These are just the assumptions of the Tien-Walker-Wolontis model. 
In addition, we shall assume a small signal applied at the input end of a 
long tube, where the beam entered unmodulated. What we are looking 
for are therefore the characteristics of the tube beyond the point at which 
the device begins to act non-linearly. Let us imagine a flow of electron 
discs. The motions of the discs are computed from the circuit and the 
space charge fields by the familiar Newton's force equation. The elec- 
trons, in turn, excite waves on the circuit according to the circuit equa- 
tion derived either from Brillouin's model^ or from Pierce's equivalent 
circuit. The force equation, the circuit equation, and the equation of 
conservation of charge in kinematics, are the three basic equations 
from which the theory is derived. 

3. FORWARD AND BACKWARD WAVES 

In the traveling-wave amplifier, the beam excites forward and back- 
ward waves on the circuit. (We mean by "forward" wave, the wave 
which propagates in the direction of the electron flow, and by "back- 
ward" wave, the wave which propagates in the opposite direction.) 
Because of phase cancellation, the energy associated with the backward 
wave is small, but increases with the beam to circuit coupling. It is there- 
fore important to compute it accurately. In the first place, the waves on 
the circuit must satisfy the circuit equation 

dH^(z,t) 2d'V{z,t) „ d'p^iz, t) ,v 

Here, V is the total voltage of the waves. Vo and Zo are respectively the 
phase velocity and the impedance of the cold circuit, z is the distance 
along the tube and t, the time, p^ is the fundamental component of the 
linear charge density. V and p„ are functions of z and /. The complete 
solution of (1) is in the form 

Viz) = Cre'^'' + (726 "^"^ 

+ e —-y— J e " po,{^) dz ^2) 

+ e " —^ j e p^{z) dz 



352 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

where the common factor e^"' is omitted. To = j{co/vo), j = \/— 1 and w 
is the angular frequency. Ci and C2 are arbitrary constants which will 
be determined by the boundary conditions at the both ends of the beam. 
The first two terms are the solutions of the homogeneous equation (or 
the complementary functions) and are just the cold circuit waves. The 
third and the fourth terms are functions of electron charge density and 
are the particular solution of the equation. 

Let us consider a long traveling-wave tube in which the beam starts 
from z = and ends at 2; = D. The motion of electrons observed at any 
particular position is periodic in time, though it varies from point to 
point along the beam. To simplify the picture, we may divide the beam 
along the tube into small sections and consider each of them as a current 
element uniform in z and periodic in time. Each section of beam, or each 
current element excites on the circuit a pair of waves equal in ampli- 
tudes, one propagating toward the right (i.e., forward) and the other, 
toward the left. One may in fact imagine that these are trains of waves 
supported by the periodic motion of the electrons in that section of the 
beam. Obviously, a superposition of these waves excited by the whole 
beam gives the actual electromagnetic field distribution on the circuit. 
One may thus compute the forward traveling wave at z by summing all 
the waves at z which come from the left. Stated more specifically, the 
forward traveling energy at z is contributed by the waves excited by the 
current elements at the left of the point z. Similarly the backward travel- 
ing energy, (or the backward wave) at z is contributed by the waves 
excited by the current elements at the right of the point z. It follows 
obviously from this picture that there is no forward wave at 2 = 
(except one corresponding to the input signal), and no backward wave 
at 2 = D. (This implies that the output circuit is matched.) With these 
boundary conditions, (1) is reduced to 



z) = Finput e " + e ° — -— / e " po,{z) 

Z Jo 



dz 



+ /-^J e-%.(.) 



(3) 



dz 



Equations (2) and (3) have been obtained by Poulter.^ The first term of 
(3) is the wave induced by the input signal. It propagates as though the ; 
beam were not present. The second term is the voltage at z contributed 
by the charges between 2 = and 2 = 2. It is just the voltage of the 
forward wave described earlier. Similarly the third term which is the 
voltage at 2 contributed by the charges between z = z and 2 = D is the 
voltage of the backward wave at the point 2. Denote F and B respec- 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 353 

tively the voltages of the forward and the backward waves, we have 
F{z) = Fi„put e-'"^ + e-^»^ ^« r e'^' p^z) dz (4) 

Z Jo 

Biz) = e^- ^° £ e-^-p„(e) dz (5) 

It can be shown by direct substitution that F and B satisfy respectively 
the differential equations 



dz Vo dt 2 (9^ 


(6) 


dB(z, t) 1 a5(2, Zo ap„(2, 

(92 1^0 di 2 dt 


(7) 



We put (4) and (5) in the form of (6) and (7) simply because the differ- 
ential equations are easier to manipulate than the integral equations. 
In fact, we should start the analysis from (6) and (7) if it were not for a 
physical picture useful to the understanding of the problem. Equations 
(6) and (7) have the advantage of not being restricted by the boundary 
conditions at 2; = and D, which we have just imposed to derive (4) 
and (5). Actually, we can derive (6) and (7) directly from the Brillouin 
model in the following manner. Suppose Y, I and Zo are respectively 
the voltage, current and the characteristic impedance of a transmission 
line system in the usual sense. (V + /Zo) must then be the forward wave 
and {V — IZo) must be the backward wave. If we substituted F and B 
in these forms into (1) of the Brillouin' s paper,^^ we should obtain exactly 
(6) and (7). 

It is obvious that the first and third terms of (2) are respectively the 
complementary function and the particular solution of (6), and similarly 
the second and the fourth terms of (2) are respectively the comple- 
mentary function and the particular solution of (7). From now on, we 
shall overlook the complementary functions which are far from syn- 
chronism with the beam and are only useful in matching the boundary 
conditions. It is the particular solutions which act directly on the elec- 
tron motion. With these in mind, it is convenient to put F and B in the 
form 

Fiz, t) = -j~ [aiiij) cos <p - aiiy) sin ^] (8) 

B{z, t) = -^ [hiiy) cos ip - h^iy) sin 9?] (9) 



354 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

where ai(y), 02(2/), hi(y) and 62(2/) are functions of y. y and <p are inde- 
pendent variables and have been used by Nordsieck to replace the vari- 
ables, z and t, such as 

y = C — Z 

(f = w [ — — t ] 

\Vo / 

Here as defined earlier, I'o is the phase velocity of the cold circuit and Vq 
the average velocity of the electrons. They are related by the parameter 
h defined by Pierce as 

Uo 1 



vo (1 - hC) 
C is the gain parameter also defined by Pierce, 

^3 _ ZqIo 

in which, Vo and 7o are respectively the beam voltage and current. 
Adding (6) to (7), we obtain an important relation between F and B, 
that is, 

dFjz, t) _^ 1_ dF{z, t) ^ dBjz, t) _j_ l_ dBjz, t) ^^Q^ 

dz Vo dt dz Vo dt 

Substituting (8) and (9) into (10) and carrying out some algebraic 
manipulation, we obtain 

'"'^^ = " 2(1 + bC) I ^'^^'> + "'-^"^^ 

(11) 

"'^^'^ = 2(1 + bC) ly '"'^^^ + '"^^^' 



or 



B{z, t) = 



ZqIo C 

dMy) + bM) ,„, ^ + diaM+ b.(,j)) ^.^ - 
dy dy 



[ 



For better understanding of the problem, we shall first solve (12a) ap- 
proximately. Assuming for the moment that hiiy) and h^^y) are small 
compared with ai{ij) and a^iy) and may be neglected in the right-hand 



A LARGE SIGNAL THEORY OP TRAVELING-WAVE AMPLIFIERS 355 



member of the equation, we obtain for the first order solution 



iKz, t) ^ 



ZqIo I (^ 



sin <p + — ^^^ cos <p 



40 \ 2(1 + bC) I dij ^ ' dy 



(12b) 



Of course, the solution (12b) is justified only when hi(y) and ?)2(?y) thus 
obtained are small compared with ai(y) and aoiy). The exact solution 
of B obtained by successive approximation reads 



Biz, t) 

+ 



ZqIo I c 



4(7 V 2(1 + bC) 



4(1 + hC) 
It may be seen that the term involving 



dai(y) . , da2(ij) 

-^ sm <p + , cos 
_ dy dy 

■ ] 



•] 



'^^' cos<p + — f^sm^ + 



(12c) 



dy- dy- 



4(1 + bcy 

and the higher order terms are neglected in our approximate solution. 
For C equal to few tenths, the difference between (r2b) and (12c) only 
amounts to few per cent. We thus can calculate the backward wave by 
(12b) or (12c) from the derivatives of the forward wave. To obtain the 
complete solution of the backward wave, we should add to (12b) or 
(12c) a solution of the homogeneous equation. We shall return to this 
point later. 

4. WORKING EQUATIONS 

With this discussion of the backward wave, we are now in a position 
to derive the working equations on which our calculations are based. In 
Nordsieck's notation, each electron is identified by its initial phase. 
Thus, (p(y, (fo) and Cuow(y, <po) are respectively the phase and the ac 
velocity of the electron which has an initial phase (fo . It should be remem- 
bered that y is equal to 

and is used by Nordsieck as an independent variable to replace the vari- 
al)le z. Let us consider an electron which is at Zo when /, = and is at 
z (or ?/) when t = /. Its initial phase is then 

OiZo 

<Po = — 



356 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



and its phase at y is 



<p(y,<po) = oj f- - tj 



i 




The velocity of the electron is expressed as 

dz 



dt 



= Wo[l + Cw{ij, ip^)] 



where Uo is the average velocity of the electrons, and, Cuow(y, tpo) as men- 
tioned earlier, is the ac velocity of the electron when it is at the position 
y. The electron charge density near an electron which has an initial phase 
cpo and which is now at y, can be computed by the equation of conserva- 



tion of charge, it is 



p(y, <Po) = - 



Wo 



d(po 



d(p{y, <po) 



1 



1 + Cw(y, ifo) 



(13) 



One should recall here that h is the dc beam current and has been de- 
fined as a positive quantity. When several electrons with different initial 
phases are present at y simultaneously, a summation of 

d<po 



of these electrons should be used in (13). From (13), the fundamental 
component of the electron charge density is 



pMt) = --- 



sm 






d<po 



sin (fiy, <po) 
1 + Cw{y, <pq) 

r^" , cos <p{y, <po) 
+ cos <p I d(po 
Jo 



(14) 



1 + Cw(y, ifo)/ 

These are important relations given by Nordsieck and should be kept 
in mind in connection with later work. In addition, we shall frequently 
use the transformation 

I = t s = ^"(' + ^'"(^-» 1^ 

which is written following the motion of the electron. Let us start from 
the forward wave. It is computed by means of (6). After substituting 
(8) and (14) into (6), we obtain by equating the sin <p and the cos v' 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 357 

terms 

dax{y) ^ _2 T^" ^ sin <p(y, cpo) .^. 

dy IT h " 1 + Cwiv, (po) 

da.Xy) 2 f^" cos<p{y,<po) ..^n 

— 1 — = ~- / d(po , r (.lb; 

dy IT Jo 1 + Cw{y, <po) 

Next we shall calculate the electron motion. We express the acceleration 
of an electron in the form 

d'z „ /I I /o / ^^ dw{y, <po) 
^ = Cuod + Cw{y, M -^^ 

and calculate the circuit field by differentiating F in (8) and B in (12c) 
with respect to z. One thus obtains from Newton's law 

2[1 + Cw{y, <po)] ^^'^J' ^°^ = (1 + hOMy) sin <p + a,{y) cos <p\ 

dy 

+ ^-^ r^ «in ^ + ^^ cos J - -^ ^. 
4(1 + 6C) L ^Z/- c?^^ J WomwC^ 

Here Eg is the space charge field, which will be discussed in detail later. 
Finally a relation between w{y, (po) and <p{y, ^o) is obtained by means of 
(13) 

difiy, <po) _ ^ ^ ^^(y, <Po) QgN 

dy 1 + Cw{y, <pq) 

Equations (15), (16), (17) and (18) are the four working equations 
which we have derived for finite C and including space charge. 

Instead of writing the equations in the above form, Rowe, ignoring 
the backward wave, derives (15) and (16) directly from the circuit 
equation (1). He obtains an additional term 

C d^tti 



2 dy"" 



for (15) and another term 



C d"ai 

2df 

for (16). It is apparent that the backward wave, though generally a 
small quantity, does influence the terms involving C. 



358 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



5. THE SPACE CHARGE EXPRESSION 



We have mentioned earlier that the space charge field is computed 
from the disc-model suggested by Tien, Walker and Wolontis. In their 
calculation, the force excited on one disc by the other is approximated 
by an exponential function 



F. = 



— [a(z'— z)/ro] 



27rro-eo 



Here ro is the radius of the disc or the beam, q is the charge carried by 
each disc, and eo is the dielectric constant of the medium. The discs are 
supposed to be respectively at z and z . a is a constant and is taken 
equal to 2. 

Consider two electrons which have their initial phases <pq and ^o and 
which reach the position ij (or z) at times t and t' respectively. The time 
difference, 



* - / = 1 

00 



wt — — Z — [bit — — Z) 

Vo \ Vq J 



CO 



multiplied by the velocity of the electron i<o[l + Cw(y, (po )] is obviously 
the distance between the two electrons at the time t. Thus 

(z - z)t=t = - y(y, <Po) - <p(y, <Po)]uo[l + Cw(y,ipo)] (19a) 

In this equation, we are actually taking the first term of the Taylor's 
expansion, 



(z — z)t=t = 



dzjij, cpo) 
dt 



t=t 



(t _ /^ j_ ^ c?^2(y, <pq) 



it - ty 



t=t 



(19b) 



+ 



It is clear that the electrons at y may have widely different velocities 
after having traveled a long distance from the input end, but changes in 
their velocities, in the vicinity of y and in a time-period of around 2 tt, 
are relatively small. This is why we must keep the first term of (19b) 
and may neglect the higher order terms. From (19a) the space charge 
field Es in (17) is 



2e 



Es = 






/+00 



-k]ip(.y ,<po+<t>)—<p(.U ,<Po) 1 li+Cw(y,(po+<t>)] 



d(f> sgn (<p(<po -\- <p) - <t>iy, <po)) 
Here, e/m is the ratio of electron charge to mass, cop is the electron 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 359 

angular plasma frequency for a beam of infinite extent, and k is 

2 



k = 



a 

0) CO 

— ro — ro 

Uo Wo 



(20) 



In the small C theory, th^e distribution of electrons in time or in time- 
phase at z is approximately the same as the distribution in z (also ex- 
pressed in the unit of time-phase) at the vicinity of z. This is, however, 
not true when C becomes finite. The difference between the time and 
space distributions is the difference between unity and the factor 
(1 -}- Cw{y, <po )). We can show later that the error involved in con- 
; sidering the time phase as the space phase can easily reach 50 per cent 
or more, depending on the velocity spread of the electrons. 



6. NUMERICAL CALCULATIONS 

Although the process of carrying out numerical computations has 
been discussed in Nordsieck's paper, it is desirable to recapitulate here 
I a few essential points including the new feature added. Using the work- 
ing equations (15), (16), (17) and (18), 

dai da 2 dw , dcp 

dy ' dy ' dy dy 

\ are calculable from ai , a^ , w and <p. The distance is divided into equal 
I intervals of A?/, and the forward integrations of Oi , ao , w and (p are per- 
f formed by a central difference formula 



ax{y + A?/) = ax{y) -f 



dy 



y+y2&y 



■Ay 



In addition. 



d^ai 
dy^ 



and 



d 02 

df 



in (17) are computed from the second difference formula such that 
d''ai 



- At/ 



_ dtti da\ 

dy^ j/=j/ \_dy y+l/2i,y dy y-^/2^y_ 

We thus calculate the behavior along the tube by forward integration 
j made in steps of Ay, starting from y = 0. At ?/ = the initial condi- 
tions are determined from Pierce's linearized theory. Because of its 
complications in notation, this will be discussed in detail in Appendix I. 
j Numerical calculations were carried out using the 701-type I.B.M. 



Table I 



a; 
U 


QC 


k 


c 


6 


Ml 


MJ 


ycsAT.) 


<! 
01! 


H 
•i 

i 

a. 
1 


1 


0.1 


2.5 


0.05 


0.455 


m max. 
0.795662 


-0.748052 


5.6 


1.26 


0.415 


2 


0.1 


2.5 


0.1 


0.541 


Ml max. 
0.827175 


-0.787624 


5.2 


1.24 


0.482 


3 


0.1 


2.5 


0.1 


1.145 


0.941;ui max. 
0.778535 


-1.05370 


5.6 


1.31 


0.820 


4 


0.1 


2.5 


0.1 


1.851 


0.66jui max. 
0.550736 


-1.37968 


7.0 


1.36 


1.05 

J 


5 


0.1 


2.5 


0.2 


0.720 


m max. 
0.900312 


-0.873606 


4.8 


1.02 


0.726 


6 


0.2 


1.25 


0.1 


0.875 


jui max. 
0.769795 


-1.04078 


5.9 


1.22 


0.570 


7 


0.2 


1.25 


0.1 


1.422 


0.951^1 max. 
0.724527 


-1.29469 


6.0 


1.30 


0.803 


8 


0.2 


1.25 


0.1 


2.072 


0.666mi max. 
0.512528 


-1.60435 


7.6 


1.35 


1.08 


9 


0.2 


2.5 


0.05 


0.765 


Ml max. 
0.731493 


-0.973376 


6.2 


1.30 


0.412 


10 


0.2 


2.5 


0.1 


0.875 


Ml max. 
0.769795 


-1.04078 


5.8 


1.22 


0.490 


11 


0.2 


2.5 


0.1 


1.422 


0.941mi max. 
0.724527 


-1.29469 


6.0 


1.26 


0.720 


12 


0.2 


2.5 


0.1 


2.072 


0.666mi max. 
0.512528 


-1.60435 


7.2 


1.25 


0.92 


13 


0.2 


2.5 


0.1 


2.401 


0.300mi max. 
0.230930 


-1.76243 


12.4 


1.24 


1.36 

j 


U 


0.2 


2.5 


0.15 


0.976 


Ml max. 
0.812900 


-1.10656 


5.4 


1.11 


0.572 


15 


0.2 


2.5 


0.15 


1.549 


0.941mi max. 
0.765101 


-1.37540 


5.8 


1.14 


1.03 


16 


0.2 


2.5 


0.15 


2.2311 


0.666mi max. 
0.541234 


-1.70180 


7.0 


1.12 


1.22 


17 


0.2 


2.5 


0.15 


2.575 


0.300mi max. 
0.243864 


-1.86844 


10.8 


1.04 


1.34 


18 


0.4 


2.5 


0.05 


1.25 


Ml max. 
0.653014 


-1.36746 


7.6 


1.26 


0.315 


19 


0.4 


2.5 


0.1 


1.38 


Ml max. 
0.701470 


-1.47477 


6.6 


1.11 


0.674 


20 


0.4 


2.5 


0.1 


1.874 


0.941mi max. 
0.660223 


-1.71341 


7.8 


1.19 


1.05 


21 


0.4 


2.5 


0.1 


2.458 


0.666mi max. 
0.467038 


-1.99840 


8.6 


1.09 


1.25 



l> 



360 



A LARGE SIGNAL THEORY OF TRAVELING- WAVE AMPLIFIERS 361 

equipment. The problem was programmed by Miss D. C. Legaus. The 
cases computed are listed in Table I in which m and m2 are respectively 
Pierce's .xi and iji , and A,(d — iny) and tj at saturation will be discussed 
later. All the cases were computed with A^ = 0.2 using a model based 
on 24 electron discs per electronic wavelength. To estimate the error 
involved in the numerical work, Case (10) has been repeated for 48 elec- 
trons and Cases (10) and (19) for Ay = 0.1. The results obtained by 
using different numbers of electrons are almost identical and those ob- 
tained by varying the inter\'al A// indicate a difference in A (y) less than 
1 per cent for Case (10) and about 6 per cent for Case (19). As error 
generally increases with QC and C the cases listed in this paper are 
limited to QC = 0.4 and C = 0.15. For larger QC or C, a model of more 
electrons or a smaller interval of integration, or both should be used. 

7. POWER OUTPUT AND EFFICIENCY 

Define 

A(ij) = HVa,(yy + aM' 

-0(y)=i^n-'^-^ + by ^^^^ 

aiiy) 



We have then 



F{z,t) = ^A{y) cos 



^ -^t- e{y) 

Uo 



(22) 



The power carried by the forward wave is therefore 

2CA'hVo (23) 



(f) = 

\Z/o/ average 



and the efficiency is 



Eff. = ?£^^ = 2CA' or ^ = 2CA' (24) 

In Table I, the values of A(y), 6{y) and y at the saturation level are 
listed for every case computed. We mean by the saturation level, the 
distance along the tube or the value of y at which the voltage of the 
forward wave or the forward traveling power reaches its first peak. 
The Eff./C at the saturation level is plotted in Fig. 1 versus QC, for 
k = 2.5, h for maximum small-signal gain and C = small, 0.05, 0.1, 0.15 
and 2. It is also plotted versus h in Fig. 2 for QC = 0.2, k = 2.5 and 
C = small, 0.1 and0.15, and in Fig. 3 for QC = 0.2, C = 0.1 and k = 1.25 
and 2.50. In Fig. 2 the dotted curves indicate the values of h at Avhich 



1 



362 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195G 

4.5 




0.5 



Fig. 1 — The saturation eff./C versus QC, for k = 2.5, h for maximum small- 
signal gain and C = small, 0.1, 0.15 and 0.2. 

ixx = Ml (max), 0.94 jui(max), 0.67 iui(max) and 0.3 /ii(niax), respectively. 
It is seen that Eff./C decreases as C increases particularly when h is 
large. It is almost constant between k = 1.25 and 2.50 and decreases 
slowly for large values of C when QC increases. 

The (Eff./C) at saturation is also plotted versus QC in Fig. 4(a) for 
small C, and in Fig. 4(b) for C = 0.1. It should be noted that for C = 0.1 
the values of Eff./C fall inside a very narrow region say between 2.5 to 
3.5, whereas for small C they vary widely. 

8, VELOCITY SPREAD 

In a traveling-wave amplifier, when electrons are decelerated by the 
circuit field, they contribute power to the circuit, and when electrons 
are accelerated, they gain kinetic energy at the expense of the circuit 
power. It is therefore of interest to plot the actual velocities of the fastest 
and the slowest electrons at the saturation level and find how they vary 
with the parameters QC, C, b and k. This is done in Fig. 5. These veloci- 
ties are also plotted versus y for Case 10 in Fig. 6, in which, the A(y) 
curve is added for reference. 

9. THE BACKWARD WAVE AND THE FUNDAMENTAL COMPONENT OF THE 
ELECTRON CHARGE DENSITY 

Our calculation of efficiency has been based on the power carried by 
the forward wave only. One may, however, ask about the actual power 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 363 



6.0 
5.5 

5.0 
4.5 
4.0 

3.5 

EFFI. 

C 3.0 
(SAT.) 

2.5 
2.0 
1 .5 
1.0 
0.5 

















1 




QC = 0.2 










1 






1 

A- — 




K=2.5 










\ 


SMALI " 


*^r^ 


















\ 
1 


Sa 


y^ 


1 

1 


















\ 


_/^ 




I 


















Ji 


A 




1 


















/^ 


\ 




1 
\ 
















/ 




\ 




t 














\ 


/ 




\ 




\ 














^ 


/ 




\ 




\ 














X 


f 




\ 
( 
\ 




\ 
\ 

\ 














^ ' 


























\ 




\ 


C = 0.1 


'\ 


\ 












\ 

\ 


, 


lyj 






\ 






\ 






\ 








JT"^ 










\ 


\ 


















V C=0.15 


\ 
\ 
\ 


\ 

\ 
\ 








>"1 = 




1 
AX) 


/"1=C 


K 1 1 
).94/Z.(MAX) 


.at^ 










/ 1 


\^ 


















/t/i = 0.67//i(MAX) 


\ 






















//, = 0.3//i(MAX) 











































































0.5 



1.0 



1.5 

b 



2.0 



2.5 



3.0 



Fig. 2 — The saturation eff./C versus fe, for k = 2.5, QC = 0.2, and C = small, 
0.1 and 0.15. The dotted curves indicate the values of h for m = \, 0.94, 0.67, and 
0.3 of ;ui(max) respectively. 

output in the presence of the backward wave. For simphcity, we shall 
use the approximate solution (12b) which can be written in the form 

B{z, t) ^ Real Component of 

ZqIq c 



4C 2(1 + hC) 




dax(y)Y ^ (da,{y)\- j^^-v,.-,y+j^\ (12d) 



with 



tan ^ = 



dij 



(laiiyT 
dy , 



dy 



dchiyY 
dy , 



As mentioned earlier that the complete solution of (6) is obtained by 
adding to (12b) a complementary function such that 



-yu 1+ r Qz 



ZqIq 



+ 



c 



4C 2(1 + bC) 




dy:) ^\dy ) ' 



-hy+ji 



(25) 



364 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



EFFI. 

c 

(SAT.) 3 

















QC = o.2 
C = 0.1 




























J<_=K25. 




3- 






2.50 

























































0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 

b 

Fig. 3 — The saturation eff./C versus b, for QC = 0.2 C = 0.1 and k = 1.25 
and 2.50. 

If the output circuit is matched by cold measurements, the backward 
wave must be zero at the output end, z = D. This determines Ci , that is, 



„ ZqIq c 

^1 = ~^rPT 



or 



Cie 



jut+Toz 



4C 2(1 + bC) 



ZqIq C 



//dai(t/)Y I / da2{y) Y ro(2+bc)D+ji 




dai{y)V / da2{y) y 



4C 2(1 + 6C) y \ dy )z=o \ dy Jz^d (26) 

The backward wave therefore consists of two components. One compo- 



o 

7 


(a) 


C = SMALL 


^- 


Ml = 0.67 


/U,(MAX) 




D 

5 


^^;;:^ 


^^ 






EFFI. 
C 4 


/U, = 0.94//i(MAX) 


^^^ 


1 





(SAT.) ^ 
2 


■"Zr^AtlC^AX) 


" 




























(b) 




C = o.i 




















= 0.94//, (MAX) 

1 




Xj 


fea,^_^-VZi = 0.67 /Z, (MAX) 


>U, = /i|(MAX)- 


1 —===3 


^^^ 



















0.1 



0.2 

QC 



0.3 0.4 0.1 



0.2 

QC 



0.3 0.4 



Fig. 4 — The saturation eff./C versus QC for h corresponding jui = 1, 0.94 and 
0.67 of Mi(max), (a) for C = small, (b) for C = 0.1. 



A LARGE SIGNAL THEORY OF TRAVELING- WAVE AMPLIFIERS 365 



nent is coupled to the beam and has an amplitude equal to 

Zolo C 
IC 2(1 + bC) 



VX^'Y + 



K^y / 



\dy) 



which generally grows with the forward wave. It thus has a much larger 
amplitude at the output end than at the input end. The other component 
is a wave of constant amplitude, which travels in the direction opposite 
to the electron flow with a phase velocity equal to that of the cold cir- 
cuit. At the output end, 2 = Z), both components have the same ampli- 
tude but are opposite in sign. One thus realizes that there exists a re- 
flected wave of noticeable amplitude, in the form of (26), even though 
the output circuit is properly matched by cold measurements. Under 
j such circumstances, the voltage at the output end is the voltage of the 
forward wave and the power output is the power carried by the forward 
wave only. This is computed in (23). 
Since (26) is a cold circuit wave it may be eliminated by properly ad- 



c[-w], 



■C[w], 



5.0 



4.5 



4.0 



3.5 



5 3.0 

2 

o 

9- 2.5 



1.5 



1.0 



0.5 



(a) 






; 








/ 






y 


/ 






/ 




( 


r' 












,.--- 


( 


L"1 


.-'■ 





























(b) 




j^ 


V 




/ 


/ 






/ 






y 


1 




















Qw 


'"--^ 




^-"^ 



























(c) 




J 


i 






/ 








/ 








^ 






/ 








/ 


,''^ 


1 




/ ( 

f 


r 






1 
1 
1 

1 






< 


f 















0.1 0.2 0.3 0.4 0.5 1.0 

QC 



1.5 2.0 2.5 
b 



0.05 0.10 0.15 0.20 



Fig. 5 — Cw(y, <po) of the fast and the slowest electrons at the saturation level, 
(a) versus QC for k = 2.5, C = 0.1 and b for maximum small-signal gain; (b) versus 
6 for A; = 2.50, C = 0.1 and QC = 0.2; and (c) versus C for A- = 2.50, QC = 0.2 
and b for maximum small-signal gain. 



366 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1950 



3.5 



3.0 



2.5 



2.0 



9- 

^1.5 
U 

1.0 



0.5 























> 


r\ 
























/ 




^ 


\ 


/ 


\ 








CASE 10 
QC = 0.2 
C = 0.1 
b = 0.875 
k = 2.5 






MAXC(-W) / 




,-' 




''s 




\ 
























\ 
















/A(y) 






\ 
\ 
















, 


1 

1 






/ 


/\ 


S 


















// 






y 


/ 




X- 


















./ 




,^ 


"^AXCW 


















i 
/ 

/ 


/ 


y 


r 
























■7 


/ 




























/ 


























A 


y 
























^-' 


^ 


^ 






















:z=^ 


— ** — 


^ 

























1.4 



1.2 



1.0 



0.8 



ID 
< 



0.6 



0.4 



0.2 




0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 

y 

Fig. 6 — Cw{y, (pa) of the fast and the slowest electrons versus y for Case 
(10). A{y) is also plotted in dotted lines for reference. 

justing the impedance of the output circuit. This may be necessary in 
practice for the purpose of avoiding possible regenerative oscillation. In 
doing so, the voltage at 2 = D is the sum of the voltage of the forward 
wave and that of the particular solution of the backward wave. In every 
case, the output power is always equal to the square of the net voltage 
actually at the output end divided by the impedance of the output cir- 
cuit. 

We find from (14), (15) and (16) that the fundamental component of 
electron charge density may be written as 

f s. \ h ( . dai{y) . da2(y)\ 



= Real component of 



1/0 




dai{y) 
dy , 



+ 



doM 
dy 



(26) 



jo)—Toz—by+Ji 



) 



where —Io/uq is the dc electron charge density, po . 

If (26) is compared with (12d) or (12c), it might seem surprising that 
the particular solution of the backward wave is just equal to the funda- 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 367 



1.6 
t.5 
1.4 
1.3 
1.2 
1.1 
1.0 

Pq 0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 



1.2 
1.1 
1.0 
0.9 

Pq 0.7 
0.6 
0.5 
0.4 
0.3 
0.2 

0.1 





CASES 2, 


10,19 










(a) 


k = 2.5 
- C = 0.1 
b-»MAX n^ 






/' 


\ 






r 




\ 










J 


' 




^ 












r 


\ 


r\ 


\ 








/ 


f 


V 


1] 


s — 






QC=o.i/ 


/ 
















f 

0.2 














// 


r 


0.4 












// 


1 


V 












// 




\ 












7 




\ 


<^ 


L 








/ 








\ 




A 












\ 




^ 














^ 

















CASES 9, 


0,14 










(c) 


QC=0.2 
- k=2.5 
b-»MAX//| 


















































rv 


-\ 








1 






u 


\ 














r 








C=o,s/// 




\ 










/ rf-o 10 




\ 










///o.05 












i 


II 














k 






























A 


f 














/ 














^ 

















8 



4 

y 



CASES 1C 


,11,12 


iH 


r 


V 


r\ 
















(b) 


QC = o.2 
C = o.i 

k = 2.5 


r 


k\ 


(A 


\ 
















// 


\ / 


y 


^ 






c 


\ 
















/ 


\f 


A 








\ 


\ 










/^, = >U,MAx/^ 


' 


/ 


A 








/ 


\ 














// 




/ 


\ 






/ 


f 


\ 




\ 












A 


/ 






I 




/ 


















J 


/09« / 






11 


y 


/ 






\ 


\ . 








// 


/ 


/ 






11 


/ 








\J 




17 








// 


Ai.^i 




^^ 


"w 


1 
















/' 


y 







^^ 


.^^ 


















-^ 


>^ 




'^1 = 0.3X/,MAX 
1 1 





















7 8 

y 



10 11 



12 13 14 15 



Fig. 7(a) — p^/po versus ?/, (a) using QC as the parameter, for A; = 2.5, C = 0.1, 
and 6 for maximum small-signal gain (Cases 2, 10, and 19) ; (b) using h as the param- 
eter, for k = 2.50, C = 0.1 and QC = 0.2 (Cases 10, 11, 12 and 13); and (c) using 
C as the parameter, for k = 2.50, QC = 0.2 and h for maximum small-signal gain 
(Cases 9, 10 and 14). 



368 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



mental component of the electron charge density of the beam multiplied 
by a constant 

/ Zq/o C 2uo 



2wo\ 
h) 



(27) 



V 4C 2(1 + hC) 
The ratio of the electron charge density to the average charge density, 

P«(2) 



Po 



2319^21 
5 17/^,1 9 




^ +e 



Fig. 8(a) — y versus <f - hrj for QC = 0.2, k = 2.5, b for mi = 0.67 
C = small. 



Ml (max) and 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 369 

is plotted in Fig. 7 versus y, using QC, h and C, as the parameters. They 
lare also the curves for the backward wave (the component which is 
! coupled to the beam) when multiplied by the proportional constant given 
in (27). It is interesting to see that the maximum values of p^/po are 
between 1.0 and 1.2 for QC = 0.2 and decrease as QC increases. The 
peaks of the curves do not occur at the saturation values of y. 

10. y VERSUS ((p — by) diagrams 

To study the effect of C, b, and QC on efficiency y versus (<p — by) 
diagrams are plotted in Figs. 8(b), (c), (d) and (e) for Cases (21), (16), 
(10) and (21), respectively. {<p — by) here is ($ + 6) in Nordsieck's nota- 
tion. In these diagrams, the curves numbered from 1 to 24 correspond to 
the 24 electrons used in the calculation with each curve for one electron. 
Only odd numbered electrons are presented to avoid possible confusion 
arisen from too many lines. The reciprocal of the slope of the curve as 




-10 -9 -8 



jo-by 



Fig. 8(b) — y versus <p 
C = 0.1 (Case 12). 



bij for QC = 0.2, k = 2.5, b for mi = 0.67Mi(max) and 



370 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



given by (18) is proportional to the ac displacement of electron per unit 
of ij. (In small-C theorj^ it is proportional to the ac velocity of the elec- 
tron.) Concentration of curves is obviously proportional to the charge- 
density distribution of the beam. In the shaded regions, the axially di- 
rected electric field of the circuit is negative and thus accelerates elec- 
trons in the positive z direction. Electrons are decelerated in the un- 
shaded regions where the circuit field is positive. The boundaries of these 
regions are constant phase contours of the circuit wave. (They are con- 
stant $ contours in Nordsieck's notation.) 

These figures are actuallj' the "space-time" diagrams which unfold 
the historj^ of every electron from the input to the output ends. The 
effect of C can be clearly seen by comparing Figs. 8(a), (b) and (c). 
These diagrams are plotted for QC = 0.2, A; = 2.5, h for jui = 0.67 
jui(max) and for Fig. 8(a), C = small, for Fig. 8(b), C = 0.1, and for 
Fig. 8(c), C = .15. It may be seen that because of the velocity spread of 
the electrons, the saturation level in Fig. 8(a) is 9.3 whereas in Figs. 8(b) 
and (c), it is 7.2 and 7.0, respectively. It is therefore not surprising that 
Eff./C decreases as C increases. 

The effects of h and QC may be observed by comparing Figs. 8(d) and 
(b), and Figs. 8(b) and (e), respectively. The details will not be de- 
scribed here. It is however suggested to study these diagrams with those 
given in the small-C theory. 



7.2 


5 




1 




23 9 


11 

i'5 


7 3 


1719 21 13 


23 


15 


1719 21 












^" 




^-^^ ?ny 


J^ 


V 


^v:\ 


S 


\| 






A 


\- 


I 


SATURATION 


6.8 

6.4 
6.0 
5.6 
6.2 




.«,^ 


^ 


*tf 




LEVEL 


" 








vK 


sL- 


^ 


^N 
^ 


V 


\ 








^ 


■I 




















^ 


^ 


L^ 


i 




^ 


^ 






y 


r 


\ 


rt 
















'a 


[ \ 


1 


rt 


^VL 




/ 




1 


V 


«x 


















t 


/ 

< 
$ 




^W 


I / 


/ 


-^ 




















\ 
\ 

\ 


w 


t 


ll 1 


'— T 


^ 


kU\ 












4.4 

4.0 

3.6 

3.2 

2.8 

2.4 
?0 
















'^ 




t 






^ 


\\\^ 
































\ \ V 


\ 




I 1 


/ 




\\ 
























1 
- 1 


\ \ 
1 \ 
1 \ 


r- 




1 


/, 


1 


\ 


\\\ 


















IS _H 








1 1 

li 


V-' 


\ \ 












\ 


' 
















% 






1 

■ 1 


1 1 
1 1 

i 1 


> 1 
1 


\ 


\ 




1 


j 


























i 

r 
1 


ll 


i i 




































i5!r 

1 




ti23l 
11 1 


,3 


_ 

-1 


9 in 

L. J 


3 15 17 ig/sii 
1 i i 


23 









-10 -9 



-8 -7 



-4 -3 



1 



10 



Fig. 8(c) — y versus <p — by for QC = 0.2, k — 2.5, b for^i = 0.67^1 (max) and , 
C = 0.15 (Case 16). 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 371 
11. A QUALITATIVE PICTURE AND CONCULSIONS 

We have exhibited in the previous sections the most important non- 
linear characteristics of the traveling wave ampUfier. Xumerical compu- 
tations based on a model of 24 electrons have been carried out for more 
than twenty cases covering useful ranges of design and operating parame- 
ters. The results obtained for the saturation Eff./C may be summarized 
as follows: 

(1) It decreases with C particularly at large values of QC. 

(2) For C = 0.1, it varies roughly from 3.7 for QC = 0.1 to 2.3 for 
i}C = 0.4, and only varies slightl}^ with h. 

(3) For C = 0.15, it varies from 2.7 to 2.5 for QC from 0.1 to 0.2 and 
\i corresponding to the maximum small-signal gain. It varies slightly 
with h for QC = 0.2. 

(4) It is almost constant between k — 1.25 and 2.50. 

In order to understand the traveling-wave tube better, it is important 
to have a simplified qualitative picture of its operation. It is obvious that 
to obtain higher amplification, more electrons must travel in the region 
where the circuit field is positive, that is, in the region where electrons 



6.8 
6.4 
6.0 


17 


3 
51 9i7 


13 15 


11 




23 21 


7 
J 19 








13 


5 


11 




















O^ 


N 

N 


cl. 




\ 


vV 




\ 














vn 


^ 


. ^^vV ^ 


\ 




Vv 


\ 






\ 


^A 


TURAT 


lOM 






^ 


v^§^ 




\ 


\ 


\j 






l\ 




LEVEL 


5.6 
5.2 
4.8 
4.4 
4.0 
3.6 
3.2 
2.8 
2.4 


* 


3" - 










. 


/ 


^ 




\ 




\ 






N 














/ 




/ 

/ 

/ 


'V 


\\ 


\ 


K 




l\ 












\ 

1 
1 

1 


1 


/ 

/ 
1 


/i^ 


\\ 


-^A 








"^ 










\ 

\ 
t 


/ 
/ 
1 


r 
1 


■ /A\\\ 




f 


f 


fX 










V 

v 


1 
1 

\ 


1 
I 
1 


/ 


N 


\\v 


\\ \ 


/ 


t 


/ 


1 


\ 








^ 


\° 1 


I 

i 


/ 
f 






\\\ 




f 


1 


/ 












\ \ 

\ \ 


f 

1 
1 


( 

1 




r 








f 


// 














\\ 


1 
1 








1 


\\ 










/ 












\ \ 

11«13» 


151 17 






1 3 


5 


] 


g\ inisl 15 




7 


19 


21/ 2 


3 




?n 








1 1 


1 










1 \ li\ 


1 i 


t J iJ 





-1 



SP-by 



Fig. 8(d) — V versus <p — by for QC = 0.2, k = 2.5, b for m = mi (max) and 
'' = 0.1 (Case 10). 



372 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



8.8 


11 








5 3 15 9 


7 21 


17? 


23 


1 




15 




21 




23 










^-~ 


-^ 




R 


StiC^ 1 


^^ 




/ 


/' 


1- 


SATURATION 


8.4 
8.0 
7.6 
7.2 
6.8 
6.4 
6.0 
6.6 
5.2 
4.8 
4.4 
4.0 
3.6 
3.2 
2.8 
2.4 
?,0 


IQ** 






:^ 


^ 


■" LEVEL 




~ - 






1 


r4- 




■~3 


N, , 






/ 












\ 

\ 


\ 


\, 


7H 
1 


"^^ 


d 


\ 




/ 












^v.-— ^.-. 


""^ 




\ 
\ 




t 




y- 


[V\ 


fe 


\ 


^ 


\ 


















>^ 


\ — p 

\ \ 
\ \ 


v\ 


I 




:: 








''^: 










N\ 




"t 






^ 


:\ 






















\ 
\ 


-^^ 


K\l 


1 


H. 


'■.-■; 




















1 

1 
1 






Ui 




y 




V 




















t 
1 
J. 




1 ^W 


^\ 




/ 


/ 


)) 




















r 
i 
t 




l-^i 


\V 


d 


\ 


r 




// 




















1 
( 
1 




i : 

1 


\\ 


; 






















1 




/ 


i 


w 


\\ 


, :' 


















■,J; 


W 


I 

1 
1 
1 




— p- 

rl - 

1 \ 


/ 




] 




\ 




















-— - 


1 

1 
1 








1 


\\\ 


1 












% 






1 




1 1 
1 1 
































1 

1 
t 




n 


, 


9 1 


1 
1 


























15; 

f 




,'21 123 

/ !l 


V' 


3 15 17 jl9 


21 23 






-1 


- 


9 - 


8 - 


7 - 


6 - 


5 - 


4 - 


3 - 


2 - 


1 





1 : 


3 


3 


- 


i 5 


6 


7 


3 9 



y-by 

Fig. 8(e) — y versus <p — by for QC = 0.4, k = 2.5, b for in = 0.67ui(max) and 
C = 0.1 (Case 21). 

are decelerated by the circuit field. At the input end of the tube, elec- 
trons are uniformly distributed both in the accelerating and decelerating 
field regions. Bunching takes place when the accelerated electrons push 
forward and the decelerated ones press backward. The center of a bunch 
of electrons is located well inside the decelerating field region because 
the circuit wave travels slower than the electrons on the average (6 is 
positive). The effectiveness of the amplification, or more specifically the ! 
saturation efficiency, therefore depends on (1), how tight the bunching :' 
is, and (2), how long a bunch travels inside the decelerating field region 
before its center crosses the boundary between the accelerating and 
decelerating fields. 

For small-C, the ac velocities of the electrons are small compared with 
the dc velocity. The electron bunch stays longer with the decelerating 
circuit field before reaching the saturation level when h or QC is larger. 
On the other hand, the space charge force, or large QC or k tends to dis- 
tort the bunching. As the consequence, the saturation efficiency increases , 
with h, and decreases as k or QC increases. When C becomes finite how- 



A LARGE SIGNAL THEORY OF TRAVELING-WAVE AMPLIFIERS 373 

ever, the ac velocities of the electrons are no longer small as compared 
I with their average speed. The velocity spread of the electrons becomes 
, an important factor in determining the efficiency. Its effect is to loosen 
the bunching, and consequently it lowers the saturation level and re- 
duces the limiting efficiency. It is seen from Figs. 5 and 6 that the 
. velocity spread increases sharply with C and also steadily with b and QC. 
\ This explains the fact that in the present calculation the saturation 
Eff./C decreases with C and is almost constant with h whereas in the 
1 1 small-C theory it is constant with C and increases steadily with b. 

12. ACKNOWLEDGEMENTS 

The writer wishes to thank J. R. Pierce for his guidance during the 
course of this research, and L. R. Walker for many interesting discus- 
sions concerning the working equations and the method of calculating 

I the backward wave. The writer is particularly grateful to Miss D. C. 
Leagus who, under the guidance of V. M. Wolontis, has carried out the 

^ numerical work presented with endless effort and enthusiasm. 

APPENDIX 

The initial conditions at i/ = are computed from Pierce's linearized 
theory. For small-signal, we have 

ai(?/) = 4A(y) cos (6 -f ^2)2/ (A-1) 

«2(2/) = -4A(y) sin (6 + ju2)y (A-2) 

A(y) = ee"'' (A-3) 

Here e is taken equal to 0.03, a value which has been used in Tien-Walker- 
' Wolontis' paper. Define 

; ^ = wiy, <po) (A-4) 'X = pe-^'" + p*e^'^'> (A-5) 

dy 

where p* is the conjugate of p. After substituting (A-1) to (A-5) into the 
working equations (15) to (18) and carrying out considerable algebraic 
work, we obtain exactly Pierce's equation. 

2 (1 + jC/i)(l + bC) innn \ ah \^ r\ r\ 

(j - >iCfi -h j}/ibC)(ti + jb) 



provided that 



+ CO 

—k\((>(.y ,<po+<t>)—<p(.y ,Vo)l['^+Cw(.y ,ipo+(t>)] 




(A-7) 
• di^ sgn (^(?/, .i?o + «/)) - 9?(^, <Po)) = 8eQC 

(1 -f 3Cy){ii ^ jb) I e''" cos (arg [(1 -f jCm)(m + jb)] + my - ^0) 



374 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Here ^ = Mi + JM2 or Pierce's rri + jiji . From (A-7) the value of Up is 
determined for a given QC. The ac velocities of the electrons are derived 
from (A-4), such as, 



= -26 



M 



M + jb 
1 + jcn 



e"^" cos ( arg 



M 



M + jb 



rvi+iCM/j 



+ M22/ — <Po 



(A-8) 



(A-1), (A-2), (A-7) and (A-8) are the expressions used to calculate the 
initial conditions at y = 0, Avhen fn and jU2 are solved from Pierce's equa- 
tion (A-6). 

From (12c), the particular solution of the backward wave at small- 
signal is found to be 

j^, . ., -2iC(l+iC/x)(M+ib) 



^Ml!/ 



2j — CfjL -\- icb 

r [-2jC{\ - 



cos 



+ iCM)(M+i6)' 



Cn + jcb 



+ M2y — ^0 



which agrees with Pierce's analysis 



17 



3. 

4. 



REFERENCES 

1. J. R. Pierce, Traveling-Wave Tubes, D. Van Nostrand Co., N.Y., 1950, p. 160. 

2. R. L. Hess, Some Results in the Large-Signal Analysis of Traveling-Wave 

Tubes, Technical Report Series No. 60, Issue No. 131, Electronic Research 
Laboratory, University of California, Berkeley, California. 

C. K. Birdsall, unpublished work. 

J. J. Caldwell, unpublished work. 

5. P. Parzen, Nonlinear Effects in Traveling-Wave Amplifiers, TR/AF-4, Radia- 

tion Laboratory, The Johns Hopkins University, April 27, 1954. 

6. A. Kiel and P. Parzen, Non-linear Wave Propagation in Traveling-Wave 

Amplifiers, TR/AF-13, Radiation Laboratory, The Johns Hopkins Univer- 
sity, March, 1955. 

7. A. Nordsieck, Theory of the Large-Signal Behavior of Traveling-Wave Ampli- 
fiers, Proc. I.R.E., 41, pp. 630-637, May, 1953. 

H. C. Poulter, Large Signal Theory of the Traveling-Wave Tube, Tech. Re- 
port No. 73, Electronics Research Laboratory, Stanford University, Cali- 
fornia, Jan., 1954. 

P. K. Tien, L. R. Walker and V. M. Wolontis, A Large Signal Theory of Trav- 
eling-Wave Amplifiers, Proc. LR.E., 43, pp. 260-277 March, 1955. 

J. E. Rowe, A Large Signal Analysis of the Traveling-Wave Amplifier, Tech. 
Report No. 19, Electron Tube Laboratory, University of Michigan, Ann 
Arbor, April, 1955. 
11. P. K. Tien and L. R. Walker, Correspondence Section, Proc. I.R.E., 43, 
p. 1007, Aug., 1955. 

Nordsieck, op. cit., equation (1). 

L. Brillouin, The Traveling-Wave Tube (Discussion of Waves for Large 
Amplitudes), J. Appl. Phys., 20, p. 1197, Dec, 1949. 

Pierce, op. cit., p. 9. 

Nordsieck, op. cit., equation (4). 

Pierce, op. cit., equation (7.13). 
17. J. R. Pierce, Theory of Traveling-Wave Tube, Appendix A, Proc. I.R.E. 
35, p. 121, Feb., 1947. 



8. 



10 



12 
13 

14 
15 
16 



A Detailed Analysis of Beam Formation 
with Electron Guns of the Pierce Type 

By W. E. DANIELSON, J. L. ROSENFELD,* and J. A. SALOOM 

(Manuscript received November 10, 1955) 

The theory of Cutler and Hines is extended in this paper to permit an 
analysis of heam-spreading in electron guns of high convergence. A lens 
correction for the finite size of the anode aperture is also included. The Cutler 
and Hines theory was not applicable to cases where the effects of thermal 
velocities are large compared with those of space charge and it did not include 
a lens correction. Gun design charts are presented which include all of these 
effects. These charts may he conveniently used in choosing design parameters 
to produce a prescribed beam. 

