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THE BELL SySTj^ivt;: (
TECHNICAL JOURNAL
A JOURNAL DEVOTED TO THE
SCIENTIFIC AND ENGINEERING
ASPECTS OF ELECTRICAL
COMMUNICATION
EDITORIAL BOARD
F. R. Kappel O. E. Buckley
H. S. Osborne M. J. Kelly
J. J. PiLLioD A. B. Clark
R. BOWN D. A. QUARLES
F. J. Feely
J. O. Perrine, Editor P. C. Jones, Associate Editor
TABLE OF CONTENTS
AND
INDEX
VOLUME XXIX
1950
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
NEW YORK
I'kiNii.n IN U.S.A.
THE BELL SYSTEM
TECHNICAL JOURNAL
VOLUME XXIX, 195
o
Table of Contents
January, 1950
Traveling- Wave Tubes — /. R. Pierce 1
Communication in the Presence of Noise — Probability of Error for Two
Encoding Schemes — S. 0. Rice 60
Realization of a Constant Phase Difference — Sidney Darlington 94
Conversion of Concentrated Loads on Wood Crossarms to Loads Dis-
tributed at Each Pin Position — R. C. Eggleston 105
The Linear Theory of Fluctuations Arising from Diifusional Mecha-
nisms— An Attempt at a Theory of Contact Noise — /. M. Richard-
son 117
April, 1950
Error Detecting and Error Correcting Codes — R. W. Hamming 147
Optical Properties and the Electro-optic and Photoelastic Effects in
Crystals Expressed in Tensor Form — W. P. Mason 161
Traveling- Wave Tubes [Second Installment] — /. R. Pierce 189
Factors Affecting Magnetic Quality — R. M. Bozorth 251
JULY, 1950
Principles and Applications of Waveguide Transmission — G. C. South-
worth 295
Memory Requirements in a Telephone Exchange — C. E. Shannon .... 343
Matter, A Mode of Motion— 7^. V. L. Hartley 350
The Reflection of Diverging Waves by a Gyrostatic Medium -R. V . L.
Hartley 369
Traveling- Wave Tubes [Third Installment] — /. R. Pierce 390
iii
iv bell system technical journal
': /: ; * \ '/: '".'', ':..'* ^ }• . ■ ^Octobsr, 1950
l^eory D.f 'Relatian Jjetwqej^ tiol-e Concentration and Characteristics
ol Cjefmanium i\n.nl.i^6niVLQti—J. Bardeen 469
Design Factors of t.!ie.Bell Telephone Laboratories 155v3 Triode — /. A.
Morion and^^.^M"- Ryder 496
A New Microwave Triode: Its Performance as a Modulator and as an
Ampliller — A. E. Bowen and W. W. Mumford 531
A Wide Range Microwave Sweeping Oscillator — M. E. Hines 553
Theory of the Flow of Electrons and Holes in (iermanium and Other
Semiconductors — W. van Roosbroeck 560
Traveling-Wave Tubes [Fourth Installment] — /. R. Pierce 608
Index to Volume XXIX
A
Amj)lifier. A New Microwave Triode: Its Performance as a Modulator and an Amplifier,
.1. E. Binveii and W. W. Mioiiford, page 531.
B
Bardeen, J., Theory of Relation between Hole Concentration and Characteristics of Ger-
manium Point Contacts, page 469.
Bowen, A. E. and W. W . Mumford, A New Microwave Triode: Its Performance as a Modu-
lator and as an Amplifier, page 531.
Bozorth, R. M., Factors Affecting Magnetic Quality, page 251.
Codes, Error Detecting and Error Correcting, R. W. Hamming, page 147.
Communication in the Presence of Noise — Probaljility of Error for Two Encoding Schemes,
5. O. Rice, page 60.
Contacts, Germanium Point, Theory of Relation between Hole Concentration and Char-
acteristics of, /. Bardeen, page 469.
Crossarms, Wood, Conversion of Concentrated Loads on to Loads Distributed at Each
Pin Position, R. C. Eggleston, page 105.
Crystals, Optical Properties and the Electro-optic and Photoelastic Effects in. Expressed
in Tensor Form, IF. P. Mason, page 161.
D
Darlington, Sidney, Realization of a Constant Phase Difference, page 94.
Design Factors of the Bell Telephone Laboratories 1553 Triode, J. A. Morion and R. M.
Ryder, page 496.
Eggleston, R. C, Conversion of Concentrated Loads on Wood Crossarms to Loads Distri-
buted at Each Pin Position, page 105.
Electrons and Holes in Germanium and Other Semiconductors, Theory of the Flow of,
IF. van Roosbroeck, page 560.
Electro-optic and Photoelastic Effects in Crystals Expressed in Tensor Form, Optical
Properties and the, IF. P. Mason, page 161.
Error for Two Encoding Schemes, Probability of — Communication in the Presence of
Noise, 5. 0. Rice, page 60.
Error Detecting and Error Correcting Codes, R. W. Hamming, page 147.
Exchange, Telephone, Memory Requirements in a, C. E. Shannon, page 343.
Flow of Electrons and Holes in Germanium and Other Semiconductors, Thcor\- of the.
W. van Roosbroeck, page 560.
Fluctuations Arising from Diffusional Mechanisms, The Linear Theory of — .\n Allemj)t
at a Theory of Contact Noise, ./. .1/. Richardson, page 117.
G
Germanium and Other Semiconductors, Theory of the Flow of Electrons and Holes in,
\V. van Roosbroeck, page 560.
Germanium Point Contacts, Theory of Relation between Hole Concentration ami Charac-
teristics of, /. Bardeen, page 469.
V
vi BELL SYSTEM TECHNICAL JOURNAL
H
Ilammhig, R. W., Error Detecting and Error Correcting Codes, page 147.
Ihirtley, R. V. L., Mat ler, A Mode of Motion, page 350. The Redection of Diverging Waves
l)y a Gyrostatic Medium, page 369.
nines, M. E., A Wide Range .Microwave Sweeping Oscillator, i)age 553.
Hole Concentration and Characteristics of Germanium Point Contacts, Theory of Rela-
tion lietween, ./. Hardecn. i)agc 469.
Holes, Theory of the Flow of IClectrons and, in (Jcrmatiium and Other Semiconductors,
ir. vdii kcoshrot'ck, page 560.
Loads. Concentrated, Conversion of on Wood Crossarms to Loads Dislrilmted at Each
Pin Position, R. C. Eggleston, [^age 105.
M
Magnetic Quality, Factors AtTecting, R. M. Bozoiili, ])age 251.
Mason, IF. P., Optical Properties and the Electro-optic and Photoelastic Etiects in Crystals
Exjjressed in Tensor Form, page 161.
Matter, A Mode of .Motion, R. ]'. L. Hartley, page 350.
Medium, Gyrostatic, The Rclleclion of Diverging Waves by a, R. F. L. Hartley, page 369.
Memory Requirements in a Telephone E.xchange, C. E. Shannon, page 343.
Microwave Sweeping Oscillator, .\ Wide Range, .1/. E. Hines, page 553.
Microwave Triode, .\ New: Tts Performance as a Modulator and as an Amplifier, A. E.
Bowen and W. IF. Mumford, page 531.
Modulator. A New Microwave Triode: Its Performance as a Modulator and as an .Ampli-
fier. A. E. Bowen and W. IF. Mumford, page 531.
Morton, J. A. and R. M. Ryder, Design Factors of the Bell Telephone Laboratories 1553
Triode, page 496.
Mumford, W. IF. and A. E. Bowen. A New Microwave Triode: Its Performance as a IModu-
lator and as an Amplifier, page 531.
N
Noise, Communication in the Presence of — Probability of Error for Two Flncoding
Schemes, 5. O. Rice, page 60.
Noise, Contact, An Attempt at a Theory of — The Linear Theory of Fluclualions .\rising
from DilTusional Mechanisms, /. .1/. Richardson, page 117.
Optical Properties and the Electro-optic and Pholoelaslic Effects in Crystals Exjjressed
in Tensor Form, IF. P. Mason, page 161.
Oscillator, .\ Wide Range Microwave Sweejiing, .1/. E. Hines, page 553.
Phase DifTerence, Constant, Realization of a, Sidney Darlini^ton, page 94.
Photoelastic I'".ffecls in Crjstals lOxpressed in Tensor Form, Optical Properties and the
Electro-optic and, IF. P. Mason, i)age 161.
Pierce, J. R., 'Fraveling-Wave Tuijes, page 1. Traveling- Wave Tubes [Second Installment]
page 189. 'Fraveling-Wave Tubes [Third Installment], i)age 390. Traveling-Wave
'Ful)es [Fourth Installment], page 608.
Probai)ility of F-rror for Two Encoding Schemes -Communication in the Presence of
Noise, .S". O. Rice, page 60.
(^)uaiity, Magnetic, Factors .MTecling, A'. ,1/. Bozortli, jjage 251.
INDEX
R
Rice, S. O., Communication in the Presence of Noise — Probability of Error for Two En-
coding Schemes, page 60.
Richardson, J. M., The Linear Theor}' of Fluctuations Arising from DifYusional Mecha-
nisms— An Attempt at a Theory of Contact Noise, page 117.
Ryder, R. M. and J. A. Morton, Design Factors of the Bell Telephone Laboratories 1553
Triode, page 496.
Semiconductors, Germanium and Other, Theor>- of the Flow of Electrons and Holes in,
\V. van Roosbroeck, page 560.
Shannon, C. E., Memory Rec]uirenients in a Telephone Exchange, page 343.
Soiithivorth, G. C, Principles and Applications of Waveguide Transmission, page 295.
Transmission, Waveguide, Principles and Applications of, G. C. Soiithivorth, \mg& 295.
Traveling- Wave Tubes, J . R. Pierce:
First Listallment, page 1. Second Installment, page 189. Third Installment, page 390.
Fourth Installment, page 608.
Triode, Bell Telephone Laboratories 1553, Design Factors of the, /. A. Morton and R. M
Ryder, page 496.
Triode, Microwave, A New: Its Performance as a Modulator and as an Amplifier, .4. E.
Boiven and W . W . Mumford, page 531.
Tubes, Traveling-Wave, /. R. Pierce — See installments listed above, under "Traveling-
Wave Tubes."
van Roosbroeck, W., Theory of the Flow of Electrons and Holes in Germanium and Other
Semiconductors, page 560.
W
Waveguide Transmission, Principles and Applications of, G. C. Southworth, page 295.
Waves, Diverging, The Reflection of by a Gyrostatic Medium, R. V. L. Hartley, page 369.
Wide Range Microwave Sweeping Oscillator, A., M. E. Hlnes, page 553.
VOLUME XXIX JANUARY, 1950 no. i
THE BELL SYSTEM '^ -
■2.3 Cn
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
Traveling-Wave Tubes J. R. Pierce 1
Communication in the Presence of Noise — ^Probability of
Error for Two Encoding Schemes ;S. 0. Rice 60
Realization of a Constant Phase Difference
Sidney Darlington 94
Conversion of Concentrated Loads on Wood Crossarms to
Loads Distributed at Each Pin Position
R. C. Eggleston 105
The Linear Theory of Fluctuations Arising from Diffusional
Mechanisms — ^An Attempt at a Theory of Contact
Noise J. M. Richardson 117
Abstracts of Technical Articles by Bell System Authors 142
Contributors to this Issue 146
50^ Copyright, 1950 $1.50
per copy American Telephone and Telegraph Company per Year
THE BELL SYSTEM TECHNICAL JOURNAL
Published quarterly by the
American Telephone and Telegraph Company
195 Broadway, New York, N. Y.
EDITORIAL BOARD
F. R. Kappel O. E. Buckley
H. S. Osborne M. J. Kelly
J. J. Pilliod A. B. Clark
F. J. Feely
J. O. Perrine, Editor P. C. Jones, Associate Editor
SUBSCRIPTIONS
Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each.
The foreign postage is 35 cents per year or 9 cents per copy.
»i»i»ii»^ ■■»■■■
PRINTED IN U. S. A»
The Bell System Technical Journal
Vol. XXIX January, 1950 JSfo. 1
Copyright, 1950, American Telephone and Telegraph Company
Traveling-Wave Tubes
By J. R. PIERCE
Copyright, 1950, D. Van Nostrand Company, Inc.
The following material on traveling-wave tubes is taken from a book which
will be published by Van Nostrand in September, 1950. Substantially the
entire contents of the book will be published in this and the three succeeding
issues of the Bell System Technical Journal.
This material will cover in detail the theory of traveling-wave amplifiers. In
addition, brief discussions of magnetron amplifiers and double-stream amplifiers
are included. E.xperimental material is drawn on in a general way only, as in-
dicating the range of validity of the theoretical treatments.
The material deals only with the high-frequency electronic aiul circuit be-
havior of tubes. Such matters as matching into circuits are not considered;
neither are problems of beam formation and electron focusing, which have been
dealt with elsewhere.^
The material opens with the presentation of a simplified theory of the travel-
ing-wave tube. A discussion of circuits follows, including helix calculations, a
treatment of filter-type circuits, some general circuit considerations ivhich show
that gain will be highest for low group velocities and low stored energies, and a
justification of a simple transmission line treatment of circuits by means of an
expansion in terms of the normal modes of propagation of a circuit. Then a de-
tailed analysis of overall electronic and circuit behavior is made, including a
discussion of various electronic and circuit waves, the fitting of boundary con-
ditions to obtain overall gain, noise figure calculations, transverse motions of
electrons a) id field solutions appropriate to broad electron streams. Short treat-
ments of the magnetron amplifier and the double-stream amplifier follow.
^ For instance, "Theory and Design of Electron Beams," J. K. Pierce, Van Nostrand,
1949.
BELL SYSTEM TECHNICAL JOURNAL
CHAPTER I
INTRODUCTION
ASTRONOMERS are interested in stars and galaxies, physicists in
' atoms and crystals, and biologists in cells and tissues because these
are natural objects which are always with us and which we must under-
stand. The traveling- wave tube is a constructed complication, and it can
be of interest only when and as long as it successfully competes with older
and newer microwave devices. In this relative sense, it is successful and
hence important.
This does not mean that the traveling-wave tube is better than other
microwave tubes in all respects. As yet it is somewhat inefficient compared
with most magnetrons and even with some klystrons, although efficiencies
of over 10 per cent have been attained. It seems reasonable that the effi-
ciency of traveling-wave tubes will improve with time, and a related device,
the magnetron amplifier, promises high efficiencies. Still, efficiency is not the
chief merit of the traveling-wave tube.
Nor is gain, although the traveling-wave tubes havebeenbuilt with gains
of over 30 db, gains which are rivaled only by the newer double-stream
amplifier and perhaps by multi- resonator klystrons.
In noise figure the traveling-wave tube appears to be superior to other
microwave devices, and noise figures of around 12 db have been reported.
This is certainly a very important point in its favor.
Structurally, the traveling-wave tube is simple, and this too is impor-
tant. Simplicity of structure has made it possible to build successful ampli-
fiers for frequencies as high as 48,000 megacycles (6.25 mm). When we con-
sider that successful traveling-wave tubes have been built for 200 mc, we
realize that the traveling-wave amplifier covers an enormous range of fre-
quencies.
The really vital feature of the traveling-wave tube, however, the new
feature which makes it different from and superior to earlier devices, is its
tremendous bandwidth. I
It is comparatively easy to build tubes with a 20 per cent bandwidth at |
4,000 mc, that is, with a bandwidth of 800 mc, and L. M. Eield has reported '
a bandwidth of 3 to 1 extending from 350 mc to 1,050 mc. There seems no
reason why even broader bandwidths should not be attained.
As it happens, there is a current need for more bandwidth in the general
fiekl of communication. For one thing, the rate of transmission of intclli-
TRAVELING-WAVE TUBES 3
gence by telegraph, by telephone or by facsimile is directly proportional
to bandwidth; and, with an increase in communication in all of these fields,
more bandwidth is needed.
Further, new services require much more bandwidth than old services.
A bandwidth of 4,000 cycles suffices for a telephone conversation. A band-
width of 15,000 cycles is required for a very-high-fidelity program circuit.
A single black-and-white television channel occupies a bandwidth of about
4 mc, or approximately a thousand times the bandwidth required for te-
lephony.
Beyond these requirements for greater bandwidth to transmit greater
amounts of intelligence and to provide new types of service, there is cur-
rently a third need for more bandwidth. In FM broadcasting, a radio fre-
quency bandwidth of 150 kc is used in transmitting a 15 kc audio channel.
This ten-fold increase in bandwidth does not represent a waste of frequency
space, because by using the extra bandwidth a considerable immunity to
noise and interference is achieved. Other attractive types of modulation,
such as PCM (pulse code modulation) also make use of wide bandwidths
in overcoming distortion, noise and interference.
At present, the media of communication which have been used in the past
are becoming increasingly crowded. With a bandwidth of about 3 mc,
approximately 600 telephone channels can be transmitted on a single coaxial
cable. It is very hard to make amplifiers which have the high quality neces-
sary for single sideband transmission with bandwidths more than a few times
broader than this. In television there are a number of channels suitable for
local broadcasting in the range around 100 mc, and amplifiers sufficiently
broad and of sufficiently good quality to amphfy a single television channel
for a small number of times are available. It is clear, however, that at these
lower frequencies it would be very difficult to provide a number of long-haul
television channels and to increase telephone and other services substan-
tially.
Fortunately, the microwave spectrum, wh'ch has been exploited increas-
ingly since the war, provides a great deal of new frequency space. For in-
stance, the entire broadcast band, which is about 1 mc wide, is not sufficient
for one television signal. The small part of the microwave spectrum in the
[ wavelength range from 6 to 7^ cm has a frequency range of 1,000 mc, which
' is sufficient to transmit many simultaneous television channels, even when
broad-band methods such as FM or PCM are used.
In order fully to exploit the microwave spectrum, it is desirable to have
! amplifiers with bandwidths commensurate with the frequency space avail-
i able. This is partly because one wishes to send a great deal of information
in the microwave range: a great many telephone channels and a substan-
tial number of television channels. There is another reason why very broad
4 BELL SYSTEM TECHNICAL JOURNAL
bands are needed in the microwave range. In providing an integrated nation-
wide communication service, it is necessary for the signals to be ampUfied
by many repeaters. Amplification of the single-sideband type of signal used
in coaxial systems, or even amplification of amplitude modulated signals,
requires a freedom from distortion in amplifiers which it seems almost
impossible to attain at microwave frequencies, and a freedom from inter-
fering signals which it will be very difiicult to attain. For these reasons, it
seems almost essential to rely on methods of modulation which use a large
bandwidth in order to overcome both amplifier distortion and also inter-
ference.
Many microwave amplifiers are inferior in bandwidth to amplifiers avail-
able at lower frequencies. Klystrons give perhaps a little less bandwidth than
good low-frequency pentodes. The type 416A triode, recently developed at
Bell Telephone Laboratories, gives bandwidths in the 4,000 mc range some-
what larger than those attainable at lower frequencies. Both the klystron
and the triode have, however, the same fundamental limitation as do other
conventional tubes. As the band is broadened at any frequency, the gain is
necessarily decreased, and for a given tube there is a bandwidth beyond
which no gain is available. This is so because the signal must be applied by
means of some sort of resonant circuit across a capacitance at the input of
the tube.
In the traveling-wave tube, this limitation is overcome completely. There
is no input capacitance nor any resonant circuit. The tube is a smooth trans-
mission line with a negative attenuation in the forward direction and a
positive attenuation in the backward direction. The bandwidth can be
limited by transducers connecting the circuit of the tube to the source and
the load, but the bandwidth of such transducers can be made very great.
The tube itself has a gradual change of gain with frequency, and we have seen
that this allows a bandwidth of three times and perhaps more. This means
that bandwidths of more than 1,000 mc are available in the microwave
range. Such bandwidths are indeed so great that at present we have no means
for fully exploiting them.
In all, the traveling-wave tube compares favorably with other microwave
devices in gain, in noise figure, in simplicity of construction and in fre-
quency range. While it is not as good as the magnetron in efficiency, reason-
able efficiencies can be attained and greater efliciencies are to be expected.
Finally, it does provide amplification over a bandwidth commensurate with
the frequency space available at microwaves.
The purpose of this book is to collect and present theoretical material
which will be useful to those who want to know about, to design or to do
research on traveling-wave tubes. Some of this material has appeared in
print. Other parts of the material are new. The old material and the new
material have been given a common notation.
TRAVELING-WAVE TUBES 5
The material covers the radio-frequency aspects of the electronic behavior
of the tube and its internal circuit behavior. Matters such as matching into
and out of the slow-wave structures which are described are not considered.
Neither are problems of producing and focusing electron beams, which
have been discussed elsewhere/ nor are those of mechanical structure nor of
heat dissipation.
In the field covered, an effort has been made to select material of practical
value, and to present it as understandably as possible. References to vari-
ous publications cover some of the finer points. The book refers to experi-
mental data only incidentally in making general evaluations of theoretical
results.
To try to present the theory of the traveling- wave tube is difficult with-
out some reference to the overall picture which the theory is supposed to
give. One feels in the position of lifting himself by his bootstraps. For this
reason the following chapter gives a brief general description of the travel-
ing-wave tube and a brief and specialized analysis of its operation. This
chapter is intended to give the reader some insight into the nature of the
problems which are to be met. In Chapters III through VI, slow-wave cir-
cuits are discussed to give a qualitative and quantitative idea of their na-
ture and limitations. Then, simplified equations for the overall behavior of
the tube are introduced and solved, and matters such as overall gain, inser-
tion of loss, a-c space-charge effects, noise figure, field analysis of operation
and transverse field operation are considered. A brief discussion of power
output is given.
Two final chapters discuss briefly two closely related types of tube; the
traveling-wave magnetron amplifier and the double-stream amplifier.
' loc. cit.
BELL SYSTEM TECHNICAL JOURNAL
CHAPTER II
SIMPLE THEORY OF
TRAVELING-WAVE TUBE GAIN
Synopsis of Chapter
IT IS difficult to describe general circuit or electronic features of traveling-
wave tubes without some picture of a traveling-wave tube and traveling-
wave gain. In this chapter a typical tube is described, and a simple theoret-
ical treatment is carried far enough to describe traveling-wave gain in terms
of an increasing electromagnetic and space-charge wave and to express the
rate of increase in terms of electronic and circuit parameters.
In particular, Fig. 2.1 shows a typical traveling- wave tube. The parts of
this (or of any other traveling-wave tube) which are discussed are the elec-
tron beam and the slow-wave circuit, represented in Fig. 2.2 by an electron
beam and a helix.
In order to derive equations covering this portion of the tube, the proper-
ties of the helix are simulated by the simple delay line or network of Fig. 22,
and ordinary network equations are applied. The electrons are assumed to
flow very close to the line, so that all displacement current due to the pres-
ence of electrons flows directly into the line as an impressed current
For small signals a wave- type solution of the equations is known to exist,
in which all a-c electronic and circuit quantities vary with time and dis-
tance as exp(_;co/ — Tz). Thus, it is possible to assume this from the start.
On this basis the excitation of the circuit by a beam current of this form is
evaluated (equation (2.10)). Conversely, the beam current due to a circuit
voltage of this form is calculated (equation (2.22)). If these are to be con-
sistent, the propagation constant V must satisfy a combined equation (2.2vS).
The equation for the propagation constant is of the fourth degree in F,
so that any disturbance of the circuit and electron stream may be expressed
as a sum of four waves.
Because some quantities are in j)ractical cases small compared with others,
it is p()ssil)le to obtain good values of the roots by making an approximation.
This reduces the cciuation to the third degree. The solutions are expressed
in the form
TRAVELING-WAVE TUBES 7
Here fSg is a phase constant corresponding to the electron velocity (2.16)
and C is a gain parameter depending on circuit and beam impedance (2.43).
A solution of the equation for the case of an electron speed equal to the
speed of the undisturbed wave yields 3 values of 8 which are shown in Fig.
2.4. These represent an increasing, a decreasing and an unattenuated
wave. The increasing wave is of course responsible for the gain of the tube.
A different approximation yields the missing backward unattenuated wave
(2.32).
The characteristic impedance of the forward waves is expressed in terms
of 0c, C, and 8 (2.36) and is found to differ little from the impedance in the
absence of electrons.
The gain of the increasing wave is expressed in terms of C and the length
of the tube in wavelengths, N'
G = 47.3 CN db (2.37)
It will be shown later that the gain of the tube can be expressed approxi-
mately as the sum of the gain of the increasing wave plus a constant to take
into account the setting up of the increasing wave, or the boundary condi-
tions (2.39).
Finally, the important gain parameter C is discussed. The circuit part of
this parameter is measured by the cube root of an impedance, (E^/^^P)'^,
which relates the peak field E acting on the electrons, the phase constant
/3 = (jo/v, and the power flow. {E-f^-P)^ is a measure of circuit goodness
as far as gain is concerned.
We should note also that a desirable circuit property is constancy of
phase velocity with frequency, for the electron velocity must be near to the
circuit phase velocity to produce gain.
Evaluation of the effects of attenuation, of varying the electron velocity
and many other matters are treated in later chapters.
2.1 Description of a Traveling-Wave Tube
Figure 2.1 shows a typical traveling- wave tube such as may be used at
frequencies around 4,000 megacycles. Such a tube may operate with a
( athode current of around 10 ma and a beam voltage of around 1500 volts.
There are two essential parts of a traveling- wave amplifier; one is the helix,
which merely serves as a means for producing a slow electromagnetic wave
with a longitudinal electric field; and the other is the electron flow. At the
input the wave is transferred from a wave guide to the helix by means of a
short antenna and similarly at the output the wave is transferred from the
liclix to a short antenna from which it is radiated into the output wave
ii;uide. The wave travels along the wire of the helix with approximately the
speed of light. For operation at 1500 volts, corresponding to about x? the
8 BELL SYSTEM TECHNICAL JOURNAL
speed of light, the wire in the helix will be about thirteen times as long as the
axial length of the helix, giving a wave velocity of about iV the speed of
light along the axis of the helix. A longitudinal magnetic focusing field of a
few hundred gauss may be used to confine the electron beam and enable it
to pass completely through the helix, which for 4000 megacycle operation
may be around a foot long.
Fig. 2.1 — Schematic of the traveUng-wave amplifier.
ELECTRON
BEAM
it
ELECTROMAGNETIC WAVE TRAVELS
FROM LEFT TO RIGHT ALONG HELIX
li
¥'\g. 2.2 — Portion of the traveling-wave amplifier pertaining to electronic interaction
with radio-frequency fields and radio-frequency gain.
In analyzing the operation of the traveling-wave tube, it is necessary to
focus our attention merely on the two essential parts shown in Fig. 2.2, the
circuit (helix) and the electron stream.
2.2 'Vnv. 'Iypk of Analysis Used
A mathematical treatment of the traveling-wave tube is very important,
not so much to give an exact numerical prediction of operation as to give a
picture of the operation and to enable one to predict at least qualitatively
the effect of various ])hysical variations or features. It is unlikely that all of
I
TRAVELING-WAVE TUBES 9
the phenomena in a traveling-wave tube can be satisfactorily described in
a theory which is simple enough to yield useful results. Most analyses, for
instance, deal only with the small-signal or linear theory of the traveling-
wave tube. The distribution of current in the electron beam can have an
important influence on operation, and yet in an experimental tube it is often
difficult to tell just what this distribution is. Even the more elaborate analy-
ses of linear behavior assume a constant current density across the beam.
Similarly, in most practical traveling-wave tubes, a certain fraction of the
current is lost on the helix and yet this is not taken into account in the
usual theories.
It has been suggested that an absolutely complete theory of the traveling-
wave tube is almost out of the question. The attack which seems likely to
yield the best numerical results is that of writing the appropriate partial
differential equations for the disturbance in the electron stream inside the
helix and outside of the helix. This attack has been used by Chu and Jackson^
and by Rydbeck.^ While it enables one to evaluate certain quantities which
can only be estimated in a simpler theory, the general results do not differ
qualitatively and are in fair quantitative agreement with those which are
derived here by a simpler theory.
In the analysis chosen here, a number of approximations are made at the
very beginning. This not only simplifies the mathematics but it cuts down
the number of parameters involved and gives to these parameters a simple
physical meaning. In terms of the parameters of this simple theory, a great
many interesting problems concerning noise, attenuation and various bound-
' ary conditions can be worked out. With a more complicated theory, the work-
i ing out of each of these problems would constitute essentially a new problem
I rather than a mere application of various formulae.
i There are certain consequences of a more general treatment of a traveling-
jwave tube which are not apparent in the simple theory presented here.
Some of these matters will be discussed in Chapters XII, XIII and XIV.
rhe theory presented here is a small signal theory. This means that the
I equations governing electron flow have been linearized by neglecting certain
I quantities which become negligible when the signals are small. This results
■in a wave-type solution. Besides the small signal Umitation of the analyses
'.presented here, the chief simplifying assumption which has been made is
ithat all the electrons in the electron flow are acted on by the same a-c field,
or at least by known fields. The electrons will be acted on by essentially the
same field when the diameter of the electron beam is small enough or when
- L.J. Chu and J. D.Jackson, "Field Theory of Traveling-Wave Tubes," Froc. I. R. E.,
\n\. 36, pp. 853-863, July 1948.
^ Olof E. H. Rvdbeck, "The Theory of the Traveling-Wave Tube," Ericsson Technics,
Vo. 46, 1948.
10 BELL SYSTEM TECHNICAL JOURNAL
the electrons form a hollow cylmdrical beam in an axially symmetrical cir-
cuit, a case of some practical importance.
Besides these assumptions, it is assumed in this section that the electrons
are displaced by the a-c field in the axial direction only. This may be ap-
proximately true in many cases and is essentially so when a strong magnetic
focusing field is used. The efTects of transverse motion will be discussed in
Chapter XIII.
In this chapter an approximate relation suitable for electron speeds small
compared to the velocity of light is used in computing interaction between
electrons and the circuit.
A more general relation between impressed current and circuit field, valid
for faster waves, will be given in Chapter VI. Non-relativistic equations of
motion will, however, be used throughout the book. With whatever speed
the waves travel, it will be assumed that the electron speed is always small
compared with the speed of light.
We consider here the interaction between an electric circuit capable of
propagating a slow electromagnetic wave and a stream of electrons. We can
consider that the signal current in the circuit is the result of the disturbed
electron stream acting on the circuit and we can consider that the disturbance
on the electron stream is the result of the fields of the circuit acting on the
electrons. Thus the problem naturally divides itself into two parts.
2.3 The Field Caused by an Impressed Current
We will first consider the problem of the disturbance produced in the
circuit by a bunched electron stream. In considering this problem in this sec-
tion in a manner valid for slow waves and small electron velocities, we will
use the picture in Fig. 2.3. Here we have a circuit or network with uniformly
I M M M i i ^-^^'A
* T T T T T T T
Fig. 2.3 — E(|uivalent circuit of a traveling-wave tube. The distributed inductance
and capacitance are chosen to match the jihase velocity and field strength of the field act-
ing on the electrons. The impressed current due to the electrons is —dj/dz, where / is the
electron convection current.
distributed series inductance and shunt capacitance and with current / and
voltage V. The circuit extends infinitely in the z direction. An electron con-
vection current i flows along very close to the circuit. The sum of the dis-
placement and convection current into any little volume of the electron
beam must be zero. Because the convection current varies with distance in
II
TRAVELING-WAVE TUBES 11
the direction of flow, there will be a displacement current / amperes per
meter impressed on the transmission circuit. We will assume that the elec-
tron beam is very narrow and very close to the circuit, so that the displace-
ment current along the stream is negligible compared with that from the
stream to the circuit. In this case the displacement current to the circuit will
be given by the rate of change of the convection current with distance.
If the convection current i and the impressed current / are sinusoidal
with time, the equations for the network shown in Fig. 2.3 are
^ = -jBV + J (2.1)
dz
i- = -jXI (2.2)
oz
Here / and V are the current and the voltage in the line, B and X are the
shunt susceptance and series reactance per unit length and / is the im-
pressed current per unit length.
It may be objected that these "network" equations are not valid for a
transmission circuit operating at high frequencies. Certainly, the electric
field in such a circuit cannot be described by a scalar electric potential.
We can, however, choose BX so that the phase velocity of the circuit of
Fig. 2.3 is the^ame as that for a particular traveling-wave tube. We can
further choose X/B so that, for unit power flow, the longitudinal field acting
on the electrons according to Fig. 2.3, that is, —dV/dz, is equal to the true
field for a particular circuit. This lends a plausibility to the use of (2.1) and
(2.2). The fact that results based on these equations are actually a good ap-
proximation for phase velocities small compared with the velocity of light
is established in Chapter VI.
We will be interested in cases in which all quantities vary with distance
as exp(— Fs). Under these circumstances, we can replace differentiation
with respect to z by multiplication by — F. The impressed current per unit
length is given by
J= -^2 =Ti (2.3)
dz
Equations (2.1) and (2.2) become
-TI = -jBV -\- Ti (2.4)
-TV = -jXI (2.5)
If we eliminate /, we obtain
Vir~ + BX) = -jTXi (2.6)
12 BELL SYSTEM TECHNICAL JOURNAL
Now, if there were no impressed current, the righthand side of (2,6) would
be zero and (2.6) would be the usual transmission-line equation. In this case,
r assumes a value Fi , the natural propagation constant of the line, which
is given by
Ti = jVBX (2.7)
The forward wave on the line varies with distance as exp(— Fiz) and the
backward wave as exp(4-riz).
Another important property of the line itself is the characteristic im-
pedance A', which is given by
K = \^XjB (2.8)
We can express the series reactance X in terms of Fi and K
X = -jKT, (2.9)
Here the sign has been chosen to assure that X is positive with the sign
given in (2.7). In terms of Fi and K, (2.6) may be written
-VTiKi
V = (f.-ff) (2.10)
In (2.10), the convection current i is assumed to vary sinusoidally with
time and as exp( — Fs) with distance. This current will produce the voltage
V in the line. The voltage of the line given by (2.10) also varies sinusoidally
with time and as exp(— Fs) with distance.
2.4 Convection Current Produced by the Field
The other part of the problem is to find the disturbance produced on the
electron stream by the fields of the line. In this analysis we will use the
quantities listed below, all expressed in M.K.S. units."
■q — charge-to-mass ratio of electrons
77 = 1.759 X 10" coulomb/kg
Wo — average velocity of electrons
Vq — voltage by which electrons are accelerated to give them the velocity
«o. Mo = s/lriVQ
/o — average electron convection current
Po — average charge per unit length
po = —h/uo
V — a-c component of velocity
p — a-c component of linear charge density
i — a-c component of electron convection current
* Various physical constants are listed in Appendix I.
TRAVELING-WAVE TUBES 13
The quantities v, p, and i are assumed to vary with time and distance as
exp(;c<j/ — Tz).
One equation we have concerning the motion of the electrons is that the
time rate of change of velocity is equal to the charge- to-mass ratio times
the electric gradient.
d(uo + v) dV
dt ^ dz
(2.11)
In (2.11) the derivative represents the change of velocity observed in fol-
lowing an individual electron. There is, of course, no change in the average
velocity uo . The change in the a-c component of velocity may be expressed
dv . . . .
in terms of partial derivatives, — , which is the rate of change with time of
dt
dv . . . .
the velocity of electrons passing a given point, and — , which is variation of
dz
electron velocity with distance at a fixed time.
dv dv , dv dz dV ,^ . -X
— - = — + — =17 — l^-l-^j
dt dt dz dt dz
Equation (2.12) may be rewritten
^, + — (mo + t') = fl -^ (2.13)
dt dz dz
Now it will be assumed that the a-c velocity v is very small compared with
the average velocity Mq, and v will be neglected in the parentheses. The reason
for doing this is to obtain differential equations which are linear, that is,
in which products of a-c terms do not appear. Such linear equations neces-
sarily give a wave type of variation with time and distance, such as we
have assumed. The justification for neglecting products of a-c terms is that
we are interested in the behavior of traveling-wave tubes at small signal
levels, and that it is very difficult to handle the non-linear equations. When
we have linearized (2.13) we may replace the difi'erentiation with a respect
to time by multiplication byj'co and difi'erentiation with respect to distance
by multiplication by — F and obtain
(yw - Mor)i' - -r]VV (2.14)
We can solve (2.14) for the a-c velocity and obtain
. = -^^^ , (2.15)
<j^e - r)
[where
0, = oi/uo (2.16)
14 BELL SYSTEM TECHNICAL JOURNAL
We may think of (i,. as the phase constant of a disturbance traveUng with
the electron velocity.
We have another equation to work with, a relation which is sometimes
called the equation of continuity and sometimes the equation of conserva-
tion of charge. If the convection current changes with distance, charge
must accumulate or decrease in any small elementary distance, and we see
that in one dimension the relation obeyed must be
^i = -^^ (2.17)
dz dt
Again we may proceed as before and solve for the a-c charge density p
-Ti = -joip
p= zJIi (2.18)
CO
The total convection current is the total velocity times the total charge
density
-/n+ i = («o+ f)(po+ p) (2.19)
Again we will linearize this equation by neglecting products of a-c quanti-
ties in comparison with products of a-c quantities and a d-c quantity. This
gives us
i = pqv + Uop (2.20)
We can now substitute the value p obtained from (2.18) into (2.20) and solve
for the convection current in terms of the velocity, obtaining
Using (2.15) which gives the velocity in terms of the voltage, we obtain
the convection current in terms of the voltage
iVoU^e - r)^
2.5 OVKKAI.L ClkCUlT AND ElKCTKOMC EQUATION
In (2.22) we have the convection current in terms of the voltage. In (2.10)
we have the voltage in terms of the convection current. Any value of F for
which both of these equations are satisfied represents a natural mode of
TRAVELING-WAVE TUBES 15
propagation along the circuit and the electron stream. When we combine
(2.22) and (2.10) we obtain as the equation which F must satisfy:
1 = JJM^ , (2.23)
2 Mr; - r')Ui3. - rf
Equation (2.23) applies for any electron velocity, specified by ^3^,, and any
wave velocity and attenuation, specified by the imaginary and real parts of
the circuit propagation constant Fi . Equation (2.23) is of the fourth degree.
This means that it will yield four values of F which represent four natural
modes of propagation along the electron stream and the circuit. The circuit
alone would have two modes of propagation, and this is consistent with the
fact that the voltages at the two ends can be specified independently, and
hence two boundary conditions must be satisfied. Four boundary conditions
must be satisfied with the combination of circuit and electron stream. These
may be taken as the voltages at the two ends of the helix and the a-c velocity
and a-c convection current of the electron stream at the point where the
electrons are injected. The four modes of propagation or the waves given by
(2.23) enable us to satisfy these boundary conditions.
We are particularly interested in a wave in the direction of electron flow
which has about the electron speed and which will account for the observed
gain of the traveling- wave tube. Let us assume that the electron speed is
made equal to the speed of the wave in the absence of electrons, so that
-Fi = -j^e (2.24)
As we are looking for a wave with about the electron speed, we will assume
that the propagation constant differs from /3e by a small amount ^, so that
-r = -jn. + f
Using (2.24) and (2.25) we will rewrite (2.23) as
1 = -Klo^li-^l - 2j0e^ -f- f) .
Now we will find that, for typical traveling-wave tubes, | is much smaller
jthan (Se ; hence we will neglect the terms involving j8e^ and ^^ in the numera-
jtor in comparison with /J^- and we will neglect the term ^- in the denominator
|in comparison with the term involving l3e^. This gives us
e = -M ^ (2.27)
While (2.27) may seem simple enough, it will later be found very convenient
1«
BELL SYSTEM TECHNICAL JOURNAL
to rewrite it in terms of other parameters, and we will introduce them
now. Let
Kh/Wo = O
(2.28)
C is usually quite small and is typically often around .02. Instead of ^ we
will use a quantity or a parameter b
In terms of b and C, (2.27) becomes
i-jY" = (
J(2^~l|•2)1r^l/3
(2.29)
(2.30)
This has three roots which will be called 5i , ^2 and 63 , and these represent
three forward waves. They are
8, = e-'"" = V3/2 - j/2
h =
-:bir/6
7V/2
= -V3/2-J/2
(2.31)
83 = e' = J
Figure 2.4 shows the three values of 8. Equation (2.23) was of the fourth
degree, and we see that a wave is missing. The missing root was eliminated
-0.866 -J 0.5
0.866 -J 0.5
Fig. 2.4 — There are three forward waves, with fields which vary with distance as
exp(— jjSe -\- 0eC5)z. The three values of 8 for the case discussed, in which the circuit is
lossless and the electrons move with the phase velocity of the unperturbed circuit wave,
are shown in the figure.
by the approximations made above, which are valid for forward waves only.
The other wave is a backward wave and its propagation constant is found
to be
r = j^e
(■-?)
(2.32)
As C is a small quantity, C^ is even smaller, and indeed the backward wave
given by (2.32) is practically the same as the backward wave in the absence
of electrons. This is to be e.xpected. In the forward direction, there is a cumu-
lative interaction between wave and the electrons because both are moving
TRAVELING-WAVE TUBES 17
at about the same speed. In the backward direction there is no cumulative
action, because the wave and the electrons are moving in the opposite
directions.
The variation in the z direction for three forward waves is as
exp —Tz = exp —jjSeZ exp dC^^^ (2.33)
We see that the first wave is an increasing wave which travels a little more
slowly than the electrons. The second wave is a decreasing wave which
travels a little more slowly than the electrons. The third wave is an un-
attenuated wave which travels faster than the electrons. It can be shown
generally that when a stream of electrons interacts with a wave, the electrons
must go faster than the wave in order to give energy to it.
It is interesting to know the ratio of line voltage to line current, or the
characteristic impedance, for the three forward waves. This may be obtained
from (2.5). We see that the characteristic impedance Kn for the nth. wave is
given in terms for the propagation constant for the nth. wave, r„, by
Kn = V/I = yX/r„ (2.34)
In terms of 5„ this becomes
A',. = K{J - l3eC8n/ry) (2.35)
Kn = K{1 - jC8n) (2.36)
We see that the characteristic impedance for the forward waves differs from
the characteristic impedance in the absence of electrons by a small amount
proportional to C, and that the characteristic impedance has a small reactive
component.
We are particularly interested in the rate at which the increasing wave
increases. In a number of wave lengths N, the total increase in db is given by
20 logio exp [(\/3/2)(C)(27riV)] db
= 47.3 CN db ^^-^^^
We will see later that the overall gain of the traveling-wave tube with a
uniform helix can be expressed in the form
G = A -\- BCN db (2.38)
Here yl is a loss relating voltage associated with the increasing wave to
the total applied voltage. This loss may be evaluated and will be evaluated
later by a proper examination of the boundary conditions at the input of
the tube. It turns out that for the case we have considered
G = -9.54 + 47.3 CN db (2.39)
18 BELL SYSTEM TECHNICAL JOURNAL
In considering circuits for traveling-wave tubes, and in reformulating
the theory in more general terms later on, it is valuable to express C in terms
of parameters other than the characteristic impedance. Two physically sig-
nificant parameters are the power flow in the circuit and the electric field
associated with it which acts on the electron stream. The ratio of the square
of the electric field to the power can be evaluated by physical measurement
even when it cannot be calculated. For instance, Cutler did this by allowing
the power from a wave guide to flow into a terminated helix, so that the
power in the helix was the same as the power in the wave guide. He then
compared the field in the helix with the field in the wave guide by probe
measurements. The field strength in the wave guide could be calculated in
terms of the power flow, and hence Cutler's measurements enabled him to
evaluate the field in the helix for a given power flow.
The magnitude of the field is given in terms of the magnitude of the
voltage by
E= \VV\ (2.40)
Here E is taken as the magnitude of the field. The power flow in the circuit
is given in terms of the circuit voltage by
P = \V \y2K (2.41)
A quantity which we will use as a circuit parameter is
£V/32P = 2K (2.42)
Here it has been assumed that we are concerned with low-loss circuits, so
that T\ can be replaced by the phase constant 0^. Usually, /3 can be taken
as equal to /3e, the electron phase constant, with small error, and in the
preceding work this has been assumed to be exactly true in (2.23).
In terms of this new quantity, C is given by
C = i2K)iIo/SVo) = (E'/0-P){Io/SVo) (2.43)
If we call Vo/h the beam impedance, C^ is j the circuit impedance divided
by the beam impedance. It would have been more sensible to use E-/20-P
instead of Er/0P. Unfortunately the writer feels stuck with his benighted
first choice because of the number of curves and pubUshed equations which
make use of it.
Besides the circuit impedance, another important circuit parameter is
the phase velocity. As the electron velocity is made to deviate from the
phase velocity of the circuit, the gain falls off. An analysis to be given later
^ C. C. Cutler, "Experimental Determination of Helical-Wave Properties," Proc. IRE,
Vol. 36, pp. 230-233, February 1948.
TRAVELING-WAVE TUBES 19
discloses that the allowable range of velocity Av is of the order of
A^ ;:^ ± Cuo (2.44)
Thus, the allowable difference between the phase velocity of the circuit and
the velocity of the electrons increases as circuit impedance and beam current
are increased and decreases as voltage is increased.
We have illustrated the general method of attack to be used and have
introduced some of the important parameters concerned with the circuit
and with the overall behavior of the tube. In later chapters, the properties
of various circuits suitable for traveling-wave tubes will be discussed in
terms of impedance and phase velocity and various cases of interest will be
worked out by the methods presented.
20 BELL SYSTEM TECHNICAL JOURNAL
CHAPTER III
THE HELIX
Synopsis of Chapter
ANY circuit capable of propagating a slow electromagnetic wave can be
used in a traveling-wave tube. The circuit most often used is the helix.
The helix is easy to construct. In addition, it is a very good circuit. It has a
high impedance and a phase velocity that is almost constant over a wide
frequency range.
In this chapter various properties of helices are discussed. An approximate
expression for helix properties can be obtained by calculating the properties,
not of a helix, but of a heUcally conducting cylindrical sheet of the same
radius and pitch as the helix. An analysis of such a sheet is carried out in
AppendLx II and the results are discussed in the text.
Parameters which enter into the expressions are the free-space phase con-
stant |So = w/c, the axial phase constant /3 = w/v, where v is the phase
velocity of the wave, and the radial phase constant 7. The arguments of
various Bessel functions are, for instance, yr and 7c, where r is the radial
coordinate and a is radius of the helix. The parameters /So, jS and 7 are
related by
/32 = /35 -f 7'
For tightly wound helices in which the phase velocity v is small compared
with the velocity of light, 7 is very nearly equal to jS. For instance, at a
velocity corresponding to that of 1,000 volt electrons, 7 and /? differ by
only 0.4%.
Figure 3.1 illustrates two parameters of the helically conducting sheet,
the radius a and pitch angle ^l/. For an actual helix, a will be taken to mean
the mean radius, the radius to the center of the wire.
Figure 3.2 shows a single curve which enables one to obtain 7, and hence
/3, for any value of the parameter
coa cot i/'
Po a cot \p = .
c
This parameter is proportional to frequency. The curve is an approximate
representation of velocity vs. frequency. At high frequencies 7 approaches
TRAVELING-WAVE TUBES 21
00 cot ^ and /3 thus approaches /3o/sin ^; this means that the wave travels
with the velocity of Hght around the sheet in the direction of conduction.
In the case of an actual helix, the wave travels along the wire with the
velocity of light.
The gain parameter C is given by
C = (ro/8V,y'\E''/l3'Py"
Values of (E^/0'^PY'^ on the axis may be obtained through the use of Fig. 3.4,
where an impedance parameter F(ya) is plotted vs. ya, and by use of (3.9).
For a given helix, {E?/0^PY ^ is approximately proportional to F(ya). F{ya)
falls as frequency increases. This is partly because at high frequencies and
short wavelengths, for which the sign of the field alternates rapidly with
distance, the field is strong near the helix but falls ofif rapidly away from the
helix and so the field is weak near the axis. At very high frequencies the field
falls off away from the helix approximately as exp(— 7Af), where Ar is dis-
tance from the helix, and we remember that y is very nearly proportional to
frequency. (E^/0^P) measured at the helix also falls with increasing
frequency.
In many cases, a hollow beam of radius r (the dashed lines of Fig. 3.5
refer to such a beam) or a solid beam of radius r (the solid lines of Fig. 3.5
refer to such a beam) is used. For a hollow beam we should evaluate £- in
{E-/0"Py ^ at the beam radius, and for a solid beam we should use the mean
square value of E averaged over the beam.
The ordinate in Fig. 3.5 is a factor by which (E^/ff^Py^ as obtained from
Fig. 3.4 and (3.9) should be multiplied to give (E-/0^Py'^ for a hollow or
solid beam.
The gain of the increasing wave is proportional to F{ya) times a factor
from Fig. 3.5, and times the length of the tube in wavelengths, N. N is very
nearly proportional to frequency. Also y, and hence ya, are nearly propor-
tional to frequency. Thus, F(ya) from Fig. 3.4 times the appropriate factor
from Fig. 3.5 times ya gives approximately the gain vs. frequency, (if we
assume that the electron speed matches the phase velocity over the fre-
quency range). This product is plotted in Fig. 3.6. We see that for a given
helix size the maximum gain occurs at a higher frequency and the band-
width is broader as r/a, the ratio of the beam radius to the helix radius,
is made larger.
It is usually desirable, especially at very short wavelengths, to make the
helix as large as possible. If we wish to design the tube so that gain is a maxi-
mum at the operating frequency, we will choose a so that the appropriate
curve of Fig. 3.6 has its maximum at the value of ya corresponding to the
operating frequency. We see that this value of a will be larger the larger is
r/a. In an actual helix, the maximum possible value of r/a is less than unity,
22 BELL SYSTEM TECHNICAL JOURNAL
since the inside diameter of the heUx is less than a by the radius of the wire.
Further, focusing difficulties preclude attaining a beam radius equal even to
the inside radius of the helix.
Experience indicates that at very short wavelengths (around 6 milli-
meters, say) it is extremely important to have a well-focused electron beam
with as large a value of r/a as is attainable.
A characteristic impedance Kt may be defined in terms of a "transverse"
voltage Vt, obtained by integrating the peak radial field from a to oo , and
from the power flow. In Fig. 3.7, (v/c) Kt is plotted vs. ja. A "longitudinal"
characteristic impedance Kf is related to Kt (3.13). For slow waves Kf
is nearly equal to Ki. The impedance parameter E~/^~P evaluated at the
surface of the cylinder is twice Kf. We see that Ke falls with increasing
frequency.
A simplified approach in analysis of the helically conducting sheet is that
of "developing" the sheet; that is, slitting it normal to the direction of con-
duction and flattening it out as in Fig. 3.8. The field equations for such a
flattened sheet are then solved. For large values of ya the field is concentrated
near the helically conducting sheet, and the fields near the developed sheet
are similar to the fields near the cylindrical sheet. Thus the dashed line
in Fig. 3.7 is for the developed sheet and the solid Hue is for a cylindrical
sheet.
For the developed sheet, the wave always propagates with the speed of
light in the direction of conduction. In a plane normal to the direction of
conduction, the field may be specified by a potential satisfying Laplace's
equation, as in the case, for instance, of a two-wire or coaxial line. Thus,
the fields can be obtained by the solution of an electrostatic problem.
One can develop not only a helically conducting sheet, but an actual
helix, giving a series of straight wires, shown in cross-section in Fig. 3.9.
In Case I, corresponding to approximately two turns per wavelength, suc-
cessive wires are — , +, — , + etc.; in case II, corresponding to approxi-
mately four turns per wavelength, successive wires are +,0, — , 0, -f , 0 etc.
Figures 3.10 and 3.11 illustrate voltages along a developed sheet and a
developed helix.
Figure 3.13 shows the ratio, R^''\ of {E-f^-Py^ on the axis to that for a
developed helically conducting sheet, plotted vs. d/p. We see that, for a
large wire diameter d, {E?/0^Py'^ may be larger on the axis than for a heli-
cally conducting sheet with the same mean radius and hence the same pitch
angle and phase velocity. This is merely because the thick wires extend nearer
to the axis than does tlie sheet. The actual helix is really inferior to the
sheet.
We see this by noting that the highest value of {E'/ff'Py^ for a helically
conducting sheet is that at the sheet {r = a). With a finite wire size, the
TILiVEUNG-WAVE TUBES 23
largest value r can have is the mean helix radius a minus the wire radius.
In Fig. 3.14, the ratio of (E-/^-Py'^ for this largest allowable radius to
(E'/^'Py^ at the surface of the developed sheet is plotted vs. d/p. We see
that, in terms of maximum available field, {E-/(3-Py'^ is no more than 0.83 as
high as for the sheet for four turns per wavelength and 0.67 as high as for the
sheet for two turns per wavelength. We further see that there is an optimum
ratio of wire diameter to pitch; about 0.175 for four turns per wavelength
and about 0.125 for two turns per wavelength. Because the maxima are so
broad, it is probably better in practice to use larger wire, and in most tubes
which have been built, d/p has been around 0.5.
In designing tubes it is perhaps best to do so in terms of field on the axis
(Fig. 3.13), the allowable value of r/a and the curves of Fig. 3.6.
Figure 3.15 compares the impedance of the developed helix with that of
the developed sheet as given by the straight line of Fig. 3.7.
There are factors other than wire size which can cause the value of E'/jS-F
for an actual helix to be less than the value for the helically conducting
sheet. An important cause of impedance reduction is the influence of di-
electric supporting members. Even small ceramic or glass supporting rods
can cause some reduction in helix impedance. In some tubes the helix is
supported inside a glass tube, and this can cause a considerable reduction
in helix impedance.
When a field analysis seems too involved, it may be possible to obtain
some information by considering the behavior of transmission lines having
parameters adjusted to make the phase constant and the characteristic im-
pedance equal to those of the helix. For instance, suppose that the presence
of dielectric material results in an actual phase constant ^d as opposed to a
computed phase constant /S. Equation (3.64) gives an estimate of the con-
sequent reduction of {E~/ff-Pyi^ on the axis.
This method is of use in studying the behavior of coupled helices. For
instance, concentric helices may be useful in producing radial fields in tubes
in which transverse fields predominate in the region of electron flow (see
Chapter XIII). A concentric helix structure might be investigated by means
of a field analysis, but some interesting properties can be deduced more
; simply by considering two transmission lines with uniformly distributed self
and mutual capacitances and inductances, or susceptance and reactances.
iThe modes of propagation on such lines are affected by coupling in a manner
similar to that in which the modes of two resonant circuits are afifected by
coupling.
' If two lines are coupled, their two independent modes of propagation are
mixed up to form two modes of propagation in which both lines participate.
If the original phase velocities differ greatly, or if the coupling between the
ines is weak, the fields and velocity of one of these modes will be almost
24 BELL SYSTEM TECHNICAL JOURNAL
like the original fields and velocity of one line, and the fields and velocity of
the other mode will be almost like the original fields and velocity of the other
line. However, if the coupling is strong enough compared with the original
separation of phase velocities, both lines will participate almost equally in
each mode. One mode will be a "longitudinal mode" for which the excitations
on the two lines are substantially equal, and the other mode will be a "trans-
verse" mode for which the excitations are substantially equal and opposite.
The ratios of the voltages on the lines for the two modes are given by
(3.75). Here it is assumed that the series reactances A' and shunt susceptances
B of the lines are almost equal, differing only enough to make a difference
AFo in the propagation constants. Bn and X12 are the mutual susceptance
and reactance. We see that to make the voltages on the two lines nearly
equal or equal and opposite, B12 and Xn should have the same sign, so that
capacitive and inductive couplings add.
Fig. 3.1 — A helically conducting sheet of radius a. The sheet is conducting along hehcal
paths making an angle xp with a plane normal to the axis.
Increasing the coupling increases the velocity separation between the two
modes, and this is desirable. When there is a substantial difference in ve-
locity, operation in the desired mode can be secured by making the electron
velocity equal to the phase velocity of the desired mode.
To make the capacitive and inductive coupHngs add in the case of con-
centric helices (Fig. 3.17), the helices should be wound in opposite directions.
3.1 The Helically Conducting Sheet
In computing the properties of a helix, the actual helix is usually replaced
by a helically conducting cylindrical sheet of the same mean radius. Such a
sheet is illustrated in Fig. 3.1. This sheet is perfectly conducting in a helical
direction making an angle ^, the pitch angle, with a plane normal to the
axis (the direction of propagation), and is non-conducting in a helical direction
normal to this \p direction, the direction of conduction. Appropriate solutions
of Maxwell's equations are chosen inside and outside of the cylindrical sheet.
At the sheet, the components of the electric field in the \}/ direction are made
zero, and those normal to the \p direction are made equal inside and outside.
Since there can be no current in the sheet normal to the ^ direction, the
TRAVELING-WAVE TUBES
25
components of magnetic tield in the \f/ direction must be the same inside and
outside of the sheet. When these boundary conditions are imposed, one can
solve for the propagation constant and E^l^-P can then be obtained by
integrating the Poynting vector.
The hehcally conducting sheet is treated mathematically in Appendix II.
The results of this analysis will be presented here.
2.2
2.0
1.8
\
^..,:^^\b7
^-\
\
^^^^^
\
\
>
\
^^
0 12 3 4 5 6
y3o a coT^
Fig. 3.2— The radial propagation constant is 7- = {^^ — /3o)^'^. Here (/So/t) cot ^ is
plotted vs /Sofl cot i/', a quantity proportional to frequency. For slow waves the ordinate is
roughly the ratio of the wave velocity to the velocity the wave would have if it traveled
along the helically conducting sheet with the speed of light in the direction of conduction.
3.1a The Phase Velocity
The results for the helically conducting sheet are expressed in terms of
three phase or propagation constants. These are
/So = oi/c, jS = 03/v
7 = /^Vl - {v/cy
(3.1)
(3.2)
Here c is the velocity of light and v is the phase velocity of the wave. /3o is
the phase constant of a wave traveling with the speed of light, which would
vary with distance in the s direction as exp(— j/Sos). The actual axial phase
constant is /3, and the fields vary with distance as exp(— j/Ss).
7 is the radial propagation constant. Various field components vary as
modified Bessel functions of argument 7r, where r is the radius. Particularly,
the longitudinal electric field, which interacts with the electrons, varies
as h{yr).
I For the phase velocities usually used, 7 is very nearly equal to ji, as may
he seen from the following table of accelerating voltages Vq (to give an elec-
I Iron the velocity v), v/c and 7/jS.
26
BELL SYSTEM TECHNICAL JOURNAL
V
Vl c
y/0
100
.0198
1.000
1 ,000
.0625
.998
10,000
.1980
.980
Figure 3.2 gives information concerning the phase velocity of the wave
in the form of a plot of (/So/t) cot i/' as a function of /3o a cot yp.
The ratio of the phase velocity v to the velocity of light c may be expressed
v/c = /3n//3 = (y/(3)i^o/y) cot i^ tan rp
v/c = (7//3) tan i/- [(/^o/t) cot ^ ]
(3.4)
0.3
0.2
O.t
^\
0.08
0.06
\ \ \ \
\
0.04
0.03
0.02
-S- 0 0'
\v
v\
N\
Y
^>^\
:^
^
2 0.008
V
xx^
V
01 0.006
\
Xk.^«v ^v
JV
1 0 004
>\^ 0.003
0 002
0.001
V N?\
\^^
V^
^^^
V
TAN I//
^^NX
^/^\.
c::::;;;-
0 0008
Vv ^v^
0,0006
^v^^
^v_ ^^
"^^^Jo-^
"^--^i-^
^^
^"^^
0.0004
0.0003
0,0002
O.OOOI
N...^/^"-^
,^O07^
^^^^
^^"»«N^
^05
' '
--
"^
f^oa COT x/j
Fig. 3.3 — From these curves one can ol)tain v/c, the ratio of the phase velocity of the
wave to the velocity of light, for various values of tan \p and /3oa cot \^.
From Fig. 3.2 we see that, for large values of (ioa cot \J/, (i^o/t) cot \}/ ap-
proaches unity. For slow waves y/0 approaches unity. Under these circum-
stances, very nearly
v/c = tan \f/
i^.S)
If the wave traveled in the direction of conduction with the speed of hglil
we would have
v/c = sin yp
TRAVELING-WAVE TUBES
27
This is essentially the same as (3.5) for small pitch angles 4^. Thus, for large
values of the abscissa in Fig. 3.2, the phase velocity is just about that corre-
sponding to propagation along the sheet in the direction of conduction with
the speed of light and hence in the axial direction at a much reduced speed.
For helices of smaller radius compared with the wavelength, the speed is
greater.
The bandwidth of a traveling-wave tube is in part determined by the
range over which the electrons keep in step with the wave. The abscissa of
Fig. 3.2 is proportional to frequency, but the ordinate is not strictly propor-
tional to phase velocity. Hence, it seems desirable to have a plot which does
show velocity directly. To obtain this we can assign various values to cot rp.
1.0
0.8
~b 0.4
U- 0.3
0.1
0.08
0.06
0.04
0.03
X
F (7a) = 7.154e-0-66647'a
\
N
\
-
"S.
-
^S^^
-
^
V.
N
X
k
\
\
-
\
-
\
-
\^
1
\
01 23456789
7a
Fig. 3.4 — A curve giving the impedance function F{ya) vs. ya. On the axis, {E^/0rPy^ =
(/3/^u)"KT//3)^"f(Ta).
The ordinate (1S0/7) cot \p then gives us y/^Q and from (3.2) we see that
v/c = /3o/^ = (1 + (y/^oY-)-'" (3.6)
We have seen that, for large values of /3oo cot \}/, (/Sq/t) cot ^ approaches
unity, and v/c approaches a value
v/c = (1 + cot2 ^py'" = sin ^
(3.7)
To emphasize the change in velocity with frequency it seems best to plot the
difference between the actual velocity ratio and this asymptotic velocity
ratio on a semi-log scale. Accordingly, Fig. 3.3 shows (v/c) — sin ip vs. /3oO
cot 7 for tan \p = .05, .075, .1, .15, .2.
For large values of the abscissa the velocities are those corresponding to
28
BELL SYSTEM TECHNICALVOURNAL
about 640 volts (tan ^l^ = .05), 1,400 volts (.075), 2,500 volts (.1), 5,600 volts
(.15), 9,800 volts (.2).
3.1b The Impedance Parameter (Er/^-P)
Figure 3.4 shows a plot of a quantity Fiya) vs. ya. This quantity is com-
puted from a very complicated expression (Appendi.x II), but it is accurately
given over the range shown by the empirical relation
Fiya) = 7.154 e-''''"" (3.8)
50
/
/
40
.-' 1
/
30
/
/
HOLLOW BEAM
a; =
r
•
f
■''/
^
OLID BEAM
a
f
y .
CE 20
O
1-
■"
O/-
/
/:■'/
<
O 9
^ 5
i
/
/
/
r ,
X .^
X
^y'^y
•
A
/
^•^
^ ,y
X^^
•
/
/y
'^v
Jj-
•
.^
^/>
^ y
'',''^,
^
•
^^ ,
K^-^
'X'
X
•
^ /
^^
lu 4
Q-
1
,^ ,
' ,
^>^
<x'
x'^ ^
y
•
,/
.^'
'^
-^
5
3
'"^
^^
^
t^
•
y
' y'
^
k;^
/
2
y
€^
P^
x^
P'
^^^y
^^"ly-
^
^
'^
1
-^
^^^
m
^
^
0 0.5 10 1.5 2.0 2.5 30 35 4.0 4.5 5.0 55 60 6.5 7.0 7.5 8.0
ra
Fig. 3.5 — Factors by which (Z^/S/^P)^'^ on the axis should he muUipiied to give the cor-
rect value for hollow and solid beams of radius r.
For the field on the axis of the helix,
{Ey^'^pyi' = {i3/0oy"{y/0y"F(ya)
(3.9)
We should remember that (3/0o = c/v and that y/0 is nearly unity for veloci-
ties small compared with the velocity of light.
In the expression for the gain parameter C, the square of the field E is
multiplied by the current /o (2.28). If we were to assume that two electron
TRAVELING-WAVE TUBES 29
streams of diflferent currents, /i and lo , were coupled to the circuit through
transformers, so as to be acted on by fields Ei and Eo, but that the streams
did not interact directly with one another, we would find the effective value
of C^ to be given by
C' = (£l//3-P)(/i/8Fo) + (El/ 13' P) (1 2/SV0)
Thus, if we neglect the direct interaction of electron streams through fields
due to local space charge, we can obtain an effective value of C^ by integrat-
ing £Wo over the beam. If we assume a constant current density, we can
merely use the mean square value of E over the area occupied by electron
flow.
The axial component of electric field at a distance r from the axis is Io{yr)
times the field on the axis. Hence, if we used a tubular beam of radius r, we
should multiply {E^/^'^PY^^ as obtained from Fig. 3.4 by[ I^iyr)]^'^. The quan-
tity [/o(7'')]^'* is plotted vs. 7a for several values of r/a as the dashed lines
in Fig. 3.5.
Suppose the current density is uniform out to a radius r and zero beyond
this radius. The average value of £- is greater than the value on the axis by
a factor \Il{yr) — l]{'Yr)\ and {E-/fi'Py^ from Fig. 3.4 should in this case
be multiplied by this factor to the \ power. The appropriate factor is plotted
vs. ya as the solid lines of Fig. 3.5.
We note from (2.39) that the gain contains a term proportional to CN ,
where N is the number of wavelengths. For slow waves and usual values of
ya, very nearly, N will be proportional to the frequency and hence to 7,
while C is proportional to (E-/l3~Py'^. We can obtain {E~/^"Py'^ from Figs.
3.4 and 3.5. The gain of the increasing wave as a function of frequency will
I thus be very nearly proportional to this value of {E^/^'^Py^ times 7, or,
; times ya if we prefer.
In Fig. 3.6, yaF{ya) is plotted vs. ya for hollow beams of radius r for
various values of r/a (dashed lines) and for uniform density beams of
radius r for various values of r/a (solid lines). If we assume that the electron
speed is adjusted to equal the phase velocity of the wave, we can take the
[ordinate as proportional to gain and the abscissa as proportional to
'frequency.
We see that the larger is r/a, the larger is the value of ya for maximum
jgain. For one typical 7.5 cm wavelength traveling-wave tube, ya was about
'2.8. For this tube, the ratio of the inside radius of the helix to the mean radius
of the helix was 0.87. We see from Fig. 3.6 that, if a solid beam just filled
I this helix, the maximum gain should occur at about the operating wave-
jlength. As a matter of fact, the beam was somewhat smaller than the inside
diameter of the helix, and there was an observed increase of gain with an
increase in wavelength (a higher gain at a lower frequency). In a particular
30
BELL SYSTEM TECHNICAL JOURNAL
tube for 0.625 cm wavelength, it was felt desirable to use a relatively large
helix diameter. Accordingly, a value of ya of 6.7 was chosen. We see that,
unless r/a is 0.9 or larger, this must result in an appreciable increase in gain
at some frequency lower than operating frequency. It was only by use of
great care in focusing the beam that gain was attained at 0.625 cm wave-
length, and there was a tendency toward oscillation, presumably at longer
wavelengths. This discussion of course neglects the effect of transmission
,•''
y
(UNI
HOLLOW BEAM
SOLID BEAM
CURRENT density)
y
•
y
FORI^
r ^'
r " '
''
/
4
n.9
— — —
••""
X
•
^''■
.--'
•
^.'-
.^''"'
. 1
^^^^
-—
■"^
•
"''
^^
=j^
^rr"
:rr.
0.8_
/
/
^
rrr
0.9
—
?^^
J — ^
"^■^
"■~«_
1 '
^^
' "
■
U;8
.
2
^
"^
LU
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 55 6.0 6.5 7.0 7.5 8.0
Fig. 3.6 — The ordinate is yaF{ya) times the parameters from Fig. 3.5. For a fixed cur-
rent and voltage it is nearly proportional to gain per unit length, and hence the curves
give roughly the variation of gain with frequency.
loss or gain. Usually the loss decreases when the frequency is decreased,
and this favors oscillation at low frequencies.
3.1c Impedance of the Helix
No impedance which can be assigned to the helically conducting sheet
can give full information for malching a heli.x to a waveguide or transmission
line. As in the case of transducers between a coaxial line and a waveguide or
between waveguides of different cross-section, the impedance is important,
I
TRAVELING-WAVE TUBES
31
but discontinuity effects are also important. However, a suitably defined
helix impedance is of some interest.
Figure 3.7 presents the impedance as defined on a voltage-power basis.
The peak "transverse" voltage Vt is obtained by integrating the radial elec-
tric field from the radius a of the helically conducting sheet to oo . The
"transverse" characteristic impedance Kt is defined by the relation
P = (mvVKd
80
70
60
50
40
30
20
X.
>|o
Ij, 10
6
5
■■■
s
\
\
N
s
s,
S
\,
'
N
\
.0\ ^
7i \
\
S
s
s.
>
^
\.
\
s
\
\
s,
s
s,
S
\,
\
\
\
\
0.4 0.6 O.S 1.0 2 3 4 5 6 7 S 10 20 30
Fig. 3.7 — Curves giving the variation of transverse impedance, Kt , with ya.
The impedance is found to be given by
noil
©
(yay
h'r^)'''
/o/o)
+
(rJ('+'/f:)<^-«^''-"'^r
(3.10)
The /'s and K's are modified Bessel functions of argument ya.
' The dashed line on Fig. 3.7 is a plot of 30/70 vs. ya. It may be seen that,
lor large values of ya, very nearly
Kt = i^/my/m^O/ya)
(3.11)
32
BELL SYSTEM TECHNICAL JOURNAL
and in the whole range shown the impedance differs from this value by a
factor less than 1.5.
We might have defined a "longitudinal" voltage Vi as half of the integral
of the longitudinal component of electric field at the surface of the helically
conducting sheet for a half wavelength (between successive points of zero
field). We find that
Vt = Vl - {v/cY V, = (7//3)F,
and, accordingly, the "longitudinal impedance" K( will be
K( = [1 - {v/cy]K, = (ym'Kt
(3.12)
(3.13;
Our impedance parameter, E~/0~P, is just twice this "longitudinal
impedance."
^CIRCUMFERENTIAL
''; CIRCLES
b \ b b
DIRECTION
OF AXIS ^.,
-2 7ra SIN t/^
Fig. 3.8 — A "developed" helically conducting sheet. The sheet has been slit along a
line normal to the direction of conduction and flattened out.
The transverse voltage Vi is greater than the longitudinal voltage Vf
because of the circumferential magnetic flu.x outside of the heli.x. For slow
waves V.C is nearly equal to Vt and the fields are nearly curl-free solutions of
Laplace's equation. In this case the circumferential magnetic flux is small
compared with the longitudinal flux inside of the helix.
For the circuit of Fig. 2.3 the transverse and longitudinal voltages are
equal, and it is interesting to note that this is approximately true for slow
waves on a helix. For very fast waves, the longitudinal voltage becomes small
compared with the transverse voltage.
For a typical 4,000-megacycle tube, for which ya = 2.8, Fig. 5 indicates a
value of Ki of about 150 ohms.
3.2 The Developed Helix
For large helices, i.e., for large values of ya, the fields fall off very rapidly
away from the wire. Under these circumstances we can obtain quite accurate
results by slitting the helically conducting sheet along a spiral line normal
TRAVELING-WAVE TUBES 33
to the direction of conduction and flattening it out. This gives us the plane
conducting sheet shown in Fig. 3.8. The indicated coordinates are z to the
right and y upward: x is positive into the paper. The fields about the de-
veloped sheet approximate those about the helically conducting sheet for
distances always small compared with the original radius of curvature.
The straight dashed line shown on the helix impedance curve of Fig. 3.7
can be obtained as a solution for the "developed helix." We see that it is
within 10% of the true curve for values of ya greater than 2.8. We might note
that a 10% error in impedance means only a 3^% error in the gain
parameter C.
In solving for the fields around the sheet, the developed surface can be
extended indefinitely in the plus and minus y directions. In order that the
fields may match when the sheet is rolled up, they must be the same at
y = 0, 3 = lira sin \J/ and y = lira cos \p, z — Q. The appropriate solutions
are plane electromagnetic waves traveling in the y direction with the speed
of light.
For positive values of .v, the appropriate electric and magnetic fields are
£. = jE,e~''' e-''' e''^'' (3.14)
£, = 0
We should note that the x and s components of the field can be obtained
as gradients of a function
^ = -{E,h)e-'' e-^'' e-^^'" (3.15)
I where
-Ex = -d^/dz
(3.16)
£. = -d^/dy
d'^^/dx' + d'-^/dz'' = 0 (3.17)
i
Thus, in the xz plane, $ satisfies Laplace's equation.
The magnetic field is given by the curl of the electric field times j'/w/x.
Its components are:
IXC
H, = — Eoe-'^'e-'^'e''^'" (3.18)
uc
^ Maxwell's equations are given in Appendix I.
34 BELL SYSTEM TECHNICAL JOURNAL
The fields in the — .v direction may be obtained by substituting exp(7x)
for exp(— 7.v).
If the sheet is to roll up properly, the points a on the bottom coinciding
with the points b on the top, we have
lirya sin ^ — l-w^^^a cos ^|/ = linr (3.19)
where n is an integer.
The solution corresponding most nearly to the wave on a singly-wound
helix is that for n = 0. The others lead to a variation of field by // cycles
along a circumferential line. These can be combined with the n = 0 solu-
tion to give a solution for a developed helix of thin tape, for instance. Or,
appropriate combinations of them can represent modes of helices wound of
several parallel wires. For instance, we can imagine winding a balanced trans-
mission line up helically. One of the modes of propagation will be that in
which the current in one wire is 180° out of phase with the current in the
other. This can be approximated by a combination of the n = -\-\ and
11= —1 solutions. This mode should not be confused with a fast wave, a
perturbation of a transverse electromagnetic wave, which can exist around
an unshielded helix.
Usually, we are interested in the slow wave on a singly-wound helix, and
in this case we take n = 0 in (3.19), giving
7 sin i/' — /3o cos ^ = 0
(3.20)
tan yp = /3o/7
sin i/' = / 2 , ^2x1/2 (3.21)
(7 + Po)
Let us evaluate the propagation constant in the axial direction. From Fig.
3.8 we see that, in advancing unit distance in the axial direction, we pro-
ceed a distance cos \p in the z direction and sin \p in the y direction. Hence,
the phase constant jS in the axial direction must be
/3 = i8o sin i/' + 7 cos ^ (3.23)
Using (3.18) and (3.19), we obtain
^ = (/35 + yY' (3.24)
7 = 0' - 0iy" (3.25)
These are just relations (3.2, 33).
TRAVELING-WAVE TUBES 35
The power flow along the axis is that crossing a circumferential circle,
represented by lines a-b in Fig. 3.8. As the power flows in the y direction,
this is the power associated with a distance lira sin \[/ in z direction. Also,
the power flow in the +.v region will be equal to the power flow in the —x
region. Hence, the power flow in the helix will be twice that in the region
X = 0 to .V = + 20 , c = 0 to s = lira sin i/'.
P = 2 / / (l)(E,H* - E,Ht) dx d% (3.26)
Jz=0 •'2=0
This is easily integrated to give
P = 27ra sin ^pE, ^3 27)
The magnitude E of the axial component of field is
£ = £o cos yp (3.28)
Using (3.21), (3.22), (3.24) and (3.28) in connection with (3.27) we obtain
(£y/3'^P) = (7//3)H/3/i3o)(M^/27r7a) (3.29)
We have
Thus
nc = m/a/wc = vW^ = 377 ohms
£2//32p = (7//3)''(^/^o)(60/7a) (3.30)
The longitudinal impedance is half this, and the transverse impedance is
(8/7)- times the longitudinal impedance.
Z.?) Effect of Wire Size
An actual hehx of round wire, as used in traveling-wave tubes, will of
course differ somewhat in properties from the helically conducting sheet
for which the foregoing material applies.
One might expect a small difference if there were many turns per wave-
length, but actual tubes often have only a few turns per wavelength. For
instance, a typical 4,000 mc tube has about 4.8 turns per wavelength, while
a tube designed for 6 mm operation has 2.4 turns per wavelength.
If the wire is made very small there will be much electric and magnetic
energy very close to the wire, which is not associated with the desired field
component (that which varies as exp(— jjSs) in the z direction). If the wire
is very large the internal diameter of the helix becomes considerably less
than the mean diameter, and the space available for electron flow is reduced.
As the field for the helically conducting sheet is greatest at the sheet, this
36
BELL SYSTEM TECHNICAL JOURNAL
means that the maximum available field is reduced. Too, the impedance
will depend on wire size.
It thus seems desirable to compare in some manner an actual helix and the
helically conducting sheet. It would be very difiicult to solve the problem
of an actual helix. However, we can make an approximate comparison by
a method suggested by R. S. Julian.
In doing this we will develoj) the lielix of wires just as the helically con-
CASE n
Fig. 3.9 — The wires of a developed helix with about two turns per wavelength (case I)
and about four turns per wavelength (case II). In the analysis used, the wires are not
quite round.
BOTTOM
I I I I
III,
/'n, >7>, /'^, ,'i; /'•',
r I i
Fig. 3.10 — Voltages on a developed hehcally conducting sheet for two turns per wave-
length.
ducting sheet was developed, by slitting it along a helical line normal to the
wires. We will then consider two special cases, one in which the wires of the
developed helix are one half wavelength long and the other in which the
wires are one quarter wavelength long.
The waves propagated on the developed helix are transverse electromag-
netic waves propagated in the direction of the wires, and the electric fields
normal to the direction of propagation can be obtained from a solution of
Laplace's equation in two dimensions (as in (3.15)-(3.17)).
TRAVELING-WAVE TUBES 37
It is easy to make up two-dimensional solutions of Laplace's equation
with equipotentials or conductors of approximately circular form, as shown
in Fig. 3.9. In case I, the conductors are alternately at potentials — V,-\-V,
— V, etc.; and in case II, the potentials are —V, 0, -fF, 0, —V, 0, -\-V,
etc. Far away in the x direction from such a series of conductors, the field
will vary sinusoidally in the z direction and will vary in the same manner
with \- as in the developed helically conducting sheet. Hence, we can make
the distant fields of the conductors of cases I and II of Fig. 3.9 equal to the
distant fields of developed helically conducting sheets, and compare the
E~/(3^P and the impedance for the different systems. Case I would correspond
to a helix of approximately two turns per wavelength and case II to four
turns per wavelength.
3.3a Two Turns per Wavelength
Figure 3.10 is intended to illustrate the developed helically conducting
sheet. The vertical lines indicate the direction of conduction. The dashed
slanting lines are intersections of the original surface with planes normal to
the axis. That is, on the original cylindrical surface they were circles about
the surface, and they connect positions along the top and bottom which
should be brought together in rolling up the flattened surface to reconsti-
tute the helically conducting sheet.
Waves propagate on the developed sheet of Fig. 3.10 vertically with the
speed of light. The vertical dimension of the sheet is in this case taken as
X/2, where X is the free-space wavelength. The sine waves above and below
Fig. 3.10 indicate voltages at the top and the bottom and are, of course,
180° out of phase. As is necessary, the voltages at the ends of the dashed
slanting lines, (really, the voltages at the same point before the sheet was
slit) are equal.
A wave sinusoidal at the bottom of the sheet, zero half way up and 180°
out of phase with the bottom at the top would constitute along any horizon-
tal line a standing wave, not a traveling wave. Actually, this is only one com-
ponent of the field. The other is a wave 90° out of phase in both the horizon-
tal and vertical directions. Its maximum voltage is half-way up, and it is
indicated by the dotted sine wave in Fig. 3.10. The voltage of this com-
ponent is zero at top and bottom. It may be seen that these two compo-
nents propagating upward together constitute a wave traveling to the right.
The two components are orthogonal spatially, and the total power is twice
the power of either component taken separately.
Figure 3.11 indicates an array of wires obtained by developing an actual
' Section 3.3a is referred to as "two turns per wavelength." This is not quite accurate;
it is in error by the difference between the lengths of the vertical and the slanting lines in
Fig. 3.10.
38
BELL SYSTEM TECHNICAL JOURNAL
helix which has been slit along a helical line normal to the wire of which the
helix is wound. The clashed slanting lines again connect points which were
the same point before the helix was slit and developed. Again we assume a
height of a half wavelength. Thus, if the polarities are maximum +,—,+,
— etc. as shown at the bottom, they will be maximum —,+,—,+,—,+
etc. as shown at the top, and zero half-way up. In this case the field is a
standing wave along any horizontal line, and no other component can be
introduced to make it a traveHiig wave. Half of the field strength can be re-
garded as constituting a component traveling to the right and half as a
component traveling to the left.
TOP
Fig. 3.11 — Voltages on a developed heli.x for two turns per wavelength.
The equipotentials used to represent the field about the wires of Fig. 3.9,
Case I and Fig. 3.10 belong to the field
V -f j^p = In tan (s + jx) (3.31)
Here V is potential and i/' is a stream function. There are negative equi-
potentials about z — X = 0 and positive equipotentials about .v = 0, s =
±t/2. For an equipotential coinciding with the surface of a wire of c-diam-
eter, 2 Swire , d/p is thus
at X = 0, c < 7r/4
ats = 0
d/p = '-^
7r/4
F = hi tan c
V — In tanh .v
(3.32)
(3.34)
Hence, for an equipotential on the wire with an z-diameter 2z, the .v-diani-
eter 2.x- can be obtained from (3.33) and (3.34) as
2.V = 2 tanh-i tan z (3.35)
Of course, the ratio of the x-diameter di to the pitch is given by
di/p =
x/4
(3.36)
where x is obtained from (3.35).
TRAVELING-WAVE TUBES
39
In Fig. 3.12, di/d is plotted vs. d/p by means of (3.35) and (3.36). This
shows that for wire diameters up to d/p — .5 (open space equal to wire diam-
eter) the equipotentials representing the wire are very nearly round.
The total electric flux from each wire is lire and the potential of a wire of
2-diameter 2zisV= — In tan z. Hence, the stored energy Wi per unit length
per wire, half the product of the charge and the voltage, is
Wi = -7re In tan ^ (3.37)
1.6
:l
1.4
A
f
y
1.2
1.0
^^
^^
on
1
0 0.1 02 0.3 0.4 0.5 06 0.7 0.8 0.9
d/p
Fig. 3.12 — Ratio of the two diameters of the wire of a hehx for two turns per wave-
length (see Fig. 3.9) vs. the ratio of one of the diameters to the pitch.
The total distant field and the useful field component are given by ex-
panding (3.31) in Fourier series and taking the fundamental component,
giving
V - -2cos22g^'^ (3.38)
The — sign applies for x > 0 and the -f sign for x < 0. Half of this can be
regarded as belonging to a field moving to the right and half to a field moving
to the left.
For a field equal to half that specified by (3.38), which might be part of
the field of a developed helically conducting sheet, the stored energy Wo
per unit depth can be obtained by integrating {El + Ex) e/2 from .v =
— 00 to .T =: -f ^ and from z = — 7r/4 to +7r/4, and it turns out to be
W2= hire (3.39)
If we add another field component similar to half of (3.38), but in quadra-
ture with respect to z and /, we will have the traveling wave of a helically
conducting sheet with the same distant traveling field component as given
by (3.31). Hence, the ratio R of the stored energy for the developed sheet
to the stored energy for the developed helix is
1
R = 2W2/W1 = -
In tan z
(3.40)
40
BELL SYSTEM TECHNICAL JOURNAL
R is the ratio of tlie stored energies, and hence of the power flows (since
the waves both propagate with the speed of Hglit) of a developed helically-
conducting sheet and a developed helix with the same distant traveling fun-
damental field com{)onents. Hence, at a given distance (Er/0-Py^^ for the
helix is R^'^ times as great as for the helically conducting sheet. In Fig.
3.13, /?'/•' is plotted vs. d/p.
1.5
1.0
CASE I
2 TURNS PER WAVELENGTH
^^
-"
y
i=-— -"^"^
CASE n
4 TURNS PER WAVELENGTH
0.8
0.7
0.6
0.5
0.4
0 y,
,.
r"'' ^
/',
1 /
1 /
11
l/
0 0 1 0.2 0.3 0.4 0 5 0.6 0.7 0.8 0.9 1.0
d/p
Fig. 3.13 — Ratio R^'^ of {E^/0^PY'^ for a helix to the value for a helically conducting
sheet for the distant field.
1.0
0.9
0.8
, ^
-r- y
0.7
- OJ
^il
0.6
O LU
1- m
< <
OS
a _]
LU <
^ 5
z <
04
< 1-
UJ ID
1 1 1
L
CASE n
,.,.._4TURNS PER WAVELENGTH
/
■
A-
--.^
V
....^^^
~\^
f
CASE I
2 TURNS PER WAVELEN
STH^\
"^
d/p
Fig. 3.14 — Ratio R^'^ of {E^/0^Py'^ for a helix to the value for a helically conducting ^
sheet, field at the inside diameter of the helix or sheet.
TRAVELING-WAVE TUBES
41
The maximum available field for the developed helically conducting sheet
(equation (3.38)) is that for x = 0. The maximum available field for the
developed helix (equation (3.31)) is that for an electron grazing the helix
inner or outer diameter, that is, an electron at a value of x given by (3,35).
The fundamental sinusoidal component of the field varies as exp(— 2x)
for both the sheet and the helix, and hence there is a loss in £'- by a factor
e.xp(— 4.v) because of this. We wish to make a comparison on the basis of
E^ and power or energy. Hence, on basis of maximum available field squared
we would obtain from (3.40)
^ -'^ (3.41)
R = -
In tan z
where x is obtained from (3.35). Figure 3.14 was obtained from (3.32),
(5.35) and (3.41).
3.0
2.0
0.8
0.6
0.5
0.4
0.3
\
V
V ^, 2 TURNS PER WAVELENGTH
^
>
X
\
\,
N^ '
N
^s
.\
- — i , 4 TURNS PER WAVELENGTH^
\,
^
^
\
\
0 01 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9
d/p
Fig. 3.15 — The transverse impedance of helices with two and four turns per wavelength
vs. the ratio of wire diameter to pitch.
In a transmission line the characteristic impedance is given by
K = y^l (3.42)
Here L and C are the inductance and capacitance per unit length. This im-
pedance should be identified with the transverse impedance of the helix.
We also have for the velocity of propagation, which will be the velocity of
fight, c,
c = —^^^ = —^ (3.43)
Vlc V\
He
42 BELL SYSTEM TECHNICAL JOURNAL
From (3.42) and (3.43) we obtain
Kt = V^e/C = Vu7e(e/C)
= 377 e/C
Now C is the charge Q divided by the voltage V. Hence
Kt = 377 eV/Q (3.45)
(3.44)
In this case we have
337e bi tan z
J^t = 7i
Zire
Kt = —60 In tSLUg
(3.46)
To obtain the impedance of the corresponding helically conducting sheet
we assume, following (3.30)
Kt = (ym (7//?o) (30/7a) (3.47)
and assuming a slow wave, let 7 = /3, so that
K, = 30/|8oa (3.48)
If we are to have n turns per wavelength, and the speed of light in the
direction of conduction, then we must have
0oa = 1/w (3.49)
whence
Kt = 30n (3.50)
For n = 2 (two turns per wavelength), K = 60. In Fig. 3.15, the charac-
teristic impedance Kt as obtained from (3.46) divided by 60 (from (3.50))
is plotted vs. d/p.
3.3b Four Turns per Wavelength
In this case there are enough wires so that we can add a quadrature com-
ponent as in Fig. 3.10 and thus produce a traveling wave rather than a stand-
ing wave. Thus, we can make a more direct comparison between the de-
veloped sheet and the developed helix.
For the developed helix we have
V + j^|y = In tan (s + jx) + ^f^ . . (3.15)
cos 2 (2 -\- JX)
TRAVELING-WAVE TUBES 43
If we transform this to new coordinates Zi, Xi about an origin at 2 = 0,
.V = 7r/4 we obtain
r + i^ = ,„ (;+tanfa+ix.)\ _ / A \
\1 — tan (zi -h JXi)/ \sm 2 (21 -\-jxi)/
We can now adjust A to give a zero equipotential of diameter 2zi about x —
xi = 0, Zi = 0 (z = x/4) by letting
/I + tan zA
\1 — tan Si/
A = (sin 221) In ^ ^^" "^ (3.53)
\1 — tan Si/
If /I is so chosen, there will be roughly circular equipotentials of 2-diameter
Jzi about 2 = ± 7r/4, etc. There will also be roughly circular equipotentials
I of the same 2-diameter about z = 0, ±7r/2, etc., of potential ±F. That
! about 2=0 has a potential
V = In (i±i^A _A^ (3.54)
\1 — tan 21/ cos 2zi
I where A is taken from (3.53).
j The distance between centers of equipotentials is p — it/A, so that the
ratio of 2-diameter of the equipotentials to pitch is
d/p = 22i/(7r/4) = 2i/(7r/8) (3.55)
The :v-diameter of the equipotential about 2 = 0 (and of those about 2 =
± etc.) can be obtained as 2.v by letting V have the value given by (3.54)
i 2
i and setting ;: = 0 in (3.51), giving
i 1' = In tanh x + f , (3.56)
; cosh 2x
\
The ratio of this .v-diameter to the pitch, di/p, is
! d,/p = y/{T/S), (3.57)
j .T is obtained from (3.56).
[ To obtain the ;r-diameter of the 0 potential electrodes we take the deriva-
tive (3.52) with respect to 21, giving the gradient in the 2 direction
dV .dxp sec^ (21 -\- jxi) sec" (zi + jxi)
+ ^ ^^ = ^ I .-- /- I •■■^ +
dzi dzi 1 + tan (si +y.vi) 1 — tan (si + 7.V1)
(3.58)
_ 2A cos 2(zi -\r jxi)
sin 2(21 -\- jxi)
44
BELL SYSTEM TECHNICAL JOURNAL
We then let ^i = 0 and find the value of Xi for which dV/dzi = 0. When Zi
0, (3.58) becomes
A = s!nh 2x1 tanh 2.vi
(1 — tanh" .Vi)
(1 + tanh2 xi)
(3.59)
As A is given by (3.53), we can obtain x, from (3.57), and the ratio of the
x-diameter d^ to the pitch is
d-Jp = :ti/(7r/8)
Figure 3.16 shows di/d and d^/d vs. d/p.
(3.60)
^bi
o
0 0.1 0.2 0.3 0.4 0.5 0.6 0 7 0.8 0.9
d/p
Fig. 3.16 — Ratios of the wire diameters for the four turns per wavelength analysis.
The ratios R and the impedance are obtained merely by comparing the
power flow for the developed sheet with a single sinusoidally distributed
component with the power flow for case II for the same distant field. In a
comparison with the helically conducting sheet, n = 2 is used in (3.50). The
results are shown in Figs. 3.13, 3.14, 3.15. We see that on the basis of the
largest available field, the best wire size is d/p = .19.
3.4 Transmission Line Equations and Helices
It is of course possible at any frequency to construct a transmission line
with a distributed shunt susceptance B per unit length and a distributed
shunt reactance X per unit length and, by adjusting B and X to make the
phase velocity and E-f^-P the same for the artificial line as for the heli.x.
In simulating the helix with the line, B and X must be changed as frequency
is changed. Indeed, it may be necessary to change B and .Y somewhat in
simulating a lieli.v with a forced wave on it, as, the wave forced by an elec-
tron stream. Nevertheless, a qualitative insight into some problems can bo
obtained by use of this type of circuit analogue.
TRAVELING-WAVE TUBES 45
3.4a Effect of Dielectric on Helix Impedance Parameter
One possible application of the transmission line equivalent is in estimating
the lowering of the helix impedance parameter {E?/ff^Py^^.
In the case of a transmission line of susceptance B and reactance X per
unit length, we have for the phase constant /3 and the characteristic imped-
ance K
~^ (3.61)
(3.62)
Now, suppose that B is increased by capacitive loading so that /3 has a
larger value /3d. Then we see that A' will have a value Ka
Kd = (l3/l3d}K (3.63)
Where should K be measured? It is reasonable to take the field at the
surface of the helix or the helically conducting sheet as the point at which
the field should be evaluated. The field at the axis will, then, be changed
by a different amount, for the field at the surface of the helix is h{ya) times
the field at the axis.
Suppose, then, we design a helix to have a phase constant /3 (a phase
velocity co/)3) and, in building it, find that the dielectric supports increase
the phase constant to a value ^d giving a smaller phase velocity aj/(8d. Sup-
pose |S//So is large, so that 7 is nearly equal to /8. How will we estimate the
actual axial value of {E-f^-PY'^} We make the following estimate:
(£'/.'P)r = (£)""(gg)^"(i^V.= «- (3.64)
Here the factor (13/ ^dY'^ is concerned with the reduction of impedance
measured at the helix surface, and the other factor is concerned with the
greater falling-off of the field toward the center of the helix because of the
larger value of 7 (taken equal to 13 and /3d in the two cases).
The writer does not know how good this estimate may be.
3.4b Coupled Helices
Another case in which the equivalent transmission line approach is par-
ticularly useful is in considering the problem of concentric helices. Such
configurations have been particularly suggested for producing slow trans-
verse fields. They can be analyzed in terms of helically conducting cylinders
or in terms of developed cylinders. A certain insight can be gained very
quickly, however, by the approach indicated above.
We will simulate the helices by two transmission lines of series impedances
jXi and JX2, of shunt admittances jB^ and JB2 coupled by series mutual
46 BELL SYSTEM TECHNICAL JOURNAL
impedance and shunt mutual admittance jXn and jByi . If we consider a
wave which varies as exp(— jTz) in the z direction we have
r/i - jB^Vi - jBnV. = 0 (3.65)
TVi - jXJy - jXnh = 0 (3.66)
r/2 - JB0V2 - jBnVi = 0 (3.67)
TV2 - jX2h - 7X12/1 = 0 (3.68)
If we solve (3.65) and (3.67) for /i and A and eUminate these, we obtain
Fo -(r ^ XrBi + XnBr^
Vi Xi B12 + B2 X12
(3.69)
(3.70)
(3.71)
V2 X2 B12 4- Bi X12
Multiplying these together we obtain
r^ + (Xi B, + Xo B2 + 2X12 Bu)r
+ (Xi Xo - X^2) (Bi B2 - B^2) = 0
We can solve this for the two values of F-
P = _i(Xi 5i + Xo B2 + 2X12 ^12)
± I [(Xi 5i - Xo ^2)-^ + 4 (Xi B, + X2 ^o) (X12 Bu) (3.72)
+ 4 (Xi Xo Bu' + Br B2 Xio2)]i/2
Each value of T'~ represents a normal mode of propagation involving both
transmission lines. The two square roots of each F- of course indicate waves
going in the positive and negative directions.
Suppose we substitute (3.72) into (3.69). We obtain
- (Xi Bi - X2 B^ ± [(Xi Bx - X2 B.f
V2 ^ + 4(Xi Bi + X2 ^2)(Xi2 ^12) + 4(Xi X2 Bil + Bi B2 XA)]"- . .
V, 2{X,Bn + B2Xu) ^ - '^^
We will be interested in cases in which Xi^i is very nearly equal to X2B2.
Let
^Tl = Xi/^i - X2B2 (3.74)
and in the parts of (3.73) where the difference of (3.74) does not occur use
Xi = .Y2 = X
(3.75)
B^^ B2 =^ B
I
TRAVELING-WAVE TUBES 47
Then, approximately
(3.76)
Let us assume that AF- is very small and retains terms up to the first
power of AF-
h = ^l 4- ^ (3 77)
Fx "^^ ^ 2{XBu + BXu) ^ ^
Let
Fo = - XB (3.78)
K?=±i_ ^rg/ro (379)
Let us now interpret (3.79). This says that if AFo is zero, that is, if XiBi =
X2B2 exactly, there will be two modes of transmission, a longitudinal mode
in which F2/F1 = +1 and a transverse mode in which V2/V1 = — L If
we excite the transverse mode it will persist. However, if AFo 9^ 0, there
will be two modes, one for which V2 > W and the other for which F2 < Fi;
in other words, as AF5 is increased, we approach a condition in which one
mode is nearly propagated on one helix only and the other mode nearly
propagated on the other helix only. Then if we drive the pair with a trans-
verse field we will excite both modes, and they will travel with different
speeds down the system.
We see that to get a good transverse field we must make
AFo
-F « 2(Bn/B + Xn/X) (3.80)
1 0
In other words, the stronger the coupling (^12, X12) the more the helices
can afford to differ (perhaps accidentally) in propagation constant and the
pair still give a distinct transverse wave.
Thus, it seems desirable to couple the helices together as tightly as pos-
sible and especially to see that Bn and .Y12 have the same signs.
Let us consider two concentric helices wound in opposite directions, as in
Fig. 3.17. A positive voltage Vi will put a positive charge on helix 1 while a
positive voltage F2 will put a negative charge on helix 1. Thus, Bn/B is
negative. It is also clear that the positive current I2. will produce flux link-
ing helix 1 in the opposite direction from the positive current 7i, thus mak-
I ing Xii/X negative. This makes it clear that to get a good transverse field
between concentric helices, the helices should be wound in opposite direg-
48 BELL SYSTEM TECHNICAL JOURNAL
tions. If the helices were wound in the same direction, the "transverse"
and "longitudinal" modes would cease to be clearly transverse and longitu-
dinal should the phase velocities of the two helices by accident differ a little.
Further, even if the phase velocities were the same, the transverse and longi-
tudinal modes would have almost the same phase velocity, which in itself
may be undesirable.
Field analyses of coupled helices confirm these general conclusions.
Fig. 3.17 — Currents and voltages of concentric helices.
3.5 About Loss in Helices
The loss of helices is not calculated in this book. Some matters concern-
ing deliberately added loss will be considered, however.
Loss is added to heUces so that the backward loss of the tube (loss for a
wave traveling from output to input) will be greater than the forward gain.
If the forward gain is greater than the backward loss, the tube may oscillate
if it is not terminated at each end in a good broad-band match.
In some early tubes, loss was added by making the helix out of lossy wire,
such as nichrome or even iron, which is much lossier at microwave frequen-
cies because of its ferromagnetism. Most substances are in many cases not
lossy enough. Iron is very lossy, but its presence upsets magnetic focusing.
When the helix is supported by a surrounding glass tube or by parallel
ceramic or glass rods, loss may be added by spraying aquadag on the in-
side or outside of the glass tube or on the supporting rods. This is advan-
tageous in that the distribution of loss with distance can be controlled.
It is obvious that for lossy material a finite distance from the helix there
is a resistivity which gives maximum attenuation. A perfect conductor would
introduce no dissipation and neither would a perfect insulator.
If lossy material is placed a little away from the helix, loss can be made
greater at lower frequencies (at which the field of the helix extends out
into the lossy material) than at higher frequencies (at which the fields of
TRAVELING-WAVE TUBES 49
the helix are crowded near the helix and do not give rise to much current in
the lossy material. This construction may be useful in preventing high-
frequency tubes from oscillating at low frequencies.
Loss may be added by means of tubes or collars of lossy ceramic which fit
around the helix.
50 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX I
MISCELLANEOUS INFORMATION
This appendix presents an assortment of material which may be useful
to the reader.
Constants
Electronic charge- to-mass ratio:
Tj = e/m = L759 X 10^^ Coulomb /kilogram
Electronic charge: e = 1.602 X 10~ Coulomb
Dielectric constant of vacuum: e = 8.854 X 10" Coulomb/meter
Permitivity of vacuum: n = 1.257 X 10~ Henry/meter
Boltzman's constant: k = 1.380 X UF"^ Joule/degree
Cross Products
(^'x/i'o. = AyA': - a: Ay
{A' xA")y = a'. A': - a', a':
{A'XAn^ = a'. Ay - Ay A'^
Maxwell's Equ.ations: Rectangular Coordinates
dE, BEy . „ dH, dHy ■ r, , J
-— — = -joinHx -^ - ~7r ^ jweiix + Jx
dy oz dy dz
a£x dE, . „ dih dH, ■ J, , T
-:r- - -:r- = —joijiHy "^ " ^— = J^^^Ey + Jy
dz dx dz dx
dEy dE^ . ^ dHy dih • r , ,
dx dy dx dy
Maxwell's I^quations: Axlally Symmetrical
dz ' "^ " d
dEp dEz • TJ ^^
dz dp dz dp
— ipE^) = —joinpllz —
dp dp
dEp dEz . jj dllp dllz • 77 1 r
= - JoinlU ^- - ^- = J'^^^'P + -^*
d" '^'^
d d
ipE^) = —jcoixpllz — (p//v) = pijc^^Ez + J.,)
TRAVELING-WAVE TUBES 51
Miscellaneous Formulae Involving /„(x) and Kn(x)
1. /.^i(Z) - /.+i(Z) = ^I.iZ), K^^AZ) - K,+dZ) = - yKXZ)
2. h.,{Z) + /,+i(Z) = ll'XZ), K.^,{Z) + A',+i(Z) = - IK'XZ)
3. Z/;(Z) + j'/.CZ) = Zh,^{Z), ZK'JZ) + ^A'.(Z) = - ZA'._i(Z)
4. Z/;(Z) - vI^Z) = ZL+iiZ), ZK'XZ) - vK^Z) - - ZA%+i(Z)
= (-)"'Z'-"'A_(Z)
/ j/ \[L{Z)\ ^ LUZ) /_^\
\Z^Z/ 1 Z- J Z''+'« ' vz</z/ t z
7. /o(Z) = /i(Z), Ao(Z) = -Ai(Z)
KXZ)\^ ^_ynK. + ,n(Z)
8. /_.(Z) = /.(Z), A_.(Z) = A,(Z)
^1/2
.2Z,
9. Ai/2(Z) = ( — ) e-
10. /.(Ze'"'') = r'"^7.(Z)
11. A.(Zr"') = e-'-^'K^Z) - i ^J^L^^ IXZ)
sm vir
12. /.,(Z) A.+i(Z) + /„+i(Z) A.,(Z) = 1/Z
For small values of X :
13. h{X) = 1 + .25 X- + .015625 X' -^ • • •
14. h{X) - .5X + .0625 .Y=' + .002604 X' + • •
15. Ao(X) = -I7+ In
(|)}u..) + i.v= + Ax' +
/
16. A-.(.Y) = {. + ,„ g)} 7.(.Y) + J^ _ 1 X _ ^ .V» + . . .}
7 = .5772 . . . (Euler's constant)
For large values of A' :
e"" f, , .125 , .0703125 , .073242
"■ ^"(■^■' - (2^-- i' + ^ + '-^^ + -^ +
52
18. /i(.V)
BELL SYSTEM TECHNICAL JOURNAL
.375 _ .1171875 .102539
(27rX)"2
...«,v)~(3^)"'.-{.-f + ■.
X2
0703125
20. Ki{X)
X2
1171875
X3
)
.073242
X3
u
.102539
+
• X X2 ' X3
Fig. Al.l shows Io{X) (solid line) and the first two terms of 13 and the
first term of 17 (dashed Hnes).
Fig. A1.2 shows Ii{X) (solid fine) and the first term of 14 and the first
term of 18 (dashed lines).
7 + 1
n(f)}.(:
Fig. A 1.3 shows A'o(X) (solid line) and
first term of 19 (dashed lines).
Fig. A1.4 shows Ki(X) (solid line) and It + In (j)\hO
the first term of 20 (dashed lines).
(X) and the
(X) + 1/X and
100
80
/
-
/
-
/
/
-
/
/
/
f
/
/
■
/
■
/
-
/
^
Fig. Al.l — The coriTcl vuluc of /o(.V) (solid line), the lirst two terms of the series
expansion 13 (dashed line from origin), and the first term of the asymptotic series 17
(dashed lino to right)
TRAVELING-WAVE TUBES
53
100
/
.
/
/
/
f/
/
10
8
6
^ 3
2
1.0
//
y
[
>
/ /
/ /
/ /
/ /
/ /
f /
/ /
/
0.6
0.4
0.3
02
0.1
//
/
/
/
/
Fig. A1.2 — The correct value of /i(X) (solid line), the first term of the series ex-
pansion 14 (lower dashed line), and the first term of the asymptotic series 18 (upper
dashed line).
54
BELL SYSTEM TECHNICAL JOURNAL
1.0
o.a
0.4
0.3
0.2
0.1
0.08
0.06
0.04
0.03
0.01
0.008
0.004
0.003
0.002
0.001
^
\
^^
\
\
\
\
\
\
\
\
\
s
-
\
-
\
-
\
\
\
\
\
-
\
-
\
-
\
\
Fig. A1.3— The correct value of A'o(A') (solid line), - J7 + In (^)? /o(A') from the
series expansion 15 (left dashed line), and the first term of the asymptotic series 19
(right dashed line).
TRAVELING-WAVE TUBES
55
to
0.8
0.6
0.4
0.3
0.1
0.08
0.06
'^ 0.04
J^ 0.03
0.01
0.008
0.004
0.003
r TT^
\ \
v-\
^ \
\J, -.
\\
l\
vV
\\
\\
vV
\\
\\
a\
\\
^\
\\
'\
\\
: vt
A
\
A
\\
Kc
-S\
\\
v\
\\
^\^
\\
\\
\\
V
. ^^^^
V
] \
\
\
\
\
3 4
X
Fig. A1.4 — The correct value of A'i(A') (solid line), <7 + In (y )[ hiX) from the
series expansion 16 (upper dashed line), and the first term of the asymptotic series 20
(lower dashed line).
56 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX II
PROPAGATION ON A
HELICALLY CONDUCTING CYLINDER
The circuit parameter important in the operation of traveUng-wave tubes
is:
(eWpY" (1)
/3 = Wv. (2)
Here Ez is the peak electric field in the direction of propagation, P is the
power flow along the helix, and v is the phase velocity of the wave. The
quantity Ez/l3'P has the dimensions of impedance.
While the problem of propagation along a helix has not been solved, what
appears to be a very good approximation has been obtained by replacing
the helix with a cylinder of the same mean radius a which is conducting
only in a helical direction making an angle ^ with the circumference, and
nonconducting in the helical direction normal to this.
An appropriate solution of the wave equation in cylindrical co-ordinates
for a plane wave having circular symmetry and propagating in the z direc-
tion with velocity
' = i' «
less than the speed of light c, is
Ez = [Aloiyr) + BK,(yr)W''''-^'' (4)
where /o and A'o are the modified Bessel functions, and
y' = ^' - m = ^' - ^0. (5)
The form of the z (longitudinal) components of an electromagnetic field
varying as e' "'^ ' and remaining everywhere finite might therefore be
Ihx = B,h{yr)e''"'-^'' (6)
£z3 = B,h{yr)e'''"-^'' (7)
inside radius a, and
J(ut-ez)
//.2 = B,K,{yr)e'"''-'"' (8)
TRAVELING-WAVE TUBES 57
£.4 = B,K,{yr)e'''''-'" (9)
outside radius a. Omitting the factor g'^"^"^'^ the radial and circumferential
components associated with these, obtained by applying the curl equation,
are, inside radius a,
(10)
(11)
(12)
7
£.3 = bJ^ hiyr) (13)
^03 -
= B,^^ h(yr)
7
Hrl -
= B.^^hiyr)
7
E<i>i =
7
and outside radius a
H^i= -B,^—K,(yr) (14)
7
Hr2 - -Bj-^Kiiyr) (15)
E,,= B,^^K,{yr) (16)
7
£,4 = -Bj-^Kiiyr). (17)
7
The boundary conditions which must be satisfied at the cylinder of radius
a are that the tangential electric field must be perpendicular to the helix
direction
E,3 sin ^ + £^1 cos ^ - 0 (18)
£,4 sin >^ + £^2 cos ^ = 0, (19)
the tangential electric field must be continuous across the cylinder
£23 = £.-4 (and £^1 = £^2), (20)
and the tangential component of magnetic field parallel to the helix direc-
tion must be continuous across the cylinder, since there can be no current
in the surface perpendicular to this direction.
H^i sin ^ + //d,3 cos "^ = H^1 sin ^
(21)
+ H^i cos ^.
58 BELL SYSTEM TECHNICAL JOURNAL
These equations serve to determine the ratios of the /3's and to determine
7 through
^^^' n — ^^^7 — \ "" (/So a cot St') . (22)
ii(7Q;)Ai(7a)
We can easily express the various field components listed in (6) through
(17) in terms of a common amplitude factor. As such expressions are useful
in understanding the nature of the field, it seems desirable to list them in
an orderly fashion.
Inside the Helix:
E, = BIoiyr)e'^'''~^'^ (23)
Er = j^ ^ hMe'^"'-^'^ (24)
7
h^ = -B — — - — -^ h{yr)e' (25)
■'1(70) cot yp
"■ = -' I «" '¥\ 7^, 'M'""-'-' (26)
k (80 /i(7«) cot ^p
^' = 11 r^! 4-, hiyrV'"-'" (27)
k /3o hiya) cot r^
//* = ;• f ^^ I{yr)e'''''-''\ (28)
^ 7
Outside the Helix:
E, = B —-. — - Ko{yr)e'
Ko{ya)
(29)
Here
E.= -jB^-^^K,{yr)e^^^^-^'^ (30)
7 An(7«)
£* = -^ i?-? — N — ^, Ai(7r)e (31)
Ki{ya) cot ^
k po Ki{ya) cot ^
k Ki{ya) cot ^
Z/* =-_;-- — - — - Ki{yr)e' (34)
yfe 7 A 0(70)
ife = Vm/c = 120 TT ohms (35)
TRAVELING-WAVE TUBES
The power associated with the propagation is given by
P = ^Re f EX H*dT
taken over a plane normal to the axis of propagation. This is
P = TrRe
or
P = tEKO)
[ (ErHt - E^H*)rdr + ! (ErHt - E^Ht)rdr
Jo Ja
1 + c
7°^ ) / I\hr)r dr
iiAo/ ''0
+
^2(0)f/-^L
1 + ^' ) (/; - /./=)
-ti Ao
+
(0(-^-^:)'— 4
where k = 120 tt ohms.
Let us now write
{K/^'Py" = WfioY'^y/^rFiya)
where
+ (|y(A-.A,-Al)(.+^»)]}-
59
(36)
(37)
(38)
(39)
(40)
We can rewrite the expression for F(ya) by using relations, Appendix I:
Communication in the Presence of Noise — Probability
of Error for Two Encoding Schemes
By S. O. RICE
Recent work by C. E. Shannon and others has led to an expression for the
maximum rate at which information can be transmitted in the presence of ran-
dom noise. Here two encocHng schemes are described in which the ideal rate is
approached when the signal length is increased. Both schemes are based upon
drawing random numbers from a normal universe, an idea suggested bj'
Shannon's observation that in an efficient encoding system the typical signal
will resemble random noise. In choosing these schemes two requirements were
kept in mind: (1) the ideal rate must be approached, and (2) the problem of
computing the probability of error must be tractable. Although both schemes
meet both requirements, considerable work has been recjuired to put the expres-
sion for the probability of error into manageable form.
1. Introduction
In recent work concerning the theory of communication it has been
shown that the maximum or ideal rate of signaling which may be achieved
in the presence of noise is (1, 2, 3, 4, 5)
Ri=F \og2 (1 + Ws/Wj,) bits/sec. (1-1)
In this expression F is the width of the frequency band used for signaling
(which we suppose to extend from 0 to Z*" cps), PFs is the average signaling
power and Wn the average power of the noise. The noise is assumed to be
random and to have a constant power spectrum of W^/F watts per cps
over the frequency band (0, F).
This ideal rate is achieved only by the most efficient encoding schemes
in which, as Shannon (1, 2) states, the typical signal has many of the proj)-
erties of random noise. Here we shall study two different encoding schemes,
both of them referring to a bandwidth F and a time interval T. By making
the jjroduct FT large enough the ideal rate of signaling may be a{)i:)roached
in either case* and we are interested in the probabihty of error for rates
of signaling a little below the rate (1-1). The work given here is closely
associated with Section 7 of Shannon's second paper (2).
In the first encoding scheme the signal corresponding to a given message
lasts exactly T seconds, but (because the signal is /cero outside this assigned
interval of duration) the power spectrum of the signal is not exactly zero
for frequencies exceeding F. In the second encoding scheme, the signal
* A recent analysis by M. J. E. Golay {Proc. I. K. E., Se])t. 1949, p. 1031) indicates
that the ideal rate of signaling may also be approached by quantized PPM under
suitable conditions.
60
COMMUNICA TION IN PRESENCE OF NOISE 61
power spectrum is limited to the band (0, F) but the signal, regarded as a
function of time, is not exactly zero outside its allotted interval of length T.
It turns out that both schemes lead to the same mathematical problem
which may be stated as follows: Given two universes of random numbers
both distributed normally about zero with standard deviations a and v,
respectively. Let the first universe be called the a (signal) universe and
the second the v (noise) universe. Draw 2iV + 1 numbers A_!^, A-i^+i, • • • ,
Ao^\ • ■ • , Ai?^ at random from the a universe. These 2N -\- \ numbers
may be regarded as the rectangular coordinates of a point Pn in 2.Y + 1-
dimensional space. Draw 2N + 1 numbers B-n, • ■ • , Bi), ■ ■ • , B^ at
random from the v universe and imagine a (hyper-) sphere S of radius .Vo
= Po(?, where
1/2
•To
= Z 5; = p^, (1-2)
centered on the point Q whose coordinates are .4„ + Bn, n = —N,---,
0, • • • , iV. Return to the a universe, draw out A' sets of 2N -\- 1 numbers
each, denote thekth set by .4i^^, • • • , /lo^"\ • • • , .4^^' and the associated
point by Pa-.
What is the probability that none of the A' points Pi, • • • , Pk lie within
the sphere 5? In other words what is the probability, which will be denoted
by "Prob. (PiQ, • • • , PkQ > PoQ)," that the A distances P^Q, • ■ • , PkQ
will all exceed the radius PqQ? In terms of the ^„'s and ^„'s we ask for
the probability that all K of the numbers .ri, xo, ■ ■ ■ , Xk exceed .Vo where
Xk
= Z (Ai'' - Air - Bj' = P,Q^ (1-3)
Expression (1-2) for .vo is seen to be a special case of (1-3). The relationship
between the points Po, Q, Pi, P2, • • • , P/,, • • • , Px is indicated in Fig. 1.
The answer to this problem is given by the rather complicated expression
(4-12) which, when written out, involves Bessel functions of imaginary
argument and of order N — 1/2. When N and A become very large the
work of Section 5 shows that the probability in question is given by
Prob. {PiQ, ••■ ,PkQ> P,Q)
= (1 + erf H)/2 -f 0(1/A) + 0(7V-i/2 log''" N) (1-4)
where, with r = v'/a",
H = (^-4^ ')''' [(^V + 1/2) log. (1 + \/r) - log. (A + 1)
, 1 , 27rA^(l -f 2r)"
+ 2^°^^ (1 + .)^
(1-5)
62
BELL SYSTEM TECHNICAL JOURNAL
The symbol 0{N-'^'^ log''^ N) stands for a term of order N'^''^ log3/2 N, i.e.,
a positive constant C and a value No can be found such that the absolute
value of the term in question is less than CN~'^'~ log^'^ N' when N > .Yi,.
In order to obtain actual numerical values for C and A^o, considerably
more work than is given here would be required. The term 0(1/A') is of
the same nature. The "order of" terms have been carried along in the work
of Section 5 in order to guard against error in the many approximations
which are made in the derivation of (1-4).
0 is the origin of coordinates
a'-^Iv-Ao^V-A^n^ of point Pk
in space in 2m + t dimensions
Fig. 1 — Diagram indicating relalionshi]) between points Po, Q, and Pk corresponding
to signal, signal plus noise, and k^^ signal not sent (k > 0), respectively.
The last term within the bracket in (1-5) has been retained even though
it gives terms of order A'^"*'' log A^ when (1-5) is j^ut in (1-4) and could
thus be included in 0(.V""'' log^'- N). As shown by the table in the next
paragraph, inclusion of this term considerably imjjroves the agreement
between (1-4) and values of I^rob. (PiQ, • ■ ■ , PkQ > P^Q) obtained by
integrating the exact expression (4-12) numerically. This suggests that
the term ()(A' "- log^'- A') in (1-4) is unnecessarily large.
Although the "order of" terms in (1-4) give us some idea of the accuracy
of the apj)roximati()n expressed by (1-4) and (1-5), a better one is desirable.
With this in mind the lengthy task of computiiTg the exact ex[)ression (4-12)
for Prob. (PiQ, ■ • • , PkQ > PoQ) by numerical integration was undertaken.
COMMUNICATION IN PRESENCE OF NOISE 63
The values obtained in this way are Usted in the second column of the
following table. The values of Prob. (PiQ, • • ■ , PkQ > PoQ) obtained
from (1-4) (in which the "order of" terms are ignored) and (1-5) are given
in the third column. Column IV lists values obtained from (1-4) and a
simplified form of (1-5) obtained by omitting the last term in (1-5). These
values are less accurate than those in the third column. The values in
Column V are computed from (1-5) and a modified form of (1-4) obtained
by adding the correction term shown in equation (5-53) (with B — H).
The values in Column V are presumably the best that can be done with the
approximations made in Section V of this paper, although the first entry
renders this a little doubtful.
Prob. {PiQ, ■■■ , PkQ > PoQ) for N = 99.5 & r = 1
A- + 1
Numerical
Integration
(1-4) & (1-5)
Col. IV
Col. V
r,100 -30
2 e
.994
.9995
.9987
1.0001
^100 -15
2 e
.962
.9650
.9337
.9710
^100
.603
.621
.5000
.605
nlOO 15
2 e
.1196
.1159
.0663
.1176
2 e
.0065
.00347
.0013
.00586
become
apparent later that the value iv + 1 =
= 2^°" corresponds
to the ideal rate of signaling. The non-integer value of 99.5 for N is ex-
plained by the fact that the calculations were started before the present
version of the theory was worked out. It will be noticed that for A' -f- 1 =
9100^-30 ^^ Qf ^]r^Q approximate values exceed the .994 obtained by numerical
integration. I am in doubt as to whether the major part of the discrepancy
is due to errors in numerical integration (due to the considerable difficulty
encountered) or to errors in the approximations.
In both encoding schemes, the point Po corresponds to the transmitted
signal, Q to the transmitted signal plus noise, and Pi, P^, • • • Pk to K
other possible signals. The average signal power turns out to be (A^ + 1/2)<t-
and the average noise power to be (^V + l/2)j'-. Furthermore,
.vo = twice the average power in the noise.
Xk = " " " " " " " plus the ^th signal.
Prob. (PiQ, ■ ■ ■ PkQ > PoQ) = Probability that none of the K other
signals will be mistaken for the signal sent,
i.e., the probability of no error.
The random numbers A „ are taken to be distributed normally instead
of some other way because this choice makes the encoding signals (in our
two schemes) resemble random noise, a condition which seems to be neces-
sary for efficient encoding (1, 2).
64 BELL SYSTEM TECHNICAL JOURNAL
Both of the encoding schemes are concerned with sending, in an interval
of duration T, one of A' + 1 different messages. According to communica-
tion theory (1, 2, 3) this corresponds to sending at the rate of T~^^ loga
{K + 1) bits per second. However, instead of discussing the rate of trans-
mission, it is more convenient, from the standpoint of (1-4), to deal with
the total number of bits of information sent in time T. Thus, selecting and
sending one of the A' + 1 possible messages is equivalent to sending
M = logo(A + 1) (1-6)
bits of information. M, or one of the adjacent integers if M is not an integer,
is the number of "yes or no" questions required to select the sent message
from the A + 1 possible messages (divide the A -|- 1 messages into two
equal, or nearly equal, groups; select the group containing the sent message
by asking the person who knows, "Is the sent message in the first group?";
proceed in this way until the last subgroup consists of only the sent mes-
sage). The amount of information which would be sent in time T at the
ideal rate Ri defined by (1-1) is
Mj = TRr = FT log2 (1 + l/r)= (N -\- 1/2) logs (1 + 1/r) (1-7)
where use has been made of Wf^/Ws = v'/tr- = r, and the relation N <
FT < iV^ -t- 1 (which turns out to be common to both encoding schemes)
has been approximated by A^ + 1/2 = FT.
When (1-6) and (1-7) are used to eliminate N and A from (1-5) the
result is an expression for the actual amount M of information sent (in
time T) in terms of (1) the amount Mi which is sent by transmitting at
the ideal rate (1-1) for a time T, (2) the ratio r of the noise power to the
signal power, and (3) the probability of no error in sending M bits of in-
formation in time T, this probability being given as (1 -j- erf H)/2:
M = Ml- qMY'H + b (1-8)
where
a = 2
r iog2 e Y
L(l + r) log. (1 + l/r)J '
, 1, r 27r(l + 2r)Mj ]
' - 2 ^°^^ L(l + OMog.(l + l/.)J
(1-9)
Here the "order of" terms in (1-4) have been neglected together with
similar terms which arise when N -f 1/2 is used for A'' in com])uting a and
b. The term b is usually small compared to aM,II.
The more slowly we send, the less chance there is of error. The relation-
shij) between M, Mi and the ])r()l)ability of no error, as cominited from
COMMUNICATION IN PRESENCE OF NOISE 65
(1-8), is shown in the following table. The probability of no error is de-
noted by p and the terms are given in the same order as on the right of
(1-8) in order to show their relative importance. The ratio M/Mi{=R/Ri)
for r = 0.1 is shown as a function of M in Fig. 2.
For r = Ws/Ws = 0.1
Mi bits M for p = .5 M for p = .99 M for p = .99999
102 ^^ _ 0 + 3.75 Ml - 24.3 + 3.75 Mj - 44.6 + 3.75
W " '' + 7.07 " - 243+ 7.07 " - 446+ 7.07
106 u "4-10.38 " -2430+10.38 " -4460+10.38
For r = W^/Ws = 1
102 Mj-O-i- 4.44 M7-33.4+ 4.44 Af7-61.2+ 4.44
10^ *' "+ 7.76 " - 334+ 7.76 " - 612+ 7.76
10' " "+11.08 " -3340+11.08 " -6120 + 11.08
There may be some question as to the accuracy of the values for p = .99999,
especially for Mi = 100, since this corresponds to points on the tail of
the probability distribution where the "order of" terms in (1-4) become
relatively important.
Of course, for a given bandwidth, the ideal rate of signaling Ri (given by
(1-1)) for r = .1 exceeds that for r = 1 in the ratio (logo ll)/(log2 2) =
3.46.
The above results agree with the statement that, by efficient encoding,
the rate of signaling R can be made to approach the ideal rate Ri = Mi/T
given by (1-1). As applied to our two schemes, the term "efficient encoding"
means using a very large value oi FT or N. To see this, divide both sides of
(1-8) by Ml and rearrange the terms:
1 - M/Mi = aH M'"' + 0(M7' log Mi) (1-10)
When Mi is replaced by RiT in M/Mj, the fraction M/T occurs. We shall
set R = M/T and call R the rate of signaling corresponding to some fixed
probability of error (which determines H). Thus, when (1-7) and the defini-
tion (1-9) for a are used, (1-10) goes into
^^^ ~ ^^ = ^ A- 0((\o<y FT) /FT) (1-11)
Ri [(1 + r)FTY'Uoge(l + 1/r) + ^^lo, /^ i |, /^ i ; U ii;
Equation (1-11) shows that when r and H are fixed (i.e. when the noise
[)Ower/signal power and the probability of error are fixed) R/Ri approaches
unity as FT — » 00 . This is shown in Fig. 2 for the case r = 0.1. S'mceR/Ri =>
M/Mi, M/Mj must approach unity and consequently M as well as Mi in-
66
BELL SYSTEM TECHNICAL JOURNAL
creases linearly with FT. Thus, for efficient encoding M is large and, from
(1-6), so is A'.
It should be remembered that equation (1-8) has been established only
for the two encoding schemes of this article. The question of how much
faster M/T approaches Ri for the more efficient encoding schemes mentioned
at the end of Section 2 still remains unanswered.
0.6
h-^
.5
^
'
~
0.£
P-
**
/
/
/
/
/^.99999
y
/
>
/
/
AS M— i-OO
t-R/Rl~aH//M
WITH "a" DEFINED
BY EQUATION (1-9)
r
/
/
y
/
/
f
1
3 i
i
> i
■> i
s
\
3 i
}
\
i
i
I i
\ e
i
10^ lO'*
M = BITS OF INFORMATION IN MESSAGE
105
10®
Fig. 2 — Curves showing the approach of /?//?/ (= M I Mi) to unity as the message
length increases and the probabihty of no error remains tixed. R is the rate of signaHng
at which the probabihty of no error is p and Rj is the ideal rate.
It gives me pleasure to acknowledge the help I have received in the prepa-
ration of this memorandum from conversations with Messrs. H. Xyquist,
John Riordan, C. E. Shannon, and M. K. Zinn. I am also indebted to Miss
M. Darville for comj)uting the tables shown above and for checking a num-
ber of the equations numerically.
2. The First Encoding Scheme
Suppose that we have A' + 1 different messages any one of which is to
be transmitted over a uniform frequency band extending from zero to the
nominal cut-off frequency F in a time interval of length T. The adjective
"nominal" is used because the sudden starting and stopping of the signals
given by the first encoding scheme produces frequency components higher
COMMUNICATION IN PRESENCE OF NOISE 67
than F. A shortcoming of this nature must be accepted since it is impos-
sible to have a signal possessing both finite duration and finite bandwidth.
The first step of the encoding process is to compute the integer A^ given
by
N < FT < N + 1 (2-1)
We assume that FT is not an integer in order to avoid borderhne cases.
Let W s be the average signal power available for transmission and define
the standard deviation a of the o- universe introduced in Section 1 by
(A^ + 1/2)0-- = W s- To encode the first message, draw 2N -\- 1 numbers
A-N, • • •,Ao'\- ■ • A^^^ at random from the a universe. The signal correspond-
ing to the first message is then taken to be
/o(/) - 2-''l4r + Z (Ai'^ COS iTTUt/T + A^-'l sin 2x»//r) (2-2)
71 = 1
The remaining A' messages are encoded in the same way, the signal repre-
senting the ^th message being
hit) - 2"^'-^^'"' -H Z Un^ coslirni/T + A^-l sin 2Tni/T). (2-3)
n=I
It is apparent that each signal consists of a d-c term plus terms corre-
sponding to A' discrete frequencies, the highest being N/T < F, and that
the average power (assuming hiO to flow through a unit resistance) in the
^th signal is
T-' f' Ilit) dl = 2-\Al''f + Z 2-\{A':'f + {A'Jlir-] (2-4)
''- 7-/2 71=1
Since the .I's were drawn from a universe of standard deviation cr, the ex-
pected value of the right hand side is (2A" + \)a~/2 which is equal to the
average signal power W s, as required.
We pick one of the A' + 1 messages at random and send the correspond-
ing signal over a transmission system subject to noise. We choose our nota-
tion so that the sent signal is represented by /o(/) as given by {2-2). Let the
noise be given by
.V
/(/) = 2~"-B^ -f Z {Bn cos 2irnt/T + 5_„ sin 27r»//r) (2-5)
74 = 1
where 5_.v, • • • , 7?n, • • • , B^^ are (2A^ + 1) numbers drawn at random from
the normally distributed v universe mentioned in the introduction. The
standard deviation v of the universe is given by (A^ + l/2)v~ — W x, Wn
being the average noise power. We call ./(/) simply "noise" rather than
68 BELL SYSTEM TECHNICAL JOURNAL
"random noise" to emphasize that (2-5) does not represent a random noise
current unless N and T approach infinity.
The input to the receiver is /o(/) + /(/). Let the process of reception
consist of computing the K -\- \ integrals
Xk = 2T~' f [h(!) - /o(/) - /(/)]' dl, k = 0,1, ■■■ ,K (2-6)
J-T/2
and selecting the smallest one (all of the A' + 1 encodings have been carried
to the receiver beforehand). If the value of k corresponding to the smallest
integral happens to be 0, as it will be if the noise /(/) is small, no error is
made. In any other case the receiver picks out the wrong message.
When the representations (2-2), (2-3), and (2-5) are put in (2-6) and the
integrations performed, it is found that
x,= i; (A['' - Ai'' - B.f, Xo= t Bl (2-7)
n=—N n=— .V
which have already appeared in equations (1-2) and (1-3). If, as in Section
1, Pk is interpreted as a point in 2xV + 1 — dimensional Euclidean space with
coordinates .1-a-, • • • , Ao''\ • • • , A^-' and Q is the point A-^ + B_x, • • • ,
Ao ^ + Bg, . . . ,A]^' + Bx, then Xk is the square of thedistance between points
Pk and Q. Point Po corresponds to the signal actually sent, points Pi, • • • ,
Pk to the remaining signals, and point Q to the signal plus noise at the
receiver. The expected distance between the origin and Pa- is (t(2X + 1)^'-
= (2ir.s)^'-, that between P„ and (J is vi2X + D^'- = (2W.s-yi-\ and that
between the origin and Q is
(a- + i.2)i/2(2^r ^ 1)1/2 ^ (2ir.,. + IWsY"
No error is made when Xo is less than every one of .vi, .vo, • • • , .Va-, i.e.,
when none of the points Pi, • • • , Pk lies within the sphere S of radius .vi'"
centered on Q and passing through Pn. Therefore the probability of obtain-
ing no error when the first encoding scheme is used is equal to the probability
denoted by Prob. (PiQ, • • • , PkQ > PoQ) in the mathematical problem of
Section 1.
One might wonder why probability theory has played such a prominent
part in the encoding scheme just described. It is used because we do not
know the best method of encoding. In fact, it would not be used if we knew
how to solve the following problem:* Arrange A' + 1 points Pq, • • • Pk on
the hyj)er-surface of the 2M + 1 — dimensional sphere of radius (2irs)^'^
* C. E. Shannon has commented that although the solution of this problem leads to a
good code, it may not be the best possiljle, i.e., it is not obvious that the code obtained
in this way is the same as the one obtained by choosing a set of points so as to minimize
the probability of error (calculated from the given set of points and some given W\)
averaged over ail A' -|- 1 points.
COMMUNICATION IN PRESENCE OF NOISE 69
in such a way that the smallest of the A' (A' + l)/2 distances Pa -P^, k,( = 0, 1,
• • • ,K,k 9^ /, has the largest possible value. This would maximize the dif-
ference (as measured by the distance between their representative points)
between the two (or more) most similar encoding signals.!
In this paper we have been forced to rely on the randomness of probability
theory to secure a more or less uniform scattering of the points Po, • • • , Pr.
In our work they do not lie exactly on a sphere of radius (2Wsy'' but this
causes us no trouble.
3. The Second Encoding Scheme
The second of the two encoding schemes is suggested by one of Shannon's
(2) proofs of the fundamental result (1-1). In this scheme the A -+- 1 mes-
sages are to be sent over a transmission system having a frequency band ex-
tending from zero to F cycles per second, and are to be sent during a time
interval of nominal length T.
The first few steps in the encoding process are just the same as in the first
scheme. N is still given by (2-1) and a by (N + l/2)(r- = Ws- After drawing
A -f 1 sets of ^'s, with 2N + 1 in each set, the A -|- 1 messages are
encoded so that the signal corresponding to the ^th message, ^ = 0, 1, • • •,
A, is
/.(/) = (FT-)"' ± A':"'" ;'jl^' - f (3-1)
„=_Ar TT^lPt — n)
From (3-1), the value of lk{0 at / = n/{2F) is zero if the integer n exceeds
A^ in absolute value. If the integer n is such that | w | < N, the corresponding
value of Ik{t) is (FTY'^An''. The energy in the ^th signal is obtained by
squaring both sides of (3-1) and integrating with respect to /. Thus
r lliOdl = 2-'T i: A'!:'' (3-2)
J-aa n=-N
which has the expected value {N -f- l/2)<r-r. The average power developed
when this amount of energy is expended during the nominal signal length
r is (iV -h 1/2)(T- which is equal to W s, as it should be.
The noise introduced by the transmission system is taken to be
J(t) = (fr)'« t B.'^f^^'-f (3-3)
„ — N t{2FI — n)
t Possibly if A' + 1 discrete unit charges of electricity were allowed to move freely
on the sphere, their mutual repulsion would separate them in the required manner. In
2N -\- 1 dimensions this leads to the problem of minimizing the mutual potential energy
■where N >\ and the summation extends over k, I = 0,\, . . . K with k 9^ (. However,
this problem also appears to be difficult.
70 BELL SYSTEM TECHNICAL JCURXAL
where the v universe from which the B's are drawn has, as before, standard
deviation v given by (.V + 1 '2)v- = Ws. When the signal /o(/) is sent, the
input to the receiver is /,)(/) + J{l) and the process of reception consists of
selecting the smallest of the A' + 1 .v^'s
x, = 2T-' f [IdO - /o(/) - J{t)?dl (3-4)
•'-00
= i: (at - Air -bs-
n=—S
The second expression for .v/, is the same as the one given by (2-7) for the
first encoding scheme, and the discussion in Section 2 following (2-7) may
also be applied to the second encoding scheme. In particular, the probabiUty
of obtaining no error in transmitting a signal through noise is the same in
both systems of encoding, and is given by the Prob. {P\Q, • • • , PkQ > PoQ)
of the mathematical problem of Section 1.
4. Solution of the Mathematical Problem
We shall simplify the work of solving the mathematical problem stated
in Section 1 by taking a = 1 and v-/(r- = r. First regard the 4X + 2 numbers
An , Bn, n = — .Y, • • • , N as fixed or given beforehand. Geometrically, this
corresponds to having the points Pq and Q given. Select a typical set of
random variables A,, , w = —N,---, N, k > 0 and consider the associated
set of variables
y„ = ^f - A'y - 5„ = .4i" -f y„. (4-1)
y„ is a random variable distributed normally about its average value
% = -Ai'^ - Bn (4-2)
with standard deviation a = \. The quantity .ya-, defined by (1-3) and repre-
senting the square of the distance between Pk and Q, may be written as
N
n=—N
Thus Xk is the sum of the squares of 2.V + 1 independent and normally
distributed variates, having the same standard deviation but different
average values. The probability density of such a sum is remarkable in
that it does not depend upon the y„'s individually but only on the smu of their
squares which we denote by
«= z ill = i; u\r -^ Bnf
n=-S n=-\
(4-4)
_ 1 fEnergv' in sent signal + Encrgyl
|_in noise J
COMMUNICATION IN PRESENCE OF NOISE 71
This behavior follows from the fact that the probability density of Pk has
spherical symmetry about the origin (because all the .4^ 's have the same
a). For the probability that Xk is less than some given value x is the prob-
ability that Pk lies within a sphere of radius .t^''' centered on Q, and this,
because of the symmetry, depends only on .v and the distance «^'- of Q from
the origin. Accordingly, we write p{x, u)dx for the probability that
X < Xk < X + dx when the a'„'s (and hence n) are fixed.
The probability density p{x, u) may be obtained from its characteristic
function:
dz
r JL.
(4-5)
p{x, u) = {lir) ' / e '"""[ave. e''^]
J— 00
[A- -1
iz 2^ y'n
n =—N
= IT ave. exp [izy'n] = (1 — 2iz)~^~^'~ e.xp [ms(l — 2/3)"^]
where we have used (4-3) and, since y„ is distributed normally about Vn,
ave. exp [tzy„\ = {Itt) \ e "^ - djn
= (1 — 2is)~^'" exp [y'ni^i'^ — 2/z)~']
Hence
= (2x)"' f (1 - 2/c)-'''-''- exp fi3«(l - lizV' - izxldz
(4-6)
pix,u) =(2x)"' /* (1 - lizy-"- exp [izuil - lizT' - hx\ dz
O-l/ / \2V/2-l/4 J- r/ \l/2i -(«+x)/2
where it is to be understood that .v is never negative. The Bessel function
of imaginary argument appears when we change the variable of integra-
tion from z to / by means of 1 — 2iz = 2t/x, and bend the path of integra-
tion to the left in the / plane (6). This expression for the probability density
of the sum of the squares of a number of normal variates having the same
standard deviation but different averages has been given by R. A. Fisher
(7).
We are now in a position to solve the following problem which is somewhat
simpler than the one stated in Section 1: Given the 2X -\- 1 coordinates
AI'' oi the point Pq and the 2A" -\- 1 numbers J5„ so that the coordinates
A n -f- Bn of the point Q are given. What is the probability that none of the
K points Pi, P2, ■ ■ ■ , Pk, whose coordinates A [ are drawn at random from
a universe distributed normally about zero with standard deviation a = 1,
be inside the sphere centered on the given point Q and passing through the
other given point Po? In other words, what is the probability that all K of the
72 BELL SYSTEM TECHNICAL JOURNAL
independent random variables .vi, xo, ■ • • , Xk will exceed the given value
0^0 when ii has the value defined by (4-4) together with the given values of
the An^^^ and BnS>} The variables .ti, X2, • • • , Xr have the probability-
density p{x, n) shown in (4-6) and .vo is defined by (1-2) and the given values
of the ^„'s.
The answer to the above problem follows at once when we note that the
probability of any one of xi, • • • , Xk, say .Vi for example, being less than
Xo IS
' p{x, u) dx. (4-7)
0
The probability of Xi exceeding xo is then 1 — P(xo, u) and the probability
of all A' of Xi, • ■ • , Xk exceeding Xo is
[1 - P(xo, u)]'' (4-8)
Instead of being assigned quantities, Xo and u are actually random varia-
bles when we consider the problem of Section 1. Now we take up the problem
of finding the probability density of u when xo is fixed. Thus, from (4-4),
we wish to find the probability density of
«= i: u':' + bS' (4-9)
n=—N
in which the 2.V -f- 1 numbers An are drawn at random from a universe
distributed normally about zero with standard deviation a — \ and the
numbers B-n, • • • , Bq, ■ • ■ , Bn are given. It is seen that u is the sum of
the squares of 2N + 1 normal variates all having the standard deviation
0" = 1. The n\h variate, .4^"^ + Bn, has the average value i?„. This is just
the problem which was encountered at the beginning of this section. Equa-
tion (4-9) is of the same form as (4-3) and we have the following correspond-
ence:
Equation (4-3) Equation {4-9)
Xk u
yn Air + Bn
% ^ Bn
n = Zlyn Xo = Z^B'n
The probability that u lies in the interval u, u + du when .vo is given is there-
fore p{u, Xo) du where p{u, Xo) is obtained by putting u for x and xq for u
in the probability density p(x, u).
Until now xo has been fixed. At this stage we regard B-n, • • - , Bq, • • • , Bn
as random variables drawn from a normal universe of average zero and
standard deviation i> = ar^^- — r^'-. If the standard deviation were unity,
COMMUNICATION IN PRESENCE OF NOISE 73
the probability density of .vo could be obtained directly from p{x, u) by
letting M -^ 0 in (4-6). As it is, the a-'s appearing in the resulting expression
must be divided by r to obtain the correct expression. Thus, the probabiUty
of finding xo between .Tq and .vo + ^.Vo is
which is of the x" type frequently encountered in statistical theory.
It follows that the probability of finding u in {u, u -f du) and Xo in
(.To, .Vo + (/.Vo) at the same time is Pq{u, .Vq) du dxo where
po(u, .Vo) = p(u, Xo)po{Xo)
1 /uxoY'-'-^'* ^ ^, .„., _,„^,„„^,,.„,, (4-11)
4rr(iV + 1/2)
(..,. \ Ar/2-1/4
The replacement of (.v, «) in (4-6) by (m, .Vo) should be noted.
Now that we have the probability density of u and .Vo we may combine it
with the probability (4-8) that all A' of .Ti, • • • , Xk exceed .Vo when Xq and u
are fixed. The result is the answer to the problem stated in Section 1 :
Prob. (Pi(2, •••,Px<2>/^o0
= / du \ dxnp^{u,x^\\ — /'(.Vo, 7^]
Jo ♦'0
This result is more complicated than it seems, for ^o(;/, .Vo) is given by (4-11)
and P{xi^, ii) is obtained by integrating p{x, u) of (4-6) from .t = 0 to a; = x^
in accordance with (4-7). The remaining portion of the paper is concerned
with obtaining an approximation to (4-12) which holds when N and K are
very large numbers.
5. Behavior of Prob. {P\Q, • • • , PrQ > PoQ) as X and A' Become Large
In this section we introduce a number of approximations which lead to a
manageable expression for Prob. (PiQ, • ■ ■ , PrQ > PoQ) when N and K
become large.
Since u and Xo are sums of independent random variables, namely
n=-N
A'o = z^ B„ ,
the central limit theorem tells us that the probability density Po{u, Xo) ap-
proaches a two-dimensional normal distribution centered on the average
74
values
BELL SYSTEM TECHNICAL JOURNAL
n= Y. avcUi"'- + Bl] = {2X + 1)(1 + r)
n=-N
N
xo = E ave. Bl = (2.V + l)r
(5-2)
Here we keep the convention a = 1, v'/a- = r used in Section 4. The same
sort of reasoning as used to establish (5-2) shows that the spread about these
average values is given by
ave. (u - u)- = (4iV + 2)(1 + rY
ave. (xo — a-,))- = (4.V + 2)r-
ave. (u — u){xo — xo) = (4.V + 2)r-
(5-3)
If the parameters A^, K, and r in the integral (4-12) are such that its value
is appreciably different from zero, most of the contribution comes from the
region around I'l and .vo where />o(w, .Vo) is appreciably different from zero.
However, instead of taking m and fo as reference values, we take the nearby
values
u.2= u - 2 - 2r = (2N - 1)(1 -\- r) = 2q(l + r)
Xo = .vn — 2r = {2A — l)r = 2qr
as these turn out to be better representatives of the center of the distribu-
tion. We have introduced the number
q = N
1/2
(5-5)
in order to simplify the writing of later equations. We assume ^ > 1.
First, we shall show that
Prob. (PyQ, ■■■ ,F^Q> PoQ)
= / ' du ' dxopoiu, .Vo)[l - P{xo , li)]" + ^1
J uo—a Jx'j—b
(5-6)
where a = 2(1 + r){2q log </)'/-, b = 2r{2q log </)'/'- and Ri is of order l/q
(denoted by 0(1/^)), i.e. a constant C and a value (/n can be found such that
I i?i I < C/q when q > qo. From (4-12) it is seen that Ri is positive and less
than
f du -\- du / dxopo(u,Xo)
0 J U2+a J •'0
+
' r/.vii + / (/.\„ / di<pu{u, :
0 Jx-y^li J ''0
(5-7)
r„)
COMMUNICATION IN PRESENCE OF NOISE 75
Since p^^iii, Xn) is the joint probability density of ti and .Vo, the integration
with respect to .Vo in the first part of (5-7) yields the probability density of
u, and the integration with respect to ii in the second part gives the prob-
ability density /Jci(.Vo) (stated in (4-10)) of .Vu. Thus (8)
i
dxopoiu, .Vo) =
[«/2(l + rWe
q -«/2(l + r)
'o "■—"••- 2(l + r)r(g-f 1)
(5-8)
e
I
dupo(n, .To)
:.vo/2H'r"'°''^
'o "" ' 2rT(q + 1)
Setting (S-S) in (5-7) and putting u = 2(1 -f r)y and .Vo = 2ry in the two
parts of (5-7) reduces them to the same form. Thus (5-7) is equal to
2 -
r(<7 +
T)Ly'^~''y (5-^)
with / = (2q log qY^-. In order to show that (5-9) is 0(1 '9) we use the ex-
pansion
-y -f g log y = -q -\- qlogq - (y - q)-/(2q) + (y - qY/{3q-)
- (v - qYqlq + (y - q)d\-'/4:
where 0 ^ 6 ^ 1. Let v represent the sum of the (y — qY and (y — qY terms,
and expand exp r as 1 + v plus a remainder term. The integral of exp —
(y — q)-/(2q), taken between the hmits q ±: (, can be shown to be of the
form 1 — 0(1 'q) by integrating by parts as in obtaining the asymptotic
expansion for the error function. The term in (y — qY vanishes upon integra-
tion and the remainder terms may be shown to be of 0(1/ q). In all of this
work a square root of q comes in through the fact that
1 > i2irqY'-q''e-''/r(q -f 1) > exp [- \/il2q)] (5-10)
We have just shown that the error introduced by restricting the region of
integration as indicated by (5-6) introduces an error of order 1 'q which
vanishes as 5 ^ ^c . The normal law approximation to po(ii, Xo) predicted by
the central limit theorem holds over this restricted region. However, instead
of appealing to the central limit theorem to determine the accuracy of the
approximation, we prefer to deal directly with the functions involved.
Consideration of (5-4) and the behavior of p(,{u, .Vu) suggests the substitu-
tion
.To = 2r{q -I- a)
u = 2(1 -f r){q -f /3)
(5-11)
76 BELL SYSTEM TECHNICAL JOURNAL
where a and 13 are new variables whose absolute values never exceed
(2 g log qY'- in the restricted region of integration of (5-6). From (4-11)
p.iu'x.) dn dx, = iL±-^ (^^^"'' Uz''')e-'^^''''^^"'-'' da d& (5-12)
in which
s = ux, = 4r(l -f r){q + a)iq + fi) (5-13)
In Appendix II it is shown that
/ (^^''-) = <l'^"'e-'z^''exp[(q' + zr--\-V]
r{q + i)(g2 + zyi^iq + (^2 + 2)i/2]« ^^■''*^
where | P' | < 1/(2^ — 1) when ^ > 1. Upon using (5-10) and (5-14) the right
hand side of (5-12) may be written as
da di3{27ry"'a + r)(2r)-''((/2 -f z)-'" exp [- (1 + r)(2q + a i- /3) (5-15)
+ /(s) - log T{q + 1) + 0(l/q)]
with
f(z) = q\ogz- q log [q + (^^ -f zY''] + (^^ + s)!/^ (5-16)
The value Zo of z corresponding to the central point («2, -Vj) of />o(«, -Vo) is
obtained by putting a = /3 = 0 in (5-13):
So = 4^1 + 0?-
(5-17)
2 - So = 4K1 + r)[5(« + iS) + ai3].
Since we are interested in the form of Pq{u, Xq) in the restricted region of
integration of (5-6) we expand /(s) about s = Z2 in a Taylor's series plus a
remainder term.
