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THE BELL SYSTEM
TECHNICAL JOURNAL
A JOURNAL DEVOTED TO THE
SCIENTIFIC AND ENGINEERING
ASPECTS OF ELECTRICAL
COMMUNICATION
EDITORS
R. W. King J. O. Perrine
EDITORIAL BOARD
W. H. Harrison O. E. Buckley
O. B. Blackwell M. J. Kelly
H. S. Osborne A. B. Clark
J. J. PiLLioD F. J. Feely
TABLE OF CONTENTS
AND
INDEX
VOLUME XXVI
1947
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
NEW YORK
PRINTED IN U. S. A.
'^^H^^'Vx. cS=
THE BELL SYSTEM
TECHNICAL JOURNAL
VOLUME XXVI 1347
Table of Contents
January, 1947
Development of Silicon Crystal Rectifiers for Microwave Radar Re
ceivers — /. H. Scaff and R. S. Ohl 1
End Plate and Side Wall Currents in Circular Cylinder Cavity Reso
nator — J. P. Kinzer and 1 . G. Wilson 31
First and Second Order Equations for Piezoelectric Crystals Expressed
in Tensor Form — W. P. Mason 80
The Biased Ideal Rectifier — W. R. Bennett 139
Properties and Uses of Thermistors — Thermally Sensitive Resistors —
/. A. Becker, C. B. Green and G. L. Pearson 170
April, 1947
Radar Antennas — H. T. Friis and W. D. Lewis 219
Probability Functions for the Modulus and Angle of the Normal Com
plex Variate — Ray S. Hoyt 318
Spectrum Analysis of Pulse Modulated Waves — /. C. Lozier 360
July, 1947
Telephony by Pulse Code Modulation— If . M. Goodall 395
Some Results on Cylindrical Cavity Resonators — /. P. Kinzer and
l.G. Wilson 410
Precision Measurement of Impedance Mismatches in Waveguide —
Allen F. Pomeroy 446
Reflex Oscillators — J. R. Pierce and W. G. Shepherd 460
iii
126^^40 MI\R 9 1348
iv bell system technical journal
October, 1947
The Radar Receiver — L. W . Morrison, Jr 693
High\'acuum OxideCatliode Pulse Modulator Tubes — C. E. Fay . . . . 818
Polyrod Antennas — G. E. Mueller and W . A. Tyrrell 837
Targets for Microwave Radar Navigation — Sloan D. Robertson 852
Tables of Phase Associated with a SemiInhnite Unit Slope of Atten
uation — D. E. Thomas 870
Index to Volume XXVI
Analysis, Spectrum, of Pulse Modulated Waves, /. C. Lozier, page 360.
Antennas, Polyrod, G. E. Mueller and W . A . Tyrrell, page 837.
Antennas, Radar, E. T. Frits and W. D. Lewis, page 219.
Attenuation, Tables of Phase Associated with a SemiInfinite Unit Slope of, D. E. Thomas,
page 870.
B
Becker, J. A., C. B.Green and G. Z.Pear^ow, Properties and Uses of Thermistors — Therm
ally Sensitive Resistors, page 170.
Bennett, W. R., The Biased Ideal Rectifier, page 139.
Cavity Resonator, Circular Cylinder, End Plate and Side Wall Currents in, /. P. Kinzer
and I. G. Wilson, page 31 .
Cavity Resonators, Cylindrical, Some Results on, /. P. Kinzer and I. G. Wilson, page 410.
Code Modulation, Pulse, Telephony by, W. M. Goodall, page 395.
Crystal, Silicon, Rectifiers for Microwave Radar Receivers, Development of, /. H. Scaff
and R. S. Ohl, page 1.
Crystals, Piezoelectric, Expressed in Tensor Form, First and Second Order Ecjuations for,
W. P. Mason, page 80.
Fay, C. E., HighVacuum OxideCathode Pulse Modulator Tubes, page 818.
Friis, H. T. and W. D. Lewis, Radar Antennas, page 219.
Goodall, W. M., Telephony by Pulse Code Modulation, page 395.
Green, C. B.,G. L.PearsonandJ . A. Seeder, Properties and Uses of Thermistors — Therm
ally Sensitive Resistors, page 170.
H
Hoyt, Ray S., Probability Functions for the Modulus and Angle of the Normal Complex
Variate, page 318.
Impedance Mismatches in Waveguide, Precision Measurement of, Allen F.Pofneroy, page
446.
K
Kinzer, J. P. and /. G. Wilson, End Plate and Side Wall Currents in Circular Cylinder
Cavity Resonator, page 31.
Kinzer, J. P. and I. G. Wilson, Some Results on Cylindrical Cavity Resonators, page 410
Lewis, W . D. and H. T. Friis, Radar Antennas, page 219.
Lozier, J . €., Spectrum Analysis of Pulse Modulated Waves, page 360.
M
Mason, W . P., First and Second Order Equations for Piezoelectric Crystals Expressed in
Tensor Form, page 80.
vi BELL SYSTEM TECHNICAL JOURNAL
Microwave Radar Navigation, Targets for, Sloan D. Robertson, page 852.
Microwave Radar Receivers, Development of Silicon Crystal Rectifiers for, /. H. Scajf
and R. S. Ohl, page 1 .
Mismatches, Impedance, in Waveguide, Precision Measurement of, Allen F. Pomeroy,
page 446.
Modulated Waves, Pulse, Spectrum Analysis of, /. C. Lozier, page 360.
Modulation, Pulse Code, Telephony by, W. M . Goodall, page 395.
Modulator Tubes, HighVacuum OxideCathode Pulse, C. E. Fay, page 818.
Morrison, Jr., L. W ., The Radar Receiver, page 693.
Mueller, G. E. and W . A. Tyrrell, Polyrod Antennas, page 837.
N
Navigation, Microwave Radar, Targets for, Sloan D. Robertson, page 852.
O
Ohl, R. S. and J. H. Scaf, Development of Silicon Crystal Rectifiers for Microwave Radar
Receivers, page 1 .
Oscillators, Reflex,/. R.Pierce and W . G. Shepherd, page 460.
P
Pearson, G. L., J. A . Becker and C. B. Green, Properties and Uses of Thermistors — Therm
ally Sensitive Resistors, page 170.
Phase, Tables of, Associated with a SemiInfinite Unit Slope of Attenuation, D. E. Thomas,
page 870.
Pierce, J. R. and W. G. Shepherd, Reflex Oscillators, page 460.
Piezoelectric Crystals Expressed in Tensor Form, First and Second Order Equations for,
W. P. Mason, page 80.
Polyrod Antennas, G. E. Mueller and W. A . Tyrrell, page 837.
Pomeroy, Allen F., Precision Measurement of Impedance Mismatches in Waveguide, page
446.
Probability Functions for the Modulus and Angle of the Normal Complex Variate, Ray S.
Hoyt, page 318.
Pulse Code Modulation, Telephony by, W. M. Goodall, page 395.
Pulse Modulated Waves, Spectrum Analysis of,/. C. Lozier, page 360.
Pulse Modulator Tubes, HighVacuum OxideCathode, C. E. Fay, page 818.
R
Radar: High Vacuum OxideCathode Pulse Modulator Tubes, C. E. Fay, page 818.
Radar: End Plate and Side Wall Currents in Circular Cylinder Cavity Resonator, /. P.
Kinzer and I. G. Wilson, page 31.
Radar: Some Results on Cylindrical Cavity Resonators, /. P. Kinzer and I. G. Wilson,
page 410.
Radar: Polyrod Antennas, G. E. Mueller and W. A . Tyrrell, page 837.
Radar: Reflex Oscillators, /. R. Pierce and W. G. Shepherd, page 460.
Radar Antennas, H. T. Friis and W. D. Lewis, page 219.
Radar Navigation, Microwave, Targets for, Sloan D. Robertson, page 852.
Radar Receiver, The, L. W. Morrison, Jr., page 693.
Radar Receivers, Microwave, Development of Silicon Crystal Rectifiers for, /. H. Scaff
and R. S. Ohl, page 1.
Receiver, Radar, The, L. W. Morrison, Jr., page 693.
Receivers, Microwave Radar, Development of Silicon Crystal Rectifiers for,/. H. Scaff
and R. S. Ohl, page 1.
Rectifier, Biased Ideal, The, W. R. Bennett, page 139.
Rectifiers, Silicon Crystal, for Microwave Radar Receivers, Development of,/. E. Scaff
and R.S. Ohl, page 1 .
Reflex Oscillators, /. R. Pierce and W. G. Shepherd, page 460.
Resistors, Thermally Sensitive — Properties and Uses of Thermistors, /. A . Becker, C. B.
Green and G. L. Pearson, page 170.
Resonator, Circular Cylinder Cavity, End Plate and Side Wall Currents in, /. P. Kinzer
and I. G. IFz/50M,page31.
Resonators, Cylindrical Cavity, Some Results on, /.P. Kinzer and LG. H^i/^OM, page 410.
Robertson, Sloan D., Targets for Microwave Radar Navigation, page 852.
INDEX
Scajf, J. B. and R. S. Ohl, Development of Silicon Crystal Rectifiers for Microwave Radar
Receivers, page 1 .
Shepherd, W . G. and J. R. Pierce, Reflex Oscillators, page 460.
Silicon Crystal Rectifiers for Microwave Radar Receivers, Development of, J. E. Sea ff and
R. S. Ohl, page 1.
Spectrum Analysis of Pulse Modulated Waves, /. C. Lozier, page 360.
Tensor Form, First and Second Order Equations for Piezoelectric Crystals Expressed in»
W. P. Mason, page 80.
Thermistors, Properties and Uses of — Thermally Sensitive Resistors, /. A . Becker, C. B.
Green and G. L. Pearson, page 170.
Thomas, D. E., Tables of Phase Associated with a SemiInfinite Unit Slope of Attenuation,
page 870.
Tyrrell, W. A . and G. E. Mueller, Polyrod Antennas, page 837.
V
Vacuum, High, OxideCathode Pulse Modulator Tubes, C. E. Fay, page 818.
W
' Waveguide, Precision Measurement of Impedance Mismatches in, Allen F. Pomeroy, page
446.
Wilson, I. G. and J. P. Kinzer, End Plate and Side Wall Currents in Circular Cylinder
Cavity Resonator, page 31.
Wilson, I. G. and J. P. Kinzer, Some Results on Cylindrical Cavity Resonators, page 410.
VOLUME XXVI JANUARY, 1947 no. i
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
Development of Silicon Crystal Rectifiers for Microwave
Radar Receivers J. H. Scaff and R. S. Ohl 1
End Plate and Side Wall Currents in Circular Cylinder
Cavity Resonator J. P. Kinzer and I. G. Wilson 31
First and Second Order Equations for Piezoelectric Crys
tals Expressed in Tensor Form W. P. Mason 80
The Biased Ideal Rectifier W. R, Bennett 139
Properties and Uses of Thermistors — Thermally Sensitive
Resistors . .J.A. Becker, C. B. Green and G. L. Pearson 170
Abstracts of Technical Articles by Bell System Authors. . 213
Contributors to This Issue 217
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
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THE BELL SYSTEM TECHNICAL JOURNAL
Published quarterly by the
American Telephone and Telegraph Company
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EDITORS
R. W. King J. O. Perrine
EDITORIAL BOARD
W. H. Harrison
O. B Blackwell
H. S. Osborne
J. J. PiUiod
O. E. Buckley
M. J. KeUy
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Copyright, 1947
American Telephone and Telegraph Company
PRINTED IN U. S A
CORRECTION FOR ISSUE OF OCTOBER, 1946
In the article SPARK GAP SWITCHES FOR RADAR,
lines 214 inclusive on page 593 should have appeared be
tween lines 10 and 11 on page 588.
The Bell System Technical Journal
Vol. XXVI Ja72uary, 1947 No. i
Development of Silicon Crystal Rectifiers for
Microwave Radar Receivers
By J. H. SCAFF and R. S. OHL
Introduction
TO THOSE not familiar with the design of microwave radars the exten
sive war use of recently developed crystal rectifiers^ in radar receiver
frequency converters may be surprising. In the renaissance of this once
familiar component of early radio receiving sets there have been develop
ments in materials, processes, and structural design leading to vastly
improved converters through greater sensitivity, stability, and ruggedness
of the rectifier unit. As a result of these developments a series of crystal
rectifiers was engineered for production in large quantities to the exacting
electrical specifications demanded by advanced microwave techniques and
to the mechanical requirements demanded of combat equipment.
The work on crystal rectifiers at Bell Telephone Laboratories during
the war was a part of an extensive cooperative research and development
program on microwave weapons. The Office of Scientific Research and
Development, through the Radiation Laboratory at the Massachusetts
Institute of Technology, served as the coordinating agency for work con
ducted at various university, government, and industrial laboratories in
this country and as a liaison agency with British and other Allied organiza
tions. However, prior to the inception of this cooperative program, basic
studies on the use of crystal rectifiers had been conducted in Bell Telephone
Laboratories. The series of crystal rectifiers now available may thus be
considered to be the outgrowth of work conducted in three distinct periods.
First, in the interval from 1934 to the end of 1940, devices incorporating
point contact rectifiers came into general use in the researches in ultra
highfrequency and microwave communications techniques then under
way at the Holmdel Radio Laboratories of Bell Telephone Laboratories.
' A crystal rectifier is an assymmetrical, nonlinear circuit element in which the seat of
rectification is immediately underneath a point contact applied to the surface of a semi
conductor. This element is frequently called "point contact rectifier" and "crystal de
tector" also. In this paper these terms are considered to be S3'nonymous.
1
2 BELL SYSTEM TECHNICAL JOURNAL
At that time the improvement in sensitivity of microwave receivers employ
ing crystal rectiliers in the frequency converters was clearly recognized, as
were the advantages of rectifiers using silicon rather than certain well
known minerals as the semiconductor. In the second period, from 1941
to 1942, the advent of important war uses for microwave devices stimulated
increased activity in both research and development. During these years
the pattern for the interchange of technical information on microwave
devices through government sponsored channels was established and was
continued through the entire period of the war. With the extensive inter
change of information, considerable international standardization was
achieved. In view of the urgent equipment needs of the Armed Services
emphasis was placed on an early standardization of designs for production.
This resulted in the first of the modern series of rectifiers, namely, the
ceramic cartridge design later coded through the Radio Manufacturers
Association as type 1N21. In the third period, from 1942 to the present
time, process and design advances accruing from intensive research and
development made possible the coding and manufacture of an extensive
series of rectifiers all markedly superior to the original 1N21 unit.
It is the purpose of this paper to review the work done in Bell Telephone
Laboratories on sihcon point contact rectifiers during the three periods
mentioned above, and to discuss briefly typical properties of the rectifiers,
several of the more important applications and the production history.
Crystal Rectifiers in the Early Microwave Research
The technical need for the modern crystal rectifier arose in research on
ultrahigh frequency communications techniques. Here as the frontier
of the technically useful portion of the radio spectrum was steadily advanced
into the microwave region, certain limitations in conventional vacuum
tube detectors assumed increasing importance. Fundamentally, these
limitations resulted from the large interelectrode capacitance and the
finite time of transit of electrons between cathode and anode within the
tubes. At the microwave frequencies (3000 megacycles and higher), they
became of first importance. As transit time effects are virtually absent
in point contact rectifiers, and since the capacitance is minute, it was logical
that the utility of these devices should again be explored for laboratory use.
The design of the point contact rectifiers used in these researches was
dictated largely, of course, by the needs of the laboratory. Frequently
the rectifier housing formed an integral part of the electrical circuit design
while other structures took the form of a replaceable resistorlike cartridge.
A variety of structures, including the modern types, arranged in chrono
logical sequence, are shown in the photograph, Fig. 1. In general, the
SILICON CRYSTAL RECTIFIERS 3
principal requirements of the rectifiers for laboratory use were that the
units be sensitive, stable chemically, mechanically, and electrically, and
^v^
1934
Ti
1937
i^ «~ Jt*
Fig. 1— Point contact rectifier structures. 19341943. Approximately f actual size.
that they be easily adjusted. Considering the known vagaries of the device's
historical counterpart, it was considered prudent to provide in the structures
means by which the unit could be readjusted as frequently as might prove
necessary or desirable.
4 BELL SYSTEM TECHNICAL JOURNAL
As the properties of various semiconductors were known to vary widely,
an essential part of the early work was a survey of the properties of a number
of minerals and metalloids potentially useful as rectifier materials. There
were examined and tested approximately 100 materials, including zincite,
molybdenite, galena, iron pyrites, silicon carbide, and silicon. Of the
materials investigated most were found to be unsuitable for one reason
or another, and iron pyrites and silicon were selected as having the best
overall characteristics. The subsequent studies were then directed toward
improving the rectifying material, the rectifying surface, the j^oint contact
and the mounting structure.
Fig. 2 — Rectilicr inserts untl contact jxjints lor use in early 3(K)t) megacycle converters.
Overall length of insert ^inch approximately.
i'"()r use at freciuencies in the region of .^OOO megac}cles standard demount
able elements, consisting of rectitier "inserts" and contact points, were
develojied for use in various housings or mounting blocks, depending upon
the j)articular circuit requirements. The rectitier "inserts" consisted of
small wafers of iron pyrite or silicon, soldered to hexagonal brass studs as
shown in Fig. 2a. In these devices the surface of the semiconductor was
prei)ared by grinding, polishing, and etching to develop good rectification
characteristics. Our knowledge of the metallurgy of silicon had acKanced
by this time to the stage where a uniformly acti\e rcctilier surface could
be j)roduccd and searching for active spots was not nccessar\'. l'\irther
more, it was jiossible to ])repare inserts of a jiositive or negative \ariety,
signifying that the easy direction of current llow was obtained with the
silicon i)ositive with respect to the point or \ice \ersa. Owing to a greater
noiilincarity of the current \oUage characteristic, the nt)"pe or negative
SILICON CRYSTAL RECTIFIERS 5
insert tended to give better performance as microwave converters while
the ptype, or positive insert, because of greater sensitivity at low voltages,
proved to be more useful in test equipment such as resonance indicators in
frequency meters. In certain instances also, it was advantageous for the
designer to be able to choose the polarity best suited to his circuit design.
In contrast, however, to the striking uniformity obtained with the silicon
processed in the laboratory, the pyrite inserts were very nonuniform.
Active rectification spots on these natural mineral specimens could be
found only by tediously searching the surface of the specimen. More
over, rectifiers employing the pyrite inserts showed a greater variation in
properties with frequency than those in which silicon was used.
In addition to providing a satisfactory semiconductor, it was necessary
also to develop suitable materials for use as point contacts. For this use
metals were required which had satisfactory rectification characteristics
with respect to silicon or pyrites and sutBcient hardness so that excessive
contact areas were not obtained at the contact pressures employed in the
rectifier assembly. The metals finally chosen were a platinumiridium
alloy and tungsten, which in some cases was coated with a gold alloy.
These were employed in the form of a fine wire spot welded to a suitable
spring member. The spring members themselves were usually of a wedge
shaped cantilever design and were made from coin silver to facilitate elec
trical connection to the spring. Several contact springs of two typical
designs are shown in the photograph, Figs. 2b and 2c.
A typical mounting block arranged for use with the inserts and points
}: is shown in Fig. 1 (1940) and in Fig. 3. This block was so constructed that
I it could be inserted in a 70 ohm coaxial line without introducing serious
l discontinuities in the line. The contact point of the rectifier was assembled
1 in the block to be electrically connected to the central conductor of the
I coaxial radio frequency input fitting, while the crystal insert screwed into
I a tapered brass pin electrically connected to the central conductor of the
} coaxial intermediate frequency and dc output fitting. The tapered pin
I fitted tightly into a tapered hole in a supporting brass cylinder, but was
: insulated from the cylinder by a few turns of polystyrene tape several
; thousandths of an inch thick. This central pin was thus one terminal of a
i coaxial highfrequency bypass condenser. The capacitance of this con
' denser depended upon the general nature of the circuits in which the block
was to be used, and was generally about 15 mmfs. The arrangement of
the point, the crystal insert and their respective supporting members was
I such that the point contact could be made to engage the surface of the
silicon at any spot and at the contact pressure desired and thereafter be
clamped firmly in a fixed position by set screws. Typical direct current
characteristics of the positive and negative silicon inserts and of pyrite
inserts assembled and adjusted in this mounting block are shown in Fig. 4.
BELL' SYSTEM TECHNICAL JOURNAL
INSULATING BEAD
BRASS BLOCK 
TAPERED BRASS PIN
POLYSTYRENE
TAPE
INTERMEDIATE
FREQUENCY ^
AND
DIRECT CURRENT
OUTPUT
DETAIL OF
COAXIAL CONDENSER
ASSEMBLY ,
I
Fig. 3 — Schematic diagram of one of the early crystal converter blocks.
The inserts and points in appropriate mounting blocks were widely used
in centimeter wave investigations prior to 1940. The principal laboratory
uses were in frequency converter circuits in receivers, and as radio fre
2 G. C. Southworth and A. P. King, "Metal Horns as Directive Receivers of Ultra
short Waves," Proc. L R. E. v. 27, pp. 95102, 1939; Carl R. Englund, "Dielectric Con
stants and Power Factors at Centimeter Wave Lengths," Bell Sys. Tech. Jour., v. 23, pp.
114129, 1944; lirainerd, Koehler, Reich, and WoodrulT, "Ultra High Frequency Tech
niques," D. Van Nostrand Co., Inc., 2504th Avenue, New York, 1942.
SILICON CRYSTAL RECTIFIERS 7
quency instrument rectifiers. They were also used to a relatively minor
extent in some of the early radar test equipment. Moreover, the avail
ability of these devices and the knowledge of their properties as microwave
converters tended to focus attention on the potentialities of radar designs
employing crystal rectifiers in the receiver's frequency converter. Similarly,
the techniques established for preparation of the inserts tended to orient
subsequent manufacturing process developments. For example, the
methods now generally used for preparing silicon ingots, for cutting the
rectifying element from the ingot with diamond saws, and for forming the
lOi
102
103
^
.'
y
<
^
/
R^
''
,^'
^^ y^
NEGATIVE SILICON
GOLD ALLOY POINT
/
•
•
'^^' F.
— ■
" z
i
^'
./'
f X
/^A
/
•
/ POSITIVE SILICON >
' PLATINUM *yr
ALLOY point/]
r
F = FORWARD CURRENT
R= REVERSE CURRENT
V
/iron PYRITES
/gold ALLOY POINT
108
106
105 lO"'^
CURRENT IN AMPERES
103
102
101
Fig. 4 — Directcurrent characteristics of silicon and iron pyrite rectifiers
fabricated as inserts, 1939.
back contact to the rectifying element by electroplating procedures, are
still essentially similar to the techniques used for preparing the inserts in
1939. As a contribution to the defense research effort, this basic informa
tion, with various samples and experimental assemblies, was made available
to governmental agencies for dissemination to authorized domestic and
foreign research establishments.
Development of the Ceramic Type Cartridge Structure
The block rectifier structure previously described was well adapted to
various laboratory needs because of its flexibility, but for large scale utiliza
tion certain Umitations are evident. Not only was it necessan^ that the
parts be accurately machined, but also the adjustment of the rectifier in
8 BELL SYSTEM TECHNICAL JOURNAL
the block structure required considerable skill. With recognition of the
military importance of silicon crystal rectifiers, effort was intensified in
the development of standardized structures suitable for commercial pro
duction.
In the 19401941 period, contributions to the design of silicon crystal
rectifiers were made by British workers as a part of their development of
new military implements. For these projected military' uses, the problem
of replacement and interchangeability assumed added importance. The
design trend was, therefore, towards the development of a cartridge type
structure with the electrical adjustment fixed during manufacture, so that
the unit could be replaced easily in the same manner as vacuum tubes.
In the latter part of 1941 preliminary information was received in this
country through National Defense Research Committee channels on a
rectifier design originating in the laboratories of the British Thomson
Houston Co., Ltd. A parallel development of a similar device was begun
in various American laboratories, including the Radiation Laboratory at
the Massachusetts Institute of Technology, and Bell Telephone Labora
tories. In the work at Bell Laboratories, emphasis was placed both on
development of a structure similar to the British design and on explora
tion and test of various new structures which retained the features of
socket interchangeability but which were improved mechanicalh and
electrically.
In the work on the ceramic cartridge, the external features of the British
design were retained for reasons of mechanical standardization but a number
of changes in process and design were made both to improve performance
and to simplify manufacture. To mention a few, the position of the silicon
wafer and the contact point were interchanged because measurements
indicated that an improvement in performance could thereby be obtained.
To obviate the necessity for searching for active spots on the surface of
the silicon and to improve performance, fused high purity silicon was
substituted for the "commercial" silicon then employed by the British.
The rectifying element was cut from the ingots by diamond saws, and
carefully polished and etched to develop optimum rectification character
istics. Similar improvements were made in the prej^aration of the point
or "cats whisker", replacing hand operations l:)y machine techniques. To
protect the unit from mechanical shock and the ingress of moisture, a sjiecial
imjjregnating comjjound was de\'eloped which was completely satisfactory
even under conditions of rapid changes in temperature from —40° to 470°C.
All such improvements were directed towards ini]iro\ing quality and
establishing techniques for mass production.
In this early work time was at a jircmium because of the need for prompt
standardization of the design in order that radar system designs might in
SILICON CRYSTAL RECTIFIERS 9
turn be standardized, and that manufacturing facilities might be estabhshed
to supply adequate quantities of the device. The development and initial
production of the device was accomplished in a short period of time. This
was possible because process experience had been acquired in the insert
development, and centimeter wave measurements techniques and faciUties
were then available to measure the characteristics of experimental units
at the operating frequency. By December 1941, a pattern of manufacturing
techniques had been established so that production by the Western Electric
Company began shortly thereafter. This is believed to have been the
first commercial production of the device in this country.
As a result of the basic information on centimeter wave measurements
techniques which was available from earlier microwave research at the
Holmdel Radio Laboratory, it was possible also, at this early date, to
propose to the Armed Services that each unit be required to pass an ac
ceptance test consisting of measurement of the operating characteristics
at the intended operating frequency. This plan was adopted and standard
test methods devised for production testing. Considering the complexity
of centimeter wave measurements, this was an accomplishment of some
magnitude and was of first importance to the Armed Services because it
assured by direct measurement that each unit would be satisfactory for
field use.
The cartridge structure resulting from these developments and meeting
the international dimensional standards is shown in Fig. 5. It consists
of two metal terminals separated by an internally threaded ceramic insu
lator. The rectifying element itself consists of a small piece of silicon (p
type) soldered to the lower metal terminal or base. The contact spring or
"cats whisker" is soldered into a cylindrical brass pin which slides freely
into an axial hole in the upper terminal and may be locked in any desired
position by set screws. The spring itself is made from tungsten wire of an
appropriate size, formed into an S shape. The free end of the wire, which
in a finished unit engages the surface of the silicon and establishes rectifica
tion, is formed to a coneshaped configuration in order that the area of
contact may be held at the desired low value.
The silicon elements used in the rectifiers are prepared from ingots of
fused high purity silicon. Alloying additions are made to the melt when
required to adjust the electrical resistivity of the silicon to the value desired.
The ingots are then cut and the silicon surfaces prepared and cut into small
Dieces approximately 0.05 inch square and 0.02 inch thick suitable for use
n the rectifiers. The contact springs are made from tungsten wire, gold
Dlated to facilitate soldering. Depending upon the application, the wires
10
BELL SYSTEM TECHNICAL JOURNAL
may be 0.005 inch, 0.0085 inch, or 0.010 inch in diameter. After forming
the spring to the desired shape, the tip is formed electrolytically.
In assembUng the rectifier cartridge, the two end terminals, consisting
of the base with the silicon element soldered to it, and the top detail con
taining the contact spring, are threaded into the ceramic tube so that the
free end of the spring does not engage the silicon surface. An adhesive
wfifflSBtfSS^SSJ^ ■ i A ■ . I . ^M
CERAMIC TUBE
POINT ASSEMBLY— I
TERMINAL
Fig. 5 — Ceramic cartridge rectifier structure and parts.
Overall length of assembled rectifier is approximately finch.
is employed to secure the parts firmly to the ceramic. The rectifier is then
"adjusted" by bringing the point into engagement with the silicon surface
and establishing optimum electrical characteristics. Finally the unit is
impregnated with a special compound to protect it from moisture and from
damage by mechanical shock. Units so prepared are then ready for the
final electrical tests.
The adjustment of the rectifier is an interesting operation for at this
SILICON CRYSTAL RECTIFIERS 11
stage in the process the rectification action is developed, and to a considerable
degree, controlled. If the point is brought into contact with the silicon
surface and a small compressional deflection applied to the spring, direct
current measurements will show a moderate rectification represented by
the passage of more current at a given voltage in the forward direction than
in the reverse. If the side of the unit is now tapped sharply by means of
a small hammer, the forward current will be increased, and, at the same
time, the reverse current decreased.^ With successive blows the reverse
current is reduced rapidly to a constant low value while the forward current
increases, but at a diminishing rate, until it also becomes relatively constant.
The magnitude of the changes produced by this simple operation is rather
surprising. The reverse current at one volt seldom decreases by less than a
factor of 10 and frequently decreases by as much as a factor of 100, while
the forward current at one volt increases by a factor of 10. Paralleling
these changes are improvements in the highfrequency properties, the
conversion loss and noise both being reduced. The tapping operation is
not a haphazard searching for better rectifying spots, for with a given
silicon material and mechanical assembly the reaction of each unit to tapping
is regular, systematic and reproducible. The condition of the sihcon surface
also has a pronounced bearing on "tappability" for by modifications of
the surface it is possible to produce, at will, materials sensitive or insensitive
in their reaction to the tapping blows.
In the development of the compounds for filling the rectifier, special
problems were met. For example, storage of the units for long periods
of time under either arctic or tropical conditions was to be expected. Also,
for use in airborne radars operating at high altitudes, where equipment
might be operated after a long idle period, it was necessary that the units
be capable of withstanding rapid heating from very low temperatures.
The temperature range specified was from —40° to 70°C. Most organic
materials normally solid at room temperature, as the hydrocarbon waxes,
are completely unsuitable, as the excessive contraction which occurs at
i low temperatures is sufficient to shift the contact point and upset the precise
adjustment of the spring. Nor are liquids satisfactory because of their
tendency to seep from the unit. However, special gel fillers, consisting
of a wax dispersed in a hydrocarbon oil, were devised in Bell Telephone
Laboratories to meet the requirements, and were successfully applied by
the leading manufacturers of crj^stal rectifiers in this country. Materials
of a similar nature, though somewhat different in composition, were also
used subsequently in Britain. Further improvements in these compounds
have been made recently, extending the temperature range 10°C at low
' Southworth and Kin^; loc. cit.
12
BELL SYSTEM TECHNICAL JOURNAL
temperatures and about 30°C at high temperatures in response to the design
trend towards operation of the units at higher temperatures. The units
employing this compound may, if desired, be repeatedly heated and cooled
rapidly between — 50°C and +100°C without damage.
Use of the impregnating compound not only improves mechanical stability
but prevents ingress or absorption of moisture. Increase of humidity
would subject the unit not only to changes in electrical properties such as
variation in the radio frequency impedance, but also to serious corrosion,
for the galvanic couple at the junction would support rapid corrosion of the
metal point. In fact, with condensed moisture present in unfilled units
corrosion can be observed in 48 hours. For this reason alone, the develop
ment of a satisfactory filling compound was an important step in the suc
cessful utilization of the units by the Armed Services under diverse and
drastic field conditions.
Table I
Shelf Aging Data on Silicon Crystal Rectifiers of the Ceramic Cartridge Design
Initial Values
Values After
Storage for 7 Months
Storage Conditions
Conversion
Loss
(Median;
(L)
Noise
Ratio
(Median)
(Nr)
Conversion
Loss
(median)
(L)
Noise
Ratio
(median)
(Nr)
75°F. 65% Relative Humidity
110°F. 95% Relative Humidity
 40°C . ...
db
6.8
6.9
7.0
dh
3.9
3.9
3.9
dh
6.7
6.9
6.8
db
4.3
4.3
3.9
The large improvement in stability achieved in the present device as
compared with the older crystal detectors may be attributed to the design
of the contact spring, correct alignment of parts in manufacture and to
the practice of filling the cavity in the unit with the gel developed for this
purpose. Considering the apparently delicate construction of the device,
the stability to mechanical or thermal shock achieved by these means is
little short of spectacular. Standard tests consist of drojiping the unit
three feet to a wood surface, immersing in water, and of ra])idly lieating
from —40 to 7()°C None of these tests im])airs the quality of the unit.
Similarly the unit will withstand storage for long periods of time under
adverse conditions. Table I summarizes the results of tests on units
which were stored for approximately one year under arctic ( — 40°), tropical
(114°F — 95% relative humidity), and temi)erate conditions. Though
minor changes in the electrical characteristics were noted in the accelerated
tropical test, none of the units was inoperative after this drastic treatment.
SILICON CRYSTAL RECTIFIERS 13
Development or the Shielded Rectifier Structure
Rectifiers of the ceramic cartridge design, though manufactured in very
large quantities and widely and successfully used in military apparatus,
have certain well recognized limitations. For example, they may be ac
cidentally damaged by discharge of static electricity through the small
point contact in the course of routine handhng. If one terminal of the
unit is held in the hand and the other terminal grounded, any charge which
may have accumulated will be discharged through the small contact.
Since such static charges result in potential differences of several thousand
volts it is understandable that the unit might suffer damage from the dis
charge. Although damage from this cause may be avoided by following
a few simple precautions in handling, the fact that such precautions are
needed constitutes a disadvantage of the design.
Certain manufacturing difficulties are also associated with the use of
the threaded insulator. The problem of thread fit requires constant
attention. Lack of squareness at the end of the ceramic cyhnder or lack
of concentricity in the threaded hole tends to cause an undesirable eccen
tricity or angularity in the assembled unit which can be minimized only by
rigid inspection of parts and of final assemblies. At the higher frequencies
(10,000 megacycles), uniformity in electrical properties, notably the radio
frequency impedance, requires exceedingly close control of the internal
mechanical dimensions. In the cartridge structure where the terminal
connections are separated by a ceramic insulating member, the additive
variations of the component parts make close dimensional control inherently
difficult.
To eliminate these difficulties the shielded structure, shown in Fig. 6,
was developed. In this design the rectifier terminates a small coaxial
line. The central conductor of the line, forming one terminal of the rec
tifier, is molded into an insulating cylinder of silicafilled bakelite, and
has spot welded to it a 0.002inch diameter tungsten wire spring of an
offset C design. The free end of the spring is cone shaped. The rectifying
element is soldered to a small brass disk. Both the disk, holding the
rectifying element, and the bakelite cylinder, holding the point, are force
fits in the sleeve which forms the outer conductor of the rectifier. By
locating the bakelite cylinder within the sleeve so that the free end of the
central conductor is recessed in the sleeve, the unit is effectively protected
from accidental static damage as long as the holder or socket into which
the unit fits is so designed that the sleeve establishes electrical contact with
the equipment at ground potential before the central conductor. The
sleeve also shields the rectifying contact from effects of stray radiation.
The radio frequency impedance of the shielded unit can be varied within
certain limits by modifying the diameter of the central conductor. For
14
BELL SYSTEM TECHNICAL JOURNAL
example, in the 1N26 unit, which was designed for use at frequencies in
the region of 24,000 megacycles, a small metal slug fitting over the central
conductor makes it possible to match a coaxial line having a 65ohm surge
impedance. For certain circumstances this modification in design is
advantageous, while in others it is a disadvantage because the matching
slug is effective only over a narrow range of frequencies.
IS
POINT ASSEMBLY
OUTER
CONDUCTORn
METAL
DISC
Fig. 6 — Shielded rectifier structure and parts. Overall length of assembled rectifier is
approximately  inch.
The shielded structure was developed in 1942 and since it was of a sim
plified design with reduced hazard of static damage, it was proposed to the
Armed Services for standardization in June of that year. However, because
of the urgency of freezing the design of various radars and because the
British had aheady standardized on the outhne dimensions of the ceramic
type cartridge, Fig. 5, the Services did not consider it advantageous to
standardize the new structure when first proposed. In deference to this
international standardization program, plans for the manufacture of this
'i
SILICON CRYSTAL RECTIFIERS 15
structure were held in abeyance during 1942 and 1943. However, an
opportunity for realizing the advantages inherent in the shielded design
was afforded later in the war and a sufficient quantity of the units was pro
duced to demonstrate its soundness. As anticipated from the construc
tional features, marked uniformity of electrical properties was obtained.
Types, Applicatioks, akd Operating Characteristics
Various rectifier codes, engineered for specific military uses, were manu
factured by Western Electric Company during the war. These are listed
in Table II. The units are designated by RMA type numbers, as 1N21,
1N23, etc., depending upon their properties and the intended use. Letter
suflixes, as 1N23A, 1N23B, indicate successively more stringent perform
ance requirements as reflected in lower allowable maxima in loss and noise
ratio, and, usually, more stringent power prooftests. In general, different
codes are provided for operation in the various operating frequency ranges.
For example, the 1N23 series is tested at 10,000 megacycles while the 1N21
series is tested at 3,000 megacycles and the 1N25 at 1000 megacycles,
approximately. Since higher transmitter powers are frequently employed
at the lower frequencies, somewhat greater power handling ability is provided
in units for operation in this range.
One of the more important uses of sihcon crystal rectifiers in military
equipment was in the frequency converter or first detector in superheter
odyne radar receivers. This utilization was universal in microwave re
ceivers. In this application the crystal rectifier serves as the nonlinear
circuit element required to generate the difference (intermediate) frequency
between the radio frequency signal and the local oscillator. The inter
mediate frequency thus obtained is then amplified and detected in conven
tional circuits. As the crystal rectifier is normally used at that point in
the receiving circuit where the signal level is at its lowest value, its perform
ance in the converter has a direct bearing on the overall system performance.
It was for this reason that continued improvements in the performance of
crystal rectifiers were of such importance to the war effort.
For the converter application, the signaltonoise properties of the unit
at the operating frequency, the power handling ability, and the uniformity
of impedance are important factors. Tlie signaltonoise properties are
measured as conversion loss and noise ratio. The loss, L, is the ratio of
the available radio frequency signal input power to the available inter
mediate frequency output power, usually expressed in decibels. The
noise ratio, Nr, is the ratio of crystal output noise power to thermal (KTB)
noise power. The loss and noise ratio are fundamental properties of the
16
BELL SYSTEM TECHNICAL JOURNAL
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SILICON CRYSTAL RECTIFIERS
17
» * CRYSTAL
PARTS I I RECTIFIER
(ENLARGED)
RETAINING PLUG
Fig. 7 — Converter for wave guide circuits as installed in the radio frequency unit of
the AN/APQ13 radar system. This was standard equipment in B29 bombers for
radar bombing and navigation.
18
BELL SYSTEM TECHNICAL JOURNAL
converter. From these data and other circuit constants, the designer may
calculate* expected receiver performance.
For operation as converters,^ crystal rectifiers are employed in suitable
holders. These may be arranged for use with either coaxial line or wave
guide circuits, depending upon the application. Figure 7 shows a converter
for wave guide circuits installed in the radio frequency unit of an airborne \
radar system. A typical converter designed for use with coaxial lines is [
shown in the photograph Fig. 8. A schematic circuit of this converter '.
is shown in Fig. 9. In such circuits the best signaltonoise ratio is realized
when an optimum amount of beating oscillator power is supplied. The
optimum power depends, in part, on the properties of the rectifier itself,
and, in part, on other circuit factors as the noise figure of the intermediate
Fig. 8 — Converter for use at 3000 megacycles. The crystal rectifier is located
adjacent to its socket in the converter.
frequency amplifier. For a well designed intermediate frequency amplifier
with a noise figure of about 5 decibels, the optimum beating oscillator
power is such that between 0.5 and 2.0 milliamperes of rectified current
flows through the rectifier unit. Under these conditions and with the unit
matched to the radio frequency line, the beating oscillator power absorbed
by the unit is about one milliwatt. For intermediate frequency amplifiers
■" The quantities L and Ni? are related to receiver performance bj' the relationship
F^ = Z.(N/?  1 + FiF)
where Fr is the receiver noise figure and Fip is the noise figure of the intermediate fre
quency amplifier. All terms are expressed as power ratios. A rigorous definition of
receiver noise figure has been given l)v H. T. Friis "Noise Figures of Radio Receivers,"
Proc. L R. E., vol. 32, pp. 419422; July, 1944.
* C. F. Edwards, "Microwave Converters," presented orally at the Winter Technical
Meeting of the /. R. E., January 1946 and submitted to the /. R. E. for publication.
SILICON CRYSTAL RECTIFIERS
19
with poorer noise figures, the drive for optimum performance is higher
than the figures cited above. Conversely, for intermediate frequency
amphfiers with exceptionally low noise figures, optimum [performance is
obtained with lower values of beating oscillator drive. If desired, somewhat
higher currents than 2.0 milliamperes may be employed without damage
to the crystal.
The impedance at the terminals of a converter using crystal rectifiers,
both at radio and intermediate frequencies, is a function not only of the
rectifier unit, but also of the circuit in which the unit is used and of the
SILICON
RECTIFIER
BY PASS
CONDENSER
y^^
SIGNAL
INPUT
Fig. 9 — Schematic diagram of crystal converter.
power level at which it is operated. Consequently the specification of an
impedance for a crystal rectifier is of significance only in terms of the circuit
in which it is measured. Since the converters used in the production testing
of crystal rectifiers are not necessarily the same as those used in the field,
and since in addition there are frequently several converter designs for
the same type of unit, a specification of cr>'stal rectifier impedance in pro
duction testing can do little more than select units which have the same
impedance characteristic in the production test converter. The impedances
at the terminals of two converters of different design but using the same
crystal rectifier may vary by a factor of 3 or even more, with the inter
mediate frequency impedance generally varying more drastically than the
radio frequency impedance. The variation is also a function of the con
20 BELL SYSTEM TECHNICAL JOURNAL
version loss. Crystals with large conversion losses are less susceptible
to impedance changes from reactions in the radio frequency circuit than are
low conversion loss units.
The level of power to which the rectifiers can be subjected depends upon
the way in which the power is applied. The application of an excessive
amount of power or energy results in the electrical destruction of the unit
by ru{)ture of the rectifying material. Experimental evidence indicates
that the electrical failure may be in one of three categories. The total
energ}^ of an applied pulse is responsible for the impairment when the
pulse length is shorter than 10~' seconds, the approximate thermal time
constant of the crystal rectifier as given by both measurement and calcula
tion. For pulse lengths of the order of 10~^ seconds the peak power in the
pulse is the determining factor, and for continuous wave operation the
limitation is in the average power.
In performance tests in manufacture all units for which burnout tolerances
are specified are subjected to prooftests at levels generally comparable
with those which the unit may occasionally be expected to withstand in
actual use, but greater than those to be employed as a design maximum.
The power or energy is applied to the unit in one of two types of prooftest
equipment. The multiple, long time constant (of the order of 10" seconds)
pulse test is applied to simulate the plateau part of a radar pulse reaching
the crystal through the gas discharge transmitreceive switch.^ This test
uses an artificial line of appropriate impedance triggered at a selected
repetition rate for a determined length of time. The power available to
the unit is computed from the usual formula,
4Z'
where P is the power in watts, V is the potential in volts to which the pulse
generator is charged, and Z is the impedance in ohms of the pulse generator.
In general, where this test is employed, a line is used which matches the
impedance of the unit under test at the specified voltage.
The second type of test is the single discharge of a coaxial line through
the unit to simulate a radar pulse spike reaching the crystal before the
transmitreceive switch fires. The pulse length is of the order of 10~^
second. The energy in the test si)ike mav be computed from the relation
where E is the energy in ergs, C the capacity of the coaxial line in farads,
and r the potential in volts to whicli the line is charged.
"A. L. Samuel, J. W. Clark, and W. W. Mumford, "The Gas Discharge Transmit
Receive Switch," Bell Sys. Tech. Jour., v. 25 No. 1, pp. 48101. Jan. 1946.
SILICON CRYSTAL RECTIFIERS 21
Specification prooftest levels are, of course, not design criteria. Since
the units are generally used in combination with protective devices, such
as the transmitreceive switch, it is necessary to conduct tests in the circuits
I of interest to establish satisfactory operating levels.
I In general, however, the units may be expected to carry, without deteriora
tion, energy of the order of a third of that used in the single dc spike proof
; test or peak powers of a magnitude comparable with that used in the multiple
I flattop dc pulse prooftest. The upper Hmit for applied continuous wave
i signals has not been determined accurately, but, in general, rectified currents
i below 10 milliamperes are not harmful when the self bias is less than a few
tenths of a volt.
' The service life of a crystal rectifier will depend completely upon the
; conditions under which it is operated and should be quite long when its
! ratings are not exceeded. During the war, careful engineering tests con
! ducted on units operating as first detectors in certain radar systems revealed
j no impairment in the signaltonoise performance after operation for several
[ hundred hours. A small group of 1N21B units showed only minor impair
I ments when operated in laboratory tests for 100 hours with pulse powers
I (3000 megacycles) up to 4 watts peak available to the unit under test.
Another important military application of silicon crystal rectifiers was
as lowpower radio frequency rectifiers for use in wave meters or other
items of radar test equipment. Here the rectification properties of the
unit at the operating frequency are of primary interest. Since units which
are satisfactory as converters also function satisfactorily as highfrequency
rectifiers special types were not required for this application.
Units were also used in military equipment as detectors to derive directly
the envelope of a radio frequency signal received at low power levels.
These signals were modulated usually in the video range. The lowlevel
performance is a function of the resistance at low voltages and the direct
current output for a given lowpower radio frequency input. These may
be combined to derive a figure of merit which is a measure of receiver
performance.^
Typical directcurrent characteristics of the silicon rectifiers at tempera
tures of —40°, 25° and 70°C are given in Fig. 10. It will be noted in these
curves that both the forward and reverse currents are decreased by reducing
the temperature and increased by raising the temperature. The reverse
current changes more rapidly with temperature than the forward current,
however, so that the rectification ratio is improved by reducing the tempera
ture, and impaired by raising the temperature. The data shown are for
typical units of the converter type. It should be emphasized, however,
'' R. Beringer, Radiation Laboratory Report No. 6115, March 16, 1943.
22
BELL SYSTEM TECHNICAL JOURNAL
that by changes in processing routines the directcurrent characteristics
shown in Fig. 10 may be modified in a predictable manner, particularly
with respect to absolute values of forward current at a particular voltage.
Modern Rectifier Processes
When the development of the type 1N21 unit was undertaken, the scien
tific and engineering information at hand was insufficient to permit inten
tional alteration or improvement in electrical properties of the rectifier.
In these early units, the control of the radio frequency impedance, power
handling ability and signaltonoise ratio left much to be desired. Within
a short time, some improvements in performance were realized by process
improvements such as the elimination of burrs and irregularities from the
point contact to reduce noise. Substantial improvements were not obtained,
I0'
1 1
1 1
REVERSE
CURRENT
FORWARD
CURRENT
^^
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/
^
'r'
^^
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cu
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1
an
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JT 1
ics
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RE
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silicon (
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:ryE
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J
however, until certain improved materials, processes, and techniques were
developed.
In the engineering development of improved cr>'stal rectifier materials
and jjrocesses, basic data have been acquired which make it possible to
alter the properties of the rectifier in a predictable manner so that tlie units
may now be engineered to the specific electrical requirements desired by
the circuit designer in much the same manner as are modern electron tubes.
This has led not only to improvements in performance but also to a diver
sification in types and applications.
The simplified equivalent circuit for the point contact rectifier, shown
in Fig. 11, provides a basis for consideration of the various process features.
In Fig. 11, Cb represents the electrical capacitance at the boundary between
the point contact and the semiconductor, Rn the nonlinear resistance at
this boundary, and /^s is the spreading resistance of the semiconductor
SILICON CRYSTAL RECTIFIERS 23
proper, that is the total ohmic resistance of the siHcon to current through
the point. The capacitance Cb being shunted across the rectifying bound
ary, decreases the efficiency of the device by its bypass action because the
current through it would be dissipated as heat in the resistance Rs. Losses
from this source increase rapidly with increased frequency because of the
enhanced bypass action. It would appear, therefore, that to improve effi
ciency it would be important to minimize both Rs and Cb by some method
such as reducing the area of the rectifying contact and lowering the body
resistance of the silicon employed. For a given silicon material, the imped
ances desired for reasons of circuitry and considerations of mechanical stabiUty
place a limit on the extent to which performance may be improved by
reducing the contact area. Rs may be reduced by using silicon of lower
resistivity, but this generally results in poorer rectification. This impair
ment is due apparently to some subtle change in the properties of the
rectifying junction resulting from decreasing the specific resistance of the
silicon material.
Rg (NONLINEAR
BARRIER RESISTANCE)
Rs I WV
(SPREADING RESISTANCE)
vw
Cb
(barrier capacity)
Fig. 11 — Simplified equivalent circuit of crystal rectifier.
The answer to this apparent dilemma lies in the application of an oxidizing
heat treatment to the surface of the semiconductor. This process derives
from researches conducted independently in this country and in Britain,
though there was considerable interchange of information between various
interested laboratories. In the oxidizing treatment, apparently the im
purities in the silicon which contribute to its conductivity diffuse into the
adhering silica film, thereby depleting impurities from the surface of the
silicon. When the oxide layer is then removed by solution in dilute hydro
fluoric acid, the underlying silicon layer is exposed and remains intact as
the acid does not readily attack the silicon itself.
Since decreasing the impurity content of a semiconductor increases its
resistivity, the silicon surface has higher resistivity after the oxidizing
treatment than before. Thus by oxidation of the surface of low resistance
silicon it is possible to secure the enhanced rectification associated with
the high resistance surface layer, while by virtue of the lower resistivity
of the underlying material the PR losses through Rs are reduced.
24 BELL SYSTEM TECHNICAL JOVRXAL
In actual practice the i)roperties of the rectifier are governed by the
resistivity of the silicon material, the contact area, and the degree of oxida
tion of the surface. By the controlled alteration of these factors units
may be engineered for specific applications. The body resistance of the
silicon is controlled by the kind and quantity of the impurities present.
Aluminum, beryllium or boron may be added to purified silicon to reduce
its resistivity to the desired level. Boron is especially effective for this
purpose, the quantity added usually being less than 0.01 per cent. As little
as 0.001 per cent has a very pronounced effect upon the electrical properties.
The contact area is determined by the design of contact spring employed
and the deflection applied to it in the adjustment of the rectifier. The
degree of oxidation is controlled by the time and temperature of the treat
ment and the atmosphere employed.
In the development of the present rectifier processes, certain experimental
relationships were obtained between the performance and the contact area
on the one hand, and the power handling ability and contact area on the
other. These show the manner in which the processes should be changed
to produce a desired change in properties. For example. Fig. 12 shows the
relationship between the spring deflection applied to a unit and the conver
sion loss at a given frequency. The apparent contact area, (i.e., the area of
the flattened tip of the spring in contact with the silicon surface, as measured
microscopically) also increases with increasing spring deflection. It will be
seen in Fig. 12 that for a given silicon material, the conversion loss at 10,000
megacycles increases rapidly with the contact area. The curves tend to
reach constant loss values at the higher spring deflections. It is believed
that this may be ascribed to the fact that for a given spring size and form,
the increment in contact area obtained by successive increments in spring
deflection would diminish and finally become zero after the elastic limit of
the spring is exceeded.
The losses plotted in Fig. 12 were measured on a tuned basis, that is, the
converter was adjusted for maximum intermediate frequency output at a
fixed beating oscillator drive for each measurement. Were these measure
ments made on a fixed tuned basis, that is, with the converter initially ad
justed for maximum intermediate frequency output for a unit to which the
minimum spring deflection is applied, and the units with larger deflections
then measured without modification of the converter adjustment, even
greater degradation in conversion loss than that shown in Fig. 12 would be
observed. This results from the dependence of the radio frequency imped
ance upon the contact area. In loss measurements made on the tuned basis,
changes in the radio frequency impedance occasioned by the changes in the
contact area do not affect the values of mismatch loss obtained, while on the
SILICON CRYSTAL RECTIFIERS
25
fixed tuned basis they would result in an increase in the apparent loss be
cause of the mismatch of the radio frequency circuits.
While the conversion loss is degraded by increasing the contact area, the
power handling ability^ of the rectifiers is improved, as shown in Fig. 13.
FREQUENCY =
10,000 MEGACYCLES
Q
^^^
A
y
""'^
unitC
(
) /^
y
— 
J
/A
Y
^
i
I
\y
n
1 2 3 4 5 6 7 8
SPRING DEFLECTION IN THOUSANDTHS OF AN INCH
Fig. 12 — Relationshi]) between sjjring deflection and conversion loss in
silicon crystal rectifiers.
This is not surprising because the larger area contact gives a wider current
distribution and thus minimizes the localized heating effects near the con
tact. Generally, therefore, in the development of units for operation at a
*The measurement of power handling ability of crystal rectifiers by application of
radio freciuency jwwcr is comi)licated by the fact that the impedance of the unit under
test varies with power level. If a unit is matched in a converter at a lowpower level
and ]iower at a higher level is then applied, not all of the j^ower available is absorbed by
the unit but a portion of it is reflected (due to the change in impedance). This factor
has been called the self protection of the unit and it necessitates the distinction between
the powei absorbed hy and the power available to the unit under test. The data for
Fig. 13 were acquired by first matching the unit in converters at low powers (about 0.3
milliwatts CW 30C0 mc's) and then exposing it for a short period to successively higher
levels of pulse power cf sc[uare wave form of 0.5 microseconds width at a rei:)etition rate
of 20CO pulses per seccnd, measuring the loss and noise ratio after each power application.
The power handling ability is then expressed as the available peak power required to
cause a 3 db impairment in the conveision loss or the receiver noise figure. This method
was employed because in ladar receivers the units are matched for lowpower levels. In
this lespect the method simulates field operating conditions, but the "spike" of radar
pulses is absent.
The increase in power handling abilit\' with increasing area shown in Fig. 13 is confirmed
by similar measure ments with radio frequenc> pulse power with the unit matched at
highlevel powers, b\ directcurrent tests, and by simple 60cycle continuous wave tests.
The magnitude of the increase depends, however, upon the particular method employed
for measurement.
26
BELL SYSTEM TECHNICAL JOURNAL
given frequency, a compromise must be effected between these two impor
tant performance factors. Because of increased condenser bypass action a
smaller area must be used to obtain a given conversion loss at a higher fre
quency. For this reason the power handling ability of units designed for
use at the higher frequencies is somewhat less than that of the lowerfre
II) —
 >
uj O
Q. t
I UJ
2i
<l
UJ Z
<<
100
80
60

FREQUENCY=
3000 MEGACYCLES
•
•
»
•
•

•
• a
(

«
••

•
•
• •
1
•

1
•
■•>••
•
•
>
••
•
a
•
•


4
> t

• •
•

1
1
1
1
1
1
1
0.02
0.04 0.06 0.1 0.2 0.4 0.6 0.8 1.0
APPARENT CONTACT AREA IN SQUARE INCHES
XIO"
Fig. 13
Correlation between power handling ability measured with microsecond radio
frequency pulses and contact area in silicon crystal rectifiers.
quency units because emphasis has been placed upon achieving a given sig
naltonoise performance in each frequency band .
Use of the improved materials and processes produced rather large im
provements in the dc rectification ratio, conversion loss, noise, power
handling ability, and uniformity. Typical directcurrent rectification char
acteristics of units produced by both the old and the new processes are shown
in Fig. 14. These curves show that reverse currents at one volt were de
creased by a factor of about 20 while the forward currents were increased by
SILICON CRYSTAL RECTIFIERS
27
a factor of approximately 2.5 giving a net improvement in rectification ratio
of 50 to 1. The parallel improvement in receiver performance resulting from
process improvements is shown in Fig. 15. A comparison in power handling
ui a.
UJ u.
a. r
D \iS
il<
Q
UJ UJ
o tr
ZUJ
cnz
UJ (J
cr UJ
102
REVERSE
CURRENT
FORWARD
CURRENT
s/
^"^
'
,J
"^
y
' y
■x"*
■^
^
'■^
/
■^'
\0^ lO"'* 10"
CURRENT IN AMPERES
102
Fig. 14 — Improvement in the directcurrent rectification characteristics of
sihcon crystal rectifiers in a fouryear period.
10,000 MEGACYCLES
3000 MEGACYCLES
16
15
■ (/5 .■
■ UJ •■
• 1 '.
,■ CL ■
: </) ■■
14

■ 2 ;■
•; 1 •
;.cn.
. UJ ■•
• •Q ■•
■• o ■
•■ 2 •.
.■ <ri :.
■' J .•
■ ' tu ■'
• ; Q ■
\i
"
■.a. ■•
; ■ o . •
■.■••:•
■ 1 '■'■
•. O ■
'■ •" ■'.
;•<•;
■.. z .
• UJ •
■ ' t ■ .'
■ D •■
•■' CO ■. ■
12
'. tr •■
■.o■.•.
;(D •.
.< ■•
; ; J ._
:■' Q".
: z ■':
•■. Z '■
■■o:
;q: ;..
■! .■•
•z ..•
■<J ■■■
2
.' Q.
•■o ■.■
;•_!■.
•. UJ •■
■■ > ;■
.•■UJ .•
;■. Z ■■
• UJ ■,
■. 2 ■ •
.■• Q. ."
:.0 ■■
.. _i •
•• UJ ■,
'. Z '■
•' 2 ■
■,' 1 ■'■
:• o .■
:■ t ■:
:• z:
■■ ^.■■
.'z'..
; o ••
.'z ■■■
. O . ■'
■ ■ to
II
,■.■■ 01 :
>t
ctz
'OZ>
h ■
10
■ UJ • .
. H ■
:• u ;•
• UJ •:
'.!■.
•; Q •
• o •;
irz.
. ■ Q •.
■. _J ■.
■• cr. ■■
■. > •
•. UJ ■
;■ Q ;.
■ : > ■
•• O .
■ a.
■: Q .'•
. u ■•
• 1 ■•.
. o ■■■
m
ao
: < ■■
. _j ■
: <!■
• Q/
■ Q ■•■.;
orr
'.■ ^ .
•.■ cc '.
•: t^ '■'.
' a. ■
• CEO
■•' o ."
:o v
(CK
:o(r
.cr • ;
<z
.. UJ ■
, o ■
•o ■■
.cDi:
y

lO'
• o .
■■o •:
<^
■X
• o ..
vo ::•
.. o
■•fO .',
■_]0.
8
• • • 1 ■ ■
'.'.■.■•.•
". • ■ .
,
OCT
1942
DEC
1942
MAR APR
1945
SEPT
JAN JULY
SEPT
NOV
APR
1941
1942
1943
1944
1945
DATE
* note: — 6 DECIBELS IS THE MINIMUM RECEIVERNOISE
FIGURE ATTAINABLE WITH A DOUBLE DETECTION RE
CEIVER EMPLOYING A CRYSTAL CONVERTER AND A
5DB INTERMEDIATEFREQUENCY AMPLIFIER.
Fig. 15 — Effect of continued improvement in the crystal rectifier on the
microwave receiver performance. The noise figures plotted are average values.
ability of the 3000megacycle converter types made by the improved pro
cedures and the older procedures is shown in Fig. 16.
The flexibility of the processes may be illustrated by comparison of two
28
BELL SYSTEM TECHNICAL JOURNAL
very different units, tlie 1X26 and the 1N25. Though direct comparison of
power handling ability is complicated by the fact that the burnout test
methods employed in the de^•elopment of the two codes were widely different,
it may be stated conservatively that while the 1X26 would be damaged after
absorbing something less than one watt peak pulse power, the 1X25 unit
will withstand 25 watts peak or more. The 1X26 unit is, however, capable
of satisfactory operation as a converter at a frequency of some 20 times that
of the 1X25. These two units have been made by essentially the same pro
cedures, the difference in properties being principally due to modification of
alloy composition, heat treatment, and contact area.
u 6.0
a. 7.0
(j 8.0
6.2
7.2
8.2
9.2
notes: I. TEST FREQUENCY = 3000 MEGACYCLES
2. NOISE RATIO IS THE RATIO OF THE AVAILABLE OUTPUT
NOISE POWER OF THE CRYSTAL RECTIFIER TO KTB
IMPROVED PROCESS
""■^
~_
■^ V
1
1 1
1
1
I—
1 1
1
1 1
1
3.1 ^
 .
, III > 1 V 1 1 1 ^ ^ 1 111
■^ 6.3
> 7.3
O 8.3
^ 9.3
INITIAL PROCESS
1
^^ T^ =
:v
1
1 i
1
1 , 1,
v^
1
1 1
1
2.3
3.3
0.1 0.2 0.4 0.6 1.0 2 4 6 8 10 20 40 60 100 200
AVAILABLE PEAK PULSE POWER IN WATTS
Fig. 16 — Comparison of Uie radio fre([uency power handling aljilit_\ of silicon crystal
rectifiers prepared by different processes.
Prior to the process developments described above, in the interests of
simplifying the field supply problem one general purpose unit, the type 1X21 ,
had been made available for field use. However, it became obvious that the
advantages of having but a single unit for field use could be retained only at a
sacrifice in either power handling ability or highfrequency conversion loss.
Since the higher power radar sets operated at the lower microwave
frequencies, it seemed quite logical to employ the new processes to improve
power handling ability at the lower microwave frequencies and to impro\e
the loss and noise at the higher frequencies. A recommendation accordingly
w^as made to the Services that different units be coded for operation at v^OOO
megacycles and at 10, ()()() megacycles. The decision in the matter was
SILICOX CRYSTAL RECTIFIERS
29
INCREASING POWERHANDLING ABILITY
IN25
(14.7)
IN2IB
(12.2)
IN28
(13.2)
IN26
(15.2)
IN23B
(12.7)
IN23A
(14.2)
NEW PROCESS INTRODUCED
IN2IA
(14.6)
IN23
(17.1)
SELECTION
IN21
(16.4)
— NOTE— I
NUMBERS IN PARENTHESES ARE
RECEIVER NOISE FIGURES IN
DECIBELS CALCULATED FOR THE
POOREST UNIT ACCEPTABLE UNDER
EACH SPECIFICATION AND BASED
ON AN INTERMEDIATEFREOUENCy
AMPLIFIER NOISE FIGURE OF 5
DECIBELS '
24,000
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 17 — Evolution of coded silicon crystal rectifiers.
24