CONTENTS 

1 . Introduction 377 

2. Present Status of Gun Design; Limitations 378 

3. Treatment of the Anode Lens Problem 379 

A. Superposition Approach 379 

B. Use of a False Cathode 382 

C. Calculation of Anode Lens Strength by the Two Methods 383 

4. Treatment of Beam Spreading, Including the Effect of Thermal Electrons 388 

A. The Gun Region 388 

B. The Drift Region 392 

5. Numerical Data for Electron Gun and Beam Design 402 

A. Choice of Variables 402 

B. Tabular Data 402 

C. Graphical Data, Including Design Charts and Beam Profiles 402 

D. Examples of Gun Design Using Design Charts 403 

6. Comparison of Theory with Experiment 413 

A. Measurement of Current Densities in the Beam 413 

B. Comparison of the Experimentally Measured Spreading of a Beam with 
that Predicted Theoretically 416 

C. Comparison of Experimental and Theoretical Current Density Distri- 
butions where the Minimum Beam Diameter is Reached 418 

D. Variation of Beam Profile with T 418 

7. Some Additional Remarks on Gun Design 418 



* Mr. Rosenfeld participated in this work while on assignment to the Labora- 
tories as part of the M.I.T. Cooperative Program. 

375 



376 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

GLOSSARY OF SYMBOLS 

Ai , 2 anode designations 

B, C anode potentials 

Ci , 2 functions used in evaluating cr+' 

dA increment of area 

dl, dz increments of length 

e . electronic charge, base of natural logarithms 

En electric field normal to electron path 

F modified focal length of the anode lens 

Fd focal length of the anode lens as given by Davisson^ 

Fn force acting normal to an electronic path 

Fr , a fraction of the total current which would flow through 

a circle of radius r, a 

/, Id total beam current 

It beam current within a radius, r, of the center 

J current density 

k Boltzman's constant 

K - a quantity proportional to gun perveance 

m electronic mass 

P gun perveance 

P{r) probability that a thermal electron has a radial posi- 

tion between r and r -\- dr 

r radial distance from beam axis 

Va , c anode, cathode radii 

r^ distance from beam axis to path of an electron emitted 

with zero velocity at the edge of the cathode 

rgs radius of circle through which 95% of the beam cur- 

rent would pass 

f distance from center of curvature of cathode; hence, 

fc is the cathode radius of curvature and (fc — fa) 
is the distance from cathode to anode 

re+' slope of edge nonthermal electron path on drift side of 

enode lens 

Te-' slope of edge nonthermal electron path on gun side of 

anode lens 

R a dummy integration variable 

t time 

T cathode temperature in degrees K 

u longitudinal electron velocity 

Vc , X , y transverse electron velocities 

V, Va , f , X beam voltages with cathode taken as ground 



BEAM FORMATION WITH ELECTRON GUNS 377 

V(f, /■), Vc.(f, potential distributions used in the anode lens study 
r), etc. 

V' voltage gradient 

z distance along the beam from the anode lens 

2n,in distance to the point where rgs is a minimum 

( — a) Langmuir potential parameter for spherical cathode- 

anode gun geometry 

7 slope of an electron's path after coming into a space 

charge free region just beyond the anode lens 

r the factor which divides Fd to give the modified anode 

focal length 

5 dimensionless radius parameter 
€o dielectric constant of free space 
f dimensionless voltage parameter 

6 slope of an electron's path in the gun region 
r} charge to mass ratio for the electron 

fx normalized radial position in a beam 

a the radial position of an electron which left the cathode 

center with "normal" transverse velocity 
(T+' slope of o--electron on drift side of anode lens 

a J slope of (T-electron on gun side of anode lens 

^ electric flux 

1. INTRODUCTION 

During the past few years there have been several additions to the 
family of microwave tubes rec}uiring long electron beams of small diame- 
ter and high current density. Due to the limited electron current which 
can be "drawn from unit area of a cathode surface with some assurance 
of long cathode operating life, high density electron beams have been 
produced largely through the use of convergent electron guns which 
increase markedly the current density in the beam over that at the 
cathode surface. 

An elegant approach to the design of convergent electron guns was 
provided by J. R. Pierce^ in 1940. Electron guns designed by this method 
are known as Pierce guns and have found extensive use in the produc- 
tion of long, high density beams for microwave tubes. 

]\Iore recent studies, reviewed in Section 2, have led to a better under- 
standing of the influence on the electron beam of (a) the finite velocities 
with which electrons are emitted from the cathode surface, and (b) the 
defocusing electric fields associated with the transition from the ac- 
celerating region of the gun to the drift region beyond. Although these 
two effects have heretofore been treated separately, it is in many cases 



378 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

necessary to produce electron beams under circumstances where both 
effects are important and so must be dealt with simultaneously and more 
precisely than has until now been possible. It is the purpose of this paper 
to provide a simple design procedure for typical Pierce guns which in- 
cludes both effects. Satisfactory agreement has been obtained between 
measured l^eam contours and those predicted for several guns having 
per\'eances (i.e., ratios of beam current to the ^^ power of the anode 
voltage) from 0.07 X 10-« to 0.7 X 10"^ amp (volt)-3/2. 

2. PRESENT STATUS OF GUN DESIGN — LIMITATIONS 

Gun design techniques of the type originally suggested by J. R. Pierce 
were enlarged in papers by SamueP and by Field^ in 1945 and 1946. 
Samuel's work did not consider the effect of thermal velocities on beam 
shape and, although Field pointed out the importance of thermal veloci- 
ties in limiting the theoretically attainable current density, no method 
for predicting beam size and shape by including thermal effects was 
suggested. The problem of the divergent effect of the anode lens was 
treated in terms of the Davisson"* electrostatic lens formula, and no 
corrections were applied.* 

More recently. Cutler and Hines^ and also Cutler and Saloom^ have 
presented theoretical and experimental work which shows the pro- 
nounced effects of the thermal velocity distribution on the size and shape 
of beams produced by Pierce guns. Cutler and Saloom also point to the 
critical role of the beam-forming electrode in minimizing beam distor- 
tion due to improper fields in the region where the cathode and the 
beam-forming electrode would ideally meet. With regard to the anode 
lens effect, these authors also show experimental data which strongly 
suggest a more divergent lens than given by the Davisson formula. The 
Hines and Cutler thermal velocity calculations have been used"' "^ to 
predict departures in current density from that which should prevail in 
ideal beams where thermal electrons are absent. Their theory is limited, 
however, by the assumption that the beam-spreading caused by thermal 
velocities is small compared to the nominal beam size. 

In reviewing the various successes of the above mentioned papers in 
affording valuable tools for electron beam design, it appeared to the 
present authors that significant improvement could be made, in two 
respects, by extensions of existing theories. First, a more thorough in- 



* It is in fact erroneously statoci in Reference 5 that the lens action of an actual 
structure must be somewhat weaker than i)re(licted by the Davisson formula so 
that the beam on leaving the anode hole is more convergent than would be calcu- 
lated by llie Davisson method. This cjuestion is discussed further in Section 3. 



BEAM FORMATION WITH ELECTRON GUNS 379 

vestigation of the anode lens effect was called for; and second, there was 
a need to extend thermal velocity calculations to include cases where 
the percentage increase in beam size due to thermal electrons was as 
large as 100 per cent or 200 per cent. Some suggestions toward meeting 
this second need have been included in a paper by M. E. Hines.* They 
have been applied to two-dimensional beams by R. L. Schrag.^ The 
particular assumptions and methods of the present paper as applied to 
the two needs cited above are somewhat different from those of Refer- 
ences 8 and 9, and are fully treated in the sections which follow. 

3. TREATMENT OF THE ANODE LENS PROBLEM 

Using thermal velocity calculations of the type made in Reference 6, 
it can easily be shown that at the anode plane of a typical moderate 
perveance Pierce type electron gun, the average spread in radial posi- 
tion of those electrons which originate from the same point of the cathode 
is several times smaller than the beam diameter. For guns of this type, 
then, we may look for the effect of the anode aperture on an electron 
beam for the idealized case in which thermal velocities are absent and 
confidently apply the correction to the anode lens formula so obtained 
to the case of a real beam. 

Several authors have been concerned with the diverging effect of a 
hole in an accelerating electrode where the field drops to zero in the 
space beyond, ^° but these treatments do not include space charge effects 
except as given by the Davisson formula for the focal length, Fd , of 
the lens: 

F. = -^ (1) 

where V would be the magnitude of the electric field at the aperture if 
it were gridded, and V would be the voltage there. 

In attempting to describe the effect of the anode hole with more ac- 
curacy than (1) affords, we have combined analytical methods with 
electrolytic tank measurements in two i-ather different ways. The first 
method to be given is more rigorous than the second, hut a modification 
of the second method is much easier to use and gives essentially the 
same result. 

A. Siipcrposition Approach to the Anode Lens Problem 

Special techniques are required for finding electron trajectories in a 
space charge limited Pierce gun having a non-gridded anode. M. E. 



380 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Hines has suggested* that a fairly accurate description of the potential 
distribution in such guns can be obtained by a superposition method as 
follows: 

By the usual tank methods, find suitable beam forming electrode and 
anode shapes for conical space charge limited flow in a diode having! 
cathode and anode radii of curvature given by fc and f„i , respectively, 
as shown in Fig. 1(a). Using the electrolytic tank with an insulator along 
the line which represents the beam edge, trace out an equipotential 
which intersects the insulator at a distance fa2 from the cathode center 
of curvature. Let the cathode be at ground potential and let the voltage 
on anode Ai be called B. Suppose, now, that we are interested in electron 
trajectories in a non-gridded gun where the edge of the anode hole is a 
distance fai from the center of curvature of the cathode. Let the voltage, 
C, for this anode be chosen the same as the value of the equipotential 
traced out above for the case of cathode at ground potential and A\ 
at potential B. If we consider the space charge limited flow from a 
cathode which is followed by the apertured anode, Ai , and the full 
anode, Ai , at potentials C and B, respectively, it is clear that a conical 
flow of the type which would exist between concentric spheres will re- 
sult. The flow for such cases was treated by Langmuir,^ and the associ- 
ated potentials are commonly called the "Langmuir potentials." 

If we operate both Ai and A2 at potential C, however, the electrons 
will pass through the aperture in anode A2 into a nearly field-free region. . 
If the distance, fa2 — Tai , from A2 to Ai is greater than the diameter of 
the aperture in A2 , the flow will depend very little on the shape of Ai 
and the electron trajectories and associated equipotentials will be of the 
type we wish to consider except in a small region near Ai . We will shortly 
make use of the fact that the space charge between cathode and A 2 is 
not changed much when the voltage on Ai is changed from B to C, but 
first we will define a set of potential functions which will be needed. 

In order to obtain the potential at arbitrary points in any axially sym- 
metric gun when space charge is not neglected, w^e may superpose po- 
tential solutions to 3 separate problems where, in each case, the boundary 
condition that each electrode be an equipotential is satisfied. We will 
follow the usual notation in using f for the distance of a general point 
from the cathode center of curvature, and r for its radial distance from 
the axis of symmetry. Let Vdr, r), Vh(r, >') and Vsdr, r) be the three 
potential solutions where: (1) Vaif, r) is the solution for the case of no 
space charge with Ai and cathode at zero potential and A 2 at potential 
C, (2) Vb{r^ r) is the solution for the case of no space charge with A2 



* Verbal disclosure. 



BEAM FORMATION WITH ELECTRON GUNS 



381 



and cathode at zero potential and Ai at potential B, and (3) Vsc(f, r) is 
the soUition when space charge is present but when Ax , A^ , and cathode 
are all grounded. 

If the configuration of charge which contributes to Vs<-(f, r) is that 
corresponding to ideal Pierce type flow, then we can use the principle 
of superposition to give the Langmuir potential, VL(r, r): 



VUr, r) = Vcif, r) + V,{f, r) + V..{f, r) 



(2) 



Furthermore, the potential configuration for the case where ^i and A2 
are at potentical C can be written 



V =V.-\-^V, + F(.c)' 



(3) 



where the functional notation has been dropped and F(sc)' is the po 
icntial due to the new space charge when Ai and A2 are grounded. 
We are now ready to use the fact that F(sc)' may be well approximated 
1)3' Fsc which is easily obtained from (2). This substitution may be 
justified by noting that the space charge distribution in a gun using a 
\'oltage C for Ai does not differ significanth^ from the corresponding dis- 
tribution when Ai is at voltage B except in the region near and beyond 
A-i where the charge density is small anyway (because of the high electron 
velocities there). Substituting Fsc as given by (2) for F(sc)' in (3) then 
gives 



V 



Vi 



1 



B, 



V, 



(4) 



We have thus obtained an expression, (4), for the potential at an arbi- 



ANODE A2 

v=c 



ANODE A, 
V = B 



CATHODE 




Fig. 1(a) — ■ Electrode configuration for anode lens evaluation in Section 2>A. 



382 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

■i 

trary point in our gun in terms of the well known solution for space 
charge limited flow between two concentric spheres, Vl , and a potential 
distribution, Vb , which does not depend on space charge and can there- 
fore be obtained in the electrolytic tank. Once the potential distribution 
is found, electron trajectories may be calculated, and an equivalent lens 
sj^stem found. Equation (4) is used in this way in Part C as one basis for 
estimating a correction to the Davisson equation. (It will be noted that i 
(4) predicts a small but finite negative field at the cathode. This is be- 
cause the space charge density associated with Fsc is slightly greater 
near the cathode than that associated with F(sc)' , and it is this latter 
space charge which will make the field zero at the cathode under real 
space charge limited operation. Equation (4), as applied in Part C of this 
section, is used to give the voltage as a function of position at all points 
except near the cathode where the voltage curves are extended smoothly 
to make the field at the cathode vanish.) 

B. Use of a False Cathode in Treating the Anode Lens Problem 

Before evaluating the lens effect by use of (4), it will be useful to de- 
velop another approach which is a little simpler. The evaluation of the 
lens effect predicted by both methods will then be pursued in Part C 
where the separate results are compared. 

In Part A we noted that no serious error is made in neglecting the dif- 
ference between the two space charge configurations considered there 
because these differences were mainly in the very low space charge 
region near and beyond A2 . It similarly follows that we can, with only 1 
a small decrease in accuracy, ignore the space charge in the region near 
and beyond A2 so long as we properly account for the effect of the high 
space charge regions closer to the cathode. To place the foregoing obser- 
vations on a more quantitative basis, we may graph the Langmuir po- 
tential (for space charge limited flow between concentric spheres) versus 
the distance from cathode toward anode, and then superpose a plot of 
the potential from LaPlace's equation (concentric spheres; no space 
charge) which will have the same value and slope at the anode. The La- 
Place curve will depart significantly from the Langmuir in the region of 
the cathode, but will adequately represent it farther out." Our experi- 
ence has shown that the representation is "adequate" until the difference 
between the two potentials exceeds about 2 per cent of the anode voltage. 
Then, since space charge is not important in the region near the anode 
for the case of a gridded Pierce gun, corresponding to space charge 
limited flow between concentric spheres, it can be expected to be similarly 
unimportant for cases where the grid is replaced by an aperture. Let us 



I 



BEAM FORMATION WITH ELECTRON GUNS 



383 



therefore consider a case where electrons are emitted perpendicularly 
and with finite velocity from what would be an appropriate spherical 
equipotential between cathode and anode in a Pierce type gun. So long 
as (a) there is good agreement between the LaPlace and Langmuir curves 
at this artificial cathode and (b) the distance from this artificial cathode 
to the anode hole is somewhat greater than the hole diameter, we will 
liiid that the divergent effect of the anode hole will be very nearly the 
same in this concocted space charge free case as in the actual case where 
space charge is present. (The quantitative support for this last state- 
ment comes largely from the agreement between calculations based on 
this method and calculations by method A.) The electrode configura- 
tion is shown in Fig. 1(b), and the potential distribution in this space 
charge free anode region can now be easily obtained in the electrolytic 
j tank. This potential distribution will be used in the next section to pro- 
^•ide a second basis for estimating a correction to the Davisson equation. 

C. Calculation of Anode Lens Strength by the Two Methods 

The Davisson equation, (1), may be derived by assuming that none 
of the electric field lines which originate on charges in the cathode-anode 
region leave the beam before reaching the ideal anode plane where the 
voltage is F, and that all of these field lines leave the beam symmetrically 
and radially in the immediate neighborhood of the anode. Electrons 
I are thus considered to travel in a straight line from cathode to anode, 
and then to receive a sudden radial impulse as they cross radially diverg- 
ing electric field lines at the anode plane. A discontinuous change in 



CATHODE 




ANODE A2 
V = C 



ANODE A, 

v = c 



(b) 



^ FALSE 
CATHODE 



Fig. 1(b) — The introduction of a false cathode at the appropriate potential 
lUows the effect of space charge on the potential near the anode hole to be satis- 
:ictorily approximated as discussed in Section 3i?. 



384 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

slope is therefore produced as is common to all thin lens approximations. 
The diverging effect of electric field lines which originate on charges 
which have passed the anode plane is then normally accounted for by 
the universal beam spread curve/" In our attempt to evaluate the lens 
effect more accurately, we will still depend upon using the universal 
beam spread curve in the region following the lens and on treating the ; 
equivalent anode lens as thin. Consequently our improved accuracy 
must come from a mathematical treatment which allows the electric 
field lines originating in the cathode-anode region to leave the beam grad- 
ually, rather than a treatment where all of these flux lines leave the beam , 
at the anode plane. In practice the measured perveances, P(= I/V^'^), 
of active guns of the type considered here have averaged within 1 or 2 
per cent of those predicted for corresponding gridded Pierce guns. There- 
fore the total space charge between cathode and anode is much the 
same with and without the use of a grid, even though the charge dis- 
tribution is not the same in the two cases. The total flux which must 
leave our beam is therefore the same as that which will leave the cor- , 
responding idealized beam and we may write 

yp = I EndA = TT/VFidea/ (5) 

w^here En is the electric field normal to the edge of the beam, ra = rdfa/fc) 
is the beam radius at the anode lens, and Videai is the magnitude of the 
field at the corresponding gridded Pierce gun anode. 

To find the appropriate thin lens focal length we will now find the 
total integrated transverse impulse which would be given to an elec- 
tron which follows a straight-line path on both sides of the lens (see Fig. 
2), and we will equate this impulse to wAw where An is the transverse 
velocity given to the electron as it passes through the equivalent thin 
lens. In this connection we will restrict our attention to paraxial elec- 
trons and evaluate the transverse electric fields from (4) and from the 
tank plot outlined in Section B, respectively. The total transverse im- 
pulse experienced by an electron can be written 

f Fn dt = e [ —dl (()) 

J Path J Path U 

where u is the velocity along the path and Fn is the force normal to the 
path. 

We will usually find that the correction to (1) is less than about 20 
per cent. It will therefore be worthwhile to put (6) in a form which in 
effect allows us to calculate deviaiions from Fu as given by (1) instead 



BEAM FORMATION WITH ELECTRON GUNS 



385 



1 of deriving a completely new expression for F. In accomplishing this piir- 
f pose, it will be helpful to define a dimensionless function of radius, 6, by 



- = 1 + 5, 
r 

and a dimensionless function of voltage, f, by 




(7a) 



(7b) 



where Ta is the radius at the anode lens when the lens is considered thin, 
and T^'x is a constant voltage to be specified later. (Note that the quan- 
tities 5 and f are not necessarily small compared to 1.) Using u = \/2r]V, 
and substituting for -y/V from (7b) we obtain 



f En dl 4 r , 

= 7~7tW / ^"^1 + r + 5 + rs) ^z 



(8) 



where use has also been made of (7a) in the form 1 = r(l + d)/ra . Now, 
as outlined above, we equate this impulse to 771 An, and we obtain 



^» = WW. (/ ''■'' '' + / ''"'■'^ + ' + ^'' 'i 



(9) 



CATHODE 




Fig. 2 — The heavy line represents an electron's path when the effect of the 
.•mode hole may be represented by a thin lens, and when space charge forces are 
iihsent in the region following the anode aperture. For paraxial electrons, the 
(negative) focal length is related to the indicated angles by (y = + Ta/F). 



386 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




CENTER OF 

~~ CURVATURE 

OF CATHODE 

SURFACE 



Fig. 3 — The gun parameters used in Section SC for comparing two methods of 
evaluating the effect of the anode lens. 

The first integral can be obtained from (5) ; hence, if we are able to choose 
Vx so that the second integral vanishes, we may write: 



Au = 



raV'2riVx 



The reciprocal of the thin lens focal length is therefore 

i _ ^ _ ^' 

F ~ ~raUf ^ ~^VWf 



(10) 



where w/ and F/ are the final velocity and voltage of the electron after 
it leaves the lens region. 

The real task, then, is to use the potential distribution in the gun as 
obtained by the methods of Part A or Part B above to find the value of 
V X which causes the last integral in (9) to vanish : To compare the two 
focal lengths obtained by the methods of Part A and B respectively, a 
specific tank design of the type indicated in Fig. 1 was carried out. The 
relevant gun parameters are indicated in Fig. 3. Approximate voltages 
on and near the beam axis were obtained as indicated in Parts A and B, 
above, with the exception that in the superposition method, A, special 
techniques were used to subtract the effect of the space charge lying in 
the post-anode region (because the effect of this space charge is accounted 
for separately as a divergent force in the drift region*). From these data, 

* See Section 4B. 



BEAM FOKMATION WITH ELECTRON GUNS 



387 




800 805 810 815 820 825 830 835 840 845 850 855 860 



Fig. 4 — Curves for finding the value of Fx to be used in equation (10) for the 
set of gun parameters of Fig. 3. 



l)oth the direction and magnitude of the total electric field near the 
beam axis were (with much labor) determined. Once these data had 
been obtained, a trial value was selected for Vx , and the corresponding 
local length was calculated by (10). This enabled the electron's path 
through the associated thin lens to be specified so that, at this point in 
the procedure, both r and V were known functions of ^, and the quan- 
tities 8 and f were then obtained as functions of € from (7). Finally the 
second integral in (9) was evaluated for the particular Vx chosen, and 
then the process was repeated for other values of Vx . Fig. 4 shows curves 
whose ordinates are proportional to this second integral and whose 
abscissae are trial values for Vx . As noted above, the appropriate value 
for Vx is that value which makes the ordinate vanish, so that we obtain 
T'c = 813 and 839 for methods A and B, respectively. The percentage 
difference in the focal lengths obtained by the two methods is thus only 
1 .6 per cent, and the reasonableness of making calculations as outlined 
in Part B is thus put on a more quantitative basis. 

Even calculations based on the method of Part B are tedious, and we 
naturally look for simpler methods of estimating the lens effect. In this 
fonnection we have found that Vx is usually well approximated by the 
\alue of the potential at the point of intersection between the beam axis 
and the ideal anode sphere. The specific values of the potential at this 
point as obtained by the methods of Parts A and B were 814 and 827, 
respectively. It will be noted that these values agree remarkably well 
with the values obtained above. Furthermore, very little extra effort is 
required to obtain the potential at this intersection in the false cathode 
case: 



I 

388 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Electrolytic tank measurements are normally made in the cathode- 
anode region to give the potential variation along the outside edge of 
the electron beam (for comparison with the Langmuir potential) ; hence, 
by tracing out a suitable equipotential line, the shape of the false cathode 
can easily be obtained. With the false cathode in place and at the proper 
potential, the approximate value for Vx is then obtained by a direct tank 
measurement of the potential at an axial point whose distance from the 
true cathode center is (fc — fa) as outlined above. Although finite elec- 
tron emission velocities typically do not much influence the trajectory 
of an electron at the anode, they do nevertheless significantly alter the 
beam in the region beyond. It is in this affected region where experi- 
mental data can be conveniently taken. We must, therefore, postpone a 
comparison of lens theory with experiment until the effect of thermal 
velocities has been treated. At that time theoretical predictions com- 
bining the effects of both thermal velocities and the anode lens can be 
made and compared with experiment. Such a comparison is made in 
Section 6. 

4. TREATMENT OF BEAM SPREADING, INCLUDING THE EFFECT OF THERMAL 
ELECTRONS 

Jn Section 2 the desirability of having an approach to the thermal 
spreading of a beam which would be applicable under a wide variety of 
conditions was stressed. In particular, there was a need to extend ther- 
mal velocity calculations to include the effects of thermal velocities even 
when electrons with high average transverse velocities perturb the beam 
size by as much as 100 or 200 per cent. Furthermore, a realistic mathe- 
matical description which would allow electrons to cross the axis seemed 
essential. The method described below is intended adequately to answer 
these requirements. 

A. The Gun Region 

The Hines-Cutler method of including the effect of thermal velocities 
on beam size and shape leads one to conclude that, for usual anode 
voltages and gun perveance, the beam density profile in the plane of 
the anode hole is not appreciably altered by thermal velocities of emis- 
sion. (This statement will be verified and put on a more quantitative 
basis below.) Under these conditions, the beam at the anode is ade- 
quately described by the Hines-Cutler treatment. We will therefore find 
it convenient to adopt their notation where possible, and it will be 
worthwhile to review their approach to the thermal problem. 



BEAM FORMATION WITH ELECTRON GUNS 389 

It is assumed that electrons are emitted from the cathode of a therm- 
ionic gun with a IMaxwelhan distribution of transverse velocities 

ZTTfC 1 

where Jc is the cathode current density in the z direction, T is the cath- 
jode temperature, and v^: and Vy are transverse velocities. The number 
iof electrons emitted per second with radially directed voltages between 

V and V + dV is then 



-(.Ve/kT) 



(S) 



^J. = /.e— -^^^(^^j (12) 

Now in the accelerating region of an ideal Pierce gun (and more generally 
I in any beam exhibiting laminar flow and having constant current density 
()\'er its cross section) the electric field component perpendicular to the 
axis of symmetry must vary linearly with radius. Conseciuently Hines 
and Cutler measure radial position in the electron beam as a fraction, 
^, of the outer beam radius (re) at the same longitudinal position, 

r = fire (13) 

The laminar flow assumption for constant current densities and small 
beam angles implies a radius of curvature for laminar electrons which 
so varies linearly with radius at any given cross section so that 



a 



Substituting for r from (13), (14) becomes 

rfV , /2 dre\ dfj. 

d^^VcTt)dt=^ ^^^^ 

where Ve and dr /dt can be easily obtained from the ideal Langmuir 
solution. Since the eciuation is linear in /x, we are assured that the radial 
position of a non-ideal electron that is emitted with finite transverse 
velocity from the cathode center (where ^ = 0) will, at any axial point, 
be proportional to dii/dt at the cathode. 

Let us now define a quantity "o-" such that n = a/re is the solution 
to (15) with the boundary conditions /Xr = and 

_ 1 
where the subscript c denotes evaluation at the cathode surface, k is 





390 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Boltzman's constant, T is the cathode temperature in degrees Kelvin, 
and m is mass of the electron. For the case ixc = 0, but with arbitrary 
initial transverse velocity, we will then have 

/^\ 

^^nl_ /kf ^^^'^ 

Tc y m 

Plence we can express a in terms of the thermal electron's radial po- 
sition (r), and its initial transverse velocity, Vc , 



y m _ y 



. . - . /kT 

dt } f 



The quantity a can now be related to the radial spread of thermal 
electrons (emitted from a given point on the cathode) with respect to 
an electron with no initial velocity: By (11) we see that the number 
of electrons leaving the cathode with dji/dt = Vc/ve is proportional to Vc 
exp —Vcm/2kT. Suppose many experiments were conducted where all 
electrons except one at the cathode center had zero emission velocity, 
and suppose the number of times the initial transverse velocity of the 
single thermal electron were chosen as Vc , is proportional to Vc exp 
— Vcm/2kT. Then the probability, P{r), that the thermal electron 
would have a radial position between r and r -\- dr when it arrived at the 
transverse plane of interest would be proportional to Vc exp —Vc^(m/2kT). 
Here Vc is the proper transverse velocity to cause arrival at radius r, and 
by (17) we have 

a y m 
so that the probability becomes 

Pir) = J.e-^^'''-'^ d (^Q (18) 

We therefore identify cr with the standard deviation in a normal or 
Gaussian distribution of points in two dimensions. At the real cathod(\ 
thermal electrons are simultaneously being emitted from the cathode 
surface with a range of transverse velocities. However, if a as definml 
above is small in comparison with r,. , the forces experienced by a ther- 
mal electron when other thermal electrons are present will be very nearly 



BEAM FORMATION WITH ELECTRON GUNS 



391 



2.0 
1.8 
1.6 



> 1.4 
t 1.2 



\%y 



1.0 



0.8 



0.6 



0.4 






1.0 



1.2 



1.4 



1.6 



1.8 2.0 2.2 2.4 



2.6 



2.8 3.0 3.2 3.4 3.6 3.8 4.0 



Fig. 5 — Curves useful in finding the transverse displacement of electron tra- 
i jectories at the anode of Pierce-type guns. 

i 

tlie same as the forces involved in the equations above. Thus if o- <3C J'e , 

(18) may be taken as the distribution, in a transverse plane, of those 
electrons which were simultaneously emitted at the cathode center. 
I Furthermore, the nature of the Pierce gun region is such that electrons 
emitted from any other point on the cathode will be similarly distributed 
\\ ith respect to the path of an electron emitted from this other point 
w ith zero transverse velocity (so long as they stay within the confines 
, of the ideal beam). Hines and Cutler have integrated (15) with n^ = 
' and {dn/dt)c = 1 to give g/ {fc\/kT/'2eV^ at the anode as a function of 
; /", /fo . This relationship is included here in graphical form as Fig. 5. 
, For a large class of magnetically shielded Pierce-type electron guns, 
including all that are now used in our traveling wave tubes, Ve/a at the 
anode is indeed found to be greater than 5 (in most cases, greater than 
10) so that evaluation of a at the anode of such guns can be made with 
considerable accuracy by the methods outlined above. One source of 
error lies in the assumption that electrons which are emitted from a 
point at the cathode edge become normally distributed about the cor- 
responding non-thermal (no transverse velocity of emission) electron's 
path, and with the same standard deviation as calculated for electrons 
from the cathode center. In the gun region where Ve/a tends to be large 
this difference between representative a- values for the peripheral and 
central parts of the beam is unimportant, but it must be re-examined in 
tlie drift region following the anode. 





392 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

We have already investigated the region of the anode hole in some 
detail in Section 3 and have found it worth while to modify the ideal 
Davisson expression for focal length of an equivalent anode lens. In 
particular, let us define a quantity F by 

F = focal length = Fd/T (19) 

where Fd is the Davisson focal length. Thus T represents a corrective 
factor to be applied to Fd to give a more accurate value for the focal 
length. In so far as any thin lens is capable of describing the effects of 
diverging fields in the anode region, we may then use the appropriate 
optical formulas to transfer our knowledge of the electron trajectories 
(calculated in the anode region as outlined above) to the start of the drift 
region. In particular, 

-f (20) 

where {dr/dz)i and {dr/dz)^ are the slopes of the path just before and 
just after the lens, and r is the distance from the axis to the point where 
the ideal path crosses the lens plane. 

B. The Drift Region 

Although Te/a- was found to be large at the anode plane for most guns 
of interest, this ratio often shrinks to 1 or less at an axial distance of 
only a few beam diameters from the lens. Therefore, the assumption that 
electron trajectories may be found by using the space charge forces 
which would exist in the absence of thermal velocities of emission (i.e., 
forces consistant with the universal beam spread curve) may lead to very 
appreciable error. For example, if ecjual normal (Gaussian) distributions 
of points about a central point are superposed so that the central points 
are equally dense throughout a circle of radius Te , and if the standard de- 
viation for each of the normal distributions is cr = r^ , the relative density 
of points in the center of the circle is only about 39 per cent of what it 
would be Avith a < (re/5). 

In order to minimize errors of this type we have modified the Hines- 
Cutler treatment of the drift space in two ways: (1) The forces influenc- 
ing the trajectories of the non- thermal electrons are calculated from a 
progressive estimation of the actual space charge configuration as modi- 
fied by the presence of thermal electrons. (2) Some account is taken of 
the fact that, as the space charge density in the beam becomes less uni- 
form as a function of radius, the spread of electrons near the center of 
the beam increases more rapidly than does the corresponding spread 



BEAM FORMATION WITH ELECTRON GUNS 393 

farther out. Since item (1) is influenced by item (2), the specific as- 
sumptions involved in the latter case will be treated first. 

When current density is uniform across the beam and its cross section 
changes slowly with distance, considerations of the type outlined above 
for the gun region show that those thermal electrons which remain 
within the beam will continue to have a Gaussian distribution with re- 
spect to a non-thermal electron emitted from the same cathode point. 
When current density is not uniform over the cross section, we would 
still like to preserve the mathematical simplicity of obtaining the current 
density as a function of beam radius merely by superposing Gaussian 
distributions which can be associated with each non-thermal electron. 
To lessen the error involved in this simplified approach, we will arrive 
at a value for the standard deviation, a (which specifies the Gaussian 
distribution), in a rather special way. In particular, a at any axial po- 
sition, z, will be taken as the radial coordinate of an electron emitted 
from the center of the cathode with a transverse velocity of emission 
given by, 



ve = y- 



— (21) 

m 



It is clear from (17) that for such an electron, r = o- in the gun region. 
From (18), the fraction of the electrons from a common point on the 
cathode which will have r ^ a in the gun region is 



2 



fraction = [ e'^'-'"-''^ d ^= I - e'"' = 0.393 (22) 

If re denotes the radial position of the outermost non-thermal electron 
and if 0- > /■,, , the "a--electron" will be moving in a region where the 
space charge density is significantly lower than at the axis. We could, 
of course, have followed the path of an electron with initial velocity 
equal to say 0.1 or 10 times that given in (21) and called the correspond- 
nig radius O.lcr or lOo-. The reason for preferring (21) is that about 0.4 
or nearly half of the thermal electrons emitted from a common cathode 
point will have wandered a distance less than a from the path of a non- 
thermal electron emitted from the same cathode point, while other 
thermal electrons will ha\'e wandered farther from this path; conse- 
quently, the current density in the region of the o--electron is expected 
to be a reasonable average on which beam spreading due to thermal 
\elocities may be based. With this understanding of how a is to be cal- 
culated, we can proceed to the calculation of non-thermal electron 
trajectories as suggested in item (1). 



394 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

The non-thermal paths remain essentially laminar, and with r^ de- 
noting the radial coordinate of the outermost non-thermal electron, we 
will make little error in assuming that the current density of non-ther- 
mal electrons is constant for r < Ve . Consequently, if equal numbers of 
thermal electrons are assumed to be normally distributed about the cor- 
responding non-thermal paths, the longitudinal current density as a 
function of radius can be found in a straightforward way by using (18). 
The result is 

J ^ ^_(..,,..) n" R ^-(«^/2.^)^^ frR\ ^ /R\ ^23) 

Jd Jo a \a^/ \(t/ 

where /o is the zero order modified Bessel function and the total current 
is Id = TTVe Jd ' Equation (23) was integrated to give a plot of Jr/Jo 
versus r/a, with re/a as a parameter and is given as Fig. 6 in Reference 
6. It is reproduced here as Fig. 6. Since the only forces acting on elec- 
trons in the drift region are due to space charge, we may write the equa- 
tion of motion as 

where Er is the radial electrical field acting on an electron with radial 
coordinate r. Since the beam is long and narrow, all electric lines of force 
may be considered to leave the beam radially so that Er is simpl}^ ob- 
tained from Gauss' law. Equation (24) therefore becomes 

-— = --^— / 2irp dr = -— ! — / ■ Iirr dr 

dt^ zireor Jo Zireor Jo \/2t]V a. 



(25) 
2irenr Jo 



27reor 

From (23) we note that the fraction of the total current within any 
radius depends only on fe/o- and j'/ct: 



:il 



dr 



^ / J0')2irr ar / xo ,/o 

r = - = H-) f 

'- r.J(r)2.rdr ^''''° (2«) ' 

Jo 



■•r I a 

C 




'^dV^]^Fr-j- 

\(X a t 



\ 



BEAM FORMATION WITH ELECTRON GUNS 



395 




Fig. 6 — Curves showing the current density variation with radius in a beam 
I which has been dispersed by thermal velocities. Here r« is the nominal beam radius, 
I r is the radius variable, and <t is the standard deviation defined in equation 17. 

A family of curves with this ratio, Fr , as parameter has been reproduced 
: from the Hines-Cutler paper and appears here as Fig. 7. Using this no- 
tation, (25) becomes 



dV ^ Vr,/{2V.) j^ Fr 
di^ 27reo r 



or 



d r 
dz^ 



jn_ lo Fr^ Fr 

27r€0 (27,7a)3/2 J. J. 



(27) 



where we have made use of the dc electron drift velocity to make dis- 
tance the independent variable instead of time, and have defined a 
quantity K which is proportional to gun perveance. We can now apply 
(27) to the motion of both the outer (edge) non-thermal electron and 
the cr-electron. From (26) we see that Fr, and Fg depend only on re/a] 



396 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

12 



11 



10 



a 

LLI 
< 

u 

B 1 

z 

z 
o 

< 



o ^ 





















/^ 


















,/;^ 


^ 
















// 


P> 


^ 












Fr = 


0.995/ 


^ 


^ 


/^ 












^ 




^ 


^ 


/^ 










/ 


rz 


^ 




/ >> 


/ 








^ 


z:^ 


^ 


:^ 






/ 




y. 


^ 


/y 


'A 


%: 








^ 


;^ 


^ 


Xy 


'^. 




^^ 


^^ 










^^ 


w 


i^ 


/^ 




oao^ 








^ 




1^ 


^- 






oo^ 






= 




^^ 

















10 



re/0- 



Fig. 7 — Curves showing the fraction, Fr , of the total beam current to be found 
within any given radius in a beam dispersed by thermal velocities as in Fig. 6. 

consequently the continuous solution for r^ and r„ (= a) as one moves 
axially along the drifting beam involves the simultaneous solution of two 
equations : 



(fve 

d~a 
d^ 



KFr./re 
KFJa 



(28) 



BEAM FORMATION WITH ELECTRON GUNS 



397 



0.36 
0.32 
0.28 
0.24 

0.16 
0.12 
0.08 
0.04 



\ 
















\ 
































































1 


\ 
















\ 
















\ 
















\ 


V 
















V. 


--- 


— 


■ 







8 



10 



12 



14 



16 



I Fig. 8 — A curve showing the effect of a quantity related to the space charge 
• force (in the drift region) on a thermal electron with standard deviation a. (See 
'equation 28.) 



which are related by the mutual dependence of Fr^ and Fa on re/a. F„ 
and Frjve are plotted in Figs. 8 and 9. 

We may summarize the treatment of the drift region, then, as follows: 
1 (a) The input values of r^ and rgJ at the entrance to the anode lens 
jare obtained from the Pierce gun parameters r^ and 6, while the value 
of a and aJ at the lens entrance can be obtained as mentioned above 
by integrating (15) from the cathode, where Mc = and (dfx/dt)c = 1, 
to the anode plane. (The minus subscripts on r' and a' indicate that 
these slopes are being evaluated on the gun side of the lens; a plus sub- 
script will be used to indicate evaluation on the drift region side of the 
lens.) The values of Ve and a on leaving the lens will of course be their 
entrance values in the drift region, and the effect of the lens on r/ and 
a' is simply found in terms of the anode lens correction factor T by use 
of (20). The value of a at the anode can be obtained from (17) if n is 
known there. In this regard, (15) can be integrated once to give 



= 1_/M dt 

" " r\dt)c{r,/r,y 



(29) 



398 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



LL 



0.9 



0.8 



0.7 



0.6 



0.5 



0.4 



0.3 



0.2 



0.1 





















. — 


— 


















■-^ 














\ 






X 


^ 


















\ 


I 


/ 
























\ 

\ J 

\ / 


/ 
























A 
7 \ 


























/ \ 
/ \ 




























\ 
\ 
\ 


























\ 
\ 
\ 




























\ 




























f.,/(reA) 






















\ 
\ 
\ 


s 




























^, 


























> 






























"-. 


'"-.. 






























^-*^^ 


■•—.^ 






















































1/ 
1 



























0.38 



0.36 



0.34 



0.32 



0.30 



0.28 



0.26 



0.24 



0.22 



0.20 



0.18 1? 



0.1 6 



0.14 



0.12 



0.10 



0.08 



0.06 



0.04 



0.02 



6 7 8 9 



10 



11 12 13 



14 



Fig. 9 — Showing quantities related to the effect of the space charge force in 
the drift region on the outermost non-thermal electron. (See equation 28.) 



i 



We can now substitute for transit time in terms of distance and Lang- 
muir's well known potential function/^ —a. The value of this parameter, 
for the case of spherical cathode-anode geometry in which we are in- 
terested, depends only on the ratio fe/f which is equal to Vc/rg . (Because 
of their frerjuent use in gun design, certain functions of —a are included 
here as Table I.) In terms of —a, then, the potential in the gun region 



BEAM FORMATION WITH ELECTRON GUNS 



399 



Fable I 



Table of Functions of —a Often Used in Electron 
Gun Design 















fc/f 


(-«)2 


(- a)V3 


(- a)2/3 




difc/r) 


1.0 


0.0000 


0.0000 


0.0000 


0.0000 




1.025 


0.0006 


0.0074 








1.05 


0.0024 


0.0179 


0.134 






1.075 


0.0052 


0.0306 


0.173 






1.10 


0.0096 


0.0452 


0.212 


1.392 


0.590 


1.15 


0.0213 


0.0768 


0.277 






1.20 


0.0372 


0.1114 


0.334 


1.767 


0.716 


1.25 


0.0571 


0.1483 


0.385 






1.30 


0.0809 


0.1870 


0.432 


2.031 


0.790 


1.35 


0.1084 


0.2273 


0.476 






1.40 


0.1396 


0.2691 


0.519 


2.243 


0.874 


1.45 


0.1740 


0.3117 


0.558 






1.50 


0.2118 


0.3553 


0.596 


2.423 


0.886 


1.60 


0.2968 


0.4450 


0.667 


2.583 


0.915 


1.70 


0.394 


0.5374 


0.733 


2.725 


0.939 


1.80 


0.502 


0.6316 


0.795 


2.855 


0.954 


1.90 


0.621 


0.7279 


0.853 


2.975 


0.970 


2.00 


0.750 


0.8255 


0.908 


3.087 


0.982 


2.10 


0.888 


0.9239 


0.961 


3.192 


0.993 


2.20 


1.036 


1.024 


1.012 


3.292 


1.003 


2.30 


1.193 


1.125 


1.061 


3.388 


1.012 


2.40 


1.358 


1.226 


1.107 


3.481 


1.020 


2.50 


1.531 


1.328 


1.152 


3.570 


1.028 


2.60 


1.712 


1.431 


1.196 


3.655 


1.034 


2.70 


1.901 


1.535 


1.239 


3.738 


1.039 


2.80 


2.098 


1.639 


1.280 


3.817 


1.044 


2.90 


2.302 


1.743 


1.320 


3.894 


1.048 


3.00 


2.512 


1.848 


1.359 


3.968 


1.052 


3.1 


2.729 


1.953 


1.397 


4.040 


1.056 


3.2 


2.954 


2.059 


1.435 


4.111 


1.059 


3.3 


3.185 


2.164 


1.471 


4.180 


1.062 


3.4 


3.421 


2.270 


1.507 


4.247 


1.064 


3.5 


3.664 


2.376 


1.541 


4.315 


1.066 


3.6 


3.913 


2.483 


1.576 


4.377 


1.068 


3.7 


4.168 


2.590 


1.609 


4.441 


1.070 


3.8 


4.429 


2.697 


1.642 


4.501 


1.072 


3.9 


4.696 


2.804 


1.674 


4.563 


1.074 


4.0 


4.968 


2.912 


1.706 


4.621 


1.076 



400 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



may be written 



df df i-aaY'^ 



dt 



'V2nV a/2;^ {-a) 



2/3 



(30) 
(31) 

(32) 



so that upon substitution from (29) and (31), (17) becomes 
Fig. 5, which has been referred to above, shows 

O-a . /2eVa 
'fcV 'kf- 

as a function of {fc/fa) as obtained from (32), and allows o-„ to be de- 
termined easily. Using (20), the value of re+' is given by 



/ Tea , 



F 



-,.= -^^_,. = ,/_g-l) (33) 



where dg is the half-angle of the cathode (and hence the initial angle 
which the path of a non-thermal edge electron makes with the axis). 
We may write for 1/Fd 

1 V fe /d(-aY"\ 



Fo 4F 4(-aa)^/VV\rf(fc/r-) 7a 



(34) 



In Fig. 10 we plot —falFr, as a function of fjfa for easy evaluation of 
re+' in (33). Taking the first derivative of (32) with respect to ^, we ob- 
tain an expression for aJ. Using this in conjunction with (20) and (34) 
we find 



0-+ = 




Y (r<^i + C2) 



I 



(35) 



where 






cira 



d{fc/f) 



/3 



and 



^-i/f. ft -(-''"/ 



(-a)2/3_ 



! 



Ci and C2 are plotted as functions of fc/fa in Fig. 11. 

(b) After choosing a specific value for r and evaluating K = rj/c/ . 