/(s) = q log 2rq -f (y(l -f 2r) + {z - z^M{Arq)
(5-18)
(2 - z,f (z - 2,)^ r(^3 + g)^(3^3 - y)-
32rY(l -f 2r) "^ 3! [ 8 s' ^3
In the last term S3 = S2 + (2 — 20)6, 0^6^ 1, ^3 = </'■+ -3. The work of
obtaining this expansion is simplified if (q- -\- s)''- is replaced by ^ in (5-16)
before differentiating. For example, by using 2^'| = 1, it can be shown that
f'(z) is simply (q + |)/(2s). When the extreme values of a and f3 are put in
(5-17), it is seen that s — Zo does not exceed 0{(f'- log^'- q) in the restricted
region of integration. In the last term of (5-18) Z3 is 0(5^), ^3 is O(^) and con-
sequently the last term itself is 0(</~^/- log^'- q).
COMMUNICA TION IN PRESENCE OF NOISE 77
When the expression (5-17) for (2 — 22) is put in (5-18) an expression for
/"(:;) is obtained. This expression, together with
log T{q + 1) = (g + 1/2) \ogq-q+ (1/2) log Itt + 0(l/g),
enables us to write the argument of the exponential function in (5-15) as
q log 2r - (1/2) log lirq - Q(a, /3) + 0(q-'i'- log^/'-^ q) where Q{a, 0) denotes
the quadratic function
(5-19)
Qia, 13) = [(1 + rYia- -f 13'-) - 2r(l + r)al3]D
D = \/[2q{\ + 2r)]
Similar considerations show that
(^2 _|. .)-i/4 = ^-1/2(1 + 2r)-i/2[l + 0(^-1/2 logi/2 q)] (5-20)
When the above results are gathered together it is found that (5-12)
may be written as
p,{u, xo) dii dxo = D, exp [-Q(a, (3) + 0(q-''-' log^/^ q)] da d(3 (5-21)
where
Expression (5-21) is valid as long as | a | and | iS | do not exceed
(2q log qr\
Expression (5-21) differs from the one predicted by the central limit
theorem (and (5-2) and (5-3)) in that it is not quite centered on the average
values Xo, «, which correspond to a = 1, /3 = 1, respectively. Also, q enters
in place of ^ -f- 1. However, these differences amount to 0{q^^- log^'- q)
at most, as may be seen by putting a — 1 and (3 — 1 for a and /3 in (5-19).
By using relations (5-6) and (5-21), it may be shown that
Prob. (PiQ, ••• ,PkQ> PoQ)
r« r* (5-23)
= / da d0D,e~'''-''['L-Pixo,u)]'' + O{q-'"log'''q)
J—q J—q
where it is understood that .vo and u in P(xo, u) depend on a and /3 through
(5-11). The term 0{q~^'- log^'- q) in (5-23) represents the sum of three con-
tributions. The first is Ri in (5-6) which is 0(1 g). The second arises from
the fact that when the factor exp [^{q~^'- log^'- q)] in (5-21) is neglected in
integrating (5-21) over -^ < a < I, - C < fi < /", where /" =
{2q log qY'-, the resulting integral is in error by 0(^~^'- log^'- q). The third
is due to the contributions of the integral from the region \a\> (■,\^\> (.
78 BELL SYSTEM TECHNICAL JOURNAL
By introducing polar coordinates a = p cos 6, ^ = p sin 6 it can be shown
that the region p > ( more than covers the region in question and that
Q{a,^)^ (1 + Op-^ (5-24)
Upon integrating with respect to p and setting in the lower limit (, it is
seen that the third contribution is 0(^~^/-).
We now assume K to be large. Since 0 ^ ^(-Vo, u) ^ 1 we have
0 ^ e~'''' - (1 - P)'' ^ KP-e'"''' < l/K (5-25)
The last inequality follows from .v- exp (— .v) < 1 for x ^ 0. A proof of the
remaining portions will be found in "Modern Analysis" by Whittaker and
Watson, Cambridge University Press, Fourth Edition (1927), page 242.
When we observe that replacing [1 — P(.Vo, w)] by 1/K in the right hand
side of (5-23) gives an integral whose value is less than 1/K, we see that
Prob. (P,Q, ■•' ,PkQ> PoQ) (5-26)
J— q J^ q
We now take up the problem of expressing the cumulative probability
density /*(.Vo, u) in terms of a and /3. When .Vo and u lie in the restricted re-
gion of integration shown in (5-6) they are near their average values .fo =
(2X + l)r and u = (2X + 1)(1 + r). On the other hand the average value
X of .V and the mean square value o-; of (.v — x)- as computed from (4-6), or
directly, are 2N + 1 + « and 4.Y + 2 + -iu, respectively. Thus we see that
X — .Vo is of the same magnitude as 4iV and becomes much larger than ax as
A' -^ 2c . The asymptotic development of Appendix I may therefore be used.
In Appendix / (equations (Al-27) and (Al-29)) it is shown that when
M(= 2m = 2.Y + 1) is a large number and 1 < < (.f — .Vo) a^
P(.Vo, n) = (iirmbo)-"' (1 + ()(l/w)) exp [wF(n)] (5-27)
where we have introduced the number m = N + 1/2 = g -\- \ to save writ-
ing X + 1/2 or (/ -j- 1 repeatedly and where
2b2 = (1 - lAi)-(l + 4siyi'
V, = [! + (! + 4siyiy2s .
F(v,) = (1 + 4siy' - s - I - logn ^^"-'^
.v„ = 2ms = (2A' + l)s, u = 2ml = (2 X + 1)/
Comparison of tlie last line in (5-28) with (5-11) shows that ms and ml
are equal to r(ij -j- a) — r(m -|- a — 1) and
(l-^r){q + (3) = (l + r)(w + /3- 1),
COMMUNICATION IN PRESENCE OF NOISE
79
(5-29)
respectively. It is convenient to introduce the notation
7 = «-l, 5 = ^-1
s = r{\ + tA"), t ^ (1 + r)(l + b/m).
It is seen that for the restricted region in which | a \ and 1 18 | are less than
/ — (2q log qY'-, I 7 I and | 8 \ are at most
0(q''- logi/- q) = 0(wi/2 logi/2 m).
Hence s, /, (1 + 4siy''\ vi differ at most from r, 1 + r, 1 + 2r, 1 + l/r,
respectively, by terms of order nr'^'- log^'- m. Similar considerations show
that
(4Trmb2)-"' = {2Trqy'Di[l + 0(w-i/2 logi/2 m)]
(5-30)
The argument of the exponential function in (5-27) must be expanded in
powers of y and 5. It turns out that when y and 8 lie in the restricted region,
powers above the second may be neglected. For the sake of convenience we
rewrite (5-13) and introduce zii
z = Xou = -im-st = 4r(l -\- r)(m + y)(tn -f 8)
Gi = 4^(1 + r)m- (5-31)
z- z, = 4;'(1 + r)[m(y + 5) + 7^]
so that 3 — Si is 0(m^'- \og^'- m). Then
(1 + Astyi-' = (1 + z/m'Y'-'
= (1 -f z,/in^'i- + (s - si)(l -f Si/w2)-i/V(2w2) (5-32)
- (g - 3i)-(l + z,/m-)-'i-/{Sni') + Ro
where R-: is of the same order as (z — ZiY/m , or m~^'~ log^/- m. It follows
that
(1 -i- 45/)'
1 -f- 2^ L ^" w-_
2r-(l -F rf (7 -f 5)-
(1 + 2ry m'
+ Q{m~^'- \og'- m)
i\ =
(1 + r)
1 +
r(l -j- y/m
_ r\\ +r){y + 8f
m\\ + 2ry
r Yy + 8 yf\
\ -\- 2r\_ m wz-J
(5-33)
+ 0{nr^'~ log''' w)
80 BELL SYSTEM TECHNICAL JOURNAL
Combining these and a similar expression for log vi leads to
mF(vi) — — m log (1 + 1/r) + 7 — 5
-[(1 + r)y - r5]V[2w(l + 2r)] + 0(w-'/2 iog3/2 ^„)
(5-34)
= -{q + 1) log (1 + \/r) + a - /3 - [(1 + r)a - r^fD
+ Q{q-'i-' log3/2 q)
Substitution of (5-30) and (5-34) in (5-27) gives the result we seek:
P(xo, w) = (1 + l/rY'^-KlT^qyi'D,
(5-35)
exp (a - ^ - [(1 + r)a - r&\-D + 0(g-i/-' log^^^ ^))
Since P(.Vo, «) occurs only in the product KP{xo, u) in (5-26) we set, in
view of (5-35),
KP{x% u) = A\(oc, (3) exp S(a, (3) (5-36)
where \(a, /3) stands for the terms denoted by exp [0(^~''- log^'- q)] in (5-35)
and
A = A'(l + lA)-«-i(27r^)i/-/)i
(5-37)
S{a, 13) = a- ^-[{l + r)a- r/3]-/>
As long as I a I < i and | /3 | < f,\{a, (3) is nearly unity and we write
Xi < X(a, /3) < X2
(5-38)
Xi = 1 - e, X2 - 1 + €, e = Cq-"'- log^'^ ^
where C is a positive constant large enough to make e dominate the terms
of order q~^''' log^'- q in (5-35). q is supposed to be so large that e is very small
in comparison with unity.
Setting (5-36) in (5-26) gives
Prob. (PiQ, ■■■ , PkQ > PoQ) = / + 0(1/A-) -f 0(r'/- Iog'^2 q) (5-39)
where the contribution of the region outside | a | < /, | /S | < I has been
returned to the terms denoted by 0(^~''"' log^'- q) (we could have stayed in
the region | a | < f,\f3\ < ( from (5-23) onward, but didn't do so because
we wanted to show that the results coming from (5-25) were not restricted
to this region) and
( (
I = j da j (//3 Di exp [- Q{a, /3) - ^X(a, ^)e'^"'''^] (5-40)
Let L(X) denote the integral obtained by replacing the function X(a, /3) in
/ by the [)ositive constant X (which we shall take to be either Xi or X2 dehned
COMMUNICATION IN PRESENCE OF NOISE 81
by (5-38)). Then, since A exp S{a, 0) is positive, it follows from (5-40) that
L(Ai) > / > ^(Xo) (5-41)
Also since exp [— ^X exp S(a, /3)] lies between 0 and 1 for all real values of
a and ^ it may be shown from (5-24) that i>(X) is equal to /(X) -\- Q{q~^'~)
where
da / d^ A exp [- Qia, /3) - ^Xe^^"'^^] (5-42)
■CO •'—00
Here X is a constant and Q{a, 0), A, S{a, (3) are defined by (5-19) and (5-37).
From (5-39) and (5-41) we obtain
Prob. {PiQ, •■■ , PkQ > PoQ) = /(I) + e[/(Xi) - /(I)] (5-43)
+ (1 - d)[Ji\,) - /(I)] + 0(1/A0 + 0(5-1/2 log3/2 q)
where 0 < 0 < 1. It will be shown later that /(Xi) and /(X2) differ from /(I)
by terms which are certainly not larger than 0{q~^'^).
The problem now is to evaluate the integral (5-42) for /(X). It turns out
that exp [— ^X exp S(a, (3)] acts somewhat like a discontinuous factor which
is unity when S{a, 13) + log ^X is negative and zero when it is positive. In
order to investigate this behavior we make the change of variable
a — (3 — w a — y — rw
{I -{- r)a - rl3 = y (3 = y - (1 -{- r)w (5-44)
da dl3 = dw dy
From (5-19), (5-37), and (5-42)
Q{a, (3) = [f- + (1 + 2rmD = fD + ^^~/2q
S{a, /3) = w - y^Z) (5-45)
/oo »00
dy / dw Dx exp [- fD - ^'/2q - A\e"'~"''']
■00 •'—00
Here and in the following work j3 is to be regarded as a function of iv and y.
Split the interval of integration with respect to w into the two subintervals
(— 00 , Wo) and (wo, ^) where
li'o = f-D - log ^X (5-46)
and y is temporarily regarded as constant. In the first interval
/wo
exp[- ^'/2q- g"""'"] Jw
(5-47)
e-^'"" dw - (1 - exp [- ^-"'0])^-^^/=" dw
82 BELL SYSTEM TECHNICAL JOURNAL
Splitting the interval of integration (— <», w^) into (— cc, — log ^X) and
(— log A\, icq) in the first integral on the right of (5-47) shows that its con-
tribution to /(X) is
dy d%ce-''''-^'''''^ A / dy \ dw e~''''-^""' (5-48)
00 •'—00 «'— oo J— log A\
Integrating with respect to y, after inverting the order of integration, shows
that the value of the first integral is
tT'" j r" dl = {1+ erf B)/2 (5-49)
where, from (5-37) and the definition (5-22) of Di,
B = -Kl + ry'\-^l'~ log^X
1/2 -1/2 ,_ XA>(1 -f l/rT" (5-50)
= -i(l+.)-.-^'Mog
[27r9(l -f 2r)]i/2
That the value of /(X) differs from (5-49) by 0(5"^''') may be seen as
follows. Since 0 < exp \—^"/2q] < 1, the integral over (wo, °^) (mentioned
just above (5-46) and obtained by taking the limits of integration to be Wo
and <=o in the left side of (5-47)) is positive and less than
/ expl-e'-'lrfw = / e-'dx/x = .219... (5-51)
Likewise, the second integral on the right side of (5-47) is less than
["" (1 - exp [- e"-"'°J) dw = f (1 - e-') dx/x = .796... (5-52)
Therefore the contribution of the first integral on the right of (5-47) differs
from /(X) by a quantity less than
[ A ^"'"'(.219 -1- .796) dy = 0(^"''')
''—00
in absolute value. The contribution of the first integral on the right of (5-47)
differs from (5-49) by the second integral in (5-48) which is 0(^~^''-) because
it is less than
r Ih{yD)e-'''' dy
•'-co
The factor {y'-D) arises from 7<:'o — (— log .IX) when the mean value theorem
is applied to the integral in iv. Hence /(X) differs from (5-49) by 0(^"'''-).
Although (5-49) is a sufiicicntly accurate expression of ./(X) for our pur-
COMMUNICATION IN PRESENCE OF NOISE 83
poses, it seems worthwhile to set down approximate expressions for the
terms which have been dismissed as 0(g~^'^). From the above work,
J(\) = (1 + erf B)/2 + A C dy r^'^//"" T"''"' exp [- e"'"'^"] dw
•'-00 l.*'tt'0
/Wo
e-^"%l - exp[-e'"-^''])dw
00
/"'" 2 1
log A\ j (5-53)
^ (1 + erf B)/2 + A T dy T*''' { -.577.. + y'D}e-^\"'
= (l+erf5)/2+(l^^y'[-.577...+
4-1(1 + r)-^l + (2 + 4r)52}]e-^'
where /3i = y + (1 + r) log A\ and we have made use of the fact that
jSy^? changes relatively slowly in comparison with w when q is large.
Since J(\) differs from (1 + erf B)/2 by 0(5-1/2), and since the three B's
for X equal to Xi, 1, and X2 differ by not more than 0(q~^/^ log (X2/X1)) =
0(5-1 log^/2 gj^ fj-om (5-50) and (5-38), it follows that the terms involving
/(Xi) and /(X2) in (5-43) may be included in the term 0(q-^'^ log'/^ ^) jj^
using our result it is more convenient to deal with N and K -{- 1 instead of
q = N — 1/2 and K. Hence instead of B we deal with H defined by
_ 1 (1 + rY" (K -^ \)(l + l/rr-\l + r)
2 (5 + 1/2)1/2 «S [27r(5 + 1/2)(1 + 2r)]i/2 ' ^^'^^^
The difference B — H, with X = 1 and H finite, may be shown to be (with
considerable margin) 0{l/K) -f 0(5-1/2). From (5-43), as amended by the
first sentence in this paragraph, it follows that
Prob. (PiQ, ■■■ ,PkQ> PoQ) = (1 + erf H)/2 + 0(1/A') + 0(5-1/2 log3/2 5)
(1-4)
where the difference between erf B and erf H has been absorbed by the
"order of" terms. When 5 + 1/2 is replaced by N in (5-54) the result is ex-
pression (1-5) for H.
84 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX I
Cumulative Distribution Function for a Sum of Squares of Normal
Variates
Let .V be a random variable defined by
M
x= Y. Jn (AM)
71 = 1
where y,, is a random variable distributed normally about its average value
jn with unit standard deviation. In writing {A\ — \) we have been guided
by (4-3), where M = 2N + 1, but here we shall let M be any positive integer.
In much of the following work M/2 occurs and for convenience we put
m = M/2 (Al-2)
From the work of Section 4 it follows that the probability density p{x, «)
of X is given by Fisher's expression
p{:x, u) = 2-'{x/u)^i-'-^i-' /^_:[(zix)i/2]e-("+-)/2 (^1-3)
where u is the constant
n
E fn (Al-4)
71 = 1
Here we are interested in the cumulative distribution function, i.e., the
probability that x is less than some given value xq,
P(xo, «) = [ p(x, n) dx (A 1-5)
as M becomes large. In this case the central limit theorem tells us that
p{x, u) approaches a normal law with average x = M -\- n and variance =
ave. (x — x)- = 2M + 4u. The function P(.Vo, u) has been studied by J. I.
Marcum in some unpublished work, and by P. K. Bose(9). In i)articular,
Marcum has used the (iram-Charlier series to obtain values for P(.Vo, u) in
the vicinity of x for large values of M. However, since I have not been able
to find any previous work covering the case of interest here, namely values
of P(xo, u) when ;Vo is appreciably less than x, a separate investigation is
necessary and will be given here.
Integrating the general expression (4-5) with rcsj)cct to .v between — A^
and .Vr,, letting X— > co, and discarding the portions of the integrand which
oscillate with infinite rapidity gives
COMMUNICATION IN PRESENCE OF NOISE 85
P(xo, ti) = — r— . / z ^e "'" [ave. e"'] <^3
Zirl •/— 00, abcvcO
(Al-6)
= 1- -^. s-V^'Mave. e"lJ2
2tI J-oo. below 0
where the subscripts "above 0" and "below 0" indicate that the path of
integration is indented so as to pass above or below, respectively, the pole
at 3 = 0. The value of ave. exp (izx) may be obtained by setting N + 1/2
= m in (4-5). The new notation
xo = Ms = 2ms, u = 2ml, 2z = ^ (Al-7)
enables us to write ^
1 r°°
P{xo,u) = - ^ . r~' exp m[-is^ - log (1 - /f)
Zirl J-oo.aboveO (A 1-8)
- t + t(l- 7f)"'] d^.
The further change of variable
I - it = V (A 1-9)
carries (Al-8) into
P{x,,u) = ~ f {{ - vr'exp[mF(v)]dv (Al-10)
2Tri Jk
where the path of integration K is the straight line in the complex v plane
running from l + Zxtol — joo with an indentation to the right of z' = 1,
and
F{i') = sv - log V + t/v - s - f. (Al-11)
The K used here should not be confused with the K denoting the number
of messages in the body of the paper. We have run out of suitable symbols.
An asymptotic expression for (Al-10) will now be obtained by the method
of "steepest descents." The saddle points are obtained by setting the
derivative
F'{v) = s - \/v - l/v"- (Al-12)
to zero and are at
^1 = [1 + (1 + 4siy'V2s
1,2 = [1 - (1 -f- 4sty'']/2s (Al-13)
Xo = 0
X
00
5 = 0
1 + /
00
Vi = <x>
1
0
t'2 = — /
-//(I + /)
0
86 BELL SYSTEM TECHNICAL JOURNAL
As Xo and 5 increase from 0 to oo , w and / of course being fixed, we have the
following behavior:
(Al-14)
[t is seen that vi ^ 0 and V2 ^ 0.
Putting aside for the moment the factor (1 — z')"^ in (Al-10), the path of
steepest descent through the saddle point vi is one of the two curves specified
by equating the imaginary part of F(v) to zero. Introducing polar coordi-
nates gives
ie
V = pe
Real F(v) = (sp + l/p) cos 6 — log p — s — t
Imag. F(v) = (sp — l/p) sin 0 — 0
At I'l, Q = ^, p = vx. Imag. F{v-^ = 0 and, from (Al-12),
Real F{v^ = (25Z'i — 1) — log zji — 5 — /
= (1 + 4^/)i/2 _ log 1,1 - 5 - /
(Al-15)
(Al-16)
The path of steepest descent through Vx may be obtained in polar form
by solving
{sp - t/p) = e/s,\n e (AM 7)
for p as a function of 6. Setting ip = d esc 6 and taking the positive value of
p leads to
P = [^ + (^* + Asiyi']/2s (Al-18)
As 6 increases from 0 to tt, v? increases from 1 to co , and p starts from vx (as
it should) and ends at oo . Thus, the path of steepest descent through vx
comes in from v = — ^ -\- iir/s (when0 is nearly tt, p ~ ip/s, <p ~ 7r/(7r — B)
and p(7r — 0) ?^ tt/^), crosses the positive imaginary v axis and bends down
to cut the real positive v axis (at right angles) at z'l, and then goes out to
i) = — 00 — i-k/s along a similar path in the lower part of the plane. It thus
avoids the branch cut (which we take to run from — oo to 0) in the v plane
necessitated by the term log v in F{v). Since yn and 5 are positive the path of
integration K in (Al-10) may be made to coincide with the path of steepest
descent when vx> \. This corresponds to the case in which x^ C x as (Al-14)
COMMUNICATION IN PRESENCE OF NOISE 87
shows. When 0 < Vi < \, i.e., oo > xo> x, the two paths may still be made
to coincide but it is necessary to add the contribution of the pole dX v = 1
as K is pulled over it. This is equivalent to passing from the first to the
second of equations (Al-6). The path 6 = 0 which makes Imag. F{v) of
(Al-15) zero turns out to be the curve of "steepest ascent" and hence need
not be considered. As (Al-13) shows, the saddle point V2 does not enter into
our considerations because it lies on the negative real v axis and the path
of integration K in (Al-10) cannot be made to pass through it without
trouble from the singularity of F{v) at i) = 0.
We now suppose Xq < x so that 5 and / are such as to make z'l > 1. In
order to remove the factor {\ — v) from the denominator of the integrand
in (Al-10), we change the variable of integration from v to w:
V — \ = e"^, (1 — v)~Hv = —dw
P(xo, u) = — ;r—. I exp [mF(l + e")] dw
(Al-19)
As :; comes in along the path of steepest descent, the path of integration L
for w comes in from w = 'x> -{- iw and dips down towards the real w axis
as arg v decreases from ir. L crosses the real w axis perpendicularly at the
point
wi = log {vi - 1) (Al-20)
and then runs out to w = oo — iw along a curve which tends to become
parallel to the real w axis, wi may be either positive or negative. When xq
is almost as large as £-, wi is large and negative.
Since F{v) is real along the path of steepest descent, F{\ + e"") is real
along L. This real value is — oo at the ends of L and attains its maximum
value F{v-^, given by (Al-16), at w = Wi. Wi is a saddle point in the complex
w plane because
-^ F{\ + en = F'il + e^e"' = F'We" (Al-21)
dw
vanishes at w = wi.
Instead of F(l + e^) itself we shall be concerned with
T = F(l + e"^) - F{1 + e") (Al-22)
so that (Al-19) may be written as
_ exp|mF(l + .")] f ^-„„ ^^ (^j.23)
2in Jl
The variable t is real on the path of integration L, is zero at wi, and in-
creases to -f 00 as we follow L out to w = =© zt iir. It is convenient to split
88 BELL SYSTEM TECHNICAL JOURNAL
K into two parts (10). The first part connects <x + m to Wi and the second
part connects Wi to oo — i-jr. The values of iv on these two parts will be
denoted by Wj and W//, respectively. Corresponding to each value of r there
is a value Wi and a value wu (in fact it turns out that Wn is the conjugate
complex of Wj). Changing the variable of integration in (Al-23) from iv to
r, and remembering that K starts at oo + tV, gives
-PCvo, u) = -^ — ^-r^ / e \ wi -— wjj dr (Al-24)
liri Jo [_aT dr J
Since m is large, most of the contribution to the value of the integral
comes from around r = 0 or w = wi. In order to obtain an expression for
the integrand in this region we note that, because F'(vi) = 0, the Taylor
series for (Al-22) is of the form
T = —b'ziw — wi)- — bsiw — wiY — bi(w — WiY — • • • (Al-25)
The circle of convergence of this series is centered on ic<i and extends out to
w = dziir, these points being the nearest singularities of F(l + c"^') as may
be seen by setting t) = 1 + c"' in (Al-11) and observing that the singulariiies
of log V — t/v in the finite portion of the w plane occur at odd multiples of
dziir. We imagine the branch cuts associated with log v to run out to the
right from these points along lines parallel to the real w axis. Since (Al-25)
has a non-zero radius of convergence, the same is true of the two series ob-
tained from it by inversion, namely
■T-l/2 1/2 , , /T,2
Wi — Wl = 102 T + OzT/ 102
+ i[b7% - 5b7'bl/4y/2bl" + • • •
and the series for Wn — wi obtained from (Al-26) by changing the sign of
i. Differentiation of these two series gives a series for d{u'i — Wu)/dT which
also converges for sufficiently small | r | (putting aside the term in t~^'-),
and which, when put in (Al-24), leads to
That this is an asymptotic expansion holding for large values of m follows
from a lemma given by Watson (11). The conditions of the lemma hold
since we have already shown that the series for d{ivi — Wii)/dT converges
for I T I small enough. Furthermore, d{wi — iVii)/dT is bounded for c ^ t
where t is real and 0 < a ^ the radius of convergence of (Al-26). This
follows the fact that
*" [3 ' = '-'■"(! + ^">"i"
di
COMMUNICATION IN PRESENCE OF NOISE 89
is bounded except near w = Wi (i.e., r = 0) and, indeed, decreases to zero
like —e~^/s as w ^ oo ± t'tt (i.e., r ^ oo).
The values of b-2, bs, hi obtained by expanding (Al-22) and comparing
the result with (A 1-25) are
b, = [F"'(v,)e"''' + 3F"(v{)e'''']/6 (Al-28)
b, = [F""(vy'' + 6F"'(vy"' + 7K(z'i)e'"'']/24
F"{v) = D-2 + 2/^-^ F"'(v) = -2v-' - 6/^-^ F"''(v) = 6v-' + 24/z)-5
Our asymptotic expression for P(xo, u), when .vo < x, is given by (Al-28)
and (Al-27). Only the leading term of (Al-27) is used in the paper. Some-
times the following expressions are more convenient than the ones which
have already been given.
b, = vT\v, + lOi'^'ll = v\\v, + 2/)(ri - 1)V2
= (1 - lAi)2(l + 4s/)i/V2 (Al-29)
F{v^ = (1 + 45/)i/2 _ ^ _ / _ log Vx.
In all of these formulas v\ is given in terms of 5 and / by (Al-13) and s and
/ in terms of Xo and u by (Al-7).
When .To > X, the saddle point x\ lies between 0 and 1 in the v plane. As
V follows the path of steepest descent (discussed just below equation (Al-18))
arg {v — 1) now stays close to tt. From (Al-19) Imag. w stays close to x on
the new path of steepest descent in the w plane, and the saddle point W\
now lies on the negative real portion of the line Imag. \v = tt. The new path
starts at w = =o + it, swings down a little as it comes in, swerves up to
pass through wi and then goes out to 2ei = °o + iir above the branch cut
joining w = iir iow = «^ + itt. The analysis goes along much as for V\ > 1
except that instead of being 0 the imaginary part of Wi is jtt. This causes
the terms in bz and 64 containing exp {iw]) to change sign. The numerical
values of bo and F{vi) are computed by the formulas (Al-29) as before. The
fact that Z>2 contains the factor exp {H-k) shows up only in changing the sign
of h\ to give the minus sign in the leading term :
P(.i-o, u) ^\ - (47rw| b-2 \)-'i~ exp [wF(i'i)]
which holds for .Vq > x. The one arises from the pole at i) = 1 and is the
same as the one in the second of equations (Al-6).
In order to see how (Al-27) breaks down near .vo = x, we set .to — x =
2m{s — 1 — /) = — 2me or s = 1 + / — € where e is a small positive number
90 BELL SYSTEM TECHNICAL JOURNAL
Using (Xx — ave. (x — xY = 4(m + m) = 4m (1 + 2/) it is found that
vi= \ + e/(l + 2/) = 1 - 2(.To - x)al
mFivi) = -me'/il + 4/) = -(.To - xY/lal
Imbo = m(vi — 1)-(1 + 2/) = (xq — xf/di
and that, since ii\ — > — <» , Jj — > 62 and 64 — > Ihi/Vl. When these values are
put in (Al-27) the leading term becomes
P{x^, u) - (27r)-i/2((r./2) exp [-sV2(rI]
and the term within the braces in (Al-27) reduces to 1 — allz where z = x
— xo > 0. Since the asymptotic expansion is useful only in the region where
the second term within the braces is small in comparison with the first term,
which is unity, x — .tq must be several times as large as Ox before we can use
(Al-27). It will be noticed that the above expression for P(to, w) is closely
related to the asymptotic expansion of the error function.
APPENDIX II
An Approximation for \ii{oc)
When 2 in the Bessel function Jq{qz) is imaginary a formula given by
Meissel (12) becomes
T (n'.,\ - (?y)' exp {qw -\r V) , .
^'^^^^ - en\q + 1)^1/^(1 + ^a^y ^^^'^^
where w = (1 + y^)^!- and F is a function of y and q which, when q is large,
has the formal expansion
^2,,,6
24^ ( w^ ] 16q^w'
1 f _ 16 -I- 1512/ - 3654/ -j- 375y\
5760^3 \ w' j ^
(A2-2)
Here we shall show that for y ^ 0 and ? > 1
\V\ < 1/(29 - 1) (A2-3)
Consideration of (A2-2) and also of the method used to establish (A2-3)
indicates that the inequality is very rough. It doubtlessly can be greatly
improved (but not beyond the l/(l2q) obtained by letting y and 9 — > 00 in
(A2-2)). Incidentally, it may be shown that the constant terms which re-
main in (A2-2) when y = 00 are associated with the asymptotic expansion
of log T(q -f 1).
COMMUNICATION IN PRESENCE OF NOISE 9l
When (A2-1) is substituted in Bessel's differential equation, which we
write as
^' dy"- ^'^^-^^ '^ ^dy ^"'"^^^ ~ ^'^^ "*" ^'^^"^^^^ ^ ^'
we obtain a differential equation for V:
V" = (4 - y)w-V4 - {2qu' + w-^)y-W - V'"- (A2-4)
Here the primes denote differentiation with respect to y. The constants of
integration associated with (A2-4) are to be chosen so that
y _^ 3,2/(4^ + 4) as y -> 0. (A2-5)
This condition is obtained by comparing the limiting form of (A2-1), in
which w -^ 1 + >'V2, with
Condition (A2-5) completely determines V since substitution of the
assumed solution
F = 4->(5 + i)-y + ciy + C2/ + . • .
in (A2-4) leads to relations which determine Ci, Ci, • • • successively.
Let V = V. Then (A2-4) becomes
v' = c - 2bv - v^ (A2-6)
where c and b are known functions of y defined by
c = (4 - y'~)w-'/4, b = (qw-\- w-^/2)y-' (A2-7)
From (A2-5), v -^ y/{2q + 2) as y -^ 0 and therefore
V
I vdy (A2-8)
•'0
We first show that \v\ < l/(2q — 1) when q > 1. The (y, v) plane may
be divided into regions according to the sign of v'. The equations of the
dividing lines between these regions are obtained by setting ii' = 0 in (A2-6).
Thus, for a given value of y, v' is positive if V2 < v < vi and negative if
V > viOT V < Vi where
v,= -b+ (b' + cy = c/[b + (62 + cyi^]
v^= -b+ (62 + cyi^ (A2-9)
When y > 0 we have b ^ q. A plot of c versus y shows that | c | ^ 1. Hence,
92 BELL SYSTEM TECHNICAL JOURNAL
when q > \,
b-'+c^ q^-\> {q-iy
\v,\< \/{2q - 1) (A2-10)
V2 < -2q+ \
The curve obtained by plotting vi as a function of y plays an important
role because, as we shall show, the maxima and minima of the curve for v
lie on it. Therefore, the maximum value of | i) | cannot exceed the maximum
value of \vi\. The maxima and minima must lie on either the Vi or the v^
curve since v' vanishes only on these curves. In order to show that it is the
Vi curve we note from (A2-9) that, near y = ^, Vi behaves like y/{2q -\- 1).
Consequently both the Vi and v curves start from i' = 0 at ;y = 0 but for a
while vi lies above v which behaves like y/{2q -\- 2). Here v lies in a v' > 0
region and continues to increase until it intersects vi (as it must do before
y reaches 2 because v\ = 0 at y = 2) at which point v' = 0, Vi ^ 0, and v
has a maximum which is less than the maximum of | Vi | so 7' < \/{2q — 1)
when q> \. Upon passing through vi, v enters a v' < 0 region and decreases
steadily until it either again intersects the Vi, curve or else approaches some
limit as y ^ 00. In either case | v | does not exceed l/(2q — 1), since, in the
first case v would have a minimum at the intersection and in the second
Vi — >^ 0 as y ^ 00. The same reasoning may be applied to the remaining
points of intersection, if any, of the v and vi curves.
In order to obtain an inequality for V itself we rewrite (A2-6) as
v' = c - {2b + v)v (A2-11)
The solution of this equation which behaves like y/i2q + 2) as y -^ 0 also
satisfies the relation
v{y) = f c{x) exp - [ [2b(0 + vm d^ dx.
Jo {_ Jx
as may be verified by making use of the relations r(.v) -^ 1 as .v — > 0 and
2b{0 -^ (2q + 1)/^, t(0 -^ ^/{2q + 2) as ^ -> 0. For then
- [ [2b(^) + v(0] d^ -^ (2q + 1) log x/y
Jx
v(y) -> r (x/yY"'-' dx = y/{2q + 2)
Hence, from (A2-8)
Viyi) = f dy £ dx) t-^P [-£ 12M^) + vm dn dx
COMMUNICATION IN PRESENCE OF NOISE 93
and
I y{yi) I < £' dy jf' I c{x) I exp T-Jj [lh{^) - \ v{0 il dp\ dx.
From b'^ qsind\v\ < l/(2q - 1) it follows that 2b(^) - \ v{^) | > 2^ - 1
when q > I. This and | c(x) | ^ (4 + x~)(l + x-)~V4 gives
! Viyi) \ < [ dy [ (4 + .v-)(l + .v-)"-4-' exp [-(2^ - l)(j - x)] dx
Jo •'0
Stt 1
16(2g - 1) 2q - I
which is the result we set out to establish. The double integral may be
reduced to a single integral by inverting the order of integration and inte-
grating with respect to y. Incidentally, most of the roughness of our result
is due to the use of the inequality for | c(x) |.
References
1. C. E. Shannon, A Mathematical Theory of Communication, Bell Sys. Tech. Jour., 27,
379-423, 623-656 (1948) See especially Section 24.
2. C. E. Shannon, Communication in the Presence of Noise Proc. I .R.E., 37 , 10-21 (1949).
3. \V. G. Tuller, Theoretical Limitations on the Rate of Transmission of Information
Proc. I.R.E., 37, 468-478 (1949).
4. N. Wiener, Cybernetics, John Wiley and Sons (1948).
5. S. Goldman, Some Fundamental Considerations Concerning Noise Reduction and
Range in Radar and Communication, Proc. I.R.E., 36, 584-594 (1948).
6. G. N. Watson, Theory of Bessel Functions, Cambridge University Press (1944),
equation (1) p. 181.
7. R. A. Fisher, The General Sampling Distribution of the Multiple Correlation Coeffi-
cient. Proc. Roy. Soc. of London (A) Vol. 121, 654-673 (1928). See in particular
pages 669-670.
8. Reference (6), equation (4) p. 394.
9. P. K. Bose, On Recursion Formulae, Tables and Bessel Function Populations Asso-
ciated with the Distribution of Classical D^ — Statistic, Sankhya, 8, 235-248 (1947).
10. Compare with §8.4 of reference (6).
11. Reference (6), p. 236.
12. Reference (6), p. 227.
Realization of a Constant Phase Difference
By SIDNEY DARLINGTON
This paper bears on the problem of splitting a signal into two parts of like am-
plitudes but different phases. Constant phase differences are utilized in such cir-
cuits as Hartley single sideband modulators. The networks considered here are
pairs of constant-resistance phase-shifting networks connected in parallel at one
end. The first part of the paper shows how to compute the best approximation
to a constant phase difference obtainable over a prescribed frequency range
with a network of prescribed complexity. The latter part shows how to design
networks producing the best approximation.
A PERENNIAL problem is that of designing a circuit to split a signal
into two parts which are the same in amplitude but which differ in
phase by a constant amount. A 90-degree phase difference is needed, for
example, in the single sideband modulation system due to R. V. L. Hartley.^
It is well known that it is not possible to obtain exactly equal amplitudes
and exactly constant phase differences at all frequencies except in the
trivial special case of a 180-degree phase difference. Various methods have
been devised, however, for approximating these characteristics over finite
frequency ranges. The most obvious method is to use a pair of constant
resistance phase shifting sections in parallel at one end and with separate
terminations at the other end^ as indicated in Fig. 1,
This paper is devoted to the problem of obtaining approximately constant
phase differences under the specific assumption that pairs of constant re-
sistance phase shifting networks are to be used. The paper has been written
with two objects in mind. The first is the development of a method for
determining the best approximation to a constant phase difference which
can be obtained over a prescribed frequency range with a pair of phase
shifting networks of a prescribed total complexity. The second object is
the description of a straightforward design procedure by means of which
the networks can be designed to give this best possible approximation.
The problem under consideration is typical of those usually described
as problems in network synthesis. In other words, a network of a prescribed
general type is to be designed to approximate as closely as possible an ideal
operating characteristic of a prescribed form. The same procedure will be
followed as that appropriate for most such problems. The procedure begins
with the development of a mathematical expression representing the most
^U. S. Patent 1,666,206, 4/17/28, Modulation System.
* Another common method uses reactance shunt branches between effectively infi-
nite impedances, such as the plate and grid impedances of screen grid tubes.
94
CONSTANT PHASE DIFFERENCE
95
general characteristics which can be obtained with the prescribed type of
inetwork. This is followed by the determination of particular choices of the
arbitrary constants in the expression, which will lead to the best approx-
imation to the prescribed ideal characteristic. The next step is to deter-
imine formulae for the degree of approximation to the ideal, which will be
PHASE-SHIFTING
NETWORKS
Fig. 1 — Phase-shifting networks for approximation to a constant phase diffLrence.
Z 5
>
7
y
1
1
/
/
n
'
/
J
1
/
/
/
/
/
/
1
/
)
/
/
/
1
1
/
/
)
/
/
/
1
i
/
/
/
/
X
/
A
/
/
/
y
y
/
_^
-^
>^
y
^
/
^
^
y
-
-^
^
10^
--
1 2 3 4 5 6 7 8 10 20 30 40 60 80 100 200 400
FREQUENCY RATIO , Wa/^i
Fig. 2— Variation in phase difference, when average is 90°, with a network of n sections.
obtained with those particular values of the constants. The final step is
the development of a method for determining corresponding actual net-
works.
From the optimum choice of constants, curves can be calculated which
show what can be done with a network of any given complexity (Fig. 2).
Then the complexity needed for any particular application can be read
directly from the curves. The special choice of constants also leads to special
96 BELL SYSTEM TECHNICAL JOURNAL
formulae for element values of corresponding networks, using tandem sec-
tions of the simplest all-pass type (Fig. 3).
Form of tiie tan { - ) Function I
\2/ I.
If fix and /So represent the phase shifts through the two constant resistance I
networks of Fig. 1, then tan ( -^ 1 and tan ( ^ j must both be realizable I
as the reactances of physical reactance networks. In other words, these
quantities must be odd rational functions of w with real coefficients and
must also meet various other special restrictions. If /3 is used to represent
the phase difference 182 — /3i , the function tan ( - 1 must also be an odd
rational function of oj with real coefficients. Because of the minus sign
Fig. 3 — Simplest all-pass section.
associated with ^i in the definition of /3, however, tan ( - J does not have to
meet the additional restrictions which must be imposed upon tan I J ) and
tan ( ^ )• In a later part of the paper a method will be described by which
a pair of physical phase shifting networks can be designed to produce any I
tan ( - j function which is an odd rational function of co with real coefficients. j|
In any range where the phase difference /3 approximates a constant, the |
function tan [ - I will also approximate a constant. Hence, the present 1
problem is really that of ai)proximating a constant over a given frequency
range with an odd rational function of w with real coefficients. In this prob- i'
lem, the degree of the function must be assumed to be prescribed as well
as the frequency range in which a good approximation is to be obtained,
for the degree of the function determines the complexity of the correspond-
ing network.
W. Cauer shows how functions of certain types can be designed to approx-
CONSTANT PHASE DIFFERENCE
97
imate unity in prescribed frequency ranges.^ These functions, however, are
not odd rational functions of frequency but are irrational functions appro-
priate to represent filter image impedances or the hyperbolic tangents or
cotangents of filter transfer constants. It turns out, however, that they
can be transformed into odd rational functions of the desired type by a
simple transformation of the variable.
Each of Cauer's functions is said to approximate a constant in the Tcheby-
cheff sense, which means that in the prescribed range of good approximation
the maximum departure from the approximated constant is as small as is
permitted by the specifications on the frequency range and the degree of
the function. Each function also has the property of exhibiting series of
equal maxima and equal minima in the range of good approximation, such
as those indicated in the illustrative /3 curve'* of Fig. 4.
LU
O
lllUJ
u. O
IL LU
60 80 100
200 400 600 1000 2000
FREQUENCY IN CYCLES PER SECOND
10,000
Fig. 4 — Example of a phase difference characteristic.
Of the various forms in which Cauer's Tchebycheff functions F can be
expressed, the following form is the one appropriate for showing how odd
rational functions of frequency can be obtained:
When 11 is odd
'2s -
(1)
F = U\/\ - X2 it ^
When n is even
F =
U
n['-™'(^'^''*)^
n
■[l-.„'gA-.*).V']
' "Ein Interpolationsprohlem niit Funktionen mit Positivem Realteil," Mathematische
Zeitschrift, 38, 1-44 (1933).
* The data for the illustrative curve were obtained from a trial design carried out by
P. W. Rounds.
98 BELL SYSTEM TECHNICAL JOURNAL
In these equations, the symbol sn indicates an elHptic sine, of modulus
k, while A' represents the corresponding complete elliptic integral. U is
merely a constant scale factor, while n is an integer measuring the complex-
ity of corresponding networks. In the case of phase-difference networks,
n represents the total number of sections of the type indicated in Fig. 3,
which are included in the two phase-shifting networks or their tandem sec-
tion equivalents.
In Cauer's filter theory, the variable X represents a rational function of
CO which permits F to be an image impedance or a coth ( - ] function. In
order that F may be an odd rational function of oj, however, as is required
when it is to represent tan ( - j , X must be defined by the relation
(2) (0 = C02V1 - X\
Cauer shows that F approximates a constant in the Tchebycheff sense in
the range 0 < X < ^ . Hence, in terms of o, the range of approximation
is coi < CO < C02 , where coi and 002 are arbitrary provided the modulus k is
assumed to be determined by the relation
Vol -
(3) k = ^ "^ ~ "■ .
Alternative Expression for the tan ( - j Function
While equations (1) are the most convenient form of F to use in deriv-
ing the transformation of the variable, an alternative more compact form
is more suitable for determining the degree of approximation to a constant
phase difference and the element values of corresponding networks. When
F represents tan I - j and hence co and X are related as in (2), the equivalent
expression is as follows:^
tan I - 1 = Udnxnu-—
(4) \2/ \ A
CO = C02 dn{u, k).
In this expression, dn represents a so-called "</«" function, the third type of
Jacobian elliptic function usually associated with the elliptic sine, or sn
function, and the elliptic cosine, or en function. The symbol ii represents
^ This expression depends on a so called modular transformation of elliptic functions
not found in the usual elliptic function text. The transformation theory may be found in
"An Elementary Treatise on Elliptic Functions," Arthur Cayley, G. Bell & Sons, Lon-
don, 1895.
CONSTANT PHASE DIFFERENCE 99
a "parametric variable" which would be eliminated on forming a single
equation from the two simultaneous equations indicated. The modulus ki,
of the dn function corresponding to tan ( ;^ ) is related to the modulus k,
of the dn function corresponding to co, in the manner indicated below. The
constant Ki, of course, represents the complete integral of modulus k-[,
just as K represents the complete integral of modulus k.
Corresponding to any modulus k there is a so-called modular constant q.
Using ^1 to represent the corresponding modular constant of modulus ki,
it is here required that
(5) qi = q\
One modulus can be computed from the other by means of this relation-
ship and tabulations of logio q vs sin~^ k which are included in most elliptic
function tables."
Degree of Approximation to a Constant Phase Difference
When M is real and varies from zero to infinity, the corresponding value
of CO as determined by (4) merely oscillates back and forth between the values
0)1 and C02. In other words, it merely crosses back and forth across the range
in which tan ( - j approximates a constant. Similarly, when u is real and
increases from zero to infinity, tan ( - j oscillates between U\/l — kj and
U. The equal ripple property of the curve illustrated in Fig. 4 is explained
by the fact that the period of oscillation of tan ( - j with respect to u is
(9
merely a fraction of that of co, so that tan ( - ) passes through several ripples
while the value of co moves from coi to co2.
Combining the formulae for the maximum and minimum values of
tan l-j gives the relation
(6) tan('^U^('-^/'"^9
2/ 1 + UWl - kl
^ When k is extremely close to unity, it may be easier to obtain accurate computations
by using the additional relation
logio iq) logic iq')
\!oge (10;/
OJl
where q' is the modular constant of modulus y/l — k^
100 BELL SYSTEM TECHNICAL JOURNAL
in which 5 represents the total variation of the phase difference jS in the
approximation range. Similarly, the average value /3a of )3 in the approxi-
mation range is given by^
(7) tan m = ^:(L±Vl^).
1 - £/Vl - k\
If the phase variation 5 is reasonably small, (6) and (7) can be replaced
by the approximate relationships
sin (^a) ,2 ,.
6 = — ^ — ki radians
/3a
tanj^^') = I' v^l - k\.'
A still further modification is obtained by replacing k\ by the quantity 16(/i,
which is an approximate equivalent when kl is small, and by then replacing
q\ by the equivalent q" of (5). This gives
(9) 5 = 8 sin (/3„)5"
tan r| j = V \/l - 165".
When combined with (3) and tabulations of sin~^(^) vs logio(9) , these
formulae can be used to compute 5 when the parameters coi, C02, /3a and n are
prescribed. Curves of 5 are plotted against ^2/^1 in Fig. 2, assuming ^a to
be 90 degrees.
Determination' of a Network Corresponding to a General
Phase Difference Function
Since tan ( - 1 must be an odd rational function of co, it can be expressed
in the form
(10) tan (f) = "1
\Z/ A.
in which A and B are even polynomials in co. This requires
(11) ^ = arg {A + iu^B).
' More exactly, /3„ is the average of the maximum and minimum values of (3 occurring
in the range of api)roximation.
' In the important sjjecial case in which the average phase dilTerence /3a is 90°, this
expression for tan ( ) is exact rather than approximate.
CONSTANT PHASE DIFFERENCE 101
Similarly, if attention is focused on the phase shifts of the individual
phase-shifting networks rather than on the phase difference, the following
odd rational functions can be introduced:
-(f) = t
(12)
tan
©■
in which Ai, Bi, A2, and B2 are additional even polynomials in co. This
requires
(13)
It also requires
I' = arg(^i + zco5i)
arg (.42 + ioi^'i).
(14) -^^ = arg (A, - icB,).
Since the argument of a product is the sum of the arguments of the sep-
arate factors, (13) and (14) require
(15) ^ = ^^^' = ^^g (^- + ^■'^^2)(^i - ico^i).
This permits us to write
(16) (A2 + io:B2){Ai - ic^B,) = H(A + ic^B)
in which ^ is a real constant.
When tan ( - j is prescribed, a corresponding polynomial of the form
{A -\r io}B) can readily be derived. The problem is then to factor it into
the product of two polynomials (.42 + 100^2) and (.4i — iwBi) such that
Ai, Bi, A2, and Bo determine physically realizable phase shifts through
(12). Two factors of the general form (A2 + icioB2) and (.4i — iooBi) can
readily be obtained in a number of ways. The only question is how to obtain
them in such a way that the corresponding phase characteristics will be
physical. A procedure meeting this requirement is described below.
The variable co is first replaced in (.4 + iuB) by p representing ico. This
leaves a polynomial in p with real coefficients, since A and B represent
polynomials in co-, while p~ represents -co''. Suppose all the roots of the poly-
nomial A -\- pB are determined. Then this polynomial can be split into
102 BELL SYSTEM TECHNICAL JOURNAL
two factors by assigning various of the roots to each of the two factors.
It turns out that physically realizable phase characteristics will be obtained
if all those roots with positive real parts are assigned to the factor (^4 1 — pB\)
which appears in (16) when ico is replaced by p, all other roots being assigned
to the factor (^2 + pB^.
The physical realizability of the above division of the roots follows from
pB
a theorem which states that -j^ is realizable as the impedance of a two-
terminal reactance network whenever Ax and B^ are even polynomials in
p with real coefficients such that Ax+ pBx has no roots with positive real
parts.^ From this theorem and the fact that the evenness oi Ax and Bx
causes them to remain unchanged when p is reversed in sign, it follows that
^— ^ will also be the impedance of a physical two-terminal reactance net-
Ax
work whenever Ax — pBx has no roots with negative real parts. Thus, by
(12) the above division of the roots oi A -\- pB makes tan ( ^ j and tan
( — ) realizable as the impedances of two-terminal reactance networks.
These reactance networks and their inverses are merely the arms of unit
impedance lattices producing the phase characteristics defined by (12).
The above argument merely shows that each of the two phase-shifting
networks can at least be realized as a single lattice when tan ( - 1 and
tan I ^ 1 are determined by the method described. Actually, they can be
broken into tandem sections directly as soon as the roots of (^1 — pB-^
and {A2 + PB2) have been determined. From (^1 — pB^ , the quantity
(^1 + pBi) can be found by merely reversing the signs of the roots. Then
by using the principle that the argument of a product is the sum of the
arguments of the separate factors, phase-shifting networks can be designed
corresponding to various factors or groups of factors as determined from
the known roots of (^1 + pBi) and {A2 + pB-i) . There can be a separate
section for each real root and each conjugate pair of complex roots.^"
Determination of a Network Corresponding to a Tcheby-
CHEFF Type of Phase Difference Characteristic
The procedure described above for determining a network corresponding
to a general phase difference characteristic is complicated by the necessity
» See "Synthesis of Reactance 4-Poles which Produce Prescribed Insertion Loss Char-
acteristics," Journal of Mathematics and Physics, Vol. XVIII, No. 4, September, 1939—
page 276.
i» See II. W. Bode, "Network Analysis and Feedback Amplifier Design," D. Van
Nostrand Company, New York, 1945, Page 239, §11.6.
CONSTANT PHASE DIFFERENCE 103
of determining the roots of the polynomial A -{- pB . In the case of the
Tchebycheff type of characteristic described in the first part of the paper,
the required roots can be determined by means of special relationships.
In the first place, the roots oi A -]- pB are the roots of ( 1 + i tan ;^ ) • In
other words, by equation (4) they are the roots of 1 -f- iU dni nu— , kij \.
The values of u at the roots turn out to have an imaginary part iK', where
K' is the complete elliptic integral of modulus \/l — k^. If a new variable
u' is defined by
(17) u= u' + iK'
the roots can be shown to correspond to the values of u' determined by
cn\nu -^, ki j
If it is assumed that the phase variation is small in the range of approx-
imation to a constant, it can be shown that one value of u' determined
by the above relation is given approximately by
(19) ^ = -/3.
where I3a is the average phase difference for the range of approximation as
before (in radians). After this value of u' has been computed, all the roots
.,[
\ -\- iU dnxnu
[nu— , kij
can be found by computing the values of to
hj
corresponding to this value of u' and to those values obtained by adding
2K / K \
integral multiples of the real period — of dni nu — ^ , ^i J. This gives the
following formula for the roots in terms oi p = iw.
(2aK . .
en I h Mo
(20) ^'^'"'{ii ^' <r = Q,---,{n-\)
sn I + Mo
in which «o is the value of u' determined by (19).
Finally, instead of using the above elliptic function formula directly, one
may replace the elliptic functions by equivalent ratios of Fourier series
expansions of 6 functions. This gives
,o.x . / cos (X,,) + q^ cos (3XJ + if cos (5XJ • • •
(,21j pc = Vcoia)2 -^ — TT-y- r—- — /o. N I 1—. — tfTn
sm (Xa) — q^ sm (3X») + q^ sm (5X,) • • •
104 BELL SYSTEM TECHNICAL JOURNAL
in which the angle X^ is defined by
(22) X, ^. ^-^SO" - 2^" degrees, cr = 0, •••,(«- 1).
Because all the paS are real in this Tchebycheff case, corresponding net-
works can be made up of sections of the simple type indicated in Fig. 3.
In one of the two phase-shifting networks there will be one section for each
positive pa, and it will be given by
L= ^ C = ^
pa Ropa
where i?o is the image impedance. Similarly, in the second phase-shifting
network there will be one section for each negative p„, and it will be given by.
L = - - C = ^
pa Ropa'
Conversion of Concentrated Loads on Wood Crossarms to
Loads Distributed at Each Pin Position
By RICHARD C. EGGLESTON
ONE of the most important requisites in all fields of engineering endeavor
is knowledge of the strength of materials. The development of testing
machines and techniques to study the basic properties of metals, plastics
and wood products to withstand breaking forces has been a distinctive
achievement during the last half century. All materials, whether they be
part of a bridge, a building, a shipping crate, a telephone pole or a crossarm
on a telephone pole, break under an excessive stress. To have accurate
knowledge of the strength of the millions of crossarms used to carry the
regular load of wires, which are frequently subjected to the extra loads of
wind and ice, is most important in electrical communication.
When strength tests of crossarms are made, the information most gen-
erally sought is how great a vertical load equally distributed at each insulator
pin hole will the arms stand. In the past many crossarm tests have been
made by the concentrated load method, where the arm is either supported
at each end and loaded at the center, or supported at the center and loaded
at the ends until failure occurs (Fig. 1, a and b). Some have been made by
the distributed load method by placing, manually and simultaneously, 50-
pound weights in wire baskets suspended from each pin hole, and continuing
such load applications until the arm fails. The method is objectionable
chiefly because, in many of the tests, the loading is inadvertently carried
past the maximum loads the arms will support. This objection was over-
come in recent tests made by the Bell Telephone Laboratories^, where the
loads were also distributed at each pin position. However, instead of sub-
jecting the 10-pin test arms to sudden 500-pound load increments (viz. 50
pounds at each of the 10 pin holes), the loads were applied gradually by a
hydraulic testing machine (Fig. 1, c). But, in spite of the advantages of
this machine method of distributed load application, it is probable that, be-
cause of the less elaborate apparatus involved in simple beam tests, there
will continue to be tests made by the concentrated load method.
Where tests have been made by the concentrated load method, the ques-
tion arises how can the results be converted to a load-per-pin basis? A
conversion is needed before a fair comparison can be made of all test results,
and also to furnish the information generally most wanted, which is, as
'Bell System Monograph No. B-1563, Strength Tests of Wood Crossarms.
105
106
BELL SYSTEM TECHNICAL JOURNAL
. *"
—
c.
rt
^
3
''^
C
a;
rt
a
o
O
C
rt
o
^
a
S
r^
cd
i2 ^"o
en 60 O.
O t- '^
1- 1) s
i S E'
CROSSARM LOAD CONVERSION 107
previously stated, what load per pin will the arm support? There are more
than twenty million crossarms in the pole lines of the Bell System and each
year about a million arms are added. A complete understanding of every
problem associated with this important item of outside plant material is
manifestly worth while. This paper is intended to contribute to that end.
It presents a solution of the problem of converting concentrated vertical
loads to comparable loads distributed at each insulator pin position.
The location of the critical section in crossarms is a basic factor in a study
of the problem. The critical section of a crossarm is the section at which the
fiber stress is greatest when the arm is loaded. It is the section where the
arm may be expected to break if overloaded. To determine its location, the
bending moment at various sections along the arm is divided by the section
modulus of the respective sections. The quotient in each instance is the
fiber stress for each section investigated. The location showing the greatest
fiber stress is the critical section. Since horizontal shear is not the control-
ling stress in crossarm failures under loads distributed at each pin hole,
bending stresses only were considered in this analysis.
Because of the diflferences in arm shape and in the spacing of pin holes,
the location of the critical section is not the same in all arms. It is es-
timated that at least three fourths of the arms in the Bell System are lOA
and lOB crossarms.^ Both are 10 feet long and 3.25" x 4.25" in cross section.
In the lOA arm (Fig. 3), the space between the pin holes is 12 inches, except
between the pole pin holes, where the space is 16 inches. In the lOB (Fig.
4) the pin hole spacing is 10 inches with a 32-inch space between the pole
pins. Both types are bored for wood pins. Most of the arms now in the
plant are ''roofed", that is, the top surface of the arm, except the center foot
of length, is rounded on a radius of about 4.25 inches. Under the current
design, however, the top surface of Bell System arms is fiat, except for the
edges, which are beveled.
Previous studies of both roofed and beveled arms of various types have
shown that the critical section of clear arms under vertical loads is either at
the center or at the pole pin hole sections. This study is confined to those
sections of clear lOA and lOB crossarms of nominal dimensions, both roofed
and beveled. Moreover, it was assumed for the purpose of load analysis,
that the crossarms are supported at the center only; since, under loads on
each side of the pole, the standard crossarm braces provide no significant
support when the loads are suflacient to break the arm.
Roofed lOA Arm
Let it be assumed that the breaking load concentrated in each end pin
hole of a roofed lOA arm is 800 pounds. As shown in Calculation 1 in the
*10A and lOB crossarms were formerly known as Type A and Type B crossarms,
respectively.
108 BELL SYSTEM TECHNICAL JOURNAL
appendix, the bending moment at the center of the arm from the assumed
loads would be 44,800 pound-inches, and the fiber stress at the center would
be 4600 psi. That calculation also shows that the bending moment at the
pole pin holes would be 38,400 pound-inches, and the fiber stress at the pole
pin holes 7515 psi. Since the stress at the pole pin holes is greater than that
at the center, the critical section of a roofed lOA arm is at the pole pin holes
when the arm is subjected to a breaking load at each end pin hole.
The information wanted, however, is what load at each of the ten pin
positions would have produced the same moment and same fiber stress at
the critical section? A tentative answer is found by dividing the 38,400
bending moment by the "total-per-pin" lever arm*, 120" (see Calculation
1) or 320 pounds. Checking to determine whether the location of the criti-
cal section changes under loads of 320 pounds at each pin position, Calcula-
tion 1 shows that the fiber stress at the pole pin holes and at the center
would be 7515 psi and 5257 psi, respectively. Since the stress at the pole
pin holes is greater, it is clear that the critical section is there also under
equal loads at each pin position; and the 320-pound load per pin is com-
parable to the concentrated load of 800 pounds at each end of the arm.
If a similar investigation were made of a roofed lOB arm and of a beveled
lOA arm, it would be found that the pole pin hole section is the critical
section of these arms; and that the load per pin comparable to concentrated
loads at the arm ends would, like the roofed lOA arm, be equal to the bending
moment at the pole pin hole section due to the concentrated load divided by
the total per pin lever arm to that section. Figure lb shows a roofed lOA
arm that broke under test at a pole pin hole (critical) section from concen-
trated loads at the ends of the arm.
Beveled lOB Arm
For the investigation of this arm, let a breaking load of 1000 pounds at
each end pin hole be assumed. Incidentally, it should be noted that so far
as this analysis is concerned, the magnitude of the assumed concentrated
loads is of no importance. However, since both computations and tests
show the lOB arm to be stronger than the lOA, it seemed appropriate to
assume a larger concentrated load for the lOB arm.
As shown in Calculation 2 of the appendix, the bending moment at the
center due to the 1000-pound load would be 56,000 pound-inches and the
fiber stress 5882 psi, while at the pole pin holes the bending moment would
be 4(),0()() ])()un(l-inches and the liber stress 6885 psi. Here again, under
concentrated loads at each end pin hole, the critical section is at the pole
I)in holes.
■'Hy l()t;il-|)er-i)in lever arm is meant the summalion of llie dislances from each ]iin
j)osilio)i to the section concerned -in this instance (o the pole pin hole section.
CROSSARM LOAD CONVERSION 109
Calculating the load for each of the 10 pin positions in the same manner
as for the roofed lOA arm, we have, tentatively, a load per pin of 400 pounds.
However, in checking to determine whether the location of the critical sec-
tion changes under loads of 400 pounds at each pin position, we obtain re-
sults quite different from those in Calculation 1 ; for Calculation 2 indicates
a liber stress of 6885 psi at the pole pin holes, but a higher stress (7563 psi)
: at the center, which shows that the location of the critical section does
change. Moreover, this change would occur whether the loads were 400
pounds per pin or 4 pounds per pin. But let us now consider the 400-
I pound load.
If a concentrated load of 1000 pounds results in a fiber stress at the pole
pin hole section of 6885 psi and causes failure, that stress is the maximum
ultimate fiber stress for the arm. It is, therefore, not reasonable to suppose
that the same arm would have endured a higher stress (7563 psi) at the
center if it had been loaded at each pin position. If 6885 psi is the maxi-
mum stress for the arm, the maximum moment it would endure at its center
would be 65,500 pound-inches (viz. 6885 multiplied by 9.52, the section
modulus of the center section). The maximum load per pin would be 364
pounds (viz. 65,500 divided by 180, the total-per-pin lever arm to the center) ;
and this load of 364 pounds, not 400 pounds, distributed at the 10 pin posi-
tions is comparable to the 1000-pound concentrated load. Thus, while the
critical section of a beveled lOB arm is at the pole pin holes when the load is
concentrated at the arm ends, it shifts to the center when the load is dis-
tributed at each pin position; and, moreover, the load is less than the load
per pin tentatively computed.
A graphic illustration of this shift of the critical section is shown in Fig. 2.
Graph 1 in this figure is the graph of the resisting moments of a clear,
straight-grained beveled lOB arm, 3.25" x 4.25" in cross-section, and having
an assumed ultimate fiber strength in bending of 6000 psi. Each point
in the graph is equal to the section modulus of the section under considera-
tion multiplied by 6000 psi. Graph 2, which is the graph of a concentrated
load at the end pin position, was drawn from the zero moment under the
end pin to the point of greatest moment possible without intersecting re-
sisting moment Graph 1. Since the point of coincidence between Graphs
1 and 2 is the pole pin hole section, that section is the critical section for a
concentrated load at the end pin. The magnitude of this concentrated
load is equal to the resisting moment at the pole pin hole, 34,860 pound-
inches (viz. 5.81 inches^ x 6000 psi) divided by the 40" lever arm, or 871.5
pounds. The load per pin, tentatively figured, would be 34,860 pound-
inches divided by 100 inches or 348.6 pounds. Graph 3 is the graph of a
load (P) of 348.6 pounds at each pin hole. Under such loading, however,
the bending moment at the center of the crossarm would be 62,748 pound-
110
BELL SYSTEM TECHNICAL JOURNAL
inches (viz. 348.6 x 180), which exceeds the 57,120 pound-inches resisting
moment at the center (viz. 9.52 inches^ x 6000 psi). This means that the
1
i
u
1
[
-0--
1 1 1
1 1 ill
1 1 1 ^
1
(
60
1
( 1
^ 1
871 1/2
1 LBS
1
1 1 :
\
1
GRAPH 1 1
1
1
1
55
50
^45
I
o
?40
Q
Z
D
O 35
Q.
Q
Z
< 30
in
D
O
I
t- 25
Z
5 20
O
2 15
10
5
0
r \
\ \
\ \
\ >
\
\
\
Vx
^
\
\
. \
\v ^
\^
\,
\
nN,
GRAPH 2
^
^
\
^V N
;\3
\
\
^x^>
^v
\
V
N
«=^
^
^
0 5 10 15 20 25 30 35 40 45 50 55 60
DISTANCE FROM CENTER IN INCHES
Fig. 2 — Moment diagrams for a beveled lOB crossarm:
Graph 1 — Resisting moments of a clear,straight grained, 3.25" x 4.25" arm. Fiber
stress assumed to be 6000 psi ;
Graph 2 — Bending moments from a concentrated load of 871.5 pounds at end pin hole;
Graph 3 — Bending moments from a load of 348.6 pounds at each pin hole; and
Graph 4 — Bending moments from a load of 317.3 pounds at each pin hole.
arm would fail under such loading; and that the critical section of the arm
under loads distributed at each pin hole is not at the pole pin holes but at
the center of the arm. The maximum load per pin that the arm would
CROSSARM LOAD CONVERSION 111
endure is the resisting moment at the center divided by the total-per-pin
lever arm, or 57,120 pound-inches divided by 180 inches or 317.3 pounds.
Graph 4 is the bending moment graph of this 317.3-pound maximum load
per pin.
Summary
Let W = Concentrated load,
P = Load per pin,
Mp = Bending moment at pole pin hole section,
fc = Fiber stress at center section,
fp = Fiber stress at pole pin hole section,
Sc = Section modulus of center section, and
Sp = Section modulus of pole pin hole section.
Using this notation, the results of the analyses may be summarized as
follows:
For lOA arms both roofed and beveled:
Mp = 48W (for concentrated loads) , and
Mp = 120P (for pin loads). Therefore,
48W
P = TFo = "■'''
For lOB arms-roofed:
Mp = 40W (for concentrated loads) , and
Mp = lOOP (for pin loads). Therefore,
40W
P = — = 0.4W
For the beveled lOB arm, however, where the critical section is at the
center, the value P = 0.4W does not apply. The value of P would be such
as to produce the same fiber stress at the center section as the fiber stress
resulting from the concentrated load (W) at the pole pin hole section. Thus
r 180P
ic = —7— and
Sc
fp =
Equating these, we have
180P 40 W
and
40W
Sp
Sc Sp
40W Sc _ 2ScW _ 2 X 9.52W „ ._..
^--SF^180--W~ 9 X 5.81 = ^-^^^^
112 BELL SYSTEM TECHNICAL JOURNAL
Therefore, under the conditions assumed, and only under such conditions,
we may say that the loads per pin (P) comparable to the assumed concen-
trated loads (W) would be
I lOA arms — roofed
P = 0.4W for < lOA arms— beveled
[lOB arms — roofed
and
P = 0.364W for lOB arms— beveled
While these results are restricted to the four arm types listed, the same
principles followed in arriving at these results may be applied to other types
and sizes of arms, and to other conditions of loading. Whether the conver-
sion of single concentrated loads to loads per pin is performed by the method
illustrated in Calculations 1 and 2 of the appendix, or is done by a moment
diagram, as in Fig. 2, the procedure recommended is as follows:
Step 1. Determine the critical section under the concentrated load.
Step 2. Divide the bending moment at the critical section by the total-per-
pin lever arm to the critical section to determine the load per pin.
Step 3. Check the tiber stress (under such loads per pin) at various sections
to see whether the location of the critical section differs under load
per pin.
Step 4. If it does differ, proceed as shown for the beveled lOB arm (viz.,
the comparable load per pin is equal to the resisting moment of
the critical section divided by the total-per-pin lever arm to the
critical section). If it does not differ, the load per pin as deter-
mined in Step 2 is the comparable load per pin sought.
Conclusions
(1) The location of the critical section under loads distributed at each
pin position must be determined before undertaking the conversion of
concentrated loads to distributed loads.
(2) The location of the critical section of a crossarm under a given condi-
tion of loading may or may not be the same under a different condition of
loading.
(3) The load per pin comparable with a given concentrated load is equal
to the resisting moment of the critical section divided by the total-per-pin
lever arm to the critical section.
(4) While the results shown are confined to the conversion of concentrated
vertical loads to distributed loads for lOA and lOB arms only, the principles
of the study may be applied to other types and sizes of arms and to other
conditions of loading.
CROSSARM LOAD CONVERSION
113
APPENDIX
Calculation 1. Bending Moments and Fiber Stresses in a Roofed 10 A
Crossarm — (See Figure 3)
Notation:
W = 800 pounds concentrated load
P = Load per pin
Mc = Bending moment at arm center
Mp = Bending moment at pole pin hole
fc = Fiber stress at center
fp = Fiber stress at pole pin hole
Sc = Section modulus of center section**
Sp — Section modulus of pole pin hole section^
^'POLE PIN
HOLE
12" >
w
-^-
'T
12"
56"-
Fig. 3 — Loading diagram for a roofed lOA crossarm.
Concentrated Load:
Mc = W X 56 = 800 X 56 = 44,800 pound-inches
Mp = W X 48 = 800 X 48 = 38,400 pound-inches
fc = Mc ~ Sc = 44,800 -^ 9.74 = 4600 psi
fp = Mp -^ Sp = 38,400 -^ 5.11 = 7515 psi
Load per Pin:
Mc = 56P -f 44P + 32P + 20P + 8P = 160P
Mp = ASP + 36P -f 24P + UP - 120P
(Note: The total-per-pin lever arms are 160" to center and 120" to the
pole pin hole).
Since under IF load/ is maximum at pole pin hole, the P load that would
result in same/ is P - 38,400 -^ 120 = 320 pounds. Thus
fp = nop -^ Sp ^ (120 X 320) ^ 5.11 = 7515 psi
fc = 160P H- Sc = (160 X 320) h- 9.74 = 5257 psi
Conclusion:
Under both IF loads and P loads, the critical section is the pole pin hole
section.
'Sc = 9.74 and Sp = 5.11 for clear roofed 3.25" x 4.25" crossarms. (See Pages 27 and
28 of Bell Sys. Tecli. Jour., Jan. 1945).
114
BELL SYSTEM TECHNICAL JOURNAL
Calculation 2. Bending Moments and Fiber Stresses in a Beveled lOB
Crossarm — (See Figure 4)
Notation:
W = 1000 pounds concentrated load
P = Load per pin
Mc = Bending moment at arm center
Mp = Bending moment at pole pin hole
fc = Fiber stress at center
fp = Fiber stress at pole pin hole
Sc = Section modulus of center section*
Sp = Section modulus of pole pin hole section^
nuLC
-4-
w
-lr-
- 10" >K — 10" — >K — 10- — *■
-1*
r-
^^*jj
40
Fig. 4 — Loading diagram for a beveled lOB crossarm.
Concentrated Load:
Mc = W X 56 = 1000 X 56 = 56,000 pound-inches
Mp = W X 40 = 1000 X 40 = 40,000 pound-inches
fc = Mc ^ Sc = 56,000 -^ 9.52 = 5882 psi
fp = Mp -¥ Sp = 40,000 -^ 5.81 = 6885 psi
Load per Pin:
Mc = 56P -f 46P -f 36P + 26P -f 16P = 180P
Mp = 40P + SOP + 20P + lOP = lOdP
P = 40,000 ^ 100 = 400 pounds
fP =
lOOP 100 X 400
sp
5.81
= 6885 psi
, 180 180 X 400 _,- .
^'=31= 9.52 =75^3P^^
Conclusion:
Critical section shifts under P loads, and arm will not support 400 pounds
per pin.
^Sc = 9.52 and Sp = 5.81 for clear beveled 3.25" x 4.25" crossarms. (See Calculation
3 of this appendix.)
CROSSARM LOAD CONVERSION
0.3248"
CENTER SECTION
115
POLE
PIN HOLE
SECTION
i
k-0.3248"
^M
"
^
1
Ri— ^^^
]
^ 1
C
4.
1
d
^r
i
'
r
25
t
R2
f h
1
,
4.0625"
c
PIN HOLE
1.25"
r
1
2
r 1
'
1
' )
< W - 3.25"-
< V — *■
»«
Fig. 5 — Beveled crossarm sections showing significance of the notation used in Calcu-
lation 3 of this appendix.
Calculation 3. Section Modulus — Clear Beveled Sections
{The notation used in this calculation is shown in Fig. 5)
Section width (W) Inches
Section depth (D) Inches
V = (IF - 1.25") ^ 2 Inches
U = V - .3248" Inches
A reas:
T Sq. Ins.
Rl Sq. Ins.
R2 Sq. Ins.
R3 Sq. Ins.
Total 1
/ = h+ (.1875" -^ 3)
r\ = h+ (.1875" -r 2)
r2 = h- (1.59375" -h 2)
r3 = 1.78125" -^ 2
Moments about MM:
Tt
R\r\
Rlrl
RlrZ
Total 2
c = Total 2 -^ Total 1
d/ = / - c
dr\ = r\ — c
drl = r2 — c
dr3 — c — r3
Moments of Inertia:
IT
IRl
IR2
IRS
T{dty
Rlidriy
R2{dr2Y
RSidrSy
I Ins."
y = D — c Inches
Section Modulus:
S = I ^ y Ins.«
Center Section
3.25
4.25
(2r)
.0609
.4876
5.1797
5.7891
20.3239
2.1359
9.52
Pole Pin Hole
Section
3.25
4.25
1.00
.6752
.0304
.1266
4.0625
Sq. Ins.
11.5173
4.2145
Inches
4.1250
4.1250
Inches
4.1563
4.1563
Inches
3.2656
i^h) 2.0313
Inches
.8906
—
Ins.'
{2Tl) .2512
.1254
Ins.3
2.0266
.5262
Ins.3
16.9148
8.2522
Ins.3
5.1558
24.3484
—
Ins.3
8.9038
Inches
2.1141
2.1127
Inches
2.0109
2.0123
Inches
2.0422
2.0436
Inches
1.1515
(c - r2) .0812
Inches
1.2151
—
Ins."
(2/r) .0001
.00006
Ins."
.0014
.0003
Ins."
1.0964
5.5873
Ins."
1.5307
—
Ins."
[2Tidty] .2463
.0612
Ins."
2.0336
.5287
Ins."
6.8680
.0268
Ins."
8.5474
—
6.2044
2.1373
ihS) 2.9029
S = 5.81
116
The Linear Theory of Fluctuations Arising from Diffusional
Mechanisms — An Attempt at a Theory of Contact Noise
By J. M. RICHARDSON
The spectral density is calculated for the electrical resistance when it is linearly
coupled to a diffusing medium (particles or heat) undergoing thermally excited
fluctuations. Specific forms of the spectral density are given for several types of
coupling which are simple and physically reasonable. The principal objective is
the understanding of the frequency dependence of the resistance fluctuations in
contacts, rectifying crystals, thin films, etc.
1. Introduction
WHEN a direct current is passed through a granular resistance such as
a carbon microphone or a metallic-film grid leak, or through a single
contact, there is produced a voltage fluctuation possessing a component
called contact noise which is differentiated from the familiar thermal noise
component by the fact that its r.m.s. value in any frequency band is roughly
proportional to the magnitude of the average applied voltage, and is differ-
entiated from shot noise by the strong frequency dependence of its spectral
density. One may regard this component of the voltage fluctuation as aris-
ing from the spontaneous resistance fluctuations of the element in question
if one is wiUing to allow the resistance to have a slight voltage dependence.
This effect has been the subject of numerous experimental investigations, ~
among which we mention in particular that of Christensen and Pearson
on granular resistance elements. These authors (henceforth abbreviated
as CP) arrived at an empirical formula, to be discussed presently, connect-
ing the contact noise power per unit frequency band with the applied volt-
: age, the resistance, and the frequency for several types of granular resistance.
Their measurements covered a range of frequency from 69 to 10,000 cps,
; and involved the variation of several other parameters, i.e., pressure. More
I recently, Wegel and Montgomery^'' have measured the noise power arising
' H. A. Frederick, Bell Telephone Quarterly 10, 164 (1931).
2 A. W. Hull and N. H. Williams, Phys. Rev. 25, 173 (1925).
^ R. Otto, Hoc'/freqiienzleclinlkund Elektroakuslik 45, 187 (1935).
^G. W. Barnes. Jour. Franklin Inst. 219, 100 (1935).
' Erwin Mever and Heinz Thiede, E. N. T. 12, 237 (1935).
8 F. S. Gjucher, Jour. Franklin Inst. 217, 407 (1934). Bell Sys. Tech. Jour. 13, 163
(1934).
^J. Bernamont, Annales de P'lys., 1937, 71-140.
8 M. Surdin, R. G. E., 47, 97-101 (1940).
' C. J. Christensen and G. L. Pearson, Bell Sys. Tech. Jour. 15, 197-223 (1936).
•0 Private communication.
117
118 BELL SYSTEM TECHNICAL JOURNAL
from single contacts and have obtained results in agreement with the CP
empirical formula down to frequencies of the order of 10""^ — 10~- cps.
Significant theoretical work upon this problem has not been attempted
until recently. G. G. Macfarlane" has advanced a theory based upon a
non-linear mechanism containing one degree of freedom which seems to be
in agreement with the CP law. W. Miller^- has worked out a general theory
of noise in crystal rectifiers. His theory is linear, contains essentially an
infinite number of degrees of freedom, and is equivalent in many respects
to the theory discussed in this paper; however, he has not succeeded in ob-
taining agreement with the experimental data on crystal rectifiers (which
satisfy approximately the CP law) for any of the specific models he used.
The purpose of this paper is the calculation of the spectral density of the
fluctuations of the electrical resistance when it is linearly coupled to a diffus-
ing medium (particles or heat), or, mathematically speaking, is equal to a
linear function of the concentration deviations of this diffusing medium.
This diffusing medium undergoes thermally excited fluctuations and thereby
causes fluctuations in the resistance. The motive behind this investigation
was the understanding of the frequency dependence of contact noise dis-
cussed in the following paragraphs, but at the present time it is apparent
that this treatment in addition may apply to rectifying crystals, thin films,
transistors, etc. The quantitative details of the coupling between the resist-
ance and the diffusing medium are not considered here; in consequence of
which, this work can hardly pretend to give a complete explanation of con-
tact noise. However, important results are given concerning the relation
between the spectral density of the resistance, on one hand, and the geom-
etry of the coupling and the dimensionality of the diffusion field on the
other.
Now let us consider the CP empirical formula in detail. Let R be the
average resistances^ of the contact (we will henceforth consider only contacts
and will regard a granular resistance as a contact assemblage) and let Ri
(/) be the instantaneous deviation from the average. By theorems 1-3 of
Appendix I, we can express the m.s. value of R\ as a sum of the m.s. values
of Ri in each frequency interval as follows :
R\ = f 5(co) do:, (1.1)
•'0
" G. G. Macfarlarie, Proc. Phys. Soc. 59, Pi. 3, 366-374 (1947).
'^ To ho. published.
"The resistance of a contact is composed of two jiarls: the "gap resistance" and the
"spreading resistance." The term "gip resistance" is seU'-explanatory. The "spreading
resistance" is the resistance involved in driving the electric current through the body of
the contact material along paths converging near the area of lowest gap resistance. The
measured contact resistance is the sum of these two parts. In some of the particular
physical models considered in Section 5, fi is taken to be the gap resistance necessitating
ad hoc arguments relating gap resistance and total resistance.
LINEAR THEORY OF FLUCTUATIONS 119
where 5(w) is called the spectral density of Ri and co is the frequency in
radians per second. Now in our notation the CP formula may be expressed
5(co) = KV-^R^+yoi, (1.2)
where V is the applied d-c voltage across the contact, ii^ is a constant de-
pending upon the temperature and the nature of the contact, and a and b
are constants having values of about 1.85 and 1.25 respectively. CP state
that the constant K is equal to about 1.2 X 10~'" in the case of a single
carbon contact at room temperature.
In this paper we will regard the nonvanishing of a — 2 as arising from
a non-linear effect which should become negligible at a sufficiently low
voltage, although this interpretation does not seem completely justified on
the basis of the work of CP. Consequently we assume that a -^ 2 as F — > 0
in such a way that F"~- -^ 1. This is in keeping with the idea that the
resistance fluctuations are truly spontaneous — at least for small applied
voltages.
Although Eq. (1.2) may represent the observations over a large range of
frequency it must break down at very high and very low frequencies in
order that the noise power be finite (or, in other words, in order that the
integral (1.1) converge).
One has several clues to be considered in looking for an underlying mecha-
nism of the resistance fluctuations. First of all, the mechanical action of
the thermal vibrations in the solid electrodes of the contact seems to be
unimportant because of the following reasons: (1) there are no resonance
peaks in S{w) at the lowest characteristic frequencies of mechanical vibra-
tion of the contact assembly; (2) S{w) becomes very large far below the
lowest characteristic frequency; and (3), according to CP, Rl is strictly
proportional to F^ when the fluctuations are produced by acoustic noise
vibrating the contact, whereas Ri is proportional to V°~-, a '^ 1.85, when
the fluctuations arise from the dominant mechanism existing in the macro-
scopically unperturbed contact. One of the obvious mechanisms left is a
diffusional mechanism. Such a mechanism does not violate any of the ob-
servations to date and, furthermore, possesses a sufficient density of long
relaxation times to give large contributions to S{w) near zero frequency.
Evidence that diffusion of atoms (or ions) can be important in modulating
a current is provided by the "flicker effect" in which the emission of elec-
trons from a heated cathode is caused to fluctuate by the fluctuations in
concentration of an adsorbed layer. We might suppose that contact noise
is a different manifestation of the basic mechanism involved in the flicker
effect.
In view of these considerations it seems worthwhile to investigate in a
120 BELL SYSTEM TECHNICAL JOURNAL
general way a large class of models involving resistance fluctuations arising
from diffusional mechanisms. In the next section we propose a general
mathematical model embracing a class of linear diffusional mechanisms.
In Sections 3 and 4 the consequences of the general mathematical model
are obtained by the "Fourier" and "Smoluchowski" methods, respectively,
these alternative methods leading to identical results. In Section 5, the
general results are speciahzed to several physical cases, some of which are
introduced only for the purpose of providing some insight into the relations
between the possible physical mechanisms and the resultant resistance
fluctuations, and one of which along with its refinement is a successful'*
attempt to provide a theory of Eq. (1.2). Section 6. is a summary.
2. The General Mathematical Model
The physical models which we consider in this paper are concerned with
the fluctuations of contact resistance arising from a diffusional process. We
are consequently led to consider the following general mathematical model
embracing a rather extensive class of the physical models as special cases:
Let us consider the instantaneous contact resistance R{l) to be related to
the intensity cir, i) of some diffusing quantity as follows'^:
G{R{t)) = j F{r, c{r, /)) dr, (2.1)
where r is a vector in two or three dimensional space depending on whether
the diffusion takes place on a surface or in a volume, and dr is correspond-
ingly a differential area or volume. The intensity c{r, t) may be either a
concentration (in the case of diffusion of material in two or three dimensions)
or a temperature (in the case of heat flow in three dimensions). In writing
Eq. (2.1) we have evidently assumed that the contact resistance R{t) is
independent of the applied voltage. Eq. (2.1) may of course allow a de-
pendence on voltage through the quantity c; however, we will consider no
processes involving a dependence of c on the voltage. These restrictions,
strictly speaking, make the model applicable only in the limit of low applied
voltages.
Before proceeding further let us limit the treatment to the case in which
the deviations of R and c from their average values are sufficiently small
for higher powers of these deviations to be neglected. Let
R{1) = R-\- R,{t), (2.2)
c{r, t) = c + c,{r, 0, (2.3)
'* That is, successful in so far as agreement with the form of Eq. (1.2) is concerned.
"A relation more general than R{t) — SF{t, c{t, /)) dr is retiuired as one can see from
considering the special case of a total resistance composed of a parallel array of resistive
elements.
LINEAR THEORY OF FLUCTUATIONS 121
where R and c are the average^^ values of R{l) and c{r, t) respectively.
Evidently, Ri{t) = 0, and Ci(r, /) = 0. Introducing the expressions (2.2)
and (2.3) into Eq. (2.1), expanding in terms of Ci(r, /), and neglecting terms
of the order of Ci, we get
Ri{l) = f m cr(r, I) dr, (2.4)
where
The function /(r) defines the linear coupling between Ri and Ci and depends
upon the specific physical model used. The non-linear terms neglected in
Eq. (2.4) may be of importance under some conditions; however, we will
not consider them here. Nevertheless, non-linear effects in the behavior of
Ci itself are possibly important in determining the form of the power spec-
trum of Ri{l) in the neighborhood of zero frequency.
3. The Fourier Series Method of Solution
In this section we consider the state of the diffusing system to be defined
by the Fourier space-amplitudes Ck(i) of Ci(r, /)• The time behavior of Ck{t)
will be described by an infinite set of ordinary differential equations con-
taining random exciting forces according to the conventional theory of
Brownian motion.^^ This method yields the spectral density of Ri(l) directly.
Now the diffusion process is assumed to occur in a rectangular area A2 =
Li X L2 or in a rectangular parallelopiped of volume ^3 = Zi X ^-2 X Z-s •
In regions of the above types, if we apply periodic boundary conditions ,
ci(ro/) may be expanded in Fourier space-series as follows:
c,{r, t) = E' CkiDe'"'-' (3.1)
k
where the components of k take the values
ki = linti/Li , i = \, • • ■ ,v, (3.2)
in which rii are integers and v is the number of dimensions. The prime on
the summation indicates that the term for fe = 0 is to b3 omitted. This is
required by the equivalence of the time and space averages of Ci (true for
A, sufficiently large) and by the vanishing of the time average of Ci (by
definition).
'^ The average values here may be considered as either time or ensemble averages but
not space averages.
" See Wang and Uhlenbeck, Rev. Mod. Pliys., 17, 323-342 (1945).
" If the final results are given by integrals over k- space they will be insensitive to the
boundary conditions.
122 BELL SYSTEM TECHNICAL JOURNAL
Before proceeding to the solution itself let us consider what it is that we
wish to know about Ck{t). Expanding the function /(r) of Eq. (2.4) in a
Fourier space-series in the region A^ ,
f(r) = Zfke'"-', (3.3)
k
we can write Eq. (2.4) in the form
RiU) = A.Z'ftck(i) (3.4)
k
where /fe is the conjugate oifk.
The spectral density 5(w) of Ri(t) is then
Sic) = A;Z'Ckk'(o:)ftfk', (3.5)
kk'
where Ckk' («) is the spectral density matrix for the set Ck{i) given by
C*fe'(co) = 2ir Lim - [ckico, T)ck'{o3, r) + Ck( — c», T)ct'( — o}, r)] (3.6)
7-»W T
in which
1 /.+T/2
Cki<^, r) = ^ Ckide-''"' dl. (3.7)
ZTT J-t/2
For a full discussion of spectral densities and spectral density matrices see
Appendix I. Consequently our objective in this section is the calculation
of the maxtix Ckk' {<^) defined by Eq. (3.6).
Now we assume that Ci(r, /) satisfies the diffusion equation
^ c,{r, I) = Z)v'ci(r, /) + g{r, i) (3.8)
ot
where Z) is a constant, V^ is the Laplacian operator in two or three dimen-
sions, and where g{r, t) is a random source function, whose Fourier space-
amplitudes gk{l) possess statistical properties to be discussed presently.
The random source function g is required for exciting a sufficiently to main-
tain (he fluctuations given by equilibrium theory. In the case of material
diffusion the random source function g may be discarded in favor of a ran-
dom force term of the form —D/x T-^-[f{c + Ci)], where V/ = Jn/, x
is the Boltzmann constant, T is the temperature, and / is the random force;
however, in the linear approximation these two procedures will give identical
final results. In the case of heat flow it is understood that the diffusion
constant \% D = K/pC where A' is the thermal conductivity, p the density,
and C the specific heat. Eq. (3.8) as written is valid only for D a constant
and Ci small.
LINEAR THEORY OF FLUCTUATIONS 12^
Introducing the expansion (3.1) and the expansion
gir, t) Z' gk{t)e'^-' (3.9)
k
into Eq. (3.8) we obtain the infinite set of ordinary differential equations
|cfe(/) = -Dk'ckd) +gfe(/), (3.10)
I describing the time behavior of the Fourier space-amphtudes Ck(t). The
Fourier space-amphtudes gkO) are assumed to be random functions of /
possessing a white (flat) spectral density matrix Ckk > independent of fre-
quency. Multiplying Eq. (3.10) by 7— «"'"', integrating with respect to time
from — ^r to -\-^t, and neglecting the transients at the end points of the
T-interval, we obtain
^ / N gk(^, t) ^2Ai\
where Cfe(w, r) is given by Eq. (3.7) and gfe(co, r) is given by an analogous
equation. Forming the spectral density matrices we get for the diagonal
elements
Ckk'io,) = . ^''.^ (3.12)
The matrix G^k' can now be evaluated by the thermodynamic theory of
fluctuations (See Appendix II). This theory gives
CkiOct'iO = ^' (3.13)
where
(3.14)
s and e being the entropy and energy, respectively, per unit area or volume,
T the average temperature, and % the Boltzmann constant. In the case
where c is the concentration of particles whose configurational energy is
constant, 5" = x/c. If c be the temperature T then s" = C/T~ where C
124 BELL SYSTEM TECHNICAL JOURNAL
is the heat capacity per unit area or volume. Now by a general theorem
concerning spectral density matrices (see Appendix I) we have
Jo
giving finally by combination with (3.12) and (3.13),
and
Gkk' -, — 7/ ^kk' , (3.15)
The spectral density 5(w) of R\{i) then becomes
(3.17)
If we are concerned with frequencies greater than a characteristic fre-
quency
coo = AttW/L" (3.18)
where L is the smallest of Li , t = 1, • • • , i', then the summation in (3.17)
may be replaced by an integration giving
^( ^ _ ..+1 .-1 XD f \f(k)\'kUk
Sic.) -2 rr ^ j -^^r^TDn^ ^^'^^^
where
f^^^ = rr^^ [ f(r)e-"'-'' dr. (3.20)
The integration in Eq. (3.19) is carried out over the entire iz-dimensiona?
/j-space. If the range of the function /(r) is sufficiently small compared with
the region ^, , or if we let A^ become indefinitely large, then the integration
in Eq. (3.20) may be extended to all of v-dimensional r-space.
It is perhaps revealing to rephrase Eqs. (3.17) and (3.19) in terms of
distributions of relaxation times. In the theory of dielectrics we speak of
the real part of the dielectric constant being equal to a series of terms
summed over a distribution of relaxation times: 2,fltTi/(l + riw'), if the
distribution is discrete, or / a(T)rJr/(l + tW), if the distribution is con-
tinuous. In the above, o, is the weight for the relaxation time tj , and, in
LINEAR THEORY OF FLUCTUATIONS 125
the case of a continuous distribution, a(T)dT is the weight for the relaxation
times in the range dr containing r. In these terms Eq. (3.17) becomes
>S(a;) =Z-4^^,, (3.17a)
k 1 + TftOJ
where
Eq. (3.19) becomes
«* = -4, IM'- (3.17b)
TT S
where
5(») = f fMlJ^ (3.19a)
«« = -M^. I \f(i/VD-r) r d^. (3.19b)
in which / is the unit vector in the direction of k, dUy is the differential
"solid" angle in the I'-dimensional /j-space, and the integration is over the
total solid angle (lir in 2 dimensions, or 4t, in 3).
It is of interest to calculate the self-co variance Ri(t)Ri(t + «). In Appendix
I, it is shown that the self-covariance above is related to the spectral density
S{o}) as follows:
Ri{i)I^i(t + u) = I S(o}) cos tico do). (3.21)
Using S(o)) in the form (3.17), Eq. (3.21) gives
RiiORiii + u) = xA,/^" • Z I/* P ^~^"'', (3.22)
k
u > 0;
whereas with S(oi}) in the form (3.19) we get
Ri{t)Ri(l + u) = {ItY x/s" f I m \' r""'' dk. (3.23)
The method of the next section yields the self-covariance directly.
4. Smoluchowski Method or Solution
We call the procedure employed in this section the "Smoluchowski
method" because it is based on an equation very closely analogous to the
well-known Smoluchowski equation forming the basis of the theory of
126 BELL SYSTEM TECHNICAL JOURNAL
Markoff processes. ^^ We set out directly to calculate the self-covariance for
Ri{l) which, by Eq. (2.4), is given by
R,{l)R,{t + u) = fffir')f{r)c,{r't)c,{r',t + u) dr' dr. (4.1)
Thus the problem is now reduced to the calculation of Ci(r', t)ci{r, t + u).
The quantity Ci{r' , t)ci{r, t + u) is calculated in two steps. First we find
Ci{t, t -\- u) , the average value of Ci at the point r at the time / + u
with the restriction that Ci is known at every point r' with certainty to be
Ci(r', /) at the time / (assuming, of course, that u > 0). Then we find that
the required self-covariance for d, is given by multiplying the above
(t+u)
Ci(r, t -\- u) by Ci{r', i) and averaging over-all values of Ci{r, t) at
time /; thus:
ci(r', /)ci(r, / + w)"-^"' = ci(r', Oci(r, t -f- u). (4.2)
.{t+u)
Now we assume that C\{r, / + m) is related to c{r\ t) by an integral
equation, analogous to the Smoluchowski equation, as follows:
(<+«) /•
ci(r, t -\- u) = j p(\r - r' \, u)ci(r', t) dr'. (4.3)
In the case that c represents a concentration as in the diffusion of particles,
p(| r — r' 1, ii) dr is the probability that a particle be in the j^-dimensional
volume element dr at time / + u when it is known with certainty to be at
r' a time /. Now the number of particles in dr' at r' at time / is evidently
[c + Ci{r' , t)] dr'; consequently, the probable number of particles in dr at
time / + u which were in dr' at time / is p(| r — r' |, u)[c + Ci{r', t)] dr dr' .
Integration over dr' gives the total probable number
{l+u)
(c -\- Ci(r, t -\- u) ) dr of particles in dr equal to
( / p(| r — r' I, u)[c + Ci(r', /)] dr' j dr which reduces to
( c + I p(\r — r' \, u)ci(r', I) dr' I dr. Division by dr and subtraction of f
from both sides of the equality leads directly to Eq. (4.3). For the case of
heat flow in crystal lattices the above picture can be used approximately if
one uses the concept of phonons.-" For a diffusional process p(| r — r' |, w)
is the normalized singular solution of the diffusion equation-^; thus
p{\r-r'\,u) = ^^, exp [- 1 r - r' \'/4Du] (4.4)
i» ix)c cit.
20 J. Weigle, Experientia, 1, 99-103 (1945).
2' Chandrasekhar, Rea. Mod. Phys. 15, 1 (1943).
LINEAR THEORY OF FLUCTUATIONS 127
where v, as previously defined, is the number of dimensions of the region in
which the process occurs.
Combining Eqs. (4.2) and (4.3) we get
/. (0
ci(r', Oci(r, t + u) = j c,{f, t)c,{r", t) p{\ r - r" |, u) dr". (4.5)
Now using the fact that
(0
c,{r', l)c,{r", t) = cyir', t)cy{r", l) (4.6)
and using the relation
c,(r', l)c,{r", I) = ^, 5(r' - r") (4.7)
iproved in Appendix II, Eq. (4.5) reduces to
ci(r', t)ci(r, t-\-u) = ^,pi\r - r'\, u). (4.8)
Introducing the expression (4.8) into Eq. (4.1) we obtain at once the desired
result
'Ri{t)Ry{t + u) = ^, fffir)f{r')p{\ r - r' \, u) dr dr'
(4.9)
For the sake of comparison with Eq. (3.23) it is necessary to write (4.9)
in terms of the Fourier space-transforms of the pertinent quantities. We
write
/W = ff(.k)e"''
where
Also, we write
P(| r - r' I, u) = ^^J^, exp [-\r - r' f/4Du]
= (2^. / e^P [-Duk' + ik- (r - r')] dk.
After introduction of these expressions into (4.9) a short calculation yields
the result
R,(OR,il + u) = (27r)'' ^, f \f{k) 1%-^"^-' dk (4.10)
128 BELL SYSTEM TECHNICAL JOURNAL
which is identical with Eq. (3.23) (provided that we let ^^ -^ <» in the lat-
ter). Thus the methods of approach used in Section 3 and in this Section
are completely equivalent.
5. Special Physical Models
In the previous two Sections we have developed by two different methods
the consequences of the general mathematical model discussed in Section 2.
Here we apply the general results to some special physical cases. In this
task we will be principally concerned with finding the form of the function
/(r) and establishing the number of dimensions v of the diffusion field. The
main objective here is to provide some orientation on what mechanisms are
or are not reasonable and to find at least one mechanism leading to the
observed spectral density (inversely proportional to the frequency).
a. A General Class of Models. Here we consider all at once mechanisms
which can be adequately represented by having /(r) a j/-dimensional Gaussian
function of the form
where aJ'^ is the "width" of the function measured along the t-th coordinate
Xi. This form of /(r) can represent approximately several types of localiza-
tion of the coupling between R\ and Ci , as will be seen in the special examples
later. Now if we work with Av= oo , we will then have to consider the Fourier
space-transform of /(r), which is readily shown to be
f^'^-Tk^l^^^^^-^'"''
(5.2)
Inserting this result into Eq. (3.19) we obtain immediately
exp ( — 2^ ^i ^i ) k~ dk
^w^^rnss/
2xD _
7r5" VfJi 27ry J «2 + I>2^* {S.2,)
1=1
Inserting this expression (5.2) into Eq. (3.23) gives
LINEAR THEORY OF FLUCTUATIONS 129
In order that (5.3) give the observed S(ui) a 1/w as a result, the integral
.iu£t reduce to scmething proportional io I k dk/(cj^ + D^k*). It is clearly
impossible that any choice of v and any set of A,- can achieve this result.
Furthermore the seif-covariance i?i(/)i?i(/ + ti) corresponding to the ob-
served S(o)) should depend explicitly on the way S(o}) deviates from l/w
as CO goes to zero. The expression (5.4) is finite for all w > 0 and does not
depend upon any cut-off phenomena in S(o:) at low frequencies. Therefore
we can exclude any physical mechanisms belonging to the class considered
here. However, since several mechanisms that have been proposed do fall
into this class, we consider them below:
(f) Scl.iffs Mechanism. Schiff^^ considered tentatively that the fluctua-
tions in contact resistance may be due to the variation in concentration of
diffusing ions (atoms, or molecules) in a high resistance region bounded by
parallel planes of very small separation. Schiff arrived at a noise spectrum
proportional to 1/co but at the expense of disagreeing in a fundamental way
with thermodynamic fluctuation theory. Here we will show what the cor-
rect consequences of this mechanism are.
Consider that the high resistance region is bounded on either side by planes
parallel to the {xi , X2)-plane and that the thickness in the Xs-direction is
very small. Now this is obviously a case of the general model just considered
in which we take v = 3 and
Ai,2»Du,\
(5.5)
As «Du,]
where l/u is of the order of magnitude of the frequencies of interest. It is
then a matter of algebraic manipulation to show that
i2 22 72 -.00 ,2
and
X
b\ ^2 63
1
i5i'2 s"
26/2
x^Al'^A
1/2 ^1/2
a2 l2 7,2
Ol 02 bz
1
> 0.
{^irf'Ar
A^^
(5.6)
R.{l)Rr{t + u)= -^^ . .. .3;.:m.i/2 -TXF^ « > 0- (5.7)
Thus we see that Shiff's mechanism leads to a noise spectrum proportional
to l/oj^'^, not l/cj. The explanation of the singularity of the self-covariance
(5,7) at « = 0 lies in the inequalities (5.5).
«L. I. Schiff, BuSldps Contract NObs-34144, "Tech. Rpt. j^3", (1946). Before the
publishing of this paper, Schifl informed the writer that he has discarded this mechanism.
130
BELL SYSTEM TECHNICAL JOURNAL
The above treatment could just as well be applied to the case in which
the diffusing quantity is heat instead of ions.
(«) Resistance of a Localized Contact Disturbed by a Dif using Surface
Layer. Here we consider the case of two conductors covered with diffusing
surface layers. It is supposed that the conduction from one conductor to
the other is distributed Gaussianly with a width A^'^. Finally, it is supposed
that the conductivity through the above area varies with the surface con-
centration of the surface layer in that region. This situation is well repre-
sented by the above general model by taking v = 2, Ai = A2 = A, and
bi = bi = b.
The self-covariance is readily calculated with the result
RMRAI +«)-fr 4,(^ + On)
(5.8)
The corresponding spectral density is
ZTT 5 Jo
COS uoi du
A + Du
27r2 s"D
-COS {coA/D)Ci{coA/D)
+ sin (coA/£>) (1 - Si{aiA/D)
(5.8a)
where Ci(x) and Si{x) are the cosine and sine integrals-^ respectively. When
to « D/A
Sic)^ -l-,j^\og{8coA/D),
ZTT" 5 U
8 = 0.5772,
(5.8b)
and when w >>> D/A
5(co) c^
1 xb D 1
2^2 5"A2 0)2
(5.8c)
Thus we see that this case does not lead to the experimental form of the
spectral density. It must be remarked that here S{co) is very sensitive to
the form of the self-covariance for small u.
b. Contact between Relatively Large Areas of Rough Surfaces Covered with
Diffusing Surface Layers. We consider this case in detail since it leads to
results in agreement with experiment. Furthermore, the more detailed
consideration of this case will illustrate more fully the use of the general
mathematical model, which may be of use in studying other diffusional
mechanisms should they be postulated at some future time. This mechanism
does not fall into the class just considered.
^ See Jahnke and Emde, "Tables of Functions," p. 3, Dover (1943).
LINEAR TUEORY OF FLUCTUATIONS 131
Suppose that the contact in an idealized form consists of two rough sur-
faces close together. Let positions on the surfaces be measured with respect
to a plane between the surfaces, which we will call the mid-plane. Let the
coordinate system be oriented so that the .Ti and X2 axes lie in the mid-
plane. Furthermore let the region in the mid-plane corresponding to close
proximity of the rough surfaces be a rectangular area A2 = LyX Lo. Now,
for convenience, we describe positions on the mid-plane by a two dimensional
vector r = (.ti , X2), and henceforth it will be understood that all vector
expressions refer to this two-dimensional space. Let the distance between
the upper and lower surfaces at r be denoted by h{r). The geometry of the
above model is illustrated in Fig. I.
Now suppose that each surface is covered by a diffusing absorbed layer,
such that the sum of the concentrations on both surfaces is c(r, t) at the
time / in the neighborhood of r. Now consider the conduction of current
between the surfaces. Let us assume that the conductance per unit area
(of mid-plane) is a function of the separation // of the surfaces and the total
concentration c of absorbate near the point in question, i.e., F{h, c). The
total conductance will be the sum of the conductances through each element
of area: hence, the instantaneous resistance R{l) at time / will be given by
\/R{l) = f F{h{r), c{r, /)) dr (5.9)
''A 2
where dr is the differential area on the mid-plane and the integration extends
over the rectangle A2 = Li X L2 . Behind the above statements lies the
tacit assumption that the radii of curvature of the rough surfaces are gen-
erally considerably larger than the values of //. However, we will not explic-
itly concern ourselves with this implied restriction.
At this point it is expedient to imagine that we have an ensemble of con-
tacts identical in all respects except for different variations of the separa-
tion h{r). If we have any function of h, \f/{h) say, which we wish to average
with respect to the variations of h, we simply average the function over the
(e)
above ensemble giving a result which we denote by \l/(h) .
Now let us write
and, as before,
h(r) = h''' + //i(r), (5.10)
R{t) = R + R,0), ]
(5.11)
c(r,l) = c-}-cj(r, /).
We assume that the ensemble average h''" and the time averages R and c
are constants independent of r and /. Let us also assume that the integrals
132
BELL SYSTE}f TECHNICAL JOURNAL
of hi(r) and Ci(r, /) over Ao vanish. Inserting (5.10) and (5.11) into (5 9) and
expanding, we get
l/R - Ri{l)/U' + . . . = A,F{fi"\ c)
+
'ji)^fjM.ir,6Jr + i{gy}jirUr
dhd
+
(5.12)
where the super zero on the derivatives indicates that they are evaluated at
h = ^*'^ and c = c. In accordance with previous approximations in this
memorandum we neglect-^ terms of the order of Ci and Rl . We also neglect
terms of the order of hi . After taking the time average of (5.12) and sub-
tracting the result from (5.12) we get
RiiO = f f{r)ci{r, i) dr,
•I A 2
fix) = arf-lhir),
\dhdcj
(5.13)
Thus we now have a special case of our general mathematical model for the
number of dimensions v = 2, provided that we assume that the total con-
centration c on both of the rough surfaces fluctuates in the same manner as
the concentration of a single adsorbed layer confined to a plane rectangular
surface. The spectral density S(co) of Ri{l) is then given by Eq. (3.17) which
we repeat here
5(a;) = -
2 xAo.D
(5.14)
where /j is a two-dimensional vector whose components take the values
ki = l-n-tii/Li , Hi = 0, ± 1, ± 2, • • • , and where /^ are the Fourier space-
amplitudes of /(r) given by
/* = ^r f /(r)e-'*'-
JA2
dr.
It may be appropriate at this point to consider the quantity s'' in detail
for this particular case. If the energy e per unit area is independent of c,
d s
we have s" = — — „ evaluated at c = c where s is here the entropy of the
absorbate per unit area. For the sake of illustration let us consider a single
layer of absorbate in which the molecules are non-interacting. If c, the sur-
^■' Wc neglect these terms not because they are small compared with Ci or hiCi but,
because they are non-fluctuating (in time), are hence to be compared with l/R.
LINEAR THEORY OF FLUCTUATIONS 133
face concentration of the absorbate, be measured in molecules (atoms, or
ions) per unit area, then, for the ideal system above, it follows that s = —x
c log c and finally that s" = xli- However, in the mechanism discussed in
this part we have a compound system consisting of two separate layers on
the upper and lower surfaces respectively. Nevertheless, a detailed analysis
reveals that with c equal to the sum of the concentrations of both layers
we still have ^ = x/^ even though s itself is no longer given by an expres-
sion the same as that above. In conclusion let us consider the factor xA"
in Eq. (5.14). This factor is under the above idealization simply equal to
c. That is, the spectral density 5(co) is directly proportional to the average
concentration of absorbed molecules, meaning simply that each molecule
makes its contribution to the resistance fluctuations independently of the
others. Of course, in any real system this will not be quite true; however,
the existence of interactions will be manifested only by making xh" not
equal to c in Eq. (5.14).
The results quoted thus far apply to a system with a given hiy). Now we
shall average the right-hand side of Eq. (5.14) over the ensemble of varia-
tions of //(r), it being supposed that S{<ji) itself on the left-hand side will
be negligibly affected by this operation. This amounts to replacing |/fe |2
(e)
by I /ftp . We then have
I
where /^^ are the Fourier space-amplitudes of hxif)
We now consider more closely the problem of calculating | hk |^ . We want
to assume that h\{y) is a more or less random function of r. If h\{y) were a
random function of r in the same way that the thermal noise voltage is a
(e)
random function of the time /, then | hk ^ would be a constant independent
(e)
of k and the self-co variance hi{r)/h(r') would vanish for r 9^ r'. This
clearly cannot be so, since the function hi(r) with such statistical properties
would represent a highly discontinuous type of surface incapable of physical
existence. We then fall back upon the more reasonable assumption that the
gradient of hi possesses statistical properties of the above type. This notion
is precisely formulated by means of the following equations:
V//i(r) =/>(r) (5.16)
where
fkf = a^R' Ihkl''"' (5.15)
-.M
and
f p{r) dr = 0, (5.17)
p{r)p{r'f^ = /3U(r - r'). (5.18)
134 BELL SYSTEM TECHNICAL JOURNAL
In Eq. (5.18) /3 is a parameter (with the dimensions of area) characterizing
the ampUtude of the surface roughness, and J; is the unit tensor in two di-
mensions. Expressing (5.16) in terms of the Fourier space-amplitudes hk
and pk of //i and p respectively, we have
-ikhk = pk, (5.19)
giving finally
\lik\r =kPkPt ' 'k/k' (5.20)
Expressing (5.18) in terms of Fourier space-ampHtudes we get
(e)
Pktt' = l3A,'Ukk', (5.21)
which, when inserted into (5.20) gives the following desired result:
-(e) ^^_i ^_2
|//*p = I3A2 k~\ (5.22)
-(e)
Now replacing 1/^ \^ by|/fe |- in Eq. (5.14) and substituting the ex-
pression (5.22) with the use of Eq. (5.15), we obtain
Sic.) = ? • ^ -^crR' E . _/ ^,,, (5.23)
If the frequencies of interest are larger than a certain characteristic frequency
coo = 47r2 D/L^ where L is the smaller of Zi and Zo , the summation in
(5.23) may be replaced by an integration giving finally
k dk
Si.) = ^D/rs":WM- I -,_^^,^,
(5.24)
This result is in agreement with experiment in most respects. The dependence
on frequency is, of course, that experimentally observed by all investigators.
The non-dependence on the voltage applied across the contact is implied
by the basic assumptions common to all of the mechanisms considered here,
and is in approximate agreement with the results of Christensen and Pearson
(see Eq. (1.2)). For our result to agree with the results of CP as regards the
dependence on the average resistance-^ R, the factor c-R^ A^ must be pro-
portional to 7^2+'' where b '~ 1.25. These authors also imply that some of
''^It must be remembered that the resistance Ti in the CP formula is the total contact
resistance equal to sum of the gap resistance and the spreading resistance, whereas the R
in our theory evidently should be considered the gaj) resistance. For the purposes of com-
parison we make the ad hoc assumption that the gap resistance is proportional to the total
contact resistance.
LINEAR THEORY OF FLUCTUATIONS 135
the parameters necessary to complete the description of a contact between
given substances at a given temperature show up impHcitly only through
/?. According to our theory the factor a- W -42 does not depend in any unique
way upon R; it matters by what means R is varied. If the resistance R is
changed by altering the contact area A^ , keeping other parameters fixed,
we would find that RA2 is constant so that the factor in question would be
proportional to R^, that is, 6 = 1. However, if R is changed by varying the
contact pressure, the effect would show up through the factor a^, (/3 also,
to some extent, perhaps) and, since one would expect a to increase somewhat
with pressure whereas R decreases with pressure, the factor of interest would
probably depend upon some power of R between 3 and 4, that is, 1 < 6 < 2.
The theory formulated here suffers from the difficulty that the integral
of the power spectrum with respect to frequency is logarithmically divergent
at 0 and 00 , that is
/ S{(S) d<j} = I do) oj= log (CO2/CO1) — > 00 as coi — >0 and
J 01 1 ''oil
C02-
The divergence at 00 does not bother us as much as the divergence at 0
since, with only a divergence at 00, the self-covariance Ri{t)Ri(t + u) exists
for all non- vanishing values of u; whereas, with a singularity at 0, the self-
covariance does not exist for any value of m. For this reason we cannot con-
sider the self-covariance here. In Part c of this Section we consider a possible
way of removing the divergence at 0, and consequently, then, we are able
to calculate the self-covariance for non-vanishing values of u.
c. Refinement of the Theory of Part b. Here we propose a simple modifi-
cation of the model of Part b, removing the divergence of the integral of
S{u)) at CO = 0. The modification considered here, although it is one of several
possibilities any one of which is sufficient for removing the divergence
(See Section 6.), is perhaps the only one that is sufficiently simple to treat
in a memorandum of this scope. The results of this section are thus intended
to be only provisional and suggestive.
Let us reconsider the statistics of the function h{r) giving the separation
between the surfaces near a point r on the mid-plane. The distribution of
/?'s considered in the last section is open to several criticisms: (1) it possesses
no characteristic length parallel to the mid-plane; and (2) the self-covariance
h\{r)h\{r'Y^^ does not exist for any value of r — r' .
To correct partially for these difficulties we replace Eq. (5.22) by
\h^fl= ^^ , (5.25)
(e)
where / is a new characteristic length. The self-covariance h\{r)lh{r')
based upon (5.25) now exists for all values of r — r' except 0. Thus we still
136
BELL SYSTEM TECHNICAL JOURNAL
— (e) ... . ,
have the objection that the variance/zj is infinite; however, this will
cause us no trouble.
With Eq. (5.25) instead of (5.22) the spectral of density Ri takes the form
-
1
eii'
k dk
+ (^ k"' C02 + Z)2 k^
Q(y) =
= {x/'^Ts")-0a'R'Arl/o:-Q(y),
y(y - -\ogy)
1 -\-f
y = ro}/D.
(5.26)
In obtaining the above equation we have made the usual assumption that
the frequencies of interest are larger than ojo = ^ir'^D/D, and have replaced
the original sum by an integral. The function Q(y) has the following proper-
ties:
Q(y) a^ —- y log y for y <K 1
IT
Qiy) ~ 1 for y » 1
(5.27)
Hence for w <<C D/^, S(u) ^ log w, the integral of which converges as a; — > 0;
whereas, for w ^ D/(^, 5(co) differs negligibly from that given by the unre-
fined theory (Eq. (5.24))^
The self-covariance Ri{l) R\{l -f u) now exists for all non- vanishing u and
is given by
R,{t)R,{t +u) = (x/2Ts")-^a'R*A, • j j
-,2 —Duk'^kdk
I e
JO 1 + P ^2
= (x/47r5'0 '^a' R'A2-e'""^\-Ei(-Du/f)] J
(5.28)
where
— Ei{—x) = I e " dv/v,
J X
<^ —log yx for X <K 1,
c^ — for X » 1,
X
7 = 0.5772.
Thus for u « ^yZ), Ri{t)Ri(t + u) a - log {jDu/f') and for u » ^/D,
Ri(l)Riit + u) a 1/u.
LINEAR TUEORY OF FLUCTUATIONS 137
Thus we have illustrated how one modification of the model has removed
the divergence at co = 0.
It appears from the treatment here and in part b that roughness and
diffusion in two dimensions are essential (at least in a linear treatment)
features in obtaining S(ui)a 1/co. In the case of a non-linear coupling (to be
considered in a later paper) a "self-induced" roughness effect may occur
without introducing roughness ab initio as an intrinsic feature of the model.
6. Summary
(a) If the resistance deviation i?i(/) is related to the concentration devia-
tion Ci(r, t) of a diffusing medium (particles or heat) by the linear functional
RxU) = f firMr, 0 dr, (6.1)
where r is a vector and dr a volume element in a j^-dimensional space of
volume A^ , then the spectral density S{co) of Ri(t) is
, . 2xA.D k'lfkf ,...
where D is the diffusion constant, s" is defined by Eq. (3.14), x is the Boltz-
mann constant, w is the frequency (in radians per sec), k is the wave number
vector in v-dimensional /j-space limited to a discrete lattice of points (defined
by Eq. (3.2)) over which the summation is taken, and/^ is the /jth Fourier
component of /(r) (Eq. {2>2)).
(a) If the important terms in (6.2) vary slowly between lattice points in
/j-space (true if co > coo given by Eq. (3.18)), then (6.1) can be replaced by
the integral
where the integration extends over the entire /j-space and where f{k) is
given by (Eq. 3.20)
(b) Let oj' be a frequency in the middle of a wide range. Suppose [ f{k) \^
averaged over ths total solid angle in j'-dimensional /j-space is proportional
to k" ', where n is an integer, in a wide range of k with k = \/o:'/D in its
middle. It follows then that S(ui) a D'"""'^ ^-i+«+W2 ^^ ^^^^^ as — 1 < 2w
+ j'+l<3. Asa consequence, we see that with n an integer (as is true for
the simple cases considered in Section 5) v must be 2 — ^the only even di-
138 BELL SYSTEM TECHNICAL JOURNAL
mensionality — in order that S{u)) be inversely proportional to co in agree-
ment with experiment. In this case the only allowed value oinis — 1.
(c) From (b) we have the interesting result that S{(ji) is independent of D
when it is inversely proportional to oj. This means that very slowly diffusing
substances can contribute as much to contact noise as rapidly diffusing
substances. This result can be derived on quite dimensional grounds and is
not dependent upon the special assumptions underlying our treatment.
(d) A system comprising a high resistance layer modulated by the three-
dimensional diffusion of particles or heat gives 5(co) « co~' ^. See Case a.(i)
in Section 5.
(e) In a system composed of a localized contact disturbed by a diffusing
surface layer (See Case a.(ii), Section 5), the self-covariance Ri{t)Ri{l -\- u)
is inversely proportional to A + Du where A may be considered the contact
area. We have S{w) oc — log a -\- const, for co « Z)/A and 5'(co) a ijT''
for w » D/^.
(f) In a system involving the contact between relatively large areas of
rough surfaces covered with diffusing surface layers (Cases b. and c, Section
5), we have been successful in obtaining S{<ji) «■ co~\ and also in obtaining
a reasonable dependence upon the average resistance.
APPENDIX I
Spectral Density and the Self-Covariance
Here we consider in detail the spectral density, the self-covariance, and
the relation between these two quantities, first for the case of a single random
variable. The treatment is subsequently extended to the case of a set of
random variables which necessitates the consideration of the spectral density
matrix and the covariance matrix.
Let y(/)be a real random variable whose time average vanishes, y{l) = 0.
Now the m.s. value of y can be defined
3^ = Lim ^ f y\l, r) dl (I-l)
where y(t, r) ^ y{t) in the interval —-</<- and vanishes outside this
interval. Evidently y{t, t) can be expressed by the Fourier integral
.+00
where
.+{T/2)
y{l, t) = [ z{o,, Tje'-" do, (1-2)
''-00
z{w, t) = -— / y{l)e^'"^ dl,
l-K J-{tI2)
UN EAR THEORY OF FLUCTUATIONS 139
By Parseval's theorem we obtain
y'{t, t) dl = Itt I I yico, t) I" Joj,
00 *'— 00
which, when combined with (I-l), gives finally the desired result (using
the fact that | y(co, r) |^ is an even function of co)
Til) = f F(a,) d^ (1-3)
Jo
where
F(co) = 47r Lim - | y(a), r) |' (1-4)
T-»oo T
is the spectral density.
By a procedure not very different from the preceding, one can show that
yiOyil + m) = / Y(ui) cos COM dw, (1-5)
Jo
Y(u) = — I y{l)y{l + u) cos co?^ du. (1-6)
TT J
The quantity (y(/)y(/ -|- «) is called the self-covariance.
Now let us suppose that we have a set of random variables yiit) which are
in general complex and whose time averages vanish. We are then led to
consider, instead of (1-3), relations of the form
y.{i)y*{i) = /" I\;(C0) 6fcO (1-7)
Jo
where now
riy(co) = 27r Lim - [y,(co, T)y*(w, t) + Jz(-co, r)y*(-aj, r)] (1-8)
T->00 T
in which
J /. + (t/2)
>'i(w, r) = — \ yXl)e"^ dt.
iTT J-(rl2)
Instead of self-covariances like y{l)y{L -{• n) we have to consider a covari-
ance matrix of the form y,(/)y; (/ + u). Since we shall not have occasion in
this paper to consider the relation between the spectral density matrix and
the covariance matrix we will not consider the derivation of the analogue of
Eq.(I-5).
140 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX II
Thermodynamic Theory of Fluctuations
The value of the quantity fi(r, i)ci{r^, t) or {ck{l)ck^{l)) is determined from
equilibrium considerations. Before going into the above continuum problem
let us first consider the problem for the case of a system described by a finite
set of variables. More specifically let us suppose that the state of the system
subject to certain restraints (i.e. fixed total mass and energy) is described
by the set of variables Xi , • • ■ , Xn . Let the equilibrium state be given by the
values Xi , ■ • • , x„ , and let
Xi = Xi -{- ai . (ITl)
If the system is constrained to constant average energy E, the entropy of the
non-equilibrium state S = S^ -{- AS will be less than vS", the entropy of the
equilibrium state, by an amount
AS = -liHSijaiUi, (II-2)
where
d'^S
I ( d'E\
r^\dxidxj.
Obviously, AS must be the negative of a positive definite quadratic form,
otherwise the equilibrium state would not be a state of maximum entropy.
The probability distribution''*' for the a's is given by
P{a,,--- ,«„) = Ne''"'' (II-3)
where TV is a normalization factor and x is the Boltzmann constant. Averag-
ing the products cci a; we find that
2_/^ij«j«/fc = X^ik- (II-4)
i
Multiplying (II-4) by the arbitrary set ji and summing over i we get
22 yiSijOijaK- = XTi- (II-5)
The generalization to a system described by a continuous set of variables
is not difficult on the basis of (II-5). Now suppose that, in a i^-dimensional
space ^^ , we have a system whose state at time / is defined by the continuous
set of values of the variable c{r, l) = c -{- Ci{r, t) ; we have
AS = -\l s"c\(r,/)dr (II-6)
«« H. B. G. Casimir, Rev. Mod. Pliys. 17, Nos. 1 and 3, 343-4 (1945).
LINEAR THEORY OF FLUCTUATIONS 141
where
when 5 and e are the entropy and energy, respectively, per unit volume (of
jz-dimensional space). In calculating (II-6) it was assumed that
/ ci{r, /) dr = 0,
J A,
expressing the fact that the system is closed. In order to put (II-6) into a
form strictly analogous to (II-2) we write it
AS = -I- f f s"6{r - r')ci{r, t)ci{r', l) dr dr'. (II-7)
2. J A, J A,
We see that the equation analogous to Eq. (II-5) must be
I [ y{r')s"d{r - r")ci(r", t)c,{r, t) dr dr' = x7(r) (II-8)
JAy J Ay
where 7(r) is an arbitrary function. Integrating (II-8) with respect to f
and using the fact that the delta function is defined by
fy{r')8(r' -r) dr' = y{r)
we readily arrive at the result
ci(r, l)cy{r', t) = ^, b{r - r'). {11-9)
Using the Fourier space-expansions of Ci and 6(r)
k
Kr) =~T.e''\
A, k
in the region A;, = LiX • • • XL„ with ki = lirni/Li , we can write (II-9)
over into the equivalent expression:
^ u s
where
fl if ft = k',
bkk' = {
0 otherwise.
Abstracts of Technical Articles by Bell System Authors
Audio-Frequency Measuremenls} f W. L. Black* and H. H. Scott. This
paper indicates the theory involved in making measurements of gain, fre-
quency response, distortion, and noise at audio frequencies, with particular
emphasis on such measurements made on high-gain systems. There are
also discussed techniques of measurement and factors affecting the accu-
racy of results. This subject is not new art but has not previously been
pubUshed in correlated form, to the knowledge of the authors.
Growing Quartz Crystals} f E. Buehler and A. C. Walker. The Bell
Telephone Laboratories started an investigation of this subject in March
1946, based on information gleaned from several investigators who visited
Germany after the war, particularly Mr. J. R. Townsend of these Labora-
tories, and Professor A. C. Swinnerton of Antioch College. After a relatively
few experiments made with equipment similar to that used by Professor
Richard Nacken in Germany, and with the process he described, it became
apparent that Nacken had made substantial progress in the art of growing
quartz at temperatures and pressures near the critical state of water, i.e.,
about 374°C, and 3,200 pounds per square inch. This report summarizes
further progress that has been made in the Laboratories since March 1946.
Corrosion of Telephone Outside Plant Material} f K. C. Compton and
A. Mendizza. Problems resulting from corrosion in the telephone outside
plant are many and varied. In this article an attempt is made to give a
broad overall picture of these problems and the manner in which they are
met and solved by the telephone plant engineer.
Magnetic Recording in Motion Picture Techniques} John G. Frayne and
Halle Y Wolfe. Development of magnetic recording at the Bell Telephone
Laboratories is described with the application of such facilities to Western
Electric recording and reproducing systems. A method of driving 35-mm.
magnetic film with a flutter content not greater than 0.1 per cent is de-
scribed, as is a multigap erasing head.
Semi-Conducting Properties in Oxide Cathodes} f N. B. Hannay, D.
MacNair, and A. H. White. It has been widely assumed, without ade-
' Proc. I. R. E., V. 37, pp. 1108-1115, October 1949.
* Of Bell Tel. Labs.
^Sci. Monllily, v. 69, pp. 148-155, September 1949.
^Corrosion, v. 5, pp. 194-197, June 1949.
".S". M. P. E. Jour., V. 53, pp. 217-234, September 1949.
"•Jour. Applied Pliysics, v. 20, pp. 669-681, July 1949.
t A reprint of this article may be obtained by writing to llie Editor of the Bell System
Technical Journal.
142
ABSTRACTS OF TECHNICAL ARTICLES 143
quate experimental verification, that barium-strontium oxide, as used in
the oxide cathode, is an excess electronic semi-conductor. Accordingly, the
electrical conductivity of (Ba,Sr)0 has been studied as a function of tem-
perature before and after activation with methane, extensive precautions
being taken to exclude spurious effects. The increase in conductivity ob-
tained characterizes (Ba,Sr)0 as a "reduction" semi-conductor, and hence
very probably as an electronic semi-conductor whose conduction electrons
arise from a stoichiometric excess of (Ba,Sr) atoms in solid solution.
A basic prediction of the semi-conductor theory has been tested quan-
titatively with the finding that the electrical conductivity and the thermionic
emission of a (Ba,Sr)0 cathode are directly proportional through three
orders of magnitude of activation; well-defined chemical and electrical
activation and deactivation procedures were used in obtaining this result.
It may be concluded that activation represents an increase in the chemical
potential of the electrons in the oxide, little or no change in the state of the
surface occurring. It has also been found that deviations from the propor-
tionality of conductivity and emission may be expected under conditions
leading to inhomogeneity in the oxide, in agreement with the semi-conduc-
tor theory also.
Electron Microscope and Difractlon Sludy of Metal Crystal Textures by
Means of Thin Sections.^ f R. D. Heidenreich. Bethe's dynamical theory
of electron diffraction in crystals is developed using the approximation of
nearly free electrons and Brillouin zones.
The use of Brillouin zones in describing electron diffraction phenomena
proves to be illurr. inatin j since the energy discontinuity at a zone boundary
is a fundamental quantity determining the existence of a Bragg reflection.
The perturbation of the energy levels at a corner of a Brillouin zone is
briefly discussed and the manner in which forbidden reflections may arise at
a corner pointed out. It is concluded that the kinematic theory is inadequate
for interpreting electron images of crystalline films.
An electrolytic method for preparing thin metal sections for electron
microscopy and diffraction is introduced and its application to the structure
of cold-worked aluminum and an aluminum-copper alloy demonstrated.
It is concluded that cold-worked aluminum initially consists of small, in-
homogeneously strained and disoriented blocks about 200A in size. These
blocks are not revealed by etching but would contribute to line broadening
in conventional diffraction experiments. By means of a reorientation of the
blocks through a nucleation and growth process, larger disoriented domains
about l-3;u in size found experimentally could be accounted for. It is sug-
^Jour. Applied Pliysics, v. 20. pp. 993-1010, October 1949.
t A reprint of this article may be obtained by writing to the Editor of the Bell System
Technical Journal.
144 BELL SYSTEM TECHNICAL JOURNAL
gested that such a nucleation and growth reorientation phenomenon is re-
sponsible for self-recovering in cold-worked metals.
The formation of CuAU precipitate particles is demonstrated with both
electron micrographs and diffraction patterns. A fine lamellar structure found
in the quenched Al-4 per cent Cu alloy is at present unexplained.
Path-Length Microwave Lenses?] Winston E. Kock. Lens antennas for
microwave applications are described which produce a focusing effect by
physically increasing the path lengths, compared to free space, of radio
waves passing through the lens. This is accompUshed by means of baffle
plates which extend parallel to the magnetic vector, and which are either
tilted or bent into serpentine shape so as to force the waves to travel the
longer-inclined or serpentine path. The three-dimensional contour of the
plate array is shaped to correspond to a convex lens. The advantages over
previous metallic lenses are: broader band performance, greater simplicity,
and less severe tolerances.
Refracting Sound Waves. ^] Winston E. Kock and F. K. Harvey.
Structures are described which refract and focus sound waves. They are
similar in principle to certain recently developed electromagnetic wave lenses
in that they consist of arrays of obstacles which are small compared to the
wave-length. These obstacles increase the effective density of the medium
and thus effect a reduced propagation velocity of sound waves passing
through the array. This reduced velocity is synonymous with refractive
power so that lenses and prisms can be designed. When the obstacles ap-
proach a half wave-length in size, the refractive index varies with wave-
length and prisms then cause a dispersion of the waves (sound spectrum
analyzer). Path length delay type lenses for focusing sound waves are also
described. A diverging lens is discussed which produces a more uniform
angular distribution of high frequencies from a loud speaker.
Double-Stream Amplifiers?] J. R. Pierce. This paper presents expressions
useful in evaluating the gain of a double-stream amplifier having thin con-
centric electron streams of different velocity and input and output gaps
across which both streams pass.
Direct Voltage Performance Test for Capacitor Paper }^] H. A. Sauer and
D. A. McLean. Performance of capacitors on accelerated life test may vary
over a wide range depending upon the capacitor paper used. Indeed, at
present a life test appears to be the only practical means for evaluating
■^ Proc. I. R. E., V. 37, pp. 852-85.S, Augu?t 1949.
^ Acous. Soc. Amer. Jour., v. 21, pp. 471-481, September 1949.
' Proc. I. R. E., V. 37, pp. 980-985, Septeml)er 1949.
'« Proc. I. R. E., V. 37, pp. 927-931, August 1949.
t A reprint of this article may be obtained by writing to the Editor of the Bell System
Technical Journal.
ABSTRACTS OF TECHNICAL ARTICLES 145
capacitor paper, since, within the Umits observed in commercial material,
the chemical and physical tests usually made do not correlate with life.
Lack of correlation is ascribed to obscure physical factors which have not
yet been identified.
Generally, several weeks are required to evaluate a paper by life tests of
the usual severity. Unfortunately, the duration of these tests is too long for
quaUty control of paper.
The desire for a life test which requires no more than a day or two for
evaluation led to the development of a rapid d-c. test. The philosophy of
rapid life testing is based upon the experimental evidence that the process
of deterioration under selected temperature and voltage conditions is prin-
cipally of a chemical nature, and also upon the well-known fact that rates
of chemical reaction increase exponentially with temperature.
Life tests on two-layer capacitors conducted at 130°C. provide an ac-
celeration in deterioration many fold more than that obtained in the lower-
temperature life tests, and correlate well with these tests.
Contributors to this Issue
Sidney Darlington, Harvard University, B.S. in Physics, 1928; Massa-
chusetts Institute of Technology, B.S. in E.E., 1929; Columbia University,
Ph.D. in Physics, 1940. Bell Telephone Laboratories, 1929-. Dr. Darlington
has been engaged in research in applied mathematics, with emphasis on
network theory.
Richard C. Eggleston, Ph.B., 1909 and M.F., 1910, Yale University;
U. S. Forest Service, 1910-1917; Pennsylvania Railroad, 1917-1920; First
Lieutenant, Engineering Div., Ordnance Dept., World War I, 1918-1919.
American Telephone and Telegraph Company, 1920-1927; Bell Telephone
Laboratories, 1927-. Mr. Eggleston has been engaged chiefly with problems
relating to the strength of timber and with statistical investigations in the
timber products field.
J. R. Pierce, B.S. in Electrical Engineering, California Institute of
Technology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936-. En-
gaged in study of vacuum tubes.
S. O. Rice, B.S. in Electrical Engineering, Oregon State College, 1929;
California Institute of Technology, 1929-30, 1934-35. Bell Telephone Lab-
oratories, 1930-. Mr. Rice has been concerned with various theoretical
investigations relating to telephone transmission theory.
J. M. Richardson, B.S., California Institute of Technology, 1941; Ph.D.,
Cornell, 1944. Bell Telephone Laboratories, 1945-49. Dr. Richardson at
these Laboratories had been mainly associated with studies of ferroelectric
materials, noise contacts, and contact erosion. At present he is with the
Bureau of Mines at Pittsburgh.
116
&
VOLUME XXIX APRIL, 1950 no. 2
Kansas. City, jyio
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
Error Detecting and Error Correcting Codes
R. W. Hamming 147
Optical Properties and the Electro-optic and Photo-
elastic Effects in Crystals Expressed in Tensor
Form W. P. Mason 161
Traveling-Wave Tubes [Second Installment] J. R. Pierce 189
Factors Affecting Magnetic Quality R,M. Bozorth 251
Technical Articles by Bell System Authors Not Appear-
ing in the Bell System Technical Journal 287
Contributors to this Issue 294
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The Bell System Technical Journal
Vol. XXVI April, 1950 No. 2
Copyright, 1950, American Telephone and Telegraph Company
Error Detecting and Error Correcting Codes
By R. W. HAMMING
1. Introduction
THE author was led to the study given in this paper from a considera-
tion of large scale computing machines in which a large number of
operations must be performed without a single error in the end result. This
problem of "doing things right" on a large scale is not essentially new; in a
telephone central office, for example, a very large number of operations are
performed while the errors leading to wrong numbers are kept well under
control, though they have not been completely eliminated. This has been
achieved, in part, through the use of self-checking circuits. The occasional
failure that escapes routine checking is still detected by the customer and
will, if it persists, result in customer complaint, while if it is transient it will
produce only occasional wrong numbers. At the same time the rest of the
central office functions satisfactorily. In a digital computer, on the other
hand, a single failure usually means the complete failure, in the sense that
if it is detected no more computing can be done until the failure is located
and corrected, while if it escapes detection then it invalidates all subsequent
operations of the machine. Put in other words, in a telephone central office
there are a number of parallel paths which are more or less independent of
each other; in a digital machine there is usually a single long path which
passes through the same piece of equipment many, many times before the
answer is obtained.
In transmitting information from one place to another digital machines
use codes which are simply sets of symbols to which meanings or values are
attached. Examples of codes which were designed to detect isolated errors
are numerous; among them are the highly developed 2 out of 5 codes used
extensively in common control switching systems and in the Bell Relay
147
148 BELL SYSTEM TECHNICAL JOURNAL
Computers/ the 3 out of 7 code used for radio telegraphy ,2 and the word
count sent at the end of telegrams.
In some situations self checking is not enough. For example, in the Model
5 Relay Computers built by Bell Telephone Laboratories for the Aberdeen
Proving Grounds/ observations in the early period indicated about two
or three relay failures per day in the 8900 relays of the two computers, repre-
senting about one failure per two to three million relay operations. The self-
checking feature meant that these failures did not introduce undetected
errors. Since the machines were run on an unattended basis over nights and
week-ends, however, the errors meant that frequently the computations
came to a halt although often the machines took up new problems. The
present trend is toward electronic speeds in digital computers where the
basic elements are somewhat more reliable per operation than relays. How-
ever, the incidence of isolated failures, even when detected, may seriously
interfere with the normal use of such machines. Thus it appears desirable
to examine the next step beyond error detection, namely error correction.
We shall assume that the transmitting equipment handles information
in the binary form of a sequence of O's and I's. This assumption is made
both for mathematical convenience and because the binary system is the
natural form for representing the open and closed relays, flip-flop circuits,
dots and dashes, and perforated tapes that are used in many forms of com-
munication. Thus each code symbol will be represented by a sequence of
O's and I's.
The codes used in this paper are called systematic codes. Systematic codes
may be defined^ as codes in which each code symbol has exactly n binary
digits, where m digits are associated with the information while the other
k = n — m digits are used for error detection and correction. This produces
a redundancy R defined as the ratio of the number of binary digits used to
the minimum number necessary to convey the same information, that is,
R = n/m.
This serves to measure the efficiency of the code as far as the transmission
of information is concerned, and is the only aspect of the problem discussed
in any detail here. The redundancy may be said to lower the effective channel
capacity for sending information.
The need for error correction having assumed importance only recently,
very little is known about the economics of the matter. It is clear that in
' Franz Alt, "A Bell Telephone Laboratories' Computing Machine" — I, II. Mathe-
matical Tables and Other Aids to Computation, Vol. 3, pp. 1-13 and 60-84, Jan. and
Apr. 1948.
* S. Sjjarks, and R. G. Kreer, "Tape Relay System for Radio Telegraph Operation,"
/2.C./1. /^m'ew, Vol. 8, pp. 393-426, (especially p. 417), 1947.
' In Section 7 this is shown to be equivalent to a much weaker appearing definition.
ERROR DETECTING AND CORRECTING CODES 149
using such codes there will be extra equipment for encoding and correcting
errors as well as the lowered effective channel capacity referred to above.
Because of these considerations applications of these codes may be expected
to occur first only under extreme conditions. Some typical situations seem
to be:
a. unattended operation over long periods of time with the minimum of
standby equipment.
b. extremely large and tightly interrelated systems where a single failure
incapacitates the entire installation.
c. signaling in the presence of noise where it is either impossible or un-
economical to reduce the effect of the noise on the signal.
These situations are occurring more and more often. The first two are par-
ticularly true of large scale digital computing machines, while the third
occurs, among other places, in "jamming" situations.
The principles for designing error detecting and correcting codes in the
cases most likely to be applied first are given in this paper. Circuits for
implementing these principles may be designed by the application of well-
known techniques, but the problem is not discussed here. Part I of the paper
shows how to construct special minimum redundancy codes in the follow-
ing cases:
a. single error detecting codes
b. single error correcting codes
c. single error correcting plus double error detecting codes.
Part II discusses the general theory of such codes and proves that under
the assumptions made the codes of Part I are the "best" possible.
PART I
SPECIAL CODES
2. Single Error Detecting Codes
We may construct a single error detecting code having n binary digits
in the following manner: In the first n — \ positions we put n — \ digits of
information. In the w-th position we place either 0 or 1, so that the entire n
positions have an even number of I's. This is clearly a single error detecting
code since any single error in transmission would leave an odd number of
I's in a code symbol.
The redundancy of these codes is, since m = n — 1,
i? = -^ = 1 + ^
w — 1 n — \'
It might appear that to gain a low redundancy we should let n become very
large. However, by increasing w, the probability of at least one error in a
150 BELL SYSTEM TECHNICAL JOURNAL
symbol increases; and the risk of a double error, which would pass unde-
tected, also increases. For example, if /> « 1 is the probability of any error,
then for ;/ so large as \/p, the probability of a correct symbol is approxi-
mately l/e = 0.3679 . . . , while a double error has probability l/2e =
0.1839
The type of check used above to determine whether or not the symbol
has any single error will be used throughout the paper and will be called
a parity check. The above was an even parity check; had we used an odd
number of I's to determine the setting of the check position it would have
been an odd parity check. Furthermore, a parity check need not always
involve all the positions of the symbol but may be a check over selected posi-
tions only.
3. Single Error Correcting Codes
To construct a single error correcting code we first assign m of the n avail-
able positions as information positions. We shall regard the m as fixed, but
the specific positions are left to a later determination. We next assign the k
remaining positions as check positions. The values in these k positions are
to be determined in the encoding process by even parity checks over selected
information positions.
Let us imagine for the moment that we have received a code symbol, with
or without an error. Let us apply the k parity checks, in order, and for each
time the parity check assigns the value observed in its check position we
write a 0, while for each time the assigned and observed values disagree
we write a \. When written from right to left in a line this sequence of k O's
and I's (to be distinguished from the values assigned by the parity checks)
may be regarded as a binary number and will be called the checking number.
We shall require that this checking number give the position of any single
error, with the zero value meaning no error in the symbol. Thus the check
number must describe m -\- k -\- \ different things, so that
2" >m+k+\
is a condition on k. Writing n = m -\- k we find
2"
2'" <
n -\- 1
Using this inequality we may calculate Table T, which gives the maximum
m for a given w, or, what is the same thing, the minimum ;/ for a given m.
We now determine the positions over which each of the various parity
checks is to be applied. The checking number is obtained digit by digit,
from right to left, by applying the parity checks in order and writing down
the corresponding 0 or 1 as the case may be. Since the checking number is
ERROR DETECTING AND CORRECTING CODES
151
Table I
n
m
Corresponding k
1
0
1
2
0
2
3
1
2
4
1
3
5
2
3
6
3
3
7
4
3
8
4
4
9
5
4
10
6
4
11
7
4
12
8
4
13
9
4
14
10
4
15
11
4
16
11
5
Etc.
to give the position of any error in a code symbol, any position which has
a 1 on the right of its binary representation must cause the first check to
fail. Examining the binary form of the various integers we find
1 = 1
3 = 11
5 = 101
7 = 111
9 = 1001
Etc.
have a 1 on the extreme right. Thus the first parity check must use positions
1,3,5,7,9, ••• .
In an exactly similar fashion we find that the second parity check must
use those positions which have I's for the second digit from the right of their
binary representation,
2 = 10
3 = 11
6 = 110
7 = 111
10 = 1010
11 = 1011
Etc.,
152
BELL SYSTEM TECHNICAL JOURNAL
the third parity check
4
=
100
5
=
101
6
=
110
7
=
111
12
=
1100
13
=
1101
14
=
1110
15
=
nil
20
=
10100
Etc.
It remains to decide for each parity check which positions are to contain
information and which the check. The choice of the positions 1, 2, 4, 8, • • •
for check positions, as given in the following table, has the advantage of
making the setting of the check positions independent of each other. All
other positions are information positions. Thus we obtain Table II.
Table II
Check Number
Check Positions
Positions Checked
1
2
3
4
1
2
4
8
1,3,5,7,9, 11, 13, 15, 17,- • •
2,3,6,7, 10, 11, 14, 15, 18,- • •
4,5,6,7, 12, 13, 14, 15, 20,---
8,9, 10, 11, 12, 13, 14, 15, 24,---
As an illustration of the above theory we apply it to the case of a seven-
position code. From Table I we find for n = 7, w = 4 and k = ?>. From
Table II we find that the first parity check involves positions 1, 3, 5, 7 and
is used to determine the value in the first position; the second parity check,
positions 2, 3, 6, 7, and determines the value in the second position; and
the third parity check, positions 4, 5, 6, 7, and determines the value in posi-
tion four. This leaves positions 3, 5, 6, 7 as information positions. The results
of writing down all possible binary numbers using positions 3, 5, 6, 7, and
then calculating the values in the check positions 1, 2, 4, are shown
in Table III.
Thus a seven-position single error correcting code admits of 16 code sym-
bols. There are, of course, 1' — 16 = 112 meaningless symbols. In some ap-
plications it may be desirable to drop the first symbol from the code to
avoifl the all zero combination as either a code symbol or a code symbol plus
a single error, since this might be confused with no message. This would still
leave 15 useful code symbols.
ERROR DETECTING AND CORRECTING CODES
153
Table III
Position
Decimal Value of
Symbol
2
3
4
5
6
7
0
0
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
0
2
0
0
0
0
1
3
0
0
1
0
0
4
0
1
0
0
1
0
5
1
0
0
1
1
0
6
0
0
0
1
1
7
1
0
0
0
0
8
0
0
0
0
9
0
0
1
0
10
0
1
0
0
1
11
0
1
1
0
0
12
0
0
1
0
13
0
0
0
1
1
0
14
1
1
1
15
As an illustration of how this code "works" let us take the symbol
0 11110 0 corresponding to the decimal value 12 and change the 1 in
the fifth position to a 0. We now examine the new symbol
0 1110 0 0
by the methods of this section to see how the error is located. From Table II
the first parity check is over positions 1, 3, 5, 7 and predicts a 1 for the first
position while we find a 0 there ; hence we write a
1 .
The second parity check is over positions 2, 3, 6, 7, and predicts the second
position correctly; hence we write a 0 to the left of the 1, obtaining
0 1 .
The third parity check is over positions 4, 5, 6, 7 and predicts wrongly; hence
we write a 1 to the left of the 0 1, obtaining
10 1.
This sequence of O's and I's regarded as a binary number is the number 5;
hence the error is in the fifth position. The correct symbol is therefore ob-
tained by changing the 0 in the fifth position to a 1.
4. Single Error Correcting Plus Double Error Detecting Codes
To construct a single error correcting plus double error detecting code we
begin with a single error correcting code. To this code we add one more posi-
154 BELL SYSTEM TECHNICAL JOURNAL
tion for checking all the previous positions, using an even parity check. To
see the operation of this code we have to examine a number of cases:
1. No errors. All parity checks, including the last, are satisfied.
2. Single error. The last parity check fails in all such situations whether
the error be in the information, the original check positions, or the last
check position. The original checking number gives the position of the
error, where now the zero value means the last check position.
3. Two errors. In all such situations the last parity check is satisfied, and
the checking number indicates some kind of error.
As an illustration let us construct an eight-position code from the previous
seven-position code. To do this we add an eighth position which is chosen
so that there are an even number of I's in the eight positions. Thus we add
an eighth column to Table III which has:
Table IV
0
0
1
1
1
1
0
0
1
1
0
0
0
0
1
1
PART II
GENERAL THEORY
5. A Geometrical Model
When examining various problems connected with error detecting and
correcting codes it is often convenient to introduce a geometric model.
The model used here consists in identifying the various sequences of O's and
I's which are the symbols of a code with vertices of a unit //-dimensional
cube. The code points, labelled .v, y, z, ■ ■ • , form a subset of the set of all
vertices of the cube.
Into this space of 2" j^oints we introduce a dislcDicc, or, as it is usually
called, a metric, D{x, y). The delinition of the metric is based on the observa-
tion that a single error in a code point changes one coordinate, two errors,
two coordinates, and in general d errors produce a diflference in d coordinates.
ERROR DETECTING AND CORRECTING CODES 155
Thus we define the distance D{x, y) between two points x and y as the num-
ber of coordinates for which x and y are different. This is the same as the
least number of edges which must be traversed in going from x to y. This
distance function satisfies the usual three conditions for a metric, namely,
D(x, y) = 0 if and only if x = y
D(x, y) = D(y, x) > 0 iix ^ y
D{z, y) + D(y, z) > D(x, z) (triangle inequality).
As an example we note that each of the following code points in the three-
dimensional cube is two units away from the others,
0 0 1
0 1 0
1 0 0
111.
To continue the geometric language, a sphere of radius r about a point x
is defined as all points which are at a distance r from the point x. Thus, in
the above example, the first three code points are on a sphere of radius 2
about the point (1, 1, 1). In fact, in this example any one code point may be
chosen as the center and the other three will lie on the surface of a sphere
of radius 2.
If all the code points are at a distance of at least 2 from each other, then it
follows that any single error will carry a code point over to a point that is
not a code point, and hence is a meaningless symbol. This in turn means that
any single error is detectable. If the minimum distance between code points
is at least three units then any single error will leave the point nearer to the
correct code point than to any other code point, and this means that any
single error will be correctable. This type of information is summarized in
the following table:
Table V
Minimum
Distance
Meaning
uniqueness
single error detection
single error correction
single error correction plus double error detection
double error correction
Etc.