22
■M^:
20

18

16

14

12

10

8
6

4

2

* EXTRAPOLATED
_
[
■:\
1942 1943 1944 1945 *
YEAR
Fig. 18 — Relative annual production of silicon crystal rectifiers at the
Western Electric Company 19421945.
affirmative. The importance of this decision may be appreciated from the
fact that it permitted the coding and manufacture of units such as the 1X2 IB
and 1N28, high burnout units with improved performance at 3000
30 BELL SYSTEM TECHNICAL JOURNAL
megacycles, and the 1N23B unit which was of such great importance
in 10,000 megac3xle radars because of its exceptionally good performance.
From this stage in the development the diversification in types was quite
rapid. The evolution of the coded units, of increasing power handling
ability for a given performance level at a given frequency, and of better per
formance at a given frequency is graphically illustrated in Fig. 17. The
large improvements in calculated receiver performance are again evident,
especially when it is considered that the receiver performances given are
for the poorest units which would pass the production test limits.
Extent of Manufacture and Utilization
An historical resume of the development of crystal rectifiers would be
incomplete if some description were not given of the extent of their manu
facture and utilization. Commercial production of the rectifiers by Western
Electric Company started in the early part of 1942 and through the war years
increased very rapidly. Figure 18 shows the increase in annual production
over that of the first year. By the latter part of 1944 the production rate
was in excess of 50,000 units monthly. Production figures, however, reveal
only a small part of the overall story of the development. The increase in
production rate was achieved simultaneously with marked improvements in
sensitivity, the improvements in process techniques being reflected in manu
facture by the ability to deliver the higher performance units in increasing
numbers.
The recent experience with the silicon rectifiers has demonstrated their
utility as nonlinear circuit elements at the microwave frequencies, that they
may be engineered to exacting requirements of both a mechanical and elec
trical nature, and that they can be produced in large quantities. The defi
ciencies of the detector of World War I, which limited its utility and contribu
ted to its retrogression, have now been largely eliminated. It is a reasonable
expectation that the device will now find an extensive application in commu
nications and other electrical equipment of a nonmilitary character, at
microwave as well as lower frequencies, where its sensitivity, low capacitance,
freedom from aging effects, and its small size and lowpower consumption
may be employed advantageously.
Acknowledgements
Tlie development of crystal rectifiers described in this paper required the
cooperative effort of a number of the members of the staff of Bell Telephone
Laboratories. The authors wish to acknowledge these contributions and in
j)articular the contributions made by members of the Metallurgical group
and the Holmdel Radio Laboratory with wliom they were associated in the
development.
End Plate and Side Wall Currents in Circular Cylinder
Cavity Resonator
By J. p. KINZER and I. G. WILSON
Formulas are given for the calculation of the current streamlines and in
tensity in the walls of a circular cylindrical cavity resonator. Tables are
given which permit the calculation to he carried out for many of the lower
order modes.
The integration of / '.,,,'" dx is discussed; the integration is carried out for
Jo ^^'*'"»
C = \,2 and 3 and tables of the function are given.
The current distribution for a number of modes is shown by plates and figures.
Introduction
In waveguides or in cavity resonators, a knowledge of the electromagnetic
field distribution is of prime importance to the designer. Representations
of these fields for the lower modes in rectangular, circular and elliptical
waveguide, as well as coaxial transmission line, have frequently been de
scribed.
For the most i)art, however, these representations have been diagram
matic or schematic, intended only to give a general physical picture of the
fields. In actual designs, such as high Q cavities for use as echo boxes,^
accurately made plates of the distributions were found necessary to handle
adequately problems of excitation of the various modes and of mode sup
pression.
One use of the charts is to determine where an exciting loop or orifice
should be located and how the held should be oriented for maximum coup
ling to a particular mode. Optimum locations for both launchers and ab
sorbers can be found. Naturally, when attention is concentrated on a
single mode these will be located at the maximum current density points.
! If, however, two or more modes can coexist, and only one is desired, com
I promise locations can sometimes be found which minimize the unwanted
phenomena.
Also, in a cylindrical cavity resonator of high Q with diameter large com
pared with the operating wavelength, there are many high order modes of
j oscillation whose resonances fall within the design frequency band. Some
I of these are undesired and one of the objectives of a practical design is to
! reduce their responses to a tolerable amount. This process is termed
! ' "High Q Resonant Cavities for Microwave Testing," Wilson, Schramm, Kinzer,
I B.S.T.J., July 1946.
I 31
32
BELL SYSTEM TECIIMCAL JOURNAL
"suppression of the extraneous modes". In this process, an exact knowledge
of the distribution of the currents in the cavity walls has been found highly
useful.
For example, it has been found experimentally that annular cuts in the
end pliUes of the cylinder give a considerable amount of suppression to many
types of extraneous modes with very little effect on the performance of the
desired TE Oln mode. These cuts are narrow slits concentric with the axis
of the cylinder and going all the way through the metallic end plates into a
dielectric beyond. The physical explanation is that an annular slit cuts
through the lines of current fiow of the extraneous modes, and thereby
interrupts the radial component of current and introduces an impedance
which damps, or suppresses, the mode. For the TE Oln mode, the slits
TE Modes
TM Modes
Ph
C
W
II, = \'j'({k,p) COs(d
K 1
kl kip
kip
He = J'fikip) cos (d
1
v.
„ r .hJfikiD/2)''
^^'l^krkVDrr _
[sin (Q cos ^3 2I
IL ^ Jf(ki DID cos iQ sin ^3 s
He = J'f(ki D/2) cos (6 cos ^3 2
//. =
k = ^ ^ kl^ kl
A
^1
2r , _ nv
D ' ~ L
r = ;;;"' root of J f{x) = for TM Modes.
= m"' root of /;.(.v) = for TE Modes.
D = cavity diameter
L = cavity length
Fig. 1 — Components of H vector at walls of circular c_\ Under cavity resonator.
are parallel to the current streamlines and there is no such interruption;
presumably there is a slight increase in current density alongside the slit,
2 Similar cuts through the side wall of tlie cylinder in planes i)erpendicular to the
cylinder axis are also henctkial, hut are more troublesome mechanically.
CIRCULAR CYLINDER CAVITY RESONATOR 33
as the current formerly on the surface of the removed metal crowds over
onto the adjacent metal, but this is a secondorder effect.
To determine the best location of such cuts, therefore, it is necessary to
know the vector distributions of the wall currents for the various modes.
This current vector, /, is proportional to and perpendicular to the mag
netic vector, //, of the field at the surface. Expressions for the components
of the //vector at the surfaces of the end plates and side walls are given in
Fig. 1.
End Plate: Contour Lines
At the end plates, the magnitude of the //vector at any point is given by:
IP = H,' + lie'. (1)
Xow substitute values of Hp and He from Fig. 1 into (1); drop any constant
factors common to Hp and He as these can be swallowed in a final propor
tionality constant; introduce the new variable x:
X = kip = r ^. (2)
where R = D/2 = cavity radius. Thus is obtained;
//' = [J fix) cos (df +
 J fix) sin (6
X
(3)
Now Jf and Jf, are expressed in terms of Jf^i and Jf^i and a further re
duction leads to.
//"' = (//_ cos (d)' + iJf+ sin Cey (4)
where
Jf. = Jf.,ix)  Jf^.ix) (5)
and
Jf+ = Jf.r(x) + Jf,:ix) (6)
The formulas (4) to (6) apply to both TE and TM modes. The values
obtained depend on r, which is different for each mode.
When ^ = 0, / is proportional to Jf. and when 6 — ir/lf, I is proportional
to Jf+ . Relative values of / are thus easily calculated for these cases,
once tables of // are available. Such tables have been prepared and are
attached. For TE modes, when d = 0, He — 0, and the currents are all
in the 6 direction. For TM modes, when 6 = 0, Hp = 0, and the currents
are all in the pdirection. When d = tt/K, the converse holds.
Figures 3 to 18 are a set of curves showing the relative magnitude of H
(or /) for several of the lower order TE and TM modes. The abscissae
34 BF.Ll. SYSTEM TKCHNICAI. JOURNAL
are relative radius, i.e., p/R; the ordinates are relative magnitude referred
to the maximum value. The drawings also give r = ttD/Xc for each mode,
where Xc is the cutoff wavelength in a circular guide of diameter D. Values
for any point of the surface of the end plate can be calculated by using these
curves in Conjunction with equation (4).
In general, for each mode there are certain radii at which the current
flow is entirely radial, (/« =0). At these radii, which correspond to zeros
of Jt(x) or Jf(x), the annular cuts mentioned in the introduction are quite
effective. However, the maxima of Ip do not coincide with the zeros of
fe; and a more sophisticated treatment gives the best radius as that which
maximizes pip. X'alues of the relative radius for this last condition are
given in Table IV.
Contour lines of equal relative current intensity are obtained by setting
H^ constant in (4), which then expresses a relation between x and 6. The
easiest and quickest way to solve (4) is graphically, by plotting H vs. x for
different values of 6.
End Plate: Current Streamlines
It is easy to show that the equations of the current streamlines are given
by the solutions of the differential equation
Ie^~'Hp ^^^
In the case of the TE modes, (7) is easily solved by separation of the vari
ables, leading to the final result:
J((x) cos fd = C (8)
in which C is a i)arameter whose value depends on the streamline under
consideration. In the TE modes, the £lines in the interior of the cavity
also satisfy (8), hence a {)lot of the current streamlines in the end plate
serves also as a plot of the E lines.
In the case of the TM modes, (7) is not so easily solved. Separation of
the variables leads to:
f fJ({x)
logsm^^ = j ^j'^^dx. (9)
The righthand side of (9) can be reduced somewiiat, yielding
log sin te = log [xJt{x)\ + \ i/, dx (10)
J Jf(x)
but no further reduction is possible. The remaining integral represents a
new function which must be tabulated. Its ev^aluation is discussed at
CIRCULAR CYLINDER CAVITY RESONATOR 35
length in the Appendix, where it is denoted by Fi{x). Table II of the Ap
pendix gives its values (for ( — \, 2 and 3) and also those of G({x) where
Fi{x) = \ogG({x) (11)
Thus (10) becomes
log sin (d = log [x Jt{x)/G({x)] + C (12)
and the final equation for the current streamlines is
[xJt{x)/Gl{x)] sin (d ^ C (13)
where C is a parameter as before.
It is not difficult to show that G({x)/Jc{x) has zeros at the zeros of J((x).
For these values of x, sin €6=0 whatever the value of C, and all stream
lines converge on (or diverge from) 2(m points on the end plate.
The flow lines of (13) are orthogonal to the family (8) and could readily
be drawn in this manner. However, better accuracy is obtained by plotting
(13).
End Plate: Distributions
The 32 attached plates show the distribution of current in the end plates
of a circular cylinder cavity resonator for a number of modes.
In the first set of 21, the scaling is such that the diameters of the figures
are proportional to those of circular waveguides which would have the
same cutoff frequency. This group is of particular interest to the wave
guide engineer.
In a second group of 11, the scaling is such as to make the outside diam
eters of the cylinders uniform. This group is of particular interest to a
cavity designer.
This distribution is a vector function of position; that is, at each point in
the end plate the surface current has a different direction of flow and a dif
ferent magnitude or intensity. The variation in current intensity is repre
sented by ten degrees of background shading. The lightest indicates re
gions of least current intensity and the darkest greatest intensity. The
direction of current flow is shown by streamlines. Streamlines are lines
such that a tangent at any point indicates the direction of current flow at
that point.
The modes represented are the
r£ 01, 02, 03 TM 01,02, 03
r£ 11, 12, 13 TM U, 12, 13
TE 21, 22, 23 TM 21, 22
TE3l,32 TM3l,32
36 BELL SYSTEM TECHNICAL JOURNAL
in the nomenclature which has become virtually standard. In this system,
TE denotes transverse electric modes, or modes whose electric Lines lie
in planes perpendicular to the cylinder axis; TM denotes transverse mag
netic modes, or modes whose magnetic lines lie in transverse planes. The
first numerical index refers to the number of nodal diameters, or to the order
of the Bessel function associated with the mode. The second numerical
index refers to the number of nodal circles (counting the resonator boundary
as one such) or to the ordinal number of a root of the Bessel function asso
ciated with the mode. On the end plates, the distribution does not depend
upon the third index (number of half wavelengths along the axis of the cylin
der) used in the identiiication of resonant modes in a cylinder. This con
siderably simplifies the problem of presentation. The orientation of the
field inside the cavity and hence the currents in the end plate depend on
other things; thus the orientation of the figures is to be considered arbitrary.
The plates also apply to the corresponding modes of propagation in a cir
cular waveguide as follows: The background shading represents the in
stantaneous relative distribution of energy across a cross section of guide.
For TE modes, the current streamlines depict the E lines; for the TM
modes, they depict the projection of the E lines on a plane perpendicular
to the cylinder axis.
Side Wall:
The current distribution in the side walls is easily obtained from the
field equations of Fig. 1. For TM modes, the currents are entirely longi
tudinal; their magnitudes vary as cos (6 cos nirz/ L. This distribution is so
simple as not to require plotting.
For TE modes, the situation is more complicated, since both Hz and He
exist along the side wall. The current streamlines are given by the solu
tions of the differential equation
dz DHe ,.,.
de~2H/ ^^^^
By .separation of the variables, the solution is found to be
Contour lines of constant magnitude of the current are given by
\k\D
In the above, C and A' are j)arameters, different values of which correspond
to difTerent streamlines or contour lines, respectively.
log (C cos (6) =
log cos ksZ. (15)
2 ^ sin (d cos ksZj \ (cos fd sin k^z)' = K\ (16)
CIRC ULA R C I UNDER CA VIIY RESONA TOR 37
Since both streamlines and contours are periodic in z and 6, it is not
essential to represent more than is covered in a rectangular piece of the side
wall corresponding to quarter periods in :: and d. These are covered in a
L . ttD
length T~ along the cavity and in a distance ~t~ around the cavitv. If
2h 4'
such a piece of the surface be rolled out onto a plane it forms a rectangle
irnD
of proportions ~. .
The ditliculty in depicting the side wall currents of TE modes, as com
pared with the end plate currents, is now apparent. For the end plate, the
"proportions" are fixed as being a circle. Furthermore, for a given f, as
m increases the effect is merely to add on additional rings to the previous
streamline and contour plots. Here, however, the proportions of the rec
tangle are variable, in the first place. And for a given rectangle the stream
lines and contours both change as ( and )n are varied. Another way of ex
pressing the same idea is that for end plates the current distribution does
not depend upon the mode index n, and varies only in an additive way with
the index m, whereas for the side walls the distribution depends in nearly
equal strength on f, m and ;/.
Some simplification of the situation is accomplished by introducing two
new parameters, the "shape" and the "mode" parameters, defined by:
irnD (
S = — M=^ (17)
and two new variables
Z = hz <f> = (d. (18)
Substitution of the above, and also the expressions for k\ and ^3 (see Fig.
1) into (15) and (16) yields
cos Z = C(cos (/)) (streamlines) (19)
T2 2 . ni/2
cos Z
{S^M^ sin2 4>  cos <^).
(contours). (20)
For given proportions S, one can calculate the streamlines and contours for
various values of M. Thus a "square array" of side wall currents can be
prepared, such as shown on Fig. 2.
The mode parameter, if, in the physical case takes on discrete values
which depend on the mode. Some of its values are given in the following
table. They all lie between and 1 and there are an infinite number of
them.
38
BELL SYSTEM TECHNICAL JOURNAL
Valxjz ot a/ = l/r FOR TE Modes
t
1
2
3
4
5
6
10
15 '
20
m= 1
2
3
4
.5432
.1875
.1172
.0854
.6549
.2982
.2006
.1519
.7141
.3743
.2644
.2057
.7522
.4309
.3154
.2506
.7793
.4753
.3575
.2888
.8000
.5113
.3930
.3219
.8495
.6080
.4945
4209
.8813
.6774
.5730
.9001
For any given mode in any given cavity, the values of S and M can be
calculated from (17). In general, these values will not coincide with those
which have been plotted, but by the same token, they will lie among a group
of four combinations which have been plotted. Since the changes in dis
tribution are smooth, mental twoway interpolation will present no difficulty.
Acknowledgment
The final plates depicting the current distributions are the result of the
efforts of many individuals in plotting, spray tinting of the background,
inking of the streamlines on celluloid overlay, and photographing. Special
mention must be made, however, of the contribution of Miss Florence C.
Larkey, who carried out all the lengthy calculations of the tables hereto
attached and of the necessary data for the plotting.
1 d.
1 \j
/ 1
4\
j
4
'"^ ^^
\
/
/ 1
"'"7/
yi
\
or TE modes (/ > 0)
CIRCULAR CYLINDER CAVITY RESONATOR
39
U. UJ
oir
ZqC
r.o
0.8
^^
::^
~—
^^
.^...^
"^v^^
^V
Sw
He OR I^"*^".
^^^
0.6
\
AT e = 90"
^^^^
Hp OR leN.
0.4
0.2
ATe=0" >
V
^^=1.841
^C
N
\
\
V
N^
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 3 — End plate currents in TE 11 mode.
^y
'"^
He OR I,^ ■
(Hp 0Rle=0)
^
y
y
X
/
^° 2.405
Ac
/
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 4 — End plate currents in TM 01 mode.
Oa
^
"'^
He OR Ip
AT e = 45»
^
^
y
/
Hp OR le
AT e = 0»
X
^ = 3.054
^
/
\
0.3
0.7
Fig. 5
0.4 0.5 0.6
RELATIVE RADIUS
End plate currents in TE 21 mode
40
BELL SYSTEM TECHNICAL JOURNAL
"^^
"n
1
\
X
\
N
\
N
N
N
\
\
V HpOR le
^\aT e = 90°
\
^
\^
s
V Hq or Ip
\ AT e = 0"
\
s.
ID  3.832
K
\
v^
•J 0.4
0.4 0.5 O.a
RELATIVE RADIUS
Fig. 6 — End plate currents in TM 11 mode.
/
HpOR I^^V^
(He OR lp=0)
X
J
/
N
\
/
N
\
A
/
Ac
\
\
/
\
4
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 7 — End plate currents in TE 01 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
41
^
^'^HeOR Ip
ATG30»
^
im. 4.201
Xc
/
/
/
/^
•"y^po^. le
ATe^O"
V
y\
^
\
V
^^
y
K
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
RELATIVE RADIUS
Fig. 8 — End plate currents in T£ 31 mode.
/
^
/
/^
HpOR le
^ AT e=45«
\f
\
\
\
/
f
N
V He OR Ip
\ ATe:^
\
s.
/
\
\
N
\
^ = 5.,36
Ac
\
\
\
s.
V
. .
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 9 — End plate currents in TM 21 mode.
42
BELL SYSTEM TECHNICAL JOURNAL
2z
"■a
A
^
AT e22'/2°
^ = 5.3,e
Ac
/
/
/
/^
Hp OR le^
AT e=o°
V
.^'
^
y^
K
\ .
^
^
\
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 10 — End plate currents in TE 41 mode.
2 0.2
» 0.4
^
\^
\
\
\
\
""
He OR Ip
Sw AT 090"
\ ^P
OR le
e=o»
\
^~
/
N
\
/
/
''0=5.332
Ac
\
/
\^
y
/
0.4 0.5 0.6
RELATIVE RADIUS
1.0
Fig. 11 — End plate currents in TE 12 mode.
* ^
\
/
r
\
\
/
\
^
\
\
>
\ HeORlp
TTD
^r — 5.520
Ac
\
V
K
(HpOB Ie=0)
0.2
0.3
0.7
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 12 — End plate currents in TM 02 mode.
/
X
HpOR le
ATe = 30«
/
\
A
/ ^
"^
He OR Ip
V AT 6=0°
\
y
r
\
\
\
y
\
\
\
^ = 6.3eo
Ac
\
\
k
\
\
^^
0.3
0.7
Fig. 13
0.4 . 0.5 0.«
RELATIVE RADIUS
End plate currents in TM 31 mode.
43
0.8
0.9
1.0
44
BELL SYSTEM TECHNICAL JOURNAL
t 0.4
/
/^eOR Ip
AT 6 = 18*
/
/
/
/
1
/^
^
•"Hp OR le
AT 6=0"
N
\,
.
^^
^
\
0.4 0.5 0.6
RELATIVE RADIUS
Fig. 14 — End plate currents in TE 51 mode.
0.3
0.4 0.5 0.6
RELATIVE RADIUS
0.7
0.1 0.2
Fig. 15 — End plate currents in TE 22 mode.
Z 0.8
I 0.4
^
^1
\
\
He OR \p
AT e = o» .
^
\
N
\
>
\^
\
V ^P
OR le
e=9o«
/
\.
^
/
\
"^
/
^^
/
/
/
^ = 7.016
Ac
\^
^y
/
0.2 0.3
0.4 0.5 0.6 0.7 O.a 0.9 1.0
RELATIVE RADIUS
Fig. 16 — End plate currents in TM 12 mode.
5 0.2
5 0.6
/
^ ^
\
/
\
\
\
(He OR lp = 0)
Hp OR Ie\
\
\
/
\
\
J
/
P= 7.016
\
^^
y
04 0.5 0.6
RELATIVE RADIUS
Fig. 17 — End plate currents in TE 02 mode.
45
4(3
BELL SYSTEM TECHNICAL JOURNAL
< to.e
Oct
^3 0.4
0.2
y
>^eOR Ip
''^ AT 9 = 15'
il^= 7.501
J
/
/
k^
y^p OR le
AT 9=0°
"N
\
.^
k"
^
\
0.3
0.7
Fig.. 18
0.4 0.5 0.6
RELATIVE RADIUS
End plate currents in TE 61 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
47
Fig. 19— TE 01 mode.
Fig. 20— TE 02 mode.
48
BELL SYSTEM TECHNICAL JOURNAL
Fig. 21— TE 03 mode.
Fig. 22— TK 11 mode.
Fig. 23— TE 12 mode.
Fig. 24— TE 13 mode.
49
50
BELL SYSTEM TECHNICAL JOURNAL
Fig. 25^TE 21 mode.
Fig. 26 — TE 22 mode.
Fig. 27— TE 23 mode.
Fig. 28— TE 31 mode.
51
52
BELL SYSTEM TECHNICAL JOURNAL
Fig. 29 — TE il mode.
Fig. 30— TM 01 mode.
Fig. 31— TM 02 mode.
Fig. 32— TM 03 mode.
53
54
BELL SYSTEM TECHNICAL JOURNAL
Fig. 33— TM 11 mode.
Fig. 34— TM 12 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
55
Fig. 35— TM 13 mode.
a__
Fig. 37 TM 11 mode.
56
CIRCULAR CYLINDER CAVITY RESONATOR
57
Fig. 38— TM 31 mode.
58
BELL SYSTEM TECHNICAL JOURNAL
Fig. 39— TM il mode.
CIRCULAR CYLIXDER CAVITY RESONATOR
59
Fig. 40— TE 11 mode.
60
BELL SYSTEM TECHNICAL JOURNAL
Fig. 41— TE 12 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
61
Fig. 42— TE 13 mode.
62
BELL SYSTEM TECHNICAL JOURNAL
Fig. 43— TE 21 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
63
Fig. 44— TE 22 mode.
64
BELL SYSTEM TECHNICAL JOURNAL
Fig. 45— TE 31 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
65
Fig. 46— TE 32 mode
66
BELL SYSTEM TECHNICAL JOURNAL
Fig. 47— TM 11 mode.
CIRCULAR CYLINDER CAVITY RESONATOR
67
Fig. 48— TM 12 mode.
68
BELL SYSTEM TECHNICAL JOURNAL
Fig. 49— TM 21 mode
CIRCULAR CYLINDER CAVITY RESONATOR
69
Fig. 50— TM 22 mode.
70 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX
/•■'" J fix)
INTEGRATION OF / 777 dx
The discussion here is concerned only with integral values of ^ > 0. The
integral is not simply expressible in terms of known (i.e., tabulated) func
tions, hence what amounts to a series expansion is used. The method
follows Ludinegg^ who gives the details for ^ = 1.
The value of the integrand at :r = is first discussed. For ^ = 1 , /i(0) =
and /i(0) = 0.5, hence the integrand has the value zero. For I > \,
both numerator and denominator are zero, hence the value is indeterminate.
Evaluation by (f — 1) differentiations of numerator and denominator
separately leads to the result that the integrand (and the integral also) is
zero at X = for all C.
We now introduce a constant p(. and a function 4>({x) which are such
that the following equation is satisfied, at least for a certain range of values
of x:
Ji= pcij'i^^^^^^ + <i>tJl (1)
Denote the desired integral by F(.{x), i.e.:
Then substitution of (1) into (2) yields:
F( = pC
log
For X = 0, J (/ x^ ^ is indeterminate, but evaluation by difTerentiating
numerator and denominator separately (/' — 1) times gives the value
iM^l)!
If we can now arrange matters so that 4>c remains finite in the range
(0, x), its integration can be carried out, a) by expansion into a power
series and integration termbyterm, or, b) by numerical integration.
Solving (1) for (j)C one obtains
«= ^, ^^. (4):
Jf
Equation (4) becomes indeterminate at .v = 0, when (■ > \. Evaluation by
differentiating numerator and denominator separately € times shows </)^(0) = 0.
> Uoclifrcqiicnztech. u. Elckhoak., V. 62, j)]). .VS44, .Auk 1943.
CIRCULAR CYLINDER CAVITY RESONATOR 71
At the first zero of Je (the value of x at a zero of j'i will be denoted by r),
4>l is held finite by choice of the value of p( . It is clear that (4) becomes
indeterminate at x = r, if
Since // satisfies the differential equation
j7 + j(h {1  fyx')j( = (6)
X
and J(ir) = 0, one has by substitution
Values of p for several cases are:
^=1234 1 1
n = 1.841 3.054 4.201 5.318 rz = 5.331 r, = 8.536
pf = 1.418 1.751 2.040 2.303 1.036 1.014
4>iir)=0.n6 0.286 0.446 0.604 0.180 0.115
Evaluation of 4>f{r) by the usual process gives:
Mr^ ^ S^l^ (S)
Values of (f)({r) are given in the preceding table.
Since <p( is finite at the origin and at the first zero of Jf , it may be ex
panded into a Maclaurin series whose radius of convergence does not,
however, exceed the value of x at the second zero of J( . Alternatively,
by choosing p{ to keep <^f finite at the second (or'^"") zero of J( it may be
expanded into a Taylor series about some point in the interval between
the first (or (k — 1)"') and third (or (k + l)"") zeros. Expansions about the
origin are given in Table I.
Unfortunately, the convergence of these power series is so slow that they
are not very useful. Instead, equation (4) is used to calculate (l>( and
/ 4>( dx is obtained by numerical integration.
With pt fixed to hold 4>( finite at the first root, f i , of J( , it is soon found
that 4>f becomes infinite at the higher roots. This is because different values
Substitute (6) into (4) to eliminate JJ; dilTerentiate numerator and denominator
separately; use (6) to eliminate J^; allow x — > r, using J'Ar) = and value of p^ from (7).
72 BELL SYSTEM TECHNICAL JOURNAL
of p are required at the difl"erent roots, as shown for ( — 1 in the table
above. A logical extension would therefore be to make p a function of .v
such that it takes on the required values at ri , rj , rs , • • • . When this is
done and p({x) is introduced into (1) and (2), one has to integrate
/
K.v)/"(..) ,,^
and this is intractable.
Hence p{x) is made a discontinuous function, such that p has the value
pi corresponding to ;'i for values of .v from zero to a point bi between ri and
ri ; the value p2 corresponding to r^ for values of .v from bi to a point bi be
tween ri and rs; and so forth. This introduces discontinuities in </>. No
discontinuities exist, however, in the function
G( = e~'( (9)
which is given in Table II. The calculations were made by Miss F. C.
Larkey; numerical integration was according to Weddle's rule.
Within the limits of this tabulation, then, G( and F( are now considered
to be known functions.
Table I
Power Series Expansions of 4>t{x)
/ ^p\ /I 17A /7 19p\
,,,,, , (■ , _ _f j , + (^^ _ ^ j ,. + (^  _ j ...+ .,.
= 0.063813.V 0.001 178x3 0.0000358.v5 _ ...
*,W   ^) .V + (i  '^^ .V. + [^  ^^ V + . . .
= +0.15451.V +0.01648.r'  O.OO.SSO.v^  ••■
/! Sp\ ( \ 41/. \ / 13 103/> \
'^^^■^' = (i  2ij ^ + Vn  5760 j ^"^ + (,17280 " 276480 j "^ +
= +0.12210.V +0.00667.V' +0.00375.vS  ••• .
^ Unless p = b + cJ' {b and c constants), which is not of any use.
CIRCULAR CYLINDER. CAVITY RESONATOR
73
Table II
r Ji ix)
Values OF FiU) = / — dx;G,{x) = e^^i
Jo '^i(^"''
F,{x)
y
.1
.2
.3
.4
.5
1291
.6
.7
.8
.9
0050
0201
0455
0816
1887
2616
3493
4539
1
5782
7261
9036
1.1192
1.3874
1.7336
2.2103
2.9577
4.6961
4.1846
2
2.7727
2.0801
1.6199
1.2775
1.0073
7864
6018
4454
3117
1970
3
0987
0147
0564
1157
1640
2018
2296
2475
2556
2537
4
2416
2188
1845
1377
0769
+0960
2153
3646
5549
5
8060
1.1595
1.7307
3.2014
2.3851
1.4478
9635
6373
3939
2024
6
0470
0812
1879
2768
3506
4111
4594
4966
5233
5398
7
5463
5429
5292
5049
4693
4214
3598
2826
 1868
0685
8
+0789
2657
5107
8530
1.3992
2.7313
2.1565
1 . 1974
7154
3942
9
1562
0300
1802
3034
4053
4897
5590
6150
6591
6921
G,{x)
.1
•2
.3
.4
.5
.6
.7
.8
1.0000
9950
9801
9555
9216
8789
8280
7698
7052
5609
4838
4051
3265
2497
1766
1097
0519
0091
0625
1249
1979
2787
3652
4555
5478
6406
7322
9060
9854
1.0580
1.1226
1.1781
1.2236
1.2581
1.2808
1.2912
1.2733
1.2445
1.2026
1.1476
1.0799
1.0000
9085
8063
6945
4467
3136
1772
0407
0921
2351
3816
5287
6744
9541
1.0846
1.2067
1.3190
1.4200
1.5084
1.5831
1.6432
1.6877
1.7269
1.7209
1.6976
1.6568
1.5989
1.5241
1.4331
1.3265
1.2054
9241
7667
6001
4261
2468 0813
1157
3020
4890
8554
1.0304
1.1974
1.3545
1.4998
1.6318
1..7489
1.8497
1.9330
6351
0152
8212
1.2888
5741
8168
1.7157
1.0709
6742
1.9978
74
BELL SYSTEM TECHNICAL JOURS A L
Valuks ok Fi{x)
rMx)
Jo J^ix]
dx; (l,{x) = e"':
F,{x)
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
0025
0100
0226
0403
0632
0914
1251
1645
2097
1
2612
3192
3840
4563
5365
6253
7236
8323
9528
1.0866
2
1.2357
1.4008
1.5913
1.80C1
2.0541
2.3456
2.0972
3.1380
•3.7263
4.6110
3
6.4527
6.7644
4.7528
3.8572
3.2808
2.8597
2.5316
2.2658
2.0451
1.8590
4
1.7002
1.5641
1.4470
1.3466
1.2607
1.1881
1 . 1275
1.0783
1.0396
1.0112
5
9928
9843
9858
9974
1.0190
1.0530
1.0985
1.1573
1.2311
1.3223
6
1.4345
1.5726
1.7447
1.9040
2.2555
2.6743
3.3910
6.5119
3.5122
2.7144
7
2.2595; 1.9432
1.7034
1.5131
1.3579
1.2294
1 . 1223
1.0328
.9586
.8977
S
.84901 .8115
.7846
.7679
.7612
.7615
.7779
.8020
.8372
.8845
y
.9452; 1.0212
1
1.1149
1.2301
1.3725
1.5512
1.7817
2.0950
2.5660
3.4864
.V
1.0000
.1
9975
.2
.3
.4
.5
.6
.7
.8
8483
.y
9900
9777
9605
9388
9127
8824
8108
1
7701
7267
6811
6336
5848
5351
4850
4350
3856
3373
2
2906
2459
2036
1643
1282
0958
0674
0434
0241
0099
3
0017
0012
0086
0211
0376
0573
0795
1037
1294
1558
4
1826
2093
2353
2601
2834
3048
3238
3402
3536
3638
5
3705
3737
3731
3688
3607
3489
3334
3143
2920
2665
6
2383
2075
1747
1403
1048
0690
0337
0015
0298
0662
7
1044
1432
1821
2202
2572
2925
3255
3560
3834
4075
8
4278
4442
4563
4640
4671
4656
4593
4484
4329
4129
9
8886
3602
3280
2923
2535
2120
1683
1231
0768
0306
CIRCULAR CYLINDER CAVITY RESONATOR
75
Values ok Fs(x)
Gi{x) = e'^i
X
.1
.2
.3
0152
.4
.5
.6
0604
.7
.K
M
0017
0067
0268
0420
0826
1081
1373
1
1703
2070 2476
2922; 3410
3942
4518
5141
5814
6539
2
7319
8158 9060
1.00281 1.1070
1.2192
1.3401
1.4706
1.6118
1.7650
3
1.9321
2.1150
2.3165
2.5402 2.7908
3.0752
3.4034
3.7905
4.2624
4.8669
4
5.7117
7.1373
16.2303
7.2383 5.8409
5.0409
4.4852
4.0843
3.7292
3.4543
5
3.2239
3.0282
2.8605
2.7160 2.5913
2.4838
2.3914
2.3128
2.2467
2.1922
6
2.1487
2.1156
2.0927
2.0798
2.0768
2.0838
2.1012
2.1293
2.1685
2.2208
7
2.2864
2.3674
2.4664
2.5868
2.7340
2.9159
3.1460
3.4491
3.8790
4.5950
8
6.9408
4.9414
4.0348
3.5348
3.1912
2.9324
2.7276
2.5608
2.4227
2.3074
9
2.2108
2.1302
2.0637
2.0097 1.9676
1.9361
1.9147
1.9036
1.9025
1.9115
G,{x)
X
.1
9983
.2
.3
.4
9734
..s
9589
.6
.7
.8
.9
1.0000
9933
9849
9413
9208
8975
8717
1
8434
8130
7806
7466
7110
6742
6365
5980
5591
5200
2
4810
4423
4041
3668
3305
2955
2618
2298
1995
1712
3
1448
1206
0986
0789
0614
0462
0333
0226
0141
0077
4
0033
0008
0000
0007
0029
0065
0113
0172
0240
0316
5
0398
0484
0572
0661
0749
0834
0915
0990
1057
1117
6
1166
1206
1233
1250
1253
1244
1223
1189
1143
1085
7
1016
0937
0849
0753
0650
0542
0430
0318
0207
0101
8
0010
0071
0177
0292
0411
0533
0654
0772
0887
0995
9
1096
1188
1270
1340
1398
1443
1474
1490
1492
1479
Table III
Bkssel FiNCTioN.s OF The First Kind
/o{x)
X
.0
.1
.2
.3
.4
9604
.5
.6
.7
.8
8463
.9
+ 1.0
9975
9900
9776
+9385
9120
8812
8075
1
+7652
7196
6711
6201
5669
+5118
4554
3980
3400
2818
?.
+2239
1666
1104
0555
0025
0484
0968
1424
1850
2243
3
2601
2921
3202
3443
3643
3801
3918
3992
4026
4018
4
3971
3887
3766
3610
3423
3205
2961
2693
2404
2097
5
1776
1443
1103
0758
0412
0068
+0270
+0599
+0917
+ 1220
fi
+ 1506
1773
2017
2238
2433
+2601
2740
2851
2931
2981
7
+3001
2991
2951
2882
2786
+2663
2516
2346
2154
1944
8
+ 1717
1475
1222
0960
0692
+0419
0146
0125
0392
0653
9
0903
1142
1367
1577
1768
1939
2090
2218
2323
2403
Jdx)
+0
+4401
+5767
+3391
0660
3276'
6 I 2767
7 0047
8 +2346
9 +2453
.1
.2
.3
.4
1960
.5
.6
2867
.7
.8
0499
0995
1483
+2423
3290
3688
4709
4983
5220
5419
+5579
5699
5778
5815
5683
5560
5399
5202
+4971
4708
4416
4097
30()i)
2613
2207
1792
+ 1374
0955
0538
0128
1033
1386
1719
2028
2311
2566
2791
2985
3371
3432
3460
3453
3414
3343
3241
3110
2559
2329
2081
1816
1538
1250
0953
0652
+0252
+0543
+0826
+ 1096
+ 1352
1592
1813
2014
2476
2580
2657
2708
+2731
2728
2697
2641
2324
2174
2004
1816
+ 1613
1395
1166
0928
1
4059
5812
3754
0272
3147
2951
0349
2192
2559
0684
Jiix)
X
.0
.1
.2
.3
.4
.5
.6
.7
0588
.8
.9
+0
0012
0050
0112
0197
0306
0437
0758
0946
1
+ 1149
1366
1593
1830
2074
2321
2570
2817
3061
3299
?,
+3528
3746
3951
4139
4310
4461
4590
4696
4777
4832
3
+4861
4862
4835
4780
4697
4586
4448
4283
4093
3879
4
+3641
3383
3105
2811
2501
2178
1816
1506
1161
0813
5
+0466
0121
0217
0547
0867
1173
1464
1737
1990
2221
6
2429
2612
2769
2899
3001
3074
3119
3135
3123
3082
7
3014
2920
28(X)
2656
2490
2303
2097
1875
1638
1389
8
1130
0864
0593
0320
0047
+0223
0488
0745
0993
1228
9
+ 1448
1653
1840
2008
2154
2279
2380
2458
2512
2542
Jz{x)
X
.0
.1
.2
.3
0006
.4
.5
.6
0044
.7
0069
.8
0102
.9
+0
0002
0013
0026
0144
1
+0196
0257
0329
0411
0505
0610
0725
0851
0988
1134
2
+ 1289
1453
1623
1800
1981
2166
2353
2540
2727
2911
3
+3091
3264
3431
3588
3734
3868
3988
4092
4180
4250
4
+4302
4333
4344
4333
4301
4247
4171
4072
3952
3811
5
+3648
3466
3265
3046
2811
2561
2298
2023
1738
1446
6
+ 1148
0846
0543
0240
0059
0353
0641
0918
1185
1438
7
1676
1896
2099
2281
2442
2581
2696
2787
2853
2895
8
2911
2903
2869
2811
2730
2626
2501
2355
2190
2007
9
1809
1598
1374
1141
0900
0653
0403
0153
+0097
+0343
76
Ja{x)
X
.0
.1
.2
.3
.4
.5
.6
.7
0006
.8
.9
+0
0001
0002
0003
0010
0016
1
+0025
0036
0050
0068
0091
0118
0150
0188
0232
0283
2
+0340
0405
0476
0556
0643
0738
0840
0950
1037
1190
3
+ 1320
1456
1597
1743
1892
2044
2198
2353
2507
2661
4
+2811
2958
3100
3236
3365
3484
3594
3693
3780
3853
5
+3912
3956
3985
3996
3991
3967
3926
3866
378S
3691
6
+3576
3444
3294
3128
2945
2748
2537
2313
2077
1832
7
+ 1578
1317
1051
0781
0510
0238
0031
0297
0557
0810
8
1054
1286
1507
1713
1903
2077
2233
2369
2485
2581
9
2655
2707
2736
2743
2728
2691
2633
2553
2453
2334
J,(x)
X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
+0
0001
0001
1
+0002
0004
0006
0009
0013
0018
0025
0033
0043
0055
2
+0070
008S
0109
0134
0162
0195
0232
0274
0321
0373
3
+0430
0493
0562
0637
0718
0804
0897
0995
1098
1207
4
+ 1321
1439
1561
1687
1816
1947
2080
2214
2347
2480
5
+2611
2740
2865
2986
3101
3209
3310
3403
3486
3559
6
+3621
3671
3708
3731
3741
3736
3716
3680
3629
3562
7
+3479
3380
3266
3137
2993
2835
2663
2478
2282
2075
8
+ 1858
1632
1399
1161
0918
0671
0424
0176
0070
0313
9
0550
0782
1005
1219
1422
1613
1790
1953
2099
2229
/6(X)
X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
0001
0001
0002
0002
0003
0005
0007
0009
2
0012
0016
0021
0027
0034
0042
0052
0065
0079
0095
3
0114
0136
0160
0188
0219
0254
0293
0336
0383
0435
4
0491
0552
0617
0688
0763
0843
0927
1017
1111
1209
5
1310
1416
1525
1637
1751
1868
1986
2104
2223
2341
6
2458
2574
2686
2795
2900
2999
3093
3180
3259
3330
7
3392
3444
3486
3516
3535
3541
3535
3516
3483
3436
8
3376
3301
3213
3111
2996
2867
2725
2571
2406
2230
9
2043
1847
1644
1432
1215
0993
0768
0540
0311
0082
J7{X)
X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
0001
0001
0001
2
0002
0002
0003
0004
0006
0008
0010
0013
0016
0020
3
0025
0031
0038
0047
0056
0087
0080
0095
0112
0130
4
0152
0176
0202
0232
0264
0300
0340
0382
0429
0479
5
0534
0592
0654
0721
0791
0866
0945
1027
1113
1203
6
1296
1392
1491
1592
1696
1801
1908
2015
2122
2230
7
2336
2441
2543
2643
2739
2832
2919
3001
3076
3145
8
3206
3259
3303
3337
3362
3376
3379
3371
3351
3319
9
3275
3218
3149
3068
2974
2868
2750
2620
2480
2328
77
V'i(x)
J
.0
.1
.2
4925
.3
.4
.5
.6
.7
.8
.9
+5000
4981
4832
4703
4539
4342
4112
3852
3565
1
+3251
2915
2559
2185
1798
1399
0992
0581
0169
0241
2
0645
1040
1423
1792
2142
2472
2779
3060
3314
3538
3
3731
3891
4019
4112
4170
4194
4183
4138
4059
3948
4
3806
3635
3435
3210
2962
2692
2404
2100
1782
1455
5
1121
0782
0443
0105
+0227
+0552
0867
1168
1453
1721
6
+ 1968
2192
2393
2568
2717
2838
2930
2993
3027
3032
7
+3007
2955
2875
2769
2638
2483
2307
2110
1896
1666
8
+ 1423
1169
0908
0640
0369
0098
0171
0435
0692
0940
9
1176
1398
1604
1792
1961
2109
2235
2338
2417
2472
J'2ix}
X
.0
.1
.2
0497
.3
.4
.5
.6
.7
1610
.8
.9
+0
0250
0739
0974
1199'
1412
1793
1958
1
+2102
2226
2327
2404
2457
2485
2487
2463
2414
2339
2
+2239
2115
1968
1799
1610
1402
1178
0938
0685
0422
3
+0150
0128
0409
0691
0971
1247
1516
1777
2026
2261
4
2481
2683
2865
3026
3165
3279
3368
3432
3469
3479
5
3462
3419
3349
3253
3132
2988
2821
2632
2424
2199
6
1957
1702
1436
1161
0879
0592
0305
0018
+0266
+0544
7
+0814
1074
1321
1553
1769
1967
2144
2300
2434
2543
8
+2629
2689
2725
2734
2719
2679
2614
2526
2415
2283
9
+2131
1961
1774
1572
1358
1133
0899
0659
0416
0170
A{x)
X
.0
.1
.2
.3
0056
.4
0098
.5
.6
.7
.8
0374
.9
+0
0006
0025
0152
0217
0291
0465
1
+0562
0665
0772
0881
0991
1102
1210
1315
1415
1508
2
+ 1594
1671
1737
1792
1833
1861
1875
1873
1855
1821
3
+ 1770
1703
1619
1519
1403
1271
1125
0965
0793
0609
4
+0415
0212
0003
0213
0432
0653
0874
1094
1310
1520
5
1723
1918
2101
2272
2429
2570
2695
2801
2889
2956
6
3003
3028
3031
3013
2973
2911
2828
2724
2600
2457
7
2296
2118
1925
1719
1500
1270
1033
0789
0540
0289
8
0038
+0211
+0457
+0696
+0928
+ 1150
1360
1557
1739
1904
9
+2052
2180
2288
2376
2441
2485
2507
2506
2483
2438
J[{x)
X
.0
.1
.2
0001
0003
.4

0007
.5
.6
0022
.7
.8
.9
+0
0013
0034
0051
0071
1
+0097
0126
0161
0201
0246
0296
0350
0409
0473
0539
2
+0610
0GS2
0757
0833
0909
0985
1060
1133
1203
1269
3
+ 1330
1385
1434
1475
1508
1532
1545
1549
1541
1522
4
+ 1490
1447
1391
1323
12431
1150
1045
0929
0802
0665
5
+0518
0363
0200
0030
0145
0324
0506
0690
0874
1057
6
1237
1412
1582
1745
1900
2045
2178
2299
2407
2500
7
2577
2638
2683
2709
2718!
2708
2679
2633
2568
2485
8
2385
2267
2134
1986
1824'
1649
1462
1265
1060
0847
9
0629
0408
0184
+0039
+0261
+0480
0694
0900
1098
1286
78
CIRCULAR CYLINDER CAVITY RESONATOR
79
/5(X)
X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
+0
0001
0002
0003
0005
0008
1
+0012
0018
0025
0034
0045
0058
0073
0092
0113
0137
2
+0164
0194
0228
0265
0305
0348
0394
0443
0494
0548
3
+0603
0660
0718
0777
0836
0895
0952
1008
1062
1113
4
+1160
1203
1242
1274
1301
1321
1333
1338
1335
1322
5
+1301
1270
1230
1180
1120
1050
0970
0881
0782
0675
6
+0559
0435
0304
0166
0023
0126
0278
0433
0591
0749
7
0907
1064
1217
1368
1513
1652
1783
1906
2020
2123
8
2215
2294
2360
2412
2449
2472
2479
2470
2446
2405
9
2349
2277
2190
2088
1972
1842
1700
1546
1382
1208
A{x)
X
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
+0
0001
1
+0001
0002
0003
0004
0006
0009
0012
0016
0021
0027
2
+0034
0043
0053
0065
0078
0094
0111
0130
0152
0176
3
+0202
0231
0262
0295
0331
0368
0408
0450
0493
0538
4
+0585
0632
0680
0728
0776
0823
0870
0916
0959
1000
5
+1039
1074
1105
1132
1155
1172
1183
1188
1187
1178
6
+ 1163
1139
1108
1069
1022
0967
0904
0833
0753
0666
7
+0572
0470
0362
0247
0127
0002
0128
0261
0397
0535
8
0674
0813
0952
1088
1222
1352
1478
1597
1710
1816
9
1912
2000
2077
2143
2198
2240
2270
2287
2290
2279
Table IV
Relative Radius for Maximum of pll
Mode
TE 11
.737
12
.982
.254
13
.993
.613
.159
21
.894
22
.988
.407
23
.995
.664
.274
31
.937
32
.991
.491
41
.956
42
.993
.548
51
.967
CI
.974
TM 01
.901
02
.983
.393
03
.993
.627
.250
11
.961
12
.989
.525
13
.995
.682
.362
21
.977
22
.992
.596
31
.984
32
.994
.643
41
.988
51
.990
61
.992
1
First and Second Order Equations for Piezoelectric
Crystals Expressed in Tensor Form
By W. P. MASON
Introduction
AEOLOTROPIC substances have been used for a wide variety of elastic
piezoelectric, dielectric, pyroelectric, temperature expansive, piezo
optic and electrooptic effects. While most of these effects may be found
treated in various publications there does not appear to be any integrated
treatment of them by the tensor method which greatly simplifies the method
of writing and manipulating the relations between fundamental quantities.
Other short hand methods such as the matrix method can also be used for
all the linear effects, but for second order effects involving tensors higher
than rank four, tensor methods are essential. Accordingly, it is the purpose
of this paper to present such a derivation. The notation used is that agreed
upon by a committee of piezoelectric experts under the auspices of the Insti
tute of Radio Engineers.
In the first part the definition of stress and strain are given and their inter
relation, the generalized Hookes law is discussed. The modifications caused
by adiabatic conditions are considered. When electric fields, stresses, and
temperature changes are applied, there are nine first order effects each of
which requires a tensor to express the resulting constants. The effects are
the elastic effect, the direct and inverse piezoelectric effects, the temperature
expansion effect, the dielectric effect, the pyroelectric effect, the heat of
deformation, the electrocaloric effect, and the specific heat. There are
three relations between these nine effects. Making use of the tensor trans
formation of axes, the results of the symmetries existing for the 32 types of
crystals are investigated and the possible constants are derived for these
nine effects.
Methods are discussed for measuring these properties for all 32 crystal
classes. By measuring the constants of a specified number of oriented cuts
for each crystal class, vibrating in longitudinal and shear modes, all of the
elastic, dielectric and piezoelectric constants can be obtained. Methods
for calculating the properties of the oriented cuts are given and for deriving
the fur.damental constants from these measurements.
1 For example Voigt, "Lehrl)uch der Kiistall Physik," B. Tcul)ner, 1910; Wooster,
"Crystal Physics," Cainl)ridge Press, 1938; Cady "Piezoelectricity" McGraw Hill, 1946.
* The matrix method is well described 1)V W. L. Bond "The Mathematics of the Ph\sical
Properties of Crystals," B. S. T. J., Vol. 22, pp. 172, 1943.
80
PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 81
Second order effects are also considered. These eflfects (neglecting second
order temperature eflfects) are elastic constants whose values depend on
the applied stress and the electric displacement, the electrostrictive eflfect,
piezoelectric constants that depend on the applied stress, the piezooptical
effect and the electrooptical effect. These second order equations can
also be used to discuss the changes that occur in ferroelectric type crystals
such as Rochelle SaU, for which between the temperature of — 18°C. and
f24°C.,a spontaneous polarization occurs along one direction in the crystal.
This spontaneous polarization gives rise to a first order piezoelectric deforma
tion and to second order electrostrictive effects. It produces changes in
the elastic constants, the piezoelectric constants and the dielectric constants.
Some measurements have been made for Rochelle Salt evaluating these
second order constants.
Mueller in his theory of Rochelle Salt considers that the crystal changes
from an orthorhombic crystal to a monoclinic crystal when it becomes
spontaneously polarized. An alternate view developed here is that all of
the new constants created by the spontaneous polarization are the result of
second order eflfects in the orthorhombic crystal. As shown in section 7
these produce new constants proportional to the square of the spontaneous
polarization which are the ones existing in a monoclinic crystal. 0.i this
view "morphic" eflfects are second order eflfects produced by the spontaneous
polarization.
1. Stress and Strain Relations in Aeolotropic Crystals
I.I. Specification of Stress
The stresses e.xerted on any elementary cube of material with its edges
along the three rectangular axes X, Y and Z can be specified by considering
the stresses on each face of the cube illustrated by Fig. 1. The total stress
acting on the face ABCD normal to the X axis can be represented by a
resultant force R, with its center of application at the center of the face,
plus a couple which takes account of the variation of the stress across the
face. The force R is directed outward, since a stress is considered posi
tive if it exerts a tension. As the face is shrunk in size, the force R will be
proportional to the area of the face, while the couple will vary as the cube of
the dimension. Hence in the limit the couple can be neglected with respect
to the force R. The stress (force per unit area) due to R can be resolved
into three components along the three axes to which we give the designation
Here the first letter designates the direction of the stress component and the
second letter x^ denotes the second face of the cube normal to the X axis.
Similarly for the first X face OEFG, the stress resultant can be resolved
82
BELL SYSTEM TECHNICAL JOU R^AL
into the compo7ients 7„, , Ty,, , T,., , which are oppositely directed to
those of the second face. The remaining stress components on the other
four faces have the designation
Face OABE
CFGD
OADG
bcfp:
r.
n
n
(2)
Fig. 1. — Cube showing method for specifying stresses.
The resultant force in the X direction is obtained by summing all the forces
with components in the X direction or
F\ = (n.,  r„J dydz + {T^y,  T.y,) dxdz + (n„  T^,) dxdy. (3)
But
Tzxt — ~~Txx^ 4" —^ — dx; iiyj — J xyj
+ 'I^'.,r. r„.= r„,+^v.
(4)
and equation (3) can be written in the form
/dTxx , dTxv 1 dT;
J''
l^' + j^^ + ''±^^dxdydz.
\ dx dy dz )
(5)
Similarlv the resultant forces in the other directions are
(6)
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
83
We call the components
r
T.
T
T21, 7^22, T,,
T31 , Tz2 , T33
(7)
the stress components exerted on the elementary cube which tend to deform
it. The rate of change of these stresses determines the resultant force on
the cube. The second form of (7) is commonly used when the stresses are
considered as a second rank tensor.
Fiff. 2. — Shearing stresses exerted on a cube.
It can be shown that there is a relation between 3 pairs of these compo
nents, namely
T = T
1 TV 1 ■
T = T
T = T
(8)
To show this consider P'ig. 2 which shows the stresses tending to rotate the
elementary cube about the Zaxis
the cube about the Z axis by producing the couple
The stresses Ty^^'dnd Ty^^ tend to rotate
Tyx dx dy dz
(9)
The stresses Tjy^ and T^y.^ produce a couple tending to cause a rotation in
the opposite direction so that
^ {Tyj, — T:ry) dx dy dz = couple
I (hi
(10)
is the total couj^ie ter.ding to produce a rotation around the Z axis.
But from dynamics, it is known that tliis cou])le is equal to the product of
the moment of inertia of the section times the angular acceleration. This
moment of inertia of the section is proportional to the fourth power of the
cube edge and the angular acceleration is fmite. Hence as the cube edge
M
84
BELL SYSTEM TECHNICAL JOURNAL
approaches zero, the right hand side of (10) is one order smaller than the
left hand side and hence
T = T
(11)
The same argument applies to the other terms. Hence the stress com
ponents of (7) can be written in the symmetrical form
r.
T.
T.
n.
Tn,
Tn,
Tu
T.,
n,
n
Ty^
=
Tn,
T22 ,
Tiz
=.
Te,
T2,
T,
r„
Tn ,
T,,,
Tiz
T,,
T,,
Tz
(12)
The last form is a short hand method for reducing the number of indices
in the stress tensor. The reduced indices 1 to 6, correspond to the tensor
indices if we replace
llbyl; 22 by 2; 33 by 3; 23 by 4; 13 by 5; 12 by 6.
This last methcd is the mcst common way for writing the stresses.
1.2 Strain Component,
The types of strain present in a body can be specified by considering two
points P. and ^ of a medium, and calculating their separation in the strained
condition. Let us consider the point P at the origin of coordinates and the
point Q having the coordinates x, y and z as shown by Fig. 3. Upon strain
Fig. 3. — Change in length and position of a hne due to strain in a solid body.
ing the body, the points change to the positions P', Q'. In order to specify
the strains, we have to calculate the difTerence in length after straining, or
have to evaluate the distance P'Q'P Q. After the material has stretched
the point P' will have the coordinates ^i , 7?i , f 1 , while Q' will have the
coordinates v + I2 ; v + 772 ; 2 + ^> . But the displacement is a continuous
function of the coordinates .r, y and z so that we have
^2 = ^1 + ^ X + / >' + ^ 3
dx dy dz
PIEZOELEC TRIG CR YS TA LS IN TENSOR FORM 85
Similarly
. dr} , drj drj
ox oy dz
(13)
i ^' = ^'^dx'^dyy^dz'
' Hence subtracting the two lengths, we iind that the increases in separation
\ in the three directions are
5x = .T ^ + V / + S ^
I dx dy dz
I
' dr] dq drj ,...
5v = ^^+>'t+2^ (14)
ox dy dz
dx dy dz
d^
The net elongation of the line in the x direction is x — and the elongation
dx
. d^ . . .
per unit length is —^ which is detined as the linear strain in the x direction.
dx
We have therefore that the linear strains in the x. y and s directions are
5, = f; S.^p; 53 = ^f. (15)
dx dy dz
The remaining strain coefficients are usually defined as
oy dz dz dx dx dy
and the rotation coefficients by the equations
_ d^ dtf _ d^ d^ _ drj d^
dy dz dz dx dx dy
Hence the relative displacement of any two j.oints can be expressed as
h = xS, + y [~^) + z [^)
(17)
(18)
86
BELL SYSTEM TECIINICA L JOI 'UNA L
which represents the most general type of disj^lacement that the Hne P Q
can undergo.
As discussed in section 4 the definition of the shearing strains given by
equation (16) does not allow them to be represented as part of a tensor.
If however we defined the shearing strains as
25,3 = S, =
\dy dzj
253 — Si,
= i^ + ^i •
dz dx '
25. = S. = p + 'J
dx ay
(19)
they can be expressed in the form of a symmetrical tensor
S(, 65
^11
S\2
012
'S13
S22
'*>'23
=
s,.
^^33
Si
Se
S2
s,
2
2
s.
s,
S;
2
2
(20)
For an element suffering a shearing strain S^ — 2Si2 only, the displace
ment along X is proportional to y, while the displacement along y is propor
tional to the X dimiension. A cubic element of volum.e will be strained into
a rhombic form, as shown by Fig. 4, and the cosine of the resulting angle 6
Fig. 4. — Distortion due to ;i shear! iig strain.
measures the shearing deformation. For an element suffering a rotation
ccz only, the dis])lacement along x is proj;ortional to y and in the negative
y direction, while the dis])laccmcnt along y is in the ]>ositive .v direction.
Hence a rectangle has the displacement shown by lig. 5, which is a pure
rotation of the body without change of form, about the z axis. For any
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
87
body in equilibrium or in nonrotational vibration, the co's can be set equal
to zero.
The total potential energy stored in a general distortion can be calculated
as the sum of the energies due to the distortion of the various modes. For
fih
example in expanding the cube in the x direction by an amount — dx =
ox
Si dx, the work done is the force times the displacement. The force wil
Fig. 5. — A rotation of a solid body.
be the force Ti and will be Ti dy dz. Hence the potential energy stored in
this distortion is
T\ dSi dx dy dz
For a shearing stress T^ of the type shown by Fig. 4 the displacement
dS(,dx
7r» T
times the force T^ dy dz and the displacement — ^^ times the force T(, dx dz
equals the stored energy or
AP^e = \ (dS^Te + dSeT^) dx dy dz = dS^T^ dx dy dz.
Hence for all modes of motion the stored potential ener gy is equal to
APE = [Ti dSi +■ Ti dS2 + Ti dSi + Ti dSi + T^, dSs
(21)
+ Tt dSe] dx dy dz.
1 .3 Generalized Hookers Law
Having specified stresses and strains, we next consider the relationship
; between them. For small displacements, it is a consequence of Hooke's
I Law that the stresses are proportional to the strains. For the most un
I symmetrical medium, this proportionality can be written in the form
(22)
88 BELL S YSTEAf TECH NIC A L JOURNA L
T\ = CnSi + C12S2 f C13S3 \ CuSi \ Ci^Si \ CioSe
T2 = C21S1 + C22S2 + C23S3 + C24S4 \ C2bSs + ^26^ 6
7^3 = ^31'5*1 + CS2S2 + ^33^3 + €3484 + ^35^6 + ^36^6
Ti = C41S1 + €4282 + r43'5'3 + CiiSi \ €4^3 f, \ ^46^6
Tt = Cr,iSi + f52^2 + ^53^3 + C^Si + Ci^S;, + ^56.5 6
7^6 = CeiSl \ f 62'?2 + f e3'S'3 + C64Si + f 65^5 + ^66^6
where Cn for example is an elastic constant expressing the proportionality
between the Si strain and the Ti stress in the absence of any other strains.
It can be shown that the law of conservation of energy, it is a necessary
consequence that
C12 = C21 and in general c,, = Cji. (23)
This reduces the number of independent elastic constants for the most
unsymmetrical medium to 21. As shown in a later section, any symmetry
existing in the crystal will reduce the possible number of elastic constants
and simplify the stress strain relationship of equation (22).
Introducing the values of the stresses from (22) in the expression for the
potential energy (21), this can be written in the form
2PE = cnSl + 2C12S1S2 + IcnSiSs + 2fi4^i54 + 2cuSiS\ + Ici&SiS^
+ ^22^2 + 2r23^2'S'3 + 2C24'S'25'4 + 2f25'S'2^5 + 2C2oS'26'6
+ C33S3 { IcsiSsSi \ IczffSzSi, f IcsgS^S^
+ f44'^4 + 2r45^4^'5 + ICi^'iSfi (24)
The relations (22) thus can be obtained by differentiating the potential
energy according to the relation
c)PF c)PF
It is sometimes ad\antageous to exi)ress the strains in terms of the stresses.
This can be done by solving the equations (22) simultaneously for the
strains resulting in the equations
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
S9
Si = 511^1 + 512^2 + SuTz + SuTi + 51575 + Sy^Ti,
Si = S21T1 \ S22T2 + 523^3 + S^iTi + 5257^6 + 526^6
53 = S31T1 + 532^2 + 533^3 + 53474 + 53575 + 53676
54 = 54i7i + 54272 + 54373 + 5447i + 54575 + 54676
'^'5 = S^iTi \ Sf,iTl \ 55373 + 55474 + 55575 + ^6676
Si = 56l7i + 56272 + 56373 + 56474 + 56575 + 56676
(26)
Inhere
i+i
Sii =
_(i)'"^A:y
(27)
for which A*^ is the determinant of the dj terms of (28) and'A^y the minor
obtained by suppressing the ith andjth columo
A'^ =
<"ll Ci2 Ci3 Cu '"15 <^16
^12 ^22 <r23 Cu C25 ^26
Cl3 C23 ^33 C34 <"36 ^36
ri4 C24 C34 f44 C45 C46
^15 ^25 <"35 Cib Cbb ^56
^16 <^26 ''36 ^46 C{,( Ce6
(28)
Since c.y = cy, it follows that 5,y = 5y,. The potential energy can be
expressed in the form.
27£ = 5ii7? + 2S12T1T2 + 25i37\73 + IsuTiTi + 25i57i76 + 25i67i76
+ 52272 + 2S23T2T3 + 25247274 + 2S26T2T5 + 2S2iT2T ^
+ •^3373 + 253^X3X4 \ 2S3bT3Tb + 25367376
+ 54474 + 25457475 + 25467476 (29)
+ 55575 + 2sb%Ti,Ti
\ SbbTe
The relations (26) can then be derived from expressions of the type
5i =
dPE
S, =
dPE
(30)
dTi ' ' "" 576
1.4 Isothermal and Adiabatic Elastic Constants
We have so far considered only the elastic relations that can be measured
statically at a constant temperature. The elastic constants are then the
isothermal constants. For a rapidly vibrating body, however, there is no
90 BELL SYSTEM TECHNICAL JOURNAL
chance for heat to equalize and consequently the elastic constants operative
are the adiabatic constants determined by the fact that no heat is added
or subtracted from any elemental volume. For gases there is a marked
difference between the adiabatic and the isothermal constants, but for
piezoelectric cr^'stals the difference is small and can usually be neglected.
To investigate the relation existing we can write from the first and second
laws of thermodynamics, the relations
dV = [Ti dSi 4 T2 dS2 + T3 dSs
(31)
+ T, dSi + Ts dS, + 7^6 dS,] \ed(r
which expresses the fact that the change in the total energy U is equal to
the change in the potential energy plus the added heat energy dQ = Q da
where is the temperature and cr the entropy. Developing the strains and
entropy in terms of the partial differentials of the stresses and temperature,
we have
dS, = ^^ dT, + ?i^ dT, + ^' dTs
dTi dT2 ST.
oli 01^ die oQ
dS, = '^Ut. h ^^' dT. + §' dn
oil 01 2 alz
(32)
do = l^ AT. + If AT, + If dT^
all 01 2 01 i
^^dT, + ^dT, + ^ dT, + ^dQ.
dTi an  dTe ae
The partial derivatives of the strains with regard to the stresses are readily
seen to be the isothermal elastic compliances. The partial derivatives of
the strains by the temperatures are the six temperature coefficients of ex
pansion, or
dSi dSi ...
ae ' ae
To evaluate the partial derivatives of the entropy with re.^pect to the
stresses we make use of the fact that U is a perfect difTerential so that
dS\ da dS^ da ,,..
PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 91
Finally multiplying through the last of equation (32) by 9 we can write
them as
Si = snTi + 512^2 + suTz + SuTi + Si^T^ + suT^ + oci dQ
Si = SieTi + ^267^2 + ■^36^3 + SisT4 + 5667^6 + •^662^6 + OC^ dO
dQ = Q d(T = 6[aiTi + q:27'2 + otsTs + 0474 + ai,T^ + a^Te] + pCpdQ
since ©t^ is the total heat capacity of the unit volume at constant stress,
which is equal to pCp, where p is the density and Cp the heat capacity at
constant stress per gram of the material.
To get the adiabatic elastic constants which correspond to no heat loss
from the element, or dQ = 0, dQ can be eliminated from (35) giving
^1 = s'nTi + 5127^2 + SnTs + 3^X4 + s[f,Tf, + s'^Tf, + (ai/pCp) dQ
(36)
Se = s'uTi + sIbT^ + SuTs + s'teTi + sl^T^ + s^Te + (as/pCp) dQ
where
,, = s%  «i^. (37)
pLp
For example for quartz, the expansion coeffxients are
ai = 14.3 X 10"V°C; 02 = 14.3 X 10"V°C; a, = 7.8 X 10"V°C;
The density and specific heat at constant pressure are
p — 2.65 grams/cm ; Cp= 7.37 X 10^ergs/cm^
Hence the only constants that differ for adiabatic and isothermal values are
•^11 = 522 ; .^12 ; ^13 ; ^33 •
Taking these values as
sn = 127.9 X 10~'* cmVdyne; Su = 15.35 X 10"'';
su = 11.0 X 10"'*; 533 = 95.6 X 10"''.
We find that the corresponding isothermal values are
sfi = 128.2 X 10"'*; 5?2 = 15.04 X 10"'*;
5?3 = 10.83 X 10"'*; s% = 95.7 X 10"'* cmVdyne
^See "Quartz Crystal Applications" Bell System Technical Journal, Vol. XXII>
No. 2, July 1943, W. P. Mason.
92 BELL SYSTEM TECHNICAL JOURNAL
at 25°C. or 298° absolute. These differences are probably smaller than
the accuracy of the measured constants.
If we express the stresses in terms of the strains by solving equation (35)
simultaneously, we find for the stresses
(38)
7^6 = Ci^Si \ c^^S'i \~ Cz^Si + Cif,SA + CjfrSs + Ces'S'e — Xe dQ
where
The X's represent the temperature coefficients of stress when all the strains
are zero. The negative sign indicates that a negative stress (a compression)
has to be applied to keep the strains zero. If we substitute equations (38)
in the last of equations (35), the relation between increments of heat and
temperature, we have
dO = Qda = e[\iSi + MSi + XsSs + XiS^ + X565 + Xe^e]
(39)
+ [pCp — 0(aiXi + 012X2 + 0:3X3 + 0:4X4 + 0:5X5 + a^X6)]dQ.
If we set the strains equal to zero, the size of the element does not change,
and hence the ratio between dQ and dB should equal p times the specific
heat at constant volume C„. We have therefore the relation
p[Cp — Cv] = B[a:iXi + 02X2 + 0:3X3 + 04X4 + 0:5X5 + osXe]. (40)
The relation between the adiabatic and isothermal elastic constants Cij
thus becomes
c'j = cl + ^'. (41)
Since the difference between the adiabatic and isothermal constants is so
small, no differentiation will be made between them in the following sections.
2. Expression for The Elastic, Piezoelectric, Pyroelectric and
Dielectric Relations of a Piezoelectric Crystal
When a crystal is piezoelectric, a potential energy is stored in the crystal
when a voltage is applied to the crystal. Hence the energy expressions of
(31) requires additional terms to represent the increment of energy dl'.
If we employ C(iS units which have so far been most widely used, as applied
PIEZOELECTRIC CR VST A LS IN" TENSOR FORM 93
to piezoelectric cn^stals, the energy stored in any unit volume of the crystal is
dU = Ti dSi + T2 dS2 + T3 dS^ + Ti dS, + Ts dS, + Te dSe
, J, dD, , ^ dD, , _, dDi ,^, (42)
■iir 47r ■iir
where Ei , E2 and £3 are the components of the field existing in the crystal
and Di , A and D3 the components of the electric displacement. In order
to avoid using the factor l/4ir we make the substitution
The normal component of 5 at any bounding surface is fo the surface charge.
On the other hand if we employ the MKS systems of units the energy of
any component is given by Zn^/^^n directly and in the following formulation 5
can be replaced by D.
There are two logical methods of writing the elastic, piezoelectric, pyro
electric and dielectric relations. One considers the independent variables
as the stresses, fields, and temperature, and the dependent variables as the
strains, displacements and entropy. The other system considers the strains,
displacements and entropy as the fundamental independent variables and
the stresses, fields, and temperature as the independent variables. The
first system appears to be more fundamental for ferroelectric types of
crystals.
If we develop the stresses, fields, and temperature in terms of their partial
derivatives, we can write
i/d.<t 0^2/ D.a OCis/D.tr OOi/ D,a
\/ s,a da /a
Obz/S.a Off Js.D
T, = ^^\ dS,^^^") dS2\^^^ ^^3 + ^^^^ dS,
(44 A)
O'Jl/D.a 002/ D.a OOs/D.a 004/0,0
a^)5/D,o O0(,/D,a O0\ / s.a 002/ S,a
003 /s.a dcr /S,D
94 BELL S YSTEM TECH NIC A L JOURNA L
£x = £i = ^^ ) dS, +
)>b/D,<r O06/D,o OOi/s.a OOi / S.a
+ f) ,,, + fl) ,.
Oh/a.a OCT /S.D
£. = £, = ^A ''51 + ^sl) ''5, + ^') ''•Ss + lf) i&
OOi/Cff U02/ D,a OOZ/ D,a OOi/D.a
+ ^^) ,5. + f) .6^, + f) .a, + f) .a,
OOf,/ D,a OO^/D.a OOl/s.a OO2 / S,a
dds/s.a OCT /a,D
,e=f) .5. + 11) .5, + 11) .53 + 11) </5,
OOl/D.a OOi/D.a OOs/D.a OO4/ D.a
J D,a 00%/ D,a OOl/s.o O02/S,a
) d5z+f) da.
:/S,a Off/s.D
883/8,0
The subscripts under the partial derivatives indicate the quantities kept
constant. A subscript D indicates that the electric induction is held
constant, a subscript a indicates that the entropy is held constant, while a
subscript 5 indicates that the strains are held constant.
Examining the first equation, we see that the partial derivatives of the
stress Ti by the strains are the elastic constants c,, which determine the
ratios between the stress Ti and the appropriate strain with all other strains
equal to zero. To indicate the conditions for the partial derivatives, the
superscripts D and a are given to the elastic constants and they are written
c^j'. The partial derivatives of the stresses by 5 = D/^t are the piezo
electric constants //,/ which measure the increases in stress necessary to
hold the crystal free from strain in the presence of a displacement. Since
if the crystal tends to expand on the application of a displacement, the
stress to keep it from exi)anding has to be a compression or negative stress,
the negative sign is given to the /{"a constants. As the only meaning of
the // constants is obtained by measuring the ratio of the stress to 5 = D/iir
at constant strains, no superscript S is added. However there is a difference
I.etween isothermal and adiabatic piezoelectric constants in general, so
PIEZOELECTRIC CR VST A LS IN TENSOR FORM 95
that these piezoelectric constants are written Z/"^^. Finally the last partial
derivatives of the stresses by the entropy a can be written
dT
'da
") ^' = 1,^P) Q^'^^ST^") 'iQ = yrdQ (45)
• /s,D 6 da /s,D 6 oa /s.d
where dQ is the added heat. We designate 1/6 times the partial derivative
as — Yn and note that it determines the negative stress (compression)
necessary' to put on the cr>'stal to keep it from expanding when an increment
of heat dQ is added to the crystal. The electric displacement is held
constant and hence the superscripts S, and D are used. The first six equa
tions then can be written in the form
(46)
— h'nxhi — /U'Jo — h'na^s — y^f dQ.
To evaluate the next three equation? involving the fields, we make use of
the fact that the expression for dU in equation (42) is a perfect differential.
As a consequence there are relations between the partial derivatives,
namely
(47)
ar„. _
a£„.
dT^
ae .
dEn
_ dQ
dbn
dSj
da
dSm
da
dhn
We note also that
dEA
d8n / S.a
=
47r/3f;;
(4.S)
where /3 is the so called "impermeability" matrix obtained fiom '.he dielectric
matrix e„m by means of the equation
&r.n = ^^ (40)
where A is the determinant
fll ,
fl2 ,
CIS
€12 ,
fno
COS
fKi ,
^s ,
e.s3
(5(!)
and a"''" the minor obtained by suppressing the wth row and ;/th column.
The partial derivatives of the fields by the entropy can he written
dE^
da
A . 1 dE„\ 1 dE„,\ .s,z. ,,, ....
/S.D U da /S.D 6 da /s n
where q'n is a pyroelectric constant measuring the increa:£e in field required
to produce a zero charge on the surface when a heat /() is added to the
96 BELL SYSTEM TECH NIC A L JOURNA L
crystal. Since the voltage will be of opposite sign to the charge generated
on the surface of the crystal in the absence of this counter voltage a nega
• • • , S,D
tive sign is given to g „ .
Finally the last partial derivative
6e\ , 1 ae\ _ , i ae\ ._ dQ
aa/s.D U OCT /s.D U da /s,d pC„
represents the ratio of the increase in temperature due to the added amount
of heat dQ when the strains and electric displacements are held constant.
It is therefore the inverse of the specific heat at constant volume and constant
electric displacement per gram of material times the density p. Hence
the ten equations of equation (44) can be written in the generalized forms
— h'nlh — llnlh — il'nzh " In dQ
Em, = —h\mSl — him^l ~ I'Sm'^S ~ ^UmSi — ll^mS^ — IlimSt ~\~ iTrfSml^l
+ ^Tr&^ + iw^^  qlf dQ (53)
Je=— e[7i ^1 + 72 02 + 73 03 + 74 04 + 75 OS + 76 OeJ
—Q[qi 5i + 92 ^2 + 93 53] + ttd •
11= 1 to 6; m = 1 to 3
If, as is usually the case with vibrating crystals the vibration occurs
with no interchange of heat between adjacent elements dQ — and the
ten equations reduce to the usual nine given by the general forms
Tn = CnlSl + Cn^Si + CnsSi + CniSi + CnbSf, + C ntS e
— hni5i — hnih — hnzh
Em = —JllmSl — IhviSi — IhmSz — /74m'S'4 — IhmSb — /'Cm'S'e
+ 47r/3mi5i + 4T/3I262 + 47ri3'l3 53.
(54)
In these equations the superscript a has been dropj^ed since the ordinary
constants are adiabatic. The tenth equation of {S3>) determines the increase
in temperature caused by the strains and displacements in the absence of
any flow of heat.
If we introduce the e.xpression of equations (53) into equation (42) the
total energy of the crystal is per unit volume.
PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 97
21 = rii 61 + 2fio ^1^2 + 2^13 oiJs + ^i^H 'Ji'J4 + 2ri5 ^165 + 2ci6 oiOe
+ r.?i'5l + 2c^fS,S, + 2r?4'^^2>S4 + 2c^_,''SoS, + 2f?6%56
^33 J3 "T" ^^^34 03^4 "T" ^^'35 0305 f Z('36 03O6
(■44 O4 i Zf45 O4O5 j Zr46 O4O6
+ D,(T ^2 I rj Z).ff o O '
+ f66''^'6 (55)
(2//Ii5,5'i + 2/;I,5i52 + 2//l35i.93 + 2/;l45i54 + 2li%5,S, + 2//l65i56)
(2//2l5,5l + 2J1U2S2 + 2//235253 + 2//24^2^^4 + 2111^^3^ + 2//26526'6)
(2//3l53.Si + 2hl.MS2 + 2//33^3^3 + 2/;345354 + 2//35636'5 + 2//3653^6)
(27i'%^/() + 272'%f/<3 + 2yl'^SsdQ
+ 274'''6'4fi?(? + 275'°55rf() + 2y'l''S,dQ)
+iirWiUl + 2/3^;r6if2 + 2(Sf,'d,bs + /3^;r62 + 2f32zdod, + /Sf^^i]
(29f%r/C' + 2qt''5,dQ + 2gt''''W0 + ~§r".
Equations (53) can be derived from this expression by employing the partial
1 derivatives
i The other form for writing the elastic, f)iezoelectric, pyroelectric and di
j electric relations is to take the strains, displacements, and entropy as the
! fundamental variables and the stresses, fields and temperature increments
■ as the dependent variables. If we develop them in terms of their partial
j derivatives as was done in (44), use the relations between the partial deriva
t tives shown in equation (57).
(57)
and substitute for the partial derivatives their equivalent elastic, piezo
electric, pyroelectric, temperature expansions, dielectric and specific heat
constants, there are 10 equations of the form
ddm
_ dSn .
dSr, _
da
d5^
da
dTn
dEm '
dQ ~
dT„ '
60
dE„
98 BELL SYSTEM TECH NIC A L JOURNA L
+ ^2^2 + (tzEz + a^Je
5m = (iimTl + dirnTi. + d^mTz + dimTi + d^^Th + d^^Te
+ l£, + ^l £, + !pi £3 + /'Ic/e (58)
47r 47r 47r
</^ = 9 (/o = 6[ai Ti + Q!2 7^2 + af Ts + af 7^4 + af Ts + af rej
+ eiplE, + Pa'^Es + plE,] + />C^(/e.
w = 1 to 6, m = 1 to 3
The superscripts E, 0, and T indicate respectively constant field, constant
temperature and constant stress for the measurements of the respective
constants. It will be noted that the elastic compliance and the piezo
electric constants d^n are for isothermal conditions. The a^ constants are
the temperature expansion constants measured at constant field, while the
p^ constants are the pyroelectric constants relating the ratio of 5 == D/47r
to increase in temperature ^6, measured at constant stress. Since there is
constant stress, these constants take into account not only the "true" pyro
electric effect which is the ratio of 5 = Z>/47r to the temperature at constant
volume, but also the so called "false" pyroelectric effect of the first kind
which is the polarization caused by the temperature expansion of the crystal.
This appears to be a misnomer. A better designation for the two effects
is the pyroelectric effect at constant strain and the pyroelectric effect at
constant stress. Cp is the specific heat at constant pressure and constant
field.
If we substitute these equations into equation (42), the total free energy
becomes
!^ = E Z s^nTmTn + 2 ^^ Xl d'toT^Eo 4 2 i; a'„Tje
n = l 0=1
3 T,e
+ Z E ^ £o£, + 2 E PoEpde + ^^ ^e.
0=1 p=i 47r 0=1 t)
Equation (58) can then be obtained by partial derivatives of the sort
at/ _ d£ _dQ dU
(59)
dTn' dEp' e d(de)'
By tensor transformations the expression for U in (59) can be shown to
be equal to the expression for U in (55).
The adiabatic equations holding for a rapidly vibrating crystal can be
PIEZOELECTRIC CR VST A LS IN TENSOR FORM 99
obtained by setting dQ equal to zero in the last of equations (58) and elim
inating dQ from the other nine equations. The resulting equations are
Bm = dim Ti + d^m T2 + dzm Ti + dim Ti (60)
+ d,m n ^ d^T,+ '^ El + ^' £2 + '^^ £3
47r 4t 47r
where the symbol a for adiabatic is understood and where the relations
between the isothermal and adiabatic constants are given by
E E (^ B .T f^ T,a T,Q l.T .T r\
Hence the piezoelectric and dielectric constants are identical for isothermal
and adiabatic conditions provided the crystal is not pyroelectric, but differ
if the crystal is pyroelectric. The difference between the adiabatic and
isothermal elastic compliances was discussed in section (1.4) and was shown
to be small. Hence the equations in the form (60) are generally used in
discussing piezoelectric crystals.
Two other mixed forms are also used but a discussion of them will be
delayed until a tensor notation for piezoelectric crystals has been discussed.
This simplifies the writing of such equations.
3. General Properties of Tensors
The expressions for the piezoelectric relations discussed in section 2 can
be considerably abbreviated by expressing them in tensor form. Further
more, the calculation of elastic constants for rotated crystals is considerably
simplified by the geometrical transformation laws established for tensors.
Hence it has seemed worthwhile to express the elastic, electric, and piezo
electric relations of a piezoelectric crystal in tensor form. It is the purpose
of this section to discuss the general properties of tensors applicable to
Cartesian coordinates.
If we have two sets of rectangular axes (Ox, Oy, Oz) and (Ox', Oy' , Oz)
having the same origin, the coordinates of any point P with respect to the
second set are given in terms of the first set by the equations
x' — (iX \ miy \ Jhz
y' = lix \ m^y + «22 (61)
z' = I3X + m^y \ HiZ.
100 BELL SYSTEM TECH NIC A L JOURNA L
The quantities (^i , • • • , ;/3) are the cosines of the angles between the various
axes; thus A is the cosine of the angle between the axes Ox', and Ox; n^ the
cosine of the angle between Oz' and Oz, and so on. By solving the equations
(61) simultaneously, the coordinates .v, y, z can be expressed in terms of
.t', y' , z' by the equations.
X = l,x' + t^' + t,z'
y = mix' + Woy' + nviz' (62)
z = nix' + n<iy' + r^z' .
We can shorten the writing of equations (61) and (62) considerably by
changing the notation. Instead of x, y, z let us write .Ti , x? , Xz and in place
of x' , y' , z' we write X\ , X2 , Xs. We can now say that the coordinates with
respect to the first system are .Ti , where i may be 1, 2, 3 while those with
respect of the second system are Xj , where / = 1, 2 or 3. Then in (61)
each coordinate Xj is expressed as the sum of three terms depending on the
three x, . Each x, is associated with the cosine of the angle between the
direction of x, increasing and that of x, increasing. Let us denote this
cosine by c , y . Then we have for all values of j,
3
x'j = aijXi + a2jX2 + asjXs = ^ aijXi. (63)
Conversely equation (62) can be written
3
Xi = XI ^•■y'^y (64)
y=i
where the a ,; have the same value as in (63), for the same values of i and 7,
since in both cases the cosine of the angle is between the values of x; and x;
increasing. Such a set of three quantities involving a relation between two
coordinate systems is called a tensor of the first rank or a vector.
We note that each of the equations (63), (64) is really a set of three equa
tions. Where the suffix i or j appears on the left it is to be given in turn
all the values 1, 2, 3 and the resulting equation is one of the set. In each
such equation the right side is the sum of three terms obtained by giving j
or / the values 1, 2, 3 in turn and adding. Whenever such a summation
occurs a suffix is repeated in the expression for the general term as dijXj .
We make it a regular convention that whenever a suffiix is repeated it is
to be given all possible values and that the terms arc to be added for all.
Then (63) can be written simply as
x^ = a,;X,
the summation being automatically understood by the convention.
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
101
There are single quantities such as mass and distance, that are the same
for all systems of coordinates. These are called tensors of the zero rank
or scalars.
Consider now two tensors of the first rank «, and Vk ■ Suppose that each
component of one is to be multiplied by each component of the other, then
we obtain a set of nine quantities expressed by Ui Vk , where i and k are
independently given all the values 1. 2, 3. The components of «; Vk with
respect to the Xj set of axes are Uj V( , and
tijVi = (aijtii) (aicfk) = anQkiUiVk
(65)
The suffixes / and k are repeated on the right. Hence (65) represents nine
equations, each with nine terms. Each term on the right is the product
of two factors, one of the. form a ijOki, depending only on the orientation of
the axes, and the other of the form UiVk , representing the products of the
components referred to the original axes. In this way the various Uj Vf can
be obtained in terms of the original UiVk . But products of vectors are not
the only quantities satisfying the rule. In general a set of nine quantities
IV ik referred to a set of axes, and transformed to another set by the rule
^';Y = OijQki u>ik
(66)
is called a tensor of (he second rank.
Higher orders tensors can be formed by taking the products of more
vectors. Thus a set of n quantities that transforms like the vector product
XiXj • • • Xp is called a tensor of rank /?, where n is the number of factors.
On the right hand side of (66) the / and k are dummy suffices; that is,
they are given the numbers 1 to 3 and summed. It, therefore, makes no
difference which we call i and which k so that
^^'j7
jakfiCik — OkjaifCkf
(67)
Hence Wk( transforms by the same rule as u' ik and hence is a tensor of the
second rank. The importance of this is that if we have a set of quantities
li'n
U'i2 U'i3
W21
K'22 'iC'23
■Z^'31
li'SO ICi^
fthe
second ra
Wn
K'21 ^C'31
«'12
1^22 W'32
"d'n
K'23 "^£'33
which we know to be a tensor of the second rank, the set of quantities
(68)
(69)
is another tensor of the second rank. Hence the sum (idk + i^'ki) and the
difference (^c',k — iVk,) are also tensors of the second rank. The first of
102 BELL SYSTEM TECH NIC A L JOURNA L
these has the property that it is unaltered by interchanging i and k and
therefore it is called a symmetrical tensor. The second has its components
reversed in sign when i and k are interchanged. It is therefore an antisym
metrical tensor. Clearly in an antisymmetric tensor the leading diagonal
components will all be zero, i.e., those with i = k will be zero. Now since
Wik= \ {wik + Wki) + h (u'ik — Wki) (70)
we can consider any tensor of the second rank as the sum of a symmetrical
and an antisymmetrical tensor. Most tensors in the theory of elasticity
are symmetrical tensors.
The operation of putting two suffixes in a tensor equal and adding the
terms is known as contraction of the tensor. It gives a tensor two ranks
lower than the original one. If for instance we contract the tensor ut Vk
we obtain
UiVi = UiVi + U2V2 + U3V3 (71)
which is the scalar product of u i and Vk and hence is a tensor of zero rank.
We wish now to derive the formulae for tensor transformation to a new
set of axes. For a tensor of the first rank (a vector) this has been given
by equation (61). But the direction consines A to «3 can be expressed in
the form
(72)
_ dx' _ axi
dx dxi '
dx' dxi
Wi = — = — ;
dy dxt
dx'
dz
dxs
_ dy' _ dX2
dx dxi '
dy' dX2
W2 = ^ = r— ;
dy 6x2
dy'
dz
_ dX2
8x3
_ dz' _ dx'z
dx dxi '
dz' dx'i
dy dX2
dz'
dz
_ dx's
dxs
Hence equation {61) can
be
expressed in the tensor form
X
/ dXj
dXi
(73)
Similarly since a tensor of the second rank can be regarded as the product
of two vectors, it can be transformed according to the equation
/ / /dXj \ /dXf \ dXj dXf .»,v
\dXi / \dxis / dXi dXk
which can also be expressed in the generalized form
/ dXj dXf /rv
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
103
In general the transformation equation of a tensor of the ;zth rank can be
written
xi
OXj^ OXj., a.V/„
(76)
4. Application of Tensor Notation to the Elastic, Piezoelectric
AND Dielectric Equations of a Crystal
Let us consider the stress components of equation (7)
T T T
^ XX ^ xy •* J2
T T T
^ yx ^ yy •'2/2
T,x T,y r,,
from which equation (8) is derived
■i xy I yx ] ^ xz i zi , ^ yz •* zj/
and designate them in the manner shown by equation (77) to correspond
with tensor notations
(77)
by virtue of the relations of (8). We wish to show now that the set of 9
elements of the equation constitutes a tensor, and by virtue of the relations
of (8) a symmetrical tensor.
The transformation of the stress components to a new set of axes x', y', z'
has been shown bv Love to take the form
Tn
Tn
Tn
T21
T22
Toa
=
Tn
T,2
7^33
Tn
Tn
Tn
Tn
T22
T2,
Tn
^23
7^33
T^x = fl T^j, + rn\Tyy ~\ nlT,, + lliMiT^y + 2(iUiTj,z + ImiUiTy,
(78)
Txy = (ifiTjcx + fnitnoTyy'r nin2T,,{ (Awo + limi)T^y + (A«2 + hnifT^^
+ {mini + niniiiTy^
where A to 113, are the direction cosines between the axes as specified by
equation (61). Noting that from (72)
«3 =
dXj
dx3
the first of these equations can be put in the form
^ See "Theory of Elasticity," Love, Page 80.
104
BELL SYSTEM TECHNICAL JOURNAL
, /dx'i^\ dx[ dx'i
\ dxi I d.Ti 0x2
+ ''P '^ Tn + (g)
8x2 dxi
dxi dxi
dXi dxs
dXi dxi _ dxi dxi
i 22 r T—  ^— i 23 — r —  — 1 k(
0X2 0X3 OXk dX(
(79)
5xi dxi dxi dxi ( dx
~r X — z — i 31 "T r — 7 — / 32 "rl r
d.T3 dxi dX3 dX2 \0iC3
:)■
while the last equation takes the form
/ _ dxi 8x2 . dxi dx2 „ , dxi 8x2 „
■t 12 — ^ — z — i 11 ~r 7, — ;:— i 12 r r — r — i 13
dxi 0X1 dxi 6x2 oxi 0x3
dxi dxo ™ , dxi 6x2 rp , dxi dxo ^ _„„,„...
1 7. — z — i 21 "t" r —  — 1 22 ~r r —  — i 23 — r — 'Z — i kf
0X2 oXi 0X2 0X2 d.Vo 0X3 ax/c oXf
, dxi 6x2 „ , dxi 6x2 „ ,
~r ~ — ;; — i 31 "T T—  r — i 32 "T*
0X3 d.V] 0x3 0x2
The general expression for any component then is
r' . = ^^ f
'' dXk dxf
dxi 8x2
dxf
dxi dx'2
dxs 6x3
(80)
(81)
which is the transformation equation of a tensor of the second rank. Hence
the stress components satisfy the conditions for a second rank tensor.
The strain components
•J XX '^xy "Jxz
•^yx '^yy '^yz
>J zx ^ zy ^ zz
do not however satisfy the conditions for a second rank tensor. This is
shown by the transformation of strain components to a new set of axes,
which have been shown by Love to satisfy the equations
Sxy — 2A^2'5'ii + 2viim2Syy + luirioSzz + (Aw2 + ^2Wi)5'j
(82)
+ (A"2 + fl\(2)S^z + (Wl"2. + m2lh)S:c
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
If, however, we take the strain components as
105
c _ c _ ^^
■'11 — 'In — TT 1
ax
S,2
dr)
By '
c _ c _ ^f
O33 — 'Jjz — :r
dz
2 \dx dx/ '
(83)
Sii — Siy> —
1 (dj
dy
+
dr,\
dzj
the nine components
^n
.SV2
A'l3
.V21
.Vo,
.V23
.V31
.S'32
A'33
(83)
will form a tensor of the second rank, as can be sh(jwn by the transformation
equations of (82).
The generaUzed Hooke's hiw given by equation {22) becomes
'/'..=
CijkfSkt
(84)
CijkC is a fourth rank tensor. The right hand side of the equation being
the product of a fourth rank tensor by a second rank tensor is a sixth rank
tensor, but since it has been contracted twice by having k and ^ in both
terms the resultant of the right hand side is a second rank tensor. Since
dm is a tensor of the fourth rank it will, in general, have 81 terms, but on
account of the symmetry of the T , j and Sic( tensors, there are many equiva
lences between the resulting elastic constants. These equivalences can be
determined by expanding the terms of (84) and comparing with the equiva
lent expressions of (22). For example
+ ^1121621 f ril22'S'22 + ("1123»^23
+ <"n3 Al + <"1132S'32 + CU33'S33 •
(85)
Comparing this equation with the tirst of (22) noting that Su — S21 =
— ', etc., we have
t'UU — C\\ ; ('1112 — ("1121 — '"in ; <"1133 — '"iS ', f'llU
^^1122 = fl2 ; f'll23 = t'll32 = 6"l4 •
t1131
(86)
106 BELL SYSTEM TECH NIC A L JOURNA L
In a similar manner it can be shown that the elastic constants of (22)
correspond to the tensor elastic constants djui according to the relations
C\\ = fun ; Cl2 = <'1122 = C22II ; Cl3 — Ca33 = f33n ', ^14 = ^1123 = ^132 =
Cnn = C32U ; Cib = diw = ^1131 = ^'isu = Cun ', Cu = fiii2 = Cn2i = <^i2ii =
^2111 ', C22 — <^2222 ', C2Z — <^2233 — ^3322 ', ^24 = ^2223 = ^2232 = ^2322 = ^3222 ',
C2b — ^2213 = <"2231 = '"1322 = ^3122 ', <^26 = <^2212 = <^2221 = <'l222 = ^2122 ', C33 =
C3333 ; C34 = ^3323 = ^3332 = ^2333 = ^3233 ', ^36 = 3313 = ^3331 = '^1333 — ^3133 J
(87)
^36 = ^3312 — C3321 — C1233 — ^2133 ', ^44 — ^2323 — ^2332 — ^3223 — f3232 y ^46 —
^2313 — ^2331 = ^3213 = <^3231 = 1323 = 1332 = ^3132 = ^3123 ', ''46 = ^2312 =
£"2321 — C32I2 = C322I — ^1223 = C1232 = C2I23 — C2132 ; C55 = C1313 = C1331 =
f3U3 = ^^3131 ; Cb6 — fl312 = 0321 = ^3112 = C3121 = fl213 = ^1231 = ^2113 =
£"2131 ) f 66 = f 1212 = <"1221 = ^2112 — <^2121 •
Hence there are only 21 independent constants of the 81 djkf constants
which are determined from the ordinarily elastic constants c,/ by replacing
1 by 11 ; 2 by 22; 3 by 33; 4 by 23; 5 by 13; 6 by 12 (88)
and taking all possible permutations of these constants by interchanging
them in pairs.
The inverse elastic equations (26) can be written in the simplified form
Sij = SijkfTk(. (89x
By expanding these equations and comparing with equations (26) we can
establish the relationships
_ _ Su _ _ _ _
Sn = ^1111 ; ^12 = 51122 — ^2211 ; ^13 — 51133 — ^3311 ; "y — ^1123 — 51132 — 52311 —
■^16 _ _ ■^16 _ _ _ _
•^3211 ; W — 51113 — 51131 — 51311 — 53111 ', y — 5lll2 — ■51121 — 51211 — 52111 ;
•522 —
•52222 ;
523 = 52:
233 =
53322 ;
2
1 _
Sr.
!23 —
•52232 =^
52322 =
53222 ;
526 _
2
^2213 =
= 52231
= 51322 =
= 53122
5? 6
'■' 2
5221
2 =
: 5222
1 = ^51222
= 52122
; 533 =
= 53333
(90 A)
^34 _
2
53323 '■
= 53332 =
52333
= 5.^233 ;
2
=
53313
= 53331 
= 51333 =
= 5;tl33
. 536
' 2