BEAM FORMATION WITH ELECTRON GUNS 



401 



Q 

LL 



lU 



I.O 

1.4 
1.2 
1.0 
0.8 
0.6 
0.4 
0.2 



































\ 






























\ 


V 






























\ 
































\ 


































~~~- 


■--- 















































































1.0 12 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 40 

rcAa 

Fig. 10 — Curve used in finding ?•«+', the direction of a nonthermal edge elec- 
tron as it enters the drift region. (See equation 33.) 

(27r€o(277 Fa) ''), (28) is integrated numerically using the BTL analog com- 
puter to obtain a and r^ as functions of axial distance along the beam, 
(c) Knowing a and Ve , other beam parameters such as current dis- 
tribution and the radius of the circle which would encompass a given 
percentage of the total current can be found from Figs. 6 and 7. 



X 

tvi 

U 



20 

15 

10 

5 



-5 

-10 

-15 

-20 

-25 

-30 





POLYNOMIAL REPRESENTATION 
(ACCURATE WITHIN 2°/o) 


-OR 


c, & 


C2 










,'''' 




C, = 4.13 fc/ra + 2.67 

C2 = 0.635(r^/faf-13.56 rc/fa + 19-33 




, ,-' 


.' ' 


.-''' 






















.'-' 


,'-' 












\ 


^^ 








-' 


^^-' 


< 


















^v 


■^ 


X 


,.-' 






















^** 


,^-' 


''H 




'^ 


































"^ 


^ 


^ 




































^^ 
































\> 


^ 

































20 
18 
16 
14 

12 

rO 
O 

10 X 

(J 

8 



2 


10 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 

tc/fa 



Fig. 11 — Curves used in evaluating o-+', the slope of the trajectory of a thermal 
electron with standard deviation a as it enters the drift region. (See equation 35.) 



402 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

5. NUMERICAL DATA FOR ELECTRON GUN AND BEAM DESIGN 

A. Choice of Variables 

Except for a scaling parameter, the electrical characteristics of an 
ideal Pierce electron gun are completely determined when three param- 
eters are specified, e.g., fc/fa , perveance, and Va/T. Also, for the simp- 
lest case r is equal to 1 so that (since K depends only on gun perveance) 
in this case no additional parameter is needed. This implies that nor- 
malized values of ?-/, a, a', and K at the drift side of the anode lens are 
not independent. If, however, the value of F at the anode lens is taken 
as an additional variable, four parameters plus simple scaling are re- 
quired before complete predictions of beam characteristics can be made. 
In assembling analog computer data which would adequately cover 
values of fc/fa , perveance, and Va/T which are likely to be of interest 
to us in designing future guns, we chose to present the major part of 
our data with T fixed at 1.1. This has seemed to be a rather typical value 
for r, and by choosing a specific value we decrease the total number of 
significant variables from 4 to 3. (The effect of variations in T on the 
minimum radius which contains 95 per cent of the beam is, however, 
included in Fig. 16 for particular values of Va/T and perveance.) Al- 
though the boundary conditions for our mathematical description of the 
beam in a drift space are simplest when expressed in terms of Vg , r/, a 
and ct', we have attempted to make the results more usable by express- 
ing all derived parameters in terms of fc/fa , s/Va/T, and the perveance, 
P. 

B. Tabular Data 

The rather extensive data obtained from the analog computer for the 
r = 1.1 case and for practical ranges in perveance, Ve/T, and fc/fa 
are summarized in Tables IIA to E where the parameters r^ and a which 
specify the beam cross section are given as functions of axial distance 
from the anode plane. Some feeling for the decrease in accuracy to be 
expected as the distance from the anode plane increases can be obtained 
by reference to Section 6B where experiment and theory are compared 
over a range of this axial distance parameter. 

C. Graphical Data, Including Design Charts and Beam Profdes 

In typical cases, the designer of Pierce electron guns is much more 
concerned with the beam radius at the axial position where it is smallest 
(and in the axial position of this minimum) than he is in the general 






BEAM FORMATION WITH ELECTRON GUNS 403 



jspreadiug of the beam with distance. This is true because, in microwave 
beam tubes, the beam from a magnetically shielded Pierce gun normally 
enters a strong axial magnetic field near a point where the radius is a 
minimum, so that magnetic focusing forces largely determine the beam's 
subsequent behavior. The analog computer data has therefore been re- 
processed to stress the dependence of the beam's minimum diameter and 
the corresponding axial position of the minimum on the basic design 
iparameters fdfa , perveance, and s/Va/T. As a first step in this direc- 
tion, the radius, rgs , of a circle which includes 95 per cent of the beam 

: I current is obtained as a function of axial position along the beam. Such 
idata are shown graphically in Fig. 12. Finally, the curves of Fig. 12 are 

. lused in conjunction with the tabular data to obtain the "Design Curves" 
of Fig. 13 where all of the pertinent information relating to the beam 
at its minimum diameter is presented. 

\D. Example of Gun Design Using Design Charts 

Assume that we desire an electron gun with the following properties : 
anode voltage Va = 1,080 volts, cathode current Ip = 7.1 ma, and mini- 
mum beam diameter 2(r95)min = 0.015 inches. Let us further assume a 
cathode temperature T = 1080° Kelvin, an available cathode emission 
density of 190 ma per square cm, and an anode lens correction factor 
of r = 1.1. From these data we find -x/Va/T = 1.0, perveance P = 
0.2 X 10"^ amps/(volts)^''" and (r95)min/''c = 0.174. Reference to the de- 
sign chart, Fig. 13, now gives us the proper value for fc/fa : using the 
upper set of curves in the column for y/Va/T =1.0 we note the point 
of intersection between the horizontal line for {rgr^^i^/rc = 0.174 and 
the perveance line P = 0.2, and read the value of fc/fa (= 2.8) as the 
corresponding abscissa. The convergence angle of the gun, de , is now 
simply determined fi'om the equation^^ 

de = cos-^ {\ - t|^ X 10^) (37) 

{Qe is found to be 13.7° in this example) and the potential distribution 
in the region of the cathode can be obtained from (30). 

When this point has been reached, the gun design is complete except 
for the shapes of the beam forming electrode and the anode, which are 
determined with the aid of an electrolytic tank in the usual way. The 
radius of the anode hole which will give a specified transmission can be 
found by obtaining (re/a)a through the use of Fig. 5, and then choosing 
the anode radius from Fig. 7. In practical cases where (rf/a)a > 3.0, 



^' 



O 



O 

z 
o 

I-H 

e 
o 

w 

o 

O 

CO 

a 

Q 
O 
12; 

< 



O 
> 

m 

IS 
< 



Q o 



P 

a, 

o 
O 



o 

< 

o 

m 



PQ 







"5 

d 
II 

> 




1 

I-H 1 

II ■! ' 


o 

-1" 


bl>? 




^H (M lO O C35 O M C^ t^ C^ GO 
GOt^cOCDiOcOCOr^GO'Cl^COCO 
(M C^l C^l C^) 'M C^J !M C^l C<l (N CO CO CO 

ooooooooooood 


ooooooooooo 


vib 


T-H 1-H 1—1 O O O O 1"^ 1"^ »~^ c^ 

1 i 1 II 1 


COiO^HOd-fiOiO ^HCO-1"-f^H 

iooo^co»o^(MOt^coc;-t<co 

COIMtMi-HOOO^^CSMCO-* 

1 1 1 II 1 1 


''k" 


,— 1 1— 1 1— 1 T-H 1-H M 


00?--HcDCOOOO-l'*COOOO'ti 

T-H T-H T-H T-H T-H C^ O'l 




J.« 


(M GO GO OO (N OO CO lO »0 lO 

l0>0^'*-rt<l010C0t^G0Ort 


COCOCO':t<OCO<MOi^(MOCO<N 

t^t^r^t^ooGoooiT-HcoGOcooj 

C^C^iM(N!M(M(MC^COCOCO'+-rf' 


ooooooooooo o 


oc ooooooooooo 


."It, 


^ 05C0C0OOOO >oco 

GOiO^OOiOC^rt-tl^OO-liO 


(M U3 ,-H CO lO lO (M CO CD CO t^ 

COOC0t^O'f--H--HCO'-HG0'*'Q0 


^Hi — ^r-HOOOOOO' — ^' — ^1 — 1 

1 II M 1 


COCOC^i-h^OOOO^Ht-hiMOI 

J 1 1 1 1 1 


«| a 


O(N^CD00O(M-*<£>Q0OC<) 

T— 1 1-H T— t T— 1 T-H C^ C^l 


0(M^COGOOrt(N^cDO'*00 

t-Ht-Ht-Hi-H^05(MIM 


q 


b|=? 


O IC O to iO lO cr. (M C-j t^ C^ C-j 
-f-fOiOcOt^QOOi— iiOOCS 


OCOI-^O-IOOt^ OOCSl COUO 
I- t^ l-^ CCi O) C~. ^ C^ C^ f 1^ O CD CO 
!MCM(MC^llMO-1COCOCOCOCO-*-flO 


O OOOOOOOOOOO 


OOO OOOOOOOOOOO 


>,"|b 


lOOiOOOC^OOiOCOOt^-HtO 

coocooooioco.-H^-rooc2 


OiO(M t^COiCOC^tXCOl-- 

t^T— lCDT-HCDT-HCD^!MT-H-fCDT-HTt< 


^,-,^^000 0000 

1 1 1 1 


COCOC^(M--HrtOOOOOO^^ 

1 1 1 1 1 


"k= 


OiM'Tt<CCiOOO(M'*COO'#CD 
^ --1 ^ --KM Ol (N 


O(NTt<CO00O(MC0^C000O'*Q0 

t-Ht-Ht-Ht-Ht-Ht-hC<1(MC^ 




bl>? 


t^iOOOOt— iiOOO>000 
C^-t^CDGOOiM-t'l^OiMCOO 


■*CO^(MCOGO^'fiM'*COO'-H 
COt^GOClO'-HCO-flOt^C^IOOCD 
(M(M(MC^C0C0C0COC0C0-*-rt-»O 


OOOOOOOOOOOO 


OOOOOOOOOOOOO 


-"lb 


OGOlO-*rtlcOClC^OOlO(M 
O^'^IMOOC'OiOCOC^^ 


co<Mr^coao^QOcoocooocoo 

I-- CO ^X' -f O t^ CO C^ O GO CO O iM 
COCO(M<M(Mi-ht-h,-ht-iOOOO 

1 


^,__,-H^OOOOOOO 


"U» 


0(M't"X)G0005^COOOOiM 

T— 1 T— t T— 1 1 1 T-H C^ C^ 


O<MTtHcD00OiMC0'rf<cDO'*G0 

t-Ht-Ht-Hi-Ht-H(>JCM(M 


q 


bk" 


CO 00 1-H CO O rH C5 rH 

-H04^k0t^0 coo 
lOiCiOiCiOO cOt^ 

dddodo dd 


COOiO(NOOOOOOCi(NcOCO(M 

lOt^GOO^COiCt^OiOT-HGO 

<MC^(Mcococococo-r^»cio 


K"lb 


iOcOOOt^O'*iOOOUO 

OOOt^COCOTfCOT-HO 


t-hioOOIi-hiOt— iOt-hO^iO 

C5»0(MGOcoco^ait^-*T-HO 
cococo(M(Noaoq.-Hi-HT-H,-HO 


«l « 


0^(MCO-t<COGOO<M 

1-H 1 — ( 


OCa'*COOOOiMTt<cDO'*<00 
,-H ,_, ,_, ^ (M CKj C^l 


II 


bk« 


>0 iCCi 005 

O5C0 t^c^ CO 

Tfi lO lO CO CO 

o o oo o 


OOt^OOOC^OO<MOOiO(McOGO 
■^COCXl^COiOCOOCOOiOGO 
O^C^OJCOCOCOCO-^'^'^lOiO 


."lb 


(M O CD lOiO 
O 05 t^ CO lO 

C<l T-H 1-H T-H T-H 


COTti00kCCD00C0O:h-CDO5(M 

oi>-*iMOoot^iOTt<oaoo 

^COCOCOCO(N(MC^C<I(N<M<M 


«|w» 


0<M -^CDOO 


OCq-^cOOOOCI-^cDOTttCD 

^ 1-H T-H r-H(M (N C^ 



404 



o 

(N 






i-i(NOiOOt^iO'-<>0 


ooooooooooo 


00 iC O (M O O lO lO lO o 
OlMOCOrCOCOiMiMGOCO 


1 1 1 1 1 1 1 


rt ^ ^ .— 1 C^5 C^l (M 


00>OO00^^(MOiOC;»O-f 

dooooooooooo 


lO lO O (M O lO IC 


1 1 1 1 1 1 1 


T-lrHi-l.-H,-l(N(M(MCO 



lOQOCOOCiOC^lC^COiO'— 10? OJ 

I-Hi— I.— Ir-Hr-Hi— It— I,— IC^CO-^IOCO 

dooooooooooo o 

O COOOCOOCOCiOOt-^COt^ 
Tt< CO IM -* O «3 CO O 1^ IM lO t^ 00 

r^^ccoi— i^^oooO' — ^1 — ^1 — ^^H 

I I I I I 



O^OClMCOTt^iOcOO'^COiMCO 

1— ii— ii— irHi— itNoac^coco 


INO— <000(MO(NO'*COC^ 
CO-^iOCOOOO^COCiCOiOiO 

,-ii-irtT-Hrt^c^c^c^co-r'O 


OOOOOOOOOOOO 


t^i^asrHt-c^cioooiOGOrt 

lO t^ rt 00 CO t^ O-l C-.' CO O --H CO 


I>»C^<M(M.-i.-iOOOOO 

1 1 


O-^OOIM-^COOOO^OOC^O 
.-(.-irtrtC^ir^lMCOCO 



00(MCiO^O»C>O00t^-*OO 
(N-*»000OC0«5O'O^00CC>^ 
1-1.— iT-^,— (C^l(M(MCOCOTt<'*>OCO 

dodoooooooooo 

>— I IC "—I 

00-*C^(MiCO-tiOt^iCC004^ 

t^CCHCThlCOC^C^CSIrtrt^rtr-l 



O-rt<0C<McCiO^00(Mc0O-^C0 

i-Hi-io^c^c^coco^^^ 


'^iCOrtOOcOCOOiO-tiiO 
rHi-lT-(T-HC<IC^(MCOCOCO-* 


OOOOOOOOOOO 


"3 Tti (M 1— 1 O C^ (M t-- 1— 1 

OC5'-i»00'X>COOOOcC>»0 


00<»COlO4OTt<^':fCOCOCO 



O-*i00(MCDO'<ti00(MCDO 
i-ii— iiM(MC<)COCOTti 



O 

fa 



fa 

o 
o 

fa 

o 
O 

O 
> 



fa 

o 
o 
;z 
<1 



1/3 

PL, 



^^ 

|x 

1—1 
go 
g II 

CL, 

o 
O 

o 
< 



63 

o 

fa 
> 

fa 
IS Ph 

fa 
o 

w. 



pq 






bi « 



;ib 



;ib 



b| e 



«h 



^1 b 



wl b 



d 



> 



I^OOiOOOOt^t^COCAiOC^O 

ot^cocoio»oiocc!r^cx:ic:o 

COCOCOCOCOCOCOCOCOCOCO-* 
OOOOOOOOOOOO 



(MC0i0iO'*i0c000«5OC0l^ 
iOOOO«5(MrtiOco^C:C^iO 

Csj^T-HOOOOi-ifMiMCOCO 
I I I I I I I 



0!M-1">OOt^CC'OiM-f»OCO 



OC^(M00(MC5iO>O-t<I^rtO-Ht^ 
OJOOCCOOOiCnO^HC^^cOCOCifMCO 
COCOCOCOCOCO'^^-r-f'+^-^'ClO 



OOOOOOOOOO! 



! o o o 



COiOO»C>»C»0(M^Ht^M<0<MC10 
>OClC05C'COOC<liOt^O»C01C^'^ 

Cvj^,-HOOoddo^^^(N!N 

I I I I I I I I 



oiM-^cor^oooiO^o^-^cooooj 



(M^^cooor^ot^ioocx) 

C«0C'C5O(M-^«5OC0t^(MCC) 
COC0C0^-*'*^iO»O>OCOCO 

dd dooooooooo 



O^COOOCOGCC^C<lt^O^OO 
^O»0OOrtrti0G0O<M-^ 

iMiM^^OOOOO^rtrt" 

I I I I I I 



O(M-<*<cD00OC<lTt^c000O(M 
1— I T— I I— < 1— I .— I (N (M 



iM^>O00iMC^C2t-^^HOO-^ 
t^GOOC^iOOO'— iiOO-^0»C 
COCO-t''^'^-^iOkOCOOt~-t~- 

oooooooooooo 



oioc^icor^t^ocococj-*!^ 

0(MCC-rOt-~iOiMOOC^CO 



(MC^rtrti-iOOOOOOO 



O(NTt^cC>00O(M'titr>00O(M 

.— ( T— I ^H r-l ^H C^ C^ 



COO^';D(M-*'CiiOiOO 
COC2(M>CO-. COt^C^t^CO 

coco-^-^-^fiOioasot-- 
oooooooood 



lOI^-^tOlN'— iCOt^-^CO 

t^COOt^iOCO-— iC500I^ 



CaiMC^'— ii— ii— (1— lOOO 



OlM^O00O(M-*O00 



b|^ 


OOi^OOiOcOOOCO 
iO00COt^<Ml^C^00 
COCO-^^iOiCtOCO 

'6> <~) <Z! <Z> <6 <Z) di <S 


C^l b 


lOt^iMCOiCIMOO 

OOiOCOi-iOiOOt-i© 

(MC^(MC<i.-i.-i.-(.-i 


"1^ 


O<N-*«500O(M'* 

1— ) T— 1 I— ( 



405 



Q 

O 

O 

PQ 



pq 









o 

II 

> 




O 
CM 

II 

E~i 

a 

> 


o 


''U" 


OOiCOOOiO(NOOiOiC<lOO 


o^i-^oi'^coocoaio^o-^i^io 

C2C5<35C3aCO^-*'GC'fOI^^OO 
O0000'-^^'^^(M(MCOCO 


oooooooooooo 


ooooooooooooo 


>?ib 




ooooooot^LOiooaiooiic 

>— iC^tMCOOIM'^OOCOOCM-^ 
Ot^-^fM^O^COCO-^-^iOiC 

1 1 1 1 1 1 1 1 


looooq— icooc-ieocc*^ 

1 1 1 1 1 1 1 


«k« 


. — ^ 1 — ^ 1— * 1 — 1 ^^ 


OlM'^'OCOt^OOOiM^cOCOO 

1 — ^ ^H 1-H ^H 1 — ^ C^ 




bk« 




00(M^OCO^'#C^CDI^(MOGOt^ 
CTsOiOOO^Cq^OOCOCiiOOcO 
OOO^^'-^rt'— irt(N<MCO'+iTt< 

dddddddddddddd 


ooooooooooooo 


^ib 


»CeOiMr-iOOOO'-i(M(NC^CC 

1 1 1 1 1 1 1 


iO(M lOiOOO lOOO lOOO CO 
(NcOt-HCD-^^COOOiOOOOlMCO 

di>»o<M^oo^^(Mcicococo 

1 II 1 1 1 1 1 


"k« 


OOJ-^CDt^OOCyiOIMrfHcOGOO 


O<N-*CDt^Q00lO(N^CD00O(M 

T-H 1-H 1— 1 f— 1 1— 1 C^ CM 


q 


bU« 


c; CT> o ^ c^) CO t-^ ^ CD ^ r^ CO c--' 

^1— iC^(MC^C<l(MCOCO-f<-rt<lCliO 

ddddddddddddd 


OiOO^OiO-*00-*OOCOCX)0 
OiOOO^^CMCOiOOiCO-^rt 
OOr-Hrtrt^rt^rtCM(MCO»Ct^ 

oddddddddooddd 


^1 b 


COcDOt^iOOO^O«5CO'*OCO 

c^^rtT-it^cocar^ocoiot^oo 

1 1 1 1 1 1 1 


oooooooc<iooooooco»oco 

-rtH(M(MOq';f<«5Cr-COOCOOTt<CDI^ 

dcocO'^iM'-HOodd'-i'-i.-i.-i 
1 1 M 1 1 


«1 « 




OCM-<*H«DG0C:Or-i(M-^cOO'*00 

.— 1.— 1,— 11— It— ((MCMCM 




bk' 


CDrtO(MOOOOCiCOOCOOlOO 
rtrtlMC^(MC^(MC^COCO^iOI^ 


coiOr-Hr^iocoi— i^HTtit^iot^Oi— 1 

C-, OOOi— iCM'+iOCOOlOCMOOiO 

OOi— 11— li— I'-Hr-H,— l,-(,-HlM^lOI>. 


OOOOOOOOOOOOOO 


^1 b 


t^C2CO'fflt^iOC»-t<CO^OiO(M 
COTfCOGOi— IIOOJOOCO^COIO 


•ooot^coiccMCMr^ooiooO'— 1 

t^OiMCDCO^r-Ht^COOC'CMO'-HCM 

oo6i>iO'*co(M'-i^ododd 

III 


'O^COOliM'— ii— i^OOOOO 

1 1 1 


>'!>? 


O<M^CDGi0O^(M-tiOO^00 


0<M^tDOOOiMCO-*COO-*00(N 

.— (rHi— 1.— I^HC^CMCMCO 


q 


bl>? 


(M>OOC500t^OOO-f<t^»OCiOOO 
OOas-^^iMCO-tt^OliCCOtMCOCO 
.-ir-H(M(MiMC^O](M(MCO-t<»OcOt^ 


t— iiOCOt— lOC^iCOCO'^'COOO'— lie 
OOlO-HCMCO^CDt^O^t^COCMO 

00--l.-i-H,-H,-H,-l,-l,-^(M!MCO'+CO 


oooooooooooooo 


OOOOOOOOOOOOOO o 


^1 b 


00 lO CO »0 OS CO ^ »0 O Oi o t^ 
-^t^'-HOO'OCOOcOCOt^COOOJt^ 

lO'*-*iCOCOCOCO(M(Mrti— irtOO 


OO CO lO »o O »OC0 o o 
O-^T-iOOCOCDOiOi-HOOCMOt^iO 


^OOOt^cDiO-^-^COCOCMC^i-i^.— 1 
1— 1 


"1^° 


O(N'*tCiC0t^00O<MOO'*00(M 
i-H — 1 rt CM (M (M CO 


OlM-ttOOOOIM^CDOOO^GOCMcO 

T-Hr-Hi— It— Ir-HCMCMC^COCO 


II 

1^ 


bk« 


iOCOCO-^(M-+'-t<^^^ 
^Hi— ii— ((MIMCOCO-^iiOiO 


OOCOCDQOOCOCOi»<X)-*COCCI^ 
OOOiO^COTtiiOOO.— IIOOCOIO 

OOi— IrH.— Ii-lrt,— (<M(M(MCOCO 


oooooooooo 


OOOOOOOOOOOOO 


^Ib 


(MC2O(Mt-i,-hc0O000^ 

t^co'-H<riC2i<Of^"tiTt< 

lOiOiO-^cOCOCOC^OlIM 


iCiO>CO»0(MO'CiOiO»C>CiO 

coO'-Hcot^o^oo.-icocMot^co 


--lOCSOOt^t^cOCO'OiC'*^^ 

T— 1 1—1 


"k"* 


O.-i(NTt<C0(MCDO^':0 
.-H ,-(<N(N<M 


O(N-<H5O00O(NCDO-*C0(M-* 
1— l.-l.-l<MIM(MCOCO 



406 









d 

II 

o 

> 




O 

II 

a 

> 




o 

CO* 

II 

> 


o 


*>l^ 




^■^iCOO^OiOiOO 

^cococo^cc>o»o^ 
T— ti— It— irHi — ^^HC^^(^^co 

ooooooooo 


iMOOOO-HOiMlM'OOO 

t^coor^ooooc^cico 

OOOOOi-H^C^C-lCO 


oooooooooo 


^Ib 


Mill 


lO lO <M lO C5 lO 


loooo 

Oli-HO-^iOOiOiMOiCO^ 

eoooi-HOOiM-^-*»oio 

1 1 1 1 1 1 


COTtHOOO^CO^iC 


"1^ 


O(N-*-*'>i:i<r>G0O(M 


OfN-^M^iOCDOOOlM 


OCO'^-'J^iCiOOOOlM^ 

1-H 1-H r-H 


re 


H^ 


00 QO 00 GCi CI (M I^ Ol 


t-iOO00^'*(M00 
■*'*'rri'#0«COO 
^^^^(MOICO-^ 

d d d d d d d d 


rt^i-H^iCOOOOiOt^iM 

t^t^t^r^ooo-*c^ooi05 
oooooi-Hi-Hc^icoco^ 


oooooooo 


OOOOOOOOOOO 


^Ib 


•Ot^-^iOOOOOOS 

iccc-— iiooor^eo 

1 1 1 


1-Ht^OlOiOCOOO''— 1 
t-^'rt^(Md'-i<M(MCO 

1 1 1 1 


00 >OiO 

iMcq'^.-Hc^oooicaiOi-H 


-^Ol^C^O^i-HIMiMfOCO 

1 1 1 1 1 1 


«l^ 




O(M^5O00O(M't< 

1-H i-H 1— ( 


OIN^iOCOt^OOOiM'^CO 

1-H 1-H 1-H 1-H 




H2 


COQOiCOO^OOiOOO 
l^f^OOO'tiCsiO'^OO 


oooiooooor-Hococoo 

CO-*'^COt^OCOCOC<l'— 1.— 1 
1— i,-i^rtrt010)CO-t<iOcO 


050C<)i-H-^Oi^r-HlOlOi-H 

COt^t^OOOi-HOlOsOlOiM 
OOOOOi-Hi-HiMCOiO^D 


ooooooooo 


OOOOOOOOOOO 


OOOOOOOOOOO 


^1 b 


COIMIMiOOOOl^OOiO 
COiO-^-^COOiOQOt-h 


icoiocaiMtMr^cocoioo 
CMOooo—i'^'iMr^oiMTfi 


iCOt^OiO'*-rt<0000'H 


1 1 1 1 1 


I^lOlMOOOrH-HC^C^C^ 

1 1 1 1 1 1 


•^OiOIMOO^^i-HiMCq 

'^ 1 1 1 1 1 1 


«1^ 


1 — ^ 1 — 1 .— 1 ,-H 


O!M-tc0t^00O(M^O00 

I— t T— ( T— 1 T-H 1— t 


OC^-^«3t^00O<M^O00 

1-H 1-H r-H 1-H r— < 


H.» 


COIMiC^OOOO'OO 

coooocor^csic«^(>i^ 


C0(MC0OiCt^02»0OOO 
CO^iOt^OiCOOt^t^t^OO 

.-H,— li-H,— lrtC^C^COTf<lO^ 


l^OiOCOiOOiOiCOiOOOiOO 
Ot^t^COO^COCOCOCOiOOOlM 
OOOOO^i-H(MCO-*iO«500 


oooooooooo 


OOOOOOOOOOO 


OOOOOOOOOOOOO 


^ib 


t-QOOCOr-COOfM^CO 


O lO lOO O "OOI 


ocDcoiO'+iooooeoi-Hi-Hi-Hi-HfM 


COiMC^T-iOOOOOO 

1 1 1 


t^iC^iM^OOOOOO 

1 1 1 1 


iOi-HOO»OCO^OOOOOOO 

1 1 1 1 


«I2 


0(M'*<COOOO(M'*<CDOO 
t— 1 1 — I 1— 1 1 — 1 1 — I 


0<M^COOOO(M'*0000 

I— ( T-H 1— 1 1-H 1-H C^ 


OC^'^OOOOCOtJhcCOOOC^M* 

i-Hi-Hi-Hi-Hi-HlMC^fM 




b|^ 


COOC^OQC'C. COiOiO 


Ot^COiMlO-^QO^GOlQlO 
CjTfCOOOOCOCOO'Ot^C^ 
i-Hi— ii-Hi— (C<I(M(MC0C0'*O 


'O-t-OO-HiC'H^HiOOCD'OiC 
CD I-^ OC' C-. O i-H CO lO t^ CO -H rH CO 
OOOO^'-Hi-H.-Hrt(MC0'*>0 


oooooooooo 


OOOOOOOOOOO 


OOOOOOOOOOOOO 


^1 b 


03C^O--tt^-*0iO00f- 


1000»0 lOiOOOlMO 


iOiCCO»OrtO^^OC<lt^'*'C^ 


COCO(MC^^^rtt-iOO 


t^COiO-^COC^lC^Mrti-Hrt 


i0iMOC0t--C0i0-*iC0C0C~^01C^ 

1 — ^ ^H ^H 


-12 


OIM'^OOOOCO^CDOO 

T-H T-H I— t T— ( T— ( 


O(N-^«D00O(M-*COO'* 

1-H T-H 1-H .— 1 IM 01 


OiMTtHCDOOOIM-^OO^OOC^ 

^,_l,.Hl-Hc^l^^c^^co 


II 




M2 


t^OiOOlMOO^ 

'i^ Ci CO 00 CO c; lO -H 

(MC^COCO'^'^'CCO 
OOOOOOOO 


COiOOC<IO-*'+'CC> OC' 

c<) ^ CO c; ^ -^ r- o -*< t^ <M 

^ rt ^ rt C^j C^] (>J CO CO CO -Tf 


COCO^COO-^OCD-^CO-^COOO 
COt^OOO^C^COiOt^O^COiO 

OOOO^i-Hi-Hi-Hi-Hr-HC^OllM 


OOOOOOOOOOO 


OOOOOOOOOOOOO 


^1 b 


O lO ^ t^ lO CO rt Ci 
-*COCO<N<M(N(Mt-i 


1— lOi-HiOOtDCOOX'^iO 


iC lO »0 lO 

0'y:>'-Hoooo.-H»oocDco'-Hocr- 


oor^cDioio-^-^-^cococo 


!OCO(M00500000I--t^l:^OcO 

1-H 1 — 1 ^S 1-H 


"12 


OC^-^COOOOOlTfi 


OtM-^COOOOlM-'t'COOOO 

f-H 1— t 1-H I-H r— 1 (M 


OIM^COOOO<M-<*<COOOOC<I'* 

.-Hi-Hi-Hi-Hi-HIM(NC^ 



407 



• 

II 








U5 

d 

II 

> 


q 

l-H 

11 

s 

> 




o 

II 

E-. ! 

=3 

> 


q 


bk« 


<£>00iMCD'-hOC^JOOO 

oo:c3c;'0'-Hcococ2!N 


CO r^ lO X' ^ CO to CO 00 o 

O O CI Cj Ol lO C CO t^ c^ 
^OOO^^'-H(M(NC0 


(MOXOcOOI^tOiOtC 1 
lOtO-f-fl^iMCD^CO^ 
O O OOO T-H l-H (M (NCO ; 


o 


oooooooooo 


OOOOOOOOOO 


OOOOOOOOOO i 


g 


.rib 




Id O »0 lO t^ X X 
t^Xt^tOCi-^COCl^cO 


to 

tO-^-^COcDi-Ht^i-HCOiO 


o 


1 1 1 1 1 1 


C3COCOO^CO-+-fiOiO 

1 1 1 1 1 1 


C-. COt^^C^^-HiOiOlO 

^^ 1 1 1 1 II 




«k" 




Oi-HIMCO-^iOCOt^XO 


O l-H (M CO -* to CD t^ X OJ 


o 
O 

i_J CO 




bk« 


OiC(MCOO-HiOOOO 
O CT. CI 0-. -H CO CO O -^ 00 
C^4^rt^C^lC^(MCOC0CO 

OOOOOOOOOO 


OXcDCi-t'iO >0(MX 

o o c; o o l-H >o o TO c-j 

1-hOOO' — ^1 — ^i^^hC^cO 


C5XO(M<MCOOOrHC 

iO-t<-t<-ftocOOcO<>)Xi- 
OOOOOOr-Hrt(MiMr| 


OOOOOOOOOO 


ooooooooooq 


.?! b 


icr^t'-c^ioootDio 

t^COC;iMCO^t^(McO 


lO O (M 

OiOXi-HXCOCiXt^i-H 


O to to IOC" 

cocMXcoco-r^tocooi 
o^a.coT-HO(Neocoeoe<i 

CCl-H II 1 1 1 




lOCOC^OOi-HIMfMCOCO 

1 1 1 I 1 1 


Ot^^<MOO^(MCO'* 

1 1 1 1 1 


0-1 


"k« 


O^iMCO'^iCCOt^OOOl 


»o 

o 

Ot-hC^COCO-*iiOcOXi-h 


to ! 

Oi-HC^icoco^iocor^xc 

r 


< < 


r4 


bk« 


COOOOiCOOO»OiO 
00(>4CSlt^cOCDt^^ 
-— I'-HOlC^liMCO-t'iOI^ 


C-jCi^iC»0-+i^O 
OO^H^HfMcOiOI^ 


X O CO(M --t< CO 

'^ to to X 1^ CO Ol c^ 

OOOO^COtot^ ' 

<Zi cii <6 <6 d><6 cS <=) 


Analog Computer Data foe 
Perveance = 0.4 X 10~* 


ooooooooo 


OOOOOOOO 


;?1 b 


OiOOiOOOiOiO^O 
>— iC^iOt^'— iCOl— i-:fCO 

iOCC'—iOOOi— '^-^i— 1 

1 1 1 1 


COCOC^iOcOOCO-* 

OCOCOOOi-Hi-Hi-H 

1 1 1 1 


o o to 

t^ to t^ O to X O O i 
O (N to l-H O O O rt 

^^ 1 1 II 


"k' 


OC^'*iOCOCOOiM-r 

l-H l-H l-H 


OiM-t<cOXOC<I^ 

l-H l-H T-H 


OOJ-rcOXOlM-* 

T-H i-H l-H 1 




bk« 


OOOOOi-HOOiOO 

ooocor — tiict-^oo 

^(MC-llMCO'-t^iOt^OO 

ooooooooo 


■* o o >o iC to 

OOi-HCOXt^O-*OX 

o^^rtrtC^^ior^x 


t^ o to O X to to 
"TfiOtOl^OCOCOOt^tO'' 

ooooo^c^-t^toix: 


OOOOOOOOOO 


ooooooooooe 


.-"lb 


(MOOOfflt^OOfNiO 
(^S«5COC^'*Or-(COTt< 


<M X O O to 

t^C^lO^i-H^i-HOi-Hi-H 


to O O lOC 

cor^coiMto(Mr^co-<ti-*^ 


iOCO(N^OOOOO 

1 1 1 


Ot^^lMi-HOOOOO 

1 1 


-H-t'cttooai-Hoooot 

(N T-H 


"k« 


O(N^t000O<M'*O 

l-H l-H l-H l-H 


OfN'^COXOiM^cOX 

l-H l-H T-H l-H l-H 


OfM^COXOC^Tt^cDXC 

l-H T-H l-H T-H T-H C 


o 


q 


bk" 


'^OiMOOOiOOO 
i-H(M(M(MCO-:t-iOCOt^ 

ddddddddd 


(Mco-rr^c--OOC<iiOio 

O O C^) -t t-- C^ X lO -* t^ 
O^i-Hrt^OJiMCO-fcO 


cOOOOCOOOiOiMOC 
-ftOcOI^X^-tt^iMtOT 
OOOOOi-Hi-Hi-HCv^CO' 




O O O O O O' o o o o 


ooooooooooe 


p 
xn 


.-•1 b 


iOO»OiOOO^»Ot-< 
'*OC2'^cO(M05t-«5 

lo -t- c^ (m' ^' -h' o o d 


X "ti 

C~ji-<OiO-*c001^tO(N 


to oc 

X to to (M CO CO O to 05t 


OXCO-rt^COC^C^i-li-Hi-H 


■-(l:^(MOit^tO-*COCO(Mt 

C<) T-H T-H 




«k» 


O(N^cDQ0O(M^C0 

1— t r— 1 1 — \ r~< 


OlM^cOXO(NTt<COO 

T-H T-H T-H l-H C<l 


OfN-^COXOiM-^CDO- 

T-H l-H l-H T-H C^ C 


l-H 


>o 

II 

u 
1 K 


bk= 


CO O CO ^ (M (M CO 
t^ (M CO 05 GO "O iM 
^ C^ C^ CO CO -t< »o 

<~> <6i ^ Q Q <6> i^i 


XXC^OXOCOCOCOO 
XOCOtOXC^iOOCOX 

Or-Hrtrtr^C^CaC<ICOCO 


-^tM-^XiMXCOCOXO 
rhiOCOt^C20<M-^COa5 

OOOOOl-Hl-HT-H^l-H 


1^ 


OOOOOOOOOO 


OOOOOOOOOO 


r1 


^1 b 


CI ic iM l-H cs o cr. 

CO CO O ■* O 'X lO 
lO ■<*< CO CO CO (N (M 


-*COOCl<Mt^COOXCO 


OCiOiOt^(MCOcD t^eo 


tn 


l-H 


(MX-^CO<Mi-h00005 

OJi-Ht-Hi-Hi-Hi-Ht-Ht-H 




"lv» 


O (M -t COOO O (N 

T— t T-H 


OC^'t'COXOiM'^cOX 

l-H l-H l-H l-H l-H 


OtM-^COXOC^l^cOXi 

T-H T-H T-H l-H l-H ™ 



408 










• 


d 

II 

> 




o 

T-H 

II 



> 




o 

II 

> 


q 


bk« 


^ OQ C' C-l CO -+ o o 
iO-t'^~vuril-j—i<:0'—i 


COCNO'-H»CO<MO>0 
t^ I^ t^ t^ t^ 00 ^ t^ (>) 00 

OOOOOOt-i,-i(M(M 


OOCOiO>OO^COiO»C 
COCOCOCOlOXi-*-— (00 
OOOOOO^C^Ol 




OOOOOOOOO 


oooooooooo 


OOOOOOOOO 


.!lb 




O O lO 

C^^OC-.O-*'— il^iOO 


t^ t^ CD 1-1 05 i-H CD 00 

lo Tt< r^ 1-1 im' CO lO lo ic 

^^ 11,11 


OCO(NOO(NCO'*>0 
1 II 1 1 


COl^^OOr-ico^iOCO 

1 1 1 1 1 


«I2 


OT-Hi-KNC^eOTtHiCiCD 


»0 (N lOOOOO 
O.-irtiM(N(Ne0Tj<iCCD 


Oi-i^iMoqcO'^oco 




bl2 


COOiOiOOOOO^O 


CO CO ^ CO lO lO »o 

t^t^l^OCiC^lcDCOi-^C 

OOOCCOr-Hr-lC^CO'* 


coco cDi— i^Hi— icocOO 
COCO-^>OCiCO(Mi— i^M 
OOOOOi-HC^COrfiO 


OOOOOOOOO 


OOOOOOOOOO 


OOOOOOOOOO 


K-lb 


Qccoco-r>oocccor- 


^ lO lO 00 

r^o-^ocot^'i^^ioco 


lOt^O-^t^i-iCOOi— iC-lCO 


COTfi^OOiMIMCOCO 

Mill 


COOOCOi— IOt— KNCOCOCO 

1 1 M 1 1 


t^CO^-IMOfNC^COCOCOCO 

^^ 1 1 1 1 1 1 


«l^ 


Or-l(^^(^^|^^TflOcD^^ 


O'-i(M(MC0C0'*i0c0t^ 


0'-i(MC<icOCO'*iOCOt^OO 


p 


bl^ 


OOOO^iOiCiOOiC 
TtHi0c0Q0O«5Tt<C005 

1— IT-Hr-H,— I(NC^COIO;0 


O'-HIOIOO'— iO»CiiOOO»C 

r^t^t^oooc^ooO'— ICO 

OOOOi— i^OJC^'^iiOcO 


»C»Ol^(M 00 1-1 00 ira (M 00 

COCOCO'rtHlOCOiOCOOiCOt^ 
OOOOOOi— IIMCOICCO 


OOOOOOOOO 


OOOOOOOOOOO 


OOOOOOOOOOO 


w'lb 


10>0 O O iO<M IC 
T— iCJOOC^COOiOO'— 1 


O O C5 Ol •— 1 
COiOCOOCOfMi— ilCCOt^OO 


>00 lOiCOO 
lOCDOOiMO r-((McOT}H 


t^(MT-HOOT-Hr-H(M<M 

1 1 1 1 1 


'*iOiO(MOO'— i^i-Hi-Hi— 1 

1 1 1 1 1 1 


ooooocoi— lOi— i^^i— ii— (1— 1 

^^^ 1 1 1 II 


«k« 


OlMOOeO-^iOcOQOOl 


O'-i(MC0C0TjHiCcDt^Q0a> 


OT-i(Ncoeo-*»OcDl:^ooc:i 




bk» 


itSCO CD OOCl 
C0»Ct^0:>'*O'X>00— 1 

1-Hl— IrHi— lC^C«5-*CDO 


1--CO COOOOO 
CDt^O(Nt^^t^CO 

OOt— irti— iCOiCOO 


■<:t<COOC(NC5i-iC2iCCOOI^ 

COCOCO-*iTt<COOO-*(MCO'*i-l 

OOOOOOOi-i(MCO-*<t^ 

oooooooddodd 


OOOOOOOOO 


ooo ooooo 


^Ib 


MiO lO'O'-iOOOiM 
"^Ot^iOt^C^KMiO^ 

t^'^O^'^OOOOO 

1 1 1 


OOOOO CO 

OOOli— iCDb-OOi-H 

■^t^eoT-HOOOO 

1 1 


lO O O 00 o 
t^CDiOiC-^'^t^OcOiC'*-^ 

Oi-iioococoi-Hooood 

(M (M 1-1 1—1 


«k 


O(MC0-*iCCD00O<N 
1— t 1— t 


O<N-*i0C000O(N 

I— 1 r-1 


Oi-i(MCO-*iOCDt-OOOiO(N 
1—1 1—) 


q 


bk= 


000(MOiOOO»C 

cc!Ooooro<ocooiTt< 

,-lrHrt(MC^(MCO-^CO 


1000»00^00 10 00 
COOOO"— iCOiOQOiO-^t^rt 
OO"— ii— 11— ii— It— '(MCO^CO 




OOOOOOOOO 


OOOOOOOOOOO 


K-ib 


t^OOCOt^C^COOlO 


TtH CO lO IC Ci CO CO ^ Oi !>■ 




t^iCTtHCO(M(Mi— ii— (1— 1 


»OOCD»C^COCO(M(M--H.-H 


«k" 


O(Me0TfU5cD00O(N 


0(N-*iOCOl^OOOlM-*iCO 

I— ( j-H 1— t I— 1 




II 
e 


bk= 


lO lO kC iC o o 
(M CO 1-1 t- -t^ cq 
.-1 .-1 0:i C^ fO Tt< 

o ooooo 


(N coo O lOiOCl 

CO 00 i-H '^^ t^ rt lO 

O O T-H --1 ^ C^ (M 

d c> d d d d d 


^HC^cDi— lOlM'+'O 

CO^iCt^O^COCO 
OOOOOi— ii— ii— < 


OOOOOOOO 


K^b 


O lO 1— 1 O Oi 
O i-H O <M 00 Tfi 

OO CO Tt< Tj^ CO CO 


O rH 00 lO CO r-l t^ 

CD (N d 00 t^ t- CO 
1—i 1—i 


.-l(MtXN-<*<(M^OO 

(m' ^ d i>^ "5 -^ CO im' 

CO<Mi— (1— ii— ti— <i— ii— t 


"k« 


O (M Tt<CD0CiO 

I— ( 


O iM ""l^ CO 00 O (M 

I-H 1—1 


OC^'^COOOOfNr}* 
1— 1 1— ) f— 1 



409 



o 



> 



J 


\ 


l\\ 






\ 


\^ 


\ 


y 






^ 


^V 


A 








\ 




\ 


\ 


\ 






\ 


. 


'S 




^ 


1 




ii 


f 


10/ / 

1 "v V 






^// 


r 


^0 

y 








# 







\s 


\\ 


\ 






\; 


\ 


\ 








V 


\ 






> 


A. 


N 


\ 






^ 


N 


M^ 






vl 


'^i 

i 


^ 


); 












i 


^ 


^ 





t? 



N 



<D 


















V 


"^sVs 


\ 








\ 




\ 


\ 










s 


1 


^ 








i 


A 






<o\ 


A 


/' 








r 







\ 

». 


\. 








^ 


.\ 


\ 






<s 


^ 


\^ 










\ 


^ 


K 








1 


\ 


^ 






ii 




I J 


m 






ii7 


^ 


y 





ti 



<D 



ID 

d 






*^ 










V 


\ 












\ 


\ 














I 



II 






w 

















\ 










^ 










\ 










V > 




^ 






II 


01 lO 

oil fVi, ' 


m 










^ 







(\l 



>? 



t>J 



o 



CO 

6 



d 



5_0l X SOO=d 



o o 



(\J 






00 


'- 









_0l 


X 


10 


= 



d 






410 













\ 


\ 








\ 


x^ 


\ 






N 














^^ 


\ 




""''^ 


'^ 






\ 






V/ 


^ 


^ 






U 


^ 


/ 























V 


\, 








-^ 


\ 


\ 






\ 


l1 




Vv 




"^^^^\" 




fe 




o> = 


^«^^-"~. 

'^^ 






k 








^ 


^ 



\ 










\ 


\ 








^\ 


t^ 










.-^ 


\- 






°^ 


^ 




^ 








\ / 


^ 


\ 






W 


^ 















V 












\v. 








"0^ 

'■^ 

3 


k^ 


V 






^ 


^ 






0-^ 


Ij^ 


M^ 




Jj 






w 


^ 


^ 













\J 










^ 

^r^^ 


^ 








\ 


vs; 








^. 


\ "A 


<r)l 
1 "^'/ 


t-V 






\V \ 

A 


k 


'<'> 





o 



<\J 



CO 


^ 


o o 


o 


o 





(\J 



OO 

d 



d 



CO 


&-; 




Q 




t- 


CO 


> 




T) 




fl 


'J 


c3 


N 






I5~ 








1^ 


o 
(\J 


ST 




^^ — ' 




a) 




a 

0) 




> 




Lh 




«1) 


OJ 


a 




a 




• p— 1 




r^ 










CO 


o 








-t^ 




rt 








Si 




-il 


^ 


!> 




^ 




o 




%-. 




OJ 


o 


^3 


<D 


O 








<A 




C 




3 


^ 


tJU 


OJ 














o 




;_ 




5<-l 


O 
CM 


a; 



N 























^ 


s^ 








^ 


^ 




(O 
















o 




'^ 








9_0l X 2-0 = d 



9.01 X fO = d 



C 



CO 
CO 

a; 
> 

a 

o 

<D 

>o 

CO 

.2 












O 

-C 

CO 
CO 

> 

3 
o 



fci) 

• r-t 
fa 



411 



q 



> 



> 













\ 












\ 








1 


\ 




\ 






^ 




\ 


. 




^-- 


t^ 


^ 


J^ 




IJ?£!\ 










I 


^ 



tsl 



> 



V 


\ 








\ 




\ 






\ 


\"^ 


^ 


L 




^^ 


\ 


^ 

^V. 


K 






^ 


C> 


■- > 








^ 


•i 






\& 


^ 





^N 










K\ 


■^ 


^ 






d 
II 
1— 


^ 


^ 


V 


s. 




> 




^ 




^ 








1^ 


\ y ~A^ 


K 










^ 


> 



o 



N CO 

9_01X 90:=c) 

412 



d 



|5- 



o 
d 
> 



C3 
O 
• i-t 

03 

c3 
> 



o 

OS 

c 
a 

faC 



o 

u 

0) 

o 

o 

o3 



3 
CO 

> 

C 
o 

o3 









o3 



O 

-a 

EC 

a> 
> 

O 



O 

.— I 
bb 



BEAM FORMATION WITH ELECTRON GUNS 413 

we find less than 1 per cent anode interception if 

anode hole radius = 0.93 r^a + 2o-a (38) 

Additional information about the axial position of (r95)min and the cur- 
rent density distribution in the corresponding transverse plane is con- 
tained in Fig. 13. The second set of curves in the \/Va/T = 1 column 
gives Zm\n/Tc — 2.42 for this example, so that we would predict 

Zmin = distance from anode to (r95)inin = 0.104'' 

The remaining 3''^ and 4*^^ sets of curves in the ■\/Va/T = 1 column 
allow us to find o- and re/a- at ^min . In particular we obtain a = 0.0029" 
and I'e/o = 0.8, and use Fig. 6 to give the current density distribution at 
2min .* Section VI contains experimental data which indicate a some- 
what larger value for 2m in than that obtained here. However the pa- 
rameter of greatest importance, (r95)niin , is predicted with embarrassing 
precision. 

For those cases in which additional information is required about the 
beam shape at axial points other than ZnVin , the curves of Fig. 12 or the 
data of Table II may be used. 

6. COMPARISON OF THEORY WITH EXPERIMENT 

In order to check the general suitability of the foregoing theory and 
the usefulness of the design charts obtained, several scaled-up versions 
of Pierce type electron guns, including the gun described in Section 5D, 
were assembled and placed in the double-aperture beam analyzer de- 
scribed in Reference 7. 

A. Measurement of Current Densities in the Beam 

Measurements of the current density distributions in several trans- 
verse planes near Smin were easily obtained with the aid of the beam 
analyzer. The resulting curve of relative current density versus radius 
at the experimental 2min is given in Fig. 14 for the gun of Section 52). 
(This curve is further discussed in Part C below.) For this case, as well 
as for all others, special precautions were taken to see that the gun was 
functioning properly : In addition to careful measurement of the size and 
position of all gun parts, these included the determination that the dis- 
tribution of transverse velocities at the center of the beam was smooth 



* When j'c/o- < 0.5, the current density distribution depends almost entirely on 
a, and, in only a minor way, on the ratio Te/a- so that in such cases this ratio need 
not be accurately known. 



q 










































/ 


/ 


/ 


/ 


/ 


/ 












/ A 


y 




V 


/ 


i 












/ 


^ 


/ 


y 


y 


/ 






/a 


^ 


^ 


^ 


4 


<^^ 






























1 












1 


1 



























/ 


'/ 








// 


1 




/ 


YfA/. 


h 




(^ 


Y 


// 










^*^.. 














00 


<o in 


'i- 


ro 


(\j 


— 


(D 


<o 


in 


^ 


n 


ry 


o 


O 


o o 


6 


O 


d 


o 


o 
d 


o 
d 


o 
d 


o 
d 


o 
o 


o 
d 





^J/''(N'^J) 



Dj/Nm^ 



o 



m IS-. 
(\j 



C\J o — 



II 








































// 


/ 


/ 


/ 


/ 














o/ ox o_ 




/^ 














A 


i^ 


^ 


/ 












c 


J 


A 


^ 


/ 










i 


i 


^ 


s^ 


^ 














1 


1 












1 









O 00 <D in 
-■ d do 



ro 

d 



rv; 

d 





















/, 


// 


// 




y 


<• 


^ 


// 

\7/ 


i 


/ 


<. 


/: 


^ 


A 




c 


,(. 


cc 


^(f 










^>:-5> 




^^~5 


~^^^^r 















if'i^ 



— CO ID m ^ 

do o o o 

d d d d 



p 

n It] 

it- 



in IS- J 






d 

II 






<!. LU 

LU Q 

mto 
^, xz 

a Oo 
LU tr 
> u. 

o 







inlnj 




















O 
> 

n 






















/ 


/ 


V/ 


1 






2 
< 




/ 


/ 


// 


// 








1 




/ 


/ 


A 


/ / 






1 

D 


X 

1 "^ 
1 O 




/ 


\ 


^ 


4 




/ 


^ 


'^ 


^ 


^ 


/■ 








-f 


1 


1 












1 



3 CO 


ID in 


'3- 


(O 


(\J 


- d 


O O 


O 


o 


o 



















/ 


A 


/ / 


// 






A 




y oy 

1 / / 


fo 






Y 




/ 








\ 




/ 












^^-^ 


~~, 















0j/S6(N,^j) 



- <0 



o 

^ i 

It'-' 2 

m 11-. ' 



4 

« 



Tj- fO (M 



o- 



414 



















/ 


1, 


V/ 


' 




/ 


/ 


/ 


// 






A 


/ o 


1 


d /d 




( 


\ 


V 


u 






1 


V 


\ 


\^ 


^^ 


\ 


ll;.:: 













o 

CM 

d 



d d 

NIIAI iV 



CO 

o 

d 



o 
d 



o <o 




^■j./d 



O 00 (O 


































\ \«1 

1i> 
















1 
















J 










CM 


^^ 


J 


f 




00 

6 


'J- 

d 




o 

\ 




/§ 






' 















If) (D 
O — 

d 



O CO <o 

Nl^^l XV d/^j. 



q 



<o It, 


































1 
















I III 
ijiii 














ii^ 


f 






^ 




_,^^ 


^ 


^ 






00 

d _ 


1 


> 


^ d 


/d 




















is«p 



o 

CO (D 
lU 

•^ lii^ 

ry 



o 



NIW IV 



in ^ fO 

N I IN IV i?/^J 



fvi 



415 



416 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



12 
11 

10' 



z 

UJ 

o 

5 
UJ 

> 









































^ 


\PREDICTED 






























^■o-.. 


-IN 


\ 






























measured"' 


^ 


V 




































4 




































\ 


^. 




































N 


V 




































\ 


k 




































\ 


"V. 




































\ 


\ 


} 




































^ 


^.. 




































P" 


^-Gr.^ 


==^ 



01 23456789 

RADIUS IN MILS 

Fig. 14 — Current density distribution in a transverse plane located where the 
95 per cent radius is a minimum. The predicted and measured curves are normal- 
ized to contain the same total current. (The corresponding prediction from the 
universal beam spread curve would show a step function with a constant relative 
current density of 64.2 for r < 1.2 mils and zero beyond.) The gun parameters are 
given in Section 5D. 

and generally Gaussian in form, thereby indicating uniform cathode 
emission and proper boundary conditions at the edge of the beam near 
the cathode. The ejffect of positive ions on the beam shape was in every I 
case reduced to negligible proportions, either by using special pulse 
techniques, or by applying a small voltage gradient along the axis of 
the beam. 

B. Comparison of the Experiinentally Measured Spreading of a Beam with 
that Predicted Theoretically 

From the experimentally obtained plots of current density versus 
radius at several axial positions along the beam, we have obtained at 
each position (by integrating to find the total current within any radius) 
a value for the radius, rgs , of that circle which encompasses 95 per cent 
of the beam. For brevity, we call the resulting plots of rgs versus axial 
distance, "beam profiles". The experimental profile for the giui de- 
scribed in Section 5D is shown as curve A in Fig. 15(a). Curve B shows 
the profile as predicted by the methods of this paper and obtained from 
Fig. 12. Curve C is the corresponding profile which one obtains by the 
Hines-Cutler method, and Curve D represents Tq^ as obtained from the 



BEAM FORMATION WITH ELECTRON GUNS 



417 



CO 



20 
18 
16 
14 
12 



t- 8 



2 


I 

50 
45 
40 
35 



if) 30 



Z 25 

l? 20 

15 

10 

























(a) 

GUN PARAMETERS: 
fc/fa=2.8 


s 
















/ 






^ 


\, 












(C)j 


/ 
1 




/ 




e = i3.7° 


VVa/T-i.o 


\ 


^> 


k 










/ 
/ 
/ 




[B]/ 


r 


rc = 0.043" 

(A) EXPERIMENT 

(B) METHODS OF 
THIS PAPER 

(C) HINES-CUTLER 
METHOD 

(D) UNIVERSAL BEAM 
SPREAD CURVE 




\ 


^^ 


V 








/ 
/ 
/ 


/ 


/ 








\ 


N 

s. 


<; 


>^ 


^ 


'4 


/ 


^ 


<. 








\ 


\, 


"^ 


^>3e 


















\ 


\, 










y 


/(D) 




















\ 


\ 






y 


y 


























"~~- 


^^ 


^ 















40 



80 120 160 200 240 

Z, DISTANCE FROM IDEAL ANODE IN MILS 



280 



320 













(b) 










i 


/(C) 








y 








GUN PARAMETERS: 
f c/fa = 2.5 






1 
1 

1 








/ 


/ 










e = 


8° 
1.0 




/ 


1 
* 






y 


^B) 




^/V, /T- 


\ 


x^ 




V a/ 

rc = 0.150" 




/ 

/ 
/ 
f 






y 


/^ 






V 


^ 

V 


^ 


^***^^ 








•■ 


• 

• 






} 


\ 










X 


X 




"^ 







, -» 


-^ 


<^ 




— ■^ 
(A) 




















\ 


^^ 




































.^ 
















y 




















^-- 


^^ 






^ 




(D) 





































100 200 300 400 500 600 

Z, DISTANCE FROM IDEAL ANODE IN MILS 



700 



800 



Fig. 15 — Beam profiles (using an anode lens correction of r = 1.1 and the gun 
parameters indicated) as obtained (A) from experiment, (B) bj^ the methods of this 
paper, (C) Hines-Cutler method, (D) by use of the universal beam spread curve. 

universal l^eam spread curve'" (i.e., under the assumption of laminar 
flow and gradual variations of beam radius with distance) . Note that in 
each case a value of 1.1 has been used for the correction factor, r, repre- 
senting the excess divergence of the anode lens. The agreement in 
(/'95)min as obtaiucd from Curves A and B is remarkably good, but the 
axial position of (r95)min in Curve A definitely lies beyond the correspond- 



418 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

ing inininumi position in Curve B. Fortunately, in the gun design stage, 
one is usually more concerned with the value of (r95)min than with its 
exact axial location. The principal need for knowing the axial location of 
the minimum is to enable the axial magnetic field to build up suddenly 
in this neighborhood. However, since this field is normally adjusted ex- 
perimentally to produce best focusing, an approximate knowledge of 
2m in is usually adequate. 

In Fig. 15b we show a similar set of experimental and theoretical beam 
profiles for another gun. The relative profiles are much the same as in 
Fig 15a, and all of several other guns measured yield experimental 
points similarly situated with respect to curves of Type B. 

C. Comparison of Experimental and Theoretical Current Density Dis- 
tributions where the Minimum Beam Diameter is Reached 

In Fig. 14 we have plotted the current density distribution we would 
have predicted in a transverse plane at ^min for the example introduced 
in Section 5Z). Here the experimental and theoretical curves are nor- 
malized to include the same total currents in their respective beams. 
The noticeable difference in predicted and measured current densities 
at the center of the beam does not appreciably alter the properties such 
a beam would have on entering a magnetic field because so little total 
current is actually represented by this central peak. 

D. Variation of Beam Profile with T 

All of the design charts have been based on a value of T = 1.1, which 
is typical of the values obtained by the methods of Section 3. When 
appreciably different values of F are appropriate, we can get some feel- 
ing for the errors involved, in using curves based on T = 1.1, by refer- 
ence to Fig. 16. Here we show beam profiles as obtained by the methods 
of this paper for three values of F. The calculations are again based on 
the gun of Section 5D, and a value of just over 1.1 for F gives the ex- 
perimentally obtained value for (r95)min . 

7. SOME ADDITIONAL REMARKS ON GUN DESIGN 

In previous sections we have not differentiated between the voltage 
on the accelerating anode of the gun and the final beam voltage. It is 
important, howovei', that the separate functions of these two voltages 
be kept clearly in mind: The accelerating anode determines the total 
current drawn and largely controls the shaping of the beam; the final 
beam voltage is, on the other hand, chosen to give maximum interaction 
between the electron beam and the electromagnetic waves traveling 
along the slow wave circuit. As a consequence of this separation of func- , 



BEAM FORMATION WITH ELECTRON GUNS 



419 




0.006 



0.02 



0.18 



0.20 



0.22 



0.04 0.06 0.08 0.10 0.12 0.14 0.16 

Z, DISTANCE FROM IDEAL ANODE IN INCHES 

Fig. 16 — Beam profiles as obtained by the methods of this paper for the gun 
parameters given in Section bD. Curves are shown for three values of the anode 
lens correction, viz. T = 1.0, 1.1, and 1.2. 

tions, it is fouiicl that some beams which are difficult or impossible to 
obtain with a single Pierce-gun acceleration to final beam voltage may 
be obtained more easily by using a lower voltage on the gun anode. The 
acceleration to final beam voltage is then accomplished after the beam 
has entered a region of axial magnetic field. 

Suppose, for example, that one wishes to produce a 2-ma, 4-kv beam 
with (rgs/rc) = 0.25. If the cathode temperature is 1000°K, and the gun 
anode is placed at a final beam voltage of 4 kv, we have \^Va/T = 2 
and P = 0.008. From the top set of curves under \^Va/T = 2 in Fig. 
13, we find (by using a fairly crude extrapolation from the curves shown) 
that a ratio of fc/fa'^ 3.5 is required to produce such a beam. The value 
of {ve/o-) at Zmin IS therefore less than about 0.2 so that there is little 
x'mblance of laminar flow here. On the other hand we might choose 
r, = 250 volts so that a/fT^ = 0.5 and P = 0.51. From Fig. 13* 
we than obtain fc/fa = 2.6 and (re/o-)min = 0.8 for the same ratio of 
'■'joAc(= 0.25). While the flow could still hardly be called laminar, it is 
(•(jnsiderably more ordered than in the preceding case. Here we have in- 
cluded no correction for the (convergent) lens effect associated with the 
post-anode acceleration to the final beam voltage, F = 4 kv. 

Calculations of the Hines-Cutler type will always predict, for a given 
set of gun parameters and a specified anode lens correction, a minimum 
beam size which is larger than that predicted by the methods of this 
])aper. Nevertheless, in many cases the difference between the minimum 
sizes predicted by the two theories is negligible so long as the same anode 
lens correction is used. The extent to which the two theories agree ob- 



420 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

viously depends on the magnitude of Velo. When rel(T as calculated by 
the Hines-Cutler method (with a lens correction added) remains greater 
than about 2 throughout the range of interest, the difference between 
the corresponding values obtained for rgs will be only a few per cent. 
For these cases where rja does not get too small, the principal advan- 
tages of this paper are in the inclusion of a correction to the anode lens 
formula and in the comparative ease with which design parameters may 
be obtained. In other cases r^la may become less than 1, and the theory 
presented in this paper has extended the basic Hines-Cutler approach 
so that one may make realistic predictions even under these less ideal 
conditions where the departure from a laminar-type flow is quite severe. 

ACKNOWLEDGMENT 

We wish to thank members of the Mathematical Department at 
B.T.L., particularly H. T. O'Neil and Mrs. L. R. Lee, for their help in 
programming the problem on the analog computer and in obtaining the 
large amount of computer data involved. In addition, we wish to thank 
J. C. Irwin for his help in the electrolytic tank work and both Mr. Irwin 
and W. A. L. Warne for their work on the beam analyzer. 

REFERENCES 

1. Pierce, J. R., Rectilinear Flow in Beams, J. App. Phys., 11, pp. 548-554, Aug., 

1940. 

2. Samuel, A. L., Some Notes on the Design of Electron Guns, Proc. I.R.E., 33, 

pp. 233-241, April, 1945. 

3. Field, L. M., High Current Electron Guns, Rev. Mod. Phys., 18, pp. 353-361, 

July, 1946. 

4. Davisson, C. J., and Calbick, C. J., Electron Lenses, Phys. Rev., 42, p. 580, 

Nov., 1932. 

5. Helm, R., Spangenburg, K., and Field, L. M., Cathode-Design Procedure for 

Electron Beam Tubes, Elec. Coram., 24, pp. 101-107, March, 1947. 

6. Cutler, C. C, and Hines, M. E., Thermal Velocity Effects in Electron Guns, 

Proc. I.R.E., 43, pp. 307-314, March, 1955. 

7. Cutler, C. C, and Saloom, J. A., Pin-hole Camera Investigation of Electron 

Beams, Proc. I.R.E., 43, pp. 299-306, March, 1955. 

8. Hines, M. E., Manuscript in preparation. 

9. Private communication. 

10. See for example, Zworykin, V. K., et al.. Electron Optics and the Electron 

Microscope, Chapter 13, Wiley and Sons, 1945, or Klemperer, O., Electron 
Optics, Chapter 4, Cambridge Univ. Press, 1953. 

11. Brown, K. L., and Siisskind, C., The Effect of the Anode Aperature on Po- 

tential Distribution in a "Pierce" Electron Gun, Proc. I.R.E., 42, p. 598, 
March, 1954. 

12. See, for example, Pierce, J. R., Theory and Design of Electron Beams, p. 147, 

Van Nostrand Co., 1949. 

13. See Reference 6, p. 5. 

14. Langmuir, I. L., and Blodgett, K., Currents Limited by Space Charge Be- 

tween Concentric Spheres, Phys. Rev., 24, p. 53, July, 1924. 

15. See Reference 12, p. 177. 

16. See Reference 12, Chap. X. 



Theories for Toll Traffic Engineering in 

the U.S.A.* 

By ROGER I. WILKINSON 

(Manuscript received June 2, 1955) 

Present toll trunk traffic engineering practices in the United States are 
reviewed, and various congestion formulas compared with data obtained on 
long distance traffic. Customer habits upon meeting busy channels are noted 
and a theory developed describing the probable result of permitting subscribers 
to have direct dialing access to high delay toll trunk groups. 

Continent-wide automatic alternate routing plans are described briefly, 
in which near no-delay service will permit direct customer dialing. The 
presence of non-random overflow traffic from high usage groups co7nplicates 
the estimation of correct quantities of alternate paths. Present methods of 
solving graded multiple problems are reviewed and found unadaptable to the 
variety of trunking arrangements occurring in the toll plan. 