Conversely, it is evident that, if we are to effect the detection and correc-
tion listed, then all the distances between code points must equal or exceed
the minimum distance listed. Thus the problem of finding suitable codes is
156 BELL SYSTEM TECHNICAL JOURNAL
the same as that of finding subsets of points in the space which maintain at
least the minimum distance condition. The special codes in sections 2, 3,
and 4 were merely descriptions of how to choose a particular subset of points
for minimum distances 2, 3, and 4 respectively.
It should perhaps be noted that, at a given minimum distance, some of
the correctability may be exchanged for more detectability. For example, a
subset with minimum distance 5 may be used for:
a. double error correction, (with, of course, double error detection).
b. single error correction plus triple error detection.
c. quadruple error detection.
Returning for the moment to the particular codes constructed in Part I
we note that any interchanges of positions in a code do not change the code
in any essential way. Neither does interchanging the O's and I's in any posi-
tion, a process usually called complementing. This idea is made more precise
in the following definition:
Definition. Two codes are said to be equivalent to each other if, by a finite
number of the following operations, one can be transformed into the other:
1. The interchange of any two positions in the code symbols,
2, The complementing of the values in any position in the code symbols.
This is a formal equivalence relation (~') since A '^ A\ A ^^ B implies
B '^ A\ and A ^^ B, B '~^ C implies A ^^ C. Thus we can reduce the study
of a class of codes to the study of typical members of each equivalence class.
In terms of the geometric model, equivalence transformations amount to
rotations and reflections of the unit cube.
6. Single Error Detecting Codes
The problem studied in this section is that of packing the maximum num-
ber of points in a unit w-dimensional cube such that no two points are closer
than 2 units from each other. We shall show that, as in section 2, 2"~ points
can be so packed, and, further, that any such optimal packing is equivalent
to that used in section 2.
To prove these statements we first observe that the vertices of the n-
dimensional cube are composed of those of two {n — l)-dimensional cubes.
Let A be the maximum number of points packed in the original cube. Then
one of the two (w — l)-dimensional cubes has at least A/2 points. This cube
being again decomposed into two lower dimensional cubes, we find that one
of them has at least A/2^ points. Continuing in this way we come to a two-
dimensional cube having A/2''~ points. We now observe that a square can
have at most two points separated by at least two units; hence the original
w-dimensional cube had at most 2"~^ points not less than two units apart.
ERROR DETECTING AND CORRECTING CODES 157
To prove the equivalence of any two optimal packings we note that, if
the packing is optimal, then each of the two sub-cubes has half the points.
Calling this the first coordinate we see that half the points have a 0 and half
have a 1. The next subdivision will again divide these into two equal groups
having O's and I's respectively. After {n — 1) such stages we have, upon re-
ordering the assigned values if there be any, exactly the first n — \ positions
of the code devised in section 2. To each sequence of the first n — \ coordi-
nates there exist n — \ other sequences which dififer from it by one co-
ordinate. Once we fix the n-ih. coordinate of some one point, say the origin
which has all O's, then to maintain the known minimum distance of two
units between code points the w-th coordinate is uniquely determined for all
other code points. Thus the last coordinate is determined within a comple-
mentation so that any optimal code is equivalent to that given in section 2.
It is interesting to note that in these two proofs we have used only the
assumption that the code symbols are all of length n.
7. Single Error Correcting Codes
It has probably been noted by the reader that, in the particular codes of
Part I, a distinction was made between information and check positions,
while, in the geometric model, there is no real distinction between the various
coordinates. To bring the two treatments more in line with each other we re-
define a systematic code as a code whose symbol lengths are all equal and
1. The positions checked are independent of the information contained
in the symbol.
2. The checks are independent of each other.
3. We use parity checks.
This is equivalent to the earlier definition. To show this we form a matrix
whose i-\h. row has I's in the positions of the i-th parity check and O's else-
where. By assumption 1 the matrix is fixed and does not change from code
symbol to code symbol. From 2 the rank of the matrix is k. This in turn
means that the system can be solved for k of the positions expressed in
terms of the other n — k positions. Assumption 3 indicates that in this
solving we use the arithmetic in which 1+1 = 0.
There exist non-systematic codes, but so far none have been found which
for a given n and minimum distance d have more code symbols than a sys-
tematic code. Section 9 gives an example of a non-systematic code.
Turning to the main problem of this section we find from Table V that a
single error correcting code has code points at least three units from each
other. Thus each point may be surrounded by a sphere of radius 1 with no
two spheres having a point in common. Each sphere has a center point and
158 BELL SYSTEM TECHNICAL JOURNAL
11 points on its surface, a total of « + 1 points. Thus the space of 2" points
can have at most:
n + 1
spheres. This is exactly the bound we found before in section 3.
While we have shown that the special single error correcting code con-
structed in section 3 is of minimum redundancy, we cannot show that all
optimal codes are equivalent, since the following trivial example shows that
this is not so. For « = 4 we find from Table I that m = \ and ^ = 3. Thus
there are at most two code symbols in a four-position code. The following
two optimal codes are clearly not equivalent:
0
0 0 0
and
0 0 0 0
1
1 1 1
0 111
8. Single Error Correcting Plus Double Error Detecting Codes
In this section we shall prove that the codes constructed in section 4 are
of minimum redundancy. We have already shown in section 4 how, for a
minimum redundancy code of ;/ — 1 dimensions with a minimum distance
of 3, we can construct an n dimensional code having the same number of
code symbols but with a minimum distance of 4. If this were not of minimum
redundancy there would exist a code having more code symbols but with
the same n and the same minimum distance 4 between them. Taking this
code we remove the last coordinate. This reduces the dimension from ;/ to
n — 1 and the minimum distance between code symbols by, at most, one
unit, while leaving the number of code symbols the same. This contradicts
the assumption that the code we began our construction with was of mini-
mum reduncancy. Thus the codes of section 4 are of minimum redundancy.
This is a special case of the following general theorem: To any minimum
redundancy code of N points in n — 1 dimensions and having a minimum
distance of 2^ — 1 there corresponds a minimum redundancy code of A^
points in n dimensions having a minimum distance of 2k, and conversely.
To construct the n dimensional code from the n — \ dimensional code we
simply add a single w-th coordinate which is fixed by an even parity check
over the n positions. This also increases the minimum distance by 1 for
the following reason: Any two points which, in the n — \ dimensional code,
were at a distance 2^—1 from each other had an odd number of differences
between their coordinates. Thus the parity check was set oppositely for the
two points, increasing the distance between them to 2k. The additional co-
ordinate could not decrease any distances, so that all points in the code are
now at a minimum distance of 2k. To go in the reverse direction we simply
ERROR DETECTING AND CORRECTING CODES 159
drop one coordinate from the n dimensional code. This reduces the minimum
distance of 2k to 2^ — 1 while leaving N the same. It is clear that if one
code is of minimum redundancy then the other is, too.
9. Miscellaneous Observations
For the next case, minimum distance of five units, one can surround each
code point by a sphere of radius 2. Each sphere will contain
1 + du, 1) + C(n, 2)
points, where C{n, k) is the binomial coefficient, so that an upper bound on
the number of code points in a systematic code is
2" 2"+^
1 + C{n, 1) + C{n, 2) n^ + « + 2
> T
This bound is too high. For example, in the case of n — 7, we find that
w = 2 so that there should be a code with four code points. The maximum
possible, as can be easily found by trial and error, is two.
In a similar fashion a bound on the number of code points may be found
whenever the minimum distance between code points is an odd number.
A bound on the even cases can then be found by use of the general theorem
of the preceding section. These bounds are, in general, too high, as the above
example shows.
If we write the bound on the number of code points in a unit cube of dimen-
sion n and with minimum distance d between them as B{ii, d), then the
information of this type in the present paper may be summarized as follows:
B{n, 1) = l""
Bin, 2) = r-'
Bin, 3) = 2™ <
2"
Bin, 4) = 2" <
I,
Bin - 1, 2/^ - 1) = Bin, 2k)
Bin, 2/^ - 1) = 2" <
n + 1
2«-i
1 + cin, 1) + . . . + c(«, k - i;
While these bounds have been attained for certain cases, no general
methods have yet been found for contructing optimal codes when the mini-
mum distance between code points exceeds four units, nor is it known
whether the bound is or is not attainable by systematic codes.
160 BELL SYSTEM TECHNICAL JOURNAL
We have dealt mainly with systematic codes. The existence of non-sys-
tematic codes is proved by the following example of a single error correcting
code with n — 6.
0 0 0 0 0 0
0 10 10 1
10 0 110
1110 0 0
0 0 10 11
111111.
The all 0 symbol indicates that any parity check must be an even one.
The all 1 symbol indicates that each parity check must involve an even num-
ber of positions. A direct comparison indicates that since no two columns
are the same the even parity checks must involve four or six positions. An
examination of the second symbol, which has three I's in it, indicates that
no six-position parity check can exist. Trying now the four-position parity
checks we find that
12 5 6
2 3 4 5
are two independent parity checks and that no third one is independent of
these two. Two parity checks can at most locate four positions, and, since
there are six positions in the code, these two parity checks are not enough
to locate any single error. The code is, however, single error correcting since
it satisfies the minimum distance condition of three units.
The only previous work in the field of error correction that has appeared
in print, so far as the author is aware, is that of M. J. E. Golay.'*
* M. J. E. Golay, Correspondence, Notes on Digital Coding, Proceedings of the LR.E.,
Vol. 37, p. 657, June 1949.
optical Properties and the Electro-optic and Photoelastic
Effects in Crystals Expressed in Tensor Form
By W. p. MASON
I. Introduction
THE electro-optic and photoelastic effects in crystals were first investi-
gated by Pockels,^ who developed a phenomenological theory for these
effects and measured the constants for a number of crystals. Since then not
much work has been done on the subject till the very large electro-optic
effects were discovered in two tetragonal crystals ammonium dihydrogen
phosphate (ADP) and potassium dihydrogen phosphate (KDP). With these
crystals light modulators can be obtained which work on voltages of 2000
volts or less. Their use has been suggested^ in such equipment as light valves
for sound on film recording and in television systems. Furthermore, since
the electro-optic effect depends on a change in the dielectric constant with
voltage, and the dielectric constant is known to follow the field up to 10^"
cycles, it is obvious that this effect can be used to produce very short light
pulses which may be of interest for physical investigations and for strobo-
scopic instruments of very high resolution. Hence these crystals renew an
interest in the electro-optic effect.
In looking over the literature on the electro-optic effect and photoelastic
effect in crystals, there do not seem to be any derivations that give them
in terms of thermodynamic potentials, which allow one to investigate the
condition under which equalities occur between the various electro-optic
and photoelastic constants. Hence it is the purpose of this paper to give such
a derivation. Another object is to give a derivation of Maxwell's equations
in tensor form, and to apply them to the derivation of the Fresnel ellipsoid.
The first sections deal with the optics of crystals, and derive the Fresnel
cHipsoid from Maxwell's equations. Other sections give a derivation of the
two effects, discuss methods for measuring them by determining the bi-
refrigence in various directions and give the constants for the two effects in
terms of crystal symmetries. The final section discusses the application of
the photoelastic effect for measuring strains in isotropic media.
' F. Pockels, Lehrbuck Der Kristalloptic, B. Teubner, Leipzig, 1906.
- See Patent 2,467,325 issued to the writer; "Light Modulation by P type Crystals,"
(ieorge D. Gotschall, Jour. Soc. Motion Picture Engineers, July, 1948, pp. 13-20; B. H.
Hillings, Jour. Opt. Soc. Am., 39, 797, 802 (1949).
161
162 BELL SYSTEM TECHNICAL JOURNAL
II. Solution of Maxwell's Equations In Tensor Form
In tensor notation, Maxwell's equations for a nonmagnetic medium with
no free charges take the form
1 dDi _ dHj 1 dHj _ dEk dDi _ . dHj _ . ,^.
V dt dx'k V dt dXi dXi dXj
where Dt is the electric displacement, H_, the magnetic field, Eu the electric
field, V the velocity of light in vacuo and ^ijk a tensor equal to zero when
i = j or k ox j — k, but equal to 1 or — 1 when all three numbers are different.
If the numbers are in rotation, i.e. 1, 2, 3; 2, 3, 1; 3, 1, 2 the value is +1
while, if they are out of rotation, the value is —1.
We assume the electric vector to be representable by a plane wave whose
planes of equal phase are taken normal to the unit vector »j . Then
£, = Eo.e^"^'""'"'"'^ (2)
where Eo^ are constants representing the maximum values of the field along
the three rectangular coordinates and 7 — v — 1. Substituting (2) in the
second of equations (1), noting that Eof. are not functions of the space co-
ordinates, we have
1 dHj Jiii r -, ji^U-XiUilv] ,2\
Tr^T = -~ [ejkiEo^mle . (3)
V dt V
Integrating with respect to the time
Hj — — [ejiciEoi^nile = Ha-e . (.4;
Hence,
^oy = - ka-£oi«i] (5)
V
and therefore the magnetic vector is normal to the plane determined by
£oi and Ui .
Next, using the first of equations (1),
dt dXk dXk
(6)
— [(ijh lit), iit\e
V '
Integrating with respect to time,
OPTICAL PROPERTIES IN CRYSTALS 163
Inserting the value of Hq. from (5), this equation takes the form
7-v r / c ^ 1 io>\t—xiniiv\
Lfi = r- Leiifc WAi -c-Oi WiMiJe
and, in general,
V
Di = -[ei]k{ejkiEkfii)nk\. (9)
V-
Kx])an(ling the inner parenthesis, we have the components
(£2^3 - £3^2)1; (£3«i - £1^3)2; {Eifh — E2ni)-i. (10)
Then
€,■^^-[(£2^3 — E-jvi); (E-iHi — Eiih); (EiUo — E2ni)]nk gives
A = — r [(-E3W1 — Eins)m — (£1^2 — £2«i)w2]
= [(£3^3 + £2^2 + Eini)ni - Ei{nl + «2 + th)]
V
Di = —-— [{Ein-2 — Eifi^fh — (£2^3 — £3^2)^3]
V'
(11)
= [(£3W3 + £2^2 + £iwi)«2 — £2(^1 + nl + nl)]
A = — "V [(-E2W3 — Ezn2)n-2 — (E^ni — Eins)ni]
= [(£3^3 + £2;i2 + EiHiJth — Ez{nl + lii + nl)].
Xow, since n\ -\- nl + ^3 = 1 because n is a unit vector, we have
V' V
Di ^ —- [Ei - {Ejnjjuil or — Di - Ei - (Ejnj)ni = 0. (12)
v^ V-
This equation states that Di , £, and «, are in the same plane, VLj being
normal to the plane as shown by Fig. 1. The energy flow vector
Si = —- eijkEjHk (13)
47r
also lies in the plane since it is perpendicular to £ and H. It is at the same
angle 6 with n that £ is with D. The velocity of energy flow is Vcos 6. The
energy velocity is called the ray velocity and the energy path the ray path.
Next, from the relation for a material medium, that
Di = KijEj or conversely £_,■ = ^jiDi (14)
164
BELL SYSTEM TECHNICAL JOURNAL
where Ka are the dielectric constants measured at optical frequencies and
/3yt are the impermeability constants determined from the relations
where
(15)
A^x
Kl2
Kn
A^ =
Ki2
K22
K23
Ku
K23
Ksz
and A^* the determinant obtained by suppressing the/ row and t* column,
we can eliminate Ei from equation (12) and obtain
2
Tr^D2 = ^12 Di + ^22 D2 + (32zDz
(Ej nj)n2
y^ Dz = pnDi + /323l>2 + 033 Dz - {EjnjW
This can be put in the form
{Ejnj)ni = D,\fin - i^l^X + ^12^ + ^izDz
(Ejnj)n2 = ^i2A + 0322 - v^/V')D2 + (SizDz
{Ejnj)nz = /3i3A + 1823^ + (^33 - v^/V')Ds.
Solving for Di , A and Dz
D, = [(^22 - t;VF2)(^33 - V^/V') - As][Ejnj\n,
D2 = [(^u - vyV'){^zz - vyV') - 0U[Eini\n2
Dz = [(^n - ^VF2)(^22 - vyV') - ^iME^nz-
Now, since D and n are at right angles,
A«i + D2ih + Dziiz = 0.
Hence,
0 = [(/322 - t'VF)(/333 - ^VF) - /3L3]«1
+ [(/3n - ^Vn(/333 - ^VF) - ^i3]"2
(16)
(17)
(18)
(19)
(20)
+ [(|Sn - t'Vn(^22 - vyV'~) - fnWz.
OPTICAL PROPERTIES IN CRYSTALS
165
Fig. 1 — Position of electric, magnetic and normal vectors for an electromagnetic plane
wave in a crystal.
Fig. 2 — Rotated axes and angles for relating them to unrotated axes.
liy choosing the original x, y, z axes so that /3i2 = /3i3 = 1S23 = 0 and using
the values jSn = |5i , ^ti = ^1 , 1833 = 1S3 this gives the equation
2 2 2
2 I 2 l~ 2
V ^ a ^
0.
(21)
For transmission along the X axis «i = \,rh = fh = 0 and the two velocities
are given by
^,2 = ^2^2 = b\ v' = /33F2 = c\ (22)
166 BELL SYSTEM TECHNICAL JOURNAL
Similarly the third velocity v- = /3iF- = a- can also be used and equation
(21) reduces to
"' + tA-, + -A-, = 0. (23)
q2 — ^2 ^2 — ,y2
This is a quadratic equation for the velocities v in terms of the principal
velocities a, b and c which are usually taken so that a > b > c.
Solving for the velocities, we obtain the quadratic equation
V* - v''[nl(b^ + c2) + nlia'^ + c'~) + nl{a' + 6^)]
2. 0 ^ ' (24)
+ nib~c~ + «2a'C" + Wsfl^^- = 0.
Letting L = ni(b- — c~), M = ihic- — a?), N = nl(a- — b~) the solutions
for the velocities become
Iv' = ni{b + c~) + Jioic" + a") + n^ia" + ft")
/ . (25)
This equation can be put into a simpler form if we change to the coordinate
system shown by Fig. 2. Here the rotated system is related to the original
system by three angles 9, (p, \f/. 6 is the angle between the Z axis and the
Z axis, <p is the angle the plane containing Z and Z makes with the X axis
while xj/ represents a rotation of the primed coordinate systems about the
Z axis. The direction cosines for the primed system with respect to the
normal system are designated by the matrix
(26)
where, in terms of 0, (f and \p, these direction cosines are,
£i — cos 6 cos (p cos \p — sin (p sin \p,
nil — cos 6 sin (p cos xp -\- cos (p sin \p, «i = — sin 6 cos xp
ti = —cos 6 cos <p sin \p — sin (p cos \p,
m-i = cos (p cos \p — sin v? sin xp cos 0, «2 = sin 6 sin i/'
•^3 = cos (p sin ^, Mi = sin (^ sin 9, rh — cos ^. (27)
If we take Z' as the direction of the wave normal, then in equation (25)
ill = k , ih = m-i , Hz = «3
X
Y
Z
X'
(l
mi
ni
I"
l1
mi
ni
Z'
u
m-i
W3
OPTICAL PROPERTIES IN CRYSTALS 167
and the equation for the velocities becomes
2v~ = a'isin (p sin' 6 + cos"" 6) + ft"(cos" (p sin' 6 -\- cos' d) + c' sin' d
/(o2 - 62)2(cos2 0 cos'^ (^ + sin2 <py + 2(a2 - b-'){c^ - b'-)
^ y sin2 ^(cos2 d cos2 ^ _ gin^ ^) + (c^ - b^y sin* ^
(28)
A very elegant construction for the wave-velocities and the directions of
vibration is the Fresnel index ellipsoid. Consider the ellipsoid
aV + by- + ch'- =1 (29)
Then FresneP showed that, for any diametral plane perpendicular to the
wave normal, the two principal axes of the ellipse were the directions of the
two permitted vibrations, while the wave velocities were the reciprocals of
the principal semi-axes.
We wish to show now that the maximum and minimum values of the im-
permeability constants in a plane perpendicular to the direction of the
wave normal determine the directions of vibration and the values of the two
velocities. To show this we make use of the fact that /?,•,■ is a second rank
tensor and transforms according to the tensor transformation formula
0;, = p ^ ft, (,,0)
dXk dxt
where the partial derivatives are the direction cosines
dxi
— — c 1 .
dXi
dx,
— = «1
axi ''
dT2
dxs
dx'i
7- -fa,
dxi
dX2
= W2,
dX2
dXi
—- = W2
dXi
dx'i
dxi
dx'z
dXo
dx's
T- = W3
OXs
Expanding equation (30) the six transformation equations become
/3ll = <^l/3u + 2AWl/3i2 + 2A»l/3l3 + Wi/?22 + 2oti»ijS23 + "1/333
0'u = ('Mn + (dm, + w/2)/5l2 + (A«2 + w/2)/3i3 + mim^22
+ (niirh + WlW2)i323 + "l«2^33
+ ("1W3 + WiH3)/323 + "l"3iS33 (31)
' See for example 'Thotoelasticity," Coker and Filon, Caml)riclge University Press,
pages 17 and 18.
168 BELL SYSTEM TECHNICAL JOURNAL
^22 = fi^n + 2i2nhl3n + ^(^fh^n + ^2^22 + ^m^th^^z + nl^zz
/323 = (iCz^n + (4w3 + W2^3)iSi2 + (/2W3 + ^Jz)l3n + W2W31822
+ (W2W3 + n2mz)^23 + rhnz^zz
/333 = 4/3u + 2Czmil3i2 + 2fznz^n + W3/?22 + 2mznz^2z + ^3^33.
Now, if the axes refer to the axes of a Fresnel eUipsoid, /3i2 = /3i3 = 1823 = 0
and one of the impermeability constants for any direction, say 1833 , can be
expressed in the form
/333 = (l^i + ^3^2 + nl^z (32)
If r, which hes along Z' of Fig. 2, is the radius vector of the Fresnel ellipsoid,
then the direction cosines 4 , niz and M3 are
f X y z
^3 = -, mz = -, nz = -.
r r r
From equation (24) /3i = a''/V\ ^2 = byV\ jSj = cVF^ and equation (32)
becomes
2Tr2o' 2 2 I ,2 2 I 2 2 ^
Hence the square of the radius vector of the Fresnel ellipsoid is 1/F"j833
and the radius vector of the impermeability ellipsoid agrees with that of the
Fresnel ellipsoid. Hence, the directions of vibration can be determined from
the principal axes of the impermeability ellipsoid for any diametral plane.
When light transmission occurs along Z', the direction for maximum and
minimum impermeability can be obtained by evaluating jSn and deter-
mining the angle xp for which it has an extreme value. Inserting the direction
cosines d , mi and Wi from equation (27), we find
o' o r 2„ 2 2 , sin 2^ sin 21/^ cos ^ , .2 • 2 ."^
Pi\ = Pi cos 6 cos ^ cos xp — -~- + sm ^ sm xp
, Q \ 2 /J • 2 2 , , sin 2^7 sin 2i/' COS 0 , 2 . 2 ,
+ 182 cos 6 sm ^ cos \p -\- — —^ [- cos (p sm \p
+ 183 sin" 6 cos" yp.
03)
Differentiating with respect to \p and setting the resultant derivative equal
to zero, the value of }p that will satisfy the equation is given by
o / (182 — (3i) sin 2<p cos d
tan 2\p =
(/3i - 182) (cos2 d cos2 <p - sin2 <p) + (^83 - ^2) sin^ 6
{b — a) sin 2<p cos 6
(^2 - 62) (cos2 d cos2 <p - sin2 <p) + {c^ - b^) sin^ d '
OPTICAL PROPERTIES IN CRYSTALS 169
For a given value on the right-hand side there are two values of \p, 90° apart,
that will satisfy the equation and hence we have two directions of vibration
at right angles to each other. Inserting (34) in {?>?)) the values of /3u and
/3ii for these two directions are
2|5ii = /3i(sin- ip sin'' d-\- cos- 6) + /32(cos- <p sin- d + cos- 6) + /33 sin- 6
, , /(/3i - 182)2 (cos2 d cos2 ^ + sin2 ^y + 2(/3i - ^2)(^3 - ^2)
=*" y • sin2 ^ (cos2 0 cos2 <p - sin^ <p) + (^3 - ^^Y sin^ d.
Since /3i corresponds to a^, etc., this equation agrees with the two velocities
given in equation (28) and shows that the directions of vibration correspond
with the maximum and minimum values of /3u .
It can also be shown that the two directions of electric displacement co-
incide with the two values of i// given by equation (34). Transforming the
electrical displacements to the X\ Y', Z' set of axes we have
d[ = pD,i- p D2 -i-pD, = hD, + m,D, -f mD^
dxi 6X2 0X3
D2 = pD, + pD, + pD3 = l,D, + m,D, + n,D^ (35)
ox\ 0X2 ox?
Dz = pDr + P D2 -VpD^- UD, + mzD2 + nM.
dxi 0x2 dxz
Hence, inserting the values of A, A, D^ from equation (18), we find
D[ = IM?2 - /3n)(^3 - /3n) + mmzifix ~ ^'nWz - ^n)
+ nM^i - /3n)(^2 - iSn)
D', = (.M^2 - /3n)(/33 - ^n) + nvmzi&i - I3n)((3s - /3n)
+ fhtpM - ^n)(02 - /3ii)
Ds = (1(132 - ^n)(^3 - /3n) + ml(0i - /3n)(/33 - /3n)
+ «3(ft - /3n)^2 - ^11).
From equation (20) with /3i2 = 1813 = 1823 = 0, it is evident that the Dz com-
ponent vanishes and hence the two values of electric displacement lie in a
plane perpendicular to Z'. By inserting the values of jSn and the value of
\p found from equation (34) we find that A = 0 and hence the electric dis-
placement lies along the directions of the greatest value of (3n • Similarly,
from the second value of (3n , A vanishes and hence the second wave is per-
pendicular to the first and in the direction of the smallest value of jSn .
(36)
170 BELL SYSTEM TECHNICAL JOURNAL
III. Location of Optic Axes in a Crystal
When the expression in the radical of equation (28) vanishes the two
velocities are equal and an optic axis exists. Since the expression inside the
radical can be written
Ua' - h')(co^' dcos^ ^ + sin^so) - (b'- - c'-)sm^ 6]^
(37)
— 4(a- — b-) (c- — b-) sin- ^sin- tp = ()
then, since the square is always positive and since (a- — b-) > 0 and
(b^ — c^) > 0, the equation can vanish only if ^ = 0. But ^ = 0 indicates
that the two optic axes always lie in a plane perpendicular to the inter-
mediate velocity b. With (p = Q then the square vanishes when
If (a- — b'-} < ib- — c~) the value of the tan 6 is less than unity and the
crystal is called a positive crystal. For this case the two axes approach more
closely the Z axis having the velocity c than they do the X axis. If
(a^ — b^) > (b- — c~) the crystal is negative.
li a — b or b = c the crystal has a single optic axis and is respectively a
positive or negative uniaxial crystal. For the first case the two velocities
are given by
vi = a = b, V2 = Va- cos^ 6 -}- c- sin^ 6. (39)
The first velocity is that of the ordinary ray while that of the second is that
of the extraordinary ray. Since a > c, the ordinary ray will have a velocity
greater than the extraordinary ray except along the optic axis where they
are equal. Since c < a, the maximum axis for any ellipse, formed by inter-
secting the Fresnel ellipsoid at an angle to the optic axis, will lie in the plane
formed by the normal and the c axis and hence the direction of polarization
of the extraordinary ray will lie in the c, n plane. The polarization of the
ordinary ray will be perpendicular to this plane.
\i h = c the a axis is the optic axis and the velocities of the two rays are
again
vi = c and V2 = a (1-sin" 0cos" <p) + c"'(sin" ^cos' <p) (40)
Hence, when d— 90°, (p = 0°, the two velocities are equal and a is the optic
axis. In this case the velocity of the extraordinary ray is greater than that
of the ordinary ray except along the a axis, and the crystal is a negative
uniaxial crystal. The polarization of the extraordinary ray lies again in the
OPTICAL PROPERTIES IN CRYSTALS 171
plane of the normal and the optic axis while the ordinary ray is perpendicu-
lar to it.
IV. Derivation or the Electro-optic and Photoelastic Effects
In a previous paper and in the book "Piezoelectric Crystals and Their
Application to Ultrasonics", D. Van Nostrand, 1950, it was shown that the
electro-optic and photoelastic effects can be expressed as third derivatives
of one of the thermodynamic potentials. Probably the most fundamental
way of developing these properties is to express them in terms of the strains,
electric displacements and the entropy. For viscoelastic substances it has
been shown that the photoelastic effects are directly related to the strains.
In terms of the electric displacements, the electro-optic constants do not
vary much with temperature whereas, if they are expressed in terms of the
fields, the constants of a ferroelectric type of crystal such as KDP increase
many fold near the Curie temperature. The entropy is chosen as the funda-
mental heat variable, since most measurements are carried out so rapidly
that the entropy does not vary.
The thermodynamic potential which has the strains, electric displace-
ments and entropy as the independent variables is the internal energy U,
given by
dU = Tij dSij i- Em^ + & da (41)
where Sij are the strains, Tij the stresses, £„, the fields. Dm the electric dis-
placements, 0 the temperature and a the entropy. In this equation the
strains Sa are defined in the tensor form
1 /dUi dtij\
2 \dXj dXiJ
2 \dXj dXiJ
where the m's are the displacements along the three axis. In the case of a
shearing strain occurring when i ^ j, the strain is only half that usually
used in engineering practice. In order to avoid writing the factor l/47r, we
use the variable 5„,= Dm/^T. Then, from (41),
Since, for most conditions of interest, adiabatic conditions prevail, we can
set dcr equal to zero and can develop the dependent variables, the fields and
■* "First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor
Form," W. P. Mason, B.S.T.J., Vol. 26, pp. 80-138, Jan., 1947.
172
BELL SYSTEM TECHNICAL JOURNAL
the stresses in terms of the independent variables, the strains and the elec-
tric displacements. Up to the second derivatives, these are
_ dEm o, . dEm »
do ,7 dOn
+
Tkt =
+
1
dTrd
d'Em O O r ^^ E,m r- ^ r d" Em
.00,7 do gr OOijOOn OOnOOo
+
(44)
1 r rn-^
2!L55iy55
dSij d8n
d'Tkf
8ndo
+
For the electro-optic and photoelastic cases, the two tensors of interest are
dbn d8o
d^Em
d'U
_ d En
dSkid8nd8o dSk(d8o
= Airmkinc
d'U
(45)
= (4Tr) fmno.
d8n d8o d8m d8n d8o
For the first partial derivatives, we have the values
dTjd
dSij
CijkC J
dTjd
d8n
dEm
d8n
d^U
_ dEn
dSki d8n dSkC
'iTrl3mn
= -hnkl
(46)
where cljkt are the elastic stiffnesses measured at constant electric displace-
ment, hnki are the piezoelectric constants that relate the open circuit voltages
to the strains, and ^mn are the impermeability constants measured for con-
stant strain.
With these substitutions and neglecting the other second partial deriva-
tives, we have, from (44),
■tLm — •^m ij 'J ij I •'>' n
Tkt = c^ijklSij + D
s
+
_hoke . mklon D
17 2
4
(47)
This equation shows that there is a relation between the change in the im-
permeability constant due to stress in the first equation, and the electro-
strictive constant in the second equation through the tensor nijjmn ■ These
OPTICAL PROPERTIES IN CRYSTALS 173
effects, however, have to be measured at the same frequency before equahty
exists.
To obtain the changes in the optical properties caused by the strain and
the electric displacement we have to determine the fields and displacements
occurring at the high frequencies of optics. Even for piezoelectric vibrations
occurring at as high frequencies as they can be driven by the piezoelectric
effect, these frequencies are small compared to the optic frequencies / and
can be considered to be static displacements or strains. Hence, writing
Em = Eli- Eme^-", Dn = dI + D^e^"',
w
here co = lirf, the first of equation (47) can be written in the form
Jlim — Umij >^ ij 1" J-^ n
Ptmi r Wijmn •J ij I n
jat
no x^O I
(48)
Tf we develop one of the fields, say £i , this can be written in the form
E.e''^' = [^11 + mar^Sii + rnM + rn2Dl + r.uDWD.e'-"
+ [^12 + MimSij + n.M + ruoDl + ri2zDl]D2e'''' (49)
+ [/3i3 + MimSij + r^^lDl + rn2Dl + ruzDl]D^e''''
where the first number of r refers to the field, the second to the optical value
of D and the third to the static value of D. Hence, for the general case,
-Em^^"' = Dne^'^^[(3mn + mijmnSij + rmnoE>^. (50)
From the definition of the two tensors niijno and r,„„o given by equation
(45), we can show that there are relations between the various components
of the tensors. For the first tensor niijno , since Sij = Sji is a symmetrical
tensor, then
niijno ^ nijino \p^)
r) / ri^ TI \
dSij \d8n d8o/
it is obvious that we can interchange the order of 5„ and 80 so that
niiino ^^ niijon
174 BELL SYSTEM TECHNICAL JOURNAL
Since ij and no are reversible, it has been customary to abbreviate the tensor
by writing one number in place of the two in the following form:
11 = 1; 22 = 2; 33 = 3; 12 = 21 = 6; 13 = 31 = 5; 23 = 32 = 4 (52)
Since the reduced tensor is associated with the engineering strains, it is
necessary to investigate the numerical relationships between the four in-
dex symbols and the two index symbols. From equation (48), when m
7^ n, the change in the impermeability constant i8,„„ is given by
Since Sr = 2Sij = 2Sji we have the relation that
tnijmn = mrs(i,j, fti, u = I to 3, f, s, = I to 6) (54)
In equation (45) we cannot in general interchange the order of ij and no
since U does not contain product terms of strains and electric displace-
ments and hence in general
Mrs 7^ nisr. (55)
Hence in the most general case there are 36 photoelastic constants. Crystal
symmetries cut down the number of constants as shown in a later section.
The tensor r„,„o defined in equation (45) as
pfi jj
(47r)V^„<, = -■ ,■ ,■ (56)
OOmOOn OOo
shows that we can interchange the order of m and n since U contains product
terms of bm and 6„ . Hence
'mno ' nmo I,*-' ' /
and this is usually replaced by the two index symbols
rqo = r,nno(ni, n, 0 = I to 3; q = I to 6).
The so called "true" electro-optic constants are measured at constant
strain and for this case the modifications in the impermeability constants
are given by the equation
Em = DnWmn + fmnoDo]. (58)
Since m and // are interchangeable, the third rank tensor is usually replaced
by the two index symbols
rtino == rqoini, n, o = t to 3; q = \ to 6). (59)
As discussed in the next sections, these constants can be determined by
applying an electric field of a frequency high enough so that the principal
resonances and their harmonics cannot be excited by the applied field, and
measuring the resulting birefringence along definite directions in the crystal.
On the other hand if we apply a static field to the crystal, an additional effect
occurs because the crystal is strained by the piezoelectric effect and this
causes a photoelastic effect in addition to the "true" electro-optic effect. A
Em = Dne'
OPTICAL PROPERTIES IN CRYSTALS 175
better designation for these effects is the electro-optic effect at constant
strain and stress.
This latter effect can be calculated from equation (47) by setting the
stresses Tkt equal to zero and eliminating the Sij strains. After neglecting
second order corrections,
„S lis I fnijmnnoki\ t^O { AC\\
?mn + I r„,no + T, ] ^o\- loUj
Since houdcljkt = gon, the other piezoelectric constant relating the open
c ircuit voltage to the stress, the electro-optic effect at constant stress can be
written in the form
T _ .S , mijmn goij /^|N
• mno ' mno 1^ . • \^' ' /
47r
Tn terms of the two index symbols
r; = rto + ''-^ (62)
47r
since it has been shown^ that goo = gop/2 when i 9^ j, and the tensor in (61)
has ij as common symbols which involves the summations of two terms.
The electro-optic effect is usually measured in terms of an applied field.
The change in the impermeability constant (3mn for this case can be de-
termined from the first equations (47), setting Tkf equal to zero and neglect-
ing second order terms. Multiplying through by the tensor Kop of the di-
electric constants
Dl = ElKl, (63)
since the product Kopl3op = 1- Introducing this equation into (58) we have
Em = Dnl^L + rinpKopEl] = Dn[(3L + zLoEl]. (64)
where the new tensor 2^710 is equal to
S _ S T^T /z-rx
^mno 'mnp^op • V"'^/
In terms of the two index symbols
4o = rqpKop . (66)
in which the repeated index indicates a summation. The difference between
the electro-optical constant at constant stress expressed in terms of the field
and the electro-optical constant at constant strain is
T _ S 1^ fftijmngoij j^.T _ ^S , j ( (:ii\
■^mno — Zmno I ~, ^op — ■^mno "T ''lijmn '^ pij \^ > J
47r
since the piezoelectric constants dpn are related to the g constants by the
equation
d^,j = S^a^, (68)
47r
176 BELL SYSTEM TECHNICAL JOURNAL
In terms of two index symbols
zlo = 4o + mpqdopip, q = 1 to 6; 0 = 1 to 3) (69)
where a repeated index means a summation with respect to this index.
Finally the photoelastic effect is sometimes expressed in terms of the
stresses rather than the strains. As can be seen from equation (47), the new
set of constants is
TTpg = niprSrq (70)
where the Srg are the elastic compliances measured at constant electric dis-
placement.
V. Birefringence Along Any Direction In the Crystal and
Determination of the Electro-optic and
Photoelastic Constants
If we take axes along the Fresnel ellipsoid when no stress or field is ap-
plied to the crystal, the result of the electro-optic and photoelastic effects
is to change the impermeability constants by the values
(71)
^11 = /3i + Ai ; /322 = ^2 + As ; /333 = ^3 + A3
/323 = A4 ; |8i3 = As ; I3n = Ag
where
Ai = ZnEi + 212^2 + ZnEs -f mnSi + mi252 + Wi3'S'3 + muSi
A2 = Z21E1 -\- Z22Ei + ZnEz + moiSi + nhiSo + W23'S'3 + m^iSi
+ ^2555 + nh^i
A3 = ZziEi -f 232^2 + ZizEz + mziSl + W^32'S'2 + W33^3 + m^^i
A4 = ZiiEi + 242^2 + Z43-E3 + niiiSi + mioS2 + MizSz + ^44^4
A5 = Zf,iEi + 252^2 + 253-E3 + m^iSi 4- m52S2 + m^sSs + ni^iSi
-f- Ws&S's + WbeSe
Ae = ZuEi. + 262^2 + 263£3 + m^iSi + ^62^2 + m^zSz -f triuSi
+ nhbSb + nh^Se .
If we transmit light along the 2' axis which, as shown by Fig. 2, makes an
angle of 6 degrees with the 2 axis in a plane making an angle <p with the xz
plane, the birefringence can be calculated as follows: Keeping z' fixed and
rotating the other two axes about 2' by varying the angle ^, one light vector
(72)
OPTICAL PROPERTIES IN CRYSTALS
177
will occur when /3n is a maximum and the other when /3n is a minimum.
Using the transformation equations (31) and the direction cosines of (27),
we find that fin is given by the equations
Ju = fill cos 6 cos" (f cos" l/'
sin 2(p sin 2^ cos d , .2 . " ,
• + sm (f sm" \p
+ i3i2[sin 2(p cos 2\l/ — sin" ^ sin 2(p cos i/* + cos 6 sin 2i/' cos 2^]
+ iSi3[ — sin 26 cos ^ cos ;/' + sin ^ sin ^ sin 2\p\ (73)
to r 2.-2 2 , , cos ^ sin 2ip sin 2i/' , 2 • 2 , ~|
+ 1822 cos d sm ^ cos i/' + 1- cos (p s\n rj/ \
+ |823[ — sin 20 sin (y? cos" 4/ — sin 0 cos tp sin 2i/'] + fi^^^ sm 9 cos^ ^
sb'
Differentiating with respect to \p and setting — ~ = 0, we find an ex-
pression for tan 2\f/ in the form
tan 2\J/ =
— fill sin 2(p cos 6 + 2fii2 cos 6 cos 2^
+ 2(Si3 sin <p sin 9 -}- fioo cos ^ sin 2(p — 2/323 sin 6 cos (^
/3ii[cos" 0 cos (p — sin ^] + |8i2[(l + cos" 9) sin 2<p]
— fiu sin" 6 Ck s(p -\- /322(cos" ^ sin ^ — cos" (p)
— 1S23 sin 29 sin ^ + fiss sin^ ^
(74)
Inserting this value back in equation (73) we find that the two extreme values
of fill are given by the equation
2fi'ii = 2^22 + (fin - /322)(cos2 9 cos2 cp + sin2 <p) + 0333 - fi-i^) sin' 9
— ;Si2 sin^ 9 sin 2^ — fin sin 20 cos^ — 1823 sin 29 sin ^
±
1/
(/3ii - /322)'(cos2 9 cos2 ^ + sin2 cpY + 2(^ii - /?22)(/333 - ^22) sin2 9X
(cos2 9 cos2 ^ - sin2 <p) + (/333 - /322)' sin^ 9 - 2(fiii - fi2o)X
[(Si2(sin 2<p sin^ 0(cos"^ 9 cos^ ^ + sin- (p) -{- fin sin 20 cos ^X
(cos^ 0 cos <p -\- sinV) — fi2z sin 20 sin ^(1 + cos^ <p sin^ 0)]
+ 2(/333 — fe) sin^ 9\fii2 sin 2^(1 + cos- 0) — ;3i3 sin 20 cos (p
4 (75)
— ;323 sin 20 sin (p] + (2,Si2)2[sin 0 sin^ ^ cos- 93 -f cos- 0]
- 4i3i2i8i3sin2 0 sin^[cos2 0 cosV + sin"^ <p] - 4(^i2fe)
[sin 20 cos ^(sin- (p cos- 0+ cos- (p)] + (2/3i3)- sin- 0X
(cos^ 0 cos^ ^ + sin^ (p) — 4:fiizfi2z sin 2(p sin 0
-f (2/323)^ sin^ 0(cos- 0 sin^ (p + cos^ <p)
1 1
1
1
2 + 2 = Ai ;
2 ~
2 — '
V/iTo
Ml M2
Ml
M2
178 BELL SYSTEM TECHNICAL JOURNAL
The birefringence in any direction can be calculated from equation (75) ;
since (Sn = ■Vi/V, it equals l//ii where /zi is the index of refraction corre-
sponding to a light wave with its electric displacement in the (3'n direction.
Similarly, for the second solution at right angle to the first,
^11 = -o - -^ (76)
1 " M2
Hence if we designate the expression under the radical by ivo and half the
expression on the right outside the radical by Ki , we have
(77)
Since mi and fio are very nearly equal even in the most birefringent crystal,
we have nearly
3
M2 - Ml = ^ = y VkI . (78)
For special directions in the crystal, the expression for Ko simplifies very
considerably. Along the x, y and z axes, the values are
3
X, {<p = 0°,d = 90°) ; 5. = ^ V(/?3. - M' + (2,523)^
F, (^ = 90°, e = 90°); By^^ V(/3n - fe)^ + (2^^)' (79)
3
Z, {<p = 0°,d ^ 0°); B, ^^ V(/3ii - fe)^ + (2/3i2)2.
If any natural birefringence exists along these axes, (2/323)- will be very
small compared to this and
^x = - (^33 - /3o + A;i - Ao) = - ( - - -^ + ^, - A, )
/^, = -;^ (/3i - iS3 + Ai - A.) = - ( - - -1 + Ai - A J
Z Z \lJia fJ-c /
i?. = V (^1 - /32 + Ai - Ao) = - ( -2 - - + A, - A, ) .
Z Z \lla IJ-b /
Hence, for this case, measurements along the three axes will loll the ditTer-
ence between the three effects Ai , A^ and An . To get absolute values requires
a direct measurement of the index of refraction along one of the axes and
its change with fields or stresses. This is a considerably more difficult meas-
(80)
OPTICAL PROPERTIES IN CRYSTALS ll9
urement than a birefringence measurement and requires the use of an ac-
curate interferometer.
If, however, the Z axis is an optic axis as it is in ADP, for example, and
I Ai = A2 = 0, a birefringence occurs due to the term ^n . As shown in the
next section, the electro-optic constants for ADP (tetragonal 42w) are Zi\
and 063 • 063 occurs in the expression for /3i2 = Ae , as can be seen from equa-
tions (72), and hence the birefringence along the Z axis is
3
B^ = yx2/3i2 = nlzesEs. (81)
The constants ^es and Zn have been measured independently by W. L. Bond,
Robert O'B. Carpenter, and Hans Jaffe. Probably the most accurate meas-
urements, and the only one published, are those of Carpenter,^ who finds
that the indices of refraction and the ^63 and 2:41 constants for ADP and
KDP are in cgs imits
f^a Me ''63x10'' ^4jx10^
ADP 1.5254 1.4798 2.54 ± 0.05 6.25 ± 0.1
KDP 1.5100 1.4684 3.15 ± 0.07 2.58 ± 0.05
An even larger constant has been found for heavy hydrogen KDP by Zwicker
and Scherrer.® They find at 20°C that res = 6 X 10"'^. Using this constant, a
half wave retardation for a X = 5461 A° mercury line occurs for a voltage
of 4000 volts.
For tetragonal crystals of these types the only photoeleastic constant for
the z axis is mm , and the birefringence for this case is given by
Bz = MaWee^e (82)
When a natural birefringence exists for the crystal, measurements of the
other three effects A4 , As and Ae can be made by determining the bire-
fringence along other directions than the Fresnel ellipsoid axes. In a direction
of Z' lying in the XZ plane <p — 0, 9 — variable and
J, _ ^'\ /[(/3ii - 1322) cos2 e + (/333 - (822) sin2 6 - /3i3 sin^ d]^ /..n
"' ~ 2 y + [2i8i2 cos d + 2fe sin d]\ ^^-^^
When a natural birefringence exists, this reduces to
l'.y = ^ { -2 - -1 + Ai - A., ] cos' 9
2 L\Ma Mb /
+ (—-— + A3 - A2 ) sin' ^ - As sin 29
(---
I 2 — 2
Vc M6
_ (84)
5 "The Electro-optic Effect in Uniaxial Crystals of the Type XH!jP04 ," Robert O'B.
Carpenter, Jour. Opt. Soc. Am., in course of publication.
^Zwicker and Scherrer, Helv. Phys. Acta., 17, 346 (1944).
180
BELL SYSTEM TE,CHNICAL JOURNAL
and hence, by measuring at 45° between the two axes, one can evaluate the
As term.
Similarly, for the YZ plane, ip = 90°, 6 = variable and.
By
[-(/3ii - fe) + {8n - 6-n) sin2 d - fe sin 20]^
+ [2/3i2 cos d - 2/D,3 sin 6]'.
(85)
Hence, when a natural birefringence exists, we have
3
A.
l^vz
M
M - ^ + Ai - A.)
+ (-2 - -2 + A, - Ao)
2 + A, - A,
In the XF plane 6= 90°, ^ = variable and
B,
sni
— A4 si
n 2^
(S6)
Then, for natural birefringence.
^12) sin2 if — (/333 - 1S22) — /3i2 sin 2(pY
+ [2/5i3 sin v? — fe cos (^]2.
•Dxu
(A -A
.Va Mb
+ Ai - A
I sin"
I 2 — 2
\Mc M6
— ( ~ - "^ + A;; — A2 ) — Afi sin 2(p
'6
■)
(87)
(88)
Hence, with measurements at 45° between the axes and with suitably ap-
plied fields and strains, the three effects A4 , A5 and Ae can be measured.
Since the axes of the test specimen are turned with respect to the X, Y and
Z axes, suitable transformations of the effects Ai to Ae with respect to the
new axes will have to be made. These can be done as shown in reference (4)
by means of tensor transformation formulae.
Another method for measuring the constants in A4 , A5 , Ae is to measure
the amount they rotate the axes of the Fresnel ellipsoid. As an example con-
sider the 2:41 constant of ADP. For example, if we look along the X axis and
apply a field in the same direction, then, in equation (74), 6 = 90°, (p = 0 and
tan 2\J/ =
2/32
/333 - 13-2
1
ifJ-b + Mc)(m6 — Mc)
(89)
fie Mb
According to Car{)enter, the 241 electro-optic constant of ADP is 6.25 X 10"'''
in cgs units. Ha = iib = 1.5254; Mc = 1.4798; hence the angle of rotation for
a field of 30,000 volts per centimeter = 100 stat volts cm is
yp = —2.25 X 10^ radians = 7.7 minutes of arc. (90)
OPTICAL PROPERTIES IN CRYSTALS
181
VI. Electro-optic and Photoelastic Tensors for Various
Crystal Classes
Since r,„„o = r,„„„ and 2:,„„„ = Znmo are third rank tensors similar to the
//„, ij piezoelectric tensor, they will have the same components for the various
( rystal classes. For the twenty crystal classes that show the electro-optic
effect these tensors are given below. They are given with the crystal system
they belong to, and the symmetry is designated by the Hermann-Mauguin
symbol. The last number of the subscript of z designates the direction of the
applied static field.
(91)
Trichnic; 1
Monoclinic; 2
Monoclinic; 2 = m
( )rthorhombic; 222
' Orthorhomic; 2mm
'I'etragonal; 4
Zn
Zi\
231
241
251
261
Zn
Zii
232
242
252
262
Z\z
Zn
233
243
253
263
0
0
0
241
0
261
Zn
2^22
232
0
252
0
0
0
0
243
0
263
Z\\
Zoi
231
0
251
0
0
0
0
242
0
262
2i3
Z2Z
233
0
253
0
0
0
0
241
0
0
0
0
0
0
252
0
0
0
0
0
0
263
0
0
0
0
251
0
0
0
0
2 12
0
0
Zn
z-n
233
0
0
0
0
0
0
241
251
0
0
0
0
~25i
241
0
Zn
— 2^13
0
u
0
263
182
BELL SYSTEM TECHNICAL JOURNAL
Tetragonal; 4
0
0
0
241
261
0
0
0
0
251
-241
0
i
2l3
2l3
233
0
0
0
Tetragonal; 42m
0
0
0
241
0
0
0
0
0
0
241
0
0
0
0
0
0
263
Tetragonal; 422
0
0
0
241
0
0
0
0
0
0
-241
0
0
0
0
0
0
0
Tetragonal; 4mm
0
0
0
0
261
0
0
0
0
251
0
0
Ziz
Zu
233
0
0
0
Trigonal; 3
sn
-Zn
0
241
251
— 222
— 222
2-22
0
251
-241
-2ii
2l3
Zu
233
0
0
0
Trigonal; 32
Zn
—Zn
0
241
0
0
0
0
0
0
-241
— 2il
0
0
0
0
0
0
Trigonal; 3m
0
0
0
0
251
— 222
— Z22
222
0
251
0
0
2l3
2l3
233
0
0
0
Hexagonal; 6
2ll
—Zn
0
0
0
— 222
— Z22
Z22
0
0
0
-211
0
0
0
0
0
0
OPTICAL PROPERTIES IN CRYSTALS
183
Hexagonal; 6m2
Z\l
-Zn
0
0
0
0
0
0
0
0
0
-2ll
0
0
0
0
0
0
Hexagonal ; 6
0
0
0
241
261
0
0
0
0
^61
— 241
0
Zn
Zu
Zzz
0
0
0
Hexagonal; 622
0
0
0
241
0
0
0
0
0
0
-241
0
0
0
0
0
0
0
I lexagonal ; 6mm
0
0
0
0
251
0
0
0
0
251
0
0
Zn
Zn
Zzz
0
0
0
Cubic; 23 and 43m
0
0
0
241
0
0
0
0
0
0
241
0
0
0
0
0
0
241
The r tensor has similar terms.
The photoelastic constants are similar to the elastic constant tensors
except that utrs 9^ Msr in general. However, for the tetragonal, trigonal,
hexagonal and cubic systems, Pockels found that nin = Woi . This follows
from the transformation equations about the Z axis which is the n fold
axes for these groups. For a rotation of an angle 6 about Z, the direction
cosines are
,1 = — = cos 6
OXi
„ dX2
^2 = T— = -Sin
OXi
U = f^' = 0
OXi
dx\
mi = — = sin I
0x2
dXi
W2 = r — = cos
aooi
dxs
mz = —- = 0
0x2
dxi
wi = — - = 0
0x3
dX2 f.
«2 = T— = 0
OXz
dxz
«3 = ^— = 1
0X3
(92)
184
BELL SYSTEM TECHNICAL JOURNAL
(93)
(94)
Transforming the two terms mivi-i = Wri and Wnn = W21 by the tensor
transformation equation
_ dxi dXj dXk dx(
Wijh( ^~ ~ r r l^mnop
CvvyTi OXfi (jXq OXn
we find, for these two coefBcients,
W12 = (wii + m-22 — 4w66) sin- 6 cos- 6 + 2(w62 — ^le)
sin d cos' d + 2(w6i ~ Wie) sin ^ cos 6 -\- nin cos ^ + yrvn sin ^
W21 = (wii + W22 — 4w66) sin- ^ cos- 0 + 2(wi6 — W62)
sin B cos 0 + 2(w26 — Wei) sin 0 cos 6 -\- m^i cos 0 + W12 sin ^
If W12 = W21 for all angles of rotation we must have
W16 + nhe = W61 + nieo
For all the classes that W12 = nhi, either w^e = — Wie and m^o = — Wei or
else W16 = W26 = niei = m^o = 0.
Now, if Z is a four-fold axis, as it is in the tetragonal and cubic systems,
then, for a 90° rotation, the value of nin or W21 must repeat. From the first
of (92) this means that
W12 = W21 and nhi = niu
For a trigonal or hexagonal system additional relations are obtained between
mm and mn , ^^22 and mn in the usual manner. Hence the photoelastic matrices
become, for the various crystal classes,
(95)
Triclinic 36
Constant
mn
vtvi
m22
mn
;«23
?«14
;«24
W25
W16
;«j6
The TT ten-
sor is en-
tirely anal-
ntn
W32
W33
?W34
/«35
W36
ogous
niAi
WI42
'W43
J«44
w;45
»?46
mn
W62
W63
W64
;«56
W/66
mu
me,2
'«63
'«64
'/«65
>«66
Monoclinic
20 Con-
stants
mn
ni>i
»h2
Wl3
W2:i
0
0
nil'.,
1H2:,
0
0
The TT ten-
sor is en-
tirely anal-
m-n
W32
>«33
0
IH-jb
0
ogous
0
0
0
W44
0
w«
Wm
'"62
W/G3
0
»/[,5
0
0
0
0
W64
0
»»66
OPTICAL PROPERTIES IN CRYSTALS
185
Ortho-
rhombic 12
Constants
inn
0
0
0
nil' Wi3
nh-i "'33
0 0
0 0
0 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
'«66
The IT ten-
sor is en-
tirely anal-
ogous
Tetragonal
4,4,4/w9
Constants
mzi mu W33
0 0 0
0 0 0
Wei — mm 0
0
0
0
ntii
0
0
0
0
0
0
niu
0
'«16
0
0
0
W66
The IT ten-
sor is en-
tirely anal-
ogous
I'etragonal
42m, 422
■imm,
{\,'m)mm
7 Constants
mil
mi2
mn
mn
trill
miz
mn
W31
'«33
0
0
0
0
0
0
0
0
0
0
0
0
W44
0
0
0
0
0
0
W44
0
0
0
0
0
0
»M66
The IT ten-
sor is en-
tirely anal-
ogous
Trigonal
i,6 11 Con-
stants
mil mn mn
mn mil mn
msi WZ31 ms3
niii — '»4i 0
- >«52 "«62 0
0
0
0
-niu
0
?»44
- ;H45
'«2b
-W25
?«25
0
WZ45
OT44
Trigonal
wu
'»;2
mn
mn
0
M,im
,m/m) 8
?«I2
;«ii
mn
— mu
0
Constants
0
0
W31
'«31
mzi
;w4i
— Wkl
0
«?44
0
0
0
0
0
nui
0
The TT ten-
sor is anal-
0
0
W52
ogous ex-
cept that
X46 = 2ir52
7r66 = 2:r4i
^66 =
W41
(tTh — TTlo)
mn-mn
2
0
The TT ten-
sor is ana-
0
0
logous ex-
cept that
7r56 = 2X41
0
■T66 =
TTll — Xi2
0 0
mu
mn
mii — mn
186
BELL SYSTEM TECHNICAL JOURNAL
Hexagonal
6,6/«2,6
622, 6A«;
"'12
Wl2
nin
0
0
0
0
6mmAmM
'm
6 Constants
0
0
nizi
0
0
0
0
0
0
0
0
mu
0 0
0
0
0
0
0
ntn — niii
The TT ten-
sor is anal-
ogous ex-
cept that
Tree =
TTll — 7ri2
Cubic Sys-
tem 23,432
2 4 2
— 3,43w,— 3 —
m mm
3 Constants
Wll
mil
Wl2
mi2
nin
mii
m)2
mvi
Wli
0
0
0
0
0
0
0
0
0
0
0
0
mu
0
0
0
0
0
0
mu
0
0
0
0
0
0
mu
The TT ten-
sor is en-
tirely anal-
ogous
(95)
Isotropic
Systems 2
Constants
mn mi2 miz
mi2 mil Wi2
Wi2 mi2 mil
0
0 0
0 0 0
0 0 0
0
0
0
WU — Wi2
0
0
0
0
mn — mn
2
0
0
0
0
0
0
mn — mn
The IT ten-
sor is anal-
ogous ex-
cept that
T66 =
TTu — 7ri2
From measurement^ on the photoelastic effects at high pressure for cubic
crystals, it has become apparent that the second derivatives of equation
(44) are not sufficient to represent the experimental results and derivatives
up to the fourth power should be included. This extension, however, is not
considered in the present paper.
VII. Photoelasticity in Isotropic Media
The photoelastic effect in isotropic solids has been used extensively in
studying the stresses existing in machine parts and other pieces. For this
purpose a plastic model cut in the shape of the original is used and is loaded
in a similar manner to that of the machine part to be studied. Since stresses
are aj)plied, the tf, photoelastic constants are most useful. If we look along
"> H. B. Maris, Jour. Optical Society of Amer., Vol. 15, pp. 194-200, 1927.
OPTICAL PROPERTIES IN CRYSTALS 187
the Z axis, the last of equations (79) shows that the birefringence is equal
to
3
5. = I VCft + Ai - ^2 - ^,y + 4(A6)2 (96)
Since, for an isotropic substance /3i = /^2 , we have, after substituting the
\-alue of Ai and A2 , with the appropriate photoelastic constants from equa-
tion (95), (last tensor):
3
5. = I (tTu - TTis) V(7^] - 7^2)2 + 4r62 (97)
If we transform to axes rotated by an angle d about Z, the values of Tn
and r22 are given by
Tn = cos 207^1 + 2 sin 0 cos OT^ + sin^ dT^
Til = sin ""QTx — 2 sin 0 cos QT^ -f cos ^T<i,
If, now, we choose the angle Q so that Tn is a maximum, we find
(98)
tan IQ = -±^ (99)
-i 1 — -t 2
Inserting this value of tan 20 in (98) we find
t', = ^^"^ ^' - W{Ti - T,f + ^T,^
(100)
and, hence,
t[ - rj = ^y{T^ - T^y + 47^62 (101)
Hence the birefringence obtained in stressing a material is proportional to
the difference in the principal stresses. By observing the isoclinic lines of a
photoelastic picture, methods^ are available for determining the stresses
in a model. A photograph^ of a stressed disk is shown by Fig. 3. The high
concentration of lines near the surface shows that the shearing stress is
\ery high at these points. By counting the number of fines from the edge
and knowing the stress optical constant, the stress can be calculated at any
point.
If we apply a single stress Ti , the birefringence is given by the equation
3
Bz= ^ (tth - 7ri2)ri (102)
^ See Photoelasticity, Coker and Filon, Cambridge University Press, 1931.
^ This photograph was taken by T. F. Osmer.
188 BELL SYSTEM TECHNICAL JOURNAL
Instead of using the constants tth and ttu it is customary to use a single
constant C given by
B = ijie - fJio = r = CT (103)
where the constant C is called the relative stress optical constant and r the
retardation. The dimensions of C are the reciprocal of a stress and are
Fig. 3 — Photoelastic picture of a disk in compression.
measured in cm- per dyne. A convenient unit for most purposes is one of
10~^'* cmVdyne; if this is used, the stress optical coefficients of most glasses
are from 1 to 10 and most plastics are from 10 to 100. This unit so defined
has been called the "Brewster". In terms of the Brewster, the retardation is
r = CTd (104)
If C is measured in Brewstcrs, (/ in millimeters and 7' in l)ars (10^ dynes/
cm'-) then r, as given by the formula, is expressed in angstrom units,
Traveling-Wave Tubes
By J. R. PIERCE
Cop.vright, 1950, D. Van Nostrand Company, Inc.
[SECOND INSTALLMENT]
CHAPTER IV
FILTER-TYPE CIRCUITS
Synopsis of Chapter
SIDE FROM HELICES, the circuits most commonly used in traveling-
A^
wave tubes are iterated or filter-type circuits, composed of linear
arrays of coupled resonant slots or cavities.
Sometimes the geometry of such structures is simple enough so that an
approximate field solution can be obtained. In other cases, the behavior of
the circuits can be inferred by considering the behavior of lumped-circuit
analogues, and the behavior of the circuits with frequency can be expressed
with varying degrees of approximation in terms of parameters which can be
computed or experimentally evaluated.
In this chapter the field approach will be illustrated for some very simple
circuits, and examples of lumped-circuit analogues of other circuits will be
given. The intent is to present methods of analyzing circuits rather than
particular numerical results, for there are so many possible configurations
that a comprehensive treatment would constitute a book in itself.
Readers interested in a wider and more exact treatment of field solutions
are referred to the literature.^-
The circuit of Fig. 4.1 is one which can be treated by field methods. This
"corrugated waveguide" type of circuit was first brought to the writer's
attention by C. C. Cutler. It is composed of a series of parallel equally spaced
thin fins of height h projecting normal to a conducting plane. The case treated
is that of propagation of a transverse magnetic wave, the magnetic field
being parallel to the length of the fins. It is assumed that the spacing ( is
small compared with a wavelength. In Fig. 4.2, ^h is plotted vs. /3n//. Here /3
is the phase constant and /Jo = w/c is a phase constant corresponding to the
velocity of light.
1 E. L. Chu and W. W. Hansen, "The Theory of Disk-Loaded Wave Guides," Journal
of Applied Physics, Vol. 18, pp. 999-1008, Nov. 1947.
2L. Brillouin, "Wave Guides for Slow Waves," Journal of Applied Physics, Vol. 19,
l>p. 1023-1041, Nov. 1948.
189
190 BELL SYSTEM TECHNICAL JOURNAL
For small values of /3o//, that is, at low frequencies, very nearly |8 = /3o ;
that is, the phase velocity is very near to the velocity of light. The field
decays slowly away from the circuit. The longitudinal electric field is small
compared with the transverse electric field. In fact, as the frequency ap-
proaches zero, the wave approaches a transverse electromagnetic wave
traveling with the speed of light.
At high frequencies the wave falls ofif rapidly away from the circuit, and
the transverse and longitudinal components of electric field are almost equal.
The wave travels very slowly. As the wavelength gets so short that the
spacing / approaches a half wavelength (/3^ = tt) the simple analysis given
is no longer valid. Actually, ^( = tt specifies a cutoff frequency; the circuit
behaves as a lowpass filter.
Figure 4.3 shows two opposed sets of fins such as those of Fig. 4.1. Such
a circuit propagates two modes, a transverse mode for which the longi-
tudinal electric field is zero at the plane of symmetry and a longitudinal
mode for which the transverse electric field is zero at the plane of symmetry.
At low frequencies, the longitudinal mode corresponds to the wave on a
loaded transmission line. The fins increase the capacitance between the con-
ducting planes to which they are attached but they do not decrease the
inductance. Figure 4.6 shows ^h vs. /?o/^ for several ratios of fin height, //,
to half -separation, d. The greater is h/d, the slower is the wave (the larger
is /3//3o).
The longitudinal mode is like a transverse magnetic waveguide mode; it
propagates only at frequencies above a cutoff frequency, which increases
as h/d is increased. Figure 4.7 shows ^h vs. fioh = {<j}/c)h for several values
of h/d. The cutoff, for which ^C — tt, occurs for a value of ^qJi less than ir/l.
Thus, we see that the longitudinal mode has a band pass characteristic. The
behavior of the longitudinal mode is similar to that of a longitudinal mode of
the washer-loaded waveguide shown in Fig. 4.8. The circuit of Fig. 4.8 has
been proposed for use in traveling-wave tubes.
The transverse mode of the circuit of Fig. 4.3 can also exist in a circuit
consisting of strips such as those of Fig. 4.1 and an opposed conducting
plane, as shown in Fig. 4.5. This circuit is analogous in behavior to the disk-
on-rod circuit of Fig. 4.9. The circuit of Fig. 4.5 may be thought of as a
loaded parallel strip line. That of Fig. 4.9 may be thought of as a loaded
coaxial line.
Wave-analysis makes it possible to evaluate fairly accurately the trans-
mission properties of a few simple structures. However, iterated or repeating
structures have certain properties in common: the properties of filter
networks.
For instance, a mode of propagation of the loaded waveguide of Fig. 4.10
or of the series of coupled resonators of Fig. 4.11 can be represented ac-
curately at a single frequency by the ladder networks of Fig. 4.12. Further,
/
FILTER-TYPE CIRCUITS 191
if suitable lumped-admittance networks are used to represent the admit-
tances Bi and B2, the frequency-dependent behavior of the structures of
Figs. 4.10 and 4.11 can be approximated.
It is, for instance, convenient to represent the shunt admittances B2 and
the series admittances Bi in terms of a "longitudinal" admittance Bl and
a "transverse" admittance Bt . Bl and Bt are admittances of shunt resonant
circuits, as shown in Fig. 4.15, where their relation to Bi and B2 and ap-
proximate expressions for their frequency dependence are given. The res-
onant frequencies of Bl and Br , that is, wl and cot , have simple physical
meanings. Thus, in Fig. 4.10, ojz, is the frequency corresponding to equal
and opposite voltages across successive slots, that is, the x mode frequency.
wr is the frequency corresponding to zero slot voltage and no phase change
along the filter, that is, the zero mode frequency.
If w I, is greater than cot , the phase characteristic of this lumped-circuit
analogue is as shown in Fig. 4.17. The phase shift is zero at the lower cutoff
frequency cor and rises to t at the upper cutoff frequency col . If oot is greater
than col , the phase shift starts at — tt at the lower cutoff frequency wi, and
rises to zero at the upper cutoff frequency cor, as shown in Fig. 4.19. In this
case the phase velocity is negative. Figure 4.20 shows a measure of (Er/jS^P)
plotted vs. CO for col > ojt ■ This impedance parameter is zero at cor and rises
to infinity a,t wl .
The structure of Fig. 4.11 can be given a lumped-circuit equivalent in a
similar manner. In this case the representation should be quite accurate.
We find that coz, is always greater than ojt- and that one universal phase curve,
shown in Fig. 4.27, applies. A curve giving a measure of (E^/^^P) vs. fre-
quency is shown in Fig. 4.28. In this case the impedance parameter goes to
infinity at both cutoff frequencies.
The electric field associated with iterated structures does not vary sinus-
oidally with distance but it can be analyzed into sinusoidal components.
The electron stream will interact strongly with the circuit only if the elec-
tron velocity is nearly equal to the phase velocity of one of these field com-
ponents. If 6 is the phase shift per section and L is the section length, the
phase constant ^m of a typical component is
/3„ = (^ -f- 2m7r)/i:
where m is a positive or negative integer. The field component for which
m = 0 is called the fundamental; for other values of m the components are
called spatial harmonics. Some of these components have negative phase
velocities and some have positive phase velocities.
The peak field strength of any field component may be expressed
E = -M(V/L)
■ Here V is the peak gap voltage, L is the section spacing and M is a function
! of /8 (or /3m) and of various dimensions. For the electrode systems of Figs.
192
BELL SYSTEM TECHNICAL JOURNAL
4.29, 4.30, 4.31 and 4.32 M is given by (4.69), (4.71), (4.72) and (4.73),
respectively.
The factor M may be indifferently regarded as a factor by which we
multiply the a-c beam current to give the induced current at the gap, or,
as a factor by which we multiply the gap voltage in obtaining the field. We
can go further, evaluate E^/^'^P in terms of gap voltage, and use M'/o as the
effective current, or we can use the current /o and take the effective field in
the impedance parameter as
£2 = M'\V/(T-
It is sometimes desirable to make use of a spatial harmonic (w 9^ 0)
instead of a fundamental, usually to (1) allow a greater resonator spacing
(2) to obtain a positive phase velocity when the fundamental has a negative
phase velocity (3) to obtain a phase curve for which the phase angle is
nearly a constant times frequency; that is, a phase curve for which the group
velocity does not change much with frequency and hence can be matched
by the electron velocity over a considerable frequency range. Figure 4.33
shows how ^ + 27r (the phase shift per section for m = \) can be nearly a
constant times w even when 6 is not.
l-^i ^
Fig. 4.1 — A corrugated or finned circuit with filter-like properties.
4.1 Field Solutions
An approximate field analysis will be made for two very simple two-
dimensional structures. The first of these, which is shown in Fig. 4.1, is
empty space for y > 1 and consists of very thin conducting partitions in the
y direction from y = 0 to y = — //; the partitions are connected together
by a conductor in the z direction at y = — //. These conducting {)artitions
are spaced a distance C apart in the z direction. The structure is assumed to
extend infinitely in the -\-x and —x directions.
In our analysis we will initially assume that the wavelength of the propa-
gated wave is long compared with (. In this case, the effect of the partitions
is to prevent the existence of any y component of electric field below the z
axis, and the conductor at y = —h makes the s component of electric held
zero at y = —z.
In some perfectly conducting structures the waves propagated are either
transverse electric (no electric lield component in the direction of propaga-
tion, that is, z direction) or transverse magnetic (no magnetic field com-
FILTER-TYPE CIRCUITS 193
ponent in the z direction). We find that for the structure under consideration
there is a transverse magnetic solution. We can take it either on the basis
of other experience or as a result of having solved the problem that the
correct form for the x component of magnetic field for v > 0 is
H. = Hoe'-'"-'^'' (4.1)
Expressing the electric field in terms of the curl of the magnetic field, we have
. ^ dHz dHy „
ay dz
. ^ OHx dHz
J(X)et,j = — -—
dz ax
(4.2)
coe
Hoe'-'"-''" (4.3)
. dHy dHx , .
ji^iE^ = ~—^ -— (4.4)
dx dy
E.= - j 1 Hoe'-'"-'''' (4.5)
we
We can in turn express H^ in terms of Ey and E.
dEz d
dy dz
j.,H, = "-^ - ^y (4.6)
This leads to the relation
/32 _ y = co-yue (4.7)
Now, l/v jue is the velocity of light, and co divided by the velocity of light
has been called /3o , so that
/32 - 7- = /3o- (4.8)
Between the partitions, the field does not vary in the z direction. In any
space between from y = 0 to y = —h, the appropriate form for the magnetic
field is
^^^^^c^s^o(y + A) (4^^
cos /3o«
From this we obtain by means of (4.4)
E,= _igog^sin/Jo(y + /0 ^^^^^^
coe cos |So^
Application of (4.6) shows that this is correct.
194 BELL SYSTEM TECHNICAL JOURNAL
Now, at y = 0 we have just above the boundary
E, = -jlHoe-^^' (4.11)
coe
The fields in the particular slot just below the boundary will be in phase
with these (we specify this by adding a factor exp —j^z to 4.10) and hence
will be
coe
From (4.11) and (4.12) we see that we must have
^oh tan ^oh = yh
10
(4.12)
(4.13)
/3h
fii=Tr
1
/
y\
/
__
-
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Fig. 4.2 — The approximate variation of the phase constant j3 with frequency (propor-
tional to Poll) for the circuit of Fig. 4. 1 . The curve is in error as p( approaches x, and there
is a cutoff at B( = tt.
Using (4.8), we obtain
^h =
cos |So h
(4.14)
In Fig. 4.2, I3h has been plotted vs |Qo//, which is, of course, proportional to
frequency. This curve starts out as a straight line, /? = /3o ; that is, for low
frequencies the speed is the speed of light. At low frequencies the field falls
off slowly in the y direction, and as the frequency approaches zero we have
essentially a plane electromagnetic wave. At higher frequencies, /? > /3o ,
that is, the wave travels with less than the speed of light, and the field falls
off rapidly in the y direction. According to (4.14), /3 goes to infinity
at ^oh = ir/2.
As a matter of fact, the match between the fields assumed above and below
the boundary becomes increasingly bad as jS^ becomes larger. The most rapid
FILTER-TYPE CIRCUITS 195
alteration we can have below the boundary is one in which fields in alternate
spaces follow a +, — , +, — pattern. Thus, the rapid variations of field above
the boundary predicted by (4.14) for values of ^^h which make ^t greater
than TT cannot be matched below the boundary. The frequency at which
l3i = T constitutes the cutoff frequency of the structure regarded as a filter.
There is another pass band in the region x < ^oh < Sir/l, in which the ratio
oi Eto H below the boundary has the same sign as the ratio oi Eto H above
the boundary.
A more elaborate matching of fields would show that our expression is
considerably in error near cutoff. This matter will not be pursued here; the
behavior of filters near cutoff will be considered in connection with lumped
circuit representations.
We can obtain the complex power flow P by integrating the Poynting
vector over a plane normal to the z direction in the region y > 0. Let us
consider the power flow over a depth W normal to the plane of the paper.
Then
P = 1 f f {E,H* - EyHt) dx dy (4.15)
I Jo Jo
Using (4.1) and (4.3), we obtain
2 Jo W€
4 coe7
(4.16)
We will express this in terms of E the magnitude of the z component of
the field at y = 0, which, according to (4.5), is
E=^Ho (4.17)
We will also note that
coe = coVjUe/ V w/e
= {<^/c)/VljTe = ^o/V/V^
and that
VaiA = 377 ohms (4.19)
By using (4.17)-(4.18) in connection with (4.16), we obtain
£-//3'P = (4//^oTr)(T//3)' V/Ve (4.20)
We notice that this impedance is very small for low frequencies, at which
(4.18)
196
BELL SYSTEM TECHNICAL JOURNAL
the velocity of the wave is high, and the field extends far in the y direction
and becomes higher at high frequencies, where the velocity is low and the
field falls off rapidly.
We will next consider a symmetrical array of two opposed sets of slots
(Fig. 4.3) similar to that shown in Fig. 4.1. Two modes of propagation will
be of interest. In one the field is symmetrical about the axis of physical
symmetry, and in the other the fields at positions of physical symmetry are
equal and opposite.
In writing the equations, we need consider only half of the circuit. It is
convenient to take the z axis along the boundary, as shown in Fig. 4.4.
^Ly/////////////
Fig. 4.3 — A double finned structure which will support a transverse mode (no longi-
tudinal electric field on axis) and a longitudinal mode (no transverse electric field on axis).
Fig. 4.4 — The coordinates used in connection with the circuit of Fig. 4.3.
This puts the axis of symmetry at }' = +^, and the slots extend from y — 0
to y = —h.
For negative values of y, (4.9), (4.10), (4.12) hold.
Let us first consider the case in which the fields above are opposite to the
fields below. This also corresponds to waves in a series of slots opposite a con-
ducting plane, as shown in Fig. 4.5. In this case the appropriate form of the
magnetic field above the boundary is
_ cosh y{d - y) jp,
iix — -tJO \ 3 ^
cosh 7a
From Maxwell's ecjuations we then find
/3
cosh 7((/ - y) ^^jp,
cosh yd
(4.21)
(4.22)
FILTER-TYPE CIRCUITS
197
p - ^ ^ n si"h y{d - y) j^,
±Lz — —J — Ho r — -. — e
coe cosh yd
/3o = /3- - T
At y = 0 we have from (4.23) and (4.12)
E, = -j - Hoe~'^' tanh yd
E, = -i ^ ^oe"'^' tan /3o/^
coe
Hence, we must have
yh tanh {{d/h)yh) = /5o/? tan /Jq/^
(4.23)
(4.24)
(4.25)
(4.12)
(4.26)
Fig. 4.5 — The transverse mode of the circuit of Fig. 4.3 exists in this circuit also.
Here we have added parameter, (d/h). For any value of d/h, we can obtain
yh vs f^oh; and we can obtain (Sh in terms of yh by means of 4.24
0h = ({yhY + (l3ohYy"
We see that for small values of jSah (low frequencies)
7- = (I'/d) 0l
1^ ^ 1^0
h + d
(4.27)
(4.28)
(4.29)
If we examine Fig. 4.5, to which this applies, we find (4.28) easy to explain.
At low frequencies, the magnetic field is essentially constant from y = d
to y = —h, and hence the inductance is proportional to the height h + d.
The electric field will, however, extend only from y = 0 to y == ^; hence
the capacitance is proportional to \/d. The phase constant is proportional
to \/LC, and hence (4.29). At higher frequencies the electric and magnetic
fields vary with y and (4.29) does not hold.
We see that (4.26) predicts infinite values of y for j3h = -kJI. As in the
previous cases, cutoff occurs at ,3^ = tt.
198
BELL SYSTEM TECHNICAL JOURNAL
As an example of the phase characteristic of the circuit, fih from (4.26)
and (4.27) is plotted vs M for h/d = 0, 10, 100 in Fig. 4.6. The curve for
h/d = 0 is of course the same as Fig. 4.2.
If we integrate Poynting's vector from y = Q io y = d and for a distance
W in the x direction, and multiply by 2 to take the power flow in the other
half of the circuit into account, we obtain
E'/^'P = (2//3oTF)(7/^)'
sinh" 7^
sinh 'yd cosh 7^/ + yd
Vix/e (4.30)
/3h
/
1
7htanh(^)7h=/3oh tar
i/3oh
/
y9h :
- U, lUjlUU
y
/
1
= l/(7h)^ + (/3oh)^
/
/
/
f i
t'^
/
ji
^
X
1^
V
TT\
2 1
^
^
^y^
0 0
2 0
4 0
6 0
8 t
0 1.
2 1
4 1.6
73oh
Fig. 4.6 — The variation of /3 with frequency (proportional to 0oh) for the transverse
mode of the circuit of Fig. 4.3. Again, the curves are in error near the cutoff at /3^ = w.
At very low frequencies, at which (4.28) and (4.29) hold, we have
£7/3' P = (y'/^o^'Kd/W) Vm76
E'/^'P = {h/df" (1 + d/hf" {d/W) V/IA
(4.31)
At high frequencies, for which yd is large, (4.30) approaches | of the value
given by (4.20). There is twice as much power because there are two halves
to the circuit.
Let us now consider the case in which the field is symmetrical and E, does
not go to zero on the axis. In this case the appropriate field for y > 0 is
//.
^j sinh y{d - y) -jp,
sinh yd
(4.32)
FILTER-TYPE CIRCUITS
199
Proceeding as before, we find
= jSo h tan jSo h
tanh {(d/h) yh)
We see that, in this case, for small values of yh we have
^oh tanh ^oh = h/d
(4.33)
(4.33a)
There is no transmission at all for frequencies below that specified by (4.33).
As the frequency is increased above this lower cutoff frequency, yh and
hence I3h increase, and approach infinity at fSoh = x/2. Actually, of course,
the upper cutoff occurs at /3^ = x. In Fig. 4.7 I3h is plotted vs jSoh for h/d — 0,
20
/3h
7h
- /I K ^
tanh(l)rh
h .
d
/3h :
: 0,10,100
1
= l/Cyh)2+(/3ohl2
P
/
h
d
0 100 1
^
^
r
0 0
.2 0
4 0
6 0
8 1
0 1.
2 1
4 1
.6
/3oh
Fig. 4.7 — The variation of /3 with frequency (proportional to /3o//) for the longitudinal
mode of the circuit of Fig. 4.3. This mode has a band pass characteristic; the band narrows
as the opening of width 2d is made small compared with the iin height. Again, the curves
are in error near the upper cutoff at 0( = tt.
10, 100. This illustrates how the band is narrowed as the opening between
the slots is decreased.
By the means used before we obtain
E'/^'P ^ i2/MV){y/^y
cosh yd
sinh yd cosh yd — yd
)v.
fJi/e (4.34)
We see that this goes to infinity at 7 J = 0. For large values of yd it be-
comes the same as (4.30).
4.2 Practical Circuits
Circuits have been proposed or used in traveling-wave tubes which bear
a close resemblance to those of Figs. 4.1, 4.3, 4.5 and which have very similar
200
BELL SYSTEM TECHNICAL JOURNAL
properties^ Thus Field^ describes an apertured disk structure (Fig. 4.8)
which has band-pass properties very similar to the symmetrical mode of the
circuit of Fig. 4.3. In this case there is no mode similar to the other mode,
with equal and opposite fields in the two halves. Field also shows a disk-on-
rod structure (Fig. 4.9) and describes a tube using it. This structure has low-
Fig. 4.8 — This loaded waveguide circuit has band-pass properties similar to those of
Fig. 4.7.
Fig. 4.9 — This disk-on-rod circuit has properties similar to those of Fig. 4.6.
R
(a) (b)
Fig. 4.10— A circuit consisting of a ridged waveguide with transverse slots or resonators
in the ridge.
pass properties very similar to those of the circuit of Fig. 4.5, which are
illustrated in Fig. 4.6.
Figure 4.10 shows a somewhat more complicated circuit. Here we have a
rectangular waveguide, shown end on in a of Fig. 4.10, loaded by a longi-
tudinal ridged portion R. In b of Fig. 4.10 we have a longitudinal cross sec-
•■> I'". B. Llewellyn, U. S. Patents 2,367,295 and 2,395,560.
■* Lester M. Field, "Some Slow-Wave Structures for Traveling- Wave Tubes," Froc.
I.R.E., Vol. 37, pp. 34-40, Jan. 1949.
FILTER-TYPE CIRCUITS
201
tion, showing regularly spaced slots S cut in the ridge R. The slots S may be
thought of as resonators.
Figure 4.11 shows in cross section a circuit made of a number of axially
symmetrical reentrant resonators R, coupled by small holes H which act as
inductive irises.
It would be very difficult to apply Maxwell's equations directly in de-
ducing the performance of the structures shown in Figs. 4.10 and 4.11.
Moreover, it is apparent that we can radically change the performance of
Fig. 4.11 — A circuit consisting of a number of resonators inductively coupled by means
holes.
JBs
JB,
JB2
JB,
JBj
JB,
J B2
2JB,
JBa
JB,
JB,
JB,
JB,
JB, --
:b)
Fig. 4.12 — Ladder networks terminated in -w (above) and T (below) half sections. Such
networks can be used in analvzing the behavior of circuits such as those of Figs. 4.10
;ind 4.11.
such structures by minor physical alterations as, by changing the iris size,
or by using resonant irises in the circuit of Fig. 4.11, for instance.
As a matter of fact, it is not necessary to solve Maxwell's equations afresh
each time in order to understand the general properties of these and other
circuits.
4.3 Lu.MPED ITER.A.TED An.ALOGUES
Consider the ladders of lossless admittances or susceptances shown in
Fig. 4.12. Susceptances rather than reactances have been chosen because the
202 BELL SYSTEM TECHNICAL JOURNAL
elements we shall most often encounter are shunt resonant near the fre-
quencies considered; their susceptance is near zero and changing slowly but
their reactance is near infinity.
If these ladders are continued endlessly to the right (or terminated in a
reflectionless manner) and if a signal is impressed on the left-hand end, the
voltages, currents and fields at corresponding points in successive sections
will be in the ratio exp(-r) so that we can write the voltages,
Vn = Fo r"'' (4.35)
If the admittances Yx and Y^ are pure susceptances (lossless reactors), V
is either purely real (an exponential decay with distance) or purely imaginary
(a pass band). In this case F is usually replaced by 7/3. In order to avoid
confusion of notation, we will use jd instead, and write for the lossless case
in the pass band
Vn = Fo «"'■"' (4.35a)
Thus, d is the phase lag in radians in going from one section to the next.
In terms of the susceptances,*
cos 0 = 1 + ^2/251 (4.36)
We will henceforward assume that all elements are lossless.
Two characteristic impedances are associated with such iterated networks.
If the network starts with a shunt susceptance 5i/2, as in a of Fig. 4.12, then
we see the mid-shunt characteristic impedance K.^
K, = 2{-B,{B2 + 45i))-i/2 (4.37)
If the network starts with a series susceptance 2Bx we see the mid-series
characteristic impedance Kt
Kr = ±(l/2^i)(-^2 + 450/52)1/2 (4.38)
Here the sign is chosen to make the impedance positive in the pass band.
When such networks are used as circuits for a traveling-wave tube, the
voltage acting on the electron stream may be the voltage across B^ or the
voltage across Bi or the voltage across some capacitive element of B^ or
Bi . We will wish to relate this peak voltage F to the power flow P. If the
voltage across B2 acts on the electron stream
FyP = 2K, (4.39)
If the voltage across Yi acts on the electron stream
F = I/jBx
* The reader can work sucli relations out or look them up in a variety of books or hand-
books. They are in Schelkunoli's Electromagnetic Waves.
FILTER-TYPE CIRCUITS
203
where I is the current in By
P ^\P\ Kt/2
and hence
vyp = 2/Bi^Kt
VyP = -4{B,/B0(-Bo{B2 + 45i))-i/2
V'/P = -2(B2/B{)K.
(4.40)
(4.41)
(4.42)
J lere the sign has been chosen so as to make V^/P positive in the pass band.
Let us now consider as an example the structure of Fig. 4.10. We see that
two sorts of resonance are possible. First, if all the slots are shorted, or if no
\oltage appears between them, we can have a resonance in which the field
between the top of the ridge R and the top of the waveguide is constant
JB,
—
—
JB,
—
-
1
1
1
I
1
"I ■ L^
i
JB2
2
1
jB2
2
1
JB2
2
1
JB2
2
I
JBa
2
1
JB2
2
1
Fig. 4.13 — A ladder network broken up into tt sections.
all along the length, and corresponds to the cutoff frequency of the ridged
waveguide. There are no longitudinal currents (or only small ones near the
slots S) and hence there is no voltage across the slots and their admittance
(the slot depth, for instance) does not affect the frequency of this resonance.
Looking at Fig. 4.12, we see that this corresponds to a condition in which
all shunt elements are open, or B^ = 0. We will call the frequency of this
resonance cct , the T standing for transverse.
There is another simple resonance possible ; that in which the fields across
successive slots are equal and opposite. Looking at Fig. 4.12, we see that
this means that equal currents flow into each shunt element from the two
series elements which are connected to it. We could, in fact, divide the net-
work up into unconnected tt sections, associating with each series element of
susceptance Bi half of the susceptance of a shunt element, that is, Bo/2,
at each end, as shown in Fig. 4.13, without affecting the frequency of this
resonance. This resonance, then, occurs at the frequency co^ (L for longi-
tudinal) at which
Bi + B2/A = 0. (4.43)
We have seen that the transverse resonant frequency, cor , has a clear
meaning in connection with the structure of Fig. 4.10; it is (except for small
204
BELL SYSTEM TECHNICAL JOURNAL
errors flue to stray fields near the slots) the cutoff frequency of the wave-
guide without slots. Does the longitudinal frequency col have a simple
meaning?
Suppose we make a model of one section of the structure, as shown in
Fig. 4.14. Comparing this with b of Fig. 4.10, we see that we have included
the section of the ridged portion between two slots, and one half of a slot
at each end, and closed the ends off with conducting plates C. The resonant
frequency of this model is wl , the longitudinal resonant frequency defined
above.
We will thus liken the structure of Fig. 4.10 to the filter network of Fig.
Fig. 4.14 — A section which will have a resonant frequency corresponding to that for tt
radians phase shift per section in the circuit of Fig. 4.10.
B.= B, + 5i
Bt = B;
17
Bl= 2Ci_{cU-CJi)
Bt = 2Ct (co-OJt)
Fig. 4.15 — The approximate variation with frequency (over a narrow l)andj of the
longitudinal (^/J transverse (Bt) susceptances of a filter network.
4.12, and express the susceptances Bi and B2 in terms of two susceptances
Bt and Bl associated with the transverse and longitudinal resonances and
defined below
Bt = -62
Br. = 7^1 + 52/4
(4.44)
(4.45)
At the transverse resonant frequency cor , B-, ~ 0, luul at (he longitudinal
resonant frequency oi^ , Bl = 0. So far, the lumped-circuit representation
of the structure of Fig. 4.14 can be considered exact in the sense that at
any frequency we can assign values to Bt and A'/, which will give the correct
values for 6 and for V'^/P for the voltage across either the shunt or the series
elements (whichever we are interested in).
FILTER-TYPE CIRCUITS 205
We will go further and assume that near resonances these values of Bt
and Bl behave like the admittances of shunt resonant circuits, as indicated
in Fig. 4.15. Certainly we are right by our definition in saying that 5r === 0
at cor , and Bl = 0 at wi, . We will assume near these frequencies a linear
variation of Bt and Bl with frequency, which is very nearly true for shunt
resonant circuits near resonance*
Bt = 2Cr(co - cor) (4.46)
Bl = 2Cx.(co - coO (4.47)
Here Ct can mean twice the peak stored electric energy per section length
for unit peak voltage between the top of the guide and the top of the ridge R
when the structure resonates in the transverse mode, and Cl can mean twice
the stored energy per section length L for unit peak voltage across the top
Fig. 4.16 — Longitudinal and transverse susceptances which give zero radians phase
shift at the lower cutoff (w = wt) and ir radians phase shift at the upper cutoff (w = cot).
of the slot when the structure resonates in the longitudinal mode.
In terms of Bt and Bl , expression (4.36) for the phase angle d becomes
We see immediately that for real values of 6 (cos 6 < 1), Bt and Bl must
have opposite signs, making the denominator greater than the numerator.
Figure 4.16 shows one possible case, in which cor < ool • In this case the
pass band {6 real) starts at the lower cutoff frequency co = cor at which Bt
is zero, cos ^ = 1 (from (4.48)) and ^ = 0, and extends up to the upper
cutoff frequency co = wz, at which Bl = 0, cos 6 = —\ and 6 = w.
* In case the filter has a large fractional bandwidth, it may be worth while to use the
accurate lumped-circuit forms
Bt = corCrCw/wr — wy/w) (4.46a)
Bl = ulCl(.Wul - wl/«) (4.46b)
206
BELL SYSTEM TECHNICAL JOURNAL
The shape of the phase curves will depend on the relative rates of varia-
tion of Bt and Bl with frequency. Assuming the linear variations with fre-
quency of (4.46) and (4.47) the shapes can be computed. This has been done
for Cl/Ct = 1, 3, 10 and the results are shown in Fig. 4.17.
/
\
//
/
/
7
y
y
/
^
3,
y
,y
r
^
^
,^
^
^
—
—
Fig. 4.17 — Phase shift per section, Q, vs radian frequency w for the conditions of Fig. 4.16.
Fig. 4.18 — Longitudinal and transverse susceptances which give — tt radians phase
shift at the lower cutoff (co = col) and 0 degrees phase shift at the upper cutoff (w = cor).
This means a negative phase velocity.
It is of course possible to make oj/, > cor • In this case the situation is as
shown in Mg. 4.18, the pass band extending from co/, to cot . At co — wl ,
cos B = —\, 6 = —IT. At CO = cor , cos 0=1 and 6 — O.ln Fig. 4.19, as-
suming (4.46) and (4.47), 0 has been plotted vs co for CJCt = 1, 3, 10.
The curves of Figs. 4.17 and 4.18 are not exact for any physical structure
of the type shown in Fig. 4.10. Tn lumped circuit terms, they neglect coupling
FILTER-TYPE CIRCUITS
207
between slots. They will be most accurate for structures with slots longitu-
dinally far apart compared with the transverse dimensions, and least ac-
curate for structures with slots close together. They do, however, form a
valuable guide in understanding the performance of such structures and in
evaluating the effect of the ratio of energies stored in the fields at the two cut-
off frequencies.
^
^
Cl_
10^
^
^
/
3
^
^
/
/
/
.^
^
//
/
/
/
/
//
f
/
f
Fig. 4.19 — Phase shift per section, Q, vs radian frequency, w, for the conditions of Fig
4.18.
It is most likely that the voltages across the slots would be of most in-
terest in connection with the circuit shown in Fig. 4.10. We can rewrite
(4.41) in terms of Bt and Bi,
r-/p =
1
2(1 - ^BJBt){-BtB,)
1/2
(4.49)
We see that V'^/P goes to 0 at 5^ = 0 (w = wr) and to infinity at jB/, = 0
(w = coi,). In Fig. 4.20 assuming (4.46) and (4.47), (FVP)(coz,CLWrC7-) is
plotted vs CO for CJCt =1,3, 10.
Let us consider another circuit, that shown in Fig. 4.11. We see that this
consists of a number of resonators coupled together inductively. We might
draw the equivalent circuits of these resonators as shown in Fig. 4.21. Here
L and C are the effective inductance and the effective capacitance of the
resonators without irises. They are chosen so that the resonant frequency
Wo is given by
COo
(4.50)
208 BELL SYSTEM TECHNICAL JOURNAL
and tlie variation of gap susceptance B with frequency is
dB/dui = 2C
(4.51)
The arrows show directions of current flow when the currents in the gap
capacitances are all the same.
1.0
0.9
0.8
.-. 0.7
!_, 0.6
3,
0.5
O
U
^
(\j\
0.4
0.3
0.2
0.1
0
\
\
/
1 ,
'
A
^
/
/
J
/
:rr
•^
y
U)j
Fig. 4.20 — A quantity proportional to {E?/^P) vs w for the conditions of Figs. 4.16
and 4.17.
TJW^pMF
-W^f-^WT
TW^pWT
Y'lg. 4.21 — A representation of the resonators of Fig. 4.11.
We can now represent the circuit of Fig. 4.11 by interconnecting the
circuits of Fig. 4.21 by means of inductances Lm of Fig. 4.22. This gives a
suitable representation, but one which is open to a minor objection: the
gap capacitance does not appear across either a shunt or a series arm.
Tt is important to notice that there is another equall}^ good representa-
tion, and there are probably many more. Suppose we draw the resonators as
shown in Fig. 4.23 instead of as in Fig. 4.21. The inductance L and capaci-
tance C are still properly given by 4.50 and 4.51. We can now interconnect
the resonators inductively as shown in Fig. 4.24.
We should note one thing. In Fig. 4.21, the currents which are to flow in
the common inductances of Fig. 4.22 flow in opposite directions when the
FILTER-TYPE CIRCUITS
209
;4ap currents are in the same directions. In the representation of Fig. 4.23
the currents which will flow in the common inductances of Fig. 4.24 have
been drawn in opposite directions, and we see that the currents in the gap
capacitances flow alternately up and down. In other words, in Fig. 4.24,
every other gap appears inverted. This can be taken into account by adding
a phase angle — tt to ^ as computed from (4.48).
Fig. 4.22 — The resonators of Fig. 4.11 coupled inductively.
2L 2L 2L 2L 2L 2L
KTKRP-KM^H i-^WU^-r-^M^5^ KOWH-O^M^
O
O
O
O
o
o
Fig. 4.23 — Another representation of the resonators of Fig. 4.11.
2L 2L 2L 2L 2L 2L
Fig. 4.24 — Figure 4.23 with inductive coupling added.
La La LMb
I —
I--
Lb
— n
Lb
(a) ™ (b) ™
Fig. 4.25 — A r — TT transformation used in connection with the circuit of Fig. 4.24.
Now, the T configuration of inductances in a of Fig. 4.25 can be replaced
by the TT configuration, b of Fig. 4.25. Imagiiae I and II to be connected
together and a voltage to be applied between them and III. We see that
U= La+ 2LMa (4.52)
Imagine a voltage to be applied between I and II. We see that
l/La = l/U + 2/LMb (4.53)
If LMa <3C La , then Lb will be nearly equal to La and LMb ^ Li .
By means of such a, T — t transformation we can redraw the equivalent
circuit of Fig. 4.24 as shown in Fig. 4.26. The series susceptance Bi is now
210
BELL SYSTEM TECHNICAL JOURNAL
that of Li , and the shunt susceptance is now that of the shunt resonant
circuit consisting of d (the effective capacitance of the resonators) and L2 .
Fig. 4.26 — The final representation of the circuit of Fig. 4.11.
1
/
/
77
j/
/
2
^
y
^^
/
>-
/
^
/
/
7T
/
Fig. 4.27 — The phase characteristic of the circuit of Fig. 4.11.
The transverse resonance, B2 = 0, occurs at a frequency
(jiT = \/C2jL2
Near this frequency the transverse susceptance is given by
Bt = ICii^ — cor)
The longitudinal resonance occurs at a frequency
a)L = \^lC2UUI{Lx + 2L2)
and near cji, ,
Bl = Ciioj — Ul)
(4.54)
(4.55;
(4.56)
(4.57)
These are just the forms we found in connection with the structure of Fig.
4.10; but we see that, in the case of the circuit of Fig. 4.11, the effective
transverse capacitance is always twice the effective longitudinal capacitance
{Cl/Ct = 1/2 in Fig. 4.19), and that ojl > oor for attainable volume of Li.
FILTER-TYPE CIRCUITS
211
We obtain 0 vs w by adding — tt to the phase angle from 4.48, using (4.55)
and (4.57) in obtaining Bt and B^ . The phase angle vs. frequency is shown
in Fig. 4.27. As the irises are made larger, the bandwidth, co/, — cor , becomes
larger, largely by a decrease in w/. .
The voltage of interest is that across C2 , that is, that across the gap.
I-rom (4.37), (4.44), (4.45), (4.55) and (4.57) we obtain
V'/P = l/i-BrBi^y-' (4.58)
V'/P = (V2/CMo:l - a;)(co - cor))'''' (4.59)
This goes to infinity at both co = wl and w = coy • In Fig. 4.28,
(rV^)C2\/co/.ajr is plotted vs w. This curve represents the performance of
all narrow band structures of the type shown in Fig. 4.11.
9
8
7
1" 6
1
-3 5
>
3
2
\
\
\
/
\
\
J
r
\
N^
^
/
0
a
't
CiJ
>
OJ
Fig. 4.28 — A quantity proportional to {E?/^'^P) for the circuit of Fig. 4.11, plotted vs
radian frequency w.
In a structure such as that shown in Fig. 4.11, there is little coupling
between sections which are not adjacent, and hence the lumped-circuit
representation used is probably quite accurate, and is certainly more ac-
curate than in structures such as that shown in Fig. 4.10.
Other structures could be analyzed, but it is believed that the examples
given above adequately illustrate the general procedures which can be
employed.
4.4 Traveling Field Components
Filter-type circuits produce fields which are certainly not sinusoidal with
distance. Indeed, with a structure such as that shown in Fig. 4.11, the elec-
212
BELL SYSTEM TECHNICAL JOURNAL
trons are acted upon only when they are very near to the gaps. It is possible
to analyze the performance of traveling-wave tubes on this basis'. The chief
conclusion of such an analysis is that highly accurate results can be obtained
by expressing the field as a sum of travehng waves and taking into account
only the wave which has a phase velocity near to the electron velocity. Of
course this is satisfactory only if the velocities of the other components are
quite different from the electron velocity (that is, different by a fraction
several times the gain parameter C).
As an example, consider a traveling-wave tube in which the electron stream
passes through tubular sections of radius a, as shown in Fig. 4.29, and is
acted upon by voltages appearing across gaps of length ( spaced L apart.
->|ih- -AxW JiK JiU
Vn-1 Vn Vn+i Vn+2
Fig. 4.29— A series of gaps in a tube of inside radius a. The gaps are ( long and are
spaced L apart. Voltages Vn , etc., act across them.
A wave travels in some sort of structure and produces voltages across the
gaps such that that across the «th gap, F, is
Vn = V,e
-jnB
(4.60)
where n is any integer.
We analyze this field into traveling-wave components which vary with
distance as exp(-j(3mz) where
(3,n = (^ + 2nnr)/L (4.61)
where m is any positive or negative integer. Thus, the total field will be
-C' / J J^m X > •''i )
-j?m'
hiymr)
(4.62)
a J - l3o'
(4.63)
Here hiymr) is a modified Bessel function, and 7,„ has been chosen so that
(4.62) satisfies Maxwell's equations.
^ J. R. Pierce and Nelson Wax, "A Note on Filter- Type Traveling-Wave Amjjliilers,"
Froc. I.R.E., Vol. 37, pp. 622-625, June, 1949.
FILTER-TYPE CIRCUITS 213
We will evaluate the coefficients by the usual means of Fourier analysis.
Suppose we let z = 0 at the center of one of the gaps. We see that
EE* dz = Z / A„,Alll{y„,r) dz
til m=-oo J—Lhl
(4.64)
= XI AmAtjlhmr)!^
All of the terms of the form E,„Ep , p ^ m integrate to zero because the
integral contains a term exp(-j2Tr{p — m)/L)z.
Let us consider the field at the radius r. This is zero along the surface of
the tube. We will assume with fair accuracy that it is constant and has a
value —V/i' across the gap. Thus we have also at r = a,
f EE*dz= - {V/O E f Ale-'^-' h{y,„a) dz
J— lIi »«=— 00 J— (hi
{v/o z cf:)/o(7.a) (
m=— « \
g i?.^t|■2 _ ^i^,n(l'^
(4.65)
I
We can rewrite this
"' EE- dz = - (V/() ± A:h{y„.a) "^^^^^ (4.66)
L/2 m=-QO \Pmtl I
By comparison with (4.64) we see that
^,„ = - (F/L)( sin (/^„//2)/CS„//2))(l//o(7a)) (4.67)
This is the magnitude of the wth field component on the axis. The magnitude
of the field at a radius r would be loic^r) times this.
The quantity ^,J- is an angle which we will call dg , the gap angle. Usually
we are concerned with only a single field component, and hence can merely
write 7 instead of jm . Thus, we say that the magnitude E of the travelling
field produced by a voltage V acting at intervals L is
E = -M{V/L) (4.68)
^^sin(^/oW
{dg/2) lo(ya)
dg = I3(. (4.70)
The factor M is called the gap factor or the modulation coefficient*.
For slow waves, 7 is very nearly equal to ^, and we can replace yr and ya
by /3r and jSa. For unattenuated waves, ikf is a real positive number; and,
* This factor is often designated by /3, but we have used /3 otherwise.
214
BELL SYSTEM TECHNICAL JOURNAL
for the slowly varying waves with which we deal, we will always consider
If as a real number.
The gap factor for some other physical arrangements is of interest. At a
distance y above the two-dimensional array of strip electrodes shown in
Fig. 4.30
sM^)^.
(4.71)
Fig. 4.30— A series of slots dg radians long separated hy walls L long.
Fig. 4.31 — A system similar to that of Fig. 4.30 but with the addition of an opposed
conducting plane.
If we add a conducting plane a at y = //, as in Fig. 4.31,
^ ^ sin ieg/T) sinh y{h - y)
(ei/2)
sinh yh
(4.72)
For a symmetrical two-dimensional array, as shown in Fig. 4.32, with a
separation of 2 /; in the y direction and the fields above equal to the fields
below
M =
sin {Og/2) cosh yy
(4.73)
{dg/2) cosh yh
4.5 Effective Field and Effective Current
In Section 4.4 we have expressed a field component or ''effective field"
in terms of circuit voltage by means of a gap-factor or modulation coeffi-
FILTER-TYPE CIRCUITS
215
cient M. This enables us to make calculations in terms of fields and currents
at the electron stream.
The gap factor can be used in another way. A voltage appears across a
gap, and the electron stream induces a current at the gap. At the electron
1 stream the power Pi , produced in a distance Z by a convection current
i with the same ^-variation as the field component considered, acting on the
■field componciit is
Pi = -Ei*L
= -j-(MV)i*
(4.74)
Fig. 4.32 — A system of two opposed sets of slots.
At the circuit we observe some impressed current / flowing against the
voltage V to produce a power
Po = vr
(4.75)
By the conservation of energy, these two powers must be the same, and we
deduce that
/* = Mi*
or, since we take M as a real number
I = Mi
(4.76)
(4.77)
Thus, we have our choice of making calculations in terms of the beam
current and a field component or effective field, or in terms of circuit voltage
and an effective current, and in either case we make use of the modulation
coefficient M.
Our gain parameter C^ will be
a = (F/L)W2/o/8/32Fo
216 BELL SYSTEM TECHNICAL JOURNAL
where I' is circuit voltage. We can regard this in two ways. We can think
of —{V''L)M as the effective field at the location of the current /o , or we
can think of M'^h as the effective current referred to the circuit.
If we have a broad beam of electrons and a constant current density /o
we compute (essentially as in Chapter III) a value of C^ by integrating
a = (l/8/3-'Fo)/o(F/L)2 f AP da (4.78)
where da is an element of area. We can think of the result in terms of an
effective field Ee
El = (V/LY ^ (^-'^^
a
where a is the total beam area, and a total current cr/o , or we can think of
the integral (4.77) in terms of an effective current /i, given by
= Jo I M- da (4.80)
and the voltage at the circuit.
Of course, these same considerations apply to distributed circuits. Some-
times it is most convenient to think in terms of the total current and an
effective field (as we did in connection with helices in Chapter III) and
sometimes it is most convenient to think of the field at the circuit and an
effective current. Either concept refers to the same mathematics.
4.6 Harmoxic Operatiox
Of the field components making up E in (4.62) it is customary to regard
the m = 0 component, for which (S — d/L, as the fundamental field com-
ponent, and the other components as harmonic components. These are some-
times called Hartree harmonics. If the electron speed is so adjusted that the
interaction is with the m — 0 ox fundamental component we have funda-
mental operation; if the electron speed is adjusted so that we have interac-
tion with a harmonic component, we have harmonic operation.
There are several reasons for using harmonic operation in connection
with filter-type circuits. For one thing the fundamental component may
appear to be traveling backwards. Thus, for circuits of the type shown in
Fig. 4.11, we see from Fig. 4.27 that d is always negative. Now, in terms of
the velocity v
^ = a;/r = e^L (4.81)
and if 6 is negative, v must be negative. However, consider the w = 1
component
^ = 0,/^, = (27r + e)/L (4.82)
FILTER-TYPE CIRCUITS
217
We see that, for this component, v is positive.
The interaction of electrons with backward-travehng field components
will be considered later. Here it will merely be said that, in order to avoid
interaction with waves traveling in both directions, one must avoid having
the electron speed lie near both the speed of a forward component and the
speed of a backward component.
In order that the fundamental component be slow, 6 must be large or L
must be small. The largest value of d is that near one edge of the band, where
d approaches tt. Thus, the largest fundamental value of /3 is tt/L, and to make
377r 77
FILTER
CHARACTERISTIC
CONST X<*^-
— '^'""
/'^^ CONST Xo;
0 CO\ <^2
Fig. 4.33 — The variation of phase with frequency for the fundamental (0 to ir over the
band) and a spatial harmonic {Itt to 37r over the band). The dotted lines show co divided
by the electron velocity for the two cases. For amplification over a broad band the dotted
curve should not depart much from the filter characteristic.
j8 large with w = 0 we must make L small and put the resonators very close
together. This may be physically difiicult or even impossible in tubes for
very high frequencies. The alternative is to use a harmonic component,
for which /3 = (2w7r + 0) L.
Another reason for using harmonic operation is to achieve broad-band
operation. The phase of a filter-type circuit changes by tt radians between
the lower cutoff frequency coi and the upper cutoff frequency a;2t. Now,
for the wave velocity to be near to the electron velocity over a good part
of the band, /3 must be nearly a constant times w. Figure 4.33 shows how
this can be approximately true for the m = 1 component even when it ob-
viously won't be for the m = 0 or fundamental component. Similarly, for
a filter with a narrower fractional bandwidth and hence a steeper curve of
6 vs CO, a. larger value of m might give a nearly constant value of v.
t The phase of some filters changes more than this, but they don't seem good candidates
for traveling-wave tube circuits.
218 BELL SYSTEM TECHNICAL JOURNAL
CHAPTER V
GENERAL CIRCUIT CONSIDERATIONS
Synopsis of Chapter
TN CHAPTERS III AND IV, helices and filter-type circuits have been
^ considered. Other slow-wave circuits have been proposed, as, for in-
stance, wave guides loaded continuously with dielectric material. One may
ask what the best type of circuit is, or, indeed, in just what way do bad cir-
cuits differ from good circuits.
So far, we have as one criterion for a good circuit a high impedance,
that is, a high value of Er/^P. If we want a broad-band amplifier we must
have a constant phase velocity; that is, 13 must be proportional to frequency.
Thus, two desirable circuit properties are: high impedance and constancy
of phase velocity.
Now, E^/l3~P can be written in the form
Er-/(3'-P = E'/l3nVvg
where W is the stored energy per unit length for a field strength E, and Vg
is the group velocity.
One way of making E-/i3-P large is to make the stored energy for a given
field strength small. In an electromagnetic wave, half of the stored energy
is electric and half is magnetic. Thus, to make the total stored energy for a
given field strength small we must make the energy stored in the electric
field small. The energy stored in the electric field will be increased by the
presence of material of a high dielectric constant, or by the presence of large
opposed metallic surfaces, as in the circuits of Figs. 4.8 and 4.9. Thus, such
circuits are poor as regards circuit impedance, however good they may be in
other respects.
If the stored energy for a given field strength is held constant, £"- J3'-P
may be increased by decreasing the group velocity. It is the phase velocity
V which should match the electron speed. The group velocity Vg is given in
terms of the phase velocity by (5.12). We see that the group velocity may
be much smaller than the phase velocity if —dv'dw is large. It is, for in-
stance, a low group velocity near cutoff that accounts for the high imped-
ance regions exhibited in Figs. 4.20 and 4.28. We remember, however,
that, if the phase velocity of the circuit of a travehng-wave tube changes
with frequency, the tube will have a narrow bandwidth, and thus the high
GENERAL CIRCUIT CONSIDERATIONS 219
impedances attained through large values of —dv/d(j: are useful over a nar-
row range of frequency only.
If we consider a broad electron stream of current density /o , the highest
effective value of B?/^'^P, and hence the highest value of C, will be attained
if there is current everywhere that there is electric field, and if all of the
electric field is longitudinal. This leads to a limiting value of C, which is
given by (5.23). There Xo is the free-space wavelength. The nearest practical
approach to this condition is perhaps a helix of fine wire flooded inside and
outside with electrons.
In many cases, it is desirable to consider circuits for use with a narrow
beam of electrons, over which the field may be taken as constant. As the
helix is a common as well as a very good circuit, it might seem desirable
to use it as a standard for comparison. However, the group velocity of the
helix differs a little from the phase velocity, and it seems desirable instead
to use a sort of hypothetical circuit or field for which the stored energy is
almost the same as in the helix, but for which the group velocity is the same
as the phase velocity. This has been referred to in the text as a "forced
sinusoidal field." In Fig. 5.3, (E-^/(3~Py'^ for the forced sinusoidal field is
compared with {Er/^'Pyi^ for the helix.
Several other circuits are compared with this: the circular resonators of
Fig. 5.4 (the square resonators of Fig. 5.4 give nearly the same impedance)
and the resonant quarter-wave and half-wave wires of Figs. 5.6 and 5.7.
The comparison is made in Fig. 5.8 for three voltages, which fix three phase
velocities. In each case it is assumed that in some way the group velocity
has been made equal to the phase velocity. Thus, the comparison is made on
the basis of stored energies. The field is taken as the field at radius a (cor-
responding to the surface of the helix) in the case of the forced sinusoidal
field, and at the point of highest field in the case of the resonators.
We see from Figs. 5.8 and 5.3 that a helix of small radius is a very fine
circuit.
In circuits made up of a series of resonators, the group velocity can be
changed within wide limits by varying the coupling between resonators, as
by putting inductive or capacitive irises between them. Thus, even cir-
cuits with a large stored energy can be made to have a high impedance by
sacrificing bandwidth.
The circuits of Fig. 5.4 have a large stored energy because of the large
opposed surfaces. The wires of Fig. 5.6 have a small stored energy asso-
ciated entirely with "fringing fields" about the wires. The narrow strips of
Fig. 5.5 have about as much stored energy between the opposed flat sur-
faces as that in the fringing field, and are about as good as the half-wave
wires of Fig. 5.7.
An actual circuit made up of resonators such as those of Fig. 5.4 will be
220 BELL SYSTEM TECHNICAL JOURNAL
worse than Fig. 5.8 implies. Thus, there is a decrease of {Er/^'^Py^ due to
wall thickness. Thickening the tlat opposed walls of the resonators decreases
the spacing between the opposed surfaces, increases the capacitance and
hence increases the stored energy for a given gap voltage. In F"ig. 5.9 the
factor/ by which {Er/^Py^ is reduced is i)lotted vs. the ratio of the wall
thickness / to the resonator spacing L.
There is a further reduction of effective lield because of the electrical
length, 6 in radians, of the space between opposed resonator surfaces.
The lower curve in Fig. 5.10 gives a factor by which (Er/^-Py^ is reduced
because of this. If the resonator spacing, di in radians, is greater than 2.33
radians, it is best to make the opening, or space between the walls, only
2.33 radians long by making the opposed disks forming the walls very
thick.
There is of course a further loss in effective field, both in the helix and in
circuits made up of resonators, because of the falling-off of the field toward
the center of the aperture through which the electrons pass. This was dis-
cussed in Chapter IV.
Finally, it should be pointed out tliat the fraction of the stored energy
dissipated in losses during each cycle is inversely proportional to the Q of
the circuit or of the resonators forming it. The distance the energy travels
in a cycle is proportional to the group velocity. Thus, for a given Q the sig-
nal will decay more rapidly with distance if the group velocity is lowered
(to increase Er/l^P). Equations (5.38), (5.42) and (5.44) pertain to attenu-
ation expressed in terms of group velocity. The table at the end of the
chapter shows that a circuit made up of resonators and having a low enough
group velocity to give it an impedance comparable with that of a helix can
have a very high attenuation.
5.1 Group and Phase Velocity
Suppose we use a broad video pulse F{t), containing radian frequencies
p lying in the range 0 to />o , to modulate a radio-frequency signal of radian
frequency co which is much larger than po , so as to give a radio-frequency
pulse /(/)
/(/) = e'"'Fil) (5.1)
the functions P{l) and /(/) are indicated in Fig. 5.1.
F(l), which is a real function of time, can be expressed by means of its
Fourier transform in terms of its frequency components
FiO - r A{p)e'"' dp {S2)
V— Tin
GENERAL CIRCUIT CONSIDERATIONS
221
Here A{p) is a complex function of />, such that A{—p) is the complex con-
jugate of A{p) (this assures that F{t) is real).
With F{t) expressed as in (5.2), we can rewrite (5.1)
A{p)
Tin
e'^""-"" dp
(5.3)
Now, suppose, as indicated in Fig. 5.2, we apply the r-f pulse /(/) to the
input of a transmission system of length L with a phase constant ^ which
Fig. 5.1 — A radio-frequenc}' pulse varying with time as/(/). The envelope varies with
time as F(t). The pulse might be produced by modulating a radio-frequency source
with F(t).
PHASE CONSTANT /3{a))
F(t)
f(t)"
G(t)
'g(t)
Fig. 5.2 — When the pulse of Fig. 5.1 is applied to a transmission system of length L
and phase constant /3(a)) (a function of co), the output pulse g{t) has an envelope G{1).
is a function of frequency. Let us assume that the system is lossless. The
output g(t) will then be
g(t) = ['\i(p)e''''''-'''-''' dp (5.4)
J-Po
We have assumed that pn is much smaller than co. Let us assume that over
the range co — po to o) -\- po , l3 can be adequately represented by
/3 = ft + |^?
oco
In this case we obtain
g{t) = e^'"'-''>'' r A{p)
*'-Po
The envelope at the output is
G{t) = r A{p) e^"''-''^'"-'''
jp(t-(dfildw)L) 1.
dp
(5.5)
(5.6)
(5.7)
222 BELL SYSTEM TECHNICAL JOURNAL
By comparing this with (5.2) we see that
G{i) ^F^t-f^L^ (5.8)
In other words, the envelope at the output is of the same shape as at the
input, but arrives a time r later
r = 1^ i (5.9)
This implies that it travels with a velocity Vg
% = L/r = (^y (5.10)
This velocity is called the group velocity, because in a sense it is the veloc-
ity with which the group of frequency components making up the pulse
travels down the circuit. It is certainly the velocity with which the energy
stored in the electric and magnetic fields of the circuit travels; we could ob-
serv^e physically that, if at one time this energy is at a position x, a time /
later it is at a position x + Vg I.
If the attenuation of the transmission circuit varies with frequency, the
pulse shape will become distorted as the pulse travels and the group velocity
loses its clear meaning. It is unlikely, however, that we shall go far wrong
in using the concept of group velocity in connection with actual circuits.
We have used earlier the concept of phase velocity, which we have desig-
nated simply as v. In terms of phase velocity,
/? = " (5.11)
V
We see from (5.10) that in terms of phase velocity v the group velocity
Vg is
.. = » (l - " fV (5.12)
\ V dec/
For interaction of electrons with a wave to give gain in a traveling-wave
tube, the electrons must have a velocity near the phase velocity v. Hence,
for gain over a broad band of frequencies, v must not change with frequency;
and if v does not change with frequency, then, from (5.12), Vg — v.
We note that the various harmonic components in a filter-type circuit
have different phase velocities, some positive and some negative. The group
GENERAL CIRCUIT CONSIDERATIONS 223
\ velocity is of course the same for all components, as they are all aspects of
i one wave. Relation (4.61) is consistent with this:
/3. = (^ + 2w7r)/L (4.61)
1 1/t'ff = d^Jdo: = {dd/doi)/L (5.13)
5.2 Gain and Bandwidth in a Traveling- W.ave Tube
We can rewrite the impedance parameter E^/0^P in terms of stored
energy per unit length TT' for a field strength £, and a group velocity Vg .
If ir is the stored energy per unit length, the power flow P is
I P = WVg (5.14)
and, accordingly, we have
j £2//32P = pr-/^Wvg (5.15)
And, for the gain parameter, we will have
C = (P?/l3'Wvg)'''ih/8Vo}"' (5.16)
For example, we see from Fig. 4.20 that E-/^'~P for the circuit of Fig. 4.10
t^^oes to infinity at the upper cut-off. From Fig. 4.17 we see that dd/do},
and hence 1 '^'g , go to infinity at the upper cutoff, accounting for the infinite
impedance. We see also that dd do: goes to infinity at the lower cutoff, but
there the slot voltage and hence the longitudinal field also go to zero and
hence E-fff-P does not go to infinity but to zero instead.
In the case of the circuit of Fig. 4.11, the gap voltage and hence the longi-
tudinal field are finite for unit stored energy at both cutoffs. As dd do: is
infinite at both cutoffs, V- P and hence E~ff-P go to infinity at both cut-
; offs, as shown in Fig. 4.28.
i To get high gain in a traveling-wave tube at a given frequency and volt-
age (the phase velocity is specified by voltage) we see from (5.16) that we
must have either a small stored energy per unit length for unit longitudinal
field, or a small group velocity, v g .
To have ampUfication over a broad band of frequencies we must have the
phase velocity v substantially equal to the electron velocity over a broad
band of frequencies. This means that for very broad-band operation, v
j must be substantially constant and hence in a broad-band tube the group
velocity will be substantially the same as the phase velocity.
If the group velocity is made smaller, so that the gain is Increased, the
I range of frequencies over which the phase velocity is near to the electron
velocity is necessarily decreased. Thus, for a given phase velocity, as the
group velocity is made less the gain increases but the bandwidth decreases.
Particular circuits can be compared on the basis of (E-/l3''P) and band-
224 BELL SYSTEM TECHNICAL JOURNAL
width. We have discussed the impedance and phase or velocity curves in
Chapters III and W . Field' has compared a coiled waveguide structure with
a series of apertured disks of comparable dimensions. Both of these struc-
tures must have about the same stored energy for a given field strength.
He found the coiled waveguide to have a low gain and broad bandwidth
as compared with the apertured disks. We explain this by saying that the
particular coiled waveguide he considered had a higher group velocity than
did the apertured disk structure. Further, if the coiled waveguide could be
altered in some way so as to have the same group velocity as the apertured
disk structure it would necessarily have substantially the same gain and
bandwidth.
In another instance, Mr. O. J. Zobel of these Laboratories evaluated the
efifect of broad-banding a filter-type circuit for a traveling-wave tube by
w-derivation. He found the same gain for any combination of m and band-
width which made v = Vg(dv/do: = 0). We see this is just a particular
instance of a general rule. The same thing holds for any type of broad-
banding, as, by harmonic operation.
5.3 A Comparison of Circuits
The group velocity, the phase velocity and the ratio of the two are param-
eters which are often easily controlled, as, by varying the coupling between
resonators in a filter composed of a series of resonators. Moreover, these
parameters can often be controlled without much affecting the stored energy
per unit length. For instance, in a series of resonators coupled by loops or
irises, such as the circuit of Fig. 4.11, the stored energy is not much affected
by the loops or irises unless these are very large, but the phase and group
velocities are greatly changed by small changes in coupling.
Let us, then, think of circuits in terms of stored energy, and regard the
phase and group velocities and their ratio as adjustable parameters. We
find that, when we do this, there are not many essentially different configura-
tions which promise to be of much use in traveling-wave tubes, and it is
easy to make comparisons between extreme examples of these configura-
tions.
5.3a L niform Currenl Density throughout Field
Suppose we have a uniform current density /q wherever there is longi-
tudinal electric field. We might approximate this case by flooding a helix
of very fine wire with current inside and outside, or by passing current
through a series of flat resonators whose walls were grids of fine wire.
' Lester M. Field, "Some Slo\v-\V;ive Structures for Traveling-Wave Tuijes," Proc.
I.R.E., Vol. 37, J)]). 34-40, January 1949.
GENERAL CIRCUIT CONSIDERATIONS 225
In the latter case, if resonators had parallel walls of very fine mesh normal
to the direction of electron motion there would be substantially no trans-
verse electric field. All the electric field representing stored energy would
act on the electron stream. In this case, we would have
W
= \l E'dZ (5.17)
Here dH is an elementary area normal to the direction of propagation. W
given by this expression is the total electric and magnetic stored energy
per unit length. Where E is less than its peak value, the magnetic energy
makes up the difference.
In evaluating £-/o in (5.16) we will have as an effective value
(£/o)eff = JoJEd^ (5.18)
Hence, we will have for the gain parameter C
Jo j E' dl
C =
(^)^ (^ / FJ dz) z,(8Fo)
/ T \ 1/3
(5.19)
C =
4 - eVg Vo
It is of interest to put this in a slightly different form. Suppose Xo is the
free-space wavelength. Then
^ = ^^ (5.20)
V Xo V
where c is the velocity of light
c = 3 X 10^° cm/sec - 3 X 10^ m/sec
Further, we have for synchronism between the electron velocity m
and the phase velocity v
i^ = 2r,Vo (5.21)
Also
c = l/V^
e = l/cVJ^e (5.22)
r
\ ix/f: = 377 ohms
226
BELL SYSTEM TECHNICAL JOURNAL
Using (5.20), (5.21), (5.22) in connection with (5.19), we obtain
= 11.16 {J,\<?/v,yi'
(5.23)
We have in (5.23) an expression for the gain parameter C in case longi-
tudinal fields only are present and in case there is a uniform current density
/o wherever there is a longitudinal field.
In a number of cases, as in case of a large-diameter helix, or of a resonator
with large apertures, the stored energy due to the transverse field is about
equal to that due to the longitudinal field and C will be 2~^i^ times as great
as the value of C given by (5.23). Thus, the value of C given by (5.23), or
even 2~^'^ times this, represents an unattainable ideal. It is nevertheless
of interest in indicating how limiting behavior depends on various parame-
ters. For instance, we see that if the wavelength Xo is made shorter, a higher
current density must be used if C is not to be lowered; for a constant C
the current density must be such as to give a constant current through a
square a wavelength on a side.
In the table below, some values of C have been computed from (5.23)
for various wavelengths and current densities. The broad-band condition
of equal phase and group velocities has been assumed, and the voltage has
been taken as 1,000 volts.
\
\
WavelengthX Amp/cm^
Cm \
\
.1
1
5
.060
.130
.5
.013
.028
For larger voltages, C will be smaller. C can of course be made larger by
making the group velocity smaller than the phase velocity.
Of course, if the electron stream does not pass through some portions of
the field, C will be smaller than given by (5.23). C will also be less if there
are "harmonic" field components which do not vary in the z direction as
exp(yco2/t)).
5.3b Narroiv Beams
Usually, no attemj)t is made to iill tlie entire field with electron flow even
though this is necessary in getting a large value of C for a given current
density. Instead a narrow electron beam is shot through a region of high
GENERAL CIRCUIT CONSIDERATIONS
227
field. We then wish to relate the peak field strength to the stored energy in
comparing various circuits.
Let us first consider a helically conducting sheet of radius a. The upper
curve of Fig. 5.3 shows {F^/ff^Py^iv/cyi^ vs. j8a. In obtaining this curve it
was assumed that v <$C c, so that 7 can be taken as equal to /3. The field E
is the longitudinal field at the surface of the helically conducting cylinder.
Figure 5.3 can be obtained from Fig. 3.4 by multiplying F{ya) by {Io(ya)Y'^
to give a curve valid for the field at r = a.
The helix has a very small circumferential electric field which represents
"useless" stored energy. The lower curve of Fig. 5.3 is based on the stored
electric energy of an axially symmetrical sinusoidal field impressed at the
radius a.f This field has no circumferential component but is otherwise the
A
B
~-
^
^
^^^
HELIX
FORCED y^^**
SINUSOIDAL
FIELD
"^
-^
C
1
5 6 7 8 9 to
(3a
Fig. 5.3 — The impedance parameter (Er/p-P)^'^ compared for a helically conducting
sheet (A) and a forced sinusoidal lield (B) with a group velocity equal to the phase ve-
locity. The helix has a higher impedance because the phase velocity is higher than the
Kioup velocit}' by a radio showni to the j power by curve C.
same as the electric field of the helix (again assuming r <<C c). We can imagine
such a field propagating because of an inductive sheet at the radius a,
which provides stored magnetic energy enough to make the electric and
magnetic energies equal. The quantity plotted vs. I3a is {Er/fS^Py^ (v/cY^^
The forced sinusoidal field is not the field of some particular circuit for
which a certain group velocity Vg corresponds to a given phase velocity i'.
Hence, the factor (vg/vY'^ is included in the ordinate, so that the curve will
be the same no matter what group velocity is assumed. For the helically
conducting sheet, a definite group velocity goes with a given phase velocity.
In Fig. 5.3, the ordinate of the curve for the helically conducting sheet
does not contain the factor (vg/vy^. If, for instance, we assume Vg = v
t See Appendix III.
228
BELL SYSTEM TECHNICAL JOURNAL
in connection with the curve for the forced sinusoidal field, then the two
ordinates are both {E?/0^Pyi^ {v/cY'^ and the curve for the sheet is higher
than that for the forced field because, for the helicallv conducting sheet
(a) (b)
Fig. 5.4 — Pillbox and rectangular resonators. When a number of resonators are coupled
one to the next, a filter-type circuit is formed.
T
Q:
a
a
I
^^
®
b>
Vg < V for small values of -ya. Curve C shows {v/vgY-'^
for the sheet vs. ^a. Aside from the influence of group
velocity, we might have expected the curve for the
sheet to be a little lower than that for the forced field
because of the energy associated with the transverse
electric field component of the sheet. This, however,
becomes small in comparison with the transverse mag-
netic component when v « r, as we have assumed.
Various other circuits will be compared, using
the impressed sinusoidal field as a sort of standard
of reference.
One of the circuits which will be considered is a
series of fiat resonators coupled together to make a
filter. Figure 5.4a shows a series of very thin pill-
boxes with walls of negligible thickness. A small cen-
tral hole is provided for the electron stream, and the
field E is to be measured at the edge of this hole.
The diameter is chosen to obtain resonance at a
wavelength Xo . Figure 5.4b shows a similar series
of flat square resonators.
For the round resonators it is found that*
Fig. 5.5 — Resonators
with the opposing paral-
lel surfaces reduced to
lower stored energ\- and
increase impedance.
(E^/lS^Py = 5.36 (v/cyi' {v/vgY'^
for the square resonators*
{E^/^H^yi-' = 5.33 {v/cyi^ {v/v„yi^
For practical purposes these are negligibly diiTerent.
* See Appendix Til.
(5.24)
GENERAL CIRCUIT CONSIDERATIONS
229
Suppose we wanted to improve on such circuits by reducing the stored
energy. An obvious procedure would be to cut away most of the fiat opposed
surfaces as shown in Fig. 5.5. This reduces the energy stored between the
resonator walls, but results in energy storage outside of the open edges,
energy associated with a "fringing lield."
Going to an extreme, we might consider an array of closely spaced very
fine wires, as shown in Fig. 5.6. Here there are no opposed fiat surfaces,
and all of the electric field is a fringing field; we have
reached an irreducible minimum of stored energy in
paring down the resonator.
The structure of Fig. 5.6 has not been analyzed
exactly, but that of Fig. 5.7 has. InFig. 5.7, wehave
an array of fine, closely spaced half-wave wires be-
tween parallel planes.* This should have roughly
twice the stored energy of Fig. 5.6, and we will esti-
mate (Er/ff^Py^ for Fig. 5.6 on this basis. We obtain
in Appendix III:
For the half-wave wires.
i^/^^pyi' = 6.20 {v/vgy^
(5.25)
Fig. 5.6— Quarter-wave
wires, which have a min-
imum of stored energy.
and hence for the quarter-wave wires, approximately
(£V/52p)i/3 = 7_8i (^,/i,Ji/3 (5.26)
As we have noted, (v/c), which appears in the expression for {E-f^-Py^
for the sinusoidal field impressed at radius a and in (5.24) and (5.25), is a
Fig. 5.7 — Half-wave wires between parallel planes. The stored energy can be calculated
for this configuration, assuming the wires to be very fine. The circuit does not propagate a
wave unless added coupling is provided.
function of the accelerating voltage. Figure 5.8 makes a comparison be-
tween the sinusoidal field impressed at a radius a, curve A ; the flat resona-
tors, either circular or square, B; the half-wave wires, C; and the quarter-
* There is no transverse magnetic wave propagation along such a circuit unless extra
coupling or loading is provided. Behavior of nonpropagating circuits in the presence of an
electron stream is considered in Section 4 of Chapter XTV.
230
BELL SYSTEM TECHNICAL JOURNAL
wave wires C . In all cases, it is assumed that the coupling is so adjusted as
to make (r„ z') = 1 (broad-band condition).
What sort of information can we get from the curves of Fig. 5.8? Con-
sider the curves for 1,000 volts. Suppose we want to cut down the opposed
areas of resonators, as indicated in Fig. 5.5, so as to make them as good as
half-wave wires (curve C). The edge capacitance in Fig. 5.5 will be about
equal to that for quarter-wave wires (curve C). Curve C is about 3.7 times
as high as curve B, and hence represents only about (1/3.7)'^ = .02 as much
capacitance. If we make the opposed area in Fig. 5.5 about .01 that in Fig.
5.4a or b, the capacitance* between opposed surfaces will equal the edge
>!?
Q. 8
rvj
6
\
100 VOLTS
A IMPRESSED SINUSOIDAL FIELD
B CIRCULAR RESONATORS
\
C HALF-WAVE WIRES
C' QUARTER-WAVE WIRES
\
1000 VOLTS
10,000 VOLTS
\
c'
\
:\
c'
c
c
^
-\-
c
^
^
\
\A
B
B
^■^^ —
B
/3a
4
fla
fla
Fig. 5.8 — Coni]5arisons in terms of impedance parameter of an im|)ressed sinusoidal
field (.'1 ), circular resonators (B), half-wave wires (C) andquarter-wave wires (C) assuming
the group and phase velocities to equal the electron velocity. The radius of the impressed
sinusoidal field is a.
capacitance and the total stored energy will be twice that for quarter-wave
wires, or equal to that for half-wave wires. This area is shown appro.xi-
mately to scale relative to Fig. 5.4 in Fig. 5.5. Thus, at 1,000 volts the
resonant strips of Fig. 5.5 are about as good as fine, closely spaced half-
wave wires.
Suppose again thai we wish at 1,000 volts to make the gain of the reso-
nators of Fig. 5.4 (or of a coiled waveguide) as good as that for a helix with
(ia — 3. For /3a = 3 the helix curve .1 is about 3.2 limes as high as ihc resona-
* This takes into account a difference in field distriljution — thai in I'Ik. 5.4h.
GENERAL CIRCUIT CONSIDERATIONS
231
(or curve B. As {E^/^'^Py^ varies as {v/vgY^^, we must adjust the coupling
between resonators so as to make
Vg = V(3.2)3 _ 031 z;
in order to make {Er/^-Py^ the same for the resonators as for the helix.
I'Yom (5.12) we see that this means that a change in frequency by a frac-
tion .002 must change v by a fraction .06. Ordinarily, a fractional variation
of V of ±.03 would cause a very serious falling off in gain. At 3,000 mc the
total frequency variation of .002 times in v would be 6 mc. This is then a
measure of the bandwidth of a series of resonators used in place of a helix
lor which ^a = 3 and adjusted to give the same gain.
0.8
0.6
0.4
0.2
\
\
\
Fig. 5.9 — The factor/ by which (£?//3^P)^'^ for a series of resonators such as those of
I ig. 5.4 is reduced because of wall thickness t, in relation to gap spacing L.
5.4 Physical Limitations
In Section 3.3b the resonators were assumed to be very thin and to have
walls of zero thickness. Of course the walls must have finite thickness, and
it is impractical to make the resonators extremely thin. The wall thickness
and the finite transit time across the resonators both reduce E'l^P.
?.4a Effect of Wall Thickness
Consider the resonators of Fig. 5.4. Let L be the spacing between resona-
tors (1/Z resonators per unit length), and / be the wall thickness. Thus, the
gap length is (L — /). Suppose we keep L and the voltage across each
232
BELL SYSTEM TECHNICAL JOURNAL
resonator constant, so as to keep the field constant, but vary /. The capaci-
tance will be proportional to (L — t)~^ and, as the stored energy is the
voltage squared times the capacitance, we see that {E?/0^P) ^'^ will be re-
duced by a factor /,
/ = (1 - //L)i/3
The factor/ is plotted vs. t/L in Fig. 5.9.
(5.27)
\
V
V
\
\
^^
\
^
^45/eO'/^
\
\
\
/'s.N (6/2)^/3
\
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 2 4 6 8 10 12 14 16 18 20 22
TRANSIT ANGLE IN RADIANS
Fig. 5.10— The lower curve shows the factor by which E?/^P is reduced by gap length,
d in radians. If the gap spacing is greater than 2.33 radians, it is best to make the gap 2.33
radians long. Then the upper curve applies.
5.4b Transit Time
As it is impractical to make the resonators infinitely thin, there will be
some transit angle dg across the resonator, where
dg = ^t (5.28)
Here (, is the space between resonator walls, or, the length of the gap.
If we assume a uniform electric field between walls, the gap factor M,
that is, the ratio of peak energy gained in electron volts to peak resonator
voltage, or the ratio of the magnitude of the sinusoidal field component
produced to that which would be produced by the same number of infinitely
thin gaps with the same voltages, will be (from (4.69) with r = a)
sin {dg/2)
M =
dg/2
(5.29)
GENERAL CIRCUIT CONSIDERATIONS 233
For a series of resonators dg long with infinitely thin walls E?/fi^P will be
less than the values given by (5.24) and (5.25) by a factor M"^'^. This is
plotted vs. dg in Fig. 5.10.
5.4c Fixed Gap Spacing
Suppose it is decided in advance to put only one gap in a length specified
by the transit angle dt . How wide should the gap be made, and how much
will F?/^^P be reduced below the value for very thin resonators and infi-
nitely thin walls?
Let us assume that all the stored energy is energy stored between parallel
planes separated by the gap thickness, expressed in radians as 6 or in dis-
tance as L
9t = i3e
dg = /3L
Here ^ is the gap spacing and L is the spacing between resonators.
From Section 4.4 of Chapter IV we see that if V is the gap voltage, the
field strength E is given by
E = MV/L
The stored energy per unit length, W, will be
W = W^VyiL (5.30)
Here Pf^o is a constant depending on the cross-section of the resonators.
Thus, for unit field strength, the stored energy will be
W = WoL/m^
(5.31)
W = Wo(d^/dg)(dg/2y/smHdg/2)
We see that Wo is merely the value of W when dt = 9g and dg = 0, or,
for zero wall thickness and very thin resonators. Thus, the ratio W/Wo re-
lates the actual stored energy per unit length per unit field to this optimum
stored energy for resonators of the same cross section.
For dt < 2.33, W/Wo is smallest (best) for dg = dt (zero wall thickness).
For larger values oi dt , the optimum value of dg is 2.33 radians and for
this optimum value
(Wo/Wy = (lASO/dtY'^ (5.32)
If 0i < 2.33, it is thus best to make dg = dt. Then {F?/l3^Py'^ is re-
duced by the factor [sm{d/2)/{d/2)Y'\ which is plotted in Fig. 5.10. If
dt > 2.33, it is best to make d = 2.33. Then {E?/0^Pyi^ is reduced from the
234 BELL SYSTEM TECHNICAL JOURNAL
value for thin resonators with infinitely thin walls by a factor given by
(5.32), which is plotted vs. di in Fig. 5.10.
If there are edge effects, the optimum gap spacing and the reduction in
{F?/^Pyi^ will be somewhat different. However, Fig. 5.10 should still be a
useful guide.
In case of wide gap separation (large dt), there would be some gain in
using reentrant resonators, as shown in Fig. 4.11, in order to reduce the
capacitance. How good can such a structure be? Certainly, it will be worse
than a helix. Consider merely the sections of metal tube with short gaps,
which surround the electron beam. The shorter the gaps, the greater the
capacitance. The space outside the beam has been capacitively loaded,
which tends to reduce the impedance. This capacitance can be thought of
as being associated with many spatial harmonics in the electric field, which
do not contribute to interaction with the electrons.
5.5 Attenuation
Suppose we have a circuit made up of resonators with specified unloaded
Q.\ The energy lost per cycle is
W^ = IwWs/Q (5.33)
In one cycle, however, a signal moves forward a distance L, where
L = vjj (5.34)
The fractional energy loss per unit distance, which we will call 2q', is
la = ^1-^ \ (5.35)
whence
0^ = 7^ (5.36)
So defined, a is the attenuation constant, and the amplitude will decay
along the circuit as exp( — as).
The wavelength, X, is given by
X = v/f = 27rVco (5.37)
'J1ie loss per wavelength in db is
db/wavelength = 20 logio exp(«X)
db/wavelength = ~
t Disregarding coupling losses, the circuit and the resonantors will both have this
same Q.
GENERAL CIRCUIT CONSIDERATIONS 235
We see that, for given values of v and Q, decreasing the group velocity,
which increases E-/0^P, also increases the attenuation per wavelength.
5.5a Attenuation of Circuits
For various structures, Q can be evaluated in terms of surface resistivity,
R, the intrinsic resistance of space, v yu/^ ~ >^^^ ohms, and varous other
parameters. For instance, Schelkunoff- gives for the Q of a pill-box resona-
tor
1 -1- a/h
Here a is the radius of the resonator and h is the height. If we express the
radius in terms of the resonant wavelength Xo {a — 1.2Xo/7r), we obtain
(1 + h/a)n
Here n is the number of resonators per wavelength (assuming the walls
separating the resonators to be of negligible thickness); thus
n = h/\ = (VXo)(c/^) (5.41)
From (5.40) and (5.38) we obtain for a series of pill-box resonators
db/ wavelength = ^M{R/\^^e){c/vg){\ + h/a)n (5.42)
In Appendix III an estimate of the Q of an array of fine half-wave paral-
lel wires is made by assuming conduction in one direction with a surface
resistance R. On this basis, Q is found to be
Q = (VMR){v/c) (5.43)
and hence
db/wavelength = 27.3{R/\/M{c/v,) (5.44)
For non-magnetic materials, surface resistance varies as the square root
of the resistivity times the frequency. The table below gives R for copper
and db/wavelength for pill-box resonators for h/a « 1 (5.42) and for wires
(5.44) for several frequencies
f, mc R, Ohms (db/wavelength)/ (c/vg)
Pill-box Resonators Wires
i.i X 10-^w 10.3 X 10-^
6.0 X 10-% 18.1 X 10-^
10.4 X 10-% 32.6 X 10-^
In Section 3.3b a circuit made up of resonators, with a group velocity
.031 times the phase velocity, was discussed. Suppose such a circuit were
2 Electromagnetic Waves, S. A. Schelkunoff, Van Nostrand, 1943. Page 269.
3,000
.0142
10,000
.0260
30,000
.0450
236 BELL SYSTEM TECHNICAL JOURNAL
used at 1,000 volts {c/v = 16.5), were 40 wavelengths long, and had three
copper resonators per wavelength. The total attenuation in db is given below
f, mc Attenuation, db
3,000 21
10,000 38
30,000 67
CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 237
CHAPTER VI
THE CIRCUIT DESCRIBED IN TERMS OF
NORMAL MODES
Synopsis of Chapter
IN CHAPTER II, the field produced by the current in the electron stream,
which was assumed to vary as exp {—Tz), was deduced from a simple
model in which the electron stream was assumed to be very close to an ar-
tificial line of susceptance B and reactance X per unit length. Following
these assumptions, the voltage per unit length was found to be that of
equation (2.10) and the field E in the z direction would accordingly be V
times this, or
E = ^f~f2 i (6.1)
Here we will remember that Fi is the natural propagation constant of
the line, and K is the characteristic impedance.
We further replaced K hy a. quantity
£2//32p = 2K (6.2)
where E is the field produced by a power flow P, and /3 is the phase constant
of the line. For a lossless line, Fi is a pure imaginary and
&■= -Vl (6.3)
From (6.1) and (6.2) we obtain
2(rf - F^) ' ^^-^^
To the writer it seems intuitively clear that the derivation of Chapter
II is correct for waves with a phase velocity small compared with the
velocity of light, and that (6.4) correctly gives the part of the field asso-
ciated with the excitation of the circuit. However, it is clear that there are
other field components excited; a bunched electron stream will produce a
field even in the absence of a circuit. Further, many legitimate questions
can be raised. For instance, in Chapter II capacitive coupling only was
considered. What about mutual inductance between the electron stream
and the inductances of the line?
238 BELL SYSTEM TECHNICAL JOURNAL
The best procedure seems to be to analyze the situation in a way we know
to be vahd, and then to make such approximations as seem reasonable. One
approximation we can make is, for instance, that the phase velocity of the
wave is quite small compared with the speed of light, so that
|ri|2»|3o = {o^/cY (6.5)
In this chapter we shall consider a lossless circuit which supports a group
of transverse magnetic modes of wave propagation. The tinned structure of
Fig. 4.3 is such a circuit, and so are the circuits of Figs. 4.8 and 4.9 (assum-
ing that the fins are so closely spaced that the circuit can be regarded as
smooth). It is assumed that waves are excited in such a circuit by a current
in the z direction varying with distance as exp {—Tz) and distributed normal
to the z direction as a function of x and y,J{x, y). Such a current might
arise from the bunching at low signal levels of a broad beam of electrons
confined by a strong magnetic field so as not to move appreciably normal
to the z direction.
The structure considered may support transverse electric waves, but these
can be ignored because they will not be excited by the impressed current.
In the absence of an impressed current, any field distribution in the struc-
ture can be expressed as the sum of excitations of a number of pairs of nor-
mal modes of propagation. For one particular pair of modes, the field dis-
tribution normal to the z direction can be expressed in terms of a function
Tn{x, y) and the field components will vary in the z direction as exp(±r„2;).
Here the + sign gives one mode of the pair and the — sign the other. If
r„ is real the mode is passive; the field decays exponentially with distance.
If r„ is imaginary the mode is active; the field pattern of the mode propa-
gates without loss in the z direction.
An impressed current which varies in the z direction as exp(— ^^) will
excite a field pattern which also varies in the z direction as exp(— F^'), and
as some function of .v and y normal to the z direction. We may, if we wish,
regard the variation of the field normal to the z direction as made up of a
combination of the field patterns of the normal modes of propagation, the
patterns specified by the functions 7r„(.f, y). Now, a pattern specified by
TTnC^") y) coupled with a variation exp(±rn5;) in the z direction satisfies
Maxwell's equations and the boundary conditions imposed by the circuit
with no impressed current. If, however, we assume the same variation with
.V and y but a variation as exp(— ^^) with z, Maxwell's equations will be
satisfied only if there is an impressed current having a distributioii normal
to the z direction which also can be ex-j^jressed by the function 7r,j(.v, y).
Su[)p()sc we add up the various forced modes in such relative strength
and i)hasc that the total of tlic imjjresscd currents associated witli them is
equal to the actual impressed current. Then, tlie sum of the fields of these
CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 239
modes is the actual field produced by the actual impressed current. The
field is so expressed in (6.44) where the current components /„ are defined
by (6.36).
If it is assumed that there is only one mode of propagation, and if it is
assumed that the field is constant over the electron flow, (6.44) can be put
in the form shown in (6.47). For waves with a phase velocity small compared
with the velocity of light, this reduces to (6.4), which was based on the simple
circuit of Fig. 2.3.
Of course, actual circuits have, besides the one desired active mode, an
infinity of passive modes and perhaps other active modes as well. In Chapter
VII a way of taking these into account will be pointed out.
Actual circuits are certainly not lossless, and the fields of the helix, for
instance, are not purely transverse magnetic fields. In such a case it is per-
haps simplest to assume that the modes of propagation exist and to cal-
culate the amount of excitation by energy transfer considerations. This has
been done earlier^, at first subject to the error of omitting a term which
later- was added. In (6.55) of this chapter, (6.44) is reexpressed in a form
suitable for comparison with this earher work, and is found to agree.
Many circuits are not smooth in the z direction. The writer believes that
usually small error will result from ignoring this fact, at least at low signal
levels.
6.1 Excitation of Transverse Magnetic Modes of Propagation by
A Longitudinal Current
We will consider here a system in which the natural modes of propagation
are transverse magnetic waves. The circuit of Fig. 4.3, in which a slow wave
is produced by finned structures, is an example. We will remember that the
modes of propagation derived in Section 4.1 of Chapter IV were of this
type. We will consider here that any structure the circuit may have (fins,
for instance) is fine enough so that the circuit may be regarded as smooth
in the z direction.