PIEZOELECTRIC CRYSTALS IN TENSOR FORM 107
^44 ^45
^3312 — •^3321 — ^1233 — 52133 ', J — ■^2323 — ^2332 — ^3223 — ^3232 ', — — ^2313 =
_ _ _ _ _ _ ^46 _
■^2331 — •^3213 — ^3231 — ^1323 — ^1332 — .^3123 — ^3132 ', J — ^2312 — ^2321 =
(90 B)
_ _ _„_ _ ■>55_ _
^3212 — ■^3221 — ^1223 — J1232 — ^2123 — ^2132 ; ~J — ■^1313 — ^1331 — •^3113 =
•^56 _ _ .
•^3131 ; J" ~ "^^^12 ~ "^131 ~ "^3112 — •^3121 — ^1213 — ^1231 — ■^2113 — •^2131 ',
•^66 _ _ _ _
; '■ •^1212 — •^1221 — ^2112 — 52121 •
4
Here again the SijkC elastic constants are determined from the ordinary
elastic constants 5,y by replacing
1 by 11, 2 by 22, 3 by 33, 4 by 23, 5 by 13, 6 by 12.
However for any number 4, 5, or 6 the elastic compliance Sij has to be di
vided by two to equal the corresponding SijkC compliance, and if 4, 5 or
6 occurs twice, the divisor has to be 4.
The isothermal elastic compliance of equations (39) can be expressed
in tensor form
Si,^slk(T,c + a,,dQ (91)
1 where as before a,; is a tensor of the second rank having the relations to
the ordinary coefficients of expansion
Oil = «ii ; 02 = "22 ; "3 = «33 ,* y = ^23 i
oib ae
The heat temperature equation of (35) is written in the simple form
I dQ = + akt Tut e + pCp de. (92)
' . . .
ii By eliminating dO from (92) and substituting in (91) the adiabatic constants
!i are given in the simple form
SijkC = SijkC  —^ — . (93)
The combination elastic and piezoelectric equations (60) can be written
in the tensor form
T
Sii = S^jkCTkC + d^ijEm ; hr, = ~ Eyn + dnkCTkC (94)
4ir
108
BELL SYSTEM TECH NIC A L JOVRNA L
Here d^ij is a tensor of third rank and €,„„ one of second rank. The dmi)
constants are related to the eighteen ordinar}" constants (/,/ by the equations
du = d\n ; dn — d\oo ; dy
di6
2
"133 ; — "123 — "132 , — — "113 — "131 ;
'^222 ; ^^23 — '^233 ', Z '/223
</o32
T — dnu = ^231 ; ~r = 'A>i2 = fi'221 ; '/31
2
(})h)
^34 _ , _ , ^35
— "323 — "332 ; ^
</313 — dz
' 2
— "311 ; "32 — "322 ; "33 — "333
= "312 — "321 •
The tensor equations (94) give a simple method of expressing the piezo
electric equations in an alternate form which is useful for some purposes.
This involves relating the stress, strain, and displacement, rather than the
applied field strength as in (94;. To do this let us multii')ly through the
right hand equation of (94) by the tensor 47r,S,L, , obtaining
AK'Sl „ 5 „ = e J, ntimnEm + 47r(/ „ kt l^m n T k(
(96)
where /il,, is a icn:or of the "free" dielectric impermeability obtained from
the determinant.
^L = (1)'
,.yJ.
*r .
whe e A is the determinant
fu
€12
fl3
T
fl2
T
C22
r
€23
r
ei3
T
€23
T
€33
(97)
(98)
and Am,, the minor obtained from this by suppressing the wth row and nth
column. If we take the i)roduct el„ /i„.„ for the three values of w, we have
as multiijliers of E\ , Eo , E^ , respectively
€11 Pn + €12 Pl2 + €13 Pl3 = 1
€21 P2I + €22 P22 + €23 P23 == 1
€31 P31 r €32 P32 "T" €33 P33 — 1
(99):
Bui by virtue of equations (97) and (98) it is obvious that the value of
each term of (99) is unity. Hence we have
E„ — Aw0mn 5„ — (47r dnkt iSmn) 'i\t
(100)
i
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
109
Since the dummy index n is summed for the values 1, 2, and 3, we can set
the value of the terms in brackets equal to
and equation (100) becomes
Em = 47r (3mn 5„ — gmkC Tkl .
Substituting this equation in the first equations of (94) we have
where
Si,k( = Sijkf. — d„ni gmkl = Sijkt — 4:X[j8„„ d nkt dmij\.
(101)
(102)
(103)
By substituting in the various values of i,j, k and ^ corresponding to the 21
elastic constants, the difference between the constant displacement and
constant potential elastic constants can be calculated. If equations (102)
and (103) are expressed in terms of the Si, ■ , S^ strains and Ti, ■ •, T^
stresses, the gnij constants are related to the gij constants as are the corre
sponding dij constants to the (/„,/ constants of equation (95).
Another variation of the piezoelectric equations which is sometimes em
ployed is one for which the stresses are expressed in terms of the strains
and field strength. This form can be derived directly from equations (9i)
by multiplying both sides of the first equation by the tensor c^jkC for the
elastic constants, where these are defined in terms of the corresponding
s^j elastic compliances by the equation
4 = (i)^'"^^a:;/a
(104)
where A is the determinant
A^ =
^11
5l2
SlZ
5i4
^15
5l6
.f.
5^2
E
•^23
E
524
E
525
E
526
E
E
•^23
E
533
•^34
sl.
536
E
E
S2i
E
Sz\
E
544
E
545
54%
515 525 535 545 555 556
516 526 536 546 566 566
and A*y in the minor obtained by suppressing the /th row and^'th column.
Carrying out the tensor multiplication we have
Cijkt Sij = djkt Sijkt Tkf + dmij cjkC E„
(105)
no BELL SYSTEM TECHNICAL JOURNAL
As before \vc find that the tensor product of cijk( Si,k( is unity for all values
of k and (. Hence equation (105) can be written in the form
Tu(= clu(Si, e„.uE„, (106)
where Cmk( is the sum
CmkC = d,„ij cljkl (107)
surrn ed for all values of the dummy indices / and 7. If we substitute the
equation (106) in the last equation of (94) we lind
s
bn=^PEm + er^^Sij (108)
where e"™,, the clamped dielectric constant is related to the free dielectric
constant emn by the equation
ein ^ tin MdnUtemkt]. (109)
Expressed in two index piezoelectric constants involving the strains ^u • Svi
and stresses Tw • • T12 the relation between the two and three index piezo
electric constants is given by the equation
en = ^ni ; ^12 = ^122 ; ^13 = ^133 ; ^14 — ^123 = ^132 ; ^15 = ^U3 = ^131
e\e = «U2 = em ; ^21 = ^211; ^22 = ^222 ; ^23 = ^233 ; ^24 = ^223 = ^232
e25 = ^213 — ^231 ; ^26 = ^212 = ^221 ", ^31 = ^3U ; <'32 = <'322 ', «33 = ^333
^34 = ^323 = ^332 ', ^35 = ^313 = ^331 i ^36 = €312 = ^321 •
(110)
Finally, the fourth form for expressing the piezoelectric relation is the
one given by equation (53). Expressed in tensor form, these equations
become
TkC = c'^]k(S,j — h„ktb„ ; Em = 47r^'l„ bn — hmijSij (111)
In this equation the three index piezoelectric constants of equation (HI) are
related to the two index constants of equation (53) as the e constants of
(110). These equations can also be derived directly from (106) and (108)
by eliminating Em. from the two equations. This substitution yields the
additional relations
h„k( = ^T^ernkf (imn \ ^ikf = cfjkf + C„,k( I'mrj = C^ijkl
(112)
+ 47r emk( Cnij 0mn
where
i3L = (i)^"'*"'a:;Va''
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
111
in which
s
en
S
€12
S
€13
s
ei2
S
€22
.S
€23
s
€13
6'
€2.f
.S
€33
The four forms of the piezoelectric equations, and the relation between
them are given in Table I.
Table I
Four Forms of the Elastic, Dielectric, and Piezo Electric Equations
AND their Interrelations
Form
Elastic Relation
5,, = Si,k(Tu( + d„,,E,,
Electric Relation
bn = ~ En+ dnkfTkf
47r
2
Sii = Si,kfTk( + gn^jSn
E„, = 4x^„n5„  gmk(Tk(
3
Tk( = Cij(kSi,  emk(E„
s
iTT
4
Tk( = CijkfSu  h„kfbn
Em = iTT^ijn  hmiiSii
Form
Relation Between
Elastic Conjlaii.j
Relation Between
Piezoelectric Constants
Relation Between
Dielectric Constants
1
<*^= ^O^Z'^W^mAf
g^,(= 47r^l,d,,f
^L
= (i)""+">A^yA*^
2
cf^ = (1)(' + ^'a^^^/a«^
e,nkt = d„,,cf^^^
'tn
= e^  ■i^idnkfe„kf)
3
'iikf. = 'f,kf+''n>'f/^'"'i
k„k( = 47r^'L.'',„i/'
^L
 ^T ^ Rnkthn^kt
mn 4^
4
cO,= (1) (' + '■' A'^^/A^""
hnk( = SniiC'^,,(
^t.
= (1)('"+")A^V^'^
I 5. Effect of Symmetry and Orientation on the Dielectric Piezo
j electric and Elastic Constants of Crystals
j All crystals can be divided into 32 classes depending on the type of sym
1 metry. These groups can be divided into seven general classifications
il depending on how the axes are related and furthermore all il classes can
^ be built out of symmetries based on twofold (binary) axes, threefold (irig
1 onal) axes, fourfold axes of symmetry, sixfold axes of symmetry, planes of
j' reflection symmetry and combinations of axis reflection symmetry besides
a simple symmetry through the center. Each of these types of symmetry
1 1 2 BELL SYSTEM TECH NIC A L JOURNA L
result in a reduction of the number of dielectric, piezoelectric, and elastic
constants.
Since the tensor equation is easily transformed to a new set of axes by
the transformaion equations (76) this form is particularly advantageous
for determining the reduction in elastic, piezoelectric and dielectric con
stants. For example consider the second rank tensors, c^^ and ak( for the
dielectric constant and the expansion coefficients. Ordinarily for the most
general symmetry each tensor, since it is symmetrical, requires six inde
pendent coefficients. Suppose however that the X axis is an axis of twofold
or binary symmetry, i.e., the properties along the positive Z axis are the
same as those along the negative Z axis. If we rotate the axes 180° about
the A' axis so that f Z is changed into — Z, the direction cosines are
(113)
/  ^^1  1 .
,dxi
bxx ^
Wi = — = ;
dx2
dxi „
dX3
dX2 .
9X2 n
„2 = =
dxs
^3 = f^0;
dx\
dx's
"•'  a., ~ " ■
dx's
«3 = ^ = 1
dxs
transformation
equations for a second
/ dx'i dxj
dxk dxt
rank tensor are
(114)
Applying (113) to (114) summing for all values of k and / for each value of
i, and J we have the six components
' ' _ ' _ ' _ ' _ ' _ ('1 1 \
€11 — CU ; «12 — ~ €12 ; tl3 — — ei3 ; €22 — €22 ; ^23 — ^23 ', ^33 — ^33 • \ll^)
Since a crystal having the A' axis a binary axis of symmetry must have the
same constants for a \Z direction as for a — Z direction, this condition
can only be satisfied by
€12 = €13 = 0. (116)
The same condition is true for the expansion coefficients since they form a
second rank tensor and hence
«12 = «13 = 0. (117)
In a third rank tensor such as dijk , enk , gnh , I' nk , we similarly find that
of the eighteen independent constants
hm = //le ; //ii3 = //i5 ; /?2ii = /'2i ; //222 = /'22 ; //223 = hi ;
(118)
//233 = /'23 ; /'311 = //31 ', /'322 = /'32 ', Ihi^i — ll'M ', //333 — "33 •
are all zero. The same terms in the dijk , ^nk , gnk tensors are also zero.
PIEZOELECTRIC CR VST A LS IN TENSOR FORM 113
In a fourth rank tensor such as Cijk(, Sijkt, applying the tensor trans
formation equation
_ dXi dXj dXk dxe . .
'^*^tn ^'^n v'V'o ""vp
and the condition (113) we similarly find
Cl6 = Cl6 = ^25 = C26 = C35 = C36 = C45 = Ca = 0. (120)
If the binary axis had been the Y axis the corresponding missing terms
can be obtained by cyclically rotating the tensor indices. The missing
terms are for the second, third and fourth rank tensors, transformed to
two index symbols.
Cu , Cl6 , C24 , C26 , C34 , C36 , C45 , C55 .
Similarly if the Z axis is the binary axis, the missing constants are
ei3 , fi2 ; hn , hn , Ihz , hn , hi , h^ , ha , A26 , hzi , hzf, ;
(121)
(122)
Cu , CiB , C2A , C25 , Czi , C35 , C46 , Cb6 •
Hence a cr>'stal of the orthorhombic bisphenoidal class or class 6, which
has three binary axes, the X, Y and Z directions, will have the remaining
terms,
Cu , ^22 , ^33 ; hu , ^25 , ^'36 ', Cn , Cn , Cl3 , C21 , C23 , C33 , C44 , C55 , Cee (123)
with similar terms for other tensors of the same rank. Rochelle salt is a
crystal of this class.
If Z is a threefold axis of symmetry, the direction cosines for a set of
axes rotated 120° clockwise about Z are,
f I =  =  .5 ; wi = — =  .866 ; «i = t— =
oxi 0X2 dXz
^3 = ^^ = .866; m2=^=.5; «2 = ^^ = (124)
0x1 0x2 0x3
, dx'z dx'z ^ dx'z
4 = — =0; m3=— = 0; riz = ^— = 1.
dxi 0X2 0X3
Applying these relations to equations (114) for a second rank tensor, we
find for the components
€11 = .25eii+ .433ei2+ •75e22 ; ei2 = —. 433 cu + .25 €12 + .433^22
ei3 = — Seis — .866e23 ; €22 = .75€u — .433ei2 + .25c22 (125)
€23 = .866 en — .5e2j ; €33 = €33 •
114 BELL SYSTEM TECHNICAL JOURNAL
For the third and tifth equations, since we must have ei3 = cis ; €23 = f2;>
in order to satisfy the symmetry relation, the equations can only be satis
fied if
e.3 = eo3 = 0. (126)
Similarly solving the lirst three equations simultaneously, we find
fl2=0;6u= 622. (127)
Hence the remaining constants are
en = 622 ; 633 • (128)
Similarly for third and fourth rank tensors, for a crystal having Z a trigonal
axis, the remaining terms are
hn , hu = —lh\ , hn = 0; hu , //15 , /'le = — /'22
/?21 = — /'22, //22 , /'23 = 0, //24 = /'l5 ] hb = — hi , hi = " /?I1 (129)
//31 ; ^32 = //31 ; //33 ; /'34 = 0; /;35 = 0; //36 =
cn ; ^12 ; ^13 ; cu ; fis = ~<^25 ; ^le =
c\2 ; C21. — c\\ ; C23 = c\i ; C24 = — '"14 ; C25 ; ^26 =
Cn ; C20 = C\3 ; f33 ; ("34 = 0; czh — ^\ C36 =
(130)
Cu ; ^24 — ~Cu ;czi — ^; cu ; f45 — 0; C46 — c\^
C\i = —^25; ^25 ; <'35 = 0; f45 = 0; Css = C44 ; C56 = Cu
C16 = 0; ^26 — 0; r36 = 0; C46 — C21, ; ("56 "^ Cu ; fee = 2 vn~Ci2)
If the Z axis is a trigonal axis and the X a binary axis, as it is in quartz,
the resulting constants are obtained by combining the conditions (116),
(118), (120) with conditions (128), (129), (130) respectively. The resulting
second, third and fourth rank tensors have the following terms
611 ; 612 = 0; 613 =
612 = 0; 622 = 6U ; 623 = (131)
613 = 0; €23 = 0; €33
flu ; fin = — //ii ; //13 = 0; //i4 ; //I5 = 0; //i6 =
//21 = 0; //22 = 0; //23 = 0; //24 = 0; h, = hu ; //26 = hn (132)
//3l = 0; //32 = 0; //33 = 0; //34 = 0; hy,  0; //36 
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
115
(133)
Cn ; Ci2 ; C\3 ; Cu ; cis = 0; cie =
Cn 5 ^22 = ^11 ; ^23 = C\3 ; C24 = Ci4 ; C25 =0; C26 =
Ci3 ; ^23 = Ciz ; (^33 ; C34 = ; C35 = ; Cae =
fi4 ; C24 = — Ci4 ; r34 = 0; <:44 ; C45 = 0; r46 =
C15 = 0; ^26 = 0; f35 = 0; C45 = 0; C55 = ("44 ; C56 = Cu
<^i6 = 0; C26 = 0; f36 = 0; Css = 0; C55 = ru ; Cee = 2 (<^ii~fi2)
vS.l Second Rank Tensors for Crystal Classes
The symmetry relations have been calculated for all classes of crystals.
For a second rank tensor such as e,/, the following forms are required
Triclinic Classes 1 and 2 eu , €12 , €13
ei2 , ^22 , C23
«13 , «23 , ^33
fU , , €13
, €22 ,
ei3 , , €33
€11,0 ,0
, 622 , (134)
,0 , €33
€11,0 ,0
, €„ ,
,0 , €33
€11,0 ,0
, €„ ,
0,0, €„
5.2 Third Rank Tensors of the Piezoelectric Type for the Crystal Classes
hn , hu , his , /'i4 , /'15 , /'le
Monoclinic sphenoidal, 1' a binary axis, Class 3
MonocHnic domatic, Y a plane of symmetry. Class 4
Monoclinic prismatic, Center of symmetry, Class 5
Orthorhombic
Classes 6, 7, 8
Tetragonal, Trigonal
Hexagonal
Classes 9 to 27
Cubic
Classes 28 to 32
Triclinic Assymetric (Class 1) No
Symmetry
//21 , ^/22 , //23 , //24 , //26 , ^'26
/'31 , hsi , /?33 , //34 , //35 , hzr,
116
BELL SYSTEM TECHNICAL JOV RNAL
Triclinic pinacoidal, (center of symmetry) h = (Class 2)
,0 ,0 , //14 , , /?16
hii , lin , fhz ,0 , //26 ,
,0 ,0 , //34 , , /;,6
hn , Ih2 , hn ,0 , /7i6 ,
,0 ,0 , /724 , ,ht
hi , /'32 , /'33 , , hsB ,0
Monoclinic prismatic (center of symmetr>0 h = (Class 5)
,0 ,0 , /7i4 , ,0
,0 ,0 ,0 ,//26,0
,0 ,0 ,0 ,0 ,//36
,0 ,0 ,0 ,/;i6,0
,0 ,0 , //24 , ,0
/?31 , //32 , //33 , ,0 ,0
Orthorhombic bipyramidal (center of s}mmetr>) // = (Class 8)
, 0,0, liu , liib ,
Monoclinic Sphenoidal (Class 3) Y is
binary axis
Monoclinic domatic (Class 4) Y plane
is plane of symmetry
Orthorhombic bisphenoidal (Class 6)
X, Y, Z binary axes
Orthorhombic pyramidal (Class 7) Z
binary, X, Y, planes of s\Tnmetry
Tetragonal bisphenoidal (Class 9)
Z is quaternar}^ alternating
Tetragonal pyramidal (Class 10) Z
is quaternar}'
, 0,0, //15, /7l4,0
//31 , /'31 , , 0,0, //36
,0 ,0 , Ihi , //15 ,
,0 ,0 ,//l5, //i4,0
//31 , //31 , //33 , , ,0
Tetragonal scalenohedral (Class 11) / I ,0 ,0 , liu ,0 ,0
quaternar\'. A' and I' binary ,^ ,^ ,, ,, , n
^ ' , , , , //i4 ,
,0 ,0 ,0 ,0 , //36
Tetragonal trapezohedral (Class 12) jO ,0 ,0 , Im , ,0
Z quaternar^^ A' and F binar^^ 0.0,0,0, /;. ,
I , , , , 0,0
(135)
PI EZOELECTKTC CRYSTALS TN TENSOR FORM
117
Ditetragonal pyramidal (Class 14) Z
quaternary, X and 1' planes of
sy mmet ry
Tetragonal bipyramidal (center of symmtery) h — Q (Class 13)
, () ,0 ,0 , /;,5 ,
.0 .0 ,//i5,0 ,0
/■■U , /?.l , //33 , ,0 ,0
Ditetragonal bipyramidal (center of symmetry) // = (Class 15)
Trigonal pyramidal (Class //u , — //u , , hu , /?i5 , —fi'n
16) Z trigonal axis / a / / /
— //22, /^22 , , /;i5 —hu,—fin
hn , //;u , //3.3 , , ,
Trigonal rhombohedral (Class 17) center of symmetry, // =
Trigonal trapezohedral (Class
18), Z trigonal, .Y binary
Trigonal bipyramidal (Class
19), / trigonal, plane of
symmetry
Ditrigonal pyramidal (Class
20) Z trigonal, Y plane of
symmetry
Ditrigonal bipyramidal (Class
22) Z trigonal, Z plane of sym
metry and 1' plane of symmetry
Hexagonal pyramidal (Class 2i)
Z hexagonal
Hexagonal trapezohedral (Class
24) Z hexagonal, .Y binary
//u,
//u ,
/?14 ,
,
()
,
,0
,
Ihi ,
hn
,
f) ,
,
//ll.
//11,()
,
, — A22
//22,
//22 ,
,
, hn
,
,
,
,
,
(» ,
,
//15
//22
//22,
//?2 ,
//15,
hu ,
Ihl , //33
,
.
1) center of symmetry.
// =
hn,
//u ,0
, ,
,
,
,0
,0 ,
, hn
,
,
,0 ,
,
,
,
, Ihi ,
//15
,
,
, flu ,
— hu
,
hi ,
//31 , //33
, ,
,
,
,
, //14 ,
, (•
,
,0
,0 ,
//14
,
,
,0
,0 ,
,
1 18 BELL SYSTEM TECH NIC A L JOURNA L
Hexagonal bipyramidal (Class 25) center of symmetry, /? =
Dihexagonal pyramidal (Class 26) .Y
hexagonal Y plane of symmetry
,0 ,0 ,0 ,/7i5,0
,0 ,0 ,/7i5,0 ,0
h\ , //31 , /'33 , ,0 ,0
Dihexagonal bipyramidal (Class 27) center of symmetry, h =
Cubic tetrahedralpentagonaldedo
cahedral (Class 28) A', V, Z binary
,0 ,0 ,hu,0 ,0
,0 ,0 ,0 , //i4 ,
,0 ,0 ,0 ,0 ,/;,4
Cubic pentagonalicositetetrahedral (Class 29) ^ =
Cubic, dyakisdodecahedral (Class 30) center of symmetry, // =
Cubic, hexakisletrahedral (Class 31)
X, I', / quaternary alternating
,0 ,0 , /;i4 , ,0
,0 ,0 ,0 , //i4 ,
,0 ,0 ,0 ,0 ,/7i4
Cubic, hexakisoctahedral (Class 32) center of symmetry, // =
This third rank tensor has been expressed in terms of two index symbols
rather than the three index tensor symbols, since the two index symbols
are commonly used in expressing the piezoelectric effect. The relations
for the // and e constants are
// 14 , /' i5 , // lb are equivalent to // ,23 , // 113 , /' 112
(136)
in three index symbols, whereas for the d ij and gij constants we have the
relations
</,4 fl,5
1 ' T'
dit
are equivalent to r/,23 , d,n, ^,12
(137)
Hence the </, relations for classes 16, 18, 19, and 22 will be somewhat dif
ferent than the // symbols given above. These classes will be
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
119
Class 16
Class 18
Class 19
Class 22
dn —dn du dn —Id^i
— dvt d^i </i5 —du —2dn
dn dsi d33
^u dn du
du 2dn
dn dn 2^22
da (/22 2dn
^11 dn
2dn
(138)
5.3 Fourth Rank Tensors of the Elastic Type for the Crystal Classes
Triclinic System
cn
C\2
^13
Cu
Cl5
^6
The 5 tensor is
(Classes 1 and 2) 21
moduli
Cn
Coo
Cos
Coi
^25
C06
entirely analo
gous
Cl3
Cos
C33
C34
<"35
C36
Cli
C2i
C34
^44
a 5
f46
fl5
<:26
C3&
C45
f55
Cb6
("16
^20
C36
C46
^56
^66
(139)
Monoclinic System
Cn
C\o
Cn
fl5
The s tensor is
(Classes 3, 4 and 5) 12
moduli
Cl2
Co.i
C03
C2b
entirely analo
gous
C\3
Co.3
C33
Csb
Cii
C4f,
Cl5
f"25
<"36
(^55
C46
^66
120 BEl
Rhombic System
(Classes 6, 7 and 8)
9 moduli
Tetragonal system, Z
a fourfold axis (Classes
9, 10, 13) 7 moduli
Tetragonal system, Z a
fourfold axis, X a two
fold axis (Classes 11,
12, 14, 15) 6 moduli
Trigonal system, Z a
twofold axis, (Classes
16, 17) 7 moduli
L SYSTEM 7
^ECH
.V/Cl /
JOIRNAL
'11
Cu
('i.i
The s tensor is
en
C22
C23
{)
entirely analo
gous
Cn
C23
C33
C44
("55
CbG
Cn
C\2
Cn
{)
Cu
The s tensor is
C\2
Cn
Cn
— Cu
entirely analo
gous
Cn
Cn
C33
C44
Cu
^16
C16
Cf,e,
Cn
C\2
Cn
The i tensor is
Cu
Cn
Cn
entirely analo
gous
("13
Cn
C33
()
Cii
(44
("6fi
C\\
Cl2
Cn
("14 
t25
The 5 tensor is
cu
Cn
Cn
— ("14
("26
(^
analogous ex
cept that 546 =
Cn
Cn
C33
n
2^25 , ■^56 = 2^14 ,
Cu
(14
(■44
'25
^66 = 2 (511 — ^12)
— f26
(25
("44
("14
("25
("14
"11 — ^12
•
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
121
Trigonal system, Z a
trigonal axis, X a
binary axis (Classes
18, 20, 21) 6 moduli
Hexagonal system, Z a
sixfold axis, X a two
fold axis (Classes 19,
22, 23, 24, 25, 26, 27)
5 moduli
Cubic system (Classes
28, 29, 30, 31, 32) 3
moduli
Isotropic bodies,
moduli
Cl3
Cu
Cn
Cu — Ci4
Cn
C\1
C\3
Cxi
Cn
Cn
Cn
Cn
Cn
Cn
C\z
Cn
Cn
Cn
Cn
Cn
Cn
Cn Cu
Cn — Cu
C33
Cn
Cn
C3Z
Cn
Cn
Cn
Cn
Cn
Cn
C44
C44
C44
Cu
Cu
Cn
Cn
2
Cii
Cn
Cn Cn
C44
^11 ~ Cn
2
Cn
Cn ~ Cn
Cn — Cn
The 5 tensor is
analogous ex
cept that 556 =
2^14 , Stt —
2(511— 512)
The 5 tensor is
analogous ex
cept 566 =
2 (511 — 512)
The 5 tensor is
entirely analo
gous
The . 5 tensor
analogous ex
cept last three
diagonal terms
are 2 (511 — 512)
122 BELL SYSTEM TECH NIC A L JOURNA L
5.4 Piezoelectric Equations for Rotated Axes
Another application of the tensor equations for rotated axes is in deter
mining the piezoelectric equations of crystals whose length, width, and thick
ness do not coincide with the crystallographic axes of the crystal. Such
oriented cuts are useful for they sometimes give properties that cannot be
obtained with crystals h'ing along the crystallographic axes. Such proper
ties may be higher electromechanical coupling, freedom from coupling to
undesired modes of motion, or low temperature coefficients of frequency.
Hence in order to obtain the performance of such crystals it is necessary to
be able to express the piezoelectric equations in a form suitable for these
orientations. In fact in first measuring the properties of these crystals a
series of oriented cuts is commonly used since by employing such cuts the
resulting frequencies, and impedances can be used to calculate all the pri
mary constants of the crystal.
The piezoelectric equations (111) are
Tkl = CijkfSij — hnkC^n ; Em. = ^TTPmn^ n ~ hmijSij . (HI)
The first equation is a tensor of the second rank, while the second equation is
a tensor of the first rank. If we wish to transform these equations to another
set of axes x', y', z', we can employ the tensor transformation equations
, ^ dx[dx^ ^ dxldxf
dxk dX( dxk dx(
[CukfSn \~ 2Ci2k(Sl2 \ 2Cl3t^5'l5 + C22k(S22
+ 2c23ktSa + C33ktS3z]  ' —[hikth + h2k(b2 + hklh] (140)
axk oxf
EL = 47r p^ [/3li5i + ^':2 62 + ^isd^]  ^'
OXm dx„,
[hmllSl] + lllmuS 12 + 2llml3Sli + Am22«S'22 + 2(1^23^23 "f" hmSiSzs].
These equations express the new stresses and fields in terms of the old strains
and displacements. To complete the transformation we need to express
all quantities in terms of the new axes. For this purpose we employ the
tensor equations
dXi dXj , dXn ,
where ~r~i are the direction cosines between the old and new axes. It is
OXi
dx ■ 3x ■
obvious that — ' = — ^ and the relations can be written
OXi dx i
PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 123
A =
Wi = ;^ ; ^"2 = —/ ; ^3 = ^ (142)
Hence substituting equations (141) in equations (140) the transformation
equations between the new and old axes become
dxi
dxi
dxi
dxi
(2 — ^ ' \
dXi
dXi
dXi
dX2
dx[ '
dX2
dX2
W3 =  /
dxz
dxz
dxs
dX2
dX3
dxz
rp' _ D dXk dXf dXi dXj , _ dx^ dxf dx^ /
dXk ax I dXi axj dXk dXf dxn
(143)
These equations then provide means for determining the transformation of
constants from one set of axes to another.
As an example let us consider the case of an ADP crystal, vibrating longi
tudinally with its length along the xi axis, its width along the X2 axis and
its thickness along the X3 axis, which is also the X3 axis, and determine the
elastic, piezoelectric and dielectric constants that apply for this cut when
Xi is 9 = 45° from xi . Under these conditions
dx'i dxi
A = z— = 3/ = cos 9;
dxi 0x1
dxi dXi .
mi = — = —, = sm 8; W2
0x2 OXl
dxi dxs
Ml =— = —,= 0; Hi =
6x3 dxi
SX2
_ dxi .
dxi
, bill 17,
bX2
dxi dx'z
_ 6x2
_ dX2 .
dx2
, cost;,
0X2
dxs dx2
dX2 0x3
6X2
 ^""^  0
dXs
dx^ "'
dx'z dxz
dXz dx3
(144)
Since ADP belongs to the orthorhombic bisphenoidal (Class 6), it will have
the dielectric, piezoelectric and elastic tensors shown by equations (134),
(135), (139). Applying equations (143) and (144) to these tensors it is
124 BELL SYSTEM TECH NIC A L JOURNA L
readily shown that the stresses for 6 = 45° are given by the equations ex
pressed in two index symbols
^38 5 a
r =
(cfl +
2
+ 2c?«) ^,
Ol
+
((:fi +
Cl2
2
~ 2C66) c'
02
+
C\zSz
(rfl +
2
~ 2(;66) e'
Ol
4
(cfi +
Cl2
+ 2C?6) ^'
+
D c,'
Ci9 Oa
(145)
Tz — CizSl 4" Cl3 02 4" C33O3
r; = Cf4 5l + //14 62 ; £1 = /?145b + 47rLSuai']
Te = cf4 ^5  /?i4 5i ; £2 = h^'x + 47r[)Su52]
J,, ^ icn  c^2 ) _^^ . £^ ^ _^^^f^ „ 5^j ^ 4x1/333 53].
For a long thin longitudinally vibrating crystal all the stresses are zero
except the stress Ti along the length of the crystal. Hence it is more ad
vantageous to use equations which express the strains in terms of the
stresses since all the stresses can be set equal to zero except Ti . All the
strains are then dependent functions of the strain Si and this only has to
be solved for. Furthermore, since plated cjystals are usually used to
determine the properties of crystals, and the field perpendicular to a plated
surface is zero, the only field existing in a thin crystal will be £3 if the thick
ness is taken along the ^3 or Z axis. Plence the equations that express the
strains in terms of the stresses and fields are more advantageous for calcu
lating the properties of longitudinally vibrating crj^stals. By orienting
such crystals with respect to the crystallographic axis, all of the elastic
constants except the shear elastic constants can.be determined. All of
the piezoelectric and dielectric constants can be determined from measure
ments on oriented longitudinally vibrating crystals.
For such measurements it is necessary to determine the appropriate
elastic, piezoelectric, and dielectric constants for a crystal oriented in any
direction with respect to the crystallographic axes. We assume that the
length lies along the Xi axis, the width along the .T2 axis and the thickness
along the Xz axis. Starting with equations of the form
O t; ^^ Sxjlc(llcC ~l d i jmt'm
T (146)
47r
k
PIEZOELECTRIC CR YSTA LS IX TENSOR FORM 125
and transforming to a rotated system of axes whose direction cosines are
given by (142), the resulting equation becomes
(147)
, _ £ dx'i dx'j dXk dxt rp' , J dXi dx, dXm j^i
»'■ ~ ^''''^ ^. ^ f)r[ f)r'. ^^ '"* ax ax 'ax' '
OXi OXj OXk OXf UXt VXj UXm
./ _ emn dXn dXm 77' i j , ^^n dXk dx( f
47r daPn OX,n OXn OXk dX(
All the stresses except Tn can be set equal to zero and all the fields except
Ez vanish. Furthermore, all the strains are dependently related to ^n .
Hence for a thin longitudinal crystal the equation of motion becomes
, _ £ dx[ dx'i dXk dxt rp' , . dx[ dxi dx„ /
"^*'^'^ dx dx dx'y dx^ '"" dxdXdx{ '
. , , (148)
./ _ c^ ^ 5^ p' a: J dxzdxkdx( ,
47r 5x;, ^jcs " 5x„ 5a;i dx'i
In terms of the two index symbols for the most general type of crystal, we
have
E' E' £ /)4 I /^ E I E \ i)2 2 1 /T £ I E \ el 2
51111 = ^11 = SiiW + (2^12 + 566)^1^^1 + (2^13 + 55b)4Wi
+ 2{Sii + 5f6)^iWl«l + Isf^Vh + 25f6AWi + 5^2^!
+ /0 £ I jB \ 2 2 I r. E 3 , r,/ E , E\ Iff
(isiz + summi + isufmni + 2(^25 + 546)wi^i?h
2s26fniCi + 533W1 + IsziHinii + 2536^1^1
+ 2(5^6 + 5f5)«iAwi
(149)
! din = dn = dn^sd + du^ml + ^is^^i + dutzmiiti + dif^t^itii
I + dwtilinii + diinizli + doomm + d^sntsfii + dumsmifh
I + </25«3A"i + di^niztinii 4 c?3i"3^i + dsiUzml + dzs,mn\
i + dummini + dziUzkni + dz&nz^inii
\ €33 = «ii4 + leiitzmz + 2i.iz(znz + €22^3 + 2€23W3W3 + €33^3
I Hence by cutting 18 crystals with independent direction cosines 9 elastic
constants and 6 relations between the remaining twelve constants can be
I determined. All of the piezoelectric constants and all of the dielectric
constants can be determined from these measurements. These constants
can be measured by measuring the resonant and antiresonant frequencies
\ and the capacity at low frequencies. The resonant frequency Jr is deter
I mined by the formula
h = Yi V^ ^^^^
^^ y psii
126
BELL SYSTEM TECHNICAL JOURNAL
for any long thin crystal vibrating longitudinally. Hence when the density
is known, Sn can be calculated from the resonant frequency and the length
of the crystal. Using the values of Sn obtained for 15 independent orienta
tions enough data is available to solve for the constants of the first of
equations (149). The capacities of the different crystal orientations meas
ured at low frequencies determine the dielectric constant 633 and si.x orienta
tions are sufficient to determine the six independent dielectric constants
tmn ■ The separation between resonance and antiresonance Af = /a — Jr
determines the piezoelectric constant dn according to the formula
d\i =
;1/
£33
4^
^11
(151)
The \alues of dn measured for 18 independent orientations are sufficient
to determine the eighteen independent piezoelectric constants.
The remaining six elastic constants can be determined by measuring long
thin crystals in a face shear mode of motion. Since this is a contour mode
of motion, the equations are considerably more complicated than for a
longitudinal mode and involve elastic constants that are not constant field
or constant displacement constants. It can be shown that the fundamental
frequency of a crystal with its length along x\ , width (frequency determining
direction) along .Vo and thickness (direction of applied field) along xs , will be
1 / c.E I c,E , a// c.E c,E\2 1 . c.
{ = — i/ ^ 22 \ C66 ± V (C22 — ^66 ) + 4C26
^ 2C y 2p
(152)
where the contour elastic constants are given in terms of the fundamental
elastic constants by
E E £2
c.E ^ll •^66 ■^16
C21 = ;
E E E E
c,E ^12 ■^16 •^11 ^26
C26 = 1
E E £2
c.B _ SnS22 ^12
C66 — :
(153)
where A is the determinant
A =
Su ,
SV2 ,
Sl6
E
S\2 ,
B
S22 }
E
526
E
Sl6 ,
E
^26 ,
E
■^66
(154)
Since all of the constants except svi and ^ee can be determined by measure
ments on longitudinal crystals and the value of (25f2 + ^ee) has been de
' This is proved in a recent paper "Properties of Dipotassium Tartrate (DKT) Crys
tals," Phys. Rev., Nov., 1946.
PIEZOELECTRIC CR VST A LS IN TENSOR FORM 127
termined, the measurement of the lowest mode of the face shear crystal
gives one more relation and hence the values of 5i2 and S6& can be uniquely
determined.
Similar measurements with crystals cut normal to Xi and width along Xs
and with crystals cut normal to X2 and width along Xi determine the constants
SAi , 523 and 555 , Siz respectively. The equivalent constants are obtained
by adding 1 to each subscript 1, 2, 3 or 4, 5, 6 for the iirst crystal with the
understanding that 3+1 = 1 and 6+1 =4. For the second crystal 2
is added to each subscript.
Finally the remaining three constants can be determined by measuring
the face shear mode of three crystals that have their lengths along one of
the crystallographic axes and their width (frequency determining. axis)
45° from the other two axes.
Any symmetry existing in the crystal will cut down on the number of
constants and hence on the number of orientations to determine the funda
mental constants.
6. Temperature Effects in Crystals
In section 2 a general expression was developed for the effects of tempera"
ture and entropy on the constants of a crystal. Two methods were given,
one which considers the stresses, field, and temperature differentials as the
independent variables, and the second which considers the strains, displace
ments and entropy as the independent variables. In tensor form the 10
equations for the first method take the form
Em= — hm i jS i J + ■iir^m'n 5 n " qll dQ (155)
The piezoelectric relations have already been discussed for adiabatic condi
tions assuming that no increments of heat dQ have been added to the
crystal.
If now an increment of heat dQ is suddenly added to any element of the
crystal, the first equation shows that a sudden expansive stress is generated
S.D
proportional to the constant X;t^ which has to be balanced by a negative
stress (a compression) in order that no strain or electric displacement shall
be generated. This effect can be called the stress caloric effect. The
second equation of (155) shows that if an increment of heat dQ is added to
the crystal an inverse field Em has to be added if the strain and surface
charge are to remain unchanged. This effect may be called the field caloric
128 BELL SYSTEM TECH NIC A L JOURNA L
effect. Finally the third equation of (155) shows that there is a reciprocal
efifect in which a stress or a displacement generates a change in temperature
even in the absence of added heat dQ. These effects can be called the strain
temperature and charge temperature effects.
The second form of the piezoelectric equations given by (58) are more
familiar. In tensor form these can be written
Sij = sfjktT.cl + dZijEm + afy do
8n = dlk( Tk( + '4^E^ + pi dQ (156)
47r
dQ = eda ^ QatcTut + QplErr, + pCl dS
The afy are the temperature expansion coefficients measured at constant
field. In general these are a tensor of the secjnd rank having six com
ponents. The constants pn are the pyroelectric constants measured at
displacements which relate the increase in polarization or surface charge
due to an increase in temperature. They are equal to the socalled "true"
pyroelectric constants which are the polarizations at constant volume caused
by an increase in tempeiature plus the "false" pyioelectric effect of the
first kind which represents the polarization caused by a uniform temperature
expansion of the crystal as its temperature increases by dQ. As mentioned
previously it is more logical to call the two effects the pyroelectric effects
at constant stress and constant strain. By eliminating the stresses from
the first of equations (156) and substituting in the second equation it is
readily shown that
Pn = Pn — OC^,enij (157)
Hence the difference between the pyroelectric effect at constant stress and
the pyroelectric effect at constant strain is the socalled "false" pyroelectric
effect of the first kind a^je^a .
The first term on the right side of the last equation is called the heat of
deformation, for it represents the heat generated by the application of the
stresses TkC ■ The second term is called the electrocaloric effect and it
represents the heat generated by the application of a field. The last term
is p times the specific heat at constant pressure and constant field.
The temperature expansion coefficients a.y form a tensor of the second
rank and hence have the same components for the various crystal classes
as do the dielectric constants shown by equation (134).
The pyroelectric tensor pn and /?'„ are tensors of the first rank and in
general will have three components pi , p2 , and Ps . In a similar manner
to that used for second, third and fourth rank tensors it can be shown that
the various crystal classes have the following comi)onents for the first rank
tensor />,. .
FIEZOELECTKIC CRYSTALS IN TENSOR FORM 129
Class 1 : components pi , pi , ps ■
Class 3 : I' axis of binary symmetry, components 0, p2 ,0 (158)
Class 4: components pi , 0, ps .
Classes 7, 10, 14, 16, 20, 23, and 26: components 0, 0, pz ; and Classes
2, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 19, 21, 22, 24, 25, 27, 28, 29, 30, 31, and
32: components 0, 0, 0, i.e., /> = 0.
For a hydrostatic pressure, the stress tensor has the components
Tn = T22= Tss^ —p = pressure; T12 = Tn = ^23 = (159)
Hence the displacement equations of (156) can be written in the form
K = '4^ Em <^np + pldQ (160)
where
<^np = dnlJn + d n22T22 + <^n33?'3.3
that is the contracted tensor d nkkTkk ■ This is a tensor of the tirst rank
which has the same components as the pyroelectric tensor pn for the various
cPv'stal classes.
7. Second Order Effects in Piezoelectric Crystals
We have so far considered only the conditions for which the stresses and
tields are linear functions of the strains and electric displacements. A
number of second order effects exist when we consider that the relations are
not linear. Such relations are of some interest in ferroelectric crystals such
as Rochelle salt. A ferroelectric crystal is one in which a spontaneous
polarization exists over certain temperature ranges due to a cooperative
effect in the crystal which lines up all of the elementary dipoles in a given
"domain" all in one direction. Since a spontaneous polarization occurs in
such crj'stals it is more advantageous to develop the equations in terms of
the electric displacement rather than the external field. Also heat effects
are not prominent in second order effects so that we develop the strains and
potentials in terms of the stresses and electric displacements D. By means
of McLaurin's theorem the first and second order terms are in tensoi form
_ dSij dSij 1 r d'^Sij
^'' ~ dTkf ^'^ ^ a5„ ^" + 21 IdTkCdT^r ^'^^'^
+ 2 „„ „j TkCdn + rr^r 6„5o + ■ • • higher terms
d'E„
(161)
dTktdTn
TklT^r
d^Em d^E„, 1
+ 2 ^^T^T ^i<^^" + ^777 5„5o + • • • higher terms
dTktdSn d5„d5o
whereas before 8 = D/4ir
130 BELL S YSTEM TECH NIC A L JOURNA L
In this equation the linear partial differentials have already been discussed
and are given by the equations
dSij y dSij dEn dEm T
where s^nkt are the elastic compliances of the crystal at constant displace
ment, gijn the piezoelectric constants relating strain to electric displacement
/At, and /3l„ the dielectric "impermeability" tensor measured at constant
stress. We designate the partial derivatives
dTddT,r ^'''^''■' dT,m„, dn^dT^r y'^"
d Sij _ d'En _ ^D . d'Em __(^D
ddnddo dTijdSo dSndSo
(163)
The tensors N, M, Q, and are respectively tensors of rank 6, 5, 4 and 3
whose interpretation is . discussed below. Introducing these definitions
equations (161) can be written in the form
Em = Tkflgmkf.+ h^^ikfnTqr + Qkfmn^,] + SnlATT^mn + ^Omno^
Written in this form the interpretation of the second order terms is obvious.
N'ijkfgr represents the nonlinear changes in the elastic compliances s^jj
caused by the application of stress Tgr . Since the product of N nklqrTqr
represents a contracted fourth rank tensor, there is a correction term for
each elastic compliance. The tensor M'^jkfn can represent either the non
linear correction to the elastic compliances due to an applied electric dis
placement Dn or it can represent the correction to the piezoelectric constant
gijn due to the stresses Tk( . By virtue of the second equation of (162),
the second equivalence of (163) results. The fourth rank tensor ^Qnno
represents the electrostrictive effect in a crystal" for it determines the strains
existing in a crystal which are proportional to the square of the electric
displacement. Twice the value of the electrostrictive tensor ^Q^j„o , which
appears in the second equation of (164) can be interpreted as the change
in the inverse dielectric constant or "impermeability" constant. Since a
change in dielectric constant with applied stress causes a double refraction
of light through the crystal, this term is the source of the piezooptical effect
in crystals. The third rank tensor Omno represents the change in the "im
permeability" constant due to an electric field and hence is the source of
the electrooptical effect in crystals.
These equations can also be used to discuss the changes that occur in
ferroelectric type crystals such as Rochelle Salt when a spontaneous polariza
PIEZOELECTRIC CR YSTA LS IN TENSOR FORM 131
tion occurs in the crystal. When spontaneous polarization occurs, the
dipoles of the crystal are Uned up in one direction in a given domain. For
Rochelle salt this direction is the ±X axis of the crystal. Now the electric
displacement Dz is equal to
47r 47r 47r
= f:^^ = f:f + P,„ + P,^ = ^o p,^ (165)
where Px^ is the electronic and atomic polarization, and Px^ the dipole
polarization The electronic and atomic polarization is determined by the
field and hence can be combined with the field through the dielectric constant
eo , which is the temperature independent part of the dielectric constant.
When the crystal becomes spontaneously polarized, a field E^ will result, but
this soon is neutralized by the flow of electrons through the surface and
volume conductance of the crystal and in a short time Ez = 0. Hence for
any permanent changes occurring in the crystal we can set
8x = — =PxD = dipole polarization (166)
47r
which we will write hereafter as Pi .
In the absence of external stresses the direct effects of spontaneous polari
zation are a spontaneous set of strains introduced by the product of the
spontaneous polarization by the piezoelectric constant, and another set
produced by the square of the polarization times the appropriate electro
strictive components. For example, Rochelle salt has a spontaneous
polarization Pi along the Xi axis between the temperatures — 18°C to
+ 24°C. The curve for the spontaneous polarization as a function of
temperature is shown by Fig. 6. The only piezoelectric constant causing
a spontaneous strain will be ^14/2 = gnz • Hence the spontaneous polariza
tion causes a spontaneous shearing strain
S, = guPz = 120 X 10"' X 760 = 9.1 X 10~* (167^
if we introduce the experimentally determined values. Since .5'4 is the
cosine of 90° plus the angle of distortion, this would indicate that the right
angled axes of a rhombic system would be distorted 3.1 minutes of arc.
This is the value that should hold for any domain. For a crystal with
several domains, the resulting distortion will be partly annulled by the
different signs of the polarization and should be smaller. Mueller measured
an angle of 3'45" at 0°C for one crystal. This question has also been
* This has been measured by measuring the remanent polarization, when ail the domains
are lined up. See "The Dielectric Anomalies of Rochelle Salt," H. Mueller, Annals of
the N. Y. Acad. Science, Vol. XL, Art. 5, page 338, Dec. 31, 1940.
^ "Properties of Rochelle Salt," H. Mueller, Phys. Rev., Vol. 57, No. 9, May 1, 1940.
132
BELL SYSTEM TECH NIC A L JOURNAL
investigated by the writer and Miss E. J. Armstrong by measuring the
temperature expansion coefficients of the Y and Z axes and comparing their
average with the expansion coefficient at 45° from these two axes. The
difference between these two expansion coefficients measures the change
in angle between the Y and Z axes caused by the spontaneous shearing
strains. The results are shown by Fig. 7. Above and below the ferro
electric region, the expansion of the 45° crystal coincides with the average
expansion of the Y and Z axes measured from 25°C as a reference tempera
ture. Between the Curie temperatures a difference occui^ indicating thai
the Y and Z crystallographic axes are no longer at right angles. The dif
ference in expansion per unit length at 0°C (ihe maximum point) corresponds
to 1.6 X 10"* cm per cm. This represents an axis d istortion of 1 .1 minutes
700
600
t3
No
<5 500
II
Q.O
a.
m
400
DO
^ai 300
0<J 200
O 7
100
20 16 12 84 4 8 12 16 20 24 28
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 6. — Spontaneous polarization in Rochelie Salt along the X axis.
of arc. Correspondingly smaller values are found at other temperatures
in agreement with the smaller spontaneous polarization at other tempera
tures. It was also found that practically the same curve resulted for either
45° axis, indicating that the mechanical bias put on by the optometer used
for measuring expansions introduced a bias determining the direction of the
largest number of domains.
The second order terms caused by the square of the spontaneous polariza
tion is given by the expression
S,i = QlnP\ (168)
Since Q is a fourth rank tensor the possible terms for an orthorhombic
bisphenoidal crystal (the class for Rochelie salt) are
5u = QinxPl ; ^22 = Q2inP\ ; ^33 = QunPl (169)
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
133
In an effort to measure these effects, careful measurements have been made
of the temperature expansions of the three axes X, Y and Z. The results
are shown by Table II. On account of the small change in dimension from
(lO*
O 16
at 18
.'
'
/
/
/
/
f
J
i
V
>
y
/
V
J
f
1
>
V
^
}
•
M.n' OF ARC
J
•
r /
^
I
•
•
^
t
A
r
/
f
X
/
y
Y'
r
40 35 30 25 20 15 10 5 5 10 15 20 25 30 35
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 7. — Temperature expansion curve along an axis 45° between Y and Z as a
function of temperature.
the Standard curve it is difiScult to pick out the spontaneous components
by plotting a cur\e. By expressing the expansion in the form of the
equation
AL
^ = ai(r25) + 02(^25)' + ^3(^25)'
(170)
134
BELL SYSTEM TECHNICAL JOURNAL
Table II
Measured Temperature Expansions for the Three Crystalographic .\xes
Temperature Expansion
Temperature
in °C.
Expansion
X lOi
Y Axis
Temperature
in °C.
Expansion
in °C.
X io«
a: Axis
X 10«
Z Axis
39.6
38.7
35.2
10.2
9.46
6.96
+ 35.0
30.3
25.25
4.45
2.5
0.2
1,34.9
29.9
25.05
+4.9
2.5
+ .05
30.2
27.2
26.2
3.63
1.41
0.71
23.9
22.9
19.35
0.42
0.88
2.4
24.0
19.95
14.95
.5
2.62
5.11
25.15
24.0
23.0
0.06
0.71
1.39
14.9
10.0
5.4
4.25
6.25
8.18
+9.75
+4.9
7.55
9.9
12.31
21.8
16.0
15.2
2.37
6.5
7.05
+0.3
9.7
16.3
10.15
13.98
16.41
6.35
10.5
15.0
15.3
17.29
19.42
4.9
+0.3
4.7
14.12
17.28
20.3
20.85
25.1
30.3
17.94
19.22
—20.8
18.0
23.2
25.1
20.86
23.08
23.96
10.7
15.3
20.7
24.0
26.6
30.2
35.0
39.7
53.2
22.23
23.54
27.60
31.1
35.0
40.0
26.59
28.28
30.4
25.7
30.1
34.7
32.7
35.2
37.85
40.7
45.0
50.5
41.25
44.0
47.0
and evaluating the constants by employing temperatures outside of the
ferroelectric range, a normal curve was established. For the X, Y, and Z
axes these relations are
AL
lO/T, i\3
AL
= 69.6 X 10""'(r25) + 7.4 X 10""(T25)'  3.13 X 10 "'(T25)
{X direction)
= 43.7 X 10~*(T25) + 8.16 X 10''(T25)'  3.60 X 10~'''(T25)'
(I' direction)
= 49.4 X 10~'(r25) + 1.555 X 10"'(r25)'  2.34 X nr'\T25)
{Z direction)
(i7i;
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
135
The difference between the normal curves and the measured values in the
Curie region is shown plotted by the points of Fig. 8. The solid and dashed
curves represent curves proportional to the square of the spontaneous
polarization and with multiplying constants adjusted to give the best fits
for the measured points. These give values of Qim , Qizn , Qasn equal to
Qnu = 86.5 X 10^^'; Q^,u = +17.3 X 10~'';
Q^^n=2A.2xm'' (172) ^^^^^
Another effect noted for Rochelle salt is that some of the elastic constants
suddenly change by small amounts at the Curie temperatures. This is a
consequence of the tensor Mfy^^,,, for if a spontaneous polarization P
5 15
o 40
A

'\
r
A
^^
^S22
^
^^
>''
A
^r^
^
'\
^
.n"'''^
^
"^^
D
D
%>'
''''
1
\
"*~^
1
1
—''■
n
/
\
/
o
\
/
\
/
\
/
\o
/
\
/
°X
/
4 4 8 12 16
TEMPERATURE IN DEGREES CENTIGRADE
Fig. 8. — Spontaneous electrostrictive strain in Rochelle Salt along the
three crystallographic axes.
occurs, a sudden change occurs in some of the elastic constants as can be seen
from the first of equations (164). The second equation of (164) shows
that this same tensor causes a nonlinear response in the piezoelectric con
stant. Since a change in the elastic constant is much more easily deter
mined than a nonlinear change in the piezoelectric constant, the first effect
is the only one definitely determined experimentally. Since all three crys
tallographic axes are binary axes in Rochelle salt, it is easily shown that
the only terms that can exist for a fifth rank tensor are terms of the types
Mxxm ; Mf2223 ; iWf2333 (173)
with permutations and combinations of the indices. Hence when a spon
taneous polarization l\ occurs, the elastic constants become
s%kt  MtikdPx (174)
136
BELL SYSTEM TECHNICAL JOURNAL
Comparing these with the relation of (90) we see that the spontaneous
polarization has added the elastic constants
D {Minn + Minii + Mnni + Mz2ni)Pi
(175)
014 
2
<r" 
(Af 22231 + M2232I + M2322I + 3/32221) fl
J24 
2
V
(M^3331 + Mf2331 + ^33321 + MiZ2ii)Pi
Sb6 —
(iWf21bl + M'f32U + M3II2I + M312II
+ Mf2i3i + Mf23ii + Mnni + A/^i3ii)/^i
between the two Curie points. Hence while the spontaneous polarization
Pi exists, the resulting elastic constants are
^11 ,
5l2,
5l3 ,
Sh ,
,
•^12 ,
522,
523,
524 ,
,
•^13 ,
■^23 ,
533 ,
534 ,
,
•^14 ,
■^24 ,
Sm ,
544 ,
,
,
,
,
,
555 ,
5&6
,
,
,
,
556 ,
566
(176)
Comparing this to equation (139) which shows the possible elastic constants
for the various crystal classes, we see that between the two Curie points,
the crystal is equivalent to a monoclinic sphenoidal crystal (Class 3) with
the X axis the binary axis. Outside the Curie region the crystal becomes
orthorhombic bisphenoidal. This interpretation agrees with the tempera
ture expansion curves of Fig. 7.
The sudden appearance of the polarization 1\ will affect the frequency
of a 45° ,Ycut crystal, for with a crystal cut normal to the .Y axis and with
the length of the crystal at an angle B with the Y axis, the value of the
elastic compliance 522 along the length is
522' = 5^2 cos* G f 25^4 cos^ B sin B + (25^3 + 54*4) sin B cos B
(177)
+ 2^34 sin B cos B + Sn sin B
Hence for a crystal with its length 45° between the Y and Z axes, elastic
compliance becomes
'« _ 522 ~1~ 2(524 ~1~ 523 + 534) + ^44 + 533
S21 —
(178)
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
137
For a 45*^ Xcut crystal we would expect a sudden change in the value of
522 as the crystal becomes spontaneously polarized between the two Curie
points due to the addition of the s^i and s^^ elastic compliances. Such a
change has been observed for Rochelle salt* as shown by Fig. 9 which shows
the frequency constant of a nonplated crystal for which the elastic com
pliances s^j should hold.
uj 217
5
Z 216
UJ
u
QC 215
O 209
2 208
q 207
\^
\
^
*.^
^
X
■^
/^l
_Q_
1
^
\
"x"
*x^
1
\
\
^
>>
N^
^^^
1
1
\
V
^
V_
1
1
\
1
1
^*>
^
\
\
\
\
^_^
,
,
">s^
N
\
\
\
^/'
^'
N
\ ,
V
n\
V
,^_y
V
\\
\
u.
o
6 Hi
3
4
3
2
1
20 16 12
84 4 8 12 16 20 24 28 32
TEMPERATURE IN DEGREES CENTIGRADE
36 40 44 48
Fig. 9. — Frequency constant and Q of an unplated 45°X cut Rochelle Salt ctystal
plotted as a function of temperature.
Hence the sudden change in the elastic constant is a result of the two
second order terms s^ f s^i , which are caused by the spontaneous polariza
tion. The value of the sum of these two terms at the mean temperature
of the Curie range, 3°C is
•^24 + ^34 = 4.1 X 10 cm"/ dyne
(179)
Crystals cut normal to the Y and Z axes should not show a spontaneous
change in their frequency characteristic since the spontaneous terms Su ,
524 , 534 and 5b6 do not affect the value of Young's modulii in planes normal
to Y and Z. Experiments on a 45° Fcut Rochelle salt crystal do not show
a spontaneous change in frequency at the Curie temperature, although there
is a large change in the temperature coefficient of the elastic compliance
between the two Curie points. This is the result of third order term and is
' "The Location of Hysteresis Phenomena in Rochelle Salt Crystals," W. P. Mason,
Phys. Rev., Vol. 50, p. 744750, October 15, 1940.
gn
^12
gl3
gl4
138 BELL S YSTEM TECH NIC A L JOURNA L
not considered here. The spontaneous ^ae constant affects the shear con
stant ^66 for crystals rotated about the A' axis and could be detected experi
mentally. No experimental values have been obtained.
The effects of spontaneous polarization in the second equation of (164)
are of two sorts. For an unplated crystal, a spontaneous voltage is gen
erated on the surface, which, however, quickly leaks off due to the surface
and volume leakage of the crystal. The other effects are that the spon
taneous polarization introduces new piezoelectric constants through the
tensor Qkfmn , changes the dielectric constants through the tensor Omno and
introduces a stress effect on the piezoelectric constants through the tensor
Mkfmqr ■ Siuce piezoelectric constants are not as accurately measured as
elastic constants, the first effect has not been observed. The additional
piezoelectric constants introduced by the tensor Qkfmn are shown by equa
tion (180)
g2, g26 (180)
Since the only constants for the Rochelle salt class, the orthorhombic
bisphenoidal, are gu , g2b , gse , this shows that between the two Curie points
the crystal becomes monoclinic sphenoidal, with the A' axis being the
binary axis. The added constants are, however, so small that the accuracy
of measurement is not sufficient to evaluate them. From the expansion
measurements of equation (172) and the spontaneous polarization values,
three of them should have maximum values of
gn = 6.6 X 10"^ gu = +1.3 X 10"'; gn = 1.8 X 10"' (181)
These amount to only 6 per cent of the constant gu , and hence they are
not easily evaluated from piezoelectric measurements.
The effect of the tensor Omuo is to introduce a spontaneous dielectric
constant €23 between the Curie temperatures so that the dielectric tensor
becomes
en, ,
0, e,,, 623 (182)
, €23 , f33
As discussed at length by Mueller'"* this introduces a spontaneous bire
fringence for light passing through the crystal along the A', 1' and Z axes
which adds to the birefringence already present.
« "Proi)crtics of Rochcilc Salt I and IV," Phvs. Rev. 47, 175 (1935); 58, 805 November 1,
1940.
i
The Biased Ideal Rectifier
By W. R. BENNETT
Methods of solution and specific results are given for the spectrum of the
response of devices which have sharply defined transitions between conducting
and nonconducting regions in their characteristics. The input wave consists
of one or more sinusoidal components and the operating point is adjusted by bias,
which may either be independently applied or produced bv the rectified output
itself.
Introduction
THE concept of an ideal rectifier gives a useful approximation for the
analysis of many kinds of communication circuits. An ideal rectifier
conducts in only one direction, and by use of a suitable bias may have the
critical value of input separating nonconduction from, conduction shifted
to any arbitrary value, as illustrated in Fig. 1. A curve similar to Fig. 1
might represent for example the current versus voltage relation of a biased
diode. By superposing appropriate rectifying and linear characteristics
with different conducting directions and values of bias, we may approximate
the characteristic of an ideal limiter. Fig. 2, which gives constant response
when the input voltage falls outside a given range. Such a curve might
approximate the relationship between flux and magnetizing force in certain
ferromagnetic materials, or the output current versus Signal voltage in a
negativefeedback amplifier. The abrupt transitions from nonconducting
to conducting regions shown are not realizable in physical circuits, but the
actual characteristics obtained in many devices are much sharper than can
be represented adequately by a small number of terms in a power series
or in fact by any very simple analytic function expressible in a reasonably
small number of terms valid for both the nonconducting and conducting
regions.
In the typical communication problem the input is a signal which may
be expressed in terms of one or more sinusoidal components. The output
of the rectifier consists of modified segments of the original resultant of the
individual components separated by regions in which the wave is zero or
constant. We are not so much interested in the actual wave form of these
choppedup portions, which would be very easy to compute, as in the fre
quency spectrum. The reason for this is that the rectifier or limiter is
usually followed by a frequencyselective circuit, which delivers a smoothly
varying function of time. Knowing the spectrum of the chopped input
to the selective network and the steadystate response as a function of
139
140
BELL SYSTEM TECHNICAL JOURNAL
BIAS
APPLIED VOLTAGE
Fig. 1. — Ideal biased linear rectifier characteristic.
(1)
LINEAR
CHARACTERISTIC
(2)
BIASED POSITIVE
RECTIFIER
(3)
BIASED NEGATIVE
RECTIFIER
bi
^
(4)
BIASED IDEAL
LIMITER
I'ig. 2. — Synthesis of liniiter characteristic.
THE BIASED IDEAL RECTIFIER 141
frequency of the network, we can calculate the output wave, which is the
one having most practical importance. The frequency selectivity may in
many cases be an inherent part of the rectifying or limiting action so that
discrete separation of the nonlinear and linear features may not actually
be possible, but even then independent treatment of the two processes
often yields valuable information.
The formulation of the analytical problem is very simple. The standard
theory of Fourier series may be used to obtain expressions for the amplitudes
of the harmonics in the rectifier output in the case of a single applied fre
quency, or for the amplitudes of combination tones in the output when two
or more frequencies are applied. These expressions are definite integrals
involving nothing more compUcated than trigonometric functions and the
functions defining the conducting law of the rectifier. If we were content
to make calculations from these integrals directly by numerical or mechanical
methods, the complete solutions could readily be written down for a variety
of cases covering most communication needs, and straightforward though
often laborious computations could then be based on these to accumulate
eventually a suflficient volume of data to make further calculations un
necessary.
Such a procedure however falls short of being satisfactory to those who
would like to know more about the functions defined by these integrals
without making extensive numerical calculations. A question of consider
able interest is that of determining under what conditions the integrals may
be evaluated in terms of tabulated functions or in terms of any other func
tions about which something is already known. Information of this sort
would at least save numerical computing and could be a valuable aid in
studying the more general aspects of the communication system of which
the rectifier may be only one part. It is the purpose of this paper to present
some of these relationships that have been worked out over a considerable
period of time. These results have been found useful in a variety of prob
lems, such as distortion and crossmodulation in overloaded ampUfiers,
the performance of modulators and detectors, and efifects of saturation in
magnetic materials. It is hoped that their publication will not only make
them available to more people, but also stimulate further investigations of
the functions encountered in biased rectifier problems.
The general forms of the integrals defining the amplitudes of harmonics
and side frequencies when one or two frequencies are applied to a biased
rectifier are written down in Section I. These results are based on the
standard theory of Fourier series in one or more variables. Some general
relationships between positive and negative bias, and between limiters and
biased rectifiers are also set down for further reference. Some discussion is
given of the modifications necessary when reactive elements are used in the
circuit.
142 BELL SYSTEM TECHNICAL JOURNAL
Section 11 summarizes specific results on the singlefrequency biased
rectitier case. The general expression for the amplitude of the typical
harmonic is evaluated in terms of a hypergeometric function for the power
law case with arbitrary exponent.
Section HI takes up the evaluation of the twofrequency modulation
products. It is found that the integerpowerlaw case Tan be expressed in
finite form in terms of complete elliptic integrals of the first, second, and
third kind for almost all products. Of these the first two are available in
tables, directly, and the third can be expressed in terms of incomplete
integrals of the first and second kinds, of which tables also exist. No direct
tabulation of the complete elliptic integrals of the third kind encountered
here is known to the author. They are of the hyperbolic type in contrast
to the circular ones more usual in dynamical problems. Imaginary values
of the angle /3 would be required in the recently published table by Heuman .
A few of the product amplitudes depend on an integral which has not
been reduced to elliptic form, and which is a transcendental function of two
variables about which little is known. Graphs calculated by numerical
integration are included.
The expressions in terms of elliptic integrals, while finite for any product,
show a rather disturbing complexity when compared with the original
integrals from which they are derived. It appears that elliptic functions
are not the most natural ones in which the solution to our problem can be
expressed. If we did not have the elliptic tables available, we would prefer
to define new functions from our integrals directly, and the study of such
functions might be an interesting' and fruitful mathematical exercise.
Solutions for more than two frequencies are theoretically possible by the
same methods, although an increase of complexity occurs as the first few
components are added. When the number of components becomes very
large, however, limiting conditions may be evaluated which reduce the
problem to a manageable simplicity again. The case of an infinite number
of components uniformly spaced along an appropriate frequency range has
been used successfully as a representation of a noise wave, and the detected
output from signal and noise inputs thus evaluated . The noise problerri
will not be treated in the present paper.
I. The General Problem
Let the biased rectifier characteristic, Fig. 1, be expressed by
/ 0, E < b\
I = (1.1)
\f{E b), b < eJ
1 Carl Heuman, Tables of Comi)letc Ellii)tic Integrals, Jour. Math, and Phvsics, Vol.
XX, No. 2, pp. 127206, April, 1941.
. ^ W. R. Hcnnctt, Response of a Linear Rectifier to Signal and Noise, Jour. Acous. Soc.
Amer., Vol. 15, pp. 164172, Jan. 1944.
THE BIASED IDEA L RECTIFIER
143
Then if a single frequency wave defined by
E = P cos pt,  P < b < P, (1.2)
is applied as input, the output contains only the tips of the wave, as shown
in Fig. 3. It is convenient to place the restrictions on P and b given in
Eq. (1.2). The sign of P is taken as positive since a change of phase may
be introduced merely by shifting the origin of time and is of trivial interest.
If the bias b were less than —P, the complete wave would fall in the con
ducting region and there would be no rectification. If b were greater than
,«Pcos pt
Fig. 3. — Response of biased rectifier to singlefrequencj' wave.
P, the output would be completely suppressed. Applying the theory of
Fourier series to (1.1) and (1.2), we have the results
Oo
2 r
a„ = 
If Jo
2 n=l
arc cos h/P
\ Zli (^n COS n pt
f(P COS X — b) cos nx dx
(1.3)
(1.4)
When two frequencies are applied, the output may be represented by a
double Fourier series. The typical coefficient may be found by the method
explained in an earlier paper by the author^. The problem is to obtain the
double Fourier series expansion in x and y of the function g{x,y) defined by:
/O, P cos x \ Q cos y < b \
Six, V) = (1.5)
\f{P cos T + () COS y — b), b < P cos .v + Q cos v/
' W. R. Bennett, New Results in the Calculation of Modulation Products, B. 5. T./.,
Vol. XII, pp. 228243, April, 1933.
144
BELL SYSTEM TECHNICAL JOURNAL
We substitute the special values x = pt,y = qt after obtaining the expansion.
Let
^1 = Q/P, h = b/P (1.6)
The most general conditions of interest are comprised in the ranges:
0<y^i<l, 2<^o<2'* (1.7)
To P
J
/;
\
\ CASE 1
1
n
TT "2
/ ^
1 °
2
\ CASE n
TT
\
V
X— >.
/case hi
/
\:
/
Fig. 4. — Regions in x3'plane bounded by ^o + cos x ■\ k\ cos )» = 0.
The regions in the x^plane in which g{x,y) does not vanish are bounded
by the various branches of the curve :
^0 + cos :v + ^1 COS T = (1 .8)
We need to consider only one period rectangle bounded by x = ±x, y = zLir,
since the function repeats itself at intervals of lir in both x and y. The
shape of the curve (1.8) within this rectangle may have three forms, which
are depicted in Fig. 4. In Case I, ^o \ ki > k, ko — ki < 1, the curve
divides into four branches which are open at both ends of the x and yaxes.
In Case (2), )^o + ^i <1, ^o — ^i > —1, the curve has two branches open
THE BIASED IDEAL RECTIFIER 145
at the ends of the yaxis. In Case (3), —1 < ^o + ^i < 1, ^o — ^i < —1,
a single closed curve is obtained. The limits of integration must be chosen
to fit the proper case. The Fourier series expansion of g{x,y) may be
written :
00 00
g(^) y) = zL ^ O'mn COS mx cos ny (1.9)
where amn is found from integrals of the form:
A = ^^ / dy I j{P cos X \ Q cos y — b) cos mx cos ny dx (1.10)
Here, as usual, «„ is Neumann's discontinuous factor equal to two when m
is not zero and unity when m is zero. The values of the limits for the dif
ferent cases are :
Case I, flmn = Ai\ A2
({xi = 0, X2 = arc cos (—^0 — ki cos y)
1/^0 I (^^^^
yi = arc cos — , y2 — tt
(ari = 0, :i:2 = X
1 _ ^j, I (1.12)
yi = 0, ^2 == arc cos — —
Xo = arc cos ( — ^0 ~ ^1 cos y
y2 = TT
X2 = arc cos (—^0 — ^1 cos y)
(1.13)
y2 = arc cos
{'^)
(1.14)
For a considerable variety of rectifier functions/, the inner integration may
be performed at once leaving the final calculation in terms of a single definite
integral.
A somewhat different point of view is furnished by evaluating the integral
(1.4) for the biased singlefrequency harmonic amplitude, and then replacing
the bias by a constant plus a sine wave having the second frequency. When
each harmonic of the first frequency is in turn expanded in a Fourier series
146
BELL 5VSTEM TECHNICAL JOURNAL
in the second frequency, the twofrequency modulation coefficients are ob
tained. Some early calculations carried out graphically in this way are
the source of the curves plotted in Figs. 18 to 21 inclusive, for which I am
indebted to Dr. E. Peterson.
If reactive elements are used in the rectifier circuit, the voltage across the
rectifying element may depart from the input wave shape applied to the
complete network. The solution then loses its explicit nature since the
rectifier current is expressed in terms of input voltage components which in
turn depend on voltage drops produced in the remainder of the network
by the rectifier currents. Practical solutions can be worked out when
relatively few components are important.
n
In+ Ii
BIASED RECTIFItR
UNIT
EInR
effective: bias on
Fig. 5. — Biased rectifier in series with RC network.
As an example consider the familiar case of a parallel combination of
resistance R and capacitance C in series with the biased rectifier, Fig. 5.
If C has negligible impedance at all frequencies of importance in the rectifier
circuit except zero, we may assume that the voltage across R is constant and
equal to loR, where /o is the dc. component of the rectifier current. The
voltage across the rectifier unit is then E — loR The effect is a change
in the value of bias from b io b \ IqR. If the dc component in the output
is calculated for bias b + IqR, we obtain the value of /o in terms of 6 f IqR,
an implicit equation defining Io If this equation can be solved for /n, the
bias b + !oR can then be determined and the remaining modulation products
calculated.
A more imj)ortant case is that of the socalled envelope detector, in which
the imjjcdance of the condenser is very small at all frequencies contained in
the input signal, but is very large at frequencies comparable with the band
width of the s[)cctrum of the input signal. These are the usual conditions
prevailing in the detection oi audio or video signals from modulated rf or
if waves. The sf)lution dei)en(ls on writing the input signal in the form
of a slowly varying positive valued envelope function multiplying a rapidly
THE BIASED IDEAL RECTIFIER 147
oscillating cosine function. That is, if the input signal can be repre
sented as
E= A (0 COS0 (/), (1.15)
where .1 (/) is never negative and has a spectrum confined to the frequency
range in which lirfC is negligibly small compared with 1/7?, while cos 0(/)
has a spectrum confined to the frequency range in which \/R is negligibly
small compared with 2irfC, we divide the components in the detector output
into two groups, viz.:
1. A lowfrequency group /;/ containing all the frequencies comparable
with those in the spectrum of .1 (/). The components of this group flow
through R.
2. A highfrequency group Ihf containing all the frequencies comparable
to and greater than those in the spectrum of cos (f) (/). The components
of this group flow through C and produce no voltage across R.
The instantaneous voltage drop across R is therefore equal to Ii/R, and
hence the bias on the rectifier is 6 + Ii/R. If .1 and </> were constants, we
could make use of (1.3) and (1.4) to write:
.arc cos [(b+Ii/R)/A]
I If + hf = :^" + 2 <^n cos nd (1.16)
rt pare COS 1(011 If a )i A i
On =  I f{A cos X — b — IifR) cos nx dx (1.17)
TV Jo
If .4 and (f) are variable, the equation still holds provided Ii/R < .1 at all
times. Assuming the latter to be true (keeping in mind the necessity of
checking the assumption when /;/ is found), we note that terms of the form
fln cos n d consist of high frequencies modulated by low frequencies and hence
; the main portion of their spectra must be in the highfrequency range.
I Hence we must have as a good approximation when the envelope frequencies
ii are well separated from the intermediate frequencies,
\ ■> /«arc cos [(6+/;/K)/4]
1 hf = ^ =  \ f{A cosx  b  IifR)dx (1.18)
1 I TT Jq
jl This equation defines /;/ as a function of A, and if it is found that the
! condition b \ IifR<\ is satisfied by the resulting value of Ii/, the problem
j is solved. If the condition is not satisfied, a more complicated situation
, exists requiring separate consideration of the regions in which b + Ii/R < A
' and 6 f IifR > A .
I To be specific, consider the case of a linear rectifier wnth forward con
ductance a = l/R, and write V — Ij/R. Then
'^'V ^ Va  {b a Vy  (b A V) arc cos ^^tZ (1.19)
XV A
148 BELL SYSTEM TECHNICAL JOURNAL
When 6 = (the case of no added bias), this equation may be satisfied by
setting
V = cA,() <c < 1, (1.20)
which leads to
R yd'
1 — arc cos c, ^ (121)
defining c as a function of Ro/R The value of c approaches unity when
the ratio of rectifier resistance to load resistance approaches zero and falls
off to zero as Ro/R becomes large. The curve may be found plotted else
where . This result justifies the designation of this circuit as an envelope
detector since with the proper choice of circuit parameters the output
voltage is proportional to the envelope of the input signal.
The equations have been given here in terms of the actual voltage applied
to the circuit. The results may also be used when the signal generator
contains an internal impedance. For example, a nonreactive source inde
pendent of frequency may be combined with the rectifying element to give a
new resultant characteristic. If the source impedance is a constant pure
resistance tq throughout the frequency range of the signal input but is
negligibly small at the frequencies of other components of appreciable size
flowing in the detector, we assume the voltage drop in ro is roCi cos (/).
We then set n — 1 in (1.17) and replace ai by {Aq — A)/rQ, where Aq is
the voltage of the source. The value of lu in terms of A from (1.18) is
then substituted, giving an implicit relation between A and Ao .
A further noteworthy fact that may be deduced is the relationship be
tween the envelope and the linearly rectified output. By straightforward
Fourier series expansion, the positive lobes of the wave (1.15), may be
written as:
(E, £>0\ p
£r  =4(/)  + ' cos 4>{t)
\ 0, E <0 /
TT 2
2 Y^ ( — )"* cos 2m 0(/)
(1.22)
7rm=i 4w2 — 1
Hence if we represent the lowfrequency components of Er by Ei/, we have:
£,/ = ^ (1.23)
IT
or
A (/) = wE,f (1.24)
* See, for example, the top curve of Fig. 925, p. 311, H. J. Reich, Theory and AppHca
tions of Electron Tubes, McGrawHill, 1944.
THE BIASED IDEAL RECTIFIER 149
Equation (1.23) expresses the fact that we may calculate the signal com
ponent in the output of a halfwave linear rectifier by taking I/tt times the
envelope. Equation (1.24) shows that we may calculate the response of
an envelope detector by taking t times the lowfrequency part of the
Fourier series expansion of the linearly rectified input. Thus two procedures
are in general available for either the envelope detector or linear rectifier
solution, and in specific cases a saving of labor is possible by a proper choice
between the two methods. The final result is of course the same, although
there may be some difficulty in recognizing the equivalence. For example,
the solution for linear rectification of a twofrequency wave P cos pt ^ Q
cos qt was given by the author in 1933', while the solution for the envelope
was given by Butterworth in 1929^ Comparing the two expressions for
the directcurrent component, we have:
 2P o
Elf = y[2E — (1 — k") K], where K and £are complete elliptic integrals
of the first and second kinds with modulus k = Q/P
— 2P
A {t) = — (1 + k) El, where Ei is a complete elliptic intregal of the
TT
second kind with modulus ki = 2 \/k/{\ + k). Equation (1.24) implies
the existence of the identity
(1 +k)Ei^ 2E {\  k') K (1.25)
The identity can be demonstrated by making use of Landen's transforma
tion in the theory of elliptic integrals.
2. SingleFrequency Signal
The expression for the harmonic amplitudes in the output of the rectifier
can be expressed in a particularly compact form when the conducting part
of the characteristic can be described by a power law with arbitrary ex
ponent. Thus in (1.4) if /(c) = az\ we set X = b/P and get
•arc cos X
2 73" /<arc cos a
aP i , ^ y ,
I fln = / (cos X — A) COS nx ax
TT Jo
!
2^T{p + DaPW  X)"^^
I
* S. Butterworth, Apparent Demodulation of a Weak Station by a Stronger One
Experimental Wireless, Vol. 6, pp. 619621, Nov. 1929.
150 BELL SYSTEM TECHNICAL JOURNAL
The equation holds for all real values of v greater than —1. The symbol
F represents the Gaussian hypergeometric function*:
f (a, 6; .; .) = , + "*. + °(" + D ^^^ + D ,.+ ... (2.2)
c 1! c(c + 1) 2!
The derivation of (2.1) requires a rather long succession of substitutions,
expansions, and rearrangements, which will be omitted here.
When V is an integer, the hypergeometric function may be expressed in
finite algebraic form, either by performing the integration directly, or by
making use of the formulas:
F{yi/2, — n/2; 1/2; z) — cos (^i arc sin z),
(2.3)
sin (fi arc sin z)
.(i±M^i.0
HZ
together with recurrence formulas for the f'f unction. When p is an odd
multiple of one half, the /'function may be expressed in terms of complete
elliptic integrals of the first and second kind with modulus [(1 — X)/2] " by
means of the relations,
F(hh;i;k') =K,
IT
F{h^;^;k') =E,
(2.4)
and the recurrence formulas for the /''function. For the case of zero bias,
we set X = 0, and applv the formula
F{a. Xac; 1/2) = ^J^T]^^M+Zzj\ ^
obtaining the result:
We point out that the above results may be applied not only when the
api)lied signal is of the form P cos pt with P and p constants, but to signals
" For an account of the ])roi)crties of the hypergeometric function, see Ch. XIV of
Whittaker and Watson, Modern Analysis, Cambridge, 1940. A discussion of elliptic
integrals is given in ("h. XXII of the same hook.
THE BIASED IDEAL RECTIFIER 151
in which F and p are variable, provided that P is always positive. We thus
can apply the results to detection of an ordinary amplitudemodulated wave
or to the detection of a frequencymodulated wave after it has passed through
a slope circuit.
A case of considerable practical interest is that of an amplitudemodulated
wave detected by a diode in series with a parallel combination of resistance
R and capacitance C. The value of C is assumed to be sufficiently large so
that the voltage across R is equal to the ao/2 component of the current
through the diode multiplied by the resistance. This is the condition for
envelope detection mentioned in Part 1. The diode is assumed to follow
Child's law, which gives v = 3/2. We write
V _ r(5/2)(l X^aP^'' (, ...lA .2.
where X = V/P. Note that K is a constant equal to the directvoltage
output if P is constant. If P varies slowly with time compared with the
highfrequency term cos pt, V represents the slowly varying component of
the output and hence is the recovered signal.
But
Hh, h 3; k') = i^ [2(2^=^  1)£ + (2  3k'){l  k')K] (2.8)
where A' and E are complete elliptic integrals of the first and second kind
with modulus k. Hence
37r (1 t 3X)(1 f X)
^:vp = ^ = — I — ^^^ <"'
where the modulus of A' and E is \/(l — X)/2. This equation defines p
as a function of X, and hence by inversion gives X as a function of p. The
resulting curve of X vs. p is plotted in Fig. 6 and may be designated as the
function X = g (p). If we substitute X = V/P we then have
V = P g{3Tr/Ra V2P) (2.10)
This enables us to plot V as a function of P, for various values of Ra, Fig. 7.
Since P may represent the envelope of an amplitudemodulated (or diflf
erentiated FM) wave, and V the corresponding recovered signal output
voltage, the curves of Fig. 7 give the complete performance of the circuit
as an envelope detector. In general the envelope would be of form P —
Po[l f c s{l)\, where s{t) is the signal. We may substitute this value of P
directly in (2.10) provided the absolute value of c s{t) never exceeds unity.
152
BELL SYSTEM TECHNICAL JOURNAL
Fig. 6. — The Function X = g{f>) defined by Eq. (2.9).
Fig. 7. — Performance of 3/2 — powerlaw rectifier as an envelope detector with lowimped
ance signal generator.
To express the output in terms of a source voltage f o in series with an
impedance equal to the real constant value ro at t,he signal frequency and
zero at all other frequencies, we write
ra
ai
3C,P3/2(1 _ X)2
:^7— /^(^f, I;3r^j (2.11)
THE BIASED IDEAL RECTIFIER
153
or
Po =
(i+^)p.
(2.12)
where
E =
3i?a(l  X)2/'i
4\/2
= ?^
/ ^ 1 *X \
P 1 1> "ij ^' 2 /
(2.13)
V2P[2(1  ife' + )fe')£  (2  yfe')(l  k')K].
1.4
1.2
15 20 25
Pq in volts
Fig. 8. — Performance of 3/2 — powerlaw rectifier as an envelope detector with impedance
of signal generator low except in signal band.
By combining the curves of Fig. 7 giving V in terms of P with the above
equations giving the relation between P and Pq, we obtain the curves of
Figs. 8, 9, 10, giving F as a function of Pq. The curves approach linearity
as Ra is made large. On the assumption that the curves are actually linear,
we define the conversion loss D of the detector in db in terms of the ratio
of maximum power available from the source to the power delivered to the
load:
D = 10 log!
Po/8ro
vyR
= 10 logi
m
R^
Sro
(2.14)
Curves of D vs r^/R are given in Figs. 11 and 12. The optimum relation
between r^ and R when the forward resistance of the rectifier vanishes has
long been known to be r^/R. — .5. The curves show a minimum in this
154
BELL SYSTEM TECHNICAL JOURNAL
region when Ra is large. In the limit as Ra approaches infinity, we may
show that the relation between f o and V approaches:
(2.15)
15 20 25
P„ IN VOLTS
Fig. 9. — Performance of 3/2 — powerlaw rectifier as an envelope detector with impedance
of signal generator low except in signal band.
Fig. 10. — Performance of 3/2 — powerlaw rectifier as an envelope detector with impedance
of signal generator low except in signal l)and.
The corresponding limiting formula for D is
(2.16)
THE BIASED IDEAL RECTIFIER
155
The minimum value of D is then found to occur at tq = R/2 and is zero
db. We note from the curves that the minimum loss is 1.2 db when Ra =
10 and 0.4 when Ra = 100.
This example is intended mainly as illustrative rather than as a complete
tabulation of possible detector solutions. The methods employed are
sufficiently general to solve a wide variety of problems, and the specific
evaluation 'process included should be sufficiently indicative of the proce
dures required. Cases in which various other selective networks are asso
ciated with the detector have been treated by Wheeler^.
Fig. 11. — Conversion loss of 3/2 — powerlaw rectifier as envelope detector with impedance
of signal generator low except in signal band.
m 14
o
Z 12
to 10
<n
3 8
Ra =
10 1
X — xRa=100 1
\. 27= VOLTS OUTPUT
— =
\ ! 1 ^
ssss*
V^=i2_5_^^^
y —
^^S^^iSr^^^HO . 3
Fig. 12. — Conversion loss of 3/2 — powerlaw rectifier as envelope detector with impedance
of signal generator low except in signal band.
3. TwoFrequency Inputs
The general formula for the coefficients in the twofrequency case depends
on a double integral as indicated by (1.10). In many cases one integration
may be performed immediately, thereby reducing the problem to a single
definite integral which may readily be evaluated by numerical or mechanical
' H. A. \Mieeler, Design Formulas for Diode Detectors, Proc. I. R. E., Vol. 26, pp.
745780, June 1938.
156
BELL SYSTEM TECHNICAL JOURNAL
means. It appears likely in most cases that the expression of these results
in terms of a single integral is the most advantageous form for practical
purposes, since the integrands are relatively simple, while evaluations in
terms of tabulated functions, where possible, often lead to complicated
terms. Numerical evaluation of the double integral is also a possible method
in cases where neither integration can be performed in terms of functions
suitable for calculation.
One integration can always be accomplished for the integer powerlaw
case, since the function / (P cos x \ Q cos y — J) in (1.12) then becomes a
polynomial in cos x and cos y. Cases of most practical interest are the
zeropower, linear, and squarelaw detectors, in which /(z) is proportional
to z", z , and z" respectively. The zeropowerlaw rectifier is also called a
total limiter, since it limits on infinitesimally small amplitudes. We shall
tabulate here the definite integrals for a few of the more important loworder
OS pt+Qcos qt
RESPONSE OF LIMITER
_A_
I
m\mm
■kw////////A
m
"^ TIME ». ^
f
Fig. 13. — Response of biased total limiter to twofrequency wave.
coefficients. To make the listing uniform with that of our earlier work, we
express results in terms of the coefhcient Amn, which is the amplitude of the
component of frequency mp ± nq. The coefl&cient Amn is half of «„„ when
neither m nor n is zero. When w or » is zero,, we take Amn = a^n and drop
the component with the lower value of the i sign. When both m and n
are zero, we use the designation Aqq/I for ooo, the dc term. In the tabula
tions which follow we have set/(z) = otz' with v taking the values of zero
and unity.
We first consider the biased zeropowerlaw rectifier or biased total
limiter. This is the case in which the current switches from zero to a
constant value under control of two frequencies and a bias as illustrated
by Fig. 13. The results are applicable to saturating devices when the
driving forces swing through a large range compared with the width of the
linear region. It is also to be noted that the response of a zeropowerlaw
rectifier may be regarded as the Fourier series expansion of the conductance
THE BIASED IDEAL RECTIFIER
157
of a linear rectifier under control of two carrier frequencies and a bias.
The results may therefore be applied to general modulator problems based
on the method described by Peterson and Hussey**. We may also combine
the Fourier series with proper multiplying functions to analyze switching
between any arbitrary forms of characteristics. We give the results for
positive values of ^o The corresponding coefficients for —ko can be ob
tained from the relations:
(3.1)
^00 ^00
Here we have used plus and minus signs as superscripts to designate co
efficients with bias +^o and — ^o respectively. We thus obtain a reduction
in the number of different cases to consider, since Case III consists of nega
tive bias values only, and these can now be e'xpressed in terms of positive
bias values falling in Cases I and II. It is convenient to define an angle 6
by the relations:
^ T^ ^^^^ k,> \,h h<\ . (Case I) \
,h + h<\,h k,> \ (Case 11)/
arc cos 
ZeroPower Rectifier or TotalLimiter Coefficients
Setting y(2) = a in (1.10),
—^ = 1 — — / arc cos (^o + ki cos y) dy.
2a r Je
— = 4 f Vl  {ko + kr cos yy dy
An ^ 2h r sin^ y dy
a TT^ ie \/l — (^0 + ^1 cos yY
— = — / cos Vl — (^0 + ^1 cos y) dy \ {2>3)
a TT^ Je
— = — ^ / (^0 + ki COS y) Vl  (^0 + ki cos yy dy
a TT^ J e
Aw. _ 2^1 r sin^ y cos y dy
a TT^ h Vl — (^0 + ^1 cos yy
—  — — / (^0 + ^1 COS y) COS y Vl — (^o + ki cos y)' dy
a TT^ Jft J
' E. Peterson and L. W. Hussey, Equivalent Modulator Circuits, B. S. T. J., Vol. 18,
pp. 3248, Jan. 1939.
158
BELL SYSTEM TECHNICAL JOURNAL
Similarly for a linear rectifier:
1 +
2 2
Au — aP — AiQ
^01 = aQ — Aq\
Amn ^^ \ ) A
mn J
W + « > 1
(.3.4)
We have shown in Fig. 2 how an ideal limiting characteristic, which trans
mits linearly between the upper and lower limits, may be synthesized from
two biased linear rectification characteristics. Equation (3.4) shows how
to calculate the corresponding modulation coefficients, when the coefficients
for bias of one sign are known. The limiter characteristic is equal to az—
h (2)  h (2), where
/i (2) = oc
z  bi,
0,
z > —bi
z < —bi
z > bi
1 /2 (2) = a I
0, z < bxj \z + 62
The expression for/2 (zj may also be written:
'z — ( — 62), 2 > —bi
0, Z < ^2
ji (z) = a (z + 62) — a
)
(3.5)
(3.6)
Hence the modulation coefficient A^n for the limiter may be expressed in
terms of y4„,„ (61) and A^n ( — 62) as follows:
(61) + {T^^'Amn (62), m ^ n 7^ \ (3.7)
A mn — A 1
If the limiter is symmetrical {b\ = 62), the even order products vanish and
the odd orders are doubled. The terms aP, aQ are to be added to the
dexter of (3.7) for .4 10, ^01 respectively. The odd Hnearrectifier coefficients,
when multiplied by two, thus give the modulation products in the output
of a symmetrical limiter with maximum amplitude ^0, as may be seen by
substituting fti = 62 = —^0 in (3.7). For the fundamental components
aP and aQ respectively must be subtracted from twice the Aio and Aoi co
efficients for ^n
Linear Rectifier Coefficients
D.C.
^00
2
/aP = ko\ \ f [Vl  (*o + ki cos 3)^
(3.8)
— (^0 + ki cos y) arc (cos ^0 + ^1 cos y)] dy
THE BIASED IDEAL RECTIFIER
FXJNDAMENTALS
159
(3.9)
(3.10)
(3.11)
(3.12)
AWaP = 1 + ^ f f(^o + ^1 COS y) Vl  {h + ^1 cos yY
— arc cos (^o + ^i cos y)] dy
Aoi/aP = ki^f [Vl  (ko + ki COS yy
•K^ J e
— {ko + ^1 COS y) arc cos (^o + ^i cos y)] cos y dy
Sum and Difference Products — Second Order
^11 = ^ / [(^0 + ki cos y) Vl  (^0 + ki cos yy
— arc cos (^o "1" ki cos y)\ cos y dy
Sum and Difference Products — Third Order
A21 = ^ I [1 — (^0 + ki cos yYf~ cos y dy
6t~ Je
The above products are the ones usually of most interest. Others can
readily be obtained either by direct integration or by use of recurrence
formulas. The following set of recurrence formulas were originally derived
by Mr. S. O. Rice for the biased linear rectifier:
2n Amn + ^1 (« — m — 3) Am+l,nl
{ ki (m \ n { 3)Am+i,n~i + 2kon .4„,+i,„ =
2» Amn + kl (n j m — 3) Aml.n+l
+ ^1 (w — w + 3) A „,!,„+! + 2kon Am\.n =
2m ki Amn \ {m — n — 3) Aml,n^l
+ (m f n + 3)A„.+i,„+i + 2^ow ^m,„+i =
2 m h Amn + {m ]r n — 3) Ami.n\
\ {m — n \ 3)Am+l,nl + 2^oW A^.nl =
By means of these relations, all products can be expressed in terms of .4 00,
^10, Aoi, and An. The following specific results are tabulated:
.^20 = 3(^00 ~ 2kiAn ~ 2^0^10)
_ 1 \ (3.14)
A02 — TT (^1^1 no ~ 2^4 11 — 2^0 4 01) '
3ki )
(3.13)
160 BELL SYSTEM TECHNICAL JOURNAL
1 [ (3.15)
An = jr {kiAio — yloi — ^0^11)
^30 = —^0^20 — ^1^21 1
1 (3.16)
^03 = — r (^0^02 + ^112)
ki J
The thirdorder product A21 is of considerable importance in the design
of carrier ampHfiers and radio transmitters, since the (2/> — 9)product is
the crossproduct of lowest order falling back in the fundamental band when
overload occurs. Figure 14 shows curves of .I21 calculated by Mr. J. O.
Edson from Eq. (3.12) by mechanical integration.
We point out also that the Unearrectifier coefficients give the Fourier
series expansion of the admittance of a biased squarelaw rectifier when two
frequencies are applied.
We shall next discuss the problem of reduction of the integrals appearing
above to a closed form in terms of tabulated elliptic integrals^. This can
be done for all the coefficients above except the dc for the zeropower law
and for the dc and two fundamentals for the linear rectifier. These contain
the integral
H(i^o , ^1) = / arc cos (^0 + ^1 cos y) dy (3.17)
which has been calculated separately and plotted in Fig. 22. When the
arc cos term is accompanied by cos wy as a multiplier with m ?^ 0, an integra
tion by parts is sufficient to reduce the integrand to a rational function of
cos y and the radical \^\ — {ko + ki cos yY, which may be reduced at once
to a recognizable elliptic integral by the substitution z = cos y. It is
found that all the integrals except that of (3.17) appearing in the results
can be expressed as the sum of a finite number of integrals of the form:
• cos 9 gm ^2
By differentiating the expression z""" V'(l — z)^[l — {ko f kiz"] with
respect to 2, we may derive the recurrence formula:
^rn = —7 7Tr2 K2W — 3)^0^12™!
[m — l)ki
+ (w  2){kl  k\ 1)Z^_2 (3.19)
 (2W  S)hkiZra3 + (W  3)(1  kl)Z^i\
' Power series expansions of coefficients such as treated here have been given by A. G.
Tynan, Modulation Products in a Power Law Modulator, Proc. L K. E., Vol. 21, pp.
12031209, Aug. 1933.
/cc
1
I
THE BIASED IDEAL RECTIFIER
161
o o
V A ■=
UJ u / '
O 9
II
o
T
r— A^A
3
~ II
• aQ
^
\. 1 —
^^ >*
^
/
^ ^/^
<^
PCOS
QCOS
/>
^
^
^
^^/
/
i
y ^
y
1
Zi
v
1
/"
(
\
%
\
\
^
o<x
^^^s^
^
\^
\
\
^*!a
^
^
onla ~
H
d/'^V
1+ 1+ 1+ 1+
ionaoad( b+cl2) jo sanindwv
162
BELL SYSTEM TECHNICAL JOURNAL
It thus is found that the value of Zm for all values of m greater than 2 can be
expressed in terms of Zq, Z\, and Z2.
Eq. (3.18) may be written in the form:
z'^dz
Z3 =
Zi
The substitution
V(Z  2i) (S  Z2)(Z3  z)(Zi  2)
Zl = — (1 + ^o)Al , Zo = — 1
/ (1  /to)Ai , Case I) \
\ 1, Case II /
(1, Case I \
(1  /feo)//fei, Casell, /
Z2CZ3 — Zl) — Zi(Z3 — Z2)U^
Z =
reduces the integral to
^m —
Z3 — 2i — (Z3 — Z2)m2
— Zl) *'o /
du
h V(24  Z2)(23  zO h ^7 (73 ,2)(1 _ ^^2 )
where:
tl =
Z3 — Z2
23 — Zl
2 (Z4 — Zi)(23 — Z2)
X =
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
(3.25)
(24  Z2)(23  2i)
Hence if A', E and 11 represent respectively complete elliptic integrals of
the first, second, and third kinds with modulus «, and in the case of third
kind with parameter —t], we have immediately:
2K
Zo =
Zl =
z,=
kl\/(Zi — Z2)(23 — 2i)
2 [2i K \ {Z2 2i) n]
ki\/{Zi  22) (Z3  2i^
^lV(24l)(232i) [^^ ^' + '^'(^^  ^^^"
(3.26)
(3.27)
(3.28)
THE BIASED IDEAL RECTIFIER
163
To complete the evaluation of Z2, assume a relation of the following type
with undetermined constants Ci, C2, C3, C4:
I (1 
dii
h (1  T/w')' V(l  «') (1  k' w2)
c
'io V(l  m2)(1 k'u^
du
u^ ^«^ + C3 y^ (J _ ^^2) ^^j _ ^^^ ^^ _ ^3^,^
+ C4
z y/Cl  z^) (1  K^ 2^)
1  1722
(3.29)
2 1.2
UJ
<
o
Z 1.0
5 0.2
<
0.2
O
H0.4
<
N. 1
— ^\s
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.5 IS 2.0
RATIO OF BIAS TO LARGER FUND^MENTAL
Fig. 15. — Fundamentals and (Ip ± q) — product from fullwave biased zeropowerlaw
rectifier with ratio of applied fundamental amplitudes equal to 0.5. Fi = larger funda
mental, F2 = smaller fundamental, F3 = (2/> ± q) — product.
Differentiate both sides with respect to z, set z = 1, and clear fractions.
Equating coefificients of like powers of z separately then gives four simul
taneous equations in G, C2, C3, C4. Solving for C], C2, C3 and setting z = 1
in (3.29) gives
r^ du 1 r
i (1  vuf \/(l  W) (1 ^?^) " 2(r,  1) [^ "^
_j_ (2,?  3) k"  7,(77  2) jjl
77^
.2
(3.30)
164
BELL SYSTEM TECHNICAL JOURNAL
u
Q _)
i\
O
o q:
a. uj
Q. o
u
a.
0.8
0.6
0.4
0.2
0.2
^2
Q2 0.4 0.6 08 1.0 1.2 1.4 1.6 1.8 2
^0
RATIO OF BIAS TO LARC^TD FiJMD.tMENTAL
Fig. 16. — Fundamentals and (2/> ± q) — product from fullwave biased zeropowerlaw
rectifier with equal applied fundamental amplitudes.
<
t<0
1
1
i
"^
^
^0
«_^
A
V
3^
^
^
^.^i^___^
0.2
0.4
0.8
Fig. 17. — The integral Zm with ^i = 0.5.
Since the necessary tables of FI are not available, we make use of Legendre's
Transformation, which in this case gives:
'" Legendre, Traites dcs Fonctions EUiptiques, Paris, 182528, Vol. I, Ch. XXIII.
THE BIASED IDEAL RECTIFIER
165
2.0
Aqo
^
•^Or^.O
^
^
—
0,
■^
^
^
1
^
^
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Fig. 18. — Dc. term in linear rectifier output with two applied frequencies.
1.0
0.8
0.6
Aqi
0.4
02
K,= 1
^
0.8
0.6
0.4
■
0.2
0.2 0.4 0.6 0.8 1.0 12 1.4 1.6 1.8 2.0
«0
Fig. 19. — Smaller fundamental in biased linear rectifier output.
n = ir +
tan <^
V 1 — K sin^ ^
1/2
= arc sin
Jo
£(0) = f vn^
K
dd
Vl  K^ sin2 e
2 sin2 e dd
(3.31)
(3.32)
(3.33)
(3.34)
The functions F(0) and E(0) are incomplete elliptic integrals of the first
and second kinds. They are tabulated in a number of places. Fairly good
tables, e.g. the original ones of Legendre, are needed here since the difference
between KE(«^) and EF(0) is relatively small.
166
BELL SYSTEM TECHNICAL JOURNAL
1
1
M
M
■^
/ / /
M
//.
^
/ \
///
///
//
i\
/,
/V
/ /
7
1
o
/Ol /CO /
f d/ 6/
6/ o
•O / yt \ (0/
o/ d/ d/
o
1
d
1
/ 1
\
/ f
THE BIASED IDEAL RECTIFIER
167
V /
^
/
/ /oy
i^
/ / /
^
^ V. \.
l.
^
V X >
^^^
^
^
\^
^^^
s_
\ ^
\
O C
d C
i l
>
^ c
> c
i c
1 r
) ■■*.
) C
■>!■
3
3 d
•^ •?;
PlH
168
BELL SYSTEM TECHNICAL JOURNAL
_l 4
<
O 2
UJ
3
_l
^ 1
^° = °'o^2
0.5
0.8
,
_JJ—
_
^
■'
,
,.
K4_^
Ji—
■^
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
•^1
Summarizing:
Fig. 22. — Graph of the integral E (^o ^i).
Case I, ^0 + ^1 > 1, ^0 — ^1 < 1
K
Zo =
V^i
Zi =  [KE{4>)  EFm  Z,,
Z2
^ib
— ko
'^^A
K =
v^
n
<P 
 arc sin
fv
2*1
+ *o + *1
Case II, ^0 + ^1 < 1, ^0 — ^1 > — 1
Zo =
2K
V(i + hY  kl
Zx = I [ir£(<^)  EFm  Zo
^2 = ^2 (1 + ^I  i^o)Zo  2^o)^iZ,  2£ V(l + ^i)'  ^5
a/ 4^1
= arc sin
/'
— ^0 + ^1
(3.35)
(3.36)
THE BIASED IDEAL RECTIFIER
169
The values of the fundamentals and thirdorder sum and difference
products for the biased zeropowerlaw rectifier have been calculated by the
formulas above for the cases ki — .5 and ^i = 1. The resulting curves are
shown in Fig. (15) and (16). The values of the auxiliary integrals Zo, ^i ,
and Zo are shown for ^i = .5 in Fig. (17). These integrals become infinite
at kit = I — ki so that the formulas for the modulation coefficients become
indeterminate at this point. The limiting \alues can be evaluated from
the integrals {^.3), etc., directly in terms of elementary functions when the
relation ^o = 1 — ^i is substituted, except for the Hfunction.
Limiting forms of the coefficients when k„ is small are of value in calcu
lating the effect of a small signal superim[)osed on the two sinusoidal com
ponents in an unbiased rectifier. By straightforward powerseries expan
sion in ^oi we find :
ZeroPower Law Rectifier, ko Small:
Aro = „£ 
2£
7r2(l  k'')
kl +
Aoi = ~ [£  (1  kl)K\ + ^^^
r^^)^» +
[ (3.37:
A21 =  ,r, [(1
+
TT'^l
K 
2k\)E
1  2k\
1  k\
(1  k\)K\
'^kl
+
In the above expressions, the modulus of K and E is ki. When k^ = 0,
these coefficients reduce to half the values of the fullwave unbiased zero
powerlaw coefficients, which have been tabulated in a previous publication.
Acknowledgment
In addition to the j)ersons already mentioned, the writer wishes to thank
Miss M. C. Packer, Miss J. Lever and Mrs. A. J. Shanklin for their assistance
in the calculations of this paper.
" R. M. Kalb and W. R. Bennett, Ferromagnetic Distortion of a TwoFrequency
Wave, B. S. T. J., Vol. XIV, .\pril 1935, Eq. (21), p. 336.
Properties and Uses of Thermistors — Thermally
Sensitive Resistors '
By J. A. BECKER, C. B. GREEN and G. L. PEARSON
A new circuit element and control device, the thermistor or thermally sensitive
resistor, is made of solid semiconducting materials whose resistance decreases
about four per cent per centigrade degree. The thermistor presents interesting
opportunities to the designer and engineer in many fields of technology for ac
complishing tasks more simply, economically and better than with available
devices. Part I discusses the conduction mechanism in semiconductors and the
criteria for usefulness of circuit elements made from them. The fundamental
physical properties of thermistors, their construction, their static and dynamic
characteristics and general principles of operation are treated.
Part II of this paper deals with the applications of thermistors. These include :
sensitive thermometers and temperature control elements, simple temperature
compensators, ultrahigh frequency' power meters, automatic gain controls for
transmission systems such as the Types K2 and LI carrier telephone systems,
voltage regulators, speech volume limiters, compressors and expandors, gas pres
sure gauges and flowmeters, meters for thermal conductivity determination of
liquids, and contactless time delay devices. Thermistors with short time con
stants have been used as sensitive bolometers and show promise as simple com
pact audiofrequency oscillators, modulators and amplifiers.
PART I— PROPERTIES OF THERMISTORS
Introduction
THERMISTORS, or thermsMy sensitive resistors, are devices made of
solids whose electrical resistance varies rapidly with temperature.
Even though they are only about 15 years old they have already found im
portant and large scale uses in the telephone plant and in military equip
ments. Some of these uses are as time delay devices, protective devices,
voltage regulators, regulators in carrier systems, speech volume limiters,
test equipment for ultrahighfrequency power, and detecting elements for
very small radiant power. In all these applications thermistors were
chosen because they are simple, small, rugged, liave a long life, and require
little maintenance. Because of these and other desirable properties, ther
mistors promise to become new circuit elements which will be used exten
sively in the fields of communications, radio, electrical and thermal
instrumentation, research in physics, chemistry and biology, and war tech
nology. Specific types of uses which will be discussed in the second part
of this paper include: 1) simple, sensitive and fast responding thermometers,
* Published in Elec. Engg., November 1946.
The authors acknowledge their indebtedness to Messrs. J. H. Scaff and H. C. Theuercr
for furnishing samples for most of the curves in Fig. 4, and to Mr. G. K. Teal for the data
for the lowest curve in that figure.
170
PROPERTIES AND USES OF THERMISTORS 171
temperature compensators and temperature control devices; 2) special
switching devices witiiout moving contacts; 3) regulators or volume limiters;
4) pressure gauges, flowmeters, and simple meters for measuring thermal
conductivity in liquids and gases; 5) time delay and surge suppressors; 6)
special oscillators, modulators and amplifiers for relatively low frequencies.
Before these uses are discussed in detail it is desirable to present the physical
principles which determine the properties of thermistors.
The question naturally arises "why have devices of this kind come into
use only recently?" The answer is that thermistors are made of semi
conductors and that the resistance of these can vary by factors up to a
thousand or a million with surprisingly small amounts of certain impurities,
with heat treatment, methods of making contact and with the treatment
during life or use. Consequently the potential application of semiconduc
tors was discouraged by experiences such as the following: two or more
units made by what appeared to be the same process would show large
variations in their properties. Even the same unit might change its re
sistance by factors of two to ten by exposure to moderate temperatures or
to the passage of current. Before semiconductors could seriously be con
sidered in industrial applications, it was necessary to devote a large amount
of research and development efifort to a study of the nature of the conduc
tivity in semiconductors, and of the effect of impurities and heat treatment
on this conductivity, and to methods of making reliable and permanent
contacts to semiconductors. Even though Faraday discovered that the
resistance of silver sulphide changed rapidly with temperature, and even
i though thousands of other semiconductors have been found to have large
\ negative temperature coefficients of resistance, it has taken about a century
i of effort in physics and chemistry to give the engineering profession this
j new tool which may have an influence similar to that of the vacuum tube
I and may replace vacuum tubes in many instances.
If thermistors are to be generally useful in industry:
! 1) it should be possible to reproduce units having the same character
istics;
I 2) it should be possible to maintain constant characteristics during use;
the contact should be permanent and the unit should be chemically
inert ;
3) the units should be mechanically rugged;
4) the technique should be such that the material can be formed into
various shapes and sizes;
5) it should be possible to cover a wide range of resistance, temperature
coefficient and power dissipation.
Thermistors might be made by any method by which a semiconductor
172
BELL SYSTEM TECHNICAL JOURNAL
could be shaped to definite dimensions and contacts applied. These meth
ods include: 1) melting the semiconductor, cooling and solidifying, cutting
to size and shape; 2) evaporation; 3) heating compressed powders of semi
conductors to a temperature at which they sinter into a strong compact
mass and firing on metal powder contacts. While all three processes have
been used, the third method has been found to be most generally useful
for mass production. This method is similar to that employed in ceramics
or in powder metallurgy. At the sintering temperatures the powders
recrystallize and the dimensions shrink by controlled amounts. The powder
process makes it possible to mix two or more semiconducting oxides in
varjnng proportions and obtain a homogeneous and uniform solid. It is
thus possible to cover a considerable range of specific resistance and tem
Fig. 1. — Thermistors made in the form of a bead, rod, disc, washer and flakes.
perature coefficient of resistance with the same system of oxides. By ''
means of the powder process it is possible to make thermistors of a great
variety of shapes and sizes to cover a large range of resistances and power
handling capacities.
Figure 1 is a photograph of thermistors made in the form of beads, rods,
discs, washers and flakes. Beads are made by stringing two platinum alloy
wires parallel to each other with a spacing of five to ten times the wire diam
eter. A mass of a slurry of mixed oxides is applied to the wires. Surface
tension draws this mass into the form of a bead. From 10 to 20 such beads
are evenly spaced along the wires. The beads are allowed to dry and are
heated slightly until they have sufficient strength so that the string can be
handled. They then are passed through the sintering furnace. The oxides I
shrink onto the i)latinum alloy wires and make an intimate and permanent, I
electrical contact. The wires then are cut to separate the individual beads. i(
PROPERTIES AND USES OF THERMISTORS 173
The diameters of the beads range from 0.015 to 0.15 centimeters with wire
diameters ranging from 0.0025 to 0.015 centimeters.
Rod thermistors are made by mixing the oxides with an organic binder
and solvent, extruding the mixture through a die, drying, cutting to length,
heating to drive out the binder, and sintering at a high temperature. Con
tacts are applied by coating the ends with silver, gold, or platinum paste
as used in the ceramic art, and heating or curing the paste at a suitable
temperature. The diameter of the rods can ordinarily be varied from 0.080
to 0.64 centimeter. The length can vary from 0.15 to 5 centimeters.
Discs and washers are made in a similar way by pressing the bonded
I powders in a die. Possible disc diameters are 0.15 to ,^ or 5 centimeters;
l thicknesses from 0.080 to 0.64 centimeter.
Flakes are made by mixing the oxides with a suitable binder and solvent
to a creamy consistency, spreading a film on a smooth glass surface, allowing
! the film to dry, removing the film, cutting it into flakes of the desired size
and shape, and firing the flakes at the sintering temperatures on smooth
\ ceramic surfaces. Contacts are applied as described above. Possible
dimensions are: thickness, 0.001 to 0.004 centimeter; length, 0.1 to 1.0
! centimeter; width, 0.02 to 0.1 centimeter.
! In any of these forms lead wires can be attached to the contacts by solder
' ing or by firing heavy metal pastes. The dimensional limits given above
, are those which have been found to be readily attainable.
In the design of a thermistor for a specific application, the following
characteristics should be considered: 1) Mechanical dimensions including
^ those of the supports. 2) The material from which it is made and its prop
; erties. These include the specific resistance and how it varies with tem
I perature, the specific heat, density, and expansion coefficient. ^) The
i dissipation constant and power sensitivity. The dissipation constant is
I the watts that are dissipated in the thermistor divided by its temperature
[ rise in centigrade degrees above its surroundings.. The power sensitivity is
I the watts dissipated to reduce the resistance by one per cent. These con
stants are determined by the area and nature of the surface, the surrounding
'medium, and the thermal conductivity of the supports. 4) The heat ca
j.pacity which is determined by specific heat, dimensions, and density. 5)
:The time constant. This determines how rapidly the thermistor will heat
[or cool. If a thermistor is heated above its surroundings and then allowed
to cool, its temperature will decrease rapidly at first and then more slowly
until it finally reaches ambient temperature. The time constant is the time
! required for the temperature to fall 63 per cent of the way toward ambient
i temperature. The time constant in seconds is equal to the heat capacity
tin joules per centigrade degree divided by the dissipation constant in watts
174
BELL SYSTEM TECHNICAL JOURNAL
per centigrade degree. 6) The maximum permissible power that can be
dissipated consistent with good stability and long life, for continuous opera
tion, and for surges. This can be computed from the dissipation constant
and the maximum permissible temperature rise. This and the resistance
temperature relation determine the maximum decrease in resistance.
Properties of Semiconductors
As most thermistors are made of semiconductors it is important to discuss
the properties of the latter. A semiconductor may be defined as a substance
io«
10*
2
I
O .
I 10'
ill
o
z
,<
y,n2
KT'
\
v\
\
\
\
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^
\
1
1
c^
v^
^
Cr
^ ^
■^,
PL
.ATir
guM