Evidence is given that the principal fluctuation characteristics of overflow- 
type of non-random traffic are described by their mean and variance. An 
approximate probability distribution of simultaneous calls for this kind of 
non-random traffic is developed, and found to agree satisfactorily with theo- 
retical overflow distributions and those seen in traffic simidations. 

A method is devised using ^^ equivalent random''^ traffic, which has good 
loss predictive ability under the "lost calls cleared" assumption, for a diverse 
field of alternate route trunking arrangements. Loss comparisons are made 
with traffic simulation residts and with observations in exchanges. 

Working curves are presented by which midti-alternate route trunking 
systems can be laid out to meet economic and grade of service criteria. Exam- 
ples of their application are given. 

Table of Contents 

1 . Introduction 422 

2. Present Toll Traffic Engineering Practice 423 



* Presented at the First International Congress on the Application of the 
Theory of Probability in Telephone Engineering and Administration, Copen- 
hagen, June 21, 1955. 

421 



422 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

3. Customers Dialing on Groups with Considerable Delay 431 

3.1. Comparison of Some Formulas for Estimating Customers' NC Service 

on Congested Groups 434 

4. Service Requirements for Direct Distance Dialing by Customers 436 

5. Economics of Toll Alternate Routing 437 

6. New Problems in the Engineering and Administration of Intertoll Groups 
Resulting from Alternate Routing 441 

7. Load-Service Relationships in Alternate Route Systems 442 

7.1. The "Peaked" Character of Overflow Traffic 443 

7.2. Approximate Description of the Character of Overflow Traffic 446 

7.2.1. A Probability Distribution for Overflow Traffic 452 

7.2.2. A Probability Distribution for Combined Overflow Traffic Loads 457 

7.3. Equivalent Random Theory for Prediction of Amount of Traffic Over- 
flowing a Single Stage Alternate Route, and Its Character, with Lost 

Calls Cleared 461 

7.3.L Throwdown Comparisons with Equivalent Random Theory on 

Simple Alternate Routing Arrangements with Lost Calls 

Cleared 468 

7.3.2. Comparison of Equivalent Random Theory with Field Results 

on Simple Alternate Routing Arrangements 470 

7.4. Prediction of Traffic Passing Through a Multi-Stage Alternate Route 

Network 475 

7.4.1. Correlation of Loss with Peakedness of Components of Non- 
Random Offered Traffic 481 

7.5. Expected Loss on First Routed Traffic Offered to Final Route 482 

7.6. Load on Each Trunk, Particularly the Last Trunk, in a Non-Slipped 
Alternate Route 486 

8. Practical Methods for Alternate Route Engineering 487 

8.1. Determination of Final Group Size with First Routed Traffic Offered 
Directly to Final Group 490 

8.2. Provision of Trunks Individual to First Routed Traffic to Equalize 
Service 491 

8.3. Area in Which Significant Savings in Final Route Trunks are Real- 
ized by Allowing for the Preferred Service Given a First Routed 
Traffic Parcel 494 

8.4. Character of Traffic Carried on Non-Final Routes 495 

8.5. Solution of a Typical Toll Multi-Alternate Route Trunking Arrange- 
ment : Bloomsburg, Pa 500 

9. Conclusion 505 

Acknowledgements 506 

References 506 

Abridged Bibliography of Articles on Toll Alternate Routing 507 

Appendix I: Derivation of Moments of Overflow Traffic 507 

Appendix II: Character of Overflow when Non-Random Traffic is Offered 

to a group of Trunks 511 



1. INTRODUCTION 

It has long been the stated aim of the Bell System to make it easily 
and economically possible for any telephone customer in the United 
States to reach any other telephone in the world. The principal effort 
in this direction by the American Telephone and Telegraph Company 
and its associated operating companies is, of course, confined to inter- 
connecting the telephones in the United States, and to providing com- 
munication channels between North America and the other countries of 
the world. Since the United States is some 1500 miles from north to 
fSOuth and 3000 miles from east to west, to realize even the aim of fast 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 423 

and economical service between customers is a problem of great magni- 
tude; it has engaged our planning engineers for many years. 

There are now 52 million telephones in the United States, over 80 per 
cent of which are equipped with dials. Until quite recently most telephone 
users were limited in their direct dialing to the local or immediately sur- 
rounding areas and long distance operators were obliged to build up a 
circuit with the aid of a "through" operator at each switching point. 

Both speed and economy dictated the automatic build-up of long toll 
circuits without the intervention of more than the originating toll oper- 
ator. The development of the No. 4-type toll crossbar switching system 
with its ability to accept, translate, and pass on the necessary digits (or 
lujuivalent information) to the distant office made this method of oper- 
ation possible and feasible. It was introduced during World War II, and 
now by means of it and allied equipment, 55 per cent of all long distance 
calls (over 25 miles) are completed by the originating operator. 

As more elaborate switching and charge-recording arrangements were 
developed, particularly in metropolitan areas, the distances which cus- 
tomers themselves might dial measurably increased. This expansion of 
the local dialing area was found to be both economical and pleasing to 
the users. It was then not too great an effort to visualize customers 
dialing to all other telephones in the United States and neighboring 
countries, and perhaps ultimately across the sea. 

The physical accomplishment of nationwide direct distance dialing 
which is now gradually being introduced has involved, as may well be 
imagined, an immense amount of advance study and fundamental plan- 
ning. Adequate transmission and signalling with up to eight intertoll 
trunks in tandem, a nationwide uniform numbering plan simple enough 
to be used accurately and easily by the ordinary telephone caller, pro- 
^ ision for automatic recording of who called whom and how long he 
talked, with subsequent automatic message accounting, are a few of 
man}^ problems which have required solution. How they are being met is 
a romantic story beyond the scope of the present paper. The references 
given in the bibliography at the end contain much of the history as well 
as the plans for the future. • 

2. PRESENT TOLL TRAFFIC ENGINEERING PRACTICE 

There are today approximately 116,000 intertoll trunks (over 25 miles 
in length) in the Bell System, apportioned among some 13,000 trunk 
groups. A small segment of the 2,600 toll centers which they interconnect 
is shown in Fig. 1. Most of these intertoll groups are presently traffic 
engineered to operate according to one of several so-called T-schedules: 
T-8, T-15, T-30, T-60, or T-120. The number following T (T for Toll) is 



424 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




KEY 

O TOLL CENTERS 

INTERTOLL TRUNK GROUPS 



Fig. 1 — Principal intertoll trunk groups in Minnesota and Wisconsin. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 425 




4 5 6 7 8 9 10 
NUMBER OF TRUNKS 



30 40 50 



Fig. 2 — Permitted intertoU trunk occupancy for a 6.5-minute usage time 
per message. 

the expected, or average, delay in seconds for calls to obtain an idle 
trunk in that group during the average Busy Season Busy Hour. In 1954 
the system "average trunk speed" was approximately 30 seconds, re- 
sulting from operating the majority of the groups at a busy-hour trunk- 
ling efficiency of 75 to 85 per cent in the busy season. 

The T-engineering tables show permissible call minutes of use for a 



426 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

wide range of group sizes, and several selections of message holding 
times. They were constructed following summarization of many obser- 
vations of load and resultant average delays on ringdown (non-dial) 
intertoll trunks.^ Fig. 2 shows the permissible occupancy (efficiency) of 
various trunk group sizes for 6.5 minutes of use per message, for a va- 
riety of T-schedules. It is perhaps of somfe interest that the best fitting 
curves relating average delay and load were found to be the well-known 
Pollaczek-Crommelin delay curves for constant holding time — this in 
spite of the fact that the circuit holding times were far indeed from 
having a constant value. 

A second, and probably not uncorrected, observation was that the 
per cent "No-Circuit" (NC) reported on the operators' tickets showed 
consistently lower values than were measured on group-busy timing de- 
vices. Although not thoroughly documented, this disparity has generally 
been attributed to the reluctance of an operator to admit immediately 
the presence of an NC condition. She exhibits a certain tolerance (very 
difficult to measure) before actually recording a delay which would 
recjuire her to adopt a prescribed procedure for the subsequent handling 
of the call.* There are then two measures of the No-Circuit condition 
which are of some interest, the "NC encountered" by operators, and the 
"NC existing" as measured by timing devices. 

It has long been observed that the distribution of numbers n of simul- 
taneous calls found on T-engineered ringdown intertoll groups is in re- 
markable agreement with the individual probability terms of the Erlang 
"lost calls" formula, 

f n — a ' 

a e 



fin) = ^-^^ (1) 

e 



E- 



n=o n! 

where c = number of paths in the group, 

a' = an enhanced average load submitted such that 
a'[l — Ei^c(a')] = L, the actual load carried, and 
Ei^cid') = fie) = Erlang loss probability (commonly called Er- 
lang B in America). 
An example of the agreement of observations with (1) is shown in Fig. 
3, where the results of switch counts made some years ago on many 
ringdown circuit groups of size 3 are summarized. A wide range of "sub- 



* Upon finding No-Circuit, an operator is instructed to try again in 30 seconds 
and GO seconds (before giving an NC report to the customer), followed by addi- 
tional attempts 5 minutes and 10 minutes later if necessary. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 427 




0.10 0.2 0.5 1.0 2 

AVERAGE "submitted" LOAD IN ERLANIGS 



Fig. 3 — Distributions of simultaneous calls on three-trunk toll groups at 
.\lbany and Buffalo. 

I nit ted" loads a' to produce the observed carried loads is required. On 
Fig. 4 are shown the corresponding comparisons of theory and obser- 
vations for the proportions of time all paths are busy ("NC Existing") 
for 2-, 4-, 5-, 7-, and 9-circuit groups. Good agreement has also been ob- 
served for circuit groups up to 20 trunks. This has been found to be a 
stable relationship, in spite of the considerable variation in the actual 
practices in ringdown operation on the resubmission of delayed calls. 
Since the estimation of traffic loads and the subsequent administration 
of ringdown toll trunks has been performed principally by means of 
Group Busy Timers (which cumulate the duration of NC time), the 
Erlang relationship just described has been of great importance. 



428 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



With the recent rapid increase in operator dialed intertoll groups, it 
might be expected that the above discrepancy between " % NC encoun- 
tered" and "% NC existing" would disappear — for an operator now 
initiates each call unaware of the momentary state of the load on any 
particular intertoll group. By the use of peg count meters (which count 
calls offered) and overflow call counters, this change has in fact been 
observed to occiu'. ]\Ioreo^'er, since the initial re-trial intervals are com- 
monly fairly short (30 seconds) subsequent attempts tend to find some 
of the previous congestion still existing, so that the ratio of overflow to 
peg count readings now exceeds slightly the "% NC existing." This 
situation is illustrated in Fig. 5, which shows data taken on an operator- 



1.0 




AVERAGE SUBMITTED LOAD 



Fig. 4 — Observed proportions of time all trunks were busy on Albany and 
Buffalo groups of 2, 4, 5, 7, and 9 trunks, 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 429 



u 

z 

o 

z 
I- 

UJ 
HI 

5 

ul 
_i 
_i 
< 
o 

U- 

o 

z 
o 

(- 
cc 
o 
a. 
o 
tr 
a. 




0.001 



12 14 

L = LOAD CARRIED IN ERLANGS 



18 



Fig. 5 — Comparison of NC data on a 16-trunk T-engineered toll group with 
various load versus NC theories. 



dialed T-engineered group of 16 trunks between Newark, N. J., and 
Akron, Ohio. Curve A shows the empirically determined "NC encoun- 
tered" relationship described above for ringdown operation; Curve B 
gives the corresponding theoretical "NC existing" values. Lines C and D 
give the operator-dialing results, for morning and afternoon busy hours. 
The observed points are now seen generally to be significantly above 
Curve B.* 

At the same time as this change in the "NC encountered" was occur- 
ring, due to the introduction of operator toll dialing, there seems to have 
l)een little disturbance to the traditional relationship between load 

* The observed point at 11 erlangs which is clearly far out of agreement with 
the remainder of the data was produced by a combination of high-trend hours 
and an hour in which an operator apparently made many re-t^rials in rapid suc- 
cession. 



430 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



u. 


10 

z 
















































































































o 


o 














































































































































































































































































5 

o 
m 

rvj 

ti 

_r 
< 

(- 

z 

LU 

z 
o 




z 
o 

i 

tr 

UJ 

I/) 

§ 

o 


























































































«--- LIMIT OF 
OBSERVED 
DATA 
































i 


[ 






oiT 
























/ 








































/ 






/ 


































/ 






/ 
































/ 


/ 




/ 


































/ 
/ 






/ 
































y 


/ 


/ 






































/' 


































^•^ 




/^ 
































^ 


^ 


««- 






























^ 


^ 


Tt^^ 


























^•^^ 


^ 


s:;^ 





































8 



If) 



o 

0> 






o 

(0 



(O 



o 



If) 



o 



in 



o 






o 



in 
tvj 



o in 



SBIONII^ 1- a3AO SidlAjaiiV dO iN3D«3d 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 431 

carried and " % NC existing." C. J. Truitt of the A.T. & T. Co. studied 
i a number of operator-dialed T-engineered groups at Newark, New Jersey, 
in 1954 with a traffic usage recorder (TUR) and group-busy timers, and 
found the relationship of equation (1) still good. (This analysis has not 
been published.) 

A study by Dr. L. Kosten has provided an estimate of the probability 
that when an NC condition has been found, it will also appear at a time 
T later." When this modification is made, the expected load-versus-NC 
relationship is shown by Curve E on Fig. 5. (The re-trial time here was 
taken as the operators' nominal 30 seconds; with 150-second circuit-use 
time the return is 0.2 holding time.) The observed NC's are seen to lie 
slightly above the E-curve. This could be explained either on the basis 
that Kosten's analysis is a lower limit, or that the operators did not 
strictly observe the 30-second return schedule, or, more probably, a 
combination of both. 

3. CUSTOMERS DIALING ON GROUPS WITH CONSIDERABLE DELAY 

It is not to be expected that customers could generally be persuaded to 
wait a designated constant or minimum re-trial time on their calls which 
meet the NC condition. Little actual experience has been accumulated 
on customers dialing long distance calls on high-delay circuits. However, 
it is plausible that they would follow the re-trial time distributions of 
customers making local calls, who encounter paths-busy or line-busy 
signals (between which they apparently do not usually distinguish). 
Some information on re-trial times was assembled in 1944 by C. Clos by 
observing the action of customers who received the busy signal on 1,100 
local calls in the City of New York. As seen in Fig. 6, the return times, 
after meeting "busy," exhibit a marked tendency toward the exponential 
distribution, after allowance for a minimum interval required for re- 
dialing. 

An exponential distribution with average of 250 seconds has been 
I fitted by eye on Fig. 6, to the earlier ■ — and more critical — customer re- 
turn times. This may seem an unexpectedly long wait in the light of indi- 
vidual experience; however it is probably a fair estimate, especially 
since, following the collection of the above data, it has become common 
practice for American operating companies in their instructional lit- 
erature to advise customers receiving the busy signal to "hang up, wait 
a few minutes, and try again." 

The mathematical representation of the situation assuming exponen- 
I tial return times is easily formulated. Let there be .r actual trunks, and 



432 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

imagine y waiting positions, whore y is so large that few calls are re- 
jected.* Assume that the offered load is a erlangs, and that the calls have 
exponential conversation holding times of unit average duration. Finally \ 
let the average return time for calls which have advanced to the waiting > 
positions, be 1/s times that of the unit conversation time. The statistical j 
equilibrium equation can then be written for the probability j\m, n) (j 
that m calls are in progress on the x trunks and n calls are waiting on 
the y storage positions: ■ 

/(w, n) = aj{m — 1, n) dt + s(w + l)/(m — 1, n + 1) dt ''■) 

+ (m + \)J{m + 1, n) dt + a/(.r, n - 1) dH^ (2) 

+ [1 - (a*** + sn**) dt - m dt]f(m, n) ^ 

where ^ m ^ .-r, ^ w ^ //, and the special limiting situations are 
recognized by: 

■* Include term only when m — x 

**■ Omit sn when m = x 

*** Omit a when m = x and n = y 

Equation (2) reduces to 

(a*** + snifif + m)f{m; n) = af{m — 1, n) 1 

+ s(n + l)/(m - 1, w + 1) (3) 

+ (m + l)/(w + 1, n) + af(x, n - !)•, 

Solution of (3) is most easily effected for moderate values of x and y 
by first setting f(x, ?/) = 1 .000000 and solving for all other /(/?? , ?? ) in 

X y 

terms of /(o:, ?/). Normahzing through zl 11f(m, n) = 1.0, then gives 

m=0 n=0 

the entire f(m, n) array. 

The proportion of time "NC exists," will, of course be 

Z Six, n) (4) 

n=0 

and the load carried is 

L = Xl X wi/(m, n) (5) 

The proportion of call attempts meeting NC, including all re-trials 



* The quant itjr y can also be chosen so that some calls are rejected, thus roughly 
describing those calls abandoned after the first attempt. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A, 433 



will be 



W{x, a, s) = 



Expected overflow calls per unit time 
Expected calls offered per unit time 

Z (a + sn)jXx, n) - , ./ ^ ^^^ 

sn -\- af{x, y) 



n=0 



X y 



S 2 (« + sn)f(m, n) 



a -{- sn 



m=0 71=0 



X y 



in which n = ^ 2^ nf(7n, n). And when y is chosen so large that/(.r, y) 

7H = 71=0 

is negligible, as we shall use it here, 



L = a 



W(x, a, s) = 



sn 



a -\- sn 



(5') 
(6') 



1^ 0.5 

< 

"^O 0.4 
ilZ 

Oo 
ZZ 0.3 

Ol- 

pllJ 

o5 0.2 

Q. 

o 

? 0.1 



6 TRUNKS 



/ // APOISSON 
' ^1 P(C,L) 



5=0.6 




2 4 6 8 

L=LOAD CARRIED IN ERLANGS 




APOISSON 
P(C,L) 



fly >^- 

f I6j _, 



8 10 12 14 

L = LOAD CARRIED IN ERLANGS 



Fig. 7 — ■ Comparison of trunking formulas. 



434 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



I 



This formula provides a means for estimating the grade of service 
which customers might he expected to receive if asked to dial their calls 
over moderate-delay or high-delay trunk groups. For a circuit use length 
of 150 seconds, and an average return time of 250 seconds (as on Fig. 6), 
both exponential, the load-versus-proportion-NC curves for 6 and IG 
trunks are given as curves (3) on Fig. 7. For example with an offered 
(= carried) load of a = 4.15 erlangs on 6 trunks we should expect to find 
27.5 per cent of the total attempts resulting in failure. 

For comparison with a fixed return time of NC-calls, the IF-formula 
curves for exponential returns of 30 seconds (s = 5) and 250 seconds 
(s = 0.6) averages are shown on Fig. 5. The first is far too severe an 
assumption for operator performance, giving NC's nearly double those 
actually observed (and those given by theory for a 30-second constant 
return time). The 250-second average return, however, lies only slightly 
above the 30-second constant return curve and is in good agreement with 
the data. Although not logically an adequate formula for interpreting 
Peg Count and Overflow registrations on T-engineered groups under 
operator dialing conditions, the IF-formula apparently could be used for 
this purpose with suitable s-values determined empirically. 

3.1. Comparison of Some Formulas for Estimating Customers' NC Service 

on Congested Groups 

, 1 
As has been previously observed, a large proportion of customers who 

receive a busy signal, return within a few minutes (on Fig. 6, 75 per cent 
of the customers returned within 10 minutes). It is well known too, that 
under adverse service conditions subscriber attempts (to reach a par- 
ticular distant office for example) tend to produce an inflated estimate 
of the true offered load. A count of calls carried (or a direct measurement 
of load carried) will commonly be a closer estimate of the offered load 
than a count of attempts. An exception may occur when a large propor- 
tion of attempts is lost, indicating an offered load possibly in excess even 
of the number of paths provided. Under the latter condition it is diffi- 
cult to estimate the true offered load by any method, since not all the 
attempts can be expected to return repeatedly until served; instead, a 
significant number will be abandoned somewhere through the trials. In 
most other circumstances, however, the carried load will prove a reason- 
ably good estimate of the true offered load in systems not provided with 
alternate paths. 

This is a matter of especial interest for both toll and local operation 
in America since principal future reliance for load measurement is ex- 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 435 

pected to be placed on automatically processed TUR data, and as the 
TUR is a switch counting device the results will be in terms of load 
carried. Moreover, the quantity now obtained in many local exchanges 
is load carried.* Visual switch counting of line finders and selectors off- 
normal is widely practiced in step-by-step and panel offices; a variety of 
electromechanical switch counting devices is also to be found in crossbar 
offices. It is common to take load-carried figures as equal to load-offered 
when using conventional trunking tables to ascertain the proper pro- 
vision of trunks or switches. Fig. 7 compares the NC predictions made by 
a number of the available load-loss formulas when load carried is used as 
the entry variable. 

The lowest curves (1) on Fig. 7 are from the Erlang lost calls formula 
El (or B) with load carried L used as the offered load a. At low losses, 
say 0.01 or less, either L or a = L/[l — Ei(a)] can be used indiscrimi- 
nately as the entry in the Ei formula. If however considerably larger 
losses are encountered and calls are not in reality "cleared" upon meet- 
ing NC, it will no longer be satisfactory to substitute L for a. In this 
circumstance it is common to calculate a fictitious load a' to submit to 
the c paths such that the load carried, a'[I — Ei^dd')], equals the desired 
L. (This was the process used in Section 2 to obtain " % NC existing.") 
The curves (2) on Fig. 7 show this relation ; physically it corresponds to 
an initially offered load of L erlangs (or L call arrivals per average hold- 
ing time), whose overflow calls return again and again until successful 
but without disturbing the randomness of the input. Thus if the loss 
from this enhanced random traffic is E, then the total trials seen per 
holding time will be L(l + ^ + ^' -f • • •) = L/(l - E) = a', the ap- 
parent arrival rate of new calls, but actually of new calls plus return 
attempts. 

The random resubmission of calls may provide a reasonable descrip- 
tion of operation under certain circumstances, presumably when re-trials 
are not excessive. Kosten^ has discussed the dangers here and provided 
upper and lowxr limit formulas and curves for estimating the proportions 
of NC's to be expected when re-trials are made at any specified fixed 
leturn time. His lower bounds (lower bound because the change in con- 
gestion character caused by the returning calls is ignored) are shown by 
open dots on Fig. 7 for return times of 1.67 holding times. They lie above 
curves (2) (although only very slightly because of the relatively long 
return time) since they allo\\- for the fact that a call shortly returning 



* In fact, it is difficult to see how any estimate of offered load, other than carried 
load, can be obtained with useful reliability. 



436 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

after meeting a busy signal will have a higher probability of again find- 
ing all paths busy, than would a randomly originated call. 

The curves (3) show the TF-formula previously developed in this sec- 
tion, which contemplates exponential return times on all NC attempts. 
The average return time here is also taken as 1 .67 holding times. These 
curves lie higher than Kosten's values for two reasons. First, the altered 
congestion due to return calls is allowed for; and second, with exponential 
returns nearly two-thirds of the return times are shorter than the aver- 
age, and of these, the shortest ones will have a relatively high probability 
of failure upon re-trying. If the customers were to return with exponen- 
tial times after waiting an average of only 0.2 holding time (e.g., 30 
seconds wait for 150-second calls) the TT^-curves would rise markedly to 
the positions shown by (4). 

Curves (5) and (6) give the proportions of time that all paths are busy 
(equation 4) under the T'F-formula assumptions corresponding to NC 
curves (3) and (4) respectively; their upward displacement from the 
random return curves (2) reflects the disturbance to the group congestion 
produced by the non-random return of the delayed calls. (The limiting 
position for these curves is, of course, given by Erlang's E2 (or C) delay 
formula.) As would be expected, curve (6) is above (5) since the former 
contemplates exponential returns with average of 0.2 holding time, as 
against 1.67 for curve (5). Neither the (5)-curves nor the open dots of 
constant 30-second return times show a marked increase over curves (2). 
This appears to explain why the relationship of load carried versus "NC 
existing" (as charted in Figs. 3 and 4) was found so insensitive to vari- 
able operating procedures in handling subsequent attempts in toll ring- 
down operation, and again, why it did not appreciably change under 
operator dialing. 

Finally, through the two fields of curves on Fig. 7 is indicated the 
Poisson summation P{c, L) with load carried L used as the entering 
variable. The fact that these values approach closely the (2) and (3) sets 
of curves over a considerable range of NC's should reassure those who 
have been concerned that the Poisson engineering tables were not useful 
for losses larger than a few per cent.* 

4. SERVICE REQUIREMENTS FOR DIRECT DISTANCE DIALING BY CUSTOMERS 

As shown by the TF-curves (3) on Fig. 7, the attempt failures by cus- 
tomers resulting from their tendency to re-try shortly following an NC 

* Reference may be made also to a throwdown by C. Clos (Ref. 3) using the 
return times of Fig. 6; his "% NC" results agreed closely with tlie Poisson pre- 
dictions. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 437 

would be expected to exceed slightly the values for completely random 
re-trials. These particular curves are based on a re-trial interval of 1.67 
times the average circuit-use time. Such moderation on the part of the 
customer is probably attainable through instructional literature and 
other means if the customer believes the "NC" or "busy" to be caused 
by the called party's actually using his telephone (the usual case in local 
practice). It would be considerably more difficult, however, to dissuade 
the customer from re-trying at a more rapid rate if the circuit NC's 
should generally approach or exceed actual called-party busies, a con- 
dition of which he would sooner or later become aware. His attempts 
might then be more nearly described by the (4) curves on Fig. 7 cor- 
responding to an average exponential return of only 0.2 holding time — or 
e\en higher. Such a result would not only displease the user, but also 
result in the requirement of increased switching control equipment to 
handle many more wasted attempts. 

If subscribers are to be given satisfactory direct dialing access to the 
iiitertoll trunk network, it appears then that the probability of finding 
XC even in the busy hours must be kept to a low figure. The following 
engineering objective has tentatively been selected: The calls offered to 
the ^'final" group of trunks in an alternate route system should receive no 
more than 3 per cent NC(P.03) during the network busy season busy hour. 
(If there are no alternate routes, the direct group is the "final" route.) 

Since in the nationwide plan there will be a final route between each 
of some 2,600 toll centers and its next higher center, and the majority 
of calls offered to high usage trunks will be carried without trying 
their final route (or routes), the over-all point-to-point service, while 
not easy to estimate, will apparently be quite satisfactory for cus- 
tomer dialing. 

5. ECONOMICS OF TOLL ALTERNATE ROUTING 

In a general study of the economics of a nationwide toll switching plan, 
made some years ago by engineers of the American Telephone and Tele- 
graph Company, it was concluded that a toll line plant sufficient to give 
ihe then average level of service (about T-40) with ordinary single-route 
procedures could, if operated on a multi-alternate route basis, give the 
desired P.03 service on final routes with little, if any, increase in toll line 
investment.* On the other hand to attain a similar P.03 grade of service 
by liberalizing a typical intertoll group of 10 trunks working presently 



* This, of course, does not reflect the added costs of the No. 4 switching equip- 
I nient. 



438 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

at a T-40 grade of service and an occupancy of 0.81 would recjuire an 
increase of 43 per cent (to 14.3 trunks), with a corresponding decrease 
in occupancy to 0.57. The possible savings in toll lines with alternate 
routing are therefore considerable in a system which must pro\'ide a 
service level satisfactory for customer dialing. 

In order to take fullest advantage of the economies of alternate rout- 
ing, present plans call for five classes of toll offices. There will be a large 
number of so-called End Offices, a smaller number of Toll Centers, and 
progressively fewer Primary Centers (about 150), Sectional Centers 
(about 40) and Regional Centers (9), one of which will be the National 
Center, to be used as the "home" switching point of the other eight 
Regional Centers.* Primary and higher centers will be arranged to per- 
form automatic alternate routing and are called Control Switching 
Points (CSP's). Each class of office will "home" on a higher class of 
office (not necessarily the next higher one) ; the toll paths between them 
are called "final routes." As described in Section 4, these final routes will 
be provided to give low delays, so that between each principal toll point 
and ever}' other one there will be available a succession of approximatelj' 
P.03 engineered trunk groups. Thus if the more direct and heavily loaded 
interconnecting paths commonly provided are busj- there will still be a 
good chance of making immediate connection over final routes. 

Fig. 8 illustrates the manner in which automatic alternate routing will 
operate in comparison with present-day operator routing. On a call from 
Syracuse, X. Y., to Miami, Florida, (a distance of some 1,250 miles), 
under present-day operation, the Syracuse operator signals Albany, and 
requests a trunk to Miami. With T-schedule operation the Syracuse- 
Miami traffic might be expected to encounter as much as 25 per cent NC 
during the busy hour, and approximately 4 per cent NC for the whole 
day, producing perhaps a two-minute over-all speed of serA-ice in the 
busy season. 

With the proposed automatic alternate routing plan, all points on the 
chart will have automatic switching systems. f The customer (or the 
operator until customer dialing arrangements are completed) will dial a 
ten-digit code (three-digit area code 305 for Florida plus the listed 
Miami seven-digit telephone number) into the Jiiachine at Syracuse. 
The various routes which then might conceivably be tried automatically 



* Sec the hihlio^rajjliy ( i);irticulMily Pilliod and Truitt) for details of tlie 
general trunkinji plan. 

t The notation uscmI on the diagram of Fig. 8 is: Opon firclo — Primary Center 
(Syracuse, Miamij; Triangle — Sectional Center (All)an\-, Jacksonville); Sqviare 
— Regional Center (White Plains, Atlanta, St. Louis; St. Louis is also the Na- 
tional Center). 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 439 



PRESENT OPERATOR 

ROUTING '^^ 




AUTOMATIC ALTERNATE 
ROUTING 



white Plains 
N. Y.) 



Miami 




Miami 



Fig. 8 — Present and proposed methods of handling a call from Syracuse, N. Y., 
to Miami, Florida. 



are shown on the diagram numbered in the order of trial; in this par- 
ticular layout shown, a maximum of eleven circuit groups could be tested 
for an idle path if each high usage group should be found NC. Dotted 
lines show the high usage roiites, which if found busy will overflow to the 
final groups represented by solid lines. The switching ecjuipment at each 
point upon finding an idle circuit passes on the required digits to the 
next machine. 

While the routing possibilities shown are factual, only in rare instances 
would a call be completed over the final route via St. Louis. Even in the 
busy season busy hour just a small portion of the calls would be expected 
to be switched as many as three times. And only a fraction of one per 
cent of all calls in the busy hour should encounter NC. As a result the 
service will be fast. When calls are handled by a toll operator, the cus- 



440 



THE BELL SYSTEM TECHNICAL JOURNAL; MARCH 1956 



tomer will not ordinarily need to hang up when NC is obtained. When 
he himself dials, a second trial after a short wait following NC should 
have a high probability of success. 

Not many situations will be as complex as shown in Fig. 8; commonly 
several of the links between centers will be missing, the particular ones 
retained having been chosen from suitable economic studies. A large 
number of switching arrangements Avill be no more involved than the 
illustrative one shown in Fig. 9(a), centering on the Toll Center of 
Bloomsburg, Pennsylvania. The dashed lines indicate high usage groups 
from Bloomsburg to surrounding toll centers; since Bloomsburg "homes" 
on Scranton this is a final route as denoted by the solid line. As an exam- 
ple of the operation, consider a call at Bloomsburg destined for Williams- 
port. Upon finding all direct trunks busy, a second trial is made via 
Harrisburg; and should no paths in the Harrisburg group be available, 
a third and final trial is made through the Scranton group. 

In considering the traffic flow of a network such as illustrated at 
Bloomsburg it is convenient to employ the conventional form of a two- 
stage graded multiple having "legs" of varying sizes and traffic loads 
individual to each, as shown in Fig. 9(b). Here only the circuits im- 
mediately outgoing from the toll center are shown; the parcels of traffic 



(a) GEOGRAPHICAL LAYOUT 
WILLIAMSPORT I 



SCRANTON 




BLOOMSBURG 
HARRISBURG PA. 

(b) GRADED MULTIPLE SCHEMATIC 



FRACKVILLE 

HAZLETON 

WILKES- 
BARRE 



PHILADELPHIA 



FINAL GROUP TO SCRANTON 



H.U. GROUP TO HARRISBURG 



.1 M t 



I 



NO. TRUNKS IN H.U. GROUPS I [T] [jF] [^ [A] [T] [28 1 rsl m 

LOAD TO AND FROM ^^^ .^. ^^ ^ 

DISTANT OFFICE (CCS) "^^^ '^' ^^ ^'^^ ^^' '^0 '^3 836 228 154 

DISTANT OFFICE SCRN HBG PTVL SHKN SNBY WMPT FKVL HZN WKSB PHLA 



Fig. 9 
liiirg, Pa. 



Aulonialic ;ilU'riiaie routing for direct distance dialing at Blooms- 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 441 

calculated for each further connecting route will be recorded as part of 
the offered load for consideration when the next higher switching center 
is engineered. It is implicitly assumed that a call which has selected one 
of the alternate route paths will be successful in finding the necessary 
paths available from the distant switching point onward. This is not 
quite true but is believed generally to be close enough for engineering 
piu'poses, and permits ignoring the return attempt problem. 

6. NEW PROBLEMS IN THE ENGINEERING AND ADMINISTRATION OF INTER- 
TOLL GROUPS RESULTING FROM ALTERNATE ROUTING 

With the greatly increased teamwork among groups of intertoll trunks 
which supply overflow calls to an alternate route, an unexpected increase 
or flurry in the offered load to any one can adversely affect the service to 
all. The high efficiency of the alternate route networks also reduces their 
overload carrying ability. Conversely, the influence of an underprovision 
of paths in the final alternate route may be felt by many groups which 
overflow to it. With non-alternate route arrangements only the single 
groups having these flurries would be affected. 

Administratively, an alternate route trunk layout may well prove 
easier to monitor day by day than a large number of separate and in- 
dependent intertoll groups, since a close check on the service given on 
the final routes only may be sufficient to insure that all customers are 
being served satisfactorily. When rearrangements are indicated, how- 

SIMPLE PROGRESSIVE 

GRADED MULTIPLE GRADED MULTIPLE 

(a) (b) 



t t t t t t tt t t tl 

ILLUSTRATIVE INTERLOCAL AND INTERTOLL 
ALTERNATE ROUTE TRUNKING ARRANGEMENT; 

(c) (d) 



t t t t t = ,-"" ^ 

tttl It ttl 1 t 

Fig. 10 — Graded multi])los .•nid altornaic route trunking nrrangeinoiits. 



I 



442 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

ever, the determination of the proper place to take action, and the de 
sirable extent, may sometimes be difficult to determine. Suitable traffic 
measuring devices must be provided with these latter problems in mind 
For engineering purposes, it will be highly desirable: 

(1) To be able to estimate the load-service relationships with any 
specified loads offered to a particular intertoll alternate routing network; 
and 

(2) To know the day-to-day busy hour variations in the various 
groups' offered loads during the busy season, so that the general grade of 
service given to customers can be estimated. 

The balance of this paper will review the studies which have been made 
in the Bell System toward a practicable method for predicting the grade 
of service given in an alternate route network under any given loads. 
Analyses of the day-to-day load variations and their effects on customer 
dialing service are currently being made, and will be reported upon later. 

?; 

7. LOAD-SERVICE RELATIONSHIPS IN ALTERNATE ROUTE SYSTEMS 

In their simplest form, alternate route systems appear as symmetrical 
graded multiples, as shown in Fig. 10(a) and 10(b). Patterns such as 
these have long been used in local automatic systems to partially over- 
come the trunking efficiency limitations imposed by limited access 
switches. The traffic capacity of these arrangements has been the sub- 
ject of much study by theory and "throwdowns" (simulated traffic 
studies) both in the United States and abroad. Field trials have sub- 
stantiated the essential accuracy of the trunking tables which have 
resulted. 

In toll alternate route systems as contemplated in America, however, 
there will seldom be the symmetry of pattern found in local graded 
multiples, nor does maximum switch size generally produce serious 
limitation on the access. The ''legs" or first-choice trunk groups will vary 
widely in size; likewise the number of such groups overflowing calls 
jointly to an alternate route may cover a considerable range. In all cases 
a given group, whether or not a link of an alternate route, will have one 
or more parcels of traffic for which it is the first-choice route. [See the 
right-hand parcel of offered traffic on Fig. 10(c).] Often this first routed 
traffic will Ijc the bulk of the load offered to the group, which also serves 
as an alternate I'oute for other traffic. 

The simplest of the approximate formulas developed for solving the 
local graded multiple problems are hopelessly unwieldy when applied 
to such arrangements as shown in Fig. 10(d). Likewise it is impracticable i 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 443 

to solve more than a few of the infinite variety of arrangements by means 

of "throwdowns." 

However, for both engineering (planning for future trunk provisions) 
I and administration (current operating) of trunks in these multi-alternate 

routing systems, a rapid, simple, but reasonably accurate method is 
(required. The basis for the method which has been evolved for Bell 

System use will be described in the following pages. 

7.1. The "Peaked" Character of Overflow Traffic 

The difficulty in predicting the load-service relationship in alternate 
route systems has lain in the non-random character of the traffic over- 
flowing a first set of paths to which calls may have been randomly 
offered. This non-randomness is a well appreciated phenomenon among 
traffic engineers. If adecjuate trunks are provided for accommodating 
the momentary traffic peaks, the time-call level diagram may appear 
as in Fig. 11(a), (average level of 9.5 erlangs). If however a more limited 
j number of trunks, say a: = 12, is provided, the peaks of Fig. 11(a) will be 
Ichpped, and the overflow calls will either be "lost" or they may be 
j handled on a subsequent set of paths y. The momentary loads seen on 2/ 
then appear as in Fig. 11(b). It will readily be seen that a given average 
i load on the y trunks will have quite different fluctuation characteristics 
i than if it had been found on the x trunks. There will be more occurrences 
of large numbers of calls, and also longer intervals when few or no calls 
are present. This gives rise to the expression that overflow traffic is 
"peaked." 

Peaked traffic requires more paths than does random traffic to operate 
at a specified grade of delayed or lost calls service. And the increase in 
paths required will depend upon the degree of peakedness of the traffic 
involved. A measure of peakedness of overflow traffic is then required 
which can be easily determined from a knowledge of the load offered and 
the number of trunks in the group immediately available. 

In 1923, G. W. Kendrick, then with the American Telephone and 
I Telegraph Company, undertook to solve the graded multiple problem 
■through an application of Erlang's statistical eciuilibrium method. His 
i principal contribution (in an unpublished memorandum) was to set up 
I the equations for describing the existence of calls on a full access group 
\oi X -{- y paths, arranged so that arriving calls always seek service first 
iu the .T-group, and then in the ^/-group when the x are all busy. 

Let f{m, n) be the probability that at a random instant m calls exist 
j on the x paths and n calls on the y paths, when an average Poisson load 



444 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



of a erlangs is submitted to the x -\- y paths. The general state equation 
for all possible call arrangements, is 



(a* + m + n)f{m, n) = (w + l)/(m + 1, n) 

+ (n + l)/(m, w + 1) + ajim — 1, n) + aj{x, n — 1)% 



(7) 



in which the term marked {%) is to be included only when m = x, and 
* indicates that the a in this term is to be omitted when in -\- n = x -{- y. 
m and n may take values only in the intervals, -^ m ^ x;Q -^ n -^ y. 
As written, the equation represents the "lost calls cleared" situation. 



(a) RANDOM TRAFFIC 




10 00 AM 



< I 

if) Q. 



a. 



2 
to 



10 00 A M 



10 30 
TIME OF DAY 



(b) PEAKED OVERFLOW TRAFFIC 









PI 




-^ 



























10 30 
TIME OF DAY 



Fig. 11 — Production of peakedness in overflow traffic. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 445 

By choosing x -]- y large compared with the submitted load a a "lost 
calls held" situation or infinite-overflow-trunks result can be approached 
as closely as desired. 

Kendrick suggested solving the series of simultaneous equations (7) by 
determinants, and also by a method of continued fractions. However 
little of this numerical work was actually undertaken until several years 
later. 

Early in 1935 Miss E. V. Wyckoff of Bell Telephone Laboratories be- 
came interested in the solution of the (x -\- 1)(^/ + 1) lost calls cleared 
simultaneous equations leading to all terms in the /(m, n) distribution. 
She devised an order of substituting one equation in the next which pro- 
vided an entirely practical and relatively rapid means for the numerical 
solution of almost any set of these equations. By this method a con- 