Any transverse electric modes which may exist in the structure will not
be excited by longitudinal currents, and hence may be disregarded.
The analysis presented here will follow Chapter X of Schelkunoff's
Electromagnetic Waves.
The divergence of the magnetic field H is zero. As there is no z component
of field, we have
'J. R. Pierce, "Theory of the Beam-Type Traveling-Wave Tube," Rroc. I.R.E. Vol.
35, pp. 111-123, February, 1947.
^ J. R. Pierce, "Effect of Passive Modes in Traveling-Wave Tubes," Froc. I.R.E.,
Vol. 36, pp. 993-997, August, 1948.
240 BELL SYSTEM TECHNICAL JOURNAL
^^ + ^^ = 0 (6.6)
dx dy
This will be satisfied if we express the magnetic field in terms of a "stream
function", t
H. = g (6.7)
H.= -^ (6.8)
dx
IT can be identified as the z component of the vector potential (the vector
potential has no other components).
We will assume x to be of the form
T = f (x, y)e'^' (6.9)
Here w (x, y) is a function of x and y only, which specifies the field dis-
tribution in any x, y plane.
We can apply Maxwell's equations to obtain the electric fields
dH^ dHy . ^
dy az
Using (6.7) and (6.8), and replacing dififerentiation with respect to z by
multiplication by — F, we find
£. = -S^ ^ (6.10)
ue ox
Similarly
E.-'^'~ (6.11)
coe dy
We see that in an x, y plane, a plane perpendicular to the direction of propa-
gation, the field is given as the gradient of a scalar potential V
V = (-yr/aje)7r (6.12)
This is because we deal with transverse magnetic waves, that is, with waves
which have no longitudinal or z component of magnetic field. Thus, a closed
path in an x, y plane, which is normal to the direction of propagation, will
link no magnetic flux, and the integral of the electric field around such a
path will be zero.
We can apply the curl relation and obtain E^
dHy dH^ . ^
dx dy
(6.14)
coe Xdx^ dy"^)
CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 241
Applying Maxwell's equations again, we have
dEz dEy
3 ^ = ico/x£?x
ay dz
j d fd T d''7t \ _|_ ir ^x _ . dit
coe dy ydx^ dy^ / coe dy dy
(6.15)
This is certainly true if
■^2
iSo = cov^ = co/c (6.17)
We find that this satisfies the other curl E relations as well.
From (6.16) and (6.14) we see that
E. = (-i/coe)(r2 + ^l^^^^ 3,),-r^ (^ Ig)
For a given physical circuit, it will be found that there are certain real
functions 7r„(x, 3') which are zero over the conducting boundaries of the
circuit, assuring zero tangential field at the surface of the conductor, and
which satisfy (6.16) with some particular value of F, which we will call r„ .
Thus, as a particular example, for a square waveguide of width W some
(but not all) of these functions are
T^n(x, y) = cos (mry/W) cos (utx/W) (6.19)
where n is an integer. We see from (6.10), (6.11) and (6.18) that this makes
Ex , Ey and Ez zero at the conducting walls x = ±:W/2, y = ±W/2.
Each possible real function Ttn{x, y) is associated with two values of
r„ , one the negative of the other. The r„'s are the natural propagation
constants of the normal modes, and the tt^'s are the functions giving their
field distribution in the x^ y plane. The 7r„'s can be shown to be orthogonal,
at least in typical cases. That is, integrating over the region in the x, y
plane in which there is field
/ / Ttn{x, y) 7r„(x, y) dx dy ^ 0
(6.20)
n 9^ m
For a lossless circuit the various field distributions fall into two classes:
those for which r„ is imaginary, called active modes, which represent
waves which propagate without attenuation; and those for which r„ is
real, which change exponentially with amplitude in the z direction but do
not change in phase. The latter can be used to represent the disturbance
in a waveguide below cutoff frequency, for instance.
242 BELL SYSTEM TECHNICAL JOURNAL
If r„ is imaginary (an active mode) the power flow is real, while if r„ is
real (a passive mode) the power flow is imaginary (reactive or "wattless"
power).
The spatial distribution functions 7r„ and the corresponding propagation
constants r„ are a means for si)ecifying the electrical properties of a physical
structure, just as are the physical dimensions which describe the physical
structure and determine the various 7r„'s and r„'s. In fact, if we know the
various 7r„'s and r„'s, we can determine the response of the structure to an
impressed current without direct reference to the physical dimensions.
In terms of the 7r„'s and r„'s, we can represent any unforced disturbance
in the circuit in the form
Y.^n{x, y)[Ane-^"' + ^„/"n (6.21)
n
Here An is the complex amplitude of the wave of the ni\i spatial distribu-
tion traveling to the right, and Bn the complex amplitude of the wave of
the same spatial distribution traveling to the left.
It is of interest to consider the power flow in terms of the amplitude, An
or Bn . We can obtain the power flow P by integrating the Poynting vector
over the part of the .v, y plane within the conducting boundaries
(6.22)
P -\ll (^-^* - E^y^*) dx dy
By expressing the fields in terms of the stream function, we obtain
---"'^wK-y+fex
dx dy (6.23)
We can transform this by integrating by parts (essentially Green's
theorem). Thus
I — — dx — TTn — — / TT,, - ^ dx (6.24)
Jxi ox ax dx xi Jxi dx'
Here Xi and x^ , the limits of integration, lie on the conducting boundaries
where 7r„ = 0, and hence the first term on the right is zero. Doing the same
for the second term in (6.23), we obtain
CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 243
By using (6.16), we obtain
Pn = AnAt 0~\ (r; + /35) ff (wnY dx dj (6.26)
It is also of interest to express the z component of the nth mode, Ezn ,
expHcitly. For the wave traveling to the right we have, from (6.18),
£.„ = An (^j (r'„ + /3n)7^„(.^^ y) (6.27)
Let the field at some particular position, say, x = y = 0, be E^no ■ Then
^"- (rl + /3^)x„(0,0) ^^-2^^
and from (6.26)
^•' = (^'""^"'"*^ 2,l(oro)(rl'+ gg) // l*"(^' ^)'' '"" ''y ('^■2")
We can rewrite this
E^noE^nO* 27^l(0, 0)(rl + /3o)
{-Tl)Pn . ^ , ^2^ /Tr. / M2, , (6.30)
jwer„( — r^„) // [7r„(:r, y)f dx dy
For an active mode in a lossless circuit, r„ is a pure imaginary, and the
negative of its square is the square of the phase constant. Thus, for a par-
ticular mode of propagation we can identify (6.30) with the circuit parame-
ter E?/0^P which we used in Chapter II.
Let us now imagine that there is an impressed current J which flows in
the z direction and has the form
/ = J(x, y)g~J (6.31)
According to Maxwell's equations we must have
dx dy
Now, we will assume that the fields are given by some overall stream func-
tion TT which varies with x and y and with z as exp(— F^).
In terms of this function tt, Hx , Hy and Ex , Ey will be given by relations
(6.7), (6.8), (6.10), (6.11). However, the relation used in obtaining Ez is
not valid in the presence of the convection current. Instead of (6.16) we
have
dHy dHx • z. , r
dx dy
(6.33)
244 BELL SYSTEM TECHNICAL JOURNAL
Again applying the relation
dE^ dEy
dy dz
we obtain
ft + ^= -(t' + ^Dt-J (6.34)
We will now divide both tt and / into the spatial distributions charac-
teristic of the normal unforced modes.
Let
J{x, y) = ^ JnTTnix, y) (6.35)
n
// J{x, y)Trn{x, y) dx dy
Jn = (6.36)
// [ftnix, y)] dx dy
This expansion is possible because the 7r„'s are orthogonal. Let
Tt = e~ ' zl CnTtn{x, y) (6.37)
n
Here there is no question of forward and backward waves; the forced ex-
citation has the same ^-distribution as the forcing current.
For the wth component, we have, from (6.16),
dTrn{x,y) dirn{x,y) , i , ^2w / n
dx^ dy-
From (6.34) we must also have
^ /d'TTnix, y) d^TTnix, y)
(6.38)
= — C„(r" +;So)7r„(x, y) — JnTTnix, y)
Accordingly, we must have
The overall stream function is thus
7r = e-"E^#^" (6.41)
n i n i
From (6.33) and (6.34) we see that
E, = ^ (r'' + fiDir (6.42)
CIRCUIT DESCRIBED IN TERMS OP NORMAL MODES 245
So
E. = e E ^^(p. _ r^) (6.43)
£. = Zi[i:±^ ,- Z !^|^ C6.44)
coe 1 ;, — 1 -^
6.2 Comparison with Results of Chapter II
Let us consider a case in which there is only one mode of propagation,
characterized by 7ri(:K, 3^), Fi, and a case in which the current flows over a
region in which 7ri(x, y) has a constant value, say, 7ri(0, 0). This corre-
sponds to the case of the transmission line which was discussed in Chapter
II.
We take only the term with the subscript 1 in (6.44) and (6.30). Combin-
ing these equations, we obtain for the field at 0, 0
E, =
{E^/P'P)(T' + ^l) ^^-^^ // f"^^^' ^^^' ^^ ^^
(VI + ^i) 2f^i(o, 0)
We have from (6.36)
7ri(0, 0)
Ji =
If IHx, y)]^ dx dy
(6.45)
(6.46)
From (6.45) and (6.46) we obtain
2(r? + /3^)(r? - r')
Let us compare this with (6.4), which came from the transmission line
analogy of Chapter II, identifying Ez and / with E and i. We see that,
for slow waves for which
iSo « I r'x I (6.48)
j8o « I r' I (6.49)
(6.47) becomes the same as (6.4). It was, of course, under the assumption
that the waves are slow that we obtained (2.10), which led to (6.4).
6.3 Expansion Rewritten in Another Form
Expression (6.44) can be rewritten so as to appear quite different. We
can write
r' + /3'o = r' - tI + r'„ + ^l
246 BELL SYSTEM TECHNICAL JOURNAL
Thus, we can rewrite the expression for Ez as
77 .-r^ // •/ N Y^ (!"" + 0o)7tn(x, y)Jn
E. = e ^( -jM Z _ ^,
(6.50)
+ (i^f) S 7r„(.v, y)Jn)
The second term in the brackets is just j/coc times the impressed current,
as we can see from (6.35). The first term can be rearranged
i-jMivl + 0l)Jn
{-j/o:e)ivl + /3o) // Tnix, y)J(x, y) dx dy {(),^\)
11 [Ttnix, y)f dx dy
Referring back to (6.29), let ^„ be twice the power P„ carried by the
unforced mode when the field strength is
I £^no I = 1 (6.52)
Further, let us choose the7r,j's so that, at some specified position, x = y = 0,
„(0, 0) = 1 (6.53)
Then
Using this in connection with (6.51), we obtain
TnTTnix, y) 1 1 TTnix, y)j{x, y) dx dy
E^ = e-'\ - E
^«(r?. - n)
+ (i/we)y(x, y)
(6.55)
An expression for the forced field in terms of the parameters of the nor-
mal modes was given earlier '". In deriving this expression, the existence of
a set of modes was assumed, and the field at a point was found as an in-
tegral over the disturbances induced in the circuit to the right and to the i
left and propagated to the point in question. Such a derivation applies for
lossy and mixed waves, while that given here applies for lossless transverse-
magnetic waves only.
CIRCUIT DESCRIBED IN TERMS OF NORMAL MODES 247
The earlier derivation^ leads to an expression identical with (6.55) except
that ^n appears in place of ^„ . In this earlier derivation a sign was im-
plicitly assigned to the direction of flow of reactive power (which really
doesn't flow at all!) by saying that the reactive power flows in the direction
in wliich the amplitude decreases. If we had assumed the reactive power to
flow in the direction in which the amplitude increases, then, with the same
definition of ^n , for a passive modern would have been replaced by — '^„
which is equal to S^„ (for a passive mode, ^n is imaginary).
In deriving (6.55), no such ambiguity arose, because the power flow was
identified with the complex Poynting vector for the particular type of wave
considered. In any practical sense, ^ is merely a parameter of the circuit,
and it does not matter whether we call Im SE' reactive power flow to the right
or to the left.
The existence of a derivation of (6.55) not limited in its application to
lossless transverse magnetic waves is valuable in that practical circuits often
have some loss and often (in the case of the heUx, for instance) propagate
mixed waves.
6.4 Iterated Structures
Many circuits, such as those discussed in Chapter IV, have structure in
the z direction. Expansions such as (6.55) do not strictly apply to such struc-
tures. We can make a plausible argument that they will be at least useful
if all field components except one differ markedly in propagation constant
from the impressed current. In this case we save the one component which
is nearly in synchronism with the impressed current and hope for the best.
248 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX III
STORED ENERGIES OE
CIRCUIT STRUCTURES
A3.1 Forced Sinusoidal Field
If i; <3C c, the field can be very nearly represented inside the cylinder of
radius a by
_ T. hW jp^ _ E /o(/3r) y^.
and outside by
Inside
F = Fo ^^ e-^'^ (2)
K{ya)
^=-y^;_^.-^^Fo (4)
Outside
^=_y,|^.-Fo (6)
Because there is a sinusoidal variation in the z direction, the average stored
electric energy per unit length will be
"'"' " (0C2) /=o ^''^'"'^^'^' + (£.max)'j(27rr dr) (7)
Here Er max and Ez max are maximum values at r = a. The total electric
plus magnetic stored energy will be twice this. This gives
{E'/^'P)
W =
W =
1/3
APPENDIX III
ireiya)' \ lo — hh
■weya
L n
+
i^Tn iTo — K.
2-1
0 T^2
K
E-
{c/vY'^ivM
1/3
120
ni/3
249
(8)
(9)
A3. 2 Pill-Box Resonators
Schelkunoff gives on page 268 of Electromagnetic Waves an expression
for the peak electric energy stored in a pill-box resonator, which may be
written as
.135 7r€a2/;£2
Here a is the radius of the resonator and h is the axial length. For a series
of such resonators, the peak stored electric energy per unit length, which is
also the average electric plus magnetic energy per unit length, is
For resonance
Whence
W = .135 7rea2£2
a = 1.2Xo/7r
W = .0618 €Xo2£2
And
(EP/^'-Pyi^ = 5.36 (v/vgY'^ {v/cy'
The case of square resonators is easily worked out.
A3.3 Parallel Wires
(10)
(11)
(12)
(13)
Let us consider very fine very closely spaced half-wave parallel wires with
perpendicular end plates.
If z is measured along the wires, and y perpendicular to z and to the
direction of propagation, the field is assumed to be
Ej: — E cos 8xe cos — z
Ao
Ey = E sin I3xe cos — z
Xo
(14)
Here the + sign applies for y < 0 and the — sign for y > 0. We will then
find that
250 BELL SYSTEM TECHNICAL JOURNAL
W = 2W, = '^
W = — ° E'
4/3
Jo
(15)
and
(F?/l3^Py'' = 6.20 (v/vsY" (16)
The surface charge density a on one side of the array of wires (say, y > 0)
is given by the y component of field at y = 0.
2x
0" = eEy = eE sin /5.r cos — z (17)
This is related to the current 7 (flowing in the z direction) per unit distance
in the .v direction by
dz di
From (18) and (17) we obtain for the current on one side of the array
I = — —z — E sm i5x sm — z (19)
/TT Xo
If we use the fact that a;Xo/27r = c and c e = l/\//x/€, we obtain
—jE, . . 2x
/ = '/=F sin ^x sin — z (20)
VM/f Xo
If R is the surface resistivity of either side (y > 0, y < 0) of the wires, when
the wires act as a resonator (a standing wave) the average power lost per
unit length for both sides is
P = ii?Xo£7(MA) (21)
In this case the stored electric energy is half the value given by (15), and
we find
Q = (ViuA/i?) {v/c) (22)
Factors Affecting Magnetic Quality*
By R. M. BOZORTH
IN THE preparation of magnetic materials for practical use it is impor-
tant to know how to obtain products of the best quahty and uniformity.
In the scientific study of magnetism the goal is to understand the relation
between the structure and composition on the one hand and the magnetic
properties on the other. From both standpoints it is necessary to know the
principal factors which influence magnetic behavior. These are briefly
reviewed here.
The properties depend on chemical composition, fabrication and heat-
treatment. Some properties, such as saturation magnetization, change only
slowly with chemical composition and are usually unaffected by fabrication
or heat treatment. On the contrary, permeability, coercive force and hystere-
sis loss are highly sensitive and show changes which are extreme among all
the physical properties. Properties may thus be divided into slruclure-
sensilive and structiire-inseusitive groups. As an example. Fig. 1 shows mag-
netization curves of permalloy after it has been (a) cold rolled, (b) annealed
and cooled slowly, and (c) annealed and cooled rapidly. The maximum
permeability varies with the treatment over a range of about 20 fold, while
the saturation induction is the same within a few per cent. Structure sensi-
tive properties such as permeability depend on small irregularities in atomic
spacings, which have little effect on properties such as saturation induction.
Some of the more common sensitive and insensitive properties are listed
in Table I. The principal physical and chemical factors which affect these
properties are listed in column 3. Their various effects will now be briefly
discussed and illustrated.
Phase Diagram
Some of the most drastic changes in properties occur when the fabrication
or heat treatment has brought about a change in structure of the material.
For this reason the phase diagram or constitutional diagram is of the ut-
most importance in relation to the preparation and properties of magnetic
materials. As an example consider the phase diagram of the binary iron-
cobalt alloys of Fig. 2. Here the various areas show the phases, of different
*This article is the substance of Chapter II of a Iraok entitled "Ferromagnetism" to
be published early in 1951 by D. Van Nostrand Company, Inc.
251
252
BELL SYSTEM TECHNICAL JOURNAL
composition or structure, which are stable at the temperatures and com-
positions indicated. The a phase has the body-centered-cubic crystal struc-
ture characteristic of iron. At 910°C it transforms into the face-centered
phase 7, and at 1400° into the 5 phase, which has the same structure as the
a phase. At about 400°C cobalt transforms, on heating, from the e phase
(hexagonal structure) into the 7 phase.
10
14
70 PERMALLOY
If-^ ANNEALED AND
^^' COOLED RAPIDLY
12
10
8
6
4
2
0
/
"
A
/.\ ANNEALED AND
( \°) COOLED SLOWLY
/
^
^
/
^
^
_^
/
(a) COLD ROLLED
L
y
0 2 4 6 8 10 12 14 16 18 20
FIELD STRENGTH, H, IN OERSTEDS
Fig. 1 — Effect of mechanical and heat treatment on the magnetization curve of 70
permalloy (70% Ni, 30% Fe).
Table I
Properties Commonly Sensitive or Insensitive to Small Changes in Structure, and Some of the
Factors which Effect Such Changes
Structure-Insensitive Properties
Structure-Sensitive
Properties
Factors Affecting the
Properties
/, , Saturation Magnetization
6, Curie Point
Xs , Magnetostriction at Saturation
K, Crystal Anisotropy Constant
M, Permeability
He Coercive Force
Wh Hysteresis Loss
Composition (gross)
Impurities
Strain
Temperature
Crystal Structure
Crystal Orientation
The dotted lines indicate the Curie point, at which the material becomes
non-magnetic.
In between the areas corresponding to the single phases a, 7, 8 and e
there are two-phase regions in which two crystal structures co-exist, some
of the crystal grains having one structure and others the other. Such a two-
phase structure is usually evident upon microscopic or X-ray examina-
FACTORS AFFECTING MAGNETIC QUALITY
253
ATOMIC PER CENT COBALT
30 40 50 60
Co
^7 + MELT
'^
7 + MELT
7 (face-centered)
MAGNETIC
TRANSFORMATION ,-
TRANSFORMATION
400 -
«^ + 7 I 7+
7+e
(b).
I , I (
Fe
40 50 60
PER CENT COBALT
Co
Fig. 2 — -Phase diagram of iron-cobalt alloys.
0.02% CARBON
0.06 *Vo CARBON
, -\-- ' • ■■■■■-
r. ■■ ■
'
«*!..- •. -
•*
/--^_
^ *, - ,
/ • "'•^--- /- • ■
> . ■'.,
>,
/ ~-.'
\t'
/ ■; , . A.
/ ■
/
/
IRON-COBALT-
MOLYBDENUM
Fig. 3— Photomicrographs of remalloy (12% Co, 17% Mo, 71% Fe) showing the pre-
cipitation of a second phase in the specimen containing an excess of carbon (0.06%)
Courtesy of E. E. Thomas. Magnification: (a) 50 times, (b) 200 times.
254
BELL SYSTEM TECHNICAL JOURNAL
tion. Microphotographs of a single-phase alloy and a two-phase alloy of
iron-cobalt-molybdenum are reproduced in Fig. 3 (a) and (b).
The diagram of Fig. 2 shows several kinds of changes that afifect the mag-
netic properties. At (a) the material becomes non-magnetic on heating,
without change in phase. At (b) there is a change of phase, both phases
;^ 400
(J
2 100
Ni
H = ABOUT
150 OERSTEDS
n
Jf
\
V
^J
If
V
y
IRON-COBALT
1
\
200 400 600 800 1000
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 4 — Effect of phase transformation of cobalt on magnetization with a constant
field of 150 oersteds. Both phases magnetic. Masunioto.
5 16
— >.
^
^^^\
\,
IRON-COBALT
\
0 200 400 600 800 1000 1200
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 5 — Phase transformation in iron-cobalt alloy (50% Co). High-temperature phase
is non -magnetic.
being magnetic. Figure 4 shows the changes in magnetic properties that
occur during this latter transition; they are due partly to the high local
strains that result from the change in structure, and partly to the difference
in the crystal structures of the two phases. At (c) there is a change from a
ferromagnetic to a non-magnetic phase, and Fig. 5 shows the rapid change
in magnetization that occurs when the temperature rises in this area. At
FACTORS AFFECTING MAGNETIC QUALITY
255
(d) the a phase becomes ordered on cooling, i.e., the iron and cobalt atoms
tend to distribute themselves regularly among the various atom positions
so that each atom is surrounded by atoms of the other kind. This phenome-
non is especially important in connection with the properties of iron-alumi-
num and manganese-nickel alloys.
The transition at (e) is entirely in the non-magnetic region but it has
its influence on the properties of iron at room temperature. If iron is cooled
very slowly through (e), the internal strains caused by the change in struc-
ture will be relieved by diffusion of the metal atoms, but if the cooling is too
rapid there will not be sufficient time for strain relief. Practically this means
that to obtain high permeability in iron it must be annealed for some time
below 900°C, or cooled slowly through this temperature so that diffusion
will have time to occur. In most ferromagnetic materials diffusion occurs
at a reasonably rapid rate only at temperatures above about 500 to 600°C.
10^x16
:^\2
>'
-TENSION,(7'=8KG/Mm2
Z'
1
/"
TENSION, 0" = 0
/
68 PERMALLOY
FIELD STRENGTH, H, IN OERSTEDS
Fig. 6 — Effect of tension on the magnetization curve of 68 permalloy.
The effect of a homogeneous strain on the magnetization curve can be
observed in a simple way, as by applying tension to an annealed wire and
then measuring B and H. The efifect of tension on some materials is to
increase the permeability and on other materials to decrease it, as shown
in Fig. 6. Compression usually causes a change in the opposite sense.
The internal strains resulting from plastic deformation of the material,
brought about by stressing beyond the elastic hmit, as by pulling, rolling
or drawing, almost always reduce the permeability. The material is then
under rather severe local strains similar to those present after phase change,
and these strains are different in magnitude and direction in different places
in the material and have quite different values at points close together.
Strains of this kind can usually be relieved by annealing; therefore, metal
that has been fabricated by plastic deformation is customarily annealed to
raise its permeability. Figure 1 shows the effect of annealing a permalloy
strip that has been cold-rolled to 15 per cent of its original thickness.
256
BELL SYSTEM TECHNICAL JOURNAL
The temperature also is effective in changing permeability and other prop-
erties, even when no change in phase occurs. Figure 7 shows the rapidity
with which the initial permeability decreases as the Curie point is ap-
proached. For this material, Ferroxcube III, a zinc manganese ferrite
(ZnMnFe408), the Curie point is not far above room temperature.
The effect of impurities may be illustrated by the B vs H curves for iron
containing various amounts of carbon. Curve (a) of Fig. 8 is for a mild
5
iiJ 1000
_
■— -^
\
FERROXCUBE 3
\
20 40 60 80 100 120
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 7 — Variation of initial permeability of Ferroxcube 3, showing maximum at tem-
perature just below the Curie temperature.
lo^xie
O 6
/
-(C) <0. 001% C
^--
Z^
- —
-^-
/^
■ —
^^'
.--^
^
"
/
f
/''
z'
1
1
1
/
/[a) 0.2% C
1
1
1
/
1
f
/
MILD STEEL
1
1
7
'-{\D] 0.02% C
J^
/
0 I 23456789 10 11 12
FIELD STRENGTH, H, IN OERSTEDS
Fig. 8 — -Effect of impurities on magnetic properties of iron. Annealing at 1400''C in
hydrogen reduces the carbon content from about 0.02 per cent to less than 0.001 per cent.
steel having 0.2 per cent carbon, (b) is for the iron commonly used in elec-
tromagnetic apparatus — it contains about 0.02 per cent carbon and is an-
nealed at about 9()0°C. When this same iron is purified by heating for several
hours at 1400°C in hydrogen, the carbon is reduced to less than 0.001 per
cent and other impurities are removed, and curve (c) is obtained.
Finally, Fig. 9 shows that large differences m permeability may be found
by simply varying the direction of measurement of the magnetic properties
in a single specimen. The material is a single crystal of iron containmg about
4 per cent silicon, and the directions in which the properties are measured
FACTORS AFFECTING MAGNETIC QUALITY
257
are [100] (parallel to one of the crystal axes), and [111] (as far removed as
possible from an axis). The magnetic properties in the two directions are
different because different "views" of the atomic arrangement are ob-
tained in the two directions.
Production of Magnetic Materials
In the preparation of magnetic materials for either laboratory or commer-
cial use there are many processes which influence the chemical and physical
10^X650
bSU
SINGLE CRYSTAL
A
OF SILICON IRON
1
'\
500
450
400
350
300
/
\
/
\
I
/
f
\[100]
/
\
/
\
/
\
250
/
\
/
\
200
150
100
/
\,
/
\
\
/
\
/
{
/
[110]
50
/
[111]
\.
\
u,
V.
^
"0 2 4 6 8 10 12 14 16 18X10^
INTRINSIC INDUCTION, B-H, IN GAUSSES
Fig. 9 — Dependence of permeability on crystallographic direction. Williams.
structure of the product. The selection of raw materials, the melting and
casting, the fabrication and the heat treatment, are all important and must
be carried out with a proper knowledge of the metallurgy of the material. A
brief description of the common practices is now given. For further dis-
cussion the reader is referred to more detailed metallurgical books and ar-
ticles.
258
BELL SYSTEM TECHNICAL JOURNAL
Melling and Casting
For experimental investigation of magnetic materials in the laboratory,
the raw materials easily obtainable on the market are generally satisfactory.
When high purity is desirable specially prepared materials and crucibles
must be used and the atmosphere in contact with the melt must be con-
trolled. The impurities that have the greatest influence on the magnetic
properties of high permeability materials are the non-metallic elements,
Vig. 10 — Induction furnace designed for small niclts in controlled atmosphere, as de-
signed hy J. H. Scaff and constructed by the Ajax Northrup Company.
j)articularly oxygen, carbon and sulfur, and the presence of these im[)urities
is therefore watched carefully and their analyses are carried out with special
accuracy. Impurities are likely to change in important respects during the
melting and pouring on account of reactions of the melt with the atmos-
phere, the slag or the crucible lining, or because of reactions taking place
among the constituents of the metal.
Melting of small lots (10 pounds) is best carried out in a high-frequency
induction furnace, l-'igure 10 shows such a furnace designed for melting ten
to lifty pounds, and casting by tilting the furnace, the whole operation being
FACTORS AFFECTING MAGNETIC QUALITY
259
carried out in a controlled atmosphere. High-frequency currents (usually
1,000 to 2,000 cycles/sec but sometimes much higher) are passed through
the water-cooled copper coils, and the alternating magnetic field so produced
Fig. 11— .-^rc luniace lor large Lumiiicrcial nit-lls. CoiulcbV ul" J. S. Marsh of the Bethle-
hem Steel Company.
heats the charge by inducing eddy currents in it. Crucibles are usually com-
posed of alumina or magnesia.
On a commercial scale melts of silicon-iron are usually made in the open
260
BELL SYSTEM TECHNICAL JOURNAL
hearth furnace, in which pig-iron and scrap are refined and ferro-siUcon
added. The furnace capacity may be as large as 100 tons. Sometimes siUcon-
iron, and usually iron-nickel alloys, are melted in the arc furnace, in
amounts varying from a few tons to 50 tons. A photograph of such a fur-
nace, in the position of pouring, is shown in Fig. 11. The heat is produced
in the arc drawn between large carbon electrodes immersed in the metal,
the current sometimes rising to over 10,000 amperes. By tipping the fur-
nace the melt is poured into a ladle, and from this it is poured into cast-iron
molds through a valve-controlled hole in the ladle bottom. Special-purpose
alloys, including permanent magnets, are prepared commercialh^ in high-
Table II
Heats of Formation and Other Properties of Some Oxides {Sachs and Van Horn'^)
Oxide
Heat of formation
(Kilo-cal per gram
atom of metal)
Melting Point (°C)
Density (g/cm')
CaO
152
144
144
141
127
116
109
101
95
94
91
89
85
73
68
66
58
>2500
>2500
2800
>1700
2050
1970
1640
*
1670
580
1650
2700
*
*
1130
1420
**
3.4
BeO
3 0
MgO
UiO
3.65
2 0
AI2O3
3.5
V2O2
4.9
Ti02
NajO
4.3
2 3
Si02
2.3
B2O3
MnO
ZrOz
ZnO
P2O6
1.8
5.5
5.5
5.5
2 4
Sn02
FeO
6.95
5.7
NiO
7.45
* Sublimes.
** Decomposes before melting.
frequency induction furnaces or in arc furnaces in quantities ranging from a
fraction of a ton to several tons.
Slags are commonly used when melting in air, both to protect from oxi-
dation and to reduce the amounts of undesirable impurities. Common pro-
tective coverings are mixtures of lime, magnesia, silica, fluorite, alumina,
and borax in varying proportions. In commercial production different slags
are used at different stages, to refine the melt; e.g., iron oxide may be used
to decarburize and basic oxides to desulfurize.
Melting in vacuum requires special technique that has been described in
some detail by Yensen.^ Commercial use has been described by Rohn^ and
others.' Melting in hydrogen has been used on an experimental scale in both
•T. D. Yensen, Trans. A.I.E.E. 34, 2601-41 (1915).
2 VV. Rohn, Heraeus Vacuumsclimelze, Alberlis, Hanau, 356-80 (1933).
nV. Hessenbruch and K. Schichlel, Zeits. f. Metallkunde 36, 127-30 (1944).
FACTORS AFFECTING MAGNETIC QUALITY
261
high-frequency and resistance-wound furnaces. In commercial furnaces Rohn
has used hydrogen and vacuum alternately before pouring, for purification
in the melt, in low-frequency induction furnaces having capacities of several
tons.
Just before casting a melt of a high-permeability alloy such as iron nickel,
a deoxidizer may be added, e.g. aluminum, magnesium, calcium or silicon,
in an amount averaging around 0.1 per cent. The efficacy of a deoxidizer is
measured by its heat of formation, and this is given for the common ele-
- 240
V)
m 200
160
uj 80
<
!"•/
9
^2
r^.
A^^v
'->4
^>-,
)
y
^\
,<^'^
j^
/
>^^^
o>/
9
^
y
^^
^
-::
600 800 1000 1200 1400 1600
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 12 — Solubility of some gases in iron and nickel at various temperatures. Sieverts.
ments in Table II, taken from Sachs and Van Horn.^ Also several tenths of a
per cent of manganese may be put in to counteract the sulfur so that the
material may be more readily worked; the manganese sulfide so formed col-
lects into small globular masses which do not interfere seriously with the
magnetic or mechanical properties of most materials.
Ordinarily a quantity of gas is dissolved in molten metal, and this is likely
to separate during solidification and cause unsound ingots. The solubilities
of some gases in iron and nickel have been determined by Sieverts^ and
others and are given in Fig. 12, adapted from the compilation by Dushman.®
The characteristic decrease of solubility during freezing is apparent. Most
* G. Sachs and K. R. Van Horn, Practical Metallurgy, Am. See. Metals, Cleveland
(1940).
6 A. Sieverts, Zeits.f. Metallkimde 21, 37-46 (1929).
^S. Dushman, Vacuum Technique, Wiley, New York (1949).
262
BELL SYSTEM TECHNICAL JOURNAL
of the gases given off by magnetic metals during heating are formed from
the impurities carbon, oxygen, nitrogen and sulfur; CO is usually given off
in greatest amount from cast metal, and some No and H2 are also found.
Refining of the melt is therefore of obvious advantage, and the furnace of
Fig. 10 is especially useful for this purpose.
Small ingots are sometimes made by cooling in the crucible. Usually,
however, ingots are poured into cast iron molds for subsequent reduction
by rolling, etc.; permanent magnet or other materials are often cast in si;nd
Fig. 13 — Design of rolls in a blooming mill for hot reduction of ingots to rod. Carnegie
Illinois Steel Corp.
in shapes which require only nominal amounts of machining or grinding
for use in apparatus or in testing. Special techniques are used for specific
materials.
Other considerations important in the melting and pouring of ingots are
proper mixing in the melt, the temperature of pouring, mold construction,
inclusions of slag, segregation, shrinkage, cracks, blow holes, etc.
Fabricalioii
Magnetic materials require a wide variety of modes of fabrication, which
can best be discussed in connection with the specific materials. The methods
include hot and cold rolling, forging, swaging, drawing, pulverization, elec-
FACTORS AFFECTING MAGNETIC QUALITY 263
trodeposition, and numerous operations such as punching, pressing and
spinning. In the commercial fabrication of ductile material it is common
practice to start the reduction in a breakdown or blooming mill (Fig. 13)
after heating the ingot to a high temperature (1200° to 1400°C). Large ingots,
of several tons weight, are often led to the mill before they have cooled
below the proper temperature. The reduction is continued as the metal
cools, in a rod or flat rolling mill, depending on the desired form of the final
product. When the thickness is decreased to 0.2 to 0.5 inch the material has
usually cooled below the recrystallization temperature. Because of the diffi-
culty in handhng hot sheets or rod of small thickness, they are rolled at or
near room temperature, with intermediate annealings if necessary to soften
or to develop the proper structure. In experimental work, rod is often
swaged instead of rolled.
In recent years the outstanding trends in methods of fabricating materials
have been toward the construction of the multiple-roll rolling mill for roll-
ing thin strip, and the continuous strip mill for high-speed production on
a large scale. Figure 14 shows the principle of construction of a typical 4-high
mill ((a) and (b)), and of two special mills ((c) and (d)). In the 20-high
Rohn'^ mill and 12-high Sendzimir^ mill the two working rolls are quite
small (0.2 to one inch in diameter). These are each backed by two larger
rolls and these in turn by others as indicated. In the Rohn mill (c), power is
supplied to the two smallest rolls and the final bearing surfaces are at the
ends of the largest rolls. In the Sendzimir mill (d) the power is suppHed to
the rolls of intermediate size and the bearing surfaces are distributed along
the whole length of the largest rolls so that no appreciable bending of the
rolls occurs. The small rolls reduce the thickness of thin stock with great
efficiency, and the idling rolls permit the application of high pressure.
In the Steckel mill power is used to pull the sheet through the rolls, which
are usually 4-high with small working rolls.
The continuous strip mill is an arrangement of individual mills such that
tlie strip is fed continuously from one to another and may be undergoing
reduction in thickness in several mills simultaneously. Figure 15 shows a
mill of this kind, used for cold reduction, with 6 individual mills in tandem.
For magnetic testing numerous forms of specimens are required for vari-
ous kinds of tests; these include strips for standard tests for transformer
sheet, rings or parallelograms for conventional ballistic tests, "pancakes"
of thin tape spirally wound for measurement by alternating current, ellip-
soids for high field measurements, and many others. The various forms are
■^ W. Rohn, Heraeus Vacuumschmelze, Albertis, Hanau, 381-7 (1933).
8T. Sendzimir, Iron and Steel Engr. 23, 53-9 (1946).
264
BELL SYSTEM TECHNICAL JOURNAL
required lo study or eliminate the effects of eddy-currents, demagnetizing
lields and directional effects and to simulate the use of material in apparatus.
Most of the needs arizing in commerce and in experimental investigation
are filled by strips or sheets of thicknesses from 0.002 inch to 0.1 inch from
(a) 4-HIGH MILL, SIDE VIEW
(b) 4-HIGH MILL, END VIE\,
(C) 20-HIGH ROHN MILL {d)l2-HIGH SENDZIMIR MILL
Fig. 14 — Arnmgement of rolls ia mills used for reduction of ihin sheet: (a) and (h) con-
ventional 4-high mill; (c) Rohn 20-high; (d) Sendzimir 12-high.
which coils can be wound or parts cut, by rods from which relay cores or
other forms can be made, by powdered material used for pressing into cores
for coils for inductive loading, and by castings for permanent magnets or
other objects which may be machined or ground to final shape.
FACTORS AFFECTING MAGNETIC QUALITY
265
266
BELL SYSTEM TECHNICAL JOURNAL
Heal-Trealmenl
High permeability materials are annealed primarily to relieve the internal
strains introduced during fabrication. On the contrary permanent magnet
materials are heat-treated to introduce strains by precipitating a second
phase. Heat-treatments are decidedly characteristic of the materials and
their intended uses and are best discussed in detail in connection with them.
1300
1200
900
800
700
600
500
400
300
200
100
/
-— «
/
PUB
IFICA'
noN
"~^
\
/
\
//
w
7
^
1
\ ^ — DOUBLE TREATMENT
Vf (MAY BE COOLED RAPIDLY
\\ TO ROOM TEMPERATURE
\\ AND REHEATED TO 600°C)
/
1
'>H
r
1
\
\
/
1 1
g^i^p 1
1
1
J
\
/
1
>
1
1
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AIR QUENCH A
»
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1
K
s
V FURNACE
^v^^ COOL
1
1
'\
V
-^
\
\
TIME IN HOURS
Fig. 16 — Some common heat treatments for magnetic materials.
Figure 16 shows some of the commonest treatments in the form of tempera-
ture-time curves. The purpose of these various heating and cooling cycles,
and typical materials subjected to them, may be listed as follows:
(1) Relief of internal strains due to fabrication or phase-changes (furnace
cool). Magnetic iron.
(2) Increase of internal strains by precipitation hardening (air quench
and bake). Alnico type of permanent magnets.
FACTORS AFFECTING MAGNETIC QUALITY
267
(3) Purification by contact with hydrogen or other gases. SiUcon-iron
(cold rolled), hydrogen- treated iron, Supermalloy.
There are also special treatments, such as those used for "double-treated"
permalloy, "magnetically annealed" permalloy, and perminvar.
Occasionally it is necessary to homogenize a material by maintaining the
lO^x 32
MAXIMUM PERMEABILITY,
Mm
^A
l\
;;_i^^
SifOAy
r^
pl.^^^
[iON^
hi
>\
,_cu
^E PC
)INJT,
9
■'^—- -»
-•--.
1
J
V
y
/
\
ri r
"""■"
400 I
O
-^^
900 9
600 (/I
500 S',
400
^
<I)
^
H
Z
o
tl
200
111
0 1 23456789 to
PER CENT SILICON IN IRON
Fig. 17 — Variation of some properties of iron-silicon alloys with composition: B,, satura-
tion intrinsic induction; 0, magnetic transformation point; p, electrical resistivity; /xm,
maximum permeability as determined by Miss M. Goertz.
temperature just below the freezing point for many hours. Heat-treatments
also may affect grain size and crystal orientation.
Furnaces for heat-treating have various designs that will not be considered
here. A modern improvement has been the use of globar (silicon carbide)
heatmg elements that permit treatment at 1300 to 1350°C in an atmosphere
of hydrogen or air.
268
BELL SYSTEM TECHNICAL JOURNAL
Further discussion of "Metallurgy and Magnetism" is given in an exxel-
lent small book of this title by Stanley. '■•
Effect of Composition
Gross Chemical Composilioii
The effect of com])osition on magnetic properties will now be considered,
using as examples the more important binary alloys of iron with silicon,
10^x24
^'
\
\
-■--.
^"^
\
/
^
"X
<i
\
\
/
\
\
i
y
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\
N
/
/
/
1
\
\
\
\
a
/
/
f
y
\
/
/
'/
/
/
1
500 qT
30 40 60 60 70 80
PER CENT NICKEL IN IRON
Fig. 18 — Variation of Bs and B wilh the composition of iron-nickel alloys.
nickel or cobalt, on which are based the most useful and interesting mate-
rials. The iron-silicon alloys are used commercially without additions, the
iron-nickel and iron-cobalt alloys are most useful in the ternary form; and
many special alloys, for example material for permanent magnets, contain
four or live components.
J'"igure 17 shows four im])ortant properties of the iron-silicon alloy's of low
silicon content, after they have been hot rolle<l and annealed. The commer-
cial alloys (3 to 5% silicon) are the most useful because they have the best
' J. K. Stanley, Metallurgy and Magnetism, Am. Soc. Metals, Cleveland (1949).
FACTORS AFFECTING MAGNETIC QUALITY
269
combination of properties of various kinds. The properties shown in the
figure are important in determining the best balance: the maximum per-
meabih'ty, ^im , only indirectly (it is a good measure of hysteresis loss and
maximum field necessary in use), and the Curie point, d, only in a minor
role. The saturation Bs , permeabiUty, and resistivity p, should all be as
high as possible. Bs , 6 and p are structure insensitive, and vary with com-
position in a characteristically smooth way, practically independent of
heat treatment; jum depends on heat treatment (strain), impurities and
crystal orientation. There are no phase changes to give sudden changes with
composition of properties measured at room temperatures.
4x10^
\ K
"\
A
/ \
/
'TN
\
/ \
/ \
/ »
/ »
/
\
/~-
K
N
\
r
\
\
20 40 60 80
PER CENT NICKEL IN IRON
Fig. 19 — Variation of saturation magnetostriction, Xs, and crystal anisotropy, A', with
the composition of iron-nickel alloys.
Some of the properties of the iron-nickel alloys are given in Figs. 18 and
19. The change in phase from a to 7 at about 30 per cent nickel is responsible
for the breaks at this composition. The permeabilities, yuo and Hm , (Fig- 20)
show characteristically the effect of heat treatment. The maxima are closely
related to the points at which the saturation magnetostriction, X,. , and crys-
tal anisotropy. A', pass through zero (Fig. 19).
Additions of molybdenum, chromium, copper and other elements are
made to enhance the desirable properties of the iron-nickel alloys.
The iron-cobalt alloys, some properties of which are shown in Fig. 21, are
usually used when high inductions are advantageous. The unusual course of
the saturation induction curve, with a maximum greater than that for any
other material, is of obvious theoretical and practical importance. The sud-
270
BELL SYSTEM TECHNICAL JOURNAL
10^X10
o
(a)
r
/
1
1
RAPID
COOL
\
/
/
/
I
\
V
r
\
SLOW \
COOL-s, \
V
/
"■---_
— -'''
^xV
£2 50
2
D 100
X
<
2 c«
(b)
/^
MAGNETIC/
ANNEAL /
\
\
/
1 RAPID/^
COOL/
\
_^
y^
1
\ SLOW
J<iQ.OOL
V"''
~-^
r** — i
40 50 60
PER CENT NICKEL
Fig. 20 — Dependence of the initial and maximum permealiilities (yuo, Mm) of iron-nickel
alloys on the heat treatment.
oz
__Bs
.<»*'
6 "'
*•
r^v^
/
oc
^ 1
1
«i o
0 20 40 60 80 100
PER CENT COBALT IN IRON
Fig. 21 — Variation of B, and Q of iron-cobalt alloys with composition.
FACTORS AFFECTING MAGNETIC QUALITY
271
den changes in the Curie point curve are associated with a, 7 phase boun-
daries, as mentioned earlier in this chapter. The peak of the permeability
curve (Fig. 22) occurs at the composition for which atomic ordering is stable
at the highest temperature (see also Fig. 2). The sharp decline near 95 per
cent cobalt coincides with the phase change y,e at this composition. Addi-
tions of vanadium, chromium and other elements are used in making com-
mercial ternary alloys.
Some useful alloys based on the binary iron-sihcon, iron-nickel and iron-
cobalt alloys are described in Table III.
The hardening of material resulting from the precipitation of one phase in
another is often used to advantage when magnetic hardness (as in per-
manent magnets), or mechanical hardness, is desired. To illustrate this
10^x2.0
ftr 1.2
-jUJ
= 0
5 "0.8
0.4
A
k
■ — ■
r\^
/
\
/
'""'-x
•<•
1
1
\
1
1
600
20 40 60 80
PER CENT COBALT IN IRON
Fig. 22 — Variation of permeability at H = 10 oersteds, and of the critical temperature
of ordering, with the composition of iron-cobalt alloys.
process consider the binary iron molybdenum alloys, a partial phase dia-
gram of which is given in Fig. 23. The effect of the boundary between the
a and a -j- e fields is shown by the variation of the properties with composi-
tion (Fig. 24a). Saturation magnetization and Curie point are affected but
little, the principle change in the former being a slight change in the slope
of the curve at the composition at which the phase boundary crosses 5(X)°C,
the temperature below which diffusion is very slow. The Curie point curve
has an almost imperceptible break at the composition at which the phase
boundary lies at the Curie temperature. The changes of maximum per-
meability and coercive force are more drastic ; Hm drops rapidly as the amount
of the second phase, e, increases and produces more and more internal strain
(Fig. 24b), and He increases at the same time. The experimental points
correspond to a moderate rate of cooling of the alloy after annealing.
272
BELL SYSTEM TECHNICAL JOURNAL
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FACTORS AFFECTING MAGNETIC QUALITY
273
ATOMIC PER CENT MOLYBDENUM
10 20
< 1600
2 800
10 20 30
PER CENT MOLYBDENUM
Fig. 23 — Phase diagram of iron-rich iron-molybdenum alloys, showing solid solubility
curve important in the precipitation-hardening process.
CD 25
o t"
0
10-'x25
< 15
"T'
d
V
(a)
1
1
^
'f^
^^
<5»
G
"-^
^
/
/
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A
/
''He
(b)
1 }
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/
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/
An
\
/
/
/
A— li— t?n
/
hrCl-ii—
Y
Q
<i>o
10 15 20 25 30 35
PER CENT MOLYBDENUM IN IRON
Fig. 24 — Change of structure-insensitive properties {d and Bs) and structure-sensitive
properties (/in, and //c) with the composition when precipitation-hardening occurs.
274 BELL SYSTEM TECHNICAL JOURNAL
When the amount of the second phase is considerable (as in the 15% Mo
alloy) it is common practice to quench the alloy from a temperature at which
it is a single phase (e.g. 1100 or 1200°C) and so maintain it temporarily as
such, and then to heat it to a temperature (e.g., 600°C) at which diffusion
proceeds at a more practical rate. During the latter step the second phase
separates slowly enough so that it can easily be stopped at the optimum
point, after a sufficient amount has been precipitated but before diffusion
has been permitted to relieve the strains caused by the precipitation. A
conventional heat treatment for precipitation-hardening of this kind, used
on many permanent magnet materials, has already been given in Fig. 16.
In some respects the development of atomic order in a structure is like
the precipitation of a second phase. When small portions of the material
become ordered and neighboring regions are still disordered, severe local
strains may be set up in the same way that they are during the precipitation
hardening described above. The treatment used to estabhsh high strains is
the same as in the more conventional precipitation hardening. The decom-
position of an ordered structure in the iron-nickel-aluminum system has
been held responsible, by Bradley and Taylor,^" for the good permanent
magnet qualities of these alloys.
Some of the common permanent magnets, heat treated to develop in-
ternal strains by precipitation of a second phase, or by the development of
atomic ordering, are described in Table IV.
The changes in properties to be expected when the composition varies
across a phase boundary of a binary system are shown schematically by the
curves of Fig. 25.
Impurities
The principle of precipitation hardening, as just described, apphes also
to the lowering of permeability by the presence of accidental impurities.
For example, the solubilities of carbon, oxygen and nitrogen in iron, de-
scribed by the curves of Fig. 26, are quite similar in form to the curve sep-
arating the a and a -\- e areas of the iron-molybdenum system of Fig. 23 ;
the chief difference is that the scale of composition now corresponds to con-
centrations usually described as impurities. One expects, then, that the
presence of more than 0.04 per cent of carbon in iron will cause the perme-
ability of an annealed specimen to be considerably below that of pure iron.
The amount of carbon present in solid solution will also affect the magnetic
properties.
Because the amounts of material involved are small, it is difficult to carry
out well defined experiments on the effects of each impurity, especially in
"> A. J. Bradley and A. Taylor, Proc. Roy. Soc. (London) 166, 353-75 (1938).
FACTORS AFFECTING MAGNETIC QUALITY
275
w ^
H s
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276
BELL SYSTEM TECHNICAL JOURNAL
a [ / cr+/3
/3
1-
z
of
Q.
\
<-> 0
in
ZCD
O ^
0
\
^^
o
O LU "
cro
ujcc
oo
ou.
0
^
<5.
>
1-
> f
W
m
q:
0
/
^
<
Li]
tr
a 0
^
PER CENT ALLOYING ELEMENT
Fig. 25 — Diagrams illustrating the changes in various properties that occur when a
second phase precipitates.
the absence of disturbing amounts of other impurities. Two examples of the
effect of impurities will be given, in addition to Fig. 8. In Fig. 27 Yensen and
Ziegler" have plotted the hysteresis loss as dependent on carbon content,
" T. I). Ycnscn and N. A. Ziegler, Traus. Am. Soc. Mctah 24, .i?7-58 (1936).
FACTORS AFFECTING MAGNETIC QUALITY
277
the curve giving the mean values of many determinations. The hysteresis
decreases rapidly at small carbon contents, when these are of the order of
magnitude of the solid solubility at room temperature.
^ 600
D 300
0 0.01 0.02 0.03
SOLUBILITY IN PER CENT BY WEIGHT
Fig. 26 — Approximate solubility curves of carbon, oxygen and nitrogen in iron.
% 2400
ui 2000
o
-C— -
> o
0.04 0.08 0.12
PER CENT CARBON IN IRON
Fig. 27 — Effect of carbon content on hysteresis in iron. Yensen and Zieglcr.
Cioffi'- has purified iron from carbon, oxygen, nitrogen and sulfur by
heating in pure hydrogen at 1475°C, and has measured the permeabiUty
12 p. p. Cioffi, Phys. Rev. 39, 363-7 (1932).
278
BELL SYSTEM TECHNICAL JOURNAL
at different stages of purification. Table V shows that impurities of a few
thousandths of a per cent are quite effective in depressing the maximum
permeabiUty of iron.
Carbon and nitrogen, present as impurities, are known to cause "aging"
in iron — that is, the permeability and coercive force of iron containing these
elements as impurities will change gradually with time when maintained
somewhat above room temperature. As an example, a specimen of iron was
maintained for 100 hours first at 100°C, then 150°C, then 100°C, and so on.
Table V
Maximum permeability of Arnico iron with different degrees of purification, effected by heat
treatment in pure hydrogen at 147 5° C for the times indicated {P. P. Cioffi).
Analyses from R. F. Mehl (private communication to P. P. Cioffi).
Time of Treatment
in Hours
f«m
Composition in Per Cent
C
s
0
N
Mn
P
0
1
3
7
18
7000
16000
30000
70000
227000
0.012
.005
.005
.003
.005
0.018
.010
.006
<.O03
0.030
.003
.003
.003
.003
0.0018
.0004
.0003
.0001
.0001
0.030
.028
0.004
.004
Precision of analysis .
.001
.002
.002
.0001
The corresponding changes in coercive force are given in the diagram of
Fig. 28. A change of about 2-fold is observed.
Some Important Physical Properties
There are many physical characteristics that are important m the study
of ferromagnetism from both the practical and the theoretical point of view.
These include the resistivity, density, atomic diameter, specific heat, ex-
pansion, hardness, elastic limit, plasticity, toughness, mechanical damping,
specimen dimensions, and numerous others. In a different category may be
mentioned corrosion, homogeneity and porosity. Most of these properties
are best discussed in connection with specific materials or properties; only
the most important characteristics will be mentioned here. A table of the
atomic weights and numbers, densities, melting points, resistivities and
coefficients of thermal expansion of the metallic elements, is readily avail-
able in the Metals Handbook.
Dissolving a small amount of one element in another increases the re-
sistivity of the latter. To show the relative effects of various elements, the
common binary alloys of iron and of nickel are shown in Figs. 29 and 30.
From a theoretical standpoint it is desirable to understand (1) the relatively
FACTORS AFFECTING MAGNETIC QUALITY
279
high resistivity of the ferromagnetic elements compared to their neighbors
in the periodic table and (2) the relative amounts by which the resistivity
of iron (or cobalt or nickel) is raised by a given atomic percentage of vari-
ous other elements. From a practical standpoint, a high resistivity is usually
AGING TEMPERATURE IN DEGREES CENTIGRADE
O
IL
Uj 0.8
<-ie)0-»-
■^100-»j*-150-»
«-100-»4-«-160^
IRON /
\
/
V
/
\
/
\
/
\
/
^
(
1
200 300
TIME IN HOURS
Fig. 28 — Effect of nitrogen impurity on the coercive force of iron annealed successively
at 100 and 150°C.
U 22
2 16
<^
> 12
a 10
/
'/
^
^
//
/
^
#
V
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f"'
^
/^
^M ^
— -^
^^
^
^
^
^
COBALT
1
06 0.8 1.0 1.2 1.4
PER CENT OF ALLOYING ELEMENT IN IRON
to
Fig. 29 — Dependence of resistivity on the addition of small amounts of various elements
iron.
desirable in order to decrease the eddy-current losses in the material, and
so decrease the power wasted and the lag in time between the cause and
effect, for example, the time lag of operation of a relay.
Knowledge of the atomic diameter is important in considering the effects
280
BELL SYSTEM TECHNICAL JOURNAL
O 16
d: 8
/
/
/
■^
/
^
o
or
A
,r.<=,^.
■^
!/(/
^
.
— -
\^
^
■
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
PER CENT OF ALLOYING ELEMENT IN NICKEL
Fig. 30 — Resistivity of various allo}-s of nickel.
1
0
Rb
n K
4.5
6 Pa
~os
r
4.0
}
ca
1
9n
fO-
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oNa
i^
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!S3.5
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r^
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e>^
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OSL
B Nl
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Ge
2.0
L_
1.5
^c
1.0
20 30 40 50 60 70 80
ATOMIC NUMBER
Fig. 31 — .Vtomic diameter of various nietallic elements.
FACTORS AFFECTING MAGNETIC QUALITY
281
of alloying elements, and values for the metallic and borderline elements are
shown in Fig. 31. Most of the values are simply the distances of nearest
approach of atoms in the element as it exists in the structure stable at room
temperature. Atomic diameter is especially important in theory because the
very existence of ferromagnetism is dependent in a critical way on the dis-
tance between adjacent atoms. This has been discussed more fully in a
previous paper. ^^
Even when no phase change occurs in a metal, important changes in struc-
ture occur during fabrication and heat treatment, and these are compli-
cated and imperfectly understood. When a single crystal is elongated by
tension, slip occurs on a limited number of crystal planes that in general
are inclined to the axis of tension. As elongation proceeds, the planes on
which slip is taking place tend to turn so that they are less inclined to the
axis. In this way a definite crystallographic direction approaches parallelism
(a) ROLLED
(b) RECRYSTALLIZED
(C) DRAWN
Fig. 32 — The preferred orientations of crystals in nickel sheet and wire after fabrication
and after recrj-stallization.
with the length of the specimen. In a similar but more complicated way,
any of the usual methods of fabrication cause the many crystals of which it
is composed to assume a non-random distribution of orientations, often
referred to as preferred or special orientations, or textures. Some of the tex-
tures reported for cold rolled and cold drawn magnetic materials are given
in Table VI, taken from the compilation by Barrett.''' The orientations of
the cubes which are the crystallographic units are shown in Fig. 32 (a) and
(c) for cold rolled sheets and cold drawn wires of nickel.
Since the magnetic properties of single crystals depend on crystallographic
direction (anisotropy), the properties of polycrystalline materials in which
there is special orientation will also be direction-dependent. In fact it is
difficult to achieve isotropy in any fabricated material, even if fabrication
involves no more than solidifying from the melt. The relief of the internal
'3R. M. Bozorth, Bell Sys. Tecli. Jl. 19, 1-39 (1940).
i-i C. S. Barrett, Structure of Metals, McGraw Hill, New York (1943).
282
BELL SYSTEM TECHNICAL JOURNAL
strains in a fabricated metal by annealing proceeds only slowly at low
temperatures (up to 600°C for most ferrous metals) without noticeable grain
growth or change in grain orientation, and is designated recovery. The prin-
ciple change is a reduction in the amplitude of internal strains, and this can
be followed quantitatively by X-ray measurements. Near the point of com-
plete relief distinct changes occur in both grain size and grain orientation,
and the material is said to recrystallize. At higher temperatures grain growth
increases more rapidly. The specific temperatures necessary for both re-
covery and recrystallization depend on the amount of previous deformation,
as shown in Fig. 33. Special orientations are also present in fabricated mate-
rials after recrystallization, and some of these are listed in Table VI and illus-
trated for nickel in Fig. 32 (b).
As an example of the dependence of various magnetic properties on direc-
tion, Fig. 34 gives data of Dahl and Pawlek'^ for a 40 per cent nickel iron
Table VI
Preferred Orientations in Drawn Wires and Rolled Sheets, Before and After Recrystalliza-
tion, and in Castings (Barrett^^)
The rolling plane and rolling direction, or wire axis, or direction of growth, are designated
Crystal
Structure
Drawn wires
Rolled Sheets
As
Metal
As Drawn
Recrys-
tallized
As Rolled
Recrystallized
Cast
Iron
Cobalt
Nickel
BCC
HCP
FCC
[110]
[ill] and
[100]
[110]
(001), [110] and
others
(001)
(110), [112] and
others
(001), 15° to
[110]
(100), [001]
[100]
alloy reduced 98.5 per cent in area by cold rolling and then annealed at !■
11(X)°C. After further cold rolling (50 per cent reduction) the properties
are as described in Fig. 35.
The mechanical properties ordinarily desirable in practical materials are
those which facilitate fabrication. Mild steel is often considered as the
nearest approach to an ideal material in this respect. Silicon iron is limited
by its brittleness, which becomes of major importance at about 5 per cent
silicon; this is shown by the curve of Fig. 36. Permalloy is "tougher" than
iron or mild steel and requires more power in rolling and more frequent
annealing between passes when cold- rolled, but can be cold-worked to smaller
dimensions. If materials have insufficient stiffness or hardness, parts of
apparatus made from them must be handled with care to avoid bending
and consequent lowering of the permeability. If the hardness is too great
the material must be ground to size. This is the case with some permanent
magnets.
" O. Dahl and F. Pawiek, Zeits. f. Metallkunde 28, 230-3 (1936).
FACTORS AFFECTING MAGNETIC QUALITY
283
).3
0.2
/
\
T ^
15 20 30 50
PER CENT REDUCTION IN THICKNESS
Fig. a — Dependence of the grain size of iron on the amount of deformation and on the
temperature of anneal. Kenyan.
90"
/
120°^
SS DIRECTION \
60°
,50V^
/ \
Y
\
\ \^°°
7\
/ J
X ----- O
\^
\ .-^>V
/
"^^ / ^^
/A'^ cc
Vfc_ \ /'"^'^ \
1.
/^\
/^\ u
^7^
\.^
^^\"\\
/
/ / /
\ •'j^---"
^
.LING DIRECTION ^
f
/ / /
/ /
ROt
0°|
/ 1
1 1
1 1
1 n '
LO
5.0 2.5 0 2.5 5.0
COERCIVE FORCE, He, IN OERSTEDS
lO^xlO.O
5.0 2.5 0 2.5 5.0
RESIDUAL INDUCTION, Br, IN GAUSSES
7.5 10-0 X103
40 30 20 10 0 10 20 30 40
PERMEABILITY, /i, (FOR MEDIUM FIELDS)
Fig. 34 — Variation of magnetic properties with the direction of measurement in a sheet
of iron-nickel alloy (40% Ni) severely rolled (98.5%) and annealed at 1100°C.
The eflfect of size of a magnetic specimen is often of importance. This is
well known in the study of thin films, a,nd fine powders in which the smallest
284
BELL SYSTEM TECHNICAL JOURNAL
dimension is about 10"^ cm or less. Many studies have been made of thin
electrodeposited and evaporated films. Generally it is found that the per-
meability is low and the coercive force high. The interpretation is uncertain
\zo° ^.-^""'^
9
0°
^^^T"""-^-^
s
60°
150y
/
/ ISOPERM \
/ 40% NICKEL \ ^^
60% IRON >\
COLD ROLLED \
/ K^<Z
li 1-
U "J
1 III
a.
^^ Q
</)
■J ,'-H^
\ \30°
/
"■' \l >s.
7/\
W ROLLING DIRECTION
180°
U 1 , > u"
5 0 5
COERCIVE FORCE, He, IN OERSTEDS
tO^x 15
10 5 0 5 10
RESIDUAL INDUCTION, Bp, IN GAUSSES
15 X 10^
50 0 50
INITIAL PERMEABILITY,//,)
100
10^x3
1 0 1
MAXIMUM PERMEABILITY, fJL^
3X103
Fig. 35 — Properties of the same material as that of Fig. 34, after it has been rolled,
annealed, and again rolled.
I 2 3 4 5 6
PER CENT SILICON IN IRON
Mg. 36 -Variation of the breaking strength of iron-silicon alloys, showing the onset of
brittlcness near 4 per cent silicon.
because it is difficult to separate the effects of strains and air gaps from the
intrinsic effect of thickness, though it is known that each one of these vari-
ables has a definite effect. As one example of the many experiments, we
FACTORS AFFECTING MAGNETIC QUALITY
285
will show here the effect of the thickness of electrodeposited films of cobalt.
Magnetization curves are shown in Fig. 37 according to previously un-
published work of the author.
I 10
I
CD
THICKNESS IN MICRONS— ^^
(l MICRON=1 CMx 10-'*J Q
^^''
--— '
,"Z-
L-
.—
—=-_-=.
—
\0^
o7
—
- —
,^
^
/
/
1 .'
1
1
1
0 '^
^
^
■^
/
/
^
^
//
1
1
1 A
M
/
/
/
^^
^
j/l
1^
^
^
-^
I
/ 1
/ 1
/ /
' 1
^^
/
^^
6°^
f
//,
^
y"
^
x^
^
^
0
10
20
30
100
110
120 130
40 50 60 70 80 90
FIELD STRENGTH, H, IN OERSTEDS
Fig. 37 — Dependence of the magnetization curves of pure electrodeposited cobalt films
on the thickness.
10^x12
, 6
\
MnBL
N
\
k
^--^
— o
• 20 40 60 80
PARTICLE DIAMETER IN MICRONS
(l MICRON = lO"'* cm)
Fig. 38 — Dependence of coercive force on the particle size of MnBi powder. Giiillaitd.
'J'he high coercive force obtained in fine powders by GuillaucU^ is one of
the most clear cut e.xamples of the intrinsic effect of particle size. The coer-
cive force increases by a factor of 15 as the size decreases to 5 X 10~'* cm
(Fig. 38).
i«C. Guillaud, Thesis, Strasbourg (1943).
286 BELL SYSTEM TECHNICAL JOURNAL
Properties Affected by Magnetization
In addition to the magnetization, other properties are changed by the
direct apphcation of a magnetic field. Some of these, and the amounts by
which they may be changed, are as follows:
Length and volume (magnetostriction) (0.01%)
Electrical resistivity (5%)
Temperature (magnetocaloric effect; heat of hysteresis) (1°C)
Elastic constants (20 per cent)
Rotation of plane of polarization of light (Kerr and Faraday
effects) (one degree of arc)
In addition to these properties there are others that change with tem-
perature because the magnetization itself changes. Thus there is "anoma-
lous" temperature-dependence of:
Specific heat
Thermal expansion
Electrical resistivity
Elastic constants
Thermoelectric force
and of other properties below the Curie point of a ferromagnetic material,
even when no magnetic field is applied.
Also associated with ferromagnetism are galvanomagnetic, chemical and
other effects.
Technical Articles by Bell System Authors Not Appearing
in the Bell System Technical Journal
Measurement Method for Picture Tubes. M. W. Baldwin.^ Electronics,
V. 22, pp. 104-105, Nov., 1949.
Diffusion in. Binary AIloys.'\ J. Bardeen.^ Phys. Rev., V. 76, pp. 1403-
1405, Nov. 1, 1949.
Abstract — Darken has given a phenomenological theory of diffusion in
binary alloys based on the assumption that each constituent diffuses inde-
pendently relative to a fixed reference frame. It is shown that diffusion via
vacant lattice sites leads to Darken's equations if it is assumed that the
concentration of vacant sites is in thermal equilibrium. Grain boundaries
and dislocations may act as sources and sinks for vacant sites and act to
maintain equilibrium. The modifications required in the equations if the
vacant sites are not in equilibrium are discussed.
Variable Phase-Shift Frequency-Modulated Oscillator. 0. E. de Lange.^
I.R.E., Proc, V. 37, pp. 1328-1331, Nov., 1949.
Abstract — The theory of operation of a phase-shift type of oscillator is
discussed briefly. This oscillator consists of a broad-band amplifier, the out-
put of which is fed back to the input through an electronic phase-shifting
circuit. The instantaneous frequency is controlled by the phase shift through
this latter circuit. True FM is obtained in that frequency deviation is
directly proportional to the instantaneous amplitude of the modulating sig-
nal and substantially independent of modulation frequency.
A practical oscillator using this circuit at 65 mc is described.
Erosion of Electrical Contacts on Make.\ L. H. Germer^ and F. E. Ha-
woRTH.i //. Applied Phys., V. 20, pp. 1085-1108, Nov., 1949.
Abstract — When an electric current is established by bringing two elec-
trodes together, they necessarily discharge a capacity. Unless the current
which is set up is above 1 ampere, the erosion which is produced in a low
voltage circuit is appreciable only when the capacity is of appreciable size
and when it is discharged very rapidly by an arc. When the arc occurs, its
energy is dissipated almost entirely upon the positive electrode and, when
the circuit inductance is sufficiently low, melts out a crater intermediate in
volume between the volume of metal which can be melted by the energy
t A reprint of this article may be obtained on request to the editor of the B. S.T.J.
1 B.T.L.
287
288 BELL SYSTEM TECHNICAL JOURNAL
and that which can be boiled. Some of the melted metal lands on the nega-
tive electrode and, with repetition of the phenomenon, results in a mound
of metal transferred from the anode to the cathode. This transfer, which is
about 4 X lO^^"* cc of metal per erg, is the erosion which occurs on the
make of electrical contacts.
The arc voltage is of the order of 15. If the initial circuit j)otential is
more than about 50 volts, there may be more than one arc discharge, suc-
cessive discharges being in opposite directions and resulting in the transfer
of metal in opposite directions — always to the electrode which is negative.
The occurrence of an arc is dependent upon the condition of the electrode
surfaces and upon the circuit inductance. For "inactive" surfaces an arc
does not occur for inductances greater than about 3 microhenries. Platinum
surfaces can be "activated" by various organic vapors, and in the active
condition they give arcs even when the circuit inductance is greater than
this limiting value by a factor of 10^.
The Conductivity of Silicon and Germanium as A ffected by Chemically In-
troduced Impurities. G. L. Pearson.^ Paper presented at A. I. E. E., Swamps-
cott, Mass., June 20-24, 1949. Included in compilation on semiconductors.
Elec. Engg., V. 68, pp. 1047-1056, Dec. 1949.
Absteact — Silicon and germanium are semiconductors whose electrical
properties are highly dependent upon the amount of impurities present.
For example, the intrinsic conductivity of pure silicon at room temperature
is 4 X 10~® (ohm cm)~^ and the addition of one boron atom for each million
silicon atoms increases this to 0.8 (ohm cm)~\ a factor of 2 X 10\
Although such impurity concentrations are too weak to be detected by
standard chemical analysis, the use of radioactive tracers and the Hall
effect has made it possible to make quantitative measurements at impurity
concentrations as small as one part in 5 X 10*.
Silicon and germanium are elements of the fourth group of the periodic
table with the same crystal structure as diamonds and they have respec-
tively 5.2 X lO" and 4.5 X lO-"' atoms per cubic centimeter. The addition
of impurity elements of the third group such as boron or aluminum gives
defect or p-type conductivity. Elements from the tifth group such as j^ihos-
])horous, antimony or arsenic give excess or n-type conductivity.
The conductivity at room temperature, where it has been shown that
each impurity atom contributes one conduction cliarge, is given by equa-
tion (1) where N is tlie number of solute atoms per cubic tentinicter.
(7 = A + H.\. (1)
' H.T.I..
ARTICLES BY BELL SYSTEM AUTHORS 289
The constants A and B for the various alloys investigated are given in the
following table:
Alloy
A
B
Si + B
Si + P
Ge + Sb
4 X 10-«
4 X 10-«
1.7 X 10-2
1.6 X 10-17
4.8 X 10-"
4.2 X 10-16
Equation (1) applies to solute atom concentrations as high as 5 X 10'^
per cc. At higher concentrations the mobilities are lowered due to increased
impurity scattering so that the computed conduction is higher than the
measured.
Microstructures of Silicon Ingots.^ W. G. Pfann^ and J. H. Scaff.^
Melds Trans., V. 185 (//. Metals, V. 1) pp. 389-392, June, 1949.
Increasing Space-Clmrge Waves.'\ J. R. Pierce.^ //. Applied Phys., \. 20,
pp. 1060-1066, Nov. 1949.
Abstract — An earlier paper presented equations for increasing waves in
the presence of two streams of charged particles having different velocities,
and solved the equations assuming the velocity of one group of particles to
be zero or small. Numerical solutions giving the rate of increase and the
phase velocity of the increasing wave for a wide range of parameters, cover-
ing cases of ion oscillation and double-stream amplification, are presented
here.
Traveling-Wave Oscilloscope. J. R. Pierce. ^ Electronics, \\ 22, pp. 97-99,
Nov., 1949.
.A.BSTRACT — This paper describes a 1,000 volt oscilloscope tube with a
traveling-wave deflecting system. The tube is suitable for viewing periodic
signals with frequencies up to 500 mc. A signal of 0.037 volt into 75 ohms
deflects the spot one spot diameter. A few milliwatts input gives a good
pattern, so that the tube can be used without an amplifier. The pattern is
viewed through a sixty power microscope.
P-type and X-type Silicon and the Formation of Photovoltaic Barrier in
Silicon Ingots.f J. H. Scaff,^ H. C. Theurerer^ and E. E. Schumacher.^
Metals Trans., \. 185 (//. Metals, V. 1) pp. 383-388, Jan., 1949.
Longitudinal Xoise in Audio Circuits. H. W. Augustadt^ and W. F.
Kaxxexberg.^ Audio Engg., Y. 34, pp. 22-24, 45, Jan., 1950.
Transistors. J. A. Becker.^ Compilation of three papers presented at
A. I. E. E. meeting Swampscott, Mass., June 20-24, 1949. Elec. Engg.,
V. 69, pp. 58 64, Jan., 1950.
t A reprint of this article ma\' he obtained on retjuest to the ecUlor of the B. S.T.J.
1 B.T.L.
290 BELL SYSTEM TECHNICAL JOURNAL
Application of Thermistors to Control Networks.] J. H. Bollman^ and
J. G. Kreer.i /. R. E., Proc, V. 38, pp. 20-26, Jan., 1950.
Abstract — In connection with the application of thermistors to regulat-
ing and indicating systems, there have been derived several relations be-
tween current, voltage, resistance, and power which determine the electrical
behavior of the thermistor from its various thermal and physical constants.
The complete differential equation describing the time behavior of a di-
rectly heated thermistor has been developed in a form which may be solved
by methods appropriate to the problem.
Sensitive Magnetometer for Very Small Areas. "f D. M. Chapin.^ Rev. Sci.
Instruments, V. 20, pp. 945-946, Dec, 1949.
Abstract — A vibrating wire system for measuring weak magnetic fields
is described for use in very small spaces. Quartz crystals are used for drivers
to get sufficient velocity with very small displacements. To adjust the
driving voltage to correspond exactly to the natural crystal frequency, the
crystal is also used to regulate the oscillator.
Method of Calculating Hearing Loss for Speech from an Audiogram.] H.
Fletcher.1 Acoustical Soc. Am., Jl., V. 22, pp. 1-5, Jan., 1950.
Abstract — The question frequently arises. Can one compute the hearing
loss of speech from the audiogram and thus make it unnecessary to make a
speech test after the hearing loss for several frequencies has been recorded.
This paper shows that this can be done by taking a weighted average of the
exponentials of the hearing loss at each frequency. Or if /3s is the hearing
loss for speech and j3{ the hearing loss at each frequency,
1q(^3/io) ^ j(. ioWi«df
The weighting factor G was determined by Fletcher and Gait from thresh-
old measurements of speech coming from filter systems. As specifically
applied to the case of hearing loss at the five frequencies 250, 500, 1000,
2000 and 4000 cps, the above equation is approximately equivalent to
/Js = -10 log [.01 X \0-^^'"'^ -\- .13 X 10"^''^''°^
+ .40 X IQ-^''^''"^ -f .38 X lO-^'^^'^''^ + .08 X lO-^'^^''"^]
where jSi is hearing loss at 250 cps
/32 is hearing loss at 500 cps
/Sa is hearing loss at 1000 cps
04 is hearing loss at 2000 cps
05 is hearing loss at 4000 cps
t A reprint of this article may be obtained on request to the editor of the B.S.T.J.
1 B.T.L.
ARTICLES BY BELL SYSTEM AUTHORS 291
Designing for Air Purity. A. M. Hanfmann.^ Heating & Ventilating, V.
47, pp. 59-64, Jan., 1950.
Reciprocity Pressure Response Formula Which Includes the Effect of the
Chamber Load on the Motion of the Transducer Diaphragms, f M. S. Hawley.^
Acoustical Soc. Am., Jl., V. 22, pp. 56-58, Jan., 1950.
Abstract — In order to reduce the effects of wave motion in the coupling
chamber to permit reciprocity pressure response measurements to higher
frequencies, only two of the three transducers involved are coupled at a
time to the chamber. Given for these conditions is a derivation of the pres-
sure response formula which includes the effect of the chamber load on the
motion of the transducer diaphragms.
Theory of the "Forbidden" (222) Electron Reflection in the Diamond Struc-
ture.i R. D. Heidenreich.i Phys. Rev., V. 77, pp. 271-283, Jan. 15, 1950.
Abstract — The dynamical or wave mechanical theory of electron diffrac-
tion is extended to include several diffracted beams. In the Brillouin zone
scheme this is equivalent to terminating the incident crystal wave vector
at or near a zone edge or corner. The problem is then one of determining the
energy levels and wave functions in the neighborhood of a corner. The solu-
tion of the Schrodinger equation near a zone corner is a linear combination
of Bloch functions in which the wave vectors are determined by the boundary
conditions and the requirement that the total energy be fixed. This leads to
a multipUcity of wave vectors for each diffracted beam giving rise to inter-
ference phenomena and is an essential feature of the dynamical theory.
At a Brillouin zone edge formed by boundaries associated with reciprocal
lattice points S and O the orthogonality of the unperturbed wave functions
in conjunction with the periodic potential requires that another recipro-
cal lattice point X be included in the calculation. The indices of X must be
such that (X1X2X3) = (S1S2S3) — (gig2g3) • The perturbation at the zone edge
results in non-zero amplitude coefficients Cg, Cs and Cj for the diffracted
waves irrespective of whether or not the structure factor for X , s or g van-
ishes. This is the basis of the explanation of the (222) reflection and since it
arises through perturbation at a Brillouin zone edge or corner the term
I "perturbation reflection" is advanced to replace the commonly used "for-
bidden reflection."
! The octahedron formed by the (222) Brillouin zone boundaries exhibits
j an array of lines due to intersections with other boundaries to form edges.
I This array of lines is called a "perturbation grid" and the condition for the
j occurrence of a (222) reflection is simply that the incident wave vector
I terminate on or near a grid line. Numerical intensity calculations are pre-
t A reprint of this article may be obtained on request to the editor of the B. S.T.J.
1 B.T.L.
2W. E. Co.
292 BELL SYSTEM TECHNICAL JOURNAL
sented wliich sliow that a strong (222) can be accounted for by the dynamical
theory.
An impedance network model is briefly discussed which may aid in quah-
tative considerations of the dynamical theory for the case of several
diffracted waves.
Determiiialioii of g- Values in Paramagnetic Organic Compounds by Micro-
wave Resonance. A. N. Holden/ C. Kittel/ F. R. Merritt^ and W. A.
Yager.i Letter to the Editor, Phys. Rev., V. 77, pp. 146-147, Jan. 1, 1950.
Nonlinear Coil Generators of Short Pulses.^ L. W. Hussey.^ I.R.E., Proc,
V. 38, pp. 40-44, Jan., 1950.
Abstract — Small permalloy coils and circuits have been developed which
produce pulses well below a tenth of a microsecond in duration with repeti-
tion rates up to a few megacycles.
The construction of these coils is described. Low power circuits are di-
cussed suitable for different types of drive and different frequency ranges.
Subjective Effects in Binaural Hearing. W. Koenig.' Letter to the Editor,
Acoustical Soc. Am., Jl., V. 22, pp. 61-62, Jan., 1950.
Abstract — Experiments with a binaural telephone system disclosed some
remarkable properties, notably its ability to "squelch" reverberation and
background noises, as compared to a system having only one pickup. No
explanation has been found for this subjective effect. It was also discovered
that a well-known defect in the directional discrimination of binaural sys-
tems was remedied by a mechanical arrangement which rotated the pickup
microphones as the listener turned his head.
Corrosion Testing of Buried Cables. T. J. Maitland.^ Corrosion, V. 6, pp.
1-8, Jan., 1950.
40AC1 Carrier Telegraph System. A. L. Matte.' Tel. & Tel. Age, No. 2,
pp. 7-9, Feb., 1950.
Giving New Life to Old Equipment. P. H. Miele."' Bell Tel. Mag., V. 28,
pp. 154-163, Autumn, 1949.
Thermionic Emission of Thin Films of Alkaline Earth Oxide Deposited by
Evaporation.\ G. E. Moore' and H. W. Ai>lison'.' Phys. Rev., V. 77, pp.
246-257, Jan. 15, 1950.
Abstract — Monomolecular lilms of BaO or SrO were deposited by evap-
oration on clean tungsten or molybdenum surfaces with precautions to elimi-
nate effects caused by excess metal of the oxide or by heating. Thermionic
emissions of the same order of magnitude as from commercial oxide cathodes
have been ol)taine(l from these systems. The results can be explained quali-
tatively ])y considering the adsorl^ed molecules as oriented di])oles. Although
t A re])riiil of lliis article nia\- he olilaiiiL-d on rc'(|iH'sl lo tin.' (.'dilor ol llu' 15..S.'1'.J.
' li.T.L.
■' A. T. & 'I'.
ARTICLES BY BELL SYSTEM AUTHORS 293
the results may suggest a possible mechanism for a portion of the emission
from thick oxide cathodes, there exist serious obstacles to such thin tilm
phenomena as a complete explanation.
Long Distance Finds the Way. W. H. Nunn.^ Bell Tel. Mag., V. 28, pp.
137-147, Autumn, 1949.
Private Line Services for the Aviation Lndustry. H. V. Roumfort.-^ Bell
Tel. Mag., V. 28, pp. 165-174, Autumn, 1949.
Growing and Processing of Single Crystals of Magnetic Metals.] J. G.
Walker,^ H. J. Willl\mS' and R. M. Bozorth.^ Rev. Sci. Lnstruments,
V. 20, pp. 947-950, Dec, 1949.
Abstract — Single crystals of nickel, cobalt and various alloys are grown
by slow cooling of the melt. They are oriented by optical means and by
X-rays, and ground to the desired shape using the technique described.
A Look Around — and Ahead. L. A. Wilson.^ Bell Tel. Mag., V. 28, pp.
133-136, Autumn, 1949.
t A reprint of this article may be obtained on request to the editor of the B. S.T.J.
1 B.T.L.
3A. T. & T
Contributors to this Issue
R. M. BozoRTH, A.B., Reed College, 1917; U. S. Army, 1917-19; Ph.D.
in Physical Chemistry, California Institute of Technology, 1922; Research
Fellow in the Institute, 1922-23. Bell Telephone Laboratories, 1923-. As
Research Physicist, Dr. Bozorth is engaged in research work in magnetics.
R. W. Hamming, B.S. in Mathematics, University of Chicago, 1937;
M.A. in Mathematics, University of Nebraska, 1939; Ph.D. in Mathe-
matics, University of Illinois, 1942. Dr. Hamming became interested in the
use of large scale computing machines while at Los Alamos, New Mexico,
and has continued in this field since joining the Bell Telephone Laboratories
in 1946.
W. P. Mason, B.S. in E.E., University of Kansas, 1921; M.A., Ph.D.,
Columbia, 1928. Bell Telephone Laboratories, 1921-. Dr. Mason has been
engaged principally in investigating the properties and applications of piezo-
electric crystals and in the study of ultrasonics.
J. R. Pierce, B.S. in Electrical Engineering, California Institute of Tech-
nology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936-. Dr. Pierce
has been engaged in the study of vacuum tubes.
294
unjiiu i-iorary
Kansas Citv,, M«k
VOLUME XXIX JULY, 1950 no. 3
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
Principles and Applications of Waveguide Transmission
G. C. Southworth 295
Memory Requirements in a Telephone Exchange
C. E. Shannon 343
Matter, A Mode of Motion R.V.L. Hartley 350
The Reflection of Diverging Waves by a Gyrostatic Mediimi
R. V. L. Hartley 369
Traveling-Wave Tubes (Third InstaUment) ..J. R. Pierce 390
Technical Publications by Bell System Authors Other than
in the Bell System Technical Journal 461
Contributors to this Issue 468
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The Bell System Technical Journal
Vol. XXIX July, 1950 No. j
Copyright, 1950, American Telephone and Telegraph Company
Principles and Applications of Waveguide Transmission
By GEORGE C. SOUTHWORTH
Copyright, 1950, D. \'an Nostrand Company, Inc.
Under the aliove title, D. Van Nostrand Company, Inc. will shortiv publish
the book from which the following article is e.xcerpted. Dr. Southworth is one
of the leading authorities on waveguides and was one of the lirst to foresee the
great usefulness that this form of transmission might offer. The editors of the
Bell System Technical Journal are grateful for permission to ])ublish here parts
of the preface and the historical introduction and chapter 6 in its entirety.
Preface
Though it has been scarcely fifteen years since the waveguide was pro-
posed as a practicable medium of transmission, rather important applica-
tions have already been made. The first, which was initiated several years
ago, was in connection with radar. A more recent and possibly more im-
portant application has been in television where waveguide methods pro-
vide a very special kind of radio for relaying program material cross-country
from one tower top to another. Already Boston and New York have been
connected by this means and shortly Chicago and intervening cities will
be added. Other networks extending as far west as the Pacific may be ex-
pected. It is reasonable to e.xpect that these two apj)lications will be but
the beginning of a more general use.
Interest in the subject of waveguide transmission is not limited to com-
mercial application alone. A comparable interest, perhaps less readily evalu-
ated but nevertheless extremely important, lies in its usefulness in teaching
important physical principles. For example there are many concepts that
follow from the electromagnetic theory that, in their native mathematical
form, may appear rather abstract. However, when translated to phenomena
actually observed in waveguides, they become very real indeed. As a re-
sult, these new techniques have already assumed a place of considerable
importance in the teaching of electrical engineering and applied physics both
in lecture demonstrations and in laboratory exercises. It is to be expected
295
296 BELL SYSTEM TECHNICAL JOURNAL
that they will be used even more extensively as their possibilities become
better appreciated.
Interest in waveguides has been greatly enhanced by the fact that they
brought with them a series of extremely interesting methods of measure-
ment, comparable both in accuracy and scope, with similar measurements
previously made only at the lower frequencies. This extension of the range
over which electrical measurements may be made has contributed also to
neighboring ftelds of research. One early application led to the discovery of
centimeter waves in the sun's spectrum. Another led to important new infor-
mation about the earth's atmosphere. Still another contributed to the study
of absorption bands in gases, particularly bands in the millimeter region.
Also of great importance was its contribution to our knowledge of the prop-
erties of materials for it led at a fairly early date to measurements at higher
frequencies than heretofore of the primary constants, permeability, dielectric
constant and conductivity — all for a wide array of substances ranging from
the best insulators to the best conductors and including many of the so-
called semi-conductors. It is because this new art has already attained con-
siderable stature and is already showing promise as an educational medium
that this book has been prepared.
CHAPTER I
INTRODUCTION
1.5 Early History of Waveguides
That it might be possible to transmit electromagnetic waves through
hollow metal pipes must have occurred to physicists almost as soon as the
nature of electromagnetic waves became fully appreciated. That this might
actually be accomplished in practice was probably in considerable doubt,
for certain conclusions of the mathematical theory of electricity seemed to
indicate that it would not be possible to support inside a hollow conductor
the lines of electric force of which waves were assumed to consist. Evidence
of this doubt appears in Vol. I (p. 399) of Heaviside's "Electromagnetic
Theory" (1893) where, in discussing the case of the coaxial conductor, the
statement is made that "it does not seem possible to do without the inner
conductor, for when it is taken away we have nothing left on which tubes
of displacement can terminate internally, and along which they can run."
Perhaps the first analysis suggesting the possibility of waves in hollow
pipes aj)peared in 1893 in the book "Recent Researches in Electricity and
Magnetism" by J. J. Thomson. This book, which was written as a sequel
to Maxwell's "Treatise on Electricity and Magnetism," examined mathe-
matically the hypothetical question of what might result if an electric charge
WAVEGUIDE TRANSMISSION 297
should be released on the interior wall of a closed metal cylinder. This
problem is even now of considerable interest in connection with resonance
in hollow metal chambers. The following year Joseph Larmor examined as
a special case of electrical vibrations in condensing systems the particular
waves that might be generated by spark-gap oscillators located in hollow
metal cylinders. A more complete analysis relating particularly to propaga-
tion through dielectrically-fiUed pipes both of circular and rectangular cross
section was published in 1897 by Lord Rayleigh. Later (1905) Kalahne
examined mathematically the possibility of oscillations in "ring-shaped"
metal tubes. Still later (1910) Hondros and Debye examined mathematically
the more complicated problem of propagation through dielectric wires. Trans-
mission through hollow metal pipes was also considered by Dr. L. Silberstein
in 1915.
As regards experimental verification, it is of interest that Sir Oliver Lodge
as early as 1894 approached but probably did not quite realize actual wave-
guide transmission. In a demonstration lecture on electric waves given before
the Royal Society, he used, as a source of waves, a spark oscillator mounted
inside a "hat-shaped" cylinder. An illustration pubHshed later suggests that
the length of the cylinder was only slightly greater than its diameter. There
is no very definite evidence that the short cylinder functioned as a waveguide
or that such a function was discussed in the lecture. Perhaps of greater
significance were some experiments reported a year later by Viktor von Lang
who used pipes of appreciable length and repeated for electric waves the
interference experiment that had been performed for acoustic waves by
Quincke some years earlier. Other similar experiments were later performed
by Drude and by Weber.
About 1913 Professor Zahn of the University of Kiel became interested
in this problem and assigned certain of its aspects to two young candidates
for the doctorate, Schriever and Renter by name. They had barely started
when World War I broke out, and both left for the front. Zahn continued
this work until he was called a year later. It is reported that by this time he
had succeeded in propagating waves through cylinders of dielectric, but it
is understood that he did little or no quantitative work. Renter was killed
at Champagne in the autumn of 1915, but Schriever survived and returned
to complete his thesis in 1920, using for his source the newly available
Barkhausen oscillator.
The contributions of Thomson, Rayleigh, Hondros and Debye, and
Silberstein were, of course, purely m.athematical. Those of von Lang, Weber,
Zahn and Schriever were experimental, but they were of rather limited
scope. The concept of the hollow pipe as a useful transmission element, for
example as a radiator or as a resonant circuit, apparently did not exist at
these early dates. Nothing was yet known quantitatively about attenuation,
298 BELL SYSTFAf TECHNICAL JOVRXAL
and little or nothiiif,^ of the |)resent-day experimental technique had yet
appeared. At this time, the position of this new art was perhaps com{)arahle
with that of radio prior to the time of Marconi.
The history of waveguides changed abruptly about 1933 when it was
shown that they could be put to practical use. Several patent applications
were filed/ and numerous scientific papers were published. More recently a
great many papers have appeared, too many in fact for detailed consideration
at this time. Three of the earlier papers are mentioned in the footnote
below.'' Others will be referred to in the text that follows.
The writer's interest in guided waves stems from some experiments done
in 1920 when such waves were encountered as a troublesome spurious effect
while working with Lecher wires in a trough of water. In one case there were
found, superimposed on the waves that might normally travel along two
parallel conductors, other waves having a velocity that somehow depended
on the dimensions of the trough. These may now be identified as being the
so-called dominant type. In another case, the depth of water was apparently
at or near "cut-off," and conditions were such that water waves in the
trough gave rise to depths that were momentarily above cut-off, followed a
moment later by depths that were below cut-off. This led not only to varia-
tions in power at the receiving end of the trough but also to variations in
the plate current of the oscillator supplying the wavepower. Indeed these
effects could be noted even when the wires were removed from the trough.
These waves were recognized as being roughly like those described the same
year by Schriever.^
Several years later this work was resumed and since that time a con-
tinued effort has been made to develop from fundamental principles of
waveguide transmission a useful technique for dealing with microwaves.
The earliest of these experiments consisted of transmitting electromagnetic
waves through tall cylinders of water. Because of the high dielectric con-
stant of water, waves which were a meter long in air were only eleven centi-
meters long in water. Thus it became possible to set up in the relatively
small space of one of these cylinders many of the wave configurations pre-
dicted by theory. In addition it was possible, by |)roducing standing waves,
to measure their apparent wavelength and thereby calculate their phase
velocity. Also by investigating the surface of the water by means of a probe,
' Reference is huuIl' particuhirlv lo U.S. Palenls 2,120.711 (lik-d 3/16/33, 2.12'),712
(filed 12/9/33), 2,206,923 (filed 9/12/34) and 2,106,768 (filed 9/2.S/34).
-Carson, Mead and Schelkunnff, "Hv])cr-I're(iucncv \Vavef!;uicles — Mathenialical
Theory," B.S.TJ., Vol. 15, pp 310-333, .\i)nl 1936. G. C. Southwortii, "Hyper-frequency
Wave (niides — Oeneral C'onsideralions and l'".\i)eri menial Results," /^..S'.7\/., Vol. 15, pp
284-309, April 1936. .Mso "Some I-'undamenlal ICxi)eriments with \\'ave<];uides," Proc.
r.R.R.,\o\. 25, i)p 807 822, Jul\- 1937. \\ . L. Harrow, "Transmission of Mleclromagnelic
Waves in Hollow 'I'uhes of Metal," Proc. I .R.E., Vol. 24, pp 1298 1398, October 1936.
■■' The waves actually observer! are now known as TEm waves in a reclanRular guide,
wliile those described by Schriever are now recognized as TM.ji waves in a circular guide.
WAVEGUIDE TRANS.\fISSION 299
the directions and also the relative intensities of lines of electric force in the
wave front could be mapped. It is probable that certain of these modes were
observed and identitied for the tirst time.
Shortly afterwards, sources giving wavelengths in air of fifteen centi-
meters became available and the experimental work was transferred to air-
filled copper pipes only 5 inches in diameter. At this time, a 5-inch hollow-
pipe transmission line 875 feet in length was built through which both
telegraph and telephone signals were transmitted. Measurements showed
that the attenuation was relatively small. This early work, which was done
prior to January 1, 1934, was described along with other more advanced work
in demonstration-lectures and also in papers published in 1936 and 1937.''
It was recognized at an early date that a short waveguide line might, with
suitable modification, function as a radiator and also as a reactive element.
These properties were likewise investigated experimentally, and numerous
useful applications were proposed. Descriptions may be found in the numer-
ous patents that followed. These properties were also the subject of several
experimental lectures given before the Institute of Radio Engineers and
other similar societies by the writer and his associates during the years 1937
to 1939.^ Included were demonstrations of the waveguide as a transmission
line, the electromagnetic horn as a radiator, and the waveguide cavity as a
resonator. An adaptation of the waveguide cavity was used to terminate a
waveguide line in its characteristic impedance.
From the first, progress was very substantial and by the autumn of 1941
there were known, both from calculation and experiment, the more important
facts about the waveguide. In particular, the reactive nature of discon-
tinuities became the subject of considerable study, and impedance matching
devices (transformers), microwave filters, and balancers soon followed. Also
a wide variety of antennas was devised. Similarly, amplifiers and oscillators
as well as the receiving methods followed.
As might be expected, a great many people have contributed in one way
or another to the success of this venture. Particular mention should be
made of the very important parts played by the author's colleagues, Messrs.
A. E. Bowen and A. P. King, who, during its early and less promising period,
contributed much toward transforming rather abstract ideas into practical
equipment, much of which found important military uses immediately upon
the advent of war. Also of importance were the parts played by the author's
colleagues. Dr. S. A. Schelkunoff, J. R. Carson, and Mrs. S. P. Meade, who,
in the early days of this work, provided a substantial segment of mathe-
matical theory that previously was missing. During the succeeding years,
Dr. Schelkunoff, in particular, made invaluable contributions in the form
■* A description of one of the earlier lectures appears in the Bell Laboratories Record
for March 1940. (Vol. XVIII, No. 7, p. 194.)
300 BELL SYSTEM TECHNICAL JOURNAL
of analyses which in some cases indicated the direction toward which experi-
ment should proceed and, in others, merely confirmed experiment, while,
in still others, gave answers not readily obtainable by experiment alone. In
the chapters that follow, the author has drawn freely on Dr. Schelkunoff,
particularly as regards methods of analysis.
Beginning sometime prior to 1936, Dr. W. L. Barrow, then of the Massa-
chusetts Institute of Technology, also became interested in this subject and
together with numerous associates made very substantial contributions. No
less than eight scientific papers were published covering special features of
hollow-pipe transmission lines and electromagnetic horns. For several years
the work being done at the Massachusetts Institute of Technology and at
the Bell Telephone Laboratories probably represented the major portion, if
not indeed the only work of this kind in progress, but with the advent of
World War II, hundreds or perhaps thousands of others entered the field.
For the most part, the latter were workers on various military projects.
Starting with the considerable accumulation of unpublished technique that
was made freely available to them at the outset of the war, they, along with
others in similar positions elsewhere in this country and in Europe, have
helped to bring this technique to its present very satisfactory state of de-
velopment.
CHAPTER VI
A DESCRIPTIVE ACCOUNT OF ELECTRICAL
TRANSMISSION
6.0 General Considerations
The preceding four chapters presented the more important steps in the
development of the theory of electrical transmission, particularly as it
applies to simple networks, wire lines, and waves in free space and in guides.
For the most part, the analysis followed conventional methods and made use
of the concise and accurate short-hand notation of mathematics. It had for
its principal objective the derivation of a series of equations useful in the
practical application of waveguides.
Closely associated with the theory of electricity and almost a necessary
consequence of it are the numerous concepts and mental pictures by means
of which we may explain rather simply the various phenomena observed in
electrical practice Tiiough extremely important, this aspect of the theory
was not stressed before. Instead it was deferred to the present chapter where
it could be considered by itself and from the purely qualitative point of
view. It is hoped that this arrangement of material will be of special use to
those who find it necessary to substitute for mathematical analysis, simple
WAVEGUIDE TRANSMISSION 301
models to explain the phenomena which they observe in practice. It is be-
lieved that, for these people, this chapter together with a few key formulas
taken from the earlier sections will be helpful in gaining a fairly satisfactory
understanding of the practical aspects of waveguide transmission.
At the lower frequencies, the current aspect of electricity meets most of
the needs and in comparison it is only occasionally that there is a need to
discuss lines of electric and magnetic force. In waveguide practice, on the
other hand, currents are usually not available for measurement and, al-
though we recognize their reality, they necessarily assume a secondary role.
In contrast with currents, we consider the fields present in a waveguide as
very real entities and we attach a very great importance to their orientations
as well as to their intensities.
6.1 The Nature of Fields of Force
As a suitable introduction to the discussion that follows, we shall review
some of the fundamental properties of lines of electric and magnetic force
and show pictorially the part that they play in transmission along an or-
dinary two-wire line.
The Electrostatic Field-
As is well known, the concept of the electric field was devised by Faraday
to explain the force action between charged bodies. According to his view
there exist in the space between the charged bodies, lines or tubes of electric
force terminating respectively on positive and negative charges attached to
the bodies. These tubes of force are endowed with a tendency to become
as short as possible and at the same time to repel, laterally, neighboring lines
of force. Their direction at any point is purely arbitrary, but, by subsequent
convention, the positive direction is taken from the positively charged body
to the negative. This is such that a small positive charge (proton) placed in
the field tends to be displaced in the positive sense while an electron tends to
move in a negative direction. The force exerted on the unit charge is a
measure of the magnitude of the electric intensity E. It is measured in volts
per meter and, since it has direction as well as magnitude, it is a vector quan-
tity.^ Figure 6.1-1 illustrates in a general way the arrangement of lines of
electrostatic force that are assumed to exist between two oppositely charged
spheres. Also shown is a representative vector E.
The Magnetostatic . Field
In the same way that Faraday provided a satisfactory explanation for
the forces between charged bodies, so was he able to explain the forces be-
1 Black-face type will be used when it seems desirable to emphasize the vector proper-
ties of quantities having direction as well as magnitude.
302 BEIJ. SYSTEM TECHNICAL JOURNAL
Iween magnetized bodies. In the latter case, the two kinds of electrostatic
charge are replaced by north-seeking and south-seeking magnetic poles re-
spectively. Similarly the tubes of electric force are replaced by tubes of
magnetic force. Roughly speaking, the two kinds of tubes are endowed with
analogous properties. Because these magnetic lines are at rest, it is appro-
priate to speak of them as magnetostatic lines of force and consider them as
being comparable but of course not identical with electrostatic lines already
discussed. The force exerted on a unit magnetic pole is a measure of magnetic
intensity H. Like its electric counterpart, it is a vector quantity. In the par-
Fig. 6.1-1. Arraiigemenl of lines of electrostatic force in the region between two oppositely
charged spheres.
■* 1 — r'v.
?
Fig. 6.1-2. .\rrangcment of lines of magnetostatic force in the region between two
o])l)ositely magnetized poles.
ticular system of units u.sed in this text, it is measured in amjicres i)er meter.
Figure 6.1-2 illustrates the arrangement of the lines of magnetic force that
are assumed to exist between two opposite magnetic poles.
Interrelationship of Electric and Magnetic Fields
As a result of the electromagnetic theory, there are certain properties with
which we may endow lines of electric and magnetic force and thereby ex-
])lain numerous jihenomena of clcclrical transmission. This establishes a
relationship between electric and magnetic llelds that makes them appear
WAVEGUIDE TRANSMISSION
303
at times as if tiiey were different aspects of the same thing. They are as
follows:
1. Lilies of magnetic force, when displaced laterally, induce in the space
immediately adjacent, lines of electric fcrce. The direction of the induced electric
force is perpendicular to the direction of motion and also perpendicular to the
direction of the original magnetic force. The intensity E of the induced electric
LINE OF ELECTRIC FORCE
LINE OF MAGNETIC FORCE
Fig. 6.1-3. Directions of electric vector E and magnetic vector // relative to the velocity
V of motion of such lines.
Fig. 6.1-4. Simple corkscrew rule for remembering the directions of E, 11 and v.
force is proportional to the velocity v of displacement and proportional to the
intensity H of the original lines of magnetic force.
The directions of the vectors v, E and H are shown in Fig. 6.1-3. They
are so related that, when E moves clockwise into H, it is as though a right-
hand screw had progressed in the direction of v as shown in P'ig. 6.1-4.
A convenient short-hand notation used rather generally by mathematicians
makes it possible to express these facts by the following vector equation:
E = -;u(vxH)
(6.1-1)
304 BELL SYSTEM TECHNICAL JOURNAL
The quantity )u is the magnetic permeability of the medium under considera-
tion.
2. Lines oj electric force, ivhen displaced lalerally, induce in the immediately
adjacent space lines of magnetic force. The direction of the induced magnetic
force is perpendicular to the direction of motion and also perpendicular to the
direction of the original electric force. The intensity H of the induced magnetic
force is proportional to the velocity v of displacement and proportional to the
intensity E of the original lines of electric force.
UNIT VOLUME CONTAINING
STORED ENERGY
POYNTING VECTOR
FLOW OF POWER
Z
Fig. 6.1-5. Directions of the vectors E and H relative to the Poyntlng vector P in an
advancing wave front.
Again Fig. 6.1-3 and also the right-hand or cork-screw rule apply. In the
short-hand notation these facts may be expressed by the following vector
equation:
H = €(vxE) (6.1-2)
In this equation, e is the dielectric constant of the medium.-
3. When an electric field of intensity E is translated laterally, it together with
its associated magnetic field H represents a flow of energy. The direction of the
flow of energy is perpendicular to both E and H and is therefore in the direction
of the velocity v. The magnitude of the energy flow per unit volume across a unit
area measured perpendicular to v is proportional to the product of the electric
intensity E and the magnetic intensity H. // may be designated by the vector P.
The relative directions of the vectors P, E, and H are shown in Fig. 6.1-5.
The energy per unit volume moves with a velocity expressed by
V = —r^ (6.1-3)
Viue
* The values of permeability ju and dielectric constant e appearing in these equations
are not the values found in most tables of the properties of materials. As here given p. is
smaller than the usual value ^r by a factor of 1.257 X 10 "^ while e is smaller than tr by
a factor of 8.854 X 10"'^. The use of these special values leads to certain mathematical
simplifications.
WAVEGUIDE TRANSMISSION 305
It therefore corresponds to a flow of power. In the notation just referred to,
it may be expressed by the vector equation
P = E X H (6.1-4)
4. Lines of force exhibit the properties of inertia. They therefore resist ac-
celeration.
Other principles not quite so fundamental but nevertheless useful in
application are :
5. Lines of force are under tension and at the same time are under lateral
pressure.
6. For perfect conductors there can be no tangential component of electric
force. That is to say, lines of electric force when attaching themselves to a
perfect conductor must approach perpendicularly. This is substantially
true also for common metals such as copper.
In passing it is well to point out that the first principle is really that by
which the ordinary dynamo operates. The second is, for practical purposes.
Oersted's Principle, if we assume that the Unes of electric force are attached
to charges flowing in near-by conductors. The third is known as the Poynting
Principle. It has a wide field of application contributing very materially to
the physical pictures of both radio and waveguide transmission. When ap-
plied to the very simple case of low frequencies propagated along a trans-
mission line, it gives a result that is in keeping with the usual view that the
power transmitted is equal to the product of the total voltage times the total
current. The fourth principle is useful in explaining qualitatively how radia-
tion from an antenna takes place. The usefulness of these four principles will
be made more evident by the examples that follow.
6.2 Transmission of Power along a Wire Line
Direct Current
According to the Poynting concept, one may think of an ordinary dry
cell as two conductors combined with chemical means for producing a con-
tinuous supply of lines of electric force. This need not be counter to the ac-
cepted views concerning electrolysis, for we may think of these lines of force
as being attached to ionic charges incidental to dissociation. As long as the
cell is on open circuit, these lines of electric force remain in a static condition
in which many are grouped in the neighborhood of the terminals of the cell
as shown in Fig. 6.2-1 (a). In this state of equilibrium, the forces of lateral
pressure are balanced by the forces of tension. There is no motion and hence
no flow of power. For an ordinary dry cell such as used in flashlights, the
electric intensity E will depend on the spacing of electrodes, but it may be
as much as 200 volts per meter If we attach to the dry cell two parallel
wires spaced perhaps a centimeter apart with their remote ends open, electro-
306
BELL SYSTEM TECHNICAL JOURNAL
static lines will be communicated to the wires, thereby providing a dis-
tribution roughly like that shown in Fig. 6.2-1 (b). Except at the moment of
contact, there is no motion of the lines of electric force and therefore no
magnetic field and, accordingly, there can be no flow of power. The final
configuration is to be regarded as the resultant of the forces of tension and
lateral pressure. The electric intensity, E, measured in volts per meter at
any point along the line, may be altered at will, merely by changing the
spacing.
If, next, we close the remote end of the line by substituting a conducting
wire for the particular line of force shown as a heavy line in Fig. 6.2-1 (c),
the adjacent lines of electric force will collapse on the terminating conductor,
Fig. 6.2-1. Lines-of-force conccj)! ;i|)])lic(l to ihe transmission of d-c power along a wire line-
as opposing charges unite. This removes the lateral pressure on the neighbor-
ing lines with the result that the whole assemblage starts moving forward.
Each line of force meets in its turn the fate of its forerunners, therel)y de-
livering up its energy to the resistance as heat. As soon as the lateral pressure
at the cell is relieved, chemical equilibrium is momentarily destroyed and
more lines of force are manufactured to fill the gaps of those that have gone
before. All of this is, of course, at the exj)ense of chemical action.
According to the electromagnetic theory, as set forth in the second prin-
ciple, this is but a i)art of the stor}- of transmission. We must add that the
motion of the lines of electric force from the dry cell toward the resistance
gives rise in the surrounding space to lines of magnetic force in accordance
WAVEGUIDE TRANSMISSIOX
307
with Equation 6.1-2 and furthermore the two fields together give rise to
component Poynting vectors representing power flow. Each component
vector has a magnitude at any point equal to the product of the electric and
magnetic intensities there prevaihng and a direction at right angles to the
two component forces in accordance with Equation 6.1-3. This is illustrated
in Fig. 6.2-1 (d).
Since the fields reside largely outside the conductors, we conclude that
the principal component of power flow is through the space between the
wires and not through the wires themselves. If, in the case cited above,
there is appreciable resistance in the connecting wires, then we may expect
that there will be a small component of energy flowing into the wires to be
dissipated as heat. To account for this, we may picture lines of electric force
Circle enclosing
one half
transmitted power
Dissipotive Material
(a) (b)
Fig. 6.2-2. Fields of electric and magnetic force and also direction of power flow in the
vicinity of conductors, (a) Magnified view showing power tiow along a single
dissipative wire, (h) Cross-sectional view of parallel-wire line.
which in the immediate vicinity of the conducting wire lag somewhat behind
the portions more remote. This is illustrated by Fig. 6.2-2(a) which shows a
highly idealized and greatly enlarged section of the field in the immediate
vicinity of one of the two dissipative conductors. The very small component
of power flowing into the conductor is designated as the vector P' to dis-
tinguish it from the much greater ])ower P which we shall assume is being
propagated parallel to the conductor.^
The magnetic field associated with two cylindrical conductors consists of
circles with centers on the line joining the two conductors, whereas the
electric field consists of another series of circles orthogonally related to the
^ For all metals from which conducting lines are ordinarilj- made, the component of
power flowing into the conductor is extremely small compared with the power flowing
parallel to its surface. In Fig. 6.2-2(a) therefore, we should regard vector P' as greatly
exaggerated in magnitude relative to that of vector F,
308 BELL SYSTEM TECHNICAL JOURNAL
first, and having centers on a line at right angles to the first as shown in
Fig. 6.2-2(b). The total flow of power through any plane set up perpen-
dicular to the wires is found by adding up the various component products
of E and H from the boundaries of the wires to infinity. The method by which
this is carried out is outside of the scope of this chapter, but, as already
pointed out, it leads to the same result as obtained by multiplying together
the total voltage and the total current. There are two results of this integra-
tion that are of special interest. (1) In the case of two parallel cylinders, one-
half of the total power flows through the space enclosed by a circle drawn
about the wire spacing as a diameter [see Fig. 6.2-2(b)]. The remaining half
extends from this circle on out to infinity. (2) Since both the electric and
magnetic intensities are greatest in the neighborhood of the wire, most of
the total power flow takes place in the immediate vicinity of the wire.
Transmission of A-c Power
If the simple d-c source mentioned previously is replaced by an alternat-
ing electromotive force, a variety of phenomena may take place, the more
important of which will depend on the frequency of alternation. If this fre-
quency is low (very long wavelength), the line may be relatively short com-
pared with the wavelength, with the result that changes occurring at the
source may appear very soon at the remote end. For this case, the observed
phenomena will vary sinusoidally with time everywhere along the line, in
substantially the same phase. This is the typical alternating-current power
line problem* and, except for minor details, which we shall not discuss at
this time, it does not differ materially from the simple d-c case already
covered.
If, on the other hand, the frequency is high (short wavelength), the line
may be regarded as being electrically long, with the result that sinusoidal
changes occurring at the source may not have traveled very far before the
direction of flow at the source has changed. The over-all result in extreme
cases may become very complicated indeed; for, wavepower may not only
be reflected from the remote end of the line but, if there are sharp bends in
the line or abrupt changes in spacing, it may be reflected from these points
also. The phenomenon observed is usually referred to as wave inlerference
and it often leads to standing waves. Though described above as complicated,
there are many cases where the results of wave interference may be suffi-
ciently simple to be readily visualized. Practical difliculties of various kinds
may arise from these effects, but they may also serve very useful purposes.
In fad, a substantial i)orti()n of our microwave technique is based on wave
■' The wavclcnglh corresponding to a frequency of 60 cycles per second is five million
meters. A commercial |)ower line having a length as great as 100 miles is therefore but
0.03 wavelength long. It is said to he electrically sliorl.
WAVEGUIDE TRAiVSAflSSION
309
interference. Certain specific examples will be discussed later, but first we
shall discuss a somewhat simpler case.
The Infinite Line
Let us take, for discussion, a uniform two-wire line that is infinitely long.
Waves launched on such a line are assumed to be propagated to infinity.
There are no reflected components and hence no wave interference. If the
frequency is very high, the forerunners of the lines of force sent out by the
source will not have traveled very far when the emf at the source will have
reversed its direction. This gives rise at the source to a second group of lines
CIRCLE ENCLOSING ONE HALF
TRANSMITTED POWER
(a)
■1
c o
kA
3 D ir V
■LINES OF ELECTRIC FORCE LINES OF MAGNETIC FORCE
• OUT OIN
(b)
Fig. 6.2-3. (a) Arrangement of lines of electric and magnetic force in both the longitudinal
and transverse sections of an infinitely long transmission line, (b) Space relationship
between electric vector E and magnetic vector // as observed in a plane containing
the two conductors.
of force exactly like the first except oppositely directed. This, in turn, will be
followed by a third group identical with the first and a fourth identical with
the second and so forth until equilibrium is reached. Because the lines of
electric force are in motion, we must expect them to be accompanied by
lines of magnetic force. Both are of equal importance. Therefore it is not
correct to refer to either alone as a distinguishing feature of the wave. Both
components are shown in cross section at the right in Fig. 6.2-3 (a).
The distance between successive points of the same electrical phase in a
wave is known as the wavelength X. It depends on the frequency/ of alterna-
tion and the velocity of propagation v\\ — v/f. The velocity of propagation
in turn depends on the nature of the medium between the two wires. For
310 BELL SYSTEM TECHNICAL JOURNAL
air, the velocity iv is substantially 3()0,()00,()00 meters per second (186,000
mi per sec). For other media :' = t'a/\//Xre, • Thus it will be seen that, by re-
{)lacing the air normally found between the two wires of a transmission line
by another medium such as oil (e,. = 2 and Mr = D, the wavelength will be
reduced by a factor of 1 '•\/2.
If .1.1 is the ma.ximum amplitude reached by the oscillating source during
any cycle, the amplitude at any time /, measured from an arbitrary begin-
ning, may be e.xpressed by the equation
.•1 = Au sin (co/ + 4>) = --In sin (-'^ ?■/ + 0 j (6.2-1)
where 4> is the initial |)hase of the amj)Htude relative to an arbitrary refer-
ence angle
If the transmission line is free from dissipation and we choose a datum
point in a plane at right angles to the direction of propagation and at a
distance far enough from the source that the lines of force have had an oppor-
tunity to conform to the wire arrangement and if we designate the electric
intensity at this point as E i and the corresponding magnetic intensity as
//ii, then the electric and magnetic intensities at other corresponding points
at a distance z further along the line may be represented by
E = Ei) sin — (:; — vl)
A
and
// = Ih sin — (c - vl) (6.2-2)
A
These equations are the trigonometric representations of an unattenuated
sinusoidal wave of electric intensity and magnetic intensity traveling in a
positive direction along the z axis. They are plotted in the yz and xz planes of
Fig. 6.2-3(b). An electromagnetic configuration similar to the above but
traveling in the opposite direction is given by
E - £„ sin ^'^ {z + vl)
A
// = //„ sill ^"^ (:; -f- vl) (6.2 3)
These c(|uali()iis may i)c furlluT conlirmcd by |)li)lliiig arbitrary \"alues on
rectangular-coordinalc pajjcr. In an infmitc line the magnetic intensity H
and the electric intensity E are in the same i)hase as shown in I'ig. 6.2-3.
WAVEGUIDE TRANSMISSION
311
If the wave is subject to an attenuation of a units per unit distance,
possibly due to resistance in the wires, the corresponding components of
E and H are equally attenuated. Either component may be expressed by
an equation of the type
E = E,
oe
sm — \z
\
vt)
(6.2-4)
This is a very special form of certain equations appearing in Sections 3.2
and 3.3.
a = o
distance— z
Fig. 6.2-4. Effect of attenuation on an advancing wave front.
If the attenuation is negligible, then « = 0 and the term e^"' will be unity.
Equation 6.2-4 will then reduce to 6.2-2. If, on the other hand, the attenua-
tion is considerable, the product of a times z will increase rapidly with dis-
tance, and the factor <? "^ will have the effect of reducing the electric
intensity E prevailing at various points along the line. Figure 6.2-4(a) illus-
trates the variation, with distance, of the electric intensity E for an un-
attenuated wave a = 0. There is included for comparison purposes the case,
a = 0.1. Figure 6.2-4(b) shows the effect of this rate of attenuation on waves
that have traveled for some distance. It is significant that moderate amounts
of attenuation have little or no effect on wavelength.
312 BELL SYSTEM TECHNICAL JOURNAL
At low frequencies, conductor loss is often the principal cause of attenu-
ation. At high frequency, this loss may be still more important^ and in addi-
tion there may be losses in the medium around the two conductors. The
latter is particularly true when the conductors are supported on insulators
or are embedded in insulating material. There may also be losses due to
lines of force that detach themselves from the wires and float off into the
surrounding space (radiation). All three lead to attenuation and may be
expressed in terms of an equivalent resistance. They are amenable to cal-
culation for certain special cases.
According to one view of electricity, the individual charges to which
lines of force attach themselves are unable to flow through the conductor
with the velocity of light If this is true, lines of force snap along from one
charge to the next in a rather mysterious fashion which we will not attempt
to picture at this time. This view, like others mentioned previously, tends to
relegate the charges and hence the currents to a secondary position.
Although infinitely long transmission lines cannot be constructed in prac-
tice, it is possible, by a variety of methods, to approximate this result. In
general, a resistance connected across the open end of a short transmission
line, of the kind here assumed, absorbs a portion of the arriving wavepower
and reflects the remainder. If the resistance is either very large or very small,
the reflected power may be very substantial but, by a suitable choice of inter-
mediate values of resistance, the reflected part may be made very small in-
deed. In the ideal case, the arriving wavepower is completely absorbed. A
line connected to this particular value of resistance appears to a generator
at the sending end as though it were infinitely long. The particular resistance
that can replace an infinite line at any point, without causing reflections, is
known as the characteristic impedance of the line. This quantity depends on
the dimensions and spacings of the two conductors as well as the nature of
the medium between. A parallel-wire line, in air, usually has a characteristic
impedance of several hundred ohms. A coaxial line filled with rubber often has
a characteristic impedance of a few tens of ohms. A line having characteristic
impedance connected at its receiving end is said to be match-terminated.
Reflections on Transmission Lines
If the transmission line ends in a termination other than characteristic
impedance, or if there are discontinuities, due to impedances connected
either in series or in shunt with the line, reflections of various kinds will
occur.^ Much of the practical side of microwaves has to do with these re-
flections.
"■ The losses in most conductors increase with ihe square rod of the frequency.
•> At the higher fre(|ucncies, rellcclions may also occur at points where the wire spacing
changes al)rui)liy. In some instances al)rupt changes in wire diameter may be sulVicient
to cause reflection. These discontinuities may be regarded as changes in characteristic
impedance.
WAVEGUIDE TRANSMISSION
313
A particularly simple form of reflection occurs when the high-frequency
transmission line is terminated in a transverse sheet of metal of good con-
ductivity, as for example, copper. An arrangement of this kind is shown in
Fig. 6.2-5. As it is difficult to represent a wave front moving toward the
reflecting plate, we shall substitute an imaginary thin slice or section of the
electromagnetic configuration. A slice of this kind is shown in Fig. 6.2-5(a).
Experiment shows that, at the boundary of the nearly perfect reflector,
the transverse electric force E is extremely small. This is consistent with
the sixth principle set forth in the previous section which states that there
can be no tangential component of electric force at the boundary of a per-
fect conductor. The result actually observed can be accounted for if it is
assumed that the reflecting conductor merely reverses the direction of lines
of electric force as they become incident, thereby giving rise to two sets of
*^^^^^^^^^^^' S\\
ni
^
^V \\^ \\\\\\\k\<\\<\\V'
>r ,r
i
(a) (b)
Fig. 6.2-5. (a) Propagation of an electromagnetic wave along a two-wire line terminated
by a large conducting plate, (b) Representative lines of force reflected by the
conducting plate.
lines of force as shown in Fig. 6.2-5(b), one of intensity Ei = E directed
downward in the figure and moving laterally toward the metal sheet (in-
cident wave) and the other of intensity Er = —E directed upward and mov-
ing away from the metal sheet (reflected wave). Accordingly the resultant
electric intensity at the surface is zero.
If the reflector is non-magnetic, the magnetic intensity H will be un-
affected by the reflecting material. We find by applying the right-hand rule
of Fig. 6.1-4 that the electric intensity E^ = — E when combined with H
constitutes a wave that must travel in a negative direction of v. This wave
may be represented by Equation 6.2-3. In a similar way the Poynting vector
which before reflection is represented by P = E x H now takes the form
P = (— ExH). The negative sign according to the right-hand rule of
Fig. 6.1-4 shows that the power approaching the conductor is reflected back
upon itself. If E and H are respectively equal in magnitude before and after
314 BELL SYSTEM TECHNICAL JOURNAL
incidence, the reflection is perfect, and the coeflicient of reflection is said to
be unity. Bearing in mind that H, = e(vxE) before reflection and H^ =
e( — V X — E) after reflection, it is evident that the direction of the magnetic
intensity has been unchanged by the process of reflection and that the re-
sultant magnitude at the surface of the metal is | //j | + | ^r I = 2 | ^ | .
Thus we see that, at the moment of reflection from a metallic surface, the
resultant electric force vanishes and the resultant magnetic force is doubled.
The reflection of waves at the end of the line naturally gives rise to two
oppositely directed wave trains. This is a well-known condition for standing
waves. Though a complete discussion of standing waves calls for the math-
ematical steps taken in Section 3.6, there are certain qualitative results that
may be deduced from relatively simple reasoning. Some of these deductions
will be made in the paragraphs that follow.
If an observer, endowed with a special kind of vision for individual lines
of force, were to be stationed at various points along a lossless transmission
line as shown in Fig. 6.2-5, he would observe a variety of phenomena as
follows. Near the reflector he would observe a waxing and waning of lines
of force, both electric and magnetic, corresponding to the arrival of crests
and hollows of waves. Also he would observe a similar waxing and waning
corresponding to waves leaving the reflector. The sum of the two waves
would give rise at the conducting barrier to a resultant electric intensity of
zero and to a corresponding magnetic intensity that would oscillate between
limits of plus or minus 2H. Since it is the magnetic component that is the
the more evident near the barrier, this region would appear to the observer
much like the interior of a coil carrying alternating current.
If the observer were to pass along the line to a point one-eighth wave-
length to the left of the reflector, the distance up to the reflector and back
would then be a quarter wave and he would then find that at the moment
that a wave crest (maximum intensity) was passing on its way toward the
reflector a point on the wave corresponding to zero intensity would be re-
turning from the reflector. Adding the corresponding electric and magnetic
intensities at this point, he would observe that the electric intensity would
not always be zero but instead it would oscillate between limits of plus or
minus \/2 E. Similarly the corresponding magnetic intensity would no longer
oscillate between limits of plus or minus 211, but instead it would never reach
limits greater than plus or minus \/2 //. Thus at this point the electric and
magnetic comj)onents would have the same average intensity.
If the observer were to move farther along the line, stopping this time at a
distance of one-fourth wavelength to the left of the metal plate, the total
electrical distance to the barrier and back again would be a half wave-
length and he would now fmd that at the time a crest passed on its way
toward the reflector a hollow (maxiimini negative intensity) would be pass-
WAVEGUIDE TRANSMISSION 315
ing on its return journey. This time, the resultant electric intensity would
oscillate between limits of plus or minus 2E, and the resultant magnetic
intensity would be zero at all times. To this observer then, this quarter-wave
point on the Une would have many of the characteristics of the interior of
a condenser charged by an alternating voltage.
If our observer were to move another one-eighth wave farther along the
line, he would note that the resultant electric and magnetic forces would
again be equal. Proceeding on to a point one-half wavelength from the metal
reflector, he would observe that, at the time crests (maximum positive in-
tensity) were passing on their way toward the reflector, hollows would be
returning, and accordingly upon examining the resultant electric intensity
he would find it to be zero at all times, whereas the corresponding magnetic
intensity would be oscillating between limits of plus or minus 2H. At this
point along the line, he would be unable to distinguish his electrical environ-
ment from that prevailing at the metal boundary. The half-wave line, there-
fore, has had the effect of translating the metal barrier to another point in
space a half wave removed.
If the observer were to continue still farther along the line, he would
pass, alternately, points where the resultant electric force is zero and other
points where the resultant magnetic force is zero. It is important to note
that at points in a standing wave where the magnetic force is a maximum,
the electric force is a minimum and at points where the electric force is a
maximum, the corresponding magnetic force is a minimum. It is customary
to call the points of minimum E (or H) "mins," though the term node is
sometimes substituted. Points of maximum E (or H) are known as "maxs"
with the term loop as its alternative. If the observer were to measure current
and voltage along the line, he would find that points of maximum voltage
correspond to maximum E and that points of maximum current correspond
to maximum H.
An examination of the energy associated with the incident and reflected
waves shows that, except for minor losses not to be considered here, there
is as much energy led away from the reflector as is led up to the reflector,
and that there is associated with the standing wave a stored or resident
energy. The regular arrangement of nodes and loops along a standing wave
with minima at half-wave intervals is a very important characteristic, for
such points may be located very accurately experimentally, and accordingly
wavelength may be measured with considerable precision.
If, instead of terminating the wire line in a large conducting plane as-
sumed previously, it is terminated in a relatively thin cross bar as shown in
Fig. 6.2-6, the reflection will assume a somewhat more complicated form.
First of all, the thhi cross bar will intercept, initially at least, only a portion
of the total wave front. The i)articular lines of force arriving along a plane
316
BELL SYSTEM TECHNICAL JOURNAL
containing the two wires will be the first to be reflected and they will behave
at reflection much like those already discussed, whereas those outside the
plane of the two wires will not be intercepted initially by the thin cross bar
but instead will advance for a short distance beyond the end of the line
before their forces of tension bring them to rest. These outlying lines of
force are represented by the lines designated as c in Fig. 6.2-6. After the
first lines of force have been reflected, lateral pressure will be removed from
those adjacent, with the result that they will close in and collapse on the
conductor at a slightly later time than their neighbors. One over-all result
of this process is to make the effective length of such a line slightly greater
than the true length. Effects of this kind are observed in practice and they
are referred to as fringing. Discrepancies between the wavelength as
measured in the last section of line where fringing may take place and that
measured between other minima along the same line are usually small but
J'/ //// ///////
/////,
9^ ^ / ^ / / / / / /> /•////•/
(a)
(b)
Fig. 6.2-6. (a) Representative transmission line terminated by a conductor of finite
dimensions, (b) Nature of reflection by a finite conductor.
they are nevertheless measurable It is also true that, as the wave front ap-
proaches a limited barrier of this kind, some of its energy continues on into
the space beyond and is lost as radiation. In general, the smaller the barrier,
the larger will be the losses.
Consider next a line open at its remote end, as shown in Fig. 6.2-7 In
this case, none of the lines of force of the advancing wave is intercepted by
a conductor, with the result that a very considerable number momentarily
congregate near the end of the line and, because of inertia, they e.xtend into
the space beyond as suggested by Fig. 6.2-7 (b). This process continues until
forces of tension in the lines, still clinging fast to the ends of the wires, bring
the assemblage temporarily to rest. At this moment, there is no magnetic
component; for v, in the relation H = ((v x E), is zero while the correspond-
ing electric intensity is approximately 2£. The lines of electric force, being
momentarily at rest, represent energy stored in the electric form.
WAVEGUIDE TRANSMISSION
317
This static situation is extremely temporary, for the tension momentarily
created in the lines of electric force soon forces the configuration as a whole
to move backward. As the wave front gets under way, the magnetic force
H increases in magnitude in accordance with the relation H = e(vxE).
The fact that the wave front extends momentarily for a short distance
beyond the physical end of the fine and requires time to come to rest and
get into motion in the reverse direction implies inertia or momentum in the
wave front. This is the inertia referred to in the fourth principle mentioned
in Section 6.1. In this form of reflection, fringing is usually very evident,
and because of fringing we may have an apparent reflection point that is
considerably beyond the end of the wires. Thus the distance from the end
of the wires back to the first voltage minimum is much less than the quarter
wave that otherwise might be expected.
^\\\\\\\\\\\\\v\\\ \\vy
^\\V\S\\\V\VV\v.
(a)
(b)
Fig. 6.2-7. (a) Transmission along a line open at the remote end. (b) Nature of reflection
from open end.
It is generally true that processes of reflection in which fringing takes place
are usually attended by considerable amounts of radiation. This suggests
that in the process of reflection some of this extended wavepower detaches
itself from the parent circuit and is lost. Experience shows that this lost
power may be greatly enhanced by separating the two wires or by flaring
their open ends. The so-called half-wave dipole, so familiar in ordinary radio,
is but a transmission line in which the last quarter-wave length of each wire
has been flared to an angle of 90 degrees. If we wish to minimize radiation,
we follow a reverse procedure and reduce the spacing between the two parallel
wires. This also reduces fringing, for we find that the measured distance from
the ends of the wires to the first voltage minimum is now more nearly a
quarter wave.
It is of interest to compare reflections taking place at the open end of a
transmission line with those at a closed end. When a wave front becomes
incident upon a perfect conductor, the electric force vanishes. At the same
time, the lines of magnetic force, though effectively brought to rest, are
318 BELL SYSTEM TECHNICAL JOURNAL
momentarily doubled in intensity. The energy is predominantly magnetic,
and the type of retlection may be regarded as inductive. When the wave is
reflected from the ideal open-end line, a reverse situation prevails. The lines
of magnetic force momentarily vanish while lines of electric force, though
brought to rest, are doubled in intensity. At this moment the energy is pre-
dominantly electrostatic, and the reflection may be considered as being
capacitive.
When a line is terminated in a sheet of metal of good conductivity such
as copper or silver, reflection is almost perfect. If the sheet is a poor conductor
such as lead or German silver, most of the incident power will still be re-
flected; but if a semi-conductor, such as carbon, is used as a reflector, a per-
ceptible amount of the incident power will be absorbed. It is interesting also
that the penetration into all metals at the time of reflection is very slight,
for relatively thin sheets seem to serve almost as well as thick plates. It is
therefore possible to use as reflectors extremely simple and inexpensive
materials, for example, foils or electrically deposited films fastened to a
cheaper material such as wood.^
A more general study of reflections on transmission lines shows that the
examples cited previously are special cases of a very general subject. Not
only may there be reflections from the open and closed ends of a transmission
line, but there may be reflections also when the line is terminated in an in-
ductance, in a capacitance, or in a resistance. Details concerning the re-
flections that may be observed from various combinations of these three
impedances are discussed in connection with Fig. 3.6-3. The outstanding
results of these discussions may be summarized for the ideal case as follows:
1. A pure inductance (positive reactance) connected at the end of a
transmission line always leads to a reflection coefBcient having a magnitude
of unity. The standing wave resulting from this reflection will be charac-
terized by the following: (a) If the terminating inductance is infinitely large
(reactance of positive infinity), the reflection will be identical with that from
an ideal open-end line, and the distance to the nearest voltage minimum will
be a quarter wave. [See Fig. 3.6-3(a).] (b) If the inductance is finite but very
large, the distance to the nearest voltage minimum, as measured toward the
generator, will be somewhat greater than a quarter wave. [See Fig. 3.6-3(b).|
(c) If the inductance is reduced progressively toward zero (reactance zero),
the distance to the same voltage mininnmi will approach one-half wave-
length. In this limiting case, another voltage minimum will appear at the
end of the line. [See Fig. 3.6-3(c) and 3.6-3(d).]
2. A j)ure capacitance (negative reactance) connected at the end of a
~' One convenient and inexpensive form of reflector is a kind of l)uilding paper coated
with copper or aluminum foil. Moderately good reflectors can also he made i)y covering
wood with a special ])aint containing i'lnely divided silver in susi)ension (I)u I'ont's 4817).
Most aluminum paints are unsatisfactory for this purpose.
WAVEGUIDE rRANS.\riSSION 319
transmission line also leads to a reflection coefftcient having a magnitude of
unity. In this case, the resulting standing wave will be characterized as
follows: (a) If the capacitance is zero, (reactance equal to minus infinity),
the reflection will correspond to that from the open end of a transmission
line, and a voltage minimum will be found at a distance of a quarter wave
from the end. [See Fig. 3.6-3 (g).] (b) If the capacitance is increased from
zero to a small finite value, the distance to the nearest voltage minimum
will be somewhat less than a quarter wave. [See Fig. 3.6-3(f).] (c) If the ca-
pacitance is increased progressively toward infinity (reactance zero), the
distance to the nearest voltage minimum will approach zero. [See Figs.
3.6-3(e) and 3.6-3(d).] The limiting condition, in which the terminating
capacitance is zero, is comparable with that in which the termination is an
infinitely large inductance.
3. If a pure resistance is connected at the end of a transmission line, the
magnitude of the reflection coefftcient varies with the resistance chosen.
The relations are such that: (a) If the terminating resistance is infinite, the
magnitude of the reflection coef^cient will be unity and its sign will be posi-
tive. [See Fig. 3.6-3(h).] (b) If the terminating resistance approaches the
characteristic impedance of the line, the distance to the nearest voltage
minimum will remain constant, but the magnitude of the reflection coefficient
will approach zero. [See Figs. 3.6-3(i) and 3.6-3(j).] (c) If the terminating re-
sistance is made less than characteristic impedance, the sign of the reflection
coefficient will be reversed, and, as the terminating resistance approaches
zero, its magnitude will approach unity. [See Figs. 3.6-3(k) and 3.6-3(1).]
When the terminating resistance is infinite, the reflection is comparable
with that in an ideal open-end line, and the nearest voltage minimum will
be found at a distance of a quarter wave. When the terminating resistance
is zero, the reflection is comparable with that in a closed-end line, and the
voltage minimum will appear at the end of the line and also at a point one-
half wave closer to the generator. If the line is terminated in a pure resistance
of intermediate value, the voltage minima of such standing waves as may
be present will be found at the end of the line for all values of the resistance
that are less than characteristic impedance and a quarter wave removed from
the end of the line for all values greater than characteristic impedance. When
the terminating resistance equals characteristic impedance, there is no
standing wave.
If, instead of terminating the line considered above in an inductance coil
or in a capacitance or a resistance, we assume that it continues indefinitely
into a mass of material having either a conductivity or a dielectric constant
different from that of air, similar reflections may take place at the surface.
A particular e.xample is shown in Fig. 6.2-8. In general, a part of the wave-
power arriving at the surface will be reflected and a part will be transmitted.
320 BELL SYSTEM TECHNICAL JOURNAL
One may picture a portion of the Faraday tubes of force turned back at the
interface while the remainder continue into the second medium. If one were
to reverse the direction of transmission and consider wavepower transmitted
from the second medium back into the first, a similar partial reflection would
be noted. In both cases the part turned back and returned to the source may
be regarded as a reactive component since no energy is really lost. In a similar
way, the transmitted component, since it is not returned to the source, may
be regarded as a resistive or dissipative component.
If the medium into which wavepower is transmitted is a perfect insulator,
the transmitted wave will continue indefinitely except as attenuated by the
Fig. 6.2-8. Reflection and transmission of lines of force incidental to a change of medium
along a transmission line.
wires along which it is guided. Its wavelength, X, in the dielectric will be
less than the wavelength, Xo , in air as expressed by the relation
If the second medium is somewhat conducting, the wave will be further
attenuated, the rate of attenuation being related in a rather complicated
way not only to the conductivity of the second medium but to its dielectric
constant and permeability as well. Thus far in microwave practice, little
practical use has been made of materials having permeabilities very different
from unity. However, considerable use has been made of materials having
various dielectric constants, e^, and conductivities, g. Sometimes these take
the form of plates placed across a waveguide transmission line. Examples
will appear in Section 9.8.
If a thin sheet of insulating material having a dielectric constant, €r,
and conductivity of zero is placed across a two-wire transmission line, the
percentage of power reflected is given approximately by
qw = ^ (cr - 1) (6.2-5)
Xo
A thin sheet of this kind is approximated when wires carrying very high
frequencies pass through the glass walls of a vacuum tube. If the glass
WAVEGUIDE TRANSMISSION 321
thickness, /, is small compared with the wavelength in air, Xc, the power
reflected by the glass envelope will likewise be small.
Sometimes it is not feasible to reduce the wall thickness sufficiently to
avoid serious reflections. In these instances it may be possible to make the
thickness one-half wavelength as measured in glass whereupon the wave
reflected from one face of the plate will be approximately equal in amplitude
to that from the other face and, since they are separated by one-half wave-
length, they tend to cancel.
Another case of practical interest is that in which the line is terminated
in a plate of very special dielectric constant e^, conductivity gi, and thick-
ness /. This is followed by a second plate of nearly infinite conductivity.
This arrangement is shown in longitudinal section in Fig. 6.2-9. By a proper
choice of constants, the combination may be made a good absorber of wave-
Fig. 6.2-9. A transmission line terminated in a conductor coated with a special materia
such that all of the incident wave power is absorbed.
power. It will therefore be substantially reflectionless It may be shown that
to satisfy this requirement
and
Xo = -f^ (6.2-6)
' 607rgi(2« - 1) ^VTr{2n - 1) ^^"^ ^^
where n is any integer. One common example is that in which n = 0. The
plate is then a quarter wave thick as measured in the medium.^ A reflection-
less plate of this kind when placed at the end of a transmission line appears
to the source as though the line were terminated in its characteristic im-
pedance. Devices incorporating this principle are sometimes used as match
terminators for waveguides.*
* A more complete discussion of this problem was published in 1938 by G. W. O. Howe,
"Reflection and Absorption of Electromagnetic Waves by Dielectric Strata." Wireless
Engr., Vol. 15, pp 593-595, November 1938.
* Plates of this kind may be made very simply by mixing carbon with plaster in vary-
ing proportions until the right combination is reached.
322
BELL SYSTEM TECIIXICAL JOURXAL
When a two-wire transmission line assumes the coaxial form, the lines
of electric force are radial and lines of magnetic force are coaxial circles.
The directions of these two components obey the right-hand rule. (See Fig.
6.2-10.) Since the wave configuration is completely enclosed except for a
small exposure at each end, radiation from this type of line can be made very
small.
K//////////////////////////^^^
• 1 •
LINES OF ELECTRIC FORCE LINES OF MAGNETIC FORCE
Fig. C.2-10. Arrangement of lines of electric and magnetic force associated with transmission
along a coaxial arrangement of conductors.
6.3 Radiation
Electromagnetic waves, including both light and radio waves, are not
unlike the waves that are guided along wire lines. Their difference is largely
a matter of environment. In one case they are attached to wires w^hile in
the other they have presumably detached themselves from some configura-
tion of conductors and are spreading indefinitely into surrounding space.
We shall present in this section one of several possible pictures of the launch-
ing of radio waves from a transmission line. Like other verbal pictures drawn
in this chapter, it should be regarded as highly qualitative.
-Assume a two-wire line with one end flared as shown in Fig. 6.3-1. If at
some point to the left there is a source of wavepower, there will flow from
left to right along the line a sinusoidal distribution of lines of electric and
magnetic force not unlike that shown in Fig. 6.2-7. In order to simplify our
illustration, we shall single out for examination two representative lines of
electric force a-h and c-d located a half wave apart. It is understood, of
course, that there are present many other lines both before and behind those
represented. Also there are lines of magnetic force at right angles to the
electric force. As time progresses each element of length of the line of force
a-h moves laterally with the velocity of light. In the region where the wires
are parallel, it remains straight but, upon reaching the flared section, its
two ends fall behind the central section, thereby forming a curve as shown
in Fig. 6.3-1 (c). As this line of force moves to the end of the flared section
[Fig. 6.3-1 (d)], its successor c-d follows one-half wavelength behind.
WA V ECU IDE TRANSMISSION
323
Because of the property of inertia with which all lines of force are assumed
to be endowed, the central section of a-b, which is already greatly extended
due to curvature, continues in motion for some time after the two ends, at-
tached to the conductors, have come to rest. The result is shown approxi-
mately by Fig. 6.3-1 (e). An instant later and perhaps after the two ends of
line of force a-b have started on their return journey, the line of force c-d
approaches sufficiently close to a-b that a coalescence ensues [Fig. 6.3-1 (f)].
An instant later lission takes place as illustrated in Fig. 6.3-1 (g), leaving a
portion of the energy of each a-b and c-d now shared by a radiated com-
Fig. 6.3-1. Successive epochs in a highly idealized representation of radiation from the
flared end of a transmission line.
ponent, r, and a reflected component, .r. That the two components r and x
should travel in opposite directions seems reasonable when it is noted that
lines of electric force in .v are in the same direction as in the adjacent portion
of r. They may therefore be expected to repel. The first of these components,
r, appears to the transmitter as though it were a resistance since it represents
lost energy. The second, .v, appears as a reactance since it represents energy
returned to the transmitter. The radiated component, r, will be followed by
other components ri, r-i, etc., as represented in Fig. 6.3-1 (h).
In the radiated wave front, the two components E and H are everywhere
mutually perpendicular and in the same phase. Because the wave front
324
BELL SYSTEM TECHNICAL JOURNAL
is curved, as shown in cross section in Fig. 6.3-2, the component Poynting
vectors which specify the directions in which energy is flowing will be slightly
divergent. As a result, only a portion of the total wavepower will proceed
in the preferred direction. It follows that, for best directivity, the emitted
wave front should be substantially plane, and the lines of force should be as
nearly straight as possible. There is shown in Fig. 6.3-3 a series of configura-
Direction of propagation
Fig. 6.3-2. Cross section of electromagnetic waves radiated from the flared end of a trans-
mission line. Lines of electric force lie in the plane of the illustration; lines of magnetic
force are perjiendicular to the illustration while the flow of ]:)owcr is along the divergent
arrows P.
tions based partly on speculation and partly on deductions from Huygens'
principle. They illustrate in a rough way how, by increasing the aperture
between the two wires of the elementary radiator, we may make the indi-
vidual coiTiponent Poynting vectors more nearly parallel. ^°
'" Figure 6.3-3 has been greatly oversimplified. Experiment shows that, to achieve the
result desired, the angle between the two wires of Fig. 6.3-3 must be smaller for larger
apertures than for small apertures.
WAVEGUIDE TRANSMISSION
325
Thus far, we have restricted our considerations to directivity in the plane
of the two conductors (vertical plane as here assumed). Experiment shows
that, in the plane perpendicular to that illustrated, the directivity from a
single pair of wires is slight. However, we may obtain additional directivity
by increasing the horizontal aperture. One method of accomplishing this
result is to array, at rather closely spaced intervals, identical elementary
radiators each of the kind just described. [See Fig. 6.3-4(a).] An infinite num-
ber of these elements infinitesimally spaced become two parallel plates as
shown in Fig. 6.3-4(b). If metal plates are now attached at the right and left
(a)
(b)
Fig. 6.3-3. Illustrating how radiating systems of large aperture may give rise to wave fronts
of large radius of curvature and hence lead to increased directivity.
(a) (b)
Fig. 6.3-4. Alternate ways by which the aperture of a flared transmission line radiator may
be increased.
sides, the resulting configuration will become a waveguide horn. As a general
rule, the larger the area of aperture, the more directive will be the antenna.
The highly schematic array shown in Fig. 6.3-4(a) is introduced for illustra-
tive purposes only. It is not one of the preferred forms used in microwave
work. More practicable forms will be found in Chapter X.
The wave model shown in Fig. 6.3-2 conveys but a portion of the known
facts about a radiated wave. A more accurate model is shown in skeleton
form in Fig. 6.3-5. It is assumed that the transmitted wave has been launched
with about equal directivity in the two principal planes and that the ob-
326
BKI.L SYSTEM TEC/I. \/CAL JULR.XAL
server is looking into one-half of a cut-away section of the total configuration.
In the complete configuration, the individual lines of electric force (solid
lines) and magnetic force (dotted lines) form closed loops, thereby pro-
ducing in each half-wave interval a packet of energy. The stream of projected
energy from an antenna is, according to this view, a series of these packets
one behind the other moving along the major axis of transmission. At the
transmitter each packet may have lateral dimensions that are only slightly
greater than the corresponding dimensions of the radiating antenna; but,
since the packet has curvature and since propagation is radial, the packet
spreads as it progresses so that at the distant receiver it may be very large
indeed.
p"=o
Fig. 6.3-5. Highly idealized representation of a wave-packet radiated by a typical micro-
wave source. One half of the total packet is assumed to be cut awa\'.
Around the edge of each packet there is a region where the relationship
between the vectors E, H, and v is rather involved. For example, in the vicin-
ity of point 1 in Fig. 6.3-5, there is a substantial component of E but at this
point the vector // is zero and accordingly the Poynting vector P' at that
point is also zero. (See Equation 6.1-4.) In a similar way there may be in the
vicinity of point 2 a substantial component of magnetic force H; but, since
at this point the electric force is substantially zero, we conclude that the
Poynting vector P" is again zero and again no power is propagated."
" The ])eculiar edge effects noted may be regarded as a result of a kind of wave inter-
ference not unlike that ])rc\-ailing in the regions of minimum E and // in the case of stand-
ing waves as discussed in Section 6.3. A similar kind of wave interference is cited in Section
6.5 to account for regions of low E and // in transmission along a waveguide-.
WAVEGUIDE TRANSMISSION 327
The sharpest radio beams now in general use are only a few tenths of a
degree across. We conclude that for these sharp beams a small but neverthe-
less appreciable curvature remains in the radiated wave packet. This means
that, when the wave front has arrived at a distant receiver, it is still many-
times larger than any receiving antenna it may be practicable to construct,
and accordingly the latter can intercept but a small portion of the total
advancing wavepower. This implies a considerable loss of power, which is
indeed the case.
In the process of radio reception, one may think of the antenna structure
as a device that cuts from the advancing wave front a segment of wavepower
which it subsequently guides, preferably without reflection, to the first
stages of a nearby receiver. To be efficient, the wavepower intercepted should
be large. This, in turn, calls for a receiving antenna of considerable area. It
will be remembered that a large aperture was also a necessary feature for
high directivity at the transmitter. This is consistent with the accepted view
that the processes of reception and transmission through an antenna are
entirely correlative and that a good transmitting antenna is a good receiving
antenna and vice versa. The directive properties of an antenna are some-
times specified in terms of its effective area. (See Section 10.0.)
The term uniform plane wave is a highly idealized entity assumed in
many problems for purposes of simplicity but never quite attained in prac-
tice. In an idealized wave front, the electric and magnetic components
E and H are not only everywhere mutually perpendicular but both com-
ponents are exclusively transverse. That is, there is no component of either
E or H in the direction of propagation. Such a wave belongs to a class
known as transverse electromagnetic waves (TEM). These may be com-
pared with others, to be described later, known as transverse electric waves
(TE) and transverse magnetic (TM) waves. Waves guided along parallel
conductors are also TEM waves, but except in the case of infinitely large
conductors they are not uniform plane waves.
6.4 Reflection of Space Waves from a Metal Surface
One of the early triumphs of the electromagnetic theory was its ability
to account satisfactorily for the reflection and refraction of light. This
theory was so general as to include not only a wide range of wavelengths
but also a wide range of surfaces as well. According to this theory, re-
flections may occur whenever electromagnetic waves encounter a dis-
continuity. This may happen, for example, when waves fall on a sheet of
metal, in which case the discontinuity is due to the sudden change in
conductivity. Reflection may also occur when waves are incident on a
thick slab of glass or hard rubber, in which case reflection is due to a sud-
328
BELL SYSTEM TECIIX/CAL JOURNAL
den change in dielectric constant.'" Similar reflections may theoretically
take place also at an interface where the permeability of the medium
changes suddenly. The case in which there is a change of conductivity has
an important bearing on waveguide transmission. It will therefore be dis-
cussed in considerable detail.