100
100 200
TEMPERATURE °C
300
400
Fig. 2. — Logarithm of specific resistance versus temperature for three thermistor ma
terials as compared with platinum.
whose electrical conductivity at or near room temperature is much less than
that of typical metals but much greater than that of typical insulators.
While no sharp boundaries exist between these classes of conductors, one
might say that semiconductors have specific resistances at room tempera
ture from 0.1 to 10* ohm centimeters. Semiconductors usually have high h
negative temperature coefKicients of resistance. As the temperature is
increased from O^C. to 300°C., the resistance may decrease by a factor of a
thousand. Over this same temperature range the resistance of a typical
metal such as platinum will increase by a factor of two. Figure 2 shows
how the logarithm of the specific resistance, p, varies with temperature, T,
in degrees centigrade for three typical semiconductors and for platinum.
PROPERTIES AND USES OF THERMISTORS
175
Curves 1 and 2 are for Materials No. 1 and No. 2 which have been ex'ten
sively used to date. Material No. 1 is composed of manganese and nickel
oxides. Material No. 2 is composed of oxides of manganese, nickel and
cobalt. The dashed part of Curve 2 covers a region in which the resistance
temperature relation is not known as accurately as it is at lower tempera
tures. Curve 3 is an experimental curve for a mixture of iron and zinc
2
U 10
2
5
y 10
/
r
y
/
/
—
/
/
)
/
/
^

y—
— ^
/^ —
f
v^
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/
/
/
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t 7.
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/
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/
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/
/ J
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/
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/ /'
' /
/
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3.0
xiO''
temperature: °k
Fig. 3. — Logarithm of the si)ecific resistance of two thermistor materials as a function
of inverse absolute temperature. See equation (1).
oxides in the proportions to form zinc ferrite. From Fig. 2 it is obvious
that neither the resistance R nor log R varies linearly with T.
Figure 3 shows plots of log p versus l/T, for Materials No. 1 and No. 2.
These do form approximate straight lines. Hence
BlT
Pooe or p = poe
(,bIt){bitq)
(1)
where T = temperature in degrees Kelvin; p„ — p when T = oo or \/T = 0;
P{i = p when T = To ; e = Naperian base = 2.718 and 5 is a constant equal
to 2.303 times the slope of the straight lines in Fig. 3. The dimensions of B
176 BELL SYSTEM TECHNICAL JOURNAL
are Kelvin degrees or centigrade degrees; it plays the same role in equation
(1) as does the work function in Richardson's equation for thermionic
emission. For Material No. \, B — 392()C°. This corresponds to an elec
tron energy equivalent to 3920 11600 or 0.34 volt.
While the curves in Fig. 3 are approximately straight, a more careful
investigation shows that the slope increases linearly as the temperature
increases. From this it follows that a more precise expression for p is:
, T — c PIT
p = A 1 6 or
log p = log .1  r log T + D/2.303r (2)
The constant c is a small positive or negative number or zero. For Ma
terial No. 1, log A = 5.563, < = 2.73 and D = 3100. For a particular
form of Material No. 2 log .1 = 11.514, c = 4.83 and D = 2064.
If we define temperature coefilicient of resistance, a, by the equation
a = {\/R) {(IR/dT) (3)
it follows from equation (1) that
a = B/r. (4)
For Material No. 1 and T  300°K, a  3920/90,000 = 0.044. For
platinum, a — +0.0037 or roughly ten times smaller than for semiconduc
tors and of the opposite sign. From equation (2) it follows that
«= {D/D (c/T). (5)
From equation (3) it follows that
a = (1 2.303) {(flogR'dT). (6)
For a discussion of the nature of the conductivit}^ in semiconductors,
it is simpler and more convenient to consider the conductivity, a, rather
than the resistivity, p.
a = \/p and logo = —log p. (7)
The characteristics of semiconductors are brought out more clearly if the
conductivity or its logarithm are plotted as a function of \/T over a wide
temj:;erature range. Figure 4 is such a j)lot for a number of silicon sam
ples containing increasing amounts of impurity. At high temperatures
all the samples have nearly the same conductivity. This is called the
intrinsic conductivity since it seems to be an intrinsic properly of silicon.
At low temperatures the conductivity of different sami:)les varies by large
factors. Tn this region silicon is said to be an impurity semiconductor.
For extremely i)ure silicon only intrinsic conductivity is present and the
PROPERTIES AND USES OF THERMISTORS
177
resistivity obeys equation (1). As the concentration of a particular im
purity increases, the conductivity increases and the impurity conductivity
predominates to higher temperatures. Some impurities are much more
effective in increasing the conductivity than others. One hundred parts
per million of some impurities may increase the conductivity of pure silicon
at room temperature by a factor of 10^ Other impurities may be present
7 '0
O
310
I
o
bio
o
§'0
o
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\
2
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— \—
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V
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7
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TEMPERATURE °K
Fig. 4. — Logarithm of the conductivity of various specimens of silicon as a function
of inverse absolute temperature. The conductivity increases with the amount of im
purity.
in 10,000 parts per million and have a small effect on the conductivity.
Two samples may contain the same concentration of an impurity and still
differ greatly in their low temperature conductivity; if the impurity is in
solid solution, i.e., atomically dispersed, the effect is great; if the impurity
is segregated in atomically large particles, the effect is small. Since heat
treatments affect the dispersion of impurities in solids, the conductivity of
semiconductors may frequently be altered radically by heat treatment.
Some other semiconductors are not greatly affected by heat treatment.
178
BELL SYSTEM TECHNICAL JOURNAL
The impurity need not even be a foreign element; in the case of oxides or
sulphides, it can be an excess or a deficiency of oxygen or sulphur from the
exact stoichiometric relation. This excess or deficiency can be brought
about by heat treatment. Figure 5 shows how the conductivity depends
on temperature for a number of samples of cuprous oxide, CU2O, heat
ID'
1.0
o
^io«
I
y
8io
tCT^°i
— \i
s^ —
X,
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v^
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N
X.
\^
^^
V^
V.
^
*
t^
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K
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^
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=»^
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\
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s.
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k
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\,
\
N
^i^^
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^^^
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\
\
\
— ^
\
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1
1
xlQ
temperature: °k
Fig. 5. — Logarithm of the conductivity of various specimens of cuprous oxide as a
function of inverse absolute temperature. The conductivity increases with the amount
of excess oxygen above the stoichiometric value in CuoO. Data from reference 1.
treated in such a way as to result in varying amounts of excess oxygen from
zero to about one per cent.' The greater the amount of excess o.xygen the
greater is the conductivity in the low temperature range. At high tem
peratures, all samples have about the same conductivity.
Semiconductors can be classified on the basis of the carriers of the current
into ionic, electronic, and mixed conductors. Chlorides such as NaCl and
some sulphides are ionic semiconductors; other sulphides and a few oxides
PROPERTIES AND USES OF THERMISTORS
179
such as uranium oiide are mixed semiconductors; electronic semiconductors
include most oxides such as MnsOs, FejOs, NiO, carbides such as silicon
carbide, and elements such as boron, silicon, germanium and tellurium.
In ionic and mixed conductors, ions are transported through the solid.
This changes the density of carriers in various regions, and thus changes
the conductivity. Because this is undesirable, they are rarely used in mak
ing thermistors, and hence we will concentrate our interest on electronic
semiconductors.
The theoretical and experimental physicists have established that there
are two types of electronic semiconductors which can be called N and P
type, depending upon whether the carriers are negative electrons or are
equivalent to positive "holes" in the filled energy band. In N type, the
ACCEPTOR
M PURITIES
INTRINSIC
Fig. 6. — Schematic energy level diagrams illustrating intrinsic, N and P types of semi
conductors.
carriers are deflected by a magnetic field as negatively charged particles
would be and conversely for P type. The direction of deflections is ascer
tained by measurement of the sign of the Hall effect. The direction of the
thermoelectric effect also fixes the sign of the carriers. By determining
the resistivity, Hall coefficient and therm.oelectric power of a particular
specimen at a particular temperature it is possible to determine the density
of carriers, whether they are negative or positive, and their mobility or mean
free path. The mobility is the mean drift velocity in a field of one volt per
centimeter.
The existence of these classifications is explained by the theoretical physi
cist^ . 3 , 4 j^ terms of the diagrams in Fig. 6. In an intrinsic semiconductor
at low temperatures the valence electrons completely fill all the allowable
energy states. According to the exclusion principle only one electron can
occupy a particular energy state in any system. In semiconductors and
180 BELL SYSTEM TECHNICAL JOURNAL
insulators there exists a region of energy values, just above the allowed band,
which are not allowed. The height of this unallowed band is expressed in
equivalent electron volts, A£. Above this unallowed band there exists an
allowed band; but at low temperatures there are no electrons in this band.
When a iield is applied across such a semiconductor, no electron can be
accelerated, because if it were accelerated its energy would be increased to
an energy state w^hich is either tilled or unallowed. As the temperature is
raised some electrons acquire sufficient energy to be raised across the un
allowed band into the upper allowed band. These electrons can be ac
celerated into a slightly higher energy state by the applied field and thus
can carry current. For every electron that is put into an "activated"
state there is left behind a "hole" in the normally filled band. Other
electrons having slightly lower energies can be accelerated into these holes
by the applied field. The physicist has shown that these holes act toward
the applied field as if they were particles having a charge equal to that of an
electron but of opposite sign and a mass equal to or somewhat larger than
the electronic mass. In an intrinsic semiconductor about half the con
ductivity is due to electrons and half due to holes.
The quantity A£ is related to B in equation (1) by:
2B = (A£) e/k (8)
in which B is in centigrade degrees, A£ is in volts, e is the electronic charge
in coulombs, k is Boltzmann's constant in joules per centigrade degree.
The value of e/k is 11,600 so that
A£ = Z^/5800. (8a)
The difference between metals, semiconductors, and insulators results
from the value of A£. For metals A£ is zero or very small. For semicon
ductors A£ is greater than about 0.1 volt but less than about 1.5 volts.
For insulators A£ is greater than about 1.5 volts.
Some impurities with positive valencies which may be present in the semi
conductor may have energy states such that A£i volts equivalent energy
can raise the valence electron of the impurity atom into the allowed con
duction band. See Figure 6. The electron now can take part in conduc
tion; the donator impurity is a positive ion which is usually bound to a par
ticular location and can take no part in the conductivity. These are excess
or A^ type conductors. The conductivity de[)ends on the density of dono
tors, A£i , and T.
Similarly some other impurity with negative valencies may have an
energy state A/S2 volts above the top of the lilled band. At room temi)era
ture or higher, an electron in the filled band may be raised in energy and
PROPERTIES AND USES OF THERMISTORS 181
accepted by the impurity which then becomes a negative ion and usually
is immobile. However, the resulting hole can take part in the conductivity.
In all cases represented in Fig. 6 an electron occupying a higher energy
level than a positive ion or a hole has a certain probability that in any
short interval of time it will drop into a lower energy state. However, dur
ing this same time interval there will be electrons which will be raised to a
higher energy level by thermal agitation. When the number of electrons
per second which are being elevated is equal to the number which are de
scending in energy, equilibrium prevails. The conductivity, a, is then
a = N evii P ev2 (9)
where N and P are the concentrations of electrons and holes respectively,
e is the charge on the electron, z'l and V2 are the mobilities of electrons and
holes respectively.
The above picture explains the following experimental facts which other
wise are difficult to interpret. 1) A^ type oxides, such as ZnO, when heated
in a neutral or slightly reducing atmosphere become good conductors,
presumably because they contain excess zinc which can donate electrons.
If they then are heated in atmospheres which are increasingly more oxidiz
ing their conductivity decreases until eventually they are intrinsic semi
conductors or insulators. 2) P type oxides, such as NiO, when heat treated
in strongly oxidizing atmospheres are good conductors. Very likely they
contain oxygen in excess of the stoichiometric relation and this oxygen
accepts additional electrons. When these are heated in less oxidizing or
neutral atmospheres they become poorer conductors, semiconductors, or
insulators. 3) When a P type oxide is sintered with another P type oxide,
the conductivity increases. Similarly for two N type oxides. But when a
P type is added to an N type the conductivity decreases. 4) If a metal
forms several oxides the one in which the metal exerts its highest valence is
N type, while the one in which it exerts its lowest valence will be P type.^
For several reasons it is desirable to survey the whole field of semicon
ductors for resistivity and temperature coefficient. One way in which this
might be done is to draw a line in Figure 3 for each specimen. Before long
such a figure would consist of such a maze of intersecting lines that it would
be difficult to single out and follow any one line. The information can be
condensed by plotting log po versus B in equation (1) for each specimen.^
The most important characteristics of a specimen thus are represented by
a single point and many more specimens can be surveyed in a single diagram.
Figure 7 shows such a plot for a large number of semiconductors investi
gated at these Laboratories or reported in the literature. Values for po
and B are given for T = 25 degrees centigrade. The points form a sort of
182
BELL SYSTEM TECHNICAL JOURNAL
milky way. Semiconductors having a high po are Ukely to have a high
value of B and vice versa. If a series of semiconductors have points in Fig.
7 which fall on a straight line with a slope of 1/2.37^0 , they have a common
intercept in Fig. 3 for (l/T) = 0.
10"



~

—



r

—


—

10^
,
10*
J
•

10*
o
?n
f\J
ijlO*
5
.
l03
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•
w
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10°
,
I0'
irrZ
6
xlO'^
'"0 I 2 3 4
B IN X AT 25t
Fig. 7.— Logarithm of the resistivity of various semiconducting materials as a func
tion of B in equation (I). The quantity, B, is proportional to the temperature coefiicient
of resistance as given in equation (4).
Physical Properties of Thermistors
One of the most interesting and useful properties of a thermistor is the
way in which the voltage, F, across it changes as the current, /, through
it increases. Figure 8 shows this relationship for a 0.061 centimeter diam
eter bead of Material No. 1 suspended in air. Each time the current is
PROPERTIES AND USES OF THERMISTORS
183
changed, sufficient time is allowed for the voltage to attain a new steady
value. Hence this curve is called the steady state curve. For sufficiently
small currents, the power dissipated is too small to heat the thermistor
appreciably, and Ohm's law is followed. However, as the current assumes
larger values, the power dissipated increases, the temperature rises above
ambient temperature, the resistance decreases, and hence the voltage is less
than it would have been had the resistance remained constant. At some
current, !„ , the voltage attains a maximum or peak value, Vm • Beyond
/^^
Xso
h
\
\
6(fS.
^^s^
I
100
^^^
'
..
""""^55
2
0.5
5 10
MILLIAMPERES
Fig. 8. — Static voltagecurrent curve for a typical thermistor. The numbers on the
curve are the centigrade degrees rise in temperature above ambient.
i
this point as the current increases the voltage decreases and the thermistor
is said to have a negative resistance whose value is dV/dl. The numbers on
the curve give the rise in temperature above ambient temperature in centi
grade degrees.
Because currents and voltages for different thermistors cover such a
large range of values it has been found convenient to plot log V versus log /.
Figure 9 shows such a plot for the same data as in Fig. 8. For various points
on the curve, the temperature rise above ambient temperature is given.
In a log plot, a line with a slope of 45 degrees represents a constant resist
ance; a line with a slope of —45 degrees represents constant power.
184
BELL SYSTEM TECHNICAL JOURNAL
For a particular thermistor, the position of the log V versus log I plot is
shifted, as shown in Fig. 10, by changing the dissipation constant C. This
IjO
MILLIAMPERES
Fig. 9. — Logarithmic plot of static voltagecurrent curve for the same data as in Figure
8. The diagonal hnes give the values of resistance and power.
B=3900 R= 50,000 OHMS T=300°K
100
V\/
/
K /
X
■!o4 \
\
/X
y
<
\
X
/^
X
A
y
X
X
k
/
10"' 10"" 10' ■" 10'' 10"' I 10
CURRENT IN AMPERES
Fig. lO.^Logarithmic plots of voltage versus current for three values of the dissipa
tion constant C. These curves are calculated for the constants given in the upper jiart
of tlje figure.
can be done by changing the air pressure surrounding the bead, changing
the medium, or changing the degree of thermal coupling between the thermi§
PROPERTIES AND USES OF THERMISTORS
185
tor and its surroundings. The value of C for a particular thermistor in
given surroundings can readily be determined from the V versus / curve in
either Figs. 8 or 9. For each point, V/I is the resistance while V times /
is IF, the watts dissipated. The resistance data are converted to tempera
ture from R versus T given by equation (2). A plot is then made of W
versus T. For thermistors in which most of the heat is conducted away,
W will increase linearly with T, so that C is constant. For thermistors
suspended by fine wires in a vacuum, W will increase more rapidly than pro
portional to T, and C will increase with T. For thermistors of ordinary
size and shape, in still air, C/Area = 1 to 40 milliwatts per centigrade degree
per square centimeter depending upon the size and shape factor.
B=3900
C=5X10 WATTS/DEG.
aoo'K
100
/ rfp/\
\*
Ay \
\^^
5p /y
\ /
\4/
X
10"* 10** 10"' 10^ 10"' I K)
CURRENT IN AMPERES
Fig. 11.— Logarithmic plots of voltage versus current for three values of the resistance,
Ro , at ambient temperature. These curves are calculated for the constants given in the
upper part of the figure.
The user of a thermistor may want to know how many watts can be dis
sipated before the resistance decreases by one per cent. This may be called
the power sensitivity. It is equal to C/{a X 100), and amounts to about
one to ten milliwatts per square centimeter of area in still air. Both C and
the power sensitivity increase with air velocity. The dependence of C on
gas pressure and velocity is the basis of the use of thermistors as manom
eters and as anemometers or flowmeters. Note that in Fig. 10 one curve
can be superposed on any other by a shift along a constant resistance line.
Figure 1 1 shows a family of log V versus log / curves for various values on
Ro while B, C, and To are kept constant. This can be brought about by
changing the length, width and thickness to vary Ro while the surface area
is kept constant. If the resistance had been changed by changing the am
bient temperature. To , the resulting curves would not appear very different
186
BELL SYSTEM TECHNICAL JOURNAL
from those shown. Note that one curve can be superposed on any other
curve by a shift along a constant power Hne.
Figure 12 shows a family of log V versus log / curves for eight different
values of B while C, Ra , and To are kept constant. In contrast to the curves
in Figs. 10 and 11 in which any curve could be obtained from any other
curve by a shift along an appropriate axis, the curves in Fig. 12 are each
distinct. For each curve there exists a limiting ohmic resistance for low
C=5X10"'^WATTS/DEG.
RoSQPOO OHMS
T = 300 K
1000
o*
10
10
10
10
102
CURRENT IN AMPERES
Fig. 12. — Logarithmic plots of voltage versus current for eight values of B in equation
(1). These carves are calculated for the constants given in the upper part of the figure.
currents and another for high currents. For B = these two are identical.
As B becomes larger the log of the ratio of the two limiting resistances in
creases proportional to B. Note also that for B > 1200 A'°, the curves have
a maximum. For large B values this maximum occurs at low powers and
hence at low values of T — To . This follows since W = C{T — To).
As B decreases, Vm occurs at increasingly higher powers or temperatures.
For B < 1200 K°, no maximum exists.
The curves in Figs. 10 to 12 have been drawn for the ideal case in which
the resistance in series with the thermistor is zero and in which no tempera
ture limitations have been considered. In any actual case there is always
PROPERTIES AND USES OF THERMISTORS 187
some unavoidable small resistance, such as that of the leads, in series with
the thermistor and hence the parts of the curves corresponding to low re
sistances may not be observable. Also at high powers the temperature may
attain such values that something in the thermistor structure will go to
pieces thus limiting the range of observation. These unobservable ranges
have been indicated by dashed lines in Fig. 12. The exact location of the
dashed portions will of course depend on how a completed thermistor is con
structed. In setting these limits consideration is given to temperature limi
tations beyond which aging efifects might become too great.
The curves in Figs. 9 to 12 have been computed on the basis of the follow
ing equations:
W = C(T  To) = VI (11)
For these curves the constants Rq , To , B, and C are specified. The values
of temperature, T^ , power, W^ , resistance, R^. , voltage, F„ , and current,
Im , that prevail at the maximum in the voltage current curve are given
by the following equations in which T^ is chosen as the independent param
eter. By differentiating equations (10) and (11) with respect to /, putting
the derivatives equal to zero, one obtains
Tl = B{Tm  To) (12)
whose solution is
r„ = {B/2) (1 T Vl  4To/B). (13)
The minus sign pertains to the maximum in Figs. 10 to 12 while the plus
sign pertains to the minimum. Note that Tm depends only on B and To ,
and not on R, Ro or C. From equations (4), (10) and (11) it follows that:
 a^ {T^  To) = 1 (14)
\V„. = C{T„,  To) (15)
i?,„ = Ro r^""'^" ^ Ro t'iX  (r„  To)/To +
(1/2) {(n. To)/ToV ] (16)
F„ = [C Ro {Tm  To) {e''^')]'"
= \\C Ro (r„.  To) €' [1  {Tm  To)/ To 4
(1/2) \{Tm To)/ToV WV" (17)
Jr. = [{C/Ro) {Tm  To) e''^^r
= {{{C/Ro) {Tm  To) e[\ + {Tm  To)/To +
(1/2)1 (r. To)/To}'+ ■■■ ]}V'' (18)
188
BELL SYSTEM TECHNICAL JOURNAL
Thus far the presentation has been limited to steady state conditions, in
which the power supplied to the thermistor is equal to the power dissipated
by it, and the temperature remains constant. In many cases, however, it
is important to consider transient conditions when the temperature, and
any quantities which are functions of temperature, var}^ with time. A
simple case which will illustrate the concepts and constants involved in
such problems is as follows: A massive thermistor is heated to about 150 to
200 degrees centigrade by operating it well beyond the peak of its voltage
200
100
'
\
V
80
N^
\^
60
\
'
V
o
N.
N.
Z
^
"^20
H
\
10
\,
\
8
\
4
2
k
150 200
TIME IN SECONDS
Fig. 13.— Cooling characteristic of a massive thermistor: log of temperature above
ambient versus time.
current characteristic. At time / = 0, the circuit is switched over to a con
stant current having a value so small that PR is always negligibly small.
The voltage across the thermistor is then followed as a function of time.
From this, the resistance and temperature are computed. Figure 13 shows i
a plot of log (r  Ta) versus / for a rod thermistor of Material No. 1 about
1.2 centimeters long, 0.30 centimeter in diameter and weighing 0.380 gram.
In any time interval Al, there are C(T  To) A/ joules being dissipated. 
.As a result the temperature will decrease by A7" given by
HAT = ar  Ta) A/ or (7'  7'„)  {H/C) (A7'/A/) iV))
PROPERTIES AND USES OF THERMISTORS
189
where H = heat capacity m joules per centigrade degree. The solution of
this equation is
(r  r„) = (r„  r„)
in which 2\ — T when / = and
r = H/C,
(20)
(21)
where r is in seconds. It is commonly called the time constant. From
equation (20) it follows that a plot of log {T — T a) versus t should yield a
straight line whose slope = — 1/2.303t. If // and C vary slightly with
temperature then t will vary slightly with T and /. The line will not be
perfectly straight but its slope at any t or (T — To) will yield the appro
Table I. — Values of C, t, H as Functions of T for a Thermistor of Material No. 1
ABOUT 1.2 Centimeters Long, 0.30 Centimeters in Diameter and Weighing 0.380 Gram
Ta = 24 degrees centigrade
T
Degrees Centigrade
C
Watts per C.
degree
T
Seconds
//
Joules per C.
degree
h
Joules per gram
per C. degree
44
64
0.0037
0.0037
76
74
0.28
0.27
0.75
0.72
84
104
0.0038
0.0037
71
69
0.27
0.26
0.71
0.68
124
144
0.0038
0.0038
68
67
0.26
0.26
0.67
0.67
164
; 184
0.0039
0.0041
67
66
0.26
0.27
0.69
0.71
204
0.0042
66
0.28
0.73
priate t or H/C for that T. As previously described, C can be determined
from a plot of watts dissipated versus T. For this thermistor this curve
became steeper at the higher temperatures so that C increased for higher
temperatures. Table I summarizes the values of C, r, and // at various T
for the unit in air.
When a thermistor is heated by passing current through it, conditions
are somewhat more involved since the PR power will be a function of time.
At any time in the lieating cycle the heat power liberated will be equal to
the watts dissipated or C{T — Ta) plus watts required to raise the tem
perature or HdT/dl. The heat power liberated will de})end on the circuit
conditions. In a circuit like that shown in the upper corner of Figure 14, the
current varies with time as shown by the six curves for six values of the
battery voltage E. If a relay in the circuit operates when the current
reaches a definite value, a considerable range of time delays can be achieved.
190
BELL SYSTEM TECHNICAL JOURNAL
This family of curves will be modified by changes in ambient temperature
and where rather precise time delays are required, the ambient temperature
must be controlled or compensated.
Since thermistors cover a wide range in size, shape, and heat conductivity
of surrounding media, large variations in //, C, and t can be produced.
The time constant can be varied from about one millisecond to about ten
minutes or a millionfold.
One very important property of a thermistor is its aging characteristic
or how constant the resistance at a given temperature stays with use. To
obtain a stable thermistor it is necessary to: 1) select only semiconductors
which are pure electronic conductors; 2) select those which do not change
chemically when exposed to the atmosphere at elevated temperatures;
3U
40
KEY 1
THERMISTOR^
E=£
JOVCLTS
■■^ n 1^ 1
(/I
^
70
20
1 II
tf 30
66 OSCIi^
GRAPH
ENT
//
/"
^
W
<
^0
5 20
■7
I
V
/^
^
40
^ ,n
"
30
Ld 10
h
/^
.
^
•
^
D
P
i ^
\
3 i
3
1 f
3 9
TIME IN SECONDS
Fig. 14. — Current versus time curves for six values of the battery voltage in the circuit
shown in the insert.
3) select one which is not sensitive to impurities likely to be encountered in
manufacture or in use; 4) treat it so that the degree of dispersion of the
critical impurities is in equilibrium or else that the approach to equilibrium
is very slow at operating temperatures; 5) make a contact which is intimate,
sticks tenaciously, has an expansion coefficient compatible with the semi
conductor, and is durable in the atmospheres to which it will be exposed;
6) in some cases, enclose the thermistor in a thin coat of glass or material
impervious to gases and liquids, the coat having a suitable expansion coeffi
cient; 7) preage the unit for several days or weeks at a temperature some
what higher than that to which it will be subjected. By taking these pre
cautions remarkably good stabilities can be attained.
Figure 15 shows aging data taken on threequarter inch diameter discs
of Materials No. 1 and No. 2 with silver contacts and soldered leads. These
discs were measured soon after production, were aged in an oven at 105
degrees centigrade and were periodically tested at 24 degrees centigrade.
PROPERTIES AND USES OF THERMISTORS
101
The percentage change in resistance over its initial value is plotted versus
the logarithm of the time in the aging oven. It is to be noted that most of
the aging takes place in the first day or week. If these discs were preaged
for a week or a month and the subsequent change in resistance referred to
the resistance after preaging, they would age only about 0.2 per cent in one
year. In a thermistor thermometer, this change in resistance would cor
respond to a temperature change of 0.05 centigrade degree. Thermistors
mounted in an evacuated tube or coated with a thin layer of glass age even
less than those shown in the figure. For some applications such high
stability is not essential and it is not necessary to give the thermistors special
treatment.
"
.rMM *\=.
_
MATEe\£i=^^
■
.0
^^■^^
'
i^'
y
wiATrRlAL'**^2