siderable number of /(m, n) distributions on x, y type multiples with 
varying load levels were calculated. 

From the complete m, n matrix of probabilities, one easily obtains the 
distribution 9m{n) of overflow calls when exactly m are present on the 
lower group of x trunks; or by summing on m, the d{n) distribution with- 
out regard to m, is realized. A number of other procedures for obtaining 
the/(m, n) values have been proposed. All involve lengthy computations, 
very tedious for solution by desk calculating machines, and most do not 
have the ready checks of the WyckofT-method available at regular points 
through the calculations. 

In 1937 Kosten^ gave the following expression for /(m, n) : 

/(», n) = (- l)V.fe) i (i) M^- "f^'l., (8) 



i=0 



(Pi^l{x)ipi(x) 



where 



(po{x) = 



x^—a 

a e 



xl 



; and for i > 0, 




;=o \ J / (.^• - J)i 



These equations, too, are laborious to calculate if the load and num- 
1 K^rs of trunks are not small. It would, of course, be possible to program a 
modern automatic computer to do this work with considerable rapidity. 

The corresponding application of the statistical equilibrium equations 
to the graded multiple problem was visualized by Kendrick who, how- 
ever, went only so far as to write out the equation for the three-trunk 



446 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

case consisting of two subgroups of one trunk each and one common 
overflow trunk. 

Instead of solving the enormously elaborate system of equations de- 
scribing all the calls which could simultaneously be present in a large 
multiple, several ingenious methods of convoluting the 

X 

6(n) = Z/(w, n) 

overflow distributions from the individual legs of a graded multiple have 
been devised. For example, for the multiple of Fig. 10(a), the probability 
of loss Pi as seen by a call entering subgroup number i, is approximately, 

Pi = 2 £ e.Ar)-rl^{z -r) +J: d.Ar) (9) 

r=0 z=y T—y 

in which \l/{z — r) is the probability of exactly z — r overflow calls being 
present, or wanting to be present, on the alternate route from all the 
subgroups except the zth, and with no regard for the numbers of calls 
present in these subgroups. The ^x,i(^) = jiixi , r) term, of course, con- 
templates all paths in the particular originating call's subgroup being 
occupied, forcing the new call arriving in subgroup i to advance to the 
alternate route. This corresponds to the method of solving graded mul- 
tiples developed by E. C. Molina^ but has the advantage of overcoming 
the artificial "no holes in the multiple" assumption which he made. 
Similar calculating procedures have been suggested by Kosten.* These 
computational methods doubtless yield useful estimates of the resulting 
service, and for the limited numbers of multiple arrangements which 
might occur in within-office switching trains (particularly ones of a sym- 
metrical variety) such procedures might be practicable. But it would be 
far too laborious to obtain the individual overflow distributions Q{n), 
and then convolute them for the large variety of loads and multiple 
arrangements expected to be met in toll alternate routing. 

7.2. Approximate Description of the Character of Overflow Traffic 

It was natural that various approximate procedures should be tried in 
the attempt to obtain solutions to the general loss formula sufficiently 
accurate for engineering and study purposes. The most ol^vious of these 
is to calculate the lower moments or semi-invariants of the loads over- 
flowing th(; sul)groups, and from them construct approximate fitting 

* Kosten gives the above approximation (9), which he calls Wb^, Jis an upper 
limit to the blocking. He also gives a lower limit , Wr, in which z = // throughout 
(References 4, 5). 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 447 

I distributions for 6{n) mid dx(;n). Since each such overflow is independent 
I of the others, they may be combined additively (or convokited), to ob- 
[tain the corresponchng total distribution of calls appearing before the 
, I alternate route (or common group) . It may further be possible to obtain 
I [an approximate fitting distribution to the sum-distribution of the over- 
flow calls. 

The ordinary moments about the point of the subgroup overflow 
distribution, when m of the x paths are busy, are found by 

V 

ta'im) = 2 njim, n) (10) 

When an infinite number of |/-paths is assumed, the resulting expres- 
sions for the mean and variance are found to be:* 
Number of x-paths busy unspecified :'\ 

Mean = a = a-Ei,^{a) (11) 

Variance = v = a[l — a -{- a{x -\- I -\- a — a)'^] (12) 

All x-paths occupiedi 

Mean = a^^ = a[x - a + 1 -\- aEiMf^ (13) 

Variance = v^ = ax[l — ax + 2a(x + 2 + a^ — a)~^] (14) 

Equations (11) and (12) have been calculated for considerable ranges 
1 of offered load a and paths x. Figs. 12 and 13 are graphs of these results. 
i For example when a load of 4 erlangs is submitted to 5 paths, the aver- 
I age overflow load is seen to be a = 0.80 erlang, the same value, of 
I course, as determined through a direct application of the Erlang Ei 
formula. During the time that all x paths are busy, however, the over- 
flow load wdll tend to exceed this general level as indicated by the value 
of ax = 1.41 erlangs calculated from (13). Similarly the variance of the 
overflow load will tend to increase when the x-paths are fully occupied, 

* The derivation of these equations is given in Appendix I. 
t The skewness factor may also be of interest : 






ilz 



l^i: 



3/2 



^" + "-"^"' +a^ (15) 



x+1 +a- a \x + 2\{x-a)'^^-2{x-a) + x + 2 + {x^-2-a)a 

+ 3(1 -a) I + a(l - a)(l - 2a) 



o 



K:i' \ 



. . . t i > . 


wm 


Mm 

^ 


' \'' '^ 


'mV \ 




I ■ . \ 


m 





\ 





















^. 








\ 










\ 











q 




o 

6 





















|r ly\v\\ 



\ 




. . \ 



















• ^ \ 






\ 






\ 






■ ■■ \, 


















r- 


'\ 


iD 




o 


'^ 0) '^ 


* \ 




« 










, 
























\ 








\ 



\ 
\ 





o 



































































\ 























































F?^ 



\ 




\ 




X, 




a 























v^ 








X 


V^ 


'S 


























■f 




'x^ 




















^^ 




ro 


















"^ 














■ ■■■^ 












■^ 










































."-^.^ 














"■\ 
















^^N 
















"^ 

























































































































z 
< 

_] 
a: 

LU 



q: 

LU 



a ■ 

< i 

- o : 



< ) 

ir ■ 

> ■; 

< . 



lO <o 



o 



o r» lO 'f ro (M 



in ^f 
6 6 



6 



6 



o 
6 



o 
6 



o o 



o 



o 



o o o o 



o 
6 



Re) ''''3'e=1 s^Nviaa Ni'sHivd x ONissvd avon 39Vd3A\f'=» 

448 



o 

in 



6 










\ 


\ 
\ 




\ 










, o 



'N 



CO 



IHH t t W 



It 
1 



\' , 


\ 


V 


'\ , 


■\^ 


\ 


■A' 


• ^ \ 



\ 




\ 


\ 




\, 











(M ' 



II 



\ 



\ 



\ 








\ 




^,^ 








\ 





































o 






T3 


\ 


« 








Lh 








O 




^ 

n 




c 


\, 


o3 






If) 


1^ 








o 


'^i 




"X 




z 










< 

_1 


O 






o 


a. 


T— 1 


■ 


<^ 


2 


r-< 






■~ 


+3 








Q 

OJ 

cr 




o 






\ 




00 


UJ 
LL 
IL 

o 


O 




X 




< 

O 


^ 




X 


o 








_] 


q3 






ID 
OJ 


IIJ 
0) 

< 

LU 








> 

o 








o 


■■,,. 










II 


<o 








bO 










a> 






> 

< 














(M 




1 






t\J 




T— 1 


X 




o 




bJQ 






S 




^x. 













X ■ v>r.m,^Mt«.f,.i.,sxrrfri 



o o o o 



~ p 6 6 6 d d 

S5NVla3 Nl 'SHlVd X ONISSVd QVOl 30Va3AV = » 



449 




SHlVd X ONISSVd avOI dO 30NVIHVA = A 



450 



o 



in 
6 







■ ^^ 










1:1 

i:, 






!• 
























> o o 





X 



o 
ro 



o 



\ 



X 



\ 



\ 



\ 



o 



\- 




\ 




' \ 


\ 




s 


























V 




■■V. 






V 



x 



N 



\ 



\ 















\ 






\ 











X 








\ 

















\ . 








\ 








^ 


V • 



X 



\ 



CM 
ro in 

o 
z 
< 

_l 
a. 
oi 

o 

n z 

Q 
lU 

cc 

ico li- 
ifvj o 

i a 
< 

o 

_i 



UJ 

(M ^ 

CK 

> 
< 



o 



■n \ 



\ 




', 












' 





II 



o t-- vo n r^ 

SHiVd X ONISSVd avO"l dO 30NVIbVA=A 

451 



q r~; m ^ n 

- 6 6 6 6 



6 



o 

bO 

C 



O 



O 



O 

> 

o 



o 

a 
•I— I 



1— t 
bi) 



452 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

as shown by ?; = 1.30, and Vx = 1.95. In all cases the variances v and Vx 
will exceed the variance of corresponding Poisson traffic (which would 
have variances of a and ax respectively). 

7.2.1. A Prohahility Distribution for Overflow Traffic 

It would be of interest to be able, given the first several descriptive 
parameters of any traffic load (such as the mean and variance and skew- 
ness factors of the overflow from a group of trunks), to construct an 
approximate probability distribution d{n) which would closely describe 
the true momentary distribution of simultaneous calls. Any proposed 
fitting distribution for the overflow from random traffic offered to x 
trunks, can, of course, be compared with . ^ 

X 

determined from (7) or (8). 

Suitable fitting curves should give probabilities for all possitive in- 
tegral values of the variable (including zero) , and have sufficient unspeci- 
fied constants to accommodate the parameters selected for describing 
the distribution. Moreover, the higher moments of a fitting distribution 
should not diverge too radically from those of the true distribution ; that 
is, the "natural shapes" of fitting and true distributions should be simi- 
lar. Particularly desirable would be a fitting distribution form derived 
with some attention to the physical circumstances causing the ebb and 
flow of calls in an overflow situation. The following argument and der- 
ivation undertake to achieve these desiderata.* 

A Poisson distribution of offered traffic is produced by a random arrival 
of calls. The assumption is made or implied that the probability of a new 
arrival in the next instant of time is quite independent of the number 
currently present in the system. When this randomness (and correspond- 
ing independence) are disturbed the resulting distribution will no longer 
be Poisson. The first important deviation from the Poisson would be 
expected to appear in a change from variance = mean, to variance ^ 



* A two-parameter function which has the ability to fit quite well a wide variety 
of true overflow distributions, has the form 

t(n) = Kin + l)''e-^(''+i) 

in which K is the normalizing constant. The distribution is displaced one unit 
from the usual discrete generalized exponential form, so that ^(0) 9^ 0. The ex- 
pression, however, has little rationale for being selected a priori as a suitable 
fitting function. 



I 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 453 

mean. Corresponding changes in the higher moments would also be 
expected. 

WTiat would be the physical description of a cause system with a vari- 
ance smaller or larger than the Poisson? If the variance is smaller, there 
must be forces at work which retard the call arrival rate as the number 
of calls recently offered exceeds a normal, or average, figure, and which 
increase the arrival rate when the number recently arrived falls below 
the normal level. Conversely, the variance will exceed the Poisson's 
.should the tendencies of the forces be reversed.* This last is, in fact, a 
rough description of the incidence rates for calls overflowing a group of 
trunks. 

Since holding times are attached to and extend from the call arrival 
instants, calls are enabled to project their influence into the future; that 
is, the presence of a considerable number of calls in a system at any in- 
stant reflects their having arrived in recent earlier time, and now can be 
used to modify the current rate of call arrival. 

Let the probability of a call originating in a short interval of time dt be 

Po.n = [a + (n — a)co(n)] dt 

where n = number of calls present in the system at time t, 

a = base or average arrival rate of calls per unit time, and 
w(n) = an arbitrary function which regulates the modification in 
call origination rate as the number of calls rises above 
or falls below a. 
Correspondingly, let the probability that one of n calls will end in the 
short interval of time dt be 

which will be satisfied in the case of exponential call holding times, with 
mean unity. Following the usual Erlang procedure, the general statistical 
equilibrium equation is 



(16) 



Jin) = /(n)[(l - Po.n){l - Pe,n)\ + /(« " l)Po,n-l(l " Pe.n-l) 

-Vj{n+ 1)(1 - Po.„+i)P,,„+i 
which gives 

(Po,„ + P.,„)/(n) = Po,«-i/(n - 1) + Pe,n+xKn + 1) 

i ignoring terms of order higher than the first in dt. 

* The same thinking lias been used by Vaiilot^ for decreasing the call arrival 
I rate according to the number momentaril}^ present; and by Lundquist^ for both 
increasing and decreasing the arrival rate. 



454 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



(17) 



Or, 

[a + (n — a)w(?i) + ??.]/(n) 

= [a + (n -a- l)co(n - l)]/(/^ - 1) + (n + l)/(7i + 1) 

The choice of aj(n) will determine the solution of (17). Most simply, 
co(n) = k, making the variation from the average call arrival rate directly- 
proportional to the deviation in numbers of calls present from their 
average number. In this case, the solution for an unlimited trunk group 
becomes, with a' = a{l — k), 

a (a + k) -■■ [a + {n - 1)A;] 

fin) = 



n! 



^^ , , a' (a' + k) , a' (a' + k)(a' + 2k) , 
1 + « H ^t; H ^ TT, + 



(18) 



2! ' 3! 

which may also be written after setting a" = a'/k = a(l — k)/k, as 

a'ia' + 1) • • • [a" + (n - 1)]A;" 
fin) = 



n! 



(19); 



(1 - k)- 
The generating function (g.f.) of (19) is 



Z/(n)r = 



(1 - kT)-"" 



n=0 (1 - k)--" 

which is recognized as that for the negative binomial, as distinguished 
from the g.f., 

P 



(i + ? tX 



(1/g)^ 

for the positive binomial. 

The first four descriptive parameters of /(w) are: 



Order 


Moment about Mean 


Descriptive Parameter 


1 


Ml 


= 


Mean = n = a 


(20) 


2 


M2 


= variance, v = a/(l — k) 


Std Devn, <r = [a/(l - fc)]'/2 


(21) 


3 


f^a 


a(l + k) 
(1 - fc)^ 


Skewness, \/sT — — , , 


(22) 


4 


M4 


3a2(l - A0 + a(fc2+4A; + l) 


M4 A;2 + 4fc + 1 

Kurtosis, /3., = - = 3 + —7^ ry- 

o-* a(l — k) 


(23) 


(1 - fc)3 



I 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 455 

Since only two constants, a and k, need specification in (18) or (19), 
the mean and variance are sufficient to fix the distribution. That is, with 
the mean /7 and variance v known, 

a = ,7 or a' = n(l - k) = if/v, or a" = n(l - k)/k (24) 

A: = 1 - a/y = 1 - n/v. (25) 

The probability density distribution f(n) is readily calculated from 
(19); the cumulative distribution G(^n) also may be found through use 
of the Incomplete Beta Function tables since 

G(^n) = hi7i - l,a") 

(26) 
= h(n - l,a(l - k)/k) 

The goodness with which the negative binomial of (19) fits actual dis- 
tributions of overflow calls requires some investigation. Perhaps a more 
elaborate expression for co(n) than a constant k in (17) is required. Three 
comparisons appear possible: (1), comparison with a variety of 0«(n) 
distributions with exactly m calls on the x trunks, or d{n) with m unspeci- 
fied, (obtained by solving the statistical equilibrium equations (7) for a 
divided group) ; (2), comparison with simulation or "throwdown" results; 
and (3), comparison with call distributions seen on actual trunk groups. 
These are most easily performed in the order listed.* 

Co7nparison of Negative Binomial with True Overflow Distributions 

Figs. 14 to 17 show various comparisons of the negative binomial dis- 
tribution with true overflow distributions. Fig. 14 gives in cumulative 
form the cases of 5 erlangs offered to 1, 2, 5, and 10 trunks. The true 



j = n 



distributions (shown as solid lines) are obtained by solving the difference 
equations (7) in the manner described in Section 7.1. The negative bi- 
nomial distributions (shown dashed) are chosen to have the same mean 
and variance as the several F{^n) cases fitted. The dots shown on 



* Comparison could also be made after equating means and variances respec- 
tively, between the higher moments of the overflow traffic beyond x trunks and 
the corresponding negative binomial moments: e.g., the skewness given by (15) 
can be compared with the negative binomial skewness of (22). The difficulty here 
is that one is unable to judge whether the disparity between the two distribution 
functions as described by differences in their higher parameters is significant or 
not for traffic engineering purposes. 



456 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



the figure are for random (Poisson) traffic having the same mean values 
as the /'' distributions. The negative binomial provides excellent fits 
down to cumulated probabilities of 0.01, with a tendency thereafter to 
give somewhat larger values than the true ones. The Poisson agreement 
is good only for the overflow from a single trunk, as might have been 
anticipated, the divergence rapidly increasing thereafter. 

Fig. 15 corresponds with the cases of Fig. 14 except that the true over- 
flow Fxi^n) distributions for the conditional situation of all .r-paths 
busy, are fitted. Again the negative binomial is seen to give a good agree- 
ment down to 0.01 probability, with somewhat too-high estimates for 
larger values of the simultaneous overflow calls n. 

Fig. 16 shows additional comparisons of overflow and negative bi- 
nomial distributions. As before, the agreement is quite satisfactory to 
0.01 probability, the negative binomial thereafter tending to give some- 
what high values. 

On Fig. 17 are compared the individual 6(n) density distributions for 
several cases. The agreement of the negative binomial with the true 
distribution is seen to be uniformly good. The dots indicate the random 
(Poisson) individual term distribution corresponding to the a = 9.6 case- 



1.0 


"T*^ 


;J-^ 


— 




TRUE DISTRIBUTION 




\ 


^^^^^\- 


_ 





NEGATIVE BINOMIAL 




\ 




<^ 


• 

\ 


FITTING DISTRIBUTION 

CORRESPONDING 
RANDOM TRAFFIC 


0.1 


-\ 


\ 


> 


v 


\ 




_ \ 


• \ 




•\ 


\ 


n) 


\ 


\ ^ 


\ 




» \\ \ 






\ • 


\ 




^ V \ 


0.01 


- 


\ 




V5 


• v\ 






\s:=io 




\\ 


\\ \ 




. 


^ 






V \> V 






• \ 


• 




\ ^^ \ n^ 


0.001 










_J M \ i 1 \ l> \V 1 1 



t 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
n = NUMBER OF SIMULTANEOUS CALLS 

Fig. 14 — Probability distributions of overflow traffic with 5 erlangs offered to 
1, 2, 5, and 10 trunks, fitted by negative binomial. 



I 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 457 

the agreement, of course, is poor since the non-randomness of the over- 
flow here is marked, having an average of 1.88 and a variance of 3.84. 

Comparison of Negative Binomial with Overflow Distributions Observed 
hi/ llirowdoivns and on Actual Trunk Groups 

Fig. 18 shows a comparison of the negative binomial with the over- 
How distributions from four direct groups as seen in throwdown studies, 
'ilie agreement over the range of group sizes from one to fifteen trunks is 
seen to be excellent. The assumption of randomness (Poisson) as shown 
by the dot values is clearly unsatisfactory for overflows beyond more 
than two or three trunks. 

A number of switch counts made on the final group of an operating 
toll alternate routing system at Newark, New Jersey, during periods 
when few calls were lost, have also shown good agreement with the neg- 
ative binomial distribution. 

7.2.2. A Probability Distribution for Combined Overflow Traffic Loads 

It has been shown in Section 7.2.1 that, at least for load ranges of wide 
interest, the negative binomial with but two parameters, chosen to agree 



Fx(§n) 



0.01 



0.001 



TION 

OMIAL 
BUTION 




I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
n= NUMBER OF SIMULTANEOUS CALLS 



Fig. 15 — Probability distributions of overflow traffic with 5 erlangs offered to 
1, 2, 5, and 10 trunivs, when all trunks are busy; fitted by negative binomial. 



458 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



with mean and variance, gives a satisfactory jfit to the distribution of 
traffic overflowing a group of trunks. It is now possible, of course, to 
convohite the various overflows from any number of groups of varying 
sizes, to obtain a combined overflow distribution. This procedure, how- 
ever, would be very clumsy and laborious since at each switching point 
in the toll alternate route system an entirely difl"erent layout of loads and 
high usage groups would require solution; it would be unfeasible for 
practical working. 

We return again to the method of moments. Since the overflows of 
the several high usage groups will, in general, be independent of one 
another, the iih semi-invariants Xi of the individual overflows can be 
combined to give the corresponding semi-invariants A, of their total, 



Ai — iXi + 2X1 + 



(27) 



Or, in terms of the overflow means and variances, the corresponding 
parameters of the combined loads are 

Average = A' = ai -{- az + ■ ■ ■ (28) 

Variance = V = vi + V2 + • • - (29) 



TRUE DISTRIBUTION 



NEGATIVE BINOMIAL 

FITTING DISTRIBUTION 




0.001 



2 3 4 5 6 7 8 9 10 II 12 13 14 15 
n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 16 — Probability distributions of overflow traffic: 3 erlangs offered to 2 
trunks, and 9.6 erlangs offered to 10 trunks. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 459 

With the mean and variance of the combined overflows now deter- 
mined, the negative binomial can again be employed to give an approxi- 
mate description of the distribution of the simultaneous calls (p{z) offered 
to the common, or alternate, group. 

The acceptability of this procedure can be tested in various ways. One 
way is to examine whether the convolution of several negative binomials 
(representing overflows from individual groups) is sufficiently well fitted 
by another negative binomial with appropriate mean and variance, as 
found above. 

It can easily be shown that the convolution of several negative bi- 
nomials all with the same over-dispersion (variance-to-mean ratio) but 
not necessarily the same mean, is again a negative binomial. Shown in 
Table I are the distribution components and their parameters of two 
examples in which the over-dispersion parameters are not identical. The 
third and fourth semi-invariants of the fitted and fitting distributions, are 
seen to diverge considerably, as do the Pearsonian skewness and kurtosis 
factors. The test of acceptability for traffic fluctuation description comes 
in comparing the fitted and fitting distributions which are shown on 
Fig. 19. Here it is seen that, despite what might appear alarming dis- 



0(n) 



0.01 



O.OOI 



TRUE DISTRIBUTION 



NEGATIVE BINOMIAL 

FITTING DISTRIBUTION 

• RANDOM TRAFFIC, 8=1.9 




a = 9.6 

= 3.84 



I 2 3 4 5 6 7 8 9 10 II 12 

n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 17 — Probability density distributions of overflow traffic from 10 trunks, 
fitted by negative binomial. 



460 



THE BELL SYSTEM TECHNICAL JOUENAL, MARCH 1956 



parities in the higher semi-invariants, the agreement for practical traffic 
purposes is very good indeed. 

Numerous throwdown checks confirm that the negative binomial em- 
ploying the calculated sum-overflow mean and variance has a wide range 
over which the fit is quite satisfactory for traffic description purposes. 
Fig. 20 shows three such trunking arrangements selected from a con- 
siderable number which have been studied by the simulation method. 
Approximate!}^ 5,000, 3,500, and 580 calls were run through in the three 
examples, respective!}' . Tlie overflow parameters obtained !)y experiment 
are seen to agree reasonably well with the theoretical ones from (28) 
and (29) when the numbers of calls processed is considered. 

On Fig. 21 are sliown, for the first arrangement of Fig. 20, distributions 
of simultaneous offered calls in each subgroup of trunks compared with 
the corresponding Poisson; the agreement is satisfactory as was to be 
expected. The sum distribution of the overflows from the eight subgroups 
is given at the foot of the figure. The superposed Poisson, of course, is a 
poor fit; the negative binomial, on the other hand, appears quite accept- 
able as a fitting curve. 



1.0 
0.8 
0.6 



P 2n 



1 TRUNK- a = \.22 



3 TRUNKS- a = 2.24 



0.4 - 



0.2 ■ 



1.0 



0.8 - 



0.6 
0.4 
0.2 



234501 234 

n=NUMBER OF SIMULTANEOUS CALLS 





THEORY 


OBSD 


V\ 


( ) 


( ) 




AVG 0.67 


0.63 




VAR 0.77 


0.60 




i • RANDOM TRAFFIC 




\, a = 0.67 





THEORY OBSD 

c- ) ( — 1 


u 


AVG 0.55 


0.51 




VAR 0.77 


0.63 


\\ 


• RANDOM 


TRAFFIC 




a= 


D.55 




v^^ 








P^n 



1.0 


15 TRUNKS- a 
\ THEORY 


= 11.46 
OBSD '-O 




.\ ( H 


( ) 


0.8 


*\ AVG 0.81 


0.80 '-'•® 




'A VAR 1.88 


1.42 


0.6 


"\\, • RANDOM TRAFFIC °-^ 




\l a=o.8i 




0.4 




0.4 


0.2 




0.2 





• ^'^v,.^^^^ 


_ , n 



9 TRUNKS- a = 6.21 

THEORY OBSD 
( -) ( ) 

AVG 0.52 0.46 

VAR 1.00 1.48 

. RANDOM TRAFFIC 
a = 0.52 




4 68 10 024 68 

n=NUMBER OF SIMULTANEOUS CALLS 



10 



12 



Fifj;. 18 — Ovorflow (li.-<t ril)utioiis from diroct interoffice trunk groups; negative 
binomial theory versus thrgwclowji observations. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 461 

Table I — Comparison of Parameters of a Fitting 

Negative Binomial to the Convolution of 

Three Negative Binomials 



Example No. 1 


Example No. 2 


Component 


Component 


parameters 


Component 
dist'n No. 


Component parameters 


dist'n No. 


Mean 


Variance 


Mean 


Variance 


1 


5 


5 


1 


1 


1 


2 


2 


4 


2 


2 


3 


3 


1 


3 


3 


2 


6 




8 


12 




5 


10 



Semi-Invariants A, Skewness \/pi , and Kurtosis ^2 , of Sum Distributions 



Parameter 


E.xact 


Fitting 


Parameter 


Exact 


Fitting 


Ai 


8 


8 


Ai 


5 


5 


Ao 


12 


12 


A2 


10 


10 


As 


32 


24 


As 


37 


30 


A4 


168 


66 


A4 


239.5 


130 


VFi 


0.770 


0.577 


V/3i 


1.170 


0.949 


/32 


4.167 


3.458 


/32 


5.395 


4.300 



Fig. 22 shows the corresponding comparisons of the overflow loads in 
the other two trunk arrangements of Fig. 20. Again good agreement 
with the negative binomial is seen. 



7.3. Equivalent Random Theory for Prediction of Amount of Traffic Over- 
flowing a Single Stage Alternate Route, and Its Character, with Lost 
Calls Cleared 

As discussed in Section 7.2, when random traffic is offered to a limited 
number of trunks x, the overflow traffic is well described (at least for 
traffic engineering purposes) by the two parameters, mean a and variance 
V. The result can readily be applied to a group divided (in one's mind) 
two or more times as in Fig. 23. 

Employing the a and v curves of Figs. 12 and 13, and the appropriate 
numbers of trunks a;i , Xi + 0:2 , and Xi + X2 + x^ , the pairs of descrip- 
tive parameters, ai , vi , ao , vo and a-s , v-a can be read at once. It is clear 
then that if at some point in a straight multiple a traffic with parameters 
ai , Vi is seen, and it is offered to .r2 paths, the overflow therefrom will 
have the characteristics 012 , vo . To estimate the particular values of a-y 
and v-i , one would first determine the values of the equivalent random 



462 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



P5n 




P^n 




CONVOLUTION OF 3 NEGATIVE BINOMIAL 
VARIABLES WITH PARAMETERS: 

AVG WR 

1 I 

2 3 
2 6 

, -FITTING NEGATIVE BINOMIAL 



6 8 10 12 

n= NUMBER OF CALLS PRESENT 



I 



-I I l_^ 



14 



16 



Fig. 19 — Fitting sums of negative binomial variables with a negative binomial. 



traffic a and trunks .Ti which would have produced ai and Vi . Then pro- 
ceeding in the forward direction, using a and Xi + X2 , one consults the 
a and v charts to find txi and Vz . Thus, within the limitations of straight 
group traffic flow, the character (mean and variance) of any overflow 
load from x trunks can be predicted if the character (mean and variance) 
of the load submitted to them is known. 

Curves could be constructed in the manner just described by which the 
overflow's a' and v' are estimated from a load, a and v, offered to x trunks. 
An illustrative fragment of such curves is shown in Appendix II, with an 
example of their application in the calculation of a straight trunk group 
loss by considering the successive overflows from each trunk as the 
offered loads to the next. 

Enough, perhaps, has been shown in Section 7.2 of the generally ex- 
cellent descriptions of a variety of non-random traffic loads obtainable 
by the use of only the two parameters a and v, to make one strongly 
suspect that most of the fluctuation information needed for traffic engi- 
neering purposes is contained in those two values. If this is, in fact, the 
case, we should then be able to predict the overflow a', v' from x trunks 






THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 463 

\\ith an offered load a, v which has arisen in any manner of overflow from 
earlier high usage groups, as illustrated in Fig. 24. 

This is found to be the case, as will be illustrated in several studies de- 
scribed in the balance of this section. In the determination of the charac- 
teristics of the overflow traffic a', v' in the cases of non-full-access groups, 
such as Figs. 24(b) and 24(c), the equivalent straight group is visualized 
[Fig. 24(a)], and the Eciuivalent Random load A and trunks S are found.* 
I Using A, and *S + C, to enter the a and v curves of Figs. 12 and 13, a 
, and v' are readily determined. To facilitate the reading of .1 and S, Fig. 
25 1 and Fig. 26 f (which latter enlarges the lower left corner of Fig. 25) 
have been drawn. Since, in general, a and v will not have come from a 
simple straight group, as in Fig. 24(a), it is not to be expected that *S, 

OVERFLOW THEORY OBSD 

AVERAGE 5.76 5.98 

VARIANCE 12.37 14.89 

= = _ = OST N0.1 



t t 

13.16 1024 


f 
1024 


t t 
10.18 9.22 


t t 
7.63 7.48 


0.76 ERLANGS 


OVERFLOW 
AVERAGE 
VARIANCE 




THEORY 
5.02 
9.95 


OBSD 
5.06 
7.90 




^^ 




~ ^ 


— 


OST N0.6 



t t t t t t t 

OFFER 10.66 3.24 2.44 11.46 9.81 9.59 1.42 ERLANGS 

OVERFLOW THEORY OBSD 

AVERAGE 2.83 2.87 

VARIANCE 3.35 3.34 



OST N0.14 



t t t t 1 1 t 

OFFER 2.52 1.08 0.94 0.94 0.59 1.13 0.85 ERLANGS 

Fig. 20 — Comparison of joint-overflow parameters; theory versus throwdown. 



* A somewhat similar method, commonly identified with the British Post 
Office, which uses one parameter, has been employed for solving symmetrical 
graded multiples (Ref. 9). 

t Figs. 25 and 26 will be found in the envelope on the inside back cover. 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



GROUP N0.1 

17 TRUNKS, a = 13.t6 




GROUP NO. 2 
14 TRUNKS, a: 




10 15 20 5 10 15 

r = NUMBER OF SIMULTANEOUS CALLS OFFERED TO THE DIRECT TRUNKS 



0.15 



0.10 



f(r) 



ao5 



I- 
z 

ui 

BC 

a. 



Ill 

S 
1- 



z 
o 

o 

o 
a. 
a. 



10 



GROUP NO. 3 

13 TRUNKS, a = 10.24 




q: 
< 


1- A GROUP NO. 5 

A\^^ 12 TRUNKS, a= 9.22 


10 0.10 

11 f(r) 

O 0.05 


//v. 


c 


y^ ^x^\_^ 


a 





20 



0.20 


r GROUP NO. 7 




/\ 10 TRUNKS, a = 748 


0.15 


/7X\ 


f(r) 0.10 


/ nk 


0.05 


yy \v 





---^r ^^^^^C::^— -^ 



GROUP N0.4 ' 
14 TRUNKS,; 




GROUP N0.6 
10 TRUNKS, i 



8 10 12 14 16 18 




0.15 r 



0.10 



F(n) 



0.05 



DISTRIBUTION OF OVERFLOW CALLS FROM 8 DIR 
GROUPS OFFERED TO 1ST ALTERNATE ROUTE 




THEORY 


OBSD 


AVG 5.76 


5.98 


VAR 12.4 


14.9 


THROWDOWN OBSNS 




NEGATIVE BINOMIAL 




-^^^^l!!^^^^^>^ 





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 

n = NUMBER OF SIMULTANEOUS CALLS OFFERED TO THE ALTERNATE ROUTE 



17 



Fig. 21 — Comparison of theoretical and throwdown dis(ril)utions of simul- 
taneous calls offered to direct groups and to tlieir first alternate route (OST No. 1). 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 465 

read from Fig. 25, will be an integer. This causes no trouble and S should 
be carried along fractionally to the extent of the accuracy of result de- 
sired. Reading *S' to one-tenth of a trunk will usually be found sufficient 
for traffic engineering purposes. 

Example 1: Suppose a simple graded multiple has three trunks in each 
of two subgroups, which overflow to C common trunks, where C = 1, 



P^n 




OST NO. 6 

THEORY OBSD 

AVG 5.02 5.06 

VAR 9.95 7.90 

• RANDOM TRAFFIC, a = 5.0 

-OBSD 

-NEGATIVE BINOMIAL 

2 4 6 8 10 12 14 16 18 

n = NUMBER OF SIMULTANEOUS CALLS 



P?n 




--OBSD 



OST N0.14 

THEORY OBSD 

( ) ( ) 

AVG 2.83 2.87 

VAR 3.35 3,34 

RANDOM TRAFFIC, a = 2.8 



-NEGATIVE BINOMIAL 



2 4 6 8 10 12 14 16 18 

n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 22 — Combined overflow loads off'ered to alternate-route OST trunks from 
lirect interoffice trunks; negative binomial theory vs throwdown observations. 



t«3. 



V, 



ta2,y 



2y^2 



f a,,i 



Fig. 23 — A full access group divided at several points to examine the traffic 
character at each point. 



466 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



2 or 3. A load of a erlangs is submitted to each subgroup, a having the 
values 1, 2, 3, 4 or 5. What grade of service will be given? 

Solution: The load overflowing each subgroup, when a = 1 for example, 
has the characteristics a = 0.0625 and y = 0.0790. Then A' = 2a = 0.125 
and V — 2v = 0.158. Reading on Fig. 26 gives the Ecjuivalent Random 
values oi A = 1.04 erlangs, S = 2.55 trunks. Reading on Fig. 12.1 with 
C + *S = 3.55 when C = 1, and A = 1.04, we find a' = 0.0350 and 
oi' liflx + a-^ = 0.0175. We construct Table II in which loss values pre- 
dicted by the Equivalent Random (ER) Theory are given in columns 
(3), (5) and (7). For comparison, the corresponding exact values given 
by Neovius* are sho\vn in columns (2), (4) and (6). (Less exact loss 



s 

(OR X) 



(a) 



ta,v 



(b) 

fa'.v 



ta,v 



(c) 
fa'.V 




|A fa, f; 



la, faafaa 134*35* J 



Fig. 24 — Various high usage trunk group arrangements producing the same 
total overflow a, v. 



figures were given previously by Conny Palm^°. The agreement is seen 
to be excellent for engineering needs for all values in the table. 

Example 2: Suppose in Fig. 24(b) the random offered loads and paths 
are as given in Table III; we desire the proportion of overflow and the 
overflow load characteristics from an alternate route of 5 trunks. 

Solution: The individual overflows ai , vi ; a^ , v-i ; and as , Vz are read 
from Figs. 12 and 13 and recorded in columns (4) and (5) of the table. 
The a and v columns are totalled to obtain the sum-overflow average A' 
and variance V . The Equivalent Random load A which, if submitted to 
S trunks would produce overflow A', V , is found from Fig. 26. Finally, 
with A submitted io S -\- C trunks the characteristics a' and y', of the 
load overflowing the C trunks are found. The numerical values obtained 

* Artificial Traffic Trials Using Digital Computers, a paper presented by G. 
Neovius at the First International Congress on the Application of the Theory of 
Probability on Telephone Engineering and Administration, Copenhagen, June, 
1955. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 467 



Table II^ — Calculation of Loss in a Simple Graded Multiple 
g = 2, Xi = X2 = S, ai = a2 = a = 1 to 5, C = 1 to 3 



T nafl Submitted to each 


Proportion of Each Subgroup Load which Overflows 

= a'/(.ai + ai) 


Subgroup in Erlangs 
a 


C = 1 


C = 2 


C = 3 




True 


ER 


True 


ER 


True 


ER 


(1) 
1 

2 
3 
5 
5 


(2) 

0.01737 
0.11548 
0.24566 
0.35935 
0.44920 


(3) 

0.0175 

0.115 

0.246 

0.363 

0.445 


(4) 

0.00396 
0.05630 
0.16399 
0.27705 
0.37336 


(5) 

0.0045 

0.057 

0.163 

0.279 

0.370 


(6) 
0.00077 

0.02438 
0.10212 
0.20535 
0.30308 


(7) 

0.00088 

0.024 

0.103 

0.210 

0.305 



for this example are shown in the lower section of Table III. As before, 
of course, the "lost" calls are assumed cleared, and do not reappear in 
the system. 

Example 3: A load of 18 erlangs is offered through four groups of 
10-point selector switches to twenty- two trunks which have been desig- 
nated as "high usage" paths in an alternate route plan. Which of the 
trunk arrangements shown in Fig. 27 is to be preferred, and to what 
extent? 

Solution: By successive applications of the Equivalent Random 
method the overflow percentages for each of the three trunk arrange- 
ments are determined. The results are shown in column 2 of Table IV. 
The difference in percentage overflow between the three trunk plans is 
small; however, plan 2 is slightly superior followed by plans 3 and 1 in 



Table III — Calculation of Overflows from a Simple 
Alternate Route Trunk Arrangement 



Subgroup 
Number 


Offered Load in 

Erlangs 

a 


Number of Trunks 

X 


Overflow Loads 


a 


V 


1 
2 
3 




3.5 
5.7 
6.0 

15.2 


3 
6 
9 


1.41 
1.39 
0.45 

3.25 


1.98 
2.40 
0.85 

5.23 



Description of load offered to alternate route: A' = 3.25, V = 5.23. 
]'"quivalent straight multiple: S = 5.8 trunks, A = 8.00 erlangs (from Fig. 26). 
Overflow from C = 5 alternate route trunks (enter Figs. 12 and 13 with A = 

8.0 and S + C = 10.8: a' = 0.72, v' = 1.48. 
Proportion of load to commons which overflows = 0.72/3.25 = 0.22. 
Proportion of offered load which overflows = 0.72/15.2 = 0.0475. 



468 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



PROPORTION OVERFLOWING 


N0.1 E.R THEORY NEOVIUS THROWDOWNS 


f — ^- • • • • 

—m- • . • . 

A = 18 <^ 

l^ — •- • • • » 


[ 1 




BESK PUNCHED 
1 CARDS 






■ ■ 




» 


-*- 0.123 0.118 0.114 


NO. 2 


A = 18 < 


fr: : : 1 1 1 
l~: : : n I 




■ 


1 


-^0.113 0.110 0.110 


N0.3 


f"*' ■ 1 1 n 1 1 




-»-0.118 0.113 0.111 


l::::imii 




' 



Fig. 27 — Comparison of losses on three graded arrangements of 22 trunks. 

that order. The results of extensive simulations made by Neovius on the 
three trunk plans are available for comparison.* The values so obtained 
are seen to be very close to the ER theoretical ones ; moreover the same 
order of preference among the three plans is indicated and with closely 
similar loss differentials between them. 

7.3.1. Throwdown Comparisons with Equivalent Random Theonj on 
Simple Alternate Routing Arrangements with Lost Calls Cleared 

Results of manuallj' run throwdowns on a considerable number of 
non-symmetrical single-stage alternate route arrangements are available. 
Some of these were shown in Fig. 20; they represent part of a projected 
multi-alternate route layout (to be described later) for outgoing calls 
from the local No. 1 crossbar Murray Hill-6 office in New York to all 
other offices in the metropolitan area. The paths hunted over initially are 
called direct trunks; they overflow calls to Office Selector Tandem (GST) 
groups, numbered from 1 to 17, which are located in widely dispersed 
central office buildings in the Greater New York area. 



* Loc. cit. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 469 

Table IV — Loss Comparison of Graded Arrangements 





Estimates of Percentage of Load Overflowing 


Plan Number 


ER Theory 


Neovius Throwdowns 




BESK Computer 

(262144 calls) 


Punched Cards 
(10,000 calls) 


(1) 
1 
2 
3 


(2) 
12.3 
11.3 
11.8 


(3) 
11.81 

10.98 
11.25 


(4) 
11.4 
11.0 
11.1 



Table V — Comparison of Theory and Throwdowns for the 

Parameters of Loads Overflowing the Common Trunks 

in Single-Stage Graded Multiples 



OST (Alternate) 
Route Group 


No. of 
Groups of 


Total No. 
of Trunks 


Total Load Offered to 
Direct Trunks 


Total Overflow 


Load from OST 


















Group 


No. of 

trunks 


Direct 
Trunks 


in Direct 
Groups 


Erlangs 


Approximate 
No. of Calls 


Theory 


Throwdown 


no. 




















(in 2.7 hours) 


a' 


v' 


a' 


v' 


1 


6 


8 


91 


68.91 


4950 


2.00 


5.50 


2.36 


6.52 


2 


3 


3 


45 


37.49 


2690 


2.10 


5.60 


2.05 


6.36 


3 


6 


6 


80 


60.62 


4355 


1.50 


4.00 


1.30 


5.67 


4 


3 


6 


52 


38.49 


2765 


2.30 


5.20 


2.08 


6.43 


5 


3 


3 


17 


12.51 


900 


0.45 


0.83 


0.49 


1.02 


6 


4 


7 


64 


48.62 


3490 


2.50 


5.90 


2.36 


4.88 


7 


8 


12 


78 


57.42 


4125 


2.20 


5.60 


1.71 


4.08 


8 


6 


9 


16 


12.96 


930 


0.82 


1.63 


0.81 


1.11 


9 


1 


2 


22 


16.96 


1220 


1.30 


2.60 


1.02 


1.73 


10 


5 


6 


10 


9.52 


685 


0.78 


1.40 


1.05 


2.07 


11 


8 


13 


16 


16.43 


1180 


1.90 


3.80 


2.77 


7.29 


12 


8 


9 


2 


6.88 


495 


0.70 


1.30 


0.81 


1.83 


13 


5 


15 


33 


21.42 


1540 


1.75 


3.30 


1.16 


2.01 


14 


2 


7 


11 


8.05 


580 


1.46 


2.20 


1.63 


2.14 


15 


9 


15 


8 


11.97 


860 


1.60 


3.25 


1.55 


4.12 


16 


11 


22 


34 


27.46 


1970 


1.75 


4.00 


1.34 


2.26 


17 


3 


7 


4 


5.81 


420 


1.53 


2.31 


1.43 


1.80 




26.64 


58.42 


25.92 


61.32 



In Table V are given certain descriptive data for the 17 OST trunk 
arrangements showing numbers of legs of direct trunks, total direct 
trunks, the offered erlangs and calls, and the mean and variance of the 
alternate routes' overfiovvs, as obtained by the ER theory and by 
throwdowns.* The throwdown a' and v' values of the OST overflow 



* Additional details of this simulation study are given in Section 7.4. 



470 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



i.a 



^O 0.2 



EQUIVALENT 
RANDOM THEORY 




ERLANG THEORY- 



1 2 3 4 5 6 7 8 9 10 It 12 13 14 15 16 17 
ALTERNATE ROUTE (OST) NUMBER 

Fig. 28 — Comparison of theoretical and throwdown overflows from a number 
of first alternate routes. 

were obtained by 36-second switch counts of those calls from each OST 
group which had come to rest on subsequent alternate routes. 

On Fig. 28 is shown a summary of the observed and calculated pro- 
portions of "lost" to "offered" traffic at each OST alternate route group. 
As may be seen from the figure and the last four columns of Table V, 
the general agreement is quite good ; the individual group variations are 
probably no more than to be expected in a simulation of this magnitude. 

An assumption of randomness (which has sometimes been argued as 
returning when several overflows are combined) for the load offered to 
the OST's gives the Erlang Ei loss curve on Fig. 28. This, as was to be 
expected, rather consistently understates the loss. 

Since "switch-counts" were made on the calls overflowing each OST, 
the distributions of these overflows may be compared with those esti- 
mated by the Negative Binomial theory having the mean and variance 
predicted abo\'e for the overflow. Fig. 29 shows the individual and cumu- 
lative probability distributions of the overflow simultaneous calls from 
the first two OST alternate routes. As will be seen, the agreement is 
quite good even though this is traffic which has been twice "non-ran- 
domized." Comparison of the observed and calculated overflow means 
and variances in Table V indicates that similar agreement between 
observed and theoretical fitting distributions for most of the other OST's 
would be found. 



7.3.2. Comparison of Equivalent Random Theory with Field Results on 
Simple Alternate Routing Arrangements _ 

Data were made available to the author from certain measurements 
made in 1941 by his colleague C. Clos on the automatic alternate routing 
trunk arrangement in operation in the Murray Hill-2 central office in 
New York. Mr. Clos observed for one busy hour the load carried on 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 471 

several of its OST alternate rovite groups (similar to those shown in 
Table V for the Murray Hill-6 office, but not identical) by means of an 
electromechanical switch-counter having a six-second cycle. During 
each hour's observation, numbers of calls offered and overflowing were 
also recorded. 

Although the loads offered to the corresponding direct trunks which 
()^'erflowed to the OST group under observation were not simultaneously 
measured, such measiu'ements had been made previously for several 
hours so that the relative contribution from each direct group was 
closely known. In this way the loads offered to each direct group which 
produced the total arriving before each OST group could be estimated 
with considerable assurance. From these direct group loads the character 
(mean and variance) of the traffic offered to and overflowing the OST's 
was predicted. The observed proportion of offered traffic which over- 
flowed is shown on Fig. 30 along with the Equivalent Random theory 
prediction. The general agreement is again seen to be fairly good al- 
though with some tendency for the ER theory to predict higher than 
observed losses in the lower loss ranges; perhaps the disparity on in- 



(n) 



0.5 
0.4 

0.3 

0.2 

0.1 




OST N0.1 

THEORY OBSD 



AVG 
VAR 



2.00 2.36 

5.50 6.52 




RANDOM TRAFFIC 

^--NEGATIVE BINOMIAL 
-THROWDOWN 



OST NO. 2 

THEORY OBSD 



AVG 
VAR 



2.10 
5.60 




2.05 
6.36 



>RAND0M TRAFFIC 



THROWDOWN 

-NEGATIVE BINOMIAL 



10 15 5 

n = NUMBER OF SIMULTANEOUS CALLS 



15 



p^n 



-NEGATIVE BINOMIAL 




-THROWDOWN 



-NEGATIVE BINOMIAL 




THROWDOWN 



10 15 5 

n = NUMBER OF SIMULTANEOUS CALLS 



15 



Fig. 29 — Distributions of loads overflowing from first alternate (OST) groups; 