Assume a plane wave incident obliquely upon a conducting surface as
shown in Fig. 6.4-1. The line along which the wave is progressing (wave-
normal) is referred to as the incident ray. It intersects the conducting
surface or interface at a point 0 and makes an angle d with the perpendicu-
lar OZ. After reflection, the normal to the new^ wave wave front makes an
angle 6' with the perpendicular OZ. This second wave-normal is known as the
Fig. 6.4-1. Reflection at oblique incidence from a metal plate for the particular case where
the electric vector is perpendicular to the plane of incidence.
reflected ray, and its angle with the perpendicular OZ is known as the angle
of reflection. The plane containing the incident ray and the perpendicular
OZ is known as the plane of incidence. The incident and reflected rays lie
in the same plane, and their corresponding angles of incidence and reflection
are numerically equal.
In problems of oblique incidence there are two cases of interest, depend-
ing on whether the electric or the magnetic comi)onent lies in the plane of
incidence. For our particular j)urpose, the second of these two cases is of
special interest and it will therefore be discussed in considerable detail.
The vector relations corresj)onding to this case are shown in Fig. 6.4-1.
'^ For a more general discussion of the electromagnetic thcor\- of retlectioii: L. Page
and N. 1. Adams, "Princii)lcs of I'.lectricity," 1). \'an Xostrand Co., Inc., pp 569-575,
New York 1931. R. I. Sarhacher and \V. A. I'-dson, "Hyper and Ultra-high Fretiuency
Engineering," John \\ ilcy & Sons, Inc., i)p 105-116, New York 1943.
WAVEGUIDE TRANSMISSION 329
Included are the relative directions of E and H both before and aftei
reflection.
In Fig. 6.4-2 there are shown in cross section representative lines of
electric force in an advancing plane wave front. They are numbered re-
spectively 1, 2, 3, 4, 5, 6, and 7. Each individual figure [(a), (b), (c), etc.]
represents a succeeding period of time. We shall assume that the particular
wave front singled out for illustration represents the crest of a wave
Both ahead and behind this crest there are located alternately at half-wave
intervals other crests and hollows, and their respective lines of force alternate
in direction. Each line of force m the wave front is assumed to be moving
in a direction indicated by the vector v. It is furthermore assumed that
there is also present a magnetic component, indicated by the dotted vector
// that is perpendicular to E and also to v. The vectors v and H must of
course be so directed as to be in keeping with the right-hand or cork-screw
rule, both before reflection and after reflection. Also at the point of incidence
the tangential electric force must be zero. To account for this, we assume
that as each line of electric force moves up to the conducting plane it is
reversed in direction, thereby making on the average as many lines of
electric force at the surface directed toward the observer as directed away
from the observer. Consider, for example, lines of force 3 and 5, 2 and 6,
and 1 and 7, in Fig. 6.4-2(c).
Associated with these two components of electric force which, let us say,
are E and E', there are two components of magnetic force // and H'.
These may be specified by H = e(v x E), each of which at the interface may
be resolved into two components shown in Fig. 6.4-3 sls H = H j_ + -^n
at the left and H _i_' = —H\{ at the right. Combining these four vec-
tors, assuming reflection to be perfect, we find that at the interface
H j_ — H j_' = 0 and H^^ — { — H/) = 2H, giving as an over-all result:
(1) the electric force at the interface is everywhere zero; (2) the vertical
component of the magnetic force at this point is also zero; and (3) the
tangential component of the magnetic force at the interface is 2//.
The peculiar configuration that resides close to the metal boundary is
propagated to the right as a kind of magnetic wave. It has rather inter-
esting properties which will become more evident by referring again to
Fig. 6.4-2. Two conclusions may be drawn from this figure, depending
on the point of view assumed. To a myopic observer located at the inter-
face and unable to see far beyond the point p and unable to distinguish one
line of force from another, the advancing wave front would look like a con-
figuration of amplitude H^^ = 2H and £|| = 0 moving parallel to the inter-
face with velocity v^ = I'/sin d. To this observer the apparent velocity
would increase as 6 becomes progressively smaller until, at perpendicular
incidence, Vz would approach infinity. These results follow from the geo-
330
BELL SYSTEM TECHNICAL JOURNAL
INCIDENT WAVE
FRONT ~---
>)
(b)
(c)
(d)
(e)
w.
V777-/W777777777777777777777777777777777777,
VTTZ
777777777.-^777777777777777777777777777:^^777777777-
2®
3®
®7
®6
®5
"^777777777777/^777777777777777777777,
I®
2®
3®
4 ® H /
5® ®7
REFLECTED
'""wave FRONT
77777777777777777777:
t
• V't
77////////////,>/^^////\
P'
V' =
VSIN0
V
J
SIN B
Fig. 6.4-2. Successive steps in the reflection of a single plane wave front by a metal plate.
WAVEGUIDE TRANSMISSION
331
metrical relations shown in the lower part of Fig. 6.4-2. Phenomena
similar to this are sometimes observed when water waves, coming in from
the ocean, break upon the beach. If the approach is nearly perpendicular,
the point at which the wave breaks may proceed along the beach at a
phenomenal speed. A similar effect may be produced by holding at arm's
length a pair of scissors and observing the point of intersection as the blades
are showly closed. A relatively slow motion of the blades leads to a rather
rapid motion of the point of intersection.
Since, in the case of incident waves, the apparent velocity is Vz = z'/sin 6,
the corresponding wavelength is A^ — X/sin d. Both quantities play an
important part in the picture of waveguide transmission to be drawn later.
In particular, the apparent velocity v^ will prove to be identical with a
quantity known as phase velocity.
Path of Incident
Line of Force
/
h:=h,;+h„:.2h„
h[= h^- h^= 0
e''= E - e' = 0
Electric Vector-^
Electric Vector
Path of Reflected
Line of Force
(directed away (directed toward
from observer) observer)
Fig. 6.4-3. Relationship Ijetween various components of E and H before and after reflection
by a metal plate.
A second observer located at the interface, shown in Fig. 6.4-2, endowed
with better vision and able to single out particular lines of force may obtain
a somewhat different view of reflection. If he observes a particular line of
force such as (4) in Fig. 6.4-2 for the considerable period of time, /, required
for it to approach the conducting interface [Figs, (a) to (c)] and recede to
a comparable distance [Figs, (c) to (e)], he will note that, whereas the line
of force has really traveled a total distance vl, its effective progress parallel
to the interface has been v'l = vi sin Q. (See geometrical relations in lower
part of Fig. 6.4-2.) This provides another kind of velocity {v' = v sin 0)
known as group velocity. It is the effective velocity with which energy is
propagated parallel to the metal surface. It approaches zero at perpen-
dicular incidence. It will be observed that
V = Vz sm"
332 BELL SYSTEM TECHNICAL JOURNAL
and
v'v, = v" (6.4-1)
Group velocity also plays an important part in waveguide transmission.
6.5 Waveguide Transmission
It was pointed out in an earlier chapter that each' of the various con-
figurations observed in waveguides may be considered as the resultant of a
series of plane waves each traveling with a velocity characteristic of the
medium inside, all multiply reflected between opposite walls. In the case
of certain of these waves, this equivalence may not be readily obvious, but
for the dominant mode in a rectangular guide, which is one of the more
important practical cases, it is relatively simple. It also happens that the
analysis of such waves throws considerable light on the nature of guided
waves, and furthermore it enables us to deduce many of the useful relations
used in waveguide practice — relations that might otherwise call for rather
complicated mathematical analysis.
It is assumed in Fig. 6.5-1 that we are viewing, in longitudinal section
and at successive intervals of time, a hollow rectangular pipe having
transverse dimensions of a and h measured along the x and y axes respec-
tively. In this case the illustration is in the xz plane. It is further assumed
that the electric force lies perpendicular to the larger dimension a and is
consequently perpendicular to the plane of the illustrations. We assume
in Fig. 6.5-1 (a) a particular plane wave front 1, perhaps a crest, that has
recently entered the guide from below. Let us say that its velocity is
1) — i>a/\^ iJ-rir ^nd that is it so directed as to make an angle d with the left-
hand wall as shown. ^^ Reflection at the left-hand wall will therefore be
identical with that already shown in Fig. 6.4-2. A portion of the wave front
that has just previously undergone reflection is shown immediately below
at 2 in Fig. 6.5-1 (a). We assume further that this front is made up of lines
of electric force perpendicular to the illustration together with associated
lines of magnetic force lying in the plane of the illustration. It will be
obvious presently that, like the case of reflection from a single conduct-
ing sheet discussed in the previous section, we may obtain two rather
different pictures of what takes place within the guide, depending on
whether we fix our attention on the configuration as a whole or on some
particular line of force which we may identify and follow through a con-
siderable interval of time. We shall first consider the configuration as a
whole.
'^ II is to 1)C noted that the angle 0 which the wave front makes with the metal wall is
ccjual to the angle which the wave-normal (ray) makes with the perpendicular to the metal
wall.
WAVEGUIDE TRANSMISSION
333
We show in Fig. 6.5-1 (b) the same wave front shown in Fig. 6.5-1 (a)
but at an epoch later — after it has progressed a considerable distance along
the guide. We now find the reflected portion 2 complete and a new portion
Fig. 6.5-1. The propagation of a multipl)' reflected wave front between two metal plates
[Figs, (a)-(d)] is equivalent to the transmission of a TE wave parallel to the
two plates. [Fig. (f)].
3 about to enter the guide. Following wave front 1 and at a distance of
one-half wave behind, we find, shown dotted, the "hollow" of the wave.
This we shall designate by the numeral 1'. We find here also a new portion
of the "hollow" 2' that has just undergone reflection.
334 BELL SYSTEM TECHNICAL JOURNAL
In Fig. 6.5-1 (c) and again in Fig. 6.5-1 (d) we find successive positions
of these same wave fronts as they have moved forward in the guide. We
may, if we like, think of these fronts as discrete waves moving zig-zag
through the guide or as a single large wave front folded repeatedly back
upon itself. Fixing our attention for the moment on Fig. 6.5-1 (d), we
observe that the velocity v at which any point of incidence of the wave front
(say at point 5) moves along the guide is given by the relation
V
sm Q
This particular velocity z'a is the phase velocity of the wave as seen by a
myopic observer located near a lateral wall of the guide.
Referring again to Fig. 6.5-1 (d) and fixing our attention on the geometri-
cal relation between the wavelength X and the width of the guide c, we
may construct a right triangle with X/2 and a as sides and show that
cos 0 = ;^ (6.5-1)
la
and since
sin Q = Vl - cos- 8 (6.5-2)
" = 1/1 - {ij (6.5-3)
and
m
(6.5-4)
This says that for very large guides, that is, X < 2a, Vg = v, but as X ap-
proaches 2a, Ve approaches infinity. The particular case where \ = 2a
and Vz = CO is referred to as the cut-of condition. At cut-off, it would appear
that the individual waves approach the wall at perpendicular incidence
and a kind of resonance between opposite walls prevails. At wavelengths
greater than cut-off no appreciable amount of power is propagated through
the guide.
The particular value of wavelength measured in air, corresponding to
cut-off, is referred to as the critical or cut-ojf wavelength and is designated
thus: X,; = 2a. The corresponding frequency is similarly known as the
critical or cul-ojf frequency and it is designated thus : /« = v/ X,-. It is sometimes
convenient to designate the ratio of the operating wavelength to the
critical wavelength by the symbol p. From Equation 6,5-4 it follows that
WAVEGUIDE TRANSMISSION 335
1 1 1
(6.5-5)
V^' V^'
VT^
Referring to Fig. 6.5-1 (a) we have indicated that the wave front 1 is
made up of lines of electric force directed through the plane of the illustra-
tion and hence away from the observer. There are, of course, lines of mag-
netic force and also other lines of electric force both ahead and behind the
wave front drawn, but these have purposely been omitted in order to
simplify the illustration. If we were to take the magnetic force into con-
sideration we would find as in Fig. 6.4-2 that, at the reflecting surface, a
tangential component only is present and its magnitude is twice that of
the magnetic component of the incident wave.
In the discussion of reflection of plane waves in the previous section, it
was also pointed out that the act of reflecting a wave reverses the direction
of the electric force. Applying this principle to the case at hand, we see
that if the electric force is directed downward in the section of wavefront
1 of Fig. 6.5-1 (a), it will be directed upward in 2. Carrying this idea for-
ward to Fig. 6.5-1 (e) we find that in fronts 1, 2, 3, etc., which we rather
arbitrarily called crests, the electric vector alternates in direction as shown
by the open and solid circles. Likewise the direction of the electric vector
alternates in the fronts designated as 1', 2', and 3', but in this case they
are respectively opposite in direction to 1, 2, and 3. Continuing to fix
our attention on Fig. 6.5-1 (e), it will be observed that the direction of lines
of force is the same in 1' and 2, in 2' and 3, and in 3' and 4, indefinitely along
the entire length of the guide. Thus there are regularly spaced regions
along the length of the guide where the electric vector is directed toward
the observer alternating with other regions where the electric vector is
directed away from the observer. Between the two are still other regions
where the respective component vectors are oppositely directed and hence
their sum may be zero.
Adding the foregoing effects, bearing in mind that there are lines of force
both ahead and behind the highly simplified wave fronts shown, we have
a new wave configuration moving parallel to the main axis of the guide
with a phase velocity Vz as suggested by Fig. 6.5-1 (f). Examining more
carefully the wave interference that is here taking place, it becomes evident
that if we pass laterally across the guide along the line x in Fig. 6.5-1 (e)
the instantaneous value of the resultant electric vector as shown is every-
where zero. On the other hand, if we cross the guide along a parallel line
x', the electric vector varies sinusoidally beginning at zero at either wall
and reaching a maximum in the middle of the guide. It will be observed
that if we pass along the major axis z of the guide the electric vector at
336 BELL SYSTEM TECHNICAL JOURNAL
any instant again varies sinusoidally with distance. However, at the
boundary of the guide the resultant electric vector is everywhere zero.
Since there was no component of the electric force lying along the axis s
of the guide in the component waves that gave rise to this configura-
tion, there can be no such component in the resultant. Waves in which
the electric vector is exclusively transverse are known as transverse electric,
or TE, waves.
A complete account of transmission of this kind should include, of course,
a consideration of the lines of magnetic force. From Fig. 6.4-3 it is evi-
dent that, at the point of reflection of the component plane wave on the
guide wall, there are two components of magnetic force Hj_ and ^n in
both the incident and reflected waves. When these are added, the re-
sultant of the transverse magnetic force, like that of the electric force,
differs at different points in the guides. Following alone the line .v', it
is found that for the particular condition here assumed, the magnetic
force is zero at each wall increasing sinusoidally to a maximum midway
between. At this point the magnetic component is entirely transverse.
Following along the line x, it will be found that the magnetic vector is a
maximum near each wall decreasing cosinusoidally to zero in the middle.
It is of particular interest that, at the wall of the guide, the magnetic
component lies parallel to the axis. Magnetic lines of force are, in this type
of wave, closed loops, whereas lines of electric force merely extend from
the upper to the lower walls of the guide. The arrangement of lines of
electric and magnetic force in this type of wave is shown in Fig. 5.2-1.
The quantitative relationships between the various components of E and
H are specified more definitely by Equation 5.2-1. The significance of the
wavelength X^ of this new configuration will be obvious from Fig. 6.5-1 (f).
There are certain useful results that follow from Fig. 6.5-1 (f). It may
be seen from the triangle there shown that
^ = ^ cot e (6.5-6)
From Equations 6.5-1 and 6.5-3, it will also be seen that
7^
cos 6 A _ ^
cot d = - — = / , ■ • (6.:»-/)
sm d T
2a
Therefore
K = — ^T^ = ;yf=, (6.5-8)
/-ej
WAVEGUIDE TRANSMISSION 337
Since l/vl — v' is the ratio of the apparent wavelength in the guide to
that in free space and since for hollow pipes it is greater than unity, it is
sometimes referred to as the stretching factor. It appears frequently in
quantitative expressions relating to waveguides. Since velocity is equal
to the number of waves passing per second times the length of each wave,
we have
(6.5-9)
This is equivalent to the relation shown as Equation 6.5-5.
A matter of special interest is the rate at which energy is propagated
along the guide. For present purposes, it is convenient to regard a moving
Hne of force and its associated magnetic force as a unit of propagated energy.
A knowledge of the path followed by such a line of force will therefore
shed light on the rate at which energy is propagated along a waveguide.
It was pointed out in connection with Equation 6.4 2 that, when a wave
is incident obliquely upon a metal surface, the apparent phase of the wave
progresses at a velocity v, greater than the velocity of light v, but that the
energy actually progresses parallel to the interface at a velocity v' less than
the velocity of light. It was pointed out, too, that v' = v ?>\n 6 = v, sin- d.
Because of multiple reflections between opposite walls of a waveguide, its
phase velocity is identical with v^. Also, because of these multiple reflections,
energy being carried by these component plane waves follows a rather
devious zig-zag path and will therefore progress along the axis of the guide
at a relatively slow rate. This velocity which is known as the group lelocity
is idential with v' above. From relations already given, it will be seen that
v' = v\/\ - v' (6.5-10)
also
v' = v,{l - v") (6.5-11)
It will be apparent from this relation that, at cut-off, where v = I,
energy is propagated along the guide with zero velocity. This is consistent
with the idea already set forth that, at cut-off, energy oscillates back and
forth between opposite faces of the guide. As we leave cut-off and progress
toward higher frequencies (shorter waves), the group velocity v' increases
as the phase velocity z'j decreases, until, at extremely high frequencies,
both approach the velocity v characteristic of the medium. This relation-
ship is made more evident by Fig. 6.5-2.
Reviewing again the simple analysis just made, we find that the wave
configuration that actually progresses along a conventional rectangular
waveguide may be regarded as the result of interference of ordinary uni-
338
BELL SYSTEM TECHNICAL JOURNAL
form plane waves multiply reflected between opposite walls of the guide.
This viewpoint accounts for not only the distribution of the lines of force
in the wave front but also for the velocity at which the phase progresses
and the velocity at which energy is propagated. As we shall soon see, it
accounts also for the rate of attenuation.
In the particular configuration just described the electric component is
everywhere transverse, whereas the magnetic component may be either
longitudinal or transverse, depending on the point in a guide at which
observations are made. These waves are plane waves, but, since the elec-
REGION OF LOW ATTENUATION
Fig. 6.5-2. Relative phase velocity Vz and group velocity v' for various conditions of
operation of a waveguide.
trie intensity is not uniformly distributed over the wave front, they are not
uniform plane waves.
The concept of multiply reflected waves provides a basis for calculating
the attenuation in rectangular guides as was shown by John Kemp several
years ago.''' The procedure is outlined briefly below. The reader is referred
to the published article for details.
There is shown in Fig. 6.5-3 a short section of hollow waveguide in which
we imagine multiply reflected plane waves are proj)agated. We fix our
attention on a zig-zag section cut from the guide and so directed that it
'■' John Kemp, "Electromagnetic Waves in Metal Tubes of Rectangular Cross-section,"
Jour. I.E.E., Part III, Vol. 88, No. 3, pp 213-218, September 1941.
WAVEGUIDE TRANSMISSION
339
lies parallel to the direction of propagation of the elemental wave fronts.
The top and bottom conductors so formed may be regarded as a uniform
flat-conductor transmission line with oblique reflecting plates (sections
of the side walls) spaced at regular intervals. Other transmission lines
adjacent to that under consideration behave in exactly the same way as
that singled out for examination and at the same time act as guard plates
to insure that the lines of force so propagated remain straight.
It is clear that the attenuation in each elemental transmission line will
be that incidental to losses in the upper and lower conductors plus the
losses incidental to reflection at oblique incidence from the several reflecting
Fig. 6.5-3. Elementary transmission lines terminated periodically by reflecting plates
which go to make up a rectangular waveguide.
plates. The total attenuation of the rectangular guide may then be found
by summing up over a unit length of waveguide all of the elemental lines.
This has been done with results that are equivalent to the corresponding
equations given in Chapter V. The results are plotted in Fig. 6.5-4.
Certain characteristics of these curves may be readily accounted for.
For instance, at cut-off (6 = 0), both the number of unit reflection plates
and the number of flat-plate transmission lines in a given length of wave-
guide will be infinite. As a result, the component attenuations arising in
each of these two sources will likewise be infinite. As the frequency is in-
creased above cut-off the angle 9 will increase accordingly, leading thereby
340
BELL SYSTEM TECHNICAL JOURNAL
to fewer side-wall reflections and to a shorter over-all length of zig-zag
transmission line. Thus, in this frequency range, the attenuations con-
tributed both by the side walls and by the top and bottom plates de-
crease with increasing frequency. Proceeding to frequencies far above cut-off,
where 6 approaches 90 degrees, there will not only be very few reflections
but the over-all length of zig-zag line will approach as its limit a single,
straight two-conductor line made up of the top and bottom plates alone.
Thus the attenuation due to the side walls will approach zero and that
due to the top and bottom plates will increase as the square root of the
0.0001
1
\
\
^Total otlenuation
\
_
1 x
_ _1
^^=
=J! ■
~~
\\
^
_ C
==-
\n
I— -^
1
\
B
\^ Contributed by upper
\
3ioie
3
A
\
C
sntrib
1 sid
uted
e wa
11$
\,
s.
V,
\,
\
X
'^
^
^
--
--
8
12 3 4 5 6 7
Frequency — ttiousands of megacycles
Fig. 6..v4. ComponeiU atlentuations contributed by the top and bottom plates and also
the two side walls of a rectangular waveguide.
frequency. Since the attenuation contributed by the top and bottom plates
first decreases but later increases with frequency, we may expect, be-
tween these two ranges, a region of mininuun attenuation. The attenu-
ations contributed by the upi)er and lower plates and also by the side walls
of a 7.5 cm X 15 cm coi)per guide carrying the dominant mode have been
calculated. The results have beei^ ])lotted as curves ,1 and B in Fig.
6.5-4. They follow the courses jjredicted by the preceding qualitative
reasoning.
The fact that the reflection type of attcituation, such as is c\-ident in the
side walls above, decreases with frequency, suggests that, if a kind of wave-
WAVEGUIDE TRANSMISSION
341
guide could be devised where this type of attenuation alone exists, we
could then operate the guide at extremely high frequencies and thereby
obtain relatively low attenuations. This can, in effect, be done. It calls
for a guide of circular cross section and a special configuration, known as
I I '>• . •
iLi .
o."!-
_o ^
. !i ■
9 ^.*^f' I
_o_o ^ _^ 9'
I
c-d d
■LINES OF ELECTRIC FORCE
K; -;>ir'-,^ vt^
.x\^\\^^^^^^^^^^^^^'^^
TE^ WAVE
01
LINES OF MAGNETIC FORCE
• TOWARD OBSERVER ©AWAY FROM OBSERVER
Fig. 6.5-5. The circular electric or TEoi configuration in a circular waveguide.
(d) (e)
Fig. 6.5-6. Evolution of the circular-electric wave in a circular pipe from a dominant wave
in a rectangular pipe.
the ciradar-eleclric wave. In this conliguration, the resultant electric force
is everywhere parallel to the conducting boundary as shown in Fig. 6.5-5.
That such a wave will lead to the interesting frequency characteristic
noted is made more plausible by referring to Fig. 6.5-6 and its associated
discussion. Figure 6.5-6(a) shows a conventional form of rectangular
guide in which plane waves are multiply reflected from the two short sides.
342 BELL SYSTEM TECHNICAL JOURNAL
In Fig. 6.5-6(b) the proportions of the guide have been altered some-
what, but since the Unes of electric force are still perpendicular to the top
and bottom plates, the guide may be expected to function substantially
as before. At the most, some attenuation that previously originated in
the left-hand side wall may now be transferred to the top and bottom
walls. As a second step, we may extend the width of the top and bot-
tom walls as shown in Fig. 6.5-6(c) until they intersect, thereby forming
an arc-shaped guide. The attenuation now prevailing is evidently confined
to the top and bottom walls and the right-hand wall. It is reasonable to
assume that the side wall attenuation still decreases with frequency
since incident lines of force are everywhere parallel to this wall. As
a third step, we assemble as in Fig. 6.5-6(d) a number of identical arc-
shaped guides to form a composite circular guide with radial partitions.
If, finally, we imagine the radial partitions removed as in Fig. 6.5-6(e), the
resulting configuration will not be altered and we shall have removed the
component of attenuation attributable to the top and bottom walls leaving
only the component of attenuation attributable to the one side wall, which,
as we have pointed out, becomes progressively smaller as the frequency is
injefinitely increased.
Memory Requirements in a Telephone Exchange
By CLAUDE E. SHANNON
{Manuscript Received Dec. 7, 1949)
1. Introduction
A GENERAL telephone exchange with N subscribers is indicated sche-
matically in Fig. 1. The basic function of an exchange is that of setting
up a connection between any pair of subscribers. In operation the exchange
must "remember," in some form, which subscribers are connected together
until the corresponding calls are completed. This requires a certain amount
of internal memory, depending on the number of subscribers, the maximum
calling rate, etc. A number of relations will be derived based on these con-
siderations which give the minimum possible number of relays, crossbar
switches or other elements necessary to perform this memory function.
Comparison of any proposed design with the minimum requirements ob-
tained from the relations gives a measure of the efficiency in memory utili-
zation of the design.
Memory in a physical system is represented by the existence of stable
internal states of the system. A relay can be supplied with a holding con-
nection so that the armature will stay in either the operated or unoperated
positions indefinitely, depending on its initial position. It has, then, two
stable states. A set of N relays has 2^ possible sets of positions for the arma-
tures and can be connected in such a way that these are all stable. The total
number of states might be used as a measure of the memory in a system,
but it is more convenient to work with the logarithm of this number. The
chief reason for this is that the amount of memory is then proportional to
the number of elements involved. With N relays the amount of memory is
then M = log 2^ = A'' log 2. If the logarithmic base is two, then log2 2=1
and M = N. The resulting units may be called binary digits, or more
shortly, bits. A device with M bits of memory can retain M different "yes's"
or "no's" or M different O's or I's. The logarithmic base 10 is also useful in
some cases. The resulting units of memory will then be called decimal
digits. A relay has a memory capacity of .301 decimal digits. A 10 X 10
crossbar switch has 100 points. If each of these points could be operated
independently of the others, the total memory capacity would be 100 bits
or 30.1 decimal digits. As ordinarily used, however, only one point in a
vertical can be closed. ,Vith this restriction the capacity is one decimal
digit for each vertical, or a total of ten decimal digits. The panels used in a
343
344
BELL SYSTEM TECHNICAL JOURNAL
panel type exchange are another form of memory device. If the commutator
in a panel has 500 possible levels, it has a memory capacity of log 500; 8.97
bits or 2.7 decimal digits. Finally, in a step-by-step system, 100-point selec-
tor switches are used. These have a memory of two decimal digits.
Frequently the actual available memory in a group of relays or other
devices is less than the sum of the individual memories because of artilicial
restrictions on the available states. For technical reasons, certain states are
made inaccessible — if relay A is operated relay B must be unoperated, etc.
In a crossbar it is not desirable to have more than nine points in the same
horizontal operated because of the spring loading on the crossarm. Con-
straints of this type reduce the memory per element and imply that more
than the minimum requirements to be derived will be necessary.
Fig. 1 — General telephone exchange.
2. Memory Required for any S Calls out of N Subscribers
The simplest case occurs if we assume an isolated exchange (no trunks
to other exchanges) and suppose it should be able to accommodate any pos-
sible set of 5 or fewer calls between pairs of subscribers. If there are a total
of .V subscribers, the number of ways we can select m pairs is given by
N{N - DCY - 2) • • • (.Y - 2m + 1)
N\
I'^mliN - 2m)
(1)
The numerator N{N — 1) • • • {N — 2m + 1) is the number of ways of
choosing the 2m subscribers involved out of the N. The m\ takes care of
the permutations in order of the calls and 2"' the inversions of subscribers
in pairs. The total number of possibilities is then the sum of this for m —
0, l,---,.S;i.e.
N\
In 2'"m\{N - 2m) I
(2)
The exchange must have a stabk- iiUcrnal stale corresponding to each of
these possibilities and must have, therefore, a memory capacity M where
M = log Z
AM
2"'m\{N - 2m) \'
(3)
MEMORY REQUIREMENTS IN A TELEPHONE EXCHANGE
345
If the exchange were constructed using only relays it must contain at least
log2 X^ .Yl/2"'ml{X — 2m) I relays. If 10 X 10 point crossbars are used in
the normal fashion it must contain at least — - logio ^ Nl/2'"m\(N
2m)
of these, etc. If fewer are used there are not enough stable configurations of
connections available to distinguish all the possible desired interconnections.
With N = 10,000, and a peak load of say 1000 simultaneous conversations
M = 16,637 bits, and at least this many relays or 502 10 X 10 crossbars
would be necessary. Incidentally, for numbers N and S of this magnitude
only the term m = S is significant in (3).
The memory computed above is that required only for the basic function
of remembering who is talking to whom until the conversation is completed.
Supervision and control functions have been ignored. One particular super-
visory function is easily taken into account. The call should be charged to
MEMORY RELAYS
SWITCHING
NETWORK
u
u
R2
5
A
CONTROL CIRCUIT
Fig. 2 — Minimum memory exchange.
the calling party and under his control (i.e. the connection is broken when
the calling party hangs up). Thus the exchange must distinguish between
a calling b and b calling a. Rather than count the number of pairs possible
we should count the number of ordered pairs. The effect of this is merely
to eliminate the 2"' in the above formulas.
The question arises as to whether these limits are the best possible — could
we design an exchange using only this minimal number of relays, for ex-
ample? The answer is that such a design is possible in principle, but for
various reasons quite impractical with ordinary types of relays or switching
elements. Figure 2 indicates schematically such an exchange. There are M
memory relays numbered 1, 2, . . ., M. Each possible configuration of calls
is given a binary number from 0 to 2^' and associated with the corresponding
configuration of the relay positions. We have just enough such positions to
accommodate all desired interconnections of subscribers.
The switching network is a network of contacts on the memory relays
such that when they are in a particular position the correct lines are con-
nected together according to the correspondence decided upon. The control
circuit is essentially merely a function table and requires, therefore, no
memory. When a call is completed or a new call originated the desired con-
346 BELL SYSTEM TECHNICAL JOURNAL
figuration of the holding relays is compared with the present configuration
and voltages applied to or eUminated from all relays that should be changed.
Needless to say, an exchange of this type, although using the minimum
memory, has many disadvantages, as often occurs when we minimize a
design for one parameter without regard to other important characteristics.
In particular in Fig. 2 the following may be noted: (1) Each of the memory
relays must carry an enormous number of contacts. (2) At each new call or
completion of an old call a large fraction of the memory relays must change
position, resulting in short relay life and interfering transients in the con-
versations. (3) Failure of one of the memory relays would put the exchange
completely out of commission.
3. The Separate Memory Condition
The impracticality of an exchange with the absolute minimum memory
suggests that we investigate the memory requirements with more realistic
assumptions. In particular, let us assume that in operation a separate part
of the memory can be assigned to each call in progress. The completion of
a current call or the origination of a new call will not disturb the state of the
memory elements associated with any call in progress. This assumption is
reasonably well satisfied by standard types of exchanges, and is very natural
to avoid the difficulties (2) and (3) occurring in an absolute minimal design.
If the exchange is to accommodate 5 simultaneous conversations there
must be at least S separate memories. Furthermore, if there are only this
number, each^ of these must have a capacity log — To see this,
suppose all other calls are completed except the one in a particular memory.
The state of the entire exchange is then specified by the state of this par-
ticular memory. The call registered here can be between any pair of the N'
subscribers, giving a total of NiN — l)/2 possibilities. Each of these must
correspond to a different state of the particular memory under considera-
tion, and hence it has a capacity of least log N{N — l)/2.
The total memory required is then
M ^ Slog -^-^ . (4)
If the exchange must remember which subscriber of a pair originated the
call we obtain
M = Slog NiN - 1). (5)
or, very closely when .V is large,
M = 2S log N. (6)
1 li. D. Holhrook has pointed out that l)y using more than 5 memories, each can have
for certain ratios of ^, a smaller memory, resulting in a net saving. This only occurs,
however, with unrealistically high calling rates.
MEMORY REQUIREMENTS IN A TELEPHONE EXCHANGE
347
s
The approximation in replacing (5) by (6), of the order of — log e, is equiva-
lent to the memory required to allow connections to be set up from a sub-
scriber to himself. With .V = 10,000, 6" = 1,000, we obtain M = 26,600
S INTERCONNECTING ELEMENTS
Fig. 3 — Minimum separate memory exchange.
N = 2M-
-2M = N
Fig. 4 — Interconnecting network for Fig. 3.
from (6). The considerable discrepancy between this minimum required
memory and the amount actually used in standard exchanges is due in part
to the many control and supervision functions which we have ignored, and
in part to statistical margins provided because of the limited access property.
The lower bound given by (6) is essentially realized with the schematic
exchange of Fig. 3. Each box contains a memory 2 log ;V and a contact
network capable of interconnecting any pair of inputs, an ordered pair being
associated with each possible state of the memory. Figure 4 shows such an
interconnection network. By proper excitation of the memory relays 1, 2,
• • • , M, the point p can be connected to any of the ;Y = 2'" subscribers on
the left. The relays 1', 2', ■ ■ -, M' connect p to the called subscriber on
348 BELL SYSTEM TECHNICAL JOURNAL
the right. The general scheme of Fig. 3 is not too far from standard methods,
although the contact load on the memory elements is still impractical. In
actual panel, crossbar and step-by-step systems the equivalents of the
memory boxes are given limited access to the lines in order to reduce the
contact loads. This reduces the flexibility of interconnection, but only by
a small amount on a statistical basis.
4. Rel.\tiox to Information Theory
The formula M = 2S log .V can be interpreted in terms of information
theory.- When a subscriber picks up his telephone preparatory to making
a call, he in effect singles out one line from the set of .Y, and if we regard
all subscribers as equally likely to originate a call, the corresponding amount
of information is log X . When he dials the desired number there is a second
choice from N possibilities and the total amount of information associated
with the origin and destination of the call is 2 log N. With S possible simul-
taneous calls the exchange must remember 25 log iV units of information.
The reason we obtain the "separate memory" formula rather than the
absolute minimum memory by this argument is that we have overestimated
the information produced in specifying the call. Actually the originating
subscribers must be one of those not already engaged, and is therefore in
general a choice from less than N. Similarly the called party cannot be
engaged; if the called line is busy the call cannot be set up and requires no
memory of the type considered here. When these factors are taken into
account the absolute minimum formula is obtained. The separate memory
condition is essentially equivalent to assuming the exchange makes no use
of information it already has in the form of current calls in remembering
the next call.
Calculating the information on the assumption that subscribers are
equally likely to originate a call, and are equally likely to call any number,
corresponds to the maximum possible information or "entropy" in com-
munication theory. If we assume instead, as is actually the case, that certain
interconnections have a high a priori probability, with others relatively
small, it is possible to make a certain statistical saving in memory.
This possibility is already exploited to a limited extent. Suppose we have
two nearby communities. If a call originates in either community, the
probability that the called subscriber will be in the same community is
much greater than that of his being in the other. Thus, each of the exchanges
can be designed to service its local traffic and a small number of intercom-
munity calls. This results in a saving of memory. If each exchange has N
subscribers and we consider, as a limiting case, no traffic between exchanges,
■' C. K. Shannon, "A Mathematical Theory of Communication," Bell Svstem Technical
Journal, Vol. 27, |)|). ,?70 42.^, and 62.S 6,S6, July and October 1948,
MEMORY REQUIREMENTS IN A TELEPHONE EXCHANGE 349
the total memory by (6) would be 45" log .Y, while with all 2.V subscribers
in the same exchange -iS log 2N would be required.
The saving just discussed is possible because of a group effect. There are
also statistics involving the calling habits of individual subscribers. A typical
subscriber may make ninety per cent of his calls to a particular small
number of individuals with the remaining ten per cent perhaps distributed
randomly among the other subscribers. This effect can also be used to
reduce memory requirements, although paper designs incorporating this
feature appear too complicated to be practical.
Acknowledgment
The writer is indebted to C. A. Lovell and B. D. Holbrook for some
suggestions incorporated in the paper.
Matter, A Mode of Motion
By R. V. L. HARTLEY
{Manuscript Received Feb. 28, 1950)
Both the relativistic and wave mechanical properties of particles appear to
be consistent with a i)icture in which particles are represented Idv localized oscil-
latory disturbances in a mechanical ether of the MacCullagh-Kelvin type. Gyro-
static forces impart to such a medium an elasticity to rotation, such that, for
very small velocities, its approximate equations are identical with those of Max-
well for free space. The important results, however, follow from the inherent
non-linearity of the complete equations and the time dependence of the elas-
ticity associated w-ith finite displacements. These lead to reflections which permit
of a wave of finite energy remaining localized. Because of the non-linearity, the
amplitude and energy of a stable mode, as well as the frequency, are determined
by the constants of the medium. Such a stable mode is capable of translational
motion and so is suitable to represent a particle. The mass assigned to it is de-
rived from its energy by the relativity relation. While this mass is dimensionally
the same as that of the medium it is differently related to the energy and so
need not conform to the classical laws which the latter is assumed to obey.
Exchanges of energy between particles and between a particle and radiation
involve frequency changes as in the quantum theory. The experimental detection
of a uniform velocity relative to the medium is not to be expected. Besides pro-
viding a new approach to the problems of particle mechanics, the theory ofifers
the prospect of incorporating the present pictures into a more comprehensive
one, with a material reduction in the number and complexity of the independent
assumptions.
Introduction
THE following quotation states a conclusion which is widely held: "But
in view of the more recent development of electrodynamics and optics
it became more and more evident that classical mechanics afifords an in-
sufficient foundation for the physical description of all natural phenomena."^
This impUes that classical mechanics and classical electromagnetics are so
alike that one may be condemned for the shortcomings of the other. Actu-
ally, classical electromagnetics is in open disagreement with classical mech-
anics particularly with respect to those features for which it has been most
criticized. According to the mechanical principle of relativity,- the equations
of any mechanical system are invariant under the Newtonian transformation,
X = x' -\- Vt',y = y',z = z', t = t', where F isa constant velocity in the x
direction. Since the classical electromagnetic equations are not invariant
under this transformation, they cannot describe the performance of any
classical mechanical system. Their failures, therefore, should not stand in
the way of a study of the possibilities of such systems.
The system considered here is the so-called rotational ether, suggested
> A. Einstein, The Theory of Relativity, Mcthuen & Co., Ltd., London, 1921, p. 13.
^ Haas, Introduction to Theoretical Physics, 2nd Ed., Vol. I, p. 46.
350
MATTER, A MODE OF MOTION 351
by MacCuUagh and elaborated by Kelvin, in which the stiffness is associ-
ated with gyrostatic forces. Some consideration has been given to an alter-
native model consisting of a non-viscous liquid in a high state of fine scale
turbulence. It is well known that, by virtue of the gyrostatic forces associ-
ated with it, a vortex will transmit a wave of transverse displacement along
its axis. It would appear, therefore, that a gross wave involving similar
displacements would be passed along from vortex to vortex, much as a
sound wave is passed from molecule to molecule. However, since this model
has not yet been shown to be fully equivalent to Kelvin's, attention will be
confined to the latter. While this, as developed by Kelvin, gave a satis-
factory description of electromagnetic waves in free space, it had nothing
to represent matter. This was assumed to be something different from ether,
which might or might not be pervaded by it. A closer study of the model has
indicated that the peculiar nature of its stiffness makes possible sustained
oscillatory disturbances in which the energy remains localized about a
center which may move with any velocity less than that of a free wave.
It is proposed to use such quasi-standing wave patterns to describe material
particles. Matter, then, has no existence apart from the ether, and the
motion of particles is the motion of patterns of mechanical wave motion.
While the ether itself conforms to Newtonian mechanics, the mechanics of
such a wave pattern, considered as a particle located at its center, is much
more complicated than that of the familiar mass point of particle dynamics.
This complexity provides a bridge from the older concepts of particle be-
havior to the new.
The study of this model given below reveals no insuperable obstacles such
as were encountered by the electromagnetic theory and the simpler ether
model. The properties of the wave-patterns are qualitatively consistent
with many of the concepts of modern physics, though in some cases not
with the generality of application which is now assigned to them. Among
these concepts are: the space-time of special relativity, relativistic mechanics,
de Broglie waves, proportionality of energy and frequency, energy thresh-
holds, and transfers of energy according to the quantum frequency formula.
The ether model also leads to certain concepts not found in the present
theories. It provides, for example, for a possible failure of the mass-energy
balance such as has been observed in nuclear reactions. It also suggests the
possibility of a new type of particle which, by virtue of its negative inertial
mass, is capable of exerting a binding force between other particles.
These results make it more probable that classical mechanics may, after
all, afford a sufficient "foundation for the physical description of all natural
phenomena" even though the super-structure be very different from that
contemplated by its originators. The present argument, however, is not
that this particular description is necessary, but rather that it offers distinct
352 HELL SYSTEM TECHNICAL JOURNAL
advantages. On the philosophical side, there is the prospect of greater
unilication of the basic theory through a reduction in the number of inde-
pendent assum])tions. Matter and radiation appear as wave motions which
satisfy the same equations. The apparent conflicts between current concepts
appear to be reconcilable through a more exact determination of the con-
ditions under which each applies. On the more practical side, the ether
model provides a difTerent approach and technique. It has the advantage
inherent in all models that, once one is found which fits one set of condi-
tions, a study of its properties under widely different conditions may bring
out relations which it would be difficult to postulate solely on the basis of
observations made under the second conditions. The suggested existence
of particles having negative inertia, as discussed near the end of the paper,
should it lead to anything of value, would be an example of such a relation.
Also it makes available the added relationships which are characteristic of
non-linear equations, without encountering those difficulties with respect
to absolute motion which may arise when non-linearity is introduced ar-
bitrarily. While the working out of the quantitative relations involved is
a rather formidable undertaking, any effort in that direction may well
throw new light on those problems which have not yielded to other methods.
The Gyrostatic Ether
As stated above the specific form of gyrostatic medium on which the
present discussion is based is the ether model proposed by Kelvin. This is
discussed in detail in a companion paper. ^ It is there shown that, for in-
finitesimal displacements, it is characterized by the wave equations:
vx(f)=.f
where po is the density, tjo is a generalized stiffness determined by the con-
stants of the medium, q is the vector velocity, and T is a vector torque per
unit volume, which has its origin in the torque with which a gyrostat op-
poses an angular displacement of its axis. For a plane polarized plane wave,
T .
the quantity can be interpreted as a surface tractive force per unit area,
which a layer of the medium normal to the direction of propagation exerts
on the layer just ahead. Its direction lies in the surface of separation, and
is parallel to that of the velocity q.
' R. V. I.. Hartley, "'J'hc Rctlcction of Diverging Waves by a Gyrostatic Medium" —
this issue of The Bell Svstem Technical JournaL
MATTER, A MODE OF MOTION 353
These equations become identical with those of Maxwell for free space,
- T - 1
if we replace q by £, -• by H, po by f and - by /x. Then p^q corresponds to
D and — 2</j to B where ^ is the angular displacement of an element of the
medium. Or the roles of the electric and magnetic quantities may be inter-
changed.
For present purposes, however, we are more interested in finite displace-
ments. The relations which then apply are discussed in detail in the com-
panion proper. It is there shown that changes of two kinds appear in (1)
and (2), with corresponding changes in the transmission properties of the
medium. The simple linear relations are to be replaced by non-linear ones,
which cause distortion of a wave but no reflection. In addition, a qualitative
difference appears in the nature of the elasticity, as was pointed out by
Kelvin. The restoring torque is no longer proportional to the angular dis-
placement alone. When the axis of a gyrostat is displaced it begins rotating
toward the axis of the displacement, thereby decreasing the component of
its spin which is normal to that axis. Thus the restoring torque for a con-
stant angular displacement decreases with time. The restoring torque is
therefore a function of the time as well as of the displacement. Because of
this time dependence, a disturbance of finite amplitude generates waves
which propagate both backward and forward.
Vox a plane progressive sine wave it is found that the reflected waves
interfere destructively. However, if a central generator starts sending out
a diverging sinusoidal disturbance, a part of the energy is reflected inward
as a wave of the same frequency as the generator and another smaller part
as waves the frequencies of which are odd multiples of that frequency. This
reflection attenuates the outgoing wave. If the incoming wave is reflected
rather than absorbed at the generator, it tends to set up a standing wave
pattern. As time goes on, the impedance of the medium as seen from the
generator becomes more reactive and less power is drawn from the generator.
Due to the attenuation, the energy in spherical shells of a given thickness
decreases with increasing radius, so that it and the power transmitted at the
wave front approach zero as /' approaches infinity. This falling off is some-
what similar to that suffered by a wave the frequency of which lies in the
stop band of a filter, but with one important difference. There the attenua-
tion is independent of the distance. But here, since the attenuation is a
354 BELL SYSTEM TECHNICAL JOURNAL
function of the magnitude of the disturbance and of the curvature of the
wave-front, the attenuation constant approaches zero as r increases in-
definitely.
Whether or not the total energy stored in the wave pattern will approach
a finite or infinite value depends on how fast the attenuation decreases with
distance, and a more complete solution is needed to give an exact answer.
If it does approach infinity it will do so much more slowly than for a medium
which does not reflect.
The disagreement between classical electromagnetics and mechanics, re-
ferred to above, may now be stated more explicitly. The former says that
electromagnetic waves are represented exactly by Maxwell's equations,
regardless of the magnitudes of the electromagnetic variables. When these
waves are interpreted as existing in a mechanical ether, classical mechanics
says that Maxwell's relationship is approached as a limit as the mganitudes
approach zero. Waves of finite amplitude are to be represented by the more
complicated relations.
The two systems differ in three important respects; their relation to
uniform linear motion, the linearity of their equations and the nature of
the elasticity involved. Because the classical electromagnetic equations are
not invariant under a Newtonian transformation, the set of axes to which
the equations refer are uniquely related to other sets which are moving
uniformly with respect to them. In special relativity, this condition is
avoided by modifying the classical concepts of space and time to conform
to the fact that the equations are invariant under the Lorentz transforma-
tion. The Newtonian invariance of the ether equations, however, insures
that a set of axes at rest with respect to the undisturbed ether is not unique.
Hence in the modified model, in which Ihe motions which constitule matter
conform to the laws of the ether, a uniform linear velocity of the entire
system cannot be detected. This is consistent with the accepted principle
that absolute velocity is meaningless.
We are, however, still faced with the question of the detection of uniform
motion of matter relative to the ether. This is discussed at length below,
where it is shown that the properties of the ether lead directly to an auxili-
ary space-time, which applies very closely under the experimental condi-
tions and accounts for the failure to detect the motion. This "experimental"
space-time is formally identical with that of special relativity. Thus the
modification of the space-time of classical electromagnetics which appears in
special relativity might be said to bring it into closer formal agreement
with the classical mechanics of ether wave patterns. At any rate the es-
tablishing of this theoretical connection between the space-time of special
relativity and a classical mechanical model is a step toward unification.
On the matter of linearity, proposals have been made to add arbitrary non-
MATTER, A MODE OF MOTION 355
linear terms to Maxwell's equations. While this also makes the electro-
magnetic equations more like those of the ether, an important difference
still remains. An equation obtained in this way is not necessarily invariant
under either a Newtonian or a Lorentz transformation. If, then, the axes
with respect to which it is expressed are not to be unique, it must be shown
that some transformation exists under which it is invariant. Not only is
the form of the equation important here but also the interpretation of the
dependent variables. For example, since the complete equations of the
ether contain q-V, if the mechanical variables be replaced by the analogous
electromagnetic ones, the equations will be Newtonian invariant only if
E, which replaces q, is interpreted as a velocity. It is evident, therefore,
that the fact that we are dealing with a mechanical model is an important
point in the argument. Also, unless the added terms make the effective
constants depend on the time as well as the dependent variables, there will
be no reflection of the energy in a finite disturbance and the medium will
not have the energy trapping property which is essential to the present
argument.
Stationary Wave Patterns
The first question to be considered is the possibility of setting up a sus-
tained wave pattern suitable to represent a particle at rest with respect to
the ether. The simplest procedure might seem to be to look for it as a solu-
tion of the approximate linear equations in the form of a pair of spherical
waves propagating radially, one outward and one inward, so as to form
together a standing wave pattern. However, certain difficulties are en-
countered. There is nothing in the free linear ether which can serve as
boundary conditions to fix the position or size of the pattern. Even if these
were determined, there would be nothing to fix the amplitude, and so the
energy. Most patterns, particularly those which involve a single frequency,
have one or more of the following features. Some of the variables become
infinite at the center; the total energy is infinite, energy is propagated away
radially.
These difficulties disappear, however, when we take account of the prop-
erties of the ether for disturbances of finite amplitude. Let us suppose that
the energy which is to constitute the pattern is supplied by a central gener-
ator, the impedance of which is mainly reactive, so that reflected waves
which reach it are reflected outward again. Once a standing wave pattern
has been established as described above, let the force of the generator be
reduced to zero without changing its impedance. The pattern will then
persist except for a small and decreasing damping due to the outward radia-
tion at its periphery. However, in the region near the center the displace-
ments will be very large, and the incoming reflected waves will suffer reflec-
356 BELL SYSTEM TECHNICAL JOCRXAL
tions which increase with decreasing radius. These reflections will effectively
take the place of the assumed reacti\'e impedance of the generator, and so
the latter may be discarded. The fact that the retlections take place from
a somewhat diffuse inner boundary prevents the amplitude from building
up to an infinite value at the center as it would with a linear medium.
However, the reflected wave includes components of triple and higher
frequencies and, due to the non-linearity, other frequency components will
be generated. If the entire pattern is to be stable, all of these must satisfy
the boundary conditions. Their magnitudes relative to the fundamental, for
a particular mode of oscillation, will depend on the amplitude and fre-
quency of the fundamental, as well as on the constants of the medium.
Hence the amplitude as well as the frequency of a stable pattern of a par-
ticular mode should be uniquely determined. Particles of different prop-
erties would then be expected to consist of patterns involving different
modes of oscillation.
Returning to the lack of complete reflection at the outer boundary and
the change it might be expected to make in the pattern with time, this
might be an important factor for a single particle alone in the universe.
Actually, however, a very large number of particles are present. If we con-
sider a point at a considerable distance from any one particle, a point in a
vacuum, the resultant of the disturbances produced there by all the patterns
will be very large compared with that due to any one. But the effect on a
particular pattern of its own loss by radiation will be determined by this
small component, and so will be small compared with the effect exerted on
it by the combined small fields of its neighbors. This combined field due
to a large number of patterns, randomly placed, and moving at random, will
constitute a randomly varying electromagnetic field in a vacuum, such as
has recently been postulated for other reasons. If, now, the center of a
pattern be placed at the point in question, this random field may occasion-
ally take on so large a value as to disturb the equilibrium conditions of
the pattern.
It may be argued that, in spite of the merging of a given pattern in that
of the random group, the group as a whole will suffer a progressive loss of
energy through incomplete reflection. Were this to occur the total loss of
energy would not be evenly distributed among the j)articles. As discussed
below the particles would exchange energy through the mechanism of the
non-linearities, continually forming less stable grouj) i)at terns of greater
energy, which in turn suffer transitions to more stable patterns of lower
energy. A small continuous decrease in total energy would manifest itself
as an increase in the rate of transitions downward in energy comj^ared to
those upward.
Associated with a standing wave pattern such as that described above
MATTER, A MODE OF MOTION 357
would be three regions. Near the center would be a relatively small core in
which the non-linear effects predominate and linear theory is totally inap-
plicable. Farther out the departure from linearity is only moderate, and the
variation of the constants with distance is slow enough that the reflections
are small. It should be possible to treat wave propagation in this region by
the methods developed for a string of variable density, which are sometimes
cited as analogous with those employed in wave mechanics. The analogy is
made closer by the fact that the variations in impedance which correspond
to the varying density are determined by the energy density of the pattern
itself. Still farther out the amplitudes become still smaller, the ether con-
stants become very nearly but not quite uniform, and the pattern ap-
proaches very closely to that in a linear medium.
While the nature of the pattern is determined largely by the non-linear
inner region, because of the small volume of this region most of the energy
will be located in the nearly linear region. So we might expect some at least
of the macroscopic properties of the pattern to differ very little from those
deduced from a consideration of the corresponding pattern in a linear me-
(lium. We will therefore begin by examining such a pattern. For the linear
case, when the axes are at rest with respect to the undisturbed ether, (1)
and (2) lead to the wave equation for the vector displacement s,
^'' c'v'-s. (3)
5 s 2„2
As is well known, this is satisfied by any function of the form
where
— Rx \ "y I Rz J \^)
and the constants co, ^x, ^y and h^ , are real or complex. Since an imaginary
frequency is interpreted as an exponential change with time, it is not suit-
able for representing a permanent pattern, so co will be taken to be real.
Imaginary values of k are interpreted as exponential variations with dis-
tance. But, since s is always real, we may, by a four-dimensional Fourier
analysis, represent/ as the summation of components of the form
s = J^ cos (co/ ± kxX ± kyy ± k^z), (5)
where A is a, complex vector representing the amplitude and phase of the
component, and kx , ky and k^ are real. Since each component must satisfy
(3), the new constants must satisfy (4). Each such component constitutes
a plane progressive wave traveling, with velocity c in a direction, the cosines
of which are proportional to the wave numbers kx , etc.
358 BELL SYSTEM TECHNICAL JOURNAL
As a first step in building up a stationary pattern, in which there is no
steady propagation of energy in any direction, we combine two progressive
wave components (5) which are identical, except that their directions of
phase propagation along, say, the z axis are opposite. The signs of the last^
terms are then opposite and the sum can be written
s = lA cos (co/ ± kjcX ± kyj) cos kaZ
Proceeding in the same way for x and y, we arrive at the standing wave
pattern,
s = ^A cos co/ cos kxX cos kyV cos keZ. (6)
Components of this sort, each with its own amplitude and phase, may be
combined to build up possible stationary patterns. However, we shall not
attempt here to build such patterns, but rather to deduce what information
we can from a study of a single component.
Moving Wave Patterns
In order to represent approximately a particle in uniform linear motion,
we are to look for a solution of (3) which represents a moving wave pattern
For this we make use of two functions which may readily be shown to be
such solutions,
s = g+ U{oi + VK)t - ^(h + ^ j X ± kyy ± Kzj ,
s = g_ (b{c^ - Vkjt + /3 U, - ^j X ± k,y ± k^zj ,
where co, kx , ky and k^ are real and satisfy (4), F is a real constant, and
C2
g+ represents a plane progressive wave the propagation of which along tht
X axis is in the positive direction. g_ represents one of lower frequencyJ
propagating in the negative x direction. Their wave numbers in the .v direc-j
tion differ in such a way that those in the y and z direction are the same foi
the two. In the plane wave case, where ky — kg = 0 and co = ckx , they re-
duce to
The two waves then travel in the .v direction with velocities c and
c -\- V
their frequencies are in the ratio — .
^ f - V
MATTER, A MODE OF MOTION 359
In order to derive a quasi stationary pattern we replace the functions
g+( ) and g_( ) by Bcosa{ ) and combine components in a manner
similar to that used in deriving (6). The result is
S = 9,B cos a/^co ( ' ~ — -^^ ) cos a:/3^i(.v — Vt) cos akyj cos ak^z, (7)
where 5 is a complex vector, and a may be any real scalar function of V.
When we compare this with (6) we tind that the last three factors, which
in (6) describe a fixed envelope, in (7) describe an envelope which moves in
the .V direction with velocity I'. For the same values of k^ , k^j and k^ , the
moving pattern has its dimensions in the x direction reduced relative to
those in the v and z in the ratio -. The first factor in (6) describes a sinusoidal
variation with time which is everywhere in the same phase. In (7) it de-
scribes one, the phase of which varies linearly with .v. This factor also de-
scribes a wave which progresses in the .v direction with a velocity — . The
existence of such a wave as a factor in the expression for a moving wave
pattern was commented on by Larmor."* Aside from the constant a in (7)
it will be recognized as the Lorentz transform of (6), as it should be since
the approximate equations of which it is a solution are invariant under
this transformation.
We shall take (7) to represent one component of a moving wave pattern
which represents a moving particle. If we transform this to axes moving
with the pattern by a Newtonian transformation it becomes
s = SB cos oc { - t' — — r- x' I cos q;/3^j x' cos aky y' cos ak^z', (8)
in which the envelope is at rest. This may be thought of as a stationary wave
in an ether which is movmg relative to the axes with a velocity — V. It
is a solution of the wave equation for such an ether, as obtained by trans-
forming (3) to the moving a.xes, or
dt'- dx'dt' dx-
The one dimensional form of this equation is identical with that given by
Trimmer for compressional waves in moving air, except that in one case s
is solenoidal and in the other divergent.
So far we have found no reason to associate any particular moving
pattern with the assumed stationary one, in the sense that the moving pat-
^Larmor, Ency. Brit. 11th Ed., 1910; 13th Ed., 1926, Vol. 22, p. 787.
^ J. D. Trimmer, Jour. Aeons. Sac. Am., 9, p. 162, 1937.
360 BELL SYSTEM TECHXICAL JOURNAL
tern describes the result of setting in motion the particle which is described
by the stationary pattern. Without further knowledge or assumptions re-
garding the factors which control the form of the pattern, we can go no
farther in this direction by theory alone. Rather than try to guess at these
factors, it seems preferable to investigate what properties the wave patterns
must have in order to conform to the known results of experiment.
Let us start with the Michelson-Morley experiment to which the earlier
ether theory did not conform. The entire apparatus involved in the experi-
ment is now to be considered as made up of particles each of which consists
of a wave pattern in the ether. The apparatus as a whole may be regarded
as a more complicated wave pattern. The interference pattern formed by
the light beams may, if we wish, be included in the over-all pattern. The
results to be expected in the experiment do not depend on the oscillatory
nature of the wave, nor on its amplitude or phase, but only on its spatial
distribution, which is determined by the envelope factors. It is obvious from
(8) that, for any uniform velocity — F of the ether relative to the apparatus,
the ratios of the dimensions of the envelope along the motion to those across
it are reduced, relative to their values when V is zero, in the ratio - . That is
to say the apparatus like the fringes undergo this change in relative dimen-
sions. But, as is well known, this is exactly what is required in order that
there shall be no apparent motion of the fringes. Hence any one of the
stationary patterns in a moving ether, as represented by (8), is consistent
with the experiment. This experiment therefore furnishes no basis for select-
ing any particular pattern.
More generally, in any experiment, the distances and time intervals
which are available as standards of comparison are associated with the
wave i)atterns and change with their motion. Thus we may, following the
special theory of relativity, define an auxiliary space and time, the units
of which are associated with the dimensions and cyclic interval of a par-
ticular periodic wave pattern. This pattern then plays the roles of the
"practically rigid body" and the "clock" which determine space and time in
relativity theory. An examination of (8) shows that the dimensions of the
pattern, its frequency, and its phase change with the velocity of the ether
relative to the pattern in just the way that the corresponding quantities
associated with the rigid body and clock change with velocity in the rela-
tivity theory. But there these changes arc known to be sucli that no experi-
ment can detect the velocity involved. Tt follows, therefore, that no experi-
ment in which the ai)])aratus consists of wave patterns of small amplitude
is capable of detecting the velocity V, in (8), which in tliis case is the velocity
of the ether relative to the a])])aratus. Hence any of the above j^at terns are
consistent with the taiUirc of all exj)erimciils designed to detect motion
MATTER, A MODE OF MOTION 361
relative to the ether. When account is taken of the non-Hnearity of the
ether the result to be expected should differ from that just found for the
linear case only by the small difference between the linear and non-linear
patterns, which may easily be too small to measure. Thus the principal
obstacle to the older ether theory is removed.
While the special theorj^ of relativity is usually written in the form
which corresponds to a being unity in (8), it has long been recognized that
there is no theoretical basis for this particular value. The ether patterns
are consistent with the more general formulation. In order to pin down
the value of a for the ether patterns we resort to another experiment.
Ives and Stillweir found that a molecule which emits radiation of fre-
quency CO when at rest emits a frequency - when in motion. This moving
frequency is taken relative to axes moving with the molecule, and so is to
be compared with the frequency of oscillation- co in (8). This indicates that
in order to represent a component of the pattern which results when the
fixed pattern is set in motion, we are to put a equal to unity.
Another observed relation is that the energy of a moving particle is /3
times that of the same particle at rest. This information should be useful
in checking any theory of the mechanism by which the non-lmearity of the
medium determines the energy of the pattern. All we shall do here is to point
out one relation, the significance of which from the standpoint of mecha-
nism will be discussed below. In (7), where the frequency is expressed rela-
tive to the same axes as the energy of the moving pattern, if we put a
equal to unity, the frequency also varies as /3. Hence if the pattern conforms
to experiment with respect to its energy, the energy must be proportional
to the frequency.
Obviously, if we define the mass of the particle-pattern as its energy over
C-, the particle will conform to relativistic mechanics. The mass of a particle
as so defined, while dimensionally the same as that of the ether, is in other
respects quite different. Since it is derived from the energy associated with a
disturbance of the ether, it would be zero in the undisturbed ether, while
the ether mass would be finite. The momentum of a particle would be deter-
mined by the flow of energy associated wuth it. Also within a particle, if the
mode of oscillation were such that the wave propagated continuously around
the axis in one direction, the resulting rotation of the energy would be
interpreted as an angular momentum or spin. This concept of spin was
suggested by Japolsky^ in connection with cylindrical waves in a linear
medium. There is, therefore, no a priori reason to expect that the motion
6 H. E. Ives and C. R. Stillwell, Jour. Opt. Soc. Am., 28, 215, 1938 and 31, 369, 1941.
■ N. S. Japolsky, Phil Ma^. 20, 417, 1935.
362 BELL SYSTEM TECIIMCAL JOURNAL
of particles should conform to the laws of classical mechanics. As just noted,
it should conform much more closely to those of relativistic mechanics.
Also, to the extent that the flow of energy follows the laws of wave mechan-
ics, as suggested below, the behavior of the particles will also conform to
those laws. Similar considerations apply to the mass of radiation as derived
from its energy.
Another experiment which helps to fix the required properties of the
patterns is that of Davisson and Germer, in which it is shown that a particle
moving with velocity V is diffracted as if it had a wave length X such that
A= '
(^mo V '
where h is Planck's constant and mo is the rest mass.
If, in (7) with a unity, we assume the energy frequency ratio to be equal
to //, the wavelength associated with the first factor reduces to the value
given by experiment. This does not mean that an ordinary physical wave of
this length is present in the pattern. It does mean that, at any instant, the
amplitude of the sinusoidal variation of displacement with distance, as
given by the remaining factors, varies sinusoidally with the wave length X,
and is zero as points separated by - . Hence, when the presence of equally
spaced obstacles calls for zero values of displacement at equally spaced
intervals, the distorted wave should be capable of forming a stable dif-
fraction pattern when the translational velocity of the pattern is such that
the interval between points of zero displacement has the value required by
the spacing of the obstacles.
Thus the wave pattern will conform to this experiment provided, first,
that it is characterized by a particular wave length, and second, that the
factor of proportionality between its energy and frequency is equal to //.
The first requirement implies that the wave pattern when at rest has
practically all of its energy associated with components which are all of the
same frequency, or else are confined to a narrow band near the characteristic
frequency.
At this point let us pause for a short review and discussion. Brieily, we
have replaced the "rigid body" of special relativity by an oscillatory motion
of the ether, the envelope of which is analogous with the configuration of the
rigid body. We have found that when in motion this envelope behaves as
does the rigid body, and the time relations conform to those of a moving
clock. These latter may also be interpreted as a multiplying factor which
has the form of a plane wave of the DeEroglic type. In wave mechanics,,
this is treated as a wave of a single frequency and of a variable phase veloc-
ity greater than that of light. In the ether theory this wave is interpreted
MATTER, A MODE OF MOTION 363
as one factor in the description of an interference pattern which results from
the superposition of component progressive waves of different frequencies,
each of which travels with velocity c. This difference in viewpoint leads to
other differences.
One of these has to do with the possibility of describing accurately both
the position and velocity of a particle, which is ruled out from the wave
mechanics viewpoint. An ether wave pattern, however, may have its posi-
tion accurately described by its envelope, while at the same time the pattern
moves with a definite velocity. The particle velocity may here be regarded
as a group velocity derived from two waves progressing in opposite direc-
tions, but does not depend on the presence of dispersion as does that for
waves in the same direction. It is not to be concluded from this that the
position and velocity can be measured with this accuracy, for we have still
to deal with the disturbing effect of the measurement.
From the ether viewpoint, one of the limitations of wave mechanics is
to be expected, its inability to calculate directly the position of a particle.
The information regarding this position is contained in the expression for
the envelope, while the wave factor depends only on its state of motion. A
calculation based on a solution which involves the wave factor without the
envelope would be expected to be indefinite regarding position. We should
expect, however, that it would give information as to the probability of the
presence of the particle in a given region, since this is derivable from its
state of motion.
Returning to the comparison with experiment, while wave patterns based
on the linear equations have shown close agreement so far, the next experi-
ment upsets the applecart. It has been observed that the motion of one
particle is modified by the presence of other particles in its neighborhood.
So long as the assumed equations are linear, the law of superposition holds,
and every solution is independent of every other one. So any wave pattern,
when once set up, will continue in its state of rest or of uniform motion
indefinitely, and will not be influenced by the presence of other patterns or
of free progressive waves. But these together comprise all other matter and
radiation. Hence, while we have provided for the property of inertia, there is
nothing which tends to alter the state of motion of a body, that is, there
are no forces. In this respect the present linear treatment is similar to the
special theory of relativity. So, in order to represent the interactions between
particles, account must be taken of those between patterns which result
from the non-linearity and time dependence of the ether.
Reactions between Patterns
The general problem of the effect of one pattern on another is even more
intricate than that of the stable state of a single pattern, which it includes.
364 BELL SYSTEM TECHNICAL JOURNAL
and its solution will not be attempted here. Some conclusions may, however,
be drawn. Since the amount of retiected energy generated by an element of
the medium depends on powers of the instantaneous disturbance higher
than the first, the superposition of a second pattern will alter the standing
wave pattern of the first, and vice versa. Also, as pointed out in the com-
panion paper, the propagation of both the main and reflected waves also
depends on higher powers of the instantaneous disturbance there. The result-
ing variations in the propagation will also affect the conditions for a stable
pattern. Neither pattern, then, can satisfy its stability conditions inde-
pendently of the other; but if the combined patterns are to be stable they
must together satisfy a new set of conditions common to both. How much
each is altered by such a union will depend on the degree of coupling be-
tween them, that is, on the amount of energy which must be regarded as
mutual to the two.
The effect of this coupling will be very different, depending on whether
the frequencies of the two patterns are the same or different. When they
are different the non-linear terms give rise to frequencies related to the first
two by the quatum formula. The transfer of energy to these frequencies
may, under favorable conditions, set up a new mode of oscillation the sta-
bility conditions of w'hich are better satisfied than those of the original
frequencies. The new mode might be that of an e.xcited atom. Or the fre-
quency of one or both of the patterns may be changed to that corresponding
to the particle in motion with a particular velocity. In either of these proc-
esses some of the energy may be released as radiation at one of the dif-
ference frequencies.
If, however, the frequencies of the two patterns are identical, no new
frequencies will result from their superposition. If the combined pattern is
to persist there must be a stable mode for the combination, the frequency
of which is identical with that of the separate patterns. This is hardly to be
expected. Also the oscillations of the second pattern, being of the same
frequency as those of the first, would have a much greater disturbing effect
on its conditions for stability. It would appear, then, that if it were possible
to bring two patterns of identical frequency into superposition, they would
mutually disintegrate. This does not mean that two particles of the same
type cannot exist in the same neighborhood. If they have different velocities,
for example, their frequencies will be different. The similarity of these
considerations to Pauli's exclusion principle is obvious.
If the second pattern has much greater energy than the first, as it will if
it represents a much heavier particle, its stability conditions may be little
affected by the presence of the first. The behavior of the first, an electron,
may then be discussed on the assumption that it exists in a medium, the
properties of which vary with |)ositi()n in accortlancc willi the fixed j)attern
MATTER, A MODE OF MOTION 365
of the second particle, the nucleus. Since the stability conditions for the
electron pattern particle are most strongly influenced by the effective con-
stants of the medium near its center, we would expect its energy and fre-
quency to be controlled largely by that part of the nuclear pattern which is
near its center. Let us assume that, through some external agency, the
center of the electron pattern is transferred from one position of rest to
another which is difTerently placed relative to the nucleus. Owing to the
different effect of the nuclear pattern on the effective constants of the
medium as viewed by the electron pattern, the stable energy of the latter
would be different at the second position. This change in rest energy with
position may be interpreted as a measure of the change in a field of static
potential associated with the massive nucleus. The similarity between this
relationship and that which exists between the electron and the nuclear
potential in wave mechanics is obvious.
In speaking of a change in the effective constants of the medium, we refer
to an average value taken over a number of cycles and wave lengths of the
oscillations which make up the second pattern, or nucleus. Calculations
based on this concept should not therefore be expected to give valid results
when the time intervals involved in the averages are comparable to the
//
period ;, of the second particle at rest, or the distances are comparable to
nioc-
the corresponding wave length of the pattern. For a proton this period
is 4.38 X 10~ seconds and the wave length is 1.31 X 10~ cms. If, then,
an electron is to be subject to the kind of nuclear potential field just de-
scribed, the linear dimensions of that part of it which is controlled by the
potential field of the proton must be at least of the order of 10~ cm. This is
consistent with Gamow's^ observation that "It seems, in fact, that a length
of the order of magnitude of 10 centimeters plays a fundamental role in
the problem of elementary particles, popping out wherever we try to esti-
mate their physical dimensions."
The variations in the medium due to the nucleus might be treated in
terms of their effect on the progressive wave components, the interference
of which gives rise to the wave pattern of the electron. The component waves
as so influenced should combine to form an interference pattern which
represents the behavior of the electron in the field of the nucleus. It is also
possible that a technique may be found for treating their effect on that
factor of the electron wave which is similar to the DeBroglie wave. This
should be more nearly like the techniques now used in wave mechanics.
If two particles are brought so close together that the central cores of
their patterns overlap, the departure from linearity becomes so great that
* G. Ganiow, Physics Today, 2, p. 17, Jan., 1949.
366 BELL SYSTEM TECHNICAL JOURNAL
a procedure which may be successful at intermediate separations becomes
inadequate. Relativistic mechanics breaks down and Lorentz invariance
may lose its significance. This is in agreement with the experimental result
that, in some nuclear reactions, the energy balance, as calculated from
the relativistic relations, is not satisfied. Also the difficulty which has been
encountered in calculating nuclear phenomena by the techniques of wave
mechanics suggests that the extremely non-linear condition is approached
for the separation of the particles within a nucleus. This viewpoint suggests
that an understanding of the nucleus might make possible an experimental
determination of velocity relative to the ether.
The reactions between wave patterns of appreciable amplitude may also
be viewed from a somewhat different angle. We may think of the various
wave patterns as being the analogs of the various modes of motion of, say,
an elastic plate. For very small amplitudes they have negligible effect on
one another. For larger amplitudes, where Hooke's law does not hold, the
force may be represented as a power series of the displacement. The first
power term represents the linear stiffness. If the frequencies of two modes
which are in oscillation are wi and 0)2 , the higher power terms represent
forces of frequencies ;;zcoi ± iicoo where m and n are integers or zero. These
forces set all the modes into forced oscillation at the frequencies of the
various forces, in amounts which depend on the impedance of the particular
mode for the particular frequency. When the frequency of the force coin-
cides with the resonant frequency of one of the natural modes, the forced
oscillations may be large. Thus the variation in stiffness with displacement
provides a coupling whereby energy may be transferred from one or more
modes, that is wave patterns, to other modes. But in this transfer the energy
always appears associated with a new frequency which is related to those of
the modes from which it came in accorance with the familiar formula of
quantum theory.
The theory of such energy transformations with change of frequenc}^ has
been worked out in considerable detail for vacuum tube and other variable
resistance modulators, and the results show little in common with the quan-
tum theory beyond the relations connecting the frequencies. Wlien, however,
the variation is not in a resistance but in a stiffness, as occurs in the ether
case, the situation is cjuite different. This problem has been explored both
theoretically" and experimentally. It is found tliat an oscillation of one
frequency in one mode may provide the energy to support sustamed oscil-
lations of two other lower frequencies in two other dissipative modes. For
this to occur the frequencies involved must be related through the quantum
formula. Also the amplitude of the generating oscillation must exceed a
'■• R. V. L. Harllcv, Bell Svs. Tech. Jour., 15, 424, 1936.
"' L. W. Husscy :in<i !-. R. Wralliall, Bell Sys. Tech. Jour., 15, 441, 1936.
MATTER, A MODE OF MOTION 367
threshold value which depends on the frequencies, the impedance involved,
and the constant of non-linearity. The transformed energy divides itself
between the generated modes in the ratio of their frequencies. In a non-
dissipative system, the frequencies of possible combinations of sustained
oscillations are determined by the energy of the system. Here also they are
connected by the quantu n formula.
The particle wave pattern discussed above would approximate very
closely to such a non-dissipative non-linear system. We should therefore
expect its frequency to be related to its energy through the constants of the
ether. In the more complex wave patterns associated with more than one
particle, it is unlikely that the pattern representing, say, an electron could
maintain its identity as part of some arbitrarily chosen pattern, the magni-
tudes of which are not commensurable with its own. This suggests that the
stable states of the complex pattern would be confined to a sequence of
discreet patterns which are related to one another through some property of
the electron. These possible non-dissipative combinations of energy and fre-
quency would represent the stable quantum states of the atom. The radia-
tion process would then be similar to that referred to above in which energy
from a source of higher frequency distributes itself between two lower fre-
quencies in the ratio of the frequencies. The energy in the pattern of an
excited atom would serve as the source. One of the two lower frequencies
would be that of a pattern corresponding to a lower energy state to which
the transition occurs. The other would be that of the radiating wave which
carries off the energy lost in the transition.
A Suggested New Particle
We saw above that the observed variation of the energy of a particle
with its velocity calls for a mechanism in which the energy varies directly
as the frequency. The fact that a system, in which the stiffness varies with
the displacement, is characterized by this relation suggests that the energy
of a particle pattern depends mainly on variations in the stiffness of the
ether. However, the non-linearities of the ether equations cannot all be
interpretated as variable stiffnesses. The non-linearity which appears in (1)
when the displacements are iinite is equivalent to a variable inertia. It is
in order, therefore, to inquire into the properties of a pattern in which the
energy is determined by this kind of non-linearity. The variable inductance
of an iron-core coil constitutes such a variable inertia. Theoretical and ex-
perimental studies of circuits involving these coils have shown that they
behave very much as do systems having variable stiffness, with one im-
portant exception. The energy distributes itself in the inverse ratio of the
frequencies.
If, then, we assume that the energy of a moving pattern is determined by
368 BELL SYSTEM TECHNICAL JOURNAL
a mechanism wliich conforms to this relation, it follows from (7) that its
energy will vary as - . Expanding in the usual manner we then have
W = WoC^ - i WoF2 + • • •
This says that a particle represented by such a wave pattern would have
a positive rest mass and a negative inertial mass. Its momentum is directed
oppositely to its velocity, and energy must be taken from it to set it in
motion and given to it to stop it. Such a particle, when bouncing back and
forth between two rigid walls or rotating about two centers of force, would
exert a force tending to draw them together, instead of the usual repulsion.
It is interesting to speculate that if, in an atomic nucleus, the positive charges
which are passed back and forth between other nuclear particles were
associated with particles of this type their motion would exert a binding
force on the other particles.
Conclusion
It appears, then, that the ether model is capable of sustaining wave
patterns the behavior of which is qualitativ'ely in agreement with the
results of experiment. In order to establish fully the sufficiency of classical
mechanics for the physical description of natural phenomena, it will be
necessary to work out the complicated quantitative relations whereby the
constants of the ether may be deduced from experimental measurements.
However, until a serious attempt to do this has failed for some reason other
than sheer mathematical complexity, the insufficiency of classical mechanics
can scarcely be argued.
In conclusion, I wish to acknowledge the contributions of those of my
colleagues who, through discussions over the years, have helped in develop-
ing the concepts which have been put together in the above picture.
The Reflection of Diverging Waves by a Gyrostatic Medium
By R. V. L. HARTLEY
{Manuscript Received Feb. 28, 1950)
This paper furnishes the basis for a companion one, which discusses the pos-
sibility of describing material particles as localized oscillatory disturbances in a
mechanical medium. If a medium is to support such disturbances it must reflect
a part of the energy of a diverging spherical wave. It is here shown that this
property is possessed by a medium, such as that proposed by Kelvin, in which
the elastic forces are of gyrostatic origin. This is due to the fact that, for a
small constant angular displacement of an element of this medium, the restoring
torque, instead of being constant, decreases progressively with time.
Introduction
IN A companion paper it is pointed out that it may be possible to de-
scribe the behavior of material particles as that of moving patterns of
wave motion, provided a medium can be found which is capable of sus-
taining a localized oscillator^' disturbance. In most media this is not possible,
for the energy of the disturbance would be propagated away in all directions.
Something special in the way of a medium is therefore called for. It must
be capable of trapping the wave energy released from a central source.
Kelvin proposed a mechanical medium, the equations of which, for small
disturbances, were identical with those of Maxwell for free space. The
medium derived its elasticity from gyrostats. He recognized that, for finite
disturbances, the restoring torque depends on the time as well as the angular
displacement. It is the present purpose to show that this time dependence
imparts to his medium exactly the energy trapping property required.
The GYROST.A.TIC Ether
The concept of an ether with stiffness to rotation originated with Mac-
CuUagh- in 1839, and was further developed by Kelvin^ in 1888. MacCullagh
showed that certain optical phenomena associated with reflection could not
be represented by the elastic solid ether of Fresnel, but required for their
mechanical representation a medium in which the potential energy is a func-
tion of what is now called the curl of the displacement. Fitzgerald'* remarked
in 1880 that its equations are identical with those of the electromagnetic
^ R. V. L. Hartley, Matter, a ^Mode of Motion — this issue of the Bell System Technical
Journal.
^ Collected Works of James MacCullagh, Longmans Green & Co., London, 1880, p. 145.
^ Alathematical and Physical Papers of Sir William Thomson, Vol. HI, Art. XCIX,
p. 436, and Art. C, p. 466.
'' Phil. Trans. 1880, quoted by Larmor, Ether and Matter, Cambridge U^niv. Press,
1900, p. 78.
369
370 BELL SYSTEM TECHNICAL JOURNAL
theory of optics developed by Maxwell. This conclusion is confirmed in
later discussion by Gibbs/ Larmor,^ and Heaviside.''
Kelvin, apparently unaware of MacCuUagh's work, was led by similar
considerations to the same result. He went farther and devised a physical
model which consisted of a lattice, the points of which were connected by
extensible, massless, rigid rods in such a manner that the structure as a whole
was incompressible and non-rigid. Each of these rods supported a pair of
oppositely rotating gyrostats. By a gyrostat he meant a spinning rotor
mounted in a gimbal so that it is effectively supported at its center of mass
and can have its spin axis rotated by a rotation of the mounting. The
resultant angular momentum of the rotors was the same in all directions.
This model, considered as a continuous medium, exhibits a stiffness to
absolute rotation, the nature of which can be described by comparing it
with the elasticity of a solid. A solid is characterized by a rigidity n such
that small displacements u, v, w are accompanied by a stress tensor, one
component of which is
^dv . du\
dy)'
dx
For the ether model the corresponding component is
dv du
where tp is a small angular displacement of the element about the z axis.
More generally a small vector rotation A(p is accompanied by a vector re-
storing torque per unit volume,
AT = -4wA^. (1)
The quantity in therefore represents a stiffness to angular displacement
of the element.
In the appendix it is shown that the lattice of gyrostats, treated as a
continuous medium, exhibits this kind of elasticity. It is also shown that
for infinitesimal displacements, the medium is described by the wave
equations (8a and 6a).
^5 Collected Works of |. Willard C;il)l)s, Longmans Green & Co., New York l')2S, Vol.
II, p. 232.
" Heavisidc, Klectroniai^nelic Theory, KriiesL licnii, Ltd., London, 189.i, Vol. I, j). 226.
REFLECTION OF DIVERGING WAVES 371
where po is the constant density, rjo is a generaUzed stiffness of the undis-
turbed medium, given by (7a), q is the vector velocity, and T is the torque
per unit volume. In a plane wave q is normal to the direction of propagation.
T . ....
— is a tractive force per unit area m the direction of q, which acts on a surface
normal to the direction of propagation.
If, however, the amplitude is finite the equations become much more
complicated. For present purposes we need consider only waves for which
there is no component of velocity or torque in the direction of propagation,
and we need consider only plane polarized waves for which the direction of
the velocity is the same at all times and places. Also, as will appear below,
we are concerned with the equations which describe a wave of infinitesimal
amplitude which is superposed on a finite disturbance. This description need
cover only infinitesimal ranges of time and position. It can therefore be
expressed in terms of wave equations in which the constants of the medium
have local instantaneous values which depend on the finite disturbance.
Subject to these restrictions it is shown in the appendix that (2) is to be
replaced by (23a)
where Iq is a unit vector in the fixed direction of the velocity, and p is an
instantaneous local density, defined in terms of the finite disturbance by
(20a). And, in place of (3), (22a)
where l^ is a unit vector in the direction of the axis of rotation, p is again
an instantaneous local density, c is an instantaneous local velocity derived
in the usual way from p and an instantaneous local stiffness ??, while / is a
function defined by the relation, (13a),
T = -IM^P, 0.
This function takes account of the fact that when the spin axis of the rotor
is given a constant finite displacement, the restoring torque is not constant
as in (1), but changes with time as the spin axis rotates toward the axis of
displacement, and so reduces the component of the sjjin which is normal
f>f
to the displacement axis and so is effective in producing stiffness. — 4 —
dt
represents the rate of this change in torque for a fixed angular displacement.
— 4— is to be interpreted as the rate of change of torque with angular
372
BELL SYSTEM TECHNICAL JOURNAL
displacement, when the time consumed is infinitesimal, that is when the
angular velocity is infinite. It is therefore an instantaneous local angular
stiffness from which the instantaneous local generalized stiffness -q is derived
as in (19a).
To simplify these expressions, let the direction of propagation be .v and
that of q be y. Then
V X 9 = f ^ {jq) =k^l,
ox dx
so l^ is in the direction of s, and represents a clockwise rotation about z.
(5) then becomes the scaler equation
dq_
dx
1
d T
pc- \_dt \ 2
T is also in the z direction, so
^ + 2
dt
(6)
V X
dx
T
. d
ax
But q is in the y direction, so
dx
^ dt
(7)
These, then, are the desired equations of motion, for the type of wave
under consideration.
The Generation of Reflected Waves
In this section we shall show that when a finite wave is proi)agated in
this medium each element of the medium becomes the source of auxiliary
waves which propagate in both directions from the source.
To do this we shall make use of the argument by which Riemann" sliowed
tliat this does not occur for sound waves in an ideal gas. This will first be
restated in more modern language. We consider a plane wave pro])agating
along the .v axis. We picture the finite pressure p and the longitudinal
velocity u at a jwint in the medium as having been built up by the successive
superposition of waves of infinitesimal amplitude, each propagating relative
to the medium in its condition at the time of its superposition. If the first
increment is propagating in the positive direction,
du
dp
P'"
^ I-amh, HvdnKlvnamics, Sixth Edition, p. 481. Rayleigh, Theory of SouiicI, Second
Ivlilion, Vol. II. |>.\^<S.
REFLECTION OF DIVERGING WAVES 373
where the characteristic resistance is pc. Here
2 _ dp
dp
He assumes adiabatic expansion, so that p and c are functions of p only. If
a second incremental wave of pressure dp^ also traveling in the positive
direction, be added, its velocity increment, being relative to the medium,
will add to that already present. Its value will be related to dp through a
new characteristic resistance corresponding to the modified density result-
ing from the previous increment. Hence the velocity u resulting from a
large number of such waves will be
Jo DC
pc
where w is the quantity represented by co in Lamb's version. If, then, all
of the wave propagation is in the positive direction
u — w.
Similarly, if an incremental wave is traveling in the negative direction,
du = - — ~ ,
pc
and the condition for all the propagation to be in that direction is
11 = —w.
Obviously, then, if u has some other value than one of these it results from
the addition of increments some of which propagate in each direction.
Riemann deduces from the aerodynamic equations that
I + (« + c) f) (to + u) = 0, (8)
i + (" - c) A) (. _ „) = 0, (9)
That is, the value of iv + ii is propagated in the positive direction with a
velocity of c + « and that of w — u, in the negative direction with a velocity
c — u. If, over a finite range of x, a disturbance be set up such that neither
of these quantities is zero, it must be made up of incremental waves in both
directions. However, as w + « propagates positively it will be accompanied
at any instant by a value of w — u which has been propagated from the other
direction. But, since the value of this was initially finite over a limited dis-
tance only, when all of this finite range is passed, a' — u will be zero, u will
37-1 BELL SYSTEM TECHNICAL JOURNAL
be equal to w and all of the wave will be traveling positively. A similar
argument applies at the negative side of the wave. Thus the initial disturb-
ance breaks up into two parts which travel in opposite directions without
reflection. More generally, these considerations hold for any medium in
which the stress is a function of the strain only.
For the ether model, since we have assumed the displacements are normal
to the direction of propagation, the velocity of wave propagation relative to
the medium is the same as that relative to the axes.
If now, following Riemann, we let
i» = 1 rf g) , (10)
so that now
then from (7) and (6)
= /^^©'
dq _
dt
div
dx
dw
Tt
dx
2 df
pc dt
^ives
{w + q)
_ _ 2 df
pc dt
- c —
dx/
1 («' - (?)
2 df
pcdt'
Adding and subtracting gives
d
dt
d
d't
r) f
which are to be compared with (8) and (9). Hence when — is not zero the
dt
values of zt' + (/ and w — q are not propagated without change.
To show that reflection occurs, consider a disturbance at a point .v at
time /, characterized by (/ and tv. At .v and / + A/, Ti' + </ will difi"cr from the
d ^ 2 df
value It had at .v — <A/, /, or ic + (/ — ~ {w + u)cH, bv — ^ 7" A/. The
dx ' pc dt
increment at .v in time \l is
Ak' + Ar/ = - |- (k' + q)c\t - 1 ^ A/,
dx pc dt
REFLECTION OF DIVERGING WAVES 375
and
A7£' — A(7 = — {w — (/)cA< — i^t.
dx pc dt
From which
Aw = —c-^At — At,
dx pc dt
A^ = —c — At.
dx
Hence the velocity is the same as when — is zero but iv is changed by
— — — A/. But the only way in which w can change with q constant is
pc at
by adding waves of equal amplitude propagating in opposite directions, so
that their contributions to w are equal and those to q are equal and opposite.
T f)f
From (10) this involves an increment of — of — 2 — A/ or a time rate of change
2 dt
of —2 — .This agrees with (6), from which it is evident that the presence of
dt
— alters — from what it would otherwise be by — -. But, since q is
dt dx ^ pc^dt ' ^
unchanged, the velocities at .v -| — ~ and x — — - are increased by — — ^—Ax
2 2 pc- dt
1 /)/
and — - — A.v. The first is the velocity associated with an auxiliary wave which
pc- dt ^ ^
propagates in the positive direction of x, and the second that of one which
propagates in the negative direction, that is a reflected wave. Hence the
1 ri{
medium generates a reflected wave of — - ^ per unit length in the direction
pc- dt
of propagation.
The Reflection of a Progressive Diverging Wave
So far attention has been confined to a single point. If a continuous dis-
turbance is being propagated, it is important to know how the waves reflected
at different points combine, for it is conceivable that they may interfere
destructively. From the standpoint of the application to be made of these
results in a companion paper, the case of most interest is that in which energy
is propagated outward from a central generator as a sinusoidal wave of
finite amplitude, beginning at time zero. Near the center, the wave of dis-
placement will include radial as well as tangential components. As the radius
376 BELL SYSTEM TECHNICAL JOURNAL
increases the radial components become relatively negligible. We shall
confine our attention to this outer region, where, in the absence of reflection,
the propagation differs from that of a plane wave only in that the amplitude
varies inversely as the radius. We shall neglect the effect of any reflections
on the outgoing wave, and calculate the resultant reflected wave at a radius
Ti as a function of the time and so of the radial distance r the wave front
has traveled.
If the outgoing wave were of infinitesimal amplitude, its velocity q^
could be represented by
Oo = - <3o sin {icl — kr), (11)
r
for values of r < ct, and by zero for r > ct, where Qo is the amplitude at
some reference radius ro . The sine function is chosen to avoid the necessity
of an infinite acceleration at the wave front, as would be required by a
cosine function. When the amplitude is finite this wave suffers distortion
due to the fact that k which is equal to - varies slightly with the variations
in the instantaneous value of c. However, these will be small and, since
fluctuations in velocity alone do not cause reflection, we shall neglect them.
The procedure is to make use of ^o to calculate the reflected wave incre-
ment generated in a length Ar' at a radius r', calculate the amplitude and
phase of this at a fixed point r^ <r', and at ri integrate the waves received
there for values of r' from ri to the farthest point from which reflected waves
can reach ri at the time t under consideration.
To find the reflected wave generated in a length Ar' at r', we have from
above that its velocity
Aq' = l^^Ar'.
From (21a), (19a) and (l7a)
1 Fi
2 —
pc
(1— a I (p dt 1
where tjo and a are constants of the medium given by (7a) and (15a). From
(18a)
di
80
= — arjo^ / <pdt,
REFLECTION OF DIVERGING WAVES 377
dr'
aFnp I <p dt
— a \ I <p dt \
which reduces to
jp = - a(p j (p dt,
if we neglect second powers of the variables compared with unity.
To the same accuracy, from (14a)
1 f dqo
From (11)
^0 ^ ^oQo
dr' r'
k cos (co/ — kr') + —. sin (oot — kr')
r
Here k is lir over the wavelength so, if as we have assumed ri , and therefore
also r', is large compared with the wavelength, we may neglect the second
term. Then
/
if = -J^ sin (co/ - kr'),
<p dt = J' , cos {<ut — kr'),
2cur
dq' _ a froQoY ..-.^
, , — „ . , , sin (co/ — kr') cos (w/ — kr'),
dr 8co \ cr
= __i froi^«Y [cos (co/ - /^r') + cos 3(co/ - kr')].
8co \ cr /
This, when multiplied by Ar', gives the value at r' of the wave, generated
in the interval Ar', which propagates in the negative direction of r. This is
made up of components of frequency co and 3co. We are primarily interested,
from the stand-point of reflection, in that of frequency co, so we shall confine
our attention to this component, with the understanding that the other
can be treated in exactly the same fashion. As the fundamental component
r'
propagates inward to r^ it increases in amplitude m the ratio — and suffers
a phase lag of k{r' — ri). If we call the resultant of all the reflected waves at
ri, qi , then the contribution to qi of the wave generated at r' is
378
BELL SYSTEM TECHNICAL JOURNAL
Agi
~j- cos (co/ + A-ri - 2kr')Ar'.
This is to be integrated from Vi to the farthest point from which a reflected
wave has reached ri at the instant / under consideration. This point is at
K^i + cl). So
3 ^i(ri+ct)
'■ = -87;
Here the integrand is a function of r' and / and the upper hmit of integration
is also a function of /. We therefore make use of the relation*
'Wo A f i- cos (co/ + )^ri - 2k/) dr'.
c / Jr, r-
d_
da
f'fix, a) dx = £ (J^fix, a)) dx + f(b, a) ^ - f(a, a)
da
da
Putting / for a, r' for .v we have
dqi _ a /roQo
IF ~ Sr, \c
/
i(ri+et)
2c
^, sin (o)/ + kri — 2kr')
r- CO (ri + ct)-_
which, upon integration becomes,
dt 8ri \ c / \ri
■ sin (co/ + kr,) - [C/(w/ + kt\) - Ci{2krx)\
cos (co/ + kr^)
Ic
)
co(/-i + cif
Since q\ is zero when / is - , its value at / will be found by integrating from
to /, so
--^;^^-^-^-'''^ "■
2kr^
• CO / .V/(co/ + kri) sin (co/ + kr^) dt + Si{2krx)
• [cos (co/ + A';-]) — cos 2/.';'i| — co / C'/(co/ + kr^) cos (co/ + ^rj rf/
''n/<-
+ Ci{2kr^) lsin(co/ + ^r,) - sin 2kr^\
which reduces to
'* Byerly, Integral Calculus, second edition j). W.
REFLECTION OF DIVERGING WAVES 379
,; = -JL. {'^^ (cos („/ - kr,) - ^, + 2kr,
• [-[Siicot + kr,) - Si{2kn)] cos (co/ + /trO
- [CiW + /feri) - C/(2/^ri)J sin (co/ + )^ri)
+ Si{2c^l + 2/&r,) - ^'i(4/feri)] j .
The first term represents the value at Vi of an outwardly moving wave in
phase quadrature with the main wave. The second is a transient, the value
of which is equal and opposite to that of the first term at the instant that
the main wave passes ri . The first two terms in the inner bracket are waves
which propagate inward and so are to be regarded as reflections of the
main wave. The last two terms represent a velocity which is zero when the
main wave passes fi , and subsequently oscillates about and approaches
- — Si(4^ri). Physically it appears to result from the particular form chosen
for the main wave, which starts abruptly as a sine wave. The time integral
of the impressed force, and so the applied momentum, has a component in
one direction. Presumably if the main wave built up gradually these terms
would be absent.
Returning to the reflected waves, their amplitudes are zero when the
main wave passes Vi , after which they become finite. Si{x) and Ci(x) os-
cfllate about and approach - and zero respectively as .v approaches infinity.
Hence, as / increases indefinitely, the amplitudes of the reflected waves
approach - — Si(2kri) and Ci{2kr^. For the assumed large values of 2kri
these quantities are small compared with unity. When multiplied by 2kri
1
their variation is very slow. Hence the amplitudes vary roughly as ^ ,
and approach zero as the main wave at ri approaches an ideal plane one.
However, the significant fact is not that the reflected waves are small
but that they are of finite magnitude. Because of this the main wave will
not behave exactly as we assumed above, but will decrease slightly more
rapidly with increasing radius. This should increase the reflection slightly,
for the existence of the reflected wave is dependent on the decrease in am-
plitude with distance when the radius of curvature is finite.
To describe exactly what happens when the generator begins sending out
waves from a central point would be hopelessly complicated, but we may
form a general picture. In the early stages where the curvature is consider-
able, the reflected waves would be quite large and the main wave would be
380 BELL SYSTEM TECHNICAL JOURNAL
correspondingly attenuated. The arrival of the reflected waves at the gen-
erator adds a reactive component to the impedance of the medium, as seen
from the generator, which reduces the power delivered to the medium.
Meanwhile energy is being stored as standing waves in the medium and
the rate of flow of energy in the wavefront is decreasing. The energy in
successive shells of equal radial thickness decreases with increasing r, in-
stead of being uniform as it would be in the absence of reflection. In the
limit it approaches zero, but as the rate of decrease depends on the curva-
ture, the rate of approach also approaches zero. As the rate at which energy
is stored and that at which it is carried outward at the wavefront both
approach zero, the resistance which the medium offers to the generator
approaches zero, and its impedance approaches a pure reactance.
The total energy stored in the medium depends on how the over-all at-
tenuation of the main wave is related to its amplitude. If there were no
attenuation, the impedance would remain a pure resistance, the energy in
successive shells would all be the same, and the total energy would increase
linearly with r, and so with the time, and approach infinity. If the attenua-
tion were independent of r, the total energy would approach a finite value.
The present case is intermediate between these, the attenuation being finite
but approaching zero with increasing r. If we assume it to vary as some
power of the amplitude of the velocity, then W. R. Bennett has shown that
if this power is less than the first the total energy approaches a finite value.
If it is equal to the first, the energy approaches infinity as log r, and if it is
greater than this, the power approaches infinity more rapidly. Until more is
known as to the actual variation of amplitude with distance, nothing
definite can be said about the limit of the total energy.
APPENDIX: EQUATIONS OF THE KELVIN ETHER
We are concerned with the wave properties of the model for wavelengths
long enough compared with the lattice constant so that it may be regarded
as a continuous medium. Its density is equal to the average mass of the
gyrostats per unit volume. Its elastic properties are to be derived from the
resultant of the responses of the individual gyrostats.
We shall therefore begin by considering the behavior of a single element,
which is shown schematically in Fig. 1. Here the outer ring of the gimbal,
which is rigidly connected with the lattice, lies in the .v y plane. The axis
about which the inner ring rotates is in the .v direction, and the spin axis C
of the rotor is in the 2 direction. We wish to examine the effect of a small
angular displacement <p of the lattice, that is, of the outer ring. If it is about
X or z, it will, because of the frictionless bearings, make no change in the
rotor. If it is about y it will produce an equal displacement of the spin axis
REFLECTION OF DIVERGING WAVES
381
Fig. 1 — Diagram of a gyrostat, showing its axes of rotation.
C about y. To study its effect we make use of Euler's equations for a rotating
rigid body.^
A~' - {B -C)a,2C03 = L,
at
B
It
- (C - ^)C03C01 = M,
C^ - {A - B)o}ico2 = N,
at
where wi , C02 and 0)3 are the angular velocities about three principal axes of
inertia, fixed in the rotor, the moments of inertia about which are A, B
and C, and L, M , and .V are the accompanying torques about the three axes.
They are also at any instant the values of the torques about that set of
axes, fixed in space, which, at the instant, coincide with the axes 1, 2, 3,
which are fixed relative to the body. We let the 3 axis coincide with the
spin axis C. We choose as the 1 and 2 axes, lines in the rotor which, at the
instant, are in the x and y directions respectively. Since the moments of
'Jeans, Theoretical Mechanics, Ginn and Co.. p. 308.
382 BELL SYSTEM TECHNICAL JOURNAL
inertia about these are equal, .1 and B are equal. By virtue of the frictionless ^^
bearings the external torques L and X about 1 and 3 are zero.
Introducing these relations we have
A^-^+ (C - .Oco^co, = 0, (la)
at
A^ - (C - .Ocoico;, = M, (2a)
dt
C^^ = 0. (3a)
dt
From (3a) the velocity of spin w-i remains constant. The torque M about y
is then to be found from (la) and (2a). For very small displacements,
C02 = (p.
Putting this in (la) and integrating from zero to /, assuming tp to be zero
at / = 0, gives
(2a) then becomes
C - A
COi = ~ 0)z(p.
.... (c - aY o
A(p -\- ~ 053^9 = M.
This represents an angular inertia A and stiffness -. The system
will therefore resonate at a frequency . If the frequencies in-
volved in the variation of ^ are small compared with this, the inertia torque
will be negligible, and the system will behave as a stiffness. If the displace-
ments about A associated with wi are very small the restoring torque M
will act substantially about the y a.xis. That is, the lattice will encounter a
stiffness to rotation.
Since the large number of gyrostats in an element of the model are oriented
in all directions, an angular displacement of the lattice about y will gen-
erally not be about the B a.xis for each gyrostat. If it makes an angle a with
this a.xis, then only the component (p cos a of the angular displacement will
be transmitted to the rotor. The resulting torque will then be S cos a, where
S =
iC - Afo^l
A
It will be directed about B and so will not be parallel to the applied dis-
placement. However, if a second gyrostat has the position which the first
REFLECTION OF DIVERGING WAVES 383
would have if it were rotated about y through tt, its torque along y is the
same as that of the first, and that normal to it is equal and opposite. Hence,
if the gyrostats are properly oriented, the resultant torque will be parallel
to the displacement and the medium will be isotropic. The y component of
the opposing torque will be S(p cos- a. Thus if the B axes are uniformly dis-
tributed in space the total torque will be one third what it would be if they
were all parallel to the axis of the applied displacement. Hence if there are
.V gyrostats per unit volume the vector restoring torque T per unit volume
will be
7 = — - ~ <p. (4aj
The next step is to derive the wave equations for a medium having this
stiffness to rotation. If the vector velocity q is very small,
VX9=2^, (5a)
where ^ is a vector angular displacement of an element of the medium at
the point under consideration. 2<p plays a role analagous with that of the
dilatation in compressional waves. Then, from (4a) and (5a),
where the generalized stiffness of the undisturbed medium,
^V (C - A)' 2 ,. ^
r?o = j2 "^4 '''■ ^^^^
To get the companion equation, we interpret the torque exerted by an
element in terms of the forces it exerts on the surfaces of neighboring ele-
ments. Let the x axis Fig. 2 be in the direction of the torque TAx^ which is
exerted by the medium within the small cube. This very small torque can
be resolved into the sum of two couples, one consisting of an upward force
FyAx- on the right face and an equal downward force on the left one, and
the other of a leftward force FzAx:' on the upper surface and a rightward one
on the lower one. But, if there is not to be a shearing stress, Fy and F^ must
T .
be equal, and each equal to — . Thus a torque per unit volume T is equivalent
T
to a set of tangential surface forces per unit area of — each.
Now consider the force exerted on an element by its neighbors, through
the adjoining surfaces. To take the simplest case, let T in Fig. 2 be every-
where in the x direction and independent of z but varying with y. Then
384
BELL SYSTEM TECHNICAL JOURNAL
the forces exerted on the upper and lower surfaces are equal and opposite.
That downward on the right face exceeds that upward on the left by
— ( — I A.T^. By ex-
dy\2) ^
tending the argument to three dimensions it is easily shown that the total
Ty (I ^^) ^^'
SO the force in the z direction is
Fig. 2 — Diagram showing the forces exerted by an element of the medium through
its surfaces.
force isV X
PoAr* — , so
at
i-j ^o<?. If po is
the density of the medium this force must equal
V X
T
dq
which, since q is small, reduces to
Po
dq
dl
(8a)
From this and (6a) the velocity of propagation is {vq/poY and the char-
acteristic resistance is {poVoY • In a plane wave the displacement is normal
to the direction of propagation. The stress is a tractive force per unit area
— acting in a surface normal to the direction of propagation. It is in the
direction of the velocity and in phase with it.
REFLECTION OF DIVERGING WAVES 385
However, we are also interested in the case where the amplitudes are
not negligible. We shall confine our attention to those cases where, as in
plane or spherical waves at a distance from the source, the velocity is nor-
mal to the direction of propagation and the variations in the plane of the
wave front are negligible. (5a) then becomes much more complicated.
V X 9 is, however, still a function of — , say 2Fi I — I . Then, for small varia-
dt \dt/
*i —
tions of — in the neighborhood of a particular value, we may write
dt
where Fi i — ) is a function of the particular value of — . This relation is
to take the place of (5a). Similarly, if
then, in place of (8a), we are to use, for small variations,
When we come to the transition from (5a) to (6a), however, the situation
is somewhat different. To see how this comes about, we go back to the
behavior of the single gyrostat of Fig. 1. It was assumed above that the B
axis coincided with the y axis However, when the displacement of the
rotor about A is finite, this is no longer exactly true. The situation is then
as shown in Fig. 3. A rotation (p of the lattice about y displaces A in the .r z
plane by if. The accompanying rotation of the rotor about A causes B to
make an angle 6 with y, which is independent of <p. Then
d^ a
(ji2 = -r cos d.
dt
From (la)
C - A f d<p . ,^
coi = ojs / -r cos 6 dt.
A J dt
Also
d = I oji dt,
C - A
cos U -J- cos ddtdt, (11a)
386
BELL SYSTEM TECHNICAL JOURNAL
which determines d as a function of v? and /. From (2a), neglecting the first
term as above,
M = S [ ^ cos 9 dt,
J dt
and the restoring torque about y, or
Ty — —S cos 6 I
d<p
It
cos 6 dt.
(12a)
This, together with (11a), determines Ty as a function of ip and /, instead of
<p alone as it is for infinitesimal displacements.
Fig. 3 — Diagram showing the displacement of the axes of a gyrostat.
We assumed here that, in the rest ])osition of the rotor, its B axis coin-
cides with that of the ai)i)lied displacement if. When this is not the case, the
relations arc more complicated, but (hey should be qualitatively the same.
Hence, for an element of the medium, the torcjue per unit volume should
be a function of <^ and / sirnilar to T,, , which reduces to —4tj(i(^ for very small
displacements. Since the restoring torque is in the direction of v? we may
REFLECTION OF DIVERGING WAVES
387
write
T = -IM<P, t)
(13a)
where l^ is a unit vector in the direction of the axis of rotation.
The derivation of the wave equation is much simpler if we consider only
the case of present interest where the direction of the rotation is everywhere
the same so that l^ is constant. Then (9a) can be written as
V X 9 = hlFi
dip\ dip
dt dr
(14a)
and (13a) as
T = -if(^, /).
We wish now to replace — by — I -
dt dt \2
These partial derivatives refer to a
constant position so we are interested in the total time derivatives of T as
given by (12a). To get the desired relation we need to express T explicitly
in terms of (p and /, that is, we must evaluate (p. Since the variables are
small, we neglect their products of higher order than the third. Then
cos ^ = 1
where
Putting
1
C - A
j pdt\,
C03
(15a)
T = —4770 cos 0 I —- cos d dt,
J at
in accordance with (12a) and substituting for cos 6 gives
T
■47J0
ip — a(p
\ <pdt -\- a <p- I ip dt) dt
Then
JT
^
= —47/0
1 - a
f <pdt y^ - a^' \ ^dt
When ip is constant the tirst term is zero, so the second term can be inter-
preted as the partial derivative of T with respect to /. Physically this de-
scribes the change in torque for a fixed displacement which results from the
388 BELL SYSTEM TECHNICAL JOURNAL
fact that, as the axis of the rotor rotates toward that of the appHed torque,
the component of the spin which is normal to the axis of displacement pro-
gressively diminishes. To interpret the lirst term, we let — increase in-
definitely. The second term then becomes negligible, and when we divide
through by -r- , the left side becomes -— . But the time increment which
^ ^ dt d(i>
accompanies a finite increment of (p is now infinitesimal, and so this may be
called the partial with respect to (p, with / constant.
We have then
D
dt \dif> di dt;
?here
^ = 7,0^1 - a\^j^dt y (17a)
dt
■ar)(np I <p dt. (18a)
Substituting for -^ from (16a) in (14a),
dt
d(p
dj .
We may interpret t~ as an mstantaneous stiffness to rotation and define
dip
an instantaneous local generalized stiffness by the relation
V = -p (19a)
Similarly from (10a) we may define an instantaneous density by the relation
P = F2. (20a)
Then we may speak of an instantaneous velocity c given by
.^ = ", (21a)
REFLECTION OF DIVERGING WAVES 389
and an instantaneous characteristic resistance pc. Then
(lOa) becomes
where Zg is a unit vector in the fixed direction of the velocity. These are the
equations of motion which apply to a very small disturbance superposed
on a finite disturbance.
Traveling-Wave Tubes
By J. R. PIERCE
Coinright, 1950, D. Van Nostrand Company, Inc.
[THIRD INSTALLMENT]
CHAPTER VII
EQUATIONS FOR TRAVELING- WAVE TUBE
Synopsis of Chapter
IN CHAPTER VI we have expressed the properties of a circuit in terms
of its normal modes of propagation rather than its physical dimensions.
In this chapter we shall use this representation in justifying the circuit
equation of Chapter II and in adding to it a term to take into account the
local fields produced by a-c space charge. Then, a combined circuit and
ballistical equation will be obtained, which will be used in the following
chapters in deducing various properties of traveling-wave tubes.
In doing this, the lirst thing to observe is that when the propagation con-
stant r of the impressed current is near the propagation constant Pj of a
particular active mode, the excitation of that mode is great and the excita-
tion varies rapidly as P is changed, while, for passive modes or for active
modes for which P is not near to the propagation constant P„ , the excita-
tion varies more slowly as P is changed. It will be assumed that P is nearly
equal to the propagation constant Px of one active mode, is not near to the
propagation constant of any other mode and varies over a small fractional
range only. Then the sum of terms due to all other modes will be regarded
as a constant over the range of P considered. It will also be assumed that
the phase velocities corresponding to P and Pi are small compared with
the speed of light. Thus, (6.47) and (6.47a) are replaced by (7.1), where the
first term represents the excitation of the Pi mode and the second term repre-
sents the excitation of passive and "non-synchronous" modes. In another
sense, this second term gives the field produced by the electrons in the ab-
sence of a wave propagating on the circuit, or, the field due to the "space
charge" of the bunched electron stream. Equation (7.1) is the equation for
the distributed circuit of Fig. 7.1. This is like the circuit of Fig. 2.3 save for
the addition of the cajmcitances C'l between the transmission circuit and
the electron beam. We see that, because of the presence of these capaci-
tances, the charge of a bunched electron beam will produce a field in addi-
tion to the field of a wave traveling down the circuit. This circuit is intui-
tively so appealing that it was originally thought of by guess and justified
later.
Equation (7.1), or rather its alternative form, (7.7), which gives the volt-
age in terms of the impressed charge density, can be combined with the
390
EQUATIONS FOR TRAVELING-WAVE TUBE
391
ballistical equation (2.22), which gives the charge density in terms of the
voltage, to give (7.9), which is an equation for the propagation constant.
The attenuation, the difference between the electron velocity and the phase
velocity of the wave on the circuit in the absence of electrons and the dif-
ference between the propagation constant and that for a wave traveling
with the electron speed are specified by means of the gain parameter C
and the parameters d, b and b. It is then assumed that J, b and b are around
unity or smaller and that C is much smaller than unity. This makes it pos-
sible to neglect certain terms without serious error, and one obtains an
equation (7.13) for b.
In connection with (7.7) and Fig. 7.1, it is important to distinguish be-
tween the circuit voltage Vc , corresponding to the first term of (7.7), and
the total voltage V acting on the electrons. These quantities are related
by (7.14). The a-c velocity v and the convection current i are given within
the approximation made (C « 1) by (7.15) and (7.16).
C, PER
METER
Fig. 7.1
7.1 Approxim.^te Circuit Equation'
From (6.47) we can write for a current / = / and a summation over n
modes
£. = (l/2)(r -f I5l)i E
(£V/3'i')„rl
" (r; + ^oKK - n
(6.47a)
This has a number of poles at F = F,, . We shall be interested in cases
in which F is very near to a particular one of these, which we shall call
Fi . Thus the term in the expansion involving Fi will change rapidly with
small variations in F. Moreover, even if {Er/^-P)i and Fi have very small
real components, FI — F- can be almost or completely real for values of F
which have only small real components. Thus, one term of the expansion,
that involving Fi , can go through a wide range of phase angles and magni-
tudes for very small fractional variations in F, fractional variations, as it
turns out, which are of the order of C over the range of interest.
The other modes are either passive modes, for which even in a lossy
circuit {E}/ff^P)n is almost purely imaginary, and F„ almost purely real,
392 BELL SYSTEM TECHNICAL JOURNAL
or they afe active modes which are considerably out of synchronism with
the electron velocity. Unless one of these other active modes has a propaga-
tion constant V,, such that ] {Vi — r2)/ri | is so small as to be of the order
of C, the terms forming the summation will not vary very rapidly over the
range of variation of T which is of interest.
We will thus write the circuit equation in the approximate form
E
2(ri - r') ~ coC]
(7.1)
Here there has been a simplification of notation. E is the z component
of electric field, as in Chapter II, and is assumed to vary as exp(—Tz).
{E?/^^P) is taken to mean the value for the Fi mode. It has been assumed
that jSo is small compared with | Ti | and | F- |, and /So has been neglected
in comparison with these quantities.
Further, it has been pointed out that for slightly lossy circuits, {E?/^"^?)
will have only a small imaginary component, and we will assume as a valid
approximation that (E^/^^P) is purely real. We cannot, however, safely
assume that Fi is purely imaginary, for a small real component of Fi can
aflfect the value of Fi — F- greatly when F is nearly equal to Fi .
The first term on the right of (7.1) represents fields associated with the
active mode of the circuit, which is nearly in synchronism with the elec-
trons. We can think of these fields as summing up the effect of the elec-
trons on the circuit over a long distance, propagated to the point under
consideration.
The term (— jTVcoCi) in (7,1) sums up the effect of all passive modes
and of any active modes which are far out of synchronism with the elec-
trons. It has been written in this form for a special purpose; the term will
be regarded as constant over the range of F considered, and Ci will be given
a simple physical meaning.
This second term represents the field resulting from the local charge den-
sity, as opposed to that of the circuit wave which travels to the region
from remote points. Let us rewrite (7.1) in terms of voltage and charge
density
dV
E= -^ = TV (7.2)
dz
From the continuity equation
i = (jWr)p (2.18)
-M\{E?/^^'P)
V =
_ 2(F1 - r) ^ c
']
+ n\p (7.3)
EQUA TIONS FOR TRA VELING-WA VE TUBE 393
We see that Ci has the form of a capacitance per unit length. We can, for
instance, redraw the transmission-Hne analogue of Fig. 2.3 as shown in Fig.
7.1. Here, the current / is still the line current; but the voltage V acting on
the beam is the line voltage plus the drop across a capacitance of Ci farads
per meter.
Consider as an illustration the case of unattenuated waves for which
Ti = i/3i (7.5)
r = i/3 (7.6)
where /3i and /3 are real. Then
" = L 203? - If) + cj " (^-^
In (7.7), the first term in the brackets represents the impedance pre-
sented to the beam by the "circuit"; that is, the ladder network of Figs.
2.3 and 7.1. The second term represents the additional impedance due to
the capacitance Ci , which stands for the impedance of the nonsynchronous
modes. We note that if /3 < ft , that is, for a wave faster than the natural
phase velocity of the circuit, the two terms on the right are of the same
sign. This must mean that the "circuit" part of the impedance is capacitive.
However, for /3 > ft , that is, for a wave slower than the natural phase veloc-
ity, the first term is negative and the "circuit" part of the impedance is
inductive. This is easily explained. For small values of ^ the wavelength of
the impressed current is long, so that it flows into and out of the circuit at
widely separated points. Between such points the long section of series
inductance has a higher impedance than the shunt capacitance to ground;
the capacitive effect predominates and the circuit impedance is capacitive.
However, for large values of ^ the current flows into and out of the circuit
at points close together. The short section of series inductance between
such points provides a lower impedance path than does the shunt capaci-
tance to ground; the inductive impedance predominates and the circuit
impedance is inductive. Thus, for fast waves the circuit appears capacitive
and for slow waves the circuit appears inductive.
Since we have justified the use of the methods of Chapter II within the
limitations of certain assumptions, there is no reason why we should not
proceed to use the same notation in the light of our fuller understanding.
We can now, however, regard V not as a potential but merely as a convenient
variable related to the field by (7.2).
From (2.18) and (7.3) we obtain
rrr.(g/<3'P jv^.
" = L2(r;-r') - <oC, J • ^^*
394
BELL SYSTE}f TECHNICAL JOURNAL
We use this together with (2.22)
t =
(2.22)
We obtain the overall equation
1
iVoU^e - r)
"rri(£V/3'p)
L 2(ri - r)
coCi_
(7.9)
In terms of the gain parameter C, which was defined in Chapter II,
we can rewrite (7.8)
(i/3e -
C' = (£'V/32p)(/o/8Fo)
{Ti - r
a:C,{E-/l3'P)
(2.43)
(7.10)
We will be interested in cases in which Y and Fi differ from 13^ by a small
amount only. Accordingly, we will write
(7.11)
(7.12)
The propagation constant F describes propagation in the presence of
electrons. A positive real value of 8 means an increasing wave. A positive
imaginary part means a wave traveling faster than the electrons.
The propagation constant Fi refers to propagation in the circuit in the
absence of electrons. A positive value of b means the electrons go faster
than the undisturbed wave. A positive value d means that the wave is an
attenuated wave which decreases as it travels.
If we use (7.11) and (7.12) in connection with (7.10) we obtain
[1 + C(2j8 - a')][l + C(b - >/)]
8 =
[-b + jd + j8 + Cijbd
byi + dyi + 5V2)]
_ 4/3. [(1 + C{2j8 - C8'')\C
a;Ci(£V^-^)
(7.13)
We will now assume that | 5 | is of the order of unity, that | b | and | d \
range from zero to unity or a little larger, and that C <5C 1 . We will then neg-
lect the parentheses multiplied by C\ obtaining
1
{-h+jd+j8)
4QC
Q =
a;Ci(£V^''^)
(7.14)
(7.15)
EQUATIONS FOR TRAVELING-WAVE TUBE 395
The quantity wCj has the dimensions of admittance per unit length,
^e has the dimensions of (length)"^ and (E-/'iS'P) has the dimensions of
impedance. Thus, () is a dimensionless parameter (the space-charge param-
eter) which may be thought of as relating to the impedance parameter
{E^/^-P) associated with the synchronous mode the impedance (/3,./coCi),
attributable to all modes but the synchronous mode.
At this point it is important to remember that there are not only two im-
pedances, but two voltage components as well. Thus, in (7.8), the first
term in the brackets times the current represents the "circuit voltage",
which we may call \\ . The second term in the brackets represents the
voltage due to space charge, the voltage across the capacitances Ci . The
two terms in the brackets are in the same ratio as the two terms on the right
of (7.14), which came from them. Thus, we can express the circuit com-
ponent of voltage Vc in terms of the total voltage V acting on the beam either
from (7.8) as
-['-
MTl - r^) V y (7.16)
a,Ciri(£2/^2p)J
or, alternatively, from (7.14) as
Fc = [1 - 4QCi-b + jd + j8)]-' V (7.17)
From Chapter II we have relations for the electron velocity (2.15) and
electron convection current (2.22). If we make the same approximations
which were made in obtaining (7.14), we have
{juoC/v)v = J (7.18)
0
(-2VoC'/I)i = '- (7.19)
We should remember also that the variation of all quantities with z
is as
^-;^.y,c«. (720)
The relations (7.18)-(7.19) together with (2.36), which tells us that the
characteristic impedance of the circuit changes little in the presence of
electrons if C is small, sum up in terms of the more important parameters
the linear operation of traveling-wave tubes in which C is small. The param-
eters are: the gain parameter C, relative electron velocity parameter b,
circuit attenuation parameter d and space-charge parameter Q. In follow-
396
BELL SYSTEM TECHNICAL JOURNAL
ing chapters, the practical importance of these parameters in the opera-
tion of traveUng-wave tubes will be discussed.
There are other effects not encompassed by these equations. The effect
of transverse electron motions is small in most tubes because of the high
focusing fields employed; it will be discussed in a later chapter. The dif-
ferences between a field theory in which different fields act on different elec-
trons and the theory leading to (7.14)-(7.20), which apply accurately
only when all electrons at a given ^-position are acted on by the same field,
will also be discussed.
CHAPTER VIII
THE NATURE OF THE WAVES
Synopsis of Chapter
TN this chapter we shall discuss the effect of the various parame-
-*• ters on the rate of increase and velocity of propagation of the three
forward waves. Problems involving boundary conditions will be deferred
to later chapters.
The three parameters in which we are interested are those of (7.13),
that is, b, the velocity parameter, d, the attenuation parameter and QC,
the space-charge parameter. The fraction by which the electron velocity is
greater than the phase velocity for the circuit in the absence of electrons
is bC. The circuit attenuation is 54.6 dC db/ wavelength. Q is a factor de-
pending on the circuit impedance and geometry and on the beam diameter.
For a helically conducting sheet of radius a and a hollow beam of radius
Ui , Q can be obtained from Fig. 8.12.
The three forward waves vary with distance as
-JPt(l-VC)z BeXCz
i3« = -
Wo
Thus, a positive value of y means a wave which travels faster than the
electrons, and a positive value of x means an increasing wave. The gain in
db per wavelength of the increasing waves is BC, and B is defined by (8.9).
Figure 8.1 shows x and y for the three forward waves for a lossless circuit
{d — 0). The increasing wave is described by .vi , vi . The gain is a maximum
when the electron velocity is equal to the velocity of the undisturbed wave,
or, when b = 0. For large positive values of b (electrons much faster than
undisturbed wave), there is no increasing wave. However, there is an in-
creasing wave for all negative values of b (all low velocities). For the increas-
ing wave, yi is negative; thus, the increasing wave travels more slowly
than the electrons, even ivhen the electrons travel more slowly than the circuit
wave in the absence of electrons. For the range of b for which there is an
increasing wave, there is also an attenuated wave, described by .To = — Xi
and 72 = yi • There is also an unattenuated wave described by y3(.V3 = 0).
For very large positive and negative values of b, the velocity of two
of the waves approaches the electron velocity (y approaches zero) and the
397
398 BELL SYSTEM TECH MCA L JOLRSAL
velocity of the third wave approaches the velocity of the circuit wave in the
absence of electrons (y approaches minus b). For large negative values of
b, Xi , Vi and .vo , y-i become the "electron" waves and Vs becomes the "cir-
cuit" wave. For large values of b, Vi and y^ become the "electron" waves and
yo becomes the "circuit" wave. The "circuit" wave is essentially the wave
in the absence of electrons, modified slightly by the presence of a non-syn-
chronous electron stream. The "electron waves" represent the motion of
"bunches" along the electron stream, slightly affected by the presence of
the circuit.
Figures 8.2 and 8.3 indicate the effect of loss. Loss decreases the gain of
the increasing wave, adds to the attenuation of the decreasing wave and
adds attenuation to the wave which was unattenuated in the lossless case.
For large positive and negative values of b, the attenuation of the circuit
wave (given by .V3 for negative values of b and .V2 for positive values of b)
approaches the attenuation in the absence of electrons.
Figure 8.4 shows B, the gain of the increasing wave in db per wavelength
per unit C. Figure 8.5 shows, for b = 0, how B varies with d. The dashed
line shows a common approximation: that the gain of the increasing wave
is reduced by ^ of the circuit loss. Figure 8.6 shows how, for b = 0, Xi ,
X2 and .V3 vary with d. We see that, for large values of d, the wave described
by .V2 has almost the same attenuation as the wave on the circuit in the
absence of electrons.
Figures 8.7-8.9 show .v, y for the three waves with no loss ((/ = 0) but
with a-c space charge taken into account {QC 7^ 0). The immediately
striking feature is that there is now a minimum value of b below which
there is no increasing wave.
We further note that, for large negative and positive values of 6, y for
the electron waves approaches ±2 \/QC. In these ranges of b the electron
waves are dependent on the electron inertia and the field produced by a-c
space charge, and have nothing to do with the active mode of the circuit.
As QC is made larger, the value of b for which the gain of the increasing
wave is a maximum increases. Now, C is proportional to the cube root of
current. Thus, as current is increased, the voltage for maximum gain of the
increasing wave increases. An increase in optimum operating voltage with
an increase in current is observed in some tubes, and this is at least i)artly
explained by these curves.* There is also some decrease in the maximum
value of X\ and hence of B as QC is increased. This is shown more clearly in
Fig. 8.10.
If X and B remained constant when the current is varied, then tlie gain
per wavelength would rise as C, or, as the \ power of current. However,
* Other factors include a possible lowering of electron speed because of d-c space
charge, and boundary condition eflects.
THE NATURE OF THE WAVES 399
we see from Fig. 8.10 that B falls as QC is increased. The gain per wave-
length varies as BC and, because Q is constant for a given tube, it varies as
BQC. In Fig. 8.11, BQC, which is proportional to the gain per wavelength
of the increasing wave, is plotted vs QC, which is proportional to the \
power of current. For very small values of current (small values of QC),
the gain per wavelength is proportional to the \ power of current. For
larger values of QC, the gain per wavelength becomes proportional to the
J power of current.
It would be difficult to present curves covering the simultaneous eflfect
of loss {d) and space charge (QC). As a sort of substitute, Figs. 8.13 and 8.14
show dxi/dd for (/ = 0 and b chosen to maximize Xi , and dxi/d{QC) for
QC = 0 and h = 0. We see from 8.13 that, while for small values of QC
the gain of the increasing wave is reduced by \ of the circuit loss, for large
values of QC the gain of the increasing wave is reduced by ^ of the circuit
loss.
8.1 Effect of Varying the Electron Velocity
Consider equation (7.13) in case d = 0 (no attenuation) and () = 0
(neglect of space-charge). We then have
b\b^jb)= -j (8.1)
Here we will remember that
l^e = WUo (8.2)
-Fi = -j0e{l + Cb) - ->/^-l (8.3)
Here z'l is the phase velocity of the wave in the absence of electrons, and Uo
is the electron speed. We see that
«o = (1 + Cb)vi (8.4)
Thus, (1 + Cb) is the ratio of the electron velocity to the velocity of the
undislurbed wave, that is, the wave in the absence of electrons. Hence, b
is a measure of velocity difference between electrons and undisturbed wave.
For b > 0, the electrons go faster than the undisturbed wave; for Z> < 0
the electrons go slower than the undisturbed wave. For b = 0 the electrons
have the same speed as the undisturbed wave.
li b = 0, (8.1) becomes
8' = -j (8.5)
which we obtained in Chapter II.
In dealing with (8.1), let
d = x+ jy
400 BELL SYSTEM TECHNICAL JOURNAL
The meaning of this will be clear when we remember that, in the pres-
ence of electrons, quantities vary with z as (from (7.10))
-;/3e(l+;C5)z
If V is the phase velocity in the presence of electrons, we have
coA = (a,/«o)(l - Cy) (8.7)
If Cy « 1, very nearly
V = Mo(l + Cy) (8.8)
In other words, if y > 0, the wave travels faster than the electrons; if
y < 0 the wave travels more slowly than the electrons.
From (8.6) we see that, if x > 0, the wave increases as it travels and if
.T < 0 the wave decreases as it travels. In Chapter II we expressed the
gain of the increasing wave as
BCN dh
where N is the number of wavelengths. We see that
B = 2()(2x)(logioe)x
B = 54.5x-
In terms of x and y, (8.1) becomes
(^2 _ y2^(y ^ b) -{- 2x^y +1 = 0 (8.10)
xix^ - Sy'^ - 2yb) = 0 (8.11)
We see that (8.11) yields two kinds of roots: those corresponding to
unattenuated waves, for which x = 0 and those for which
x'' = 3y2 + 2yb (8.12)
li X = 0, from (8.10)
f(y + 6) = 1
(8.13)
6 = -y + l/y^
If we assume values of y ranging from perhaps -|-4 to —4 we can find the
corresponding values of b from (8.13), and plot out y vs b for these unattenu-
ated waves.
For the other waves, we substitute (8.12) into (8.10) and obtain
2yb^ + Sy^b + 8/ + 1 = 0 (8.14)
(8.9)
THE NATURE OF THE WAVES
401
This equation is a quadratic in b, and, by assigning various values of y,
we can solve for b. We can then obtain x from (8.12).
In this fashion we can construct curves of x and y vs b. Such curves are
shown in Fig. 8.1.
VVe see that for
b < i3/2){2y"
there are two waves for which ;v ^ 0 and one unattenuated wave. The in-
creasing and decreasing waves (.r 5^ 0) have equal and opposite values of
X, and since for them y < 1, they travel more slowly than the electrons,
even when the electrons travel more slowly than the imdisturbed wave. It can be
Fig. 8.1 — The three waves vary with distance as exp (— J/3e + j0eCy + ^tCx)z. Here
the x's and y's for the three waves are shown vs the velocity parameter b for no attenua-
tion {d = 0) and no space charge {QC = 0).
shown that the electrons must travel faster than the increasing wave in
order to give energy to it.
For b > (3/2) (2) , there are 3 unattenuated waves: two travel faster
than the electrons and one more slowly.
For large positive or negative values of b, two waves have nearly the
electron speed (| y \ small) and one wave travels with the speed of the un-
disturbed wave. We measure velocity with respect to electron velocity.
Thus, if we assigned a parameter y to describe the velocity of the undis-
turbed wave relative to the electron velocity, it would vary as the 45°
hne in Fig. 8.1.
The data expressed in Fig. 8.1 give the variation of gain per wavelength
of the undisturbed wave with electron velocity, and are also useful in fitting
402 BELL SYSTEM TECHNICAL JOURNAL
boundary conditions; for this we need to know the three x's and the three
In a tube in which the total gain is large, a change in 6 of ± 1 about b =
0 can make a change of several db in gain. Such a change means a difference
between phase velocity of the undisturbed wave, i\ , and electron velocity
Uo by a fraction approximately ±C. Hence, the allowable difference between
phase velocity i\ of the undisturbed wave, which is a function of frequency,
and electron velocity, which is not, is of the order of C.
8.2 Effect of Attenuation
If we say that J ?^ 0 but has some small positive value, we mean that the
circuit is lossy, and in the absence of electrons the voltage decays with
distance as
Hence, the loss L in db/wavelength is
L = 20(27r)(logioe)Cr/
(8.15)
L = 54.5C(/ db/wavelength
or
d = .01836 {L/C) (8.16)
For instance, for C = .025, d — \ means a loss of \.^6 db wavelength.
If we assume d 9^ 0 we obtain the equations
(^2 _ y)(^ + 6) + 2.rv(.v + J) + 1 = 0 (8.17)
(x2 - /)(.v -{- d) - 2xy{y + b) = 0 (8.18)
The equations have been solved numerically for d = .5 and </ = 1, and the
curves which were obtained are shown in Figs. 8.2 and 8.3. We see that for
a circuit with attenuation there is an increasing wave for all values of b
(electron velocity). The velocity parameters yi and y-y are now distinct for
all values of b.
We see that the ma.ximum value of Xi decreases as loss is increased. This
can be brought out more clearly by showing .Vi vs b on an expanded scale.
It is perhaps more convenient to plot B, the db gain per wavelength per
unit C, vs 6, and this has been done for various values of d in Fig. 8.4.
We see that for small values of d the maximum value of .Vi occurs very
near to b = 0. If we let b = 0 in (8.17) and (8.18) we obtain
y{x^ - /) + 2xy{x + (/) -h 1 = 0 (8.19)
THE NATURE OF THE WAVES
{x^ - /)(.v + d) - 2xy2 = 0
We can rewrite (8.20) in the form
1/2
^ (l+d/xV"
J
\
N
Cl=0.5
y FOR N>N^
UNDISTURBED-'^ ^^ •
WAVE \^
^v^
\ "
^•^•*^
^
L ^3
,
''' i \
—
.
--^=z:
--CC
-S
:^
=—
— =
\\
fyp
\
^
4
\
Fig. 8.2 — The .r's and v's for a circuit with attenuation {d = .5).
403
(8.20)
(8.21)
Fig. 8.3 — The .v's and 3''s for a circuit with attenuation {d = 1).
If we substitute this into (8.19) we can solve for .v in terms of the parame-
ter d/x
a: = +
/3 + d/x\
\1 + d/x)
1/2
,3 + d/x
+ 1 + d/
x\
1/3
(8.22)
404
BELL SYSTEM TECHNICAL JOURNAL
Here we take both upper signs or both lower signs in (8.21) and (8.22).
If we assume d/x « 1 and expand, keeping no powers of d/x higher than
the first, we obtain
x= + (a/3/2)(1 - (l/3((//x))
(8.23)
The plus sign will give .Vi , which is the x for the increasing wave. Let JCjo
be the value of .Ti for J = 0 (no loss).
XiQ
= V3/2
(8.24)
6 = 0y
0.5
^
<d
1
^
^=
s
^
^
\v
::::::;
^:>--
-5-4-3-2-1 0 1 2 3 4 5
b
Fig. 8.4 — The gain of the increasing wave is BCN db, where A'^ is the number of wave-
lengths.
Then for small values of d
xi = .^10(1 - (l/3)(J/xio))
^"1 == ^10 ~ 1/3^/
(8.25)
This says that, for small losses, the reduction of gain of the increasing wave
from the gain in db for zero loss is \ of the circuit attenuation in db. The
reduction of net gain, which will be greater, can be obtained only by match-
ing boundary conditions in the presence of loss (see Chapter IX).
In Fig. 8.5, B = 54.6 Xi has been plotted vs d from (8.22). The straight
line is for Xio = d/3.
In Fig. 8.6, —Xi , x^ and .T3 have been plotted vs d for a large range in d.
As the circuit is made very lossy, the waves which for no loss are unattenu-
ated and increasing turn into a pair of waves with equal and opposite small
attenuations. These waves will be essentially disturbances in the electron
stream, or space-charge waves. The original decreasing wave turns into a
wave which has the attenuation of the circuit, and is accompanied by small
disturbances in the electron stream.
THE NATURE OF THE WAVES
8.3 Space-Charge Effects
405
J Suppose that we let d, the attenuation parameter, be zero, but consider
cases in which the space-charge parameter QC is not zero. We then obtain
GAIN WITHOUT LOSS __
LESS 1/3 OF LOSS
"v-^^
-^^^
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
d
Fig. 8.5 — For h = 0, that is, for electrons with a velocity equal to the circuit phase
velocity, the gain factor B falls as the attenuation parameter d is increased. For small
values of d, the gain is reduced by \ of the circuit loss.
2.5
1.0
/
/
y
^.
-x^^
</
/'
<.
/
-2c_^
0 0.6 1.0 1.5 2.0 2.5 3.0 3.5 4.0
d
Fig. 8.6 — How the three x's vary for 6 = 0 and for large losses.
the equations
{x^ - f-){h + );)-+- 2.^:23, -f \qc{h + v) + 1 = 0
»[(^' - /) - 2>'Cy + ^) + <2C] = 0
(8.26)
(8.27)
Solutions of this have been found by numerical methods for QC = .25,
,5 and 1; these are shown in Figs. 8.7-8.9.
406
BELL SYSTEM TECHNICAL JOURNAL
We see at once that the electron velocity for maximum gain shifts mark-
edly as QC is increased. Hence, the region around & = 0 is not in this case
worthy of a separate investigation.
\
^l
QC =0.25
V
"yi^
^
r
—
j^
N
_^
•
K
-~-
-^
)
yi,y2
<
yi
\
y2
\
Fig. 8.7 — The .t's and y's for the three waves with zero loss id = Oi hut with space
charge (QC = .25).
2
\
N^i
QC = 0.5
^
^^
^
\,
r
X, J
N
\
^
^^^
J
"^
v^2
^r
yi
yi
-2
"
^
-4
k
Kig. 8.8 The .v's and v's with greater space charge {QC = .5).
It is interesting to {)lot the maximum value of .Vi vs. the j)arameter QL\
This has, in effect, been done in Fig. 8.10, which shows B, the gain in db
per wavelength per unit C, vs. QC.
We can obtain a curve i)roportional to db per wavelength by plotting
BQC vs. QC. {Q is indei)endent of current.) This has been done in Fig.
8.11. I'or QC < 0.025, the gain in dh per wavelength varies lincarlv with
THE NATURE OF THE WAVES
407
QC. Chu and Rydbeck found that under certain conditions gain varies
approximately as the \ power of the current. This would mean a slope of f
on Fig. 8.11. A f power dashed line is shown in Fig. 8.11; it fits the upper
part of the curve approximately.
^
^y3__
QC = 1.0
■~~-
^
^
\,
t
.^
\
\
^
^Z
Ui
jy
^2
yi
"1
V
^
-5-4-3-2-1 0 1 2 3 4 5
b
Fig. 8.9 — The .r's and 3''s with still greater space charge {QC = 1).
40
30
20
10
0
\
\
^
^
0 0.25 0.50 0.75 (.00 1.25 1.50 1.75 2.0
QC
Fig. 8.10 — How the gain factor B decreases as QC is increased, for the value of h which
gives a maximum value of x\ .
If we examine Figs. 8.7-8.9 we tind that for large and small values of b
there are, as in other cases, a circuit wave, for which y is nearly equal to
— b, and two space-charge waves. For these, however, y does not approach
zero.
Let us consider equation (7.13). If b is large, the first term on the right
becomes small, and we have approximately
a = ±j2\/QC (8.28)
408
BELL SYSTEM TECHNICAL JOURNAL
These waves correspond to the space-charge waves of Hahn and Ramo, and
are quite independent of the circuit impedance, which appears in (8.28)
merely as an arbitrary parameter defining the units in which 5 is measured.
Equation (8.28) also describes the disturbance we would get if we shorted
out the circuit by some means, as by adding excessive loss.
Practically, we need an estimate of the value of Q for some typical cir-
cuit. In Appendix IV an estimate is made on the following basis: The helix
60
40
1.0
0.8
^
/
/
y
/
/
^A POWER i
f
//^
1ST POWER,
^
/
>
'/
/
/
/
r
/
/
0.04 0.06
0.1 0.2
QC
0.4 0.6 0.8 1.0
Fig. 8.11— The variation of a quantity proportional to the cube of the gain of the in-
creasing wave (ordinate) with a quantity proportional to current (abscissa). For very
small currents, the gain of the increasing wave is proportional to the \ power of current,
for large currents to the \ power of current.
of radius c is replaced by a conducting cylinder of the same radius, a thin
cylinder of convection current of radius ax and current of i exp{—jl3z) is
assumed, and the field is calculated and identified with the second term on
the right of (7.1). R. C. Fletcher has obtained a more accurate value of Q
by a rigorous method. His work is reproduced in Appendix \T, and in Fig.
1 of that appendix, Pletcher's value of () is compared with the approximate
value of Appendix IV.
In I'ig. 8.12, the value (J(j3, y)'' of Appendix I\' is plotted vs. ya for ai/a
= .9, .8, .7. For fli/a = 1, ^ = (>■ In a typical 4,()()() mc travcHng-wave
THE NATURE OF THE WAVES
409
tube, 70 = 2.8 and C is about .025. Thus, if we take the effective beam
radius as .5 times the helix radius, Q = 5.6 and QC = .14.
We note from (7.14) that Q is the ratio of a capacitive impedance to
{E-/0-P). In obtaining the curves of Fig. 8.12, the value of {E'/^-P) for a
helically conducting sheet was assumed. This is given by (3.8) and (3.9).
If {E^/l3^P) is different for the circuit actually used, and it is somewhat
different, even for an actual helix, Q from Fig. 8.12 should be multiplied
by (E^/lS^P) for the helically conducting sheet, from (3.8) and (3.9), and
divided by the value of {E-/l3~P) for the circuit used.
600
400
200
100
80
60
1
1.0
0.8
0.6
-
/ /
' /
/
-
O"/
//
/
/
J' /
T J
V /
f >/
/
/
-
/ /
/
/
-
/ /
/ ^
/
/
^ / /
'/
<bj/
/
T/
/.
y
///
V
/^
^
^
-
//>
/ /
y
''
-
///
/
y
0^
-
//y
/ .
/
^
/.
/
//
y
y^
0^2^
^
/ / y
y/
X
^^-^^
- ////
//
/^
^
7a
Fig. 8.12 — Curves for obtaining Q for a helically conducting sheet and a hollow beam.
The radius of the helically conducting sheet is a and that of the beam is a\. .
8.4 Differential Relations
It would be onerous to construct curves giving 5 as a function of h for
many values of attenuation and space charge. In some cases, however,
useful information may be obtained by considering the effect of adding a
small amount of attenuation when QC is large, or of seeing the effect of
space charge when QC is small but the attenuation is large. We start with
(7.13)
410
BELL SYSTEM TECHNICAL JOURNAL
Let us first differentiate (7.13) with respect to 5 and d
-j dd - j db
2b db =
{-b+jd-\-jbr-
(8.29)
-0.5
-0.4
^^ -0-3
-0.2
-O.t
0
/^
f
0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.0
QC
Fig. 8.13 — h curve giving the rate of change of x\ with attenuation parameter d for
J = 0 and for various values of the space-charge parameter QC. For small values of QC
the gain of the increasing wave is reduced by \ of the circuit loss; for large values of QC
the gain of the increasing wave is reduced by \ of the circuit loss.
-0.4
^-0.8
O
2-
-1.2
-1.6
-2.0
"
■
"~~~~'
^~-
-
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
d
Fig. 8.14 — A curve showing the variation of .vi with QC for QC = 0 and for various
values of the attenuation parameter d.
By using (7.13) we obtain
db =
-j2b
- 1
dd
(8.30)
(6^ -h AQCy
If we allow d to be small, we can use the values of b of Figs. 8.7-8.9 to \)\oi
the quantity
Re(dbi/dd) = dxjdd (8.31)
THE NATURE OF TEE WAVES 411
vs. QC. In Fig. 8.13, this has been done for b chosen to make Xi a maximum.
We see that a small loss dd causes more reduction of gain as QC is increased
(more space charge).
Let us now differentiate (7.13) with respect to QC
25 ^5 = f A Z^- f ^ ^.^. - 4 d{QC) (8.32)
{-h -\- J d + jbf
By using (7.13) with QC = 0 we obtain
In Fig. 8.14, dx/d{QC) has been plotted vs. d iox b ^ 0.
We see that the reduction of gain for a small amount of space charge
becomes greater, the greater the loss is increased {d increased).
Both Fig. 8.13 and Fig. 8.14 indicate that for large values of QC or d the
gain will be overestimated if space charge {QC) and loss {d) are considered
separately.
CHAPTER IX
DISCONTINUITIES
Synopsis of Chapter
WE WANT TO KNOW the overall gain of traveling-wave tubes. So
far, we have evaluated only the gain of the increasing wave, and we
must find out how strong an increasing wave is set up when a voltage is
applied to the circuit.
Beyond this, we may wish for some reason to break the circuit up into
several sections having different parameters. For instance, it is desirable
that a traveling-wave tube have more loss in the backward direction than it
has gain in the forward direction. If this is not so, small mismatches will
result either in oscillation or at least in the gain fluctuating violently with
frequency. We have already seen in Chapter VHI the effect of a uniform
loss in reducing the gain of the increasing wave. We need to know also the
overall effect of short sections of loss in order to know how loss may best
be introduced.
Such problems are treated in this chapter by matching boundary con-
ditions at the points of discontinuity. It is assumed that there is no re-
flected wave at the discontinuity. This will be very nearly so, because the
characteristic impedances of the waves differ little over the range of loss
and velocity considered. Thus, the total voltages, a-c convection currents
and the a-c velocities on the two sides of the point of discontinuity are set
equal.
For instance, at the beginning of the circuit, where the unmodulated elec-
tron stream enters, the total a-c velocity and the total a-c convection cur-
rent— that is, the sums of the convection currents and the velocities for the
three waves — are set equal to zero, and the sum of the voltages for the three
waves is set equal to the applied voltage.
For the case of no loss (d = 0) and an electron velocity equal to circuit
phase velocity (b = 0) we And that the three waves are set up with equal
voltages, each ^ of the applied voltage. The voltage along the circuit will
then be the sum of the voltages of the three waves, and the way in which
the magnitude of this sum varies with distance along the circuit is shown in
Fig. 9.1. Here C.V measures distance from the beginning of the circuit and
the amplitude relative to the applied voltage is measured in db.
The dashed curve represents the voltage of the increasing wave alone.
412
DISCONTINUITIES 413
For large values of CN corresponding to large gains, the increasing wave
predominates and we can neglect the effect of the other waves. This leads
to the gain expression
G = A^- BCN db
Here BCN is the gain in db of the increasing wave and A measures its ini-
tial level with respect to the applied voltage.
In Fig. 9.2, A is plotted vs. b for several values of the loss parameter d.
The fact that A goes to oo for c? = 0 as 6 approaches (3/2) (2) does not
imply an infinite gain for, at this value of 6, the gain of the increasing wave
approaches zero and the voltage of the decreasing wave approaches the
negative of that for the increasing wave.
Figure 9.3 shows how A varies with d for b = 0. Figure 9.4 shows how A
varies with QC ior d = 0 and for b chosen to give a maximum value of B
(the greatest gain of the increasing wave).
Suppose that for b = QC = 0 the loss parameter is suddenly changed from
zero to some finite value d. Suppose also that the increasing wave is very
large compared with the other waves reaching the discontinuity. We can
then calculate the ratio of the increasing wave just beyond the discon-
tinuity to the increasing wave reaching the discontinuity. The solid line of
Fig. 9.5 shows this ratio expressed in decibels. We see that the voltage of
the increasing wave excited in the lossy section is less than the voltage of
the incident increasing wave.
Now, suppose the waves travel on in the lossy section until the increasing
wave again predominates. If the circuit is then made suddenly lossless, we
find that the increasing wave excited in this lossless section will have a
greater voltage than the increasing wave incident from the lossy section,
as shown by the dashed curve of Fig. 9.5. This increase is almost as great as
the loss in entering the lossy section. Imagine a tube with a long lossless
section, a long lossy section and another long lossless section. We see that
the gain of this tube will be less than that of a lossless tube of the same
total length by about the reduction of the gain of the increasing wave in
lossy section.
Suppose that the electromagnetic energy of the circuit is suddenly ab-
sorbed at a distance beyond the input measured by CN. This might be
done by severing a helix and terminating the ends. The a-c velocity and
convection current will be unaffected in passing the discontinuity, but the
circuit voltage drops to zero. For d = b = QC = 0, Fig. 9.6 shows the
ratio of Vi , the amplitude of the increasing wave beyond the break, to
V, the amplitude the increasing wave would have had if there were no break.
We see that for CN greater than about 0.2 the loss due to the break is not
414 BELL SYSTEM TECHNICAL JOURNAL
serious. For CN large (the break far from the input) the loss approaches
3.52 db.
Beyond such a break, the total voltage increases with CN as shown in
Fig. 9.7, and from CN = 0.2 the circuit voltage is very nearly equal to the
voltage of the increasing wave.
Often, for practical reasons loss is introduced over a considerable distance,
sometimes by j)utting lossy material near to a helix. Suppose we use CN
computed as if for a lossless section of circuit as a measure of length of
the lossy section, and assume that the loss is great enough so that the circuit
voltage (as opposed to that produced by space charge) can be taken as zero.
Such a lossy section acts as a drift space. Suppose that an increasing wave
only reaches this lossy section. The amplitude of the increasing wave ex-
cited beyond the lossy section in db with respect to the amplitude of the in-
creasing wave reaching the lossy section is shown vs. CN, which measures
the length of the lossy section, in Fig. 9.8.
9.1 General Boundary Conditions
We have already assumed that C is small, and when this is so the charac-
teristic impedance of the various waves is near to the circuit characteristic
impedance A'. We will neglect any reflections caused by differences among
the characteristic impedances of the various waves.
We will consider cases in which the circuit is terminated in the -{-z direc-
tion, so as to give no backward wave. We will then be concerned with the
3 forward waves, for which 8 has the values 8i , 80 , Sj and the waves repre-
sented by these values of 8 have voltages Vi , V2 , V^ , electron velocities
Vi , '02 , V3 and convection currents ii , i-i , i-i .
Let V, V, i be the total voltage, velocity and convection current at 2 = 0.