5
'^ ■
— — ''
i[
AY
IV
EEK 1 MOt
ITH
6 MONTI
S 1 YEAF
^ 2YRS 5YRS
KD' 10^ lO'' 10^
TIME IN HOURS AT 105° C.
Fig. 15. — Aging characteristics of thermistors made of Materials No. 1 and No. 2
aged in an oven at rG5°C. Per cent increase in resistance over its initial value versus
time on a logarithmic scale.
Thermistors have been used at higher temperatures with satisfactory aging
characteristics. Extruded rods of Material No. 1 have been tested for stab
ility by treating them for two months at a temperature of 300 degrees
centigrade. Typical units aged from 0.5 to 1.5 per cent of their initial
resistance. Similar thermistors have been exposed alternately to tempera
tures of 300 degrees centigrade and —75 degrees centigrade for a total of
700 temperature cycles, each lasting onehalf hour. The resistance of typ
ical units changed by less than one per cent.
In some applications of thermistors very small changes in temperature
produce small changes in potential across the thermistor which then are
amplified in high gain amplifiers. If at the same time the resistance is
fluctuating randomly by as little as one part in a million, the potential
across the thermistor will also fluctuate by a magnitude which will be
192
BELL SYSTEM TECHNICAL JOURNAL
directly proportional to the current. This fluctuating potential is called
noise and since it depends on the current it is called current noise. In order
to obtain the best signal to noise ratio, it is necessary that the current noise
at operating conditions be less than Johnson or thermal noise.'^ ■* To make
noisefree units it is necessary to pay particular attention to the raw mate
rials, the degree of sintering, the grain size, the method of making contact
and any steps in the process which might result in minute surface cracks or
fissures.
POWER IN WATTS
0.1 I 10
THERMISTOR ELEMENT CURRENT IN MILLIAMPERES
100
Fig. 16. — Logarithmic plots of voltage versus current for six values of heater curren
in an indirecth' heated thermistor. Resistance and power scales are given on the diag
onal lines.
All the thermistors discussed thus far were either directly heated by the
current passing through them or by changes in ambient temperature. In
indirectly heated thermistors, the temperature and resistance of the thermis
tor are controlled primarily by the power fed into a heater thermally coupled
to it. One such form might consist of a 0.038 centimeter diameter bead of
Material No. 2 embedded in a small cylinder of glass about 0.38 centimeter
long and 0.076 centimeter in diameter. A small nichrome heater coil hav
ing a resistance of 100 ohms is wound on the glass and is fused onto it with
more glass. Figure 16 shows a plot of log V versus log / for the bead ele
ment at various currents through the heater. In this way the bead resist
ance can be changed from 3000 ohms to about 10 ohms. Indirectly heated
thermistors are ordinarily used where the controlled circuit must be iso
lated electrically from the actuating circuit, and where the power from the
latter must be fed into a constant resistance heater.
PROPERTIES AND USES OF THERMISTORS 193
PART II— USES OF THERMISTORS
The thermistor, or thermally sensitive resistor, has probably excited more
interest as a major electric circuit element than any other except the vacuum
tube in the last decade. Its extreme versatility, small size and ruggedness
were responsible for its introduction in great numbers into communications
circuits within five years after its first appUcation in this field. The next
five year period spanned the war, and saw thermistors widely used in addi
tional important applications. The more important of these uses ranged
from time delays and temperature controls to feedback amplifier automatic
gain controls, speech volume limit ers and superhigh frequency power meters.
It is surprising that such versatility can result from a temperature dependent
resistance characteristic alone. However, this effect produces a very useful
nonlinear voltampere relationship. This, together with the ability to pro
duce the sensitive element in a wide variety of shapes and sizes results in
applications in diverse fields. (The variables of design are many and inter
related, including electrical, thermal and mechanical dimensions.
The more important uses of thermistors as indication, control and cir
cuit elements will be discussed, grouping the uses as they fall under the
primary characteristics: resistancetemperature, voltampere, and current
time or d^mamic relations.
ResistanceTemperature Relations
It has been pointed out in Part I that the temperature coefficient of elec
trical resistance of thermistors is negative and several times that of the or
dinary metals at room temperature. In Thermistor Material No. 1, which
is commonly used, the coefficient at 25 degrees centigrade is —4.4 per cent
per centigrade degree, or over ten times that of copper, which is +0.39 per
cent per centigrade degree at the same temperature. A circuit element made
of this thermistor material has a resistance at zero degrees centigrade which
is nine times the resistance of the same element at 50 degrees centigrade.
For comparison, the resistance of a copper wire at 50 degrees centigrade
is 1.21 times its value at zero degrees centigrade.
The resistancetemperature characteristics of thermistors suggest their
use as sensitive thermometers, as temperature actuated controls and as
compensators for the effects of varying ambient temperature on other ele
ments in electric circuits.
Thermometry
The application of thermistors to temperature measurement follows the
usual principles of resistance thermometry. However, the large value of
temperature coefficient of thermistors permits a new order of sensitivity to
be obtained. This and the small size, simplicity and ruggedness of thermis
194
BELL SYSTEM TECHNICAL JOURNAL
tors adapt them to a wide variety of temperature measuring applications.
VV^hen designed for this service, thermistor thermometers have longtime
stability which is good for temperatures up to 300 degrees centigrade and
excellent for more moderate temperatures. A well aged thermistor used
in precision temperature measurements was found to be within 0.01 centi
grade degree of its calibration after two months use at various temperatures
up to 100 degrees centigrade. As development proceeds, the stability of
thermistor thermometers may be expected to approach that of precision
platinum thermometers. Conventional bridge or other resistance measuring
circuits are commonly employed with thermistors. As with any resistance
thermometer, consideration must be given to keeping the measuring current
sufficiently small so that it produces no appreciable heating in order that the
Table II.
— TemperatureResistance Characteristic of a
Typical Thermistor Thermometer
Temperature CoefBcients
Temperature
Resistance
B
a
25°C.
580,000 ohms
3780 C. deg.
6.1%/ C. deg.
145,000
3850
5.2
25
46,000
3920
4.4
50
16,400
3980
3.8
75
6,700
4050
i.i
100
3,200
4120
3.0
150
830
4260
2.4
200
305
4410
2.0
275
100
4600
1.5
Dissipation constant in still air, approx 4 milliwatts/C. deg.
Thermal time constant in still air, approx 70 seconds
Dimensions of thermistor, diameter approx 0.11 inch
length approx 0. 54 inch
thermistor resistance shall be dependent upon the ambient temperature
alone.
Since thermistors are readily designed for higher resistance values than
metallic resistance thermometers or thermocouples, lead resistances are
not ordinarily bothersome. Hence the temperature sensitive element can
be located remotely from its associated measuring circuit. This permits
great flexibility in application, such as for instance wire line transmission
of temperature indications to control points.
Table II gives the characteristics of a typical thermistor thermometer.
The dissipation constant is the ratio of the power input in watts dissipated
in the thermistor to the resultant temperature rise in centigrade degrees.
The time constant is the time required for the temperature of the thermistor
to change 63 per cent of the difference between its initial value and that of
the surroundings. As a sensitive thermometer, this thermistor with a
simple Wheatstone bridge and a galvanometer whose sensitivity is 2 X
PROPERTIES AND USES OF THERMISTORS
195
U
zi^
f^
196 BELL SYSTEM TECHNICAL JOURNAL
10"^° amperes per millimeter per meter will readily indicate a temperature
change of 0.0005 centigrade degree. For comparison a precision platinum
resistance thermometer and the required special bridge such as the Mueller
will indicate a minimum change of 0.003 centigrade degree with a similar
galvanometer.
Several thermistors which have been used for thermometry are shown in
Fig. 17. Included in the group are types which are suited to such diverse
applications as intravenous blood thermometry and supercharger rotor
temperature measurement. In Fig. 17, A is a tiny bead with a response
time of less than a second in air. B is a probe type unit for use in air streams
or liquids. C is a meteorological thermometer used in automatic radio
transmission of weather data from free balloons. D is a rod shaped imit.
E is a disc or pellet, adapted for use in a metal thermometer bulb. Discs
like the one shown have been sweated to metal plates to give a low thermal
impedance connection to the object whose temperature is to be determined.
F is a large disc with an enveloping paint finish for use in humid surroimd
ings. The characteristics of these types are given in Table III.
The temperature of objects which are inaccessible, in motion, or too hot
for contact thermometry can be determined by permitting radiation from
the object to be focussed on a suitable thermistor by means of an elliptical
mirror. Such a thermistor may take the form of a thin flake attached to a
solid support. Its advantages compared with the thermopile and resistance
bolometer are its more favorable resistance value, its ruggedness, and its
high temperature coefficient of resistance. It can be made small to reduce
its heat capacity so as rapidly to follow changing temperatures. Flake
thermistors have been made with time constants from one millisecond to
one second. Since the amount of radiant power falling on the thermistor
may be quite small, sensitive meters or vacuum tube amplifiers are required
to measure the small changes in the flake resistance. Where rapidly vary
ing temperatures are not involved, thermistors with longer time constants
and simpler circuit equipments can be utilized.
Temperature Control
The use of thermistors for temperature control purposes is related closely
to their application as temperature measuring devices. In the ideal tem
perature sensitive control element, sensitivity to temperature change should
be high and the resistance value at the control temperature should be the
proper value for the control circuit used. Also the temperature rise of the
control element due to circuit heating should be low, and the stability of
calibration should be good. The size and shape of the sensitive element are
dictated by several factors such as the space available, the required speed
of response to temperature changes and the amount of power which must
PROPERTIES AND USES OF THERMISTORS
197
be dissipated in the element by the control circuit to permit the arrange
ment to operate relays, motors or valves.
Because of their high temperature sensitivity, thermistors have shown
much promise as control elements. Their adaptability and their stability
at relatively high temperatures led, for instance, to an aircraft engine con
trol system using a rodshaped thermistor as the control element.^ The
Table III. — Thermistor Thermometers
A
B
C
D
E
F
Nominal Resistance, Ohms at
25°C
5,000
2,000
900
460
250
95
3.4
150
0.1
1
Bead
0.015
0.02
325,000
100,000
33,000
13,000
6,000
1,600
500
80
4.4
300
1
7
30
4
Probe
0.1
0.6
87,500
37,500
18,000
9,700
5,500
3,700
2.8
100
7
25
Rod
0.05
1.2
610,000
153,000
48,500
17,300
7,100
3,400
870
4.4
150
7
60
Rod
0.15
0.7
490
175
71
32
16
4.5
1.6
3.8
200
Disc
0.2
0.1
13,000
25
50
3,200
950
340
75
145
100
150
70
200
300
Temp. Coeff. «, %/C. deg. at
25°C
4.4
Max. Permissible Temp., °C. .
Dissipation Constant, C,
mw/C deg.
Still air
100
20
Still water
—
Thermal Time Constant,
Seconds
Still air
Still water
—
Shape
Disc
Dimensions, Inches
Diameter or Width
Length or Thickness (less
leads)
0.56
0.31
thermistor, mounted in a standard onequarter inch diameter temperature
bulb assembly, operated at approximately 275 degrees centigrade. It was
associated with a differential relay and control motor on the aircraft 28
volt dc system. The power dissipation in the thermistor was two watts.
The resistance of a typical thermistor under these high temperature con
ditions remained within ±1.5 per cent over a period of months. This
corresponds to about ± one centigrade degree variation in calibration.
Several other related designs were developed using the same control system
198 BELL SYSTEM TECHNICAL JOURNAL
with other thermistors designed for both higher and lower temperature
operation. In the lower temperature applications, typical thermistors
maintained their calibrations within a few tenths of a centigrade degree.
In general, electron tube control circuits dissipate less power in the ther
mistor than relay circuits do. This results in less temperature rise in the
thermistor and leads to a more accurate control. While the average value
of this temperature rise can be allowed for in the design, the variations
in different installations require individual calibration to correct the errors
if they are large. The corrections may be different as a result of variations
of the thermal conductivity of the surrounding media from time to time or
from one installation to another. The greater the power dissipated in the
thermistor the greater the absolute error in the control temperature for a
given change in thermal conductivity. This follows from the relation
^T = W/C (22)
where AT is the temperature rise, W is the power dissipated and C is the dis
sipation constant which depends on thermal coupling to the surroundings.
For the same reason, the temperature indicated by a resistance thermometer
immersed in an agitated medium will depend on the rate of flow if the tem
perature sensitive element is operated several degrees hotter than its sur
roundings.
The design of a thermistor for a ventilating duct thermostat might pro
ceed as follows as far as temperature rise is concerned :
1 . Determine the power dissipation. This depends upon the circuit
selected and the required overall sensitivity.
2. Estimate the permissible temperature rise of the thermistor, set by the
expected variation in air speed and the required temperature control accur
acy.
3. Solve Equation (22) for the dissipation constant and select a thermistor
of appropriate design and size for this constant in the nominal air speed.
Where more than one style of thermistor is available, the required time
constant will determine the choice.
Compensators
It is a natural and obvious application of thermistors to use them to com
pensate for changes in resistance of electrical circuits caused by ambient
temperature variations. A simple example is the compensation of a copper
wire line, the resistance of which increases approximately 0.4 per cent per
centigrade degree. A thermistor having approximately onetenth the
resistance of the copper, with a temperature coefficient of —4 per cent per
centigrade degree placed in series with the line and subjected to the same
ambient temperature, would serve to compensate it over a narrow tempera
PROPERTIES AND USES OF THERMISTORS
199
ture range. In practice however, the compensating thermistor is associated
with parallel and sometimes series resistance, so that the com.bination gives
a change in resistance closely equal and opposite to that of the circuit to be
compensated over a wide range of temperatures. See Fig. 18.
2000
40
20 20 40 60
TEMPERATURE IN DEGREES CENTIGRADE
80
Fig. 18. — Temperature compensation of a copper conductor by means of a thermistor
network.
A copper winding having a resistance of 1000 ohms at 25 degrees centi
grade can be compensated by means of a thermistor of 566 ohms at 25
degrees centigrade in parallel with an ohmic resistance of 445 ohms as shown
in Fig. 18. The winding with compensator has a resistance of 1250 ohms
constant to ± 1.6 per cent over the temperature range —25 degrees centi
grade to t75 degrees centigrade. Over this range the copper alone varies
from 807.5 ohms to 1192.5 ohms, or ± 19 per cent about the mean. The
200 BELL SYSTEM TECHMCAL JOURI^AL
total resistance of the circuit has been increased only 6.1 per cent at the
upper temperature limit by the addition of a compensator. This increase
is small because of the high temperature coefficient of the compensating
thermistor. The characteristics of such a thermistor are so stable that the
resistance would remain constant within less than one per cent for ten years
if maintained at any temperature up to about 100 degrees centigrade.
Figure 15 shows aging characteristics for typical thermistors suitable for
use in compensators. These curves include the change which occurs during
the seasoning period of several days at the factory, so that the aging in use
is a fraction of the total shown.
In many circuits which need to function to close tolerances under wide
ambient temperature variation, the values of one or more circuit elements
may var>' undesirably with temperature. Frequently the resultant overall
variation with temperature can be reduced by the insertion of a simple ther
mistor placed at an appropriate point in the circuit. This is particularly
true if the circuit contains vacuum tube amplifiers. In this manner fre
quency and gain shifts in communications circuits have been cancelled and
temperature errors prevented in the operation of devices such as electric
meters. The change in inductance of a coil due to the variation of magnetic
characteristics of the core material with temperature has been prevented by
partially saturating the coil with direct current, the magnitude of which is
directly controlled by the resistance of a thermistor imbedded in the core.
In this way the amount of dc magnetic flux is adjusted by the thermistor
so that the inductance of the coil is independent of temperature.
In designing a compensator, care must be taken to ensure exposure of the
thermistor to the temperature affecting the element to be compensated.
Power dissipation in the thermistor must be considered and either limited to
a value which will not produce a significant rise in temperature above am
bient, or offset in the design.
VoltAmpere Characteristics
The nonlinear shape of the static characteristic relating voltage, current,
resistance and power for a typical thermistor was illustrated by Fig. 9.
The part of the curve to the right of the voltage maximum has a negative
slope, applicable in a large number of ways in electric circuits. The par
ticular characteristic showTi begins with a resistance of approximately 50,000
ohms at low power. Additional power dissipation raises the temperature
of the thermistor element and decreases its resistance. At the voltage
maximum the resistance is reduced to about onethird its cold value, or
17,000 ohms, and the dissipation is 13 milliwatts. The resistance becomes
approximately 300 ohms when the dissipation is 100 milliwatts. Such
resistancepower characteristics have resulted in the use of thermistors as
sensitive power measuring devices, and as automatically variable resistances
PROPERTIES AND USES OF THERMISTORS
201
for such applications as output amplitude controls for oscillators and am
plifiers. Their nonlinear characteristics also fit thermistors for use as volt
age regulators, volume controls, expandors, contactless switches and remote
control devices. To permit their use in these applications for dc as well as
ac circuits, nonpolarizing semiconductors alone are employed in thermistors
with the exception of two early types.
Power Meter
Thermistors have been used very extensively in the ultra and superhigh
frequency ranges in test sets as power measuring elements. The particular
advantages of thermistors for this use are that they can be made small in
size, have a small electrical capacity, can be severely overloaded without
0.5
ONE
INCH
Fig. 19. — Power measuring thermistors with different sized beads.
change in calibration, and can easily be calibrated with directcurrent or
lowfrequency power. For this application the thermistor is used as a power
absorbing terminating resistance in the transmission line, which may be of
Lecher, coaxial or waveguide form. Methods of mounting have been
worked out which reduce the reflection of high frequency energy from the
termination to negligible values and assure accurate measurement of the
power over broad bands in the frequency spectrum. Conventionally, the
thermistor is operated as one arm of a Wheatstone bridge, and is biased with
low frequency or dc energy to a selected operating resistance value, for
instance 125 or 250 ohms in the absence of the power to be measured. The
application of the power to be measured further decreases the thermistor
resistance, the bridge becomes unbalanced and a deflection is obtained on
the bridge meter. A full scale power indication of one miUiwatt is customary
for the test set described, although values from 0.1 milliwatt to 200 milli
watts have been employed using thermistors with different sized beads as
shown in Fig. 19.
202 BELL SYSTEM TECH MCA L JOURNAL
Continuous operation tests of these tliermistors indicate very satisfactory
stability with an indelinitcly long life. A grouj) of eight power meter ther
mistors, normally operated at 10 milliwatts and having a maximum rating
of 20 milliwatts, were o])erated for over 3000 hours at a power input of 30
milliwatts. During this lime the room temperature resistance remained
within 1.5 per cent of its initial value, and the power sensitivity, which is the
significant characteristic, changed by less than 0.5 per cent.
When power measuring test sets are intended for use with wide ambient
temjierature variations, it is necessary to temperature compensate the ther
mistor. This is accomplished conventionally by the introduction of two
other thermistors into the bridge circuit. These units are designed to be
insensitive to bridge currents but responsive to ambient temperature. One
of the compensators maintains the zero point and the other holds the meter
scale calibration independent of the effect of temperature change on the
measuring thermistor characteristics.
Automatic Oscillator Amplitude Control
Meacham, and Shepherd and Wise" have described the use of thermis
tors to provide an effective method of amplitude stabilization of both low
and high frequency oscillators. These circuits oscillate because of positive
feedback around the vacuum tube. The feedback circuit is a bridge with
at least one arm containing a thermistor which is heated by the oscillator
output. Through this arrangement, the feedback depends in phase and
magnitude upon the output, and there is one value of thermistor resistance
which if attained would balance the bridge and cause the oscillation ampli
tude to vanish. Obviously this condition can never be exactly attained,
and the operating point is just enough different to keep the bridge slightly
unbalanced and produce a predetermined steady value of oscillation output.
Such oscillators in which the amplitude is determined by thermistor non
linearity have manifold advantages over those whose amplitude is limited
by vacuum tube nonlinearity. The harmonic content in the output is
smaller, and the performance is much less dependent upon the individual
vacuum tube and upon variations of the supply voltages. It is necessary
that the thermal inertia of the thermistor be sufficient to prevent it from
varying in resistance at the oscillation frequency. This is easily satisfied
for all frequencies down to a small fraction of a cycle per second. Figure 20
shows a thermistor frequently used for oscillator control together with its
static electrical characteristic. This thermistor is satisfactory in oscillators
for frequencies above approximately 100 cycles per second. Similar types
have been developed with response characteristics suited to lower frequencies
and for other resistance values and powers.
PROPERTIES AND USES OF THERMISTORS
203
WTiere the ambient temperature sensitivity of the thermistor is dis
advantageous in oscillator controls, the thermistor can be compensated by
Fig. 20A. — An amplitude control thermistor. The glass bulb is 1.5 inches in length.
102
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CURRENT IN MILLIAMPERES
Fig. 20B. — Steady state characteristics of amplitude control thermistor shown in
Figure 20A.
thermostating it with a heater and compensating thermistor network, as
shown in Fig. 21.
Amplifier Automatic Gain Control
Since the resistance of a thermistor of suitable design varies markedly
with the power dissipated in it or in a closely associated heater, such ther
204 BELL SYSTEM TECIIMCAL JOlh'XAL
mistors have proven to be very valuable as automatic gain controls, es
pecially for use with negative feedback ampliliers. This arrangement has
seen extensive use in wire communication circuits for transmission level
regulation, and has been described in some detail elsewhere.^ ^^' ^^ In
one form, a directly heated thermistor is connected into the feedback circuit
of the amplifier in such a way that the amount of feedback voltage is varied
to compensate for any change in the output signal. By this arrangement,
the gain of each amplifier in the transmission system is continually adjusted
to correct for variations in overall loss due to weather conditions and other
factors, so that constant transmission is obtained over the channel at all
times. In the Type K2 carrier sj^stem now in extensive use, the system
gain is regulated principally in this way. In this system the transmission
loss variations due to temperature are not the same in all parts of the pass
band. The loss is corrected at certain repeater points along the transmission
line by two additional thermistor gain controls: slope, proportional to fre
H EATER T^'PE
/T HERMISTOR
constantI /;t\ ipRi I^CCt^ to"
CURRENTS (^) ^f^2 (Nif) CONTROLLED
SOURCE T Vp^t I rV^W CIRCUIT
DISC
THERMISTOR
HEATER THERMISTOR
Fig. 21. — Circuit employing an auxiliary disc thermistor to compensate for effect of
varying ambient temperature on a control thermistor.
quency, and bulge, with a maximum at one frequency. These thermistors
are indirectly heated, with their heaters actuated by energy dependent upon
the amplitude of the separate pilot carriers which are introduced at the send
ing end for the purpose.
In this type of application, the thermistor will react to the ambient tem
perature to which it is exposed, as well as to the current passing through it.
Where this is important, the reaction to ambient temperature can be elimi
nated by the use of a heater type thermistor as shown in Fig. 21. The
heater is connected to an auxiliary circuit containing a temperature com
pensating thermistor. This circuit is so arranged that the power fed into
the heater of the gain control thermistor is just sufficient at any ambient
temperature to give a controlled and constant value of tejnjjerature in the
vicinity of the gain control thermistor element.
Another interesting form of thermistor gain control utilizes a heater
type thermistor, with the heater driven by the output of the amplifier and
with the thermistor element in the input circuit, as shown in Fig. 22. In
this arrangement the feedback is accomplished by thermal, rather tiian
electrical coupling. A broadband carrier system, Type LI, is regulated
PROPERTIES AND USES OF THERMISTORS 205
with this type of thermistor. In this system a pilot frequency is suppHed,
and current of this frequency, selected by a network in the regulator, actu
ates the heater of the thermistor to give smooth, continuous gain control.
By utilizing a heater thermistor of diflferent characteristics, the circuit
and load of Fig. 22 may be given protection against overloads. In this
application the sensitivity and element resistance of the thermistor are
chosen so that the thermistor element forms a shunt of high resistance
value so as to have negligible effect on the amplifier for any normal value of
output. However, if the output power rises to an abnormal level, the
thermistor element becomes heated and reduced in resistance. This
shunts the input to the amplifier and thus limits the output. Choice of a
thermistor having a suitable time constant permits the onset of the limiting
eflfect to be delayed for any period from about a second to a few minutes.
LOAD
THERMISTORS"^ ^HEATER
HEATER nPE THERMISTOR
Fig. 22. — Thermal feedback circuit for gain control purposes. This arrangement has
also been used as a protective circuit for overloads.
Regulators and Limiters
A group of related applications for thermistors depends on their steady
state nonlinear voltampere characteristic. These are the voltage regulator,
the speech volume limiter, the compressor and the expandor. The com
pressor and expandor are devices for altering the range of signal amplitudes.
The compressor functions to reduce the range, while the expandor increases
it. In Fig. 23, Curve 1 is a typical thermistor static characteristic having
negative slope to the right of the voltage maximum. Curve 2 is the charac
teristic of an ohmic resistance R having an equal but positive slope. Curve
3 is the characteristic obtained if the thermistor and resistor are placed
in series. It has an extensive segment where the voltage is almost inde
pendent of the current. This is the condition for a voltage regulator or
limiter. If a larger value of resistance is used, as in Curve 4, its combination
with the thermistor in series results in Curve 5, the compressor. In these
uses the thermistor regulator is in shunt with the load resistance, so that
in the circuit diagram of Fig. 23,
E = Eo = Ei IRs. (23)
Here E is the voltage across the thermistor and resistor R, Eo is the output
206
BELL SYSTEM TECHNICAL JOURNAL
voltage, and Er , I and Rs are respectively the input voltage, current and
resistance.
If the thermistor and associated resistor are placed in series between the
generator and load resistance, an expandor is obtained, and
Eg = Ej — E.
(24)
As the resistance R in series with the thermistor is increased, the degree of
expansion is decreased and vice versa.
4 8 12 16
CURRENT IN MILLIAMPERES
20
Fig. 23. — Characteristics of a simple thermistor voltage regulator, limiter or com
pressor circuit.
The treatment thus far in this section assumes that change of operating
point occurs slowly enough to follow along the static curves. For a suffi
ciently rapid change of the operating point, the latter departs from the static
curve and tends to progress along an ohmic resistance line intersecting the
static curve. For sufficiently rapid fluctuations, control action may then
be derived from the resistance changes resulting from the r.m.s. power dis
sipated in the thermistor unit. In speech volume limiters, the thermistor
is designed for a speed of response that will produce limiting action for the
changes in volume which are syllabic in frequency or slower, and that will
not follow the more rapid speech fluctuations with resulting change in wave
PROPERTIES AND USES OF THERMISTORS 207
shape or nonlinear distortion. Speech volume limiters of this type can ac
commodate large volume changes without producing wave form distor
tion. i^.i^
Remote Control Swiches
The contactless switch and rheostat are natural extensions of the uses
just discussed. The thermistor is used as an element in the circuit which is
to be controlled, while the thermistor resistance value is in turn dependent
upon the energy dissipated directly or indirectly in it by the controlling cir
cuit. By taking advantage of the nonlinearity of the static voltampere
characteristic, it is possible to provide snap and lockin action in some
applications.
Manometer
Several interesting and useful applications such as vacuum gauges, gas
analyzers, flowmeters, thermal conductivity meters and liquid level gauges
of high sensitivity and low operating temperature are based upon the
physical principle that the dissipation constant of the thermistor depends
on the thermal conductivity of the medium in which it is immersed. As
shown in Fig. 10, a change in this constant shifts the position of the static
characteristic with respect to the axes. In these applications, the unde
sired response of the thermistor to the ambient temperature of the medium
can in many cases be eliminated or reduced by introducing a second thermis
tor of similar characteristics into the measuring circuit. The compensating
thermistor is subjected to the same ambient temperature, but is shielded
from theeflfect being measured, such as gas pressure or flow. Thetwo therm
istors can be connected into adjacent arms of a Wheatstone bridge which
is balanced when the test effect is zero and becomes unbalanced when the
effective thermal conductivity of the medium is increased. In gas flow
measurements, the minimum measurable velocity is limited, as in all '*hot
wire" devices, by the convection currents produced by the heated thermistor.
The vacuum gauge or manometer which is typical of these appHcations
will be described somewhat in detail. The sensitive element of the thermis
tor manometer is a small glass coated bead 0.02 inch in diameter, suspended
by two fine wire leads in a tubular bulb for attachment to the chamber whose
gas pressure is to be measured. The voltampere characteristics of a typical
laboratory model manometer are shown in Fig. 24 for air at several absolute
pressures from 10~® millimeters of mercury to atmospheric. The operating
point is in general to the right of the peak of these curves. Electrically
this element is connected into a unity ratio arm Wheatstone bridge with a
similar but evacuated thermistor in an adjacent arm as shown in the circuit
208
BELL SYSTEM TECHNICAL JOURNAL
schematic of Fig. 25. The air pressure caHbration for such a manometer is
also shown. The characteristic will be shifted when a gas is used having a
thermal conductivity different from that of air. Such a manometer has
been found to be best suited for the measurement of pressures from 10~^
to 10 millimeters of mercury. The lower pressure limit is set by practical
considerations such as meter sensitivity and the ability to maintain the zero
setting for reasonable periods of time in the presence of the variations of
supply voltage and ambient temperature. The upper pressure measure
ment limit is caused by the onset of saturation in the bridge unbalance
4~> ^
102
4 6 8I0'
2 468! 2 46 80'
CURRENT IN MILLIAMPERES
4 6 810^
Fig. 24. — Characteristics of a typical thermistor manometer tube, showing the effect
of gas pressure on the voltampere and resistancepower relations.
voltage versus pressure characteristic at high pressures. This is basically
because the mean free path of the gas molecules becomes short compared
with the distance between the thermistor bead and the inner surface of the
manometer bulb, so that the cooling effect becomes nearly independent of
the pressure.
The thermistor manometer is specially advantageous for use in gases
which may be decomposed thermally. For this type of use, the thermistor
element temperature can be limited to a rise of 30 centigrade degrees or
less above ambient temperature. For ordinary applications, however, a
temperature rise up to approximately 200 centigrade degrees in vacuum
PROPERTIES AND USES OF THERMISTORS
209
permits measurement over wider ranges of pressure. Special models have
also been made for use in corrosive gases. These expose only glass and plati
num alloy to the gas under test.
Timing Devices
The numerically greatest application for thermistors in the communication
field has been for time delay purposes. The physical basis for this use has
4 6 60
2 4 6 8102 2 4 6 B0
PRESSURE IN MM OF MERCURY
6 8 I
Fig. 25. — Operating circuit and calibration for a vacuum gauge utilizing the thermistor
of Figure 24.
been discussed in Part I for the case of a directly heated thermistor placed
in series with a voltage source and a load to delay the current rise after
circuit closure. This type of operation will be termed the power driven
time delay.
By the use of a thermistor suited to the circuit and operating conditions,
power driven time delays can be produced from a few milliseconds to the
order of a few minutes. Thermistors of this sort have the advantage of
small size, light weight, ruggedness, indefinitely long life and absence of
contacts, moving parts, or pneumatic orifices which require maintenance
210 BELL SYSTEM TECHNICAL fOURNAL
care. Power driven time delay thermistors tre best fitted for applications
where close limits on the time interval arc not required. In some com
munications uses it is satisfactory to permit a six to one ratio between maxi
mum and minimum times as a result of the simultaneous variation from
nominal values of all the following factors which affect the delay : operating
voltage ± 5 per cent; ambient temperature 20 degrees centigrade to 40
degrees centigrade; operating current of the relay ± 25 per cent; relay
resistance zt 5 per cent; and thermistor variations such as occur from
unit to unit of the same type.
After a timing operation a power driven time delay thermistor should bs
allowed time to cool before a second operation. If this is not done, the
second timing interval will be shorter than the first. The cooling period
depends on particular circuit conditions and details of thermistor design,
but generally is several times the working time delay. In telephone relay
circuits requiring a timing operation soon after previous use, the thermistor
usually is connected so that it is short circuited by the relay contacts at the
close of the working time delay interval. This pe: nits the thermistor to
cool during the period when the relay is locked up. If this period is suffi
ciently long, the thermistor is available for use as soon as the relay drops
out. Time delay thermistors have been operated more than half a million
times on life test with no significant change in their timing action.
To avoid the limitations of wide timing interval limits and extended cool
ing period between operations usually associated with the power driven time
delay thermistor, a cooling time delay method of operation has been used.
In this arrangement, two relays or the equivalent are employed and the
thermistor is heated to a low resistancevalue by passing a relatively large
current through it for an interval short compared with the desired time
interval. The current then is reduced automatically to a lower value and
the thermistor cools until its resistance increases enough to reduce the cur
rent further and trip the working relay. This part of the operating cycle
accounts for the greater part of the desired time interval. With this ar
rangement, the thermistor is available for reuse immediately after a com
pleted timing interval, or, as a matter of fact, after any part of it. By proper
choice of operating currents and circuit values, wide variations of voltage
and ambient temperature may occur with relatively little effect upon the
time interval. The principal variable left is the cooling time of the thermis
tor itself. This is fixed in a given thermistor unit, but may vary from unit
to unit, depending upon dissipation constant and thermal capacity, as
pointed out above.
In addition to their use as definite time delay devices, thermistors have
been used in several related applications. Surges can be prevented from
PROPERTIES AND USES OF THERMISTORS 211
operating relays or disturbing sensitive apparatus by introducing a ther
mistor in series with the circuit component which is to be protected. In
case of a surge, the high initial resistance of the thermistor holds the surge
current to a low value provided that the surge does not persist long enough
to overcome the thermal inertia of the thermistor. The normal operating
voltage, on the other hand, is applied long enough to lower the thermistor
resistance to a negligible value, so that a normal operating current will flow
after a short interval. In this way, the thermistor enables the circuit to
distinguish between an undesired signal of short duration and a desired
signal of longer duration even though the undesired impulse is several timss
higher in voltage than the signal.
Oscillators, Modulators and Amplifiers
A group of applications already explored in the laboratory but not put into
engineering use includes oscillators, modulators and amplifiers for the low
and audiofrequercy range. If a thermistor is biased at a point on the
negative slope portion of the steadystate voltampere characteristic, and
if a small alternating voltage is then superposed on the direct voltage, a
small alternating current will flow. If the thermistor has a small time con
stant, T, and if the applied frequency is low enough, the alternating volt
ampere characteristic will follow the steadystate curve and dV/dl will be
negative. As the frequency of the applied ac voltage is increased, the
value of the negative resistance decreases. At some critical frequency,
/c , the resistance is zero and the current is 90 degrees out of phase with
the voltage. In the neighborhood of /c , the thermistor acts like an induc
tance whose value is of the order of a henry. As the frequency is increased
beyor.d/c , the resistance is positive and increases steadily until it approach
es the dc value when the current and voltage are in phase. The critical
frequency is given approximately by
/c = l/2r.
If T can be made as small as 5 X 10~ seconds, fc is equal to 10,000
cycles per second and the thermistor would have an approximately
constant negative resistance up to half this frequency. Point contact
thermistors having such critical frequencies or even higher have been
made in a number of laboratories. However, none of them have been
made with sufficient reproducibility and constancy to be useful to the
engineer. It has been shown both theoretically and experimentally that
any negative resistance device can be used as an oscillator, a modulator, or
an amplifier. With further development, it seems probable that thermistors
will be used in these fields.
212 BELL SYSTEM TECHNICAL JOURNAL
Summary
The general principles of thermistor operation and examples of specific
uses have been given to facilitate a better understanding of them, with the
feeling that such an understanding will be the basis for increased use of this
new circuit and control element in technology.
References
1. Zur Elektrischen Leitfahigkeit von Kupferoxydul, W. P. Juse and B. VV. K5rtschatow.
Physikalische Zeitschrift Der Sovvjetunion, Volume 2, 1932, pages 45367.
2. Semiconductors and Metals (book), A. H. Wilson. The University Press, Cam
bridge, England, 1939.
3. The Modern Theory of Solids (book), Frederick Seitz. McGrawHill Book Company,
New York, N. Y., 1940.
4. Electronic Processes in Ionic Crystals (book), N. F. Mott and R. W. Gurney. The
Clarendon Press, Oxford, England, 1940.
5. Die Elektronenleitfahigkeit von Festen Oxyden Verschiedener Valenzstufen, M. Le
Blanc and H. Sachse. Physikalische Zeitschrift, Volume 32, 1931, pages 8879.
6. Uber die Elektrizitatsleitung Anorganischer Stofle mit Elektronenleitfahigkeit, Wil
fried Meyer. Zeitschrift Fur Physik, Volume 85, 1933, pages 27893.
7. Thermal Agitation of Electricity in Conductors, J. B. Johnson. Physical Review,
Volume 32, July 1928, pages 97113.
8. Spontaneous Resistance Fluctuations in Carbon Microphones and Other Granular
Resistances, C. J. Christensen and G. L. Pearson. The Bell System Technical
Journal, Volume 15, April 1936, pages 197223.
9. Automatic Temperature Control for Aircraft, R. A. Gund. AIEE Transactions,
Volume 64, 1945, October section, pages 73034.
10. The Bridge Stabilized Oscillator, L. A. Meacham. Proc. IRE, Volume 26, October
1938, pages 127894.
11. Frequency Stabilized Oscillator, R. L. Shepherd and R. O. Wise. Proc. IRE, Vol
ume 31, June 1943, pages 25668.
12. A PilotChannel Regulator for the K1 Carrier System, J. H. Bollman. Bell Labora
tories Record, Volume 20, No. 10, June 1942, pages 25862.
13. Thermistors, J. E. Tweeddale. Western Electric Oscillator, December 1945, pages
35, 347.
14. Thermistor Technics, J. C. Johnson. Electronic Industries, Volume 4, August 1945,
pages 747.
15. Volume Limiter for LeasedLine Service, J. A. Weiler. Bell Laboratories Record,
Volume 23, No. 3, March 1945, pages 725.
Abstracts of Technical Articles by Bell System Authors
Capacitors — Their Use in Electronic Circuits} M. Brotherton. This
book tells how to choose and use capacitors for electronic circuits. It ex
plains the basic factors which control the characteristics of capacitors and
determine their proper operation. It helps to provide that broad under
std.nding of the capacitor problem which is indispensable to the efficient
design of circuits. It tells the circuit designer what he must vmderstand
and consider in transforming capacitance from a circuit symbol into a practi
cal item of apparatus capable of meeting the growing severity of today's
operation requirements.
Mica Capacitors for Carrier Telephone Systems.^ A. J. Christopher
AND J. A. Kater. Silvered mica capacitors, because of their inherently
high capacitance stability with temperature changes and with age, now are
used widely in oscillators, networks, and other frequency determining
circuits in the Bell Telephone System. Their use in place of the previous
dry stack type, consisting of alternate layers of mica and foil clamped
under high pressures, has made possible considerable manufacturing econ
omies in addition to improving the transmission performance of carrier
telephone circuits. These economies are the result of their relatively simple
unit construction and the ease of adjustment to the very close capacitance
tolerance required.
Visible Speech Translators with External Phosphors.^ Homer Dudley
AND Otto 0. Gruenz, Jr. This paper describes some experimental ap
paratus built to give a passing display of visible speech patterns. These
patterns show the analysis of speech on an intensityfrequencytime basis
and move past the reader like a printed line. The apparatus has been
called a translator as it converts speech intended for aural perception into a
form suitable for visual prception. The phosphor employed is not in a
cathoderay tube but in the open on a belt or drum.
The Pitch, Loudness and Quality of Musical Tones {A demonstration
lecture introducing the new Tone Synthesizer)} Harvey Fletcher. Re
lations are given in this paper which show how the pitch of a musical tone
» Published by D. Van Nostrand Company, Inc., New York, N. Y., 1946.
' Elec. Engg., Transactions Section, October 1946.
^Jour. Acous. Soc. Anier., July 1946.
* Amer. Jour, of Physics, July August 1946.
213
214 BELL SYSTEM TECHNICAL JOURNAL
depends upon the frequency, the intensity and the overtone structure of the
sound wave transmitting the tone. Similar relations are also given which
show how the loudness and the quality depend upon these same three
physical characteristics of the sound wave. These relationships were de
monstrated by using the new Tone Synthesizer. By means of this in
strument one is able to imitate the quality, pitch and intensity of any musi
cal tone and also to produce many combinations which are not now used in
music.
The Sound Spectrograph.^ W. Koenig, H. K. Dunx, and L. Y. Lacy.
The sound spectrograph is a wave analyzer which produces a permanent
visual record showing the distribution of energy in both frequency and time.
This paper describes the operation of this device, and shows the mechanical
arrangements and the electrical circuits in a particular model. Some of
the problems encountered in this type of analysis are discussed, particularly
those arising from the necessity for handling and portraying a wide range of
component levels in a complex wave such as speech. Spectrograms are
shown for a wide variety of sounds, including voice sounds, animal and bird
sounds, music, frequency modulations, and miscellaneous familiar sounds.
Geometrical Characterizations of Some Families of Dynamical Trajectories}
L. A. MacColl. a broad problem in differential geometry is that of
characterizing, by a set of geometrical properties, the family of curves which
is defined by a given system of differential equations, of a more or less
special form. The problem has been studied especially by Kasner and his
students, and characterizations have been obtained for various families of
curves which are of geometrical or physical importance. However, the
interesting problem of characterizing the family of trajectories of an electri
fied particle moving in a static magnetic field does not seem to have been
considered heretofore. The present paper gives the principal results of a
study of this problem.
Visible Speech CathodeRay Translator."^ R. R. Riesz and L. Schott. A
system has been developed whereby speech analysis patterns are made
continuously visible on the moving luminescent screen of a special cathode
ray tube. The screen is a cylindrical band that rotates with the tube about
a vertical axis. The electron beam always excites the screen in the same
vertical plane. Because of the persistence of the screen phosphor and the
rotation of the tube, the impressed patterns are spread out along a horizon
^ Jour. Acous. Soc. Amer., July 1946.
^ Amer. Math. Soc. Transactions, July 1946.
' Jour. Acous. Soc. Amer., July 1946.
ABSTRACTS OF TECHNICAL ARTICLES 215
tal time axis so that speech over an interval of a second or more is always
visible. The upper portion of the screen portrays a spectrum analysis and
the lower portion a pitch analysis of the speech sounds. The frequency
band up to 3500 cycles is divided into 12 contiguous subbands by filters.
The average speech energy in the subbands is scanned and made to control
the excitation of the screen by the electron beam which is swept synchro
nously across the screen in the vertical direction. A pitch analyzer pro
duces a dc. voltage proportional to the instantaneous fundamental fre
quency of the speech and this controls the width of a band of luminescence
that the electron beam produces in the lower part of the screen. The
translator had been used in a training program to study the readability
of visible speech patterns.
Derivatives of Composite Functions.^ John Riordan. The object of
this note is to show the relation of the Y polynomials of E. T. Bell, first to
the formula of DiBruno for the wth derivative of a function of a function,
then to the more general case of a function of many functions. The sub
ject belongs to the algebra of analysis in the sense of Menger; all that is
asked is the relation of the derivative of the composite function to the
derivatives of its component functions when they exist and no questions of
analysis are examined.
The Portrayal of Visible Speech.^ J. C. Steinberg and N. R. French.
This paper discusses the objectives and requirements in the protrayal of
visible patterns of speech from the viewpoint of their effects on the legibility
of the patterns. The portrayal involves an intensityfrequencytime analy
sis of speech and the display of the results of the analysis to the eye.
Procedures for accomplishing this are discussed in relation to information
on the reading of print and on the characteristics of speech and its inter
pretation by the ear. Also methods of evaluating the legibility of the
visible patterns are described.
Short Survey of Japanese Radar — 1}° Roger I. Wilkinson. The
result of a study made immediately following the fall of Japan and recently
made available for public information, this twopart report is designed to
present a quick overall evaluation of Japanese radar, its history and de
velopment. As the Japanese army and navy developed their radar equip
ment independently of each other, Part I of this article concentrates on the
army's contributions.
*Amer. Math. Soc. Bulletin, August 1946.
^ Jour. Aeons. Soc. Amer., July 1946.
^"Elec. Engg., Aug.Sept. 1946.
216 BELL SYSTEM TECHNICAL JOURNAL
A Variation on the Gain Fcrmula for Feedback Amlifters for a Certain
DrivingImpedance Configuration.^^ T. W. Winternitz. An expression
for the gain of a feedback amplifier, in which the source impedance is the
only significant impedance across which the feedback voltage is developed,
is derived. As examples of the use of this expression, it is then applied to
three common circuits in order to obtain their response to a Heaviside
unit stepvoltage input.
" Proc. LR.E., September 1946.
Contributors to This Issue
Joseph A. Becker,. A. B., Cornell University 1918; PhD., Cornell Univer
sity, 1922. National Research Fellow, California Institute of Technology,
192224; Asst. Prof, of Physics, Stanford University, 1924. Engineering
Dept., Western Electric Company, 19241925; Bell Telephone Laboratores,
1925. Mr. Becker has worked in the fields of XRays, magnetism, thermio
nic emission and adsorption, particularly in oxide coated filaments, the
properties of semiconductors, as applied in varistors and thermistors.
W. R. Bennett, B. S., Oregon State College, 1925; A.M., Columbia
University, 1928. Bell Telephone Laboratories, 1925. Mr. Bennett
has been active in the design and testing of multichannel communication
systems, particularly with regard to modulation processes and the effects
of nonlinear distortion. As a member of the Transmission Research De
partment, he is now engaged in the study of pulse modulation techniques
for sending telephone channels by microwave radio relay.
C. B. Green, Ohio State University, B.A. 1927; M.A. in Physics, 1928.
Additional graduate work at Columbia University. Bell Telephone Lab
oratories, 1928. For ten years Mr. Green was concerned with trans
mission development for telephotography and television systems and with
the design of vacuum tubes. Since 1938 he has been engaged in the developl
ment and application of thermistors.
J. P. Kinzer, M. E., Stevens Institute of Technology, 1925. B.C.E.,
Brooklyn Polytechnic Institute, 1933. Bell Telephone Laboratories, 1925.
Mr. Kinzer's work has been in the development of carrier telephone repeat
ers; during the war his attention was directed to investigation of the mathe
matical problems involved in cavity resonators.
W. P. Mason, B.S. in E.E., Univ. of Kansas, 1921; M.A., Ph.D., Co
lumbia, 1928. Bell Telephone Laboratories, 1921. Dr. Mason has been
engaged principally in investigating the properties and applications of
piezoelectric crystals and in the study of ultrasonics.
R. S. Ohl, B. S. in ElectroChemical Engineering, Pennsylvania State
College, 1918; U. S. Army, 1918 (2nd Lieutenant, Signal Corps); Vacuum
tube development, Westinghouse Lamp Company, 191921; Instructor in
217
218 BELL SYSTEM TECHNICAL JOURNAL
Physics, University of Colorado, 19211922. Department of Development
and Research, American Telephone and Telegraph Company, 192227;
Bell Telephone Laboratories, 192 7. Mr. Ohl has been engaged in various
exploratory phases of radio research, the results of which have led to nu
merous patents. For the past ten or more years he has been working on
some of the problems encountered in the use of millimeter radio waves.
G. L. Pearson, A. B., Willamette University, 1926; M. A. in Physics,
Stanford University, 1929. Bell Telephone Laboratories, 1929. Mr.
Pearson is in the Physical Research Department where he has been engaged
in the study of noise in electric circuits and the properties of electronic semi
conductors.
J. H. ScAFF, B.S.E. in Chemical Engineering, University of Michigan,
1929. Bell Telephone Laboratories, 1929. Mr. Scaff's early work in the
Laboratories was concerned with metallurgical investigations of impurities
in metals with particular reference to soft magnetic materials. During the
war he was project engineer for the development of silicon and germanium
crystal rectifiers for radar applications. At the present time, he is re
sponsible for metallurgical work on varistor and magnetic materials.
I. G. Wilson, B.S. and M.E., University of Kentucky, 1921. Western
Electric Co., EngineeringDepartment, 192125. Bell Telephone Labora
tories, 1925. Mr. Wilson has been engaged in the development of am
plifiers for broadband systems. During the war he was project engineer in
charge of the design of resonant cavities for radar testing.
VOLUME XXVI APRIL, 1947 no. 2
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
Publk Ubnn
Radar Antennas H. T. Friis and W. D. Lewis 219
Probability Functions for the Modulus and Angle of the
Normal Complex Variate Ray S. Hoyt 318
Spectrum Analysis of Pulse Modulated Waves
/. C. Lozier 360
Abstracts of Technical Articles by Bell System Authors. . 388
Contributors to This Issue 394
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
NEW YORK
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The Bell System Technical Journal
Vol. XXVI April, 1947 No. 2
Radar Antennas
By H. T. FRIIS and W. D. LEWIS
Table of Contents
Introduction 220
Part I — Electrical Principles 224
1 . General 224
2. Transmission Principles 226
2 . 1 Gain and Effective Area of an Antenna 226
Definition of Gain 226
Definition of Effective Area 226
2.2 Relationship between Gain and Effective Area 227
2.3 The Ratio G/A for a Small Current Element 227
2.4 The General Transmission Formula 230
2.5 The Reradiation Formula 230
2.6 The Plane, Linearly Polarized Electromagnetic Wave 231
3. Wave Front Analysis 232
3 . 1 The Huygens Source 233
3.2 Gain and Effective Area of an Ideal Antenna 235
i.i Gain and Effective Area of an Antenna with Aperture in a Plane and
with Arbitrary Phase and Amplitude 236
3.4 The Significance of the Pattern of a Radar Antenna 237
3.5 Pattern in Terms of Antenna Wave Front 238
3.6 Pattern of an Ideal Rectangular Antenna 239
3.7 Effect on Pattern of Amplitude Taper 240
3.8 Effect on Pattern of Linear Phase Variation 241
3 . 9 Effect on Pattern of Scjuare Law Phase Variation 242
3. 10 Effect on Pattern of Cubic Phase Variation 244
3.11 Two General Methods 245
3. 12 Arrays 246
3. 13 Limitations to Wave Front Theory 246
4. Application of General Principles 247
Part II — Methods of Antenna Construction 247
5. General 247
6. Classification of Methods 248
7. Basic Design Formulation 250
7 . 1 Dimensions of the Aperture 250
7 . 2 Amplitude Distribution 251
7 . 3 Phase Control 251
8. Parabolic Antennas 251
8.1 Control of Phase 251
8.2 Control of Amplitude 253
8 . 3 Choice of Configuration 254
8.4 Feeds for Paraboloids 258
8.5 Parabolic CyUnders between Parallel Plates 260
8.6 Line Sources for Parabolic Cylinders 262
8.7 Tolerances in Parabolic Antennas 264
9. Metal Plate Lenses 266
9. 1 Lens Antenna Configurations 269
9.2 Tolerances in Metal Plate Lenses 269
9.3 Advantages of Metal Plate Lenses 270
10. Cosecant Antennas 270
10. 1 Cosecant Antennas based on the Paraboloid 271
10.2 Cylindrical Cosecant Antennas 274
219
220 BELL S YS TEM TECH NIC A L JO URN A L
1 1 . Lobing 274
11.1 Lobe Switching 275
11.2 Conical Lolling 276
12. Rapid Scanning 276
12.1 Mechanical Scanning 277
12.2 .\rray Scanning 278
12.3 Optical Scanning 282
Part III — Military Radar Antennas Developed by the Bell Laboratories 284
13. General .' 284
14. Naval Shipborne Radar Antennas 286
14. 1 The SE Antenna 286
14.2 The SL .\ntenna 286
14.3 The SJ Submarine Radar Antenna 291
14.4 The Modified SJ/Mark 27 Radar Antenna 294
14.5 The SH and Mark 16 Radar Antennas 294
14.6 Antennas for Early Fire Control Radars 297
14. 7 \ Shipborne .\ntiAircraft Eire Control Antenna 298
14.8 The Polyrod Eire Control Antenna '. . 300
14.9 The Rocking Horse Eire Control Antenna 301
14. 10 The Mark 19 Radar Antenna 302
14. 1 1 The Mark 28 Radar Antenna 305
14. 12 A 3 cm Anti.\ircraft Radar Antenna 307
15. Land Based Radar Antennas 307
15. 1 The SCR545 Radar " Search" and "Track" Antennas 307
15.2 The AN/TPSIA Portable Search Antenna 309
16. Airborne Radar Antennas 312
16. 1 The AN/APS4 Antenna 312
16.2 The SCR520, SCR717 and SCR720 .Antennas 313
16.3 The AN/APQ7 Radar Bombsight Antenna 315
Introduction
"O ADAR proved to be one of the most important technical achieve
'^ ments of World War II. It has many sources, some as far back
as the nineteenth century, yet its rapid wartime growth was the result
of military necessity. This development will continue, for radar has
increasing applications in a peacetime world.
In this paper we will discuss an indispensable part of radar — the
antenna. In a radar system the antenna function is twofold. It
both projects into space each transmitted radar pulse, and collects from
space each received reflected signal. Usually but not always a single
antenna performs both functions.
The effectiveness of a radar is influenced decisively by the nature and
quality of its antenna. The greatest range at which the radar can de
tect a target, the accuracy with which the direction to the target can be
determined and the degree with which the target can be discriminated
from its background or other targets all depend to a large e.xtent on
electrical properties of the antenna. The angular sector which the
antenna can mechanically or electrically scan is the sector from which
the radar can provide information. The scanning rate determines the
frequency with which a tactical or navigational situation can be ex
amined.
RADAR A NTENNA S 221
Radar antennas are as numerous in kind as radars. The unique
character and particular functions of a radar are often most clearly
evident in the design of its antenna. Antennas must be designed for
viewing planes from the ground, the ground from planes and planes
from other planes. They must see ships from the shore, from the air,
from other ships, and from submarines. In modern warfare any
tactical situation may require one or several radars and each radar must
have one or more antennas.
Radar waves are almost exclusively in the centimeter or microwave
region, yet even the basic microwave techniques are relatively new to
the radio art. Radar demanded antenna gains and directivities far
greater than those previously employed. Special military situations
required antennas with beam shapes and scanning characteristics never
imagined by communication engineers.
It is natural that war should have turned our efforts so strongly in
the direction of radar. But that these efforts were so richly and quickly
rewarded was due in large part to the firm technical foundations that
had been laid in the period immediately preceeding the war. When,
for the common good, all privately held technical information was
poured into one pool, all ingredients of radar, and of radar antennas in
particular, were found to be present.
A significant contribution of the Bell System to this fund of technical
knowledge was its familiarity with microwave techniques. Though
Hertz himself had performed radio experiments in the present micro
wave region, continuous wave techniques remained for decades at longer
wavelengths. However, because of its interest in new communication
channels and broader bands the Bell System has throughout the past
thirty years vigorously pushed continuous wave techniques toward the
direction of shorter waves. By the middle nineteenthirties members
of the Radio Research Department of the Bell Laboratories were work
ing within the centimeter region.
Several aspects of this research and development appear now as
particularly important. In the first place it is obvious that knowledge
of how to generate and transmit microwaves is an essential factor in
radar. Many lower frequency oscillator and transmission line tech
niques are inapplicable in the microwave region. The Bell Laboratories
has been constantly concerned with the development of generators
which would work at higher and higher frequencies. Its broad famil
iarity with coaxial cable problems and in particular its pioneering work
with waveguides provided the answers to many radar antenna problems.
Another telling factor was the emphasis placed upon measurement.
Only through measurements can the planners and designers of equip
222
BELL SYSTEM TECHNICAL JOURNAL
ment hope to evaluate performance, to chose between alternatives or to
see the directions of improvement. Measuring technicjues employing
double detection receivers and intermediate frequency amplifiers had
long been in use at the Holmdel Radio Laboratory. By employing
these techniques radar engineers were able to make more sensitive and
accurate measurements than would have been possible with single de
tection.
Antennas are as old as radio. Radar antennas though different in
form are identical in principle with those used by Hertz and Marconi.
Consequently experience with communication antennas provided a
valuable background for radar antenna design. As an example of the
importance of this background it can be recalled that a series of experi
Fig. 1 — An Electromagnetic Horn.
ments with short wave antennas for Transatlantic radio telephone
service had culminated in 1936 in a scanning array of rhombic antennas.
The essential principles of this array were later applied to shipborne
fire control antenna which was remarkable and valuable because of the
early date at which it incorporated modern rapid scanning features.
In addition to the antenna arts which arose directly out of communi
cation problems at lower frequencies some research specifically on micro
wave antennas was under way before the war. Earl\ workers in wave
guides noticed that an open ended waveguide will radiate directly into
space. It is not suri)rising therefore that these workers developed the
electromagnetic horn, which is essentially a waveguide tapered out to
an aperture (Fig. 1).
One of the first used and simplest radio antennas is the dipole (Fig.
MDAR ANTENNAS
111
2). Current oscillating in the dipole generates electromagnetic waves
which travel out with the velocity of light. A single dipole is fairly
nondirective and consecjuently produces a relatively weak, field at
a distance. When the wavelength is short the field of a dipole in a
i^
o o
Fig. 2 — A Microwave Dipole.
Fig. 3 — x\ Dipole Fed Paraboloid.
chosen direction can be increased many times by introducing a re
flector which directs or 'focusses' the energy.
In communication antennas the focussing reflector is most com
monly a reflecting wire array. Even at an early date in radar the wave
length was so short that 'optical' reflectors could be used. These were
224 BELL S YSTEM TECH NIC A L JOURNA L
sometimes paraboloids similar to those used in searchlights (Fig. 3).
Sometimes they were parabolic cylinders as in the Mark III, an early
shipl)orne fire control radar developed at the Whippany Radio Labora
tory.
From these relatively simple roots, the communication antenna, the
electromagnetic horn and the optical reflector, radar antennas were
developed tremendously during the war. That this development in
the Bell Laboratories was so well able to meet demands placed on it was
due in large part to the solid foundation of experience possessed by the
Research and Development groups of the Laboratories. Free inter
change of individuals and information between the Laboratories and
other groups, both in the United States and Great Britain, also con
tributed greatly to the success of radar antenna development.
Because of its accelerated wartime expansion the present radar an
tenna field is immense. It is still growing. It would be impossible
for any single individual or group to master all details of this field, yet
its broad outline can be grasped without "difficulty.
The purpose of this paper is twofold, both to provide a general dis
cussion of radar antennas and to summarize the results of radar antenna
research and development at the Bell Laboratories. Part I is a dis
cussion of the basic electrical principles which concern radar antennas.
In Part II we will outline the most common methods of radar antenna
construction. Practical military antennas developed by the Bell
Laboratories will be described in Part III.
The reader who is interested in general familiarity with the over all re
sult rather than with technical features of design may proceed directly
from this part to Part III.
PART I
ELECTRICAL PRINCIPLES
1. General
Radar antenna design depends basically on the same broad principles
which underlie any other engineering design. The radar antenna designer
can afford to neglect no aspect of his problem which has a bearing on the
final product. Mechanical, chemical, and manufacturing considerations
are among those which must be taken into account.
It is the electrical character of the antenna, however, which is connected
most directly with the radar performance. In addition it is through atten
tion to the electrical design problems that the greatest number of novel
antennas have been introduced and it is from the electrical viewpoint that
the new techniques can best be understood.
An antenna is an electromagnetic device and as such can be understood
RADAR ANTENNAS 225
through the appUcation of electromagnetic theory. Maxwell's equations
provide a general and accurate foundation for antenna theory. They are
the governing authority to which the antenna designer may refer directly
when problems of a fundamental or bafHing nature must be solved.
It is usually impracticable to obtain theoretically exact and simple solu
tions to useful antenna problems by applying Maxwell's Equations directly.
We can, however, use them to derive simpler useful theories. These
theories provide us with powerful analytical tools.
Lumped circuit theory is a tool of this sort which is of immense practical
importance to electrical and radio engineers. As the frequency becomes
higher the approximations on which lumped circuit theory is based become
inaccurate and engineers find that they must consider distributed in
ductances and capacitances. The realm of transmission line theory has
been invaded.
Transmission line theory is of the utmost importance in radar antenna
design. In the first place the microwave energy must be brought to the
antenna terminals over a transmission line. This feed line is usually a
coaxial or a waveguide. It must not break down under the voltage which
accompanies a transmitted pulse. It must be as nearly lossless and reflec
tionless as possible and it must be matched properly to the antenna terminals.
The importance of a good understanding of transmission line theory does
not end at the antenna terminals. In any antenna the energy to be trans
mitted must be distributed in the antenna structure in such a way that the
desired radiation characteristics will be obtained. This may be done with
transmission lines, in which case the importance of transmission line theory
is obvious. It may be done by 'optical' methods. If so, certain trans
mission line concepts and methods will still be useful.
While it is true that transmission line theory is important it is not nec
essary to give a treatment of it in this paper. Adequate theoretical dis
cussions can be found elsewhere in several sources.^ It is enough at this
point to indicate the need for a practical understanding of transmission line
principles, a need which will be particularly evident in Part II, Methods
of Antenna Construction.
We may, if we like, think of the whole radar transmission problem in
terms of transmission line theory. The antenna then appears as a trans
former between the feed line and transmission modes in free space. We
cannot, however, apply this picture to details with much effectiveness unless
we have some understanding of radiation.
In the sections to follow we shall deal with some theoretical aspects of
radiation. We shall begin with a discussion of fundamental transmission
1 See, for example, S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Co.,
Inc., 1943, in particular. Chapters VII and VIII, or F. E. Terman, Radio Engineer's Hand
book:, McGrawHill Book Co., Inc., 1943, Section 3.
226 BELL SYSTEM TECHNICAL JOURNAL
principles. This discussion is applicable to all antennas regardless of how
they are made or used. When applied to radar antennas it deals chiefly
with those properties of the antenna which affect the radar range.
Almost all microwave radar antennas are large when measured in wave
lengths. When used as transmitting antennas they produce desired radia
tion characteristics by distributing the transmitted energy over an area or
Svave front'. The relationships between the phase and amplitude of elec
trical intensity in this wave front and the radiation characteristics of the
antenna are predicted by 'ivave front analysis. Wave front analysis is
essentially the optical theory of diffraction. Although approximate it
applies excellently to the majority of radar antenna radiation problems.
We shall discuss wave front analysis in Section 3.
2. Transmission Principles
2.1 Gain and Effective Area of an Antenna
An extremely important property of any radar antenna is its ability to
project a signal to a distant target. The gain of the antenna is a number
which provides a quantitative measure of this ability. Another important
property of a radar antenna is its ability to collect reflected power which
is returning from a distant target. The efectiie area of the antenna is a
quantitative measure of this ability. In this section these two quantities
will be defined, and a simple relation between them will be derived. Their
importance to radar range will be established.
Definition of Gain. When power is fed into the terminals of an antenna
some of it will be lost in heat and some will be radiated. The gain G of
the antenna can be defined as the ratio
G = P/Po (1)
where P is the power flow per unit area in the plane linearly polarized elec
tromagnetic wave which the antenna causes in a distant region usually in
the direction of maximum radiation and Po is the power flow per unit area
which would have been produced if all the power fed into the terminals
had been radiated equally in all directions in space.
Definition of Effective Area. When a plane linearly polarized electromag
netic wave is incident on the receiving antenna, received power Pr will be
available at the terminals of the antenna. The effective area of the antenna
is defined, by the equation
A = Pn/P' (2)
where P' is the j^ower per unit area in the incident wave. In other words
the received power is equal lo ihc j)ower flow through an area that is equal
to the effective area of the antenna.
RA DA RAN TENNA S 227
2.2 Relationship behveen Gain and Efeclive Area
Figure 4 shows a radio circuit in free space made up of a transmitting
antenna T and a receiving antenna R. If the transmitted power 7^r had
TRANSMITTING
ANTENNA
Fig. 4 — Radio Circuit in Free Space.
been radiated equally in all directions, the power flow per unit area at the
receiving antenna would be
47r(/2
Definition (1) gives, therefore, for the power flow per unit area at the
receiving antenna
P = p,Gr = ^" (4)
and definition (2) gives for the received power
^« = ''■'' = '^ (')
From the law of reciprocity it follows that the same power is transferred if
the transmitting and receiving roles are reversed. By (5) it is thus evident
that
KJTAji = QtrAt
or
Gt/At = Gr/Ar (6)
Equation (6) shows that the ratio of the gain and effective area has the
same constant value for all antennas at a given frequency. It is necessary,
therefore, to calculate this ratio only for a simple and well known antenna
such as a small dipole or uniform current element.
2.3 The Ratio G/A for a Small Current Element
In Fig. 5 are given formulas' in M.K.S. units for the free space radiation
from a small current element with no heat loss. We have assumed that
2 See S. A. Schelkunoff, Electromagnetic Waves, D. Van Nostrand Co., Inc., 1943, p. 133
228
BELL SYSTEM TECHNICAL JOURNAL
X
CURRENT ELEMENT
(LENGTH i METERS)
(i<< A)
MAGNETIC INTENSITY^H,
le ^~T^
— e ^ 5IN9
L KV
AMPERES
METER
ELECTRIC INTENSITY = Eg = 120TrH<(, ^ ^^^
I I fr^l'^ p WATTS
POWER FLOW =P = H4,Ee = 30^^— J SIN'^e y^^^^^ Z
o ■ r. .^ fr^l^ WATTS
P 15 MAXIMUM FOR 6=90. ce., P^^ =30Tr^— J jj^^r^z
U)
(i)
POWER FLOW ACROSS SPHERE OF RADIUS r OR
r^ n 61 ^
TOTAL RADIATION = W =/ P2TTr SINS rde = 80Tt2 I yJ WATTS (s)
,2
(6)
(7)
RADIATION RESISTANCE = R r^q " T? " ^°'" TJ ^^^^
BY (4) AND (5) : P
MAX anr^
W
WATTS
METERS
Fig. 5 — Free Space Radiation from a Small Current Element with Uniform Current
I Amperes over its Entire Length.
this element is centered at the origin of a rectangular coordinate system
and that it lies along the Z axis. At a large distance r from the element
RADAR ANTENNAS 229
the maximum power flow per unit area occurs in a direction normal to it and
is given by
_ 3W w^atts ,,_.
SttH meter^
where T'F is the total radiated power. If W had been radiated equally
in all directions the power flow per unit area would be
p ^ W_ watts .gv
47rr2 meters^
It follows that the gain of the small current element is
p
Gdiople = ^— = 15 (9)
The effective area of the dipole will now be calculated. When it is used
to receive a plane linearly polarized electromagnetic wave, the available
output power is equal to the induced voltage squared divided by four times
the radiation resistance. Thus
Pn = ^ Watts (10)
4i?rad
where E is the effective value of the electric field of the wave, i is the length
of the current element and i?rad is the radiation resistance of the current
element. From Fig. 5 we see that i?rad — , ohms. Since the power
A"
flow per unit area is equal to the electric field squared divided by the im
pedance of free space, in other words Po — tt— we have
u. 1 ZOir
P ^X^
^dipoie = ^ = ^ meter" (11)
We combine formulas (9) and (11) to find that
6^dipole _ 4t
■^dipole A
Since, as proved in 2.2 this ratio is the same for all antennas, it follows that
for any antenna
^=^ (12)
230 BELL S YSTEM TECH NIC A L JOURNA L
2.4 The General Transmission Formula
Transmission loss between transmitter and receiver through the radio
circuit shown in Fig. 4 was given by ecjuation (5). By substituting the
relation (12) into (5) we can obtain the simple free space transmission
formula:
Ph = Pt 4^" watts (13)
Although this formula applies to free space only it is believed to be as useful
in radio engineering as Ohm's law is in circuit engineering.
2.5 The Reradialion Formula
One further relation, the radar reflection formula is of particular interest.
Consider the situation illustrated in Fig. 6. Let Pt be the power radiated
REFLECTING OBJECT
(As= PROJECTED AREA IN
DIRECTION OF RADAR)
RADAR '
At.Gt
TRANSMITTER
h
Ar,Gr
Fig. 6 — Radar with Separate Receiving and Transmitting Antennas.
from an antenna with effective area A t, As the area of a reflecting object at
distance d from the antenna and Ph the power received by an antenna of
effective area ^k . By equation (13) the power striking As is — — — — . If
this power were reradiated equally in all directions the reflected power flow
at the receiving antenna would be — — 3—— but since the average reradiation
is larger toward the receiving antenna, the power flow per unit area there is
usually K J ,J^f where A' > 1. It follows from (2) that
4Trd*\^
r> T^r PtAtArAs (..s.
Formula (14) shows clearly why the use of large and efflcient antennas will
greatly increase the radar range.
Formula (14) applies to free space only. Application to other conditions
RADAR A NTENNAS 231
may require corrections for the effect of the "ground", and for the effect
of the transmission medium, which are beyond the scope of this paper.
2.6 The Plane, Linearly Polarized Electromagnetic Wave
In the foregoing sections we have referred several times to 'plane, linearly
polarized electromagnetic waves'. These waves occur so commonly in
antenna theory and practice that it is worth while to discuss them further
here.
Some properties of linearly polarized, plane electromagnetic waves are
illustrated in Fig. 7. At any point in the wave there is an electric field and
a magnetic field. These fields are vectorial in nature and are at right
angles to each other and to the direction of propagation. It is customary
to give the magnitude of the electric field only.
If we use the M.K.S. system of units the magnitudes of the fields are
e.xpressed in familiar units. Electric intensity appears as volts per meter
and magnetic intensity as amperes per meter. The ratio of electric to
magnetic intensity has a value of 1207r or about 377 ohms. This is the
'impedance' of free space. The power flow per unit area is e.xpressed in
watts per square meter. We see, therefore, that the electromagnetic wave
is a means for carrying energy not entirely unlike a familiar two wire line
or a coaxial cable.
Electromagnetic waves are generated when oscillating currents flow in
conductors. We could generate a plane linearly polarized electromagnetic
wave with a uniphase current sheet consisting of a network of fine wires
backed up with a conducting reflector as shown in Fig. 7. This wave could
be absorbed by a plane resistance sheet with a resistivity of 377 ohms, also
backed up by a conducting sheet. The perfectly conducting reflecting
sheets put infinite impedances in parallel with the current sheet and the
resistance sheet, since each of these reflecting sheets has a zero impedance
at a spacing of a quarter wavelength.
A perfectly plane electromagnetic wave can exist only under certain ideal
conditions. It must be either infinite in extent or bounded appropriately
by perfect electric and magnetic conductors. Nevertheless thinking in
terms of plane electromagnetic waves is common and extremely useful. In
the first place the waves produced over a small region at a great distance
from any radiator are essentially plane. Arguments concerning receiving
antennas therefore generally assume that the incident waves are plane. In
the second place an antenna which has dimensions of many wavelengths can
be analyzed with considerable profit on the basis of the assumption that it
transmits by producing a nearly plane electromagnetic wave across its
aperture. This method of analysis can be applied to the majority of micro
wave radar antennas, and will be discussed in the following sections.
232
BELL SYSTEM TECHNICAL JOURNAL
3. Wave Front Analysis
The fundamental design question is "How to get what we want?" In
a radar antenna we want specified radiation characteristics; gain, pattern
and polarization. Electromagnetic theory tells us that if all electric and
magnetic currents in an antenna are known its radiation characteristics
may be derived with the help of Maxwell's Equations. However, the es
sence of electromagnetic theory insofar as it is of use to the radar antenna
WAVE GENERATOR
REFLECTING
SHEET
A ^
CURRENT
SHEET
WAVE RECEIVER
REFLECTING
SHEET
RESISTANCE
SHEET
i 2rr ^
MAGNETIC INTENSITY = H = Ie"~?r AMPERES
METER
ELECTRIC INTENSITY= E = 120nH ^OLTS
METER
POWER FLOW = P = EH ^^'^ ^^ s
METERS
CURRENT DENSITY^I At^^^^J^^^
METER
RESISTIVITY =R = 120Tr 0.HM5
Fig. 7 — Linearly Polarized Plane Electromagnetic Waves.
designer can usually be expressed in a simpler, more easily visualized and
thus more useful form. This simpler method we call wave front analysis.
In a transmitting microwave antenna the power to be radiated is used to
produce currents in antenna elements which are distributed in space. This
distribution is usually over an area, it may be discrete as with a dipole array
or it may be continuous as in an electromagnetic horn or paraboloid. These
currents generate an advancing electromagnetic wave over the aperture of
RADAR ANTENNAS 233
the antenna. The amplitude, phase and polarization of the electric intensity
in portions of the wave are determined by the currents in the antenna and
thus by the details of the antenna structure. This advancing wave can be
called the 'wave front' of the antenna.
When the wave front of an antenna is known its radiation characteristics
may be calculated. Each portion of the wave front can be regarded as a
secondary or 'Huygens' source of known electric intensity, phase and polari
zation. At any other point in space the electric intensity, phase and polari
zation due to a Huygens source can be obtained through a simple expression
given in the next section. The radiation characteristics of the antenna can
be found by adding or integrating the effects due to all Huygens sources of
the wave front.
This procedure is based on the assumption that the antenna is transmit
ting. A basic law of reciprocity assures us that the receiving gain and radia
tion characteristics of the antenna will be identical with the transmitting
ones when only linear elements are involved.
This resolution of an antenna wave front into an array of secondary
sources can be justified within certain limitations on the basis of the induc
tion theorem of electromagnetic theory. These limitations are discussed in
a qualitative way in section 3.13.
3.1 The Huygens Source
Consider an elementary Huygens source of electric intensity £opolarized
parallel to the X axis with area dS in the XY plane (Fig. 8). This can be
thought of as an element of area dS of a wave front of a linearly polarized
plane electromagnetic wave which is advancing in the positive z direction.^
From Maxwell's Equations we can determine the field at any point of space
due to this Huygens Source. The components of electric field, are found
to be
Ee = t — — e (1 + cos 6) cos <^ , ,
Tkr (l5)
Ea, = —I —  — e (1 1 cos 6) sm </>
2Kr
where X is the wavelength.
We see at once that this represents a vector whose absolute magnitude
at all points of space is given by
\E\ =^(l^cose). (16)
^ S. A. Schelkunoff, Loc. Cit., Chap. 9.
234
BELL SYSTEM TECHNICAL JOURNAL
Here
Ef^dS
is an amplilude factor which depends on the wavelength, intensity
and area of the elementar}' source and \/r is an amplitude factor which
specilies the \ariation of field with distance. (1 + cos 6) is an amjilitude
factor which shows that the directional pattern of the elementary source is a
cardioid with maximum radiation in the direction of propagation and no
radiation in the reverse direction.
When we use the properties of the Huygens source in analyzing a micro
Fig. 8 — The Huygens Source.
wave antenna we are usually concerned principally with radiation in or near
the direction of propagation. For such radiation Equation 16 takes a par
ticularly simple form in Cartesian Coordinates
E,
.£^^_,(,WX)r.^^^Q.^^^Q_
(17)
This represents an electric vector nearly parallel to the electric vector of the
source. The amplitude is given by the factor ^ and the phase by the
RADAR ANTENNAS
235
factor i e *''^''' ^'^. With this equation as a basis we will now proceed to
study some relevant matters concerning radar antennas.
3.2 Gain and EJJective Area of an Ideal Anlenna
On the basis of (17) we can now determine the gain of an ideal antenna of
area S {S ^ X^). This antenna is assumed to be free of heat loss and to
transmit by generating an advancing wave which is uniform in phase and
amplitude in the XY plane. Let the electric intensity in the wave front of
Fig. 9 — An Ideal Antenna.
the ideal antenna be E^ polarized parallel to the X axis (Fig. 9). The trans
mitted power Pr is equal to the power flow through S and is given by
(18)
At a point Q on the Z axis the electric intensity is obtained by adding the
effects of all the Huygens sources in S. If the distance of Q from is so
great that
r = d + ^
236 BELL S YSTEM TECH NIC A L JOURNA L
where A is a negligibly small fraction of a wavelength for every point on .9
then we see from (17) that the electric vector at Q is gi\en by
Js \r Xd
The power flow per unit area at Q is therefore
1 £^5' PtS
P =
UOir \W \H'
Po the power flow per unit area at Q when power is radiated isotropically
from is found by assuming that Pt is spread evenly over the surface of a
sphere of radius d.
The gain of a lossless, uniphase, uniamplitude, linearly polarized antenna
is, by the definition of equation 1, the ratio of 19 and 20.
It follows from 12 that the effective area of the ideal antenna is
A ^ S (22)
In other words in this ideal antenna the effective area is equal to the actual
area. This is a result which might have been obtained by more direct
arguments.
3.3 Gain and Efeclive Area of an A ntenna with Aperture in a Plane and with
Arbitrary Phase and Amplitude
Let us consider an antenna with a wave front in the XY plane which has
a known phase and amplitude variation. Let the electric intensity in the
wave front be
E{x, y) = Eoaix, y)e'*^''''^ (23)
polarized parallel to the x axis. The radiated power is equal to the power
flow through 5 and is given by
_ E'o I a'{x, y) dS
P... = " J " " (24)
1207r
The input power to the antenna is
Pt = PradA (25)
RADAR AN TENNA S liT
where Z is a loss factor (< 1). At a point Q on the Z axis the electric inten
sity is obtained by adding the effects of all the Huygens sources in S. If
OQ is as great as in the above derivation for the gain of an ideal antenna then
we see from 17 that the electric intensity at Q is
£x = i ^^^ £o I a{x, y)e"^^''US; Ey = 0; E, = 0. (26)
Ad J
The power flow per unit area at Q is given by
^^T^rl^I' (27)
and Po the power flow per unit area at Q when Pt is radiated isotropically
is given by equation (3).
The power gain of the antenna, by definition 1 is therefore
Po 1207r / 47rrf2 x2
f a{x, y)6'*^
dS
/ a(x, y)
Js
(28)
dS
The gain expressed in db is given by
Gdb = 10 log.o G (29)
We combine 12 and 28 to obtain
A = L
I a{x,y)e'*^'''''dS
(30)
/ a^{x, y) dS
a formula for the effective area of the antenna.
3.4 The Significance of the Pattern of a Radar A ntenna
The accuracy with which a radar can determine the directions to a target
depends upon the beam widths of the radar antenna. The ability of the
radar to separate a target from its background or distinguish it from other
targets depends upon the beam widths and the minor lobes of the radar
antenna. The efficiency with which the radar uses the available power to
view a given region of space depends on the beam shape of the antenna.
These quantities characterize the antenna pattern. In the following sec
tions means for the calculation of antenna patterns in terms of wave front
theory will be developed, and some illustrations will be given.
238
BELL SYSTEM TECHNICAL JOURNAL
3.5 Pattern in Terms of Antenna Wave Front
If the relative phase and amplitude in a wave front are given by
E{x, y) = a(x, y)e"''^'''
(31)
the relative phase and amplitude at a distant point Q not necessarily on the
Z axis (Fig. 10) in the important case where the angle QOZ between the
direction of propagation and the direction to the point is small, is given from
(17) by adding the contributions at Q due to all parts of the wave front.
This gives
Xa Js
dS.
(32)
Fig. 10 — Geometry of Pattern Analysis.
The quantity r in (32) is the distance from any point P with coordinates .r,
y, 0, in the XY, plane to the point Q (Fig. 10). Simple trigonometry shows
that when OQ is very large
r = d — X sin a — y sin ^ (33)
where d is the distance OQ, a is the angle ZOQ' between OZ and OQ' the
projection of OQ on the XZ plane and /3 is similarly the angle ZOQ". The
substitution of 33 into 32 gives
Eo =
• i(2WX)d ^]
*^ i ^ t(2ir/X)(T8ino+i/sin/3) + i
\d
f
*(!,!/)
a{x, y) dS.
(34)
RADAR ANTENNAS 239
In most practical cases this equation can be simplified by the assumptions
cf>(x,y) = <t>'{x) + ct>"iy)
a{x,y) = a'ix)a''{y)
from which it follows that
I £q I = Fid)Fia)F(fi) (35)
where F{d) is an amplitude factor which does not depend on angle,
F{a) = j e*'^'^''^"'""+'*'^^^a'(x)^x (36)
is a directional factor which depends only on the angle a and not on the angle
(8 or d, and F(/3) similarly depends on /3 but not on a or d. The pattern of
an antenna can be calculated with the help of the simple integrals as in 36,
and illustrations of such calculations will be given in the following sections.
3.6 Pattern of an Ideal Rectangular Antenna
Let the wave front be that of an ideal rectangular antenna of dimensions
a, b ; with linear polarization and uniform phase and amplitude. The dimen
sions a and b can be placed parallel to the .Y and F axes respectively as
sketched in Fig. 9. Equation 36 then gives
F{a) = r'\''''^''^'''" dx = a'^ (37)
Jal2 W
, , X a sin a
where ^ = .
Similarly
F^0)=b'^ (38)
where i/' =
, _ TT 6 sin /3
The pattern of the ideal rectangular aperture, in other words the distribution
of electrical field in angle is thus given approximately by
F(a)F(ff) = ai'^'^ . (39)
The function is plotted in Fig. 11. It is perhaps the most useful
function of antenna theory, not because ideal antennas as defined above are
particularly desirable in practice but because they provide a simple stand
240
BELL SYSTEM TECHNICAL JOURNAL
ard with which more useful but more complex antennas can profitably be
compared.
3.7 Efect OH Pattern oj Amplitude Taper
The — — pattern which results from an ideal wave front has undesirably
high minor lobes for most radar applications. These minor lobes will be
reduced if the wave front of constant amplitude is replaced by one which
retains a constant phase but has a rounded or 'tapered' amplitude dis
tribution.
OFF AXIS
\l /APERTURE
UNIFORM PHASE
AND AMPLITUDE
ACROSS APERTURE
5n AV\ 3n 2TT no n 2n 3n 4tt 5n
^ TTO SIN a
Fig. 1 1 — Pattern of Ideal Rectangular Antenna.
If such an amplitude taper is represented analytically by the function
a'{x) = Ci + C'i cos
ttx
(40)
then equation (36) is readily integrable. To integrate it we utilize the
identity
cos — =
a 2
upon which the integral becomes the sum of three simple integrals of the
form
. ka
,an sm
e""dx = a
all
ka
y
(41)
RADAR ANTENNAS
241
We therefore obtain
, . sin ^ C2
F{a) = aCx — ^ + ^ y
sin
(.+i)^ sin(,y
U^^d (*^)
(42)
The patterns resulting from two possible tapers are given by substi
tuting Ci == 0, C2 = 1 and Ci = 1/3, C2 = 2/3 in (42). These patterns are
sin a
evidently calculable in terms of the known function . They are plotted
a
in Figs. 12 and 13.
0.8
< 0.2
 5n 4TT
■3n 2n
non
... no sma
3TT 4n
Fig. 12 — Pattern of Tapered Rectangular Antenna.
It will be observed that minor lobe suppression through tapering is ob
tained at the expense of beam broadening. In addition to this the gain is
reduced by tapering, as could have been calculated from 28. These unde
sirable effects must be contended with in any practical antenna design.
The choice of taper must be made on the basis of the most desirable com
promise between the conflicting factors.
3.8 Efect on Pattern of Linear Phase Variation
If we assume a constant amplitude and a linear phase variation
4>'{x) = —k\x
242
BELL SYSTEM TECHNICAL JOURNAL
over an aperture —a/2 < x < a/2 then 36 becomes a simple integral of
the form (41) and we obtain
sin xp" „ ira . kia . .
/' (a) = a —777— where \f/ — — sm a —  (43)
yp A 2
The physical interpretation of^(43) is simply that the pattern is identical to
the pattern of an antenna with constant amplitude and uniform phase but
rotated through an angle 6 where
sm 6 = — —
27r
 2"n
non
u,_ no SIN a
2rr
Fig. 13 — Pattern of Tapered Rectangular Antenna.
Simple examination shows that the new direction of the radiation maximum
is at right angles to a uniphase surface, as we would intuitively expect. This
phenomenon has particular relevance to the design of scanning antennas.
3.9 Effect on Pattern of Square Law Phase Variation
If we assume a constant amplitude and a square law phase variation
(t)'{x) = —kix
over the aperture a/2 < x < a/2 then the substitution
27r .
X =
1
i2 L
sm a
X +
2k
2 _
(44)
RADAR ANTENNAS
reduces (36) to the form
/ (a) =  e V '^ ^  e
k2 J
Equation (45) can be evaluated with the help of Fresnel's Integrals
[ cos X' dX, j sin X' dX
dX
243
(45)
ANGLE f
A 1
OFF AXIS /
• *
\
\
N
t
— \ <
4
Uq
1 /
i
p
J\
Ab)
/v
n 2n 2n n
y
^ _ j]_g__5iN_a
2n no T[ 2T\ 2n n o tt 2n
Fig. 14 — Patterns of Rectangular Apertures with Square Law Phase Variation.
which are tabulated^, or from Cornu's Spiral which is a convenient graphical
representation of the Fresnel Integrals.
Typical computed patterns for apertures with square law phase variations
are plotted in Fig. 14. These theoretical curves can be applied to the fol
lowing important practical problems.
(1) The pattern of an electromagnetic horn.
■• For numerical values of Fresnel's Integrals and a plot of Cornu's Spiral see Jahnke
and Emde, Tables of Functions B, G, Teubner, Leipzig, 1933, or Dover Publications, New
York Citv, 1943,
244 BELL S YS TEM TECH NIC A L JOURNA L
(2) The defocussing of a reflector or lens due to improper placing of the
primary feed.
(3) The defocussing of a zoned reflector or lens due to operation at a fre
quency off midband.
In addition to providing distant patterns of apertures with curved wave
fronts (44) provides theoretical 'close in' patterns of antennas with plane
wave fronts. This arises from the simple fact that a plane aperture appears
as a curved aperture to close in points. The degree of curvature depends
on the distance and can be evaluated by extremely simple geometrical con
siderations. When this has been done we find that Fig. 14 represents the
socalled Fresnel diffraction field.
With this interpretation of square law variation of the aperture we can
examine several additional useful problems. We can for instance justify
the commonly used relation
for the minimum permissible distance of the field source from an experi
mental antenna test site. This distance produces an effective phase curva
ture of X/16. We can examine optical antenna systems employing large
primary feeds, in particular those employing parabolic cylinders illuminated
by line sources.
3.10 Ejffed on Pattern of Cubic Phase Variation
If we assume a constant amplitude and a cubic phase variation <l>'{x) =
— kzx over the aperture from — a/2 < x < a/2 then equation (36) becomes
F{a) = f"'e"^'.e''^''''>"'°".(ix (46)
J a/2
If ksx < ~ then it is a fairly good approximation to write
e^'l^' = I  ikW  ^Af ^ ... (47)
from which it follows that (46) can be integrated since it reduces to a sum of
three terms each of which can be integrated.
Typical computed patterns for apertures with cubic phase variation are
plotted in Figs. 15 and 16. Cubic phase distortions are found in practice
when reflectors or lenses are illuminated by primary feeds which are off axis
either because of inaccurate alignment or because beam lobing or scanning
through feed motion is desired. The beam distortion due to cubic phase
variation is known in optics as 'coma' and the increased unsymmetrical lobe
which is particularly evident in Fig. 16 is commonly called a 'coma lobe'.
RADAR ANTENNAS
245
u 0.6
Q 0.4
5
> 0.2
r
\
t
V^'^^^ ANGLE
V 1 OEF AXIS
\ 1
\j >; APERTURE
1
\
1
45"/ 
1 <
I
CL
*
1
cuBic phase: variation
TO ± 45° AT EXTREMES
OF APERTURE
^,
/
^
i
\
/
^
/
^
\
V
\
J
\
y
^
^
\J
srr 4n srr  2n no n 2n sn 4n 5n
_ no SIN a
Fig. 15 — Pattern of Rectangular Antenna with Cubic Phase Variation.
O 0.4
<
/
APERT
°o
T
UREn \
^ ANGLE OFF
AXIS
\
^c
H
J
y °o
^ O)
\
I
CL
t
[
\
UNIFORM AMPLITUDE.
CUBIC PHASE VARIATION
TO + 90° AT EXT REMES
OF APERTURE
^
/
\
/
\
r
\
,^
\
\
J
\
\
1
V
/
V
\
1/
■2n
Fig. 16 — Pattern of Rectangular Antenna with Cubic Phase Variation.
3.11 Two General Methods
In sections 3.7 and 3.8 we integrated (36) by expressing a'(x)e'*'^''^ asa sum
of terms of the form e* "". Since c'(x)e' "" for finite amplitudes in a finite
246 BELL S YSTEM TECH NIC A L JOURNA L
aperture can always be expressed as a Fourier sum of this form this solution
can in princij)le always be found.
Alternatively in section 3.10 the integral was evaluated as a sum of inte
grals of the general type / x"g''"^</.v. Since d'(.v)e'* ^'^' for finite amplitudes
in a finite aperture can always be expressed in terms of a power series,
this solution can also in principle always be found.
3.12 xirrays
When the aperture consists of an array of component or unit apertures
the evaluation of (36) must be made in part through a summation. When all
of the elementary apertures are ulike this summation can be reduced to the
determination of an 'Array Factor'. The pattern of the array is given by
multiplying the array factor by the pattern of a single unit.
The pattern of an array of identical units spaced equally at distances some
what less than a wavelength can be proved to be usually almost equivalent
to the pattern of a continuous wave front with the same average energy
density and phase in each region.
3.13 Limitations to Antenna Wave Front Analysis
Through the analysis of antenna characteristics by means of wave front
theory as based on equation (17) we have been able to demonstrate some of
the fundamental theoretical principles of antenna design. The use of this
simple approach is justified fully by its relative simplicity and by its applica
bility to the majority of radar antennas. Nevertheless it cannot always be
used. It will certainly be inaccurate or inapplicable in the following cases:
(1) When any dimension of the aperture is of the order of a wavelength
or smaller (as in many primary feeds).
(2) Where large variations in the amplitude or phase in the aperture occur
in distances which are of the order of a wavelength or smaller (as in
dipole arrays).
(3) Where the antenna to be considered does not act essentially through
the generation of a plane wave front (as in an end lire antenna or a
cosecant antenna).
When the wave front analysis breaks down alternative satisfactory ap
proaches based on Maxwell's equation are sometimes but not always fruit
ful. Literature on more classical antenna theory is available in a variety of
sources. For much fundamental and relevant theoretical work the reader
is referred to Schelkunoff.''
" S. A. Schelkunoff, Loc. Cit.
RADAR ANTENNAS 247
4. Application of General Principles
In the foregoing sections we have provided some discussion of what hap
pens to a radar signal from the time that the pulse enters the antenna on
transmission until the time that the reflected signal leaves the* antenna on
reception. We have for convenience divided the principles which chiefly
concern us into three groups, transmission line theory, transmission prin
ciples and wave front theory.
With the aid of transmission line theory we can examine problems con
cerning locally guided or controlled energy. The details of the problems of
antenna construction, such as those to be discussed in Part II frequently
demand a grasp of transmission line theory. With it we can study local
losses, due to resistance or leakage, which affect the gain of the antenna.
We can examine reflection problems and their effect on the match of the
antenna. Special antennas, such as those employing phase shifters or trans
mission between parallel conducting plates, introduce many special prob
lems which lie wholly or partly in the transmission line field.
An understanding of the principles which govern transmission through
free space aids us in comprehending the radar antenna field as a whole.
Through a general understanding of antenna gains and effective areas we
are better equipped to judge their significance in particular cases, and to
evaluate and control the effects of particular methods of construction on
them.
Wave front theory provides us with a powerful method of analysis through
which w^e can connect the radiation characteristics produced by a given
antenna with the radiating currents in the antenna. Through it we can
examine theoretical questions concerning beam widths and shape, unwanted
radiation and gain.
An understanding of theory is necessary to the radar antenna designer,
but it is by no means sufficient. It is easy to attach too much importance
to theoretical examination and speculation while neglecting physical facts
which can 'make or break' an antenna design. Theory alone provides no
substitute for the practical 'know how' of antenna construction. It cannot
do away with the necessity for careful experiment and measurement. Least
of all can it replace the inventiveness and aggressive originality through
which new problems are solved and new techniques are developed.
PART II
METHODS OF ANTENNA CONSTRUCTION
5. General
Techniques are essential to technical accomplishment. An understanding
of general principles alone is not enough. The designing engineer must have
248 BELL S YSTEM TECH NIC A L JOURNA L
at his disposal or develop practical methods which can produce the results
he requires. The effectiveness and simplicity of these methods are fair
measures of the degree of technical development.
The study of methods of radar antenna construction is the study of the
means by which radar antenna requirements are met. In a broader sense
this includes an examination of mechanical structures, of the metals and
plastics from which antennas are made, of the processes by which they are
assembled, and of the finishes by which they are protected from their envi
ronment. It might include a study of practical installation and maintenance
procedures. But these matters, which like the rest of Radar have unfolded
widely during the war, are beyond the scope of this paper. An adequate
discussion of them would have to be based on hundreds of technical reports
and instruction manuals and on thousands of manufacturing drawings. The
account of methods which is to follow will therefore be restricted to a dis
cussion, usually from the electrical point of view, of the more useful and
common radar antenna configurations.
6. Classification of Methods
During the history of radar, short as it is, many methods of antenna con
struction have been devised. To understand the details of all of these
methods and the diverse applications of each is a task that lies beyond the
ability of any single individual. Nevertheless most of the methods fall into
one or another of a limited collection of groups or classifications. We can
grasp most of what is generally important through a study of these groups.
In order to provide a basis for classification we will review briefly, from a
transmitting standpoint, the action of an antenna. Any antenna is in a
sense a transformer between a transmission line and free space. More
explicitly, it is a device which accepts energy incident at its terminals, and
converts it into an advancing electromagnetic wave with prescribed amph
tude, phase and polarization over an area. In order to do this the antenna
must have some kind of energy distributing system, some means of amplitude
control and some means of phase control. The distributed energy must be
suitably controlled in phase, amplitude and polarization.
All antennas perform these functions, but different antennas perform
them by different means. Through an examination of the means by which
they are performed and the differences between them we are enabled to
classify methods of antenna construction.
To distribute energy over its aperture an antenna can use a branching
system of transmission lines. When this is done the antenna is an array.
Arrays are particularly common in the short wave communication bands,
but somewhat less common in the microwave radar bands. In a somewhat
simpler method the antenna distributes energy over an area by radiating it
RADAR ANTENNAS
249
from an initial source or 'primary feed'. This distribution can occur in
both dimensions at once, as from a point source. Alternatively the energy
can be radiated from a primary source but be constrained to lie between
parallel conducting plates so that it is at first distributed only over a long
narrow aperture or 'line source'. Distribution over the other dimension
occurs only after radiation from the line source.
In order to control the amplitude across the aperture of an array antenna
we must design the branching junctions so that the desired power division
occurs in each one. When the energy is distributed by radiation from a
primary source we must control the amplitude by selecting the proper pri
mary feed directivity.
We can control the phase of an array antenna by choosing properly the
lengths of the branching lines. Alternatively we can insert appropriate
phase changers in the lines.
When the energy is distributed by primary feeds, methods resembling
those of optics can be used to control phase. The radiation from a point
source is spherical in character. It can be 'focussed' into a plane wave by
means of a paraboloidal reflector or by a spherical lens. The radiation from
a point source between parallel plates or from a uniphase line source is
cylindrical in character. It can be focussed by a parabolic cylinder or a
cylindrical lens.
In Table A we have indicated a possible classification of methods of radar
antenna construction. This classification is based on the differences dis
cussed in the foregoing paragraphs.
Table A
Classification of Methods of Radar Antenna Construction
 Dipoles
r Arrays of
Methods
of Radar
Antenna
Construe
tion
Optical
Methods
Polyrods
Optical Elements
r Point
sources
Spherical < and
Optics
Dipole Arrays
Wave Guide
Apertures
I Spherical r^"^"^^"''^
 Elements! Lenses
 Arrays
r Line Reflectors]
Cylindrical ^0"^^^^ [ j^^^^^^
Optics < and
Cylindrical J deflectors
 Elements "1 t
Lenses
250 BELL SYSTEM TECH NIC A L JOURNA L
7. Basic Design Formulation
Certain design factors are common to almost all radar antennas. Because
of their importance it would be well to consider these factors in a general
way before proceeding with a study of particular antenna techniques.
Almost every radar antenna, regardless of how it is made, has a well de
fined aperture or wave front. Through wave front analysis we can often
examine the connections between the Huygens sources in the antenna aper
ture and the radiation characteristics of the antenna. We can, in other
words, use wave front analysis to study the fundamental antenna design
factors, provided the analysis does not violate one of the conditions of
section 3.13.
7.1 Dimensions oj the Aperture
The dimensions of the aperture of a properly designed antenna are related
to its gain by simple and general approximate relations. If the aperture is
Uniphase and has an amplitude distribution that is not too far from constant
the relation
^ 47ryl
is useful in connecting the gain of an antenna with the area of its aperture.
The effective area is related to the area of the aperture by the equation
A = rjS
where ij is an efficiency factor. In principle 77 could have any value but in
practice for microwave antennas 77 has always been less than one. Its value
for most Uniphase and tapered amplitude antennas is between 0.4 and 0.7.
In special cases, e.g. for cosecant antennas or for some scanners its value
may be less than 0.4.
The necessary dimensions for the aperture may be determined from the
required beam widths in two perpendicular directions. Beam widths are
usually specified as half power widths, that is by the number of degrees
between directions for which the one way response is 3 db below the maxi
mum response. Figure 11 shows that for an ideal rectangular antenna with
uniform phase, polarization and amplitude ap/2= 51  degrees where a^/o ==
a
half power width in degrees, a = aperture dimension and X = wavelength.
The relation ap/2 = 65  degrees is more nearly correct for the majoritv of
a
practical antennas with round or elliptical apertures and with uniform phase
and reasonably tapered amplitudes.
RADAR ANTENNAS 251
7.2 Amplitude Distribution
Except where special, in particular cosecant, patterns are desired the
principle factors affecting amplitude distribution are efficiency and required
minor lobe level. The amplitude distribution or taper of an ideal uniphase
rectangular wave front affects the minor lobe level as indicated by Figures 1 1 ,
12 and 13. Practical antennas tend to fall somewhat below this ideal picture
because of nonuniform phase and because of variations from the ideal
amplitude distribution due to discontinuities in the aperture and undesired
leakage or spillover of energy. Nevertheless a commonly used rule of thumb
is that minor lobes 20 db or more below the peak radiation level are tolerable
and will not be exceeded with a rounded amplitude taper of 10 or 12 db.
7.3 Phase Control
Uniphase wave fronts are used whenever a simple pattern with prescribed
gain, beam widths and minor lobes is to be obtained with minimum aperture
dimensions. When special results are desired such as cosecant patteri^s or
scanning beams the phase must be varied in special ways.
Mechanical tolerances in the antenna structure make it impossible to hold
phases precisely to the desired values. The accuracy with which the phases
can be held constant in practice varies with the technique, the antenna size
and the wave length. Undesired phase variations increase the minor lobes
and reduce the gain of an antenna. The extent to which phase variations
can be expected to reduce the gain is indicated in Fig. 17.
8. Parabolic Antennas
The headlights of a car or the searchlights of an antiaircraft battery use
reflectors to produce beams of light. Similarly the majority of radar anten
nas employ reflectors to focus beams of microwave energy. These reflectors
may be exactly or approximately parabolic or they may have special shapes
to produce special patterns. If they are parabolic they may be paraboloids
which are illuminated by point sources and focus in both directions, or they
may be parabolic cylinders which focus in only one direction. If they are
parabolic cylinders they may be illuminated by line sources or they may be
confined between parallel conducting plates and illuminated by point sources
to produce line sources.
8.1 Control of Phase
A simple and natural way to distribute energy smoothly in space is to
radiate it from a relatively nondirectional 'primary' source such as a dipole
array or an open ended wave guide. This energy will be formed into a direc
tive beam if a reflector is introduced to bring it to a plane area or wave front
with constant phase. If the primary source is effectively a point as far as
252
BELL SYSTEM TECHNICAL JOURNAL
phase is concerned, that is if the radiated energy has the same phase for all
points which are the same distance from a given point, then the reflector
should be parabolic. This can be proved by simple geometrical means.
In Fig. 18 let the point source .V coincide with the point .v = /", y =
of a coordinate system and let the uniphase wave front coincide with the
line X — f. Let us assume that one point of the reflector is at the origin.
Then it can be shown that any other point of the reflector must lie on the
curve
A'2 = Afx
A
square: phase variations
/
1
<1>
/
y
/
3
SAW TOOTH PHASE VARIATIONS
01
B
i /\ /\
_l
ID
J — \/ \y \/
>
Q 2
/
/
Z
/
in
If)
/
o
J
y
/
^
^
\y
B^__^
^^
n
^
^
20 40 60 80
4>= MAXIMUM PHASE VARIATION IN DEGREES
Fig. 17 — Loss due to IMiase Variation in Antenna Wave Front.
This is a parabola with focus at/, o and focal length/.
The derivation based on Fig. 18 is two dimensional and therefore in
principle applies as it stands only to line source antennas employing para
bolic cylinders bounded by parallel conducting planes (Fig. 24 and 25). If
Fig. 18 is rotated about the X axis the parabola generates a paraboloid of
revolution (Fig. 3). This paraboloid focusses energy spreading spherically
from the point source at .5 in such a way that a uniphase wave front over a
plane area is produced. Alternatively Fig. 18 can be translated in the Z
direction perjiendicular to the XY plane. The parabola then generates a
RADAR ANTENNAS
253
parabolic cylinder and the point source S generates a line source at the focal
line of the parabolic cylinder (Fig. 19). The energy spreading cylindrically
from the line source is focussed by the parabolic cylinder in such a way that
a Uniphase wave front over a plane area is again produced. Parabolic
cylinders and paraboloids are both used commonly in radar antenna practice.
In the discussion so far it has been assumed that the primary source is
effectively a point source and that the reflector is exactly parabolic. If the
primary source is not effectively a point source, in other words if it produces
waves which are not purely spherical, then the reflector must be distorted
from the parabolic shape if it is to produce perfect phase correction. When
Fig. 18 — Parabola.
this occurs the correct reflector shape is sometimes specified on the basis of
an experimental determination of phase.
8.2 Control of Amplitude
When a primary source is used to illuminate a parabolic reflector there
are two factors which affect the amplitude of the resulting wave front. One
of these is of course the amplitude pattern of the primary source. The other
is the geometrical or space attenuation factor which is different for different
parts of the wave front. In most practical antennas each of these factors
tends to taper the amplitude so that it is less at the edges of the antenna
than it is in the central region. The effective area of the antenna is reduced
by this taper.
In any finite parabolic antenna some of the energy radiated by the primary
254
BELL SYSTEM TECHNICAL JOURNAL
source will fail to strike the reflector. The effective area must also be re
duced by the loss of this 'spillover' energy.
The maximum effective area for a parabolic antenna is obtained by design
ing the primary feed to obtain the best compromise between loss due to
taper and loss due to spillover. It has been shown theoretically that this
best compromise generally occurs when the amplitude taper across the
aperture is about 10 or 12 db and that in the neighborhood of the optimum
the efficiency is not too critically dependent on the taper.
This theoretical result is well justified by experience and has been applied
to the majority of practical parabolic antennas. It applies both when the
reflector is paraboloidal so that taper in both directions must be considered
: — PARABOLIC
CYLINDER
LINE SOURCE
ANTENNA
Fig. 19 — A Parabolic Cylinder with Line Feed.
and when the reflector is a parabolic cylinder with only a single direction
of taper. It is a fortunate byproduct of a 10 or 12 db taper that it is gen
erally sufficient to produce satisfactory minor lobe suppression.
8.3 Choice of Configuration
We have shown how a simple beam can be obtained through the use of a
paraboloidal reflector with a point source or alternatively through the use
of a reflecting parabolic cylinder and a line source. The line source itself
can be ])roduced with the help of a parabolic cylinder bounded by parallel
conducting plates. We will now outline certain practical considerations.
These considerations may determine which of the two reflector types will be
' C. C. Cutler, Parabolic Antenna Design for Microwaves, paper to be [published in Proc.
of the I. R. E.
RADAR ANTENNAS 255
used for a particular job. They may help in choosing a focal length and in
determining which tinite portion of a theoretically infinite parabolic curve
should be used. Finally they may assist in determining whether reflector
technique is really the best for the purpose at hand or whether we could do
better with a lens or an array.
In designing a parabolic antenna it must obviously be decided at an early
stage whether a paraboloid or one or more parabolic cylinders are to be
employed. This choice must be based on a number of mechanical and elec
trical considerations. Paraboloids are more common in the radar art than
parabolic cylinders and are probably to be preferred, yet a categorical a
priori judgment is dangerous. It will perhaps be helpful to compare the
two alternatives by the simple procedure of enumerating some features in
which each is usually preferable to the other.
Paraboloidal antennas
(a) are simpler electrically, since point sources are simpler than line
sources.
(b) are usually lighter.
(c) are more efficient.
(d) have better patterns in the desired polarization.
(e) are more appropriate for conical lobing or spiral scanning.
Antennas employing parabolic cylinders
(a) are simpler mechanically since only singly curved surfaces are
required.
(b) have separate electrical control in two perpendicular directions.
This last advantage of parabolic cylinders is important in special antennas,
many of which will be described in later sections. It is useful where an
tennas with very large aspect ratios (ratio of dimensions of the aperture in
two perpendicular directions) are desired. It is highly desirable where con
trol in one direction is to be achieved through some special means, as in
cosecant antennas, or in antennas which scan in one direction only.
Let us suppose that we have selected the aperture dimensions and have
decided whether the reflector is to be paraboloidal or cylindrical. The
reflector is not yet completely determined for we are still free in principle to
use any portion of a parabolic surface of any focal length. In order to
obtain economy in physical size the focal length is generally made between
0.6a and 0.25a where 'a' is the aperture. For the same reason a section of
the reflecting surface which is located symmetrically about the vertex is
often chosen (Figures 3 and 19).
When a symmetrically located section of the reflector is used certain diffi
culties are introduced. These difficulties, if serious enough so that their
removal justifies some increase in size can be bypassed through the use of an
256 BELL SYSTEM TECHNICA L JOURNAL
ofifset section as shown in Fig. 20. We can comment on these difficulties as
follows :
1. The presence of the feed in the {)ath of the reflected energy causes a
region of low intensity or 'shadow' in the wave front. The effect of
this shadow on the antenna pattern depends on the size and shape of
the feed and on the characteristics of the portion of the wave front
where it is located. Its effect is to subtract from the undisturbed
pattern a 'shadow pattern' component which is broad in angle. This
decreases the gain and increases the minor lobes as indicated in Fig. 21.^
\ VFEED
Fig. 20 — Offset Parabolic Section.
2. Return of reflected energy into the feed introduces a standing wave
of impedance mismatch in the feed line which is constant in amplitude
but varies rapidly in phase as the frequency is varied. A mismatch at
the feed which cancels the standing wave at one frequency will add to
it at another frequency. A mismatch which will compensate over a
band can be introduced by placing a raised plate of proper dimensions
at the vertex of the reflector as indicated in Fig. 22, but such a jilate
produces a harmful effect on the pattern. In an antenna which must
operate over a broad band it is consequently usually better to match
' Figures 21, 22, and 23 arc taken from V. C. Cutler, loc. cit.
J
RADAR ANTENNAS
257
Fig. 21
5 5
DEGREES OFF AXIS
Effect of Shadow on Paraboloid Radiation Pattern.
Fig. 22 — Apex Matching Plate for Improving the Impedance Properties of a Parabola.
258 BELL S YSTEM TECH NIC A L JOURNA L
the feed to space and accept the residual standing wave, or if this is
too great to use an offset section of the parabolic surface.
8.4 Feeds for Paraboloids
We have seen that an antenna with good wave front characteristics and
consequently with a good beam and pattern can be constructed by illu
minating a reflecting paraboloid with a properly designed feed placed at its
focus. In this section we will examine the characteristics which the feed
should have and some of the ways in which feeds are made in practice.
A feed for a paraboloid should
a. be appropriate to the transmission line with which it is fed. This is
sometimes a coaxial line but more commonly a waveguide.
b. Provide an impedance match to this feed line. This match should
usually be obtained in the absence of the reflector but sometimes, for
narrow band antennas, with the reflector present.
c. have a satisfactory phase characteristic. For a paraboloid the feed
should be, as far as phase is concerned, a true point source radiating
spherical waves. As discussed at the end of 8.1, if the wave front is
not accurately spherical, a compensating correction in the reflector can
be made.
d. have a satisfactory amplitude characteristic. According to 8.2 this
means that the feed should have a major radiation lobe with its maxi
mum striking the center of the reflector, its intensity decreasing
smoothly to a value about 8 to 10 db below the maximum in the direc
tion of the reflector boundaries and remaining small for all directions
which do not strike the reflector.
e. have a polarization characteristic which is such that the electric vec
tors in the reflected wave front will all be polarized in the same di
rection.
f. not disturb seriously the radiation characteristics of the antenna as a
whole. The shadow efl'ect of the feed, the feed line and the necessary
mechanical supports must be small or absent . Primary radiation from
the feed which does not strike the reflector or reflected energy which
strikes the feed or associated structure and is then reradiated must be
far enough down or so controlled that the antenna pattern is as
required.
In addition to the electrical requirements for a paraboloid feed it must of
course be so designed that all other engineering requirements are met, it
must be firmly suj^ported in the required position, must be connected to the
antenna feed line in a satisfactory manner, must sometimes be furnished with
an air tight or water tight seal, and so forth.
RA DA R A NTENNA S 259
From the foregoing it is evident that a feed for a paraboloid is in itself a
small relatively nondirective antenna. Its directivity is somewhat less
than that obtained with an ordinary short wave array. It is therefore not
surprising that dipole arrays are sometimes used in practice to feed
paraboloids.
A simple dipole or half wave doublet can in itself be used to feed a parabo
loid, but it is inefficient because of its inadequate directivity. It is prefer
able and more common to use an array in which only one doublet is excited
directly and which contains a reflector system consisting of another doublet
ov a reflecting surface which is excited parasitically.
Dipole feeds although useful in practice have poor polarization charac
teristics and although natural when a coaxial antenna feed line is used are
less convenient when the feed line is a waveguide. Since waveguides are
more common in the microwave radar bands it is to be expected that wave
guide feeds would be preferred in the majority of paraboloidal antennas.
The most easily constructed waveguide feed is simply an open ended
waveguide. It is easy to permit a standard round or rectangular waveguide
transmitting the dominant mode to radiate out into space toward the parabo
loid. It will do this naturally with desirable phase, polarization and ampli
tude characteristics. It is purely coincidental, however, when this results
in optimum amplitude characteristics. It is usually necessary to obtain
these by tapering the feed line to form a waveguide aperture of the required
size and shape. The aperture required may be smaller than a standard
waveguide cross section so that its directivity will be less. In this case it
may be necessary to 'load' it with dielectric material so that the power can
be transmitted. It may be greater, in which case it is sometimes called an
'electromagnetic horn'. It may be greater in one dimension and less in the
other, as when a paraboloidal section of large aspect ratio is to be illuminated.
If a single open ended waveguide or electromagnetic horn is used to feed
a section of the paraboloid which includes the vertex, the waveguide feed
line must partially block the reflected wave in order to be connected to the
feed. To avoid this difficulty several rear waveguide feeds have been used.
In this type of feed the waveguide passes through the vertex of the parabo
loid and serves to support the feed at the focus. The energy can be caused
to radiate back towards the reflector in any one of several ways, some of
which involve reflecting rings or plates or parasitically excited doublets.
The 'Cutler' feed is perhaps the most successful and common rear feed. It
operates by radiating the energy back towards the paraboloid through two
apertures located and excited as shown in Fig. 23.
* C. C. Cutler, Loc. Cit.
260 BELL S YSTEM TECH NIC A L JOURNA L
8.5 Parabolic Cylinders beticceii Parallel Plates
In «S.O we saw thai parabolic cylinders may be illuminated by line sources
or that they may be confined between parallel plates and illuminated by
point sources to produce line sources. In either of these two cases the char
acteristics which the feed should have are specilled accurately by the con
ditions stated at the beginning of 8.4 for paraboloidal feeds with the excep
tions that condition c must be reworded so that it applies to cylindrical
rather than to spherical optics.
We will first consider parabolic cylinders bounded by parallel plates
because in doing so we describe in passing one form of feed for unbounded
parabolic cylinders. Two forms of transmission between parallel plates
are used in practice.
r!"
Fig. 23 — Dual Aperture Rear Feed Horn.
a. The transverse electromagnetic (TEM) mode in which the electric
vector is perpendicular to the plates. This is simply a slice of the
familiar free space wave and can be propagated regardless of the spacing
between the plates. It is the only mode that can travel between the
plates if they are separated less than half a wavelength. Its velocity
of propagation is independent of plate spacing.
b. The TEoi mode in which the electric vector is parallel to the plates.
This mode is similar to the dominant mode in a rectangular waveguide
and differs from it only in that it is not bounded by planes perpen
dicular to the electric vector. It can be transmitted only if the plate
spacing is greater than half a wavelength, is the only parallel mode
that can exist if the spacing is under a wavelength and is the only sym
metrical parallel mode that can exist if the plate spacing is under three
RADAR ANTENNAS
261
halves of a wavelength. Its phase velocity is determined by the plate
spacing in a manner given By the familiar waveguide formula
Va =
where 'c' is the velocity of light, e is the dielectric constant relative to
free space of the medium between the plates, X is the wavelength in
air and 'a' is the plate spacing.
The TEM mode between parallel plates can be generated by extending
the central conductor of a coaxial perpendicularly into or through the wave
space and backing it up with a reflecting cylinder as indicated in Fig. 24.
PARALLEL
PLATES
REFLECTING
CYLINDER
PARABOLIC
CYLINDER
Fig. 24 — Parabolic Cylinder Bounded by Parallel Plates. Probe Feed.
Alternatively this mode can be generated as indicated by Fig. 25 by a wave
guide aperture with the proper polarization.
The TEni mode, when used, is usually generated by a rectangular wave
guide aperture set between the plates with proper polarization as indicated
in Fig. 25. Care must be taken that only the desired mode is produced.
The TEM mode will be unexcited if only the desired polarization is present
in the feed. The next parallel mode is unsymmetrical and therefore even
if it can be transmitted will be unexcited if the feed is placed symmetrically
with respect to the two plates.
Parallel plate antennas as shown in figures 24 and 25 are useful where
particularly large aspect ratios are required. The aperture dimension per
pendicular to the plates is equal to the plate spacing and therefore small.
262 BELL SYSTEM TECHNICAL JOURNAL
It can be increased somewhat by the addition of flares. The other dimen
sion can easily be made large.
Fig. 25 — Parabolic Cylinder Bounded b>' Parallel Plates. Wave Guide Feed.
Fig. 26. — Fxperimental 7' x 32' Antenna.
8.6 Line Sources for Parabolic ( 'yliiulcrs
A line source for a parabolic cylinder is physically an antenna with a long
narrow aperture. Any means for obtaining such an aperture can be used in
{)ro(lucing a line source. Parallel plate systems as described in 8.5 have
been used as line sources in several radar antennas. The large (7' x 32')
RADAR ANTENNAS
263
experimental antenna shown in Fig. 26 was one of the first to illustrate the
practicality of this design.
The horizontal pattern of the 7' x 32' antenna is plotted in Fig. 27. The
horizontal beam width is seen to be of the order of 0.7 degrees.
The antenna illustrated in Fig. 26 is interesting in another way for it is a
good example of a type of experimental construction which was extremely
useful in wartime antenna development. Research and development engi
2 2 4 6
DEGREES
Fig. 27 — 7' X 32' Antenna, Horizontal Pattern.
neers found that they could often save months by constructing initial
models of wood. Upon completion of a wooden model electrically im
portant surfaces were covered with metal foil or were sprayed or painted
with metal. Thus, where tolerances permitted, the carpenter shop could
replace the relatively slow machine shop.
Another form of parallel plate line feed results when a plastic lens is placed
between parallel plates and used as the focussing element. A linear array
264 BELL SYSTEM TECHNICAL JOURNAL
of elements excited with the proper phase and amplitude can also be
used. Some discussion of alternative approaches will appear in the section
on scanning techniques.
8.7 Tolerances in Parabolic Antcinias
The question of tolerances will always arise in practice. Ideal dimensions
are only approximated, never reached. The ease of obtaining the required
accuracy is an important engineering factor.
The tolerances in paraboloidal antennas or in parabolic cylinders illu
minated by line sources can be divided into three general classes:
(a) Tolerances on reflecting surfaces.
(b) Tolerances on spacial relationships of feed and reflector.
(c) Tolerances on the feed.
When the actual reflector departs from the ideal parabolic curve deviations
in the phase will result. These will tend to reduce the gain and increase the
minor lobes. The effects of such deviations on the gain can be estimated
with the help of Fig. 17. We should recall that an error of a in the reflector
surface will produce an error of about 2<j in the phase front. Based on this
kind of argument and on experience reflector tolerances are generally set in
X X
practice to about ± 77 or ± ~ dependmg on the amount of beam deteriora
tion that can be permitted.
In Fig. 28 are compared some electrical characteristics of two paraboloidal
antennas, one employing a precisely constructed paraboloidal searchlight
mirror and the other a carefully constructed wooden paraboloidal reflector
with the same nominal contour. This comparison is revealing for it shows
the harm that can be done even by small defects in the reflector surface.
Although the two patterns are almost identical in the vicinity of the main
beam, the general minor lobe level of the wooden reflector remains higher
at large angles and its gain is less.
It must not, however, be assumed that a solid reflecting surface is neces
sary to insure excellent results. Any reflecting surface which reflects all
or most of the power is satisfactory provided that it is properly located. Per
forated reflectors, reflectors of woven material and reflectors consisting of
gratings with less than half wavelength spacing are commonly used in radar
antenna practice. These reflectors tend to reduce weight, wind or water
resistance and visibility. Many of them will be described in Part III of
this paper.
The feed of a parabolic reflector should be located so thai its i)hase center
coincides with the focus of the reflector. If it is located at an incorrect dis
RADAR ANTENNAS
265
tance from the vertex a circular curvature of phase results and the system
is said to be 'defocussed' (Sec. 3.9). As the feed is moved off the axis of
the reflector the first effect is a shifting of the beam due to a linear variation
of the phase (Sec. 3.8). For greater distances off axis a cubic component of
phase error becomes effective (Sec. 3.10). Phase error, whether circular,
cubic or more complex, results in a reduction in gain and usually in an in
crease of minor lobes. Although the effects of given amounts of phase curva
ture on the radiation characteristics of an antenna can be estimated by theo
retical means, it is usually easier and quicker to find them experimentally.
5
S 25
UJ
in
10 30
1
1
1
1
n
\
ENVELOPES OF
MINOR LOBE PEAKS
]j
\
A
^•^ •
A
J
\
\^
A
T
7
\
B
^/^
\^
B
45
50
55
30 25 20 15 10 5 5 10 15 20 25 30
HORIZONTAL ANGLE IN DEGREES
Fig. 28 — Effect of Small Inaccuracies in Reflector.
The tolerances on the feed itself appear in various forms, many of which
can be examined with the aid of transmission line theory and most of which
are too detailed for discussion in this paper. It is generally true here also
that experiment is a more effective guide than theory.
Experience has shown that when parallel plate systems are used, either
as complete antennas or as line feeds for other elements, tolerances on the
parallel conducting plates must be considered carefully. It is obvious that
when the TEoi mode is used the plate spacing must be held closely, since
the phase velocity is related to the spacing. This spacing can be controlled
through the use of metallic spacers perpendicular to the plates. These
266 BELL S YSTEM TECH NIC A L JOVRNA L
spacers, if small enough in cross section, do not disturb things unduly.
The velocity of the TEM mode is, on the other hand, almost independent of
the plate spacing. This mode is, however, more likely to cause trouble by
leaks through joints and cracks in the plates.
9. Metal Plate Lenses
At visible wavelengths lenses have, in the past, been far more common
than in the microwave region, due chiefly to the absence of satisfactory lens
materials. A solid lens of glass or plastic with a diameter of several feet is a
massive and unwieldy object. By zoning, which will be discussed below,
these difticulties can be reduced but they still remain.
A new lens technique, particularly effective in the microwave region was
developed by the Bell Laboratories during the war.^ It is evident that any
material in which the phase velocity is different from that of free space can
be used to make a phase correcting lens. The material which is used in this
new technique is essentially a stack of equally spaced metal plates parallel
to the electric vector of the wave front and to the direction of propagation.
Lenses made from this material are called 'Metal Plate Lenses'.
When the spacing between neighboring plates is between X/2 and X only
one mode with electric vector parallel to the plates can be transmitted.
This is the TEoi mode for which the phase velocity is given in Sec. 8.5.
When the medium between the plates is air this equation can be converted
into the expression
N= i/l
\2a[
for the effective index of refraction. Here X is the wavelength in air and a
is the plate spacing.
As a varies between X/2 and X, A' varies as indicated in Fig. 29. In the
neighborhood of a = X, N is not far from 1 and as a approaches X/2, N ap
proaches 0. Since A^ is always less than 1 we see that there is an essential
difference between metal plate lenses and glass or plastic lenses for which N
is always greater than 1. This difference is seen in the fact that a glass lens
corrects phases by slowing down a travelling wave front, while a metal lens
operates in the reverse direction by speeding it up. This means that a
convergent lens with a real focus must be thinner in the center than the
edge, the opposite of a convergent optical lens (Fig. 30).
Unless the value of A^ is considerably different from 1 it is evident that
very thick lens sections must be used to produce useful phase corrections.
For this reason values of 'a' not far from X/2 should be chosen. On the other
hand values of *a' too close to X/2 would cause undesirably large reflections
9 W. E. Kock, "Metal Lens Antennas", Proc. I. R. E., Nov., 1946.
RADAR ANTENNAS
267
from the lens surfaces and impose severe restrictions on the accuracy of plate
spacings. The compromises that have been used in practice are N = 0.5
for which a = 0.577X and N = 0.6 for which a — 0.625X.
Even with N' = 0.5 or 0.6 lenses become thick unless inconveniently lon<7
focal distances are used. Thick lenses are undesirable not only because they
occupy more space and are heavier but also because the plate spacing must
be held to a higher degree of accuracy if the phase correction is to be as
± 0.4
0.2
^
^
^/KS7
0.75X
PLATE SPACING
l.OOx
Fig. 29 — Variation of Effective Index of Refraction with Plate Spacing in a Metal
Plate Lens.
required. To get around these difficulties the technique of zoning is used.
Zoning makes use of the fact that if the phase of an electromagnetic vector
is increased or decreased by any number of complete cycles the effect of the
vector is unchanged. When applied to a metal plate lens antenna this
means simply that wherever the phase correction due to a portion of the
lens is greater than a wavelength this correction can be reduced by some
integral number of wavelengths such that the residual phase correction is
under one wavelength. If this is done it is evident that no portion of the
268
BELL SYSTEM TECHNICAL JOURNAL
lens needs to correct the phase by more than one wavelength. It follows
that no portion of the lens need to be thicker than X/(l — A^).
(0)
FEED FEED
Fig. 30 — Comparison of Dielectric and Metal Plate Lenses.
(b)
[{liMl ""i>i{ll
(0) (b)
FEED FEED
Fig. 31 — Comparison of Unzoncd and Zoned Metal Plate Lenses.
A cross section of a ty])ical metal j)hite lens before and after zoning is
illustrated in Fig. 31. This figure shows clearly why zoning reduces con
siderably the size and mass of a lens.
RADAR ANTENNAS 269
Zoning is not without disadvantages. One disadvantage is obviously
that a zoned lens which is designed for one frequency will not necessarily
work well at other frequencies. It is in principle possible to design a broad
band zoned metal plate lens corresponding to the color compensated lenses
used in good cameras. So far, however, this has not been necessary since
band characteristics of simple lenses have been adequate.
Another difficulty that zoning introduces is due to the boundary regions
between the zones. The wave front in this region is influenced partly by
one zone and partly by the other and may as a result have undesirable phase
and amplitude characteristics. This becomes serious only if especially short
focal distances are used.
9.1 Lens Antenna Configuratio7is
Any of the configurations which are possible with parabolic reflectors have
their analogues when metal plate lenses are used. Circular lenses illumi
nated by point sources and cylindrical lenses illuminated by line sources are
not only theoretically possible but have been built and used. Since a lens
has two surfaces there is actually somewhat more freedom in lens design
than in reflector design. Metal Plate Lenses have usually been designed
with one surface flat, but the possibility of controlling both surfaces is
emerging as a useful design factor where special requirements must be met.
Feeds for lenses should fulfill most of the same requirements as feeds for
reflectors. We find a difference in size in lens feeds in that they must gen
erally be more directive because of greater ratios of focal length to aperture.
A difference in kind occurs because the feed is located behind the lens where
none of the focussed energy can enter the feed or be disturbed by it. As a
result some matching and pattern problems which arise in parabolic antennas
are automatically absent when lenses are used.
In choosing a design for a lens antenna system with a given aperture one
must compromise between the large size which is necessary when a long focal
length is used and the more zones which result if the focal length is made
short. Most metal plate lenses so far constructed have had focal lengths
somewhere between 0.5 and 1.0 times the greatest aperture dimension.
9.2 Tolerances in Metal Plate Lenses
It is not difficult to see that phase errors resulting from small displace
ments or distortions of a metal plate lens are much less serious than those
due to comparable distortions of a reflector surface. This follows from the
fact that the lens operates on a wave which passes through it. If a portion
of the lens is displaced slightly in the direction of propagation it is still
operating on roughly the same portion of the wave front and gives it the
same phase correction. If a portion of a reflector were displaced in the
same way the error in the wave front would be of the order of twice the
270
BELL SYSTEM TECHNICAL JOURNAL
displacement. Quantitative arguments show that less severe tolerances
apply to all major structural dimensions of a metal lens antenna.
It is true of course that the dimensions of individual portions of the metal
lens must be held with some accuracy. The metal plate spacing determines
the eflfective index of refraction of the lens material. Where A^ = 0.5 it is
customary to require that this be held to ±X/75, and where A' = 0.6 to
±X/50. The thickness of the lens in a given region is less critical, and must
be held to ± ttt., T7\ where it is desired to hold the phase front to ±X/16.
10 (1 — A')
Fig. 32 illustrates clearly the drastic way in which the location of a lens
can be altered without seriously afifecting the pattern. It shows, inci
dentally, how a lens may behave well when used as the focussing element
in a moving feed scanning antenna.
■ TV
(b)
l^
"
5
10
\ \
' ^
1
15
20
\—f^
i\
25
Uk,
^nJ 1
vV U
Fig. 32 — Effect on Pattern of Lens Tilting.
9.3 Advantages of Metal Plate Lenses
On the basis of the above discussion we can see that metal plate lenses
have certain considerable advantages. The most important of these is
perhaps found in the practical matter of tolerances. It is a comparatively
simple matter to hold dimensions of small objects to close tolerances but
quite another thing to hold dimensions of large objects closely under the
conditions of modern warfare. This advantage emerges with increasing
importance as the wavelength is reduced.
Metal plate lenses have contributed a great degree of flexibility to radar
antenna art. When they are used two surfaces rather than one may be
controlled, and the dielectric constant can be varied within wide limits.
Independent control in the two polarizations may be applied. We can con
fidently expect that they will become increasingly popular in the radar field.
10. Cosecant Antennas
One of the earliest uses of radar was for early warning against aircraft.
i
RADAR ANTENNAS 271
The skies were searched for possible attackers with antennas which rotated
continuously in azimuth. An equally important but later use appeared
with the advent of great bombing attacks. Bombing radars 'painted' maps
of the ground which permitted navigation and bombing during night and
under even the worst weather conditions. In these radars also the antennas
were rotated in azimuth, either continuously through 360° or back and forth
through sectors.
The majority of radars designed to perform these functions provided verti
cal coverage by means of a special vertical pattern rather than a vertical
scan. It can easily be seen that such a pattern would have to be 'special.'
If we assume, for example, that a bombing plane is flying at an altitude of
10.000 feet, then the radar range must be 10,000 esc 60° = 11,500 feet if a
target on the ground at a bomb release angle of 60° from the horizontal is to
be seen. Such a range would by no means be enough to pick up the target
at say 10° in time to prepare for bombing, for then a range of 10,000 esc
10° = 57,600 feet would be required. This range is far more than is neces
sary for the 60° angle. It appears then that in the most efficient design the
radar range and therefore the radar antenna gain, must be different in dif
ferent directions.
The required variation of gain with vertical direction could be specified
in any one of several ways. It seems natural to specify that a given ground
target should produce a constant signal as the plane flies towards it at a con
stant altitude. Neglecting the directivity of the target this will occur if the
amplitude response of the antenna is given by £ = E^cscd. This same con
dition will apply by reciprocity to an early warning radar antenna on the
ground which is required to obtain the same response at all ranges from a
plane which is flying in at a constant altitude.
This condition is not alone sufficient to specify completely the vertical pat
tern of an antenna. For one thing it can obviously not be followed when
^ = 0, for this would require infinite gain in this direction. Therefore a
lower limit to the value of 6 for which the condition is valid must be set. In
addition an upper limit less than 90° is specified whenever requirements per
mit, since control at high angles is especially difficult. When the limits have
been set it still remains to specify the magnitude of the constant £o This
can be done by specifying the range in one particular direction. This speci
fication must of course be consistent with all the factors that determine gain,
including the reduction due to the required vertical spread of the pattern.
10.1 Cosecant Antennas based on the Paraboloid
It is evident that the standard paraboloidal antennas so far discussed will
not produce cosecant patterns. These patterns being unsymmetrical will
result only if the wave front phase and amplitude are especially controlled.
272
BELL SYSTEM TECHNICAL JOURNAL
On the other hand, because paraboloidal antennas are simple and common
it is natural that many cosecant designs should be based on them. These
designs can be classified into two grouj)s, those in which the reflector is
modified and those in which the feed is modified.
Some early cosecant antennas were made by introducing discontinuities
in paraboloidal reflectors as illustrated in Fig. i3. These controlled the
radiation more or less as desired over the desired cosecant pattern but pro
NORMAL
PARABOLOID /
SURFACE /
PARABOLOID
SURFACE
Fig. 33 — Some Cosecant Antennas Based on the Paraboloid (Cosecant Energy Down
ward).
duced rather serious minor lobes elsewhere. This difficulty can be overcome
through the use of a continuously distorted surface as illustrated in Fig. 34.
This reflector, flrst used at the Radiation Laboratories, is a normal parabo
loid in the lower part whereas the upper part is the surface that would be
obtained by rotating the parabola through the vertex of the upper part about
its focal j)oint.
Several types of feed have been used in combination with paraboloids to
produce cosecant patterns. These are usually arrays which operate on the
princij)lc that each element is a feed which contributes principally to one
RADAR ANTENNAS
273
region of the vertical pattern. The elements may be dipoles or waveguide
apertures fed directly through the antenna feed line or they may be reflectors
which reradiate reflected energy originating from a single primary source.
No matter how excited they must be properly controlled in phase, amplitude
and directivity.
Cosecant antennas based on the paraboloid are common and can some
times fulfill all requirements with complete satisfaction. Nevertheless they
Fig. 34 — Barrel Cosecant Antenna (Cosecant Energy Downward).
suffer from certain disadvantages. The most serious of these is that they
lack resolution at high vertical angles, that is the beam is wider horizontally
at high angles. This is to be expected for reasons of phase alone, for a
paraboloidal reflector is, after all, designed to focus in only one direction. If
phase difliculties were completely absent however, azimuthal resolution at
high angles would still be destroyed because of cross polarized components of
radiation. These components arise naturally from doubly curved reflectors,
even simple paraboloids. They are sometimes overlooked when antennas
are measured in a one way circuit with a linearly polarized test field, but
must obviously be considered in radar antennas,
274 BELL S YSTEM TECH NIC A L JOURNA L
10.2 Cylindrical Cosecant Antennas
Harmful cross polarized radiation is produced by doubly curved reflectors.
This radiation is dillicult to control and therefore undesirable where a closely
controlled cosecant characteristic at high angles is required. Although not
at first evident, it seems natural now to bypass polarization difficulties
through the use of singly curved cylindrical reflectors. These reflectors if
illuminated with a line source of closely controlled linear polarization provide
a beam which is linearly polarized. This beam has also in azimuth approxi
mately the directivity of the line source at all vertical angles. It is thus
superior in two significant respects to cosecant beams produced by doubly
curved reflecting surfaces.
A cylindrical cosecant antenna consists of a cylindrical reflector illumi
nated by a line source. Part of the cylinder is almost parabolic and con
tributes chiefly to the strong part of the beam which lies closest to the hori
zontal. This part is merged continuously into a region which departs
considerably from the parabolic and contributes chiefly to the radiation at
higher angles.
Although wave front principles can be used and certainly must not be
violated, the principles of geometrical optics have been particularly effective
in the determination of cosecant reflector shapes. The detailed application
of these principles will not be discussed here. While applying the geo
metrical principles the designer must be sure that the overall size and con
figuration of the antenna can produce the results he wants. He must design
a line source with the desired polarization and horizontal pattern and a
vertical pattern which fits in with the cosecant design. In addition he must
take particular care to reduce sources of pattern distortion to a level at
which they cannot interfere significantly with the lowest level of the cose
cant 'tail'.
11. LOBING
In many of the tactical situations of modern war radar can be used to
provide fire control information. Radar by its nature determines range and
microwave radar with its narrow well defined beams is a natural instrument
for finding directions to a target, whether the missile to be sent to that
target is a shell, a torpedo or a bomb. In fire control radar, as opposed to
search or navigational radar, two properties of the antenna deserve par
ticular attention. These are the accuracy and the rate with which direc
tion to a target can be measured.
Lobing is a means which utilizes to the fullest extent the accuracy avail
able from a given antenna aperture and which increases, usually as far as
is desired, the rate at which this information is provided and corrected.
RADAR ANTENNAS
275
A lobing antenna which is to provide information concerning one angle only,
azimuth for example, is capable of producing two beams, one at a time,
and of switching rapidly from one to the other. This process is called
Lobe Stvitclniig. The two beams are nearly coincident, differing in direction
by about one beam width. When the signals from the two beams are com
pared, they will be equal only if the target lies on the bisector between the
beams (Fig. 35). The two signals can be compared visually on an indicator
screen of the radar or they can be compared electrically and fed directly
into circuits which control the direction of fire.
ANTENNA DIRECTED
TO LEFT OF TARGET
ANTENNA DIRECTED
AT TARGET
ANTENMA DIRECTED
TO RIGHT OF TARGET
RELATIVE SIGNALS FROM TWO BEAMS
Fig. 35 — Lobe Switching.
When two perpendicular directions are to be determined, such as the
elevation and azimuth required by an antiaircraft battery, four or in prin
ciple three discrete beams can be used. Radar antennas designed for solid
angle coverage more commonly, however, produce a single beam which ro
tates rapidly and continuously around a small cone. This rotation is
known as conical lobing. A comparison of amplitudes in a vertical plane
can then be used to give the elevation of the target and a similar comparison
in a horizontal plane to give its azimuth. Here too the electrical signals
can be compared visually on an indicator screen, but an electrical comparison
will provide continuous data which can be used to aim the guns and at the
same time to cause the radar antenna to follow the target automatically.
11.1 Lobe Sivitching
Two methods of lobe switching are common. In one of these the lobing
antenna is an array of two equally excited elements. Each of these ele
276 BELL SYSTEM TECHNICAL JOURNAL
nients occujnes one half of the final antenna aperture, and provides a Uni
phase front across this half. If the two elements were excited with the same
phase the radiation maximum of the resulting antenna beam would occur
in a direction at right angles to the combined phase front. If the phase of
one element is made to lag behind that of the other by a small amount,
60° say, the phase of the combined aperture will of course be discontinuous
with a step in the middle. This discontinuous phase front will approximate
with a small error, a uniphase wave front which is tilted somewhat with
respect to the wave fronts of the elements. The phase shift will there
fore result in a slight shift of the beam away from the normal direction.
When the phase shift is reversed the beam shift will be reversed. Two
properly designed elementary antennas in combination with a means for
rapidly changing the phase will therefore constitute a lobe switching an
tenna. Such an antenna is described more in detail in Sec. 14.6.
Another method of lobe switching is more natural for antennas based
on optical principles. In this method two identical feeds are placed side
by side in the focal region of the reflector. When one of these feeds illu
minates the reflector a beam is produced which is slightly ofif the normal
axial direction. Illumination by the other feed produces a second beam
which is equally displaced in the opposite direction. The lobe of the an
tenna switches rapidly when the two feeds are activated alternately in rapid
succession. The antenna must use some form of rapid switching appropri
ate to the antenna feed line. In several applications switches are used
which depend on the rapid tuning and detuning of resonant cavities or
irises.
11.2 Conical Lobiiig
A conically lobing antenna j)roduces a beam which nutates rapidly about
a fixed axial direction. This is usually accomplished by rotating or nutat
ing an antenna feed in a small circle about the focus in the focal plane of a
paraboloid or lens. This antenna feed can be a spinning asymmetrical
dipole or a rotating or nutating waveguide aperture. It can result in a
beam with linear polarization which rotates as the feed rotates, or prefer
ably in a beam for which the polarization remains parallel to a tixed direction.
The beam itself must be nearly circularly symmetric so that the radar re
sponse from a target in the axial direction will not vary with the lobing.
The reflector or lens aj)erture is consequently usually circular.
When the antenna is small it is sometimes easier to leave the feed fixed
and to produce the lobing by moving the reflector.
12. RAPID SCANNING
A lobing radar can j)rovide range and angular information concerning
a single target rapidi)' and accurately but these things arc not always enough.
RADAR ANTENNAS 277
It is sometimes necessary to obtain accurate and rapid information from
all regions within an agular sector. It may be necessary to watch a certain
region of space almost continuously in order to be sure of picking up fast
moving targets such as planes. To accomplish any of these ends we must
use a rapid scanning radar. A rapid scanning radar antenna produces a
beam which scans continuously through an angular sector. The beam may
sweep in azimuth or elevation alone or it may sweep in both directions to
cover a solid angle. An azimuth or elevation scan may be sinusoidal or it
may occur linearly and repeat in a sawtooth fashion. Solid angle scanning
may follow a spiral or flower leaf pattern or it might be a combination of two
one way scans. A combination of scanning in one direction and lobing in
the other is sometimes used.
Scanning antennas must, unfortunately, be constructed in obedience to
the same principles which regulate ordinary antennas. The same attention
to phase, amplitude, polarization and losses is necessary if comparable
results are to be obtained. When scanning requirements are added to
these ordinary ones new problems are created and old ones made more
difficult.
An antenna in order to scan in any specified manner must act to produce
a wave front which has a constant phase in a plane which is always normal
to the required beam direction. This can be done in several different ways.
The simplest of these, electrically, is to rotate a fixed beam antenna as a
whole in the required fashion. This can be called mechanical scanning.
Alternatively an antenna array can be scanned if it is made up of suitable
elements and the relative phases of these elements can be varied appro
priately. This can be called array scanning. Thirdly, optical scanning
can be produced by moving either the feed or the focussing element of a
suitably designed optical antenna.
12.1 MecJianical Scanning
Electrical complexities of other types of rapid scanners are such that it
is probably not going too far to say that the required scan should be accom
plished by mechanical means wherever it is at all practical. This applies
to radar antenna scans which occur at a slow or medium rate. Search
antennas, whether they rotate continuously through 360° or back and forth
over a sector are scanners in a sense but the scan is usually slow enough to
be performed by rotating the antenna structure as a whole. As the scan
becomes more rapid, mechanical problems become more severe and elec
trically scanning antennas appear more attractive.
Mechanical ingenuity has during the war extended the range in which
mechanical scanners are used. One important and eminently practical
mechanical rapid scanner, the 'rocking horse' is now in common use (Fig.
36). This antenna is electrically a paraboloid of elliptical aperture illu
278
BELL SYSTEM TECHNICAL JOURNAL
minated by a liorn feed, a combination which produces excellent electrical
characteristics. The paraboloid and feed combination is made structurally
strong and is pivoted to permit rotational oscillation in a horizontal plane.
It is forced to oscillate by a rigid crank rod which is in turn driven by an
eccentric crank on a shaft. The shaft is belt driven by an electric motor and
its rotational rate is held nearly constant by a flywheel. The mechanical
arrangement described so far would oscillate rotationally in an approxi
mately sinusoidal fashion. Since every action has an equal and opposite
reaction it would, however, react by producing an oscillatory torque on its
Fig. 36 — Experimental Rocking Horse Antenna.
mounting. Since the antenna is large and the oscillation rapid this would
J reduce a ssvere and undesirable vibration. To get around this difficulty
an opposite and balancing rotating moment is introduced into the mechan
ical system. This appears in the form of a pivoted and weighted rod which
is driven from the same eccentric crank by another and almost parallel
crank arm.
Although not theoretically perfect the rotational 'dynamic' balancing
described permits the antenna to scan without serious vibration. One form
of this antenna will be described in a later section.
12.2 Array Scanning
During our discussion of general principles in Part II, we saw that an
antenna wave front can be synthesized by assembling an array of radiating
RADAR ANTENNAS 279
elements and distributing power to it through an appropriate transmission
line network. If the radiation characteristics of the array are to be as de
sired the electrical drive of each element must have a specified phase and
ampHtude. In addition each element must in itself have a satisfactory
characteristic and the elements must have a proper spacial relationship to
each other.
Such array antennas have been extremely useful in the 'short wave' bands
where wavelengths and antenna sizes are many times larger than at most
radar wavelengths but for fixed beam radar antennas they have been largely
superceded by the simpler optical antennas. Where a rapidly scanning
beam is desired, however, they possess certain advantages which were put
to excellent use in the war. These advantages spring from the possibility
of scanning the beam of an array through the introduction of rapidly vary
ing phase changes in its transmission line distributing system.
Let us first examine certain basic conditions that must be fulfilled if an
array antenna is to provide a satisfactory scan. The pattern of any array
is merely the sum of the patterns of its elements taking due account of
phase, amplitude and spacial relationships. If all elements are alike and
are spaced equally along a straight line it is not difficult to show that a
mathematical expression for the pattern can be obtained in the form of a
product of a factor which gives the pattern of a single element and an array
factor. The array factor is an expression for the pattern of an array of
elements each of which radiates equally in all directions. Since each of the
elements is fixed in direction it is only through control of the array factor
that the scan can be obtained.
If we excite all points of a continuous aperture with equal phase and a
smoothly tapered amplitude the aperture produces a beam with desirable
characteristics at right angles to itself and no comparable radiation else
where. Similarly if we excite all elements of an array of identical equally
spaced circularly radiating elements with equal phase and a smoothly
tapered amplitude the array will produce a beam with desirable charac
teristics at right angles to itself. It will also produce a beam in any other
direction for which waves from the elements can add up to produce a wave
front. Such other directions will exist whenever the array spacing is
greater than one wavelength.
In order to see this more clearly let us examine Fig. 37, where line XX'
represents an array of elements. From each element to the line AA' is a
constant distance, so A A' is obviously parallel to a wave front when the
elements are excited with equal phase. If we can find a line BB' to which
the distance from each element is exactly one wavelength more or less than
from its immediate neighbors then it too is parallel to a wavefront, for
energy reaching it from any element of the array will have the same phase
280
BELL SYSTEM TECIIXICAL JOURNAL
except for an integral number of cycles. The same will apply to a line CC ,
to which the distance from each element is exactly two wave lengths more or
less than from its immediate neighbors, or to any other line where this dif
ference is any integral number of wavelengths.
Now in no radar antenna do we desire two or more beams for they will
result in loss of gain and probably in target confusion. The array must
therefore be designed so that for all positions of scan all beams except one
will be suppressed. This will automatically occur if the array spacing is
somewhat less than one wavelength. If the array spacing is greater than
one wavelength these extra beams will appear in the array factor; they
Fig. 37 — Some Possible Wave Fronts of an Array of Elements Spaced 2.75 X.
must therefore be suppressed by the pattern of a single element. The pat
tern of an element must in other words, have no significant components
in any direction where an extra beam can occur.
Where elements with only side fire directivity are spaced more than a
wavelength apart in a scanning array it is almost impossible to obtain
adequate extra lobe sui)pression. If these elements are spaced by the
minimum amount, that is by exactly the dimensions of their apertures and
all radiate in phase the} may indeed just manage to produce a desirable
beam. A little analysis shows however that an appreciable phase variation
from element to element, e\'en though linear, will introduce a serious ex
tra lobe. To get around this difiKulty elements with some end lircdirec
livity must be used.
RADAR ANTENNAS
281
A simple end fire element, and one that has been used in practice, is the
'polyrod' (Fig. 38). A polyrod, is as its name implies, a rod of polystyrene.
This rod, if inserted into the open end of a waveguide, and if properly pro
portioned and tapered, will radiate energy entering from the waveguide
from points which are distributed continuously along its length. If the
Fig. 38— A Polyrod.
l'>x[)erimental Polyrod Array.
wave in the polyrod travels approximately with free space velocity it will
produce a radiation maximum in the direction of its axis. The radia
tion pattern of the polyrod will have a shape which is characteristic of end
fire arrays, narrower and flatter topped than the pattern of a side fire array
which occupies the same lateral dimension. This elementary pattern can
be fitted in well with the array factor of a scanning array.
Such a scanning array is shown in Fig. 39 and will be described in
282 BELL SYSTEM TECH NIC A L JOURNA L
greater detail in section 14.8. Each element of this array consists of a fixed
vertical array of three polyrods. This elementary array provides the re
quired vertical pattern and has appropriate horizontal characteristics.
Fourteen of these elements are arranged in a horizontal array with a spacing
between neighbors of about two wavelengths. Energy is distributed among
the elements with a system of branching waveguides. Thirteen rotary phase
changers are inserted strategically in the distributing system. Each phase
change is rotating continuously and shifts the phase linearly from 0° to
360° twice for each revolution. As the phase changers rotate the array
produces a beam which sweeps repeatedly linearly and continuously across
the scanning sector.
When elements of a scanning array are spaced considerably less than one
wavelength it is a very simple matter to obtain a suitable elementary
pattern, for the array factor itself has only a single beam. This advantage
is offset by the greater number of elements and the consequent greater com
plexity of distributing and phase shifting equipment. In one useful type of
scanning antenna however distributing and phase shifting is accomplished
in a particularly simple manner. Here the distributing system is merely a
waveguide which can transmit only the dominant mode. The wide dimen
sion of the guide is varied to produce the phase shifts required for scanning.
The elements are dipoles. The center conductor of each dipole protrudes
just enough into the guide to pick up the required amount of energy.
It is evident from the above discussion that such a waveguide fed dipole
array will produce a single beam in the normal direction only if the dipoles
are all fed in phase and are spaced less than a wavelength. It is therefore
not satisfactory to obtain constant phase excitation by tapping the dipoles
into the guide at successive guide wavelengths for these are greater than
free space wavelengths. Consequently the dipoles are tapped in at suc
cessive half wavelengths in the guide and reversed successively in polarity
to compensate for the successive phase reversals due to their spacing.
This type of array provides a line source which can be scanned by moving
the guide walls. In order to leave these mechanically free suitable wave
trapping slots are provided along the length of the array.
A practical antenna of this type will be described in Sec. 16.3.
12.3 Optical Scanning
With a camera or telescope all parts of an angular sector or field are viewed
simultaneously. We would like to do the same thing by radar means, but
since this so far appears impossible we do the next best thing by looking
at the parts of the field in rapid succession. Nevertheless certain points of
similarity appear. These points are emphasized by a survey of the fixed
RA DA R A NT EN N A S 2 83
beam antenna field for there we find optical instruments in abundance,
parabolic reflectors and even lenses.
It is not a very big step to proceed from an examination of optical systems
to the suggestion that a scanning antenna can be provided by moving a
feed over the focal plane of a reflector. Nevertheless experience shows
that this will not be especially profitable unless done with due caution.
The first efl'ect of moving the feed away from the focus in the focal plane of
a paraboloid is indeed a beam shift but before this process has gone far a
third order curvature of the phase front is produced and is accompanied
by a serious deterioration in the pattern and reduction in gain. This
difficulty or aberration is well known in classical optical theory and is called
coma. Coma is typified by patterns such as the one shown in Fig. 16.
It is the first obstacle in the path of the engineer who wishes to design a
good moving feed scanning antenna.
Coma is not an insuperable obstacle however. Its removal can be
accomplished by the application of a very simple geometrical principle.
This principle can be stated as follows: "The condition for the absence of
coma is that each part of the focussing reflector or lens should be located on
a circle with center at the focus."
This condition can be regarded as a statement of the spacial relation
ship required between the feed and all parts of the focussing element. It
is a condition which insures that the phase front will remain nearly linear
when the feed is moved in the focal plane. It can be applied approximately
whether the focussing element is a reflector or a lens and to optical systems
which scan in both directions as well as those which scan in one direction.
Coma is usually the most serious aberration to be reckoned with in a
scanning optical system, but it is by no means the only one. Any defect
in the phase and amplitude characteristic which arises when the feed is
moved can cause trouble and must be eliminated or reduced until it is toler
able. Another defect in phase which arises is 'defocussing'. Defocussing
is a square law curvature of phase and arises when the feed is placed at an
improper distance from the reflector or lens. Its effect may be as shown
in Fig. 14. It can in principle always be corrected by moving the feed in a
correctly chosen arc, but this is not always consistent with other require
ments on the system. In addition to troubles in phase an improper ampli
tude across the aperture of the antenna will arise when the feed is trans
lated unless proper rotation accompanies this motion.
To combat the imperfections in an optical scanning system we can
choose overall dimensions in such a way that they will be lessened. Thus
it is generally true that an increase in focal length or a decrease in aperture
will increase the scanning capabilities of an optical system. This alone
is usually not enough, however, we must also employ the degrees of free
284 BELL S YSTEM TECH NIC A L JOURNA L
dom available to us in the designing of the focussing element and the feed
motion to improve the performance. If the degrees of freedom are not
enough \vc must, if we insist on an optical solution introduce more. This
could in principle result in microwave lenses similar to the four and five
element glass lenses found in good cameras, but such complication has not
as yet been necessary in the radar antenna art.
Since military release has not been obtained as this article goes to press
we must omit any detailed discussion of optically scanning radar antenna
techniques.
PART III
MILITARY RADAR ANTENNAS DEVELOPED BY THE
BELL LABORATORIES
i3. General
In the fuial jxirt of this paper we will describe in a brief fashion the
end products of radar antenna technology, manufactured radar antennas.
Without these final practical exhibits the foregoing discussion of principles
and methods might appear academic. By including them we hope to
illustrate in a concrete fashion the rather general discussion of Parts I
and II.
The list of manufactured antennas will be limited in several ways. Severe
but obviously essential are the limitations of military security. In addition
we will restrict the list to antennas developed by the Bell Laboratories. In
cases where invention or fundamental research was accomplished elsewhere
due credit will be given. Finally the list will include only antennas manu
factured by contract. This last limitation excludes many experimental
antennas, some initiated by the Laboratories and some by the armed forces.
It is worthwhile to begin with an account of the processes by which these
antennas were brought into production. The initiating force was of course
military necessity. The initial human steps were taken sometimes by
members of the armed forces who had definite needs in mind and sometimes
by members of the Laboratories who had solutions to what they believed
to be military needs.
With a definite job in mind conferences between military and Laboratories
personnel were necessary. Some of these dealt with legal or financial
matters, others were princi})ally technical. In the technical conferences
it was necessary at an early date to bring military requirements and tech
nical {)ossibilities in line.
As a result of the conferences a program of research and development was
oflen undertaken by the Laboratories. An initial contract was signed which
RA DA R A NT EN N A S 285
called for the delivery of technical information, and sometimes for manu
facturing drawings and one or more completed models. Usually the
antenna was designed and manufactured as part of a complete radar sys
tem, sometimes the contract called for an antenna alone.
After prehminary work had been undertaken the status of the job was
reviewed from time to time. If preliminary results and current mihtary
requirements warranted a manufacturing contract was eventually drawn
up and signed by Western Electric and the contracting government agency.
This contract called for delivery of manufactured radars or antennas ac
cording to a predetermined schedule.
Research and development groups of the Laboratories cooperated in war
as in peace to solve technical problems and accomplish technical tasks.
Under the pressure of war the two functions often overlapped and seemed
to merge, yet the basic differences usually remained.
Members of the Research Department, working in New York and at the
Deal and Holmdel Radio Laboratories in New Jersey were concerned chiefly
with electrical design. It was their duty to understand fully electrical
principles and to invent and develop improved methods of meeting mili
tary requirements. During the war it was usually their responsibility to
prescribe on the basis of theory and experiment the electrical dimensions
of each new radar antenna.
A new and diificult requirement presented to the Research Department
was sometimes the cause of an almost personal competition between alter
native schemes for meeting it. Some of these schemes were soon eliminated
by their own weight, others were carried side by side far along the road to
production. Even those that lost one race might reappear in another
as a natural winner.
In the Development Groups working in New York and in the greatly
expanded Whippany Radio Laboratory activity was directed towards coor
dination of all radar components, towards the establishment of a sound,
well integrated mechanical and electrical design for each component and
towards the tremendous task of preparing all information necessary for
manufacture. It was the job of these groups also to help the manufacturer
past the unavoidable snarls and bottlenecks which appeared in the hrst
stages of production. In addition development personnel frequently
tested early production models, sometimes in cooperation with the armed
forces.
As we have intimated, research and development were indistinguishable
at times during the war. Members of the research department often found
themselves in factories and sometimes in aircraft and warships. Develop
ment personnel faced and solved research problems, and worked closely
with research groups.
286 BELL S YS TEM TECH NIC A L JO URN A L
For several years when pressure was high the effort was intense; at times
feverish. Judging by miUtary results it was highly effective. Some of the
material results of this effort are described in the following pages.
14. Naval Shipborne Radar Antennas
14.1 The SE Auleiiiia'°
Very early in the war, the Navy requested the design of a simple search
radar s3stem for small vessels, to be manufactured as quickly as possible
in order to till the gap between design and production of the more complex
search systems then in {process of develo])ment. The proposed system was
to be small and simple, to permit its use on vessels which otherwise would
be unable to carry radar equipment because of size or power supply capabil
ity. This class of vessel included PT boats and landing craft.
The antenna designed for the SE system is housed as shown in Fig. 40.
It was adapted for mounting on the top or side of a small ship's mast, and
is rotated in azimuth by a mechanical drive, hand operated. The para
boloid reflector is 42 inches wide, 20 inches high, and is illuminated by a
circular aperture 2.9 inches in diameter. In the interests of simplicity, the
polarization of the radiated beam was permitted to vary with rotation of
the antenna.
The SE antenna was operated at 9.8 cm, and fed by 1^x3 rectangular
waveguide. At the antenna base, a taper section converted from the
rectangular waveguide to 3" round guide, through a rotating joint directly
to the feed opening.
Characteristics of the SE antenna are given below:
Wavelength 9.7 to 10.3 cm
Reflector 42" W x 20" H
Gain 25 db
Horizontal Beam Width 6°
Vertical Beam Width 12°, varj'ing somewhat with polarization
Standing Wave 9.7tolO.Ocm 4.0 db
10.0 to 10.3 cm 6.0 db
14.2 The SL Radar Antenna^'
The SL radar is a simple marine search radar developed by Bell Tele
phone Laboratories for the Bureau of Ships. During the war, over
1000 of these radars were produced by the Western Electric Company and
installed on Navy vessels of various categories. The principal tield for
installation was destroyer escort craft ("DE"s). Figure 41 shows an SL
antenna installation al)oard a DE. 'J'he antenna is covered, for wind and
" Written by R. J. Phillips.
" Written by H. T. Budenbom.
RADAR ANTENNAS
287
weather protection, in a housing which can transmit 10 cm radiation.
\'isible also is the waveguide run down the mast to the r.f. unit.
The SL radar provides a simple nonstabiUzed PPI (Plan Position
Indicator) display. The antenna is driven by a synchronous motor at
18 rpm. Horizontal polarization is used to minimize sea clutter. The
'f^T"^^
Fitr. 4U — SE Antenna.
radiating structure, shown in Figure 42, consists of a 20" sector of a 42"
paraboloid. The resulting larger beam width in the vertical plane is pro
vided in order to improve the stability of the pattern under conditions of
ship roll. Figure 43 illustrates the path of the transmitted wave from the
SL r.f. unit to the antenna. It also illustrates the manner in which horizon
tally polarized radiation is obtained. The diagram shows the position of
288
BELL SYSTEM TECHNICAL JOURNAL
"ft:
RADAR ANTENNAS
289
^ %j
■^ X,
■^i
^^P'^""" ^;
Fig. 42 — SL Antenna.
/
290
BELL SYSTEM TECHNICAL JOURNAL
the electric force vector in traversing the waveguide run. The path from
the r.f. unit is in rectangular guide (TEi, o mode) through the right angle
bend, to the base of the rotary joint. A transducer which forms the base
portion of the joint converts to the TMoi mode in circular pipe. For this
mode, the electric held has radial symmetry, much as though the wave
guide were a coaxial line of vanishingly small inner conductor diameter.
PIPE CONTAINING
SPIRAL SEPTUM
TE.
INDICATES DIRECTION
OF ELECTRIC VECTOR.
INDICATES VECTOR
LIES X TO PLANE OF
PAPER.
REFLECTOR
ROTARY JOINT
I AND CHOKE
TEio
Fig. 43 — SL Radar Antenna — Wave Guide Path.
The energy passes the rotary joint in this mode; choke labyrinths are pro
vided at the joint to minimize radio frequency leakage. The energy then
flows through another transducer, from TMoi mode back to TEio mode.
The lower horizontal portion of the feed pipe immediately tapers to round
guide, the mode being now TEn. Ne.xt the energy transverses a 90° elbow,
which is a standard 9i)° pipe casting, and enters the vertical section im
RADAR ANTENNAS 291
mediately below the feed aperture. The E vector is in the plane of the
paper at this point. However, the ensuing vertical section is fitted with a
spiral septum. This gradually rotates the plane of polarization until at
the top of this pipe the E vector is perpendicular to the plane of the paper.
Thus, after transversing another 90° pipe bend, the energy emerges horizon
tally polarized, to feed the main reflector.
Specific electrical characteristics of the SL antenna are:
Polarization — Horizontal
Horizontal Half Power Beamwidth — 6°
Vertical Half Power Beamwidth — 12°
Gain — about 22 db.
14.3 The SJ Submarine Radar Antenna
It had long been expected that one of the early offensive weapons of the
war would be the submarine. It was therefore natural that early in the
history of radar the need for practical submarine radars was felt. The
principal components of this need were twofold, to provide warning of ap
proaching enemies and to obtain torpedo fire control data. The SJ Sub
marine Radar was the first to be designed principally for the torpedo fire
control function.
Work on the SJ system was under way considerably before Pearl Harbor.
When this work was initiated the advantages of lobing fire control systems
were clearly recognized, but no lobing antennas appropriate for submarine
use had been developed. Requirements on such an antenna were ob
viously severe, for in addition to fulfiUing fairly stringent electrical con
ditions, it would have to withstand very large forces due to water resistance
and pressure.
The difficulties evident at the outset of the work were overcome by an
ingenious adaptation of the simple waveguide feed. It was recognized
that a shift of the feed in the focal plane of a reflector would cause a beam
shift. Why not, then, use two waveguide feeds side by side to produce the
two nearly coincident beams required in a lobing antenna? When this was
tried it was found to work as expected.
It remained to devise a means of switching from one waveguide feed to
the other with the desired rapidity. This in itself was no simple problem,
but was solved by applying principles learned through work on waveguide
filters. The switch at first employed was essentially a branching filter
at the junction of the single antenna feed line and the line to each feed aper
ture. Both branches of this filter were carefully tuned to the same fre
quency, that of the radar. The switching was performed by the insertion
of small rapidly rotating pins successively into the resonant cavities of the
292
BELL SYSTEM TECHNICAL JOURNAL
two filters (Fig. 44). Presence of the ])ins in one of the filters detuned it
and therefore prevented ])o\vcr from Uowing through it. Rotation of the
pins accordingly produced switching as desired.
In a later modification of this switch the same general princi})les were
used but resonant irises rather than resonant cavities were employed.
The SJ Submarine Radar was in use at a comi)aralively early date in the
war and saw much ser\ice with the Pacific submarine lleet. Despite some
early doubts, submarine commanders were soon convinced of its powers.
.*<C
SWITCH UNIT
CHAMBERS
OFTUIMING
Pi MS
Fig. 44— The SJ Tuned Cavity Switch.
It is believed that in the majority of cases it replaced the periscope as the
principle fire control instrument. In addition it served as a valuable and
unprecedented aid to navigation.
It is interesting and relevant to quote from two letters to Laboratories
engineers concerning the SJ. One dated October 3, 1943, from the radar
officer of a submarine stated that there were twenty "setting sun" fiags
painted on the conning tower and asked the engineer to "let your mind dwell
on the fact that you helped to put more than 50% of those flags there".
RADAR ANTENNAS
293
The commander of another submarine wrote in a similar vein, "You can
rest assured that we don't regard your gear as a bushybrain space taker,
but a very essential part of our armament".
I'ig. 45 Tlie SJ Submarine Radar Antenna.
Figure 45 is a photograph of an SJ antenna,
characteristics are as follows:
Its principal electrical
Gain > 19 db
Horizontal Half Power Beamwidth 8°
Vertical Half Power Beamwidth 18°
Vertical Beam Character — Some upward radiation
Lobe Switching Beam Separation — approximately 5°
Gain reduction at beam crossover < 1 db
Polarization — Horizontal
294 BELL S YSTEM TECH NIC A L JOURNA L
14.4 The %rodified S J/ Mark 27 Radar Antenna
The SJ antenna described above performed a remarkable and timely fire
control job as a lobing antenna but was found to be unsatisfactory when
rotated continuously to produce a Plan Position Indicator (PPI) presenta
tion. In the PPI method of presentation range and angle are presented as
radius and angle on the oscilloscope screen. Consequently a realistic map
of the strategic situation is produced. This map is easily spoiled by false
signals due to large minor lobes of the antenna.
Since it had been established that the PPI picture was valuable for
navigation and warning as well as for target selection it was decided to
modify the antenna in a way that would reduce these undesirably high minor
lobes. These were evidently due principally to the shadowing effect of the
massively built double primary feed. Accordingly a new reflector was de
signed which in combination with a slightly modified feed provided a much
improved pattern.
The new reflector was different in configuration principally in that it was
a partially offset section of a paraboloid. The reflector surface was also
markedly different in character since it was built as a grating rather than a
solid surface. This reduced water drag on the antenna. In addition
the grating was less visible at a distance, an advantage that is obviously
appreciable when the antenna is the only object above the water.
This modified antenna was used not only on submarines as part of the
SJ1 radar but also on surface vessels as the Mark 27 Radar Antenna.
Figure 46 shows one of these antennas. Its electrical characteristics are
as follows:
Gain > 20 db
Horizontal Half Power Beamwidth = 8°
Vertical Half Power Beamwidth = 17°
Vertical Beam Character — Some upward radiation
Lobe Switching Beam Separation — approximately 5°
Gain reduction at beam crossover < 1 db
Polarization — Horizontal
14.5 The SH and Mark 16 Autenna^^
The antennas designed for the SH and Mark 16 Radar Equipments are
practically identical. The SH system was a shipborne combined fire con
trol and search system, and the Mark 16 its land based counterpart was used
by the Marine Corps for directing shore batteries.
These systems operated at 9.8 cm. The requirement that the system,
operate as a fire control as well as a search system imposed some rather
stringent mechanical requirements on the antenna. For search purposes,
the antenna was rotated at 180 rpm, and indications were presented on a
plan position indicator. For fire control data, slow, accurately controlled
motion was recjuired. Bearing accuracy is attained by lobe switching in
'^Written by R. J. Philipps.
RADAR ANTENNAS
295
much the same manner as in the SJ and SJ1 antennas previously described.
The antenna is illustrated in Fig. 47, With the SH system, the unit
is mast mounted; for the Mark 16, the unit is mounted atop a 50 foot steel
Fig. 46 — The SJ1 /Mark 27 Submarine Radar Antenna.
tower which can be erected in a few hours with a minimum of personnel.
The electrical characteristics are as follows:
Gain— 21. db
Reflector Dimensions 30" W x 20" H
Horizontal beam width — 7.5°
Vertical beam width — 12°
Lobe separation — 5° approximately
Loss in gain at lobe crossover — 1 db approximately
Scan — (1) 360°, at 180 rpm for PPI operation
(2) 360°, at approximately 1 rpm for accurate azimuth readings, with lobe
switching
296
BELL SYSTEM TECHNICAU JOURNAL
SH systems were most successfully used in invasion operations in the
Aleutians. They were installed on landing craft, and the use of the high
A
Fig. 47— SH Antenna.
speed scan enabled the craft to check constantly their relative positions
in the dense fogs encountered during the landing operations.
RADAR ANTENNAS
297
14.6 Allien lias for Early Fire Control Radars^^
The first radars to be produced in quantity for fire coiitrol on naval ves
sels were the Mark 1, Mark 3 and Mark 4 (originally designated FA, FC
and FD). These radars were used to obtain the position of the target with
sufficient accuracy to permit computation of the firing data required by the
guns. The first two (Mark 1 and Mark 3) were used against enemy surface
targets while the Mark 4 Radar was a dual purpose system for use against
both surface and aircraft targets. These radars were described in detail
in an earlier issue. ^'^ However, photographs of the antennas and per
tinent information on the antenna characteristics are repeated herein for
the sake of completeness. (See Table B and Figures 48, 49 and 50.)
Table B
Radar
Mark 1
Mark 3
Mark 4
Dimensions
6'x6'
3'xl2' 1 6'x6'
6'x7'
Operating Frequency
500 or 700 MC
680720 MC
680720 MC
Beam Width in Degrees
(Between half power points
one way.)
Azimuth
12°
6°
12°
12°
Elevation
14°
30°
14°
12°
Antenna Gain
22 db
22 db
22 db
22.5 db.
Beam Shift in Degrees
Azimuth
0°
±1.5°
±3°
±3°
Elevation
0°
0°
0°
±3°
An antenna quite similar to the Mark 3, 6 ft. x 6 ft. antenna, was also
used on Radio Set SCR296 for the Army. This equipment was similar to
the Mark 3 in operating characteristics but was designed mechanically for
fixed installations at shore points for the direction of coast artillery gun
fire. For these installations the antenna was mounted on an amplidyne
controlled turntable located on a high steel tower. The entire antenna and
turntable was housed within a cylindrical wooden structure resembling a
water tower. Equipments of this type were used as a part of the coastal
defense system of the United States, Hawaiian Islands, Aleutian Islands
and Panama.
" Written by W. H. C. Higgins.
""Early Fire Control Radars for Naval Vessels," W. C. Tinus and W. H. C. Higgins,
B. S. T. J.
298
BELL SYSTEM TECHNICAL JOURNAL
14.7 A Shipborne A nti Aircraft Fire Control Antenna}^
A Shipborne Anti Aircraft Fire Control Antenna is shown in Fig. 51.
This antenna consists of two main horizontal cylindrical parabolas in each
"^t3l ■•#*»
h:
L' Vl^ '33®' ^j^SsF ^m
jE'^^^^CI?)^^ ^^ ^^^^^ ^^^^^ ^_^ ^j^
Fig. 48— Mark 1 Antenna.
of which two groups of four halfwave dipoles are mounted with their axes
in a horizontal line at the focus of the parabolic reflectors. The four groups
of dipoles are connected by coaxial lines on the back of the antenna to a lobe
16 Written by C. A. Warren.
RADAR ANTENNAS
299
switcher, which is a motor driven capacitor that has a single rotor plate and
four stator plates, one for each group of dipoles. The phase shift intro
duced into the four feed lines by the lobe switching mechanism causes the
antenna beam to be "lobed" or successively shifted to the right, up, left
and down as the rotor of the capacitor turns through 360 degrees.
Mounted centrally on the front of the antenna at the junction of the two
parabolic antennas is a smaller auxiliary antenna consisting of two dipole
elements and a parabolic reflector, the purpose of which is to reduce the
minor lobes that are present in the main antenna beam. The auxiliary
Fig. 49 — Mark 3 Radar Antenna on Battleship New Jersey.
antenna beam is not lobe switched and is sufficiently broad in both the
horizontal and vertical planes to overlap both the main antenna beam and
the first minor lobes. The auxiliary antenna feed is so designed that its
field is in phase with the field of the main beam of the main antenna. This
causes the feed of the auxiliary antenna to "add" to the field of the main
antenna in the region of its main beam, but to subtract from the field in the
region of its first minor lobes. This occurs because the phase of the first
minor lobes differs by 180 degrees from that of the main beam. As a result,
the field of the main beam is increased and the first minor lobes are greatly
300
BELL SYSTEM TECHNICAL JOURNAL
reduced. By re(lucin<f these minor lobes to a low value, the region around
the main beam is free of lobes, thus greatly reducing the possibility of false
tracking due to "cross overs" between the main beam and the minor lobes.
14.8 The Polyrod Fire Control Antenna
The Polyrod Fire Control antenna is an arra}' scanner emplo}ing essen
tially the same principles as those used in the multii)le unitsteerable antenna
Fig. 50 — Mark 4 Radar Antenna on Ikittleship Tennessee.
system (MUSA) developed before the war for shortwave transatlantic
telephony. Some of these principles have been discussed in Sec. 12.2.
That they could be applied with such success in the microwave region was
due to a firm grounding in waveguide techniques, to the invention of the
polyrod antenna and the rotary phase changer, and especially to excellent
technical work on the part of research, development and production person
nel. It is perhaps one of the most remarkable achievements of wartimq
RADAR ANTENNAS
301
radar that the polyrod antenna emerged to fill the rapid scanning need a
early and as well developed as it did.
The Polyrod Fire Control antenna is a horizontal array of fourteen identi
cal fixed elements, each element being a vertical array of three polyrods.
Energy is distributed to the elements through a waveguide manifold. The
phase of each element is controlled and changed to produce the desired scan
by means of thirteen rotary phase changers. These phase shifters are
J 1 ly ""f~ Tf?ANSMISS10N
MINOR LOBE SUPPRESSOR ANTENNA '—MMN ANTENNA LINE
Fig. 51. — Shi[)borne AntiAircraft Fire Control Antenna
geared together and driven synchronously. Figure 52 is a schematic
diagram of the waveguide and phase changer circuits.
Figure 39 shows an experimental polyrod antenna under test at Holmdel.
Figure 53 is another view of the Polyrod antenna.
14.9 The Rocking Horse Fire Control Antenna
It was long recognized that an important direction of Radar develop
ment lay towards shorter waves. This is particularly true for fire
control antennas where narrow, easily controlled beams rather than great
ranges are needed. The Polyrod antenna had pretty thoroughly demon
302
BELL SYSTEM TECHNICAL JOURNAL
strated the value of rapid scanning, yet the problem of producing a rapid
scanning higher frequency antenna of nearly equal dimensions was a new
and different one.
Several possible solutions to this problem were known. The array
technique applied so effectively to the polyrod antenna could have been
applied here also, but only at the expense of many more elements and
greater complexity.
After much preliminary work it was finally concluded that a mechanically
scanning antenna, the "rocking horse," provided the best solution to the
higher frequency scanning problem. This solution is practical and relatively
simple.
 DELAY EQUALIZING
WAVE GUIDE LENGTHS
UNIT ANTENNAS
(VERTICAL POLYROD TRIDENTS)
WAVE GUIDE
DISTRIBUTING MANIFOLD
WITH ROTARY PHASE CHANGERS
(720° PHASE CHANGE PER REV.)
INPUT
Fig. 52. — Schematic Diagram of Poljrod Fire Control Antenna.
The operation of the rocking horse is described in Sec. 12.1. It is essen
tially a carefully designed and firmly built paraboloidal antenna which
oscillates rapidly through the scanning sector. Its oscillation is dynamically
balanced to eliminate undesirable vibration.
Figure 54 is a photograph of a production model of the rocking horse
antenna.
14.10 The Mark 19 Radar Aiilcmia^'^
In Antiaircraft Fire Control Radar Systems for Heavy Machine Guns
it is necessary to em])loy a highly directive antenna and to obtain continu
ous rapid comparison of the received signals on a number of beam positions
"Sections 14.10, 14.11 and 14.12 were written by F. E. Nimmcke.
RADAR ANTENNAS
303
304
BELL SYSTEM TECHNICAL JOURNAL
as discussed in Section 11.2. Such an antenna is also required to obtain
the high angular precision for antiaircraft fire control. These require
ments are achieved by the use of a conical scanning system. The beam
from the antenna describes a narrow cone and the deviation of the axis
of the cone from the line of sight to the target can be determined and meas
ured by the phase difference between the amplitude modulated received
signal and the frequency of the reference generator associated with the
Fig. 54. — Rocking Horse Fire Control Antenna.
antenna. This information is presented to the pointertrainer at the direc
tor in the form of a wandering dot on an oscilloscope.
The antennas described in sections 14.10, 14.11 and 14.12 were all designed
by the Bell Laboratories as antiaircraft fire control radar systems, particu
larly for directing heavy machine guns. They were designed for use on all
types of Naval surface warships.
In Radar Kquii)ment Mark 19, the first system to be associated with the
control of 1.1 inch and 40 mm anliaircraft machine guns, the antenna was
designed for operation in the 10 cm region. This antenna consisted of a
spinning half dipole with a coaxial transmission line feed. The antenna
RADAR ANTENNAS
305
was driven by 115volt, 60 cycle, single phase motor to which was coupled
a twophase reference voltage generator. The motor rotated at approxi
mately 1800 rpm which resulted in a scanning rate of 30 cycles per second.
This antenna was used with a 24inch spun steel parabolic reflector which
provided, at the 3 db point, a beam width of approximately 11° and a beam
shift of 8.5° making a total beam width of approximately 20° when scan
ning. The minor lobes were down more than 17 db (one way) from the
maximum; and the gain of this antenna was 21 db. This antenna assembly
JUNCTION BOX
'M
PARABOLOIDAL
REFLECTOR
Fig. 55— Mark 19 Ant^
was integral with a transmitterreceiver (Fig. 55) which was mounted on
the associated gun director. Consequently, the size of the reflector was
limited by requirements for unobstructed vision for the operators in the
director. As a matter of fact, for this type of radar system serious con
sideration must be given to the size and weight of the antenna and asso
ciated components.
14.11 The Mark 28 Radar Antenna
The beam from the antenna used in Radar Equipment Mark 19 was
relatively broad and to improve target resolution, the diameter of the
306
BELL SYSTEM TECBNICAL JOURNAL
reflector for the antenna in Mark 28 was approximately doubled. The
Mark 28 is a 10 cm system and employs a conical scanning antenna similar
to that described for Mark 19. The essential difference is that the spun
steel parabolic reflector is 45 inches in diameter which provides a beam
width of ai)pr<).\imately 6.5° and a beam shift of 4.5° making a total of 11°.
Fig. 56 — Mark 28 Antenna Mounted on 40 MM Gun.
The minor lobes are down more than 17 db (one way) from the maximum;
and the gain of this antenna is 26 db. It was found necessary to perforate
the reflector of this dimension in order to reduce deflection caused by gun
blast and by wind drag on the antenna assembly. The antenna assembly
for Radar Equipment Mark 28 is shown in Fig. 56. This assembly i§
shown mounted on a 40 mm Gun.
i?^ DARAN TENNA S 307
14.12 .1 3 CM AntiAircraft Radar Antenna.
To obtain greater discrimination between a given target and other targets,
or between a target and its surroundings, the wavelength was reduced to
the 3 cm region. An antenna for this wavelength was designed to employ
the conical scan principle. In this case the parabolic reflector was 30 inches
in diameter and transmitted a beam approximately 3° wide at the 3db point
with a beam shift of 1.5° making a total of 4.5° with the antenna scanning.
The minor lobes are down more than 22 db (one way) from the maximum;
and the gain of this antenna is ?)5 db.
In the 3 cm system in which a Cutler feed was used, the axis of the beam
was rotated in an orbit by "nutation" about the mechanical axis of the
antenna. This was accomplished by passing circular waveguide through
the hollow shaft of the driving motor. The rear end of the feed (choke
coupling end) was fixed in a ball pivot while the center (near the reflector)
was off set the proper amount to develop the required beam shift. This
off set was produced by a rotating eccentric driven by the motor. The
latter was a 440 volt, 60 cycle, 3 phase motor rotating at approximately 1800
rpm which resulted in a scanning rate of 30 cycles per second. The two
phase reference voltage generator was integral with the driving motor.
It was found necessary at these radio frequencies to use a cast aluminum
reflector and to machine the reflecting surface to close tolerances in order to
attain the consistency in beam width and beam direction required for
accurate pointing. An antenna assembly for the 3 cm antiaircraft radar
is shown in Fig. 57.
15. Land Based Radar Antennas
15.1 The SCR545 Radar ''Search'' and "Track" Antennas'''
The SCR545 Radar Set was developed at the Army's request to meet
the urgent need for a radar set to detect aircraft and provide accurate tar
get tracking data for the direction of antiaircraft guns.
This use required that a narrow beam tracking antenna be employed to
achieve the necessary tracking accuracy, furthermore, a narrow^ beam
antenna suitable for accurate tracking has a very limited field of view and
requires additional facilities for target acquisition. This was provided by
the search antenna which has a relatively large field of view and is provided
with facilities for centering the target in its field of view. These two an
tennas are integrated into a single mechanical structure and both radar axes
coincide.
The "Search" antenna operates in the 200 mc band and is com
" Section 15.1 was written by A. L. Robinson.
308
BELL SYSTEM TECHNICAL JOURNAL
posed of an array of 16 quarter wave dipoles spaced 0.1 wavelength
in front of a flat metal refletlor. All feed system lines and impedance
matchinj,' (Icxiccs arc made uj) of coaxial transmission line sections. The
array is divided into four quarters, each being fed from the lobe switching
mechanism. This division is required to i)ermit lobe switching in both
horizontal and vertical planes. The function of the lobe switching mecha
3C'M Anti.\irciaft Radar Antenna.
nism is to introduce a particular phase shift in the excitation of the elements
of one half of the antenna with respect to the other half. The theory of
this tyjjc of lobe switching is discussed in section 11.1. The antenna beam
spends a])j)roximately one quarter of a lobing cycle in each one of the four
lobe positions. Each of the four lobe positions has the same radiated field
intensity along the antemia axis and therefore when a target is on axis
equal signals will be received from all four lobe positions.
RADAR ANTENNAS
309
The "Track" antenna operates in the 10 cm. region and consists of a reflec
tor which is a parabola or revolution, 57 inches in diameter, illuminated by a
source of energy emerging from a round waveguide in the lobing mechanism.
Conical lobing is achieved by rotating the source of energy around the
parabola axis in the focal plane of the parabola. Conical lobing is discussed
in section 11.2. The round waveguide forming the source is filled with a
specially shaped polystyrene core to control the illumination of the para iola
and to seal the feed system against the weather. The radio frequency power
is fed through coaxial transmission line to a coaxialwaveguide transition
which is attached to the lobing mechanism.
The "Search" and "Track" antenna lobing mechanisms are synchronized
and driven by a common motor.
The radio frequency power for both antennas is transmitted through a
single specially constructed coaxial transmission line to the common antenna
structure, where a coaxial transmission line filter separates the power for
each antenna.
Figure 58 is a photograph of a production model of the SCR545 Radar
Set. The principal electrical characteristics of the antennas are tabulated
below:
Antennas
Search
Track
Gain
14.5 db
30 db
Horizontal Beamwidth
23.5°
5°
Vertical Beamwidth
25.5°
5°
Polarization
Horizontal
Vertical
Type of Lobing
Lobe switching
Conical lobing
Angle between lobe positions
10°
3°
Lobing rate
60 cycles/sec.
60 cycles/sec.
The SCR545 played an important part in the Italian campaign, particu
larly in helping to secure the Anzio Beach Head area, as well as combating
the "V" bombs in Belgium. However the majority of SCR545 equip
ments were sent to the Pacific Theater of Operations and played an im
portant part in operations on Leyte, Saipan, Iwo Jima, and Okinawa.
15.2 The AN/TPSIA Portable Search Antenna^
In order to provide early warning information for advanced units, a light
weight, readily transportable radar was designed under Signal Corps contract.
i« Written by R. E. Crane.
310
BELL SYSTEM TECHNICAL JOURNAL
«rV
RADAR ANTENNAS
311
The objective was to obtain as long range early warning as possible with
moderate accurracy of location. Emphasis was placed on detection of low
flying planes.
The objectives for the set indicated that the antenna should be built
as large as reasonable and placed as high as reasonable for a portable set.
Some latitude in choice of frequency was permitted at first. For rugged
ness and reliability reasons which seemed controlling at the time, the fre
quency was pushed as high as possible with vacuum tube detectors and
R.F. amplifiers. This was finally set at 1080 mc.
Fig. 59— AN/TPSIA Antenna.
The antenna as finally produced was 15 ft. in width and 4 ft. in height"
The reflecting surface was paraboloidal. The mouth of the feed horn was
approximately at the focus of the generating parabola. The feedhorn
was excited by a probe consisting of the inner conductor of the coaxial
transmission line extended through the side of the horn and suitably shaped.
To reduce side lobes and back radiation the feedhorn was dimensioned to
taper the illumination so that it was reduced about 10 db in the horizontal
and vertical planes at the edges of the reflector. Dimensions of probe and
exact location of feed, etc. were determined empirically to secure acceptable
impedance over the frequency band needed. This band, covered by spot
frequency magnetrons, was approximately ±2.5% from mid frequency.
Figure 59 shows the antenna in place on top of the set.
312 BELL SYSTEM TECHNICAL JOURNAL
The characteristics of this antenna are summarized below:
Gain 27.3 db.
Horizontal Half Power Bcamwidth 4.4°
Vertical Half Power Beamwidth 12.6°
Vertical Beam Characteristic Symmetrical
Polarization Horizontal
Impedance (SWR over ±2.5% <4.0db
band)
16. Airborne Rad.vr Antennas
16.1 The AX APS4 Anten)ia^
AN/APS4 was designed to provide the Navy's carrierbased planes
with a high performance high resolution radar for search against surface
and airborne targets, navigation and intercej^tion of enemy planes under
conditions of fog and darkness. For this service, weight was an all im
portant consideration and throughout a production schedule that by \"J
day was approaching 15,000 units, changes to reduce weight were con
stantly being introduced. In late production the antenna was responsible
for 19 lbs. out of a total equipment weight of 164 lbs. The military require
ments called for a scan covering 150° in azimuth ahead of the plane and 30°
above and below the horizontal plane in elevation. To meet this require
ment a Cutler feed and a parabolic reflector of 6.3" focal length and 14"
diameter was selected. Scanning in azimuth was performed by oscillating
reflector and feed through the required 150° while elevation scan was per
formed by tilting the reflector. Beam pattern was good for all tilt angles.
In early flight tests the altitude line on the B scope due to reflection from
the sea beneath was found to be a serious detriment to the performance of
the set. To reduce this, a feed with elongated slots designed for an elliptical
reflector was tried and found to give an improvement even when used with
the approximately round reflector. The elliptical reflector was also tried,
but did not improve the performance sufficiently to justify the increased
size.
As will be noted in Fig. 60, the course of the mechanical development
brought the horizontal pivot of the reflector to the form of small ears pro
jecting through the ])arabola. No appreciable deterioration of the beam
{)attern due to this unorthodox expedient was noted.
The equipment as a whole was built into a bombshaped container hung
in the bomb rack on the underside of the wing. Various accidents resulted
in this container being torn ofT the wing in a crash landing in water or
dropped on the deck of the carrier. After these mishaps, the equipment
was frequently found to be in good working order with little or no repair
required.
» Written by F. C. Willis.
RADAR ANTENNAS
313
Gain
28 db
Beamwidth
6° approx. circular
Polarization
Horizontal
Scan
Mechanical
Scanning Sector
Azimuth 150°
Scanning Sector
Elevation 60°
Scanning Rate
one per sec.
Total weight
19 lbs.
Fig. 60— AX/.\PS4 Antenna.
16.2 The SCR520, SCR717 and SCR720 Antennas'
The antenna shown in Fig. ol is typical of the type used with the SCR520
and SCR720 aircraft interception (night fighter) airborne radar equip
ment, as well as the SCR717 sea search and antisubmarine airborne radar
equipment. The parabolic reflector is 29 inches in diameter and produces a
radiation beam about 10° wide. The absolute gain is approximately 25
db. RF energy is supplied to a pressurized emitter through a pressurized
transmission line system which includes a rotary joint located on the ver
so Written by J. F. Morrison,
314
BELL SYSTEM TECHNICAL JOURNAL
tical axis and a tilt joint on the horizontal axis. Either vertical or hori
zontal polarization can be used by rotating the mounting position of the
emitter. Vertical polarization is preferred for aircraft interception work
and horizontal polarization is i)referred for sea search work.
Fig. 61— SCR520 Antenna.
For aircraft interception the military services desired to scan rapidly a
large solid angle forward of the pursuing airplane, i.e. 90° right and left, 15°
below and 50° above the line of flight. The data is presented to the opera
tor in the form of both "B" and ''C" })resentations and for this purpose
potentiometer data takeoffs are provided on the antenna. The reflector
is spun on a vertical axis at a rate of 360 rpm and at the same time it is
RADAR ANTENNAS 315
made to nod up and down about its horizontal axis by controllable amounts
up to a total of 65° and at a rate of 30° per second.
In the sea search SCR717 equipment, selsyn azimuth position data take
offs are provided which drive a PPI type of indicator presentation. The
rotational speed about the vertical axis in this case is either 8 or 20 rpm
as selected by the operator. The reflector can also be tilted about its
horizontal axis above or below the line of flight as desired by the operator.
It wUl be noted that the emitter moves with the reflector and accordingly
it is always located at the focal point throughout all orientations of the
antenna.
16.3 T/ie AN/APQ7 Radar Bombsight Antenna^^
Early experience in the use of bombingthroughovercast radar equip
ment indicated that a severe limitation in performance was to be expected
as the result of the inadequate resolution offered by the then available air
borne radar equipments. This lack of resolution accounted for gross errors
in bombing where the target area was not ideal from a radar standpoint.
To meet this increased resolution requirement in range, the transmitted
pulse width was shortened considerably. In attempting to increase the
azimuthal resolution, higher frequencies of transmission were employed.
This enabled an improvement in azimuthal resolution without resorting to
larger radiating structures, a most important consideration on modern
high speed military aircraft.
To extend the size of the radiating structure without penalizing the air
craft performance, the use of a linear scanning array which would exhibit
high azimuthal resolution was considered. This array was originally con
ceived in a form suitable to mount within the existing aircraft wing and
transmit through the leading edge. As development proceeded, the restric
tions imposed on the antenna structure as well as the aircraft wing design
resulted in the linear array scanner being housed in an appropriate separate
air foil and attached to the aircraft fuselage (Fig. 62).
The above study resulted in the development of the AN/APQ7 radar
equipment, operating at the Xband of frequencies. This equipment
provided facilities for radar navigation and bombing.
The AN/APQ7 antenna consisted of an array of 250 dipole structures
spaced at  wavelength intervals and energized by means of coupling probes
extending into a variable width waveguide. The vertical pattern was
arranged to exhibit a modified esc distribution by means of accurately
shaped "flaps" attached to the assembly.
" Written by L. W. Morrison.
*' A large part of the antenna development was carried out at the M. I. T. Radiation
Laboratory.
316
BELL SYSTEM TECHNICAL JOURNAL
ANTENNA AIRFOIL ASSEMBLY
Fig. 62— AN/APQ7 AntennaMounted on B24 ;Bomber.
CHOKE JOINT
COUPLING
SLIDING
SURFACES
Fig. 63— AN/APQ7 Antenna. Left
Expanded Wave Guide Assembly.
Contracted Wave Guide Assembly. Right —
The scanning of the beam is accomj)lished by varying the width of the
feed waveguide. This is accomplished l)y means of a motor driven actuated
cam which drives a push rod extending along the waveguide assembly back
RADAR ANTENNAS 317
and forth. Toggle arms are attached to this push rod at frequently spaced
intervals which provides the motion for varying the width of the waveguide
while assuring precise parallelism of the side walls throughout its length
(Fig. 63).
The normal range of horizontal scanning exhibited by this linear array,
extends from a line perpendicular to the array to 30° in the direction of the
feed. By alternately feeding each end, a total scanning range of ±30°
from the perpendicular is achieved. Appropriate circuits to synchronize
the indicator for this range are included.
The use of alternate end feed on the AN/APQ7 antenna requires that
the amount of energy fed to the individual dipoles is somewhat less than if a
single end feed is employed.
The AN/APQ7 antenna is 16 feet in length and weighs 180 pounds
exclusive of air foil housing.
The following data applies;
Gain = 32.5 db
Horizontal beamwidth = 0.4°
Vertical beam characteristic = modified csc^
Scan — Array scanning
Scanning Sector — ± 30° Horizontal
Scanning Rate = 45°/second
Acknowledgments
Contributors to the research and development of the radar antennas
described in this paper included not only the great number of people directly
concerned with these antennas but also the many people engaged in general
research and development of microwave components and measuring tech
niques. A complete list of credits, therefore, will not be attempted.
In addition to the few individuals mentioned in footnotes throughout
the paper, the authors would like to pay special tribute to the following
coworkers in the Radio Research Department: C. B. H. Feldman who with
the assistance of D. H. Ring made an outstanding contribution in the
development of the polyrcd array antenna; W. A. Tyrrell for his work on
lobe switches; A. G. Fox, waveguide phase changers; A. P. King, paraboloids
and horn antennas; A. C. Beck, submarine antennas; G. E. Mueller,
polyrods.
Probability Functions for the Modulus and Angle of the
Normal Complex Variate
By RAY S. HOYT
This paper deals mainly with various 'distribution functions' and 'cumulative
distribution functions' pertaining to the modulus and to the angle of the 'normal'
comy)lex variate, for the case where the mean value of this variate is zero. Also,
for auxiliary uses chiefly, the distribution function pertaining to the recijirocal
of the modulus is included. For all of these various probability functions the
paper derives convenient general formulas, and for four of the functions it supplies
comprehensive sets of curves; furthur, it gives a table of computed values of the
cumulative distribution function for the modulus, serving to verify the values
computed by a difTerent method in an earlier paper by the same author.^
Introduction
IN THE solution of problems relating to alternating current networks
and transmission systems by means of the usual complex quantity
method, any deviation of any quantity from its reference value is naturally
a complex quantity, in general. If, further, the deviation is of a random
nature and hence is variable in a random sense, then it constitutes a 'complex
random variable,' or a 'complex variate,' the word 'variate' here meaning
the same as 'random variable' (or 'chance variable' — though, on the whole,
'random variable' seems preferable to 'chance variable' and is more widely
used).
Although a complex variate may be regarded formally as a single ana
lytical entity, denotable by a single letter (as Z), nevertheless it has two
analytical constituents, or components: for instance, its real and imaginary
constituents (X and F); also, its modulus and amplitude (Z and 6).
Correspondingly, a complex variate can be represented geometrically by
a single geometrical entity, namely a plane vector, but this, in turn, has
two geometrical components, or constituents: for instance, its two rec
tangular components (X and F); also, its two polar components, radius
vector and vectorial angle (R = \ Z \ and 6).
This paper deals mainly with the modulus and the angle of the complex
variate,^ which are often of greater theoretical interest and practical im
'"Probabihty Theory and Telephone Transmission Engineering," Bell System Tech
nical Journal, January 1933, which will hereafter be referred to merely as the "1933
paper".
' Throughout the paper, I have used the term 'complex variate' for any 2dimensional
variate, because of the nature of the contemplated applications indicated in the first
318
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 319
portance than the real and imaginary' constituents. The modulus variate
and the angle variate, individually and jointly, are of considerable the
oretical interest; while the modulus variate is also of very considerable
practical importance, and the angle variate may conceivably become of
some practical importance.
The paper is concerned chiefly with the 'distribution functions'^ and the
'cumulative distribution functions' pertaining to the modulus (Sections 3
and 5) and to the angle (Sections 6 and 7) of the 'normal' complex variate,
for the case where the mean value of this variate is zero. The distribution
function for the reciprocal of the modulus is also included (Section 4).
The term 'probability function' is used in this paper generically to include
'distribution function' and 'cumulative distribution function.'
To avoid all except short digressions, some of the derivation work has
been placed in appendices, of which there are four. These may be found
of some intrinsic interest, besides faciUtating the understanding of the
paper.
1. Distribution Function and Cumulative Distribution Function
IN General: Deeinitions, Terminology, Notation, Relations,
AND Formulas
The present section constitutes a generic basis for the rest of the paper.
Let T denote any complex variate, and let p and a denote any pair of
real quantities determining r and determined by t. (For instance, p and
(7 might be the real and imaginary components of r, or they might be the
modulus and angle of t.) Geometrically, p and a may be pictured as gen
eral curvilinear coordinates in a plane, as indicated by Fig. 1.1.
Let T denote the unknown value of a random sample consisting of a
single rvariate, and p' and a' the corresponding unknown values of the
constituents of r'.
Further, let G(p, a) denote the 'areal probability density' at any point
p,a in the p,(7plane, so that G(p,a)dA gives the probability that t falls
in a differential area dA containing the point r; and so that the integral of
paragraph of the Introduction, and also because the present paper is a sort of sequel to
my 1933 paper, where the term 'complex variate' (or rather, 'complex chancevariable')
was used throughout since there it seemed clearly to be the best term, on account of the
field of applications contemplated and the specific applications given as illustrations.
However, for wider usage the term 'bivariate' might be preferred because of its prevalence
in the field of Mathematical Statistics; and therefore the paper should be read with this
alternative in view.
^The term 'distribution function' is used with the same meaning in this paper as in
my 1933 paper, although there the term ' probability law' was used much more frequently
than 'distribution function,' but with the same meaning.
320 BELL SYSTEM TECHNICA L JOURNA L
G(p,(T)dA over the entire p,oplane is equal to unity, corresponding to
certainty.
For the sake of subsequent needs of a formal nature, it will now be as
sumed that G{p,(t) = at all points p,o outside of the pi , P2 , ci , a^ quad
rilateral region in the p,oplane, Fig. 1.1, bounded by arcs of the four heavy
curv'es, for which p has the values pi and p2 and a the values ai and ao ,
with pi and en regarded, for convenience, as being less than p2 and a^ respec
tively. Further, G(p,a) will be assumed to be continuous inside of this
p+dp P^
Pa
Pi
Fig. 1.1 — Diagram of general curvilinear coordinates.
quadrilateral region, and to be noninfinite on its boundary. Hence, for
probability purposes, it will suffice to deal with the open inequalities
Pi < P < P2, (1.1) ai < a < (T2, (1.2)
which pertain to this quadrilateral region excluding its boundary; and thus
it will not be necessary to deal with the closed inequalities pi ^ p ^ P2
and (Ti ^ 0 ^ ao , which include the boundary."*
' The matters dealt with generically in this paragraph may he illustrated b> the fol
lowing two important particular cases, which occur further on, namely:
POLAR COORDINATES: p=r = 7?, <r=0 = angle of r. Then p, = A', = 0,
P2 = Ri = 'X' , <Ti = di = 0, ffi = $2 = 2ir, whence (1.1) and (1.2) become < R < oc
and Q < 6 < lir, respectively.
RECTANGULAR COORDIN.^TES : p = Re r = .v, <r = Im t = y. Then p, = .v, =
— x ^ P2 = X2 = 00, o"! = yi = — =0, 02 = vs = «= , whcucc (1.1) and (1.2) become — oo <
X < <» and — =»_< y < <«, respectively.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 321
A generic quadrilateral region contained within the quadrilateral region
Pi , P2 , 0^1 , o'2 in Fig. 1.1 is the one bounded by arcs of the dashed curves
P3 , Pi , (T3 , (Ti , where ps < p4 and as < <j\ . Here, as in the preceding
paragraph, it will evidently suffice to deal with open inequalities.
Referring to Fig. 1.1, the probability functions with which this paper
will chiefly deal are certain particular cases of the probability functions
P{p, a), P{p I 034) and Q{pz\ , C734) occurring on the right sides of the follow
ing three equations respectively:
p{p < p' < p ^ dp, (J < a' < a + d<r) = P(p,a)dpda, (1.3)
p(p < p < p ^ dp, az < a' < (Ji) = P{p I (T3i)dp, (1.4)
p{pz < p < Pi , (T3 < a' < (Ti) = Q{p3i , 034). (1.5)
These equations serve to define the abovementioned probability functions
occurring on the right sides in terms of the probabilities denoted by the
left sides, each expression p( ) on the left side denoting the probability
of the pair of inequalities within the parentheses. Inspection of these
equations shows that: P(p,(r) is the 'distribution function' for p and a
jointly; P{p \ 034) is a 'distribution function' for p individually, with the
understanding that a' is restricted to the range a^toai ; Qipsi ,o'34) is a
'cumulative distribution function' for p and a jointly.
Since the left sides of (1.3), (1.4) and (1.5) are necessarily positive, the
right sides must be also. Hence, as all of the probability functions occur
ring in the right sides are of course desired to be positive, the differentials
dp and da must be taken as positive, if we are to avoid writing  dp \ and
I (/(T I in place of dp and da respectively.
Returning to (1.3), it is seen that, stated in words, P{p,a) is such that
P{p.a)dpda gives the probability that the unknown values p' and a' of
the constituents of the unknown value r' of a random sample consisting
of a single rvariate lie respectively in the differential intervals dp and da
containing the constituent values p and a respectively. Thus, unless
dpda is the differential element of area, Pip,a) is not equal to the 'areal
probability density,' G{p,a), defined in the fourth paragraph of this section.
In general, if £ is such that Edpda is the differential element of area, then
P(p, a) = EG{p, a). (An illustration is afforded incidentally by Appendix A.)
P{p,a), defined by (1.3), is the basic 'probabiUty function,' in the sense
that the others can be expressed in terms of it, by integration. Thus
^ Thus p in p( ) may be read 'probability that' or 'probabiHty of.'
322
BELL SYSTEM TECHNICAL JOURNAL
P{p I 034) and P{(T I p3i), defined respectively by (1.4) and by the correlative
of (1.4), can be expressed as 'single integrals,' as follows*:
P(p I as,) = f * P(p,a) da, (1.6) P{a \ ps,) = H P{p,a) dp. (1.7)
(?(P34 , (T34), defined by (1.5), can be expressed as a 'double integral,' funda
mentally; but, for purposes of analysis and of evaluation, this will be replaced
by its two equivalent 'repeated integrals':
Q(p3i , Cr 3i)
f
P{p,a) da
dp
= X^ I j ^(P.<^) dp\da, (1.8)
the set of integration limits being the same in both repeated integrals
because these limits are constants, as indicated by Fig. 1.1. On account
of (1.6) and (1.7) respectively, (1.8) can evidently be written formally
as two single integrals:
Q(P34, ^34) = / P(p 1 a34) dp = / P{a\ P34) da, (1.9)
but implicitly these are repeated integrals unless the single integrations in
(1.6) and (1.7) can be executed, in which case the integrals in (1.9) will
actually be single integrals, and these will be quite unlike each other in
form, being integrals with respect to p and a respectively — though of course
yielding a com.m.on expression in case the indicated integrations can be
executed.
The particular cases of (1.4) and (1.5) with which this paper will chiefly
deal are the following three:
p{p < p' <p + dp, a, <a' < a^) = P{p  a^:) dp = P (p) dp, (1.10)
Pipi <p' <p,a,<a' < a.) = Q{< p,a,o) ^ Q{p), (1.11)
p{p <p' <p2,ai<a' < 0,)  Q{> p,an) = (?*(p). (1.12)
^ The singleintegral formulation in (1.6) can be written down directly by mere inspec
tion of the left side of (1.4). Alternatively, (1.6) can be obtained by representing the left
side of (1.4) by a repeated integral, as follows:
Pip I (^34.) dp =
pp\dp P r'Ci
•' P L"'''3
Pip, a)da
dp =
f Pip, <T)da
dp,
whence (1.6); the last equality in the above chain equation in this footnote evidently
results from the fact that, in general
fix)dx = f(x)dx, since each side of this equa
tion represents dA, the differential element of area under the graph of /(.v) from x to
X f dx.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 323
In each of these thice equations the very abbreviated notation at the ex
treme right will be used wherever the function is being dealt with exten
sively, as in the various succeeding sections. Such notation will not seem
unduly abbreviated nor arbitrary if the following considerations are noted:
In (1.10), «T]2 corresponds to the entire effective range of a, so that P(p \ o]2)
is the 'principal' distribution function for p. Similarly, in (1.11), Q(< p,on)
is the 'principal' cumultive distribution function for p. In (1.12), the star
indicates that Q*ip) is the 'complementary' cumulative distribution func
tion, since Q(p) + Q*(p) = Q(pi2 , 012) = 1, unity being taken as the measure
of certainty, of course.
For occasional use in succeeding sections, the defining equations for
the probabiUty functions pertaining to four other particular cases will
be set down here:
p{p<p' <P + dp, (Tx<a' <a) = P(p I < (t) dp, • (1.13)
p(p< p' < p{ dp, a < a' < (X2) ^ P(p \ > a) dp, (1.14)
Pip, <p' <p,a,<a' <ct) = Q{< p, < a), (1.15)
Pip <p' < p2 ,ai<a' <a) = Qi> p, < a). (1.16)
It may be noted that (1.13) and (1.14) are mutually supplementary, in the
sense that their sum is (1.10). Similarly, (1.15) and (1.16) are mutually
supplementary, in the sense that their sum is ()(p]?,< a) = Qi< (r,pi2),
which is the correlative of (1.11).
This section will be concluded with the following three simple trans
formation relations (1.17), (1.18) and (1.19), which will be needed further
on. They pertain to the probability functions on the right sides of equa
tions (1.3), (1.4) and (1.5) respectively, h and k denote any positive real
constants, the restriction to positive values serving to simplify matters
without being too restrictive for the needs of this paper.
P{hp,ka) = ^^P{p,<t), (1.17)
P{hp\k<rz,) =\Pip\ <^34), (118)
Q{hpu,kazi) = Q{pzi, (T34). (1.19)
Each of the three formulas (1.17), (1.18), (1.19) can be rather easily
derived in at least two ways that are very different from each other. One
way depends on probability inequality relations of the sort
p{t<t'<t'Vdt) = p{gt<gt'<gt^d[gt]), (1.20)
p{h<t'<U) = p{gh<gl'<gh), (1.21)
324
BELL SYSTEM TECHNICAL JOURNAL
where / stands generically for p and for a, and g is any positive real constant,
standing generically for h and for k; (1.20) and (1.21) are easily seen to be
true by imagining every variate in the universe of the /variates to be
multiplied by g, thereby obtaining a universe of (g/)variates. A second
way of deriving each of the three formulas (1.17), (1.18), (1.19) depends on
general integral relations of the sort
( f{t) di = ^^ r fit) d{gt) ^u" f () d\. (1.22)
•'« g ^ga g Jga \g/
A third way, which is distantly related to the second way, depends on the
use of the Jacobian for changing the variables in any double integral; thus,
P(p,<r)
dXdn
dpdcr
=
d{p,(T)
= 1 ^
a(p,cr)
d(X,M)
(1.23)
the first equality in (1.23) depending on the fact that the two sets of vari
ables and of differentials have corresponding values and hence are so re
lated that
p(p<p'<p\dp, a<y<(T\da) = p(\<y<X\d\ m<m'<M+^/)u), (124)
whence
P(p,a) 1 dpd<j I = Pi\,fi) I dXdfjL .
2. The Normal Complex Variate and Its Chief Probability Functions
The 'normal' complex variate may be defined in various equivalent ways
Here, a given complex variate z = x \ iy will be defined as being 'normal'
if it is possible to choose in the plane of the scatter diagram of s a pair of
rectangular axes, u and r, such that the distribution function P{u,v)
for the given complex variate with respect to these axes can be written in
the form^
P{u,v)
1
ZTTOuOv
exp
2Sl
41
2Sl\
P(u)Piv).
(2.1)
We shall call w = u \ iv the 'modified' complex variate, as it represents
the value of the given complex variate g — .t f iy when the latter is referred
to the w,raxes; P(u) and P{v) are respectively the individual distribution 1
functions for the u and r components of the modified complex variate ; and
■^ Defined by equation (L3) on setting p = it and a = v.
"This ecjuation is (12) of my 1933 paper. It can he easily verified tliat the (double)
integral of (2.1) taken over the entire n, iiplane is equal to unity.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 325
Su and Sv are distribution parameters called the 'standard deviations' of
w and V respectively. If / stands for u and for v generically, then
P(t) = 7^
vfe,^^;]' <'•'' ^' = /_j'^«'" P.3)
From the viewpoint of the scatter diagram, the distribution function
Pin,v) is, in general, equal to the 'areal probability density' at the point
u,v in the plane of the scatter diagram, so that the probabihty of falling
in a differential element of area dA containing the point ti,v is equal to
P{u,v)dA ; similarly, P{;u) and P{v) are equal to the component probability
densities. In particular, the probability density is 'normal' when P{u,v)
is given by (2.1).
Geometrically, equation (2.1) evidently represents a surface, the normal
'probability surface,' situated above the u, rplane; and P{u, v) is the ordinate
from any point u,v in the u,vp\a.ne to the probability surface.
The M,T'axes described above will be recognized as being the 'principal
central axes,' namely that pair of rectangular axs which have their origin
at the 'center' of the scatter diagram of s = x + iy and hence at the center
of the scatter diagram of u> — u \ iv, so that w = 0, and are so oriented
in the scatter diagram that m; = (whereas 2^0 and xy 9^ 0, in general).
In equation (2.1), which has been adopted above as the analytical basis
for defining the 'normal' complex variate, the distribution parameters are
Su and Sv ; and they occur symmetrically there, which is evidently natural
and is desirable for purposes of definition. Henceforth, however, it will be
preferable to adopt as the distribution parameters the quantities S and b
defined by the pair of equations
S' = Sl + Sl , (2.4) bS' = Sl  S; , (2.5)
whence
, __ Su Sy _ 1 [Sy/Su) ,r. ,,.
»Jm "r Sy 1 \ {Sy/SuJ
From (2.4), S is seen to be a sort of 'resultant standard deviation.' The
last form of (2.6) shows clearly that the total possible range of b is
— l^b^l, corresponding to '^^Sy/Su^O.
The pair of simultaneous equations (2.4) and (2.5) give
2Sl = {\ + b)S, (2.7) 2^; = (1^.)^, (2.8)
which will be used below in deriving (2.11).
'Equations (2.4) and (2.6) are respectivelj (14) and (13) of my 1933 paper.
326 BELL S YSTEM TECH NIC A L JOURNA L
With the purpose of reducing the number of parameters by 1 and of
dealing with variables that are dimensionless, we shall henceforth deal
with the 'reduced' modified variate W = U ■\ iV defined by the equation
W ^ w/S = u/S + iv/S = U + iV. (2.9)
Thus we shall be directly concerned with the scatter diagram of W =
U + iV instead of with that oi w = u \ iv.
The distribution function P(L'*,T') for the rectangular components U
and 1' of any complex variate W — U \ iV is defined by (1.3) on setting
p = i' and cr = T; thus,
p{u,v)dudv = p{U<u'<u\du,v<r'<vidV). (2.10)
When the given variate z — x \ iy is normal, so that the modified variate
11) — u {■ iv is normal, as represented by (2.1), then, since S is a mere con
stant, the reduced modified variate W — U { i]' defined by (2.9) will
evidently be normal also, though of course with a different distribution
parameter. Its distribution function P(t',l ) is found to have the formula
1 r t/2 F2 ■
where P{1) and P{V) are the component distribution functions:
t/2
= F{U)P{V), (2.11)
^(^) = vOT)^r
P(V) = ./..; . ^exp[^4
(2.12)
(2.13)
\/ir(l  b)
These three distribution functions each contain only one distribution
parameter, namely b; moreover, the variables U = u/S and 1' = v/S are
dimensionless.
' The distribution function P{R,6) for the polar components R and 6 of
any complex variate W = R{cos 6 \ i sin 6) is defined by (1.3) on setting
p = R and a — 6; thus
P{R,e)dRd9 = p{R<R'<R^dR. d<d' <d\de). (2.14)
For the case where 11' is 'normal,' it is shown in Appendix A that
R [ R'
VT
^'(^'^) = Wr^T. exp ^fi:2 (1  & cos 2d)
(2.15)
exp[L(l  6 cos 20)], (2.16)
"This formula can be obtained from (2.1) by means of (2.7), (2.8), (2.9) and (1.17)
after specializing (1.17) by the substitutions p = u,a = v and h = k = 1/5. It is (16)
of my 1933 paper, but was given there without proof.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 327
where
L= Ry{\b'). (2.17)
In P{R,d) it will evidently suffice to deal with values of 6 in the first
quadrant, because of symmetry of the scatter diagram.
The fact that P(R,6) depends on 6 as a parameter when W is 'norma]'
may be indicated explicitly by employing the fuller symbol P{R,d;b)
when desired; thus the former symbol is here an abbreviation for the latter.
In P{R,d) = P(R, 6; b) it will suffice to deal with only positive values of
b, that is, with O^b^l (whereas the total possible range of b is — l^^^l).
For (2.15) shows that changing b to —b has the same effect as changing 2d
to 7r±2e, or d to T/2±d; that is, P{R,d; b) ^ P(R, ir/2±d; b).
Seven formulas which will find considerable use subsequently are obtain
able from the integrals corresponding to equations (1.13) to (1.16), by setting
p = R and a = 6 or else p = 6 and c = R, whichever is appropriate, and
thereafter substituting for P{R,6) the expression given by (2.16), and
lastly executing the indicated integrations wherever they appear possible."
The resulting formulas are as follows:
P(R \ < d) = y^ exp(Z) / expibL cos 26) dd, (2.18)
T Jo
(2.19)
P{e \ < R) = ^^ ~ ^' 1  exp[i:(l  b cos 2d)]
2ir I — b cos 20
P(e \> R) = ^^ ~ ^' exp[£(l  b cos 29)]
2t 1 — b cos 26
(2.20)
dR (2.21)
Q{< R, < 6) =  [ \ \/l exp(L) [ exp{bL cos 26) dd
TT Jo L "^O
Vnili r" 1  exp[I(l  b cos 26)]
~27~ io 1  b cos 26 ^^' ^^^^^
Q(> R, < 6) = I VL exp(L) j exp {bL cos 26) dd dR (2.23)
Ztt Jo 1 — b cos 26
Formulas (2.21) to (2.24) are obtainable also by substituting (2.18) to
(2.20) into the appropriate particular forms of (1.9).
When a ^range of integration is 0to5(7r/2), where q = 1, 2, 3 or 4, this
" Except that in (2.22) the part 1/(1 — b cos 26) is integrable, as found in Sec. 7,
equations (7.6) and (7.7).
328 BELL S YSTEM TECH NIC A L JOURNA L
range can be reduced to 0to7r/2 provided the resulting integral is mul
tiplied by q; that is,
/«5(7r/2) ^jr/2
/ F{e)(W = q / F{e)dd, (2.25)
Jo •'0
because of symmetry of the scatter diagram.
3. The Distribution Function for the Modulus
The distribution function P{R  dv2) = F{R) for the modulus R of any
complex variate IT = R(cos 6 + / sin 0) is defined by equation (1.10) on
setting p = R, a = 9, ffi = 6] — and (r2 — 62 — 2ir; thus
P{R)dR = p(R<R'<R+dR, (xe'KlTv). (3.1)
An integral formula for F(R) is immediately obtainable from (1.6) by
setting p = R, o — 6, (Ti = ai = 61 = and 04 = a^ ~ S2 = 2x; thus
F{R) = [ F{R,d) do. (3.2)
Jo
The rest of this section deals with the case where \V = R(cos 6 + / sin 6)
is 'normal.' Since this case depends on i as a parameter, F(R) is here an
abbreviation for F{R;h). A formula for F{R;b) can be obtained by sub
stituting F{R, 6) from (2.15) into (3.2) and executing the indicated integra
tion by means of the known Bessel function formula
i:
exp(r} cos \f/) dip = 7r/o(r/), (3.3)
/o( ) being the socalled 'modified Bessel function of the first kind,' of
order zero.^' The resulting formula is found to be^^
2R
.1  d^Ti
bR^
 b'
(3.4)
This can also be obtained as a particular case of the more general formula
(2.18) by setting 6 — 2t in the upper limit of integration and then apply
ing (3.3).
In F(R;b) it will suffice to deal with positive values of b, that is, with
U^6^1, as (3.4) shows that F(R; b) = F{R;b).
12 It may be recalled that /o(c) = /o(/), and in general that /„(;) = i"Jn{i~).
In the list of references on Bessel functions, on the last page of this paper, the 'modified
Bessel function' is treated in Ref. 2, p. 20; Ref. 3, p. 102; Ref. 4, p. 41; Ref. 1, p. 77.
Regarding formula {3.3), see Ref. 1, p. 181, Eq. (4), i. = 0; Ref. 1, p. 19, Eq. (9), fourth
expression, p = 0; Ref. 2, p. 46, Eq. (10), n = 0; Ref. 3, p. 164, Eq. 103, n = 0.
^' This formula was given in its cumulative forms, / P{R; b)dR, as fornuilas (Sl.A)
and (53A) of the unpublished .\ppendix A to my 1933 paper.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATi: 32^
It will often be advantageous to express P^; 6 in terms of b and one or
the other of the auxiliary variables L and T defined by the equations
^ = r^2' (35) ^ = ^^ = 1^2 (^6)
Formula (3.4) thereby becomes, respectively,
P{R;b) = 2VLexp{L)h{bL), (3.7)
P(R;b) = 2 y^l exp[^j h{T). (3.8)
Formula (3.8) will often be preferable to (3.7) because the argument of
the Bessel function in (3.8) is a single quantity, T.
Because tables of /o(V) are much less easily interpolated than tables of
Mo(X) defined by the equation
Mo(X) = exp(X)h(X), (3.9)
extensive tables of wiiich have beeo published," it is natural, at least for
computational purposes, to write (3.4) in the form
2R r R' 1
Vl  b^
Mo
• bR'
1 b'
(3.10)
For use in equation (3.16), it is convenient to define here a function
Mi(X) by the equation
M,(X) = exp(A')/i(X), (3.11)
corresponding to (3.9) defining Mo{X). Mi(X) has the similar property
that it is much more easily interpolated than is Ii(X); and extensive tables
of Ml (A') are constituent parts of the tables in Ref. 1 and Ref. 6.
The quantity bR/{l — b') = T, which occurs in (3.4) and (3.8) as the
argument of /o( ), and in (3.10) as the argument of Mo{ ), evidently
ranges from to co when R ranges from to co and also when b ranges
from to 1. Formula (3.10) is suitable for computational purposes for all
values of the abovementioned argument bR~/(l — b'^) = T not exceeding
the largest values of X in the abovecited tables in Ref. 1 and Ref. 6. For
larger values of the argument, and partiularly for dealing with the limiting
i* Ref. 1, Table II (p. 698713), for X = to 16 by .02. Ref. 6, Table VIII (p. 272
283), for A^ = 5 to 10 by .01, and 10 to 20 by 0.1. Each of these references conveniently
includes a table of exp(A^) whereby values of /o(A') can be readily and accurately evalu
ated if desired. Values of /o(A') so obtained would enable formulas (3.4), (3.7) and (3.8)
of the present paper to be used with high accuracy without any difficult interpolations,
since the table of exp(A'') is easily interpolated by utilizing the identity exp(A'i ) A'2) =
exp(Ai) exp(A^2).
330 BELL SYSTEM TECHNICAL JOURNAL
case where the argument becomes infinite, formula (310) — and hence (3.4) —
may be advantageously written m the form
where
No{X) = V2^exp(X)/o(X) = \/2^Mo{X), (3.13)
an extensive table of which has been published.'^ The natural suitabiUty
of the function A^o(^) for dealing with large values of A' is evident from
the structure of the asymptotic series for No{X), for sufficiently large values
of X, which runs as follows:^®
iVo(X) ~ 1 + jl^ + jl^, + jl^, + . . . , (3.14)
whence it is evident that
No{oo) = 1. (3.15)
For use in Appendix C, it is convenient to define here a function A^i(A")
by the equation"
Ni{X) = V'2^exp(X)/i(X)  V2^M,{X), (3.16)
corresponding to (3.13) defining No(X), with Mi(X) defined by (3.11).
The asymptotic series for Ni{X), which will be needed in Appendix C, is^^
NiiX)  1  3
whence it is evident that
1 . 05) (l 5)(37) 1
.1!8X 2I(8X)2^ 31(8X)» ^ J' ^^ ^
Ni{oo) = 1. (3.18)
When b is very nearly but not exactly equal to unity, so that
bR" R" R"
(3.19)
1^2 162 2(1  6) '
it is seen from (3.4) that P{R;b) is, to a very close approximation, a function
15 Ref. 7, pp. 4572, for X = 10 to 50 by 0.1, 50 to 200 by 1, 200 to 1000 by 10, and
for various larger values of X.
16 Ref. 1, p. 203, with (u, m) defined on p. 198; Ref. 5, p. 366; Ref. 2, p. 58; Ref. 3, p.
163, Eq. 84; Ref. 4, pp. 48, 84.
1^ N i{X) is tabulated along with N^iX) in Ref. 7 already cited in connection with equa
tion (3.13).
" Ref. 1, p. 203, with {v, m) defined on d. 198; Ref. 5, p. 366; Ref. 2, p. 58; Ref. 3, p.
163, Eq. 84.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE
331
of only a single quantity, which may be any one of the three very nearly
equal expressions in (3.19) — but the last of them is evidently the simplest.
Fig. 3.1 gives curves of P(,R;b), with the variable R ranging continuously
Fig. 3.1 — Distribution function for the modulus {R = to 2.8).
from to 2.8 and the parameter b ranging by steps from to 1 inclusive,
which is the complete range of positive b. Fig. 3.2 gives an enlargement
(along the i?axis) of the portion of Fig. 3.1 between R — and R = 0.4,
332
BELL SYSTEM TECHNICAL JOURNAL
l/\
V\
\
0)
6
II
X)
A
1
\\
1
m
\\
l\
A
\
^
I \
\\
\=