negative binomial theory versus throwdown observations. 



472 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 195G 



dividual OST groups is within the limits one might expect for data 
based on single-hour observations and for which the magnitudes of the 
direct group offered loads required some estimation. The assumption of 
random traffic offered to the OST gives, as anticipated, loss predictions 
(Erlang £"1) consistently below those observed. 

More recently extensive field tests have been conducted on a working 
toll automatic alternate route system at Newark, New Jersey. High 
usage groups to seven distant large cities o\'erflowed calls to the New- 
ark-Pittsburgh alternate (final) route. Data describing the high usage 
groups and typical system busy hoiu- loads are given in Table ^T. (The 
loads, of course, varied considerably from day to day.) The size of the 
Pittsburgh route varied over the six weeks of the 1955 tests from 64 to 
71 trunks. Altogether the system comprised some 255 intertoll trunks. 

Observations were made at the Newark end of the groups by means 
of a Traffic Usage Recorder — making switch counts every 100 seconds 
— and by peg count and o^'erflow registers. Register readings were photo- 
graphically recorded by half-hourly, or more frequent, intervals. To 



^- 

<a 
uz 

^^ 
zz 

05 

1-0 

(T-l 
°^ 



1.0 



0.5 



0.2 



2 0.1 



0.05 



0.02 



0.01 



- 




- 




- 


Z' 


- 


^,^!^ 1^ f^-^^ 








U^ — - "^ 




Jft 




NON-RANDOM (ER) //'' 




THEORY "^xX /' 




y^ Jrf -OBSERVED 




X ^-i^sM ' 


_ 


X ^____— --sss:^*^^'^^^ / 


- 


/j^ ' ^^ ^'^-RANDOM THEORY 








^ 1 yJ 








1 




/ 




J 




^ 




• 




• 


11 





(\j 


— 


^ 


ro 


tn 


n 


— 


— 


^ 




d 








d 


d 














TANDEM 


z 


Z 


Z 


z 


z 


z 


Z 


Z 


Z 


OFFICE 





<£> 


r^ 


t: 








ro 





CI 




(^ 


m 


O) 




ro 


Q 




Q 


n 




LU 


UJ 


UJ 


LU 


LU 


CD 


LU 


(D 


03 


NO. TRUNKS 


13 


12 


8 


7 


3 


8 


3 


4 


3 


OFFERED JavG 


7.55 


7.19 


5.22 


3.81 


2.06 


7.79 


2.36 


4.09 


2.4 



LOAD |vAR 13.58 15.66 6.59 7.30 2.51 18.54 2.77 4.59 5.90 



Fig. 30 — -Observed tandem ovciflow.s in nlicriKilc 
llill-2 (New York) 1940-1941. 



loulc study at Murray 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 473 



Table VI — High Usage Groups and Typical System 

Busy Hour Loads 



High Usage Group, 
Newark to: 


Length of Direct Route 
(Air Miles) 


Nominal Size of Group 
(Number of Trunks) 


Typical Offered Load 
(erlangs) 


Baltimore 


170 

560 

395 

1375 

470 

1100 

1170 


18 

42 
27 
33 
37 
26 
5 


19 


Cincinnati 

Cleveland 

Dallas 

Detroit 

Kansas City 

New Orleans 


43 
26 
34 
36 
23 
4 



compare theory with the observed overflow from the final route, esti- 
mates of the offered load A' and its ^-ariance V are required. In the 
present case, the total load offered to the final route in each hour was 
estimated as 

A' = Average of Offered Load 

Peg Count of Calls Offered 

to Pittsburgh Group 



(Peg Count of Offered Calls) 
— (Peg Count of Overflow Calls) 



X Average Load Carried 

by Pittsburgh Group 



The variance V of the total load offered to the final route was estimated 
for each hour as 

V' = Variance of Offered Load 

7 7 

= A' — 2 «i + 2 Vi 



i=l 



where «» and Vi are, respectively, the average and variance of the load 
overflowing from the tth high usage group. (The expression. A' — 

7 

^ «i , is an estimate of the average — and, therefore of the variance 
1=1 

— of the first-routed traffic offered directly to the final route. Thus the 
total variance, V, is taken as the sum of the direct and overflow com- 
ponents.) Using A', V and the actual number, C, of final route trunks in 
service, the proportion of offered calls expected to overfloAv was calcu- 
lated for the traffic and trunk conditions seen for 25 system busy hours 
from February 17 to April 1, 1955 on the Pittsburgh route. The results 
are displayed on Fig. 31, where certain traffic data on each hour are 
given in the lower part of the figure. The hours are ordered — for con- 
venience in plotting and viewing — by ascending proportions of calls 
overflowing the group; observed results are shown by the double line 



474 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




0.001 



3 5 7 9 11 13 15 17 19 21 23 25 

NO. P'BGH TRKS 71 70 65 71 65 71 65 69 64 64 70 65 64 71 68 65 64 65 64 70 65 65 65 65 65 



HOURS BY AMT. 
OF OBS'D loss 



EST'd LOAD fAVG. 50 54 55 56 55 63 55 58 54 54 68 60 63 74 76 74 76 83 91 102 109105 101 115124 
OFFERED War. 82 95 85 89 98 101 84 98 97 89110 10588125 121140114 141 175182 170 176 179 199197 



Fig. 31 — Final route (Newark-Pittsburgh) overflows in 1955 toll alternate^ 
route study. I 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 475 

curve. The superposed single line is the corresponding estimate by EE, 
theory of the hour-to-hour call losses. As may be seen, theory and ob- 
servation are in good agreement both point by point and on the average 
over the range of losses from 0.01 to 0.50. The dashed line shows the 
prediction of final route loss for each hour on the assumption that the 
offered traffic A' was random. Such an assumption gives consistently low 
estimates of the existing true loss. 

As of interest, a series of heavy dots is included on Fig. 31. These are 
the result of calculating the Poisson Summation, P{C,L), where L is the 
average load carried on, rather than offered to, the C trunks. It is inter- 
esting that just as in earlier studies in this paper on straight groups of 
intertoll trunks (for example as seen on Fig. 7), the Poisson Summation 
with load carried taken as the load offered parameter, gives loss values 
surprisingly close to those observed. Also, as before, this summation has 
a tendency to give too-great losses at light loadings of a group and too- 
small losses at the heavier loadings. 

; 7.4 Prediction of Traffic Passing Through a Midti-Stage Alternate Route 

Network 

I In the contemplated American automatic toll switching plan, wide 
I advantage is expected to be taken of the efficiency gains available in 
i multi-alternate routing. Thus any procedure for traffic analysis and 
prediction needs to be adaptable for the . more complex multi-stage 
arrangements as well as the simpler single-stage ones so far examined. 
Extension of the Equivalent Random theory to successive overflows is 
easily done since the characterizing parameters, average and variance, 
of the load overflowing a group of paths are ahvays available. 

Since few cases of more than single-stage automatic alternate routing 
are yet in operation in the American toll plant, it is not readily possible 
to check an extension of the theoiy with actual field data. Moreover col- 
lecting and analyzing observations on a large operating multi-alternate 
route system would be a comparatively formidable experiment. 

However, in New York city's local interoffice trunking there is a very 
considerable development of multi-alternate routing made possible by 
the flexibility of the marker arrangements in the No. 1 crossbar switching 
system. None of these overflow arrangements has been observed as a 
whole, simultaneously and in detail. The Murray Hill-2 data in OST 
groups reviewed in Section 7.3.2 were among the partial studies which 
have been made. 

In connection with studies made just prior to World War II on these 



476 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Table VII — Sum of Direct Group Overflow Loads, 

Offered to OST's 



Average. 
Variance 



Theory 



86.06 
129.5 



Observed 



87.12 
127.4 



local multi -alternate route systems, a throwdown was made in 1941 on a 
proposed trunk plan for the Murray Hill-6 office. The arrangement of : 
trunks is shown on Fig. 32. Three successive alternate routes, Office 
Selector Tandems (OST), Crossbar Tandem (XBT), and Suburban: 
Tandem (ST), are available to the large majority of the 123 direct trunk 
groups leading outward to 169 distant offices. (The remaining 46 parcels 
of traffic did not have direct trunks to distant offices but, as indicated 
on the diagram, offered their loads directly to a tandem group.) A total 
of 726 trunks is involved, carrying 475 erlangs of traffic. 

A throwdown of 34,001 offered calls corresponding to 2.7 hours of 
traffic was run. Calls had approximate exponential holding times, averag- 
ing 135 seconds. Records were kept of numbers of calls and the load from 
the traffic parcels offered to each direct group, as they were carried or 
passed beyond the groups of paths to which they had access. Loads car- 
ried by each trunk in the system were also observed by means of a 36- 
second "switch-count." (The results on the 17 OST groups reported in 
Section 7.3.1 were part of this study.) 

Comparisons of observation and theory which are of interest include 
the combined loads to and overflowing the 17 OST's. Observed versus 
calculated parameters (starting with theory from the original direct 
group submitted loads) are given in Table VII. The agreement is seen 
to be very good. 

The corresponding comparison of total load from all the OST's is 
given in Table VIII. Again the agreement is highly satisfactory. 

Not all of the overflow from the OST's was offered to the 22 crossbar 
tandem trunks; for economic reasons certain parcels by-passed XBT andf 
were sent directly to Suburban Tandem.* This posed the problem of 
breaking off certain portions of the overflow from the OST's, to be added"' 
again to the overflow from XBT. An estimate was needed of the contri 
bution made by each parcel of direct group traffic to any OST's over 
flow. These were taken as proportional to the loads offered the OST by 
each direct group (this assumes that each parcel suffers the same over- 



* In the toll alternate route system by -passing of this sort will not occur. 



Tt 



U'lnnt uunu u L 



\'^\\\\\\\'' \va 



p^^^^^ 



\\nii\^ 



S 



m 



T^ 



m: 



Tr± 



7 J '''.': ; , 



±±± 



^PP 



tf+Ff- 



4^ 



:« 



tn 



^ 



'/V//'/-'//./'/''///;^ : 



fl 



NO. 16 



NO. 17 



SPECIAL 
TANDEM 



tt inttttiiittttMitiftt ttinit tit t tttttttttt ttttt 

(V OJ ^O ^^»*^u-^^OsO r- u-^Ty fv^ O OJ fVOJ O^ rNvO TO-* -t OJ C^ V\vO (V-*fV<HrH OI^C^ »H -tTOTO _*(^-*«) -*CJtO <-ITOnO C*- -* 

^O r^ O -* J- -*rj r^ -j-joj rj rH O OO 0«> C^OiJ^<*\<*\fH TO^OC^OJU^r^ ^ -^ i/n f^ OtO to r^*rfc"~'-*r^r^rsi CT^^OJi-tiH 

OO -* r^ i-H ^ O rH ^ rH.-H .-H rH rH rHrH O O O OO O O O rH (-1 ^ O O O O r^rH O O OOOOOOOOOO OOOOO 

27.46 5.81 5.44 0.31 5.99 1.96 

tOvO -JvO rH fV f'^0^_JO>JD O^ r^rH (Vr- ifNvO »A(NJryNO -*fV (NiTO Of^Of^O^ -* ^ ^ -* (N* ONf^ "^tO C^ t*^C^ <^ *'^ C^CM^^TO 

>-i t£) E-« cr: tn CO ,_j ti3 < <i<i: tH w kJ »Jo- J i-H w hJ> a. i*; M <ow»H'J<-< mo:-* > M<fHO-<>JM*s<:w mmooo 

crossbar office. 



FINAL ,0 

TANDEM 
TRUNKS 

5 



INTERMEDIATE 
TANDEM 
TRUNKS 10 



FIRST 
ALTERNATE 5 ■ 
nUTE(OST) 
TRUNKS 



SUBURBAN TANDEM 




N0.1 N0.2 N0.3 N0.4 N0.5 N0.6 



NO. 7 



N0.8 N0.9 NO.IO 



N0.11 



N0.12 



N0.I3 



N0.14 



DIRECT 

INTEROFFICE 

TRUNKS 



15 r. 
10rzE 

5:E: 

1 --- 



m ill ]M \M\ Vw timii ttimlmtt tiitifti! ti flmi tttitttttiiit tttitttft tmmmntti tiiittt tmiitittittit tmtmitmitmtni imttt tii i ititnttit ftitt 



Y I k.ni.L' IMnu ■-! fv IV iH r>j >0 J t^ i-t^.H p^tov>r^s>r^ O (v rt o (N to mr^pj O cvj ^ -tto ''^-4 O rsi tT>C'tJ't~-t^C^NO&''">-i -* (m lAi-i r\ O O- r~ f^ 'O 

[ERLAN6S) f|JOOO£>r-t^O 'OcvtJ. rHC'O'OO-Ov i^-0~0,0'0 J ■/■OO Ot^fv-^Cr'ovrH O<0 t^i-^f^r^rufOcjpHu^ pJc-j^^^^^OcJ (> 






68.91 37.49 60.62 38.49 12.51 48.62 



57.42 



DESTINATION *Cr^-*t^r>^,^ r.^>r 

OFFICES S£S253S£ gS? 



~-»f«Ot- Jr" 



mff-tO toO^iT 



^rHrHO rH p- ^ -i f-i rt -J r4 -^ ,H ^d O -4 rH r4 O O O O O O «/^ (V cJ .-* O O O -- r+^ f"* O OO O (VrHodcHO ^ddOrHOOOOOOOOOO 

12.96 16.96 9.52 16.43 6.88 21.42 8.05 11.97 27.46 5.81 5.44 0.31 5.99 1. 



T Q Ita. tfc< C3 O Z OW < f) 






Fig. 32. — Multi-alternate route trunking arrangcinenl at Murray Hill — 6 (New York) local No. 1 crossbar office. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 



477 



flow probability). The variance of this overflow portion by-passing XBT 
was estimated by assigning to it the same variance-to-average ratio as 
was found for the total load overflowing the OST. Subtracting the means 
and variances so estimated for all items by-passing XBT, left an approxi- 
mate load for XBT from each OST. Combining these corrected overflows 
gave mean and variance values for offered load to XBT, Observed values 



Table VIII - 


-Sum of Loads Overflowing OST's 




Theory 


Observed 


Avftraere 


26.64 

58.42 


25.92 


Variance 


61.32 



Table IX — Load Offered to Crossbar Tandem 



I Average. 
Variance 



Theory 



25.18 
47.67 



Observed 



25.51 
56.10 



0.10 r 



-RANDOM TRAFFIC 



-THROWDOWN 




,--NEGATIVE BINOMIAL 



to 20 30 40 50 

n = NUMBER OF SIMULTANEOUS CALLS 



P^n 







THEORY 


OBSD 


1.0 


I — - — -^ 


.^^^ ( ) 


( ) 






^X AVG 25.18 


25.51 


0.8 




^V VAR 47.67 


56.10 


0.6 




VS. 




0.4 








0.2 











, 




, 



10 20 30 40 50 

n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 33 — Distribution of load offered to crossbar tandem trunks; negative bi- 
nomial theory versus throwdown observations. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 477 

flow probability). The variance of this overflow portion by-passing XBT 
was estimated by assigning to it the same variance-to-average ratio as 
was found for the total load overflowing the OST. Subtracting the means 
' and variances so estimated for all items by-passing XBT, left an approxi- 
mate load for XBT from each OST. Combining these corrected overflows 
gave mean and variance values for offered load to XBT, Observed values 



Table VIII - 


- Sum OF Loa 


Ds Overflowing OST's 




Theory 


Observed 


Average 

Variance 


26.64 

58.42 


25.92 
61.32 



Table IX 


. — Load Offered to Crossbar Tandem 




Theory 


Observed 


Average 


25.18 
47.67 


25.51 


Variance 




56.10 



0.10 r 



^-.--RANDOM TRAFFIC 
-THROWDOWN 




--NEGATIVE BINOMIAL 



10 20 30 40 50 

n = NUMBER OF SIMULTANEOUS CALLS 

THEORY OBSD 

( ) ( ) 

AVG 25.18 25.51 

VAR 47.67 56.10 



Pin 




10 20 30 40 50 

n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 33 — Distribution of load offered to crossbar tandem trunks; negative bi- 
nomial theory versus throwdown observations. 



478 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Table X — Load Overflowing Crossbar Tandem 



Average. 
Variance , 




Observed 



6.47 
33.48 



and those calculated (in the above manner) are given in Table IX. 
Fig. 33 shows the distribution of XBT offered loads, observed and calcu- 
lated. The agreement is very satisfactory. The random traffic (Poisson) 
distribution, is of course, considerably too narrow. 

In a manner exactly similar to previous cases, the Ecjuivalent Random 
load method was applied to the XBT group to obtain estimated param- 
eters of the traffic overflowing. Comparison of observation and theory 
at this point is given in Table X. 

Fig. 34 shows the corresponding observed and calculated distributions 



0.15 



0.10 



f(n) 



0.05 



)RANDOM TRAFFIC 




THEORY OBSD 

AVG 6.55 6.47 

VAR 23.80 33.48 



^'NEGATIVE BINOMIAL 



5 10 15 20 25 30 35 

n=NUMBER OF SIMULTANEOUS CALLS 



P^n 



_^^RANDOM TRAFFIC 
--NEGATIVE BINOMIAL 




THROWDOWN 



5 10 15 20 25 30 35 

n = NUMBER OF SIMULTANEOUS CALLS 



Fig. 34 — Distribution of calls from crossbar tandem trunks; negative binomial 
theory versus throwdown observations. 



! 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 



479 



of siniiiltaneoiis calls. The agreement again is reasonably good, in spite 
of the considerable disparity in variances. 

The overflow from XBT and the load which by-passed it, as well as 
some other miscellaneous parcels of traffic, were now combined for final 
offer to the Suburban Tandem group of 17 trunks. The comparison of 
parameters here is again available in Table XI. On Fig. 35 are shown 
the observed and calculated distributions of simultaneous calls for the 
load offered to the ST trunks. The agreement is once again seen to be 
very satisfactory. 

We now estimate the loss from the ST trunks for comparison with the 
actual 'proportion of calls which failed to find an idle path, and finally 

Table XI — Load Offered to Suburban Tandem 



Average. . 
Variance . 



Theory 



15.38 
42.06 



Observed 



14.52 
48.53 



THEORY OBSD 



f(n) 




P^n 



10 20 30 40 

n = NUMBER OF SIMULTANEOUS CALLS 



I.O 


^ ^ 


\ 








0.8 


" 


^ 


, --NEGATIVE 


BINOMIAL 




0.6 






V ^-THROWDOWN 










\ \ 






0.4 












0.2 






x^^^ 











1 




" -r-^ 





10 20 30 40 

n^NUMBER OF SIMULTANEOUS CALLS 



50 



Fig. 35 — Distribution of load offered to suburban tandem trunks; negative 
linomial theory versus throwdown observations. 



480 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Table XII - 


— Grade of Service on ST Group 




Theory 


Obser- 
vation 


Observation 


Load submitted (erlangs) 

Load overflowing (er- 
langs) 

Proportion load over- 
flowing 


15.38 
3.20 
0.209 


14.52 

2.63 
0.181 


Number of calls sub- 
mitted 1057 

Number of calls over- 
flowing 200 

Proportion of calls over- 
flowing 0.189 



Table XIII — Grade of Service on the System 



Total load submitted 

Total load overflowing 

Proportion of load not served 



Theory 



Observed 



475 erlangs 
3.20 erlangs 
0.00674 



34,001 calls 
200 calls 
0.00588 



compare the proportions of all traffic offered the system which failed to 
find a trunk immediately. See Tables XII and XIII. 

After these several and varied combinations of offered and overflowed 
loads to a system of one direct and three alternate routes it is seen that 'i 
the final prediction of amount of load finally lost beyond the ST trunks 
is gratifyingly close to that actually observed in the throwdown. The 
prediction of the system grade of service is, of course, correspondingly 
good. 

It is interesting in this connection to examine also the proportions I 
overflowing the ST group when summarized by parcels contributed from 
the several OST groups. The individual losses are shown on Fig. 36; they 
appear well in line with the variation one would expect from group to 
group with the moderate numbers of calls which progressed this far 
through the multiple. 



0.4 



0.3 



octr 
o^ 0.2 

So 
a ^0.1 



,-THEORY =0.21 
._>. 



• • 



--AVG OBSD = 0.19 



12 4 6 8 10 12 14 16 18 20 
OST GROUP NUMBER 

Fig. 36 — Overflow calls on third alternate (ST) route. 






THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 



481 



7.4.1 Correlation of Loss with Peakedness of Components of Non-Ran- 
dom Offered Traffic 

Common sense suggests that if several non-random parcels of traffic 
are combined, and their joint proportion of overflow from a trunk group 
is P, the parcels which contain the more peaked traffic should experience 
overflow proportions larger than P, and the smoother traffic an overflow 
proportion smaller than P. It is by no means clear however, a priori, the 
extent to which this would occur. One might conjecture that if any one 
parcel's contribution to the total combined load is small, its loss would 
be caused principally by the aggregate of calls from the other parcels, 
and consequently its own loss would be at about the general average loss 
P, and hence not very much determined by its own peakedness. The 
Murray Hill-6 throwdowai results may be examined in this respect. The 
mean and variance of each OST-parcel of traffic, for example, arriving 
at the final ST route was recorded, together with, as noted before, its 
own proportion of overflow from the ST trunks. The variance/mean over- 
dispersion ratio, used as a measure of peakedness, is plotted for each 
parcel of traffic against its proportion of loss on Fig. 37. There is an un- 
doubted, but only moderate, increase in proportion of overflow with 
increased peakedness in the offered loads. 

It is quite possible, however, that by recognizing the differences be- 
tween the service given various parcels of traffic, significant savings in 
final route trunks can be effected for certain combinations of loads and 
trunking arrangements. Of particular interest is the service given to a 
parcel of random traffic offered directly to the final route when compared 



04 

o 

oc_l 0.3 

UJUJ 

>u 
°% 

°ia2|- 

zo 

o< 

OO0.1 

o 

a. 






• • • 



0.5 1.0 1.5 2.0 2.5 3.0 3.5 

V/a OF EACH OST PARCEL REACHING ST TRUNKS 



Fig. 37 — Effect of peakedness on overflow of a parcel of traffic reaching an 
ilternate route. 



482 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

with that received by non-random parcels overflowing to it from high 
usage groups. 

7.5 Expected Loss on First Routed Traffic Offered to Final Route 

The congestion experienced by the first-routed traffic offered to the 
final group in a complex alternate route arrangement [such as the right 
hand parcels in Figs. 10(c) and (d)] \vill be the same as encountered in a 
series of random tests of the final route by an independent observer, 
that is, it will be the proportion of time that all of the final trunks are 
busy. As noted before, the distribution of simultaneous calls n (and hence 
the congestion) on the C final trunks produced by some specific arrange- 
ment of offered load and high usage trunks can be closely simulated by 
that due to a single Equivalent Random load offered to a straight group 
of aS -f C trunks. Then the proportion of time that the C trunks are 
busy in such an equivalent system provides an estimate of the corres- 
ponding time in the real system ; and this proportion should be approxi- 
mately the desired grade of service given the first routed traffic. 

Brockmeyer has given an expression (his equation 36) for the pro- 
portion of time, Rx , in a simple S -\- C system with random offer A, 
and "lost calls cleared," that all C trunks are busy, independent of the 
condition of the *S-trunks: 

R, = f{S,C,A) 

= Ii,x,s+cKA) — — 



where 



m=o \ m / (S — m 



However, (rdS) is usually calculated more readily step-by-step using 
the formula 

<Tc{S) = aciS - 1) -f CTc-liS) , 

starting with 

crc(O) = 1 and ao(S) = A^Sl 
The average load carried on the C paths is clearly 

Lc = A[Ei,sU) - Ei,s+c{A)], (31) 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 483 

and the variance of the carried load can be shown to be* 

Vc = ALc ^ - ACEx,s+c{A) + Lc- L\ (32) 

On Fig. 38, Ri values are shown in solid line curves for several com- 
binations of A and C over a small range of S trunks. The corresponding 
losses Ri for all traffic offered the final group, where R^ = oc'/A', are 
shown as broken curves on the same figure. The R2 values are always 
above Ri , agreeing with the common sense conclusion that a random 
component of traffic will receive better service than more peaked non- 
random components. 

However, there are evidently considerable areas where the loss differ- 
ence between the two Z^'s will not be large. In the loss range of principal 
interest, 0.01 to 0.10, there is less proportionate difference between the 
R's, as the A = C paired values increase on Fig. 38. For example, at 
/?2 = 0.05, and A = C = 10, R./Ri = 0.050/0.034 = 1.47; while for 
A = C = 30, i?2/Ri = 0.050/0.044 = 1.13. Similarly for A = 2C, the 
R2/R1 ratios are given in Table XIV. Again the rapid decrease in the 
R2/R1 ratio is notable as A and C increase. 

F. I. Tange of the Swedish Telephone Administration has performed 
elaborate simulation studies on a variety of semi-symmetrical alternate 
route arrangements, to test the disparity between the Ri and R2 types 
of losses on the final route. f For example if g high-usage groups of 8 
paths each, jointly overflow 2.0 erlangs to a final route which also serves 
2.0 erlangs of first routed traffic, Tange found the differences in losses 
between the two 2-erlang parcels, i?high usage (h.u.) —Ri, shown in 
column 9 of Table XV. The corresponding ER calculations are performed 
in columns 2 to 8, the last of which is comparable with the throwdown 
\alues of column 9. The agreement is not unreasonable considering the 
sensitiveness of determining the difference between two small prob- 
abilities of loss. A quite similar agreement was found for a variety of 
other loads and trunk arrangements. 



* In terms of the first two factorial moments of n : Vc is given by 

Vc = M(2) + M(i) - M(i)*, where Mw = Lc 

(leneral expressions Mu) for the factorial moments of n are derived in an unpub- 
lished memorandum by J. Riordan. 

t Optimal Use of Both-Way Circuits in Cases of Unlimited Availability, a 
paper by F. I. T&nge, presented at the First International Congress on the Appli- 
cation of the Theory of Probability in Telephone Engineering and Administration, 
June 1955, Copenhagen. 



484 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



1.0 
0.9 
0.8 
0.7 

0.6 

0.5 

0.4 
0.3 



0.2 



D 




O 




a. 




_( 


0.1 


< 

z 


0.09 


0.08 


II 




z 
o 


0.07 


0.06 


U1 

in 


0.05 


o 




_i 


0.04 


II 




o 




7- 


0.03 


O 




(- 




a. 
o 


0.02 


n 




o 




a 




a. 




ru 




a. 


0.01 


a 


0.009 


z 


0.008 


< 


0.007 


cr 


0.006 




0.005 



0.004 



0.003 



0.002 



0.001 



























































^•>_^ 














^^ 


;-.^ 












V 


'*^^>v 


•^ 
'N^ 












Y^x. 




X 
\ 

V 








vv 


\ \ 


\ \ 

\ \ 
\ \ 




A = 30 

C = 15 
s 






\s\^ 


\ 






.' ^ 






\\v 


\ 




n, -^ ^ 


f- \ 






>■ 


\ 


\ 


\ \ 


"S^ \ 






V 




\ \ 




\ \ 






\ 


[a 


\ ""') 


\ \ 




\ 




^ 


\X' ^"' 


o\ 


\ \ 


\ \ 


\ 






\ A?T 


iO \ 


\ \ 


\ \ 










R, \'C-. 














\ \ 




I A = 20^ 

rc = io^^ 


V \ 

\ \ 
\ \ 


\ 
\ 




\ \ 


\ \ \ 




\ 


\ 








\ \ \ 




\ 


\ 






\ 


\ ^ \ 




\ 


\ 






\ 






\ 


\ 


\ 




\ 


' \^ 




\ 


\ 


\ 

\ 




\ 


A \ 




\ 




V 




A = 10,^ 
C = io \ 




\ 


\ 




\ 














\ 

















10 



15 



20 25 

S = *equivalent"number of paths 



30 



35 



Fig. 38 — Comparison of Ri and R2 losses under various load and trunk con- 
ditions. 



Table XIV— The R2/R1 Ratios for A = 2C 


A 


C 


Ri/Ri when R2 = 0.05 


10 
20 
30 


5 
10 
15 


10.6 
3.25 

2.44 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 



485 



Table XV — Comparison of E.R. Theory and Throwdowns on 

Disparity of Loss Between High Usage Overflow and 

Random Offer to a Final Group 

(8 trunks in each high usage group; 9 final trunks serving 2.0 erlangs 

high usage overflow and 2.0 erlangs first routed traffic.) 



Number of 

Groups of 

8 High Usage 

Trunks 


ER Theory {A' = 4.0) 


Tange 


V 


A 


5 


R2=a7A' 


i?i 


Rh.u, ~ 
2R-L- Ri 


Rh.u.—Rl= 

2{R2 - Ry) 


Throwdown 
Rh.u. - Ri 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


(7) 


(8) 


(9) 


1 


5.77 


7.51 


4.17 


0.0375 


0.0251 


0.0499 


0.0248 


0.0180 


2 


5.80 


7.50 


4.25 


0.0383 


0.0255 


0.0511 


0.0256 


0.0247 


3 


5.74 


7.44 


4.08 


0.0369 


0.0248 


0.0490 


0.0242 


0.0286 


4 


5.68 


7.30 


3.91 


0.0362 


0.0247 


0.0477 


0.0230 


0.0276 


5 


5.64 


7.20 


3.80 


0.0355 


0.0242 


0.0468 


0.0226 


0.0245 


6 


5.58 


7.06 


3.64 


0.0350 


0.0240 


0.0460 


0.0220 


0.0221 


7 


5.55 


7.00 


3.56 


0.0345 


0.0238 


0.0452 


0.0204 


0.0202 


8 


5.51 


6.91 


3.45 


0.0335 


0.0236 


0.0434 


0.0198 


0.0188 


9 


5.47 


6.81 


3.34 


0.0325 


0.0231 


0.0419 


0.0188 


0.0177 


10 


5.45 


6.76 


3.29 


0.0312 


0.0225 


0.0399 


0.0174 


0.0166 



Limited data are available showing the disparity of Ri and Ro in 
actual operation in a range of load and trunk values well beyond those 
for which Ri values have been calculated. Special peg count and over- 
flow registers were installed for a time on the final route during the 1955 
Newark alternate route tests. These gave separate readings for the calls 
from high usage groups, and for the first routed Newark to Pittsburgh 
calls. Comparative losses for 17 hours of operation over a wide range of 
loadings are shown on Fig. 39. The numbers at each pair of points give 
the per cent of final route offered traffic which was first routed (random). 
In general, approximately equal amounts of the two types of traffic were 
offered. 

In 6 of the hours almost identical loss ratios were observed, in 7 hours 
the overflow-from-high-usage calls showed higher losses, and in 4 hours 
lower losses, than the corresponding first routed calls. The non-random 
calls clearly enjoyed practically as good service as the random calls. This 
result is not in disagreement with what one might expect from theory. 
To compare directly with the Newark-Pittsburgh case we should need 
curves on Fig. 38 expanded to correspond to A', V values of (50, 85) 
to (120, 200). Examining the mid-range case of C = 65, A' = 70, V = 
120, we find A = 123, >S = 54. Here A is approximately 2C; extrapolat- 
ing the A = 2C curves of Fig. 38 to these much higher values of A and C 
suggests that R2/R1 w^ould be but little different from unity. 



486 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

It is clear from the above theory, throwdowns, and actual observa- 
tion that there are certain areas where the service differences given first 
routed and high usage trunk overflow parcels of traffic are significant. 
In Section 8, where practical engineering methods are discussed, curves 
are presented which permit recognition of this fact in the determination 
of final trunk requirements. 

7.6 Load on Each Trunk, Particularly the Last Trunk, in a Non-Slipped 
Alternate Route 

In the engineering of alternate route systems it is necessary to deter- 
mine the point at which to limit a high usage group of trunks and send 
the overflow traffic via an alternate route. This is an economic problem 
whose solution requires an estimate of the load which will be carried on 



1.0 



0.5 



z 

o 

il' 0.05 
a. 

UJ 

> 

o 

z 
o 
I- 
cc 
o 
a. 
O 
a. 
a- O.OiO 



0.005 



0.00)0 



6 64 
56 8' 



57 



61( 



OL 65^ 69, 
40 



56 
o 



50 



,58 



41 



58 



69 

8 



64 



52 



s 

38 



6 
66 



49 

8 



52 



O FIRST ROUTED TRAFFIC (NUMBERS INDICATE PER 
CENT OF TOTAL WHICH IS FIRST ROUTED) 

• OVERFLOW TRAFFIC FROM 7 HIGH USAGE GROUPS 



60 70 80 90 100 110 120 

A'= ESTIMATED OFFERED LOAD TO PITTSBURGH IN ERLANGS (INCLUDING RETRIALS) 



Fig. 39 — Comparison of losses on final route (Newark to Pittsburgh) for high 
usage overflow and first routed traffic. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 487 

the last trunk of a straight high usage group of any specified size, carry- 
ing either first or higher choice traffic or a mixture thereof.* 

The Equivalent Random theory readily supplies estimates of the loads 
carried by any trunk in an alternate routing network. After having found 
the Equivalent Random load A offered to *S + C trunks which corresponds 
to the given parameters of the traffic offered to the C trunks, it is a simple 
matter to calculate the expected load i on any one of the C trunks if 
they are not slipped or reversed. The load on the ith trunk in a simple 
straight multiple (or the S + jth. in a divided multiple of *S lower and C 
upper trunks), is 

A- = Is+j = A[E^,s+j-M) - Ex,s+j{A)] (33) 

where Ei,n(A) is the Erlang loss formula. A moderate range of values of 
■Ci versus load A is given on Figure 40. f 

Using this method, selected comparisons of theoretical versus observed 
loads carried on particular trunks at various points in the Murray- 
Hill-6 throwdown are shown in Fig. 41 ; these include the loads on each 
of the trunks of the first two OST groups of Fig. 32, and on the second 
and third alternate routes, crossbar and suburban tandem, respectively. 
The agreement is seen to be fairly good, although at the tail end of the 
latter two groups the observed values drop aw^ay somewhat from the 
theoretical ones. There seems no explanation for this beyond the possi- 
bility that the throwdown load samples here are becoming small and 
might by chance have deviated this far from the true values (or the 
arbitrary breakdown of OST overflows into parcels offered to and by- 
passing XBT may well have introduced errors of sufficient amount to 
account for this disparity). As is well known, (33) gives good estimates 
of the loads carried by each trunk in a high usage group to which random 
(Poisson) traffic is offered; this relationship has long been used for the 
purpose in Bell System trunk engineering. 

8. PRACTICAL METHODS FOR ALTERNATE ROUTE ENGINEERING 

To reduce to practical use the theory so far presented for analysis of 
alternate route systems, working curves are needed incorporating the 



* The proper selection point will be where the circuit annual charge per erlang 
of traffic carried on the last trunk, is just equal to the annual charge per erlang 
of traffic carried by the longer (usually) alternate route enlarged to handle the 
overflow traffic. 

t A comprehensive table of /< is given by A. Jensen as Table IV in his book 
"Moe's Principle," Copenhagen, 1950; coverage is for / ^ 0.001 erlang, z = 1(1)140; 
A = 0.1(0.1)10, 10(1)50, 50(4)100. Note that n + 1, in Jensen's notation, equals i 
here. 



488 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




6 



<0 

6 



ID 

6 6 6 

soNvibB Ni viNnai Hi-n 3hi no agiaavD avon 



= '-Tf 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 



489 



pertinent load-loss relationships. The methods so far discussed, and 
proposed for use, will be briefly reviewed. 

The dimensioning of each high usage group of trunks is expected to be 
performed in the manner currently in use, as described in Section 7.6. 
The critical figure in this method is the load carried on the last high 
usage trunk, and is chosen so as to yield an economic division of the 
offered load between high usage and alternate route trunks. Fig. 40 is 
one form of load-on-each-trunk presentation suitable for choosing eco- 
nomic high usage group size once the permitted load on the last trunk 
is established. 

The character (average a and variance v) of the traffic overflowing 
each high usage group is easily found from Figs. 12 and 13 (or equivalent 



- OST GROUP NO. 2 



1.U 


OST GROUP NO.l 


0.5 


- 










^ 


















4 5 6 

TRUNK NUMBER 



CROSSBAR TANDEM GROUP 




Z 
O 



2 4 



6 8 10 12 14 16 18 20 22 

TRUNK NUMBER 



SUBURBAN TANDEM GROUP 




12 4 6 8 10 12 14 16 

TRUNK NUMBER 



Fig. 41 — Comparison of load carried by each alternate route trunk; theory 
versus throwdowns. 



490 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



tables). The respective sums of the overflow a's and v^s, give A' and V 
by (28) and (29); they provide the necessary statistical description of 
traffic offered to the alternate route. 

According to the Equivalent Random method for estimating the alter- 
nate route trunks required to provide a specified grade of service to the 
overflow traffic A', one next determines a random load A which when 
submitted to S trunks will yield an overflow with the same character 
{A', V) as that derived from the complex system's high usage groups. 
An alternate route of C trunks beyond these S trunks is then imagined. 
The erlang overflow a', with random offer A, to S + C trunks is found 
from standard i^i-formula tables or curves (such as Fig. 12). 

The ratio R2 = a! I A' is a first estimate of the grade of service given to 
each parcel of traffic offered to the alternate route. As discussed in Sec- 
tion 7.5, this service estimate, under certain conditions of load and 
trunk arrangement, may be significantly pessimistic when applied to a 
first routed parcel of traffic offered directly to the alternate route. An 
improved estimate of the overflow probability for such first routed 
traffic was found to be R\ as given by (30). 

8,1 Determination of Final Group Size with First Routed Traffic Offered 
Directly to the Final Group 

When first routed traffic is offered directly to the final group, its 
service Ri will nearly always be poorer than the overall service given to 
those other traffic parcels enjoying high usage groups. The first routed 
traffic's service will then be controlling in determining the final group 
size. Since Ri is a function of *S, C and A in the Equivalent Random 
solution (30), and there is a one-to-one correspondence of pairs of A and 
S values with A' and V values, engineering charts can be constructed at 
selected service levels Ri which shoAv the final route trunks C required, 
for any given values of A' and V. Figs. 42 to 45 show this relation at 
service levels of Ri = 0.01, 0.03, 0.05 and 0.10, respectively.* 

* On Fig. 42 (and also Figs. 46-49) the low numbered curves assume, atjfirst 
sight, surprising shapes, indicating that a load with given average and variance 
would require fewer trunks if the average were increased. This arises from the 
sensitivity of the tails of the distribution of offered calls, to the V'/A' peaked- 
ness ratio which, of course, decreases with increases in A'. For example, with C 
= 4 trunks and fixed V = 0.52, the loss rapidly decreases with increasing A': 



A' 


V'/A' 


A 


S 


a' 


a' /A' 


0.28 
0.33 
0.40 
0.52 


1.86 
1.58 
1.30 
1.00 


6.1 
3.0 
1.42 
0.52 


10. 
5.0 
2.03 



0.0155 
0.0081 
0.0036 
0.0008 


0.055 
0.025 
0.009 
0.002 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 491 

These four Ri levels would appear to cover the most used engineering 
range. For example, if the traffic offered to the final route (including the 
first routed traffic) has parameters A' = 12 and V ^ 20, reading on 
Fig. 43 indicates that to give P = 0.03 "lost calls cleared" service to 
the first routed traffic, C = 19 final route trunks should be provided. 
(For random traffic (F' = A' = 12), 17.8 trunks would be required.) 

Other charts, of course, might be constructed from which Ri could be 
read for specific values of A', V and C. They would become voluminous, 
however, if a wide range of all three variables were required. 

8.2 Provision of Trunks Individual to First Routed Traffic to Equalize 
Service 

If the difference between the service Ri given the first routed parcel of 
traffic and the service given all of the other parcels, is material, it may be 
desirable to take measures to diminish these inequities. This may readily 
be accomplished by setting aside a number of the otherwise full access 
final route trunks, for exclusive and first choice use of the first routed 
traffic. High usage groups are now provided for all parcels of traffic. The 
alternate route then services their combined overflow. The overall grade 
of service given the ith. parcel of offered traffic in a single stage alter- 
nate route system will then be approximately 

'* 
Pi = Ei,Xi{ai)R2 = EiXiia.)^, (34) 

Thus the service will tend to be uniform among the offered parcels when 
all send substantially identical proportions of their offered loads to the 
alternate route. And the natural provision of "individual" trunks for the 
exclusive use of the first routed traffic would be such that the same pro- 
portion should overflow as occurs in the associated high \isage groups. 

This procedure cannot be followed literally since high usage group 
size is fixed b}^ economic considerations rather than any predetermined 
overflow value. The resultant overflow proportions will commonly vary 
over a considerable range. In this circumstance it would appear reason- 
able to estimate the objective overflow proportion to be used in estab- 
lishing the individual group for the first routed traffic, as some weighted 
average h of the overflow proportions of the several high usage groups. 
Thus with weights g and overflow proportions h, 

h = ^'^' + ^'^' + • ' • (35) 

^1 + ^2+ • • • 



* Although not exact, this equation can probably be accepted for most engi- 
neering purposes where high usage trunks are provided for each parcel of traffic. 











\ 














\ 


•* 






















\ 












/ 


\ 










O-l SO 


o 













N 








■-^ 


/ 




\ 
















. 


V— 


k 










-^ 


/ 


\ 






a 




— ' ^^ 


GO 

eg 











. 




r— 


^ 










"< 


\ 


b 


<M ^ 


fc 









J _ 







^, 












^ 


^ 








>\ 





;;;;::; 


"^ 


. — 


— 


r"— 


b' 


q 


to o 
6 












si 


- ■ 


-— 




[ _^ 


— — ' 


\_^^ 










1 












rj 








\ 
























>. 




s; 


imp- 






























0<V ^ 


k^ 




>^ 


S 




\ 




























O/V 






\ 


































■*> 


^ 
^ 


\ 




































2\/ 


K 





en 


"t 


o 


(\J 


7 




< 




1 


(M 


Ct 


(\J 


UJ 




Z 


o 




(\j 


1- 




-) 




C) 


00 


tr 




1 




< 


ID 


z 





or 


Al 


Lll 




11 




11 




o 


O 


Q 




< 




() 




_l 


on 


UJ 




O 




< 




(r 




LU 




> 




< 




II 


^ 


< 



tn 


It 


t\J 


o 


to 


(C 


^f 


r\j 


O 


n 


ro 


n 


ri 


(M 


(\J 


OJ 


OJ 


i\J 



ainoH nvNid Oi aaaaddo avoi dO 30NviavA=,A 



- 








\ 














y\ 












0.5 1.0 












\ 














\ 






















\ 










. \ 




\ 






















\ 








"V 




A 


\ 






X 




^___ 


,^ 




^ 








\ 










^ 




^ 




><: 


-^ 










^ 




\ 












A. 


^ 








OJ 


s^ 








--- 


"^ 




^ 


-\ 


c 


3 




d 


O 








s 


OJ ^ 


sS 




I^ 


-^ 




h. 
























OJ 




>c 


.—- - 


^ 


-;:: 


--^ 


b^ 
























'""^o 


^^.; 


-N. 

^0*-, 


^ 

^v,^ 




s 


^ 


— ^ 


\ 


















































^ 


. .'-^ 


>X 
































— 




VL 


^ 


'< 


Xj 
































^ 



D 

o 
o cr 

< 



z 



fe '■ 



DO 



< 

o 

_1 

UJ 



o 
< 

q: 

<0 UJ 

> 
< 

II 



<o 


■^ 


(\J 


o 


<o 


<o 


■<t 


ry 


o 


ri 


n 


n 


n 


(M 


<\j 


<\j 


(\) 


(M 



3inoa ivNid oi aaaaddo avcn do aoNviMVA=,A 

492 







































' 


1 




\ 
























U1 1 


§ 








\ 


. 












\ 


<*) 








Q - 

in 
d 










\ 


s 










\, 




"X^^ 










\ 








- 


^ 
















\ 


c 












\ 


. 






^ 






\ 








m 
■v. 












\ 


? 














\ 








^ 


^ 




\~ 






;; 














\ 


X 




\ 


















^^ 










^ 




\ 




1. 


! 














A 


; 


















\ 












Ok 




^^^ 
















-^ 


^^ 




CO 












, jX. in o >n c 






^\^ i 








nv 




' 




r~- 


<c 












-\ 


k 




1 










^ 
























in 


^ 

^ 




1 j 
















\^ 




1 












'\r 


K> 


' 






^ 










1 


f\i' 




















"* ;. =^-^ 




— 


k 








J 




■^A, 1 2" 


■^^ 






















O/v 


Oi 


^ 


t^ 








1 


























(OV^ 




\ 


L._ . 


































' 


^f\10<D<0^f\JO(0<0^f>JO<0(£i^OJO 
rrionrvjruryrvirvj — — — ^ — 

ainoa ivNid oi asaaddo avoi dO 30NvidVA=,A 










\ 














\ 












q - 

in 
d 

o - 












\ 














\ 























\. 












^ 


\ 








i 

r 












' 




^ 








^ 






\ 














' 


. 




N 










^ 




^ 




! 




^1 


>^ 












. 


. 




\ 














\ 






^ 
( 


J ^^'^s,. 







— - 


, 






X U1 O "1 o 








o 
rvj 


2 


N 





' — ' 








^s 






















2 


<>1 


S^ 






— 




\ 
























^% 






>==:: 


— 


■ 




^ 










i 
















04., 




^ 





—^ 






























OA/ 




V; 


a^Vfc. 




^s 


































^ 


<. 


\ 




































^ 


k 


1 

1 
1 

1 

1 


^ 


J c 


> a 

t C 


3i 


5 r 


t 

J r 
VNId 


U C 
'J 

0± Q: 


5 a 
J 

)HBdd( 


D u 
D OVO 

4 


3 •< 

n do 
93 


i 1 

30NV 


U c 

avA = 


-A 


D U 


5 ■< 


X ^ 


J o 



Tf <n 

z 
< 

rvj 



LU 



i: Z 



c3 



3 
O 

m 

ki 

ml 
> 

bO 
O 

CO 

a 

3 

-1^ 



(\J 


q: 

LU 


t-i 




U- 


, , 


o 


LL 

o 
< 


C 

to 




o 


e^-i 




_l 


o 


0} 


UJ 


r-. 




o 


o • 




< 


•t;o 


10 


LU 


.2-^ 




> 


>o 




< 


o 


■f 


II 


1 i^ 


rvj 




"5 o 


O 




bD « 


(O 




o 


Cli 




E 


(O 






<\j 




-T3 

0) 


^ 


If) 
o 


4J 




/. 


o 




< 


» 


OJ 


_l 


■4-3 


ft 


tc 


LU 


!-. 




7 


«3 


o 




OJ 


on 


LU 


> 




1- 
-5 




00 


g 


O 




-u 




_| 


to 




< 


^ 


(D 


^i 


C 




Li- 


? 




P 


-fj 


■<t 


K 








<u 




U 


-tJ 








nj 


Ct 
lU 


o 




Li- 






u. 


,_^ 




o 


o3 


o 


n 


a 




< 

n 


«a 




_) 


o 


cu 


LU 
LO 






< 


o ^ 




Ct 


.-.IC 


<o 


LU 


.^O 




i, 


>o 




II 


y II 



Ph 



c^ 



^ 


Ct-H 


'^ 


O 


. 


n> 


u 


CJ 




• fH 


t^H 


> 




tH 




0) 




<u 



494 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

A choice of all weights g equal to unity will often be satisfactory for the 
present purpose. The desired high usage group size for the first routed 
traffic is then found from standard £'i-tables showing trunks x required, 
as a function of offered traffic a and proportion overflow b. 

Since the different parcels of traffic have varying proportions h of their ' 