Then we have
V, + V,+ V^= V (9.1)
and from (7.15) and (7.16),
Oi 62 03
Fi V2 Vs
T^ + J + tI= (-2FoCV/o)e (9.3)
oi do 03
These equations yield, when solved,
Vi = \V - (60 -f 8,)(juuC v)v + 8M-2V0C- fo)i]
[(1 - 62/5,)(l - 8,8,)r'
We can ol)tain the corresponding expressions for V^ and K^ sim[)ly by inter-
DISCONTINUITIES 415
changing subscripts; to obtain Vn , for instance, we substitute subscript
2 for 1 and subscript 1 for 2 in (9.4).
9.2 Lossless Helix, Synchronous Velocity, No Space Change
Suppose we consider the case in which b = d = Q = 0, so that we have
the values of 5 obtained in Chapter II
6, = e-'""' = V3/2 - il/2
5, = e~''''" - - -v/3/2 - ./1/2 (9.5)
§3 = e/T/2 ^ j
Suppose we inject an unmodulated electron stream into the helix and
apply a voltage V. The obvious thing is to say that, at ^ = 0, r = / = 0.
It is not quite clear, however, that v = 0 Sit z — 0 (the beginning of the
circuit). Whether or not there is a stray field, which will give an initial
velocity modulation, depends on the type of circuit. Two things are true,
however. For the small values of C usually encountered such a velocity
modulation constitutes a small efTect. Also, the fields of the first part of
the helix act essentially to velocity modulate the electron stream, and hence
a neglect of any small initial velocity modulation will be about equivalent
to a small displacement of the origin.
If, then, we let v = i = Q and use (9.4) we obtain
V, = V[{1 - V5i)(l - 5,,/5:)]-' (9.6)
Fi = V/3 (9.7)
Similarly, we tind that
F2 = Vs= V/3 (9.8)
We have used T' to denote the voltage at 2 — 0. Let V^ be the voltage at z.
We have
V.= {V/3)e'-'^'''-''' (1 + 2 cosh ({V3/2)0eCz)e-''"''^''') ^"^""^^
From this we obtain
I V,/V [' = (1/9)[1 + 4 cosh2(V3/2)j8,Cz
+ 4 cos {3/2)l3eCz cosh {\/3/2)l3eCz]
(9.10)
We can express gain in db as 10 logio | V^/V |-, and, in Fig. 9.1, gain in db
is plotted vs CN, where N is the number of cycles.
We see that initially the voltage does not change with distance. This is
natural, because the electron stream initially has no convection current,
416
BELL SYSTEM TECHNICAL JOURNAL
and hence cannot act on the circuit until it becomes bunched. Finally, of
course, the increasing wave must predominate over the other two, and the
slope of the line must be
B = 47.3/CN
(9.11)
The dashed line represents the increasing wave, which starts at V^/V =
I (—9.54 db) and has the slope specified by (9.11). Thus, if we write for the
increasing wave that gain G is
G = A-{- BCN db
(9.12)
,^
/
y
/
/
f
ASYMPTOTIC ^y
EXPRESSION 7y
V
/
/
/
*
y
0 0.1 0.2 0.3 0.4 0.5 0.6
CN
Fig. 9.1 — How the signal level varies along a traveling-wave tube for the special case
of zero loss and space charge and an electron velocity equal to the circuit phase velocity
(solid curve). The dashed curve is the level of the increasing wave alone, which starts
off with \ of the applied voltage, or at —9.54 dh.
This is an asymptotic expression for the total voltage at large values of e,
where | F, | » | Fo |, | F., |, and for ^» = J = () = 0
(9.13)
A= - 9.54 db
B = 47.3
We see that (9.11) is pretty good for C.V > .4, and not too bad for C'.V > .2.
9.3 Loss IN Helix
In Chapter VIII, curves were given for 5i , 62 , b^ vs. b for QC = 0 and for
</, the loss parameter, equal to 0, 0.5 and 1. From the data from which these
curves were derived one can calculate the initial loss parameter by means
of (9.6)
A = 2()log,n| V^/V
(9.14)
DISCONTINUITIES
417
A 0
J
^
^
V
^
—
1
^
^
^"^
^
"-^
Fig. 9.2^When the gain is large we need consider the increasing wave only. Using
this approximation, the gain in db is .1 + BCN db. Here A is shown vs the velocity param-
eter b, several values of the attenuation parameter d, for no space charge (QC = 0).
-1 /
-16
-15
-14
-13
-12
-n
-10
/
/
/
/
/
/
/
/
/
/
^
/
/
/
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
d
Fig. 9.3— .4 vs ^ for 6 = 0 and QC = 0.
In Fig. 9.2, A is plotted vs b for these three values of d.
It is perhaps of some interest to plot A vs d iox b — 0 (the electron veloc-
ity equal to the phase velocity of the undisturbed wave). Such a plot is
shown in Fig. 9.3.
418
BELL SYSTEM TECHNICAL JOURNAL
9.4. Space Charge
We will now consider the case in which QC 9^ 0. We will deal with this
case only for d = 0, and for h adjusted for maximum gain per wavelength.
There is a peculiarity about this case in that a certain voltage V is applied
to the circuil at 2 = 0, and we want to evaluate the circuit voltage asso-
ciated with the increasing wave, Vd , in order to know the gain.
Atz = 0,i = 0. Now, the term which multiplies i to give the space-charge
component of voltage (the second term on the right in (7.11)) is the same
for all three waves and hence at 2 = 0 the circuit voltage is the total voltage.
Thus, (9.1)-(9.3) hold. However, after Vi has been obtained from (9.4), with
-2
-"1
-6
■
-8
^
-in
0.2
0.8
1.0
0.4 0.6
QC
Fig. 9.4 — A vs QC for d = 0 and b chosen for maximum gain of the increasing wave.
V = Vi , V — i = 0, then the circuit voltage Vd must be obtained through
the use of (7.14), and the initial loss parameter is
A = 20 1ogio| Vd/V\ (9.15)
By using the apj^ropriate values of 8, the same used in plotting Figs. 8.1
and 8.7-8.9, the loss parameter A was obtained from (9.15) and plotted vs
QC in Fig. 9.4.
9.5 Change in Loss
We might think it undesirable in introducing loss to make the whole
length of the heli.x lossy. P'or instance, we might expect the power output
to be higher if the last part of the helix had low loss. Also, from Figs. 8.2
DISCONTINUITIES 419
and 8.3 we see that the initial loss A becomes higher as d is increased. This
is natural, because the electron stream can act to cause gain only after it is
bunched, and if the initial section of the circuit is lossy, the signal decays
before the stream becomes strongly bunched.
Let us consider a section of a lossless helix which is far enough from the
input so that the increasing wave predominates and the total voltage V can
be taken as that corresponding to the increasing wave
F = Fi (9.16)
Then, at this point
{ju,CH)v = Fi/Si (9.17)
(-2FoCV/o)i- V,/b\. (9.18)
Here 5i is the value for J = 0 (and, we assume, 6 = 0). If we substitute the
values from (9.16) in (9.4), and use in (9.4) the values of 5 corresponding to
b = Q = 0, d 9^ 0, and call the value of Vi we obtain Fi , we obtain the
ratio of the initial amplitude of the increasing wave in the lossy section to
the value of the increasing wave just to the left of the lossy section. Thus,
the loss in the amplitude of the increasing wave in going from a lossless to a
lossy section is 20 logio | V\/Vi | . This loss is plotted vs d in Fig. 8.5.
This loss is accounted for by the fact that | ii/Vi \ becomes larger as the
loss parameter d is increased. Thus, the convection current injected into
the lossy section is insufficient to go with the voltage, and the volt-
age must fall.
If we go from a lossy section {d ^ 0, b — 0) to a lossless section
{d = 0, b = 0) we start with an excess of convection current and | Fi | ,
the initial amplitude of the increasing wave to the right of the discontinuity
is greater than the amplitude | Fi | of the increasing wave to the left. In
Fig. 9.5, 20 logio I Fi/Fi | is plotted vs d for this case also.
We see that if we go from a lossless section to a lossy section, and if the
lossy section is long enough so that the increasing wave predominates at
the end of it, and if we go back to a lossless section at the end of it, the net
loss and gain at the discontinuities almost compensate, and even for d — i
the net discontinuity loss is less than 1 db. This does not consider the re-
duction of gain of the increasing wave in the lossy section.
9.6 Severed Helix
If the loss introduced is distributed over the length of the helix, the gain
will decrease as the loss is increased (Fig. 8.5). If, however, the loss is dis-
tributed over a very short section, we easily see that as the loss is increased
more and more, the gain must approach a constant value. The circuit will
420
BELL SYSTEM TECHNICAL JOURNAL
be in effect severed as far as the electromagnetic wave is concerned, and any
excitation in the outi)ut will be due to the a-c velocity and convection current
of the electron stream which crosses the lossy section.
We will first idealize the situation and assume that the helix is severed
and by some means terminated looking in each direction, so that the voltage
falls from a value V to a value 0 in zero distance, while v and i remain un-
changed.
We will consider a case in which b — d = Q = 0, and in which a voltage
y
y
.'-'
^r ^
LOSS IN GOING FROM
LOSSLESS TO LOSSY
-/::-'
/
/
"^-.^ GAIN IN GOING FROM
~ LOSSY TO LOSSLESS
/
/
0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2
d
Fig. 9.5 — Suppose that the circuit loss parameter changes suddenly with distance
from 0 to (i or from d to 0. Suppose there is an increasing wave only incident at the point
of change. How large will the increasing wave beyond the point of change be? These
curves tell (6 = gc = 0).
V is applied to the helix iV wavelengths before the cut. Then, just before
the cut,
and
(jUoC/ri)vi = Vi/8i
(-2VoC'/h)h = V,/8l
etc.
(9.19)
(9.20)
DISCONTINUITIES
421
Whence, just beyond the break which makes V = Q,V,v and i are
F = 0
{ju,C/-n)v = Fi/5i + F2/52 + F3/53
(-2FoCV/o)/" = Fi/5I + V,/bl + F3/53'
(9.21)
Putting these values in (9.4), we can find V\ , the value of the increasing wave
to the right of the break. The ratio of the magnitude of the increasing wave
to the magnitude it would have if it were not for the break is then | Vi/ Vi \ ,
and this ratio is plotted vs CN in Fig. 9.6, where N is the number of wave-
lengths in the first section.
0.2
Fig. 9.6 — Suppose the circuit is severed a distance measured by CN beyond the input,
so that the voltage just beyond the break is zero. The ordinate is the ratio of the ampli-
tude of the increasing wave beyond the break to that it would have had with an unbroken
circuit (6 = QC = 0).
We see that there will be least loss in severing the helix for CN equal to
approximately j. From Fig. 9.1, we see that at CN = j the voltage is just
beginning to rise. In a typical 4,000 megacycle traveling-wave tube, CN is
approximately unity for a 10 inch helix, so the loss should be put at least
2.5" beyond the input. Putting the loss further on changes things little;
asymptotically, | Fi/F | approaches f , or 3.52 db loss, for large values of
CN (loss for from input).
It is of some interest to know how the voltage rises to the right of the cut.
It was assumed that the cut was far from the point of excitation, so that
only increasing wave of magnitude Vi was present just to the left of the cut.
The initial amplitudes of the three waves, Fi , F2 , F3 to the right of the
cut were computed and the magnitude of their sum plotted vs CN as it
varies with distance to the right of the cut. The resulting curve, expressed
in db with respect to the magnitude of the increasing wave Vi just to the
left of the cut, is shown in Fig. 9.7. Again, we see that at a distance CN = \
to the right of the cut the increasing wave (dashed straight line)
predominates.
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BELL SYSTEM TECHNICAL JOURNAL
9.7 Severed Helix With Drift Space
In actually putting concentrated loss in a helix, the loss cannot be con-
centrated in a section of zero length for two reasons. In the first place,
this is physically diilficult if not impossible; in the second place it is desirable
that the two halves of the helix l)e terminated in a refiectionless manner at
the cut, and it is easiest to do this by tapering the loss. For instance, if the
loss is put in by spraying aquadag (graphite in water) on ceramic rods sup-
porting the helix, it is desirable to taper the loss coating at the ends of the
lossy section.
Perhaps the best reasonably simple approximation we can make to such a
lossy section is one in which the section starts far enough from the input
35
30
~g 20
I-
10
5
0
/
,/i
/
y.
y
yf
/'
/
/
y/
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
CN
Fig. 9.7 — Suppose that the circuit is severed and an increasing wave only is incident
at the tjreak. How does the signal build up beyond the break? The solid curve shows
(6 = QC = 0). 0 dl) is the level of the incident increasing wave.
so that at the beginning of the lossy section only an increasing wave is
present. In the lossy section CA" long we will consider that the loss com-
pletely shorts out the circuit, so that (8.28) holds. Thus, in the lossy section
we will have onlv two values of 5, whicli we will call hi and hu .
8i = jk
5// = —jk
k = iVc
(9.21)
(9.22)
(9.23)
Let Vj and l'// be the voltages of the waves corresponding to 5/ and 6,i
at the beginning of the lossy section. Let 5| , 62 , 5:i be the values of 8 to the
left and right of the lossy section. Let I'l be the amplitude of the increasing
DISCONTINUITIES 423
wave just to the left of the lossy section. Then, by equating velocities and
convection currents at the start of the lossy section, we obtain
Fi/5x = Vi/8r + Vn^Sjj (9.24)
and, from (9.21) and (9.22)
Vr'8, = (-j/k)(Vr- Vn) (9.25)
Similarlv
So that
Vi/8l = Vi/b)j+ Vnb]i
V,/8\ = -{\^}^{Vj+ Vr
(9.26)
(9.32)
Vi = j{VJ2){k/b,){jk/h + 1) (9.27)
Vu = j{Vy/2){k/b,){jk/h - 1) (9.28)
At the output of the lossy section we have the voltages Vi and Vu
V'jj = Vne-^'^'^e-'''"'''' (9.30)
Thus, at the end of the lossy section we have
V = Vr+ V'u (9.31)
(;-«oC/77)z; = V'jlh + V'nihn
(jUoC/v)v= (-j/k){Vr- V'n)
and similarly
{-2V£Vlo)i = {-\/h?){y'j + v'n) (9.33)
From (9.27) and (9.28) we see that
y'j -f v'u = -{k/b^\+{k:b^ cos lirkCX + sin 27ry^C.Y] Fif-^^x.v (934)
V'j - v'u = j{k bi)\-{k bi) sin lirkCX + cos 2x;feCY]^i<^" '-'•'' (9.35)
Whence
V = -(k/bi)[+{k/bi) coslirkCX + smlTkCNWie-''^'^'' (9.36)
(jmC/v)v = (l'bi)[-{k,bi) sin IwkCX + COS 27rK'-V]lV-^-'-'' (9.37)
(-2FoC7/o)i = (l/5i)[(l/5i) cos IirkCX + (1 k) sin 2TrkCX]Vie~ ''"''■''' (9.38)
These can be used in connection with (9.4) in obtaining \\ , the value of
Vi just beyond the lossy section; that is, the amplitude of the component of
increasing wave just beyond the lossy section.
424
BELL SYSTEM TECHNICAL JOURNAL
In typical traveling-wave tubes the lossy section usually has a length
such that CV is j or less. In Fig. 9.8 the loss in db in going through the lossy
section, 20 logio | Fi/Fi | , has been plotted vs. CX for QC - 0, .25, .5 for
the range CV = 0 to CN = .5.
We see that, for low space charge, increasing the length of a drift space
increases the loss. For higher space charge it may either increase or decrease
the loss. It is not clear that the periodic behavior characteristic of the curves
for QC = 0.5 and 1, for instance, will obtain for a drift space with tapered
loss at each end. The calculations may also be considerably in error for
broad electron beams (7a large). The electric field pattern in the helLx differs
-2
""■^,
>
QC = 1
■>*.__
,^'
■'\
0.50/
/
J
\
0.25""
/
■/
s
^<_
y
0.2 03
CN
Fig. 9.8 — Suppose that we break the circuit and insert a drift tube of length measured
by CN in terms of the traveling-wave tube C and N . Assume an increasing wave only
before the drift tube. The increasing wave beyond the drift tube will have a level with
respect to the incident increasing wave as shown by the ordinate. Here d = 0 and h is
chosen to maximize X\ .
from that in the drift space. In the case of broad electron beams this may
result in the excitation in the drift space of several different space charge
waves having different field patterns and different propagation constants.
A suggestion has been made that the introduction of loss itself has a bad
effect. The only thing that affects the electrons is an electric field. Unpub-
lished measurements made by Cutler mode by moving a probe along a helix
indicate that in typical short high-loss sections the electric field of the
helix is essentially zero. Hence, except for a short distance at the ends,
such lossy sections should act simply as drift spaces.
9.8 Overall Behavior of Tubes
The material of Chapters VTII and IX is useful in designing traveling-
wave tubes. Prediction of the performance of a given tube over a wide range
of voltage and current is quite a different matter. For instance, in order to
predict gain for voltage or current ranges for which the gain is small, the
DISCONTINUITIES 425
three waves must be taken into account. As current is varied, the loss param-
eter d varies, and this means different x's and ^''s must be computed for
different currents. Finally, at high currents, the space-charge parameter Q
must be taken into account. In all, a computation of tube behavior under a
variety of conditions is an extensive job.
Fortunately, for useful tubes operating as intended, the gain is high.
When this is so, the gain can be calculated quite accurately by asymptotic
relations. Such an overall calculation of the gain of a helix-type tube with
distributed loss is summarized in Appendix VII.
CHAPTER X
NOISE FIGURE
Synopsis of Chapter
BECAUSE THERE IS no treatment of the behavior at high frequencies
of an electron flow with a Maxwellian distribution of velocities, one
might think there could be no very satisfactory calculation of the noise figure
of traveling-wave tubes. Various approximate calculations can be made,
and two of these will be discussed here. Experience indicates that the second
and more elaborate of these is fairly well founded. In each case, an approxi-
mation is made in which the actual multi-velocity electron current is re-
placed by a current of electrons having a single velocity at a given point but
having a mean square fluctuation of velocity or current equal to a mean
square fluctuation characteristic of the multi-velocity flow.
In one sort of calculation, it is assumed that the noise is due to a current
fluctuation equal to that of shot noise (equation (10.1)) in the current enter-
ing the circuit. For zero loss, an electron velocity equal to the phase velocity
of the circuit and no space charge, this leads to an expression for noise figure
(10.5), which contains a term proportional to beam voltage Vn times the
gain parameter C. One can, if he wishes, add a space-charge noise reduction
factor multiplying the term 80 I'oC. This approach indicates that the voltage
and the gain per wavelength should be reduced in order to improve the noise
figure.
In another approach, equations applying to single-valued-velocity flow
between parallel planes are assumed to apply from the cathode to the cir-
cuit, and the fluctuations in the actual multi-velocity stream are repre-
sented by fluctuations in current and velocity at the cathode surface. It is
found that for space-charge-limited emission the current fluctuation has no
effect, and so all the noise can be expressed in terms of fluctuations in the
velocity of emission of electrons.
For a special case, that of a gun with an anode at circuit potential I'o ,
a cathode-anode transit angle ^i , and an anode-circuit transit angle ^-j , an
expression for noise figure (10.28) is obtained. This expression can be re-
written in terms of a parameter L which is a function of P
/' = 1 + (i)(4-7r)(r,/r)(i/c)L
P = (di - do)C
426
NOISE FIGURE 427
Formally, F can be minimized by choosing the proper value of P. In Fig.
10.3, the minimum value of L, Lm , is plotted vs. the velocity parameter b
for zero loss and zero space charge (d = QC = 0). The corresponding value
of P, Pm , is also shown.
P is a function of the cathode-anode transit angle di , which cannot be
varied without changing the current density and hence C, and of anode-
circuit transit angle 6-i , which can be given any value. Thus, P can be made
very small if one wishes, but it cannot be made indefinitely large, and it is
not clear that P can always be made equal to P,„ . On the other hand, these
expressions have been worked out for a rather limited case: an anode po-
tential equal to circuit potential, and no a-c space charge. It is possible
that an optimization with respect to gun anode potential and space charge
parameter QC would predict even lower noise figures, and perhaps at attain-
able values of the parameters.
In an actual tube there are, of course, sources of noise which have been
neglected. Experimental work indicates that partition noise is very im-
portant and must be taken into account.
10.1 Shot Noise in the Injected Current
A stream of electrons emitted from a temperature-limited cathode has a
mean square fluctuation in convection current i\
T\ = 2ehBo (10.1)
Here e is the charge on an electron, /o is the average or d-c current and B i^
the bandwidth in which the frequencies of the current components whose
mean square value is il lie. Suppose this fluctuation in the beam current of
a traveUng-wave tube were the sole cause of an increasing wave
(F = V = 0). Then, from (9.4) the mean square value of that increasing
wave,, V'ls, would be
K = {8eBVlc'/Io) I 5253 H (1 - 52/5i)(l - 53/5i) \~' (10.2)
Now, suppose we have an additional noise source: thermal noise voltage
applied to the circuit. If the helix is matched to a source of temperature T,
the thermal noise power Pt drawn from the source is
Pt = kTB (10.3)
Here k is Boltzman's constant, T is temperature in degrees Kelvin and, as
before, B is bandwidth in cycles. If A'^ is the longitudinal impedance of the
circuit the mean square noise voltage Vl associated with the circuit will be
F? = kTBK( (10.4)
428 BELL SYSTEM TECHNICAL JOURNAL
and the component of increasing wave excited by this voltage, V\t , will be,
from (9.4),
Vlt = kTBKt I (1 - h/b,){\ - 8,/8i} I -2 (10.5)
The noise figure of an amplifier is defined as the ratio of the total noise
output power to the noise output power attributable to thermal noise at the
input alone. We will regard the mean-square value of the initial voltage Vi
of the increasing wave as a measure of noise output. This will be substantially
true if the signal becomes large prior to the introduction of further noise.
For example, it will be substantially true in a tube with a severed helix if
the helix is cut at a point where the increasing wave has grown large com-
pared with the original fluctuations in the electron stream which set it up.
Under these circumstances, the noise figure F will be given by
F = \^ {e/kT)(8Vlc'/IoKt) | 6,8, \^ (10.3)
Now we have from Chapter II that
C' = loKc/^Vo
whence
F = 1 + 2(eVo/kT)C I 5253 |- (10.4)
The standard reference temperature is 290°K. Let us assume b = d ^
QC = 0. For this case we have found | So | = | Ss | = 1. Thus, for these as-
sumptions we find
F = 1 + 807oC (10.5)
A typical value of Vo is 1,600 volts; a typical value of C is .025. For these
values
F = 3,201
In db this is a noise of 35 db.
This is not far from the noise figure of traveling-wave tubes when the
cathode temperature is lowered so as to give temperature-limited emission.
The noise figure of traveling-wave tubes in which the cathode is at normal op-
erating temperature and is active, so that emission is limited by space-charge,
can be considerably lower. In endeavoring to calculate the noise figure for
space-charge-limited electron flow from the cathode we must proceed in a
somewhat different manner.
NOISE FIGURE
429
10.2 The Diode Equations
Llewellyn and Peterson have published a set of equations governing the
behavior of parallel plane diodes with a single-valued electron velocity.
They sum up the behavior of such a diode in terms of nine coefficients A *-/*,
in the following equations
V^- Va = A* I + B* qa + C*r„ (10.6)
qb= D* I + E* qa + /''*Z'„ (10.7)
V, = G* I + H* qa+ I*v., (10.8)
VOLTAGE DIFFEReNCE_
(Vb-Va)
CURRENT DENSITY
lo + I
INPUT CONVECTION
CURRENT DENSITY
l^D + qa
INPUT VELOCITY
LLa+Va
OUTPUT CONVECTION
CURRENT DENSITY
ID + Qb
OUTPUT VELOCITY
Llb+Vb
a b
Fig. 10.1 — Parallel electron flow between two planes a and b normal to the flow, show-
ing the currents, velocities and voltages.
These equations and the values of the various coefficients in terms of cur-
rent, electron velocity and transit angle are given in Appendix V. The diode
structure to which they apply is indicated in Fig. 10.1. Electrons enter nor-
mal to the left plane and pass out at the right plane. The various quantities
involved are transit angle between the two planes and:
It) d-c current density to left
/ a-c current density to left
qa a-c convection current density to left at input plane a
qb a-c convection current density to left at output plane b
Ua d-c velocity to right at plane a
Ub d-c velocity to right at plane b
I'a a-c velocity to right at plane a
Vb a-c velocity to right at plane b
Vb-Va a-c potential difference between plane b and {)lane a
' F. B. Llewellyn and L. C. Peterson, "Vacuum Tube Networks," Proc. I.R.E., Vol.
32, pp. 144-166, March, 1944.
430 BELL SYSTEM TECHNICAL JOURNAL
We will notice that / and the g's are current densities and that, contrary
to the convention we have used, they are taken as positive to the left. Thus,
if the area is a, we would write the output convection current; as
i — —aqh
where qb is the convection current density used in (10.6) -(10.8).
Peterson has used (10.6)-(10.8) in calculating noise figure by replacing
the actual multi-velocity flow from the cathode by a single-velocity flow
with the same mean square fluctuation in velocity, namely,^
t,2 ^ (4 - 7r)77 {kTjh)B (10.9)
Here Tc is the cathode temperature in degrees Kelvin and h is the cathode
current.
Whatever the justification for such a procedure. Rack has shown that it
gives a satisfactory result at low frequencies, and unpublished work by
Cutler and Quate indicates surprisingly good quantitative agreement under
conditions of long transit angle at 4,000 mc.
We must remember, however, that the available values of the coeffi-
cients of (10.6)-(10.8) are for a broad electron beam in which there are
a-c fields in the z direction only. Now, the electron beam in the gun of a
traveling-wave tube is ordinarily rather narrow. While the a-c fields may
be substantially in the 2-direction near the cathode, this is certainly not
true throughout the whole cathode-anode space. Thus, the coefficients
used in (10.6)-(10.8) are certainly somewhat in error when applied to
traveling-wave tube guns.
Various plausible efforts can be made to amend this situation, as, by
saying that the latter part of the beam in the gun acts as a drift region in
which the electron velocities are not changed by space-charge fields. How-
ever, when one starts such patching, he does not know where to stop. In
the light of available knowledge, it seems best to use the coefficients as they
stand for the cathode-anode region of the gun.
Let us then consider the electron gun of the traveling-wave tube to form
a space-charge limited diode which is short-circuited at high frequencies.
If we assume com.j)lete space charge (space-charge limited emission)
and take the electron velocity at the cathode to be zero, we find that the
quantities multiplying q^ in (10.6)-(10.8) are zero.
/?* = £*=. //* = 0* (10.10)
^ L. C. Peterson, "Sjiace-Chargc and Transit-Time KtTecls on Sifjnal and Noise in Mi-
crowave Tetrodes," Proc. l.R.E., Vol. 35, pp. 1264-1272, Novcmt)er, 1947.
^ A. J. Rack, "Effect of Space Charge and Transit Time on the Shot Noise in Diodes,"
Bell System Technical Journal, Vol. 17, pp. 592-619, October, 1938.
NOISE FIGURE
431
Accordingly, the magnitude of the noise convection current at the cathode
does not matter. If we assume that the gun is a short-circuited diode as far
as r-f goes
V,- Va = 0
Then from (10.6), (10.10) and (10.11) we obtain
/ =
Va
(10.11)
(10.12)
2.0
\
1
, D*C*
r
F* A*
*
*< ..5
\
o
\
*
♦
\
o
/
y^
*
<
*
*
/
D
LL
/
1 "'^
/
/
I* A*
0
/
1
Fig. 10.2 — Some expressions useful in noise calculations, showing how they approach
unity at large transit angles.
Accordingly, from (10.7) and (10.8) we obtain
^^= 1
^'5 = 1 -
G*C*
F*Va
I*Va
(10.13)
(10.14)
In Fig. 10.2, I 1 - D*C*/F*A* j and | 1 - G*C*/I*A* \ are plotted
vs d, the transit angle. We see that for transit angles greater than about
iiT these quantities differ negligibly from unity, and we may write
More specifically, we find
qb
Qb = F*Va
Vb = I*Va
Vg h jQi e~^'
Ub
Vb — —Vae
(10.15)
(10.16)
(10.17)
(10.18)
432 BELL SYSTEM TECHNICAL JOURNAL
Here /3i is 7" times the transit angle in radians from cathode to anode. For
Va we use a velocity fluctuation with the mean-square value given by (10.9).
Suppose now that there is a constant-potential drift space following the
diode anode, of length fii/j in radians. If we apply (10.6)-(10.8) and assume
that the space-charge is small and the transit angle long, we find that qb ,
the value of qb at the end of this drift space, is given in terms of qa and Va ,
the values at the beginning of this drift space, by
q'l. = iq'a + {h/ubW'a)e~^' (10.19)
The case of Vb , the velocity at the end of this drift space, is a little dif-
ferent. The first term on the right of (10.8) can be shown to be negligible
for long transit angles and small space charge. The last term on the right
represents the purely kinematic bunching. For the assumption of small
space charge the middle term gives not zero but a first approximation of a
space-charge effect, assuming that all the space-charge field acts longitu-
dinally. Thus, this middle term gives an overestimate of the effect of space-
charge in a narrow, high-velocity beam. If we include both terms, we ob-
tain
Vb = Htq'a + e~%!, (10.20)
Here the term on the right is the purely kinematic term.*
Now, the current from the gun is assumed to go into the drift space,
so that qa is qb from (10.17) and Va is Va from (10.18). The d-c velocity at
the gun anode and throughout the drift space are both given by Ub . If
we make these substitutions in (10.19) and (10.20) we obtain
q'b = {lo/ubm - 0,)e~'^'^^'\'a (10.21)
,',= -(2|;+ l)e-'^'+^^%„ (10.22)
The term l/Si/lS^ in (10.22) is the "space-charge" term. We will in the fol-
lowing analysis omit this, making the same sort of error we do in neglecting
space charge in the traveling-wave section of the tube. If space charge in
the drift space is to be taken into account, it is much better to proceed as
in 9.7.
From the drift-space the current goes into the helix. U is now necessary
to change to the notation we have used in connection with the traveling-
wave tube. The chief difference is that we have taken currents as positive
to the right, but allowed h to be the d-c current to the left. If i and v are
* The first term has been written as shown because it is easiest to use the small space-
charge value of //* for the drift region (//*) in connection with the space-charge limited
value of /•"* for the cathode-anode region rather than in connection with (10.17).
NOISE FIGURE 433
our a-c convection current and velocity at the beginning of the hehx, and
/o and Mo the d-c beam current and velocity, and a- the area of the beam,
t = —crqb
V ^ Vb
(10.23)
/o = alo
Mo = Wo
In addition, we will use transit angles di and 62 in place of /3i and iSo
(10.24)
02 = jd2
We then obtain from (10.21) and (10.22)
q = -j{h/u,){d, - 62)6- ''''^''\a (10.25)
V = -e~^'''^''\'a (10.26)
10.3 Overall Noise Figure
We are now in a position to use (9.4) in obtaining the overall noise figure.
We have already assumed that the space-charge is small in the drift space
between the gun anode and the hehx (QC = 0). If we continue to assume
this in connection with (9.4), the only voltage is the helix voltage and for
the noise caused by the velocity fluctuation at the cathode, I'a , F = 0 at
the beginning of the helix. Thus, the mean square initial noise voltage of
the increasing wave, Ff, , will be, from (10.21), (10.22), (9.4) and (10.9),
Vl = (2(4 - 7r)kT,CBVo/Io)\ S^dsidi - d.X + (52 + h) |-
I (1 - V6i)(i - h/h) r'
(10.27)
As before, we have, from the thermal noise input to the helix
'Vlt = kTBK(\ (1 - 52/5,)(l - 53/5i) f' (10.5)
and the noise figure becomes
F = \ + V\s/V\t
F = 1 + (i/2)(4 - 7r)(r,/r)(i/c)| 5253(^1 - e2)c + (60 + h) |- (10.28)
Here use has been made of the fact that
C = KJ/Wq
434 BELL SYSTEM TECHNICAL JOURNAL
Let us investigate this for the case b = d = 0 (we have already assumed
QC = 0). In this case
5, = V^/2 - yi/2
^3 = j
and we obtain
F = l-\- (l/2)(4 - 7r)(re/r)(l/C)| {P/2 -V3/2)
(10.29) ;
- i( V3P/2 - 1/2) |2
p ^ (0^- e.)c (10.30)
For a given gun transit-angle 0i , the parameter P can be given values
ranging from diC to large negative values by increasing the drift angle
02 between the gun anode and the beginning of the helix.
We see that
F = I -\- (1/2) (4 - t){T,/T){\/C)(P' - VSP + 1) (10.31)
The minimum value of (P- — \/3P +1) occurs when
P = Vs/2 (10.32)
if the product of the gun transit angle and C is large enough, this can be
attained. The corresponding value of (P- — V^SP + 1) is \, and the cor-
responding noise figure is
^ = 1 + (1/2)(1 - 7r/4)(r,/r)(l/C) (10.33)
A typical value for 7\. is 1()20°A', and for a reference temperature of 290° A',
Tc/T = 3.5
A typical value of C is .025. For these values
F = 17
or a noise figure of 12 db.
Let us consider cases for no attenuation or space-charge but for other
electron velocities. In this case we write, as before
52 = X2 + jy2
53 = a-3 + jy:i
Let us write, for convenience,
L = 1 doSiP + 5i + 52 I' (10.34)
NOISE FIGURE
435
Then we find that
L = [(.V2.V3)- + (y.y,)- + (.V2J3)- + (x,y,y]P^
+ 2[x3(v^ + A-;) + x.(xl + yl)]P
+ (-vo + .V3)- + (y2 + ys)-
This has a minimum value for P — Pm
— [x^ixl + yl) + Xiixl + 3'3)]
P,„ =
(10.35)
(10.36)
(xiXsY + (yoysY + (xzysY + (xs^'o)^
We note that, as we are not deahng with the increasing wave, Xo and .V3
1.0
0.8
0.6
0.1
0.08
0.06
-■" — ■
•~ —
^Pm_
--.
-
~"-~,
^
-
""""
-^.^
-
\^
-
^
\
\
\
Lm^
^
;
"
-
Fig. 10.3 — According to the theory presented, the overall noise figure of a tube with a
lossless helix and no space charge is proportional to L. Here we have a minimum value
of L,a , minimized with respect to P, which is dependent on gun transit angle, and also
the corresponding value of F, Fm . According to this curve, the optimum noise figure
should be lowest for low electron velocities (low values of b). It may, however, be impos-
sible to make F equal to P^ .
must be either negative or zero, and hence Pm is always positive. For no
space-charge and no attenuation, Xs is zero for all values of b and
P _ -^^
-£ m — 2 I 2
^2 + X2
From (10.36) and (10.35), the minimum value of L, Lm , is
Lm = {x2 + XiY + iy-i + jif
_ [x-iiyl + xo) + X2(x3 + yl)Y
{xiXz)- + {y-iy^y + {x'iy-iY + {x^yiY
When X3 - 0, as in (10.37)
2 2
Lm = X2 + ^2 + ^yty-i +
2 , 2
X2 + yt
(10.37)
(10.38)
(10.39)
436 BELL SYSTEM TECHNICAL JOURNAL
In Fig. 10.3, Pm and L„, are plotted vs h for no attenuation {d = 0). We
see that Pm becomes very small as b approaches (3/2)2' ^, the value at which
the increasing wave disappears.
If space charge is to be taken into account, it should be taken into account
both in the drift space between anode and helix and in the helix itself. In
the helix we can express the effect of space-charge by means of the parameter
QC and boundary conditions can be fitted as in Chapter IX. The drift
space can be dealt with as in Section 9.7 of Chapter IX. The inclusion of
the effect of space-charge by this means will of course considerably com-
plicate the analysis, especially if 6 ?^ 0.
While working with Field at Stanford, Dr. C. F. Quate extended the
theory presented here to include the effect of all three waves in the case of
low gain, and to include the effect of a fractional component of beam cur-
rent having pure shot noise, which might arise through failure of space-
charge reduction of noise toward the edge of the cathode. His extended
theory agreed to an encouraging extent with his experimental results.
Subsequent unpublished work carried out at these Laboratories by Cutler
and Quate indicates a surprisingly good agreement between calculations
of this sort and observed noise current, and emphasizes the importance of
properly including both partition noise and space charge in predicting noise
figure.
10.4 Other Noise Considerations
Space-charge reduction of noise is a cooperative phenomenon of the whole
electron beam. If some electrons are eliminated, as by a grid, additional
"partition" noise is introduced. Peterson shows how to take this into
account.'
An electron may be ineffective in a traveling-wave tube not only by being
lost but by entering the circuit near the axis where the r-f field is weak
rather than near the edge where the r-f field is high. Partition noise arises
because sidewise components of thermal velocity cause a fluctuation in the
amount of current striking a grid or other intercepting circuit. If such side-
wise components of velocity appreciably alter electron position in the helix,
a noise analogous to partition noise may arise even if no electrons actually
strike the helix. Such a noise will also occur if the "counteracting pulses"
of low-charge density which are assumed to smooth out the electron flow
are broad transverse to the beam.
These considerations lead to some maxims in connection with low-noise
traveling-wave tubes: (1) do not allow electrons to be intercepted by various
electrodes (2) if practical, make sure that loifir) is reasonably constant over
the beam, and/or (3) provide a very strong magnetic focusing field, so that
electrons cannot move appreciably transversely.
NOISE FIGURE 437
10.5 Noise in Transverse-Field Tubes
Traveling-wave tubes can be made in which there is no longitudinal field
component at the nominal beam position. One can argue that, if a narrow,
well-collimated beam is used in such a tube, the noise current in the beam
can induce little noise signal in the circuit (none at all for a beam of zero
thickness with no sidewise motion). Thus, the idea of using a transverse-
field tube as a low-noise tube is attractive. So far, no experimental results
on such tubes have been announced.
A brief analysis of transverse-field tubes is given in Chapter XIII.
CHAPTER XI j
BACKWARD WAVES j
WE NOTED IN CHAPTER IV that, in filter-type circuits, there is an \
infinite number of spatial harmonics which travel in both directions.
Usually, in a tube which is designed to make use of a given forward com-
ponent the velocity of other forward components is enough different from
that of the component chosen to avoid any appreciable interaction with the
electron stream. It may well be, however, that a backward-traveling com-
ponent has almost the same speed as a forward-traveling component.
Suppose, for instance, that a tube is designed to make use of a given
forward-traveling component of a forward wave. Suppose that there is a
forward-traveling component of a backward wave, and this forward-travel-
ing component is also near synchronism with the electrons. Does this mean
that under these circumstances both the backward-traveling and the for-
ward-traveling waves will be amplitied?
The question is essentially that of the interaction of an electron stream
with a circuit in which the phase velocity is in step with the electrons but
the group velocity and the energy flow are in a direction contrary to that of
electron motion.
We can most easily evaluate such a situation by considering a distributed
circuit for which this is true. Such a circuit is shown in Fig. 11.1. Here the
series reactance A^ per unit length is negative as compared with the more
usual circuit of Fig. 11.2. In the circuit of Fig. 11.2, the phase shift is 0°
per section at zero frequency and assumes positive values as the frequency
is inci;eased. In the circuit of Fig. 11.1 the phase shift is —180° per section
at a lower cutoff frequency and approaches 0° per section as the frequency
approaches infinity.
Suppose we consider the equations of Chapter II. In (2.9) we chose the
sign of X in such a manner as to make the series reactance positive, as in
Fig. 11.2, rather than negative, as in Fig. 11.1. All the other equations apply
equally well to either circuit. Thus, for the circuit of Fig. 11.1, we have, in-
stead of (2.10),
V = (-^, (....) '
The sign is changed in the circuit equation relating the convection current
find the voltage. Similarly, we can modify the equations of Chapter VII,
438
BACKWARD WAVES
439
(7.9) and (7.12), by changing the sign of the left-hand side. From Chapter
VIII, the equation for a lossless circuit with no space charge is
Hd^jb) = -j (8.1)
The corresponding moditication is to change the sign preceding 5'\ giving
S--{S+jb) = +j (11.2)
-H( I l( I l( I l( I [(--
Fig. 11.1 Fig. 11.2
Fig. 11.1 — A circuit with a negative phase velocity. The electrons can be in synchron-
ism with the field only if they travel in a direction opposite to that of electromagnetic
energy flow.
Fig. 11.2 — A circuit with a positive phase velocity.
2.0
,^
N,
A
/
\
V
/
\
1.0
^
X
^
^
^
0.5
n
Fig. 11.3 — Suppose we have a tube with a circuit such as that of Fig. 11.1, in which
the circuit energy is really flowing in the opposite direction from the electron motion.
Here, for QC = rf = 0, we have the ratio of the magnitude of the voltage Vz a distance
z from the point of injection of electrons to the magnitude of the voltage V at the point
of injection of electrons. V, is reallv the input voltage, and there will be gain at values of
6 for which | V^IV\ < 1.
In (11.2), h and 5 have the usual meaning in terms of electron velocity and
propagation constant.
Now consider the equation
^\h-jk) =j (11.3)
Equations (11.2) and (8.1) apply to different systems. We have solutions
of (8.1) and we want solutions of (11.2). We see that a solution of (11.2)
440 BELL SYSTEM TECHNICAL JOURNAL
is a solution of (11.3) for k = —b. We see that a solution of (11.3) is the con-
jugate of a solution of (8.1) if we put b in (8.1) equal to k in (11.3). Thus, a
solution of (11.2) is the conjugate of a solution of (8.1) in which b in (8.1)
is made the negative of the value of b for which it is desired to solve (11.2).
We can use the solutions of Fig. 8.1 in connection with the circuit of
Fig. 11.1 in the following way: wherever in Fig. 8.1 we see b, we write in
instead —b, and wherever we see yi , y-i or y^ we write in instead —y\ ,
—yi or —yz .
Thus, for synchronous velocity, we have
5i = V3/2 + jY^
h= -Vs/2-\-jy2
^3 = -j
We can determine what will happen in a physical case only by fitting
boundary conditions so that at z = 0 the electron stream, as it must, enters
unmodulated.
Let us, for convenience, write $ for the quantity (3Cz
^Cz = $ (11.4)
We will have for the total voltage Vz at z in terms of the voltage F at 2 = 0
V, = Ve~'^'([{l - 52/50(1 - 53/50]~V
+ [(1 - 53/50(1 - 5x/50r'^"'*''^*'' (11.5)
+ [(1 - 5i/53)(l - 52/53)]-i^-^*^'/^')
We must remember that in using values from an unaltered Fig. 8.1 we use
in the 5's and as the y's the negative of the y's shown in the figure (the sign
of the x's is unchanged), and for a given value of b we enter Fig. 8.1 at —b.
In Fig. 11.3, I Vz/V I has been plotted vs 6 for $ = 2. We see that, for
several values of 6, | Fz | (the input voltage) is less than | V \ (the output
voltage) and hence there can be "backward" gain.
We note that as $ is made very large, the wave which increases with
increasing $ will eventually predominate, and | Vz | will be greater than
I F |. "Backward gain" occurs not through a "growing wave" but rather
through a sort of interference between wave components, as exhibited in
Fig. 11.3.
Fig. 11.3 is for a lossless circuit; the presence of circuit attenuation would
alter the situation somewhat.
APPENDIX IV
EVALUATION OF SPACE— CHARGE PARAMETER Q
Consider the system consisting of a conducting cylinder of radius a and
an internal cylinder of current of radius ai with a current
• jut —Tz /i\
te e . (1)
Let subscript 1 refer to inside and 2 to outside. We will assume magnetic
fields of the form
H,, = Ahiyr) (2)
H^2 = BhM + CK,M (3)
From Maxwell's equations we have,
— {rH^) = jwerE, + rJ, (4)
or
Now
f (s/i(.)) = zh{z) (5)
oz
I (zK,{z)) = -zKoiz) (6)
dz
Hence
£.1 = -^ Ahiyr) (7)
£.2 = ^ (5/0(7/-) - CKoiyr)) (8)
coe
at r = a, £22 = 0
at r = fli , Ezi = E;
C = B ^, (9)
Zl
Al^iya^ = B { /o(Tai) — 7^^ — c ^'0(7^1)
Ao(7a)
^0(7^) /o(7ai)
441
(10)
442
BELL SYSTEM TECHNICAL JOURNAL
In going across boundary, we integrate (4) over the infinitesimal radial
distance which the current is assumed to occupy
rdll^ — rJdr
iTrrJdr = i
rjdr =
Thus
(11)
Iirr Zira-i
hiyai) + ~ — ,- A'i(7ai) — /i(7ai) ( 1
B =
B
Ko(ya)
J oil a)
Io{ya)Ko{yai)
Ko(ya)I(,(yai)/_
i Koiya)
A'o(t«) /o(7<Zi)
A'i(7ai) A'oCTai)"^"^
1-Kax I^{yd)Ix{yax) L/)(7<?i) -^o(7<?i)
at r = (7i
-Ezl = -Ej2
Now
Hence
— /7\ / '" \ Koiya) h{yaCj
(joe / \2irai/ Io{ya) Ii(yai)
1 -
loiya) Kniyai)
Kiiyai) Koiyai)
Ki)iya) h{yai)/ \_h{ya\) h{yai) _
1
Vti/i
377
^0
^^V = E, = j ^ ll{yax)G{ya, yax)i
V
^(^^ lliyadG
(ya, yih)ij
G(ya, yai) — 60
Ao(7tf)1
/o(Xa) J
Ao(7ai)
_-^o(7<^i)
In obtaining this form, use was made of the fact that
1
K,{z)h(z) + Koiz)hiz) =
(12)
i
(13)
(14)
(15)
(16)
(17)
APPENDIX IV 443
Now
where (E^/^'^P) is the value of this quantity at r = Ui . In order to evaluate
Q we note that
(1}C\ coCi
A = 1' = ( I ] h
X^Ixb) ^o'^'>'^i^^^'>'^' '>'^i^
(20)
On the axis, (E/^P) has a value (E?/^P)o
At a radius di
(£^/^^» = (^0 (|^y f'^(7a)/5(7ai) (22)
Hence
APPENDIX V
DIODE EQUATIONS
FROM LLEWELLYN AND PETERSON
These apply to electrons injected into a space between two planes a and
b normal to the .v direction. Plan b is in the +.v direction from plane a.
Current density I and convection current q are positive in the — x direction.
The d-c velocities «a , «& and the a-c velocities Va , Vb are in the +.v direction.
T is the transit time. The notation in this appendix should not be con-
fused with that used in other parts of this book. It was felt that it would
be confusing to change the notation in Llewellyn's and Peterson's^ well-
known equations.
Table I
Electro>^ics Equations
Numerics Employed:
r, = 10' - = 1.77 X 10^', e - l/(367r X W) '' = 2 X 10-''
m €
Direct-Current Equations:
Potential- velocity: tjVd ^ {l/2)u- (1)
Space-charge-factor definition: f = 3(1 — To/T))
Distance: x = (1 - ^/3)(ua + Ua)T/2 > (2)
Current density: {y}/t)lD = {ua + Ub)2^/T'' J
Space-charge ratio: lo/Im = (9/4)f(l - f/3)- (3)
Limiting-current density:
r _ 2.33 iVna + Vv^bY (,.
" " 10« X' ~ ^^^
Alternating-Current Equations:
Symbols employed :
0 = ie, d = o}T, i = V^
1 F. B. Llewellyn and L. C. Peterson "Vacuum Tube Networks," Proc. I./i.E., vol 32.
pp. 144-166, March, 1944.
444
APPENDIX V
445
P= \ - e
^^ 2 3^8
2 6 24
5 = 2 - 2r^ - /3 - /3e^'' =
6 12 40 180
General equations for alternating current
q — alternating conduction-current density
II = alternating velocity
n- F„ - A*I + B*qa+C*v,
qb - D*I + E*qa + F*7'a
'"f
(5)
Vb = G*I + iy*^a + I*Vaj
Table II
Values of Alternating-Current Coefficients
1 r 1
A* ^ - Ua -\- Ub — -
e I p
E* = — [ub — ^{ua + Ub)]e
tib
[■-!('
125\
/7=t
B* =
1 T'
K(P - 0Q) - UbP
+ r(«a + tlb)P]
P
€ 2f {Ug + W ^ -^
Ma^ + r(^<a + Ub)P\
C* = - - 2f (Ma + W6) ^„
e 2 Ub
D'
2f
(Wa + Ub) P_
(1 - f)
«6
[a^a — r(z'a + %)]e
Complete space-charge, f = 1.
1 r'
1 7^
e v5p
g* =
UailP - ^Q)
446 BELL SYSTEM TECHNICAL JOURN A L
2 P
■t] /32
€ 2 («a + Mb) -^
7?* =
,2
I
ff* = 0
APPENDIX VI
EVALUATION OF IMPEDANCE AND Q FOR
THIN AND SOLID BEAMS'
Let us first consider a thin beam whose breadth is small enough so that
the field acting on the electrons is essentially constant. The normal mode
solutions obtained in Chapters VI and VII apply only to this case. The more
practical situation of a thick beam will be considered later. The normal mode
method consists of simultaneously solving two equations, one relating the
r-f field produced on the circuit by an impressed r-f current from the electron
stream and the other relating r-f current produced in the electron stream by
an impressed r-f field from the circuit.
We have the circuit equation
and the electronic equation
oK IjQKVn .
i/3e h £ (2)
The solution of these two equations gives T in terms of To , A', and Q, which
must be evaluated separately for the particular circuit being considered.
The field solution is obtained by solving the field equations in various
regions and appropriately matching at the boundaries. For a hollow beam of
electrons of radius b traveling in the z direction inside a helix of radius a and
pitch angle xp, the matching consists of finding the admittances ( W^ I inside
and outside the beam and setting the difference equal to the admittance of
the beam. Thus the admittance just outside the beam for an idealized helix
will be-
V =^ =. -"^ /i(7^>) - SKijyb) . .
' E,o ^ y h{yb) + SKoiyb) '
' This appendix is taken from R. C. Fletcher, "Helix Parameters in Traveling- Wave
Tube Theory," Proc. I.R.E., Vol. 38, pp. 4l3-il7 (1950).
^ L. J. Chu and J. D. Jackson, "Field Theory of TraveUng-Wave Tubes," I.R.E.,
Proc, Vol. 36, pp. 853-863, July, 1948.
O. E. H. Rydbeck, "Theory of the Traveling-Wave Tube," Ericsson Technics, No. 46
pp. 3-18, 1948.
447
448
BELL SYSTEA
where
8 =
1
Y^
cot ^
Kliya)
ya
^1 =
2
and
7—
- r2
-^i.
j Ii{ya)Ki{ya) — Io{ya)Ko{ya)) ,
(The /'s and A''s are modified Bessel functions). The admittance inside the
beam is
V =^' =^— h{yb) , .
' E,i y h{yh)- ^ ^
Boundary conditions require that E^o = Ez^ = Ez and Hzq — Hzi = z-^ .
Combining the boundary conditions, we see that
'''-'''-Li' (5)
where the ratio of ^r is given by (2). Thus the field method gives two equa-
tions which are equivalent to the circuit and electronic equations of the
normal mode method.
A6,l Normal Mode Par.-^meters for TraN Beam
The constants appearing in eq. (1) can be evaluated by equating the cir-
cuit equation (1) to the circuit equation (5). Thus if Yc = Vo — Vi ,
The constants can be obtained by expanding each side of eq. (6) in terms of
the zero and pole occurring in the vicinity of To . Thus if 70 and 7^ are the
zero and pole of Yc , respectively,
Y.^-(y,-yJp) (r^^'), (7)
and the two sides of eq. (6) will be equivalent if
To = -70 - I3l , (8)
-1/2
f = (-f"-^^'
7p — 7o
APPENDIX VI 449
and
7o and yp can be obtained from eqs. (3) and (4) through the implicit equations
(/3flC0t^) - (Toa) r— — — (11)
Ii{yoa)Ki{yoa)
Koiypb) Kliy^a)
ypa
and l/K is found to be
/i(7pa)A'i(7pa.) - h{ypa)Ko{ypa) ,
(12)
_1^ ^ i/i /^l + ^Y^^ ^° hiyoa) hiyoa) _ /o(7oa)
A' ^ y fx\ yl) /o(Toft) Ao(7oa) L-^o(7ofl) /i(7off)
iro(7og) _ A:i(7oa) 4 1
ifi(7oa) Ao(7oa) To a J
(13)
The equations for 70 and A' are the same as those given by Appendix II,
evaluated by solving the iield equations for the helix without electrons pres-
ent. The evaluation of yp , and thus Q, represents a new contribution. Values
(o2\ -1/2
1 + -li I are plotted in Fig. A6.1 as a function of 70a for various
75/
ratios of h/a. (It should be noted that for most practical applications the
(^2\ -1/2
1 + -;; ) is very close to unity, so that the ordinate is prac-
7o/
tically the value of Q itself.)
Appendix IV gives a method for estimating Q based on the solution of
the field equations for a conductor replacing the helix and considering the
liKOV^ ...
resultant field to be ^-— ^ — i. This estimate of Q is plotted as the dashed
Pe
lines of Fig. A6.1.
A6.2 Thick Beam Case
For an electron beam which entirely fills the space out to the radius b,
the electronic equations of both the normal mode method and the field
method are altered in such a way as to considerably complicate the solution.
In order to find a solution for this case some simplifying assumptions must
be made. A convenient type of assumption is to replace the thick beam by
an "equivalent" thin beam, for which the solutions have already been
worked out.
450
BELL SYSTEM TECHXICAL JOURNAL
TJ
Two beams will be equivalent if the value of -=^ is the same outside the
beams, since the matching to the circuit depends only on this admittance.
1000
800
600
400
100
80
60
+ 10
fN|<Q^ 6
o
1.0
0.8
0.6
-
/
/
/
/
-
/
/
/
/
-
J
V
f
J
/
/
/
-
/
/
/
/
/
/
/
«\
/
/
r
/
y
-
1
' /
/
l/
/
/
-
/
/
/
9
/
/
/
-
J
7
r
/
,/
-
/
/;
//
/
/
A
y
^
y-
/
',
//
/
/
/
f'
/
^
-
//
/
/ ,
(*
//
^
'
-
y
0,
/
y^
^^
^
-
/
//
/
'',
f
^
^^
^
-
/
^
^
•
<
^
^'
<^"
//,
////
u
''/.
/
•
^.^
y
^^^
^
0
9
rr^
- /
'/if,
^/y
' /
,^
•^
-»-'
- It
7 }
f//
//
^
^^
f
l/
^^>
^ .
y
V
^
*»'
^^
"-PIERCE'S APPROX.
''/
y
y
.''
y^'
'
^ .
t*
7o^
Fig. A6.1 — Passive mode parameter Q for a hollow beam of electrons of radius h inside
a helix of radius a and natural propagation constant yo . The solid line was obtained by
equating the circuit efjuation of the normal mode method, which defines Q, with a cor-
responding circuit equation found from the field theory method. The dashed line was
obtained in Appendix IV from a solution of the field equations for a conductor replacing
the helix.
The problem, then, of making a thin beam the equivalent of a thick beam
is the problem of arranging the position and current of a thin beam to give
the same admittance at the radius h of the thick beam. This is of course
impossil)lc for all values of 7. Tt is desirable therefore that the admittances
APPENDIX VI
451
be the same close to the complex values of 7 which will eventually solve
the equations.
The solution of the field equations for the solid beam yields the value for
7/(1)
11 ^
at the radius b as
H ^
jcoe nliinyb)
E,
7 h{nyb) '
where
«' = 1 +
1 /m ^eL
i3o y e 2x62
1
(14)
(15)
Thus the electronic equation for the solid beam which must be solved simul-
taneously with the circuit equation (given above by either the normal mode
approximation or the field solution) must be
Y = ^ - Y = i^
nliinyb) h{yb)
h(yb)_
(16)
Complex roots for 7 will be expected in the vicinity of real values of 7
By plotting Ye and Yc vs. real values of
dYe dY
for which Ye ^^ Ye and -^ ^ ~
dy 07
7, it is found that the two curves become tangent close to the value of 7 for
which n = 0, using typical operating conditions (Fig. A6.2). Our procedure
for choosing a hollow beam equivalent of the solid beam, then, will be to
equate the values of Ye and ~-^ at n — 0. This will give us two equations
dy
from which to solve for the electron beam diameter and d-c current for the
equivalent hollow beam.
If the hollow beam is placed at the radius sb with a current of th , the
TT
value of -^ at the radius b gives the value for Yen as
F.ff =
I', =
jcoeb -^ (1
y'b'll{syb)-i\
n)
lljsyb)
n{yb)
Koisyb)
K.iyb)
llo(syb) Io{yb)_\
Equating this with eq. (16) at n = 0 yields the equation
- = ie'n{sd)
'Ko(se) KM
(17)
(18)
452
BELL SYSTEM TECHNICAL JOURNAL
-0.8
- 1 y3p iT
1.4
/
1 '-"= U.i=. —r. =0.02
To a =1.55
b/a =0-55
/
^^
^
y
4
-'^
^
/^
n =0
k
^"■^"^
/
/
"/
1
/
1
YeHfFLETCHER)/
YeH (PIERCE)
1
1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56 1.58 1.60 1.62
7b
3
— )
\
/3ea =
/3. .r
4.0
f:S=
- ' L-S- \\ -^ — — = 0.01
7oa = 4.10
b/a = 0.7
>
Ye.
^^
/yc
/
/
1
y
y
1
11
!n=o
y
/
'
YgH(PIERCE)/
1
t^YgH (FLETCHER)
/
1
1
1
4.00 4.02 4.04 4.06 4.08
4.10
yb
4.12 4.14 4.16 4.18 4.20
Fig. A6.2 — Electronic admittance Ye of a solid electron beam of radius b and circuit
admittance Yc of a helix of radius a plotted vs. real values of the propagation constant
dY,, dYc
y in the vicinity of where . ' = , where complex solutions for y are expected, for two
ay ay
typical sets of operating conditions. Plotted on the same graph is the electron admit-
tance Yen for two equivalent hollow electron beams: the dashed curve (Fletcher) is matched
to F, at n = 0, while the dot-dashed curve (Pierce, Appendix IV) is matched at « = 1
(of! the graph).
APPENDIX VI 453
where 9 = yj) and ye is the value of 7 at « = 0; i.e. for 7e ]:^ )3o
^^^0 ^/3e. (19)
7e = /3e +
In the vicinity of /; = 0, n varies very rapidly with 7, and hence matching
— ^ ) is practically the same as matching -j-^ . With this approximation
eqs. (16) and (17) can be differentiated with respect to n and set equal at
0.6
N
\
\
BA
no c
^
^Dll (
H-
_
\
\
— -
" "
A
v
N
s.
\
V.
\
^f^
1
'0
S
f^
^:^\
X
K
\
7eb
Fig. A6.3 — Parameters of the hollow electron beam which is matched to the solid
electron beam of radius b and current /o at 7 = 7« c^ /3« , where n = Q. sb is the radius
and th is the current of the equivalent hollow beam.
« = 0 to yield the second relation
= d'll(d)ll{sd)
1 ^2.2,.^.2,..^ Ko(se) , K,{er''
t
L h{sd) "^ h{d) J
(20)
Equations (18) and (20) can then be solved to give the implicit equation
for 5 as
K,{sd)
KM , _A_
"T r>r2/
lo(sd) hid) ' 211(d)
and the simpler equation for /
e' nisd)
(21)
(22)
454
BELL SYSTEM TECHNICAL JOURNAL
s and / arc plotted as a function of d in V\g. A6.3. Tlie value of Yen using
these values of i- and / is compared in Fig. A6.2 with Ye in the vicinity of
where I\. is almost tangent to Ye for two typical sets of operating conditions.
1000
800
600
400
100
80
60
40
f?Ic!?; 6
1.0
0.8
0.6
0.4
-
1 ,
t
/
/
-
/
/
/
/
-
/
/
/
/
/
-
/
/
/
/
/
/
/
/
/
/
/
X
-
/ ,
1 o/
/
,/'
-
i
'/
/ ^/
/
,/
-
/
/
/
/
/
/
-
/
',
//
/
/
vr
r
^
y^
^
/
'/,
//
y
/
A
Y^'
-
/
V'
(/
/
y
--^
-
//>
//
/
/
/
^
-
//
w
/
^
-
1
'//,
^
/
^
^
^^
^_U0_
—
M
//
'/
y
^
- ////////
y
-/////// y
-m////
W/
/
III
7
7oa
Fig. A6.4 — Passive mode parameter Qs for a solid beam of electrons of radius h inside
a helix of radius a and natural propagation constant 70 , obtained from the equivalent
hollow beam parameters of Fig. 3 taken at 7^ = 70 . All the normal mode solutions which
have been found*-^' *^' for a hollow beam will be approximately valid for a solitl beam if Q
is replaced l)y Q, and A' is replaced by ATj (Fig. 5).
It is of course |)()ssible to ])ick other criteria for determining an "equiva-
lent" hollow beam. In Chapter XI\', in essence, I',, and I',// were e.xpanded
in terms of (1 — »-) and the coelTicients of the first two terms were equated.
This has been done for the cylindrical beams, and the values of 5 and / found
by this method determine values of Y,n shown in Fig. A6.2. The greater
APPENDIX VI
455
departure from the true curve of Ye would indicate that this approximation
is not as good as that described above.
It is now possible to find the values of Q^ and K^ appropriate to the solid
100
80
60
40
i 06
SI
0.10
0.08
006
0.04
0.02
-
-
k
-
\
-
>
k
1
^s
-
L\\\
-
\
-
\
\,
-
^
\
^
i^
\
\
\
s
-
S, '
S,
^x
-
A
\,
'v,
b.
-
^
>
k
^
1^'
-
\
\\
\
\
^
\
^
N
s.
h
-
\\
\
V
s
-
\
A
^-
\,
-
V
\p
\
s
\,
-
1
\
\
N
\\
\
s
\
7oa
Fig. A6.5 — Circuit impedance K, for a solid beam of electrons of radius b inside a helix
of radius a and natural propagation constant 70 , obtained from the equivalent hollow
£2 (2), (3)
beam parameters of Fig. 3 taken at >« = 70 . Ks should replace K = —^ in order
for the normal mode solutions for a hollow beam to be applicable to a solid beam.
beam. Thus if Q ( 70 a, - 1 and K iyoa, - j are the values for the hollow beam
calculated from eqs. (9), (12) and (13),
Qs = Q{yoa,s-),
(23)
456 BELL SYSTEM TECHNICAL JOURNAL
and'
Ks = lK(yoa,s-Y (24)
The / is placed in front of K in eq. (24) because //o and K appear in the
thin beam solutions only in the combination tloK. Using tK instead of K
allows us to use 7o , the actual value of the current in the solid beam in the
-1/2
70 /, , i8.
solutions instead of tlo , the equivalent current. Values of 0« - ( 1 +
^e \ To'
arnd Ks — . ( 1 -| — \) are plotted vs. 70a inFigs. A6.4 and A6.5 for different
To \ ToV
values of b/a and for values of t and s taken at t« = To • All the solutions
obtained for the hollow beam will be valid for the solid beam if Qs and A'«
are substituted for Q and K.
APPENDIX VII
HOW TO CALCULATE THE GAIN OF A
TRAVELING-WAVE TUBE
The gain calculation presented here neglects the effect at the output of
all waves except the increasing wave. Thus, it can be expected to be ac-
curate only for tubes with a considerable net gain. The gain is expressed
in db as
G = A + BCN (1)
Here A represents an initial loss in setting up the increasing wave and BCN
represents the gain of the increasing wave.
We will modify (1) to take into account approximately the effect of the
cold loss of L db in reducing the gain of the increasing wave by writing
G = A-\- [BCN - aL] (2)
Here a is the fraction of the cold loss which should be subtracted from the
gain of the increasing wave. This expression should hold even for moderately
non-uniform loss (see Fig. 9.5).
Thus, what we need to know to calculate the gain are the quantities
A, B, C, N, a, L
A7.1 Cold Loss L db
The best way to get the cold loss L is to measure it. One must be sure that
the loss measured is the loss of a wave traveling in the circuit and not loss
at the input and output couplings.
A7.2 Length of Circuit in Wavelengths, N
We can arrive at this in several ways. The ratio of the speed of light c to
the speed of an electron Uq is
c_ _ 505
uo ^ VVo ^^^
where Vo is the accelerating voltage. Thus, if ^ is the length of the circuit and
X is the free-space wavelength and X^, is the wavelength along the axis of
457
458 BELL SYSTEM TECHNICAL JOURNAL
the helix
X. = X 2? (4)
C
N = i = il (5)
K 7 «o
Also, if ^w is the total length of wire in the helix, approximately
N = ^^ (6)
A7.3 The Gain Parameter C
The gain parameter can be expressed
1/3 /^Xl/3
-(f;s¥^^^"V
Here A' is the helix impedance properly defined, h is the beam current in
amperes and Vq is the beam voltage.
A 7. 4 Helix Impedance K
In Fig. 5 of Appendix \T, A'(— j(l+f — jj is plotted vs. 70a for
values of b/a. Kg is the effective value of K for a solid beam of radius b, and
a is the radius of the helix. 70 is to be identified with 7 for present purposes,
and is given by
1/2
27r
1 - .
X
2n
7
(8)
where X^ is given in terms of X by (4). We see that in most cases (for voltages
up to several thousand)
(X«/A)'- « 1 (9)
and we may usually use as a valid approximation
27r
7o =
and
(10)
^0
As /So = 27r/X, this approximation gives
7.a = ^-^ (11)
(S)' =
1 + - = 1 +
and we may assume
APPENDIX VII 459
,+i|)y'^i (12)
Thus, we may take Ks as the ordinate of Fig. 5 multiphed by c/u^) , from
(3), for instance.
The true impedance may be somewhat less than the impedance for a
helically conducting sheet. If the ratio of the circuit impedance to that of a
helically conducting sheet is known (see Sections 3 and 4.1 of Chapter III,
and Fig. 3.13, for instance), the value of Ks from Fig. 5 can be multiplied
by this ratio.
A7.5 The Space-Charge Parameter Q
The ordinate of Fig. 4 of Appendix VI shows <3s — ( 1 + ( — ) I vs.
I^e \ VTo/ /
ya for several values of b/a. Here <2,, is the effective value of Q for a solid
beam of radius b. As before, for beam voltages of a few thousand or lower,
we may take
The quantity j8e is just
ft ='^ (13)
and from (8) we see that for low beam voltages we can take
I3e = y = 70
so that the ordinate in Fig. 4 can usually be taken as simply Qs.
A7.6 The Increasing Wave Parameter B
In Fig. 8.10, B is plotted vs. QC. C can be obtained by means of Sections
3 and 4, and Q by means of Section 5. Hence we can obtain B.
A7.7 The Gain Reduction Parameter a
From (2) we see that we should subtract from the gain of the increasing
wave in db a times the cold loss L in db. In Fig. 8.13 a quantity dxi/dd,
which we can identity as a, is plotted vs. QC.
A7.8 The Loss Parameter d
The loss parameter d can be expressed in terms of the cold loss, L in db.
460 BELL SYSTEM TECHNICAL JOURNAL
the length of tlie circuit in wavelengths, X, and C "
d = 0.0183 ^ (15)
I
A7.9 The Initial Loss .4
The quantity A of (2) is plotted vs. d in Fig. 9.3. This plot assumes
QC — 0, and may be somewhat in error. Perhaps Fig. 9.4 can be used in
estimating a correction; it looks as if the initial loss should be less with
QC 9^ 0 even when <i ?^ 0. In any event, an error in .4 means only a few db,
and is hkely to make less error in the computed gain than does an error in
B, for instance.
Technical Publications by Bell System Authors Other Than
in the Bell System Technical Journal
Progress in Coaxial Telephone and Television Systems* L. G. Abraham.^
A.I.E.E., Trans., V. 67, pt. 2, pp. 1520-1527, 1948.
Abstract — This paper describes coaxial systems used in the Bell System
to transmit telephone and television signals. Development of this system
was started some time ago, with systems working before the war between
New York and Philadelphia and later between Minneapolis, Minnesota and
Stevens Point, Wisconsin. Various stages in the progress of this develop-
ment have been described in previous papers and the telephone terminal
equipment has been recently described. This paper will outline how the
system works and discuss some transmission problems, leaving a complete
technical description for a number of later papers.
Use of the Relay Digital Computer. E. G. Andrews and H. W. Bode.^
Elec. Engg., V. 69, pp. 158-163, Feb., 1950.
Abstract — This paper is concerned primarily with the operating features
of the computer and its application to problems of scientific and engineer-
ing interest. The material herein has been derived largely from the experi-
ence gained with one of the computers during a trial period of about 5
months before final delivery. An effort was made during that time to try the
machine out on a variety of difficult computing problems of varying char-
acter to obtain experience in its operation and to establish as well as pos-
sible what its range of usefulness might be.
Longitudinal Noise in Audio Circuits. H. W. Augustadt and W. F. Kan-
NENBERG.i Audio Engg., V. 34, pp. 18-19, Feb., 1950.
Abstract — The words "longitudinal interference" have often been used
to explain the origin of unknown noise in audio circuits with little actual
regard to the source of the interference. In this respect, the usage of these
words is similar to the popular usage of the word "gremlins". We attribute
to gremlins troubles whose causes are unknown without much attempt to
delve deeper into the matter. Similarly in the audio facilities field, many
noise troubles are attributed to "longitudinal interference" or "longitudi-
nals" or even simply "line noise" without a clear understanding of the na-
ture of the trouble or the actual meaning of the terms. The noise trouble,
however, still persists irrespective of the name applied to it until its causes
are thoroughly understood and the correct remedial action is applied. This
*A reprint of this article mav be obtained on request to the editor of the B.S.T.J.
ip.T.L.
461
462 BELL SYSTEM TECHNICAL JOURNAL
paper describes ami illustrates, with representative examples, various types
of common noise induction in order to lead to an understanding of their
nature. The paper includes, in addition, a discussion of simple remedies
which may be employed for representative cases of noise troubles due to
longitudinal induction.
Mobile Radio. A. Bailey.^ A.I.E.E., Trans., V. 67, pt. 2, pp. 923-931,
1948.
Stabilized Permanent Magnets.* V. P. Cioffi.' A.I.E.E.. Trans., V. 67,
pt. 2, pp. 1540-1543, 1948.
Abstract — Permanent magnets are stabilized against forces tending to
demagnetize them, by partial demagnetization. It is shown that, after such
stabilization, the magnet operates at a point on a secondary demagnetiza-
tion curve. This curve may be treated identically as the major demagnetiza-
tion curve is treated in ordinary magnet design problems. Formulas are
developed for determining secondary demagnetization curves from the major
demagnetization curve when stabilization is achieved by magnetization of
the magnet before assembly, and by an applied magnetomotive force after
magnetization in assembly.
It will be shown that, w'hen the magnet is partially demagnetized for the
purpose of stabilization, its operating point lies on a curve which, for con-
venience, will be called a secondary demagnetization curve. The object of
this paper is to discuss the derivation of secondary demagnetization curves
for given conditions of stability against demagnetizing forces and their
applications to magnet design problems.
Relay Preference Lockout Circuits in Telephone Switching.* A. E. Joel,
Jr.! A.I.E.E., Trans., V. 67, pt. 2, pp. 1720 1725, 1948.
Abstract — Occasions arise in telephone switching, particularly at com-
mon controlled stages, where calls compete for the use of equipment com-
ponents or switching linkages. These call requests for service are received
at random by circuits which must choose among and serve them on a
one-at-a-time basis. Circuits which perform this function are known as
"preference lockouts". E.xtensive use has been made of these circuits in
manual, panel, and crossbar switching systems. This paper describes the
design philosoj^hies of relay preference lockout circuits based on some of
these applications.
Piezoelectric Crystals and Their Application, to Ultrasonics. W. P. Mason.'
Book, New York, Van Xostrand, 508 i)ages, 1950.
Television Terminals for Coa.vial Systems.* L. W. Morkisox, Jr.' Elec.
Engg., \. 69, pp. 109 115, I'^ebruary, 1950.
*A repriiU of tliis arlicic ni;i\ l)t' nhiaim-d on rt.'(|iR'st to the ociitor of the H.S.T.f.
' B.T.I..
•^ A. T. & 'I-.
ARTICLES BY BELL SYSTEM AUTHORS 463
Abstract — The broad features of operation of the LI Coaxial System for
the transmission of television have been discussed in a recent paper (L. G.
Abraham, "Progress in Coaxial Telephone and Television Systems", AIEE
Transactions, Vol. 67, pp. 1520-1527, 1948). It is the purpose of this paper
to describe, in somewhat more detail, the factors influencing the design of
the coaxial television terminals and the features of the equipment now in
service in the Bell System's Television Network. The television terminals
here described were placed into network service in 1947, but in basic form
are similar to experimental models developed prior to the war and used in
early television transmission studies over the coaxial cable.
Alternate to Lead Sheath for Telephone Cables. A. Paone.^ Corrosion, V. 6,
pp. 46-50, February, 1950.
Bridge Erosion in Electrical Contacts and Its Prevention* W. G. Pfann.^
A .I.E.E., Trans., V. 67, pt. 2, pp. 1528-1533, 1948.
Abstract — The size of the molten bridge which forms as two contacts
separate depends upon the contact material and the current. The molten
bridge has two diameters, one in each contact. By pairing dissimilar con-
tact materials an asymmetric bridge is created, in which the bridge diam-
eters are unequal and with which is associated a self-limiting transfer
tendency. Under certain conditions the use of unlike pairs can prevent the
continued transfer of material from one contact to the other.
Chess-playing Machine.* C. E. Shannon.^ Sci. Am., V. 182, pp. 48-51,
February, 1950.
Military Teletypewriter Systems of World War II.* F. J. Singer.^ Bibli-
ography. A.I.E.E., Trans., V. 67, pt. 2, pp. 1398-1408, 1948.
Abstract — This paper reviews the evolution of military teletypewriter
communications since 1941 and briefly describes some of the important sys-
tems that were developed during the war by Bell Telephone System engi-
neers for the armed forces.
Optimum Coaxial Diameters.* P. H. Smith. ^ Electronics, V. 23, pp. Hi-
ll 2, 114, February, 1950.
Abstract — The derivation of the optimum ratios is briefly described and
optimum values are indicated to one part in ten thousand. In all cases the
medium between conductors is assumed to be a gas with a dielectric con-
stant approaching unity, and any effect of inner conductor supports upon
the optimum conductor diameter ratio for a given property has been neg-
lected.
General Review of Linear Varying Parameter and Nonlinear Circuit Analy-
sis.* W. R. Bennett. 1 I.R.E., Proc, V. 38, pp. 259-263, March, 1950.
*JA reprint of this article may be obtained on request to the editor of the B. S.T.J.
iB.T.L.
^A. T. &T.
464 BELL SYSTEM TECHNICAL JOURNAL
Abstract — X'ariable and nonlinear systems are classified from the stand-
point of their significance in communication problems. Methods of solution
are reviewed and appropriate references are cited. The paper is a synopsis
of a talk given at the Symposium on Network Theory of the 1949 National
I.R.E. Convention.
Some Early Long Distance Lines in the Far West. W. Blackford, Sr.* and
J. F. HuTTON." Bell Tel. Mag., \. 28, pp. 227-237, Winter, 1949-50.
Radio Propagation Variations at VHF and UHF.* K. Bullington.*
LR.E., Proc., V. 38, pp. 27-32, January, 1950.
Abstract — The variations of received signal with location (shadow losses)
and with time (fading) greatly affect both the usable service area and the
required geographical separation between co-channel stations. An empirical
method is given for estimating the magnitude of these variations at vhf and
uhf. These data indicate that the required separation between co-channel
stations is from 3 to 10 times the average radius of the usable coverage area,
and depends on the type of service and on the degree of reliability required.
The application of this method is illustrated by examples in the mobile
radiotelephone field.
Speaking Machine of Wolfgang von Kempelen.* H. Dudley^ and T. H.
Tarnoczy. Acoustical Soc. Am., Jl., V. 22, pp. 151-166, March, 1950.
Perception of Speech and Its Relation to Telephony. H. Fletcher^ and
R. H. Galt.' Acoustical Soc. Am., JL, V. 22, pp. 89-151, March, 1950.
Abstract — This paper deals with the interpretation aspect and how it is
afifected when speech is transmitted through various kinds of telephone
systems.
Vacuum Fusion Furnace for Analysis of Gases in Metals. W. G. Guldner^
and A. L. Beach.^ Anal. Chem., V. 22, pp. 366-367, February, 1950.
Complex Stressing of Polyethylene. I. L. Hopkins,^ W. O. Baker^ and
J. B. Howard.^ //. Applied Phys., V. 21, pp. 206-213, March, 1950.
Noise Considerations in Sound-Recording Transmission Systems. F. L.
Hopper.2 References. S.M.P.E., JL, V. 54, pp. 129-139, February, 1950.
Radiation Characteristics of Conical Horn Antennas.* A. P. King.^ LR.E.,
Proc, V. 38, pp. 249-251, March, 1950.
Abstract — This paper reports the measured radiation characteristics of
conical horns employing waveguide excitation. The experimentally derived
gains are in excellent agreement with the theoretical results (unpublished)
obtained by Gray and Schelkunoff.
The gain and eflfective area is given for conical horns of arbitrary propor-
tions and the radiation patterns are included for horns of optimum design.
*A reprint of this article mav he obtained on rei|uest to the editor of the B. S.T.J.
•B.T.L.
2W. E. Co.
^Pac.T.&T.
ARTICLES BY BELL SYSTEM AUTHORS 465
All dimensional data have been normalized in terms of wavelength, and are
presented in convenient nomographic form.
I' Microwaves and Sound. W. E. Kock.^ Physics Today, V. 3, pp. 20-25,
March, 1950.
Abstract — A recent development shows that obstacle arrays, modeled
after the periodic structure of crystals, refract and focus not only electro-
magnetic waves, but sound waves as well. The behavior of periodic struc-
tures can be investigated by microwave and acoustic experiments on such
models.
Interference Characteristics of Pulse-Time Modulation. E. R. Kretzmer.^
I.R.E., Proc, V. 38, pp. 252-255, March, 1950.
Abstract — The interference characteristics of pulse-time modulation are
analyzed mathematically and experimentally; particular forms examined
are pulse-duration and pulse-position modulation. Both two-station and two-
path interference are considered. Two-station interference is found to be
characterized by virtually complete predominance of the stronger signal,
and by noise of random character. Two-path interference, in the case of
single-channel pulse-duration modulation, generally permits fairly good re-
ception of speech and music signals.
Electron Bombardment Conductivity in Diamond.* K. G. McKay. ^ Phys.
Rev., V. 77, pp. 816-825, March 15, 1950.
Perception of Television Random Noise.* P. Mertz.^ References. S.M.P.E.,
Jl., V. 54, pp. 8-34, January, 1950.
Abstract — The perception of random noise in television has been clari-
fied by studying its analogy to graininess in photography. In a television
image the individual random noise grains are assumed analogous to photo-
graphic grains. Effective random noise power is obtained by cumulating
and weighting actual noise powers over the video frequencies with a weight-
ing function diminishing from unity toward increasing frequencies. These
check reasonably well with preliminary experiments. The paper includes an
analysis of the effect of changing the tone rendering and contrast of the
television image.
Loudness Patterns — A New Approach.* W. A. Munson^ and M. B. Gard-
ner.^ Acoustical Soc. Am., JL, V. 22, pp. 177-190, March, 1950.
Bell System Participation in the Work of the A.S.A. H. S. Osborne.*
Bell Tel. Mag., V. 28, pp. 181-190, Winter, 1949-50.
New Electronic Telegraph Regenerative Repeater.* B. Ostendorf, Jr.^
Elec. Engg., V. 69, pp. 237-240, March, 1950.
Correlation of Gieger Counter and Hall Effect Measurements in Alloys Con-
* A reprint of this article may be obtained on request to the editor of the B. S.T.J.
iB.T.L.
3 A. T. & T.
466 BELL SYSTEM TECHNICAL JOURNAL
faming Germanium and Radioactive Antimony 124* G. L. Pearson/ J. D.
Struthers,^ and H. C\ Theurer.' Pliys. Rev., V. 77, pp. 809-813, March
15, 1950.
Optical Method for Measuring the Stress in Glass Bulbs* W. T. Read.^
Applied Phys., Jl., V. 21, pp. 250-257, March, 1950.
Programming a Computer for Playing Chess. C. E. Shannon.^ References.
Phil. Mag., V. 41, pp. 256-275, March, 1950.
Abstract — This paper is concerned with the problem of constructing a
program for a modern electronic computer of the EDVAC type which will
enable it to play chess. Although perhaps of no practical importance the
question is of theoretical interest, and it is hoped that a satisfactory solution
of this problem will act as a kind of wedge in attacking other problems of a
similar nature and of greater significance.
Recent Developments in Communication Theory. C. E. Sh.a.nnon.^ Elec-
tronics, V. 32, pp. 80-83, April, 1950.
Abstract — In this paper the highlights of this recent work will be de-
scribed with as little mathematics as possible. Since the subject is essentially
a mathematical one, this necessitates a sacrifice of rigor; for more precise
treatments the reader may consult the references.
A Symmetrical Notation for X umbers. C. E. Shannon.^ .-Iw. Math. Monthly,
V. 57, pp. 90-93, February, 1950.
Capacity of a Pair of Insulated ]]'ires.* W. H. Wise.^ Quart. Applied
Math., V. 7, pp. 432-436, January, 1950.
Echoes in Transmission at 450 Megacycles from Land-to-Car Radio Units.*
W. R. Young, Jr.' and L. Y. Lacy.' I.R.E., Proc, \. 38, pp. 255-258,
March, 1950.
Simplified Derivation of Linear Least Square Smoothing and Prediction
Theory.* H. W. Bode' and C. E. Shannon.' LR.E., Proc, \. 38, pp. 417-
425, April, 1950.
Abstract — In this paper the chief results of smoothing theory will be
developed by a new method which, while not as rigorous or general as the
methods of Wiener and Kohnogoroff, has the advantage of greater simplic-
ity, particularly for readers with a background of electric circuit theory.
The mathematical steps in the present derivation have, for the most part,
a direct i)hysical interi)retation, which enables one to see intuitively what
the mathematics is doing.
Helix Parameters Used in Traveling Wave-Tube Theory.* R. C Fletcher.^
LR.E., Proc, V. 38, pp. 413-417, April, 1950.
Abstr.act — Helix parameters used in the normal mode solution of the
traveling-wave tube are evaluated by comparison with the field equations
*A reprint of this article inav l)c obtained <>n request to the editor of the B. S.T.J.
'B.T.L.
ARTICLES BY BELL SYSTEM AUTHORS 467
for a thin electron beam. Corresponding parameters for a thick electron beam
are found by finding a thin beam with approximately the same r-f admit-
tance.
Effect of Change of Scale on Sintering Phenomena* C. Herring.^ //.,
Applied Phys., V. 21, pp. 301-303, April, 1950.
Abstract — It is shown that when certain plausible assumptions are ful-
filled simple scaling laws govern the times required to produce, by sintering
at a given temperature, geometrically similar changes in two or more sys-
tems of solid particles which are identical geometrically except for a differ-
ence of scale. It is suggested that experimental studies of the effect of such
a change of scale may prove valuable in identifying the predominant mech-
anism responsible for sintering under any particular set of conditions, and
may also help to decide certain fundamental questions in fields such as
creep and crystal growth.
Mode Conversion Losses in Transmission of Circular Electric Waves Through
Slightly Non-Cylindrical Guides* S. P. Morgan, Jr.^ //., Applied Phys.,
V. 21, pp. 329-338, April, 1950.
Abstract — A general expression is derived for the effective attenuation
of circular electric (TEoi) waves owing to mode conversions in a section of
wave guide whose shape deviates slightly in any specified manner from a
perfect circular cylinder. Numerical results are in good agreement with ex-
periment for the special case of transmission through an elliptically deformed
section of pipe. The case of random distortions in a long wave guide line is
analyzed and it is calculated, under certain simplifying assumptions, that
mode conversions in a 4.732-inch copper pipe whose radius deviates by 1
mil rms from that of an average cylinder will increase the attenuation of the
TEoi mode at 3.2 cm by an amount equal to 20% of the theoretical copper
losses. The dependence on frequency of mode conversion losses in such a
guide is discussed.
Acoustical Designing in Architecture. C. M. Harris^ and V. O. Knudsen.
Book, New York, John Wiley & Sons, Inc., 450 pages, 1950.
Abstract — This book is intended as a practical guide to good acoustical
designing in architecture. It is written primarily for architects, students of
architecture, and all others who wish a non-mathematical but comprehensive
treatise on this subject. Useful design data have been presented in such a
manner that the text can serve as a convenient handbook in the solution
of most problems encountered in architectural acoustics.
*A reprint of this article may be obtained on request to the editor of the B. S.T.J.
' B.T.L.
Contributors to this Issue
R. V. L. Hartley, A.B., Utah, 1909; B.A., Oxford, 1912; B.Sc, 1913;
Instructor in Physics, Nevada, 1909-10. Engineering Department, Bell
Telephone Laboratories, 1913-50. Mr. Hartley took part in the early radio
telephone experiments and was thereafter associated with research on teleph-
ony and telegraphy at voice and carrier frequencies. Later, as Research
Consultant he was concerned with general circuit problems. Mr. Hartley is
now retired from active service.
J. R. Pierce, B.S., in Electrical Engineering, California Institute of
Technology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936-. Dr.
Pierce has been engaged in the study of vacuum tubes.
Claude E. Shannon, B.S., in Electrical Engineering, University of
Michigan, 1936; S.M. in Electrical Engineering and Ph.D. in Mathematics,
M.I.T., 1940. National Research Fellow, 1940. Bell Telephone Laboratories,
1941 -. Dr. Shannon has been engaged in mathematical research principally
in the use of Boolean Algebra in switching, the theory of communication,
and cryptography.
George C. Southworth, B.S., Grove City College, 1914; Sc.D. (Hon.),
1931; Ph.D., Yale University, 1923. Assistant Physicist, Bureau of Stand-
ards, 1917-18; Instructor, Yale University, 1918-23. Editorial staff of The
Bell System Technical Journal, American Telephone and Telegraph Com-
pany, 1923-24; Department of Development and Research, 1924-34; Re-
search Department, Bell Telephone Laboratories, 1934-. Dr. Southworth's
work in the Bell System has been concerned chiefly with the development
of the waveguide as a practical medium of transmission. He is the author of
numerous papers relating to a diversity of subjects such as ultra-short waves,
short-wave radio propagation, earth currents, the transmission of micro-
waves along hollow metal pipes and dielectric wires and microwave radiation
from the sun.
468
.
VOLUME XXIX OCTOBER, 1950 no. 4
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
«*ubiJc Library
Kansas City, |J1
Theory of Relation between Hole Concentration and Char-
acteristics of Germanium Point Contacts . .. /. Bardeen 469
Design Factors of the Bell Telephone Laboratories 1553
Triode J. A. Morton and R. M. Ryder 496
A New Microwave Triode : Its Performance as a Modulator
and as an Amplifier
A. E. Bowen and W. W. Mumford 531
A Wide Range Microwave Sweeping Oscillator
M. E. Hines 553
Theory of the Flow of Electrons and Holes in Germanium
and Other Semiconductors W. van Roosbroeck 560
Traveling- Wave Tubes [Fourth Installment] ./. /2. Pierce 608
Technical Publications by Bell System Authors Other than
in the Bell System Technical Journal 672
Contributors to this Issue 674
30i Copyright, 1950 $1.50
per copy American Telephone and Telegraph Company per Year
THE BELL SYSTEM TECHNICAL JOURNA
Published quarterly by the
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<
The Bell System Technical Journal
Vol. XXIX October, 1950 No. 4
Copyright, 1950, American Telephone and Telegraph Company
Theory of Relation between Hole Concentration and
Characteristics of Germanium Point Contacts
By J. BARDEEN
(Manuscript Received Apr. 7, 1950)
The theory of the relation between the current-voltage characteristic of a
metal-point contact to w-type germanium and the concentration of holes in the
vicinity of the contact is discussed. It is supposed that the hole concentration has
been changed from the value corresponding to thermal equilibrium by hole in-
jection from a neighboring contact (as in the transistor), by absorption of light
or by application of a magnetic field (Suhl effect). The method of calculation
is based on treating separately the characteristics of the barrier layer of the con-
tact and the flow of holes in the body of the germanium. A linear relation be-
tween the low-voltage conductance of the contact and the hole concentration is
derived and compared with data of Pearson and Suhl. Under conditions of no
current flow the contact floats at a potential which bears a simple relation,
previously found empirically, with the conductance. When a large reverse
voltage is applied the current flow is linearly related to the hole concentration,
as has been shown empirically by Haynes. The intrinsic current multiplication
factor, a, of the contact can be derived from a knowledge of this relation.
I. Introduction
IN DISCUSSIONS of the theory of rectification at metal-semiconductor
contacts, it is usually assumed that only one type of current carrier
is involved: conduction electrons in »-type material or holes in /J-type
material.^ In the case of metal-point contacts to high-purity «-type
germanium, such as is used in transistors and high-back-voltage varistors,
it is necessary to consider flow by both electrons and 'holes. A large part
of the current in the direction of easy flow (metal point positive) con-
sists of holes which flow into the w-type germanium and increase the
conductivity of the material in the vicinity of the contact.^-' The con-
ductivity is increased not only by the presence of the added holes but
also by^the additional conduction electrons which flow in to balance the
positive space charge of the holes. There is a small concentration of^holes
normally present in the germanium under equilibrium conditions with no
' For a discussion of the nature of current flow in semi-conductors see the "Editorial
Note" in Bdl Sys. Tech. Jour. 28, 335 (1949).
'J. Bardeen and W. H. Brattain, Bell Sys. Tech. Jour. 28, 239 (1949).
' W. Shockley, G. L. Pearson and J. R. Haynes, Bell Sys. Tech. Jour. 28, 344 (1949).
469
470 BELL SYSTEM TECHNICAL JOURNAL
current flow. When the contact is biased in the reverse (negative) direc-
tion, these holes tend to flow toward the contact and contribute to the
current. The hole current is increased if the concentration of holes in the
germanium is enhanced by injection from a neighboring contact or by
creation of electron-hole pairs by light absorption.
Much has been learned about the effect of an added hole concentration
on the current voltage characteristics of contacts from studies with
germanium filaments. Part of this work is summarized in a recent article
of W. Shockley, G. L. Pearson and J. R. Haynes.^ These authors have
investigated the way the low-voltage conductance of a point contact to a
filament of «-type germanium varies with the concentration of holes in
the filament and have shown that there is a linear relation betw-een con-
ductance and hole concentration. They have shown that the current to a
contact biased with a large voltage in the reverse direction varies linearly
with hole concentration. Suhl and Shockley^ have shown that by applying
a large transverse magnetic field along with a large current flow holes
may be swept to one side of the filament. Changes in hole concentration
produced in this way are detected by measuring changes in the con-
ductance of a point contact.
Shockley^ has suggested that the floating potential measured by a con-
tact made to a semiconductor in which the concentration of carriers is
not in thermal equilibrium may depend on the nature of the contact and
differ from the potential in the interior. Pearson^ has investigated this
effect for point contacts on germanium filaments, and has shown that the
floating potential is related to the conductance of the contact. This effect
provides an explanation for anomalous values of floating potentials meas-
ured by Shockley^ and by W. H. Brattain.^ They found that potentials
measured on a germanium surface in the vicinity of an emitter point
biased in the forward direction may be considerably higher than expected
from the conductivity of the material.
The purpose of the present paper is to develop the theory of these rela-
tions. We are particularly interested in effects produced by changes in
hole concentration in w-type germanium resulting from hole injection or
photoelectric effects. The equations developed also apply to injected
electrons in />-type semiconductors with appropriate changes in signs of
carriers and bias voltages. The methods of analysis used are similar to
those which have been cm])loyed by l^rattain and the author in a dis-
cussion of the forward current in germanium point contacts-.
^ H. Suhl and W. Shockley, Phys. Rev. 74, 232 (1948).
" W. Shockley, Bell Sys. Tech. Jour. 28, 435 (1949), p. 468.
" Unpublished,
HOLE CONCENTRATION AND POINT CONTACTS
471
The problem may be divided into two parts, which can be treated
separately:
(a) The first deals with the current-voltage characteristics of the space
charge region of the rectifying contact. The current flowing across the
contact is expressed as the sum of the current which would flow if the
hole concentration in the interior were normal and the current which
results from the added hole concentration.
(b) The second is concerned with the current flow in the semiconductor
outside the space charge region. In general, both diffusion and conduction
y//////////////////////////////////,
AT OUTER BOUNDARY
OF SPACE -CHARGE LAYER:-
P=PbO' n = nbo = Nf+Pbo
V=V;
|l=Io(Vc)
'S^//////y////////////////////Ay/////
r
DEEP IN INTERIOR:
p=Po, n = no=Nf+po
V = 0
(a) EQUILIBRIUM CONCENTRATION OF HOLES IN INTERIOR
Jl = Io(Vc)-ePbaVaA/4
4 "//y///////////////////////////////y
'//////////////////////////////A
AT OUTER BOUNDARY
OF SPACE -CHARGE LAYER".'
P=Pbo + Pba' n = nbo + Pba
"^"^i- DEEP IN interior:
p = Po + Pa, n = no + Pa
V=o
(b) ADDED CONCENTRATION OF HOLES IN INTERIOR
Fig. 1. — Model and notation used for calculation of current flow in low-voltage case.
are important in determining the flow of carriers, although, depending on
conditions, one may be much more important than the other. In case the
applied voltage and current flow are small, holes in an »-type semi-
conductor move mainly by diffusion. This situation applies to the prob-
lems discussed in the first part of the memorandum. In Section IV we
discuss the opposite limiting case of large voltages in which the electron
current flowing is so large that the hole current is determined by the
electric field and diflfusion is unimportant.
The model which is used to investigate the low-voltage case is illus-
trated in Fig. 1. For purposes of mathematical convenience, the contact
is represented as a hemisphere extending into the germanium. Recom-
bination, both at the surface of the semiconductor and in the interior, is
472 BELL SYSTEM TECHNICAL JOURNAL
assumed to be negligible so that the lines of current flow are radial. The
spherical symmetry of the resulting problem simplifies the mathematics.
A calculation is given in an Appendix for a model in which the contact is
a circular disk and recombination takes place at the surface. The latter
does not give results which are significantly different from the simplified
model.
Figure 1(a) applies to the case in which the hole concentration deep in
the interior has its normal or thermal equilibrium value, p^. The sub-
script zero is used to denote values which pertain to this situation. Of a
voltage Vp applied to the contact, a part Vc occurs across the space-
charge barrier layer of the contact and a part Vi occurs in the body of
the semiconductor. Thus V p represents the voltage of the contact and V i
the voltage in the semiconductor just outside the barrier layer, both
measured relative to a point deep in the interior. It should be noted that
Vp does nol include the normal potential drop which occurs across the
barrier layer under equilibrium conditions with no voltage applied. In
the examples with which we shall deal in the present memorandum, the
spreading resistance is small compared with the contact resistance, so
that Vi is small compared with V p. Obviously,
Fp = F, + Vi. (1)
When a current is flowing to the contact the hole concentration, pbo,
measured just outside of the barrier layer, differs from the concentration
deep in the interior, pa. It is the concentration gradient resulting from
the difference between pm and />o which produces a flow of holes from the
interior to the contact. In the forward direction, />bo is larger than /»o;
in the reverse direction, pm is less than p^.
The total current, /o( Fc), flowing across the contact includes both elec-
tron and hole currents. It will not be necessary to distinguish between
these two contributions to the normal current flow across the barrier
layer in the subsequent analysis.
Figure 1(b) applies to the case in which the hole concentration deep in
the interior has been increased to />o + pa by adding a concentration Pa
to the normal concentration, />o. The concentration just outside the barrier
layer is increased to />m + pba- In addition to the normal current, h{V^,
flowing across the contact, there is an additional current of holes resulting
from the added hole concentration, pba, at the barrier.
The magnitude of this added hole current is determined in the follow-
ing way. It is assumed that all holes which enter the barrier region are
drawn into the contact by the field existing there. The number of holes
HOLE CONCENTRATION AND POINT CONTACTS 473
entering the barrier region per second is given by the following expres-
sion from kinetic theory:
pbVaA/A, (2)
where Va is the average thermal velocity, 2(2kT/Trmy''^, of a hole and A
is the contact area. This expression gives the average number of particles
which cross an area A from one side per second in a gas with concentra-
tion pb. It follows that the current due to the added holes is:
I pa = -ePbaVaA/4. (3)
Since, by convention, a current flowing into the semiconductor ispositive,
a current of holes flowing from the interior to the contact is negative.
The diffusion current resulting from the added holes depends on the
difference between pba and pa- We shall show in Section III that when pa
is small compared with the normal electron concentration,
Ipa = 2irrbkTlXp{pba — pa), (4)
where tb is the radial distance to the outer boundary of the barrier layer
and jXp is the hole mobility. The value of pba is found by equating (4)
and (3), i.e., the added current flowing from the interior to the barrier
layer and the current flowing across the barrier layer. This gives
where a, defined by
pba/ pa - a/(l + a), (5)
a = 4(kT/erb)tJip/va, (6)
is the ratio of the velocity acquired by a hole in a field 4kT/erb to thermal
velocity. This ratio is generally a small number so that the a in the de-
nominator of (5) can be neglected in comparison with unity. Equation
(3) then becomes:
Ipa = —eapaVaA/4: = — p akTiXpA / Yb- (7)
If pa is not assumed small, a similar procedure may be used but the
expressions for Ipa in terms of pa are more complicated than (4) and (7)
It is possible that the added hole current, Ipa, will affect the contact in
such a way as to change the normal current flowing. If there is such a
change, one might expect it to be proportional to Ipa as long as Ipa is
sufficiently small. The total current flow may then be expressed in terms
of an "intrinsic a" for the contact as follows:
/ = h{Vc) - a Ipaipa). (8)
474 BELL SySTE^r TECHNICAL JOURNAL
There is no good theoretical reason to expect that a is different from unity
for small current flow in normal contacts unless trapping is important.
Equation (8) is used as the basis for the analysis of the low-voltage
data. One important consequence of the equation is that if />„ is different
from zero, there is a voltage drop across the barrier layer even though
no net current flows to the point. The presence of the added holes in the
interior produces a floating potential on the point. The magnitude of this
floating potential, Vcf, is obtained by setting / = 0 in Eq. (8) and find-
ing the value of Vc which solves the equation. This potential can be
observed on a voltmeter and is analogous to a photovoltage.
Associated with the floating potential is a change in conductance of
the contact. The conductance near 7 = 0, given by
G = (rf//JFe)r,=V,, = (dh/dV,)y^^y^,, (9)
is just the conductance for normal hole concentration in the interior at an
applied voltage equal to Vc/. In setting the conductance equal to the
derivative of / with respect to Vc, we have neglected the difference,
Vi, between Vc, the voltage drop across the barrier, and Vf, the total
drop from the contact to the interior. This corresponds to neglecting the
spreading resistance in comparison with the barrier resistance.
Equation (8) may be used to relate the floating potential with change
of conductance of the contact. The appropriate equations, together with
applications to data of Pearson and of Brattain, are given in Section II.
In Section III we derive Eq. (4) which relates the added hole current
with the added hole concentration in the interior. This relation is used
to show that the point conductance G varies linearly with the added hole
concentration, /?„. The theoretical expression for conductance is comj^ared
with data of Pearson and of Suhl.
In section IV we discuss the dependence of the current-voltage char-
acteristic at large reverse voltages on hole concentration. I'nder these
conditions it is the electric held rather than diffusion which produces the
hole current in the body of the germanium. The electron and hole currents
are then in the ratio of the electron to hole conductivity. With introduc-
tion of an "intrinsic a" for the contact, a simple relation is derived for the
dependence of current on hole concentration for fixed voltage on the
point. This relation is used to determine a for several point contacts from
some data of J. R. Haynes.
II. Elo.\ting Potentiat- of Point Contact
In order to get analytic expressions for the floating potential and ad-
mittance, it is necessary to make some assumption about the normal cur-
HOLE CONCENTIL\TIOX AXD POINT CONTACTS 475
rent-voltage characteristic, h{\\). It is found empirically" that as long
as Vc is not too large (a few tenths of a volt for a point contact on n-
type germanium), it is a good approximation to take:
/.) {Vc) = Ic {exp(^eV,/kT) - 1), (10)
where Ic is a constant for a given contact. Except for the factor l3, this
is of the form to be expected from the diode theory of rectification. The
empirical value of 13 is usually less than the theoretical value of unity
in actual contacts.
If (10) is inserted into (8), the following equation is obtained for the
current when there is an added concentration of carriers, pa, in the in-
terior:
/ = /, (exp(l3eVe/kr) - 1) - «/,«. (11)
Setting 7 = 0 and solving the resulting equation for the floating po-
tential, Vc = Vcf, we find:
Vcf = {kT/e8) log [1 + aif.JIc)]. (12)
The floating potential may be simply related to the conductance cor-
responding to small current flow. Using Eqs. (9) and (11), we find:
G = (dh/dVc)v,^y,, = {0eIc/kT) exp (l3eVcf/kT). (13)
Since the normal low-voltage conductance is just
Go = ^elc/kT, (14)
we have
G = Go exp (j3eVcf/kT). (15)
By using (12), G can be expressed in terms of pa- This relation is given
and compared with experiment in Section III. Equation (15) may be
solved for the floating potential:
Vcf = {kT/ei3) log (G/Go). (16)
It should be noted that (16) does not involve pa directly. Thus it is pos-
sible to determine Vcf from a measurement of the change in conductance
without direct knowledge of the added hole concentration. It holds for
large as well as small pa.
The logarithmic relation (16) between floating potential and conduc-
tance has been demonstrated by an experiment of Pearson. Theexperi-
' See H. C. Torrev and C. A. Whitmer, "Crystal Rectifiers", McGraw-Hill Company,
New York, N. Y., (1949), p. 372-377.
476
BELL SYSTEM TECHNICAL JOURNAL
mental arrangement is illustrated in Fig. 2. Holes are injected into a
germanium filament by an emitter point and the circuit is closed by
allowing the current to flow to the large electrode at the left end. The
right end of the filament is left floating. Some of the injected holes diffuse
HOLES INJECTED
FLOATING POTENTIAL, Vnr,
AND CONDUCTANCE, Y,
MEASURED
Fig. 2. — Schematic diagram of experiment of G. L. Pearson to investigate relation be-
tween floating potential and impedance of point contact.
>"6
Z 3
o
< 2
_j
UJ
a.
(.5
/
/
>
/^
>^A
/
/
/
A
/
o
o y
/
A
o
/
O EMITTER CURRENT VARIED
A EMITTER DISTANCE VARIED
/
/o^
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
FLOATING POTENTIAL, Vf , IN TERMS OF KT/e
Fig. 3. — The relationship of admittance ratio to potential, measured at a point on a
germanium filament into which holes are emitted, with no current flow, from G. L. Pear-
son's data of September 21, 1948.
down the filament and increase the local concentration in the neighbor-
hood of the probe point. This concentration can be varied by changing
the emitter current and also by changing the distance between emitter
and ])robe. Hoth the floating potential and the conductance between the
prol)c point and the large electrode on the right end were measured.
L'nder the conditions of this experiment, the potential drop in the in-
HOLE CONCENTRATION AND POINT CONTACTS
477
terior of the floating end of the filament is small. The small drop which
does exist results from the difference in mobility between electrons and
holes. Almost all of the potential difference between the probe and the
right end is the floating potential, Vcf, across the barrier layer of the
probe point.
Pearson's data are plotted in Fig. 3. The data can be fitted by an equa-
tion of the form (16) with ^ = 0.5.
The difference in potential between a floating point contact and the
interior which exists under non-equilibrium conditions explains anoma-
lously high values of probe potential which were sometimes observed by
Shockley and by Brattain in the vicinity of an emitter point operating in
the forward direction. As an example of a case in which the effect is
Table I
Measurements of probe potential, Vp/, at a contact on an etched germanium surface
.005 cm from a second contact carrying a current /. The conductance of the probe point
is Gp. The voltage drop across the probe contact, Vp/ — Vi, at zero current is calculated
from Vpf - Vi = 2.5{kT/e) log (Gp/Go). Data from W. H. Brattain.
/
amps
2.0 X 10-»
1.0
0.5
0.2
0.1
-0.1
-0.2
-0.5
-1.0
volts
V»///
ohms
0.189
94
0.141
141
0.096
190
0.052
260
0.030
300
-0.0096
96
-0.0186
93
-0.044
88
-0.10
100
mhos
8.3 X 10-*
5.0
3.3
2.2
1.7
1.2
1.2
1.25
1.35
Gp/Go
log
(Gp/Co)
Vpf-Vi »
.062 log
(Kp/K.)
Vi
volts
6.9
1.93
0.120
0.069
4.2
1.435
0.090
0.051
2.8
1.030
0.064
0.032
1.8
0.588
0.037
0.015
1.4
0.336
0.021
0.009
1.0
1.0
—
VJI
ohms
35
51
64
75
90
large, some data of Brattain are given in Table I for the experimental
arrangement of Fig. 4. Two point contacts were placed about .005 cm
apart on the upper face of a germanium block. The surface was ground
and etched in the usual way. A large-area, low-resistance contact was
placed on the base. The potential, Vp, of one point, used as a probe,
was measured as a function of the current flowing in the second point. In
this case, the potential on the probe point is produced in part by the
Vcf term and in part by a potential, V i, in the interior which comes from
the IR drop of the current flowing from the emitter point to the base
electrode. Reasonable values are obtained for 7, from measurements of
Vp if a correction for Vc/ is properly made.
The first column of Table I gives the current and the second column
the probe potential, Vp, measured relative to the base. The third column
gives values of Vp/I. In the reverse direction (negative currents) Vp/I
478
BELL SYSTEM TECHNICAL JOURNAL
is approximately constant at a little less than 100. Values of V ,JI in the
forward direction are much larger, starting at 300 for / = 0.1 ma and
decreasing to <M at 1=2 ma. If anything, one would expect a decrease
rather than an increase in Vp/I in the forward direction as injection of
holes lowers the resistivity of the germanium in the vicinity of the point.
We shall show that Vi/I actually does decrease and that the anomalously
high values of Vp/I in the forward direction result from the drop, Vc/,
HOLES INJECTED
FLOATING POTENTIAL, Vnr,
AND CONDUCTANCE, Y,
MEASURED
Fig. 4. — Schematic diagram of experiment of W. H. Brattain for measuring floating
potential and admittance at point near emitter.
across the barrier layer between the contact point and the body of the
germanium. Thus,
Vi = F,
v..
(17)
Values of Vcj can be estimated from the change in conductance corre-
sponding to small currents in the probe point. The conductance increases
with increasing forward emitter current. Values of Vc/, calculated from
Vcf = 2.5 (kT/e) log (Gp/Go),
(18)
are given in column 6. The value 2.5, chosen empirically to give reason-
able values of V'i, is not far from the value 2.0 required to fit Pearson's
data in Fig. 1. Values of F, obtained from Eq. (17) are given in column 7.
The ratios Vi/I given in column 8 are reasonable. The decrease in Vi/I
with increasing forward current is caused by a decrease in the resistivity
of the germanium resulting from hole injection.
In another case, in which no such anomaly was observed in the for-
ward direction, it was found that 1',:/, calculated from the change in
conductance, was small comyxired with V,,.
'i'here have as yet been no measurements which permit a comparison
of the values of IS rc(|uired to correlate probe [)()tential and conductance
HOLE CONCENTRATION AND POINT CONTACTS 479
with values of /3 obtained directly from the current-voltage characteristic
of the probe. Such a comparison would provide a valuable test of the
theory.
III. Low Voltage Conductance of Point Contacts
In this section we calculate the hole current flowing in the body of the
germanium from diffusion and find an expression relating change of con-
ductance with added hole concentration. The results shall be applied to
data of Pearson and of Suhl. ^^'e need to derive Eq. (4) which gives the
hole current in terms of the added hole concentrations, pba, measured just
outside the barrier layer, and pa, measured deep in the interior.
The model which is used for the calculation is illustrated in ¥\g. 1.
The diffusion equation for hole flow is to be solved subject to the bound-
ary conditions that p = pb just outside the barrier layer and p = pi Sit
large distances from the contact in the interior. It is assumed that the
total current flow is zero or small.
We shall first derive the more general equations* which include flow
by the electric field as well as by diffusion in order to show the conditions
under which the electric field can be neglected. In the body of the semi-
conductor, conditions of electric neutrality require that the electron con-
centration, n, be given by:
n = Nf + p, (19)
where Xf, the net concentration of fixed charge, is the difference between
the concentrations of donor and acceptor ions. We shall assume that
Nf is constant so that
grad n = grad p. (20)
The general equations for electron and hole current densities, /„ and ip,
are:
in = M'l (f»^ -\- kT grad n) (21)
ij, = Mp (epF - kT grad p), (22)
where F is the electric field strength. By using (19) and (20), and setting
Hn = l^fJ'p, we can express /„ in the form:
/„ = Vp ie{.\r + p) F -f kT grad p). (23)
The magnitude of F for zero net current,
i = /■;, + in = 0, (24)
•* A discussion of the equations of flow is given in the article by VV. van Roosbroeck in
this issue of the Bell System Technical Journal.
480 BELL SYSTEM TECHNICAL JOURNAL
can be obtained by adding (22) and (23) and equating the result to zero.
This gives:
The field vanishes for b = \, corresponding to equal mobilities for holes
and electrons. For b greater than unity and for equal concentration
gradients of holes and electrons, the diffusion current of electrons is
larger than that of holes. The field is such as to equate these currents by
increasing the flow of holes and decreasing the flow of electrons.
If (25) is substituted into (22), the following equation is obtained for
ip = —kTup
^^-'^^ +llgrad^ (26)
lNfb + p{b+ 1)
If recombination is neglected, the hole current is conserved and
div ip = 0. (27)
Using this relation, an equation of the Laplace type can be obtained for
p which may be integrated subject to the appropriate boundary condi-
tions. This derivation is given in Appendix B. The results do not differ
significantly from those obtained below for p assumed small.
Rather than continue with the general case, we shall at this point
assume that p <K Nf so that the first term in the parenthesis of Eq.
(26) is negligible in comparison with unity. This amounts to setting F =
0 in Eq. (3) and assuming that the holes move entirely by diffusion.
This is a very good approximation in most cases of practical interest and
is valid for small i as well as for i = 0. We then have
ip = —kTupgrsid p. (28)
The condition div ip = 0 gives Laplace's equation for p:
W = 0. (29)
Equation (29) is to be solved subject to the appropriate boundary
conditions. For the model illustrated in Fig. 1 we can assume that p
depends only on the radial distance r and that
p = p^titr = n, (30)
p = piUtr = 00. (31)
The solution of (29) which satisfies (31) is:
P-- pi+ (Ip/2irkTfipr), (32)
HOLE CONCENTRATION AND POINT CONTACTS 481
in which Ip is the total hole current. The boundary condition (30) gives
the relation between Ip and Pb'.
pb = pi-{- {Ip/lirkTiXpn). (33)
Since the equations are linear, an equation of the form {i3>) applies to
the hole current due to the added holes as well as to the entire hole
current. For the former we have:
Pba = Pa+ iIpa/2TrkTHj,n), (34)
which is equivalent to Eq. (4).
In the derivation of Eq. (34) we have neglected recombination at the
surface as well as in the interior. In the Appendix we give a solution for a
contact in the form of a circular disk and assume that recombination
takes place at the surface. The hole concentration then satisfies Laplace's
equation subject to more complicated boundary conditions at the surface.
The results are not significantly different from those of the simplified
model.'
Equation (34), or rather its equivalent, Eq. (4), was used in the deriva-
tion of Eq. (12) for the floating potential, Vcf . If this value for Vc/
is inserted into Eq. (15), an equation relating the conductance directly
with the added hole concentration is obtained:
G = Go+ {ae'aVaA^Pa/^kT). (35)
This expression may be simplified by substituting for a from Eq. (6):
G = Go + a^HpeApa/n. (36)
By using the expression for the normal conductivity:
Co = bnpeno, (37)
the conductance can be given in the form:
G = Go + {a(3aoA/bn)(pa/m). (38)
If (To is in practical units (mhos/cm), G is in mhos.
We shall compare (38), which gives a linear variation between G and Pa,
with experimental data of Pearson^^ and of Suhl. The arrangement used
' In the applications, these equations are applied to situations in which the contact is
on a germanium filament and there is a flow of current along the length of the filament in
addition to the flow to the contact. A question may arise as to whether it is justified to
neglect the filament current when discussing flow to the contact. There is no difficulty
as long as pa/m is small compared with unity because the equations are then linear and
the solution giving the flow to the contact can be superimposed on the solution giving the
flow along the length of the filament. The neglect of the filament current cannot be rigor-
ously justified in case pa/m is large, as is assumed in the calculations of Appendix B. It is
not believed, however, that the exact treatment would yield results which are significantly
different.
'" See reference 3, p. 356 and Fig. 6.
482
BELL SYSTEM TECHNICAL JOURNAL
by Pearson is shown in Fig. 5. Two probe points were placed about
.009 cm apart near one end of a germanium filament. The concentration
HOLES INJECTED
POTENTIAL DIFFERENCE AND
CONDUCTANCE MEASURED
Y'\g. 5.- — Experimental arrangement used by G. L. Pearson to investigate relation be-
tween admittance and hole concentration in germanium filament.
io
^em^
o
X
0.05 ma
0.1
1 /
A
0.2
16
D
0.5
y^
y
14
13
12
II
10
9
y
v^n
y
y
y
y^ °^
/
< n
o
,x
8
^A
)
r
Xvd'
A
X
7
yt
r
6
r
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
HOLE DENSITY IN TERMS OF p/n^
I''ig. 6. — The relalionshii) i)elween jioint admittance and relative hole concentration,
for a germanium lilamcnl from (1. L. Pearson's data of Septcml)cr 28, 1948.
of holes was varied by current from an emitter point near tiie opposite
end of tiie filament. There was an additional current flowing between
HOLE CONCENTRATION AND POINT CONTACTS 483
electrodes at the two ends so that the tield pulling the holes along the
filament could be varied. The concentration of holes was determined from
the change in resistivity of that segment of the filament between the two
probes. Measurements of admittance were made by passing a small current
between the two probes connected in series. The area of the filament is
about 1.6 X 10~^ cm- and the normal resistance between the probes
about 1800 ohms. The normal conductivity is thus
ffo = .009/(1800 X 1.6 X 10-^) = 0.03 (ohm cm)-\ (39)
As shown in Fig. 6, Pearson finds a linear relation between G and pn-
The line best fitting Pearson's data is
G = Go + (8 X 10-«) {pa/m) (mhos). (40)
The theoretical value of the coefficient may be obtained from Eq. (38).
Taking
a = 1, iS = 0.5, ao = 0.03
h = 2.0, A = 10-« cm2, r = 5 X lO"" cm, (41)
we get
a^aoA/b r^ = 15 X 10-^ mhos. (42)
Pearson's data, represented by (40), apply to the conductance of two
point contacts in series, and the conductance of each one may be about
twice that given by (40). Thus the theoretical value is in good agreement
with the observed. There is no indication that a differs from unity at low
voltage.
Suhl varied the concentration of holes in the vicinity of probe points by
application of a transverse magnetic field as well as by injection from an
emitter point. The experiment is illustrated in Fig. 7. He used a filament
with a cross-section of about .025 X .025 cm. Four probe points were
placed along the length of the filament at intervals of about .04 cm. A
total current of 4 ma flowed in the filament.
In one experiment, none of this current was injected, so that the con-
centration of holes was normal in the absence of the magnetic field.
Measurements were made of the floating potentials and of the conduct-
ances of the probe points. Then a transverse magnetic field was applied
and the conductances measured again. We are interested here only in the
case of a large field (30,000 gauss) in such a direction as to sweep the
holes to the opposite side of the filament. Suhl believes that under these
conditions the concentration of holes near the probe points is practically
zero. The difference between the conductances with and without the field
484
BELL SYSTEM TECHNICAL JOURNAL
then givfes tVe contribution to the conductance from the normal concen-
tration of holes.
In a second experiment 1 ma of the current of A ma flowing in the fila-
ment was injected from an emitter point near one end of the filament.
From the probe potentials, estimates have been made of the change in
PROBE POINTS
-« — Ib=4ma
(a) NO MAGNETIC FIELD, NORMAL HOLE CONCENTRATION
(b) MAGNETIC FIELD SWEEPS HOLES TO OPPOSITE SIDE
OF FILAMENT
■* — Iti=4ma
(C) HOLES INJECTED BY EMITTER
Fig. 7. — Schematic diagram of experiment of H. Suhl to investigate relation between
hole concentration and impedance of point contacts.
resistivity and thus of the added hole concentration at the different probe
points. Changes in hole concentration from injection have been correlated
with changes in admittance of the probe points.
The filament with dimensions .025 X .025 X 0.4 cm has a resistance of
4,600 ohms. The normal resistivity, po, is then about 7.2 ohm cm. Since
the concentration of electrons corresponding to 1.0 ohm cm is about
HOLE CONCENTRATION AND POINT CONTACTS 485
1.8 X 10^^ the concentration corresponding to a resistivity of 7.2 ohm cm
is":
no - 1.8 X 10'V7.2 = 2.5 X IQiVcm^. (43)
The product of the equihbrium concentrations of electrons and holes is
about 4 X 10-^ in germanium at room temperature^-. Thus, for this sample,
Pa = 4.x 10-V2.5 X IQi-' = 1.5 X lOi-'/cm^. (44)
If there is an added concentration of holes, pa, resulting from injection,
the added conductivity is:
o-a = (1 + b) eix,,pa = 8.4 X 10-16 Pa • (45)
The resistivity is changed to:
P = PoO'o/(<ro + (To) ~ PO (1 — CTaPo), (46)
the approximate expression holding if the relative change is small. The
resistance per unit length of filament is:
R = 1.15 X 10^ (1 - o-apo). (47)
The change in voltage gradient, dV/dx = RI, resulting from hole injec-
tion is, for a current of 4 X 10'^ amps,
A{dV/dx) = d(AV)/dx = -46po(r„ . (48)
Suhl measured the change in probe potential, AF, which resulted when 1
ma of the total current of 4 ma was injected from the emitter instead of
having the entire 4 ma flowing between the ends of the filament. His
values of AF for the four probe points are given in Table II. We have
made a plot of these as a function of position and have estimated the
gradients at each of the four probe positions. Using these values we have
calculated Co from Eq. (48) and the corresponding injected hole concen-
tration from Eq. (45). These are given in the last column of the table.
Suhl's measurements of conductances, G, of the probe points are given
in Table III. Also given are differences, AG, from the normal values with
no magnetic field and no injection and also these differences multiplied by
no/pa ' Values of pa for the case of hole injection were obtained from
Table II. Values of AG{n^,/pa) are to be compared with the theoretical
value,
AG {no/ pa) = a^a^A/cn , (49)
"These values are based on taking //„ = 3500 cmVvolt sec and n,, = 1700 cmVvolt
sec, as measured by J. R. Haynes. They correspond to room temperature (295°K).
^ This value is obtained from an intrinsic resistivity of about 60 ohm cm for Ge at
room temperature and the mobility values in reference 11.
486
BELL SYSTEM TECHNICAL JOURNAL
from Eq. (38). Taking a = 1, /i = 0.5, a,, = 0.14,6 = 1.5, A = 10-« and
rft = 5 X 10"\ we get
G{ih)/ pa) '^' 100 micromhos.
(50)
This value is of the same order as the values obtained from Suhl's data
listed in Table III. There is a large scatter in the latter and the values are
Table II
Calculation of hole concentrations from probe potential measurements. A V measures
potential difference resulting from hole injection of 1 ma when total current is kept at 4
ma; data from H. Suhl.
Relative
dd.V
Point No.
Position
(cm)
W (volts)
dx
(volts/cm)
POCTo
Oo (mhos)
pa (cm-3)
§(>
0
-.04
-0.6
.013
.0018
2.2 X 1012
#s
.044
-.073
-1.10
.024
.0033
4.0
J^4
.084
-.13
-1.8
.039
.0054
6.5
*3
.12
-.21
-2.5
.055
.0077
9.0
Table III
Changes in conductance resulting from application of magnetic field and from hole
injection. Units are micromhos. Data from H. Suhl.
No Field
With-
-30,000 gaus
, field
With hole injection
Point
G
G
AG
-«,-;>
G
AG
^ I
»6
17.2
16.4
-0.8
130
22.5
7.8
880
%^
6.55
4.35
-2.2
365
7.0
0.45
28
*4
3.7
3.2
-0.5
80
5.1
1.4
54
fni
13.0
9.2
^3.8
630
19
6
165
not consistent. It has been suggested that the abnormal values may result
from local sources of holes.
IV. Hole Flow for a Collector with Large Reverse Voltage
Haynes has shown that there is a linear relation between the current
to a collector point operated in the reverse direction and the concentra-
tion of holes in the interior of a germanium filament. Under the conditions
of his experiment, the current flowing to the collector point is small com-
pared with the total current flowing down the tllament, so that the col-
lector current does not alter the concentrations very much. Moles are
injected into the filament by an emitter point placed near one end, and
the concentration is determined from the change in resistance of the
filament in the neighborhood of the collector point.
HOLE CONCENTRATION AND POINT CONTACTS
487
Haynes' measurements may be fitted by an empirical equation of the
following form:
/ = h\\ + ypa/m)\, (51)
in which /o is the normal collector current flow for a given collector volt-
age, / is the collector current flowing for the same collector voltage when
the hole concentration is increased by pa , and Hq is the normal electron
concentration. Values of h and 7 for four different formed phosphor-
bronze collector points are given in Table IV. The collector bias is —20
volts in each case. It can be seen that the variations in 7 are much less
than those in /q. It will be shown below that 7 is related to the intrinsic
a of the point contact.
COLLECTOR
Vf, = -20 VOLTS
■* lb SWEEPING CURRENT
Fig. 8. — Experimental arrangement used b}' J. R. Haynes to determine relation be-
tween hole concentration and current to collector point biased with large voltage in re-
verse direction.
In Haynes' experiment, holes are attracted to the collector by the field
produced by the electron current and diffusion plays a minor role. In
contrast to the preceding examples, the terms involving the field F in
Eqs. (21) and (22) are large and the diffusion terms represented by the
concentration gradients are small. It follows from (21) and (22) that the
ratio of electron to hole current density is then:
in/ip = bn/p, (52)
which is equal to the ratio of the electron and hole contributions to the
conductivity. If n and p do not vary with position, the ratio is the same
everywhere and equal to the ratio of total electron and hole currents,
/„ and Ip'.
Inllp = inlip = hnl p. (53)
The currents /„ and /,, can also be related to the intrinsic a for the con-
tact by use of an equation of the form:
/ = /„o + a/p, (54)
488 BELL SYSTEM TECHNICAL JOURNAL
in which /„o is the electron current for zero hole current. The electron
current is:
In = /„o + (a - l)/p . (55)
Thus we have
[n ^ /no + (« - l)/p ^bn ^ bjXf + p) ,
Ip Ip p p '
This equation may be solved for Ip to give:
Ip = pIno/{bNf + (a - 1 - b)p). (57)
The term (a — I — b)p is generally small compared with bN/ and may
be neglected. We thus have approximately for p/Nf small and Nf <^ tio,
I = Ino+ alp = /„o[l + (ap/bn,)l (58)
When expressed in terms of the normal current,
h = /nod + (apo/bno)], (59)
the equation for / is of the form (51) :
/ = /o [1 + iaPa/bm)l (60)
From a comparison of (51) and (60) it can be seen that:
7 = a/b or a = by. (61)
Values of a determined from empirical values of y for the four point
contacts of Haynes are given in Table IV. The values are of a reasonable
order of magnitude for formed collector points.
An estimate of the importance of diffusion can be obtained by compar-
ing the hole current in Haynes' experiments with the hole current which
would exist if the electron current were zero, so that holes move by diffu-
sion alone. Equations (28) to (33) apply to the latter case. In addition to
{33} we need an equation which expresses the hole current flowing into
the contact in terms of the hole concentration, pb, at the contact. If the
reverse bias is large, no holes will flow out and the entire hole current is
that from semiconductor to metal as given by an equation similar to (3) :
Ip = —epbVaA/4:. (62)
Substituting this value for Ip into equation (33) we get an equation which
may be solved for Pb, to give:
Pb = api{\ 4- c) ^ api, (63)
HOLE CONCENTRATION AND POINT CONTACTS 489
>
with a given by Eq. (6). Using (63) for pb, we get:
I J, = kTtxnpiA/n = {kT<roA/ebrb){pi/no). (64)
With kT/e = .025 volts, co = bnoe^l^ = 0.2 (ohm cm)-i, A - lO"* cm^
and fd = 5 X 10~* cm, we get for the diflfusion current:
/p = (5 X I0-^){pi/no) amps. (65)
Comparing (65) with (57) we see that diffusion of holes will not be im-
portant if
Ino » 5 X 10-« amps. (66)
This condition is satisfied in Haynes' experiments.
In the case of point contacts formed to have a high reverse resistance
as diodes, /o may be of the order of 10"'' to 10~® amps at room tempera-
ture. Diffusion of holes will then play a role, and the hole current will
Table IV
Relation between hole concentration and collector current from data of J. R. Haynes.
Data represented by
/ = /o(l + (ypa/no))
where / is current flowing to collector point biased at —20 volts and pa/tto is ratio of added
hole concentration to the normal electron concentration.
Probe Point
/.
a = 2.17
0
0.94
4.6
2
0.33
4.4
3
0.54
6.9
4
1.20
4.6
be larger than indicated by Eq. (53). As discussed in reference (4) there
is still a question as to the importance of holes in the saturation current
observed by Benzer in diodes with high reverse resistance. Experiments
similar to those of Haynes would be valuable to determine the influence
of hole concentration on reverse current.
Acknowledgment
The author is indebted to G. L. Pearson, J. R. Haynes, W. H. Brattain,
and H. Suhl for use of the experimental data presented herein; to W.
Shockley for a critical reading of the manuscript and a number of valuable
suggestions, and to W. van Roosbroeck for aid with some of the anal-
yses and for suggestions concerning the manuscript.
APPENDIX A
Diffusion of Holes with Surface Recombination
In the calculation of the diffusion of holes given in Section III of the
text it was assumed that no recombination of electrons and holes oc-
490
BELL SYSTEM TECHNICAL JOURNAL
curred. In the present calculation it is assumed that recombination occurs
at the surface, but not in the volume. This is a good approximation for a
point contact on germanium. It is further assumed that the hole concen-
tration is sufficiently small so that Laplace's equation (29) may be used.
The model which we shall use is illustrated in Fig. 9. The contact is in
the form of a circular disk of radius p on the surface of the semiconductor.
Cylindrical coordinates, r, 6, z, are used, with the origin at the center of
the disk and the positive direction of the s-axis running into the semi-
conductor. We calculate the flow due to the added holes, and shall use
the symbol p without subscript to denote the added hole concentration.
Fig. 9. — Coordinates used for calculation of hole flow to contact area in form of circu-
■'ar disk.
With recombination at the surface, it is necessary to have a gradient in
the interior which brings the holes to the surface.
It is assumed that the rate of recombination at the surface is:
sp — holes/cm"-^,
(lA)
where the factor 5 has the dimensions of a velocity and p is evaluated at
the surface z = 0. According to measurements of Suhl and Shockley, 5 is
about 1500 cm/sec for a germanium surface treated with the ordinary
etch. The current flowing to the surface is:
{iXpkT/e){dp/dz)z^o holes/cm-
(2A)
The l)oundary condition for p at the surface z — Q outside of the contact
area is obtained by ccjuating (lA) and (2A). This gives:
where
dp/dz = \p at z = 0, r > p
X = se/upkT,
(3A)
(4A)
HOLE CONCENTRATION AND POINT CONTACTS 491
has the dimensions of a length. For 5 = 1500 cm/sec and Mp = 1700
cm-/volt sec, corresponding to germanium at room temperature, X is
about 35 cm~^.
The boundary condition on the disk is similar to (3A) except that s is
replaced by vj^ (cf. Eq. (3)). Thus for r < p,
dp/dz = Xcp z = 0,r < p, (5 A)
where
X, = Vae/4:tJLpkT. (6A)
Evaluated for germanium at room temperature, X^ is about 6 X 10^.
In order to have a dependent variable which vanishes at infinity, we
replace p by:
y = pa- p + y^paz, (7A)
so that /» -^ />a for s = 0 as r ^ oc . The variable y satisfies Laplace's
equation subject to the boundary conditions:
dy/dz = \y z =- 0, r > p (8A)
dy/dz = \c {y - pa) z = 0, r < p (9A)
y = 0 r ,z^ <».' (lOA)
An exact solution of the problem is difficult. We shall obtain an approxi-
mate solution which satisfies (8A) but not (9A) and which applies when
Xp « 1 « Kp. (11 A)
This approximation is valid for a germanium point contact, since, for p '^
10-3 cm,
Xp ^ .035, \cP ~ 60. (12A)
We shall first discuss the limiting case for which X — ^ 0 and X,. — ^ =o .
The former implies neglect of surface recombination and the latter
y = pa for z = 0, r < p. (13 A)
The problem is the same as that of finding the potential due to a conduct-
ing circular disk. The solution of this problem, which is well known, is:
y = ilpjr) r e-''Mrt) '-^^ dt. (14A)
Jo t
The current flowing to the disk is obtained from integrating:
i, - kTp.p{dy/dz), (15 A)
492 BELL SYSTEM TECHNICAL JOURNAL
over the area of the disk. This gives:
I pa = -AppakTup. (16A)
The analogous expression for a hemispherical contact area of radius
fft, obtained from (7), is:
I pa = —lirrbpakTup. (17A)
If a comparison is made on the basis of equal radii, (17 A) is larger than
(16A) by a factor of t/2. On the more reasonable basis of equal contact
areas, (16A) is larger than (17 A) by a factor of -i/ir.
An approximate solution which includes surface recombination can be
obtained as follows. A solution of Laplace's equation which satisfies
(8A)^and (lOA) js:
y^^J^Te-UrD'^dt. (18A)
That (18A) satisfies (8A) may be verified by direct substitution;
4^^>
= ?L° /* j,{rt) sin ptdt = 0 for r > p. (19A)
= (2yoA)(p' - rT'" for r < p. (20A)
2yo f
TT Jo
Expression (18A) satisfies (9A) approximately if Xc is large. Using
(20A) and neglecting X in comparison with X^ we have:
y= pa- (2yoAXc)(p2 - f2)-i/2 for z = 0, r < p. (21A)
Except for r almost equal to p, the second term on the right of (21 A) is
very small. It is not possible to obtain an explicit expression for y for
r < p. For 2 = 0, r = p,
, ^"'•fjy'dl^y.FM. (22 A)
The integral, F(\p), can be evaluated from a more general integral in
Watson's Bessel Functions, p. 433. We have:
/<■(/;) = 2 r Jjhc) sin xdx ^ ^^^ ^ ^^^^^ _^ ^.^ ^ y^^^^ (22B)
TT Jo X -\- k
The factor multiplying yo is unity for Xp = 0, and decreases as Xp increases.
Since y is approximately equal to Pa, we have, approximately,
yo = Pa/FM. (23A)
HOLE CONCENTRATION AND POINT CONTACTS 493
The value of y can also be found for r = 0. For z = 0, r = 0, we have:
v=^^"p-^' = ^G(Xp). (24A)
The integral can be expressed in terms of integral sine and cosine func-
tions:
G{k) = 2 f ^^^ ^ 2 T-cos^ fsi ^ - ") + sin k Ci k
TT Jo X + X TT |_ \ 2/
(25A)
If k is not too large, G{k) is nearly equal to F{k), so that y is approximately-
constant over the area of the disk.
The total current flowing from the contact is found from integrating
kT^ip (dy/dz) over the disk:
/^ = - kT,, yof f *"^°^"^ f "' dl dr (26A)
Jo Jo / -|- A
JO / -f- A
The integral can be evaluated with use of the general integral of Watson,
to give:
I pa = -4pkTnpyo H(\p), (28A)
where
H{k) = f :^iM^^ =. -^ [cos k jm + sin k \\{k)l (29A)
Jo X -\- k I
Using (23 A) for yo, vve have:
I^a = -4pkTfjL,pAH(\p)/F(\p)]. (30A)
Except for the factor H(\p)/F(\p), this expression for the current is
identical with (16A). This factor, which gives the effect of recombination
on the current, is plotted in Fig. 10. Recombination gives an increase in
current flow, but the effect is small for the normal rate of surface recom-
bination, which corresponds to ^ = Xp -^^ .035.
APPENDIX B
Calculation of Hole Flow for Arbitrary Hole Concentration
In the text it was assumed that the concentration of holes was suffi-
ciently small so that the first term in the brackets of Eq. (26) could be
neglected in comparison with unity, yielding Eqs. (28) and (29). We give
494
BELL SYSTEM TECHNICAL JOURNAL
here the general integration of Eqs. (26) and (27) for p arbitrarily large.
Equation (26) may be written in the form:
■gradi/',
where
1^ = kTyip
L6 + 1 {b+ 1)2
Equation (27) then becomes:
W = 0
2bp bib - 1)N, ,_ r ^ (b + 1)A"
(IB)
(2B)
(3B)
The radial solution of this equation corresponding to a total current Ip
is:
lA = -Aoo + /p/2xr. (4B)
n ft.\
-77/2 [Jt (k) COS k + Y, (k) SIN k]
^
Jo(k) COS k + Yo(k) SIN k
^
^
_^
^
^
^
y'^
-^
1
1
1
1
1
1
1
i
0.01 0.02 0.04 0.06 0.1 0.2 0.4 0.6 1.0 2
Fig. 10. — Correction factor for surface recombination.
The constants Ip and ^^ are determined from the boundary conditions
(30) and (31) of the text corresponding to r = Vh and r = x . These con-
ditions give:
kTii
2bp,
2{b - 1)
.6+1 (b +
Tp = lirn (^Pin) - tAj,
= 27rrb
2b{pb - Pi) _ bib - l)Nf ^/ + (6 + Dp,'
i2 ^^ hAT
b-\r 1
ib + 1)-^
bNf + (6 + l)/>,-
(5B)
. (6B)
This equation is the appropriate generalization of Eq. iiS) of the text.
Since the equations are no longer linear, they do not apply strictly to the
added hole concentration. However, if the normal hole concentration, pa, is
small, pu will be negligible in comparison with pi,a and p„ when the equa-
HOLE CONCENTRATION AND POINT CONTACTS 49S
lions are not linear. Accordingly, to a close approximation, we may take
for the added hole current:
lirTh
"26 (^g - pa) _ b{b - 1) bXf + (^> + D^gl .^ .
. b+\ {b + \y-''^bNf+{b+ \)p„y ^
which is the generalization of Eq. (34) of the text.
The value of pba and thus of I pa may then be found by equating this
expression with that of Eq. (3) for I pa. This procedure yields the trans-
cendental equation:
Pba =
2b{pba - pa) _ bib - \)Nf bXf + (6 + Dpba
. b+ \ (6+1)2 ''^bNf+ {b+ 1)^J'
(8B)
where a is again defined by Eq. (6) of the text. This equation must be
solved in general by numerical methods for a particular case. The equa-
tion simplifies for pa either large or small compared with Nj . The latter
case is treated in the text. The opposite limiting case of large hole con-
centrations is treated below.
For pa large compared with X j , the logarithm may be neglected, so
that
p,a = -2ab{p,a - pa)/{b + 1). (9B)
If, as in the text, it is assumed that a is small in comparison with unity,
there results:
Pba = 2abpa/{b + 1), (lOB)
and, using (3):
I pa = -[2b/ {b -f \)]pakTiXpA/H . (IIB)
This differs from (7) by a factor 2b/ {b -\- 1). The equation corresponding to
(8) will have this additional factor, and also the expression for the con-
ductance, G, which, for large hole concentrations is:
G = Go + \2b/{b + \)\{ai5<j,A/bn){pa/no), (12B)
in place of (38) of the text. Equation (16) which relates floating potential
and conductance is general, and applies for arbitrary hole concentration.
Design Factors of the Bell Telephone Laboratories 1553 Triode
By J. A. MORTON and R. M. RYDER
(Manuscript Received Aug. 3, 1950)
TN DEVELOPING microwave relay systems for frequencies around
-■■ 4000 megacycles, one of the major problems is to provide an amplifier
tube which will meet the requirements on gain, power output, and dis-
tortion over very wide bands. As the number of repeaters is increased to
extend the relay to greater distances, the requirements on individual
amplifiers for the system become increasingly severe. A tube developed
for this service is the microwave triode B.T.L. 1553, the physical and
electrical characteristics of which were briefly described in a previous
article.' In the development of such a tube, both theoretical and ex-
perimental factors are involved; illustration of these factors in some
detail is the purpose of the present paper.
Ciiven the application, a number of questions arise at the outset. What
determines the tube type — why pick a triode for development, rather than
a velocity variation tube, or perhaps a tetrode? What electrode spacings
are necessary in such a tube, and what current must it draw? How is its
performance rated, and how does it compare with other tubes? To what
extent can the performance be estimated in advance? What experimental
tests can give more precise information? Some answers to these questions
were obtained by the use of figures of merit, which led up to the choice of
a triode as most promising for development, and which also led to the
subsequent method of optimizing the design for the particular system
application of microwave amplifiers and modulators.
The design process may be said to proceed by the following series of
steps:
1. Formulate the system requirements, frequently with the aid of one
or more figures of merit. The purpose here is to concentrate attention
upon the limitations inherent in the tube alone by eliminating considera-
tions of circuitry or of other parts of the system. The figure of merit
measures tube performance in an arbitrary environment, so chosen as to
be simple, and also directly comparable to the actual system requirement.
2. Make tentative choices of tube type, and analyze further to find out
' J. A. Morton, "A Microwave Triode for Radio Relay," Bell Liboralories Ricord 27,
166-170 (May 1949).
496
DESIGN FACTORS OF THE 1553 TRIODE 497
how the figure of merit depends on the internal parameters of the tube,
such as spacings, current density, and so on.
3. Optimize the internal parameters to make the figure of mer t as good
as possible, with due regard to practical limitations like cathode activity,
life, cost, etc.
4. Use enough experimental checks to make sure the estimates are
sound. Build then the type of tube which appears to fill the require-
ments best, including the practical as well as the technical limitations. The
figures of merit serve now as quantitative checks both of how well the
tube satisfies the application, and also of how accurate is the theory.
Given a good accurate design theory, the whole process could in prin-
ciple be calculated in advance. Such a theory would permit great savings
in effort, since spot checks of relatively few parameters are sufficient to
insure accuracy even when the theory is used to predict a wide range of
phenomena. The extent to which presently available microwave tube
theory meets this need is considerable, as will appear from some of the
results below.
The degree of accuracy required of a theory increases as the develop-
ment process continues. For preliminary estimates, such as deciding what
tube type to develop, the theory can be rather rough and still be satis-
factory. For complete predictions of final performance, only experimental
construction can suffice. By this means the theory can be checked, so that
it can serve future designs with improved accuracy.
The method outlined here is not new, but rather follows standard
practice fairly closely. It does, however, give more than usual quantita-
tive emphasis to the figures of merit, using them to codify the procedure;
and it incorporates a certain amount of quantitative calculation at micro-
wave frequencies. It will be seen that the theory of Llewellyn and Peter-
son needs only some semi-empirical supplementation in the low-voltage
input space, as has already been pointed out by Peterson.^
Preliminary Estimates — Choice of Tube Type
For the New York to Boston microwave relay, an output amplifier
was developed using already available velocity-modulation tubes. ^ With
four stagger-tuned stages, the amplifier proved satisfactory for this
service, and in fact tests indicated that this system could be extended to
considerably greater distances and still give good performance. It was
apparent, however, that these amplifiers would not be satisfactory for a
coast-to-coast system.
* L. C. Peterson, "Signal and Noise in Microwave Tetrodes," /. R. E. Proc. (Nov.
1947).
» H. T. Friis, "Microwave Repeater Research," B. S. T. J. 27, 183-246 (April 1948).
498 BELL SYSTEM TECHNICAL JOURNAL
When this Hmitation became clear several years ago, a study was under-
taken to determine which particular type of electron tube amplifier then
known had the best possibilities of being pushed to greater gain-band
products. The results of this study indicated that a very promising pros-
pect was to build, for operation at 4000 megacycles, an improved planar
triode, that is, one in which the active elements are on parallel planes.
In arriving at this conclusion, two general types of device were con-
sidered: velocity-modulated, as in a klystron, and current-modulated, as
in a triode. (Nowadays, such a study would of course include traveling-
wave tubes.) The conclusions were reached with the aid of the gain-band
figures of merit, along the following lines:
Gain-Band Product
The system performance requirements demand amplifiers capable of
reasonable gains and power outputs over prescribed bandwidths. How-
ever, it is known that bandwidth can be increased by complicating the
circuits (double-tuning, stagger-tuning, etc.). Such factors, being common
to whatever tube may be used, are extraneous to a discussion of tube
performance, and accordingly the tubes are rated by their performance
with simple, synchronous resonant circuits. Furthermore, even then the
bandwidth can be increased at the expense of a corresponding reduction
of gain, by simply depressing the impedance levels of the interstages.
Since the product of gain and bandwidth remains constant, it is a suit-
able figure of merit, independent of the particular choice of bandwidth,
provided the definition of gain is suited to the device.
Unfortunately there is more than one possible gain-band product, the
appropriate form depending on how many simple resonant circuits shape
the band of the amplifier stage. For example, a conventional pentode or a
velocity-variation tube is usually used in conjunction with two high-()
resonant circuits, one each on input and output. If these are adjusted to
give the same Q, then it is well known that, no matter what the band-
width, the product of voltage gain and bandwidth is constant. (See
Appendix 1)
I To I 5 = I I'2i I /2-K\/C~CZ, (1)
Here To is the mid-band voltage gain, B the bandwidth 6 db down (3 for
each circuit), F21 the stage transadmittance, and C\n and Com the total
effective capacitances of the resonant circuits, including the contribu-
tions of the tube*. It is assumed that the stage is matched into trans-
mission lines of some suitable constant admittance level Go .
In amplifiers using triodes such as the B.T.L. 1553 (or tetrodes) in
* As shown in Ai)pendix 1, all (luantilics in equations (1) and (2) are ihe values effective
at the electrodes adjacent to the electron stream.
DESIGN FACTORS OF THE 1553 TRIODE 499
grounded-grid circuits, the situation is different because the Q of the
input circuit is always very much smaller than that of the output. Here
a figure of merit independent of bandwidth is obtained from the product
of power gain and bandwidth:
iro|'5 = I I'2ir/47r6i,Cout (2)
Here Gi„ is the total conductance of the input circuit, including tube con-
tributions*. The gain is again measured with the tube matched at an
arbitrary admittance level Gq. The band, being now limited by only one
tuned circuit, is somewhat different in shape from the above, and is taken
3 db down.
While each figure of merit gives an unequivocal rating of tubes of
appropriate type, the intercomparison of the two types still depends on
the bandwidth. In particular, as the band is widened, the two-circuit type
(klystron) loses gain at the rate of 6 db per octave of bandwidth, while the
one-circuit type (triode) loses only 3 db per octave. Consequently, if the
two devices start with equal gains at some narrow bandwidth, the triode
rapidly pulls ahead in gain as the bandwidth is increased.
The figure of merit equation (1) states that improved klystron per-
formance implies either an increase in transadmittance F21 or a decrease in
the band-limiting capacitances C\n or Cout • According to the simplest
klystron bunching concept,* the transconductance of such a tube may be
increased indefinitely simply by making the drift time longer. Unfor-
tunately, this simple kinetic picture does not take account of the mutually
repulsive space-charge effects which set an upper limit to the useful drift
time by debunching the electrons.^ For a 2000- volt beam in the 4000-
megacycle range, this limit is approximately three micromhos per milli-
ampere. The 402 A tube used in the New York to Boston system has
already approached this limit within a factor of two. Since the capaci-
tances are also quite small, the prospect is quite dubious for any con-
siderable improvement in gain-band merit if the simple klystron type of
operation were to be used.
Improvements are possible in a klystron by changing the manner of
operation so as to lower the drift voltage Fo , because the aforesaid trans-
admittance limit is proportional to Fo~" .* This prospect is also relatively
unattractive. To get transadmittance values anywhere near the triode
would require low voltages and close spacings somewhat like the latter,
and would encounter space-charge difficulties involved in handling a large
current in a low-voltage drift space. Furthermore, the tube would be more
< D. L. Webster, Jour. App. Phys. 10, 501-508 (July 1939).
» S. Ramo, Proc. I. R. E. 27, 757-763 (December 1939).
* The value of n may vary between J/4 and ^^. See reference 5,
500 BELL SYSTEM TECHNICAL JOURNAL
complex, having several grids instead of one. A number of modifications
of klystron operation were considered, but all looked more complex me-
chanically and more speculative theoretically than a triode.
In a triode there is also an upper limit to the transconductance that
can be achieved by spacing cathode and grid more closely. This limit
would be reached if the spacing were so close that the velocity produced
by the grid voltage were of the same order as the average thermal velocity
of cathode emission. The triode limit of some 11,000 micromhos per
milliampere is, however, many times greater than that for ordinary
klystrons. What is still more important is the fact that previous micro-
wave triodes were still a factor of twenty to twenty-five below this limit,
leaving considerable room for improvement. Thus, if mechanical methods
could be devised for decreasing the cathode-grid spacing and at the same
time maintaining parallelism between cathode and grid, it seemed highly
probable that great improvements would be available from a new triode.
The choice to develop a triode for this application was therefore taken
not merely on the basis of simplicity, but also with the expectation that
performance improvements would be not only larger but also more cer-
tainly obtainable than by use of a modified klystron. Moreover, the
possibilities of using the triode over a wide frequency range in other
ways — as a low noise amplifier, modulator and oscillator — lent additional
weight to its choice. By translating the known requirements on gain,
bandwidth and power output into triode dimensions as discussed below,
it was found that the input spacings of existing commercial tubes would
have to be reduced by a factor of about five. In addition, cathode emis-
sion current densities would have to be increased about three times. A
design was evolved in which the required close spacings could be produced
to close tolerances by methods consistent with quantity production re-
quirements. The B.T.L. 1553 tube was the result (Fig. 1). Many of its design
features were adopted for use in the Western Electric 416A tube, which is
an outgrowth of this investigation.