w
I
\V\\
o
6
y 1
\
\
\V
\
M
\
L d\
V
\
/)
eo\
d\
\\
\\
\//
\
\
\ >
\\
A/
x
\
\V
d
A
q
\
\
V
\\\
' '
/
\
^
\
V
\^
\
d
/
\
\
\
\
\V
\
y
\
\
V
\
V
^
d
w
\
y.
\
\
\,
\
/^
k^
\
N,
\
\
\\
%
1
\
>«^
X
\
s\
1
^
Q
g: d"
d *«=
>i
^
^
^
1
^
' ■
is
^
^
DISTRIBUTION FUNCTION, P(R;b)
Fig. 3.2— Distribution function for the modulus (/^ = to 0.4).
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 333
and includes therein curves for a considerable number of additional values
of b between 0.9 and 1 so chosen as to show clearly how, with b increasing
toward 1, the curves approach the curve for 5 = 1 as a limiting particular
curve; or, conversely, how the curve iov b — 1 constitutes a limiting par
ticular curve — which, incidentally, will be found to be a natural and con
venient reference curve. This curve, iov b = 1, will be considered more
fully a little further on, because it is a limiting particular curve and be
cause of its resulting peculiarity at i? = 0, the curve iov b = 1 having at
R = a. projection, or spur, situated in the P{R;b) axis and extending from
0.7979 to 0.9376 therein (as shown a little further on).
The formulas and curves iov b = and b = 1, being of especial interest
and importance, will be considered before the remaining curves of the set.
For the case b = 0, formula (3.4) evidently reduces immediately to
F{R;0) = 2Rexp(R^). (3.20)
This case, 6 = 0, is that degenerate particular case in which the equiprob
ability curves in the scatter diagram of the complex variate, instead of
being ellipses (concentric), are merely circles, as noted in my 1933 paper,
near the bottom of p. 44 thereof (p. 10 of reprint).
For the case b = 1, the formula for the entire curve of P{R; b) = P(R;1),
except only the part at R = 0, can be obtained by merely setting b = I
in^^ (3.12) as this, on account of (3.15), thereby reduces immediately to
2_
V2^
P'iR;\) denoting the value of P{R;b) when b = 1 but i? 5^ 0, the restriction
i? 5^ being necessary because the quantity R~/(l — b^) in (3.12) — and in
(3.4) — does not have a definite value when b — 1 if i? = 0. Thus, in Figs.
3.1 and 3.2, the curve of P'(R;\) is that part of the curve iov b = 1 which
does not include any point in the P{R; b) axis (where R — 0) but extends
rightward from that axis toward R = f 00. The curve of P'{R;l) is the
'effective' part of the curve of P{R;l), in the sense that the area under the
former is equal to that under the latter, since the part of the curve of
P{R;l) at R = can have no area under it.
P(0;1) denoting (by convention) the value, or values, of P{R;b) when
R — and b — 1, that is, the value, or values, of P{R'S) when R = 0, it
is seen, from consideration of the curves of P{R;b) in Figs. 3.1 and 3.2 when
b approaches 1 and ultimately becomes equal to 1, that the curve of P(0;1)
consists of all points in the vertical straight line segment extending upward
in the PiR;b) axis, from the origin to a height 0.9376 [= Max P(i?;l)],20
'^ Use of (3.12) instead of (3.4), which is transformable into (3.12), avoids the indefinite
expression « .0.^ which would result directly from setting 6 = 1 in (3.4).
^^ As shown near the end of Appendix B, MaxP(^;l) is situated at /? = and is
equal to 0.9376.
^'(^; 1) = r7^exp^f]> (R ^ 0)> (3.21)
334 BELL S YSTEM TECH NIC A L JOURNA L
together with all points in the straight line segment extending downward
from the point at 0.9376 to the point at 0.7979 [= 2/ \/2^ = P'{R\\) for
R = 0+]. The curve of P(0; 1), because it has no area under it, is the
'noneffective' part of the curve of P{R\\).
Starting at the origin of coordinates, where i? = 0, the complete curve
of P{R\\) consists of the curve of P(0;1), described in the preceding para
graph, in sequence with the curve of P'(R;\), given by (3.21). Thus the
complete curve of P(R;\) is the locus of a tracing point moving as follows:
Starting at the origin of coordinates, the tracing point first ascends in the
P{R; b) axis to a height 0.9376 [= MaxP(i?;l)]; second, descends from
0.9376 to 0.7979 [= 2/ V2^ = P'iR'A) for R = 0\]; and, third, moves
rightward along the graph of P'(R;\) [b = l] toward i? = f co . The locus
of all of the points thus traversed by the tracing point is the complete
curve'' of P{R;l).
In addition to being the principal part ('effective' part) of the curve of
P{R;\), the curve of P'(R;\), whose formula is (3.21), has a further impor
tant significance. For the right side of (3.21), except for the factor 2, will
be recognized as being the expression for the wellknown 1 dimensional
'normal' law; the presence of the factor 2 is accounted for by the fact that
the variable i? =  i?  can have only posiive values and yet the area under
the curve must be equal to unity. This case, b = 1, is that degenerate
particular case in which the equiprobability curves, instead of being ellipses,
are superposed straight line segments, so that the resulting 'probability
density' is not constant but varies in accordance with the 1dimensional
'normal' law (for real variates), as noted in my 1933 paper, at the top of p. 45
thereof (p. 11 of reprint).
All of the curves of P{R;b), where O^b^l, pass through the origin,
the curve of PiR;\.) [b = 1] being no exception, since the part P(0;1) passes
through the origin.
Formula (3.12), supplemented by (3.15), shows that P(R; b) = at
i? = 00 ; and this is in accord with the consideration that the total area
under the curve of P{R;b) must be finite (equal to unity).
Since P{R;b) — slI R — and a.t R — co, every curve of P{R;b) must
have a maximum value situated somewhere between R ~ Q and R — oo —
as confirmed by Figs. 3.1 and 3.2. These figures show that when b increases
from to 1 the maximum value increases throughout but the value of R
where it is located decreases throughout.
The maxima of the function P{R;b) and of its curves (Figs. 3.1 and 3.2)
are of considerable theoretical interest and of some practical importance.
''^ The presence, in the curve of F{R; 1), of the vertical projection, or spur, situated in
the P{K; b) axis and extending from 0.7979 to 0.9376 therein, is somewhat remindful
(qualitatively) of the'Gibbs phenomenon' in the representation of discontinuous periodic
functions by Fourier series.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE
335
The cases b — Q and b = \ will be dealt with first, and then the general
case {b = b).
For the case J = it is easily found by differentiating (3.20) that P{R;b) =
P{R; 0) is a maximum Sit R — 1/ \^2 = 0.7071 and hence that its maximum
value is \/2exp (—1/2) = 0.8578, agreeing with the curve for 6 = in
Fig. 3.1.
For the case b = I, which is a limiting particular case, the maximum
value of P(R;b) — P(i?;l) apparently cannot be found driectly and simply,
as will be realized from the preceding discussion of this case. Near the
end of Appendix B, it is shown that the maximum value of P{R;\) occurs at
7? = (as would be expected) and is equal to 0.9376. This is the maximum
value of the part P(0;1 of P(R;1). The remaining part of P(R;l), namely
P'{R;1), whose formula is (3.21), is seen from direct inspection of that
formula to have a righthand maximum value a.t R = 0+, whence this
maximum value is 2/v 2ir = 0.7979.
For the general case when b has any fixed value within its possible positive
range (O^i^ 1), it is apparently not possible to obtain an explicit expression
(in closed form) either for the value of R at which P{R;b) has its maximum
value or for the maximum value of P(R;b); and hence it is not possible to
make explicit computations of these quantities for use in plotting curves of
them, versus b, of which they will evidently be functions. However, as
shown in Appendix B, these desired curves can be exactly computed, in an
indirect manner, by temporarily taking b as the dependent variable and
taking T, defined by (3.6), as an intermediate independent variable. For
let Re denote the critical value of R, that is, the value of R at which PiR;h)
has its maximum value; and let Tc denote the corresponding value of T,
whence, by (3.6),
Tc= bRl/ilb').
(3.22)
uj 0.8
I
UJ
O
5 0.4
gO.2
»
o
z
2
MAX P(R;b)
■;^
"^
Rr
"
Vib2
"~~~~
■^
Pc
""
\
\
0.1
0.2 0.3 0.4 0.5 0.6 0.7
PARAMETER, b
0.8 0.9 1.0
Fig. 3.3 — Functions relating to the maxima of the distribution function for the modulus.
336
BELL SYSTEM TECHNICAL JOURNAL
Then, computed by means of the formulas derived in AppendLx B, Fig. 3.3
gives a curve of Re and a curve of Max P(R;b), each versus b. Since the
curve of Re cannot be read accurately at 6 ?5r; 1, there is included also a
curve of Rc/y/l — b, from which Re can be accurately and easily com"
puted for any value of b; incidentally, the curve of Re/y/l — 6' is simul
taneously a curve of s/Telb, on account of (3.22). From Fig. ?i.7i it is
seen that Re varies greatly with b but that Max Pji;^ varies only a little,
as also is seen from inspection of Figs. 3.1 and 3.2 giving curves of P{R\b)
as function of R with b as parameter.
In Fig. }).?), the curve of Re shows that for 6 = 1 the maximum of P{R;b)
occurs ai R = 0; and the curve of Max P{R;b) shows that Max P{R;\) ^
0.94, agreeing to two significant figures with the value 0.9376 found near
the end of Appendix B. 
4. The Distribution Function for the Reciprocal of the Modulus
At first, let R denote any real variate, and P{R) its distribution function.
Also let r denote the reciprocal of R, so that r = \/R; and let P{r) denote
the distribution function for r. Then 
P{r) = R'PiR) = P{R)/r\
(4.1)
If P{R) depends on any parameters, P{r) will evidently depend on the
same parameters.
The rest of this section deals with the case where W = R(cos + i sin 6)
is 'normal.' Since this case depends on 6 as a parameter, P(R) and P(r)
are here abbreviations for P{R;b) and P{r;b) respectively.
As PiR;b) has the distribution function given by (3.4), the distribution
function for r will be
P{r;b) =
(Vl  b')r
3 exp
1
(1  &VJ "L(i  b'yy
(4.2)
obtained from the right side of (3.4) by changing R to l/r and multiplying
" For if r and R denote any two real variates that are functionally related, sa} F{r, K)
= 0, and if dr and dR are corresponding small increments, then evidently
P{r) \dr\ == P{R) \ dR \ whence
Pir)
PiR)
dR
dr
bF/br
dF/dR
In particular, if r = \/R, whence F = r — l/R, then (4.1) results immediately.
For a somewhat ditYerent and more detailed treatment of change of the variable in
distribution functions, see Thorton C. Fry, "Probability and its Engineering Uses,"
1928, pp. 1.S3155. (Cases of more than one variate are treated on pp. 155174 of the
same reference.)
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 337
the result by 1/r, in accordance with (4.1). Evidently P{r; — b)
= P(r;b).
By means of (4.1), formulas (3.7) and (3.8) give, respectively,
P{r;b) = 2(lb^')L"'exp{L)Io{bL), (4.3)
P(r;b) = 2(1  b') l^lj exp^^j /o(T), (4.4)
wherein L and T are defined by (3.5) and (3.6) respectively, but will now
be written in the equivalent forms
i = (T^ (4.5) r = Si=_^_A_, (4.6)
which are evidently more suitable for the present section.
A few particular cases that are especially important will be dealt with
in the following brief paragraph, ending with equation (4.8).
For the two extreme values of r, namely and oc , P{r;b) is zero for all
values of b in the b range (0^6^ 1).
When b = 0,
When b = I,
P{rb) = P{r;0) = ^^expfij. (4.7)
^f ]
Pir;b) = P{r;\) = ^^ ;;;, exp ~, \. (4.8)
Fig. 4.1 gives curves of P(r;b), with the variable r ranging continuously
from to 1.4 and the parameter b ranging by steps from to 1; however,
in the rrange where r is less than about 0.6, alternate curves had to be
omitted to avoid undue crowding. Fig. 4.2 gives an enlargement of the
section betwen r = 0.2 and r = 0.5, and includes therein the curves that
had to be omitted from Fig. 4.1.
In Fig. 4.1 it will be noted that with the scale there used for P(r;b) the
values of P(r;b) are too small to be even detectable for values of r less
than about 0.25. Even in the enlargement supplied by Fig. 4.2, the values
of P{r;b) are not detectable for r less than about 0.2.
The curves of P{r;b) in Figs. 4.1 and 4.2 would have had to be computed
from the lengthy formula (4.2) — or its equivalents — except for the fact
that curves of P{R;b) had already been computed in the preceding section
of the paper. The last circumstance enabled the P{r;b) curves to be
obtained from the P{R;b) curves by means of the very simple relation (4.1).
It will be observed that each curve of P{r;b) [Fig. 4.1] has a maximum
338
BELL SYSTEM TECHNICAL JOURNAL
ordinate, whose value and location depend on b. When b increases from
to 1, the maximum ordinate decreases throughout but the value of r where
it is located remains nearly constant, at about 0.82, until b becomes about
0.40
Z
O
3 0.35
tr
^ 0.30
a
0.25
0.20
0.15
^
<i
~v
k
/
Yf
^
vs
I
/^
0.6
x^
^
k
— .Ov
^
k
m
N>
^
^
/!
\
N
\
\^
li
^
c^
\
1
1
\
k^
\
X
1
. \,
\
1
j
"^
/I
b = t.a
o.ej<
0.6^
Ijl
111
oaU
/ h
////,
4
w
0.1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3
RECIPROCAL OF THE MODULUS, P
Fig. 4.1 — Distribution function for the reciprocal of the modulus (r = to 1.4).
0.7, after which the location of the maximum value moves rather rapidly
to about 0.71 for ft = 1.
For the cases 6=0 and b = 1, it is easily found, by differentiating (4.7)
and (4.8), that the maximum ordinates are located at r = \/2/3 = 0.8165
and at r = l/'\/2 = 0.7071 respectively; and hence, by (4.7) and (4.8).
that the values of these maximum ordinates are (3\/3/2 exp (—3/2) =
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE
339
0.8198 and (4/V27r) exp (1) = 0.5871 respectively. These results for
the cases 6 = and 6=1 agree with the corresponding curves in Fig. 4.1.
0.20
0.23 0.32 0.36 0.40
RECIPROCAL CF THE MODULUS, T
Fig. 4.2 — Distribution function for the reciprocal of the modulus {r = 0.2 to 0.5).
For the general case where b has any fixed value in the 6range (0^6^ 1),
it is apparently not possible to obtain an explicit expression (in closed form)
either for the value of r at which P{r;b) has its maximum value or for the
340
BELL SYSTEM TECH NIC A L JOURNAL
maximum value of P(r;b). However, as shown in Appendix C, curves of
these quantities versus b can be computed, in an indirect manner, by
temporarily taking b as the dependent variable and taking T, defined by
(4.6), as an intermediate independent variable. For let Tc denote the
critical value of r, that is, the value of r at which P(r;b) has its maximum
value; and let Tc denote the corresponding value of T, whence, by (4.6),
Tc= b/{\b')r\
(4.9)
Then, computed by means of the formulas derived in Appendix C, Fig. 4.3
gives a curve of Vc and a curve of Max P{r;b), each versus b. From these
curves it is seen that re and Max P{r\b) do not vary greatly with b, as also
is seen from inspection of Fig. 4.1 giving curves of P{r\b) as function of r
with b as parameter.
Tc
MAX F
^(ribT
^
^
—
<
g 0.4
to
z
2 0.2
t
u
z
£
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
PARAMETER, b
Fig. 4.3 — Functions relating to the maxima of the distribution function for the reciprocal
of the modulus.
5. The Cumulative Distribution Function for the Modulus
The cumulative distribution function Q{<R,di2) = Q{R) for the
modulus R of any complex variate W = R{cos 6 + i sin 6) is defined by
equation (1.11) on setting p = R, a = 6, pi = Ri ~ 0, ai = 6i — and
(72 = 6. = Itt; thus
QiR) = p{{)<R'<RA)<d'<2Tr). (5.1)
Similarly, from (1.12), the complementary cumulative distribution function
Q{>R,di2) = Q*{R) is defined by the equation
Q*{R)  p(R<R'<^^,{)<e'<2Tr).
(5.2)
Q*iR) is usually more convenient than Q{R) for use in engineering ap
plications, because it is usually mor? convenient to deal with the relatively
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE
341
small probability of exceeding a preassigned rather large value of R than to
deal with the corresponding rather large probability (nearly equal to
unity) of being less than the preassigned value of R.
A 'double integral' for Q{R), in the form of two 'repeated integrals,'
can be written down directly by inspection of the p{ ) expression in
(5.1) or by specialization of (1.8); thus
' / P{R,d) de clR = / P{R,d) dR dd. (5.3)
Evidently these can be written formally as two 'single integrals,'
Q{R) = / P{R) dR = / P{e\ < R) dd,
Jo Jn
(5.4)
by means of the distribution functions P(R) = P(R  ^i.) and P{e \ <R)
given by the formulas
P{R) = [ P{R,e) dd, (5.5) P{d\<R) = [ P{R,d)dR. (5.6)
Jo Jo
(5.5) is the same as (3.2). (5.6) is a special case of (1.6), and the left side
of (5.6) is a special case of P{p \ <a) detined by (1.13).
Similarly, from (5.2), we arrive at the following formulas corresponding
to (5.3), (5.4), (5.5), and (5.6) respectively:
dd,
Q*{R) = ■ / PiR,d) dd dR = / P(R,d) dR
J R \_Jo J •'oL'^''
^00 /.27r
Q*(R) = p{R) dR = P{d\ > R) dd,
J R Jo
P{d\ > R) = f P{R,d) dR.
J R
P{R) = [ P{R,d) dd,
Jo
(5.9)
(5.7)
(5.8)
(5.10)
The rest of this section deals with the case where W = i?(cos d \ i sin 6)
is 'normal.'^ Since this case depends on 6 as a parameter, Q{R) and Q*(R)
are here abbreviations for Q{R;b) and Q*{R;b) respectively.
A natural and convenient way for deriving formulas for Q{R) is afforded
by the general formula (5.4) together with the auxiliary general formulas
(5.5) and (5.6), beginning with the two latter.
For the 'normal' case, P{R,d) is given by (2.15). When this is sub
stituted into (5.5) and (5.6), it is found that each of the indicated integra
23 For the 'normal' case, the cumulative distribution function was treated in a very
different manner in my 1933 paper and its unpublished Appendix A. That paper included
applications to two important practical problems, and its unpublished Appendix C treated
a third such problem. (The unpublished appendices, A, B and C, are mentioned in foot
note 3 of the 1933 paper.)
342
BELL SYSTEM TECHNICAL JOURNAL
tions can be executed, giving the two previously obtained formulas (3.4)
and (2.19) for P(i?) = P(R;b) and P{d\ <R) respectively. When these
are substituted into (5.4), there result two types of singleintegral formulas
for Q{R): A prirrary type, involving an indicated integration as to R; and
a secondary tyj^e, involving an indicated integration as to 6. Formulas
of these two types for Q{R) will now be derived.
An integral formula of the primary type for Q{R) = Q{R;b) can be ob
tained by substituting P(R) = P(.R',b) from (3.4) into the first integral in
(5.4), giving
Q{R) = 2 [
Jo
X
Vl  b
exp
r ~^' 1
r ^^' 1
Li  b'i
h
Li  h'\
d\. (5.11)
This can also be obtained as a particular case of the more general formula
(2.21) by setting d = 2ir in the upper limit of integration and then apply
ing {i.2,).
In (5.11), X is used instead of R as the integration variable in order to
avoid any possible confusion wdth R as an integration limit. Thus the
integrand is a function of X with 6 as a parameter. Evidently Q{R;b) —
Q(R;—b). Formula (5.11) is evidently suitable for evaluation of ()(i?) by
numerical integration.^
By suitably changing the variable in (5.11), we arrive at the following
various additional formulas, which, though equivalent to (5.11), are very
different as regards the integrand and the limits of integration. As previ
ously, L denotes R/{\ — b).
Q{R)
1
Vl
K2 Jo
exp
■X
1
b'
dX,
Q{R) = Vl  b^ I exp(X) h{b\) dX,
Jq
Q(R) = LVi  b'~ I exp(LX) h{bLX) dX,
Jo
J PYn(—l
(5.12)
(5.13)
(5.14)
(5.15)
Q{R) = Vl  ^'M h{b log X) r/X.
Jexp{L)
These four additional formulas are of some theoretical interest, but ap
parently they are less suitable than (5.11) for numerical integration with
respect to R. A formula differing slightly from (5.11) could evidently be
obtained by taking X/\/l — 6^ as a new variable, and hence R/y/l — b^
as the upper limit of integration.
Corresponding formulas for Q*(R) = Q*{R;b) can of course be obtained
from the preceding formulas (5.11) to (5.15) inclusive for Q{R) = Q{R;b)
^* In this connection, Appendix D may be of interest.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE
343
by merely changing the integration Hmits correspondingly — for instance,
in (5.11), from 0, i? to i?, oo ; in (5.13), from 0, L to L, ^ \ and so on. How
ever, the first four formulas for Q*{K) so obtained would suffer .the disad
vantage of each having an infinite limit of integration, rendering those
formulas unsatisfactory for numerical integration purposes. This difficulty
can be avoided by making the substitution R = \/r in each of those formulas
for Q*{R). The resulting formulas are the following five, corresponding to
(5.11) to (5.15) respectively :24
()*(i?)
Vi
Q*{R) =
Q*(R)
VT
2 rj_
_ /i2 Jo X^
b' Jo
X2
exp
^lAH
.1  b~_
h
 exp
ri/A'
Ll  b\
h
b/}C
1  F
b/\
d\, (5.16)
1
]jx,
X'
Vl  b^ [
exp
exp
1
b
/n
xj
lxJ
L
/"n
~bL
X
X
dX,
dX,
expi—L)
' Io{b log X) dX
a
(5.17)
(5.18)
(5.19)
(5.20)
As a check on (5.16), it is obtainable from (4.2) by integrating the latter
as to r.
For purposes of evaluation by numerical integration, formula^ (5.11)
to (5.15) inclusive may evidently differ greatly as regards the amount of
labor involved and the nurrerical precision practically attainable. In
each of these formulas except (5.14) the integrand contains only one param
eter, b, while the integration range involves either R or L = R/{\ — b).
In (5.14) the integrand contains two independent parameters, b and L,
while the integration range is a mere constant, 0tol. Similar statements
apply to formulas (5.16) to (5.20) inclusive.
A partial check on any formula for Q(R) can be applied by setting R = <x> ^
since Q(°o) should be equal to unity (representing certainty). If, for
instance, this procedure is applied to formula (5.13), the right side is found
to reduce to unity by aid of the known relation"
exp (^X) JoiBX) dX =
}
Jo
1
(5.21)
together with Io{BX) — jQ(iBX).
An integral formula of the secondary type for Q*(R) = Q*{R;b) can be
obtained by substituting (2.20) into the last integral in (5.8), utilizing (2.25),
» Ref. 1, p. 384, Eq. (1); Ref. 2, p. 65, Eq. (2); Ref. 4, p. 58, Eq. (4.5).
344 BELL S YSTEM TECH NIC A L JOURNA L
changing the variable of integration by the substitution 6 = 0/2, and
rearranging; thus it is found that
Q*{R) = ylZ? r ^^P(^^ '^' ^) d<l>. (5.22)
7r exp L Jo 1 — b cos
This formula can also be obtained as a particular case of the more general
formula (2.24) by setting 6 = 27r in the upper limit of integration, utilizing
(2.25), and changing the variable of integration by the substitution 6 =
0/2.
Two partial checks on any general formula for Q{R) = Q{R;b) or for
Q*{R) = Q*{R;b) can be applied by setting b — and b — 1, and comparing
the resulting particular formulas with those obtained by integrating the
formulas for P{R;0) and F'{R;\) obtained in Section 3, namely formulas
(3.20) and (3.21) there. It is thus found that
Q*(R;0) = exp(R') = ( P{R;0)dR, (5.23)
Q{R; 1) = 2 J= jf^xp ^ dR^=^ [ ^'^^'^ ^^ ^^ ^^'^^
It will be recalled that the quantity between braces in (5.24) is extensively
tabulated, and that ^t is sometimes called the 'normal probability integral.'
Several of the above general formulas for QiR) = p{R'<R) and for
Q*{R) = p{R'>R) are closely connected with my 1933 paper." Indeed,
formulas (5.11), (5.14), (5.16), (5.19) and (5.22) above are the same as
(53A), (56A), (52A), (55A) and (22A), respectively, of the unpublished
Appendix A to the 1933 paper; and (5.12), (5.13), (5.15), (5.17), (5.18) and
(5.20) above were derived in the same connection, although they were not
included in the Appendix A.
Formula (5.22) was employed in the unpublished Appendix A of the 1933
paper, being (22A) there, as a basis for deriving two very different kinds
of series type formulas for computing the values of p{R'>R) = Q*{R)
underlying the values of pb.t){R'>R) constituting Table I (facing Fig. 8)
in that paper. ^
2*^ This formula, (5.22), was derived by me in a somewhat different manner in the un
pubHshed Appendix A to my 1933 paper. Later I found that an efjuivalent formula,
easily transformable into (5.22), had been given by Bravais as formula (51) in his classical
paper ".Analyse mathcmatique sur les probabilites des erreurs de situation d'un point,"
published in Mcmoires de I'Academie Royale des Sciences do I'lnstitut de FVance, 2nd
series, vol. IX, 1846, pp. 255332. (This is available in the Public Library of New York
City, for instance.)
^^ There the abbreviated symbols p(R' < R) and /)(/?' > R) were used with the same
meanings as the complete symbols on the right sides of ecjuations (5.1) and (5.2), respec
tively, of the present paper.
^^ Each of the two kinds of series type formulas comprised a finite portion of a con
vergent series plus an exact remainder term consisting of a definite integral. In the
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 345
In the present paper, formulas (5.11) and (5.16) have been used for numer
ical evaluation of QiR) = p{R'<R) and of Q*(R) = p{R'>R) by numerical
integration (employing 'Simpson's onethird rule'), aided by some of the
considerations set forth in Appendix D. However, only a moderate number
of values of these quantities have been thus evaluated — merely enough to
afford a fairly comprehensive check on Table I of my 1933 paper, by means
of a sample consisting of 60 values (about 26%) distributed in a somewhat
representative manner over that table. These new values of Q*{R) =
p{R'>R) = 1 — Q(R) are presented in Table 5.1 (at the end of this section)
in such a way as to facilitate comparison with the old values, namely those
in the 1933 paper. Thus, for any fixed value of R in Table 5.1, there are
two horizontal rows of computed values of Q*{R), the first row (top row)
coming from the 1933 paper, and the second row coming from the present
paper. The third row of each set of four rows gives the deviations of the
second row from the first row; and the fourth row expresses these deviations
as percentages of the values in the first row.
In the first row of any set of four rows, any value represents Q*{R) =
pb{R'>R) obtained, in accordance with Eq. (22) of my 1933 paper, by
adding exp (— i?) to pb^o{R'>R) given in Table I there. In the second
row of a set, any value represents Q*{R) = 1 — Q{R) as computed by for
mula (5.11) or (5.16) of the present paper: more specifically, the values for
R = 0.2, 0.4, 0.6 and 0.8 were computed by (5.11); and the values for
R = \.6 and i? = 2 by (5.16), taking r = 1/1.6 = 0.625 and r  1/2 = 0.5
respectively."
In the 1933 paper, the values of Pb{R'>R) = Q*{R;b) for J = and for
b — I were omitted as being unnecessary there because their values could
be easily obtained from the simple exact formulas to which the general
formulas there reduced, ior b = and ^ = 1. Those reduced formulas
were the same as (5.23) and (5.24) of the present paper, except that (5.24)
gives Q(R;\) instead of giving Q*{R;\) = 1  QiR;!). The values obtained
from these two formulas, exact to the number of significant figures here
retained, are given in Table 5.1 at the intersections of the first row of each
set of four rows with the columns 6 = and b = I. Therefore in these two
columns the deviations (in the third row of each set of four rows) are devia
tions from exact values; the values in the second row of each set are, as
use of such a formula for numerical computations, the expansion producing the con
vergent series was carried far enough to insure that the remainder deiinite integral would
be relatively small, though usually not negligible; and then this remainder definite integral
was evaluated sufficiently accurately by numerical integration.
2s In the work of numerical integration, ' Simpson's onethird rule' was employed for
R = 0.2, 0.4, 0.6, 0.8 and 2. For R = 1.6, so that r = 1/1.6 = 0.625, 'Simpson's one
third rule' was employed up to r = 0.620, and the ' trapezoidal rule' from r = 0.620 to
r = 0.625.
346
BELL SYSTEM TECUNICAL JOURNAL
already stated, those obtained by the methods of the present paper, employ
ing numerical integration.
From detailed inspection of Table 5.1 it will presumably be considered
that the agreement between the two sets of values of Q*{R\b) = pb(R'>R)
is to be regarded as satisfactory, at least from the practical viewpoint, the
largest deviation being less than one per cent (for R = 0.8, b — 0.9).
Table 5.1
Valxjes of Q*{R) = p{R' > R)
b
R
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
1. 00
0.2
.9608
.9590
.9574
.9550
.9516
.9463
.9372
.9168
.8930
.84148
"
.9623
.9605
.9590
.9567
.9528
.9473
.9387
.9206
.8925
.84124
"
.0015
.0015
.0016
.0017
.0012
.0010
.0015
.0038
.0005
.00024
"
.16
.16
.17
.18
.13
.11
.16
.41
.06
.03
0.4
.8521
.8462
.8410
.8335
.8228
.8071
.7830
.7420
.7127
.68916
"
.8537
.8477
.8427
.8351
.8240
.8081
.7841
.7459
.7125
. 68897
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6. The Distribution Function For The Angle
The distribution function P{d \ Rn) = P{d) for the angle 9 of any complex
variate W = R{cos 6 \ i sin 9) is defined by equation (1.10) on setting
p = 6, a = R, (Xi — Ri = and ao — R^ — 'x, ^ thus
P{9)d9 = p{d<9'<d\d9,0<R'<x). (6.1)
An integral formula for P(9) is immediately obtainable from (1.6) by
setting p — 9, a = R, as = (Xi — Ri = and 04 = ao — R2 = °o ', thus
p(e) = [ P{R, 9) (JR.
(6.2)
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 347
The rest of this section deals with the case where W = R{cos d \ i sin 6)
is 'normal.' Since this case depends on 6 as a parameter, P{d) is here an
abbreviation for P{B\b).
A formula for P{d;b) = P{d) can be obtained by substituting P{R,d)
from (2.15) into (6.2) and executing the indicated integration, which can
be easily accomplished. The resulting formula is found to be
2x(l — bcosld)
This formula can also be obtained as a particular case of either of the
more general formulas (2.19) and (2.20) by setting R = co m (2.19) or
7? = in (2.20); also by adding (2.19) to (2.20) and then utilizing (1.10).
In P{d) = P{d;b) it will evidently suffice to deal with values of 6 in the
first quadrant, because of symmetry of the scatter diagram.
In P{d;b) it will suffice to deal with only positive values of b, as (6.3)
shows that changing b to —b has the same effect as changing 26 toir±26,
or 6 to 7r/2±0; that is, P{e;b) = P{j/2±d;b).
Fig. 6.1 gives curves of P{6;b), computed from (6.3), as function of 6
with b as parameter, for the ranges 0^^^90° and Q^b^l.
The curves in Fig. 6.1 indicate that P{6;b) is a maximum at = 0° and
a minimum at 9 = 90°. These indications are verified by formula (6.3),
as this formula shows that:
Max P{d;b) = P{0°;b) = ^ \/ H^ , (6.4)
Thence
Min P{e;b) = P{90°;b) = i ^ jqj] • (65)
MmP{d;b)/MsixP(6;b) ^ (l6)/(l + 6), (6.6)
P{e;b)/MiixP{e;b) = P{d;b)/PiO°;b) = {lb)/{lb cos2d). (6.7)
The curves in Fig. 6.1 indicate also that P{d;b) is independent of d in
the case b = 0. This is verified by formula (6.3), as this formula shows that
P{6;0) = l/27r. (6.8)
Thence (6.3) can be written
P{d;b)/P{e;0) = (Vn^y2)/(l6cos2^). (6.9)
3" Beginning here, 6 will usually be expressed in degrees instead of radians, for prac
tical convenience.
348
BELL SYSTEM TECHNICAL JOURNAL
By setting cos 20 = in (6.3), so that d = 45°, it is found that
(vT^^2)/27r  P(45°';6), (6.10)
c
d
f\l rn ^ I/) (0 r 00 c* o>
ddddd dd d d
1
111
/
i
1
1
7
1//
/
f//
7
1
'1
/
i
V
/
i
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^
^
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0)
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7
If)/ T
o7 c
i/ d c
3 d o
r
"^
/
1
(
1
DISTRIBUTION FUNCTION, P (9 ; b)
Fig. 6.1 — Distribution function for the angle.
whence (6.3) can be written
P{d;b)/Pi45°;b) = 1/(16 cos 20).
(6.11)
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE
349
■
1
\
'
o
/
1
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/
1
/
/ ,
C\j/
d/
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1
/
/
m /
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y
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/en
/ o
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l///
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^
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y
F
/y
x^^
^
^
y
t
REDUCED DISTRIBUTION FUNCTION, P(e;b)/MAX P(9;b)
Fig. 6.2 — Reduced distribution function for the angle.
350 BELL SYSTEM TECHNICAL JOURNAL
In the case b — \, the curves in Fig. 6.1 suggest, by Hmiting considera
tions, that P(0;1) is zero for all 6 except d = 0°, and that P{d;\) is infinite
for 6 = 0°. These conclusions are verified by formula (6.3), as this formula
shows that:
P{d;\) = for ()°<d<mr; P{d;\) = c for 6 = 0°, 180°.
The curves in Fig. 6.1, though having the advantage of directly rep
resenting P{d;b) as function of 6 with b as parameter, are somewhat trouble
some to use because of their numerous crossings of each other. This
difficulty is not present in Fig. 6.2, which gives curves of P{d;b)/Ma,x
P(6;b), obtained by dividing the ordinates P{6;b) of the curves in Fig. 6.1
by the respective maximum ordinates of those curves, as given by (6.4),
so that the equation of the curves in Fig. 6.2 is formula (6.7).
7. The Cumulative Distribution Function for the Angle
The cumulative distribution function Q{<6,R]2) = Q{6) for the angle 6
of any complex variate TF ^ R{cos 6 + / sin 6) is defined by equation
(1.11) on setting p = d, a ^ R, pi = di =^ 0, ai = Ri = and 02 = R2 = »= ;
thus
Q{d) = p{0<d'<d, 0<R'<oo). (7.1)
A 'double integral' for Q{d), in the form of two 'repeated integrals,' can
be written down directly by inspection of the p( ) expression in (7.1)
or by specialization of (1.8); thus
Q(d) = f \ [ P(R, d)dR dd ^ I f P(R, e) dd dR. (7.2)
Ja \_Jtii J Jo L*'o J
Evidently these can be written formally as two 'single integrals,'
Q{d) = f P(9) dd = \ P{R\< d) dR, (7.3)
by means of the distribution functions P{d) = P(e\ R12) and P{R\ <d)
given by the formulas
P(d)  [ P{R, 6) dR, (7.4) P(R \ <d) = f P(R, 6) dd. (7.5)
Jo Jo
(7.4) is the same as (6.2). (7.5) is a special case of (1.6), and the left side
of (7.5) is a special case of P{p \ <a) defined by (1.13).
The rest of this section deals with the case where W = R{cos d \ i sin 6)
is 'normal.' Since this case depends on b as a parameter, Q{d) is here an
abbreviation for Q{6;b).
A natural and convenient way for deriving formulas for Q(d) is afforded
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 351
by the general formula (7.3) together with the auxiliary general formulas
(7.4) and (7.5), beginning with the two latter.
It will be convenient to dispose of (7.5) before dealing with (7.4), as (7.5)
turns out to be the less useful. For when P{R,d) given by (2.16) is sub
stituted into (7.5), the indicated integration cannot be executed in general,
as (7.5) becomes (2.18), wherin the indicated integration can be executed
only for certain special values of the integration limit 6 — by means of the
special Bessel function formula (3.i).
When PiR,d) given by (2.15), which is equivalent to (2.16) used above,
is substituted into (7.4), it is found that the indica^^ed integration can be
executed, giving the previously obtained formula (6.3) for F{d) = P{&',b).
A 0integral formula for Q{d) = Q{Q\h) can be obtained by substituting
P{e) = P{d;b) from (6.3) into the first integral in (7.3), giving
Vi  6 f' dd Vi  62 r'" d<f>
^^ ' ' 27r h \  b cos 28 47r h 1
b cos
(7.6)
This formula can also be obtained as a particular case of the more general
formulas (2.22) and (2.24) by setting i? = ^ in (2.22) or i? = in (2.24);
also by adding (2.22) to (2.24) and then utilizing (1.11).
The integral in (7.6) is of wellknown form, and the indicated integration
can be executed, yielding the following two equivalent formulas for Q{d\h):
27r
tan
1 1 cos 2^  6 n
''' L i6cos2d r
In Q{d;b) it will evidently suffice to deal with values of 6 in the first quad
rant, because of symmetry of the scatter diagram, and the resulting fact
that Q{n 90°) = n/i, where n = 1, 2, 3 or 4.
In Q{6;b) it will suffice to deal with positive values of b, as (7.7) shows
that^i
Q{e; b)
Ie i±M
Fig. 7.1 gives curves of Q{d;b) = Q{6) computed from (7.7), as function
of d with b as parameter, for the ranges 0^0^90° and 0^6^ 1.
Consideration of the scatter diagram of IF or of its equiprobability curves,
which are concentric similar ellipses, affords several partial checks on the
curves in Fig. 7.1 and on formula (7.7) from which they were plotted.
^1 This relation can also be derived geometrically from the fact that the scatter dia
gram for —b is obtainable by merely rotating that for b through 90°, as shown by (2.6),
or (2.7) and (2.8), or (2.11).
352
BELL SYSTEM TECHNICAL JOURNAL
\
\
1
k
\
\
Uvi
\v
\\J\
\\
vv
\
\\
N
\
^
\\
^ \
:^i
\
1
\
V
\\
\^
^v\
\
\
\
N]
V \ •*
\ ■f\<
^'
t$;
V
\
V
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^
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\
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V
V^
^^
^1
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4
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^\
^
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ij
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^
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;:$
NNV
^
X
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^
^ —
^
^
CUMULATIVE DISTRIBUTION FUNCTION, Q(e;b)
Fig. 7.1 — Cumulative distribution function for the angle.
Thus, the fact that the curve for ^ = is a straight Hne, whose equation is
(3(0 ;0) = e/2w = 07360°, {b = 0),
corresponds to the fact that for 6 = the equiprobability curves are circles.
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 353
The fact that the curve for 6 = 1 is the straight Hne Qid;l) = 1/4 = 0.25
corresponds to the fact that for 6 = 1 the scatter diagram has degenerated
to be merely a straight Hne coinciding with the real axis, so that no point
outside of this line makes any contribution to Q{d;\).
The fact that, at ^ = 90°, Qi9;b) = Q{90°;b) has for all b the value 1/4 =
0.25 corresponds to the fact that the area of a quadrant of the scatter
diagram is onefourth the area of the entire scatter diagram. Hence
Q(360°;b) = 4Q{90°;b) = 1, which is evidently correct.
Acknowledgment
The computations and curveplotting for this paper were done by Miss
M. Darville; those for the 1933 paper, by Miss D. T. Angell.
APPENDIX A
Derivation of Formula (2.15) for P{R,d)
(2.15) will here be derived from (2.11) by utiHzing the fact that the 'areal
probabiUty density', G, at any fixed point in the scatter diagram must be
independent of the system of coordinates; for G dA gives the probability
of faUing in any differential element of area dA, and this probabiUty must
evidently be independent of the shape of dA (assuming that all linear dimen
sions of dA are differential, of course). Thus, indicating the element of
area by an underline, we have, in rectangular coordinates,
G dUdV = P{U,V)dUdV, (Al) whence G = PiU,V). (A2)
In polar coordinates,
GRdddR  P(R,d)dRde, (A3) whence G = P{R,d)/R. (A4)
Comparing these two expressions for G shows that
P{R,e) = RP(U,V). (A5)
Thus, a formula for P(R,6) can be obtained from (2.11) by merelv multiply
ing both sides of that formula by R. However, in the resulting formula it
will remain to express U and F in terms of R and 6, by means of the relations
U ^ R cos d, (A6) V = R sin d. (A7)
The final result, after a simple reduction, is (2.15), which is thus proved.
APPENDIX B
Formulas of the Curves in Fig. 3.3
As in equation (3.22), Re will here denote the critical value of R, that is,
the value of R at which P{R) = P{R',b) has its maximum value; and Tc
'2 Formula (A5) can be easily verified by the entirely different method which utilizes
(1.23).
354 BELL SYSTEM TECHNICAL JOURNAL
will denote the corresponding value of T, whence Tc is given in terms of
Re and b by (3.22).
A formula for dP{R)/dR could of course be obtained directly from (3.4)
but it will be found preferable to obtain it indirectly from the less cumber
some formula (3.8) containing the auxiliary variable T defined by (3.6).
Evidently, since b does not depend on R,
dP{R) ^ dPjR) dT_ ^ 2bR dP{R)
dR dT dR 1  b' dT ' ^ ^
Thus, since the factor IbR/il — b") cannot vanish for any value of R (except
R = 0), the only critical value of R must be that corresponding to the value
of T at which dP{R)'/dT vanishes, namely Tc, since Tc has been defined
to be the value of T corresponding to Re (Incidentally, equation (Bl)
shows that Tc is equal to the value of T at which P(R) is an extremum
when P(R) is regarded as a function of T.) From (3.22),
Rl Tc (32)
1  b' b
Evidently Tc and Re must ultimately be functions of only b. The next
paragraph deals with Tc, which evidently has to be known before Re can
be evaluated.
From (3.8) it is found that, since dh{T)/dT = I\{T),
= nm ^ +
r_L , h{T) 1
(B3)
'12T h{T) b_
Hence, since P(i?) does not vanish for any value of R (except R = Q and
R = oo), Tc will be a root of the conditional equation obtained by equating
to zero the expression in brackets in (B3). This conditional equation is
transcendental in Te and apparently has no closed form of explicit solution
for Tc ; and its solution by successive approximation, or otherwise, would
likely be rather slow and laborious. However, the bracket expression in
(B3) shows that b can be immediately expressed explicitly in terms of Te
by the equation
^ ^ 1 + 2Teh{Tc)/h{Tc) ' ^^^^
For some purposes, the following two equations, each equivalent to (B4),
will be found more convenient:
T 2 + ^^/727)' ^^^^
l£ = IZ? (B6)
b 1  bh{Te)/h{Te) ^ ^
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 355
On account of (B2), the right sides of (B5) and (B6) are equal not only to
Tc/b but also to i?c/a6").
Since the utilization of formulas (B4), (B5) and (B6) for computing the
curves in Fig. 2).di will involve taking Tc as the independent variable and
assigning to it a set of chosen numerical values, the natural first step is to
find approximately the range of Tc corresponding to the 6range, O^^^l,
in order to be able to choose only useful values of Tc. This step will be
taken in the next paragraph.
Equation (B6) shows that Tc/b = 1/2 when 6 = 0, and hence that Tc ~
when b = 0; and this last is verified by (B4). The other endvalue of the
Tcrange, namely the value of Tc iox b = 1, cannot be found explicitly
and exactly. However, rough values of limits between which it must lie
can be found fairly easily as follows: To begin with, each of the equations
(B5) and (B6) shows that Tc^ b/2, for all values of b in O^b^l; in par
ticular, Tc > 1/2 when b = I. An upper limit for Tc for any value of
b can be found from (B5) by utilizing the power series expressions for
Ii{Tc) and lo(Tc), whereby it is found that
^ H^, (B7) where H =^ I  %' < 1. (B8)
Io{I c) ^ o
On substituting (B7) into (B5) and then solving for Tc in terms of b and
H, it is found that
Tc = b/(l + Vl  Hb'). (B9)
On account of (B8), (B9) shows that
Tc < b/{l + Vn^2), (BIO)
whence, in particular, Tc<l when b = 1. By successive approximation
or otherwise, it can now be rather quickly found that, when b — 1, Tc =
0.79 (to two significant figures).^^
From the preceding paragraph, it is seen that, when b ranges from to 1,
Tc ranges from to about 0.79; Tc/b ranges from 0.5 to about 0.79; and,
on account of (B2), Re ranges from ^/O.S = 0.707 down to 0.
The curves in Fig. 3.3 are constructed with the aid of the formulas and
methods of this appendix as follows: First, a set of values of Tc is chosen,
ranging from to 0.79 and slightly larger. Second, for each such chosen
Tc the right side of (B5) is computed, thereby evaluating Tc/b and also
Rc/{l — b^), these two quantities being equal by (B2). Third, the cor
responding value of b is found by dividing Tc by Tc/b; less easily, it could
^' Because of the special importance oi b = 1 in other connections, Tc for b = I was
later evaluated to four significant figures and found to be Tc = 0.7900; thence, by sub
stituting this value of T into (3.8), along with b = 1, it was found that Max. P{R;l)
= 0.9376, which occurs at R = Re = 0,hy (B2).
356 BELL SYSTEM TECHNICAL JOURNAL
be foun d by substituting Tc into (B4). Fourth, from Tc/b the value of
\/Tc/b is found, and thereby the value of Rc/y/l — b"^ and thence Re .
Finally, Max. P{R;b) is computed by inserting the critical values into any
of the various (equivalent) formulas for PiR;b), namely (3.4), (3.7), (3.8),
(3.10) or (3.12).
APPENDIX C
FOMULAS OF THE CURVES IN FiG. 4.3
The first six equations of this appendix are given without derivation
and almost without any comments because they correspond exactly and
simply to the first six equations, respectively, of Appendix B. Beginning
with the second paragraph of the present appendix, the close correspondence
ceases.
dP(r) _ dP{r) dT _ 2b dP(r)
dr dT dr (1  ^2);^ dT
(1  bVc ~ T •
dP(r)
dT
(CI)
(C2)
= P(r) ^ +
[l+b^ 1] (C3)
12T ^ h{T) b\ ' ^^^^
* 3 + 2r, h(T,)/Io(Tc) ' ^^^^
b 2 Io{Pc)
Tl = 3/2
b 1  bh{Tc)/Io(Tc) ' ^""^
The bracketed expression in (C3) is seen to be obtainable from that in (B3)
by merely changing T to T/3 wherever T does not occur as the argument
of a function; hence the three equations following (C3) are obtainable from
the three equations following (B3) by correspondingly changing Tc to
Tc/S. (In this appendix, as in Section 4, small c is purposely used as a
subscript to indicate a 'critical' value, whereas in Section 3 and in Appendix
B, capital C is used for that purpose.)
For use below, it will here be noted that
h{Tc)/h(Tc) = N,{Tc)/No{Tc), (C7)
as will be seen by dividing (3.16) by (3.13). On account of (3.17) and (3.14),
(C7) shows that for large values of Tc the right side of (C7) is equal to 1
as a first approximation, and to 1 — 1/2 Tc as a second approximation;
thus, for large Tc,
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 357
h(T,)/hiT,) = 1  l/2r, = 1. (C8)
The first step toward computing the curves in Fig. 4.3 is to find approxi
mately the Tcrange corresponding to the 6range, O^b^l. This is done
in the course of the next four paragraphs.
When b = 0, equation (C6) shows that Tc/b = 3/2 and hence that
Tc = 0; or, what is equivalent, b/Tc = 2/3 and hence l/Tc = oo (since
b^ 0).
When 6 = 1, Tc = CO, as can be easily verified from equation (C4),
(C5) or (C6) by utilizing (C8).
Thus, from the two preceding paragraphs, it is seen that, when b ranges
from to 1, b/Tc ranges from 2/3 to 0; Tc/b from 3/2 to cc ; and Tc from
to 00.
Since Tc = "^ when b = 1, the choosing of a set of finite values of Tc
will necessitate an approximate formula for computing Tc for values of
b nearly equal to 1 , which means for very large values of T. Such a formula
is easily obtainable from (C5) by utiUzing the approximation 1 — 1/2 Tc
in (C8), whereby it is found that, for large Tc,
Tc = b/{lb), (C9) b/Tc = l^*. (CIO)
As examples, these approximate formulas give: When b = 0.99, Tc ~ 99,
b/Tc = 0.01; when b = 0.9, Tc = 9, b/Tc = 0.1. It will be found that
even in the second example the results are pretty good approximations.
The curves in Fig. 4.3 are constructed with the aid of the formulas and
methods of this appendix as follows: First, a set of values of Tc is chosen,
ranging from to about 100 (the latter figure corresponding approximately
to b = 0.99). Second, for each such chosen Tc the right side of (C5) is
computed, thereby evaluating Tc/b and also 1/(1 — 6)^^, these two quan
tities being equal by (C2). Third, the corresponding value of b is found
by dividing Tc by Tc/b; less easily, it could be found by substituting Tc
into (C4). Fourth, from Tc/b the value of \/Tc/b is found, and thereby
the value of I/tc s/l — b^ and thence Tc . Finally, Max P{r;b) is computed
by inserting the critical values into any of the (equivalent) formulas for
P{r;b), namely (4.2), (4.3) or (4.4).
APPENDIX D
Some Simple General Considerations Regarding the Evaluation of
Cumulative Distribution Functions by Numerical Integration
This appendix gives some simple general considerations and relations
that may sometimes facilitate and render more accurate the evaluation
of cumulative distribution functions by numerical integration.
358 BELL S YSTEM TECH NIC A L JOURNA L
Some of these considerations and relations have found application in
Section 5 in the evaluation of the cumulative distribution function for the
modulus R = I ir . For this reason, the variate in the present section
will be denoted by R. though without thereby restricting R to denote the
modulus; rather, R will here denote any positive real variate, though it
should preferably be a 'reduced' variate, so as to be dimensionless, as in
equation (2.9). The restriction of R to positive values is imposed because
it is strongly conducive to simplicity and brevity of treatment, without
constituting an ultimate limitation. The reciprocal of R will be denoted
by r, as previously.^*
We may wish to evaluate numerically the cumulative distribution func
tion p{R'<R) = Q{R) or p{R'>R) = Q*{R) or both. Since these are not
independent, their sum being equal to unity, the evaluation of either one
determines the other, theoretically. However, when the evaluated one is
nearly equal to unity, the remaining one may perhaps not be evaluable
with sufficient accuracy (percentagewise) by subtracting the evaluated one
from unity. Then it would presumably be advantageous to introduce
for auxiliary purposes the variable r — 1/R, since evidently
p(R'>R) = p{\/R'<l/R) = p{r'<r), (Dl)
p(R'<R) = p{r'>r) = 1  p{r'<r). (D2)
Thus, if p{R'>R), in (Dl), is small compared to unity, it is presumably
evaluable with higher accuracy percentagewise by dealing with p{r'<r)
than with 1 — p{R'<R). Incidentally, after p{r' <r) has been evaluated,
it might be used in (D2) to arrive at a still more accurate value of p{R' <R)
than had originally been obtained directly by numerical integration.
Assuming that we have a plot (or a table) of the distribution function
P{R), we can evidently evaluate
P{R'<R') = / P{R)dR (D3)
Jo
directly by numerical integration, provided the plot is sufficiently extensive
to include R ; if not, we can, by (D2), resort to
P(R'<R') = 1  p(r'<r') = 1  / P{r)dr, (D4)
Jo
assuming that a sulficiently extensive i)lot (or table) of P{r) is available
and applying numerical integration to it.
Even if the plot of P{R) used in (D3) is sulficiently extensive to include
'■• The restriction of R, and hence of r, to positive values is seen to be absent from equa
tions (Dl), (D2), (D5) and (D6) but present in (D3), (D4), (D7) and (D8).
PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 359
R , so that (D3) could be evaluated, it might be that (D4) would result
in greater accuracy; this would presumably be the case when p{R' <R )
is nearly equal to unity.
Evidently an evaluation of
P(R'>R') = P(R)dR (D5)
directly by numerical integration would be less satisfactory than the evalua
tion of p{R' <R ) in the preceding paragraph. For, due to the presence
of the infinite limit in the integral in (D5), the plot of P{R) would have to
be carried to a large enough value of R so that the integral from there to «^
would be known to be negligible. This diflficulty can be avoided by start
ing with the relation
piR'>R') = 1  piR'KR") (D6)
and substituting therein the value of p{R' <R ) given by (D3) or (D4),
resulting respectively in the following two formulas:
p(R'>R') = I  P(R)dR, (D7)
P(R'>R') = p(r'<r') = / P{r)dr, (D8)
the integrals in which are evidently suitable for evaluation by numerical
integration, none of the integration limits being infinite. If p{R'>R'^)
is small compared to unity, (D8) would presumably be more accurate
(percentagewise) than (D7). If the plot of P(R) is not sufficiently exten
sive to include R , (D7) evidently could not be used; but, instead, (D8)
could be used if the plot of P{r) were sufficiently extensive to include r .
References on Bessel Functions
1. Watson, "Theory of Bessel Functions," 1st. Ed., 1922; or 2nd Ed., 1944.
2. Gray, Mathews and MacRobert, "Bessel Functions," 2nd Ed., 1922.
3. McLachlan, "Bessel Functions for Engineers," 1934.
4. Bowman, "Introduction to Bessel Functions," 1938.
5. Whittaker and Watson, "Modern Analysis," 2nd Ed., 1915.
6. "British Association Mathematical Tables," Vol. VI: Bessel Functions, Part I, 1937.
7. Anding, "Sechsstellige Tafeln der Bessel'schen Funktionen imaginaren Arguments,"
1911 (mentioned on p. 657 of Ref. 1).
Spectrum Analysis of Pulse Modulated Waves
By J. C. LOZIER
The problem here is to find the frequency spectrum produced by the simul
taneous application of a number of frequencies to various forms of amplitude
limiters or switches. The method of solution presented here is to first resolve the
output wave into a series of rectangular waves or pulses and then to combine the
spectrum of the individual pulses by vectorial means to find the spectrum of the
output. The rectangular wave shape was chosen here as the basic unit in order to
make the method easy to apply to pulse modulators.
Introduction
The rapidly expanding use of pulse modulation^ in its various forms is
bound to make the frequency spectrum of pulse modulated waves a subject
of increasing practical importance. The purpose of this paper is to show
how to determine the frequency spectrum of these waves by methods based
as far as possible on physical rather than mathematical considerations. The
physical approach is used in an attempt to maintain throughout the analysis
a picture of the way in which the various factors contribute to a given result.
To further this objective the fundamentals involved are reviewed from the
same point of view.
The method is used here to analyze two distinct types of pulse modulation,
namely, pulse position and pulse width modulation.^ These two cases are
especially important for illustrative purposes because their spectra can be
tied back to more familiar methods of modulation. Thus it will be shown
that, as the ratio of the pulse rate to the signal frequency becomes large,
pulse position modulation becomes a phase modulation of the various carrier
frequencies that form the frequency spectrum of the unmodulated pulse
wave, and pulse width modulation becomes a form of amplitude modulation
of its equivalent carriers. The analysis also shows certain interesting input
output relationships that may be obtained from such modulators, treating
them as straight transmission elements at the signal frequency.
These relationships are of more than theoretical interest. The pulse
position modulator has already been used as phase or frequency modulator
to good advantage.^ The use of a pulse width modulator as an amplifier is
' E. M. Deloraine and E. Labin, "Pulse Time Modulation", Electrical Communications ,
Vol. 22, No. 2, pp. 9198, Dec. 1944; H. S. Black "ANTRC6 A Microwave Relay Sys
tem", Bell Labs. Record, V. 33, pp. 445463, Dec. 1945.
2 By pulse position modulation is meant that form of pulse modulation in which the
length of each pulse is kept fixed but its position in time is shifted by the modulation, and
by pulse width modulation that form in which the length of each pulse varies with the
modulation but the center of each pulse is not shifted in position.
' L. R. Wrathall, "Frequency Modulation by Nonlinear Coils", Bell Labs. Record,
Vol. 23, pp. 445463, Dec. 1945.
360
SPECTRUM ANALYSIS OF WAVES 361
another practical application, of which the self oscillating or hunting servo
mechanism is an example.
The quantitative analysis of such systems depends on the ratio of the
pulse repetition rate to the signal frequency. When this ratio is low, the
solution can be obtained by a method shown here for resolving the modulated
waves into selected groups of effectively unmodulated components. This
technique is powerful since it can be done by graphical means whenever the
complexity of either the system or the signal warrants it. When the ratio of
pulse rate to signal frequency becomes high enough, such methods are no
longer practical. However, under these conditions other methods become
available, especially in cases like those mentioned above where the spectrum
of the modulation approaches one of the more familiar forms. An important
example of this occurs in the case of the pulse position modulator where, as
the spectrum approaches that of phase modulated waves, the solution can
often be found by the conventional Bessel's function technique used in
analyzing phase and frequency modulators.
The method proposed here for obtaining the spectrum analysis of pulse
modulated waves is based on the use of the magnitudetime characteristic
of the single pulse and its frequency spectrum as a pair of interchangeable
building blocks, so that the analysis will develop this relationship. Before
doing this the elementary theory of spectrum analysis will be reviewed
Review or the Elementary Theory of Spectrum Analysis
A complex wave may be represented in two ways. One way is by its
magnitude at each instant of time. The other way is by its frequency
spectrum, that is, by the various sinusoidal components that go to make up
the wave. The two representations are interchangeable.
The transformation from a given frequency spectrum to the corresponding
magnitude vs. time function is straightforward, for it is apparent that the
various components in the frequency spectrum must add up to the desired
magnitudetime function. The necessary additions may be difficult to
make in some cases but they are not hard to understand.
The reverse process of finding the frequency spectrum when the magni
tudetime characteristic is given is more involved, though using Fourier anal
ysis, the problem can generally be formulated readily enough. Furthermore
the mathematical procedures involved can be interpreted physically in
broad terms by modulation theory. However, these procedures become
more difficult to perform, and the physical relationships more obscure, as the
wave form under analysis becomes more complex. This is particularly
true when general or informative solutions rather than specific answers are
required. Pulse modulated waves are sufficiently new and complex to give
such difficulties.
362 BELL SYSTEM TECHNICAL JOURNAL
The process of finding the frequency spectrum of a complex wave from its
magnitudetime function has a simple mathematical basis. It depends on
the fact that the square of a sinusoidal wave has a positive average value
over any interval of time, whereas the product of two sinusoidal waves of
different frequencies will average zero over a properly chosen interval of
time."*
In theory then, as the magnitudetime function of a complex wave is the
sum of all the components of the frequency spectrum, we have only to mul
ti])ly this magnitudetime function by a sinusoidal wave of the desired
frccjuency and then average the product over the proper time interval to
find the component of the spectrum at this frequency.^
One physical interpretation of this procedure can be given in terms of
modulation theory. The product of the magnitudetime function with a
sinusoidal wave will produce the beat or sum and difference frequencies be
tween the frequency of the sinusoid and each component of the frequency
spectrum. Thus, if the spectrum contains the same frequency, a zero beat
or dc term is produced, and this term may be evaluated by averaging the
product over an interval that is of the proper length to make all the ac
components vanish.
The application of this principle for spectrum analysis is simple when the
magnitude of the wave in question is a periodic function of time. The very
fact that the wave is periodic is sufficient proof that the only frequencies
that can be present in the wave are those corresponding to the basic repeti
tion rate and its harmonics. Thus the frequency spectrum is confined to
these specific frequencies and so it takes the form of a Fourier series. Know
ing that the possible frequencies are restricted in this way, the problem of
finding the frequency spectrum of a complex periodic wave is reduced to one
of performing the above averaging process at each possible frequency. The
period of the envelope of the Complex Wave is the proper time interval for
averaging, and the integral formulation for obtaining this average is that
for determining the coefficients in a Fourier series.
The principle holds equally well when the magnitudetime function is non
periodic, but the concept is complicated by the fact that the frequency
spectrum in such cases is transformed from one having a discrete number of
components of harmonically related frequencies to one having a continuous
band of frequencies.*' Such s]:)ectra contain infinite numbers of sinusoidal
■• The i)roper time interval is generally some integral multiple of the period correspond
ing to the difference in frequency of the two sinusoid waves.
* In practice it is generally necessary to multiply by both sine and cosine functions
because of i)ossible phase differences.
8 One exception to this statement is the fact that any wave made up of two or more
incommensurate frequencies is nonperiodic. Yet such waves will have a discrete spectrum
if the number of components is finite. This incommensurate case is neglected throughout
the discussion.
SPECTRUM ANALYSIS OF WAVES 363
components, each of infinitesimal amplitude and so close together in fre
quency as to cover the entire frequency range uniformly.
' The continuous band type of frequency spectrum is just as characteristic
of nonperiodic waves as the discrete spectrum is of periodic waves. This
can be shown as a logical extension of the Fourier series representation of
periodic waves. The transition from a frequency spectrum consisting of a
series of discrete frequencies to one consisting of a continuous band of fre
quencies can be made by treating the nonperiodic function as a periodic
function in which the period is allowed to become very large. As the period
approaches infinity the fundamental recurrence rate approaches zero, so
that the harmonics merge into a continuous band of frequencies.
This does not of course change the basic realtionship between the fre
quency spectrum of a wave and its magnitudetime function. The mag
nitudetime function is still the sum of the components of the frequency
spectrum. Also the frequency spectrum can still be obtained frequency by
frequency, by averaging the product of the magnitudetime function and a
unit sinusoid at each frequency. However, the actual transformations
in the case of the nonperiodic functions require summations over infinite
bands of frequencies and over infinite periods of time and so fall into the
realm of the Fourier and similar integral transforms.
However, in any case the problem of spectrum analysis reduces to an
averaging process. The process can be performed by mathematical inte
gration in all cases where a satisfactory analytical expression for the mag
nitudetime function is available. Fourier analysis provides a very powerful
technique for setting up the necessary integrals in such cases.
This averaging process can also be done graphically. It is apparent from
the theory that if the product of the magnitudetime function and the
sinusoid is sampled at a sufficient number of points, spaced uniformly over
the proper time interval, then the average of the samples gives the desired
value. This technique is fully treated elsewhere" so that it will not be con
sidered in detail here. However, use will be made of it in a qualitative way
to augment the physical picture.
NonLinear Aspects
The use of the frequency spectrum in transmission studies is generally
limited to cases where the system in question is linear; that is, where the
transmission is independent of the amplitude of the signal. However, the
same techniques can still be used on systems employing successive linear
and nonlinear components, in cases where the transmission through the
nonlinear elements is independent of frequency. Under these conditions,
the magnitudetime representation of the wave can be used in computing
'Whittaker and Robinson, Calculus of Observations.
364
BELL SYSTEM TECHNICAL JOURNAL
llie transmission over each nonlinear section, where the transmission is
dependent only on the amplitude, and the frequency spectrum used over
each linear section, where the transmission is dependent only on the fre
quency. This a technique can be used on most pulse modulating systems
because such nonlinear elements as the modulators and limiters generally
encountered are substantially independent of frequency.
Frequency Spectrum of the Single Pulse
The single pulse is a nonperiodic function of time and so has a continuous
frequency spectrum. In this case the Fourier transforms are simple. They
are derived in Appendix A. Figure 1 gives a graphical representation of
the magnitudetime function and the frequency spectrum of the pulse.
The expressions are general and hold for pulses of any length or amplitude.
It is instructive to note that the frequency spectrum in this case can be
MAGNITUDETIME
FUNCTION, e (t)
1.0
LU
qO.6
D
1
E
3 0.4
a
n
TIME,
FREQUENCY SPECTRUM, g (f)
6C 4C 2C 2C 4C 6C
FREQUENCY,!, IN TERMS OF C (WHERE C = VaO
Fig. 1 — Magnitude time and frequency spectrum representations of a single pulse.
determined by using the graphical technique mentioned previously. For
example, consider the product of the magnitudetime function of the single
pulse with a sinusoidal wave of given frequency and unit amplitude, so
arranged in phase that its peak coincides with the center of the pulse.
Theoretically the average of this product taken over the infinite period will
give the relative magnitude of the component in the frequency spectrum
of the pulse having the same frequency as the sinusoidal wave. In this
case however, the average need only be taken over the length of the pulse,
since the product vanishes everywhere else. Thus at very low frequencies,
where the period of the sinusoidal wave is very much greater than the length
of the pulse, the average is proportional to 2EL where E is the amplitude
and 2L the length of the pulse. Then as the frequency increases, the average
of the product, and hence the relative amplitude of the component in the
spectrum, will first decrease. For the particular frequency such that the
length of the pulse is one half the period, the relative ami)litude will have
SPECTRUM ANALYSIS OF WAVES
365
2/2
fallen to 2EL X " I " being the average value of a half wave of unit ampli
tude ). Similarly when the frequency is such that the length of the pulse
is a full wavelength, the average will vanish, and when the pulse length is
one and a half times the wavelength, the average is negative, having two
negative and one positive half waves over the length of the pulse, and the
2
relative magnitude is 2EL X ^. These products are shown graphically
on Fig. 2. Since these amplitudes correspond to those given in Fig. 1,
for the spectrum components at/ = /o = 1/4Z, 2/o , and 3/o , it is apparent
that the spectrum could be determined in this way.
WHERE f =
WHERE f = Val
AVERAGE =2EL
1
E
1