loads overflowing to the final route, by equation (34) the parcel with 
the largest proportion will determine the permitted value of R2 . Thus ' 

R2 = P/&max (36) 

where P is the specified poorest overall service (say 0.03) for any parcel. 
It may be noted that on occasion some one parcel, perhaps a small one, 
may provide an outstandingly large bmax value, which will tend to give 
a considerably better than required service to all the major traffic 
parcels. Some compromise with a literal application of a fixed poorest 
service criterion may be indicated in such cases. 

An alternative and somewhat simpler procedure here is to use an 
average value b in (36) instead of ^max , with a compensating modifica- , 
tion of F, so that substantially the same R2 is obtained as before. The 
allowance in P will be influenced by the choice of weights g in (35). It 
will commonly be found in practice that overflow proportions to final 
groups for large parcels of traffic are lower than for small parcels. Choos- 
ing all weights, as unity, as opposed to weighting by traffic volumes for : 
example, tends to insert a small element of service protection for those , 
traffic parcels (often the smaller ones) with the higher prportionate high . 
usage group overflows. 

Having determined R2 , a ready means is needed for estimating the 
required number of final route trunks. Curves for this purpose are pro- 
vided on Figs. 46 to 49, within whose range, R2 = 0.01 to 0.10, it will 
usually be sufficiently accurate to interpolate for trunk engineering 
purposes. These F2-curves exactly parallel the i?i-curves for use when 
first routed traffic is offered directly to the final group without benefit 
of individual high usage trunks. If R2 is well, outside the charted range 
a run-through of the ER calculations may be required. 

8.3 Area in Which Significant Savings in Final Route Trunks are Realized 
by Allowing for the Preferred Service Given a First Routed Traffic Parcel 

Considerable effort has been expended by alternate route research 
workers in various countries to discover and evaluate those areas where 
first routed (random) traffic ofl'ered to a final route enjoj^s a substantial 
service advantage over competing parcels of traffic which have over- 



, 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 495 

flowed from high usage groups. A comparison of Figs. 42 to 45, (which 
indicate trunk provision for meeting a first routed traffic criterion Ri) 
with Figs. 46 to 49 (which indicate trunk provision for meeting a com- 
posite-load-offered-to-the-final-route criterion R2) gives a means for de- 
ciding under what conditions in practice it is important to distinguish 
between the two criteria. Fig. 50 shows the borders of areas, defined in 
terms of A' and V, the characterizing parameters of the total load 
offered to the final route, where a 2 and 5 per cent overprovision of final 
trunks would occur using R2 for Ri as the loss measure for first routed 
traffic. Thus in the alternate route examples displayed in Table XV, 
where x = S, g = 2 to 10, A' = 4.0 and V varies from 5.80 to 5.45, 
Fig. 50 shows that by failing to allow for the preferred position of the 2 
erlang first routed parcel, we should at R = 0.02 engineered loss, provide 
a little over 5 per cent more final trunks than necessary. (Actually 10.2 
and 9.9 versus 9.6 and 9.4 trunks f or gr = 2 and 10; respectively.) 

The curves of Fig. 50 for final route loads larger than a few erlangs, 
are almost straight lines. At an objective engineering base of i? = 0.03, 
for example, the 2 and 5 per cent trunk overprovision areas through 
using i?2 instead of Ri are outlined closely by: 

2 per cent overprovision occurs at Fy(A' — 1) = 1.4 
5 per cent overprovision occurs at V'/(A' — 1) = 1.8. 

Thus in the range of loads covered by Fig. 50, one might conclude that 
useful and determinable savings in final trunks can be achieved by use 
of the specialized /?i-curves instead of the more general 7?2-curves, when 
the ratio V'/(A' — 1) exceeds some figure in the 1.4 to 1.8 range, say 1.6. 
(In the examples just cited the V'/{A' — 1) ratio is approximately 1.9.) 

8.4. Character of Traffic Carried on Non-Final Routes 

Telephone traffic which is carried by a non-final route will ordinarily 
be subjected to a peak clipping process which will depress the variance 
of the carried portion below that of the offered load. If this traffic ter- 
minates at the distant end of the route, its character, while conceivably 
affecting the toll and local switching trains in that office, will not require 
further consideration for intertoll trunk engineering. If, however, some 
or all of the route's load is to be carried on toll facilities to a more distant 
point (the common situation), the character of such parcels of traffic will 
l)e of interest in providing suitable subsequent paths. For this purpose 
it will be desirable to have etimates of the mean and variance of these 
carried parcels. 

When a random traffic of "a" erlangs is offered to a group of "c" paths 











\ 














\ 












q 

6 
o 












\ 










\. 




^ 






















^ 










^N 


\ 


\ 














[^ 


^ 




.^^ 


^ 










^ 


\ 


\ 






o 

CM 


,— • 


^__„„— - 


.^ 


^ 








^ 










^ 


^ 


\\r\j 






^ 


-- — 




-^ 










\ 










1 


H 


^ 




to 




>< 


^ 








^ 




*\ »n o lo 




c 








<?, 




K 


^^ 




^ 




^ 


x> 




























^ 


^ 


, ^ 


^ 


^ 


\ 
























'^'^. 




x] 


^^ 


^ 




X 


























^vj 






y 

/ 




\ 






























^ 


^ 





\ 


































^ 


^*nJ 


^ 




































to 


k 



a> 




(v 




«> 




(M 




•f 


(d 


(\J 


o 




7 




< 


f\l 


_J 


f\l 


ir 








?■ 


O 




fVJ 


ILI 




t- 




-) 


en 


C) 




ir 




1 




< 


<o 


z 



Q 
UJ 



Q 
< 

3 



UJ 

< 

a: 

« UJ 



(0 


< 


(M 


O 


<0 


<o 


^ 


(\J 


O 


<») 


10 


M 


fO 


fVJ 


<\l 


(M 


(M 


(M 



ainob "ivNid oi a3a3ddo a\»cn jo 3DNviavA=,A 











\ 






1 








\ 












) 0.5 1.0 

1 1 1 1 1 1 












\ 












\ 


\ 






















\ 












\ 


\ 






















\ 








^ 




^ 


x 














^ 




^ 




X 












^ 


^ 










^ 


^ 




^ 


^ 




^ 












^ 


^ 




rn 


(\j ^ 


CM ^Ni 


^ 


^ 


^^ 


^ 






^ 


ri 




q 




vO c 

d 










OJ 


rvj 


>c 




^ 




^ 




\ 


























V 


-;;; 


^-^ 




^^ 


^ 
























'"Vo 


^^^. 


H,° 


">r 


^ 


/-^ 
^ 




X. 


























^ 7 


^^^^. 


'0 r 




/ 


/ 






























U/v 




^ 


l^ 


V 




































> 


V 




i^ 




































^^ 


k 



OO 




OJ 




lO 




C\J 




<* 


</l 


M 


0> 




?• 




< 


(\J 


_1 

a 

UJ 




7 


o 




(\I 


UJ 




t- 




"3 


OO 


<1 




a. 




\ 


(0 


< 
z 




u. 


2 


P 




n 




Ul 


(\1 


cr 

UJ 




u. 




u. 




CJ 


O 


n 




< 




o 




-J 


OO 


UJ 




O 




< 




fr 


(O 


UJ 




> 




4. 



K> 


■* 


fVJ 


o 


CO 


<o 


•<r 


AJ 


o 


n 


(«) 


n 


rn 


<\J 


f\J 


(\j 


(\j 


(\J 



Binoa ivNid 01 aaaaddo avon do 3DNviavA=,A 

496 



< 






. 


K 






r -' ^ 






N 










1 

O 

6 
o 


- 










■"^ 




\. 














K 










^ (J 
< 








- 




_-— -= 


\ 












\l 


\ 












-= 








-- 




\ 








"^ 




\ 


\ 







rvj uj 
O 




^ 




. 


—■ 




\ 








^^^ 


^ 


^ 


K 




. — ' 




-^ 


^^ 


-- 




\ 










^s^ 


^ 


"" _l 

< 

(0 z 


CO 


^ 


^ 


, ■ 





"^ 






\= 


q 




o 


c 


^ LL 








c 


< 


c---' 


^ 








\ 


\ 












ILI 

(\J >u 










s-i, 


^ 


^ 






^" 












- u. 

u. 

O 
O Q 














'-^'^q 


^'^. 


i^K 


"^ 


^ 


, 




N 










_J 




















Oa, 


^ 


s< 


y 












It 


























^ 


^ 






>s 






II 






























r^ 


^ 


is 




































•c' 


4^ 


^ 


^ 


r 




r 
r 


VI 


c 

r 


3 


r 




VJ 


r 


■< 

y r 






i a 


D il 


s ^ 


f 


r 


y c 


3 


a 


3 U 


a ^ 


f N 


o 



ainoa nvNid ojl oaaaddo avoi do 30NviyvA=,A 




■^ 


rvi 


o 


CO 


<o 


^ 


(NJ 


o 


fO 


ro 


m 


rvj 


(\J 


rvj 


(\j 


(M 



3inoa nvNid oi aaaajdo avon do 30NvibVA= a 

497 



498 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 




2 4 6 8 10 12 14 16 18 20 22 24 26 

A' == AVERAGE LOAD OFFERED TO FINAL ROUTE IN ERLANGS 



28 



Fig. 50 — Overprovision of final route trunks when R2 is used instead of Ri 
as service to first routed traffic. 

and overflowing calls do not return, the variance of the carried load is 
Fed = a[l - Er, , (a)] fl + aE^, , (a) - aEi, , _i(a)]* (37) 

and the ratio of variance to average of the carried load is 



V 



cd 



= 1 - a [£-1,0-1 (a) - Ei,c(a)]* 



= 1 - /c 



(38) 



These particular forms are due to P. J. Burke. 



THEORIES FOR TOLL TRAFFIC EXGINEERIXG IX THE U. S. A. 499 

From (38) it is easy to see that 

Fed = L{1 - Q 

= (Load carried by the group) (1 — load on last trunk) (39) 

This is a convenient relationship since for high usage trunk study work, 
both the loads carried (in eriangs) on the group and on the last trunk 
will ordinarily be at hand. 

If the high usage group's load is to be split in various directions at 
the distant point for re-offer to other groups, it would appear not un- 
reasonable to assign a variance to each portion so as to maintain the 
ratio expressed in eciuation (38). That is, if a carried load L is divided 
into parts Xi , X2 • • • where L = Xi -f X2 • • • , then the associated 
variances 71 , 72 . . • would be 

71 = Xi (1 - fc) 

y, = Xo (1 - fc) (40) 



If, however, the load offered to the group is non-random (e.g., the 
group is an intermediate route in a multi-alternate route system), the 
procedure is not quite so simple as in the random case just discussed. 
Equation (32) expresses the variance Vc of the carried load on a group 
of C paths whose 'offered traffic consists of the overflow from a first 
group of S paths to which a random load of A eriangs has been offered. 
Vc could of course be expressed in terms of A', V and C, and curves or 
tables constructed for working purposes. However, such are not avail- 
able, and in any case might be unwieldy for practical use. 

A simple alternative procedure can be used which jdelds a conserva- 
tive (too large) estimate of carried load variance. With random load 
offered to a divided two stage multiple of x paths followed by tj paths, a 
positive correlation exists between the numbers m and n of calls present 
simultaneously on the x and y paths, respectively. Then the variance 
V-n+n of the m -\- n distribution is greater than the sum of the individual 
variances of m and n, 

y m-\-n ^ ' m l~ ' n 



or 



Vm < y^n - Vn (41) 



Now n can be chosen arbitrarily, and if made very large, Vm+n becomes 
the offered load variance, and F„ the overflow load variance. Both of 
these are usually (or can be made) available. Their difference then, 
according to (41) gives an upper limit to F,„ , the desired carried load 



500 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Table XVI — Approximate Determination of the Variance 

OF Carried Loads; 
X lower paths, 8 upper paths; offer to upper paths = 3 erlangs 



Lower Paths, x 


Upper Paths, y 


No. 
Lower 
Paths 

X 


Random 

offered 

load 

A (= V) 


Variance 

of 
overflow 

Vn 


Estimated 
variance 

of carried 

load 
V -Vn 


True 

variance of 

carried load 

Eq (37) 


Variance 
of offer 

V (= Vn) 

(Col 3) 


Variance of 

overflow 

V" 


Estimated 

variance 

of cd load 

V - V" 


True 

variance 

of cd load 

(Brocli- 

meyer) 


(1) 



3 

6 

12 


(2) 

3.00 

5.399 

7.856 

12.882 


(3) 

3.00 
4.05 
4.98 
6.22 


(4) 


1.35 

2.88 
6.66 


(5) 



0.60 
1.418 
3.538 


(6) 

3.00 
4.05 
4.95 
6.22 


(7) 

0.035 
0.121 
0.236 
0.520 


(8) 

2.97 
3.93 
4.74 
5.70 


(9) 

2.853 
3.664 
4.175 
4.790 



variance- Corresponding reasoning yields the same conclusion when the 
offered load before the x paths is non-random. 

A numerical example by Brockmeyer" while clearly insufficient iu 
establish the degree of the inequality (41), indicates something as to the 
discrepancy introduced by this approximate procedure. Comparison with 
the true values is shown in Table XVI. 

In the case of random offer to the 0, 3, 6, 12 "lower paths," the ap- 
proximate method of equation (41) overestimates the variance of the 
carried load by nearly two to one (columns 4 and 5 of Table XVI). The 
exact procedure of (37) is then clearly desirable when it is applicable, 
that is when random traffic is being offered. For the 8 upper paths to 
which non-random load is offered (the non-randomness is suggested by 
comparing the variance of column 6 in Table XVI with the average 
offered load of 3 erlangs), the approximate formula (41) gives a not too 
extravagant overestimate of the true carried load variance. Until curves 
or tables are computed from equation (32), it would appear useful to 
follow the above procedure for estimating the carried load variance 
when non-random load is offered. 



8.5. Solution of a Typical Toll Multi- Alternate Route TrunJcing Arrange- 
ment: Bloomsburg, Pa. 

In Fig. 9 a typical, moderately complex, toll alternate route layout 
was illustrated. It is centered on the toll office at Bloomsburg, Pa. The 
loads to be carried between Bloomsburg and the ten surrounding cities 
are indicated in CCS (hundred call seconds per hour of traffic; 36 CCS = 
1 erlang). The numbers of direct high usage trunks shown are assumed 
to have been determined by an economic study; we are asked to find 



I 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 501 



the number of trunks which should be installed on the Bloomsburg- 
Harrisburg route, so that the last trunk will carry approximately 18 
CCS (0.50 erlang). Following this determination, (a) the number of final 
trunks from Bloomsburg to Scranton is desired so that the poorest 
service given to any of the original parcels of traffic will be no more than 
3 calls in 100 meeting NC. Also (6) the modified Bloomsburg-Scranton 
trunk arrangement is to be determined when a high usage group is pro- 
vided for the first routed traffic. 

Solution (a): First Routed Traffic Offered Directly to Final Group 

The offered loads in CCS to each distant point are shown in column 
(2) of Table XVII; the corresponding erlang values are in column (3). 
Consulting Figs. 12 and 13, the direct group overflow load parameters, 
average and variance, are read and entered in columns (5) and (6) re- 
spectively for the four groups overflowing to Harrisburg, and in columns 
(7) and (8) for the four groups directly overflowing to Scranton. The 
variance for the direct Bloomsburg-Harrisburg traffic equals its average ; 
likewise for the direct Bloomsburg-Scranton traffic. They are so entered 
in the table. The parameters of the total load on the Harrisburg group 
are found by totalhng, giving A' = 11.19, and V = 19.90. 

The required size Ci of the Harrisburg group is now determined by 
the Equivalent Random theory. Entering Fig. 25 with A' and V just 
determined, the ER values of trunks and load found are Si = 13.55, 
and Ai = 23.75. Ci is to be selected so that on a straight group of Si + 
Ci trunks with offered load A, the last trunk will carry 0.50 erlang. 
Reading from Fig. 40, the load carried by the 26th trunk approximates 
this figure. Hence Ci = 26 — *Si = 12.45 trunks; or choose 12 trunks. 

The overflow load's mean and variance from the Harrisburg group 
v/ith 12 trunks, is now read from Figs. 12 and 13, entering with load 
Ai = 23.75 and Ci -\- Si = 25.55 trunks. The overflow values (a' = 
2.50 and v' = 7.50) are entered in columns (7) and (8) of the table. 
The total offered load to Scranton is now obtained by totalling columns 
(7) and (8), giving A" = 16.27 and V" = 25.60. 

We desire now to know the number of trunks C2 for the Scranton 
group which will provide NC 3 per cent of the time to the poorest service 
parcel of traffic, i.e., the first routed Bloomsburg-Scranton parcel. The 
Ri = 0.03 and R2 = 0.03 solutions are available, the former of course 
being more closely applicable. A check reference to Fig. 50 shows a 
difference of approximately 4 per cent in trunk provision would result 
from the two methods. Entering Figs. 43 and 47 with A" = 16.27 and 



502 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 





J 


¥ 




















>- O 0) 


o 


05 
l-H 






CO 






^ 




lO 


• CCI O --I CO 


CO 




"O CO lO I-H >* 

,X ^ lO ^ (M 
<wIt-H Oi-H (M 


(N 


-IJ 


aj 




P I- o 




'-' 03 ■* CO U5 


<M 




I-H 




'a 

• pH 

S-i 


P^ 




(N --1 OOOO 

g X oddd 


d 




I-H 

d 

II 


§ 




"o 


"o 






o 


l-o 


3 cS 
II 




m 
m 

o 
o 


|2 

< o 


O 


o 








o 












I-H O 00 t^ 


'^ 


o 


( -* 












00 C3 CO >o 


I-H 


CO 


m 


o 

-4-* 




o 






dcO I-H I-H 


d 

T-H 






J 


OO 


"3 












-tj 


pq 


o a 




l> 










II 


£3 
03 


% 


3 o 














^ 










O. 


02 










O OOOcO 


-* 


t^ 


M 


•na 










iO(M OOOT 


I-H 


(M 


CO 


Iz; 


5g 










di-n'dd 


d 


d 


o 
o 




2^ 

c4 


t^ 


s 








I-H 


T-H 

II 


H 






c<> 










05 


P 






- 










^ 


(M 








ic ic o CO 


t^ 


o 








II 


o 


o 


rrj 




coco o t^ 

dec t^i-H 




05 
T-H 








> 


H 


§§• 


P> be 


O 






li 








ti 


H 


►J o 


E 














kC 


*< 
^ 


CO bO 










^ 








co' 

r-H 


« 




















II 


















W 


tn J^ 






O OCDCO 


t^ 


Oi 










Eh 


•n.ss 






cot^co o 


^ 


T-H 








^ 






CM 




O <m" (M .-H 


-<J< 


l-H 








^ 


2K 

43 


S bO 


in 






I-H 
11 








bD 

a 


o 


u 










^ 








CO 


^ 












^ 








o 


"S^^ 
















■* 


< 


















bi) 


J 


C S' s ^ 


rf 


T-HIO OiC 


^ 




ICGOOOIO 








P 




.-H(M 


o 




(M 




a 




■ OJ C S 
















o 

;h 

«4H 


< 
















T3 


-a 




















w 




(A 
















oPh 


t> 


bO 


bO 




(N Oi^ 


t-- 




(MC^COOO 


'^ 




lO ^— ' 


H-t 


L- 


C 


,,-^ 


l>-0^'* 


rt< 




"^i <N CO C^ 


I-H 








:3 


-2 


•^ 




• 










co'^' 




l-H 


""^ 


oiod^" 


-* 




COCOCO Tf< 


d 




I-H . 


c "J 


w 




.—1 i-H 






(M 


I-H 




II IM 


H 


oO 


















^^^ 

TJ 


OQ 




















P 


















<u -^ 


















a> 


tJ 


!U (0 


















M 


l-H 






_ 


CO O r-f O 


T-H 




CO CO 00^ 


>o 




• rH 




n 


fS 


(M '^ OCO 


CO 




(M CO <M lO 


CO 








u 




lO O T-H 


I-H 




rH 00(M I-H 


CO 




<a 


l-H 


t3 
til 


















" o 


> 


o 


















to 


XI 


















:3 
•♦J 


< 

Eh 


CJ 

•4-* 

a 

■*-» 

to 

5 


- 


■4J 

1- 

o 

(U c ft 

> ° s s 


bfi 
»-. 
3 
-Q 

u 




"H.S 

•C o 1 ^ 


o 

o 




11^ 












o3 




P^WJ^Ph 







THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 503 

I Y" = 25.60, we obtain the trunk requirements: 

! Rx Method 23.8 trunks 

i?2 Method 24.8 trunks 

Thus the more precise method of sokition here yields a reduction of 1 .0 
in 25 trunks, a saving of 4 per cent, as had been predicted. 

The above calculation is on a Lost Calls Cleared basis. Since the over- 
flow direct traffic calls will return to this group to obtain service, to as- 
sure their receiving no more than 3 per cent 'NC, the provision of the 
final route would theoretically need to be slightly more liberal. An esti- 
mate of the allowance required here may be made by adding the ex- 
pected erlangs loss A for the direct traffic (most of the final route over- 
flow calls which come from high usage routes will be carried by their 
respective groups on the next retrial) to both the A" and Y" values 
previously obtained, and recalculating the trunks required from that 
point onward. (In fact this could have been included in the initial com- 
putation.) Thus: 

A = 0.03 X 10.14 = 0.30 erlang 
A'" = 16.27 + 0.30 = 16.57 erlangs 
V" = 25.60 + 0.30 = 25.90 erlangs 

Again consulting Figs. 43 and 47 gives the corresponding final trunk 
values 

Ri Method 24.1 trunks 

R2 Method 25.1 trunks 

Of the above four figures for the number of trunks in the Scranton 
route, the i?i-Method with retrials, i.e., 24.1 trunks, would appear to 
give the best estimate of the required trunks to give 0.03 service to the 
poorest service parcel. 

Solution (h) : With High Usage Group Provided for First Routed Traffic 

Following the procedure outlined in Section 8.2, we obtain an average 
of the proportions overflowing to the final route for all offered load par- 
cels. The individual parcel overflow proportion estimates are shown in 
the last column of Table XVII; their unweighted average is 0.112. With 
a first routed offer to Scranton of 10.14 erlangs, a provision of 12 high 
usage trunks will result in an overflow of a = 1.26 erlangs, or a propor- 
tion, of 0.125 which is the value most closely attainable to the objective 
0.112. With 12 trunks the overflow variance is found to be 2.80. 



504 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Replacing 10.14 in columns 7 and 8 of Table XVII with 1.26 and 2.80, 
respectively, gives new estimates characterizing the offer to the final 
route. A" = 7.39 and V" — 18.26. We now proceed to insure that the 
poorest service parcel obtains 0.03 service. This occurs on the Phila- 
delphia and Harrisburg groups, which overflow to the final group ap- 
proximately 0.224 of their original offered loads. The final group must 
then, according to equation (34) be engineered for 

R2 = 0.03/0.224 = 0.134 service. 

This value lies above the highest R2 engineering chart (Fig. 49, R2 = 
0.10), so an ER calculation is indicated. 

The Equivalent Random average is 28.6 erlangs, and S = 23.5 
trunks. We determine the total trunks S -\- R which, with 28.6 erlangs 
offered, will overflow 0.134(7.39) = 0.99 erlang. From Fig. 12.2, 35.6 
trunks are required. Then the final route provision should be C = 35.6 — 
23.5 = 12.1 trunks; and a total of 12 + 12.1 or 24.1 Scranton trunks 
is indicated. 

Simplified Alternative Solution: In Section 8.2 a simplified approxi- 
mate procedure was described using a modified probability P' for the 
average overall service for all parcels of traffic, instead of P for the poor- 
est service parcel. Suppose P' = 0.01 is chosen as being acceptable. 
Then 

P' 01 

«' = T = am = oo^" 

Interpolating between the R2 = 0.05 and 0.10 curves (Figs. 48 and 49) 
gives with A" = 7.39 and F" = 18.26, C = 13.4, the number of final 
trunks required. Again the same result could have been obtained by 
making the suitable ER computation. It may be noted that if P' had 
been chosen as 0.015 (one-half of P), R2 would have become 0.134, 
exactly the same value found in the poorest-service-parcel method. The 
final trunk provision, of course, would have again l)een 12.1 trunks. 

Disscussion 

In the first solution above, 24.1 full access final trunks from Blooms- 
burg to Scranton were refiuired. The service on the first routed traffic 
was 0.03; however, the service enjoyed by the offered traffic as a whole 
was markedly better than 0.03. The corresponding ER calculation 
shows (.4 = 28.3, .S -\- C = 12.3 + 24.1) a total overflow of a" = 0.72 
erlangs, or an overall service of 0.72/91.21 = 0.008. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 505 

In the second solution, 12 high usage and 12.1 common final, or a total 
of 24.1, trunks were again required, to give 0.03 service to the poorest 
service parcels of offered load. The overall service here, however, was 
0.99/91.21 .= 0.011. Thus, with the same number of paths provided, 
in the second solution (high usage arrangement) the overall call loss was 
40 pes cent larger than in the first solution,* However, it may well be 
desirable to accept such an average service penalty since by providing 
high usage trunks for the first routed traffic, the latter's service cannot 
be degraded nearly so readily should heavy overloads occur momentarily 
in the other parcels of traffic. 

9. CONCLUSION 

As direct distance dialing increases, it will be necessary to provide 
intertoll paths so that substantially no-delay service is given at all times. 
To do this economically, automatic multi-alternate routing will replace 
the present single route operation. Traffic engineering of these compli- 
cated trunking arrangements will be more difficult than with simple 
intertoll groups. 

One of the new problems is to describe adequately the non-random 
character of overflow traffic. In the present paper this is proposed to be 
done by employing both mean and variance values to describe each par- 
cel of traffic, instead of only the mean as used heretofore. Numerous 
comparisons are made with simulation results which indicate that ade- 
quate predictive reliability is obtained by this method for most traffic 
engineering and administrative purposes. Working curves are provided 
by which trunking arrangements of considerable complexity can readily 
j be solved. 

A second problem requiring further review is the day-to-day variation 
i among the primary loads and their effect on the alternate route system's 
I grade of service. A thorough study of these variations will permit a re- 

I evaluation of the service criteria which have tentatively been adopted. 
j A closely allied problem is that of providing the necessary kind and 
[ amounts of traffic measuring devices at suitable points in the toll alter- 
! nate route systems. Requisite to the solution of both of these problems 
! is an understanding of traffic flow character in a complex overflow-type 

I * The actual loss difference may be slightly greater than estimated here since 
i in the first solution (complete access final trunks), an allowance was included for 

i j return attempts to the final route by first routed calls meeting an 0.03 loss, while 
1 in the second solution (high usage group for first routed traffic) no return at- 

i| tempts to the final route were considered. These would presumably be small since 

I I only 1 per cent of all calls would overflow and most of these upon retrial would be 
ij handled on their respective high usage groups. 



506 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

of trunking plan, and a method for estimating quantitatively the essential 
fluctuation parameters at each point in such a system. The present paper 
has undertaken to shed some light on the former, and to provide an 
approximate j^et sufficiently accurate method by which the latter can 
be accomplished. It may be expected then that these studies, as they are 
developed, will provide the basis for assuring an adequate direct dis- 
tance dialing service at all times with a minimum investment in intertoll 
trunk facilities. 

ACKNOWLEDGEMENTS 

The author wishes to acknowledge the technical and mathematical as- 
sistance of his associates, Mrs. Sallie P. Mead, P. J. Burke, W. J. Hall, 
and W. S. Hayward, in the preparation of this paper. Dr. Hall provided 
the material on the convolution of negative binomials leading to Fig. 19. 
Mr. Hayward extended Kosten's curve E on Fig. 5 to higher losses by a 
calculating method involving the progressive squaring of a probability 
matrix. The author's thanks are also due J. Riordan who has summarized | 
some of the earlier mathematical work of H. Nyquist and E. C. INIolina, 
as well as his own, in the study of overflow load characteristics; this 
appears as Appendix I. 

The extensive calculations and chart constructions are principally 
the work of Miss C. A. Lennon. 

REFERENCES 

1. Rappleye, S. C, A Study of the Delays Encountered bj'^ Toll Operators in Ob- 

taining an Idle Trunk, B. S.T.J. , 25, p. 539, Oct., 1946. 

2. Kosten, L., Over de Invloed van Herhaalde Oproepen in de Theorie der Blok- 

keringskausen, De Ingenieur, 59, j). 1'j123, Nov. 21, 1947. 

3. Clos, C, An Aspect of the Dialing Behavior of Subscribers and Its Effect on 

the Trunk Plant, B. S.T.J. , 27, p. 424, July, 1948. 

4. Kosten, L., Uber Sperrungswahrscheinlichkeiten bei Staffelschaltungen, 

E.N.T., 14, p. 5, Jan., 1937. 

5. Kosten, L., Over Blokkeerings-en Wachti)rol>lemen, Thesis, Delft, 1942. 

6. Molina, E. C, Appendix to: Interconnection of Telephone Systems — Graded 

Multiples (R. I. Wilkinson), B.S.T.J., 10, p. 531, Oct., 1931. 

7. Vaulot, A. E., Application du Calcul des Probabilites a I'Exploitation Tele- 

phonique. Revue Gen. de I'Electricite, 16, p. 411, Sept. 13, 1924. 

8. Lundcpiist, K., General Theorv for Telephone Traffic, Ericsson Technics, 9, 

p. Ill, 1953. 

9. Berkeley, G. S., Traffic and Trunking Principles in Automatic Telei)hony, 2nd 

revised edition, 1949, Ernest Benn, Ltd., London, Chapter V. 

10. Pahu, C., Calcul I']xact de la Perte dans les Groupes de Circuits Echelonn^s, 

lOricsson Technics, 3, ]). 41, 1936. 

11. Brockmever, 1']., The Simph> Overflow Problem in the Theory of Telephone 

Traffic! Teleteknik, 5, ji. 361, December, 1954. 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 507 



ABRIDGED BIBLIOGRAPHY OF ARTICLES ON TOLL ALTERNATE ROUTING 

Clark, A. B., and Osborne, H. S., Automatic Switching for Nationwide Telephone 
Service, A.I.E.E., Trans., 71, Part I, p. 245, 1952. (Also B.S.T.J., 31, p. 823, 
Sept., 1952.) 

Pilliod, J. J., Fundamental Plans for Toll Telephone Plant, A.I.E.E. Trans., 71, 
Part I, p. 248, 1952. (Also B.S.T.J., 31, p. 832, Sept., 1952.) 

Nunn, W. H., Nationwide Numbering Plan, A.I.E.E. Trans., 71, Part I, p. 257, 
1952. (Also B.S.T.J., 31, p. 851, Sept., 1952.) 

Clark, A. B., The Development of Telephony in the United States, A.I.E.E. 
Trans., 71, Part I, p. 348, 1952. 

Shiplev, F. F., Automatic Toll Switching Systems, A.I.E.E. Trans., 71, Part I, 
p. '261, 1952. (Also B.S.T.J., 31, p. 860, Sept., 1952.) 

Myers, O., The 4A Crossbar Toll System for Nationwide Dialing, Bell Lab. 
Record, 31, p. 369, Oct., 1953. 

Clos, C, Automatic Alternate Routing of Telephone Traffic, Bell Lab. Record, 
32, p. 51, Feb., 1954. 

Truitt, C. J., Traffic Engineering Techniques for Determining Trunk Require- 
ments in Alternate Routing Trunk Networks, B.S.T.J., 33, p. 277, March, 
1954. 

Molnar, I., Some Recent Advances in the Economy of Routing Calls in Nation- 
wide Dialing, A.E. Tech. Jl., 4, p. 1, Dec, 1954. 

Jacobitti, E., Automatic Alternate Routing in the 4A Crossbar System, Bell Lab. 
Record, 33, p. 141, April, 1955. 

Appendix I* 

DERIVATION OF MOMENTS OF OVERFLOW TRAFFIC 

This appendix gives a derivation of certain factorial moments of the 
c(iuilibrium probabilities of congestion in a di^dded full-access multiple 
used as a basis for the calculations in the text. These moments were de- 
rived independently in unpublished memoranda (1941) by E. C. Molina 
(the first four) and by H. Nyquist; curiously, the method of derivation 
here, which uses factorial moment generating functions, employs auxili- 
ary relations from both Molina and Nyquist. Although these factorial 
moments may be obtained at a glance from the probability expressions 
given by Kosten in 1937, if it is remembered that 

pw = |:(-i)'-'(';)^, (1.1) 

where p{x) is a discrete probability and M (k) is the A;th factorial moment 
of its distribution, Kosten does not so identify the moments and it may 
1)0 interesting to have a direct derivation. 

Starting from the equilibrium formulas of the text for f(;ni, n), the 
l)robability of m trunks busy in the specific group of x trunks, and n in 



Prepared by J. Riordan. 



508 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

the (unlimited) common group, namely 

{a -{- m -\- n)f(m, n) — (w + l)f(m + 1, n) 

— (n + l)/(m, n + 1) — af(in — 1, n) = 

(1-2) « 

(a -{- X -{- n)j{x, n) — af{x, n — 1) \ 

- (n -\- l)f(x, n + 1) - af(x - 1, n) = 

and 

/(m, n) = 0, m < or n < or m > x, 

factorial moment generating function recurrences may be found and 
solved. 

With m fixed, factorial moments of n are defined by 



M(fc)(m) = E {n)kf{m, n) (1.3) 

n=0 

or alternatively by the factorial moment exponential generating function 
M{m, = Z MUm)t'/k\ = £ (1 + 07K n) (1.4) ] 

fc=0 n=0 I 

In (1.3), {n)k = n{n — 1) • • • (n — /c + 1) is the usual notation for a \ 
falling factorial. 

Using (1.4) in equations (1.2), and for brevity D = d/dt, it is found 
that 

a^ m ^- tD)M{in, t) - (m + l)M{m + 1, t) 

- aM(m - l,t) = (1.5) 

(x - at -\- tD)M{x, t) - aM{x - \,t) = 

which correspond (by equating powers of t) to the factorial moment re- 
currences 

{a-\- m^ k)M^kM) - (m + l)Ma)(w + 1) 

- ailf (fc)(m - 1) = (1.6) 

(x + k)M(k)(x) - akM^k-i)ix) - aMik)(x - 1) = 

Notice that the first of (1.6) is a recurrence in m, which suggests (fol- 
lowing Molina) introducing a new generating function defined by 

Gdu) = T.M^k){m)u'^ (1.7) 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 509 

Using this in (1.5), it is found that 

(a -h k - au + (u - l)~\ GM = (1.8) 



Hence 



1 dGM^^^J^ ^j_g^ 



Gk(u) du I — u 

and, by easy integrations, 

Gk{u) = ce"" (1 - ur\ (1.10) 

with c an arbitrary constant, which is clearly identical with Gk(0) = 
M(.)(0). 

Expansion of the right-hand side of (1.10) shows that 

il/a,(m) = Ma)(0) Z "^ •^. , "" ■„ = Ma,(0)a-.(m), (1.11) 

j=o \ J / {m - j)l 



if 
<jo{m) = a'/ml and, a,(w) = ^ ( •^- ~ ) y-^ ^ri (1-12) 



The notation ak(m) is copied from Xyquist; the functions are closely 
related to the ^^^"^ used by Kosten; indeed akim) = e'ipm'''' ■ They have 
the generating function 

00 

Qkiu) = 53 (TkMu" = e""(l — u)~'' (1.13) 

from which a number of recurrences are found readily. Thus 
Qkiu) = (1 - u)gk+Xu) 

u -^ — = augkiu) + kugk+i(u) 
du 

= -agk-iiu) + (a - k)gk(u) + kgk-i(u) 

(the last by use of the first) imply 

ckim) = ak+iim) — (Tk+iim — 1) 

m(Tk(m) = ackim — 1) + k<jk+i(m — 1) 

= - a<jk-i{m) + (a - k)<Tk{m) + k<Xk+i(ni) 



510 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

The first of these leads to 

cr/^(0) + (7,(1) + • • • + cr,(x) = ak+i(x) (1.14) J 
and the last is useful in the form j 

kak+i{m) = {m + k - a)ak{m) + 0(rt_i(m) (1.15) ■- 

Also, the first along with ao(m) = a" /m\ leads to a simple calculation ; 
procedure, as Kosten has noticed. 

By (1.11) the factorial moments are now completely determined ex- 
cept for il/(A-)(0). To determine the latter, the second of (1.6) and the 
normalizing equation 

X 

E M,{m) = 1 (1.16) 

are available. 

Thus from the second of (1 .6) 

[(:r + k)<r,{x) - mu{x - l)].^/(A-)(0) = a/v(r,_i(.c)M(,_i)(0) (1.17) 

Also 

{x + k)ak{x) — acTkix — 1) 

= (x -\- k - a)ak(x) + a[(Xk{x) - (Tk(.x — 1)] 

= (x -\- k - a)(Tk{x) + a<Tk-i{x) 

= /t'o-fc+i^r), 

the last step by (1.15). Hence 

(Tk-l{x) 



MaM = a "-^=^ Ma-iM (1.18) 

<rk+i{x) 



and by iteration 



^k (7i(x)(roix) 



MaAO) = a' "^7" "7, Mo(0) (1.19) 

From (l.ll) and (1.16), and in the last step (1.14), 

t.M,(m) = i: il/o(0)cro(7n) = ilfo(0)cri(a;) = 1 (1.20) 

Hence finally 

Ma)(m) = Ma-M<rk{ni) 

, a,(x)ak(m) (1.21) 



= a 



(Tki.i{.r)<^k{x) 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 511 

^""^ Ma) = Z Ma,{m) = a'ao{x)/a,(x) (1.22) 



m=0 



Ordinary moments are found from the factorial moments by linear 
relations; thus if Wt is the A;th ordinary moment (about the origin) 

mo = M^o) nil = M^) m-i = il/(2) + il/(i) 

mz = il/(3) + 3Af (2) + il/(i) 

Thus 

mo(m) = (ro(m)/ai(x) 

mi(m) = aai(m)(To(x) / (Ti(x)(T2(x) 

vi-iim) = aa2{m)(TQ{x) / (r'2{x)(7z{x) + a(Ji{;m)<Ta{x) / <ti{x)(T2{x) 

and, in particular, using notation of the text 

mo{x) = (ro{x)/ax{x) = Ei,xia) 

mi(x) (Tiix) a 

(Xx = — ^r = a 



mo{x) <T.(x) .T - a + 1 + aEi,,{a) (1.23) 

ni2{x) 2 aaiix) , 2 

Vx = — 7-r — (Xx = ir-^ + OCx — ax , ^ 

mo{x) csix) (1,24) 

= ax[l — ax + 2a(x + 2 -\- ax - a)~^] 

X 

Finally the sum moments: nik = ^ mk{m) are 





Wo = 1 

mi = a = a(To{x)/(yi{x) = aEi_x{a) 
rrh = aaQ{x)/a2{x) -\- mi = mi[a{x -\- I -\- nii — a)~ +1] 



(1.25) 



(1.26) 



y = m2 — mi = mi[l — vh + a(.^' + 1 + nii — a) ] 
In these, Ei,x(a) = (ro(:c)/(ri(.T) is the familiar Erlang loss function. 

Appendix II — character of overflow load when non-random 

TRAFFIC IS offered TO A GROUP OF TRUNKS 

It has long been recognized that it would be useful to have a method 
by which the character of the overflow traffic could be determined when 
non-random traffic is offered to a group of trunks. Excellent agreement 
has been found in both throwdown and field observation over ranges of 
considerable interest with the "equivalent random" method of describ- 



to 

z 
< 

LJJ 



in 

z 

D 

h- 

X 



o 

en 

Q 

< 

o 



3 

LL 

cr 

LU 

> 

o 

LL 

o 

LLI 

< 

LJJ 

II 



0.04f/ AQ-- 
0.02 




TRUN 



10 .1 0:3 

TRUNKS 



.1 0.3 1.0 3 

a,= AVERAGE 
IN 




10 .1 0.3 



TRUN 



10 .1 0.3 1.0 3 

OF OFFERED TRAFFIC 
ERLANGS 



Fig. 51 — Mean and variance of overHovv load when non-random traffic is 
offered to a group of trunks. 

512 



THEORIES FOR TOLL TRAFFIC ENGINEERING IN THE U. S. A. 513 

ing the character of non-random traffic. An approximate solution of the 
problem is offered based on this method. 

Suppose a random traffic a is offered to a straight multiple which is 
divided into a lower Xi portion and an upper X2 portion, as follows: 

T «2 , V2 



X2. 



] OCl,Vi 



u 

From Nyquist's and Molina's work we know the mean and variance 



of the two overflows to be: 



ai = a-Ei^xiia) = a 



a"» 



•ril 






Vi = ai\ 1 — ai -\ ■ — - 

L Xi — a + ai + IJ 

a2 = a-Ei,xi+x2(0') 

V2 = aol I — a2 -\ j j : — r 

L xi + a;2 - a + 0:2 4- IJ 

Since ai and vi completely determine a and Xi , and these in turn, with 
X2 , determine 02 and Vo , we may express 02 and V2 in terms of only ai , 
Vi , and X2 . The overflow characteristics (0:2 and V2), are then given for a 
non-random load (ai and Vi) offered to x trunks as was desired. 

Fig. 51 of this Appendix has been constructed by the Equivalent Ran- 
dom method. The charts show the expected values of 0:2 and I'o when 
ai , Vi (or vi/ai), and X2 , are given. The range of ai is only to 5 er- 
langs, and v/a is given only from the Poisson unity relation to a peaked- 
ness value of 2.5. Extended and more definitive curves or tables could 
readily, of course, be constructed. 

The use of the curves can perhaps best be illustrated by the solution 
of a familiar example. 

Example: A load of 4.5 erlangs is submitted to 10 trunks; on the "lost 
calls cleared" basis; what is the average load passing to overflow? 

Solution: Compute the load characteristics from the first trunk when 
4.5 erlangs of random traffic are submitted to it. These values are found 
to be a\ = 3.G8, vi = 4.15. Now using ai and vi (or vi/ai = 4.15/3.68 = 
1.13) as the offered load to the second trunk, read on the chart the param- 
eters of the overflow from the second trunk, and so on. The successive 
overflow values are given in Table XVIII. 



514 



THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



The proportion of load overflowing the group is then 0.0472/4.50 = 
0.0105, which agrees, of course, with the Erlang £^i,io(4.5) value. The 
successive overflow values are shown on the chart by the row of dots 
along the a2 and V2 1-trunk curves. 

Instead of considering successive single-trunk overflows as in the ex- 
ample above, other numbers of trunks may be chosen and their over- 
flows determined. For example suppose the 10 trunks are subdivided 
into 2 + 3 + 2-1-3 trunks. The loads overflowing these groups are 
given in Table XIX. 

Again the overflow is 0.0472 erlang, or a proportion lost of 0.0105, 
which is, as it should be, the same as found in the previous example. 
The values read in this example are indicated by the row of dots marked 
1, 3, 6, 8 on the 2-trunk and 3-trunk curves. 

The above procedure and curves should be of use in obtaining an esti- 
mate of the character of the overflow traffic when a non-random load 
is offered to a group of paths. 



I 



Table XVIII — Successive Non-Random Overflows 




Characteristics of Load Offered to Trunk No. i 




(same as overflow from previous trunk) 


Trunk Number 




i 


Average 


Variance 


Ratio of variance to 
average 


1 


4.50 


4.50 


1.00 (Random) 


2 


3.68 


4.15 


1.13 


3 


2.92 


3.68 


1.26 


4 


2.22 


3.11 


1.40 


5 


1.61 


2.46 


1.53 


6 


1.09 


1.80 


1.64 


7 


0.694 


1.19 


1.72 


8 


0.406 


0.709 


1.75 


9 


0.217 


0.377 


1.74 


10 


0.106 


0.180 


1.70 


Overflow 


0.0472 


0.077 


1.64 






Table XIX — Sucessive Non-Random Overflows 



Trunlc Number 


No. Trunks in 
Next Bundle 


Offered Load Cliaracteristics 
(same as overflow from previous trunk) 


i 


Average 


Variance 


Ratio of variance to 
average 


1 

3 
6 

8 
Overflow 


2 
3 
2 
3 


4.50 

2.92 

1.09 

0.406 

0.0472 


4.50 

3.68 

1.80 

0.709 

0.077 


1.00 (Random) 

1.26 

1.64 

1.75 

1.64 



Crosstalk on Open-Wire Lines 

By W. C. BABCOCK, ESTHER RENTROP, and C. S. THAELER 

(Manuscript received September 29, 1955) 

Crosstalk on open-wire lines results from cross-induction between the 
circuits due to the electric and magnetic fields surrounding the wires. 
The limitation of crosstalk couplings to tolerable magnitudes is achieved 
by systematically turning over or transposing the conductors that 
comprise the circuits. The fundamental theory underlying the engineer- 
ing of such transposition arrangements was presented by A. G. Chapman 
in a paper entitled Open-Wire Crosstalk published in the Bell System 
Technical Journal in January and April, 1934. 

There is now available a Monograph (No. 2520) supplementing Mr. 
Chapman's paper which reflects a considerable amount of experience re- 
sulting from the application of these techniques and provides a basis for 
the engineering of open-wire plant. The scope of the material is indi- 
cated by the following: 

TRANSPOSITION PATTERNS 

This describes the basic transposition types which define the number 
and locations of transpositions applied to the individual open-wire 
circuits. 

TYPES OF CROSSTALK COUPLING 

Crosstalk occurs both within incremental segments of line and be- 
tween such segments. Furthermore, the coupling may result from cross- 
induction directly from a disturbing to a disturbed circuit or indirectly 
by way of an intervening tertiary circuit. On the disturbed circuit the 
crosstalk is propagated both toward the source of the original signal 
and toward the distant terminal. A knowledge of the relative importance 
of the various types of coupling is valuable in establishing certain time- 
saving approximations which facilitate the analysis of the total cross- 
talk picture. 

515 



516 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

TYPE UNBALANCE CROSSTALK 

Crosstalk is measured in terms of a current ratio between the disturb- 
ing and disturbed circuits at the point of observation. Crosstalk between 
open-wire circuits is also generally computed in terms of a current ratio 
(cu) but it is also convenient to refer to it in terms of a coupling loss 
(db). The coupling in crosstalk units (cu) is the product of three terms: 
a coefficient dependent on wire configuration; a type unbalance depend- 
ent on transposition patterns; and frequency. The coefficient represents 
the coupling between relatively untransposed circuits of a specified 
length (1 mile) at a specific frequency (1 kc). The type unbalance is a 
measure of the inability to completely cancel out crosstalk by intro- 
ducing transpositions because of interaction effects between the two 
halves of the exposure and because of propagation effects, primarily 
phase shift. Type unbalance is expressed in terms of a residual unbalance 
in miles and the frequency is expressed in kilocycles. 

The coefficients applicable to lines built in accordance with certain 
standardized specifications are available in tabular form. When it is 
desired to obtain coefficients for other types of line, it is possible to 
compute approximate values which may be modified by correction 
factors to indicate the relationship between the computed values and 
measurements on carefully constructed lines. 

Expressions for near-end type unbalance for certain simple types of 
exposures are developed and the formulas for all types of exposures are 
given. In addition, the values for near-end type unbalance are tabulated 
at 30° line angle intervals for lines where the propagation angle is iu 
2,880° or less. 

The principal component of far-end crosstalk between well transposed 
circuits results from compound couplings involving tertiary circuits. 
Again the expressions are developed for some of the exposures involving 
a few transpositions and the procedure for obtaining the formulas for 
any type of exposure is shown. Formulas are included for the types of 
exposures encountered in normal practice and the numerical values of 
far-end type unbalance are given at 30° intervals for line angles up to 
2,880°. 

SUMMATION OF CROSSTALK 

The procedures referred to thus far evaluate the crosstalk occurring 
within a limited length of line known as a transposition section. In 
practice, however, a line is transposed as a series of sections. It is neces- 
sary, therefore, to determine how the crosstalk arising within the several 



CROSSTALK ON OPEN- WIRE LINES 517 

sections and that arising from interactions between the sections tend to 
combine. In a series of like transposition sections there is a tendency 