Description of B.T.L. 1553 Triode*
The electrode spacings of this tube and of a 2C40 microwave triode are
shown in Fig. 2. In the 1553, the cathode-oxide coating is .0005" thick, the
cathode grid spacing is .0006", the grid wires are .0003" in diameter,
wound at 1000 turns per inch, and the plate-grid spacing is .012". It is
interesting to note that the whole ini)ut region of the 1553 including the
grid is well within the coating thickness of the older triode.
The arrangement of the major active elements of the tube is shown in
* This section is repeated from reference 1 for completeness.
DESIGN FACTORS OF THE 1553 TRIODE
501
Fig. 3. This perspective sketch has been made much out of scale so that
the very close spacings and small parts would be seen. The nickel core of
the cathode is mounted in a ring of low-loss ceramic in such a manner
that the nickel and ceramic surfaces may be precision ground flat and
coplanar. A thin, smooth oxide coating is applied to the upper surface of
Fig. 1. — The B.T.L. 1553 microwave triode with a cross-section drawing of it in the
background.
the cathode by an automatic spray machine developed especially for this
tube. With this machine, a coating of 0.0005" zb 0.00002" may be put on
under controlled and specifiable conditions. To insure long life with such
a thin coating, it was necessary to develop coatings from two to four
times as dense as those used in existing commercial practice.
The grid wires are wound around a flat, polished molybdenum frame
502
BELL SYSTEM TECHNICAL JOURNAL
ANODE
■////////////////////////////////////////////////////.
ANODE
V//////////////////////////////.
OXIDE COATING
OXIDE COATING
V77777777777777777777777777777777777777777777777777?
CATHODE METAL
CURRENT MICROWAVE TRIODE
V777777777777777777777777777777,
CATHODE METAL
B.T.L.-1553 TRIODE
Fig. 2. — Comparison of the spacings of the 1553 triode at the right with a previously
existing microwave triode at the left.
GRID FRAME
GRID SPACER
ANODE
"^^^ OXIDE
STEATITE
BACKING ROD
MOLYBDENUM
SUPPORT LEG
NICKEL
CATHODE PLUG
RIVET HOLE
STEATITE CATHODE
SUPPORT RING
Fig. 3, Perspective drawing of the active elements of the 1553 close-spaced triode.
DESIGN FACTORS OF THE 1553 TRIODE
503
that has been previously gold sputtered. The winding tension is held
within ±1 gram weight to about 15 gram weight, which is about sixty per
cent of the breaking strength of the wire. This is accomplished by means
of a small drag-cup motor brake, a new method which was developed
especially for these fine grids. The grid is then heated in hydrogen to
about 1100°C, at which point the gold melts and brazes the wires to the
frame. The mean deviation in wire spacing is less than about ten per
Fig. 4. — Physical appearance of the elements comprising the 1553 triode.
cent, and in fact these grids are fine enough and regular enough to be
diffraction gratings as is shown in Fig. 5. In this figure, a fourth order
spectrum diffracted by one of these grids can be seen. The third order,
which should be absent because the wire size is about one-third of the
pitch, is much less intense than the fourth. Proper spacing of the grid is
then obtained by a thin copper shim placed between the cathode ceramic
and the grid frame. Its thickness must be equal to the coating thickness,
plus the thermal motion of the cathode, plus the desired hot spacing.
504 BELL SYSTEM TECHNICAL JOURNAL
The cathode, spacer, and grid comprising the cathode-grid subassembly
are riveted together under several pounds of force maintained by the
molybdenum spring on the bottom of the assembly. The rivets are three
synthetic sapphire rods fired on the ends with matching glass. In Fig. 4,
the parts comprising this assembly are shown in appropriate pile-up se-
quence at the left, and the completed cathode-grid subassembly is shown
at the right between the bulb and the press. The grid-anode spacing of
.012" is easily obtained by means of an adjustable anode plug the sur-
face of which is gauged relative to the bulb grid disc.
Fig. 5. — Spectrum formed by the grid of the 1553 microwave triode.
Table I
Low-Frequency Characteristics
For Vp = 250 V, Ip = 25 ma, Vg = -0.3 V
g„ = 50,000 MHihos
M = 350
r„ = 7000 ohms
Ckg = 10 finf
Cgp = 1 . 05 ^l^if
Ckp = .005 nnf
The higher current density of 180 milliamperes per square centimeter,
the thin dense cathode coating, and the very close spacings, posed a
problem in obtaining adequate emission and freedom from particle
shorts, and had to be solved by quality control methods because of the
large number of factors involved and the precision required. Tubes, sub-
assemblies, and testers have been made in batches and studied by statis-
tical methods. To achieve a state of statistical control on emission, and
freedom from dust particles, it is necessary to process the parts and
assemble the tubes in a rigorously controlled environment. Completely air-
conditioned processing and assembly rooms operating under rigorous con-
trols have been found necessary^ Under such controlled conditions, good
production yields with satisfactory cathode activity have been obtained,
•R. L. Vance, Bell Laboratories Record, 27, 205-209 (June 1949).
DESIGN FACTORS OF THE 1553 TRIODB 505
whereas without such conditions not only was the yield low but it was
difficult to ascertain just what factors were operating to inhibit emission
and to cause cathode-grid shorts.
A summary of the pertinent low-frequency characteristics of the 1553
triode is given in Table I. It should be noticed that, at plate currents of
25 milliamperes, the transconductance per milliampere is about 2000, that
is, about one-fifth of the theoretical upper limit. At lower currents this
ratio is higher: at 10 milliamperes, for example, it is 3000 micromhos per
milliampere. Diodes with the same spacings have about twice these values
of transconductance per milliampere, showing that the grid is fine enough
to obtain fifty per cent of the performance of an ideal grid.
Triode Design Requirements
Analysis of the figure of merit can well begin by devoting attention to
the band-limiting capacitance Cout of the output circuit. First, some ques-
tion may be raised as to the applicability of the concept of a simple
L-C shunt resonant circuit at high frequencies, where the circuit parame-
ters are actually distributed, not lumped. Suppose the actual circuit
admittance is Yx = Gx + jB^. In order to represent it as a simple shunt
resonant circuit of admittance Vp = Gp -\- jo^Cp + 1/joiLp, we need only
require that the two be equal and have equal derivatives with respect to
frequency at the center frequency /o = wo/27r. Accordingly the "effective
values" of the actual admittance are given by the following equations:
Gp = Gx (cco)
Cp = \{Bx + 5x/coo) (3)
1
u
\{(ji(? Ex — OJoBx)
From this development one sees that the representation neglects Gx,
the first derivative of the conductance, but otherwise is correct to first
order as a function of frequency.
There are important cases where this representation as a simple circuit
does not hold. For example, double-tuned circuits having two local reso-
nances have a fundamentally different band shape. However, such compli-
cation of the circuits has been excluded from the figure of merit on the
ground that it is purely a circuit "broad-banding" problem: having de-
termined the performance of the tube for simple circuits, any broad-
banding (double-tuning, staggering, etc.) will give a calculable improve-
ment which does not depend upon the tube. Accordingly, to compare
tubes it is sufficient to consider standard simple circuit terminations,
tuned to the same frequency.
506 BELL SYSTEM TECHNICAL JOURNAL
The total capacitance Cout includes two contributions: from the active
electrode area inside the tube (C22) and from the passive resonating circuit
(Cp2). It is convenient to consider these separately, writing the figure of
merit as follows:
I r. fB = iKi- 7 ^-V7 T^x (4)
47rGii C22
(' + l')C + §^')
The first factor is the "intrinsic" electronic figure of merit of the active
transducer alone, while the second factor expresses the deterioration
caused by input passive circuit loss Gpi and output passive circuit ca-
pacitance Cp2, both of which should ideally be held as small as possible.
Consider the first factor, the intrinsic electronic gainband product
which depends only upon the properties of the electron stream and the
electrode dimensions in the regions occupied by the electron stream.
It is the responsibility of the tube design engineer to maximize this
product consistent with any limitations which may be imposed by me-
chanical, emission, thermal or circuital considerations.
On the other hand, in maximizing this intrinsic gain-band product, the
tube engineer must not proceed in ignorance of the effect of his actions
on the possibility of obtaining a favorable value for the second factor.
For example, he may attempt to make C22 so small (in order to maximize
the first factor) that it becomes physically impossible to obtain an effec-
tive circuit capacitance Cpi which is not large compared to €21 ■ In such a
case, the actual gain-band product would be much smaller than the in-
trinsic product of which the tube would be capable if circuit capacitance
were negligible. Such a balancing of effects will become apparent from
the subsequent discussion.
It is desired, therefore, to express the transadmittance, input conduct-
ance and output capacitance of the electronic transducer in terms of such
parameters as cathode current density, electrode dimensions, frequency
and potentials in such a way that it will become clear how a maximizing
process may be carried out b}' adjusting these parameters.
As a first approximation let us use the results of Llewellyn and Peter-
son's analysis of plane-parallel flow^, which makes the following assump-
tions:
1. All electrons are emitted with zero velocity.
2. All electrons in a given plane have the same velocity.
' F. B. Llewellyn and L. C. Peterson, "Vacuum Tube Networks," Proc. L R. £., i2, '
144-166 (1944).
DESIGN FACTORS OF THE 1553 TRIODE 507
3. The dimensions of the grid are iniinitesimal compared to the elec-
trode spacings.
4. The electrode dimensions are small compared to the wavelength.
It can be shown that the intrinsic gain-band product may be expressed
in the following two ways:
Mi = K
= K'
LdiFsidi),
(5)
[diF^\dWv;i
} where K, K' are parameters which are functions only of frequency.
i
Xi is the cathode-grid spacing in cm
di is cathode-grid transit angle and di — { ^ )
^ \j /
' j = cathode current density in amp/cm''^
6300 X2
do = grid-anode transit angle and 6^ — - — j—f
and Fi (di), 7^2(^2) and F^idi) are complicated functions of their respective
transit angles.
Consider frequency to be given as part of the specifications on the tube.
Variation with Current Density, j
In the first formulation the current densitv is involved only in the
second factor. This factor is a function only oi di — i " r ) and is shown
plotted in Fig. 6. If Xi and X are considered to be held fixed for the moment
the first maximum at ^1 ^ 0 requires j to be as large as possible consis-
tent with emission limitations and life. For the 1553 the cathode current
density is set at 180 ma/cm-.
The other maxima at larger values of di (and smaller values of j),
where Fz{di) goes through zero, correspond to transit angles where Gn — »
0 in the single-valued velocity theory. These maxima cannot be taken at
face value, however, to indicate maxima in the unequal-() gain-band
product since they violate the assumption that Qi << Q2 for which the
formula was developed. To make a study of gain-band variation in this
region therefore entails a study of gain-band product as a function of
bandwidth, as was pointed out previously in connection with comparison
of the equal-() and unequal-(3 cases. Such maxima are of interest pri-
508
BELL SYSTEM TECHNICAL JOURNAL
marily in narrow band cases so that for the present we shall concern
ourselves only with the first maximum at 9i — ^ 0 and j indefinitely large.
4.5
4.0
3.5
„ 30
<6
UL 2-^
1
2.0
\
V
/
I
\
^
J
/
0
'~~~
■ ■
^
y
TT
77
377
277
577
377
777
2
2
2
2
INPUT TRANSIT ANGLE, e, = j"'^3[— x\' A
Fig. 6. — Gain-band product dependence on current density (j), with input spacing
(xi) fixed.
M U-
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
I
\
^^
J
377
2
INPUT TRANSIT ANGLE , &, = X|'''3 ( '-^ j"'''3J
777
2
Fig. 7. — Gain-band jiroduct dependence on input spacing (xi), with current density (j)
fixed.
Vari.\tion with C.\tiiode-Grii) Sp.^cing, .Vi
Now consider that j has been fixed at the largest permissible value
according to the previous section and consider the second formulation
DESIGN FACTORS OF THE 1553 TRIODE
509
for Mi. The spacing .vi is involved only in the second factor which again
is a function only of 9i =
We again have a strong first maximum at
6 —> 0 requiring Xi to be as small as possible (Fig. 7). Other maxima are
indicated at larger values of 6i (and larger values of Xi) again at points
where Gn -^ 0 and the same remarks apply here as were made in the
previous section. For broad-band optima we are therefore interested in
minimum values of Xi.
1.6
1.4
/
^
N
/
\
1
\
1.2
/
\
/
\
S '0
/
\
/
\
LL
/
V
<£'0.8
/
\,
1
\
0.6
1
\
1
0.4
1
1
0.2
1
1
0
1 \
i
OUTPUT TRANSIT ANGLE, ©2 " ^2
377
2
6300
777 277
Fig. 8. — Gain-band product dependence on output spacing (X2).
VARI.A.TION WITH AnODE-GrID Sp.^CING, Xi
The anode-grid spacing x-i is involved only in the third factor of either
formulation. This factor is a function of output transit angle Q-i and
exhibits a maximum for Q = 2.9 radians as shown in Fig. 8. This opti-
mum at a fairly large value of Qi is due to the fact that the capacitance
C22 varies as l/xo whereas the coupling coefficient of the stream to the gap
decreases more slowly at first than the capacitance so that the ratio
y\\lCii improves as the spacing becomes moderately wide. The opti-
mum ^2 corresponds to an optimum value of Xi which of course depends
upon the plate voltage and frequency of operation. For the 1553 at 250
volts and 4000 Mc/s, the optimum output spacing is .022".
510 BELL SYSTEM TECHNICAL JOURNAL
Limitations in Choosing Optimum Parameters
Generally, there are mechanical, thermal, emission and specification
limits which prevent the realization of optimum values for all of the above
parameters simultaneously. A good design is one in which a nice balance
is effected between these various optima and their limitations.
Limitations on Emission Current Density, /
It is generally true that the life of a thermionic electron tube varies
inversely as the average cathode current density in a complicated fashion.
The maximum permissible value of j is therefore always a compromise
between our desire for highest figure of merit and long life. In the present
state of the cathode art as it has been evolved for the 1553 triode it is
possible to operate at a current density of 180 ma/cm- and obtain an
average life of several thousands of hours. It is perhaps of interest to
note that it was necessary to develop much more dense and smooth oxide
coatings in order to make possible such life in the thin coatings necessary
for operation at such close spacings.
Limitations on Cathode- Grid Spacing, xi
Consider the limitations in reaching the optimum in xi. There is, of
course, the obvious one that it is mechanically and electrically not pos-
sible at present to make Xi equal to zero and still retain the essential
features of unilateral controlled space charge flow. Granting then that the
spacing cannot be zero, we must choose the smallest value of xx for which
parallelism and reasonable tolerances can be maintained. To this end in
the 1553 a value of x\ = .0006" is very near this limit with present
structures.
There is, however, at present another limitation which is essentially
mechanical in nature but makes itself felt electrically in a way not indi-
cated in the above simplified theory. This theory has assumed that the
grid dimensions are infinitesimally thin compared to the electrode spac-
ings. However, if this is not the case then the grid has less control action
than an ideal fine grid, and the intrinsic gain band product must be
reduced by still another factor F^ which is a function of the grid trans-
mission factor a = and the ratio '- where p is the pitch distance be-
P P
tween grid wire centers and d is the diameter of grid wires. This function
has the form shown in Fig. 9.*
Thus if the grid pitch and wire diameter are mechanically limited to
some finite though small values, the optimum in input spacing Xi will
* Data transmitted informally from C. T. Goddarti and G. T. Ford.
DESIGN FACTORS OF THE 1553 TRIODE
511
still be for xi — > 0 but will not increase so strongly as x as before but
much more slowly, about as x~ . The grid dimensions should consequently
be made as small as possible while still maintaining a transmission fraction
at no less than 0.5 and at the same time not allowing mean deviations in
pitch more than about 15%.
In the 1553 our best grid techniques today have led to a stretched grid
(which does not move appreciably during temperature cycling) having a
transmission factor of approximately 0.7, a pitch distance of .001'' and a
mean deviation in pitch of less than 15%. For such a grid further de-
creases in input spacing without refining the grid will not pay oflE very
rapidly, since we are on the maximum slope portion of the function F4.
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
RATIO OF INPUT SPACING TO GRID PITCH, X,/p
Fig. 9. — Dependence of gain-band product on grid pitch.
Limitations on Anode-Grid Spacing, xt
In considering the choice of output spacing we must attain a balance
among the following considerations:
a. The optimum transit angle 62 = 2.9 radians requires a spacing which
varies with plate voltage and with frequency. For 250 volts and
4000 Mc/s, this optimum is .022".
b. The anode heat dissipation must be closely watched because the glass
seal in this type of tube is very close to the anode. For the 1553,
a maximum of 50 watts per square centimeter of anode active sur-
face is safe. With a maximum cathode current density of 180
ma/cm^, set by life considerations, heat dissipation limits the plate
voltage to 275 volts unless the current is lowered.
c. If the anode is moved too far out, keeping its voltage constant, then
in order to draw the desired current the grid must go positive,
perhaps drawing excessive grid current. The grid shielding factor ju
cannot be reduced without harming the transadmittance and feed-
512 BELL SYSTEM TECHNICAL JOURNAL
back values; accordingly the cathode current would have to be re-
duced below the maximum permissible from life considerations.
d. The circuit degradation factor (1 -f CpijC-nf'^ becomes more un-
favorable as the active capacitance C22 is reduced by widening the
output spacing. For discussion and calculation of this factor, see
Appendix 2.
e. A wider output spacing, by virtue of the reduced capacitance, per-
mits a higher maximum frequency limit on the tube.
The actual choice of output spacing in the 1553 is .012". This com-
promise between the foregoing factors appears to be suitable at 4000
Mc/s. The output transit angle of 1.6 radians gives 78% of the theoretical
optimum intrinsic gain-band product. The anode dissipation is near the
maximum safe value for the maximum allowable cathode current. The
grid runs very close to cathode potential so that grid current is small.
The circuit degradation factor has a value of about 0.8, while the upper
frequency limit of the tube is satisfactory (about 5000 Mc/s).
The optimum design just described is an attempt to get the best pos-
sible gain-band product in the resulting tube, and is based on a particular
electronic theory (that of Llewellyn and Peterson). Two points remain to
be discussed. (1) What would be the result of optimizing for other merit
figures such as power-band product or noise figure, and (2) how valid is
the theory?
Power-Band Product
The radio relay amplifier requires not only gain, but perhaps even more,
power output. In such a case, the design specification of greatest im-
portance is the bandwidth over which a certain power output can be
obtained with a specified maximum distortion, and is expressed by an
analogous figure of merit, the power-band product.
Of the many methods of specifying distortion, one which is particu-
larly useful in this connection is the "compression", that is, the amount by
which the gain is reduced from the small-signal value. In an amplitude-
modulated system, the compression would be a direct measure of non-
linear amplitude distortion in the amplifiers. In the actual relay, using
FM, compression is an indication that the amplifier is approaching its
maximum limit of power output.
The maximum power output depends not only on how much current
the tube can carry, but also on the magnitude of the load impedance
into which this current works, which in turn depends upon the band-
width of the load. To compare tubes without need of specifying any
bandwidth, one notes that the product of power output and band-
DESIGN FACTORS OF THE 1553 TRIODE 513
width is a constant, a figure of merit. The derivation is outlined in
Appendix 1.
47rCout
The numerator here is just the square of the maximum ac current; that
is, the dc current /20 , multiphed by a factor F{C) depending on the allow-
able compression C, and by the gap coupling coeflftcient F2(02) of the elec-
tron stream to the output gap. The latter is of course a function of the
output transit angle 6-2. . It is assumed that the load is a matched simple
resonant circuit and the band is taken 3 db down.
The power optimum must clearly be somewhat different from the gain
optimum previously discussed. For example, the transadmittance does not
appear here, nor does any property of the input circuit; while the magni-
tude of the direct electron current, which did not appear in the gain-
band product, is now important. The capacitance of the output circuit
appears in both figures of merit.
In terms of internal parameters of the tube, application of Llewellyn
and Peterson's theory along the lines previously discussed leads to the
following expression for power-band product:
Mi (P) = K[Af r~(C)] [02 Fl (62) VVJ (7)
where A is the electrode area, F^{C) is a function of the allowable dis-
tortion limits, K is a, constant which may depend upon frequency, and
the other symbols are as before.
Considering first the dependence on output transit angle and plate
voltage, one sees that this figure of merit has exactly the same form as
the gain-band product. It is, however, not quite safe to assume therefore
that exactly the same output configuration is still optimum, because the
factors entering into the choice of output spacing have not exactly the
same relative importance any longer; for example, a positive grid may be
less objectionable, or a higher plate voltage may be permissible. Still, as a
first approximation one may assume the output configuration to be al-
ready somewhere near optimum.
Other factors of the power-band figure of merit show considerable
difference from the gain-band product. For instance, the electrode area
enters the picture explicitly, suggesting that a larger area tube would
give more power. The current density enters squared instead of only to
the § power; the explicit dependence on input spacing is missing. The
compression function F(C) depends mostly on the input conditions in a
complicated way difficult to calculate. It can be approximated graphically
from static characteristics.
514 BELL SYSTEM TECHNICAL JOURNAL
A power tube similar to the 1553 might therefore be larger in electrode
area, might have a coarser grid and wider input spacing, and perhaps
would differ somewhat in output configuration, particularly if the plate
voltage were raised. Any cathode development permitting a higher cur-
rent density would improve the power output more than the gain, and
might well lead to a drastic anode redesign to permit larger plate dissipa-'
tion.
Similarly, a design to optimize noise figure would lead to still a third
version of the tube, in which one might consider such things as critical
relationships between input and output spacings.
For the 1553 at 4000 megacycles the following quantitative data may
be quoted in order to check the gain-band product estimates.^
I F21 I = 39.10~^ mhos
Gn = 73 • 10~ mhos
Note that the transadmittance is less than the dc value of 45 • 10~'
mhos by only about 15%, while the input conductance, instead of being
equal to the transadmittance as at low frequencies, is almost twice as
large, on account of loading of the input gap by electrons returning to the
cathode. Using the active capacitance C22 of .477 /x/x/, the intrinsic gain
band product is:
VB = F21 V47rGnC22 - 3480 megacycles.
With the somewhat optimistic capacitance degradation factor of .81 com-
puted in Appendix 2, the gain band product would be reduced to 2820
megacycles.
The experimental average value is about 1100 megacycles. The differ-
ence is probably due in part to resistive loss in the passive input circuit,
which may be calculated as follows: Neglecting feedback, the input' cir-
cuit may be represented as containing a resistance i?,, in series with the
short-circuit input admittance gn + jbn . Robertson gives the following
values for these elements:
^11 = 73 • 10~^ mhos
611 = 26-10-3 mhos
Rg ^ 7.6 ohms
Accordingly, the input degradation factor Ru/{Rii + Rs) should be
11.2/(11.2 -f 7.6) = .60, giving a computed overall gain-band product of
1690 megacycles. The best tubes sometimes exceed this figure. Tubes
« S. D. Robertson's measuremcnls at 4000 megacycles, B. S. T. J., 28, 619-655 (Oc-
tober 1949).
' A. E. Bowen and W. W. Miiniford "Microwave Triode as Modulator and Amplifier,"
this issue of B. S. T. J.
DESIGN FACTORS OF THE 1553 TRIODE 515
Vith lower values may have excessive input circuit loss or may have
narrower bandwidth on the input side than has been assumed. Further
measurements, by elucidating this point, might lead to a better design of
tube and circuit.
An entirely similar calculation can be made for the power-band product,
rhe additional assumptions required are that the compression function
F-{C) has the conservative value of |, and the output coupling coefficient
Fo((?2) is taken as 0.9. The power-band product at 4000 megacycles is then
computed to be 50 watt megacycles, which is quite close to the figures
found by Bowen and Mumford.
Refinements of the Electronic Theory
In the electronic computations above, the single-valued theory was used
because it is the simplest theory which describes the high frequency case
at all accurately. The most important discrepancy between the rigorous
theory and the actual situation is the first theoretical assumption listed
above, that the electrons are emitted from the cathode with zero velocity.
For actual cathodes the velocity of emission is not zero nor uniform but
has a Maxwellian distribution such that the average energy away from
the cathode is ^ ^ T*, or about equivalent to the velocity imparted by a
potential drop of 0.04 volt for an oxide cathode at 1000°K. There result
several efifects whose general nature is known but which have not yet
been formulated into a rigorous quantitative theory valid at high fre-
quencies.
(1) A potential minimum is formed at a distance on the order of
.001" in front of the cathode instead of at the cathode as in the
simple theory. This distance is not negligible for close-spaced
tubes; so that, for very close spacings, even perfect "physicists'
grids" approach a finite trans-conductance limit, [van der Ziel,
Philips Research Reports 1, 97-118 (1946); Fig. 2.]
(2) Because the potential minimum implies a retarding field near the
cathode many electrons emerging from the cathode are forced to
return to it. These returning electrons absorb energy from the signal
and also induce excess noise in it, both effects becoming important
at high frequencies.
The effects of initial velocities on the figures of merit can be measured
experimentally. For example, the circuit and electronic impedances of
diodes and triodes at 4000 Mc have been measured by Robertson.* Such
measurements can determine the electronic loading and noise separately
from the circuit degradation effects and are therefore a highly effective
* loc. cit.
516 BELL SYSTEM TECHNICAL JOURNAL
method of circuit design as well. Robertson found that the input circuit
structure of the LS53 produces a measurable impairment in its gain-band
product, which redesign of both tube and circuit may be able to improve.
Comparison of his results with the theory has given a better understand-
ing of the limits of high-frequency performance, and has lent some sup-
port to the following set of rules of thumb which have been in use for
some time:
1. The input loading arising from the returning electrons is consider-
able, the input conductance of these tubes at 4000 Mc being about
double the theoretical value of Llewellyn and Peterson.
2. The input noise of these close-spaced tubes checks well with what
one would expect of a low-frequency diode with Maxwellian veloci-
ties, whose solution is known. In high-frequency noise calculations,
therefore, one can use with some confidence Rack's suggestion that
cathode noise can be regarded as an effective velocity fluctuation at
the virtual cathode.^''
3. Single velocity theory seems to hold well when velocities are much
larger than Maxwellian, drift times are not more than a few cycles,
electron beams are short compared to their diameter, and no exact
cancellations of large effects are predicted. In particular it holds well
for the 1553 output space and for calculations of the high-frequency
trans-admittance.
Extensive calculations of signal and noise behavior in planar multigrid
tubes have been made by L. C. Peterson, using the single-velocity theory
except for an empirical value of input loading, and using Rack's sugges-
tion for cathode noise." The results so far checked have agreed well with
experiment.
In short, the optimum design for the tube is still given fairly closely
by the figures of merit based on the approximate theory, but the per-
formance will fall somewhat short of the predictions of the simple theory;
performance can be estimated with the aid of the experimental measure-
ments and rules of thumb just described.
Summary
From the foregoing calculations we draw a number of conclusions:
1. The figures of merit can be validly analyzed into their dependence
on more elementary properties like transadmittance, circuit capaci-
tance, input loss resistance, and so on.
' loc. cit.
'" A. J. Rack "Effects of Space Charge and Transit Time on the Shot Noise in Diodes,"
B. S. T. J., 17, 592-619 (October 1938).
" L. C. Peterson "Space Charge and Noise in Microwave Tetrodes," Proc. I. R. E., 35,
1202-1274 (November 1947).
DESIGN FACTORS OF THE 1553 TRIODE 517'
2. Even rough calculations, such as the coaxial line approximations
used in Appendix 2 are close enough to the facts to indicate whether
the design is close to an optimum with respect to such parameters
as output spacing, anode diameter, grid diameter, and the like.'
More accurate calculations and experiments can give more precise
answers to these questions.
3. Some considerations such as cathode activity, tube life, heater power
and so on have not yet been included in the analysis. However,
systematic optimization for such parameters as are treated quanti-
tatively is greatly facilitated. In general, each different figure of
merit leads to a somewhat different optimum and hence a different
version of the tube.
The design of tubes by the method of figure of merit has been outlined.
The method is very general, but in essence has just three steps:
1. Formulate the system performance of the projected device with the
aid of a figure of merit.
2. Find how the figure of merit depends upon the parameters of the
tube, such as spacings, current, etc.
3. Adjust the tube parameters, subject to physical limitations, to op-
timize the figure of merit.
Acknowledgments
The development of this microwave triode has required not only the ex-
pert and highly cooperative services of a large team of electrical, mechanical,
and chemical engineers but also the indispensable assistance of skilled tech-
nicians, all of whom worked smoothly together to develop these new ma-
terials and techniques to a point where they are specifiable and amenable
to quantity production. It is not practical to mention all those who have made
significant contributions to this development. The contributions of A. J.
Chick, R. L. Vance, H. E. Kern and L. J. Speck, however, are of such out-
standing nature that mention of them cannot be omitted.
APPENDIX 1
Derivation of the Figures of Merit
Gain-Band Figure of Merit
Let the problem be stated as the design of an amplifier tube to operate
with as large gain over as wide a frequency band as practicable. As a
standard environment, we use a single-stage amplifier working between
equal resistive impedances. For three reasons this standard is suitable: it
is simple; it corresponds closely to practicality in many cases especially
518
BELL SYSTEM TECHNICAL JOURNAL
in the microwave field; and in most cases, it turns out that performance is
limited by the same transadmittance to capacitance ratios as apply when
the source and load impedances are not purely resistive. The terminology
of high frequencies will be used but the analysis applies at all frequencies
under the conditions stated.
Consider the over-all single-stage amplifier of Fig. Al-1 consisting of
input resonator, tube and output resonator, to be a single transducer
0'
Y/////^////////A
£l
B +
INPUT
3'
^y/////////////////////y//////.
I-
'<^v^^'A'<zw<m^
I 11
H C H
Fig. Al-1. — Microwave triode amplifier.
whose gain and bandwidth we wish to relate to the geometry and other
pertinent characteristics of the circuits, bulb and electrode characteristics.
It is instructive to consider the whole transducer to be made up of three
transducers in tandem as follows:
1. The input passive transducer, extending from the externally avail-
able input terminals (jierhaps located somewhere in the driving wave
guide or coaxial line) up to the internal in[)ut electrodes right at the
boundary of the electron stream. Call this transducer T, ; in the
DESIGN FACTORS OF THE 1553 TRIODE
519
case of the grid-return triode of Fig. 1 it begins somewhere in the
input wave guide at 0-0' where only the dominant wave exists,
includes the input external cavity and that portion of the tube in-
terior right up to but not including the cathode-grid gap adjacent
to the electron stream at 1-1'.
The output passive transducer, extending from the externally avail-
able output terminals located in the output wave guide through the
output part of the bulb right up to the internal output electrodes at
the boundary of the electron stream. Call this To ; in the triode it
Y,2V2 Y2,V,
t \ 2' N2:i 3'
Fig. Al-2. — Amplifier representations.
extends from somewhere in the output wave guide at 3-3' where
only the dominant wave exists, includes the external coupling window,
resonator cavity and output portion of the bulb, right up to the
grid-anode gap adjacent to the electron stream at 2-2'.
3. The active electron transducer enclosing everything between the
internal terminals of the above two passive coupling transducers —
call this Te — in the triode it extends from the cathode-grid gap
adjacent to the electron stream at 1-1' to the grid-anode gap adja-
cent to the electron stream at 2-2'. Geometrically it includes the
stream and active portions of the electrodes. The term "active"
will be applied to the electron stream and to those portions of the
electrodes which interact directly with the stream.
520 BELL SYSTEM TECHNICAL JOURNAL
We may represent these three transducers as in Fig. Al-2a, where the
input and output transducers have each been replaced by an ideal trans-
former of turns ratio N and a shunt admittance Yp . This representation
is general enough for present purposes, provided that Yp and N are
allowed to be complex functions of frequency and provided that terminals
0-0' and 3-3' are chosen so that a potential minimum occurs at those
points when points 1-1' and 2-2' are shorted.
The short-circuit admittances for the whole transducer as seen at ter-
minals 0-0' and 3-3' are then
Y*n = N\ (Fn + Yp,)
F*22 = Nl (F,2 + F,2) (Al-1)
F*2i = A^A^2F2i
F*io = i\^A^2Fi2
where the Y a are the short-circuit admittances of the electron transducer
alone as seen at terminals 1-1' and 2-2'.
If the feedback admittance F12 is assumed negligible the insertion volt-
age gain may be written as
^, . 2N,A\Yn
Go(l + ^i)(l + CT2)
where the sigmas are admittance-matching factors:
iV?(Fu + Fpi) Y*n Al(F2o -|- Yp,) F22* ,,. .>,
T^i = "TT- , o'2 — ^ p7- V^i-'i;
The gain is maximum when a, , o-o are minimum, i.e., when tube and
circuits are resonant and losses are minimum.
We may rewrite this in terms of the total F*,> as follows:
tro(l + CTi)(l + 0-2J
Many practical cases are well approximated by the more special repre-
sentation of Fig. Al-2b, where the turns ratios of the ideal transformers
are real and independent of frequency, and the shunt admittance consists
of ordinary lumped constant circuit elements. The feedback admittance
F12 is neglected.
This representation as simple, lumped-constant elements holds very
well for any admittance, even a distributed, cavity-type microwave
circuit, or an electronic admittance, provided that the combined circuit
has no series and only one shunt resonance near the frequency band in
DESIGN FACTORS OF THE 1553 TRIODE 521
question. The "effective values" of the actual admittance are given by
equations (3) of the text, as follows:
Gp = Gx (wo)
Cp = i {B'x + 5./a;o) (Al-4)
1 , ,
T" = 2 ('•^c' Bx — COo Bx)
Let the complete admittances across nodal pairs 1-1' and 2-2' be called
Fie and Yte as in Fig. Al-2c, which is an abbreviation of Fig. Al-2b from
the point of view of the active transducer.
1
Yu = Gi 4- Gpi + Gil + iwCpi + ywCu + . ,
JwL,p\
1
= Gie + ]wCu + :
Y2e = Gi -\- Gp2 + G22 + i<^Gp2 + 7C0C22 +
jojLu (Al-5)
1
j(j}Lp2
— G-ie + j(j:C2e +
jwL^,
where Gi and G2 are the line admittances as seen from the active trans-
ducer:
Gi = G,/N\;G2 = Go/iVL
The Q's of the circuit are defined as
Qu = Cc'O Cie/Gie (Al-6)
Qle = CCo C^JGle
The insertion voltage gain (2) may be written as follows to emphasize
the manner in which it depends upon frequency:
_ 2F21 / GieGze /.. ^x
YuY-i. r (1 + Mi)(l + M2) ^ ^
Here /x == o'(coo) is the matching factor at band center. Frequently the
circuits are matched (mi = M2 = 1) to avoid standing waves in system
applications, and we shall discuss this case; but in any case mi and ^2
are constants with respect to frequency. For our standard circuits, Gie
and G^e are independent of frequency; also ordinarily the transadmittance
F21 may be considered constant for bandwidths commonly encountered.
There results then the fact that^the voltage gain (and phase) depends on
frequency Jn the sameway as (Fi^ F2e)~^
522 BELL SYSTEM TECHNICAL JOURNAL
Since the gain varies with frequency, the amplifier will give approx-
imately constant response only within a certain range of frequencies.
The band of the amplifier is defined as that frequency interval within
which the magnitude of the gain is constant within some specified toler-
ance; the bandwidth is the size of this interval. We wish to express the
gain of the amplifier in terms of its bandwidth, in the following way:
The voltage gain of this amplifier has a maximum, called To , at band
center frequency /o . Take the band of the amplifier Bk{A) as that interval
within which the voltage gain is within a factor of 1/N times the maxi-
mum.
r(co)
We can analogously define the band of a simple circuit J5„(C) by the
relation
> i- defines Bn{A) (Al-9)
"~ N
^^2«(t0o)
> i defines 5„(C). (Al-9)
n
It follows directly that
BniC) = -^ Vn^ - 1 . (Al-10)
Since the amplifier gain is inversely proportional to the product of the
circuit admittances, it follows that «i n2 = N.
The intrinsic bandwidth resulting from the tube admittance may not
be suitable for the intended application. In that case the band may be
widened by increasing Gu or Gjp with a corresponding decrease in gain.
We have then the problem of adjusting Gu and G^e for greatest band ef-
ficiency, i.e., maximum gain for a given bandwidth, with synchronous
tuning. It turns out that if the bandwidth is less than that needed, then
the circuit of higher Q should be lowered until either (a) the band be-
comes wide enough, or (b) the Q's become equal. In case (b), both Q's
should then be lowered, maintaining equality, until the band is wide
enough.
Two important limiting cases are to be considered: (a) Qu = Qie , i.e.
the band is shaped equally by the input and output circuits; and (b)
Qu << Qif , i-e. the band is shaped by only the output circuit. In the
equal-Q case we have
G\e _ Gie
n = N (Al-11)
Bs(A) = ^ Jpp^N-^
1
DESIGN FACTORS OF THE 1553 TRIODE 523
If only the output circuit is involved, then N = n^ and the band of the
amplifier, being shaped differently, is given by a different relation:
BAA) = ^ VW^^l. (Al-12)
In other words, a band shaped by only one circuit has the shape of (12),
while a band shaped by two circuits has the shape (11). The maximum
voltage gain is
2 I F21 I
I To I = I r(coo) I = ^/r r (^ -l \(\ j. ^ (Al-13)
Substituting for the G's in terms of the bandwidth, we have for the
equal-(2 case (from 11)
I r. i = ^-^ , ^^^^^ 1 (Ai-14)
27rVCieC2. V(l + Mi)(l + M2) ^^
and for the unequal-Q case (from 12)
These equations give the relationship between the gain and bandwidth
of a transmission system shaped by two or one independent circuits,
respectively. The comparison between these two cases is not quite straight-
forward. First, the band shapes (11) and (12) are different, although this
difference is small enough to be ignored for iV < 2 (6 db down). Second,
the gain varies differently as the band is widened; the equal-Q case loses
gain at 6 db per octave in bandwidth, the unequal-(2 case only 3 db
per octave. The comparison therefore depends on the bandwidth chosen.
However, these formulas are still quite useful, especially in comparing
two amplifiers of the same type or in optimizing an amplifier of one of
the types.
From the equal-<2 formula one notices that the product of insertion
voltage vain and bandwidth does not depend on the bandwidth, but is a
figure of merit by which two amplifiers of the same type (i.e. equal Q)
but different gains and bandwidths can be compared. Since
Cu = Cii + Cp\ ; Cie = C22 + Cp2
•--(w;„J(^,^c,Y,^£
(
2\/iV
V(l + Mi)(l + M2)>
(Al-16)
524 BELL SYSTEM TECHNICAL JOURNAL
This expression for the gain-band figure of merit of a two-circuit, line-
to-line amplilier is particularly useful for grounded-cathode pentodes and
klystrons. It is the product of three factors. The first may be called the
electronic figure of merit because it depends only upon electron stream
parameters (ratio of transadmittance to mean capacitance of the elec-
tronic transducer Te). The second is the degradation factor giving the
effect of adding passive circuit capacitance both inside and outside the
bulb to the active capacitance already present in the electronic transducer.
The third factor, called the matching factor, depends only on the matching
conditions and on the arbitrary definition of bandwidth. If the band is
taken 6 db down (3 db for eacy circuit) and the tube input and output
are matched, the third factor is unity.
In amplifiers using triodes and tetrodes in grid-return circuits, the Q
of the input circuit is usually very much smaller than that of the output.
Here it is appropriate to use the single-circuit limiting concept, with
Qu << Qie . Here a figure of merit independent of bandwidth is obtained
from the product of power gain and bandwidth:
Top Bs =
This expression for the gain-band figure of merit of a one-circuit, line-
to-line amplifier is also the product of three factors. The first is again the
intrinsic electronic figure of merit of the active transducer alone; the
second is the degradation produced by the addition of passive circuit
capacitance to the output and circuit loss to the input; the third is a
band-definition matching factor which is unity when the band is taken 3
db down and the tube is matched.
In the application of the figures of merit, the third factors are usually
omitted, since they depend only on the matching conditions and on the
particular definitions of bandwidth used.
Power-Band Figure of Merit
In the problem of power output amplifier stages, the design specifica-
tion of greatest importance is the bandwidth over which a certain power
output can be obtained with a specified maximum of distortion. Of the
many methods of specifying distortion, one which is particularly useful
for microwave systems is known as the "compression". If the power gain
is plotted in decibels as a function of the power output, as shown in Fig.
DESIGN FACTORS OF THE 1553 TRIODE
525
Al-3, it will normally be constant for low power levels (for which the
device is essentially linear) and equal to the low level power gain | F |'^ .
However, at some higher power level non-linearities appear in some or all
of the various short-circuit admittances, usually causing the power gain
to decrease below the small-signal value by an amount called the com-
pression, C. If Po/Pi be power gain for any power output and | F |-
0
1
1
1
9
8
-^
^
""
7
10 12 14 16
POWER OUTPUT IN DBM
Fig. Al-3. — Typical gain variation with power output.
DC OUTPUT CURRENT
Fig. Al-4. — Compression vs.
Alternating current in output
Direct current in output
the small-signal power gain, the compression C is defined in decibels as
follows:
C = 10 logio
= 10 logio
10 logio Po/Pi
(Al-18)
Pi
Naturally, the compression depends upon how hard the tube is driven.
It is therefore a function of the amount of drive, which may be con-
veniently expressed in terms of the ratio of the alternating output cur-
rent to the operating direct current, as in Fig. Al-4.
526 BELL SYSTEM TECHNICAL JOURNAL
Tlie power output depends on operating parameters thus:
As the output power level is continually raised, more and more cur-
rent is required to drive the load, until finally the non-linear distortion
limit is reached. The maximum output current is therefore limited to a
certain proportion of the direct current hn , thus:
hm = hjrFiO F,(d.,) (Al-2n)
where F{C) shows the dependence upon the compression C and will
naturally be the larger, the more the allowable compression. ^2(^2) indi-
cates a dependence upon output transit angle; it is the output gap cou-
pling coefficient.
The power output depends also upon the output circuit conductance
G2 and can be greater if G2 is smaller. However, a smaller Go implies a
smaller bandwidth. It results that the power is inversely proportional to
the bandwidth of the output circuit, or in other words, the product of
power output by the bandwidth of the output circuit is a constant— a
figure of merit of the tube. As in the case of the gain-band merit, this
also can be broken up into factors:
Po • Bn
(i\.F'{c)Fl{d,)\ ( 1 \ (iVm - l\ , .
V 47rC22 / \1 + C,,/Cj V 1 + M2 / ^ ^
This expression for the power-band figure of merit is the product of
three factors. The first is the intrinsic figure of merit of the active trans-
ducer alone; the second is the degradation caused by the addition of pas-
sive circuit capacitance to the output circuit; the third is a band defini-
tion— matching factor which is unity when the output is matched and the
band of the output circuit is taken 3 db down.
The power-band computation does not depend upon the input circuit.
Variations in the latter affect the gain of the amplifier, but not its over-
load point. Accordingly in the power band formula only properties of the
tube and its output circuit appear. When feedback has to be considered,
then the input circuit also affects the power, and the analysis becomes more
complicated.
We have now three figures of merit: namely, two gain-band products
applying to different kinds of amphfiers, and one power-band product.
They relate the performance of an amplifier to certain internal parameters.
For wide band service, the tube design should make the appropriate
figure of merit as large as practicable,
DESIGN FACTORS OF THE 1553 TRIODE
527
It should be understood that many other factors may have a bearing
on ampHfier design, such as power consumption, noise performance or
amount of feedback. Where such factors are important, they too must
be considered, and frequently appropriate merit figures like plate efficiency
or noise figure are useful.
APPENDIX 2
The Circuit Capacitance Degradation Factor
The capacitance degradation factor C-n/iC^i + Cp-i) which applies to
both gain-band and power-band products, can be calculated approximately
5
4
I '
z
-„ 2
rvj
Q.
O 1.5
1.0
0.8
0.6
O.S
0.4
0.2
0.15
0.1
\,
N
■\
THREE-QUARTER-WAVE TUNING
-
V
>
s.
\
\,
N
*\.
GAP CAPACITANCE C22
\
\,
\
\,
\
QUARTER-WAVE TUNING
N
\,
N
\
>k
1000 2000 3000 4000 5000
FREQUENCY IN MEGACYCLES PER SECOND
Fig. A2-1. — Passive circuit capacitance Cps.
as shown below. As the frequency is varied, this factor changes by con-
siderable amounts for the 1553 tube; accordingly, both figures of merit
vary with frequency, and design control has been exercised to produce
maximum merit around 4000 megacycles.
The capacitance degradation factor is just the proportion which the
active tube capacitance bears to the total capacitance of tube and circuit,
and would therefore have a maximum of unity if the circuit passive
capacitance were made zero. For the 1553, we may begin by assuming that
the plate circuit is to be tuned by a resonant coaxial line. As the fre-
quency is lowered the efifective capacitance will be increased, since the
line must be lengthened; its variation is shown in Fig. A2-1.
528
BELL SYSTEM TECHNICAL JOURNAL
The calculation is based on the following assumptions (Fig. A2-2):
1. The output cavity has inner diameter .180", outer .850", conse>j
quently a characteristic admittance Go :
Go = 7250/log J = 10,710 micromhos
(A2-1]
2. The gap capacitance is that of a parallel plate condenser of .180"
diameter and .012" spacing, namely
C22 = tnA/d= 0.477 mm/ (A2-2)
3. The effect of the glass vacuum envelope is neglected for simplicity.
/ OOij'i^
E LINES
Fig. A2-2. — Output cavity dimensions. A, B are concentric cylindrical portions. Actual
lines of electric force are partly dotted into sketch.
(Consequently the length 1 of the line is given by the well-known tuning
relation
C0C22 = Go cot 6 — Gq cot
0)1
(A2-3)
The distributed capacitance of the line is determined from the formulas
(3) of the text, which in this case reduces to the following:
?^ I 1 4- '^L^ 1
2~V "^"gT/
-CV2 = -2-11 +
o;C22
(A2-4)
The cavity distributed capacitance is thus comparatively easy to calcu-
late at high frequencies because of the simplicity of the geometry. At low
frequencies the computation of the distributed capacity of a coil is no
DESIGN FACTORS OF THE 1553 TRIODE
529
different in principle, but would be harder to carry out in practice be-
cause of the helical geometry. The value can of course in any case be
found by measurement of the tuning admittance as a function of fre-
quency. From these equations the circuit degradation factor can be calcu-
lated, and is shown in Fig. A2-3 as a function of frequency.
The accuracy of the coaxial line assumptions decreases as the cavity
becomes shorter. For 4000 and 6000 megacycles, since the length of the
cavity is less than its diameter, it would be more nearly correct to regard
it as a radial transmission line loaded by the inductive "nose" in the
0.9
15 0.4
0.3
0.1
-^
^
^
y
QUARTER-WAVE TUNING
/
/
/
/
/
/
THREE-Ql
JARTER-WAVE TUNING
-|
2000 3000 4000 5000
FREQUENCY IN MEGACYCLES PER SECOND
Fig. A2-3.— Capacitance degradation factor,
C22 +
center. The admittance of such a cavity can be calculated^^ or measured;
but the additional precision hardly warrants the effort in the present case.
The capacitance degradation factor at 4000 megacycles is indicated from
Fig. A2-3 as .81, or only 0.9 db less than the intrinsic limit of unity if the
passive capacitance were entirely negligible compared to the active 0.5
ii\ij. This indication is somewhat optimistic, as appears from Fig. A2-2.
The coaxial line formulas assume that the capacitance corresponds to a
radial electric field between concentric cylinders A and B. This capaci-
tance is found to be quite small (.11 ii\ij at 4000 Mc). The actual lines of
** S. Ramo and J. R. Whinnery, "Fields and Waves in Modern Radio," N. Y., Wiley,
1944.
530 BELL SYSTEM TECHNICAL JOURNAL
force, dotted in the figure, clearly correspond to a somewhat larger ca-
pacitance, especially when the length of the cavity is smaller than its
diameter; but this larger capacitance is probably still less than the active
capacitance Cn-
In so far as the gain-band product depends on the circuit capacitance
degradation factor (Fig. A2-3), the curve is probably fairly accurate up to
2000 megacycles and somewhat optimistic for higher frequencies where the
coaxial line predictions are evidently too small.
Above 5000 megacycles the quarter-wave tuning cannot be used for the
1553 tube since the glass would interfere with the tuning plunger. A
glance at Fig. A2-3 shows that moving the plunger back a half-wave to
the next node involves a drastic loss in gain-band product— a factor of
four at 6000 megacycles — because of the great increase in circuit passive
capacitance. Redesign of the tube for good figure of merit at 6000 mega-
cycles would therefore require the use of first-node tuning. A reduction in
outer diameter would be necessary, and the use of an internal pre-tuned
cavity might also be indicated.
A New Microwave Triode : Its Performance
as a Modulator and as an Amplifier
By A. E. BOWEN* and W. W. MUMFORD
(Manuscript Received Mar. 20, 1950)
This paper describes a microwave circuit designed for use with the 1553-
416A close-spaced triode at 4000 m.c. It presents data on tubes used as amplifiers
and modulators and concludes with the results obtained in a multistage amplifier
having 90 db gain.
Introduction
1\ /T ICROWAVE repeaters are of two general types: those that provide
-^^-''amplification at the base-band or video frequency and those that
amplify at some radio frequency. Of the latter there are two types: those
that involve no change in frequency and those that do involve a change
in frequency, that is, the radiated frequency is different from the re-
ceived frequency. The Boston-New York link^ is of this last type as is
also the New York-Chicago link. This paper deals chiefly with a discus-
sion of the application of the close-spaced triode^ in a repeater of the type
to be used between New York and Chicago.
A block diagram of this type of repeater appears in Fig. 1. The received
signal comes in at a frequency of, say, 3970 mc. It is converted to some
intermediate frequency, say 65 mc, in the first converter which is associated
with a beating oscillator operating at a frequency of 3905 mc. After ampli-
fication at 65 mc it is converted in the modulator back to another micro-
wave frequency 40 mc lower than the received signal and then it is ampli-
fied by the r.f. amplifier at 3930 mc and transmitted over the antenna
pointed toward the next repeater station. Our attention will be focussed
upon the performance of the close-spaced triode in the transmitting
modulator and in the r.f. power amplifier in this type of repeater.
The close-spaced triode was assigned the code number 1553 during its
experimental stage of development and, with subsequent mechanical im-
provements, it became the 416A. Some of the data reported herein were
taken on one type, and some on the other; references to both the 1553
and 416A tubes will be noted throughout the text. The difference in
electrical performance was not significant.
An early experimental circuit for the 1553 type tube will be described
* Deceased.
531
532
BELL SYSTEM TECHNICAL JOURNAL
in detail and the performance as amplifier and modulator will be pre-
sented. Measurements of noise figure will be included with a discussion
of the performance of multistage amplifiers.
RECEIVED
SIGNAL
FIRST
IF.
AMP
65 MC
SECOND
CONVERTER
R.F.
AMP
3930 MC
TRANSMITTED
SIGNAL
3970 MC
CONVERTER
MODULATOR
3930 MC "
BEATING
OSCILLATOR
3905 MC
BEATING
OSCILLATOR
3866 MC
Fig. 1. — Typical microwave repeater.
INPUT
WAVEGUIDE.
OUTPUT
WAVEGUIDE
COAXIAL TO
- WAVEGUIDE
TRANSDUCER
X/4 COAXIAL
TRANSFORMER
PLATE CIRCUIT
RESONANT CAVITY
INPUT CIRCUIT
RESONANT CAVITY
Fig. 2. — Microwave circuit for 1553 triode.
The Microwave Circuit
The experimental circuit which has to date met with greatest favor
consists of cavities coupled to input and output waveguides, as shown in
Fig. 2. The grid, of course, is grounded directly to the cavity walls and
separates the input cavity from the output cavity. An iris with its orifice
A NEW MICROWAVE TRIODE
533
couples the input waveguide to the input cavity and is tuned by a small
trimming screw across its opening. The metal shell of the base of the
tube makes contact to the input cavity through spring-contact fingers
around its circumference and forms a part of the input cavity. The
cathode and its by-pass condenser, located within the envelope, complete
the input circuit cavity. The heater and cathode leads, brought out
through eyelets in the base of the tube, are isolated from the microwave
Fig. 3. — Model eleven microwave circuit for the close-spaced triode looking into the
output waveguide.
energy in the input cavity by means of the internal by-pass condenser.
When the tube is used as a modulator, this by-pass condenser acts as a
portion of the network through which the intermediate frequency signal
power is fed onto the cathode.
The output circuit cavity is coupled to the output waveguide through a
coaxial transformer, a coaxial line and a wide-band coaxial-to-waveguide
transducer. The output cavity is bounded by the grid, the coaxial line
outer conductor, the radial face of the quarter-wave coaxial transformer
534 BELL SYSTEM TECHNICAL JOURNAL
and the sealed-in plate lead of the tube. The plate impedance of the tube
is transformed by the resonant cavity to a very low resistance (a frac-
tion of an ohm) on the plate lead just outside the glass seal. The quarter-
wave coa.xial transformer serves to match this low impedance to the surge
impedance of the coaxial line {AS ohms). Coarse tuning is accomplished
by moving the slug of the outer conductor; fine tuning by moving the
inner conductor. The coaxial line is supported at its end by a dielectric
washer. Plate voltage is applied to the tube through a high impedance
quarter-wave wire brought out to the low impedance probe through the
side wall of the waveguide. Both the modulator and the amplifier used
this type of circuit, which we call model eleven.
Fine tuning of the plate cavity is obtained by sliding the inner con-
ductor of the coaxial transformer up and down on the plate lead. This
movement is derived through a low-loss plastic screwdriver inserted
through the hollow probe transducer; the driving mechanism is housed
inside the inner conductor of the transformer, thus isolating the mechani-
cal design problem from the electrical design problem effectively. The
hollow stud at the top of the structure serves two purposes: screwing it
into the waveguide introduces a variable capacitive discontinuity which
serves to improve the match between the cavity and the waveguide. The
length of the hollow plug provides a length of waveguide beyond cutoff
which keeps the r.f. energy from leaking out through the plastic tuning
screwdriver.
The heater and cathode leads from the tube are housed in a cylindrical
metal can and are brought out through by-pass condensers to a standard
connector. The photograph, Fig. 3, illustrates these features.
The input face of the circuit is illustrated in Fig. 4. The long narrow
slot near the base of the rectangular block is the iris opening which
couples the input waveguide to the cathode-grid cavity. The single tuning
screw provided at the input iris is not adequate to match all of the tubes
over the whole frequency band of 500 megacycles; an auxiliary tuner
shown at the right of the circuit provides the necessary flexibility. This
tuner, described by Mr. C. F. Edwards of the Bell Telephone Labora-
tories^ is, in effect, two variable shunt tuned circuits about an eighth of a
wavelength ai)art in the waveguide. Each variable tuned circuit is made up
of a fixed inductive post (located off center in the waveguide) and a
variable capacitive screw. It is capable of tuning out a mismatch corre-
sponding to four db standing wave ratio of any phase.
As shown in Fig. 5, the tube slides into the bottom of the circuit and
the grid flange is soldered lo the wall of the cavity with low melting point
.4 NEW MICROWA VE TRIODE
533
solder.f The shell of the tube is grasped by the springy contacts around
the bottom of the input cavity. Above the tube the plate lead projects
into the cylindrical space which can be adjusted to the desired size by the
quarter wave slug seen to the right of the circuit. This makes contact to
the walls of the outer cylinder by spring fingers on each end. Contact to
the plate lead is then made through the movable slotted inner conductor,
seen on the extreme right of Fig. 5.
The input face of the circuit.
Figure 6 gives an exploded view of the details of the circuit, showing the
simplicity of the construction which permits easy assembly. The guide pin
which serves to keep the inner conductor of the transformer from ro-
tating as it slides up and down on the plate lead during the tuning process
can be seen on the third detail to the right of the main block. Also there
is provision for external resistive loading to be introduced into the plate
cavity through the small square holes in each side of the block. A screw
mechanism adjusts the penetration of the loading resistive strip into the
t The early experimental tubes were soldered into the circuits. Chiefly through the
efforts of Mr. C. Maggs and Mr. L. F. Moose, of B. T. L., who undertoo'.c the dsvelopmsnt
of the tube for production by the Western Electric Company, the present ■iI6A tubes
come with a threaded grid tiange to facilitate replacement.
536
BELL SYSTEM TECHNICAL JOURNAL
plate cavity to provide for a limited adjustment of the bandwidth of the
circuit. These are not always used, however, and most of the data to be
presented here are for the condition of no external resistive loading.
Fig. 5. — Bottom view of circuit.
Fig. 6. — Exploded view of details.
f2±f,
Fig. 7. — Elementary grounded grid converter schematic.
Modulator
The grounded grid transmitting converter shown schematically in Fig.
7 includes the two generators, a microwave beating oscillator, /o, and an
intermediate frequency signal, /i, which impress voltages on the cathode,
A NEW MICROWAVE TRIODE
537
the grid itself being grounded. The output circuit in the plate is tuned to
the sum or difference frequency, fz ± fv
By-pass condensers, traps and filters for other frequencies present in the
modulator must be considered. Besides the beating oscillator and the
signal, their sum and difference frequencies appear in both the input
circuit and the output circuit and of course bias voltage on the cathode
and plate voltage on the plate must be applied. Some of the traps and
by-pass condensers which influence the converter performance are in-
dicated in Fig. 8. It is obvious that microwave energy should be kept
from flowing into the i.f. signal circuit and vice versa if the highest con-
version gains are to be obtained. Both of these conditions are easily
achieved. It is not so readily apparent that the components of the wanted
and the unwanted sidebands present in the input circuit must be handled
f f
f,+f, " f,-f,
T TTT
^2~'fl *B +
Fig. 8. — Schematic diagram of converter with traps and filters for fundamental fre-
quencies of signal,/], beating oscillator,/;, and sidebands, /2 ±/i.
properly. Of these two, the more important is the wanted sideband and
the next figure illustrates just how necessary it is to treat it properly.
The simplest way to keep the wanted sideband component of the input
circuit from being absorbed by the beating oscillator branch is to reflect
the energy back into the converter by means of a reflection filter. This
reflected energy arrives back at the tube and may conspire to reduce the
conversion gain of the modulator if the phase is wrong. The phase de-
pends upon the spacing along the waveguide between the tube and the
filter and Fig. 9 illustrates how badly the gain is affected when the wrong
spacing is used. Data for two different tubes are given which indicate
that the correct spacing for one tube may be incorrect for another. It
should be pointed out, however, that these two tubes were early experi-
mental models and that production tubes behave more consistently.
The i.f. impedance of the modulator is also affected by the filter spac-
ing for the wanted sideband on the input. This effect can be utilized to
538
BELL SYSTEM TECHNICAL JOURN^iL
vary the i.f. impedance by small amounts to achieve a better i.f. match,
since the proper spacing for best gain is not a critically exact dimension.
That is to say, there is a fairly large range of spacings which give good
performance as far as conversion gain is concerned so that, as long as the
critical distance which gives poor gain is avoided, the i.f. impedance can
be adjusted by varying the spacing of the input filter.*
It is imi)ortant that the i.f. impedance of the modulator be adjusted to
match the impedance of the 'i.f. amplifier which drives it, since any mis-
match would cause a degradation of the system performance. In the
design of the matching transformer the inductance of the leads, the
capacity of the tube and by-pass condenser and the resistance of the elec-
a
L
r
>>v— ^
>-—
1
J. -
V
7
i 6
o
LU
^ 5
z
z
< ^
IB
03
in
a.
f2
Z
o
o
-\
/
t
1
^
\
i
/
/
i
N=--TUBE
' NO. SB 169
1
»
\
,<1
TUBE---
NO. PS 348
.,-^
>
1
0
1
0.25 0.50 0.75 1.00 1.25 1.50
FILTER SPACING IN INCHES
Fig. 9. — Data showing the effect of the sjjacing of a rejection filter for the wanted side-
band in the input circuit.
tron stream were measured at the base of the tube. A broad-band trans-
former was designed and the inductances were thrown into an equivalent
T network, thereby utilizing the lead inductance inside the tube as a
part of the transformer, absorbing it in the L2-M branch as indicated in
Fig. 10. In several experimental tubes the lead inductance was .OAnH.
The impedance match obtained with such a transformer gave less than
two db SWR over a band from 55 to 75 mc with the loop at the cusp on
the reflection coefiicient chart characteristic of slightly over-coupled tuned
transformers as shown in Fig. 11.
The broadband matching of the output circuit of the modulator re-
quired a different technique. Xot only is this filter called upon to pro-
vide a broad-band imi)edance match, but also it should provide dis-
* The spacing of the input filter also affects the plate impedance in a complicated way.
A NEW MICROWAVE TRIODE
539
crimination against the other microwave-frequency components present in
the mockilator output circuit; consideration of the beating oscillator and
Z -L
Li-M
^WT^
■L2-M-
■c,
TRANS -
) FORMER
^WT^
-LEAD INDUCTANCE
TUBE
capacity;
AND TRAP
MODULATOR ^
:C2 RESISTANCE<R2
AT 65 MC
]fi^ MODULATOR TUBE
Fig. 10. — Equivalent circuit of modulator at I. F.
^^^h
eHOtHS TOWARD GC:,^
0.38 0.36
Fig. 11. — Modulator I. F. impedance with transformer.
both sidebands is necessary. The variables at our disposal are the band-
width of the modulator output circuit and the number of cavity resona-
tors which follow it. The desired quantities are the specified transmission
540
BELL SYSTEM TECHNICAL JOURNAL
bandwidth and the attenuation required at the beating oscillator fre-
quency. With two equations and two unknowns, the maximally-flat filter
theory was applied to the circuit shown schematically in Fig. 12.'' This
indicated that an output circuit bandwidth of 84 mc (to the three db
loss points), associated with two external resonant branches having band-
widths of 42 and 84 mc respectively, were needed to obtain a 20 mc flat
band with 30 db suppression of the beating oscillator.
Such cavities were designed and attached to the output of a modulator
whose bandwidth had been adjusted by means of small resistive strips.
MODULATOR
OUTPUT CIRCUIT
3 4
n FILTER ELEMENTS
Fig. 12. — Sideband filter in waveguide.
5
(o I
\'J
1
1
)
\
\ ALONE
\
\
\
\
\
1
/'
/ITH
_TER
\
4
/
/
/
L
J
OL
4040 4050 4060 4070 4080
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 13. — Output circuit impedance match.
The resulting impedance match gave a standing wave ratio of less than
one db over a 20 mc band (the plate circuit alone without the filter was
only about 5 mc wide to corresponding points) as shown in Fig. 13, and
the beating oscillator power at the output of the filter was less than one
tenth of a milliwatt, corresponding to TiZ db discrimination.
The requirements and specifications for this particular experimental
model do not necessarily reflect out present thoughts upon the require-
ments for any particular microwave radio relay system; they are presented
here in some detail to indicate how certain specifications can be met,
rather than to express what those specifications should be.
Other factors which influence the performance of the 416A modulator
A NEW MICROWAVE TRIODE
541
are the plate voltage and the beating oscillator drive. The beating os-
cillator power affects the low level gain only slightly but has quite an
effect on the gain at high power levels, that is, when the output power
becomes comparable with the beating oscillator power. It is seen in Fig.
14 that compression becomes noticeable when the output power ap-
proaches within ten db of the beating oscillator driving power.
Varying the plate voltage on the modulator from 150V to 300V had
little effect upon the conversion gain at low levels, but more power output
U 6
BO
DRIVING POWER
^
^ —
-r^O fi<
^—c^
^
^~-^^f5^
r
"%
\
2 4 6 8 10 12 14 16 18 2l
OUTPUT POWER IN DBM
Fig. 14. — Modulator compression data for tube # PS62.
6 8 10 12 (4
OUTPUT POWER IN DBM
Fig. 15.— Modulator compression data for tube #PS348.
was obtained at the higher voltages. At 15 dbm power output, very little
difference between 200V and 300V was observed, but at 150 V the gain
was down two db, as shown in Fig. 15.
Fig. 16 shows the compression data for seven early experimental tubes,
used as modulators with 200V on the plate, 14 ma cathode current, and
200 mw of beating oscillator drive. Half of these tubes had over seven db
low level gain and only slight compression at power output levels of 13
dbm. The two poorest tubes would probably have been rejected before
shipment, according to present standards of production. Each of the
seven tubes was matched in impedance on the r.f. and i.f. inputs and also
542
BELL SYSTEM TECHNICAL JOURNAL
on the r.f. output. The curves represent unloaded gain; no external load-
ing was added to increase the bandwidth.
The performance of the close-spaced triode when used as a modulator
appears to be superior in some respects to that of the silicon crystal
motlulators which are used in the New York-Boston microwave relay
system.'
Single tubes had from 5 to 9 db gain compared with from 8 to 11 db
loss for the crystals for corresponding power outputs. To get this per-
formance the beating oscillator drive was only 200 milliwatts, compared
with about 700 milliwatts for the crystal modulator. This reduction in
r.f. power requirements means considerable simplihcation in a repeater.
6 8 10 12
OUTPUT POWER IN DBM
Fig. 16. — Compression data on seven 1553 triodes.
Plate voltage 200V
Plate current 14 Ma
B.O. Power 200 MW
Matched inputs and output
To offset this, the tube requires power suppUes which are not necessary
for the crystals, but low voltage power supplies should be cheap. The
bandwidth of the tube modulator, 60 to 80 mc is less than the very wide
(500 mc) band of the crystal modulator but it is comparable with the
band width of the e.xtra i.f. stages needed to drive the crystal modulator.
The life of the tubes, although very little data are available as yet, will
probably be less than the practically indefinite life of the silicon point
contact modulators.
Amplifier
The performance of the close-spaced triode as an amplifier can best
be described by referring to its impedance match, gain, transmission band-
width and compression.
.4 NEW MICROWAVE TRIODE
543
In some of the experimental tubes, bandwidths to the half power points
of 21 mc to 250 mc have been measured. Typical of one of the better
tubes, though not the best one, are the data contained in Fig. 17. The
bandwidth of the input circuit is about twice that of the output circuit,
and the SWR slumps outside the band on the low frequency side. The
output impedance is more regular, exhibiting the familiar standing wave
m 10
CE 6
^
p-.a..^ J
p
L
/
\
K
1
/
\
OUTPUT
/
\
/
f
\
/
1
1
(^
r '
INP
UT
s
\
/
.^
^
k
V
A
Y
>^
N
\/A
T
3940 3980 4020 4060 4100 4140
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 17. — Input and output standing wave ratio versus frequency.
■^ a
^ 6
<
3 DB DOWN
GAIN
BANC
) PRO
DUCT
= 129
0 MC
^
^^
V,
/
y
N
\
/
/^
\
\
^
3980 4020
FREQUENCY IN
4060 4100 4140
MEGACYCLES PER SECOND
Fig. 18. — Transmission characteristic of a one-stage ampiitier.
ratio of a simple single tuned resonant circuit. When the output impe-
dance is plotted on the Smith reflection coefficient chart, the circle which
results is also similar to that of a single tuned circuit. This is desirable
since it then becomes a simple matter to incorporate the plate circuit in
a maximally-flat filter of as many resonant branches as are needed, in
the same way that the modulator output circuit was treated.
The transmission bandwidth for this single stage amplifier was 203 mc
to the half power points, as shown in Fig. 18. This, with a gain of 8.05 db
544 BELL SYSTEM TECHNICAL JOURNAL
at midband, gave a gain-band product of 1290 mc. The bandwidth of 203
mc was considerably greater than the average for these tubes. Similar
results on 35 experimental tubes yielded the following averages: Low-level
gain 10 db; Bandwidth 103 mc; Gain-band product 916 mc. The 416-A
tubes produced by Western Electric Company exhibit comparable aver-
ages with much less spread; for example, a recent sample of 138 tubes had
average values and standard deviations as follows:
Table I
Gain and Bandwidth or 138 W. E. Co. 416-A Triodes
Low-level gain ....
Bandwidth
Gain-band product .
St'd. Dev.
1.1 db
9 mc
350 mc
It is indeed gratifying to realize that such a remarkable tube can be
produced with such uniformity.
o+B
o-B
Fig. 19. — Stabilizer circuit.
In operating these tubes, it has been found that small variations in
gain due to power line fluctuations and due to other disturbing influences
can be minimized by using a stabilizing bias network which provides a
large amount of negative feedback for the dc. path. This circuit is similar
to one proposed by Mr. S. E. Miller of the Bell Telephone Laboratories
for use in coxial repeaters which also use high transconductance tubes.
In this circuit, shown schematically in Fig. 19, a few volts negative are
applied to the cathode through a suitable dropping resistor. In the absence
of plate voltage, the grid draws current, being positive with respect to
the cathode. When plate voltage is applied, the drop in the cathode re-
sistor tends to bring the cathode nearer ground potential until a stable
voltage is reached. The resistor is set to a value which allows the desired
cathode current to flow and subsequent variations in gm or plate voltage
then have little effect on the total cathode current.
Maintaining the cathode current constant does have an appreciable
effect on the gain of the tube when operating at high output levels. This
is characterized by a decrease in gain as the driving power is increased.
A NEW MICROWAVE TRIODE
545
Fig. 20 illustrates this point. The low-level gain of this tube was 12.3
db but when the tube was driven so as to have an output power of 400
mw the gain was only about 3 db. At this point, retuning the circuit to
rematch the tube at the high output level increased the gain to about 5
db. Now, returning to low level, the gain was only 10 db. Presumably
in between these two points, 5 db at 500 mw output and 12.3 db at less
than one milliwatt output, the performance could have been better than
either of these two curves shows, i.e., the performance could have been
improved by rematching at each intermediate power level.
12
It
lo'
0) 9
_i
UJ
m
O 8
LU
Q
? 7
Z
<
O 6
5
4
3
2
V
\
■v
L^
\
CIRCUIT TUNED
AT LOW LEVEL
*
"t
r^
V.
1
\
^^j
CIRCUIT TUNED
;.AT HIGH LEVEL
\
■-V
^v
N
1
\
^N
V
X
\
V
V)
\
4
1
100 150 200 250 300 350 400 450 500
POWER OUTPUT IN MILLIWATTS
Fig. 20. — "Compression" in a one-stage microwave amplifier Ip = 30 ma.
This tube is not representative of all of the tubes tested. It is rather
poorer in the spread of the two curves than most. It was picked merely
to illustrate that besides a drop in gain also a detuning effect takes place
when the driving power is changed. In the example given here the cathode
current was held at or near 30 ma by the stabilizing bias circuit.
Without the stabilizing circuit, these so-called "compression" curves
would be quite different. For instance, if the bias were held constant, we
should expect that the gain would not drop as fast as indicated here,
since the plate current would rise as the drive was increased.
At any rate, in an F.M. system, we are not concerned with how much
546 BELL SYSTEM TECHNICAL JOURNAL
"static compression" exists, but rather with how much gain can be real-
ized without exceeding the dissipation ratings of the tubes.
With this in mind data were taken on 25 of the experimental tubes. In
each case they were matched to the input and output waveguides and the
cathode current was stabilized at 30 ma. After driving the tube to a
high level of output power, the circuits were rematched and the resulting
"compression" curves revealed the capabilities tabulated.
Table II
Summary of Data on 25 Experimental Close-Spaced Triodes
Low level gain
Gain (500 mw output) . . . .
Power Output (3 db gain) .
Highest
12.3 db
7.0 db
950 mw
Lowest
3.8 db
-8.0 db
50 niw
Average
7.8 db
1.82 db
455 mw
It can be seen from the table that we might expect to obtain a gain of
20 or 25 db with three or four stages with a power output of about 500
mw and a flat band of over 20 mc.
Three St.\ge Amplifier
A three-stage amplifier with 24 db gain has been assembled using an
earlier type of circuit and loop tested at low levels on the equipment of
Messrs. A. C. Beck, N. J. Pierce and D. H. Ring.^ This amplifier had a
bandwidth of about 30 mc to the 1 db points and while it does not
represent the best that can be done with the 416A tube, the results of the
loop test are interesting.
The recirculating pulse test, or loop test, is performed on a repeater
component to determine its ability to reproduce a pulse faithfully after
repeated transmissions. The output of the amplifier is connected to its
input through a long delay line and an adjustable attenuator. The overall
gain of the loop thus formed is adjusted to unity or zero db so that an
injected pulse will recirculate through the loop without attenuation but
accumulating distortion with each round trip. After allowing the pulse to
recirculate long enough the amplifier is blanked out or quenched and the
recirculating i)ulse amplitude dies out, thus preparing the loop for the
next injected pulse, when the process is repeated. With a pulse length of
one microsecond and an overall delay of two microseconds, one hundred
round trips occur in 0.2 milliseconds, thus allowing the process to be
repeated at the rate of two or three thousand times per second. A cathode
ray oscilloscope is used to examine the pulse shapes, and its sweep is
synchronized to the injected pulse so that successive corresponding pulses
are superposed, enabling the operator to examine the pulse after any
A NEW MICROWAVE TRIODE
547
number of round trips or select individually the nth round trip for in-
spection.
Fig. 21(a) shows the complete cycle between successive injected pulses,
and the individual pulses that follow cannot be resolved at this slow
sweep speed. Fig. 21(b) shows the first 26 round trips resolved so that they
are distinguishable. Figure 21(c) shows the first and second round trips
(a) PULSES RECIRCULATING
THROUGH AMPLIFIER
,b) PULSES RECIRCULATING
THROUGH AMPLIFIER
QUENCHED
INJECTED PULSE
(c) FIRST AND SECOND ROUND
TRIPS THROUGH AMPLIFIER
ROUND TRIP
(d) PULSE SHAPE AFTER
100 ROUND TRIPS
■INJECTED PULSE
Fig. 21. — Recirculating pulse test patterns.
\f/\f/\f/*u/
402-A VELOCITY MODULATION AMPLIFIER
Fig. 22. — Recirculating pulse patterns showing Ist, lOlh an 1 103th round trips for: Top:
Close-spaced triode amplitier. Bottom: 402-A velocity modulation arapliner.
through the amplifier, with little or no distortion discernible. Fig. 21(d)
gives, to the same scale as the preceding picture, the pulse shape after
100 round trips. A little overshoot and subsequent oscillation is now
visible, although the whole pulse shape is still not too bad.
In Fig. 22, these results are compared with the results of a similar test
performed on a four-stage, stagger tuned, stagger-damped amplifier using
the 402 velocity variation amplifier tubes; the first, the tenth and the
hundredth round trips are shown. Little or no distortion is seen at the
548
BELL SYSTEM TECHNICAL JOURNAL
tenth round trip, but the superiority of the 416A amplifier is clearly
shown in the hundredth round trip.
Both amplifiers were operating at low levels, and the pulse was an am-
plitude modulated one. Since these are not the conditions under which
our microwave radio relay circuits operate, conclusions should not be
drawn about how many repeater stations can now be put in tandem. The
Fig. 23. — An assembled three-stage microwave amplifier.
test merely indicates that an improvement has been made, thus corro-
borating the evidence obtained by other tests.
Still further improvement has been made since loop testing the model
ten amplifier. A three stage 416A amplifier (see Fig. 23) using model
eleven circuits had comparable gain, 23 db, but a bandwidth of 50 mc to
points 0.1 db down. These data again are for low level operation, but it
is reasonable that half a watt might be expected from four such stages
with comparable gain and slightly narrower bandwidth, surely 30 mc.
A NEW MICROWAVE TRIODE
549
Noise Figure
In a forward looking program it is well to keep in mind other possi-
bilities for this tube, such as use in a straight through type of repeater in
which all of the amplification is obtained at microwave frequencies. In
such an application the noise figure of the triode becomes one of its
limitations, since the 416A must compete with the low noise figure of
the silicon crystal converter which, for the New York-Boston circuit, is
around 14 db. Data on thirty five early experimental and production
41 6A tubes gave an average value of 18.08 db at 4060 mc* Each of the
1000
900
800
700
600
300
•
•
• 416 A
A 1553
•
•
•
A
» •
. A
•
•
•
A
A
•
•
•
k,\
•
•
•
A
A
A
A
17 1
NOISE
8 19 20 21
FIGURE IN DECIBELS
Fig. 24. — Noise figure vs gain-band product for close-spaced triode.
tubes was operated at 200 volts with 30 ma space current and was tuned
so as to present matched impedances to the input and the output wave-
guides. The best of this batch had a noise figure of 14.79 db and the
poorest 23.2 db. These measurements were made with a fluorescent light
noise source.^
An interesting correlation between noise figure and gain-band product
was uncovered during these tests, as can be seen in Fig. 24, which gives
the noise figure in db on the abscissa and the gain-band product in mega-
* More recently, a sample of twelve production 416A tubes ranged from 13.5 to 16.2
db and averaged 15.06 db noise figure, with a standard deviation of 0.8 db.
55t)~ BELL SYSTEM TECHNICAL JOURNAL
cycles along the logarithmic ordinate. The points scatter between the
extremes of 15 db noise tigure for a gain-band product of 2000 megacycles
to 23 db noise figure at 400 megacycles gain-band product. Extrapolating
from these data, a noise figure of 10 db might be achieved if the gain-
band product could be increased to 5500 mc. It is reasonable to expect
that an improvement of this amount can be achieved if the resistance and
return electron losses inside the tubes can be eliminated.'^
We may use these data to determine the expected noise figure of a
straight through amplifier, thus:
h = Fa -\- — J. — + ^ ^ • • • (1)
Li A (j-.i Uu
If, for example, we assume that all stages are alike in noise figure and
in gain, equation (1) approaches the expression, as the number of stages
increases without limit:
F/rA - 1
1 im
F = '^^^^-^ (2)
n-»oo Cr^i i
Using an average value of 10 db gain per stage, the overall noise figure
would be as follows:
(1) For Fa = 30 (best tube, 14.79 db)
F = 299 _ 33 2 or 15.2 db
(2) For Fa = 64 (average tube, 18.08 db)
F = ^^ = 7loY 18.5 db.
Straight-Through Amplifier
The actual performance of a ten-stage amplifier was about what should
be expected from the considerations above. The best tube (10 log F =
14.79 db) was used in the first stage, and the next best tube in the sec-
ond stage. The measured overall noise figure was 15.96 db. The overall
gain was 90 db and the band was flat to 0.1 db for 44 mc. Such an
amplifier with its associated power supply and individual control panels
is shown in Fig. 25.
Conclusions
A circuit is described which lends itself readily to utilizing the 416A
close-spaced triode as a modulator or a cascade amplifier for microwave
repeaters operating at 4000 mc. Data are {presented on early e.xperimcntal
models of the tube.
As modulators, single tubes hatl from 5 to 9 db gain wilh 10 to 20
A NEW MICROWAVE TRIODE
551
Fig. 25. — A ten-stage microwave amplifier operating at 4000 nic.
552 BELL SYSTEM TECHNICAL JOURNAL
mw output when driven with 200 mw of beating oscillator power. A
bandwidth of twenty megacycles was readily obtained.
As amplifiers at 4060 mc, the average gain of 60 tubes was 9 db, the
average bandwidth of 34 tubes was 103 mc to the half power points, the
average noise figure of 35 tubes was 18.08 db and the average power out-
put (for 3 db gain) was 455 mw for 25 tubes. Operating the tubes in
cascade produced an amplifier which had less distortion of pulse shape
than an earlier amplifier which used the 402-A velocity variation tube.
A ten-stage amplifier has been assembled and tested, yielding 90 db gain,
a noise figure (with selected tubes) of 15.96 db and a bandwidth of 44
mc to the 0.1 db points.
These data are for early experimental models of the tube and it is
likely that subsequent alterations may improve the performance in the
production models.
Acknowledgments
The work described in this paper took place at the Holmdel Radio
Research Laboratories. Mr. Bowen, with the able assistance of Mr. E.L.
Chinnock, was active in pursuing the problems connected with the ampli-
fier circuits. Mr. R. H. Brandt helpedwith the work in connection with
the modulator circuits and many others were helpful in designing and
constructing the circuits and facilities for testing.
References
1. "Microwave Repeater Research," H. T. Friis, B. S. T. J., Vol. 27, pp. 183-246, April
1948.
2. "A Microwave Triode for Radio Relay," J. A. Morton, Bell Labs. Record, Vol. 27, ^5,
May 1949.
3. "Microwave Converters," C. F. Edwards, Proc. L R. E., Vol. 35, pp. 1181-1191, Nov.
1947.
4. "Maximally-Flat Filters in Wave Guide," W. VV. Mumford, B. S. T. J., Vol. 27, pp.
684-713, Oct. 1948.
5. "Testing Repeaters with Circulated Pulses," A. C. Beck and D. H. Ring, Proc. /. R.E.,
Vol. 35, pp. 1226-1230, November 1947.
6. "A Broad-Band Microwave Noise Source," W. W. Mumford, B. S. T. /., Vol. 28, pp.
608-618, October 1949.
7. "Electron Admittances of Parallel-Plane Electron Tubes at 4000 Megacycles," Sloan
D. Robertson, B. S. T.J., Vol. 28, pp. 619-646, October 1949.
8. "Design Factors of the Bell Telephone Laboratories 1553 Triode," J. A. Morton and
R. M. Ryder, B. S. T. J., Vol. 29, f^4, pp. 496-530, Oct. 1950.
A Wide Range Microwave Sweeping Oscillator
By M. E. HINES
{Manuscript Received July 24, 1950)
1. Introduction
A SWEPT frequency oscillator is a useful laboratory tool for testing
wide-band circuit components. It permits an oscillographic display
of a frequency characteristic, avoiding much of the labor of point-by-
point testing at discrete frequencies. There was a particular need in the
Bell Telephone Laboratories for a sweeping oscillator to cover the com-
munications band between 3700 and 4200 megacycles to facilitate the
testing of components for radio relay repeaters.
This paper describes one type of oscillator designed to satisfy this
need. It utilizes the BTL 1553 (or the Western Electric 416A) micro-
wave triode. The tuning is accomplished mechanically so that the fre-
quency varies continuously back and forth over the band at a low audio
frequency rate. Continuous oscillations have been obtained over a 900
megacycle band from 3600 to 4500 megacycles.
2. Circuit Structure
Basically, the rf circuit consists of a tunable cavity for a grid-anode
resonant circuit, a means for feedback to an untuned grid-cathode circuit,
and a means for coupling the cavity to a waveguide output. The grid-
anode cavity is the only sharply tuned circuit, and it was found that oscil-
lations could be obtained over the entire band by changing the resonant
frequency of that cavity alone. In this application, the electronic conduct-
ance between the grid and cathode is so high that this portion of the
circuit has an inherent broad band such that separate tuning is unneces-
sary.
The necessity for continuous, rapid tuning virtually requires that there
be no sliding contacts in the tuning mechanism. A type of cavity was
chosen so that tuning could be accomplished by a simple variable capaci-
tor of the non-contacting type. Reduced to its simplest elements, it con-
sists of a short coaxial line, resonant in the half-wave mode. Actually the
line is much shorter than a half wavelength because of excess capacitance
at both ends. At one end is the capacitance of the grid-anode gap, and at
the other end is the variable capacitor used for tuning.
553
554
BELL SYSTFJf TECHNICAL JOURNAL
The actual cavity is illustrated in Fig. 1. This is somewhat more com-
plicated than a half wave line, but the mode of resonance is essentially
the same. The variable capacitor utilizes a thin-walled copper cup which is
movable vertically. This cup fits rather closely inside, and is coaxial with,
a cvlindrical hole in the main bodv of the cavitv. It forms the center
SPEAKER TYPE
MAGNET
SUPPORT
SPRINGS
PAPER TUBE
WAVEGUIDE
OUTPUT
ANODE '■
CONNECTION
I'"ifi. 1. — Construction of the oscillator.
conductor of a low impedance coaxial line approximately one-fourth
wavelength long, so that in this frequency range it is eflfectively short-
circuited to the cavity wall. Vertical motion of the cup is therefore roughly
equivalent to moving the end wall, thereby changing the capacitance
between the wall and the center conductor of the main cavity. The reso-
WIDE RANGE MICROWAVE SWEEPING OSCILLATOR 555
nant frequency is lowest when the two surfaces are nearly in contact, and
highest when the cup is fully extracted.
The recessed end of the cup fits over a protuberance on the center con-
ductor when they are nearly in contact. This special shape was designed
to give a reasonably straight curve of frequency vs. displacement. With
planar surfaces, the frequency would change more rapidly with displace-
ment at the low than at the high frequency end of the band.
The grid disk of the tube is separated from the wall of the cavity by
a narrow annular space, and contact is made across the gap by a number
of small screws. These screws act as an inductive reactance in series with
the circulating currents of the resonant cavity. The voltage developed
across this reactance is applied between the grid and the main envelope
of the tube, and in this way energy is fed into the grid-cathode space to
provide feedback.
The mechanical tuning device was adapted from an inexpensive per-
manent magnet loudspeaker of the type used in small home radios. The
construction is shown in Figs. 1 and 5. The speaker cone was removed
and the voice coil was attached to a thin-walled paper cylinder which
supports the tuning cup inside the cavity. Two sheet fiber springs support
the paper cylinder and maintain the axial alignment in the magnet and
cavity. These springs are cut with a number of incomplete circular slits to
reduce the stiffness for axial motion. With the voice coil actuated from a
small filament transformer, peak to peak motion | of inch is obtainable.
The heater and cathode connections are made at the base of the
tube which protrudes from the cavity. The grid is internally connected
to the main body of the cavity. The anode lead is brought out through a
quarter-wave choke and mica button condenser.
To prevent overheating of the anode of the tube, air must be blown
through the cavity. This is done by connecting a low pressure air hose to
the air inlet shown in Fig. 1. Excessive air flow must be avoided, as it will
cause erratic vibrations of the tuning plunger.
3. Adjustment and Operation
The degree of feedback is adjustable by changing the number and rela-
tive positions of the feedback screws which connect the cavity to the
grid ring of the tube. There are 16 possible screw positions, but only
about 5 or 6 are needed to obtain optimum feedback. Reducing the num-
ber of screws increases the amount of feedback.
Care should be taken that the spring which contacts the anode for dc
connection is not of such a length to have resonances within the band.
When such resonances exist, "holes" or other irregularities will be found
in the output spectrum. This spring can act as a helical line, and when it
556
BELL SYSTEM TECHNICAL JOURNAL
is too long, resonances will occur which can absorb power and otherwise
aflfect the cavity impedance.
When properly adjusted and sweeping, the output is continuous and
the frequency varies approximately sinusoidally back and forth over the
band of interest. The width of the sweeping band depends upon the ac
current in the voice coil of the speaker drive, and the center frequency
depends upon the mean position of the tuning plunger. The latter can be
adjusted mechanically by loosening the clamping screw and raising or lower-
ing the sweeping mechanism by hand. It is also possible to make small ad-
justments of the center frequency electrically by adding a dc component to
the voice coil driving current.
5 20
O 16
5
o
a 14
^^--
'loo VOLTS
,^^
_,y^y
^—
Illi::
170
.^'
'
^— — -
y
/ /
' 1
1
1
1
115
\
1
/
/
1
3600 3700
3800 3900 4000 4100 4200 4300
FREQUENCY IN MEGACYCLES PER SECOND
4400 4500
Fig. 2. — Power output curves at 115, 170, and 200 Volts on the anode, for a mean
anode current of 25 ma.
Typical curves of power output vs. frequency, taken at different anode
voltages, are shown in Fig. 2. The flattest curve requires a voltage con-
siderably lower than the tube rating. The feedback phase is not optimum
for best power output, a larger phase shift being desirable in this oscilla-
tor. The lowered voltage helps in this regard, increasing the electron
transit time in the tube and thereby increasing the phase shift. Efficiency
was sacrificed in this design to increase the tuning band. A longer feed-
back path would increase the power output, but would tend to narrow
the band over which oscillation could be obtained by a single tuning
adjustment.
The anode power supply should be variable between 100 and 250 volts,
but need not be regulated because this voltage is not critical. A rheostat
is used for cathode self-bias. The cathode heater and the sweeping mech-
anism are supplied from a single 6.3 volt filament transformer, with a
potentiometer control to vary the sweep range.
WIDE RANGE MICROWAVE SWEEPING OSCILLATOR
557
A crystal detector and an oscilloscope are used to view the output. It
is convenient to use a sinusoidal horizontal sweep on the oscilloscope,
driven from the same 6.3 volt transformer as the mechanical sweeping
mechanism. In this case, a phase shifter is needed to synchronize the
oscilloscope sweep with the motion of the tuning plunger, because there is
an appreciable mechanical phase shift in the loudspeaker mechanism.
:Rl
ANODE
/"Cgp
RADIAL
LINE
■^GP >Y2,(V^-Vj
HJ CATHODE
Fig. 3. — Simplified equivalent circuit of the oscillator.
Fig. 4. — The complete oscillator, showing the output coupling window and the ridged
waveguide coupling transformer.
When properly phased, the output spectrum will be displayed across the
oscilloscope screen with the minimum and maximum frequencies at oppo-
site ends of the trace. In addition, a vibrating relay (such as the Western
Electric 275 B Mercury Relay) is used to short out the input to the oscillo-
scope during half of each cycle. This converts the return trace into a
zero-signal reference line, so that the complete picture is a closed loop
with a flat bottom. The separation of the active from the reference trace
558
BELL SYSTEM TECHNICAL JOURNAL
is a direct indication of signal strength, displayed as a function of fre-
quency.
The results reported here were obtained using the BTL 1553 tube,
which is a laboratory model. Samples of the production model. Western
Electric 416A, have also been used in this oscillator with quite similar
results. To adapt the oscillator for the 416A, the grid ring should be
threaded on the inside to fit the threads on the grid disk of that tube.
4. An Equivalent Circuit
The tield configuration in the cavity of the oscillator is quite complex,
and cannot be readily described in any quantitative fashion. The formu-
lation of an equivalent circuit would require many approximations and
Fig. 5. — The complete oscillator, showing the sweeping mechanism partially dis-
mantled.
judicious guesses if values are to be specified for the various circuit
parameters. The circuit of Fig. 3 is believed to be equivalent in a qualita-
tive sense.
A portion of the circuit is within the tube itself. This is the region en-
closed by the dotted line in T^ig. 3. The T of elements which include I'n,
Cyp, nCgp and the injected currents, is the equivalent circuit of Llewellyn
and Peterson^ for the active region of a triode. Exj)erimentally deter-
mined values for these quantities are reported by Robertson'-. I'u is the
admittance of an equivalent diode between the grid and cathode, and the
injected currents indicated by the arrows are the electronic transfer cur-
rents associated with the grid voltage and the transadmittance. At high
' Llewellyn and Peterson, "Vacuum Tube Networks," LR.E. Proc, Vol. i2, pp. 144-
166 (.March 1944).
2 S. D. Robertson, "Electronic Admittances of Parallel-Plane Electron Tubes at 4000
•Megacycles," B. S. T. J., Vol. 28, p. 619, Oct. 1949.
WIDE RANGE MICROWAVE SWEEPING OSCILLATOR 559
frequencies, both the admittance Vu and the transadmittance F21, are
complex quantities which vary with frequency as shown by Llewellyn
and Peterson. The 4-pole box shown represents the passive radial line
between the glass seal at the edge of the tube and the cathode-grid gap.
This line is heavily loaded with dielectrics and is believed to be elec-
trically about a quarter wavelength long at 4000 Mc. The inductance
connected to the anode is that of the anode pin itself and the coaxial
center-conductor attached to it. A series resistor Rl is added to include
the effects of cavity losses and loading by the output coupling window.
Ct is the tuning capacitor which varies with tuning plunger position. The
inductance L/ is the feedback reactance introduced by the screws con-
nected to the grid disk.
5. Acknowledgments
I wish to acknowledge the assistance of Messrs. J. A. Morton, R. M.
Ryder, and the late A. E. Bowen for many helpful suggestions in the
design of this oscillator.
Theory of the Flow of Electrons and Holes in Germanium and
Other Semiconductors
By W. VAN ROOSBROECK
{Manuscript Received Mar. 30, 1950)
A theoretical analysis of the flow of added current carriers in homogeneous
semiconductors is given. The simplifying assumption is made at the outset that
trapping effects may be neglected, and the subsequent treatment is intended
particularly for application to germanium. In a general formulation, differential
equations and boundary-condition relationships in suitable reduced variables and
parameters are derived from fundamental equations which take into account
I the phenomena of drift, diffusion, and recombination. This formulation is special-
; ized so as to apply to the steady state of constant total current in a single car-
tesian distance coordinate, and properties of solutions which give the electro-
static field and the concentrations and flow densities of the added carriers are
discussed. The ratio of hole to electron concentration at thermal equilibrium
occurs as parameter. General solutions are given analytically in closed form for
the intrinsic semiconductor, for which the ratio is unity, and for some limiting
cases as well. Families of numerically obtained solutions dependent on a parame-
ter proportional to total current are given for w-type germanium for the ratio
equal to zero. The solutions are utilized in a consideration of simple boundary-
value problems concerning a single plane source in an infinite filament.
Table of Contents
1. Introduction 560
2. General Formulation 565
2.1 Outline 565
2.2 Fundamental equations for the flow of electrons and holes 566
2.3 Reduction of the fundamental ecjuations to dimensionless form 571
2.31 The general case 571
2.32 The intrinsic semiconductor 577
2.4 Differential equations in one dimension for the steady state of constant current
and properties of their solutions 578
3. Solutions for the Steady State 583
3 . 1 The intrinsic semiconductor 584
3.2 The extrinsic semiconductor: w-type germanium 586
3.3 Detailed properties of the solutions 588
3.31 The behavior for small concentrations 590
3.32 The zero-current solutions and the behavior for large concentrations. . . . 593
4. Solutions of Simple Boundary- Value Problems for a Single Source 594
5. Appendix 599
5 . 1 The concentrations of ionized donors and acceptors 599
5 . 2 The carrier concentrations at thermal equilibrium 600
5.3 Series solutions for the extrinsic semiconductor in the steady state 601
5 . 4 Symbols for quantities 605
1. Introduction
TN A semiconductor there are current carriers of two types: electrons
-*- in the conduction band, and positive holes in the filled valence band;
and the increase of their concentrations in the volume of the semicon-
560
FLOW OF ELECTRONS AND HOLES TN GERMANIUM 561
ductor over the concentrations which obtain at thermal equilibrium is
fundamental to a number of related phenomena, of which transistor ac-
tion is a famihar instance. In an «-type semiconductor, for example, in
which the carriers are predominantly electrons, the carrier concentrations
are increased by the introduction of holes which, through a process of
space-charge neutralization, produce additional electrons in the same
numbers and concentrations. The bulk conductivity of the semiconductor
is thereby so increased that power gain is obtainable.^ Holes can be
introduced by the local application of heat, or by irradiation with light,
X-rays, or high-velocity electrons — in fact, by any agency which trans-
fers electrons from the highest filled band to the conduction band. They
can be introduced also through an emitter, which may be a positively
biased point contact or a positively biased p — n junction^, as exemplified
in the transistor. In this case the emitter introduces holes, which flow
into the volume of the semiconductor^, by the removal of electrons from
the filled band.-' ^ Entirely analogous considerations apply to the intro-
duction of electrons into a ^-type semiconductor.^
In their flow in a semiconductor, added electrons and holes are subject
to drift under electrostatic fields and to diffusion in the presence of con-
centration gradients as a consequence of their random thermal motions.
They are subject also to recombination, which results in concentration
gradients in source-free regions even for the steady state in one dimen-
sion, or which augments those which may otherwise be associated with
the time-dependence of the flow, or with its geometry in the steady state.
From fundamental equations which take into account these phenomena of
drift, diffusion, and recombination, for the existence of each of which
there is experii ental evidence^, general differential equations and
boundary-conditiuj i.'^lationships in suitable reduced or dimensionless
variables and parameters may be derived, and solutions which give the
concentrations and flow densities of added carriers obtained for various
cases of physical interest.
This paper presents results of a theoretical analysis, along these lines,
of the flow of electrons and holes in semi-conductors. The treatment is
intended particularly for application to germanium. An initial formulation,
1 VV. Shockley, G. L. Pearson and J. R. Haynes, B. S. T. J. 28, (3), 344-366 (1949).
2 J. Bardeen and W. H. Brattain, Phvs. Rev. 74 (2), 230-231 (1948); W. H. Brattain
and J. Bardeen, Phys. Rev. 74 (2) 231-232 (1948).
3 \V. Shocklev, G. L. Pearson and M. Sparks, Phvs. Rev. 76 (1), 180 (1949); W. Shockley,
B. S. T. J. 28' (3), 435-489 (1949).
*E. J. Ryder and W. Shockley, Phys. Rev. 75 (2), 310 (1949); J. N. Shive, Phys. Rev.
75 (4), 689-690 (1949); J. R. Haynes and \V. Shockley, Phys. Rev. 75 (4), 691 (1949).
^J. Bardeen and W. H. Brattain, Phys. Rev. 75 (8), 1208-1225 (1949); B. S. T. J. 28
(2), 239-277 (1949).
6W. G. Pfann and J. H. Scaff, Phys. Rev. 76 (3), 459 (1949); R. Bray, Phys. Rev. 76
(3), 458 (1949).
562 BELL SYSTEM TECHNICAL JOURNAL
which retains, wherever convenient, such generality as is instructive per se
or of manifest utiHty, is speciaHzed so as to apply to the steady state of
constant current in a single cartesian distance coordinate. For the in-
trinsic semiconductor, general analytical solutions are obtainable in
closed form, and such solutions are given, as well as general solutions
obtained numerically for /z-type germanium in which the hole concentra-
tion at thermal equilibrium may be neglected compared to the electron
concentration. Solutions for these cases are given explicitly for each of
two recombination laws: recombination according to a mass-action law,
and recombination such that the mean lifetime of the added carriers is
constant. Methods are described for the fitting of boundary conditions,
and the following relatively simple boundary-value problems are con-
sidered: a source at the end of a semi-infinite semi-conductor filament;
and a single source in a doubly-infinite filament.
To indicate the presumed scope and application of the results obtained,
it may suffice to outline briefly the principal assumptions on which they are
based and the approximations employed: The assumption is made at the
outset that trapping effects may be neglected, which provides the im-
portant simplification that the recombination rates of holes and electrons
are equal at all times. One justification for this is the circumstance that
the fairly high hole mobilities found by G. L. Pearson from Hall-effect
and conductivity measurements' are no larger than those found by J. R.
Haynes from transit times under pulse conditions^ With hole trapping,
holes injected in a pulse would initially fill traps; and if there were subse-
quent relatively slow release of the holes from the traps, an apparent
reduction of mobility would be manifest. It is further assumed that sub-
stantially all donor and acceptor impurities are ionized. With the assump-
tion that the semi-conductor is homogeneous in its bulk, and free from
grain boundaries* or rectifying barriers, the assumption of the electrical i
neutrality of the semiconductor, or of the neglect of space charge, is in
general an excellent approximation: Small departures from electrical
neutrality in the volume would vanish rapidly, with time constant equal
to that for the dielectric relaxation of charge, which for germanium
equals 1.5 -10"^- sec per ohm cm of resistivity^ and is in general small
compared with the mean lifetime of added carriers. A uniform local de-
parture from electrical neutrality in germanium of only one per cent in
relative concentration would produce api)reciable changes in field in a
' G. L. Pearson. PItvs. Rro. 76 (1), 179-18T (19 tJ).
"G. L. Pearsun, P/ivs. Rro. 76 (3), 459 (1949); W. E. T.ivl )r a-i 1 H. Y. Fan, piper
0A5, and \. H. Odell and H. Y. Fan, paper 0A2 of the 1950 Annual Mssting of the
American Physical Society, February 3, 1950.
'■".A value of 16.6 for the dielectric constant of germanium is o!)tained from optical
data of H. B. Briggs: Phys. Rev. 77 (2), 287 (1950).
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 563
mean free path for the carriers, equal to 1.1 • 10~' cm at room temperature,
which would even preclude the applicability of the fundamental equations
employed. In qualitative terms, the conductivity of the semiconductor is
sufficiently large that the currents which commonly occur are produced
by moderate fields whose maximum gradients are relatively small. Space
charge may persist in the steady state, but then only in surface regions
whose thickness^" in germanium is generally less than about 10^* cm and
whose effects may be dealt with through suitable boundary conditions.
The steady-state solutions, in their qualitative aspects, are illustrative
of the phenomena taken into consideration. In an extrinsic semiconductor,
if the concentrations of added carriers are not too large, the solutions
for moderate and large fields are in general approximately ohmic in their
local behavior. The effect of diffusion is then comparatively small, and
the added carriers largely drift under a field which varies with distance
through the increased conductivity which these recombining carriers
themselves produce. Diffusion effects are incident in addition to this
behavior, and become pronounced for large concentrations or small ap-
plied fields. For example, solutions which specify the concentrations of
added holes as functions of distance, for different total currents or applied
fields in a source-free region, all approach a common solution for large
hole concentrations, regardless of applied field; those for the hole cur-
rent and the electrostatic field behave similarly. This behavior results from
diffusion in conjunction with the increase in conductivity. Another example
is that of the solutions for zero total current: As the result of diffusion in
conjunction with recombination, a flow of added holes can occur along a
semi-conductor filament with no flow of current. It is, of course, accom-
panied by an equal electron flow, so that the hole and electron currents
cancel, and occurs in any open-circuited semi-conductor filament which
adjoins a region in which added holes flow. It can also be realized by suit-
able irradiation of an end of a filament, with no applied field. A closely
related effect is illustrated in the flow of holes injected through a point-
contact emitter into a semi-conductor filament along which a sweeping
field is applied: Some of the holes will flow against the field, an appre-
ciable proportion, unless the current in the filament is sufficiently large.
As a further example, if the mobilities of holes and electrons were equal,
the electrostatic field would be given by Ohm's law as the total current
'" The (largest) distance over which the increment in electrostatic potential exceeds
kT/e may be expressed in units of the length Ld = {kTe/&Trn,e~)\ where lu is the thermal-
equilibrium concentration of electrons (or holes) in the intrinsic semiconductor; see the
paper of reference 3, also W. Schottky and E. Spenke, Wiss. Verijjf. Siemcns-Werken 18
(3), 1-67 (1939). This distance increases with resistivity, never exceeding the value 1.4
Ld for the intrinsic semiconductor. In high back voltage w-type germanium, it exceeds
about 0.5 Ld, and Ld for germanium is about 7.4-10^' cm at room temperature.
564 BELL SYSTEM TECHNICAL JOURNAL
divided by the local increased conductivity. With electrons more mobile
than holes, this ohmic field is modified by a contribution which is directed
away from a hole source and proportional to the magnitude of the con-
centration gradient divided by the local conductivity. This contribution
gives a non-vanishing electrostatic field for zero total current.
The intrinsic semiconductor has, as the result of a conductivity which
is everywhere proportional to the concentration of carriers of either type,
the property that the flow in it is as if the added carriers were actuated
entirely by diffusion, with only the carriers normally present drifting
under a field equal to the unmodulated applied field. The extrinsic semi-
conductor becomes in effect intrinsic if the concentrations of carriers are
sufficiently increased, by whatever means, the ohmic contribution to the
current density of either electrons or holes then becoming proportional to
the total current density and, in this case, negligible compared with the
contribution due to diffusion. It may, for example, be expected that the
transport velocity of added carriers in an extrinsic semiconductor can be
increased by an increase in the applied field only if the consequent joule
heating does not unduly modify the semiconductor in the intrinsic
direction.
General solutions for the steady state in one dimension are obtainable
analytically in closed form for a number of important special cases. Aside
from that for which diffusion is neglected, they include the general cases
for no recombination, for the intrinsic semiconductor, and for zero total
current, and the limiting cases of small and of large concentrations of
added carriers. W. Shockley has made use of small-concentration theory
in an analysis oi p — n junctions^. J. Bardeen and W. H. Brattain have
given solutions for the steady-state hole flow in three dimensions, neg-
lecting recombination, in the neighborhood of a point-contact emitter.^' "
Transient solutions are obtainable analytically for the intrinsic semi-
conductor for constant mean lifetime, and for the extrinsic semiconductor
if the concentrations of added carriers are sufficiently small that the
change in conductivity is negligible. For concentrations unrestricted in
magnitude, Conyers Herring has described a general method for graphical
or numerical construction of transient solutions in one dimension from a
first-order partial differential equation appropriate to the case for which
diffusion is neglected in the extrinsic semi-conductor, and has given some
solutions so obtained, with estimates of the effect of diffusion. Reference
might be made to his paper^^ also for discussion of various physical con-
' loc. cit.
"• loc. cit.
" See the paper of J. Bardeen in this issue.
"Conyers Herring, B. S. T. J. 28 (3), 401-427 (1949).
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 565
siderations and of certain interesting transient effects. Steady-state alter-
nating-current theory for relatively small total hole concentrations in the
w-type semiconductor has been used to describe the action of the filamen-
tary transistor''^ for which diffusion may in general be neglected.^
The steady-state solutions in one dimension apply to single-crystal semi-
conductor filaments, and for critical comparisons between theory and
experiment, the ideal one-dimensional geometry should be simulated as
closely as possible. Experimental estimates of hole concentrations and
flows are frequently obtained from measurements of potentials and con-
ductances of point contacts along a filament'. These estimates require a
knowledge of the dependence of the current-voltage characteristics of
point contacts on hole concentration. Theory for this dependence has been
presented by J. Bardeen", and the determination of hole concentrations by
means of the solutions here given should provide an essential adjunct to
this point contact theory for its comparison with experiment.
2. General formulation
2 . 1 Outline
The formulation of the general problem is initiated by writing the
fundamental equations for the time-dependent flow of holes and elec-
trons in a source-free region of a homogeneous semiconductor under the
assumption that there is no trapping. Conditions for their validity are
discussed. Neglecting changes in the concentrations of ionized donors and
acceptors, the fundamental equations are expressed in reduced or dimen-
sionless form by suitable transformations of the dependent and independ-
ent variables. They are simplified so that the general problem is formu-
lated by means of second-order partial differential equations in two de-
pendent variables, one for concentration and the other for electrostatic
potential; corresponding equations are derived for the intrinsic semi-
conductor. Various properties of the equations are adduced. For the flow
in one dimension, a differential equation in the hole concentration is
given for the n-type semiconductor, accompanied by expressions for the
electrostatic field and hole flow density, as well as by some boundary-
condition relationships involving specification of the latter. The equations
for this case are found to depend on three parameters: the ratio of elec-
tron to hole mobility; a reduced concentration of holes at thermal equilib-
rium ; and a parameter which fixes the total current density.
The recombination of holes and electrons is specified by means of a
1 loc. cit.
" loc. cit.
"W. Shockley, G. L. Pearson, M. Sparks, and W. H. Brattain, Phvs. Rev. 76 (3),
459 (1949).
566 BELL SYSTEM TECHNICAL JOURNAL
suitable function of the concentration of the added carrier, whose form is
specified for two recombination laws: recombination according to a mass-
action law, and recombination characterized by constant mean lifetime.
It is shown that essentially the same reduced equations apply to the case
for which recombination is neglected.
Second-order differential equations in the hole concentration for the
«-type semiconductor with the thermal-equilibrium value of the hole
concentration assumed negligible compared to the electron concentration,
and for the intrinsic semiconductor, are then written for the steady state
of constant current in one dimension. These are converted into first-
order equations which have, as dependent variable a reduced concentra-
tion gradient G, and as independent variable a reduced concentration of
added holes, AP. Boundary conditions are expressed as relationships
between these variables. Properties of the general solutions and of the
boundary conditions are accordingly examined in the (AP, G)-plane. It is
found that there are two intersecting solutions through the (AP, G)-
origin, which is a saddle-point of the differential equation, and that these
are the solutions for field directed respectively towards and away from
sources in semi-infinite regions which have sources only to one side. They
are called field-opposing and field-aiding solutions, and possess two degrees
of freedom. Solutions which do not intersect at the origin are asymptotic
to these, possess three degrees of freedom, and are called solutions of the
composite type. This is the general type, and applies to a finite region in
distance at both ends of which boundary conditions are specified. The
region may, for example, be one between a source and either another
source, a sink, a non-rectifying electrode, or a surface upon which re-
combination takes place. While the analysis of composite cases is straight-
forward, the present treatment is confined to the simpler cases of field
opposing and field aiding, the latter being the one most generally appli-
cable to experiments in hole injection. Also, where the differential equations
involved are linear, solutions for composite cases can be written as linear
combinations of field-aiding and field-opposing solutions.
From the properties of the curves in the (AP, G)-plane is determined the
qualitative behavior of the hole concentration at a hole source at the
end of a semi-infinite filament as the total current is indefinitely in-
creased.
2 . 2 Fundamental equations for the flow of electrons and holes
The equations for the flow in three dimensions of electrons and holes
in a homogeneous semiconductor contain, as principal dependent vari-
ables, the hole and electron concentrations, p and //, the flow densities
J,, and J„ , and ihc- electrostatic field, E, or potential, V. With no tra])i)ing,
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
567
the equations may be written in a symmetrical form, so that they are
apphcable to either an w-type, a ^-type, or an intrinsic semiconductor,
as follows:
dp
! (1)
— = - [p/Tp - go] - div ]p
— = - [n/jn - go] - div J„
at
J«-ji>
Mp
pt, — — grad p
= -f^pp grad
F + — log />
e
In = Mn
e
■iwe
kT
iiE grad n
e
— M^wgrad
— V-\- — log n
e J
div E = — [(/> - po) - (n - m) + (/)+ - Dt) - U~ - Ao)]
e
E = - grad V.
In the first two equations, which are the continuity equations for holes
and electrons written for a region free from external sources, go is a con-
stant which represents the thermal rate of generation of hole-electron
pairs per unit volume; for cases in which hole-electron pairs are produced
also by penetrating radiation, appropriate source terms in the form of
identical functions of the space and time coordinates can be included on
the right in the respective equations. The mean lifetimes of holes and
electrons, Tp and r„ , are in general considered to be concentration-de-
pendent and, since trapping is neglected, the quantities p/rp and w/r„
are equal, being the rate at which holes and electrons recombine. Evalu-
ated for the normal semiconductor, or the semiconductor at thermal
equilibrium with no injected carriers, they equal go .
The equations for Jp and J„ , which are vectors whose magnitudes equal,
respectively, the numbers of holes and of electrons which traverse unit
area in unit time, are diffusion equations of M. von Smoluchowski,
written for hole flow and for electron flow'^ Of the type frequently em-
ployed, after C. Wagner, in theories of rectification, each expresses the
dependence of the flow density on the electrostatic field and on the con-
centration gradient, the diffusion constant for holes or electrons having
been expressed in terms of the mobility, fjtp or Hn , in accordance with the
» S. Chandrasekhar, Rev. Mod. Phys. 15, 1-89 (1943).
568 BELL SYSTEM TECHNICAL JOURNAL
well-known relationship of A. Einstein^^. In them, e denotes the magnitude
of the electronic charge; T is temperature in degrees absolute; and k is
Boltzmann's constant. With transport velocity defined as flow density-
divided by concentration, the product of the mobility and the quantity in
square brackets in the expression for Jp or J„ on the extreme right gives
the corresponding velocity potential, which is thus proportional to the
sum of an electrostatic potential and a diffusion potential.
The next to last equation is Poisson's equation, which relates the di-
vergence of the field to the net electrostatic charge. Here e is the dielectric
constant; pa and «o are the concentrations of holes and electrons at
thermal equilibrium, in the normal semiconductor. The concentrations of
ionized donor and acceptor impurities at thermal equilibrium are repre-
sented by Dt and Aq , while Z)+ and A~ are dependent variables which
denote the respective concentrations in general of ionized donors and
acceptors in the semiconductor with added carriers. As shown in the
Appendix, variations in D^ and A~ may be neglected if the impurity
centers are substantially all ionized in the normal semiconductor, despite
the effect large concentrations of added carriers may have on the equilib-
ria'^.
The expression of the electrostatic field as the gradient of a potential
according to the last equation is consistent with the circumstance that
the effects of magnetic fields, with none applied, are in general quite
negligible.
Subtracting the first continuity equation from the second, it is found
that
(2) diva, - ]n) = -lip - "),
at
since, with no trapping, p 't„ equals ;//r„ . Neglecting changes in the con-
centrations of ionized donors and acceptors, this equation and Poisson's
equation give
W J„_J„ = J--^«-?; I„ + I„ = I-^^,
Aire ai Air dt
where J and I are solcnoidal vector point functions, in general time-
dependent. The latter is the total current density, and the term which
follows it in (3) gives the displacement current density.
"-A. Kinstcin, Annulai dcr Phvsil; 17, 549-560 (1905); Muller-Pouillct, Lchrbach der ,
Physilc, Braunschweig, 1933, IV (3), 316-319.
'" II has l)t'i-n found I'roni measurements of the temperature clc])cn(lence of the con- '
(luclivily and Hall coenUienl lliat the energN' of thermal ionization of the donors in H-
typc germanium of relalively iiigh jjurity is only about 10 -cI', whence most of tiie donors
are ionize<l at room temperature: G. L. Pearson and W. Shocklev, Pltvs. Rev. 71 (2), 142
(1947j.
FLOW OF ELECTRONS AND HOLES LY GERMANIUM 569
It may be well to point out that the validity of the diffusion equations
depends on two assumptions, which, while hardly restrictive in general
for homogeneous semiconductors, indicate the nature of the generaliza-
tions which might otherwise be necessary". The first assumption is that
there are no appreciable time changes in the dependent variables in the
relaxation time for the conductivity, or the time of the elementary fluctua-
tions. This is tantamount to the requirement that the carriers undergo
many colhsions in the time intervals of interest. The second assumption is
that the changes in the carriers' electrostatic potential energy over
distances equal to the mean free path are small compared with the aver-
age thermal energy. In accordance with this assumption, very large fields
in the electrically neutral semiconductor for which the carriers are not
substantially in thermal equilibrium with the lattice are ruled out. The
neglect of space charge then in general validates the two assumptions, if
the resistivity is not too small, since the neglect of changes in the de-
pendent variables which occur in the dielectric relaxation time obviates
their change in the relaxation time for conductivity; and the neglect in
the steady state of appreciable variations in electrostatic potential, and
thus in the other dependent variables, in the distance^" Ld , obviates their
variation in a mean free path. The dielectric relaxation time for ger-
manium, 1.5 -10"^^- sec per ohm cm of resistivity, in high back voltage
material exceeds the relaxation time for conductivity, which is about
1.0- 10^^' sec; and in semi-conductors in which the mobilities and the
conductivity are smaller than the comparatively large values for ger-
manium, the dielectric relaxation time may be appreciably larger than the
relaxation time for conductivity. Similarly, Ld for germanium is about 7
times the mean free path, and this ratio, which is essentially inversely
proportional to the square root of the product of mobility and intrinsic
conductivity, may be appreciably larger for other semiconductors.
If, on the other hand, it should be desired to consider space-charge
effects in germanium, the diffusion equations may be of rather marginal
applicability, and the use of their appropriate generalization indicated,
since with Ld equal to 7 mean free paths, appreciable space-charge varia-
tion of potential, corresponding to a field which is not small compared
with the free-path thermal-energy equivalent of about 3500 volt cm~\
may occur in at least one of the free paths. For example, diode theory,
rather than diffusion theory, provides the better approximation for the
characteristics of germanium point-contact rectifiers, and is particularly
applicable to those from low resistivity material for which the potential
variation is largely confined to one mean free path or less".
'" loc. cit.
" loc. cit.
" H. C. Torrey and C. A. Whitmer, "Crystal Rectifiers," New York, 1948, Sec. 4.3.
570
BELL SYSTEM TECHNICAL JOURNAL
Neglecting space charge, Poisson's equation becomes simply the con-
dition of electrical neutrality:
(4)
{p — pit) — (« — «o) = 0,
assuming substantially complete ionization of donors and acceptors.
Similarly, equations (3) become
(5)
L - Jn = J; Ip + In = I.
A\'ith electrical neutrality, the two continuity equations merge into one:
Since derivatives of p equal the corresponding ones of n,
div Jp = - [p/r,, - ^o] - —
(6)
= div Jn = — [«/t„ — gn] —
dn
dT
The neutrality condition in conjunction with the two equations obtained
by substituting for Jp and J,, from the diffusion equations in (6) thus pro-
vide three equations for the determination of p, n, and E or T^.
It is instructive to rewrite equations (6) in accordance with
(7)
div Jp = s
= div J„ = s
djp • i
gradp
grad n,
s =
dx
dp
dx
i +
dy
dp
d~yj
j +
'd]p ■ k
dz
dp
dz
k,
where i, j, and k are unit vectors in the directions of the respective axes.
The velocity s, which is given as well by the expression for electrons an-
alogous to that written for holes, may be defined alternatively as follows:
Suppose, for definiteness, that the second-order system of equations (4)
and (6) have been solved, so that the concentrations and flow densities
are known in terms of the cartesian coordinates .v, y, and z, and the time t.
The ;v-component of s is then the partial derivative with respect to p of
the x'-component of Jp in which x has been replaced by the proper func-
tion of p, y, z, and /, and similarly for the other components. Thus, with
s a known function, p or n may be considered to satisfy the first-order
partial differential equation obtained by substituting from (7) in (6), from
which it is evident that s is the velocity with which concentration transi-
ents are propagated^^. This velocity, which is here called the differential
^* The identification of s as tiiis propagation velocity follows the example of C. Herring,
in whose method for solving the transient constant-current ])rol)lem in one dimension
the velocity depends in a known manner on concentration only, through the neglect of
diffusion, so that the general solution of the differential equation in which thus neither
independent variable x nor I occurs explicitly may be obtained; cf. reference 12, pp. 412 ff.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
571
transport velocity and loosely referred to as the transport velocity of
added carriers, of course differs in general from the transport velocity
proper, defined as the ratio of flow density to concentration; its general
definition, which is applicable to the steady state, has been introduced to
facilitate later interpretations.
2.3 Reduction of the fundamental equations to dimensionless form
2.31 The general case
In order to obtain solutions in forms which exhibit such generality as
they may possess, the fundamental equations are to advantage written
in terms of dimensionless dependent and independent variables which are
the original variables measured in suitable units. Through formal con-
sideration of the equations (1), in conjunction with (3) or with (4) and (6),
these units can be so chosen that the system of reduced equations will
exhibit independent parameters on which it may be considered to depend.
The best choice of suitable units is by no means unique; those choices
which have been made are natural ones, in that they have been found to
result in greater formal simplicity and ease of interpretation in the theory
than others which may be equally valid in principle.
The choice for an n-type semiconductor consists in definitions of di-
mensionless variables and parameters as follows:
= \P,Tt
(8)
kT LIT
X = x/Lp , Y = y/Lp , Z = z/Lp ; Lp =
U = t/T
P = /»/(«o - pu); Pi) = po/{no - po) = goT '(«o - pa)
N = n/(no — po); No = «o/(«8 — po)
C ^ I 7„ = Ea/Eo = fxEa/iDp/T^; h ^ (ToEo ; £o ^ kT/eLp
Cp = Ip/Io
C,i = — I„//o
F = E/Eo
W = V/EoLp = eV/kT
_Q ^ r/Tp .
The rectangular cartesian space coordinates are .v, y, and z. The quantity
T is the mean lifetime of holes for concentrations of added holes small
compared with the thermal-equilibrium electron concentration, »o ; and
(To is the conductivity of the normal semiconductor. The hole mobility,
572 BELL SYSTEM TECHNICAL JOURNAL
originally Hp , is denoted by m for simplicity. If b is the ratio of electron to
hole mobility, o-q is given in general by
(9) ao = fJie(b)io + po) = Mo bfxe{ih - po), Mo = 1 + -^ Po,
0
the symbol .l/n being introduced for brevity. If />o < < ^^o , Mo is unity
and cTo equals byicn^ .
The independent dimensionless distance variables are .Y, Y and Z,
where the distance unit, L,, , is a diffusion length for a hole for the mean
lifetime, r, the diffusion constant for holes being Dp . This mean lifetime
is the unit for the independent dimensionless time variable, U. The hole
and electron concentrations are measured in units of the excess in concen-
tration of electrons over holes^'*, «o — po , the reduced variables being P
and X, respectively. The reduced total current C is total current density
measured in units of the current density /o which flows in the semicon-
ductor with no added carriers under the characteristic field Eo , which is
a field such that a carrier would expend the energy kT in drifting with it
through the distance Lp. A more illuminating alternative description is
that C is the ratio of the average drift velocity of holes under the applied
or asymptotic field, £„ , to the hole diffusion velocity (Dp/r)''. The field
E„ is that which produces the current density I in the semiconductor
with no added carriers. The corresponding reduced hole and electron flow
densities are Cp and C„ . The electrostatic field measured in units of Eq
is denoted by F, and W is the corresponding reduced electrostatic poten-
tial. The lifetime ratio (7 is a function of P which characterizes the re-
combination process. While it appears from experiment that the recom-
bination rate for holes depends on both physical and chemical properties
of the semiconductor, in a particular semiconductor at given temperature
it may be considered to depend on hole concentration alone.
Representative values for germanium of units in terms of which the
dimensionless quantities are defined are as follows: The mean lifetime r
may be of the order of 10~^ sec. With a mobility for holes" of 1700 cm^
volt"' sec~^ in germanium single crystals at 300 deg abs, the length Lp
is then about 2 -10"- cm; the characteristic field, Eo , 1.2 volt cm~^; and
the current density /o , 0.12 ampere cm~- for a resistivity of 10 ohm cm.
With these definitions'", the fundamental equations for a region free
from external sources, neglecting changes in the concentrations of ionized
^ loc. cit.
''■' The excess in concentration of electrons over holes is of course equal to that of
ionize<l donors over ionized acceptors.
'■'" The definitions ^iven ai)i)car best if there is a region in which F — Pu is small, with
P(, 9^ 0. Modified dellnitions of the reduced flow clensities, in which the conductivity
ffn is re|)laced hy the conductivity bjjie{no— po) due to the excess electrons alone, result in
criuations obtainable formally by setting Mo eriual to unity.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
573
donors and acceptors and neglecting space charge-^, are given in reduced
form as follows:
^^ = -[^»ModivC, + PQ - Pol
oU
dN
dU
= -[6ModivC„ -]- PQ - Pol
(10) i ^^ = m ^^'^ - °"'^^ ''^ = " 6F0 ^ ^^^^ ^''^ + ^°^ ^^
Cn = ^ [-KV - grad N] = -^^ -V grad [-TI' + log X]
{P - Po) - (N - No) = P - .¥ + 1 = 0
F = -grad W,
and the reduced form of equations (5) is
(11) Cp - c„ = c.
These reduced equations may be simplified and two differential equa-
tions in the dependent variables P and W written as follows:
\-bM, div C„ = div P grad [W + log P] = [PQ - Po] + ~
oU
(12)
I
div C = 0,
C = — S erad
W
b - 1
log I
where S is the conductivity <x in reduced form
(13)
2 ^ ^ = ^^ -^ P
(70 ^'iVo + A
Mo
l + '^P
An alternative formulation, due to R. C. Prim, which is obtained by evalu-
ating div [C„ ± b Cp], consists of the two equations,
,-*— div (1 + 2P) grad W = - ,-^ div grad [W -(1 + 2P)]
0—1 0 -\- I
(14)
[PQ - Po] +
dP
^1 It may be desirable to take space charge into account in cases involving high fre-
quencies or high resistivities. Poisson's equation and equations (3) are in reduced form,
9F
P - N + I = bMoT div F and Cp - C„ = C - T ^., where T = e/iircroT.
The term containing T may often be omitted from one of these equations, depending on
the nature of the particular case considered.
574 BELL SYSTEM TECHNICAL JOURNAL
in which the use of (1 + IP) as dependent variable may be desirable.
This variable is equal to the concentration of carriers of both kindb di-
vided by the excess of electron concentration over hole concentration,
which is a constant.
The expression in the equations which specifies the recombination
rate may be written more simply. Since the lifetime ratio Q is unity for
P = Po,
(15) PQ- p,= {P-Po)R,
where R, which will be called the recombination function, depends on P
and also equals unity for P = Po . The lifetime ratio and the recombina-
tion function which, of course, differ in general, both equal unity for the
case of constant mean lifetime. Recombination of holes and electrons at a
rate proportional to the product of their concentrations, called mass-action
recombination, and recombination characterized by a constant mean life-
time for holes are frequently of interest. For a combination of independent
mechanisms of both types, it is easily seen that
(Q ^ r/r,, = 1 + (7 (/>-/>o)/«o = 1 + a (P - P„)/(l + Po),
(16) < a = t/t, , 0 < a < 1
[P = 1 + ap/m = 1 + aP/(l + Po),
where t,. is the mean lifetime for small concentrations associated with
mass-action recombination alone, so that a = 0 for constant mean life-
time, and a = I for mass-action recombination. If both recombination
mechanisms are operative, that of mass-action recombination will, of
course, determine the mean lifetime where the concentration of added
carriers is sufficiently large.
Recent experiments have shown that the mean lifetime for holes in
n-type germanium can be increased materially, to at least 100 micro-
seconds, by minimizing surface recombination through decreases in sur-
face-to-volume ratios.^ On the other hand, comparatively short mean
lifetimes, of the order of one microsecond, occur in p-type germanium
produced, for example, from n-type by nucleon bombardment. It should be
possible to determine in various cases which recombination law would
provide the better approximation by use of the technique of H. Suhl and
W. Shockley of hole injection in the presence of a magnetic field-"' or by
the j)hotoelectric technique of F. S. Goucher-^.
' loc. cit.
=«H. Suhl and VV. Shockley, Pkys. Rev. 75 (10), 1617-1618; 76 (1), 180 (1949).
" F. S. Gouchcr, paper I 1 1 of the Oak Ridge Meeting of the American Physical So-
ciety, March 18, 1950; P/tys. Rev. 78 (6), 816 (1950).
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
575
It appears that solutions neglecting recombination furnish useful ap-
proximations for some applications. If recombination is neglected, by
assuming that the mean lifetime is infinite, the definitions (8) of the di-
mensionless quantities no longer have meaning, but essentially the same
differential equations and corresponding boundary-condition equations can
still be used. The reduced equations become essentially homogeneous in
T for T large, and it suffices to suppress the recombination terms, PQ —
Pq , retaining formally the definitions of the dimensionless quantities in
which now r, and thus Lp and Eq or /o no longer have physical significance.
One of these unitary quantities may be chosen arbitrarily. It might be
noted that if Poisson's equation is retained the length unit is advantage-
ously chosen as La , which gives a dielectric relaxation time for the time
unit.-^
In one cartesian dimension, with total current a function of time only,
W may be eliminated by means of the equation for C in (12) and, upon
substituting for it in any of the three remaining equations in (12) and
(14), a differential equation for P results which depends on b, Po , andC
as parameters. Dropping vector notation, this equation is
dP
dU
1^(1 -F 2P)(l + * + ^pj]
d'P b
- 1
~dP~
Idx]
-"'^sx
aX2 "^ b
1 +
b
2
(17)
Similarly, from (10),
(P - Po)R-
(18)
On —
MoCP - (1 + 2P) ^
bM,
1 +
'4-^^
. . ^ b - idP
F =
dX
l + '-±-'p
The expressions for F and Cp possess some interesting features. That for
the reduced field, F, is composed of two terms, the first of which expresses
Ohm's law, since C is reduced total current density and the denominator
is proportional to the local conductivity. The second term is a contribu-
tion which is directed away from a hole source, since b is greater than
576 BELL SYSTEM TECHNICAL JOURNAL
unity, or since electrons are more mobile than holes. If b were equal to
unity, the field would be independent of the concentration gradient. The
second term thus represents a departure from Ohm's law which is due to
diflfusion and which is associated with the presence of current carriers of
differing mobiUties. It gives a non-vanishing electrostatic field for the
case of zero total current. The two terms in the expression for Cp are
likewise ohmic and diffusion terms, but here the diffusion term would be
present even if the hole and electron mobilities were equal.
Boundary-condition relationships might be illustrated by some ex-
amples for this one-dimensional case. If it be specified that for f ' > 0 a
fraction / of the total current to the right of a source at the A'-origin,
say, be carried by holes, then, from (18),
(,^. dP ,, b + {b+ \)P
f-
c,
b + ib + 1)P_
X = +0, U > 0.
The solution in an A'-region to the right of the origin may be determined
by this condition and an additional one. The simplest is that for the flow
in the semi-infinite region, namely P — Po for A' = x>. This relationship
holds for some finite X for an idealized non-rectifying electrode there.
For the region between the source and a surface at X = Xa on which
there is recombination characterized by a hole transport velocity s,
which is also the differential transport velocity for ^ constant, it is clear
that C = 0, so that, for A" = Xa ,
(20) C = _J_ _a±_2^ dP ^ J_^
^^^^ ^' Mob+ (b+l)P dX bMo '
S ^ s/[Dp/t]\ s = Jp/p.
Consistently with these examples, boundary conditions may in general be
dP
expressed as relationships between P, —- , and the parameter C, for given
oX
values of A'.
A simple transformation of dimensionless c^uantities serves to extend all
of the analytical results wliich have been given for the ;/-type semi-
conductor to the p-iype semiconductor: Consider the substitutional
transformation which consists in replacing the original dimensional quan-
tities for holes by the corresponding ones for electrons, and vice versa,
and in replacing the electrostatic field by its negative. The original set of
fundamental equations (1) is invariant under this substitution, which
defines an equivalent transformation from the dimensionless quantities of
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
577
equations (8) to the desired new set, in which the ratio b of electron to
hole mobility is replaced by its reciprocal.
2.32 The Inlrinsic semiconductor.
For the intrinsic semiconductor, in which />o = Wo , the reduced concen-
trations given in (8) are inapplicable. As Pn approaches «o , these reduced
concentrations increase indefinitely, and the equations which those given
for the «-type semiconductor approach in the limit are homogeneous in
the concentration unit. These limiting equations therefore apply to the
intrinsic semiconductor in terms of a concentration unit which may be
chosen arbitrarily. The quantity ;?o will be chosen as this unit. Thus,
redefining the reduced concentration variables as
X = n/uo ;
N,
(21) P ^ p/uo ,
from equations (12) and (14) any two of the equations in the dependent
variables P and W given by
-{b -\- 1) div Cp = -^ div P grad W
(22) {
^^^ div grad P=[P()-1]+|^,
C, = -__^-^Pgrad[ir+ logP];
div C - 0,
C = -P grad
W
b+ 1
logP
and including the right-hand member which is common at least once, char-
acterize the intrinsic semiconductor'-^.
It is noteworthy that one of these equations contains only P as de-
pendent variable, W being absent; and this equation indicates that the
spatial distribution of carrier concentration is not subject to drift under
the field, but only to a diffusion mechanism with diffusion constant
2DpD„/{Dp + Dn), where D,, = bDp is the diffusion constant for elec-
trons.-^ This result is readily accounted for as being due to a conductiv-
ity in the intrinsic case which is everywhere proportional to the concen-
tration of carriers of either type, so that 3 = P. The expression for C
^^ These equations for the intrinsic case were first derived quite unambiguously as
those for the special case of the parameter po/n,, equal to unity in the general equations
written in terms of the concentration unit Ho. This unit is, however, less advantageous
than («(, — po) which, in obviating much of the formal dependence on po, makes for
greater generality.
^^ The equations for the intrinsic case might be written in somewhat simpler form b}-
redefining the length unit in terms of 2D,,D,J(Dp -{- Dn) as a diffusion constant instead of
D,, , but their relationshi]i to those of the general case would then l)e less evident.
578
BELL SYSTEM TECHNICAL JOURNAL
in (22) owes its special form simply to this circumstance, while that for
Cp applies also to the general case, and the differential equation in P is a
consequence of the equations in P and W from div C and div Cp . Or,
in more detailed terms, since the ohmic contribution to C,, must be pro-
portional to C, div Cp contains only the contribution due to diffusion.
This is evident from the relationship obtained from (22),
{2i)
v/p —
1
&+ iL
c -
2b
b+ 1
grad P
from which it foUow^s also that, despite the dependence of the local field
on concentration gradient, the ohmic contribution to the hole flow density
is the flow density of holes normally present in the intrinsic semiconduc-
tor under the unmodulated applied field.
The equations which have been given for one-dimensional flow in the
«-type semiconductor can readily be transformed, in the manner indi-
cated, into the corresponding equations for the intrinsic semiconductor.
2.4 Differential equations in one dimension for the steady state of constant
current and properties of their solutions
The steady state of constant current in one dimension will be con-
sidered explicitly for two limiting cases: the «-type semiconductor with
Po = 0, and the intrinsic semiconductor. These serve to illustrate and
delimit the qualitative features of the general case. Furthermore, the case
Po = 0 frequently applies as a good approximation^®, as does the intrinsic
case, which is of particular interest not only in itself but also because the
extrinsic semiconductor exhibits intrinsic behavior for large concentra-
tions, and because moderate increases in temperature above room tem-
perature, such as joule heating may produce, suffice to bring high back
voltage germanium into the intrinsic range of conductivity-^. The tem-
perature dependence of Po and of other reduced quantities is evaluated
for germanium in the Appendix.
The ordinary differential equations in the reduced hole concentration,
P, for the steady state in one dimension, which result from equations
(17) and (22) by equating the time derivatives to zero are as follows:
(24)
d'p
^ dX
b - 1
b
dP
_dX_
dX"-
[1 + 2P]
[1+^"^^
p
b
p
l + '-^p
1 + 2P
R
-'"' In »-typc germanium of resistivity al)out 5 oiim cm, for example, tlic electron con-
centration exceeds the equilibrium hole concentration bj- a factor of about 70.
" Germanium which is substantially intrinsic at room temperature has been produced:
R. N. Hall, paper 15 of the Oak Ridge Meeting of the American Physical Society, March
18. 1950.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 579
lor the w-type semiconductor with Po = 0, and
(25) ^ = ^ + ^ (p _ i)R
for the intrinsic semiconductor, with R given as (1 + aP) by (16); P has
the same meaning in both equations, the concentration unit being «o
for each case. With time variations excluded in this way, the parameter C
is a constant and the cHfferential equations apply to the steady state of
constant current.
Since the equations involve only the single independent variable X
which does not appear explicitly, their orders may be reduced by one, in
accordance with a well-known transformation, which consists in intro-
ducing P as a new independent variable, and
d d
as new dependent variable : Noting that -yr; is equivalent to <j~, the dif-
ferential equations become
(27) ^ =
^ ' dP
c-'-^o P
=; + -
1 + b+lp
R
[1 + m
x + '-^p
0
[1 + 2P\G
for the ;/-type semiconductor, and
dG h + \ {P - l)R
(28)
dP lb
for the intrinsic semiconductor. These are differential equations of the
first order.
The solutions sought in the semi-infinite region, A' > 0, are those for
which G = 0 for AP = 0, that is, those which pass through the (AP, G) —
origin, where AP, which denotes P — Po, equals P for the w-type semi-
conductor and P— 1 for the intrinsic semiconductor. This condition is that
the concentration gradient vanish with the concentration of added holes,
as it must for X infinite. It will be shown that the differential equations
possess singular points at the (AP, G)-origin, and the physical interpretation
of the solutions through these singular points will be examined. For this
purpose, consider equation (27) for the w-type semiconductor which, in
the neighborhood of the origin, assumes the approximate form,
^ ^ dP P ^ G'
580 BELL SYSTEM TECHNICAL JOURNAL
since R is close to unity for P small, whence
Similarly, for the intrinsic semiconductor, for P—\ small,
^^^ d{p -\) p - \ "^ y 2b '
There are thus, in each case, two solutions through the (AP, G)-origin,
one with a positive derivative and the other with a negative derivative.
Consider now the doubly-infinite region with a source at X = 0. Then,
for X > 0, the negative derivatives apply, since the concentration gra-
dient G is negative. Similarly, for X < 0, the positive derivatives apply.
Now, the value of the current parameter C will be substantially the same
in both regions, since it has been assumed that AP is small. For C posi-
tive, equation (30) for the 7^-type semiconductor indicates that the
magnitude of dG/dP for X^ < 0 exceeds that for X" > 0, and the situation
is reversed if the sign of C is changed. That is, the magnitude of the
concentration gradient increases more slowly with concentration for
field directed away from a source than for field directed towards a source,
which is otherwise plausible. For the intrinsic semiconductor, on the
other hand, equation (31) shows that corresponding magnitudes of the
concentration gradient are equal and entirely independent of C, a result
which the differential equation (28) establishes in general.
It thus appears that a differential equation for the steady state possesses
two. solutions through the (AP, G)-origin, and that one of the solutions
corresponds to the case of field directed towards a source, the other to the
case of field directed away from a source. Field directed towards a source
is called field opposing, while field directed away from a source is called
field aiding, the latter being the one commonly dealt with in hole-injec-
tion experiments. It should be noted that the cases of field opposing or
field aiding can be realized in a given X-region only if it adjoins a semi-
infinite region free from sources and sinks. In the region between two
sources, neither of these cases applies. L. A. MacColl has shown, through a
more detailed consideration of the singularity at the (AP, G)-origin, that
the two solutions through this point are the only ones through it. The
origin is thus a saddle-point of the differential equation, and there exist
families of nonintersecting solutions in the (AP, G)-plane for which the
solutions which intersect at the origin are asymptotes. A solution for an
A'-region between two sources, for e.xample, is a member of such a family,
as is in general any solution determined by boundary conditions at the
ends of a finite region in X. Such a solution will be called a solution for a
comj)osite case; it approaches asymptotically both a field-opposing and a
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
581
field-aiding solution, which is consistent with the qualitative geometry
associated with a saddle-point, and with the fact that, in the X-region, a
SLOPE =y(VC2 + 4 + c)
^C FOR C LARGE
SOLUTION CURVES
BOUNDARY CONDITION CURVES
ZERO CURRENT,
.C=0
(r:_ HOLE CURRENT '\
V TOTAL CURRENT/
REDUCED HOLE CONCENTRATION, P
Fig. 1. — Diagrammatic representation in the (P, G)-plane of solutions and boundary
conditions for the steady-state one-dimensional flow of hjles in an »-type semiconductor.
total current directed away from one source is necessarily directed to-
wards the Other. This behavior is illustrated diagramatically for the n-
type semiconductor in Fig. 1, which shows, in the (P, G) -plane, solution
582 BELL SYSTEM TECHNICAL JOURNAL
curves as well as boundary-condition curves for a source, for a given
positive value of C. Those for the intrinsic semiconductor differ only in
that the solution curves in the {P — \, G)-plane do not depend on C, all
being given by the ones for zero total current density, and the corre-
sponding boundary-condition curves are straight lines.
Once a solution, G(P), for field opposing, field aiding, or a composite
case, specifying G as a function of P has been obtained, the dependence of
P on X is determined by evaluating
in accordance with the definition of G, equation (26). For the general com-
posite case, G(P) is that one of the family of solutions for the given C such
that the integral between values of P determined by the intersections with
the boundary-condition relationships provides the correct interval in X.
If P° is determined by the condition that for X = 0, a fraction / of the
total current is carried by holes, then, from (19), P" is the point on the
solution curve which satisfies either
.„^ ^0 b+{b+ 1)P°
for the w-type semiconductor, or
ib + lY
f-
b -\- (b + 1)P\
C
m G" = -
2b
f-
1
6+ 1
C
for the intrinsic semiconductor, G being the corresponding value of G.
From the manner of derivation of the boundary conditions (33) and
(34), it is evident that they are perfectly general, holding in particular
for the cases of field opposing and field aiding, and whatever be the sign
of C. The concentration gradient G may be seen to have the correct sign
for these cases if it is taken into account that /, defined as Cp/C or Ip/I,
may assume any positive or negative value, being positive for field aiding,
and negative for field opposing, for which the hole flow is opposite to the
applied field. For / negative, the quantities in brackets in equations (33)
and (34) are negative. The general principle that the sign of the con-
centration gradient G is such as to be consistent with the flow of holes
from a source requires also that the quantities in brackets be positive
for field aiding, or whenever/ is positive. For the intrinsic semiconductor,
this requires that/ for field aiding never be less than l/(b + 1). This is
clearly a consistent requirement which holds in all generality since, for
zero concentration of added holes, or for the normal semiconductor, G"
vanishes and the ratio of hole current to total current equals l/(b -f 1).
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 583
In the case of the »-type semiconductor, / is not restricted in this way.
Consider, for this case, hole injection into the end of a semi-infinite
filament, to which the field-aiding solutions apply. As the total current is
increased indefinitely, the tangent to the solution in the (P, G)-plane at
the origin approaches the P-axis, as does the solution itself, and it is
evident from the boundary-condition curves of Fig. 1 that if f is less than
l/\b -\- I) the hole concentration P^ at the source approaches as a limit
the indicated abscissa of intersection of the appropriate boundary-
condition curve with the P-axis, or the value for which the quantity in
brackets vanishes. It is similarly evident that P" increases indefinitely
with total current in either semiconductor if / is greater than or equal
to l/{b + 1). This is a result otherwise to be expected from the qualitative
consideration that an extrinsic semiconductor becomes increasingly
intrinsic in its behavior as the concentration of injected carriers is in-
creased.
Figure 1 serves also to facilitate a count of the number of degrees of
freedom which the steady-state solutions possess: Corresponding to values
of the concentration and concentration gradient at a point in a semi-
conductor filament in which added carriers flow, there is a point (P, G)
in the half-plane, P > 0, of the figure. If the total current density is speci-
fied in addition, the value of C and the solution through the point (P, G)
are determined. This solution applies in general to a composite case,
which therefore possesses three degrees of freedom. That is to say, at a
point in a filament, any given magnitudes of both concentration and con-
centration gradient can be realized for a preassigned total current density
by a suitable disposition of sources to the right and left. The cases of
field opposing or field aiding, however, possess only two degrees of freedom,
since the given concentration and gradient determine the total current
density and the solution, which must pass through the origin; and which
of the two cases applies depends on whether the point (P, G) lies to the
left or to the right of the curves, shown in the figure, for the zero-current
solution. Thus, in a filament with a single source of holes, for example, the
concentration, concentration gradient, total current density, and any
functions of these, such as hole flow density and electrostatic field, are
all quantities the specification of any two of which at a point completely
determines the solution for a source-free X-region which includes the
point.
3. Solutions for the Steady State
For a given value of the current parameter C, solutions for the steady
state of constant current in a single cartesian distance coordinate, specify-
ing G in terms of the relative hole concentration P, and P, the reduced hole
584
BELL SYSTEM TECHNICAL JOURNAL
flow density Cp , and the reduced electrostatic field F, in terms of reduced
distance .Y are found in general by numerical means, which include nu-
merical integration and the evaluation of appropriate series expansions.
General solutions which have been evaluated numerically for w-type
germanium for a number of values of the current parameter are given in
the figures. In the limiting cases of P small and P large, analytical ap-
proximations for the extrinsic semiconductor are readily obtained, that
for P large being derived from an analytical solution for C equal to zero,
or zero current. If the steady-state problem for the extrinsic semicon-
ductor is simplified by neglecting either recombination or diffusion, solu-
tions are obtainable which, like the zero-current one, are expressible in
closed form.
For the intrinsic semiconductor, the general problem considered in this
section is solved quite simply by analytical means. The solution provides,
as physical considerations indicate it should, the same analytical approxi-
mation for large P as does the zero-current solution for the extrinsic case.
It may be well to consider first the intrinsic semiconductor which, aside
from the extrinsic semiconductor for the case of zero current, appears to
constitute the only analytically solvable steady-state case in one dimension
which has physical generality according to the present approach.
3 . 1 The intrinsic semiconduclor
Integrating the differential equation (28), it is found that
^2 ^ + 1
(35)
/(.
\)RdP,
with R given as 1 -f ai' by (16), for an arbitrary combination of the two
recombination mechanisms, assumed independent. Thus
(36)
o' = ^4i(.
1)''
(1 + a) + - a{P
1)
for the cases of field opposing or field aiding, for which G = 0 for P— 1 =
0; for a composite case, a suitable constant is included on the right-hand
side. Excluding composite cases, the root may be taken in (36) and G
replaced by its definition, which gives
(37)
d{P - 1)
dX
= ±
[P - 1]
(1 + a) -f ^
a{P - 1)1
and if the .Y-origin is selected more or less arbitrarily as the point at
which P is infinite, then (37) gives
"(1 + a){b + 1)7 ^
(38)
1 = ^(i+^ csch^
2a
Sb
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
585
provided a ± 0; for mass-action recombination a = 1. For a = 0 or for
constant mean lifetime, (37) gives an exponential dependence of P— 1 on X:
(39)
F - i = {P' - 1) exp
b + 1"
2b
X
where Po is the relative hole concentration for X = 0. Linear combina-
tions of the two solutions in (39) give solutions for composite cases, since
the differential equation from which (39) was derived is linear in P. A
similar result does not hold if there is mass-action recombination present,
and the more general procedure above referred to must then be followed.
A characteristic feature of these solutions for the intrinsic semicon-
ductor is their independence of the current parameter C, this parameter
occurring only through a boundary condition, such as the one given in
equation (34) of Section 2.4. They are symmetrical in shape about a
source, the dependence of the concentration on the magnitude of the
distance from the source being the same for field opposing as for field
aiding, which follows quite simply from the symmetrical forms of the
solutions, and the condition that the concentration is everywhere con-
tinuous.
Equations (22) and (23) of Section 2.32 provide the hole flow density
and the electrostatic field for this case. With G given for mass-action
recombination or for constant mean lifetime by the appropriate special
case of equation (36), and using the positive sign for an X-region to the
left of sources and the negative sign for an X-region to the right,
(40)
1
b+ 1
\
1 r
F =
1
P
C -
c -
b -
b-\-
b-\-
-A
The electrostatic potential, V, is readily expressed in terms of F: From
(41)
F =
eLp dV
_e_dV
kTdX
kT dP
and (40), it is found that
(42)
whence
_e_dV ^ b - 1 I _ r Jl.
kT dP b + I P GF'
(43) ^ = ^^logP-c/
kT
b+ 1
dP_
GP
1
b+ 1
log P - C
f dX
J P '
586 BELL SYSTEM TECHNICAL JOURNAL
with the integral to be evaluated for the particular case it is desired to
consider.