L +L TIME,t
WHERE f = I/2L
«>
/
\ AVERAGE =
<JJ r'
/
\
/
\
1 "'
/
1
a. (\j
 1
r^
TIME, t — »■
o
<u
;
V
<o
u
r
AVERAGE HVrr EL
TIME, t
a 4
3 rr
"^
_4
'3TT
RESULTANT SPECTRUM
^s
J
^c ^, L ,^^
^'4C
3C
FREQUENCY, f, IN TERMS OF C (WHERE C= V^O
Fig. 2 — Graphical derivation of spectrum of single pulse by averaging product of pulse
with sinusoidal waves of various frequencies.
Basic Technique
In the analysis presented here, the single pulse and its spectrum will be
used in such a way that the need for individual integral transforms for each
complex wave form under study is avoided. The theory is simple.
A complex wave form may be approximated to any desired accuracy by a
series of pulses, varying with respect to time in length, in amplitude, and
in position. Now the spectra of these individual pulses are already known.
Therefore, to find the frequency spectrum of the complex wave in question,
it is necessary only to combine properly the spectra of the various pulses
representing the complex wave.
Thus the process is theoretically complete. The procedure is first to
366 BELL SYSTEM TECHNICAL JOURNAL
break down the given complex wave into a series of single pulses. Next
the spectrum of each pulse is determined separately. Then the spectrum
of the complex wave is obtained by combining the spectra of the various
single pulses involved. One of the things to be demonstrated here is that it
is perfectly feasible in many cases to perform these summations graphically,
even tliough basically it does involve the handling of spectra each containing
an infinite number of frequency components.
There are other wave forms that could be used as the fundamental build
ing block instead of the single pulse. The unit step function is one possi
bility, since it is used in transient analysis for a similar purpose. However,
the single pulse has obvious advantages when the complex wave to be ana
lyzed is itself a series of pulses, as in pulse modulation. Again it would be
nice to be able to choose as the fundamental unit a wave that has a discrete
rather than a continuous band frequency spectrum, but it seems that any
wave flexible enough to make a satisfactory building unit is inherently non
periodic and so has a continuous frequency spectrum. However the fact
that the fundamental units have continuous spectra does not of itself compli
cate the results. If for example, the wave to be analyzed is periodic, the
sum of the spectra of the various pulses must reduce to a discrete frequency
spectrum. In the cases of interest here, when the pulse train under analysis
is repetitive, combinations of identical pulses will be found to occur with the
same fundamental period, and generally the first step in the summation of
such spectra is to group the series of pulses into periodic waves with discrete
spectra.
Manipulations of Single Pulses
In its use, the single pulse may be varied in amplitude, in length, and in
position with respect to time. These changes have independent efifects on
the frequency spectrum. A variation in the amplitude of a pulse does not
change its spectrum, except to increase proportionately the magnitudes of
all components. A change in position of a pulse with time does not change
the amplitude vs. frequency characteristic of the spectrum, but it does
shift the phase of each component by an amount proportional to the product
of the frequency and the time interval through which the pulse was shifted.
A change in the length of a pulse will change the shape of the amplitude vs.
frequency characteristic of the spectrum. Figure 3 shows this effect. How
ever, if the center point of the pulse is not shifted in time, the relative phases
of the components are not afifected by such changes in length.
The single pulse can also be modulated to aid in the resolution of more
complicated wave forms. This process is based on the use of the pulse as a
function having a value of unity over a chosen time interval and a value of
zero at all other times. Thus, to show a part of a sinusoidal wave, we need
SPECTRUM ANALYSIS OF WAVES
367
only multiply this wave by a pulse of the correct length and proper phase
with respect to the sinusoid to show only the desired piece of the wave. In
this simple case it is not difficult to derive the spectrum because what are
produced are the sum and the difference products of the modulating fre
quency with the spectrum of the pulse. This gives two single pulse spectra
shifted up and down in frequency by the frequency of the modulation. An
example of this is shown in Fig. 4, where the spectrum of a single half c>cle
is determined.
Pulse Position Modulation
For the first example, a simple form of pulse position modulation will be
analyzed. The pulse train in this case is made up of pulses spaced T seconds
U 0.2
a 0.4
\^
\
s
r^>
\ \
\ \
^x.;
puLse
3_L 2
LENGTHS:
L
4L
3
jr
\
\
N
S
>
=— ■
~'^
""'
I 2 3 4 5 ,
FREQUENCY, f, IN TERMS OF C (WHERE C = — )
Fig. 3 — Change in frequency spectrum with pulse length.
apart and the width of each pulse is a very small part of the spacing T.
Such a pulse train is shown on Fig. 5. The pulse train is modulated by ad
vancing or retarding the position (time of occurance) of the pulses by an
amount proportional to the instantaneous amplitude of the signal at sampled
instants T seconds apart. Figure 5 also shows the signal, in this case a sine
wave of frequency 1/lOr, and the resulting modulated pulse train. The
peak amplitude of the modulating sine wave is assumed to shift the position
of a pulse by 1 /iT. The length and the amplitude of the pulses are the same
since neither is affected in this type of modulation.
The first step in the analysis is to determine the spectrum of the pulse
train before modulation. Each pulse contributes a spectrum of the form
368
BELL SYSTEM TECHNICAL JOURNAL
shown on Fig 1. Now the phase of each component in such a spectrum
is so arranged that the spectrum forms a series of cosine terms all of which
have zero phase angle at the center of the pulse. From successive pulses T
SPECTRUM OF
SINGLE PULSE
UJ
Q
H
_l
Q
5
<
\
L 0^ L
TIME.t*
\
\
— ^^
V^
^•
/
X
MODULATION
. PRODUCTS
/
/
\
\
\
\
\
\ DIFFERENCE
\ TERMS
\ SUM TERMS
\
^^
^—