for the crosstalk to increase systematically, sometimes reaching in- 
tolerable magnitudes. This tendency can be controlled to a degree by 
introducing transpositions at the junctions between the sections, thus 
cancelling out some of the major components of the crosstalk. Complete 
cancellation is impossible because of interaction and propagation effects. 

ABSORPTION 

Since very significant couplings exist by way of tertiary circuits, it is 
possible for crosstalk to reappear on the disturbing circuit and thus 
strengthen or attenuate the original signal. This gives rise to the ap- 
pearance of high attenuation known as absorption peaks in the line 
loss characteristic at certain critical frequencies. The evaluation of such 
pair-to-self coupling requires the use of coefficients which differ from 
those between different pairs and these are given for standard configura- 
tions. 

STRUCTURAL IRREGULARITIES 

It is impracticable to maintain absolute uniformity in the spacing 
between wires and in the spacing of transpositions. Thus there are un- 
avoidable variations in the couplings between pairs from one transposi- 
tion interval to the next. This in turn reduces the effectiveness of the 
measures to control the systematic or type unbalance crosstalk and 
produces what is known as irregularity crosstalk. Since the occurrence 
of structural irregularities tends to follow a random distribution, it is 
possible to evaluate it statistically and procedures for doing so are in- 
cluded. In addition to this direct effect of structural irregularities, there 
is a component of crosstalk resulting from the combination of systematic 
and random unbalances. A method is developed for estimating the 
magnitude of this important component of crosstalk. 

EXAMPLES 

In order to demonstrate how the procedures and data are used in 
solving practical problems, there is included the development of a 
transposition system to satisfy certain assumed conditions. This is 
carried through to the selection of transposition types for one transposi- 
tion section and the selection of suitable junction transpositions. 

Additional examples of transposition engineering are given in the form 



518 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

of several transposition systems which have been widely used in the Bell 
System. These include: 

Exposed Line — for voice frequency service. 

CI — for voice frequency and carrier service up to 30 kc. 

J5 — for voice frequency and carrier operation up to 143 kc. 

01 ■ — for voice frequency and compandored carrier operation up to 
156 kc. 

RIC — suitable for exchange lines with a limited number of carrier 
assignments. 

Altogether, the theory, explanatory material, formulas and compre- 
hensive data included in the Monograph make it possible to estimate 
open-wire crosstalk couplings and provide the necessary background for 
the development of new transposition systems. 



P 



I 



Bell System Technical Papers Not 
Published in This Journal 

Alsberg, D. A.^ 

6-KMC Sweep Oscillator, I.R.E. Trans., PGI-4, pp. 32-39, Oct., 1955. 

Anderson, J. R.,i Brady, G. W.,^ Merz, W. J.,^ and Remeika, J. P.^ 

Effects of Ambient Atmosphere on the Stability of Barium Titanate, 
J. Appl. Phys., Letter to the Editor, 26, pp. 1387-1388, Nov., 1955. 

Anderson, 0. L.,^ and Andreatch, P.^ 

stress Relaxation in Gold Wire, J. Appl. Phys., 26, pp. 1518-1519, 
Dec, 1955. 

Anderson, P. W.,^ and Hasegawa, H.^ 

Considerations on Double Exchange, Phys. Rev., 100, pp. 675-681, 
Oct. 15, 1955. 

Anderson, P. W.^ 

Electromagnetic Theory of Cyclotron Resonance in Metals, Phys. 
Rev., Letter to the Editor, 100, pp. 749-750, Oct. 15, 1955. 

Andreatch, P., see Anderson, 0. L. 

Augustine, C. F., see Slocum, A. 

Barstow, J. M.* 

The ABC's of Color Television, Proc. I.R.E., 43, pp. 1574-1579, 
Nov., 1955. 

Bartlett, C. A.2 

Closed-Circuit Television in the Bell System, Elec. Engg., 75, pp. 
34-37, Jan., 1956. 



1. Bell Telephone Laboratories, Inc. 

2. American Telephone and Telegraph Company. 
5. University of Tokyo, Japan. 

519 



520 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Becker, J. A.^ 

Adsorption on Metal Surfaces and Its Bearing on Catalysis, Advances 
in Catalysis, 1955, Nov., 1955. 

Bommel, H. E.i 

Ultrasonic Attenuation in Superconducting and Normal-Conducting 
Tin at Low Temperatures, Phys. Rev., Letter to the Editor, 100, pp. 
758-759, Oct. 15, 1955. 

Bemski, G.^ 

Lifetime of Electrons in p-Type Silicon, Phys. Rev., 100, pp. 523-524, 
Oct. 15, 1955. 

Bennett, W. R.^ 

Steady State Transmission Through Networks Containing Periodi- 
cally Operated Switches, Trans. I.R.E., PGC.T., 2, pp. 17-21, Mar., 
1955. 

Bommel, H. E.,i Mason, W. P.,* and Warner, A. W., Jr.' 

Experimental Evidence for Dislocation in Crystalline Quartz, Phys. 
Rev., Letter to the Editor, 99, pp. 1895-1896, Sept. 15, 1955. 

Bradley, W. W., see Compton, K. G. 

Brattain, W. H., see Buck, T. M., and Pearson, G. L. 

Brady, G. W., see Anderson, J. R. 

Brown, W. L.' 

Surface Potential and Surface Charge Distribution from Semicon- 
ductor Field Effect Measurements, Phys. Rev., 100, pp. 590-591, 
Oct. 15, 1955. 

Buck, T. M.,' and Brattain, W. H.' 

Investigations of Surface Recombination Velocities on Germanium by 
the Photoelectric Magnetic Method, J. Electrochem. Soc, 102, pp. 
636-640, Nov., 1955. 

Cetlin, B. B., see Gait, J. K. 

Charnes, a., see Jacobson, M. J. 
1. Bell Telephone Laboratories, Inc. 



I 



TECHNICAL PAPERS 



521 



CoMPTON, K. G.,^ Mendizza, a./ and Bradley, W. W.' 

Atmospheric Galvanic Couple Corrosion, Corrosion, 11, pp. 35-44, 
Sept., 1955. 

CoRENzwiT, E., see Matthias, B. T. 
Dail, H. W., Jr., see Gait, J. K. 

Dillon, J. F., Jr.,^ Geschwind, S.,^ and Jaccarino, V.^ 

Ferromagnetic Resonance in Single Crystals of Manganese Ferrite, 
Phys. Rev., Letter to the Editor, 100, pp. 750-752, Oct. 15, 1955. 

Dodge, H. F.^ 

Chain Sampling Inspection Plan, Ind. Quality Control, 11, pp. 10-13, 
Jan., 1955. 

Dodge, H. F.^ 
Skip-lot Sampling Plan, Ind. Quality Control, 11, pp. 3-5, Feb., 1955. 

Fagen, R. E.,^ and Riordan, J.^ 

Queueing Systems for Single and Multiple Operation, J. S. Ind. Appl. 
Math., 3, pp. 73-79, June, 1955. 

Fine, M. E.^ 

Erratum: Elastic Constants of Germanium Between 1.7° and 80°K 
J. Appl. Phys., Letter to the Editor, 26, p. 1389, Nov., 1955. 

1 Flaschen, S. S.^ 

A Barium Titanate Synthesis from Titanium Esters, J. Am. Chem. 
Soc, 77, p. 6194, Dec, 1955. 

Fletcher, R. C.,^ Yager, W. A.,* and Merritt, F. R.^ 

Observation of Quantum Effects in Cyclotron Resonance, Phys. Rev., 
Letter to the Editor, 100, pp. 747-748, Oct. 15, 1955. 

Franke, H. C.i 

Noise Measurement on Telephone Circuits, Tele-Tech., 14, pp. 85-97, 
Mar., 1955. 



1. Bell Telephone Laboratories, Inc. 



522 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 



Galt, J. K.,1 Yager, W. A./ Merritt, F. R./ Cetlin, B. B.,» and 
Bail, H. W., .Tr.^ 

Cyclotron Resonance in Metals: Bismuth, Phys. Rev., Letter to the 
Editor, 100, pp. 748-749, Oct. 15, 1955. 

Geller, S.,^ and Thurmond, C. D.' 

On the Question of a Crystalline SiO, Am. Chem. Soc. J., 77, pp. 
5285-5287, Oct. 20, 1955. 

Geschwind, S., see Dillon, J. F. 

Harker, K. J.^ 

Periodic Focusing of Beams from Partially Shielded Cathodes, I.R.E. 
Trans., ED-2, pp. 13-19, Oct., 1955. 

Hasegawa, H., see Anderson, P. W. 

Haynes, J. R.,^ and Hornbeck, J. A.^ 

Trapping of Minority Carriers in Silicon II: n-type Silicon, Phys. 
Rev., 100, pp. 606-615, Oct. 15, 1955. 

Hornbeck, J. A., see Haynes, J. R. 

Israel, J. 0.,^ Mechline, E. B.,^ and Merrill, F. F.^ 

A Portable Frequency Standard for Navigation, I.R.E. Trans., PGI-4, 
pp. 116-127, Oct., 1955. 

Jaccarino, v., see Dillon, J. F. 

Jacobson, M. J.,' Charnes, A., and Saibel, E.^ 

The Complete Journal Bearing With Circumferential Oil Inlet, Trans. 
A.S.M.E., 77, pp. 1179-1183, Nov., 1955. 

James, D. B., see Neilson, G. C. 

KoHN, W.,^ and Scheciiter, D.^ 

Theory of Acceptor Levels in Germanium, Phys. Rev., Letter to the 
Editor, 99, pp. 1903-1904, Sept. 15, 1955. 



1. Bell Telephone Laboratories, Inc. 
4. Carnegie Institute. 



TECHNICAL PAPERS 523 

Law, J. T.,1 and Meigs, P. S.^ 

The Effect of Water Vapor on Grown Germanium and Silicon n-p 
Junction Units, J. Appl. Phys., 26, pp. 1265-1273, Oct., 1955. 

Leavis, H. W.i 

Search for the Hall Effect in a Superconductor: II — Theory, Phys. 
Rev., 100, pp. 641-645, Oct. 15, 1955. 

LiNViLL, J. G.,^ and Mattson, R. H.^ 

Junction Transistor Blocking Oscillators, Proc. I.R.E., 43, pp. 1632- 
1639, Nov., 1955. 

Logan, R. A.^ 

Precipitation of Copper in Germanium, Phys. Rev., 100, pp. 615-617, 
Oct. 15, 1955. 

Logan, R. A.,^ and Schwartz, M.^ 

Restoration of Resistivity and Lifetime in Heat Treated Germanium, 
J. Appl. Phys., 26, pp. 1287-1289, Nov., 1955. 

McCall, D. W., see Shulman, R. G. 

Mason, W. P., see Bommel, H, E. 

Matthias, B. T.,^ and Corenzwit, E.^ 

Superconductivity of Zirconium Alloys, Phys. Rev., 100, pp. 626-627, 
Oct. 15, 1955. 

Mattson, R. H., see Linvill, J. G. 

Mays, J. M., see Shulman, R. G. 

Mechline, E. B., see Israel, J. 0. 

Meigs, P. S., see Law, J. T. 

Mendizza, a., see Compton, K. G. 

Merrill, F. F., see Israel, J. 0. 

' Merritt, F. R., see Fletcher, R. C., and Gait, J. K. 

Merz, W. J., see Anderson, J. R. 
1. Bell Telephone Laboratories, Inc. 



524 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Moll, J. L.^ 

Junction Transistor Electronics, Proc. I.R.E., 43, pp. 1807-1818, 
Dec, 1955. J 

MuMFORD, W. W.,^ and Schafersman^, R. L.^ ^ 

Data on Temperature Dependence of X-Band Fluorescent Lamp Noise 
Sources, I.R.E. Trans., PGI-4, pp. 40-46, Oct., 1955. 

Neilson, G. C.,^ and James, D. B.^ 

Time of Flight Spectrometer for Fast Neutrons, Rev. Sci. Instr., 26, 
pp. 1018-1023, Nov., 1955. 

Nesbitt, E. A.,^ and Williams, H. J.^ 

New Facts Concerning the Permanent Magnet Alloy, Alnico 5, Conf . 
on Magnetism and Magnetic Materials, T-78, pp. 205-209, Oct., 1955. 

Nesbitt, E. A.,^ and Williams, H. J.^ 

Shape and Crystal Anisotropy of Alnico 5, J. Appl. Phys., 26, pp. 
1217-1221, Oct., 1955. 

OWNES, C. D.i 

Stability of Molybdenum Permalloy Powder Cores, Conf. on Mag- J 
netism and Magnetic Materials, T-78, pp. 334-339, Oct., 1955. 

Pearson, G. L.,^ and Brattain, W. H.^ 

History of Semiconductor Research, Proc. I.R.E., 43, pp. 1794-1806, 
Dec, 1955. 

Pederson, L.^ 

Aluminum Die Castings in Carrier Telephone Systems, Modern 
Metals, 11, pp. 65, 68, 70, Sept., 1955. 

Prince, M. B.^ 

High-Freauency Silicon Aluminum Alloy Junction Diode, Trans. 
I.R.E., ED-2, pp. 8-9, Oct., 1955. 

Remeika, J. P., see Anderson, J. R. 

RiORDAN, J., see Fagen, R. E. 

1. Bell Telephone Laboratories, Inc. 

6. University of British Columbia, Vancouver, Canada. 



TECHNICAL PAPERS 525 

Saibel, E., see Jacobson, M. J. 
ScHAFERSMAN, R. L., See Mumford, W. W. 
Schechter, D., see Kohn, W. 

Schelkunoff, S. A.^ 

On Representation of Electromagnetic Fields in Cavities in Terms of 
Natural Modes of Oscillation, J. Appl. Phys., 26, pp. 1231-1234, Oct., 
1955. 

Schwartz, M., see Logan, R. A. 

Shulman, R. G.,1 Mays, J. M.,i and McCall, D. W.^ 

Nuclear Magnetic Resonance in Semiconductors: I — ^ Exchange 
Broadening in InSb and GaSb, Phys, Rev., 100, pp. 692-699, Oct. 
15, 1955. 

Slocum, A.,^ and Augustine, C. F.^ 

6-KMC Phase Measurement System For Traveling Wave Tube, 
Trans. I.R.E., PGI-4, pp. 145-149, Oct., 1955. 

Thurmond, C. D., see Geller, S. 

Uhlir, a., Jr.^ 

Micromachining with Virtual Electrodes, Rev. Sci. Instr., 26, pp. 
965-968, Oct., 1955. 

Ulrich, W., see Yokelson, B, J, 

Van Uitert, L. G.^ 

DC Resistivity in the Nickel and Nickel Zinc Ferrite System, J. Chem. 
Phys., 23, pp. 1883-1887, Oct., 1955. 

Van Uitert, L. G.^ 

Low Magnetic Saturation Ferrites for Microwave Applications, J. 
Appl. Phys., 26, pp. 1289-1290, Nov., 1955. 

Wannier, G. H.^ 

Possibility of a Zener Effect, Phys. Rev., Letter to the Editor, 100, 
p. 1227, Nov., 15, 1955. 



1. Bell Telephone Laboratories, Inc. 



526 the bell system technical journal, march 1956 

Wannier, G. H.^ 

Threshold Law for Multiple Ionization, Phys. Rev., 100, pp. 1180, 
Nov. 15, 1955. 

Warner, A. W., Jr., see Bommel, H. E. 

Williams, H. J., see Nesbitt, E. A. i 

Yager, W. A., see Fletcher, R. C, and Gait, J. K. 

YoKELSON, B. J.,^ and Ulrich, W.^ 

Engineering Multistage Diode Logic Circuits, Elec. Engg., 74, p. 1079, 
Dec, 1955. 



1. Bell Telephone Laboratories, Inc. 



Recent Monographs of Bell System Technical 
I Papers Not Published in This Journal* 

Allison, H. W., see Moore, G. E. 
Baker, W. 0., see Winslow, F. H. 

Basseches, H., and McLean, D. A. 

Gassing of Liquid Dielectrics Under Electrical Stress, Monograph 
2448. 

BozoRTH, R. M., TiLDEN, E. F., and Willlams, A. J. 
Anistropy and Magnetostriction of Some Ferrites, Monograph 2513. 

Bradley, W. W., see Compton, K. G. 

CoMPTON, K. G., Mendizza, a., and Bradley, W. W. 
Atmospheric Galvanic Couple Corrosion, Monograph 2470. 

Davis, J. L., see Suhl, H. 

Fagen, R. E., and Riordan, John 

Queueing Systems for Single and Multiple Operation, Monograph 
2506. 

Fine, M. E. 

Elastic Constants of Germanium Between 1.7° and 80°K, Monograph 
2479. 

FoRSTER, J. H., see Miller, L. E. 

Galt, J. K., see Yager, W. A. 

II' Geballe, T. H., see Morin, F. J. 



* Copies of these monographs may l)e obtained on request to the Publication 
Department, Bell Telephone Laboratories, Inc., 463 West Street, New York 14, 
N. Y. The numbers of the monographs should be given in all requests. 

527 



528 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

GlANOLA, U. F. 

Use of Wiedemann Effect for Magnetostrictive Coupling of Crossed 
Coils, Monograph 2492. 

Green, E. I. 
The Story of Q, Monograph 2491. 

GuLDNER, W. G., see Wooten, L. A. 

Harrower, G. a. 

Measurement of Electron Energies by Deflection in a Uniform Electric 
Field, Monograph 2495. 

Haus, H. a., and Robinson, F. N. H. 

The Minimum Noise Figure of Microwave Beam Amplifiers, Mono- 
graph 2468. 

Hines, M. E., Hoffman, G. W., and Saloom, J. A. 

Positive-ion Drainage in Magnetically Focused Electron Beams, 

Monograph 2481. 

Hoffman, G. W., see Hines, M. E. 

Kelly, M. J. 

Training Programs of Industry for Graduate Engineers, Monograph 
2512. 

Law, J. T., and Meigs, P. S. 

Water Vapor on Grown Germanium and Silicon n-p Junction Units, 
Monograph 2500. 

McAfee, K. B., Jr. 

Attachment Coefficient and Mobility of Negative Ions by a Pulse 
Techniaue, Monograph 2471. 

McLean, D. A., see Basseches, H. 

Meigs, P. S., see Law, J. T. 

Mendizza, a., see Compton, K. G. 

Merritt, F. R., see Yager, W. A. 



MONOGRAPHS 529 

Miller, L. E., and Forster, J. H. 

Accelerated Power Aging with Lithium-Doped Point Contact Transis- 
tors, Monograph 2482. 

Miller, S. L. 
Avalanche Breakdown in Germanium, Monograph 2477. 

Moore, CI. E., see Wooten, L. A. 

Moore, G. E., and Allison, H. W. 

Adsorption of Strontium and of Barium on Tungsten, Monograph 

2498. 

MoRiN, F. J., and Geballe, T. H. 

Electrical Conductivity and Seebeck Effect in Nio.so Fe2.2o04 , Mono- 
graph 2514. 

Morrison, J., see Wooten, L. A. 

Nesbitt, E. a., and Williams, H. J. 

Shape and Crystal Anisotropy of Alnico 5, Monograph 2502. 

Olmstead, p. S. 
Quality Control and Operations Research, Monograph 2530. 

Pearson, G. L., see Read, W. T., Jr. 

Pfann, W. G. 
Temperature Gradient Zone Melting, Monograph 2451. 

Poole, K. M. 
Emission from Hollow Cathodes, Monograph 2480. 

Read, W. T., Jr., and Pearson, G. L. 

^ The Electrical Effects of Dislocations in Germanium, Monograph 
! 2511. 

RiORDAN, John, see Fagen, R. E. 

Robinson, F. N. H., see Haus, H. A. 

Saloom, J. A., see Hines, M. E. 



530 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

SCHELKUNOFF, S. A. 

Electromagnetic Fields in Cavities in Terms of Natural Modes of 
Oscillation, INlonograph 2505. 

Sears, R. W. 

A Regenerative Binary Storage Tube, jNIonograph 2527. '< 

Slighter, W. P. 

Proton Magnetic Resonance in Polyamides, Monograph 2490. 

SuHL, H., Van Uitert, L. G., and Davis, J. L. 

Ferromagnetic Resonance in Magnesium-Manganese Aluminum 
Ferrite Between 160 and 1900 mc, Monograph 2472. 

Tilden, E. F., see Bozorth, R. M. 

Treuting, R. G. 

Some Aspects of Slip in Germanium, Monograph 2485. 

Uhlir, A., Jr. 

Micromachining with Virtual Electrodes, Monograph 2515. 

Van Uitert, L. G., see Suhl, H. 

Walker, L. R. 

Power Flow in Electron Beams, Monograph 2469. 

Williams, A. J., see Bozorth, R. M. 

Williams, H. J., see Nesbitt, E. A. 

WiNSLOW, F. H., Baker, W. O., Yager, W. A. 

Odd Electrons in Polymer Molecules, Monograph 2486. 

WooTEN, L. A., Moore, G. E., Guldner, W. G., and Morrison, J. 
Excess Barium in Oxide-Coated Cathodes, Monograph 2497. 

Yager, W. A., see Winslow, F. H. 

Yager, W. A., Galt, J. K., and Merritt, F. R. 

Ferromagnetic Resonance in Two Nickel-Iron Ferrites, Monograph 

2478. 



Contributors to This Issue 

Armand 0. Adam,* New York Telephone Company, 1917-1920; West- 
ern Electric Company, 1920-24; Bell Telephone Laboratories; 1925-. 
Mr. Adam tested local dial switching systems before turning to design 

j on the No. 1 and toll crossbar systems. From 1942 to 1945 he was as- 
sociated with the Bell Laboratories School For War Training. Since 
then he has been concerned with the design and development of the 
marker for the No. 5 crossbar system. Currently he is supervising a group 

I doing common control circuit development work for the crossbar tandem 

I switching system. 

i Wallace C. Babcock, A.B., Harvard University, 1919; S.B., Harvard 
University, 1922. U.S. Army, 1917-1919. American Telephone and Tele- 

i graph Company, 1922-1934; Bell Telephone Laboratories, 1934-. Mr. 
Babcock was engaged in crosstalk studies until World War IL Afterward 

, he was concerned with radio countermeasure problems for the N.D.R.C. 

' Since then he has been working on antenna development for mobile 
radio and point-to-point radio telephone systems and military projects. 

I Member of I.R.E. and Harvard Engineering Society, 

, Franklin H. Blecher, B.E.E., 1949, M.E.E., 1950 and D.E.E., 
, 1955, Brooklyn Polytechnic Institute; Polytechnic Research and De- 
' velopment Company, June, 1950 to July, 1952; Bell Telephone Labora- 
I tories 1952-. Dr. Blecher has been engaged in transistor network de- 
I velopment. His principal interest has been the application of junction 
[ transistors to feedback amplifiers used in analog and digital computers. 

He is a member of Tau Beta Pi, Eta Kappa Nu and Sigma Xi and is an 

associate member of the I.R.E. 

W. E. Danielson, B.S., 1949, M.S., 1950, Ph.D, 1952, California 

Institute of Technology; Bell Laboratories 1952-. Dr. Danielson has been 

j engaged in microwave noise studies with application to traveling-wave 

[ tubes and he has been in charge of development of traveling-wave tubes 



* Inadvertently, Mr. Adam's biography was omitted from the January issue of 
the Journal in which his article, "Crossbar Tandem as a Long Distance Switch- 
ing Equipment," appeared. 

531 



532 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

for use at 11,000 megacycles since June of 1954. He is the author of 
articles published by the Journal of Applied Physics, Proceedings of the 
I.R.E., and the B.S.T.J., and he is a Member of the American Physical 
Society, Tau Beta Pi, and Sigma Xi. 

Amos E. Joel, Jr., B.S., Massachusetts Institute of Technology, 
1940; M.S., 1942; Bell Telephone Laboratories, 1940-. IMr. Joel's first 
assignment was in relay engineering. He then worked in the crossbar 
test laboratory and later conducted fundamental development studies. 
During World War II, he made studies of communications projects 
and from 1944 to 1945 designed circuits for a relay computer. Later he 
prepared text and taught a course in switching design. The next two 
years were spent designing AM A computer circuits, and since 1949 
Mr. Joel has been engaged in making fundamental engineering studies 
and directing exploratory development of electronic switching systems. 
He was appointed Switching Systems Development Engineer in 1954. 
Member of A.I.E.E., I.R.E., Association for Computing Machinery, and 
Sigma Xi. 

Esther M. Rentrop, B.S., 1926, Louisiana State Normal College. 
Miss Rentrop joined the transmission group of the Development and 
Research Department of the American Telephone and Telegraph Com- 
pany in 1928, and transferred to Bell Laboratories in 1934. In both com- 
panies she has been concerned principally wdth control of crosstalk, both 
in field studies and transposition design work. During World War II, 
she assisted in problems of the Wire Section, Eatontown Signal Corps 
Laboratory at Fort Monmouth, and later she worked on other military 
projects at the Laboratories for the duration of the war. Miss Rentrop is 
presently a member of the noise and crosstalk studies group of the Out- 
side Plant Engineering Department and is engaged in studies of inter- 
ference prevention. 

Jack L. Rosenfeld is a student in electrical engineering at the Mas- 
sachusetts Institute of Technology. He will receive the S.M. and S.B. 
degrees in 1957. He has been with Bell Telephone Laboratories on co- 
operative assignments in microwave tube development and electronic 
central office during 1954 and 1955. He is a student member of the I.R.E. 
and a member of Tau Beta Pi and Eta Kappa Nu. 

Joseph A. Saloom, Jr., B.S., 1948, M.S., 1949, and Ph.D., 1951, all 
in Electrical Engineering, University of Illinois. He joined Bell Labora- 
tories in 1951. Mr. Saloom worked on electron tube development at' 



CONTRIBUTORS TO THIS ISSUE 533 

Murray Hill until 1955 with particular emphasis on electron beam 
studies. He is now at the Allentown, Pa., laboratory where he is en- 
gaged in the development of microwave oscillators. Member of the 
Institute of Radio Engineers, Sigma Xi, Eta Kappa Nu, Pi Mu Epsilon. 

Charles S. Thaeler, Moravian College, 1923-25, Lehigh University 
1925-28, E.E., 1928. During the summer of 1927 he was employed by the 
Bell Telephone Company of Pennsylvania, returning there after gradua- 
tion, where he was concerned with transmission engineering and the 
Toll Fundamental Plan. In 1943 he was on loan to the Operating and 
Engineering Department of the A.T.&T. Co., working on toll transmis- 
sion studies. From 1944 to the present he has been with the Operating 
and Engineering Department and is currently engaged in toll circuit 
noise and crosstalk problems on open wire and cable systems. Mr. 
Thaeler is an Associate Member of A.I.E.E., and member of Phi Beta 
I Kappa, Tau Beta Pi, and Eta Kappa Nu. 

Ping King Tien, B. S., National Central University, China, 1942; 
M.S., 1948, Ph.D., 1951, Stanford University; Stanford Microwave 
]>aboratory, 1949-50; Stanford Electronics Research Laboratory, 1950- 
52; Bell Telephone Laboratories, 1952-. Since joining the Laboratories, 
' Dr. Tien has been concerned with microwave tube research, particularly 
t raveling- wave tubes. In the course of this research he has engaged in 
studies of space charge wave amplifiers, helix propagation, electron beam 
focusing, and noise. He is a member of Sigma Xi. 

Arthur Uhlir, Jr., B.S., M.S. in Ch.E., Illinois Institute of Tech- 

jnology, 1945, 1948; S.M. and Ph.D. in Physics, University of Chicago, 

' 1950, 1952. Dr. Uhlir has been engaged in many phases of transistor 

development since joining the Laboratories in 1951, including electro- 

I chemical techniques and semiconductor device theory. Since 1952 he 

has participated in the Laboratories' Communications Development 

I'laining Program, giving instruction in semiconductors. Member of 

American Physical Society, Sigma Xi, Gamma Alpha, and the Institute 

' of Radio Engineers. 

Roger I. Wilkinson, B.S. in E.E., 1924, Prof. E.E., 1950, Iowa State 
College; Northwestern Bell Telephone Company, 1920-21; American 
Telephone and Telegraph Company, 1924-34; Bell Telephone Labora- 
tories, 1934-. As a member of the Development and Research Depart- 
iinent of the A.T.&T. Co., Mr. Wilkinson specialized in the applica- 
tions of the mathematical theory of probability to telephone problems. 



534 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1956 

Since transferring to Bell Telephone Laboratories in 1934, he has con- 
tinued in the same field of activity and is at present Traffic Studies 
Engineer responsible for probability studies and traffic research. For two 
years during World War II, in a civilian capacity, he engaged in opera- 
tions analysis studies for the Far East Air Forces in the South Pacific, 
for which he received the Medal for Merit. He has also served as a con- 
sultant to the Air Force, the Navy and the Air Navigation Delevopment 
Board. Mr. Wilkinson is a member of A.I.E.E., American Society for 
Engineering Education, American Statistical Association, Institute of 
Mathematical Statistics, Operations Research Society of America, Amer- 
ican Society for Quality Control, Eta Kappa Nu, Tau Beta Pi, Phi 
Kappa Phi and Pi ]\Iu Epislon. 



I 



I 



i 



1 p 



1 cr 



FIG. 25 EQUIVALENT RANDOM LOAD A AND TRUNKS S, FROM NON-RANDOM LOAD A',V' 




8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 



A=AVERAGE RANDOM LOAD IN ERLANGS 



» 



Copyright 1955 by Bel] Telephone Laboratories, Incorporated 



Fig. 25 - Equivalent random load A and number of trunks S, from non-random load A', V - random loads to 50 erlangs 



FIG. 26 EQUIVALENT RANDOM LOAD A AND TRUNKS S, FROM NON-RANDOM LOAD A'V 




3 4 5 6 7 

A = AVERAGE RANDOM LOAD IN ERLANGS 



10 



Copyright 1955 by Bell Telephone Laboratories, Incorporated 



Fig 26 - Equivalent random load A and number of trunks S. from non-nuidom load A'. V - random loads to 10 erlangs 



[HE BELL SYSTEM 

Jechnical journal 

fIvOTED TO THE SC I E N T I FIC^W^ AND ENGINEERING 



[PECTS OF ELECTRICAL COMMUNICATION 

KANSAS C"^ MO' 



t*M— IM I III I 



(ILUME XXXV MAY 1956 NUMBERS 



Chemical Interactions Among Defects in Germanium and Silicon 

H. REISS, C. S. FULLER AND F. J. MORIN 535 

Single Crystals of Exceptional Perfection and Uniformity by Zone 
Leveling D. c, bennett and b, sawyer 637 

Diffused p-n Junction Silicon Rectifiers M. b. prince 661 

The Forward Characteristic of the PIN Diode d. a. kleinman 685 

A Laboratory Model Magnetic Drum Translator for Toll Switch- 
ing Offices F. J. buhrendorf, h. a. henning and o. j. murphy 707 

Tables of Phase of a Semi-Infinite Unit Attenuation Slope 

D. E. THOMAS 747 



Bell System Technical Papers Not Published in This Journal 751 

Recent Bell System Monographs 759 

Contributors to This Issue 762 



COPYRIGHT 195< AMERICAN TELEPHONE AND TELEGRAPH COMPANY 



THE BELL SYSTEM TECHNICAL JOURNAL 



ADVISORY BOARD 

F. R. KAPPEL, President, Western Electric Company 

M. J. KELLY, President, Bell Telephone Laboratories 

E. J. McNBELY, ExecutivB Vice President, American 
Telephone and Telegraph Company 

EDITORIAL COMMITTEE 



B. MCMILLAN, Chairman 

A. J. BUSCH 

A. C. DICKIESON 

B. L. DIETZOLD 
K. E. GOULD 

E. I. GREEN 



R. E. HONAMAN 
H. R. HUNTLEY 

F. R. LACK 
J. R. PIERCE 
H. V. SCHMIDT 

G. N. THAYER 



EDITORIAL STAFF 

J. D. TEBO, Editor 

M . E. 8TRIEBY, Managing Editor 

R. L. SHEPHERD, Prodvction Editor 



THE BELL SYSTEM TECHNICAL JOURNAL is pubUshed six timea a year 
by the American Telephone and Telegraph Company, 195 Broadway, New York 
7, N. Y. Cleo F. Craig, President; S. Whitney Landon, Se<»etary; John J. Scan- 
Ion, Treasurer. Subscriptions are accepted at $3.00 per year. Single copies are 
75 cents each. The foreign postage is 65 cents per year or 11 cents per copy. Printed 
in U. S. A» 



THE BELL SYSTEM 

TECHNICAL JOURNAL 

VOLUME XXXV MAY 1956 number 3 



Copyright 1956, American Telephone and Telegraph Company 



Chemical Interactions Among Defects in 
Germanium and Silicon 

By HOWARD REISS, C. S. FULLER, and F. J. MORIN 

Interactio7is among dejects in germanium and silicon have been investi- 
gated. The solid solutions involved hear a strong resemblance to aqueous 
solutions insofar as they represent media for chemical reactions. Such 
phenomena as acid-base neutralization, complex ion formation, andion pair- 
ing, all take place. These phenomena, besides being of interest in themselves, 
are uscfid in studying the properties of the semiconductors in which they 
occur. The following article is a blend of theory ami experime7it, and de- 
scribes developments in this field during the past few years. 

CONTENTS 

I . Introduction 536 

IL Electrons and Holes as Chemical Entities 537 

in. Application of the Mass Action Principle 546 

IV. Further Applications of the Mass Action Principle 550 

V. Complex Ion Formation 557 

VI. Ion Pairing 565 

VII. Theories of Ion Pairing 567 

VIII. Phenomena Associated with Ion Pairing in Semiconductors 575 

IX. Pairing Calculations 578 

X. Theory of Relaxation 582 

XI. Investigation of Ion Pairing by Diffusion 591 

535 



536 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 

XII. Investigation of Ion Pairing by Its Effect on Carrier Mobility 601 

XIII. Relaxation Studies 607 

XIV. The Effect of Ion Pairing on Energy Levels 610 

XV. Research Possibilities 611 

Acknowledgements 613 

Appendix A — The Effect of Ion Pairing on Solubility 613 

Appendix B — Concentration Dependence of Diffusivity in the Pres- 
ence of Ion Pairing 617 

Appendix C — Solution of Boundary Value Problem for Relaxation. . 619 

Appendix D —Minimization of the Diffusion Potential 623 

Appendix E — Calculation of Diffusivities from Conductances of 

Diffusion Layers 626 

Glossary of Symbols 630 

References 634 

I. INTRODUCTION 

The effort of Wagner' and his school to bring defects in solids into the 
domain of chemical reactants has provided a framework within which • 
various abstruse statistical phenomena can be viewed in terms of the 
intuitive principle of mass action.^ Most of the work to date in this field ' 
has been performed on oxide and sulfide semiconductors or on ionic com- '[ 
pounds such as silver chloride. In these materials the control of defects ■ 
(impurities are to be regarded as defects) is not all that might be desired, i 
and so with a few exceptions, experiments have been either semiquanti- . 
tative or even qualitative. i 

With the emergence of widespread interest in semi-conductors, cul- : 
minating in the perfection of the transistor, quantities of extremely pure , 
single crystal germanium and silicon have become available. In addition 
the physical properties, and even the quantum mechanical theory of the 
behavior of these substances have been widely investigated, so that a 
great deal of information concerning them exists. Coupled with the fact 
that defects in them, especially impurities, are particularly susceptible 
to control, these circumstances render germanium and silicon ideal sub- 
stances in which to test many of the concepts associated with defect I 
interactions. 

This view was adopted at Bell Telephone Laboratories a few years ago 
when experimental work was first undertaken. Not only has it been 
possible to demonstrate quantitatively the validity of the mass action 
principle applied to defects, but new kinds of interactions have been 
discovered and studied. Furthermore new techniques of measurement 
have been developed which we feel open the way for broader investiga- 
tion of a still largely unexplored field. 

In fact solids (particularly semiconductors like germanium and silicon) 



CHEMICAL INTERACTIONS AMONG DEFECTS IN Gg AND Si 537 

appear in every respect to provide a medium for chemical reactivity 
similar to liquids, particularly water. Such pehnomena as acid-base reac- 
tions, complex ion formation, and electrolyte phenomena such as Debye 
Hiickel effects, ion pairing, etc., all seem to take place. 

Besides the experiments theoretical work has been done in an attempt 
to define the limits of validity of the mass action principle, to furnish 
more refined electrolyte theories, and most importantly, to provide firm 
theoretical bases for entirely new phenomena such as ion pair relaxation 
processes. 

The consequence is that the field of diamond lattice^ semiconductors 
which has previously engaged the special interests of physicists threatens 
to become important to chemists. Semiconductor crystals are of interest, 
not only because of the specific chemical processes occurring in these 
substances, but also because they serve as proving grounds for certain 
ideas current among chemists, such as electrolyte theory. On the other 
hand renewed interest is induced on the part of physicists because chem- 
ical effects like ion pairing engender new physical effects. 

The purpose of this paper is to present the field of defect interaction 
as it now stands, in a manner intelligible to both physicists and chem- 
ists. However, this is not a review paper. Most of the experimental re- 
sults, and particularly the theories which are fully derived in the text or 
the appendices are entirely new. Some allusion will be made to published 
work, particularly to descriptions of the results of some previous theories, 
in order to round out the development. 

The governing theme of the article lies in the analogy between 
semiconductors and aqueous solutions. This analogy is useful not so 
j much for what it explains, but for the experiments which it suggests. 
: More than once it has stimulated us to new investigations. 
1 In our work we have made extensive use of lithium as an impurity. 
This is so because lithium can be employed with special ease to demon- 
strate most of the concepts we have in mind. This specialization should 
not obscure the fact that other impurities although not well suited to 
the performance of accurate measurements, will exhibit much of the 
same behavior. 



II. ELECTRONS AND HOLES AS CHEMICAL ENTITIES 



Since electrons and holes'* are obvious occupants of semiconductors 
I like germanium and silicon, and are intimately associated with the pres- 
[ence of donor and acceptor impurities,^ it is fitting to inciuire into the 
f roles they may play in chemical interactions between donors and ac- 



538 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 

ceptors. This question has been discussed in two papers,^- ® and only its 
principle aspects will be considered. 

To gain perspective it is convenient to consider a system representing 
the prototype of most systems to be discussed here. Consider a single 
crystal of silicon containing substitutional boron atoms. Boron, a group 
III element, is an acceptor, and being substitutional cannot readily dif- 
fuse^ at temperatures much below the melting point of silicon. If this 
crystal is immersed in a solution containing lithium, e.g., a solution of 
lithium in molten tin, lithium will diffuse into it and behave as a donor. 
Evidence suggests that lithium dissolves interstitially in silicon, thereby 
accounting for the fact that it possesses a high diffusivity^ at a tempera- 
ture where boron is immobile, for example, below 300°C. When the 
lithium is uniformly distributed throughout the silicon its solubility in 
relation to the external phase can be determined. Throughout this process 
boron remains fixed in the lattice. 

If both lithium and boron were inert impurities the solubility of the 
former would not be expected to depend on the presence or absence of 
the latter, for the level of solubility is low enough to render (under 
ordinary circumstances) the solid solution ideal.* On the other hand the 
impurities exhibit donor and acceptor behaviors respectively, and some 
unusual effects might exist. We shall first speculate on the simplest possi- 
bility in this direction, with the assistance of the set of equilibrium reac- 
tions diagrammed below.* , 

Li{Sn) «=± Li{Si) t± Li+ + e~ 

+ 
B{Si) :f±B- + e+ (2.1) 

Ti 
eV 

At the left lithium in tin is shown as Li(Sn). It is in reversible equilib- 
rium with Li(Si), un-ionized lithium dissolved in silicon. The latter, in 
turn, ionizes to yield a positive Li'^ ion and a conduction electron, e~. 
Boron, confined to the silicon lattice as B(Si) ionizes as an acceptor to 
give B" and a positive hole, e"*". The conduction electron, e~, may fall 
into a valence band hole, e"*", to form a recombined hole-electron pair, 
e"^e~. This process and its reverse are indicated by the vertical equilibrium 
at the right. 

All of the reactions in (2.1), occuring within the silicon crystal are 
describable in terms of tansitions between states in the energy band dia- 



A glossary of symbols is given at the end of this article. 



CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 539 



gram of silicon, exhibited in Fig. 1. The conduction band, the valence 
band, and the forbidden gap are shown. Lithium and boron both intro- 
duce localized energy states in the range of forbidden energies. The state 
for lithium lies just below the bottom of the conduction band while that 
for boron lies just over the top of the valence band. The separations in 
energy between most donors or acceptors and their nearest bands are of 
the order of hundredths of an electron volt while the breadths of the for- 
bidden gaps in germanium or silicon are of the order of one electron volt. 

Process 1 in Fig. 1 involving a transition between the donor level and 
conduction band corresponds to the ionization of lithium in (2.1). Proc- 
ess 2 is the ionization of boron while process 3 represents hole-electron 
recombination and generation. The various energies of transition are the 
heats of reaction of the chemical-like changes in (2.1). 

Proceeding in the chemists fashion one might argue as follows concern- 
ing (2.1). If e'^e' is a stable compound, as it is at fairly low temperatures, 
then its formation should exliaust the solution of electrons, forcing the 
set of lithium equilibria to the right. In this way the presence of boron, 
supplying holes toward the formation of e'^e", increases the solubility of 
lithium. In fact if e"*" is regarded as the solid state analogue of the hydro- 
gen ion in aqueous solution, and e~ as the counterpart of the hydroxyl 
ion, then the donor, lithium, may be considered a base while boron, may 
be considered an acid. Furthermore e'*"e~ must correspond to water. 
Thus the scheme in (2.1) is analogous to a neutralization reaction in 
which the weakly ionized substance is e'*"e~. 

If the immobile boron atoms were replaced by immobile donors, e.g., 
I phosphorus atoms, a reduction, rather than an increase, in the solubility 




IT 



BORON LEVELS (ACCEPTORS) 



x : ;w>/.-v v.^;i::-.:-:VX;^;;;v valence band v. 



DISTANCE 



Fig. 1 — Energy band diagram showing the chemical equilibria of (2.1). 



540 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 

of lithium might be expected on the basis of an oversupply of electrons 
(i.e., by the common ion effect^"). In that case we would have a base 
displacing another base from solution. 

The intimate comparison between this kind of solution and an aqueous 
solution is worth emphasizing not so much for what it adds to one's 
understanding of the situation but rather for the further effects it sug- 
gests along the lines of analogy. These additional phenomena have been 
looked for and found, and Mill be discussed later in this article. 

The scheme shown in (2.1) should be applicable, in principle, to other 
donors and acceptors and to germanium and other semiconductors as 
well as silicon. Furthermore the external phase may be any one of a suit- 
al)le variety, and need not even be liquid. Other systems, however, are 
not as convenient, especially in regard to the ease of equilibration of an 
impurity over the parts of an heterogeneous system. The lengths to which 
one can go in comparing electrolytes and semiconductors are discussed 
in a recent paper." 

In order to quantify the scheme of (2.1) it seems natural to invoke the 
law of mass action. Treatments in which holes and electrons are in- 
volved in mass action expressions are not new, although systems forming 
such perfect analogies to aqueous solutions do not seem to have been 
discussed in the past. For example, in connection with the oxidation of 
copper Wagner " writes 

4Cu -f O2 ^ 2CU2O -f 40" + 4e+ (2.2) 

in which D ~ is a negatively charged cation vacancy in the CU2O lattice, 
and e"^ is a hole. Wagner proceeds to invoke the law of mass action in 
order to compute the oxygen pressure dependence in this system. 

In another example Baumbach and Wagner^^ and others have investi- 
gated oxygen pressure over non-stoichiometric zinc oxide. They consider 
the possible reactions 

2ZnO ;=± 2Zn + O2 t\ 

u 

2Z?i+ i^ 2Zn++ -f 2e" (2.3) 

+ 

2e- 

and apply the law of mass action. In (2.3) the various states of Zii are 
presumably interstitial. 

Kroger and Vink have recently considered the problem in oxides and 
sulfides in a rathcM- general way. However in none of the oxide-sulfidc 
systems has it been possible to achieve really quantitative results. In 



CHEMICAL INTEKACTIONS AMONG DEFECTS IN Ge AND Si 541 

contrast silicon and germanium offer possibilities of an entirely new order. 
The advent of the transistor has not only provided large supplies of pure 
single crystal material, but it has also made available a store of funda- 
mental information concerning the physical properties of these sub- 
stances. For example, data exists on their energy band diagrams includ- 
ing impuritj^ states — also on resistivity — impurity density curves, 
diffusivities of impurities, etc. Furthermore, the amount of ionizable 
impurities can be controlled within narrow limits, and can be changed 
at will and measured accurately. Consequently it is reasonable to assume 
that experiments on germanium and silicon will be more successful than 
similar investigations using other materials. 

A t this point it is in order to examine whether or not the treatment of 
electrons and holes as normal chemical entities satisfying the law of 
mass action is altogether simple and straightforward. This problem has 
been investigated by Reiss who found the treatment permissible only 
as long as the statistics satisfied by holes and electrons remain classical. 
The validity of this contention can be seen in a very simple manner. 
Consider a system like that in (2.1). Let the total concentration of donor 
(ionized and un-ionized) be No , the concentration of ionized donor be 
D"*", the concentration of conduction electrons be n, and that of valence 
band holes be p. Let A''^ and A~ denote the concentrations of total ac- 
ceptor and acceptor ions respectively. Finally, let a be the thermody- 
namic activity'^ of the donor (lithium in (2.1)) in the external phase. 

Then, corresponding to the heterogeneous equilibrium in which lith- 
ium distributes itself between the two phases we can write 

^» - ^" = K, (2.4) 



a 



in which Ko depends on temperature, but not on composition. This as- 
sumes the semiconductor to be dilute enough in donor so that the ac- 
tivity of un-ionized donor can be replaced by its concentration. No — D^. 
For the ionization of the donor we can write the mass action relation, 



Z)+ 



n 



and for the acceptor. 



Nd - D+ 
A~p 



= Kd (2.5) 



= Ka (2.6) 



iVx - A- 
while for the electron-hole recombination equilibrium 

np = Ki (2.7) 



542 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 

In (2.5), (2.6), and (2.7) all the i^'s are independent of composition. To 
these equations is added the charge neutrality condition, 

D+ + p = A~ + 7i (2.8) 

Equations (2.4) through (2.8) are enough to determine No in its de- 
pendence on Na , «, and the various K's. Together they represent the 
mass action approach. To demonstrate their validity it is necessary to 
appeal to statistical considerations. 

Thus Nd — D^, the concentration of un-ionized donor is really the 
density of electrons in the donor level of the energy diagram for the semi- 
conductor. According to Fermi statistics this density is given by 

No- D+ = No/{l + M exp \{Eu - F)/kT]} . (2.9) 

in which Ed is the energy of the donor level, F is the Fermi level, k, 
the Boltzmann constant, and T, the temperature. Furthermore, accord- 
ing to Fermi statistics, n, the total density of electrons in the conduction 
band is 



n 



= E ^y {1 + exp [{Ei - F)/kT]} (2.10) 



where Qi is the density of levels of energy, Ei , in the conduction band, 
and the sum extends over all states in that band. Similar expressions are 
available for the occupation of the acceptor level and the valence band. 
F is usually determined by summing over all expressions like (2.9) and 
(2.10) and equating the result to the total number of electrons in the 
system. This operation corresponds exactly to applying the conserva- 
tion condition, (2.8). It is obvious from the manner of its determina- 
tion that F depends upon No — D^y n, etc. 

If we now form the expression on the left of (2.5) by substituting for 
each factor in it from (2.9) and (2.10), it is obvious that the result de- 
pends in a very complicated fashion upon F, and so cannot be the con- 
stant, Kd , independent of composition, since in the last paragraph F 
was shown to depend on composition. On the other hand if attention is 
confined to the limit in which classical statistics apply^ the unities in 
the denominators of (2.9) and (2.10) can be disregarded in comparison 
to the exponentials, and those equations become 



1 



No - /)+ = 2Noe''"\-'''"'' (2.11); 

and 

n = e 



I 



^"'' Z 9ie~"'"" (2.12) 



I 



CHEMICAL INTERACTIONS AMONG DEFECTS IN Ge AND Si 543 

respectively. Moreover, from (2.11) 

i)+ ^ Nn[l - 2e"'^e-^'"'^] = Nu (2.13) 

where the second term in brackets is ignored for the same reason as unity 
in the denominators of (2.9) and (2.10). Substituting (2.11) through 
(2.13) into (2.5) yields 

D^n _ ?^-- (2.14) 

in which the right side is truly independent of composition, since F has 
cancelled out of the expression. Similar arguments hold for (2.6) and 
(2.7). Therefore in the classical limit the law of mass action is valid, at 
least insofar as internal equili