3.2 The extrinsic semiconductor: n-type germanium
The evaluation of steady-state solutions for the extrinsic semiconductor
involves, as a first step, the determination of G as a function of P from
the differential equation (27), which is accomplished by numerical inte-
gration and by the use of series expansions. These variables are subse-
quently found in terms of X in the manner described in Section 2.4. The
series expansions, which are Maclaurin's series in P, and series in powers
of the current parameter, C, with coefficients functions of P, are given
explicitly for the «-type semiconductor in the Appendix; they readily
furnish the corresponding series for the /^-type semiconductor by means
of the transformation discussed at the end of Section 2.31. The Mac-
laurin's series in P are useful for starting the solutions at the {P, G)-
origin. As P increases, these series converge increasingly slowly, and it
becomes necessary to extend the solutions by other means. For the larger
values of C, however, the numerical integration for the important case of
field aiding becomes increasingly difficult, and it is advantageous to use
the appropriate series in the current parameter, which converges the more
rai)idly the larger is C. The first term alone in this series for field aiding
gives in closed form the solution for the case in which diffusion is neg-
lected; and the existence of the series itself was, in fact, originally sug-
gested by the form of the solution for this case^^. Series of this type are
given also for field opposing, and it seems probable that such series are
, obtainable for composite cases as well, though this has not been investi-
gated.
Solutions were evaluated numerically for ;/-type germanium, by the
means described, using the value 1.5 for the mobility ratio'-^, b. For the
case of mass-action recombination, solutions for values of the current
parameter, C, up to 50, specifying | G | in terms of P, are given in Fig. 2,
both for field opposing and field aiding. These solutions in the (F, G)-
plane are given to permit the fitting of boundary conditions at a hole
source, according to a method described in Section 4. Solutions specifying
P in terms of A' for field aiding are given in Fig. 3, with the A'-origin
chosen more or less arbitrarily at P = 100. The corresponding solutions
for the reduced hole flow density, Cp , and the reduced field, F, are given
2" The solution for this case was communicated by Conyers Herring and is given in his
paper of reference 12.
" The hole mol)ility and the value 1.5 for the mobility ratio were determined by G. L.
Pearson from the temi)erature dei)endence of the conductivity and Hall coelTicicnt in
p-^yV^ germanium. J. R. Haynes has recently ()l)tained, from drift-velocity measurements,
the same hole mobility, but the larger value 2.1 for the ratio of electron mobility in «-type
germanium to hole mobility in /"-type: Pai)er L2 of the Chicago Meeting of the American
Physical Society, November 26, 1949.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
587
lUU
-
//
50
-
/
-
f
20
10
-
^
^^
r
/
_
yC^
^^^0>^
^.^K-"""^ //J
7
5
- FIELD
OPPO.
5ING//^|^
^
""^^^
-
/.
yyy"
'^^/^f
GRADIENT, |C
b r\)
" A
\
'//
<^
w
If///
/
///
m
7///fiELD AIDING
CONCENTRATION
op p
^ /
y^ X /j
^// J
III
// / y
/^/ /
V/,
'>.
W/,
"
y V /
/ /
/ / / / / // 1 1
O
UJ
^ 0.05
Q
UJ
tr
~ / '^/ /
//
^/// ///
J
y//
r^
y/////
0.02
^/x/
vA
/ ////
0.01
0.005
//A/
//
////
-/ / / ^/
v/A
//
w/
/
// /
w
0.002
V /^
'/
0.001
/ ////
Mil
1 1 II 11 1 i
1 1
III!
1 1 1
1 1 II
O.Ot 0.02 0.05 0.1 0.2 0.5 t.O 2 5 10 20 50 100
REDUCED HOLE CONCENTRATION, P
Fig. 2. — The dependence of the reduced concentration gradient on reduced concentra-
tion for the steady-state one-dimensional flow of holes with mass-action recombination in
w-type germanium.
respectively in Fig. 4 and in Fig. 5. In accordance with equations (10),
(18), and (26), the solutions for Cp and F are found from
(44)
C, - - {bP -G) - ^ _^ (J + DP :
C-'^G
(45)
F = ~
i+'A^p
588
BELL SYSTEM TECHNICAL JOURNAL
The electrostatic potential may be evaluated from F in a manner similar
to that followed in the preceding section.
3 . 3 Detailed properties of the solutions
The general solutions given in the figures illustrate certain properties
which can be established through the analytical approximations obtain-
able for small and for large values of the relative concentration of added
holes. The principal qualitative properties evident from the figures are:
100
60
40
-
-
— \
-\
- \
- \
k.
—
i^^
m^
K$55^.^
- \^^^S:^
^^
^^
==:
C = 50
~40-.
S
N
\\^
^
^==-^
-^
-■^
30" ■
-
w
r^
^
©^--i.
,,_^
'1
— — — ,
-
\N
\'
\,
^
t""***^
^
-^-^ ,
-
\
V
s.
^v^
V-
"^^
^
-"^
c=^
^N
:\
v
^
\
--^
.
OJ 0.6
_i
O 0.4
S 0.2
u
z>
a.
0.06
0.04
0.02
0 2 4 6 8 10 12 14 16 18 20 22 24
REDUCED DISTANCE VARIABLE, X
Fig. 3, — The dependence of the reduced concentration on reduced distance for the
steady-state one-dimensional flow of holes with mass-action recombination in «-type
germanium.
The relative hole concentration, P, and the reduced hole current, Cp ,
depend exponentially on distance for small concentrations; and for large
concentrations all solutions for a given dependent variable run together,
independently of the value of the current parameter, and give compara-
tively rapid variations of hole concentration and current with distance'".
The property that a common solution independent of total current or
'"These rapid variations would account for the observation of J. R. Hiynes that
estimates, for a given emitter current, of hole concentrations or currents in a tllament at
a point contact removed from the emitter, with no additional aj^plied field, arc largely
indejiendent of changes in/, for the emitter.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
589
applied field obtains for large P results from diffusion in conjunction with
the increase of conductivity. As may be expected, the solutions for the
case of constant mean lifetime also have this property, the recombination
law merely affecting the form of the common solution.
In Fig. 6 are shown curves for P, Cp , and F for the case of constant mean
lifetime in w-type germanium, evaluated for C equal to 16.3. These curves
are intended to illustrate the qualitative differences between the solutions
for this case and those for mass-action recombination, which are manifest
100
60
40
1
:::^
,
"1^
'-''■'^"-j^]
— =
1
C = 50
^ k;
sT^
'~~
' i
m
fc
^
30~"
1
^
^
L^
—
" -20—
—
vW
"•^^
•^fc,^^
~
** —
-
^NN
h*.
15 ■
-
\\N>
"\^
v,^
^^..^
^^^
" ■
-.
-
\\
\\
^*^^
""-^
.^
"lO""-^
^--
V
\N
V ^
5*
7"^
^^
^^
"'^
■"
—
\
\
N
v
'^^
-
V S.
\,
N
••..^
•••.^.^
-
\ >
, ^
N,
^•^
— ^
■^--^
-
\
\
N
s
\.
^
C=(\
\
s
\
\
^\
LD 0.6
O 0.4
I
£ 0.2
O
g 0-1
0.04
0.02
0.01
0 2 4 6 8 10 12 14 16 18 20 22 24
REDUCED DISTANCE VARIABLE, X
Fig. 4. — The dependence of the reduced hole flow density on reduced distance for the
steady-state one-dimensional flow of holes with mass-action recombination in «-type
germanium.
primarily at the larger concentrations. The dashed curves in the figure
give the corresponding solutions for the case of mass-action recombina-
tion; and the A'-origins for the two cases have been so chosen that cor-
responding curves, which exhibit essentially the same dependence on X
for small P, coincide in the limit of small P. As the figure shows, constant
mean lifetime gives an exponential dependence of P on X for large P,
while mass-action recombination gives larger concentration gradients,
with an increase of P to indefinitely large values in the neighborhood of a
vertical asymptote.
590
BELL SYSTEM TECHNICAL JOURNAL
100
60
40
20
10
6
1.0
; 0.6
i 0.4
)
; 0.2
J
J
J O.t
J 0.06
)
) 0.04
)
J
• 0.02
0.01
0.006
0.004
0.002
0.001
—
-
-
-
"C-50
— 4
■— ^
■ '
|__30_
-20—
12
=^
7^,
7
- /A
<r^
4
'//
/^
2
W/
/"
1
^/ yi
r
Y y
^^'^""''^
-V A
^
^
\
—
\
-
\
-
\
-
\
yc=o
—
-
-
s.
-
\
\
0 2 4 6 8 10 12 14 16 18 20 22 24
REDUCED DISTANCE VARIABLE, X
Fig. 5. — The dependence of the reduced electrostatic field on reduced distance for
the steady-state one-dimensional flow of holes with mass-action recombination in M-type
germanium.
3.31 The behavior for small concentrations
The exponential dependence of P and Cp on distance for P small is
given for the w-type semiconductor by the analytical approximations,
iP = Ps exp [- h[±V0T^ - C\X]
(46) • 1 ^ ^
Cp = _[±VC2-f 4 + C]P,
where P., is a suitable constant. If C is positive, the plus sign holds for
field aiding^^ and the minus sign for field opposing. These approximate
^' It is evident from the curves for C,, in Fig. 4 that the exponential extrapolation hack
to the emitter location of estimates of hole concentrations or currents at a point contact
on a germanium filament lead to values of/, for the emitter which are too small. Using
moderately large injected currents and no additional a])plied heids, J. R. IIa\nes once
obtained in this way an apparent/, of about 0.2. From the figure, this is the apparent/,
to be expected for moderate and large values of C for the true/, equal to unity.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
591
solutions, which hold for any recombination law, are obtained quite
simply, by integration, from G in terms of P to the first term of the Mac-
laurin's expansion, given in the Appendix. It might be noted that for this
approximation the electrostatic field is equal to the applied field, so that
F equals C.
10
8
U- 6
^ 2
1.0
0.8
0.6
0.4
n
II
LIFETIME
-\
II
II
MASS-ACTION
- \
II
II
It
RECOMBINATION
C=I6.3
^.
1\
_
F
-
\\
^•^"^
-
\^
V
^^
\\
-
\
y<
'^
\ /
^
\
(
/ \
^-^^
.^Cp
\
V
\
\
^
~»»^— .
^^^^
= ~^^
-
\
\
"*~~
i
^,
\
y
UJ
^-e
y
Q.
5
*
>
<
— ....,.^2"'
■-^^
^«.^
8 10 12 14 16
REDUCED DISTANCE VARIABLE, X
Fig. 6. — The dependence of the reduced hole concentration, hole flow density, and
electrostatic field on reduced distance for steady-state one-dimensional hole flow in «-type
germanium, for the cases of constant mean lifetime and mass-action recombination.
Since F is small, the transport velocity of holes is equal to their differ-
ential transport velocity"". Writing the equation for Cp in dimensional
form, the transport velocity is found to equal
(47)
S = i[±V(M£a)2 + 4Dp/T + nEa],
with the plus sign for field aiding and the minus sign for field opposing,
if the applied field, Ea , is positive. This result is consistent with the
'^In accordance with equations (7), (8), and (10), the differential transport velocity
for the steady state in one dimension may be found from the general formula,
hM^CJdP = - (P - P,)R/G.
Its equalling the transport velocity proper for P small appears to result from the property
of non-composite cases that the dependent variables, for a given C, are all functions of P
which do not depend on any quantity determined by the boundary values, a property
which composite cases, with their additional degree of freedom, do not possess.
592
BELL SYSTEM TECHNICAL JOURNAL
equation for P, which may be written as
(48) P = P, exp {-x/st).
For a large aiding field, s reduces to the velocity of drift under this
field while, for a large opposing field, the magnitude of 5 is approximately
Dp/nEaT. For zero field, 5 equals the diffusion velocity {Dp/rY, which is a
diffusion distance for a mean lifetime divided by the mean lifetime. This
diffusion velocity can be specified in terms of its field equivalent, or the
field which gives an equal drift velocity, and for germanium it is found that
the equivalent field is about 8 volt cm~^ for r equal to one microsecond
and about 2.5 volt cm~^ for r equal to 10 microseconds.
For small concentrations of added holes in the intrinsic semiconductor,
or (P— 1) < < 1, equations (38) and (40) give the approximate solutions,
(49)
2b
P-l = (P«-l)exp[± pi + "'(*+«
]
I']
c„ =
h+ \\_
C
2b
b+ 1
b+ 1
the X-origin being selected arbitrarily at the point at which the relative
concentration is P" according to the approximation. It is evident from the
equation for Cp that, for (P— 1) small, the transport velocity is the drift
velocity under the applied field, which is the velocity of the holes norm-
ally present in the semiconductor. The differential transport velocity, ob-
tainable by differentiating the equation for Cp with respect to P and
using the differential equation (28), or by writing the exponent in the
equation for (P— 1) in the form given in (48), is, on the other hand, given by
(50)
26
L(l + a){b+ 1)J
Dr
1
2DpDn
_1 -f aDp-\- Dn
and is a diffusion velocity. This holds for holes added in any concentra-
tion if a = 0, or for constant mean lifetime, since the first of equations
(49) is then the general solution given in (39).
The nature of the flow for small concentrations of added carriers in the
general case, which depends on the parameter Po , is illustrated qualita-
tively by the w-type and intrinsic cases considered, for which Po is re-
spectively zero and infinite. Solutions for the general case are easily
evaluated analytically from the linear differential equation which results
from (17) if P — Po << ^ + Po . It can be shown from the field-aiding
steady-state solution that the ratio of the differential transport velocity
to the velocity, proportional to C, of drift under the applied field is for
O >> (1 + 2Po)Mo equal to the quantity l/Mn. This result is consistent
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
593
with those already derived: For large applied aiding fields, the differential
transport velocity changes from the drift velocity, for Po equal to zero
and Mo unity, to the diffusion velocity given in (50) as Po and Mo in-
crease indefinitely.
3.32 The zero-current solutions and the behavior for large concentrations
The solutions for the intrinsic semiconductor for the current parameter
equal to zero are, of course, the same as the general ones given in Section
3.1, since the current parameter does not occur in the differential equa-
tion. For the »-type semiconductor, the differential equation (27) be-
comes an equation of the Bernouilli type for C equal to zero, and may be
solved by quadratures. It is then linear in G', and gives, for field aiding or
field opposing,
(51)
G' = 2
h*''^T
L 1 + 2P J
Jo
P(l + P)(l + aP)
dP,
expressing the recombination function R according to equation (16) for a
combination of the two recombination mechanisms. Writing, for brevity.
(52)
/3^
b + 1
M-l+^lip,
and evaluating the integral in (51), the following result is obtained:
G" = 2/3
[r
M
[1
(53)
+ 2P.
[/3(M2 - 1) 4- ( 1- 4/3)(M - 1) - (1 - 2/3) log M]
+ a[f/32(M3 - 1) + 1(1 - 2/3) (M2 - 1)
+ (1 - 6/3 + 6/3-) (M - 1) - (1 - ^)(1 - 2/3) log M]
For P large, this solution gives the approximations,
'b + r
(54)
G = ±
2b
for constant mean lifetime, with a = 0, and
(55)
G = dz
'a(b -f 1)
3b
li
594 BELL SYSTEM TECHNICAL JOURNAL
if there is mass-action recombination present, so that a 5^ 0. The depend-
ence of P on A' for these approximations is readily obtained by integrating
the differential equations which result from writing in place of G, its
definition, dP/dX\ constant mean lifetime gives an exponential depend-
ence. An examination of (54) and (55) in conjunction with the general
differential equation (27) shows that, for P large, the dominant term in the
differential equation is independent of C. It follows that solutions for all
values of C approach a common solution for P large, which is given by
(54) or (55). The solutions run together appreciably for P sufl5ciently
large that P and M are substantially proportional, that is, for P large
compared with h/{h + 1), which is of order unity. It is to be expected
that the approximations (54) and (55) should apply equally well to the
intrinsic semiconductor, and this expectation is easily verified by evalu-
ating the integral in equation (35) for the intrinsic semiconductor, for P
large, for the two recombination cases here considered.
4. Solutions of Simple Boundary-Value Problems for a Single
Source
Among the boundary-value problems whose solutions are useful in
the interpretation of data from experiments in hole injection are the
following: the semi-infinite filament for field aiding, with holes injected at
the end, which constitutes a relatively simple case; and the doubly-
infinite filament with a single plane source, with which this section will be
primarily concerned.
Consider first the semi-infinite filament, and suppose that it starts at the
X-origin and extends over positive A^, so that the current parameter is
positive for field aiding. If two quantities are specified, namely the current
parameter and the fraction /« of the current carried by holes at the origin
or injection point, then the solution of the boundary-value problem is
completely determined. It is merely necessary to select the general field-
aiding solution for P or Cp in terms of A", for the particular value of the
current parameter, and then to determine the A'-origin, corresponding to
the source, which is simply the X at which the ratio /of Cp to C equals /« .
Use in the boundary-condition equations {i2>) and (34) of the approxi-
mate expressions given in (54) and (55) for G in terms of P, for large P,
permits the complete analytical determination of the dependence ofP"
on total current as this current is indefinitely increased. It was shown in
Section 2.4 that, if /^ is less than \/{b + 1) for the w-type semiconductor,
P" approaches as a limit the value for which G" vanishes according to the
boundary-condition equation (33); in all other cases for the ;/-type semi-
conductor, or if/, exceeds \/{b + 1) for the intrinsic semiconductor, P°
increases indefinitely with C. For/, > \/{b -f- 1), it is readily seen that
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
595
P° is proportional in the limit to C for constant mean lifetime, and to C^
for mass-action recombination; and, for/^ = l/(b + 1) in the case of the
w-type semiconductor, P° increases as C' for constant mean lifetime, and
as C for mass-action recombination.
Consider now the doubly-infinite semiconductor filament with a
source at the origin, and suppose that the total injected current at the
source is C, , in reduced form, with a fraction /« of this current carried
by holes. Denote by C~ and by C+ the reduced total currents for A^ < 0
and for .Y > 0, respectively. Since the injection of holes requires that Ce
be positive, at least one of C" and C+ must be positive, since total current
is conserved. Let /~ and /+ denote, respectively, the ratio of the hole
current at the origin to the left, Cp , to the total current C~ , and the
ratio of the hole current at the origin to the right, Cp , to the total current,
C+ . It might be noted that, for a flow of holes to the left, say, against
the field, C~ and C+ are positive and f~ is negative, and that, if C~ is
(plus) zero, /~ is (negatively) infinite, corresponding to the flow of holes
under zero applied field. Now, general boundary-condition equations of
the form of (33) or (34) hold with the sign conventions here employed,
as indicated in Section 2.4. One may be written for the flow to the left,
another for the flow to the right, making use of the condition that the
relative concentration P is everywhere continuous; G exhibits a discon-
tinuity of the first kind at the source, with a change in sign. Writing G~
for the limiting value of the reduced concentration gradient as the origin
is approached from the left, and G+ the limiting value as the origin is
approached from the right, the boundary-condition equations are, for
the «-type semiconductor,
b + (b -\- l)Po
(56)
cr =
G+= -
1 + 2P»
b + {b + DP'
[r-
1 + 2P»
For the intrinsic semiconductor, they are
ib + D" "
b-}-{b+ 1)P\
P°
6 + (6 + l)^'".
C'
C\
(57)
G~ =
G+ = -
2b
{b + 1)^
2b
r -
r
6 + ij
C'
c
There are, in addition, an equation which expresses the conservation of hole
flow, and one which expresses the conservation of total current, as follows:
+ r^+
(58)
\r c
c
re- =f.c.
596 BELL SYSTEM TECHNICAL JOURNAL
The solution of the problem is determined by/* and the three parameters
which specify the total currents: With these four quantities known, then,
from equations (56) or (57) in conjunction with (58) and the known
general solutions in the (AP, G)-plane which apply to the left and to the
right of the origin, all of the quantities P^, G~, G^,f~ and/"*" can be found
and the problem completely solved.
The technique of obtaining the solution depends on a simple funda-
mental result which may be expressed as follows:
For fixed /c and Ct , consider the sum of the magnitudes of the con-
centration gradients at a single common source from which holes flow
into a number of similar filaments in parallel, for any consistent distribu-
tion among the filaments of total currents, some of which may be pro-
duced by opposing fields. This sum is equal to the magnitude of the con-
centration gradient at the source if the entire flow, under the appropriate
aiding field, were confined to a single filament.
The total magnitude of the concentration gradient, in this sense, is an
invariant for fixed fe and C^ . Specifically, for the ;/-type semiconductor,
it follows from equations (56) and (58) that
(59) (-_cr=-* + (^+')^°
1 -\- 2P°
Similarly, for the intrinsic semiconductor,
(60) G--(r=- ^^
u-
h+{h+ i)p
]-
/.- '
^+1.
c..
The left-hand sides of these equations are the negative of the sum of the
magnitudes of the reduced concentration gradients, since G~ is always
positive and G+ always negative, and their right-hand sides are similar
in form to those of equations (56) and (57), with the quantities /« and Ci ,
.characteristic of the source, replacing/" and C" , or/+ and C+ .
The particular utility of these equations arises from their independence
of the unknowns /" and /+ . By means of equation (59) for the w-type
semiconductor the evaluation of the five unknown quantities can now be
effected as follows: With the current parameters known, the solutions in
the (P, G)-plane to the left and right of the X-origin are determined;
either both solutions are for field aiding, or else one is for field aiding and
the other for field opposing. From them, the sum of the magnitudes of
the reduced concentration gradients can be found as a function of P.
It is also given, for the origin, as a function of the unknown P" , by
equation (59). The values of the sum for the origin and of P" are ac-
cordingly found as those which satisfy both relationships. The value of
P" thus found determines both G~ and G+ from the respective solutions
FLOW OF. ELECTRONS AND HOLES IN GERMANIUM
597
in the (P, G)-plane, and/" and/+ may be obtained by solving for them in
equations (56).
For the intrinsic semiconductor, this method can be appHed analytic-
ally, and the solution so obtained serves at the same time as an approxi-
mation for large relative hole concentrations in the n-type semiconductor,
for which the method is otherwise essentially graphical or numerical in
the general case. Making use of the symmetry of the solutions for the
intrinsic semiconductor about a source, it follows from (57) and (58)
that
(b + 1)^
G+ = -G~ =
(61)
U
t'-^l]^*
(b + 1)^
2b
whence
(62)
■' b + 1 2 L"^' 6 + ij C-
/ b+1^2i^' 6 + lJc+*
It is easily verified that this result holds approximately for large relative
concentrations in the n-type semiconductor. Three simple special cases
of (62) might be considered: The first is
fc- = -C+ = - ic
(63)
I/- =/+=/«•
This is the rather trivial case of symmetrical flows from a source which
supplies all currents. A second special case is that for which C~ and C"*"
are both positive, say, and such that there is no hole flow to the left
against the field. It is readily found that, for this case,
(64)
b+ 1
/e-
1
r = 0; /■" =
b+ 1.
2
C+ -
^4-^[/"n-i-J--
/e
6+ 1/. + 1/(6+ 1)-
Here, the drift from the left under the applied field of holes normally
present in the intrinsic semiconductor just cancels the diffusion from the
source to the left.^^ A third special case is that in which the total current
^' Using the numerically obtained solutions, the validity of (64) as an approximation
for large concentrations in «-type germanium may be seen as follows: For /, equal to
unity and C( , C~ and C+ equal to 2, 1.5, and 3.5 respectively, P^ is about 0.6 and
the fraction of injected holes which flows against the field is nearly one-half; doubling
these current densities increases P" to 1.45 and decreases the fraction to about one-fourth,
and the fraction is less than about one-tenth if the current densities are increased so that
C" exceeds 15
598
BELL SYSTEM TECHNICAL JOURNAL
to the left of the source is zero, the left-hand side of the filament being
open-circuited. For this case, equations (62) are better written in the form
obtained by multiplying through by C~ or C"*" , and the special case in
question is then found to be given by
(65)
CZ = -
u
c~
6+ ij
0:
C+ = C.
r • r^ =
/. +
1
6+ 1
C.
according to which, life is equal to unity, the magnitude of the hole flow
to the left into the open-circuit end is b/{b + 2) times that into the circuit
end, to the right; or a fraction b/2(h + 1) of the holes flows to the left,
and a fraction (b + 2)/2(6 -f 1) to the right. Thus, for germanium, the
hole flow into the open-circuit end is 0.43 as large as that into the circuit
end, a fraction 0.30 flowing to the left, and 0.70 to the right. It might be
observed that the fractions of the injected holes which flow to the left
and right are, in this case, proportional to the total currents C" and C+
of the preceding case, for which there is zero hole flow to the left.
Another general limiting case for the «-type semiconductor is that for
Po small, so that the exponential approximations of Section 3.31 apply.
The restriction on the magnitude of P is P < < ^. This restriction obtains
if Ct is sufficiently small that C~ and C+ do not differ appreciably. Equa-
tion (59) then gives
(66) G+ - G- = - bf, C, .
Writing C for C~ and C+ , equations (30) and (46) result in
Icr = ilVoT^ + c]p' = bc^
[G^ = -hlVc^T^ - C]P'= bc-,
(67)
whence, solving for G^ — G and comparing with equation (66),
(68) P° = bf,c,/Vc?T^.
In accordance with (67), then.
(69)
These are the reduced hole flows to the left and right of the source.
While it has been assumed that C( is small compared with C, no re-
striction has been placed on C itself. For C small compared witli unity,
the equations indicate that the hole flows to the left and right are the
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 599
same in magnitude, while for C large compared with unity,
(70)
Cp^ ~ ^J^c,
Cp^^fiCf.
Thus, according to this approximation, C should exceed about 10 if no
more than one per cent of the holes are to flow against the field. From
(75) in the Appendix, a value of 10 for C corresponds to a current density
of about 1.2 amp cm~^ in germanium of 10 ohm cm resistivity, with r
equal to 10 Aisec. This current density is moderately large among those
which have been employed in experiments with germanium filaments.
Experimentally, the ideal one-dimensional geometry postulated in the
present treatment of the problem of the single source in an infinite fila-
ment cannot easily be reahzed, hole injection generally being accomplished
through a point contact or a side arm on one side of the actual filament.
If suitable averages are employed, non-uniformity in P at the injection
cross-section does not, however, vitiate the approximate results for AP
large and AP small, since their applicability depends largely on the validity
over the injection cross-section of the approximation assumed.
Acknowledgment
The author is indebted to a number of his colleagues for their stimu-
lating interest and encouragement; to J. Bardeen and W. Shockley for a
number of valuable and helpful comments, as well as to W. H. Brattain,
J. R. Haynes, C. Herring, L. A. MacColl, G. L. Pearson, and R. C. Prim.
J. Bardeen also suggested the numerical analysis for «-type germanium
which constituted one of the initial points of attack, and aided materially
in its inception. The rather difficult numerical integrations and associated
problems were ably handled by R. W. Hamming, Mrs. G. V. Smith and
J. W. Tukey.
5. APPENDIX
5 . 1 The concentrations of ionized donors and acceptors
While the donor and acceptor concentrations need not, of course, be
considered for the intrinsic semiconductor, for the extrinsic semicon-
ductor the fundamental equations, as they have been written, are in
principle incomplete: Two additional equations in the variables Z)+ and
A~ are required. One of the required equations is trivial, since changes in
the concentration of ionized centers which are compensated by those
which determine the conductivity type of the extrinsic semiconductor
600 BELL SYSTEM TECHNICAL JOURNAL
can certainly be neglected. For an «-type semiconductor, for example, the
term {A~ — Aq) in Poisson's equation may be suppressed. This procedure
is strictly consistent with the neglect of p^ and go , but undoubtedly holds
to an even better approximation. If D is the total donor concentration in
the «-type semiconductor, the concentration of ionized donors may be
considered to satisfy the equation,
(71) ^= H(D- D+) - KD'-n,
ot
which applies to the homogeneous semiconductor, with H and K con-
stants which characterize, respectively, the rate of ionization of unionized
donors, and the rate of recombination of an ionized donor with an electron.
If, as a result of a small thermal ionization energy, most of the donors are
ionized, so that KD/H < < 1, the change in ionized-donor concentration
for the steady state is given by (71) as
(72) u^ - Dt ^-^{n- Wo),
which is small compared with the corresponding change in electron con-
centration. In other cases, the use of the general expression obtainable
from (71) for the steady-state concentration of ionized donors in terms of
the electron concentration, or the expression for the other limiting case of
relatively few ionized donors, might provide a more precise description
provided the conditions under which solutions are sought do not involve
unduly rapid changes with time.
5.2 The carrier concentrations at thermal equilibrium
The ratio of the thermal-equilibrium values of the hole and electron
concentrations may be evaluated for «-type germanium from^
(73)
np = 3- 10^-i ^ exp I — I = «»
n — p = ns ~ Wo = l/biJLepo = 2.40-10'Vpo,
where the electron concentration excess «s corresponds to complete ioniza-
tion of the donors, and is approximately Wq at the highest temperature at
which Pq is still negligible, which may be taken as room temperature-''.
The resistivity po is that which determines Uq . Thus,
(74) Po = h [Vl + ^ni/noT - 1] ,
" loc. cit.
' loc. cit.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 601
with m , the concentration of holes or electrons in intrinsic germanium at
T deg abs, given in (73). It may be estimated that temperature rises of
less than 100 deg C will make 10 ohm cm »-type germanium substantially
intrinsic in its behavior.
The range of values of the parameter C for which the numerical solu-
tions are given corresponds, for example, to current densities up to the
order of 10 amp cm~- in germanium filaments of about 10 ohm cm re-
sistivity, for the mean lifetime r about 10 /xsec; for this mean lifetime, the
distance unit Lp is approximately 2- 10~- cm. Current densities correspond-
ing to the larger values of C will ordinarily produce appreciable joule
heating in filaments some 10~^ cm''^ in area of cross-section, cemented to a
backing, with temperature rises of the order of 100 deg C.
The effect of joule heating on Lp and C may be evaluated from
(75)
300
Lp= 6.6
C = 2.6- 102 I ::^
where r is expressed in sec, / in amp cm~^, and p is the normal resistivity
in ohm cm of the germanium at T deg abs. These are obtained from the
definitions (8), taking the hole mobility in the thermal scattering range
to be proportional to T ', with the value 1700 cm- volt~^ sec~^ at 300 deg
abs.'^
5.3 Series solutions for the extrinsic semiconductor in the steady state
Maclaurin's series for G in the relative concentration P are of the form
(76) G = aiP -f a^F' + a^P^ + . . .
for the cases of field opposing and field aiding, the solutions passing
through the {P, G)-origin. Substituting the series (76) for G in the differ-
ential equation (27) for the w-type semiconductor in the steady state, it
is found, in accordance with (30), that
(77) ai=HC±VCM^],
the sign of C being taken before the radical for field opposing, the other
sign for field aiding. The other coefficients are given in terms of ax and
^ loc. cit.
602 BELL SYSTEM TECHNICAL JOURNAL
also the b, C, and the constant, a, of the recombination function:
■ia'i
2 — , -\- a
0
(78)
2al +
C — 3ai
Wb + 1
, . ^ + 1 2
did-i -r 2. — i — ai
as =
6 + 1 [6 + 1 , . 1
C - 4ci
The series in the current parameter are series in ascending powers of the
reciprocal of C Writing, for convenience,
(79) 7 ^ 1/C,
the differential equation (27) may be put in the form,
b+ 1
7 [1 + 2P]
(80)
1 +
P
GG'
, b- 1^2
+ 7 — 1— G
G - yP
1 + '-4-' p
R = 0,
using the prime to denote differentiation with respect to P. Consider
expansions of the form,
(81) G = Z Ajy\
J=JO
in which the -4's are functions of P to be determined. Substituting in the
differential equation, there results
(82)
j=JO "I^JO l_
[1 + 2P]
00
- E An'
0
1 a' 1 ^ ~ 1 i I
.4y.4^ H j— .4,- J,
;■+'«+!
L ^
7?7 = 0.
Since the expansions are to hold for arbitrary values of 7, the .4's must,
for the cases of field opposing and field aiding, for which the solutions
pass through the {P, G)-origin, vanish identically for P equal to zero,
and be determined by equating to zero the coefficients of given powers of
7 in (82). It can, without loss of generality, be assumed that the coefficient
of the leading term in the expansion, Aj^ , is not identically zero. Then,
from (82), it is found that there is no expansion for jo = 0, that is, no
expansion starting with a term independent of 7. Formal expansions can be
obtained, however, foryo = — 1 and for 7*0 = + 1. These may be identified.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM
603
respectively, with the solutions for field opposing and field aiding, as will
be seen.
For Jo = — 1, or field opposing, (82) leads to differential equations of the
first order for the determination of the ^'s. The condition that these func-
tions vanish identically for P = 0 suppresses all .4's of even order. The
first term of the expansion is found by solving
(83) ^_i +
whence
(84)
b - 1
A-,
[1 + 2P]
l+'-^P
[1 + 2P]
1 +
b +
'-']
A-i =
1 + 2P
The second term is found from
b - I Ax
(85)
^1 +
[1 + 2P] fl + ^ +
Ul~L
l+'^P
R,
\_~ ' b J
whence, with R equal to unity and (1 + P), respectively,
(86)
A, =
P[l + P] [l + ^-^ P\
Ax
A
1 + 2P
1^-\P +\P'\\\ +
for constant mean lifetime
I-H^^]
1 + 2P
For the third term, making use of (84), (85) and (86),
b-\ As
for mass-action
recombination.
^3 +
b [l-t-2P][l + ^-±ip]
(87)
^3 +
= -[1 + P]
1 As
1 +
6 +
1 T for c
J lifeti
constant mean
lifetime
[1 + 2P]
H-'-4^P
= -[1 +P]
1 + ^P + ^P^
1 +
6 +
^'l
for mass-action
recombination
604
whence
BELL SYSTEM TECHNICAL JOURNAL
■P
A,=
^ )_46 + l^ y^b-\-?>p, I ^'+lp3
2b
3b
0
1 + 2P
for constant
mean lifetime
(88) I
-'[
J ^ 336 + 6^ ^ 70/) + 27^2 ^ 736 + 43^3
12/»
186
^ 386 + 30^4 ^ 2b + 2^5
^3 =
306
96
246
1 + 2P
for mass-action recombination.
For Jo = +1, or field aiding, the ^'s are determined somewhat more
simply, recursive relationships obtaining. The results are:
(89)
and
Ax = -P 1 +
b^'-V^]^
(90) <
^3 = [1 + 2P]
^5 = [1 + 2P]
^7 = [1 + 2P]
.+^-±ip
1 +
1 +
6
a^aU^-^aI
[AM' + 2^—r^AiAz
- P^IAM' + A^A^]
6- 1
^9 = [1 + 2P]
\+'4^P
+ 2
UM' + [AM']
AiAi, + 2^3
+ 2^-iUl.47 + -l3/lB]
0
The identification of the series in the parameter y as series for lield
opposing and field aiding is accomi)lished by evaluating them for small
P and then comparing them with the first terms of the corresponding
Maclaurin's series in P, expanded in powers of y. Further agreement is
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 605
obtained by comparing the first terms of the series in 7 with the func-
tions of P which result from evaluating the Maclaurin's series for 7
small.
5 . 4 Symbols for Quantities
a = t/tv , constant in recombination function.
aj = coefficients in the Maclaurin's expansion of G in powers of P; j an
integer.
Aj = coefficients in the expansion of G in powers of y-jj an integer.
A~ = concentration of ionized acceptors.
Ao" = thermal-equilibrium concentration of ionized acceptors.
b = ratio of electron mobility to hole mobility.
C = I/Io , reduced total current density.
Ce = reduced emitter current.
C~ = reduced total current to the origin from the left.
C+ = reduced total current from the origin to the right.
C„ ^ —I„/Io , reduced electron flow density.
Cp = Ip/Io , reduced hole flow density.
7 ^I/C.
T = €/47rcror, reduced time for the dielectric relaxation of charge.
D = total donor concentration.
ZH" = concentration of ionized donors.
Do = thermal-equilibrium concentration of ionized donors.
Dn = kTun/e, diffusion constant for electrons.
Dp ^ kTiXp/e, diffusion constant for holes.
e = magnitude of the electronic charge.
E = electrostatic field.
Ea = applied or asymptotic field.
Eo = kT/eLp , characteristic field,
e = dielectric constant.
/ = fraction of total current carried by holes.
ft = fraction of total current carried by holes at an emitter.
/"~ = fraction of total current carried by holes at a source, to the left.
/+ = fraction of total current carried by holes at a source, to the right.
F = E/Eo , reduced electrostatic field.
go = thermal rate of generation of hole-electron pairs, per unit volume.
G = dP/dX, reduced concentration gradient.
G° = value of G for X = 0.
G~ = limiting value of G at a source, approached from the left.
G^ = limiting value of G at a source, approached from the right.
H = probability of thermal ionization of an unionized donor, per unit
time.
606 BELL SYSTEM TECHNICAL JOURNAL
I = total current density.
/„ = current density of electrons,
/o = aEa , characteristic current.
Ip = current density of holes.
J = - I, total carrier flow density.
e
Jn = In, electron flow density.
e
Jp = - Ip , hole flow density.
e
k = Boltzmann's constant.
K = probability per unit time of electron capture by an ionized donor,
per unit electron concentration.
Ld ^ (kTe/HirHie-) , characteristic length associated with space charge
in the steady state.
Lp = {kTixr/e), diffusion length for holes for time t.
M = 1 +
Mo= 1 +
b
b-\- 1
b
fi = Up = mobility for holes.
Hn = mobility for electrons.
n = concentration of electrons.
Hi = thermal-equilibrium concentration of electrons (or holes) in the
intrinsic semiconductor.
«o = thermal-equilibrium concentration of electrons.
Hg = saturation concentration excess of electrons, corresponding to com-
plete ionization of donors.
A^ = n/{no — po), reduced electron concentration for an «-type semi-
conductor.
P = concentration of holes.
po = thermal-equilibrium concentration of holes.
P = p/(no — po), reduced hole concentration for an w-type semiconductor.
AP = {p — po)/{no — po), reduced concentration of added holes.
Pq = po/(no — po), reduced hole concentration at thermal equilibrium.
po - value of P for X = 0.
Q = t/tp , lifetime ratio.
R = general recombination function, equal to 1 + aP/{l + Po) for
mass-action and constant-mean-lifetime mechanisms combined.
p = volume resistivity in ohm cm.
s = differential transport velocity.
S = s/(Dp/t) , reduced differential transport velocity.
FLOW OF ELECTRONS AND HOLES IN GERMANIUM 607
(T ^ conductivity of semiconductor.
(To = normal conductivity of semiconductor, with no added carriers.
2 = a/ao = M/Mn , reduced conductivity of semiconductor.
/ = time variable.
T = temperature in degrees absolute.
T = mean lifetime for holes for small added concentrations, in an n-
type or in an intrinsic semiconductor.
r„ = mean lifetime for electrons (concentration-dependent).
Tp = mean lifetime for holes (concentration-dependent).
r,. = mean lifetime for holes, for small added concentrations in an n-
type semiconductor, due to mass-action recombination alone,
r = //r = reduced time variable,
ir = eV/kT, reduced electrostatic potential.
X = distance variable.
.Y = x/Lp , reduced distance variable.
V = electrostatic potential.
Traveling-Wave Tubes
By J. R. PIERCE
Copyright, 1950, D. Van Nostrand Company, Inc.
FOURTH INSTALLMENT
CHAPTER XII
POWER OUTPUT
A THEORETICAL EVALUATION of the power output of a traveling-
wave tube requires a theory of the non-hnear behavior of the tube.
In this book we have dealt with a linearized theory only. No attempt will
be made to develop a non-linear theory. Some results of non-linear theory will
be quoted, and some conclusions drawn from experimental work will be
presented.
One thing appears clear both from theory and from experiment: the gain
parameter C is very important in determining efficiency. This is perhaps
demonstrated most clearly in some unpublished work of A. T. Nordsieck.
Nordsieck assumed:
(1) The same a-c field acts on all electrons.
(2) The only fields present are those associated with the circuit ("neglect
of space charge").
(3) Field components of harmonic frequency are neglected,
(4) Backward-traveling energy in the circuit is neglected.
(5) A lossless circuit is assumed.
(6) C is small (it always is).
Nordsieck obtained numerical solutions for such cases for several electron
velocities. He found the maximum efficiency to be proportonal to C by a
factor we may call k. Thus, the power output P is
P = kCIoVo (12.1)
In Fig. 12.1, the factor k is plotted vs. the velocity parameter b. For an
electron velocity equal to that of the unperturbed wave the fractional
efficiency obtained is 3C; for a faster electron velocity the efficiency rises to
7C. For instance, if C = .025, 3C is 7.5% and 7C is 15%. For 1,600 volts
15 ma this means 1.8 or 3.6 watts. If, however, C = 0.1, which is attainable,
the indicated efficiency is 30% to 70%.
Experimental efficiencies often fall very far below such figures, although
some efficiencies which have been attained lie in this range. There are three
apparent reasons for these lower efficiencies. First, small non-uniformities
in wave propagation set up new wave components which abstract energy
from the increasing wave, and which may subtract from the normal output.
Second, when the a-c field varies across the electron flow, not all electrons
608
POWER OUTPUT
609
are acted on equally favorably. Third, most tubes have a central lossy sec-
tion followed by a relatively short output section. Such tubes may overload
so severely in the lossy section that a high level in the output section is
never attained. There is not enough length of loss-free circuit to provide
sufficient gain in the output circuit so that the signal can build up to maxi-
mum amplitude from a low level increasing wave. Other tubes with dis-
tributed loss suffer because the loss cuts down the efficiency.
Some power-series non-linear calculations made by L. R. Walker show that
for fast velocities of injection the first non-linear effect should be an expan-
sion, not a compression. Nordsieck's numerical solutions agree with this.
A power series approach is inadequate in dealing with truly large-signal be-
7
6
5
4
3
2
Fig. 12.1 — The calculated efficiency is expressed as kC, where fe is a function of the
velocity parameter b. This curve shows k as given by Nordsieck's high-level calculations.
havior. In fact, Nordsieck's work shows that the power-series attack, if
based on an assumption that there is no overtaking of electrons by electrons
emitted later, must fail at levels much below the maximum output.
Further work by Nordsieck indicates that the output may be appreciably
reduced by variation of the a-c field across the beam.
It is unfortunate that Nordsieck's calculations do not cover a wider range
of conditions. Fortunately, unlikely as it might seem, the linear theory can
tell us a little about what limitation of power we might expect. For instance,
from (7.15) we have
V . r}V
Mo UqoL
Uo
= -J
2Fo
(12.2)
610
BELL SYSTEM TECHNICAL JOURNAL
while from (7.16) we have
h
V
2Fo
(12.3)
We expect non-linear effects to become important when an a-c quantity is
no longer small compared with a d-c quantity. We see that because (1/5C)
is large, | i/h \ will be larger than | v/uo \ .
The important non-linearity is a sort of over-bunching or limit to bunch-
ing. For instance, suppose we were successful in bunching the electron flow
into very short pulses of electrons, as shown in Fig. 12.2 As the pulses ap-
proach zero length, the ratio of the peak value of the fundamental com-
ponent of convection current to the average or d-c current /o approaches 2.
We may, then, get some hint as to the variation of power output as various
parameters are varied by letting \i\ = 21 o and finding the variation of power
in the circuit for an a-c convection current as we vary various parameters.
TIME *"
Fig. 12.2 — If the electron beam were bunched into pulses short compared with a cycle,
the peak value of the component of fundamental frequency would be twice the d-c cur-
rent /o .
Deductions made in this way cannot be more than educated guesses, but in
the absence of non-linear calculations they are all we have.
From (7.1) we have for the circuit field associated with the active mode
(neglecting the field due to space charge)
E =
T^T,(FJ/I3'-P)
2{rl - r')
(12.4)
This relation is, of course, valid only for an electron convection current i
which varies with distance as exp(— Fs). For the power to be large for a
given magnitude of current, E should be large. For a given value of i, E will
be large if F is very nearly equal to Fi . This is natural. If F were equal to
Fi , the natural propagation constant of the circuit, the contribution to the
field by the current i in every elementary distance would have such phase
as to add in phase with every other contribution.
Actually, Fi and F cannot be quite equal. We have from (7.10) and (7.11)
-l\ = ^^{-j - jCb - Cd)
(12.5)
•r ^ 0e(-j + jCyi + Cxi)
(12.6)
POWER OUTPUT
611
For a physical circuit the attenuation parameter d must be positive while,
for an increasing wave, x must be positive. We see that we may expect E
to be greatest for a given current when d and x are small, and when y is
nearly equal to the velocity parameter b.
Suppose we use (12.4) in expressing the power
P =
^•"-{m/^'-p)
t'v\{e'/^'p)
(12.7)
Here we identify /3 with —jTi . Further, we use (2.43), (12.5) and (12.6),
and assuming C to be small, neglect terms involving C compared with unity.
We will further let i have a value
i = 2/o
(12.8)
5
4
3
2
1
0
Fig. 12.3 — An efficiency parameter k calculated by taking the power as that given by
near theory for an r-f beam current with a peak value twice the d-c beam current.
We obtain
P = kCIoVo
(12.9)
k =
(b + yy + (x + dy
(12.10)
We will now investigate several cases. Let us consider first the case of a
lossless circuit (d = 0) and no space charge (QC = 0) and plot the efficiency
factor k vs. b. The values of x and y are those of Fig. 8.1. Such a plot is
shown in Fig. 12.3.
If we compare the curve of Fig. 12.3 with the correct curve of Nordsieck,
we see that there is a striking qualitative agreement and, indeed, fair quanti-
tative agreement. We might have expected on the one hand that the electron
stream would never become completely bunched {i = 2/(i) and that, as it
approached complete bunching, behavior would already be non-linear.
This would tend to make (12.10) optimistic. On the other hand, even after i
612
BELL SYSTEM TECHNICAL JOURNAL
attains its maximum value and starts to fall, power can still be transferred
to the circuit, though the increase of field with distance will no longer be
exponential. This makes it possible that the value of k given by (12.10) will
be exceeded. Actually, the true k calculated by Nordsieck is a little higher
than that given by (12.10).
Let us now consider the efifect of loss. Figure 12.4 shows k from (12.10)
vs. diox b = QC = 0. We see that, as might be expected, the efficiency falls
as the loss is increased. C. C. Cutler has shown experimentally through un-
published work that the power actually falls off much more rapidly with d.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
d
Fig. 12.4 — The efficiency parameter k calculated as in Fig. 12.3 but for 6 = 0 (an elec-
tron velocity equal to the circuit phase velocity) and for various values of the attenuation
parameter d. Experimentally, the efficiency falls off more rapidly as d is increased.
Finally, Fig. 12.5 shows k from (12.10) vs. QC, with J = 0 and b chosen to
make Xi a maximum. We see that there is a pronounced rise in efficiency as
the space-charge parameter QC is increased.
J. C. Slater has suggested in Microwave Electronics a way of looking at
energy production essentially based on observing the motions of electrons
while traveling along with the speed of the wave. He suggests that the elec-
trons might eventually be trapped and oscillate in the troughs of the sinu-
soidal field. If so, and if they initially have an average velocity Av greater
than that of the wave, they cannot emerge with a velocity lower than the
velocity of the wave less Av. Such considerations are complicated by the
fact that the phase velocity of the wave in the large-signal region will not
POWER OUTPUT
613
be the same as its phase velocity in the small-signal region. It is interesting,
however, to see what limiting efficiencies this leads to.
The initial electron velocity for the increasing wave is approximately
i>a = VcCi- — yiC)
(i2.li:
where Vc is the phase velocity of the wave in the absence of electrons. The
quantity yi is negative. According to Slater's reckoning, the final electron
velocity cannot be less than
Vb = Veil + yiC)
(12.12)
Fig. 12.5 — The efficiency parameter k calculated as in Fig. 12.3, for zero loss and for an
electron velocity which makes the gain of the increasing wave greatest, vs the space-
charge parameter QC.
The limiting efficiency rj accordingly will be, from considerations of kinetic
energy
V =
2 2
Va — ^'6
If yiC <K 1, very nearly
4yiC
(\-y,cy
r? = 4 yiC
(12.13)
We see that this also indicates an efficiency proportional to C. In Fig.
12.6 4yi is plotted vs. b for QC = d = 0. We see that this quantity ranges
614
BELL SYSTEM TECHNICAL JOURNAL
from 2 for 6 = 0 up to 5 for larger values of b. It is surprising how well this
agrees with corresponding values of 3 and 7 from Nordsieck's work. Moreover
(12.13) predicts an increase in efficiency with increasing QC.
Thus, wc may expect the efficiency to vary with C from several points
of view.
It is interesting to consider what happens if at a given frequency we change
the current. By changing the current while holding the voltage constant we
increase both the input power and the efficiency, for C varies as l\'^. Thus,
in changing the current alone we would expect the power to vary as the 4/3
power of /o
P ^ /o'' (12.14)
4
_ 3
I 2
1
0
Fig. 12.6 — According to a suggestion made by Slater, the velocity by which the elec-
trons are slowed down cannot be greater than twice the difference between the electron
velocity and the wave velocity. If we use the velocity difference given by the linear theory,
for zero loss {d = 0) this would make the efficiency parameter k equal to — 4vi. Here
— 4yi is plotted vs h for QC = 0.
Here space charge has been neglected, and actually power may increase
more rapidly with current than (12.14) indicates.
A variety of other cases can be considered. At a given voltage and cur-
rent, C and the efficiency rise as the helix diameter is made smaller. How-
ever, as the helix diameter is made smaller it may be necessary to decrease
the current, and the optimum gain will come at higher frequencies. For a
given beam diameter, the magnetic focusing field required to overcome
space-charge repulsion is constant if /o/Fo is held constant, and hence we
might consider increasing the current as the 1/2 power of the voltage, and
thus increasing the power input as the ?>/2 power of the voltage. On the other
hand, the magnetic focusing field required to correct initial angular deflec-
tions of electrons increases as the voltage is raised.
There is no theoretical reason why electrons should strike the circuit.
Thus, it is theoretically possible to use a very high beam power in connec-
tion with a very fragile helix. Practically, an appreciable fraction of the
beam current is intercepted by the helix, and this seems unavoidable for wave
POWER OUTPUT 615
lengths around a centimeter or shorter, for accurate focusing becomes more
difficult as tubes are made physically smaller. Thus, in getting very high
powers at ordinary wavelengths or even moderate powers at shorter wave-
lengths, filter type circuits which provide heat dissipation by thermal con-
duction may be necessary. We have seen that the impedance of such cir-
cuits is lower than that of a helix for the broadband condition (group velocity
equal to phase velocity). However, high impedances and hence large values
of C can be attained at the expense of bandwidth by lowering the group
velocity. This tends to raise the efficiency, as do the high currents which are
allowable because of good heat dissipation. However, lowering group velocity
increases attenuation, and this will tend to reduce efficiency somewhat.
It has been suggested that the power can be increased by reducing the
phase velocity of the circuit near the output end of the tube, so that the
electrons which have lost energy do not fall behind the waves. This is a com-
plicated but attractive possibility. It has also been suggested that the elec-
trode which collects electrons be operated at a voltage lower than that of
the helix.
The general picture of what governs and limits power output is fairly
clear as long as C is very small. If attenuation near the output of the tube is
kept small, and the circuit is constructed so as to approximate the require-
ment that nearly the same field acts on all electrons, efficiencies as large as
40% are indicated within the limitations of the present theory. With larger
values of C it is not clear what the power limitation will be.
The usual traveling-wave tube would seem to have a serious competitor
for power applications in the traveling-wave magnetron amplifier, which is
discussed briefly in a later chapter.
CHAPTER XIII
TRANSVERSE MOTION OF ELECTRONS
Synopsis of Ch.a.pter
SO FAR WE HAVE taken into account only longitudinal motions of
electrons. This is sufficient if the transverse fields are small compared to
the longitudinal fields (as, near the axis of an axially symmetrical circuit)
or, if a strong magnetic focusing field is used, so that transverse motions are
inliibited. It is possible, however, to obtain traveling-wave gain in a tube in
which the longitudinal field is zero at the mean position of the electron beam.
For a slow wave, the electric field is purely transverse only along a plane.
The transverse field in this plane forces electrons away from the plane and
preferentially throws them into regions of retarding field, where they give up
energy to the circuit. This mechanism is not dissimilar to that in the longi-
tudinal field case, in which the electrons are moved longitudinally from their
unperturbed positions, preferentially into regions of more retarding field.
Whatever may be said about tubes utilizing transverse fields, it is cer-
tainly true that they have been less worked on than longitudinal-field tubes.
In view of this, we shall present only a simple analysis of their operation
along the lines of Chapter II. In this analysis we take cognizance of the fact
that the charge induced in the circuit by a narrow stream of electrons is a
function not only of the charge per unit length of the beam, but of the dis-
tance between the beam and the circuit as well.
The factor of proportionality between distance and induced charge can be
related to the field produced by the circuit. Thus, if the variation of V in the
X, y plane (normal to the direction of propagation) is expressed by a function
<l>, as in (13.3), the effective charge ps is expressed by (13.8) and, if y is the
displacement of the beam normal to the z axis, by (13.9) where $' is the de-
rivative of $ with respect to y.
The equations of motion used must include displacements normal to the
z direction; they are worked out including a constant longitudinal magnetic
focusing field. Finally, a combined equation (13.23) is arrived at. This is
rewritten in terms of dimensionless parameters, neglecting some small terms,
as (13.26)
62 ' (52 -\- P) '
616
TRANSVERSE MOTION OF ELECTRONS 617
Here 5 and b have their usual meanings; a is the ratio between the transverse
and longitudinal field strengths, and /is proportional to the strength of the
magnetic focusing field.
In case of a purely transverse field, a new gain parameter D is defined.
D is the same as C except that the longitudinal a-c field is replaced by the
transverse a-c field. In terms of D, b and 5 are redefined by (13.36) and
(13.37), and the final equation is (13.38). Figures 13.5-13.10 show how the
x's and 3''s vary with b for various values of / (various magnetic fields) and
Fig. 13.11 shows how Xi , which is proportional to the gain of the increasing
wave in db per wavelength, decreases as magnetic field is increased. A nu-
merical example shows that, assuming reasonable circuit impedance, a
magnetic field which would provide a considerable focusing action would
still allow a reasonable gain.
The curves of Figs. 13.6-13.10 resemble very much the curves of Figs.
8.7-8.9 of Chapter VIII, which show the effect of space charge in terms of
the parameter QC. This is not unnatural; in one case space charge forces
tend to return electrons which are accelerated longitudinally to their un-
disturbed positions. In the other case, magnetic forces tend to return elec-
trons which are accelerated transversely to their undisturbed positions. In
each case the circuit field acts on an electron stream which can itself sustain
oscillations. In one case, the oscillations are of a plasma type, and the re-
storing force is caused by space charge of the bunched electron stream; in
the other case the electrons can oscillate transversely in the magnetic field
with cyclotron frequency.
Let us, for instance, compare (7.13), which applies to purely longitudinal
displacements with space charge, with (13.38), which applies to purely
transverse fields with a longitudinal magnetic field. For zero loss (d = 0),
(7.13) becomes
1 = (j8 - 6) (62 + 4QC)
While
1 = 0'5- bW+f) (13.38)
describes the transverse case. Thus, if we let
4QC=P
the equations are identical.
When there is both a longitudinal and a transverse electric field, the equa-
tion for 8 is of the fifth degree. Thus, there are five forward waves. For an
electron velocity equal to the circuit phase velocity (b = 0) and for no at-
tenuation, the two new waves are unattenuated.
If there is no magnetic field, the presence of a transverse field component
merely adds to the gain of the increasing wave. If a small magnetic field is
618 BELL SYSTEM TECHNICAL JOURNAL
imposed in the presence of a transverse field component, this gain is some-
what reduced.
13.1 Circuit Equation
Consider a tubular electrode connected to ground through a wire, shown
in Fig. 13.1. Suppose we bring a charge Q into the tube from oo . A charge Q
will flow to ground through the wire. This is the situation assumed in the
analysis of Chapter II. In Fig. 2.3 it is assumed that all the lines of force
from the charge in the electron beam terminate on the circuit, so that the
whole charge may be considered as impressed on the circuit.
ELECTRODE
(^
Q
"WTTTT^TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTy
Fig. 13.1— When a charge Q approaches a grounded conductor from infinity and in the
end all the lines of force from the charge end on the conductor, a charge Q flows in the
grounding lead.
ELECTRODE
V7777777777777777777777777777777777777777777777.
Fig. 13.2 — If a charge Q approaches a conductor from infinity but in the end only part
of the Unes of force from the charge end on the conductor, a charge <I>Q flows in the ground-
ing lead, where <J> < 1.
Now consider another case, shown in Fig. 13.2, in which a charge Q is
brought from oo to the vicinity of a grounded electrode. In this case, not all
of the lines of force from the charge terminate on the electrode, and a charge
$() which is smaller than Q flows through the wire to ground.
We can represent the situation of Fig. 13.2 by the circuit shown in Fig.
13.3. HereC2 is the capacitance between the charge and the electrode and
C\ is the capacitance between the charge and ground. We see that the charge
4>() which flows to ground when a charge () is brought to a is
*<2 = <2C2/(Ci + Co) (13.1)
Now suppose we take the charge Q away and hold the electrode at a
potential V with respect to ground, as shown in Fig. 13.4. What is the po-
tential Va at fl? We see that it is
F. = [C2/(Ci-|-C2)]r = <i.F (13.2)
TRANSVERSE MOTION OF ELECTRONS
619
Thus, the same factor $ relates the actual charge to the "effective charge"
acting on the circuit and the actual circuit voltage to the voltage produced
at the location of the charge.
We will not consider in this section the "space charge" voltage produced
by the charge itself (the voltage at point a in Fig. 13.4).
The circuit voltage V we consider as varying as exp(— F^) in the direction
of propagation. The voltage in the vicinity of the circuit is given by
V(x, y) = W (13.3)
ELECTRODE
Fig. 13.3 — The situation of Fig. 13.2 results in the same charge flow as if the charge
were put on terminal a of the circuit shown, which consists of two capacitors of capaci-
tances Ci and C2 .
02
$ya
Fig. 13.4 — A voltage V inserted in the ground lead divides across the condensers so
that Va = *F, where * is the same factor which relates the charge flowing in the ground
lead to the charge Q applied at a in Figs. 13.2 and 13.3.
Here x and y refer to coordinates normal to z and <l> is a function of x and y.
We will choose x and y so
d^/dx =0 (13.4)
Then
Ey = -Vd^/dy = -^'V (13.5)
$' = d^/dy (13.6)
In (13.3), <l> will vary somewhat with T, but, as we are concerned with a
small range only in F, we will consider $ a function of y only.
From Chapter II we have
TViKi
V =
and
(r' - fD
(2.10)
(2.18)
620 BELL SYSTEM TECHNICAL JOURNAL
So that
In (13.7), it is assumed that$ = 1. If $ 5^ 1, we should replace p in (13.7)
by the a-c component of effective charge. The total effective charge pe is
PB = Hp + Po) (13.8)
The term pn is included because ^ will vary if the y-position of the charge
varies. To the first order, the a-c component pg of the effective charge is,
Pe = $p + po^'y (13.9)
PE = ^P - (lo/tioWy (13.9)
Here y is the a-c variation in position along the y coordinate. Thus, if $ 5^ 0,
we have instead of (13.7)
V = (p2 _ p2^ . (13.10)
This is the circuit equation we shall use.
13.2 Ballistic Equations
We will assume an unperturbed motion of velocity uq in the z direction,
parallel to a uniform magnetic focusing field of strength B. As in Chapter
II, products of a-c quantities will be neglected.
In the X direction, perpendicular to the y and z directions
dx/dl = —qBy
(13.11)
Assume that 3c = 0 at y =^ 0. Then
X = -qBy
(13.12)
In the y direction we have
dy/dt = r/(Bx - Ey)
(13.13)
From (13.5) this is
dy/dt = r,{Bx + $T)
(13.14)
dy/dl = dy/dt + (dy/dz)(dz/dt)
(13.15)
{dy/dt) = m(j% - T)y
(13.16)
We obtain from (13.16), (13.14) and (13.12)
(Pe - r)y = -uo^ly + v^'V/uo
(13.17)
13„, = vB/uq
(13.18)
TRANSVERSE MOTION OF ELECTRONS 621
Here ijB is the cyclotron radian frequency and /3„, is a corresponding propa-
gation constant.
Now
y = dy/di - {dy/dz){dz/di) (13.19)
y = uoij^e - T)y (13.20)
From (13.20) and (13.17) we obtain
y = OT' \( -Q T-N2 I o2t (13,21)
2T oKjiSe - r) + |8J
It is easily shown that the equation for p can be obtained exactly as in
Chapter II. From (2.22) and (2.18) we have
^"^'*^' (13.22)
13.3 Combined Equation
From the circuit equation (13.10) and the baUistical equations (13.21)
and (13.22) we obtain
1 =
(13.23)
The voltage at the beam is $ times the circuit voltage, so the effective
impedance of the circuit at the beam is $^ times the circuit impedance.
Thus
a = $2^/o/4n (13.24)
It will be convenient to define a dimensionless parameter / specifying ^^
and hence the magnetic field
/ = /3.//3.C (13.25)
We will also use b and b as defined earlier
-r = -j^e + ^eCb
-Fi = -j^e-j^eCb
After the usual approximations, (13.23) yields
i^ - * = ,U (^) (13.26)
«2 = (<J>V/3e$)' (13.27)
It is interesting to consider the quantity (4>'/j3e$)^ for typical fields. For
622 BELL SYSTEM TECHNICAL JOURNAL
instance, in the two-dimensional electrostatic field in which the potential
V is given by
V = Ae'^'^e-'^" (13.28)
dV/dy = -(3eV (13.29)
and everywhere
a2 = ($7^,4>)2 = 1. (13.30)
Relation (13.30) is approximately true far from the axis in an axially sym-
metrical field.
Consider a potential giving a purely transverse field at y = 0
V = Ae~'^'' sinh I3ey (13.31)
^ = l3Ae~'^'' cosh /3ey. (13.32)
dy
In this case, at y = 0
a2 = ($7/3,$)2 = 00 (13.33)
In the case of a purely transverse field we let
^ - I^ (13.34)
D' = (£j//3'P)(/o/8Ko) (13.35)
In (13.35), Ey is the magnitude of the y component of field for a power
flow P, and /3 is the phase constant.
We then redefine 8 and b in terms of D rather than C
-r = -p, + ^,D8 (13.36)
-ri= -j%-pM (13.37)
and our equation for a purely transverse field becomes
1 = (j^-bW-^p) (13.38)
In (13.38), 5 and b are of course not the same as in (13.26) but are defined
by (13.36) and (13.37).
13.4 Purely Transverse Fields
The case of purely transverse fields is of interest chiefly because, as was
mentioned in ('hapter X, it has been suggested that such tubes should have
low noise.
TRANSVERSE MOTION OF ELECTRONS 623
In terms of -v and y as usually defined
8 = X -\- jy
equation (13.38) becomes
x[(x'- -f-+ /-) - 2y(y + 6)] = 0 (13.39)
(y + bXx-' - / + /2) _^ 2x'y +1 = 0 (13.40)
From the x = 0 solution of (13.39) we obtain
X = 0 (13.41)
h = -^, - y. (13.42)
y" - P
It is found that this solution obtains for large and small values of b. For
very large and very small values of b, either
y = -b (13.43)
or
y = H (13.44)
The wave given by (13.43) is a circuit wave; that given by (13.44) repre-
sents electrons travehng down the tube and oscillating with the cyclotron
frequency in the magnetic field.
In an intermediate range of 6, we have from (13.39)
X = ±\/2y{y-\- b) - (P - y^) (13.45)
and
b = -2y ± \/p - l/2y. (13.46)
For a given value of /- we can assume values of y and obtain values of b.
Then, x can be obtained from (13.45). In Figs. 13.5-13.10, .v and y are plotted
vs. b for/- = 0, .5, 1, 4 and 10. It should be noted that .Vi, the parameter
expressing the rate of increase of the increasing wave, has a maximum at
larger values of b as/ is increased (as the magnetic focusing field is increased).
Thus, for higher magnetic focusing fields the electrons must be shot into the
circuit faster to get optimum results than for low fields. In Fig. 13.11, the
maximum positive value of .v is plotted vs. /. The plot serves to illustrate the
effect on gain of increasing the magnetic field.
Let us consider an example. Suppose
X = 7.5 cm
D = .03
624
BELL SYSTEM TECHNICAL JOURNAL
t.o
o.s
f2 = 0
^^,
y FOR --=^\
UNDISTURBED WAVE
\,
\
\
N
N
\
\
^^
3j_
\
■*■*..» _
^
.
—
y3
N
N
\
\
1
y, ANoy^'
~~,
^-^
y
^2
■—-
\
^^
*
-— •
'y'r
\
\
\
\
^^\>
.^^
V
Fig. 13.5 — The .v's and y's for the three forward waves when the circuit field is purely
transverse at the thin electron stream, for zero magnetic focusing field (/^ = 0).
1.0
0.5
\^
V
f2=0.5
\
N
s
''>
\
y3
*^
-^.
. —
ya
X,
.
-^
-
■-—
—
—
-" "
---
^s
/^
^
N
iy7
—
.,^
K-;
'Zz
-^.
}
^
-^2
■
■ *'^
• ^
-^
yi
—i
y^
•-.^
/"
^
I
\
\
\
ya
\
N
\
>
Fig. 13.6 — Curves similar to those of Fig. 13.5 for a parameter /' = 1. The parameter
/ is proportional to the strength of the magnetic focusing field.
TRANSVERSE MOTION OF ELECTRONS
625
\
f2 = I.O
\
s
X
^l
'*v.
•>-,
—
—
_..
— -
—
— ,
■~~
-^1
X,
,
^N
/
^
N
\
\v.
Jk
/
_._
"y?
—
"•■
"•^■^
— '
^
"-.
■^2
/"
^ —
\
\
\
\
\
ya
s
\
\
-3 -2 -I
Fig. 13.7 — The x's and y's for/^ = 1.0.
\
f2=4.0
^^
\s
"■"^^
--
.ya.
"■■*•
—
---.
y_2
'^x
>.
s
'\
X,
\
^.
(
k
\
\
^.
y
\
\
^2
\
\
—
yi
___-
J
^^v
.y2
yi
"-
^»>
^
\
\
\
\
.^2
\
\
-3-2-1 0 1 2
b
Fig. 13.8— The x's and y's for/^ = 2.0.
626
BELL SYSTEM TECHNICAL JOURNAL
J.D
\^
s.
f^=2.0
\^
.UaCx^o)
N
v^.
'
—
. —
—
'^ —
-1
—
— -
_y 2(^=0)
^•«.,
^■s
X,
\
r
2;^
N
\
V,
y
)>^,
^2
«.-
—
"yT
—
—
^"-
>^2
«»
y,(x=o)
^^^
--y"
\
\
\
.^2
\
\
40
s
-5 -4 -3 -2 -1
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
Fig. 13.9 — The .t's and y's for/ = 4.0.
Wa
—
—
—
'■"'
^v.
f 2 = 10.0
^s
s
s
^h
s
s
'<■.
N
\
s
^
^
N
\
\
>^,
"X^
y
\
\
\
\
^^
,y^2
■"■^
^
>
y<
'^i-^
-3-2-1 0 I 2
b
Fig. 13.10— The .%'s and y's iox p = 10.0.
TRANSVERSE MOTION OF ELECTRONS
627
These values are chosen because there is a longitudinal field tube which
operates at 7.5 cm with a value of C (which corresponds to D) of about .03.
The table below shows the ratio of the maximum value of Xi to the maximum
value of Xi for no magnetic focusing field.
Magnestic Field in Gauss /
Xi/.Tio
0 0
1
50 1.17
.71
100 2.34
.50
A field of 50 to 100 gauss should be sufiicient to give useful focusing action.
Thus, it may be desirable to use magnetic focusing fields in transverse-
0.8
0.7
0.6
0.5
0.4
0.3
0.2
O.t
0
— -
-^
^
\
V
X
\
"^
0.2 0.4 0.6
0,6 1.0 1.2 1.4 t.6 1.8 2.0 2.2 2.4
PROPORTIONAL TO MAGNETIC FIELD, f
2.6 2.8 3.0 3.2
Fig. 13.11 — Here Xi , the x for the increasing wave, is plotted vs/, which is proportional
to the strength of the focusing field. The velocity parameter b has been chosen to maxi-
mize Xi . The ordinate Xi is proportional to gain per wavelength.
field traveling-wave tubes. This will be more especially true in low-voltage
tubes, for which D may be expected to be higher than .03.
13.5 Mixed Fields
In tubes designed for use with longitudinal fields, the transverse fields
far off the axis approach in strength the longitudinal fields. The same is true
of transverse field tubes far off the axis. Thus, it is of interest to consider
equation (13.26) for cases in which a is neither very small nor very large,
but rather is of the order of unity.
If the magnetic field is very intense so that P is large, then the term con-
taining a^, which represents the effect of transverse fields, will be very small
and the tube will behave much as if the transverse fields were absent.
62S
BELL SYSTEM TECHNICAL JOURNAL
Consideration of both terms presents considerable difficulty as (13.26)
leads to fi\^ waves (5 values of 5) instead of three. The writer has attacked
the problem only for the special case of 6 = 0. In this case we obtain from
(13.26)
"1
5 = -j
52 '^ 52
^1
(13.47)
MacColl has shown^ that the two "new" waves (waves introduced when
a = 0) are unattenuated and thus unimportant and uninteresting (unless,
as an off-chance, they have some drastic effect in fitting the boundary
conditions).
Proceeding from this information, we will find the change in b as P is
increased from zero. From (13.47) we obtain
db =j
'2d8 2a 8d8 adf
Now, if/ = 0
If we use this in connection with (13.48) we obtain
db= -i,df
60
For an increasing wave
5i = (1 + ($7/3.$)M\/3/2 - j/2)
Hence, for the increasing wave
3(1 + a-)
(13.48)
(13.49)
(13.50)
(13.51)
(13.52)
This shows that applying a small magnetic field tends to decrease the gain.
This does not mean, however, that the gain with a longitudinal and trans-
verse field and a magnetic field is less than the gain with the longitudinal
field alone. To see this we assume that not/^ but {^'/(3e^y is small. Differen-
tiating, we obtain
2
db = -j
2db
2a bdb . da'
(52 -\- py "^ 52 + /2J
If a = 0
53 =. _j
(13.53)
(13.54)
* J. R. Pierce, "Transverse Fields in Traveling- Wave Tubes," Bdl System Technical
Journal, Vol. 27, pp. 732-746.
TlLiNSVERSE MOTION OF ELECTRONS 629
and we obtain
"'^IWTP)'" (13.55)
If we have a very large magnetic field (/- » | 5- 1), then
d8 = ^^da (13.57)
and the change in 5 is purely reactive. If / = 0 (no magnetic field), from
(13.55)
d8 =^- da' (13.58)
Adding a transverse field component increases the magnitude of 5 without
changing the phase angle.
CHAPTER XIV
FIELD SOLUTIONS
Synopsis of Cil^pter
SO FAR, it has been assumed that the same a-c field acts on all elec-
trons. This has been very useful in getting results, but we wonder if
we are overlooking anything by this simplification.
The more complicated situation in which the variation of field over the
electron stream is taken into account cannot be investigated with the same
generality we have achieved in the case of "thin" electron streams. The
chief importance we will attach to the work of this chapter is not that of
producing numerical results useful in designing tubes. Rather, the chapter
relates the appropriate field solutions to those we have been using and
exhibits and evaluates features of the "broad beam" case which are not
found in the "thin beam" case.
To this end we shall examine with care the simplest system which can
reasonably be expected to exhibit new features. The writer believes that
this will show quaUtatively the general features of most or all "broad
beam" cases.
The case is that of an electron stream of constant current density com-
pletely filling the opening of a double finned circuit structure, as shown in
Fig. 14.1. The susceptance looking into the slots between the fins is a func-
tion of frequency only and not of propagation constant. Thus, at a given
frequency, we can merely replace the slotted circuit members by suscept-
ance sheets relating the magnetic field to the electric field, as shown in
Fig. 14.2. The analysis is carried out with this susceptance as a parameter.
Only the mode of propagation with a symmetrical field pattern is con-
sidered.
First, the case for zero current density is considered. The natural mode
of propagation will have a phase constant jS such that Hx/Ez for the central
region is the same as IIx/Ez for the finned circuit. The solid curve of Fig.
14.3 shows a quantity proportional to IIx/Ez for the central space vs ^ =
/3J {d defined by Fig. 14.1), a quantity proportional to /?. The dashed fine
P represents Ux|E^ for a given finned structure. The intersections specify
values of B for the natural active modes of propagation to the left and to the
right, and, hence, values of the natural phase constants.
The structure also has j)assive modes of propagation. If we assume
fields which vary in the z direction as exp (^/f/)^, Ih/Ez for the central
()3(l
FIELD SOLUTIONS 631
opening varies with $ as shown in part in Fig. 14.4. A horizontal line repre-
senting a given susceptance of the finned structure will intersect the curve
at an infinite number of points. Each intersection represents a passive
mode which decays at a particular rate in the z direction and varies sinu-
soidally with a particular period in the y direction.
If the effect of the electrons in the central space is included, Hx/Ez for
the central space no longer varies as shown in Fig. 14.3, but as shown in
Fig. 14.5 instead. The curve goes off to + co near a value of d correspond-
ing to a phase velocity near to the electron velocity. The nature of the modes
depends on the susceptance of the finned structure. If this is represented
by Pi , there are four unattenuated waves; for P^ there are two unattenu-
ated waves and an increasing and a decreasing wave. P^ represents a tran-
sitional case.
Not the whole of the curve for the central space is shown on Fig. 14.5.
In Fig. 14.6 we see on an expanded scale part of the region about d = \,
between the points where the curve goes through 0. The curve goes to + oo
and repeatedly from — oo to + oo , crossing the axis an infinite number of
times as 6 approaches unity. For any susceptance of the finned structure,
this leads to an infinite number of unattenuated modes, which are space-
charge waves; for these the amplitude varies sinusoidally with different
periods across the beam. Not all of them have any physical meaning, for
near ^ = 1 the period of cyclic variation across the beam will become small
even compared to the space between electrons.
Returning to Fig. 14.1, we may consider a case in which the central space
between the finned structures is very narrow {d very small). This will have
the effect of pushing the solid curve of Fig. 14.5 up toward the horizontal
axis, so that for a reasonable value of P (say, Pi , Pi or P^ of Fig. 14.5) there
is no intersection. That is, the circuit does not propagate any unattenuated
waves. In this case there are still an increasing and a decreasing wave. The
behavior is like that of a multi-resonator klystron carried to the extreme of
an infinite number of resonators. If we add resonator loss, the behavior of
gain per wavelength with frequency near the resonant frequency of the
slots is as shown in Fig. 14.7.
One purpose of this treatment of a broad electron stream is to compare
its results with those of the previous chapters. There, the treatment con-
sidered two aspects separately: the circuit and the effect of the electrons.
Suppose that at j = <<? in Fig. 14.1 we evaluate not H^ for the finned
structure and for the central space separately, but, rather, the difference
or discontinuity in Hx . This can be thought of as giving the driving current
necessary to establish the field E^ with a specified phase constant. In Fig.
14.8, yi is proportional to this Hx or driving current divided by Ez. The
dashed curve T2 is the variation of driving current with 6 or ^ which we have
632 BELL SYSTEM TECHNICAL JOURNAL
used in earlier chapters, fitted to the true curve in slope and magnitude at
-y = 0. Over the range of B of interest in conneclion with increasing waves,
the fit is good.
The difference between HJEz for the central space without electrons
(Fig. 14.3) and Hx/Ez for the central space with electrons (Fig. 14.5) can
be taken as representing the driving effect of the electrons. The solid curve
of Fig. 14.9 is proportional to this difference, and hence represents the true
effect of the electrons. The dashed curve is from the ballistical equation
used in previous chapters. This has been fitted by adjusting the space-
charge parameter Q only; the leading term is evaluated directly in terms of
current density, beam width, /5, and variation of field over the beam, which
is assumed to be the same as in the absence of electrons.
Figure 14.10 shows a circuit curve (as, of Fig. 14.8) and an electronic
curve (as, of Fig. 14.10). These curves contain the same information as the
curves (including one of the dashed horizontal lines) of Fig. 14.5, but dif-
ferently distributed. The intersections represent the modes of propagation.
If such curves were the approximate (dashed) curves of Figs. 14.8 and
14,9, the values of 6 for the modes would be quite accurate for real inter-
sections. It is not clear that "intersections" for complex values of 6 would be
accurately given unless they were for near misses of the curves. In addition,
the compHcated behavior near 6 = \ (Fig. 14.6) is quite absent from the
approximate electronic curve. Thus, the approximate electronic curve does
not predict the multitude of unattenuated space-charge waves near 0=1.
Further, the approximate expressions predict a lower limiting electron
velocity below which there is no gain. This is not true for the e.xact equations
when the electron flow fills the space between the finned structures com-
pletely.
It is of some interest to consider complex intersections in the case of
near misses by using curves of simple form (parabolas), as in Fig. 14.11.
Such an analysis shows that high gain is to be expected in the case of curves
such as those of Fig. 14.10, for instance, when the circuit curve is not steep
and when the curvature of the electronic curve is small. In terms of physical
parameters, this means a high impedance circuit and a large current density.
14,1 The System and the Equations
The system examined is a two-dimensional one closely analogous to that
of Fig. 4.4. It is shown in Fig. 14.1. It consists of a central space extending
from y = —d\.oy= -\-d, and arrays of thin fins separated by slots ex-
tending for a distance // beyond the central opening and short-circuited at
the outer ends. An electron flow of current density Jq amperes/w^ fills the
open space. It is assumed that the electrons are constrained by a strong
magnetic field so that they can move in the z direction only.
FIELD SOLUTIONS
633
We can simplify the picture a little. The open edges of the slots merely
form impedance sheets.
From 4.12 we see that aX y = —d
Hx /coe ^ . ,
B
= -jB
■\^e/ II cot /3o//
(14.1)
(14.2)
(14.3)
Jq amp/cm2 —
Fig. 14.1 — Electron flow completely fills the open space between two finned structures.
A strong axial magnetic field prevents transverse motions.
Hx=JBE2
y=d
Hx=-jBEz
Fig. 14.2 — In analyzing the structure of Fig. 14.1, the finned members are regarded as
susceptance sheets.
for
/3o/co€ = 1/ce = ■\/ jx/e — 377 ohms
Similarly, at j = -{-d,
(14.4)
(14.5)
We can use fJ as a parameter rather than h. Thus, we obtain the picture
of Fig. 14.2. This picture is really more general than Fig. 14.1, for it applies
for any transverse-magnetic circuit outside of the beam.
634 BELL SYSTEM TECIIXICAL JOURNAL
Inside of the beam the effect of the electrons is to change the effective
dielectric constant in the z direction. Thus, from (2.22) we have for the elec-
tron convection current
• - 2Foaft - r)= ^^^^
Now
£. = - ^ = rF (14.6)
so that
' 2Fo(y/3. - r)^ ^^^-'^
The appearance of a voltage V in (2.22) and (14.6) does not mean that these
relations are invalid for fast waves. In (2.22) the only meaning which need
be given to V is that defined by (14.6), as it is the electric field as specified
by (14.6) that was assumed to act on the electrons in deriving (2.22).
Let us say that the total a-c current density in the z direction, Jz , is
Jz = jo}tiEz (14.8)
This current consists of a displacement current jcoeE^ and the current i,
so that
Hence
\ 2ecoFo(;/3e- 1)7
This gives the ratio of the effective dielectric constant in the z direction to
the actual dielectric constant. We will proceed to put this in a form which
in the long run will prove more convenient.
Let us define a quanity (3
(14.11)
(14.12)
(14.13)
(14.14)
\
T = j0
and a quantity A
A- '^"^'
2€«oFo
And quantities 6 and 9^
Be
= 134= (co/«o)'/
e
= (3d
FIELD SOLUTIONS 635
We recognize d as the half-width of the opening filled by electrons. Then
-A = 1 - (^ (14.15)
We can say something about the quantity .4. From purely d-c considera-
tions, the electron flow will cause a fall in d-c potential toward the center
of the beam. Indeed, this is so severe for large currents that it sets a limit
to the current density which can be transmitted. If we take Fo and Uq as
values at ^^ = ±d (the wall), the maximum value of A as defined by (14.12)
is 2/3, and at this maximum value the potential at y = 0 is Fo/4. This is
inconsistent with the analysis, in which Vq and Mo are assumed to be con-
stant across the electron flow. Thus, for the current densities for which the
analysis is valid, which are the current densities such as are usually used in
traveling-wave tubes
A « 1 (14.16)
In the a-c analysis we will deal here only with the symmetrical type of
wave in which £e(+y) = Ez{—y). The work can easily be extended to
cover cases for which Ez{-\-y) = —E;{—y). We assume
H,= Hosmhyye~'^' (14.17)
From Maxwell's equations
jojeEy = "^^ = —j^Hois'mh yy)e
dHx .^„ / . , N -S0Z
Ey = - ^ Hoisinh \y)e~'^' (14.18)
coe
Similarly
jueiEz = — ~—^ = — 7//o(cosh 7v)e "'^^
ay
£, = ^ Fn(cosh yy)e~'^' (14.19)
coei
We must also have
—joinUx = -— — ^—
ay dz
jcofxHoe ■'^"' sinh yy = "^-^ H„e '^' cosh 7V — -- HqC ^^' sinh 7y
coei ' we
y = (€i/e)0S2 _ ^l) (14.20)
/3o = aj2^€ = «Vc2 (14.21)
636 BELL SYSTEM TECHNICAL JOURNAL
Now. from (14.17), (14.19) and (14.20)
H. - ;a;6(6,/e) tanh Ke,/ey'\f - /3^)'''y]
(14.22)
But
Hence
E, {^x/^y'%^' - 3,'y"
H. -./VeAKciA)'"/?.. tanh [{e./eViS' - ^iVyl
E, (J- - /.V)"-
At V = </, (14.5) must apply. From (14.24) we can write
(14.24)
„_ {.,'.)'" i-^ix^h [{.,'. y''\d' - diY"] ,.,...
^ - (02 - eiY'-' ^ ^^
Here 0 is given by (14.14)
e^ = ^o<^ = {o3/c)d (14.26)
and P is given by
P = B/MV^ = B/doV^ (U.27)
Thus, 00 expresses J in radians at free-space wavelength and P is a measure
of the wall reactance, the susceptance rising as B rises.
14.2 Waves esj the Absence of Electrons
In this section we will consider (14.25) in the case in which there are no
electrons and ei/e = 1. Li this case (14.25) becomes
_ tanh (r - dlY" .
P - - (e2 - eg)i-2 (i-i-28)
Suppose we plot tlae right-hand side of (14.28) vs 6 for real values of di
corresponding to unattenuated waves. In Fig. 14.3 this has been done for
do = 1/ 10. For do > 7r/2 the behavior near the origin is dilYerent, but in
cases corresponding to actual traveling wave tubes do < tt/I.
Intersections between a horizontal line at height P and the curve give
values of 6 representing unattenuated waves. We see that for the case
which we have considered, in which do < ir/2 and do cot do> 1, there are
unattenuated waves if
P > - tan do/do (14.29)
For P = — 00 (no slot depth and no wall reactance) the system for do < ir/l
constitutes a wave guide operated below cutoff frequency for the type of
FIELD SOLUTIONS
637
wave we have considered. If we increase P {\ P \ decreasing; the inductive
reactance of the walls increasing) this finally results in the propagation of a
wave. There are two intersections, at ^ = ±0i , representing propagation
to the right and propagation to the left. The variation of di with P is such
that as P is increased (made less negative J di is increased; that is, the greater
is P (the smaller | P \), the more slowly the wave travels.
There is another set of waves for which d is imaginary; these represent
passive modes which do not transmit energy but merely decay with distance.
In investigating these modes we will let
= jf^
so that the waves vary with z as
(*w«
(14.30)
(14.31)
-e^\
j+e,
^\1
p
\^
\
Fig. 14.3 — The structure of Fig. 14.1 is first analyzefi in the absence of an electron
stream. Here a quantity proportional to Ux/Ei at the susceptance sheet is plotted vs
B = /3rf, a quantity proportional to the phase constant /3. The solid curve is for the inner
open space; the dashed line is for the susceptance sheet. The two intersections at ±di
correspond to transmission of a forward and a backward wave.
Now (14.28) becomes
p = -tan (^ + elyiyi^ + elyi'^
(14.32)
In Fig. 14.4 the right-hand side of (14.28) has been plotted vs $, again for
e, = 1/10.
Here there will be a number of intersections with any horizontal line
representing a particular value of P (a particular value of wall susceptance),
and these will occur at paired values of $ which we shall call db^n . The
corresponding waves vary with distance as exp (± ^r^/d).
Suppose we increase P. As P passes the point — (tan ^o)/^o , $" for
a pair of these passive waves goes to zero; then for P just greater than
— (tan d()/dn we have two active unattenuated waves, as may be seen
by comparing Figs. 14.4 and 14.3.
638
BELL SYSTEM TECHNICAL JOURNAL
14.3 Waves in the Presence of Electrons
In this section we deal with the equations
„ -(eiA)"'-'tanh [{e,/ey'\d' - 61)'^']
and
6iA = 1 -
{6- - eiyi'-
A
{e. - ey
(14.25)
(14.15)
We consider cases in which the electron velocity is much less than the
velocity of light; hence
e, » eo (14.33)
,
.' J
iy
\lv
\\
/'
/
\\
\\
f
\
-20 -15 -10 -5 0 5 10 15 20
'P
Fig. 14.4 — If a quantity proportional to Hx/Et at the edge of the central region is
plotted vs 4> = —jd, this curve is obtained. There are an infinite number of intersections
with a horizontal hne representing the susceptance of the finned structure. These corre-
spond to passive modes, for which the field decays exponentially with distance away from
the point of excitation.
In Fig. 14.5, the right-hand side of (14.25) has been plotted vs. 6 for
de = 10 00 , corresponding to an electron velocity 1/10 the speed of light.
Values oi 6 = 1/10 and A = 1/100 have been chosen merely for conven-
ience.* The curve has not been shown in the region from d = .9 \o 6 = 1.1,
where ei/e is negative, and this region will be discussed later.
For a larger value of 7*(| P \ small). Pi in Fig. 14.5, there are 4 intersec-
tions corresponding to 4 unaltenuated waves. The two outer intersections
obviously correspond to the "circuit" waves we would have in the absence
of electrons. The other two intersections near 6 = .9de and B = \Ade we
call electronic or space-charge waves.
* At a beam voltage Vo = 1,000 and for d = 0.1 cm, A = 1/100 means a current density
of about .S.SO ma/cm^, which is a current density in the range encountered in practice.
FIELD SOLUTIONS
639
For instance, increasing P to values larger than Pi changes d for the cir-
cuit waves a great deal but scarcely alters the two "electronic wave" values
of 6, near 6 = deil ± 0.1). On the other hand, for large values of P the values
of d for the electronic waves are approximately
6 = de±\^A (14.34)
Thus, changing A alters these values, but changing A has little effect on the
values of 6 for the circuit waves.
Now, the larger the P the slower the circuit wave travels; and, hence, for
large values of P the electrons travel faster than the circuit wave. Our
narrow-beam analysis also indicated two circuit waves and two unatten-
uated electronic waves for cases in which the electron speed is much larger
than the speed of the increasing wave. It also showed, however, that, as
the difference between the electron speed and the speed of the unperturbed
>'Pl .'P2
.P3 J
L,.
r!^riiZ,^__-
1
^— ^_-
-Is T^rTT!
^---.^^
Fig. 14.5 — When electrons are present in the open space of the circuit of Fig. 14.1, the
curves of Fig. 14.3 are modified as shown here. The nature of the waves depends on the
relative magnitude of the susceptance of the finned structure, which is represented by
the dashed horizontal lines. For Pi , there are four unattenuated waves, for P3 , two
unattenuated waves and an increasing wave and a decreasing wave. Line Pi represents a
transition between the two cases.
wave was made less, a pair of waves appeared, one increasing and one
decreasing. This is also the case in the broad beam case.
In Fig. 14.5, when P is given the value indicated by P2 , an "electronic"
wave and a "circuit" wave coalesce; this corresponds to yi and y-i running
together at 6 = (3/2) (2)''^ in Fig. 8.1. For a somewhat smaller value of P,
such as P3 , there will be a pair of complex values of 6 corresponding to an
increasing wave and a decreasing wave. We may expect the rate of increase
at first to rise and then to fall as P is gradually decreased from the value P2 ,
corresponding to the rise and fall of .Ti as b is decreased from (3/2) (2) in
Fig. 8.1.
It is interesting to know whether or not these increasing waves persist
down to P = —CO (no inductance in the walls). When P = — <», the
only way (14.25) can be satisfied is by
coth ((€i/€)i/2(^' - Gly) = 0
(14.35)
640 BELL SYSTEM TECHNICAL JOURNAL
This will occur only if
.l/2//,2 n2\I/2 •/ I TT
{ey/er\f - dlY'-' = j(nrr + ^
(€i/e)(r - 0^) = -(WTT + 2
(14.36)
Let
6 = ti -\- jw (14.37)
From (14.37), (14.36) and (14.15)
A
If we separate the real and imaginary parts, we obtain
1
((;/ + jwf - el) = -Lw + fj (14.38)
2 (14.39)
- AAuW^de - U) - [{de - UY + w'-] I UW + '" "
[{A - l){ee - U)' - {A + l)w-]iH- - W- - do~)
W{u[{d, - Uf + W-] - A[{de - Uf - W-\ + {de " «) (m' - W - dl)) = 0
(14.40)
The right-hand side of (14.39) is always positive. Because always A < I,
the first term on the left of (14.39) is always negative if w > (w + ^o),
which will be true for slow rates of increase. Thus, for very small values
of w, (14.39) cannot be satisfied. Thus, it seems that there are no waves
such as we are looking for, that is, slow waves {u « c). It appears that
the increasing waves must disappear or be greatly modified when P ap-
proaches — =o .
So far we have considered only four of the waves which exist in the
presence of electrons. A whole series of unattenuated electron waves exist
in the range
de - \/Z < e < de -\- VZ
In this range (ei/e)^'^ is imaginary, and it is convenient to rewrite (14.25)
as
P - i-^i/ey"te.nK-er'ey'\d'-eiy''] ,., ...
ie' - dlY" ^ ^
The chief variation in this expression over the range considered is that due 1
to variation in (— ei/e)''^. For all practical purposes we may write
(dl - elf
FIELD SOLUTIONS
641
Near 6 = 6,, the tangent varies with infinite rapidity, making an infinite
number of crossings of the axis.
In Fig. 14.6, the right-hand side of (14.41) has been plotted for a part of
the range 6 = 0.90 6eio6 = 1.10 ^^ . The waves corresponding to the inter-
sections of the rapidly fluctuating curve with a horizontal line representing
P are unattenuated space-charge waves. The nearer 6 is to 6e , the larger
(— ci/e) is. The amplitude of the electric field varies with y as
cosh (j{-
«l/«) (P
1/2/
,1/2.
^iry) = COS (i-e,/ey'W - ^tY'^y) (14.45)
10
J
/
V
[
0
10
?0
/
N
\
Fig. 14.6— The curve for the central region is not shown completely in Fig. 14.5. A part
of the detail around ^ = 1, which means a phase velocity equal to the electron velocity, is
shown in Fig. 14.6. The curve crosses the axis, and any other horizontal line, an infinite
number of times (only some of the branches are shown). Thus, there is a large number of
unattenuated "space charge" waves. For these, the amplitude varies sinusoidally in the y
direction. Some of these have no physical reality, because the wavelength in the y direction
is short compared with the space between electrons.
For small values of \6 — 6t\ the field fluctuates very rapidly in the y direc-
tion, passing through many cycles between y = 0 and y = d. For very
small values oi \d — 6e\ the solution does not correspond to any actual
physical problem: spreads in velocity in any electron stream, and ultimately
the discrete nature of electron flow, preclude the variations indicated by
(14.45).
The writer cannot state definitely that there are not increasing waves for
which the real part of 6 lies between 6e — y/A and 6e -\-'\/A, but he sees
no reason to believe that there are.
There are, however, other waves which exhibit both attenuation and
642
BELL SYSTEM TECHNICAL JOURNAL
propagation. The roots of (14.32) are modilied by the introduction of the
electrons. To show this effect, let $„ be a solution of (14.32), and_;($„ + b)
be a solution of (14.25), The waves considered will thus vary with distance
as
^((*„+6)/d]z
We see that we must have
vl/2
2n1/2
(6,/e)"^ ($; + 6t>y" cot ($1 + el)
,2x1/2
,2x1/2
1/2/
= {{^n + by + e'oY" cot l(6i/6)"^((*„ + bY + e'.Y'-']
(ei/e)''^ = 1
(14.43)
(14.44)
(14.15a)
X {Be - j^n + by J
As ^ <<C 1, it seems safe to neglect b in (14.15a) and to expand, writing
(,,/e)i/2 = 1 - «
A _ A[{el - $1) + 2jde^,]
2(de-j^ny
If I 6 I <C $„ , we may also write
(($« + bY + dlY" -
life + ^\f
^nb
(14.46)
(14.47)
{< + ^0^)
2^/2 + ^^\ + ^o)'" (14.48)
We thus obtain, if we neglect products of 5 and a
(1 - a) cot ($; + doY" = 1 +
(i
(4>„ + ^o)
^nb
cot (<!>; + ^o)
2\l/2
(14.49)
^($; + eW' ~ V ''' ^^' + ^"^
Solving this for 5, we obtain
(*l + dlY" Tcos ($1 + di)'" + CSC (4>l + dlY"
2\l/2
5 = -
5 =
+ i
LCOS i^l + e^)^'^ - CSC ($1 + ^o)''^J
a (14.50)
L*„(e! + *;)^ ' ' (el + $;)^
csc^ (*; + 0^)'^^ + cos (*l + 0^)^'n /K^o + ^lY"
Lcsc « + elY" - cos (ci>; + elY^'J
(14.51)
As the waves vary with distance as exp [(± <J>„ + 5)'S/<^]) this means that all
modified waves travel in the —z direction, and very fast, for the imaginary
part of b, which is inversely proportional to tlie pliase velocity, will be small.
FIELD SOLUTIONS 643
These backward-traveling waves cannot give gain in the +2 direction, and
could give gain in the —z direction only under conditions similar to those
discussed in Chapter XI.
14.4 A Special Type of Solution
Consider (14.25) in a case in which
^0 « Be (14.52)
Be « 1 (14.53)
In this case in the range
e <de- Va and e > de+ VA (14.54)
we can replace the hyperbolic tangent by its argument, giving
^=-(-A) = (^,-l. (14.55)
This can be solved for 6, giving
^ = 0, T \/A/{P + 1) (14.56)
If
P < -1
Then 0 will be complex and there will be a pair of waves, one increasing and
one decreasing. We note that, under these circumstances, there is no cir-
cuit wave, either with or without electrons.
What we have is in essence an electron stream passing through a series
of inductively detuned resonators, as in a multi-resonator klystron. Thus,
the structure is in essence a distributed multi-resonator klystron, with loss-
less resonators. If the resonators have loss, we can let
P = {-jG + B)/doV^ (14.57)
where G is the resonant conductance of the slots. In this case, (14.56) be-
comes
\-jG + {B + doVe/fjd/
Near resonance we can assume G is a constant and that B varies linearly
with frequency. Accordingly, we can show the form of the gain of the in-
creasing wave by plotting vs. frequency the quantity g
g = Im(-j 4- co/coo)-i/2 (14,59)
In Fig. 14.7, g is plotted vs. co/coq .
644
BELL SYSTEM TECHNICAL JOURNAL
14.5 Comparison with Previous Theory
We will compare our field solution with the theory presented earlier by
comparing separately circuit effects and electronic effects.
14.6a Comparison of Circuit Equations
According to Chapter VI the field induced in an active mode by the current
i should be
0.8
0.6
0.4
0.2
0
/
/'
\
^
y
\
\
\
-6-4-2 0 2 4 6
OJ/COq
Fig. 14.7 — In a plot such as that of Fig. 14.5, the horizontal line for the fins may not
intersect the solid line for the central space at all. Particularly, this will be true as the
central space is made very narrow. There will still be an increasing and a decreasing wave,
however. Suppose, now, that the finned structure is lossy. We find that the gain in db of
the increasing wave will vary with frequency as shown. Here oio is the resonant frequency
of the slots in the finned structure.
whence
E. =
jO'R
{e\ - e')
(14.60)
where i? is a positive constant proportional to {E-/I3-P).
Suppose that in Fig. 14.2 we have at y = d not only the current jBE^
flowing in the wall admittance, but an additional current i given by (14.60)
as well. Then instead of (14.28) we have
1
^oCVe/M) JE,
+ P = -
tanh id' - do)
2\l/2
id' - el)'"
For simplicity, let 9o « 6. Then we obtain from (14.61)
/— /— / ,^ tanh 0\ „
(14.61)
(14.62)
FIELD SOLUTIONS 645
We must identify this with (14.60). Thus, over the range considered, we
must have approximately
(eW - i)/R = BoV^^iP + (tanh0)/0) (14.63)
At 0 = ^1 , we must have both sides zero, so that
P = - (tanh 9i)/di and (14.64)
(1 - (di/ey)/R = V^((tanh ed/ei - (tanh d)/d) (14.65)
Taking the derivative with respect to 6
2^' a rr f sech' 0 , tanh e\ ,,.,,.
These must be equal at 0 = 0i , so that
l/R = (1/2) (00 VeAi) (^-^ - sech^ d^^ (14.67)
Thus, according to the methods of Chapter VI, our circuit equation should
be
GT^)ii. = ('/«(T'--^'^-)^'-<^'/^>^^ '^^-^
Using (14.64), the correct equation (14.62) becomes
tanh 01 tanh 6
68)
(14.69)
71 U
In a typical traveling-wave tube, we might have
01 = 2.5
In Fig. 14.8, the right-hand side of (14.69) is plotted as a solid line and
the right-hand side of (14.68) is plotted as a dashed line for dx = 2.5.
14.5b Electronic Comparison
Consider (14.25), which is the equation with electrons. For simplicity,
let do « 0, so that
B ^ ^ ^ _(6i/6)Uanh[(6i/6)i/^g] ^^^ ^^^
V c/m ^0 e
For no electrons we would have
B tanhe , ^ ,
= P = - -— — (14.71)
0oVe/V
646
BELL SYSTEM TECHNICAL JOURNAL
Thus, if we wish we may write (14.70) in the form
tanh^
where
P. = _^^!^ _ p (14.72)
Pe = (l/0)[(ei/e)i/2 tanh [(ei/e)!/^ d -tanh 6] (14.73)
The quantities on the right of (14.72) refer to the circuit in the absence of
electrons; if there are no electrons P« = 0 and (14.72) yields the circuit
0.1
//
r /
/
•
Fig. 14.8 — Suppose we compare the circuit admittance for the structure of Fig. 14.1
with that used in earlier calculations. Here the solid curve is proportional to the difference
of the Hi's for the finned structure and for the central space (the impressed current) di-
vided by £, . The dashed curve is the simple expression (6.1) used earlier fitted in mag-
nitude and slope.
waves. Thus, P, may be regarded as the equivalent of an added current *
at the wall, such that
^4-=^Ve/.P.
(14.74)
Now, the root giving the increasing wave, the one we are most interested
in, occurs a little way from the pole, where (ei/e)^'^ may be reasonably
large if Q is large. It would seem that one of the best comparisons which
could be made would be that between the approximate analysis and a very
broad beam case, for which B is very large. In this case, we may take ap-
proximately, away from 6 = 6,
(14.75)
tanh [(€i/€)i/2 6] = tanh 6 = 1
Pe = iMe)[{e,/eyi-^ - 1
A
Pe = (1/6)
1 -
(6. - ey
m
(14.76)
FIELD SOLUTIONS 647
Let us expand in terms of the quantity A /{Be — OY, assuming this to be
small compared with unity. We obtain
Pe =
1 + 77~-7^. +
20(0e - eyi 4(de - d)
(14.77)
The theory of Chapter VII is developed by assuming that all electrons
are acted on by the same a-c field. When this is not so, it is applied approxi-
mately by using an "effective current" or "effective field" as in Chapter
IV; either of these concepts leads to the same averaging over the electron
flow. An effective current can be obtained by averaging over the flow the
current density times the square of the field, evaluated in the absence of
electrons, and dividing by the square of the field at the reference position.
This is equivalent to the method used in evaluating the effective field in
Chapter III.
In the device of Fig. 14.2, if we take as a reference position y = ±d,
the effective current /o per unit depth
/o =
Jo / cosh^ (yy) dy
(14.78)
cosh- yd
{Jd/2) ("^^ + sech^ yd) (14.79)
This is the effective current associated with the half of the flow from y —
0 to y = d. Here y is the value for no electrons. For 0 « |8, 7 = ^. For
large values of 6, then
/o = Jod/2d (14.80)
Now, the corresponding a-c convection current per unit depth will be:
Here E is the total field acting on the electrons in the 2-direction. From
(7.1) we see that we assumed this to be the field due to the circuit (the first
term in the brackets) plus a quantity which we can write
£a = ^ i (14.82)
Accordingly
E = E,-\- £,1 (14.83)
648 BELL SYSTEM TECHNICAL JOURNAL
and we can write i
i = -J
^^ (-- + !/) '-•«^)
jh Be dEz
' 2Vo[K-{d,-dY] ^^^'^^^
Here i^ is a parameter specifying the value of /3VcoCi . As (14.85) need
hold over only a rather small range of /3, and C is not independent of /3,
we will regard K as a constant.
The parameter P, corresponding to (14.85) is
Pe =
[K - (de - dYY
2\/e/M Vo
Now, from (14.80), for large values of 9
h d{de/do) ^ Jo d^jde/do)
(14.86)
(14.87)
As
and
dg/da = c/ua ,
A =
Pe =
_Jo£_
2eUo Vo
A
2d[K - (de - ey]
Let us now expand (14.88) assuming A' to be very small
P. =
1 +
K
If we let
2eide - eyi ' {e^ - ey
K = A/4
+
(14.12)
(14.88)
(14.89)
(14.90)
we see that these hrst two terms agree witli the expansion of the broad-
beam expression, (14.77). The leading term was not adjusted; the space-
charge parameter K was, since there is no other way of evaluating the
parameter in this case.
In Fig. 14.9, the value of dPg as obtained, actually, from (14.73) rather
than (14.76), is plotted as a solid line and the value corresponding to the
FIELD SOLUTIONS
649
earlier theory, from (14.86) with K adjusted according to (14.88), is plotted
as a dashed Hne, for
A = 0.01
We see that (14.88), which involves the approximations made in our earlier
calculations concerning traveling-wave tubes, is a remarkably good fit to the
broad-beam expression derived from field theory up very close to the points
{de — d) = A, which are the boundaries between real and imaginary argu-
ments of the hyperbolic tangent and correspond to the points where the
ordinate is zero in Fig. 14.5.
Fig. 14.9 — These curves compare an exact electronic susceptance for the broad beam
case (solid curve) with the approximate expression used earlier (dashed curve). In the
approximate expression, the "effective current" was evaluated, not fitted; the space-
charge parameter was chosen to give a fit.
Over the range in which the argument of the hyperbolic tangent in the
correct expression is imaginary, the approximate expression of course ex-
hibits none of the complex behavior characteristics of the correct expression
and illustrated by Fig. 14.6. From (14.88) we see that the multiple excursions
of the true curve from — oo to + «2 are replaced in the approximate curve by
a single dip down toward 0 and back up again. R. C. Fletcher has used a
method similar to that explained above in computing the effective helix
impedance and the effective space-charge parameter Q for a solid beam inside
of a helically conducting sheet. His work, which is valuable in calculating
the gain of traveling-wave tubes, is reproduced in Appendix VI.
14.5c The Complex Roots
The propagation constants represent intersections of a circuit curve such
as that shown in Fig. 14.8 and an electronic curve such as that shown in Fig.
650
BELL SYSTEM TECHNICAL JOURNAL
14.9. The propagation constants obtained in Chapters II and VIII represent
such intersections of approximate circuit and electronic curves, such as the
dotted lines of Fig. 14.8 and 14.9. Propagation constants obtained by field
solutions represent intersections of the more nearly exact circuit and elec-
tronic curves such as the solid curves of Figs. 14.8 and 14.9.
If we plot a circuit curve giving
as given by (14.65) (the right-hand side of 14.75) and an electronic curve
giving
0.4
0.2
-0.2
i y
\
ee\ /
-yr
2
C
) t 2
Fig. 14.10 — The curves of Fig. 14.5 may be replaced by those of Fig. 14.6. Here the
curve which is concave upward represents the circuit susceptance and the other curve
represents the electronic susceptance (as in Fig. 14.9).
as given by (14.73) (the left-hand side of (14.72)), the plot, which is shown
in Fig. 14.10, contains the same information as the plot of Fig. 14.5 for which
00 , Q» and A are the same. In Fig. 14.10, however, one curve represents the
circuit without electrons and the other represents the added effect of the
electrons.
We have seen that the approximate expressions of Chapter VII fit the
broad-beam curves well for real propagation constants (real values of Q)
(Fig. 14.8 and 14.9). Hence, we expect that complex roots corresponding to
the increasing waves which are obtained using the approximate expressions
will be quite accurate when the circuit curve is not too far from the electronic
curve for real values of Q\ that is, when the parameters (electron velocity,
for instance) do not differ too much from those values for which the circuit
curve is tangent to the electronic curve.
Unfortunately, the behavior of a function for values of the variables far
FIELD SOLUTIONS
651
from those represented by its intersection with the real plane may be very
sensitive to the shape of the intersection with the real plane. Thus, we would
scarcely be justified by the good fit of the approximations represented in
Figs. 14.8 and 14.9 in assuming that the complex roots obtained using the
approximations will be good except when they correspond to a near approach
of the electronic and circuit curves, as in Fig. 14.10.
In fact, using the approximate curves, we find that the increasing wave
vanishes for electron velocities less than a certain lower limiting velocity.
This corresponds to cutting by the circuit curve of the dip down from -\- oo
of the approximate electronic curve (the dip is not shown in Fig. 14.9).
This is not characteristic of the true solution. An analysis shows, however.
^
/
/
e=dy_
0.5
-1.0 -0.5 0 0.5 1.0
P
Fig. 14.11— Complex roots are obtained when curves such as those of Fig. 14.10 do not
have the number of intersections required (by the degree of the equation) for real values
of the abscissa and ordinate. In this figure, two parabolas narrowly miss intersect ng.
Suppose these represent circuit and electronic susceptance curves. We find that the gain
of the increasing wave will increase with the square root of the separation at the abscissa
of equal slopes, and inversely as the square root of the difference in second derivatives.
that there will be a limiting electron velocity below which there is no in-
creasing wave if there is a charge-free region between the electron flow and
the circuit.
14.6 Some Remarks About Complex Roots
If we examine our generalized circuit expression (14.60) we see that the
circuit impedance parameter {E^/fi-P) is inversely proportional to the slope
of the circuit curve at the point where it crosses the horizontal axis. Thus,
low-impedance circuits cut the axis steeply and high-impedance circuits cut
the axis at a small slope.
We cannot go directly from this information to an evaluation of gain in
terms of impedance; the best course in this respect is to use the methods of
652 BELL SYSTEM TECHNICAL JOURNAL
Chapter VIII. We can, however, show a relation between gain and the
properties of the circuit and electronic curves for cases in which the curves
almost touch (an electron velocity just a little lower than that for which gain
appears). Suppose the curves nearly touch at 0 = 0i , as indicated in Fig.
14.11. Let
e = e^-\- p (14.91)
Let us represent the curves for small values of p by the first three terms of a
Taylor's series. Let the ordinate y of the circuit curve be given by
y=a, + hp^ cip^ (14.92)
and let the ordinate of the electronic curve be given by
y = (h+b2p-\- C2f (14.93)
Then, at the intersection
(ci - C2)/»2 + (bi - b2)p + (a, - oa) = 0
If we choose dx as the point at which the slopes are the same
bi- b2 = 0 (14.95)
and we see that the imaginary part of p increases with the square root of the
separation, and at a rate inversely proportional to the difference in second
derivatives. This is exemplified by the behavior of Xi and X2 for b a little small
than (3/2)(2)i/» in Fig. 8.1.
Now, referring to Fig. 14.10, we see that a circuit curve which cuts the
axis at a shallow angle (a high-impedance circuit curve) will approach or be
tangent to the electronic curve at a point where the second derivative is
small, while a steep (low impedance) circuit curve will approach the elec-
tronic curve at a point where the second derivative is high. This fits in with
the idea that a high impedance should give a high gain and a low impedance
should give a low gain.
CHAPTER XV
MAGNETRON AMPLIFIER
Synopsis of Chapter
^TpHE HIGH EFFICIENCY of the magnetron oscillator is attributed to
-■■ motion of the electrons toward the anode (toward a region of higher
d-c potential) at high r-f levels. Thus, an electron's loss of energy to the r-f
field is made up, not by a slowing-down of its motion in the direction of wave
propagation, but by abstraction of energy from the d-c field. ^
Warnecke and Guenard^ have published pictures of magnetron amphfiers
and Brossart and Doehler have discussed the theory of such devices.^
No attempt will be made here to analyze the large-signal behavior of a
magnetron amplifier or even to treat the small-signal theory extensively.
However, as the device is very closely related to conventional traveling-
wave tubes, it seems of some interest to illustrate its operation by a simple
small-signal analysis.
The case analyzed is indicated in Fig. 15.1. A narrow beam of electrons
flows in the -\-z direction, constituting a current /o . There is a magnetic
field of strength B normal to the plane of the paper (in the x direction), and
a d-c electric field in the y direction. The beam flows near to a circuit which
propagates a slow wave. Fig. 15.3, which shows a finned structure opposed
to a conducting plane and held positive with respect to it, gives an idea of
a physical realization of such a device. The electron stream could come from
a cathode held at some potential intermediate between that of the finned
structure and that of the plane. In any event, in the analysis the electrons
are assumed to have such an initial d-c velocity and direction as to make
them travel in a straight line, the magnetic and electric forces just cancelling.
The circuit equation developed in Chapter XIII in connection with trans-
verse motions of electrons is used. Together with an appropriate ballistical
equation, this leads to a fifth degree equation for F,
1 For an understanding of the high-level behavior of magnetrons the reader is referred to:
J. B. Fisk, H. D. Hagstrum and P. L. Hartman, "The Magnetron as a Generator of
Centimeter Waves," Bell System Technical Journal, Vol. XXV, April 1946.
"Microwave Magnetrons" edited by George B. Collins, McGraw-Hill, 1948.
^ R. Warnecke and P. Guenard, "Sur L'Aide Que Peuvent Apporter en Television Quel-
ques Recentes Conceptions Concernant Les Tubes Electroniques Pour Ultra-Hautes
Frequences," Annales de Radioelectricite, Vol. Ill, pp. 259-280, October 1948.
^ J. Brossart and O. Doehler, "Sur les Proprietes des Tubes a Champ Magnetique Con-
stant: les Tubes a Propagation D'Onde a Champ Magnetique," Annales de Radioelectricite,
Vol. Ill, pp. 328-338, October 1948.
653
654 BELL SYSTEM TECHNICAL JOURNAL
The nature of this equation indicates that gain may be possible in two
ranges of parameters. One is that in which the electron velocity is near to
or equal to (as, (15.25)) the circuit phase velocity. In this case there is gain
provided that the transverse component of a-c electric field is not zero, and
provided that it is related to the longitudinal component as it is for the
circuit of Fig. 15.3. It seems likely that this corresponds most nearly to usual
magnetron operation.
The other interesting range of parameters is that near
;8,//3i = 1 - /3„M (15.31)
Here /3g refers to the electrons, /8i to the circuit and /3« is the cyclotron fre-
quency divided by the electron velocity. When (15.31) holds, there is gain
whenever the parameter a, which specifies the ratio of the transverse to the
longitudinal fields, is not -f 1. For the circuit of Fig. 15.3, a approaccs +1
near the fins if the separation between the fins and the plane is great enough
in terms of the wavelength. However, a can be made negative near the fins
""llllillillll"^^
__MM
lo
Fig. 15.1 — In a magnetron amplifier a narrow electron stream travels in crossed electric
and magnetic fields close to a wave transmission circuit.
if the potential of the fins is made negative compared with that of the plane,
and the electrons are made to move in the opposite direction.
In either range of parameters, the gain of the increasing wave in db per
wavelength is proportional to the square root of the current rather than
to the cube root of the current. This means a lower gain than for an ordinary
traveling-wave tube with the same circuit and current.
Increasing and decreasing waves with a negative phase velocity are pos-
sible when the magnetic field is great enough.
15.1 Circuit Equation
The circuit equation will be the same as that used in Chapter XIII, that is,
T/ - -icoriJC(^p - {h/u,Wy) ,
(r2 _ Y\) ^^-^-^"^
It will be assumed that the voltage is given by
$ = {Ae~'''' + Be''^") (15.1)
so that
^'V = -jViAe''^" - Be'^"") (15.2)
MAGNETRON AMPLIFIER
655
At any y — position we can write
Ad
-jTa^V
-jTv
- Be
jVv
Ae-'^« + Be'^v
(15.3)
(15.4)
If r is purely imaginary, a is purely real, and as F will have only a small real
component, a will be considered as a real number. We see that a can range
from + CO to — =0 . For instance, consider a circuit consisting of opposed two-
dimensional slotted members as shown in Fig. 15.2. For a field with a cosh
distribution in the y direction, a is positive above the axis, zero on the axis
and negative below the axis. For a field having a sinh distribution in the y
////
Fig. 15.2 — K the circuit is as shown, the ratio between longitudinal and transverse field
will be different in sign above and below the axis. This can have an important effect on th
operation of the ampUfier.
direction, <x is infinite on the axis, positive above the axis and negative below
the axis.
We find then, that, (13.10) becomes
V =
-jcoTi^Kip + i«(/o/wo)ry)
p2 _ p2
15.2 Ballistic Equations
The d-c electric field in the y direction will be taken as —Eq. Thus
dy
(15.5)
dt
= V
£, + '-^ - B(z + „.)]
In order to maintain a rectilinear unperturbed path
£e = Buo
so that (15.6) becomes
dy ^ d(^V) _ ^^z
dt dy
(15.6)
(15.7)
(15.8)
656 BELL SYSTEM TECHNICAL JOURNAL
Following the usual procedure, we obtain
uoij% - r)
Ki->}.yj
We have also
dz d^V , „.
dt-'^ ex ^ '^y
. _ -7?r$F + -nBy
(15.10)
uoWe - r)
From (15.9) and (15.10) we obtain
„ -vT^V[{j% -T) + ja^J
(15.11)
where
/3m = (Om/Uo
(15.12)
Wm = VB
(15.13)
Here o)m is the cyclotron radian frequency.
As before, we have
rpo2
uoij^e - r)
whence
We can also solve (15.9) and (15.10) for y
Now, to the first order
dy , dy
y =
MJ^e - r)
and from (15.16) and (15.17)
(15.14)
T%h^V[(j^e - r) + ja(3j
(15.17)
-j7?rj>F[a(i/3, - r)+j/3j .^.
MAGNETRON AMPLIFIER 657
If we use (15.15) and (15.18) in connection with (15.5) we obtain
2 o -j^eVxVme - r) + 2j[a/{\ + a'')\^^\H^
1 — 1 1 —
Now let
If we assume
^ « 1 (15.23)
and neglect p in sums in comparison with unity, we obtain
/>(/3.//3l - 1 - pWe/^l -\- pf - (/3n.M)']
C;-/3. - r)[C;-/3« - D' + ^l]
(15.19)
2 (1 + a')«i>'X/o
2Fo
(15.20)
-ri = -i^i
(15.21)
-r = -;A(1 + P)
(15.22)
= _l [(,./,. _._„+_^]^.
(15.24)
We are particularly interested in conditions which lead to an imaginary
value of p which is as large as possible. We will obtain such large values of p
when one of the factors multiplying p on the left-hand side of (15.24) is
small. There are two possibilities. One is that the first factor is small. We
explore this by assuming
^,//3i -1 = 0 (15.25)
If p is very small, we can write approximately
■ ^\ (1 + a-) ^1 (15.27)
P = ±j[a/{l + a2)]i/2(^^/^Ji/2^
We see that p goes to zero if a = 0 and is real if a is negative. If we con-
sider what this means circuit-wise, we see that there will be gain with the
d-c voltage applied between a circuit and a conducting plane as shown in
Fig. 15.3.
Another possible condition in the neighborhood of which p is relatively
large is
^,M - 1 = ± iSj^i (15.28)
658
BELL SYSTEM TECHNICAL JOURNAL
In this case
-"^g^..
(*^"/^' - ^) + 0^]
H'
As pis small, we write approximately
(15.29)
(15.30)
We see that we obtain an imaginary value of p only for the — sign in (15.28)
that is, if
/3e//3i = 1 - ^V/5. (15.31)
Fig. 15.3 — The usual arrangement is to have the finned structure positive and opposed
to a conducting plane.
In this case
p = ±;M(1 - a)/(l + ay"m/^j"'H.
(15.32)
In this case we obtain gain for any value of a smaller than unity. We note
that a = 1 is the value a assumes far from the axis in a two-dimensional
system of the sort illustrated in Fig. 15.2, for either a cosh or a sinh distribu-
tion in the -{-y direction.
The assumption of — T = —j0i{l + p) in (15.22) will give forward (-{-z)
traveling-waves only. In order to investigate backward traveling-waves, we
must assume
-r = +jm-\-p)
(15.33)
where again p is considered a small number. If we use this in (15.19), we
obtain
+ 1 + /^
[(j.
+ 1 + H --I
311
1^
'2/31
j^'-'V^i
_2a/3m 1
1 + a')^i]
(15.34)
H'
MAGNETRON AMPLIFIER 659
As before we look for solutions for p where the terms multiplying p on the
left are small. The only vanishing consistent with positive values of jS« and
/3i is obtained for
^'+l = +§^. (15.35)
Under this condition (15.34) yields for p
,.1 (l + «) (P,^^ ,.,-..
Thus we can obtain backward-increasing backward-traveling waves for all
values of a except a = — 1. For the situation shown in Fig. 15.3, with a
backward wave, a is always negative, approaching —1 at large distances
from the plane electrode, so that the gain is identical with that given by
(15.32).
We note that (15.27), (15.32) and (15.36) show that p is proportional to
the product of current times impedance divided by voltage to the f power,
while, in the case of the usual traveling-wave tube, this small quantity
occurs to the \ power. The \ power of a small quantity is larger than the
\ power; and, hence for a given circuit impedance, current and voltage, the
gain of the magnetron amplifier will be somewhat less than the gain of a
conventional traveling-wave tube.
CHAPTER XVI
DOUBLE-STREAM AMPLIFIERS
Synopsis of Chapter
IN TRAVELING-WAVE TUBES, it is desirable to have the electrons
flow very close to the metal circuit elements, where the radio-frequency
field of the circuit is strong, in order to obtain satisfactory amplification.
It is, however, difficult to confine the electron flow close to metal circuit
elements without an interception of electrons, which entails both loss of
efficiency and heating of the circuit elements. This latter may be extremely
objectionable at very short wavelengths for which circuit elements are small
and fragile.
In the double-stream amplifier the gain is not obtained through the inter-
action of electrons with the field of electromagnetic resonators, helices or
other circuits. Instead, an electron flow consisting of two streams of elec-
trons having different average velocities is used. When the currents or charge
densities of the two streams are sufficient, the streams interact so as to give
an increasing wave.^'^'^'^ Electromagnetic circuits may be used to impress
a signal on the electron flow, or to produce an electromagnetic output by
means of the amplified signal present in the electron flow. The amplification,
however, takes place in the electron flow itself, and is the result of what
may be termed an electromechanical interaction.^
While small magnetic fields are necessarily present because of the motions
of the electrons, these do not play an important part in the amplification.
The important factors in the interaction are the electric field, which stores
energy and acts on the electrons, and the electrons themselves. The charge of
the electrons produces the electric field; the mass of the electrons, and their
kinetic energy, serve much as do inductance and magnetic stored energy in
electromagnetic propagation.
1 J. R. Pierce and W. B. Hebenstreit, "A New Type of High-Frequency Amplifier,"
B.S.TJ., Vol. 28, pp. 33-51, January 1949.
' A. V. Hollenberg, "Experimental Observation of Amplification by Interaction be-
tween Two Electron Streams," B.S.TJ., Vol. 28, pp. 52-58, January 1949.
* A. V. Haeff, "The Electron-Wave Tube — A Novel Method of Generation and Ampli-
fication of Microwave Energy," Proc. IRE, Vol. 37, pp. 4-10, January 1949.
*L. S. Nergaard, "Analysis of a Simple Model of a Two-Beam Growing- Wave Tube,"
R.C.A. Renew, Vol. 9, pp. 585-601, December 1948
' Some similar electromechanical waves are described in papers by J. R. Pierce, "Pos-
sible Fluctuations in Electron Streams Due to Ions," Jour. App. Phys., Vol. 19, pp. 231-
236, March 1948, and "Increasing Space-Charge Waves," Jour. App. Phys., Vol. 20
pp. 1060-1066; Nov. 1949.
660
DOUBLE-STREAM AMPUFIERS 661
By this sort of interaction, a traveling wave which increases as it travels,
i.e., a traveling wave of negative attenuation, may be produced. To start
such a wave, the electron flow may be made to pass through a resonator or a
short length of helix excited by the input signal. Once initiated, the wave
grows exponentially in amplitude until the electron flow is terminated or
until non-linearities limit the amplitude. An amplified output can be ob-
tained by allowing the electron flow to act on a resonantor, helix or other
output circuit at a point far enough removed from the input circuit to give
the desired gain.
In general, for a given geometry there is a limiting value of current below
which there is no increasing wave. For completely intermingled electron
streams, the gain rises toward an asymptotic hmit as the current is increased
beyond this value. The ordinate of Fig. 16.3 is proportional to gain and the
abscissa to current.
When the electron streams are separated, the gain first rises and then falls
as the current is increased. This effect, and also the magnitude of the in-
creasing wave set up by velocity modulating the electron streams, have been
discussed in the Hterature.®
Double-stream amplifiers have several advantages. Because the electrons
interact with one another, the electron flow need not pass extremely close to
complicated circuit elements. This is particularly advantageous at very
short wavelengths. Further, if we make the distance of electron flow between
the input and output circuits long enough, amplification can be obtained
even though the input and output circuits have very low impedance or poor
coupling to the electron flow. Even though the region of amplification is
long, there is no need to maintain a close synchronism between an electron
velocity and a circuit wave velocity, as there is in the usual traveling-
wave tube.
16.1 Simple Theory of Double-Stream Amplifiers
For simplicity we will assume that the flow consists of coincident streams
of electrons of d-c velocities % and U2 in the z direction. It will be assumed
that there is no electron motion normal to the z direction. M.K.S. units will
be used.
It turns out to be convenient to express variation in the z direction as
exp -j^z
rather than as
exp —Vz
6 J. R. Pierce, "Double-Stream Amplifiers," Proc. I.R.E., Vol. 37, pp. 980-985, Sept.
1949.
662 BELL SYSTEM TECHNICAL JOURNAL
as we have done previously. This merely means letting
r =i/3 (16 1)
The following nomenclature will be used
Ji , Ji d-c current densities
Ml , Ms d-c velocities
Poi , Po2 d-c charge densities
POI = —J\/U\ , Pj2 = —Jllu-i.
Pi , p2 a-c charge densities
I'l , Vi a-c velocities
Vi , Fi d-c voltages with respect to the cathodes
V a-c potential
^1 = Cj/Wi , ^2 = w/M2
From (2.22) and (2.18) we obtain
and
(16.3)
It will be convenient to call the fractional velocity separation b, so that
«i -f Ma
It will also be convenient to define a sort of mean velocity Mq
u, = J^. (16.5)
«1 -H Mj
We may also let Vt be the potential drop specifying a velocity uo , so that
Mo = \/27jFo (16.6)
It is further convenient to define a phase constant based on uo
CO
(16.7)
We see from (16.4), (16.5) and (16.6) that
)8i = /3.(1 - b/2) (16.8)
0i = /3.(1 + ^>/2) (16.9)
DOUBLE-STREAM AMPLIFIERS
663
We shall treat only a special case, that in which
Ji Ji
/o
3 — 3 — 3 '
U\ «2 Wo
(16.10)
Here /o is a sort of mean current which, together with wo , specifies the ratios
J\.lu\ and J-Jui^ which appear in (4) and (5).
In terms of these new quantities, the expression for the total a-c charge
density p is, from (16.2) and (16.3) and (16.6)
d2
P = Pi + P2 =
/or
2 Mo Fo
1
1
-|
r /
h\
-
2 +
['■(-0
2
/3a (^1
V
-iS
- /3'
(16.11)
Equation (16.11) is a ballistic equation telling what charge density p is
produced when the flow is bunched by a voltage V. To solve our problem,
that is, to solve for the phase constant /3, we must associate (16.11) with a
circuit equation which tells us what voltage V the charge density produces.
We assume that the electron flow takes place in a tube too narrow to propa-
gate a wave of the frequency considered. Further, we assume that the wave
velocity is much smaller than the velocity of light. Under these circumstances
the circuit problem is essentially an electrostatic problem. The a-c voltage
will be of the same sign as, and in phase with the a-c charge density p. In
other words the "circuit effect" is purely capacitive.
Let us assume at first that the electron stream is very narrow compared
with the tube through which it flows, so that V may be assumed to be con-
stant over its cross section. We can easily obtain the relation between V
and p in two extreme cases. If the wavelength in the stream is very short
(J3 large), so that transverse a-c fields are negligible, then, from Poisson's
equation, we have
^z (16.12)
p = e^W
If, on the other hand, the wavelength is long compared with the tube radius
(S small) so that the fields are chiefly transverse, the lines of force running
from the beam outward lo the surrounding tube, we may write
= CV
(16.13)
Here C is a constant expressing the capacitance per unit length between the
region occupied by the electron flow and the tube wall.
664
BELL SYSTEM TECHNICAL JOURNAL
We see from (16.12) and (16.13) that, if we plot p/V vs. ^/^e for real values
of |S, p/V will be constant for small values of ^ and will rise as /S^ for large
values of ^, approximately as shown in Fig. 16.1.
Now, we have assumed that the charge is produced by the action of the
voltage, according to the baUistical equation (16.11). This relation is plotted
in Fig. 2, for a relatively large value of Jo/utsV^ (curve 1) and for a smaller
value of /o/«o^o (curve 2). There are poles at jS//3, = 1
- , and a minimum
between the poles. The height of the minimum increases as J^lu{/Vt is in-
creased.
A circuit curve similar to that of Fig. 16.1 is also plotted on Fig. 16.2.
We see that for the small-current case (curve 2) there are four intersections,
giving /oMr real values of /3 and hence /owr unattenuated waves. However, for
1
^^0—
Fig. 16.1 — Circuit curves, in which the ordinate is proportional to the ratio of the charge
per unit length to the voltage which it produces. Curve 1 is for an infinitely broad beam;
curve 2 is for a narrow beam in a narrow tube. Curve 3 is the sum of 1 and 2, and approxi-
mates an actual curve.
the larger current (curve 1) there are only two intersections and hence two
unattenuated waves. The two additional values of ^ satisfying both the
circuit equation and the baUistical equation are complex conjugates, and
represent waves traveling at the same speed, but with equal positive nega-
tive attenuations.
Thus we deduce that, as the current densities in the electron streams are
raised, a wave with negative attenuation appears for current densities above
a certain critical value.
We can learn a little more about these waves by assuming an approximate
expression for the circuit curve of Fig. 1. Let us merely assume that over
the range of interest (near /3/i8, = 1) we can use
P = a'i^W
(16.14)
DOUBLE-STREAM AMPLIFIERS
665
the
s a^ IS a factor greater than unity, which merely expresses the fact that
charge density corresponding to a given voltage is somewhat greater
man if there were field in the z direction only for which equation (16.12) is
valid. Combining (16.14) with (16.11) we obtain
(-(-0-)"(-(-O-)
^eU-
(16.15)
Fig. 16.2 — This shows a circuit curve, 3, and two electronic curves which give the
sum of the charge densities of the two streams divided by the voltage which bunches them.
With curve 2, there will be four unattenuated waves. With curve 1, which is for a higher
current density than curve 2, there are two unattenuated waves, an increasing wave and
a decreasing wave.
where
W =
/o
2a ejSeWoFo'
(16.16)
In solving (16.15) it is most convenient to represent (i in terms of )3« and
a new variable h
^ = /3,(1 + h)
Thus, (16.15) becomes
i'-th'i)
1^
(16.17)
(16.18)
666
BELL SYSTEM TECHNICAL JOURNAL
Solving for h, we obtain
" = ±1 2-
f^-ff
)/(?F^r- <-->
The positive sign inside of the brackets always gives a real value of h
and hence unattenuated waves. The negative sign inside the brackets gives
unattenuated waves for small values of U/b. However, when
©
?v > I
(16.20)
1 2 4 6 8 10 20 40 60 80 100 200 400 600 1000
Fig. 16.3 — The abscissa is proportional to d-c current. As the current is increased, the
gain in db per wavelength approaches 27.3b, where h is the fractional separation in ve-
locity. If the two electron streams are separated physically, the gain is lower and first rises
and then falls as the current is increased.
there are two waves with a phase constant /3« and with equal and opposite
attenuation constants.
Suppose we let U m be the minimum value of U for which there is gain.
From (16.20)
U\ = &V8 (16.21)
From (16.19) we have, for the increasing wave,
1112
(16.22)
DOUBLE-STREAM AMPUFIERS 667
The gain in db/wavelength is
db/wavelength = 20(27r)logioe"''
= 54.6 1 h I (16.23)
We see that, by means of (16.22) and (16.23), we can plot db/wavelength
per unit b vs. {U/U mY- This is plotted in Fig. 16.3. Because W is propor-
tional to current, the variable (U/UmY is the ratio of the actual current to
the current which will just give an increasing wave. If we know this ratio,
we can obtain the gain in db/wavelength by multiplying the corresponding
ordinate from Fig. 16.3 by b.
We see that, as the current is increased, the gain per wavelength at first
rises rapidly and then rises more slowly, approaching a value 27.36 db/wave-
length for very large values of (U/UmY-
We now have some idea of the variation of gain per wavelength with
velocity separation b and with current (U/UmY- A more complete theory
requires the evaluation of the lower limiting current for gain (or of Uu) in
terms of physical dimensions and an investigation of the boundary conditions
to show how strong an increasing wave is set up by a given input signal.^- '
16,2 Further Considerations
There are a number of points to be brought out concerning double-stream
amplifiers. Analysis shows® that any physical separation of the electron
streams has a very serious effect in reducing gain. Thus, it is desirable to
intermingle the streams thoroughly if possible.
If the electron streams have a fractional velocity spread due to space
charge which is comparable with the deliberately imposed spread b, we may
expect a reduction in gain.
Haeff* describes a single-stream tube and attributes its gain to the space-
charge spread in velocities. In his analysis of this tube he divides the beam
into a high and a low velocity portion, and assigns the mean velocity to
each. This is not a valid approximation.
Analysis indicates that a multiply-peaked distribution of current with
velocity is necessary for the existent increasing waves, and gain in a "single
stream" of electrons is still something of a mystery.
CHAPTER XVII
CONCLUSION
ALTHOUGH THIS BOOK contains some descriptive material con-
cerning high-level behavior, it is primarily a treatment of the linearized
or low-level behavior of traveling-wave tubes and of some related devices.
In the case of traveling-wave tubes with longitudinal motion of electrons
only, the treatment is fairly extended. In the discussions of transverse fields,
magnetron amplifiers and double-stream amplifiers, it amounts to little
more than an introduction.
One problem to which the material presented lends itself is the calcula-
tion of gain of longitudinal-field traveling-wave tubes. To this end, a sum-
mary of gain calculation is included as Appendix VII.
Further design information can be worked out as, for instance, exact gain
curves at low gain with lumped or distributed loss, perhaps taking the space-
charge parameter QC into account, or. a more extended analysis concerning
noise figure.
The material in the book may be regarded from another point of view as
an introduction, through the treatment of what are really very simple cases,
to the high-frequency electronics of electron streams. That is, the reader may
use the book merely to learn how to tackle new problems. There are many
of these.
One serious problem is that of extending the non-linear theory of the
traveling-wave tube. For one thing, it would be desirable to include the
effects of loss and space charge. Certainly, a matter worthy of careful in-
vestigation is the possibility of increasing efficiency by the use of a circuit
in which the phase velocity decreases near the output end. Nordsieck's work
can be a guide in such endeavors.
Even linear theory excluding the effects of thermal velocities could profit-
ably be extended, especially to disclose the comparative behavior of narrow
electron beams and of broad beams, both those confined by a magnetic field,
in which transverse d-c velocities are negligible and in which space charge
causes a lowering of axial velocity toward the center of the beam, and also
those in which transverse a-c velocities are allowed, especially the Ikillouin-
type flow, in which the d-c axial velocity is constant across the beam, but
electrons have an angular velocity proportional to radius.
Further problems include the extension of the theory of magnetron ampli-
fiers and of double-stream amplifiers to a scope comparable with that of the
668
CONCLUSION 669
theory of conventional traveling-wave tubes. The question of velocity dis-
tribution across the beam is particularly important in double-stream ampli-
fiers, whose very operation depends on such a distribution, and it is important
that the properties of various kinds of distribution be investigated.
Finally, there is no reason to suspect that the simple tubes described do
not have undiscovered relatives of considerable value. Perhaps diligent work
will uncover them.
BIBLIOGRAPHY
1946
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Blanc-Lapierrc, A. and Lapostolle, P. Contribution a I'etude des amplificateurs a ondes
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Kompfner, R. Traveling wave valve — new amplifier for centimetric wavelengths. Wireless
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Pierce, J. R. Beam traveling-wave tube. Bell Lab. Record, v. 24, pp. 439-442, Dec, 1946.
1947
Bernier, J. Essai de theorie du tube 61ectronique a propagation d'onde, Ann. de Radioelec,
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Kompfner, R. Traveling-wave tuge as amplifier at microwaves, I.R.E., Proc, v. 35, pp.
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Kompfner, R. Traveling-wave tube — centimetre-wave amplifier. Wireless Engr., v. 24,
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Pierce, J. R. Theory of the beam-type traveling-wave tube, I.R.E., Proc, v. 35, pp. 111-
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Pierce, J. R. and Field, L. M. Traveling-wave tubes. I.R.E., Proc, v. 35, pp. 108-111,
Feb., 1947.
Roubine. E. Sur le circuit k h61ice utilis6 dans le tube k ondes progressives, Onde Elec,
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Shulman, C. and Heagy, M. S. Small-signal analysis of traveUng-wave tube, R.C.A. Rev.,
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1948
Brillouin, L. Wave and electrons traveling together — a comparison between traveling
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Brossart, J. and Doehler, O. Sur les propri^tes des tubes a champ magnetique constant.
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Cutler, C. C. Experimental determination of helical-wave properties, I.R.E., Proc, v. 36,
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Chu, L. J. and Jackson, J. D. Field theory of traveling-wave tubes, I.R.E., Proc, v. 36,
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Doehler, O. and Kleen, W. Phenom^nes non lineaires dans les tubes a propagation d'onde.
Ann. de RadioSlec, v. 3, pp. 124-143, Apr., 1948.
Doehler, O. and Kleen, \V. Sur I'influence de la charge d'espace dans le tube a propaga-
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Blanc-Lapierre, A., Kuhner, M., Lapostolle, P., Jessel, M. and Wallauschek, R. Etude
et realisation d'amplificateurs a h6iice. Ann. des Telecomm., v. 3, pp. 257-308, Aug.-
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Lapostolle, P. Les i)h6nonuncs d'interaction dans le tube a onde progressive, Theorie et
verifications cxperimcntaies, pp. 265-291.
Jessel, ^L and Wallauschek, R. t^tude experimentaie de la propagation de long d'une
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Lapostolle, 1'. Istude des divcrses ondes susceptibles de se propager dans une ligne en
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BIBLIOGRAPHY 671
Pierce, J. R. Effect of passive modes in traveling wave tubes, I.R.E., Proc, v. 36, pp.
993-997, Aug., 1948.
Pierce, J. R. Transverse fields in traveling-wave tubes, Bdl Sys. Tech. JL, v. 27, pp. 732-
746, Oct., 1948.
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1948.
Tomner, J. S. A. Experimental development of traveling-wave tubes. Acta Polytech.,
Elec. Engg., v. 1, no. 6, pp. 1-21, 1948.
Nergaard, L. S. Analvsis of a simple model of a two-beam growing-wave tube, RCA Rev.
vol. 9, pp. 585-601, Dec. 1948.
1949
Doehler, O. and Kleen, W. Influence du vecteur electrique transversal dans la ligne a
retard du tube a propagation d'onde, Ann. de Radioelec, v. 4, pp. 76-84, Jan., 1949.
Bruck, L. Comparison des valeurs mesurees pour le gain lineaine du tube k propagation
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Field, L. M. Some slow-wave structures for traveling-wave tubes. I.R.E., Proc, v. 37,
pp. 34-40, Jan., 1949.
Guenard, P., Berterattiere, R. and Doehler, O. Amplification par interaction electronque
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1949.
Laplume, J. Th6orie du tube a onde progressive. Onde Elec, v. 29, pp. 66-72, Feb., 1949.
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N.E.C. (Chicago), v. 4, p. 253, 1948.
Pierce, J. R. and Hebenstreit, W. B. A new type of high-frequency amplifier, Bell System
Technical Journal, v. 28, pp. 33-51, January, 1949.
Haeff, A. V. The electron-wave tube — a novel method of generation and ampUfication of
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action in valves without circuits, A^m. Radioelec, v. 4, pp. 171-177, July, 1949.
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Field, L. M. Some slow wave structures for traveling-wave tubes, Proc. I.R.E., v. 37,
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V. 37, pp. 622-625, June, 1949.
Technical Publications by Bell System Authors Other Than
in the Bell System Technical Journal
Circuits for Cold Cathode Gloiv Tubes.* W. A. Depp^ and W. H. T. Holdex.'
Elec. Mfg., V. 44, pp. 92-97, July, 1949.
Equipment for the Determination of Insulation Resistance at High Humidi-
ties. A. T. Chapman.2 A.S.T.M. Bull., no. 165, pp. 43-45, Apr., 1950.
Twin Relationships in Ingots of Germanium. W. C. Ellis. ^ //. Metals, v.
188, p. 886, June, 1950.
Magnetic Cores of Thin Tape Insulated by Cata phoresis.* H. L. B. Gould. ^
Elec. Engg., v. 69, pp. 544-548, June, 1950.
Slip Markings in Chromium. E. S. Greiner.^ //. Metals, v. 188, pp.
891-892, June, 1950.
Xew Porcelain Rod Leak. H. D. Hagstrum^ and H. W. Weinhart.^ Rev.
Sci. Instruments, v. 21, p. 394, Apr., 1950.
Significance of Nonclassical Statistics. R. \. L. Hartley.^ Science, v. Ill,
pp. 574-576, May 26, 1950.
Comment on Mobility Anomalies in Germanium. G. L. Pearson,' J. R.
H.AYNEs' and W. Shockley.' Letter to the editor. Phys. Rev., v. 78, pp.
295-296, May 1, 1950.
Dislocation Models of Crystal Grain Boundaries. W. T. Re.ad' and W.
Shockley.i Phys. Rev., v. 78, pp. 275-289, May 1, 1950.
Zero-Point Vibrations and Superconductivity. J. Bardeen.' Letter to the
editor. Phys. Rev., v. 79, pp. 167-168, July 1, 1950.
Some Observations on Industrial Research. O. E. Buckley.' Bell Tel. Mag.,
V. 29, pp. 13-24, Spring, 1950.
Properties of Single Crystals of Xickel Ferrite. J. .K Galt,' B. T. Mathl\s'
and J. P. Remeika.' Letter to the editor. Phys. Rev., v. 79, pp. 391-392,
July 15, 1950.
Photon Yield of Electron-Hole Pairs in Germanium . F. S. (tOUCHEk.'
Letter to the editor. Phys. Rev., v. 78, p. 816, June 15, 1950.
Data on Porcelain Rod Leak J. P. Molx.ar' and C. D. Hartman.' Rev. Sci.
instruments, v. 21, pp. 394-395, Apr., 1950.
Magnetic Susceptibility of aFei(\ and a¥e-2P-i with Added Titanium. V. J.
Morix.' Letter to the e(Utor. Phys. Rev., v. 78, pp. 819-820, June 15, 1950.
* A reprinl of this arliclc may 1)C ol)tainecl on request to the editor of the B. S.T.J.
' B.T.L.
2 VV.E.CO.
672
ARTICLES BY BELL SYSTEM AUTHORS 673
Alternating Current Conduction in Ice. E. J. Murphy.' Letter to the editor.
Phys. Rev., v. 79, pp. 396-397, July 15, 1950.
Ferromagnetic Resonance in Manganese Ferrite and the Theory of the Fer-
rites. W. A. Yager,' F. R. Merritt,' C. Kittel' and C. Guillaud.' Letter
to the editor. Phys. Rev., v. 79, p. 181, July 1, 1950.
Conductivity Pulses Induced in Diamond by Alpha Particles. A. J. Ahearn.^
AEC, Brookhaven conference report, BNL-C-1 High speed counters and short
pulse techniques, Aug. 14-15, 1947. 1950. p. 7.
Behavior of Resistors at High Frequencies. G. R. Arthur' and S. E.
Church.' T. V. Engg., v. 1, pp. 4-7, June, 1950.
Note on ''The Application of Vector Analysis to the Wave Equation". R. V.
L. Hartley.' Letter to the editor. Acoustical Sac. Am. Jl., v. 22, p. 511,
July, 1950.
Number 5 Crossbar Dial Telephone Switching System.* F. A. Korn' and
James G. Ferguson.' Elec. Engg., v. 69, pp. 679-684, Aug., 1950.
Traveling-Wave Tube as a Broad Band Amplifier. J. R. Pierce.' AEC,
Brookhaven conference report, BNL-C-1 High speed counters and short pulse
techniques, Aug. 14-15, 1947. 1950. p. 41.
[* A reprint of this article may be obtained on request to the editor of the B.S.T.J.
' B.T.L.
Contributors to this Issue
John Bardeen, University of Wisconsin, B.S. in E.E., 1928; M.S., 1930.
(iulf ResearchandDevelopmentCorporation, 1930-33; Princeton University,
1933-35, Pli.D. in Math. Phys., 1936; Junior Fellow, Society of Fellows,
Harvard University, 1935-38; Assistant Professor of Physics, University of
Minnesota, 1938-41; Prin. Phys., Naval Ordnance Laboratory, 1941-45.
Bell Telephone Laboratories, 1945-. Dr. Bardeen is engaged in theoretical
problems related to semiconductors.
A. E. BowEN, Ph.B., Yale University, 1921; Graduate School, Yale Uni-
versity, 1921-24. American Telephone and Telegraph Company, Depart-
ment of Development and Research, 1924-34. Bell Telephone Laboratories,
1934-42. U. S. Army Air Force, 1942-45. Bell Telephone Laboratories,
1945-48. With the American Telephone and Telegraph Company, Mr.
Bowen's work was concerned principally with the inductive coordination of
power and communications systems. From 1934 to 1942 he was engaged in
work in the ultra-high-frequency field, particularly on hollow waveguides.
He became a Major and later a Colonel while serving with the U. S. Army
Air Force from 1942 to 1945 on a special mission to Trinidad and subse-
quently in the Pentagon. After returning to Bell Telephone Laboratories in
1945 he was engaged in the problems of microwave repeater research until
his death in 1948.
M. E. HiNES, B.S. in Applied Physics, California Institute of Technology,
1940; B.S. in Meteorology, 1941; M.S. in Electrical Engineering, 1946.
U. S. Air Force Weather Service, 1941-45. Bell Telephone Laboratories,
1946-. Mr. Hines has been engaged in the development of vacuum tubes.
Jack A. Morton, B.S. in Electrical Engineering, Wayne University,
1935; M.S.E., University of Michigan, 1936. Bell Telephone Laboratories,
1936-. Mr. Morton joined the Laboratories to work on coaxial cable and
microwave amplifier circuit research; during the war he was at first a member
of a group engaged in improving the signal-to-noise performance of radar
receivers. In 1943 he transferred to the Electronic Development Department
to work on microwave tubes for radar and radio relay. Since 1948 he has
been Electronic Apparatus Development Engineer responsible for the de-
velopment of transistors and other semiconductor devices.
William W. Mumford, B.A., Willamette University, 1930. Bell Tele-
phone Laboratories, 1930-. Mr. Mumford has been engaged in work that is
chiefly concerned with ultra-short-wave and microwave radio communica-
tion.
674
CONTRIBUTORS 675
J. R. Pierce, B.S. in Electrical Engineering, California Institute of Tech-
nology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936-. Dr. Pierce
has been engaged in the study of vacuum tubes.
Robert M. Ryder, Yale University, B.S. in Physics, 1937; Ph.D., 1940.
Bell Telephone Laboratories, 1940-. Dr. Ryder joined the Laboratories to
work on microwave amplifier circuits, and during most of the war was a
member of a group engaged in studying the signal-to-noise performance of
radars. In 1945 he transferred to the Electronic Development Department
to work on microwave oscillator and amplifier tubes for radar and radio
relay applications. He is now in a group engaged in the development of
transistors.
W. VAN RoosBROECK, A.B., Columbia College, 1934; A.M., Columbia
University, 1937. Bell Telephone Laboratories, 1937-. Mr. van Roos-
broeck's work at the Laboratories was concerned during the war with carbon-
film resistors and infra-red bolometers and, more recently, with the copper
oxide rectifier. In 1948 he transferred to the Physical Research Department
where he is now engaged in problems of solid-state physics.
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