\^
v^
^
~~~ —
RESULTANT
SPECTRUM
Q
D
Q
5
r'/^
r\
T
"^
\
'/
N
\
N
1/2 SUM +
1/2 DIFFERENCE
L L
TIME.t— ♦
\^
— ^^
2C 3C 4C 5C
FREQUENCY, f, IN TERMS OF C (WHERE C^^t)
Fig. 4 — Determination of spectrum of single half sine wave by modulation of single pulse
spectrum with cos licet.
seconds apart, the component at any given frequency will have the same
amplitudes, but the relative phases will be 1kJT radians apart. It is appar
ent that frequencies for which lirjT is 2x or some multiple of 27r radians
SPECTRUM ANALYSIS OF WAVES
369
apart, the contributions from all pulses add in phase. These are the fre
quencies nc, where n = 1,2,3 and c "^ Tj. It is also apparent that at fre
quencies for which the phase differences between the components are not an
exact multiple of 2ir radians apart, the contributions from enough pulses
must be spread in phase over an effective range of to 2x radians in such a
way as to cancel one another. For example, take the particular frequency
for which the difference in phase between pulses is 361° instead of 360°.
1 1
1 1 1
1 1 1 1 1
1 1 1 1
<u
TIME.t— »■
u I
inT ' '
o
D
1
p.—
^\ 62
'°^:v '"" "
H ^_^^
a
<
^^\ \ \ "^^ ^^\
1 i
:i :i 1
TIME,t— »•
1 i 1 1 1
1 i 1 1
;AT
►1 U J UAT2
TIME.t — »•
^
^■
UNMODULATED
PULSE TRAIN
(PERIOD T)
MODULATING
FUNCTIOM
OR SIGNAL
(PERI0D=10T)
POSITION
MODULATED
PULSE TRAIN
(AT, ~e,ETC)
(REFERENCE)
(AT,orO)
TIME.t — »■
Fig. 5 — Formation of pulse position modulated pulse train and its resolution into subsidiary
unmodulated pulse trains.
The contribution from each preceding pulse will be effectively advanced in
phase 1° with respect to its successor, so that the contributions from pulses
180 periods apart will be exactly 180° out of phase. Therefore over a
sufBcient number of pulses, the net contribution is zero.
The spectrum of the unmodulated pulse train is thus made up of a do
term plus harmonics of the frequency C = \/T. The dc term is the average,
and therefore is equal to £ X 2L/T, where E is the magnitude of the pulse.
All of theother components have the same relative magnitudes that they have
370
BELL SYSTEM TECHNICAL JOURNAL
in the single pulse spectrum. This gives a spectrum like that shown on
Fig. 6. Figure 6 also shows for comparative purposes the spectrum of the
subsidiary pulse wave consisting of every 6th pulse.
Thus in the unmodulated case, the pulses have a uniform recurrence rate
and the resultant spectrum, found by adding those of the individual pulses,
reduces to a train of discrete frequencies comprised only of the harmonics of
the recurrence rate of the pulses. The fundamental frequency, correspond
WHERE PULSE LENGTf
\ = 1/6 PERIOC
) LENGTH
1.0
.
o
D
~ ~ I
E
i
0.8
~^"^
21 4T 6T 8T lOT 12T
TlME.t
0.6
UJ 0.4
""^^ FREQUENCY
^^^^ SPECTRUM
O
1
^^^
O0.2
Hi
OC
cc
UJ
1.0
2 0.8
0.6
0.4
0.2
C 2C 3C 4C 50
FREQUENCY, f, IN TERMS OF C (WHERE C =!/j)
WHERE PULSE LENGTH = 1/36 PERIOD LENGTH
FREQUENCY SPECTRUM
TITTTITfTTITrTTrrnrTTnTr.r
2V 4V 6V 8V lOV 12V 18V 24V 30V
FREQUENCY, f, IN TERMS OF V (WHERE V = l/gC = l/gT)
36V
Fig. 6 — Frequency spectrum of pulse trains where the spacing between the pulses is 6 and
36 times the pulse length respectively.
ing to the recurrence rate, and its harmonics will be called the carrier fre
quencies of the pulse train. The effect of modulating the pulse train is to
modulate each of these carriers, producing sidebands of the signal about
them.
When the pulse train is position modulated, the pulses are shifted in posi
tion by an amount AT, corresponding to the instantaneous ami^litudes of
the modulating function. The spectrum of each pulse is unchanged, since
the pulse length remains constant. However, components of successive
SPECTRUM ANALYSIS OF WAVES 371
pulses at the carrier frequency c and its harmonics will no longer add directly,
because of the phase shifts that accompany the change in position. This
phase shift is equal to AT, the shift in position, times the radian frequency
of the component in question.
However, when the signal function is periodic, each pulse will have the
same shift in position as any other pulse that occurs at the same relative
instant in a later modulating cycle. Furthermore, when the carrier fre
quency is an exact multiple of the signal frequency i.e., c = nv, there will
be a pulse recurring at the same relative instant in each cycle of v. Under
these conditions, the pulse position modulated wave can be broken down into
a group of unmodulated waves, each being made up of that series of pulses
that recur at a given part of each modulating cycle, as shown in Fig. 5.
These subsidiary waves are eflfectively unmodulated because, as each pulse
recurs at the same instant in the modulating cycle, they are shifted to the
same extent and hence will be uniformly spaced. This uniform spacing
between pulses in a given wave is equal by definition to the period of the
modulating function, and there will be as many of these unmodulated pulse
trains as there are pulses in a single cycle. Thus, if c = nv, there will be n
such pulse trains.
The reason for grouping the pulses into these unmodulated pulse tarns is
that unmodulated periodic trains have spectra of discrete frequencies. Since
the pulse widths are all equal, and since the spacing between pulses is the
same for each wave, the spectra of these unmodulated waves will all be
identical. Furthermore, these spectra will be the same as that of the
original carrier wave of pulses before modulation, except for two factors.
First, the fundamental frequency is now i', corresponding to the modulating
period, so that there are n times as many components as before. Secondly
the amplitudes are reduced by the factor  because there is only one pulse
in these new waves to every n pulses in the original wave. Thus, instead
of having a spectrum made up of the carrier frequency and its harmonics,
we now have one made up of harmonics of v. Since c = nv, such frequencies
as c, c, ± t, c ± 2v, etc., are included. An example of the spectra of both
the subsidiary and original pulse waves is shown on Fig. 6, for the case
where n = 6.
Thus the problem of finding the spectrum of such a pulse position modu
lated wave is reduced by this procedure to adding up the ;/ equal components
at each of the frequencies of interest, such as c and c dz v, allowing for the
phase difference between components corresponding to the position of one
pulse with respect to that of the other nl pulses in one modulating cycle.
As an example, suppose n = 10 and the frequency to be computed is c + ^•
Now < + I) is 10% higher in frequency than c. Thus in the unmodulated
372
BELL SYSTEM TECHNICAL JOURNAL
case, when the n pulses are equally spaced, they are 360° apart at c and
consequently 360° + 36 or 396° at c + v. Therefore in the unmodulated
case, each component would be advanced in phase 36° with respect to the
previous one, so that the diagram of the 10 components would form the
FREQUENCY C+V
(a) ZERO MODULATION
10
(b) 50 PER CENT MODULATION
10
FREQUENCY CV
(C) ZERO MODULATION
(d) 50 PER CENT MODULATION
9 8 10 2 I
Fig. 7 — Vector pattern of subsidiary pulse components.
vector pattern shown on Fig. 7A. The successive components are numbered
1 to 10. The sum in this unmodulated case is of course zero.
Now the effect of modulation is to shift the relative jjhascs of these compo
nents by an amount determined by the shift in position of the corresponding
pulses. When these relative phase shifts are such as to spoil the can
SPECTRUM ANALYSIS OF WAVES 373
cellation of the 10 components, a net component of this frequency is pro
duced in the frequency spectrum of the pulse wave. Taking the example
shown in Fig. 5, the 10 components in Fig. 7A would be shifted to the posi
tions shown in Fig. 7B. These shifts in relative phase are determined in the
following way. Figure 5 shows that the number 1 pulse is retarded an
amount AT^i equal to 15% of T, the normal spacing between pulses. Thus
at the carrier frequency c, the phase shift between the component from tkis
retarded pulse and the reference pulse is 15% more than 360° or 414°.
Thus the component at the carrier frequency c from the first subsidiary
pulse train is shifted 54° from its unmodulated position.
. At c f V, since the frequency is 10% higher, the net shift is 10% more than
at c or 59.5°. Thus the number 1 component on the vector diagram of
Fig. 7B is rotated 59.5° clockwise from its unmodulated position shown on
Fig. 7A.
Similarly pulses 2 and 3 are each shifted in position by equal amounts,
AT2 and AT3 . These shifts in position give 85° phase shift at the carrier
frequency. Hence components 2 and 3 Sit c \v are each rotated 10% more
or 93.5° from their respective unmodulated reference positions shown on
Fig. 12 A. Component number 4 is shifted 59.5° clockwise just as number 1 .
Component 6 and 9 are also shifted 59.5° each, but in this case the modulat
ing function has the reverse polarity so that the components are rotated
counterclockwise. Similarly components 7 and 8 are rotated 93.5°
counterclockwise.
The sum of these components in the vector diagram of Fig. 7B gives a
resultant that is negative with respect to the reference direction and the
magnitude that is 58% of the reference magnitude, where the reference mag
nitude and direction are those for the carrier c with no modulation.
This gives the relative magnitude and phase of the c\v term produced by
pulse position modulation for the case where the modulating function is a
sine wave of frequency v — c/10 with a peak amplitude just large enough to
shift a pulse by 1/4 of T, where T is the spacing between unmodulated pulses.
A shift of this magnitude will be defined here as 50% modulation on the
basis that 100% modulation should be 1/2 T, the maximum displacement
that can be used without possible interference between pulses.
In the same way the other component frequencies in the spectrum such as
c,c — v,c±2v,etc., have been computed for the above case of 50% modulation,
and for other peak ampUtudes of the modulating sine wave giving 25%,
70% and 100% modulation. In all cases the frequency of the modulating
function was held at z; = c/10. This information is plotted on Fig. 8, show
ing V, c and the various components of the frequency spectrum that represent
the sidebands about the carrier frequency c, as a function of the peak %
modulation.
374 BELL SYSTEM TECHNICAL JOURNAL
The above solution assumed a special case where c was an exact multiple
of V. The purpose of this assumption was to simplify the problem to the
extent that the periodicity of the modulated wave would be the same as
that of the modulating function. There are two other possible cases. For
one, the ratio of c to v could be such that a pulse would occur at the same
instant of the modulating period only once every so many periods. The
actual periodicity of the modulated pulse wave would be reduced accordingly
because it would make the same number of periods of the modulating func
tion before the modulated pulse train is repeated. This is a result of the
fact that pulse modulation provides for a discrete sampling rather than a
continuous measure of the modulating wave. The technique of spectrum
analysis demonstrated above is just as applicable to this case as it was to the
simpler one. However, there will be comparatively more terms to be
handled. The other possible case is the one where c and v are incommen
surate.^ In this case, the resulting modulated wave is nonperiodic. How
ever, on the basis that the spectrum is practically always a continuous
function of the signal frequency, this case has received no special attention
here.
At frequencies for which c is very much greater than v, so that the number
of component pulse trains becomes too numerous to handle conveniently in
the above fashion, the sidebands about each carrier or harmonic of the
switching frequency can be computed by the standard methods for phase
modulation, as the next section will demonstrate. This result follows
directly from the theorem that as the carrier frequency c becomes large with
respect to v, pulse position modulation merges into a linear phase modulation
of each of the carriers.
Pulse Position Modulation vs Phase Modulation
When a pulse, in a pulse position modulated wave, is shifted by 1/2 the
spacing between pulses (100% modulation) it is apparent from the previous
discussion that the component of the carrier in the frequency spectrum of the
pulse is shifted by 180°. Therefore to compare the spectrum of a pulse
position modulated wave like that on Fig. 8 with the equivalent spectrum of
a phase modulated wave, what is needed is Fig. 9, showing the frequency
spectrum of a phase modulated wave of the form Cos{ct — k sin vt) as a func
tion of k for values of ^ up to zr radians or 180°. The computation of the
frequency spectrum of such a phase modulated wave has been adequately
covered elsewhere and all that is done here is to give the brief development
shown in appendix B.
* Mr. W. R. Bennett has pointed out that this incommensurate case is the general one.
It requires a double Fourier series, which reduces to a single series when the signal and
carrier frequencies are commensurate. This analysis is based on the single Fourier series.
SPECTRUM ANALYSIS OF WAVES
375
A comparison of the spectra on Figs. 8 and 9 shows that the sidebands
have the same general pattern. However comparative sidebands are not
40 50 60 70
MODULATION IN PER CENT
Fig. 8 — Spectrum of pulse position modulated wave for case where the carrier frequency
C is 10 times the signal frequency v.
quite equal in the two cases. In fact comparable upper and lower side
bands in the case of the pulse modulated wave shown on Fig. 8 are not
376
BELL SYSTEM TECHNICAL JOURNAL
equal in absolute magnitude to each other. This lack of symmetry is due
to the fact that c is ()nl\' 10 limes v.
115 3
8 2 8 4
PEAK PHASE SHIFT IN RADIANS
Fig. 9— Spectrum of phase modulated wave cos {ct + k sin vt) as function of peak phase
shift k for values of ^ up to tt radians.
One way of proving this is to go through the process of computing the
c — V term in this pulse modulated wave just as ihc c\v term was computed
SPECTRUM ANALYSIS OF WAVES 377
earlier. Since the frequency c— I'is 10% less thane, the unmodulated pattern
of the 10 subsidiary components, as shown on Fig. 7C, is the mirror image of
that for c + ^ in 7A, for the first component is now 360° less 10% or 324°,
and subsequent components are each retarded 36° with respect to the pre
vious one. When the pulse train is modulated the effect is similar to the
case for c \ v and, for the same per cent modulation, the Vector pattern
of Fig. 7D is formed. The resultant in this case differs from that of 7B
in sign as well as in magnitude. The difference in sign comes from the fact
that, since component 1 in 7A corresponds to component 9 in 7C and com
ponent 2 in 7A to component 8 etc., the modulation in the case of c — t; rotates
these corresponding components in opposite directions. The difference in
magnitude is due to the fact that since c — v is an appreciably lower fre
quency than c \ v\\\ this case (approx. 20%), the phase shift corresponding
to a given shift in pulse position is proportionately less. Thus the corre
sponding Vector components are not shifted the same number of degrees.
Thus the absolute magnitudes of c f i' and c — v are not equal in this case.
It is apparent that this difference in magnitudes oi c \r v and c — v be
comes smaller as the carrier frequency c becomes larger with respect to v.
In the limiting case of c very much greater than v, c \ v and c — v would
each be shifted the same number of degrees as c itself. If this more or less
compromise shift of c is used to compute the c ± i', c ± 2v, and c db 3i; terms,
then the resulting frequency spectrum is that of the phase modulated carrier
on Fig. 9.
The higher harmonics of c in the pulse position wave are similarly phase
modulated and the interesting point is that 2c is modulated through twice as
many degrees phase shift and 3c 3 times as many degrees, etc. Thus a
single pulse position modulator could be designed to produce a harmonic of
c with almost any desired degree of phase modulation. This is a useful
method for obtaining a phase modulated wave, or with a 6 db per octave
predistortion of the signal, a frequency modulated wave.
Figure 8 also shows a term in v itself, which has been neglected so far in
the discussion. It is apparent that the components at v contributed by the
10 subsidiar}' unmodulated waves must form the same kind of vector pattern
as those oi c \ v in Fig. 7. However, in this case c \ v\% eleven times v in
frequency, so that the components of v are rotated only one eleventh as
much for a given pulse diplacement. Thus the magnitude of v at 100%
modulation is equal to that oi c \ v at approximately 9% modulation. For
different frequency ratios of c to v the relationship of the v term io c \ v will
vary, and it is apparent that for c very much greater than v, the v term will
vanish. The relationship is such that the amplitude of the v component out
of the modulator at a given per cent modulation is directly proportional to
its own frequency v for all frequencies less than approximately one quarter
378 BELL SYSTEM TECHNICAL JOURNAL
of c, and the phase is 90° with respect to the input. Thus the modulator
puts out a signal component that is the derivative of the input signal.
To summarize the case of pulse position modulation, the frequency spec
trum may be determined by the methods based on subdividing the modu
lated pulse train into a series of unmodulated ones when the ratio oi c ta v
is small, and by treating each harmonic of the carrier as a phase modulated
wave of the form Cos n (ct \ 6), where 6 is the modulating function, when the
ratio of c to D is large. In the case treated here, the modulating function was
a simple sinusoidal wave. Of course the analysis holds for more complicated
wave shapes having frequency spectra of their own. In this event however
the restriction on the relative magnitudes of the frequencies v and c should
be taken as one on c and the highest frequency in the modulating spectrum.
The complexity of the modulating function does not affect the analysis when
it is done by this technique of subdividing the pulse train, since all that need
be known is how much each pulse is shifted, and this can be done graphically.
The analysis given here has neglected the length of the individual pulses.
This was done when it was assumed that the individual contributions from
the various pulse trains had the same amplitude at all frequencies. For any
finite pulse width, the relative magnitudes of the various components must
silt X
be modified by the factor of the single pulse, as shown on Fig. 6.
As mentioned in the introduction, a complex wave could be analyzed by
multiplying its magnitudetime characteristic by unit sinusoids at each
frequency in question, sampling the product at a sufficient number of points
uniformly spaced over a cycle of the envelope of the complex wave, and then
averaging the values of the product thus obtained. This technique is par
ticularly applicable to the analysis of pulse position modulated waves since,
by taking the centers of the pulses of the modulated wave as the sampling
instants, it is possible, with a finite number of samples (same as the number of
pulses) to get the same results as though a very much greater number of
uniformly spaced samples were taken. The interesting thing to note here
is that the actual computations that would be involved in applying this
sampling method of analysis to a pulse position modulated wave are almost
identically the same calculations as required by the technique of resolving
the pulse train into unmodulated subsidiary pulse trains used here.
Pulse Width Modulation
Pulse Width Modulation as defined here could also be termed "pure"
pulse length modulation. The pulse train in the reference or unmodulated
condition is a recurrent square wave, and the lengths of the pulses will be
varied by the modulation without changing the position of the centers of
the pulses. The term "pure" pulse length modulation is appHcable to this
SPECTRUM ANALYSIS OF WAVES 379
special case where the phase relationship between spectra of adjacent pulses
does not change with modulation because the centers of the pulses are not
shifted by the modulation. The conventional form of pulse length modula
tion, where one end of the pulse is fixed in position, combines both this
pulse width modulation and the pulse position modulation previously ana
lyzed. The interest in this case of pulse width modulation arose in con
nection with the analysis of ''hunting" ser\^omechanisms, and the analysis
provides a basis for a general solution of the response of a twoposition
switch or ideal limiter to various forms of applied voltages.
Since the unmodulated wave is a square wave with pulses of length 2L
recurring at intervals of T = 4L, it has the familiar square wave spectrum
including a dc term, a fundamental term or carrier of frequency c = l/T, a
3rd harmonic with a negative ampUtude 1/3 that of the fundamental, etc.
Figure 10 shows clearly that this spectrum is the sum of single pulses of
width 2L spaced T = AL seconds apart. In the summation, all frequencies
cancel except harmonics of c and, since they all add directly in phase, the
component frequencies in the resultant spectrum have the same relative
amplitudes as they have in one single pulse.
When this pulse train is modulated, the width of each pulse becomes
2{L\ AL), where the magnitude of AL depends in some specified way on the
magnitude of thhe modulating function at the instant corresponding to the
center of the pulse. For simplicity, the case will be taken where AL is
proportional to the magnitude of the modulating function. For 100%
modulation, AL will be assumed to vary from — L to +L. Figure 3 shows
how the relative amplitude of the components of the frequency spectrum of
a pulse vary for 3 different values of AL , along with the equation that gov
erns these amplitudes.
If the modulating function has a periodicity v such that c = lOz', then
every lOth pulse, recurring at the same instant in each modulating cycle,
will be widened to the same extent and so can be formed into a subsidiary
unmodulated pulse train, as was done on Fig. 5 for the pulse position
modulated wave.
Again vector diagrams like those in Fig. 7 may be formed showing the
contribution of each of these subsidiary pulse trains at various frequencies
such as c, r + v and c — v. ^^1len the waves are unmodulated, the vector
diagrams for the same frequencies will be the same as those for the pulse
position modulated case, except for the absolute amplitudes of the com
ponents, as long as c = lOr in each case. When the pulse width system is
modulated, however, the modulation does not rotate the individual vector
components as in the pulse position case since the spacing between pulses is
not changed. What the pulse width modulation does is to change the
length of the individual component vectors exactly as it does in the case of
380
BELL SYSTEM TECHNICAL JOURNAL
the single pulses shown on Fig. 3. This change of magnitude, of course, can
spoil the cancellation of the ten unmodulated components at some frequency
like c \ 2v just as effectively as rotating them did in the case of the pulse
position modulated wave, thus ])r<)during a sj)ectrum component at that
frequency.
As an example, the case will be taken where the modulating function is a
0.4
3
h. 0.2
5
^o.sr
Q
3
5^0,4
<
liJ 0.3
>
o
h
Q.
<
^
E
1
\
L L
TIME,t — *
^
"V
y
^
UlJ
Q
Q
<
\
\
\
\
\
\
\
5L 3L L L 3L 5L 7L 9L
TIME,t —*
\
\
\
\
\
\
,'""
— ^^
\
\
^^^~ —
IC 2C 3C 4C 5C 6C
FREQUENCY, f, IN TERMS OF C (WHERE C =^)
Fig. 10 — Comparative sjiectra of square wave and single pulse.
sinusoid of frequency v. Then the change in width with modulation is
given bv the formula
^L
— k sin vl.
Since c = lOr, the successive subsidiary pulse trains will be modulated an
amount! — 1^ = ^sin( 1k — las ;;/ lakes on the values from 1 to 10. Thus
the spectra of these subsidiary pulse trains with ])ulses of length 2(L +
SPECTRUM ANALYSIS OF WAVES
381
AZ,,„) recurring every l/v seconds will be a Fourier series of harmonics of v.
The amplitude of the nth term of this series will be
J^n = 77. — sm
TTll
1 + ^ sin
27rw
lo"
This expression may be found from appendix C, equation (5a). Combining
^ 0.6
^^
^
^.
^
X
.''
^^'
^ ^^
•
' ^s
Y'''
y
y
'^
^^
•
•
•
y
^
y
y
• ^
•
•
X
2CV
^
•
y.^ ^^^
2C7^
y^^
'^^
^ _,''
"
'J^
^ ^ '^ ^
^i^^**^
y
^
ci3^
_, '"
^.^
•
•
•
y
'
""^^^^
^:^2^
•
X
•
^
40 50 60 70
MODULATION IN PER CENT
Fig. 11 — Spectrum of pulse width modulated wave for case where carrier frequency C is
10 times the signal frequency v.
the 10 such components at each frecjuency, as shown on Fig. 7 for the case
of the pulse position modulated wave, the spectrum for this case of Pulse
Width Modulation on Fig. 11 is produced. This spectrum is comparable
to that on Fig. 8 for the pulse position modulated case.
Pulse Width vs Amplitude Modulation
That pulse width modulation is a form of amplitude modulation of the
carriers of the unmodulated pulse train is shown mathematically by Equa
382
BELL SYSTEM TECHNICAL JOURNAL
0.9 1.0
Fig. 12 — Response of ideal limiter to simultaneously applied isosceles triangle wave and
sine wave inputs, k is the ratio of the peak amplitudes of sinusoidal and triangular
waves at the input. .
tion (8) in Appendix C, where the spectrum is developed as a Fourier series
in harmonics of the pulse rale c with the modulation affecting only the
amplitude of the coefficients.
This mathematical analysis is continued in Appendix D where the fre
SPECTRUM ANALYSIS OF WAVES 383
quency spectrum is determined for AL = k sin vl. The spectrum thus
computed is shown in Fig. 12. L
An example of this type of pulse modulator is given by a two position
switch or ideal limiter when the signal to be modulated is applied simul
taneously to the limiter with an isosceles triangle wave as carrier. The
carrier should have a higher peak amplitude than the signal and a recurrence
rate based on the desired carrier frequency. Figure 12 is arranged to show
the output spectrum for such a limiter in terms of k, when k is the ratio
of the peak amplitudes of the sinusoidal signal and triangular carrier wave
inputs.
A comparison of this spectrum with that on Fig. 11 shows that the
two spectra have almost the same form, c and v have the same amplitude
characteristics in each case. The c ± 2v and 2c ± v terms have differences
that are like those found before in comparing the pulse position modulated
wave on Fig. 8 and the phase modulated carrier on Fig. 9. As in that case,
when c becomes very much greater than v the differences vanish.
Application of Pulse Width Modulator
Practical interest in this case lies in the fact that the signal is present
in the output spectrum with a linear characteristic that makes such a
modulator a linear amplifier. The "onoff" or "hunting" servomechanism
is based on a modified form of such an amplifier in which the carrier is sup
plied by the self oscillation of the system. The term modified form is used
because the self oscillations in general are more nearly sinusoidal than
triangular in form and so do not give a linear change in pulse length over
as wide a range of input amplitudes as does a triangular carrier. No
attempt will be made to analyze such a system here since it has been handled
elsewhere.^ However the above method is applicable to such problems
regardless of the shape of the carrier or the signal.
Other Forms of Pulse Modulation
Another form of pulse modulation of interest is that of pulse length modu
lation in which either the start or the end of each pulse is fixed, so that the
centers of the pulses vary in position with the length. This is a combination
of both the pulse position and the pulse width modulations described above
and can be analyzed by a combination of the methods developed.
These same methods are also applicable to the analysis of frequency and
phase modulated waves after they have been put through a limiter, as they
generally are before detection.
9 See L. A. Macall, "The Fundamental Theory of Servomechanisms" D. Van Nostrand
Company, 1945.
384 BELL SYSTEM TECIIMCAL JOURNAL
APPENDIX A
Fourier Transforms For Single Pulse
The amplitude g{f) of the component of frequency/ in the spectrum of the
Complex Magnitudetime function e{t) is given by the dc component of the
Moduhition products of c{t) and cos IttJI, found by averaging the product
over the period of the comi)lex wave.
Thus, for nonperiodic waves, where the period is from — x to + x , the
ampHtude of the spectrum at / is
g(f) ^ f e(l) cos 2x/7 dt. (1)
For the single pulse, where e{l) = £ for — L < / < L and e{l) = for all
other values of /, equation (1) reduces to
gif) ~ f E cos lirft dt. (2)
Integrating,
g(/) ^ :—. sin lirfi
IttJ
g{f)^. .sin Itt/L. (3)
Equation (3) is the expression for g(f) plotted on Fig. 1.
Similarly, in the case of the single pulse, each increment in frequency df
contributes a factor proportional to g{f) cos 27r// df to the composition of
e{t), so that
e(l) = f g(f) cos 27r// df. (4)
Substituting in (4) the expression for g{f) given by equation (3), this becomes
/A ^. E /""sin 27r/Z, ^ ,^ ,. ,_,
e(/) ^  / ^^ cos 27r// df. (5)
7r Joo /
APPENDIX B
Frequency Spectrum Or Phase Modulated Wave
The Pliase Modulated Wave in this case is given by
cos ((■/ — k sin vl) = cos {ct) cos (k sin vt) f sin (ct) sin (k sin vt)
Now cos (ct) cos (k sin ct) = Jo (k) cos {ct)
+ Jo (k) cos (c  2v) t
SPECTRUM ANALYSIS OF WAVES 385
+ Jo (k) COS {c \ 2v) t + ■■■
and sin (ct) sin {k sin cl) — Ji (k) cos (c — v) t
 Ji (k) cos {c \ v) t
+ /s (k) COS {c  3v) I
 /s (^) COS (c + 3v) t + •••
.'. COS (f/ — k sin ?'/) = Jq (k) COS (c/)
+ 7] (^) COS (c — z;) /
 /i (y^) cos (c + v) t
+ /z (/^) cos (c  2tO /
+ J2 (k) cos (c + 2tO /
+ /s (k) cos (c  3z') t
 J3 (k) cos (c + 3zO / H
APPENDIX C
In this Appendix the spectrum of a train of rectangular pulses of length
2(L + AL) recurring every T seconds, will be found from the spectrum of a
single pulse of this train.
For the single pulse at any frequency/,
gin ^ .sin 2^f{L + AL). (1)
x/
For a series of such pulses recurring with a spacing T — 1/c, then the sum of
spectra of the individual pulses form a Fourier series of harmonics of c. Thus
e(t) = ^0 + Z) ^n cos liritd, (2)
n = l
where An is the sum of an iniinite number (one from each pulse) of infinitesi
mal terms g(;/c) and g{ — nc), shown in (1). Thus
^„ ^ 22 — sin 2Trnc{L + AL) (3)
Tvnc
Now to put an absolute value to the amplitudes g(/) shown in equation (1),
it is necessary to average them over the recurrence period of the single pulse,
making them infinitesimals. However, in the train of pulses recurring
every T — \/c seconds, the amplitude of An can be determined by averaging
the terms in (1) over an interval T. Then
An = ^^sin 2Tvnc{L + AZ). (4)
irncT
When T = 4L = l/c, (4) reduce to
2E .
— sm _,
wn 2
, 2E . nK (. . aA ...
y4„ = — sm — ( 1 + — j (5)
386
BELL SYSTEM TEC/LMCAL JOIKNAL
For the example taken in the text, when the pulse train was subdivided
into 10 subsiding pulse trains, the period T = 1/v = 10/c = 40L. Thus in
this case, the Fourier coefficients of the harmonics of v are
2E . TTii / AL\
(5a)
The expression for .1,, in equation (5) can be put in simpler form by using
the formula for the sin of tlie sum of t wo angles. In this way, we get
An —
IE
irn
TTll
sm I — I cos
/irn AL
L\ , /7r//\ . /irn AL
(6)
Now, for // odd, sia — alternately assumes the value ± 1 and cos — vanishes.
(?)
and for ii even, cos ( —  ) alternatelv assumes the value ±1 and sin
irn
vanishes. The A o term, being the dc average of the pulse train, is given by
E/2{L + AL) ^E (. , AL
T 2 V T
(7)
If the pulse train is transformed by shifting the zero so that it alternates
between db£/2 instead of and E, the first term in equation (7) vanishes
and (2) becomes, from (6) & (7),
e(t) = Ao A Ai cos 27rf/
+ Ai cos 2x 2cl + •
Where
etc.
A, =
A. =
m
2E /t
1 = — cos ( 
TT \Z
¥)
2L; . ML
^^ = 2. "" " U
A, =
2E Stt /AL\
3. ^^^ T \l)
(8)
APPENDIX D
The purjjose of this section is to comi)ule the si)ectrum of the carrier given
by e(ualion (S) in A])pendix C as their amplitudes vary with  = k sin vl.
SPECTRUM ANALYSIS OF WAVES 387
For the Jc term,
, EAL £, . ,
9 T' "^ ? sin vt.
For the fundamental or c term,
2E /tt . \
Ai cos ItcI = — ■ cos KT^k sin z;^ ) cos Iwcl
Using the Bessel's expansion of cos (2 sin 6), we get,
\Jo{k) cos 27rc
_g +/2(^) cos 27r(c — 2v)t
Ai cos 27rc/ = —
^ +/2(/^) COS 27r(c + 2v)t
[] etc.
In a similar fashion, the other terms can also be computed, giving the
spectrum shown on Fig. 12, where Joik) becomes the amplitude of c, J2{k)
the amplitude of either c { 2votc — 2v, etc.
Abstracts of Technical Articles by Bell System Authors
Commercial Broadcasting Pioneer. The WEAF Experiment: 19221926}
William Peck Banning. WEAF, the radio call letters which for nearly a
quarter of a century designated a broadcasting station famous for its
pioneering achievements, ceased last November to have its old significance.
WNBC are the new call letters. This book is an excellent record of the
four years during which this station was the experimental radio broad
casting medium of the American Telephone and Telegraph Company.
The author indicates that the WEAF experiment aided the development
of radio broadcasting in three ways:
First, in the scientific and technological field.
Second, in the emphasis of a high standard for radio programs.
Third, in determining the means whereby radio broadcasting could
support itself.
When TF£/1 F changed hands from the American Telephone and Telegraph
Company to new ownership, public reaction to almost every type of broad
cast had been tested, network broadcasting had been established and the
economic basis upon which nationwide broadcasting now rests had been
founded. A trail had been blazed that thereafter could be followed without
hesitation.
In so far as radio broadcasting is concerned, this book is a significant
chapter in communication history.
A Multichannel Microwave Radio Relay System} H. S. Black, J. W.
Beyer, T. J. Grieser, F. A. Polkinghorn. An 8channel microwave
relay system is described. Known to the Army and Navy as AN/TRC6,
the system uses radio frequencies approaching 5,000 megacycles. At
these frequencies, there is a complete absence of static and most manmade
interference. The waves are concentrated into a sharp beam and do not
travel along the earth much beyond seeing distances. Other systems
using the same frequencies can be operated in the near vicinity. The
transmitter power is only one fourmillionth as great as would be required
with nondirectional antennas. The distance between sets is limited but
by using intermediate repeaters communications are extended readily to
longer distances. Short pulses of microwave power carry the intclHgence
of the eight messages utilizing pulse position modulation to modulate the
1 Published by Harvard University Press, Cambridge, Massacliusetts, 1946.
^ Elec. Engg., Trans. Sec, December 1946.
388
ABSTRjiCTS OF TECHNICAL ARTICLES 389
pulses and time division to multiplex the channels. The eight message
circuits which each AN/TRC6 system provides are highgrade telephone
circuits and can be used for signaling, dialing, facsimile, picture transmission,
or multichannel voice frequency telegraph. Twoway voice transmission
over radio links totaling 1,600 miles, and oneway over 3,200 miles have
been accomplished successfully in demonstrations.
Further Observations of the Angle of Arrival of Microivaves? A. B.
Crawford and William M. Sharpless. Microwave propagation measure
ments made in the summer of 1945 are described. This work, a continua
tion of the 1944 work reported elsewhere in this issue of the Proceedings of
the I.R.E. and Waves and Electrons, was characterized by the use of an
antenna with a beam width of 0.12 degree for angleofarrival measurements
and by observations of multiple path transmission.
The Ejffect of NonUniform Wall Distributions of Absorbing Material on the
Acoustics of Rooms} Herman Feshbach and Cyril M. Harris. The
acoustics of rectangular rooms, whose walls have been covered by the non
uniform application of absorbing materials, is treated theoretically. Using
appropriate Green's functions a general integral equation for the pressure
distribution on the walls is derived. These equations show immediately
that it is necessary to know only the pressure distribution on the treated
surfaces to predict completely the acoustical properties of the room, such
as the resonant frequencies, the decay constants, and the spatial pressure
distribution. The integral equation is solved approximately using (1)
perturbation method, and (2) approximate reduction of the integral equation
to an equivalent transmission line. Criteria giving the range of validity of
these approximations are derived. It was found useful to introduce a new
concept, that of ^^efective admittance,''^ to express the results for the resonant
frequency and absorption for then the amount of computation is reduced
and the accuracy of the results is increased. The absorption of a patch of
material was found as a function of the position of the absorbing material
and was checked experimentally for a convenient case, an absorbing strip
mounted on the otherwise hard walls of a rectangular room. Particular
attention is given to the case where the acoustic material is applied in the
form of strips. The results may then be expressed in series which converge
very rapidly and are, therefore, amenable to numerical calculation. Ap
proximate formulas are obtained which permit estimates of the diffusion
of sound in a nonuniformly covered room. In agreement with experience,
these equations show that diffusion increases with frequency and with the
^ Proc. I.R.E. and Waves and Electrons, November 1946.
^Joiir. Aeons. Soc. America, October 1946.
390 BELL SYSTEM TECIIMCA L JOl RXA L
number of nodes on the treated walls. The "interaction effect" of one
strip on another is shown to decrease with an increase of the number of
nodes. The results are then applied to the case of ducts with nonuniform
distribution of absorbing material on its walls. Results are given which
permit the calculation of the attenuation per unit length of duct. The
methods of this paper hold for any distribution of absorbing material and
also if the admittance is a function of angle of incidence.
High Current Electron Guns J' L. M. Field. This j)aper presents a
survey of some of the problems and methods which arise in dealing with
the design of high current and high currentdensity electron guns. A
discussion of the general limitations on all electron gun designs is followed
by discussion of single and multiple potential guns using electrostatic fields
only. A further discussion of guns using combined electrostatic and mag
netic fields and their limitations, advantages, and some possible design
procedures follows.
Reflection of Sound Signals in the Troposphere^' G. W. Gilman, H. B.
CoxHEAD, and F. H. Willis. Experiments directed toward the detection
of nonhomogeneities in the first few hundred feet of the atmosphere were
carried out with a low power sonic "radar." The device has been named
the sodar. Trains of audiofrequency sound waves were launched vertically
upward from the ground, and echoes of sufficient magnitude to be displayed
on an oscilloscope were found. Strong displays tended to accompany
strong temperature inversions. During these periods, transmission on a
microwave radio path along which the sodar was located tended to be
disturbed by fading. In addition, relatively strong echoes were received
when the atmosphere was in a state of considerable turbulence. There was
a welldefined fineweather diurnal characteristic. The strength of the
echoes was such as to lead to the conclusion that a more complicated distribu
tion of boundaries than those measured by ordinary meteorological methods
is required in the physical picture of the lower troposphere.
A CathodeRay Tube for Vieiving Continuous Patterns? J. B. Johnsox.
A cathoderay tube is described in which the screen of persistent phosphor
is laid on a cylindrical portion of the glass. A stationary magnetic field
bends the electron beam on to the screen, while rotation of the tube produces
the time axis. When the beam is deflected and modulated, a continuous
pattern may be viewed on the screen.
6 Rev. Mod. Pliys., July 1946.
^ Jour. Acous. Soc. Amer., October 1946.
''Jour. Applied Physics, November 1946.
ABSTRA CTS OF TECHNICA L A RTICLES 391
The Molecular Beam Magnetic Resonance Method. The Radiofrequency
Spectra of Atoms and Molecules.^ J. B. M. Kellogg and S. Millman. A
new method known as the "Magnetic Resonance Method" which makes
possible accurate spectroscopy in the low frequency range ordinarily known
as the "radiofrequency" range was announced in 1938 by Rabi, Zacharias,
Millman, and Kusch (R6, R5). This method reverses the ordinary pro
cedures of spectroscopy and instead of analyzing the radiation emitted by
atoms or molecules analyzes the energy changes produced by the radiation
in the atomic system itself. Recognition of the energy changes is accom
plished by means of a molecular beam apparatus. The experiment was
first announced as a new method for the determination of nuclear magnetic
moments, but it was immediately apparent that its scope was not limited
to the measurement of these quantities only. It is the purpose of this
article to summarize the more important of those successes which the
method has to date achieved.
MetalLens Antennas.^ Winston E. Kock. A new type of antenna is
described which utilizes the optical properties of radio waves. It consists
of a number of conducting plates of proper shape and spacing and is, in
effect, a lens, the focusing action of which is due to the high phase velocity
of a wave passing between the plates. Its field of usefulness extends from
the very short waves up to wavelengths of perhaps five meters or more.
The paper discusses the properties of this antenna, methods of construction,
and applications.
Underwater Noise Due to Marine Life}^ Donald P. Loye. The wide
spread use of underwater acoustical devices during the recent war made
it necessary to obtain precise information concerning ambient noise condi
tions in the sea. Investigations of this subject soon led to the discovery
that fish and other marine life, hitherto generally classified with the voiceless
giraffe in noisemaking ability, have long been given credit for a virtue they
by no means always practice. Certain species, most notably the croaker
and the snappingshrimp, are capable of producing noise which, in air,
would compare favorably with that of a moderately busy boiler factory.
This paper describes some of the experiments which traced these noises to
their source and presents acoustical data on the character and magnitude
of the disturbances.
Elastic, Piezoelectric, and Dielectric Properties of Sodium Chlorate and
Sodium Promote}^ W. P. Mason. The elastic, piezoelectric, and di
8 Rev. Mod. PItys., July 1946.
^ Proc. I.R.E. and Waves and Electrons, November 1946.
^^ Jour. Aeons. Soc. America, October 1946.
iip/m. Rev., October 1 and 15, 1946.
392 BELL SYSTEM TECH NIC A L JOURNA L
electric constants of sodium chlorate (NaClOs) and sodium bromate
(NaBrOs) have been measured over a wide temperature range. The value
of the piezoelectric constant at room temperature is somewhat larger than
that found by Pockels. The value of the Poisson's ratio was found to be
positive and equal to 0.23 in contrast to Voigt's measured value of —0.51.
At high temperatures the dielectric and piezoelectric constants increase
and indicate the presence of a transformation point which occurs at a
temperature slightly larger than the melting point. A large dipole piezo
electric constant (ratio of lattice distortion to dipole polarization) results
for these crystals but the electromechanical coupling factor is small because
the dipole polarization is small compared to the electronic and ionic polariza
tion and little of the applied electrical energy goes into orienting the dipoles.
Paper Capacitors Containing Chlorinated Impregnants. Effects of Sulfur.^'
D. A. McLean, L. Egerton, and C. C. Houtz. Sulfur is an effective
stabilizer for paper capacitors containing chlorinated aromatics, in the
presence of both tin foil and aluminum foil electrodes. Sulfur has unique
beneficial effects on power factor which are especially marked when tin
foil electrodes are used. The value of R (Equation 4) can be used as an
index of ionic conductivity in the impregnating compound. Diagnostic
power factor measurements on impregnated paper are best made at low
voltages. Electron diffraction studies give results in line with the previously
published theory of stabilization. Several previous findings are reaffirmed:
(a) the importance of all components of the capacitor in determining its
initial properties and aging characteristics, (b) the superiority of kraft
paper over linen, and (c) widely different behavior of capacitors employing
different electrode metals.
A New Bridge PhotoCell Employing a PJwioConductive Effect in Silicon.
Some Properties of High Purity SiliconP G. K. Teal, J. R. Fisher, and
A. W. Treptow. a pure photoconductive effect was found in pyrolytically
deposited and vaporized silicon films. An apparatus is described for
making bridge type photocells by reaction of silicon tetrachloride and
hydrogen gases at ceramic or quartz surfaces at high temperatures. The
maximum photosensitivity occurs at 84008600A with considerable re
sponse in the visible region of the spectrum. The sensitivity of the cell
appears about equivalent to that of the selenium bridge and its stability
and speed of response are far better. For pyrolytic films on porcelain there
are three distinct regions in the conductivity as a function of temperature.
At low temperatures the electronic conductivity is given by the expression
'^ Indus. & Eugg. Cliemislry, Noveni1)er 1946.
^^ Jour. Applied Pliysics, Novcmljcr 1946.
ABSTRACTS OF TECHNICAL ARTICLES 393
<r = Af(T)exp(E/2kT). At temperatures between 227°C and a higher
temperature of 4(10 500°C a = Aexp—{E/2kT), where £ lies between 0.3
and 0.8 ev; and at high temperatures a = Aexp—(E/2kT), where E = 1.12
ev. The value 1.12 ev represents the separation of the conducting and
nonconducting bands in silicon. The long wave limit of the optical absorp
tion of silicon was found to lie at approximately 10,500 A (1.18 ev). The
data lead to the conclusion that the same electron bands are concerned in
the photoelectric, optical, and thermal processes and that the low values
of specific conductances found (1.8X10~* ohm~^ cm~^) are caused by the
high purity of the silicon rather than by its polycrystalline structure.
NonUniform Transmission Lines and Reflection Coefficients}^ L. R.
Walker and N. Wax. A firstorder differential equation for the voltage
reflection coefficient of a nonuniform line is obtained and it is shown how
this equation may be used to calculate the resonant wavelengths of tapered
lines.
^*Jour. Applied Physics, December 1946.
Contributors to this Issue
Harald T. Friis, E.E., Royal Technical College, Copenhagen, 1916;
Sc.D., 1938; Assistant to Professor P. D. Pedersen, 1916; Technical Advisor
at the Royal Gun Factory, Copenhagen, 191718; Fellow of the American
Scandinavian Foundation, 1919; Columbia University, 1919. Western
Electric Company, 192025; Bell Telephone Laboratories, 1925. Formerly
as Radio Research Engineer and since January 1946 as Director of Radio
Research, Dr. Friis has long been engaged in work concerned with funda
mental radio problems. He is a Fellow of the Institute of Radio Engineers.
Ray S. Hoyt, B.S. in Electrical Engineering, University of Wisconsin,
1905; Massachusetts Institute of Technology, 1906; M.S., Princeton, 1910.
American Telephone and Telegraph Company, Engineering Department,
190607. Western Electric Company, Engineering Department, 190711.
American Telephone and Telegraph Company, Engineering Department,
191119; Department of Development and Research, 1919 34. Bell
Telephone Laboratories, 1934. Mr. Hoyt has made contributions to the
theory of loaded and nonloaded transmission lines and associated apparatus,
theory of crosstalk and other interference, and probability theory with
particular regard to applications in telephone transmission engineering.
W. D. Lewis, A.B. in Communication Engineering, Harvard College,
1935; Rhodes Scholar, Wadham College, Oxford; B.A. in Mathematics,
Oxford, 1938; Ph.D. in Physics, Harvard, 1941. Bell Telephone Labora
tories, 1941. Dr. Lewis was engaged in radar antenna work in the Radio
Research Department during the war; he is now engaged in microwave
repeater systems research.
J. C. LoziER, A.B. in Physics, Columbia College, 1934; graduate physics
student, Princeton University, 193435. R.C.A. \'ictor Manufacturing
Company, 193536; Bell Telephone Laboratories, Inc., 1936. Mr. Lozier
has been engaged in transmission development work, chiefly on radio
telephone terminals. During the war he was concerned primarily with
the theory and design of servomechanisms.
394
VOLUME XXVI JULY, 1947 NO. 3
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
Telephony by Pulse Code Modulation W. M. Goodall 395
Some Results on Cylindrical Cavity Resonators
J. P. Kinzer and I. G. Wilson 410
Precision Measurement of Impedance Mismatches in
Waveguide Allen F. Pomeroy 446
Reflex Oscillators J. R. Pierce and W. G. Shepherd 460
Abstracts of Technical Articles by Bell System Authors. . 682
Contributors to This Issue 691
■*y
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
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Copyright, 1947
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PRINTED IN U. S. A.
The Bell System Technical Journal
Vol. XXVI July, 1947 No. 3
Telephony By Pulse Code Modulation*
By W. M. Goodall
An experiment in transmitting speech by Pulse Code Modulation, or PCM,
is described in this paper. Each sample amplitude of a pulse amplitude modula
tion or PAM signal is transmitted ])y a code group of OXOFF pulses. 2"
amplitude values can be represented by an n digit binary number code. For a
nominal 4 kc. speech band these n OXOFF pulses are transmitted 8000 times a
second. Experimental efuipment for coding the PAM pulses at the transmitter
and decoding the PCM pulses at the receiver is described. Experiments with
this equipment indicate that a threeunit code appears to be necessary for a
minimum grade of circuit, while a six or sevenunit code will provide good
quality.
Introduction
THIS paper describes an experiment in transmitting speech by PCM,
or pulse code modulation. The writer is indebted to his colleagues in
the Research Department, C. E. Shannon, J. R. Pierce and B. M. Oliver,
for several interesting suggestions in connection with the basic principles
of PCM given in this paper. Work on a dififerent PCM system was carried
on simultaneously in the Systems Development Department of the Bell
Laboratories by H. S. Black. This in turn led to the development of an
8channel portable system for a particular application. This system is being
described in a forthcoming paper by H. S. Black and J. O. Edson.^ A
method for pulse code modulation is proposed in a U. S. Patent issued to
A. H. Reeves.2
The material now presented is competed of three parts. The first deals
with basic principles, the second describes the experimental PCM system,
while the last discusses the results obtained.
Basic Principles
PCM involves the application of two basic concepts. These concepts
are namely, the timedivision principle and the amplitude quantization
* Paper presented in part at joint meeting of International Scientific Radio Union and
Inst. Radio Engineers on May 5, 1947 at Washington, D. C.
^ Paper presented on June 11, 1947 at A. I. E. E. Summer General Meeting, Mont
real, Canada. Accepted for publication in forthcoming issue of A. I. E. E. Trans
actions.
2 A. H. Reeves. V . .S. Patent Hl.lllfilQ, Feb. 3, 1942, assigned to International Stand
ard Electric Corp.; also, French patent * 852, 183, October 23, 1939.
395
396
BELL SYSTEM TECHNICAL JOURNAL
principle. The essence of the timedivision principle is that any input wave
can be represented by a series of regularly occurring instantaneous samples,
provided that the sampling rate is at least twice the highest frequency in the
input wave.^ For present purposes the amplitude quantization principle
states that a complex wave can be approximated by a wave having a finite
number of amplitude levels, each differing by one quantum, the size of the
quantum jumps being determined by the degree of approximation desired.
Although other arrangements are possible, in this paper we will consider
the application of these two basic principles in the following order. First
the input wave is sampled on a timedivision basis. Then each of the
samples so obtained is represented by a quantized amplitude or integer
number. Each of these integer numbers is represented as a binary number
of n digits, the binary number system being chosen because it can readily be
ENVELOPE OF
AUDIO SIGNAL
NO AUDIO SIGNAL
Fig. 1 — Pulses in a PAM System.
represented by ONOFF or twoposition pulses. 2" discrete levels can be
represented by a binary number of n digits.* Thus, PCM represents each
quantized amplitude of a timedivision sampling process by a group of
ONOFF pulses, where these pulses represent the quantized amplitude in a
binary number system.
The discussion so far has been in general terms. The principles just
discussed will now be illustrated by examples.
Multiplex transmission of speech channels by sending short pulses
selected sequentially from the respective speech channels, is now well known
in the telephone art and is called timedivision multiplex. When the pulses
consist simply of short samples of the speech waves, their varying amplitudes
directly represent the speech waves and the system is called pulse amplitude
modulation or PAM. In PAM the instantaneous amplitude of the speech
wave is sampled at regular intervals. The amplitude so obtained is trans
' This is because the DC, fundamental and harmonics of the wave at the left in Fig. 1
all become modulated in the wave at the right, and if the highest modulating frequency
exceeds half the sampling rate, the lower sideband of the fundamental will fall in the
range of the modulating frequency and will not be excluded l)y the lowpass filter. The
result is distortion.
■• In a decimal system the digits can have any one of 10 values, to 9 inclusive. In a
binary system, the digits can have only two values, either or 1.
TELEPHONY BY PULSE CODE MODULATION 397
mitted as a pulse of corresponding amplitude. In order to transmit both
positive and negative values a constant or dc value of pulse amplitude can
be added. (See Fig. 1.) When this is done positive values of the informa
tion wave correspond to pulse amplitudes greater than the constant value
while negative values correspond to pulse amplitudes less than the constant
value. At the receiver a reproduction of the original speech wave will be
obtained at the output of a lowpass filter.
The PCM system considered in this paper starts with a PAM system and
adds equipment at the terminals to enable the transmission of a group of
ONOFF pulses or binary digits to represent each instantaneous pulse
amplitude of the PAM system. Representation of the amplitude of a single
PAM pulse by a finite group of ONOFF pulses or binary digits requires
quantization of the audio wave. In other words, we cannot represent the
actual amplitude closer than ^ "quantum". The number of amplitude
levels required depends upon the grade of circuit desired. The disturbance
which results from the quantization process has been termed quantizing
noise. For this type of noise a signaltonoise ratio of 33 db would be ob
tained for 32 amplitude levels and this grade of circuit was deemed suffi
ciently good for a preliminary study. These 32 amplitude levels can be
obtained with 5 binary digits, since 32 = 2^.
Figure 2 shows how several values of PAM pulse amplitude can be
represented by this binary code. The first column gives the digit pulses
which are sent between the transmitter and receiver while the second column
shows the same pulse pattern with each pulse weighted according to its
assigned value, and the final column shows the sum of the weighted values.
The sum, of course, represents the PAM pulse to the nearest lower amplitude
unit. The top row where all the digits are present shows, in the middle
wave form, the weighted equivalent of each digit pulse. By taking different
combinations of the five digits all integer amphtudes between 31 and can
be represented. The examples shown are for 31, 18, 3, and 0.
Referring to Fig. 3 sampling of the audio wave (a) yields the PAM wave
(b). The PAM pulses are coded to produce the code groups or PCM
signal (c) . The PCM pulses are the ones sent over the transmission medium .
For a sampling rate of 8000 per second, there would be 8000 PAM pulses
per second for a single channel. The digit pulse rate would be 40,000 pps
for a fivedigit code. For a timedivision multiplex of N channels both of
these pulse rates would be multiplied by N.
Wave form (d) shows the decoded PAM pulses where the amplitudes are
shown under the pulses. The original audio wave is repeated as wave
form (e). It will be noted that the received signal is delayed by one PAM
pulse interval. It is also seen that the decoded pulses do not fit exactly on
this curve. This is the result of quantization and the output of the lowpass
398
BELL SYSTEM TECHNICAL JOVRXAL
filter will contain a quantizinjj; disturbance not shown in (e) which was not
present in the input signal.
A signal that uses regularly occurring ONOFF pulses can be "regener
ated" and repeated indefinitely without degradation. A pulse can be
"regenerated" by equipment which transmits an undistorted pulse provided
a somewhat distorted pulse is received, and transmits nothing otherwise.
BINARY NUMBER
I I I i I I I I I i
DECODED NUMBER
WEIGHTED EQUIVALENT
16 i 8 ; 4 ; 2 1 1
I i I ; 1 I I
J — L
16 ; I 1 2 ;
n =^
Fis;. 2 — Binar\' and decimal equivalents.
Thus, the received signal at the output of the final decoder is of the same
quality as one produced by a local monitoring decoder. To accomplish
this result, it is necessary, of course, to regenerate the digit pulses before
they have been too badly mutilated by noise or distortion in tlie transmission
medium.
The regenerative ])roperty of a quantized signal can be of great importance
in a long repeated system. I'"or example, with a con\cntional system each
repeater link of a lOOlink system must huNc a signaltonoise ratio 20 db
1>
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399
400 BELL SYSTEM TECHNICAL JOURNAL
better than the complete system. For PCM, however, with regenerative
repeaters the required signaltonoise ratio in the radio part of the system
is independent of the number of links. Hence, we have a method of trans
mission that is ideally suited to long repeated systems.
At this point we might consider the bandwidth required to send this type
of signal. For a 5digit code the required band is somewhat less than 5
times that required for a PAM system. It is somewhat less than 5 times be
cause in a multiplex system crosstalk becomes a serious problem. In a PAM
system this crosstalk would add up on a long system in somewhat the same
manner as noise. In order to reduce the crosstalk it would probably be
necessary to use a wider band for the PAM repeater system than would be
required for a singlelink system. For PCM, on the other hand, by using
regeneration the whole system requirement for crosstalk can be used for
each link. In addition, a relatively greater amount of crosstalk can be
tolerated since only the presence or absence of a pulse needs to be determined.
Both of these factors favor PCM. This is a big subject and for the present
we need only conclude that from considerations of the type just given the
bandwidth penalty of PCM is not nearly as great as might first be expected. .
The same two factors that were mentioned in connection with crosstalk
also apply to noise, and a PCM signal can be transmitted over a circuit
which has a much lower signaltonoise ratio than would be required to
transmit a PAM signal, for example.
Hence, we conclude that PCM for a long repeated system has some
powerful arguments on its side because of its superior performance even
though it may require somewhat greater bandwidth. There are other fac
tors where PCM differs from more conventional systems but a discussion of
these factors is beyond the scope of this paper.
The previous discussion may be summarized as follows: One begins with
a pulse amplitude modulation system in which the pulse amplitude is
modulated above and below a mean or dc value as indicated in Fig. 1.
It is assumed that it will be satisfactory to limit the amplitude range to be
transmitted to a definite number of amplitude levels. This enables each
PAM pulse to be represented by a code group of ONOFF pulses, where the
number of ampUtude levels is given by 2^, n being the number of elements
in each code group. With this system the digit pulses can be "regenerated"
and the quality of the overall transmission system can be made to depend
upon the terminal equipment alone.
Experimental PCM Equipment
The experimental coder used in these studies might be designated as one
of the "feedback subtraction type". It functions as follows: Each PAM
pulse is stored as a charge on a condenser in a storage circuit. (See Fig. 4.)
TELEPHONY BY PULSE CODE MODULATION
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The voltage across this condenser is compared with a reference voltage. The
magnitude of this reference voltage corresponds to the d^c i)ulse amplitude
of Fig. 1. The voltage has a magnitude of 16 units. If the magnitude of
the condenser voltage exceeds the magnitude of the 16unit voltage, a
positive pedestal voltage is obtained in the output of the comparing circuit.
This pedestal voltage is amplified, limited and applied to the pedestal
modulator. The pedestal modulator serves as a gate for timing pulses from
the timing pip generator. If the pedestal voltage and timing pulse are
applied simultaneously to the pedestal modulator, a pulse is obtained in the
output. In the jjresent case this pulse corresponds to the presence of the
16unit digit in the code group which represents this PAM pulse. This digit
pulse after amplification and limiting is (1) sent out over the line (PCM out)
and (2) fed back through a suitable delay circuit to a subtraction circuit.
The function of the subtraction circuit is to subtract a charge from the con
denser corresponding to the 16unit digit. The charge remaining on the
condenser is now compared with a new reference voltage which is h the
magnitude of the first reference voltage or 8 units. If the magnitude of the
voltage across the condenser exceeds this new reference voltage the above
process is rei)eated and the second digit pulse is transmitted and another
charge, this time corresponding to the 8unit digit, is subtracted from the
remaining charge upon the condenser.
If the magnitude of the voltage across the condenser is less than the
reference voltage, in either case above, then no pedestal will be produced and
no digit pulse be transmitted. Since no pulse is transmitted, no charge
will be subtracted from the condenser. Thus the charge remaining
upon the condenser after each operation represents the part of the orig
inal PAM pulse remaining to be coded. The reference voltage wave
consists of a series of voltages each of which is ^ of the preceeding one.
There is one step on the reference voltage function for each digit to be
coded.
A better understanding of the coding process can be had by reference to
the various wave forms involved. For completeness, wave forms from
audio input to the coded pulse signal are shown for the transmitter in Figs.
^ and 5 and from the coded pulse signal to audio output for the receiver in
Figs. 7 and 3. In the diagram the abscissas are time and the ordinates are
amplitudes. Some of these wave forms have already been discussed in
connection with Fig. .^. Since the coder functions in the same manner for
each PAM pulse the detailed wave forms of the coding and decoding proc
esses are shown for only two amplitudes. The block schematic for the
transmitter is given on Fig. 4, while that for the receiver is given in Fig. 6.
The letters on Figs. 4 and 6 refer to the wave forms on Fig. 3, while the
numbers refer to the wave forms in Figs. 5 and 7.
7. CODE ELEMENT TIMING PIPS
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11
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8. CODE GROUPS
Fig. 5 — Detailed wave forms for PCM Transmitter (amplitude vs. time).
403
404
BELL SYSTEM TECHNICAL JOURNAL
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1. DELAYED CONTROL PULSE
2. STEP TIMING PIPS
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3. REFERENCE STEP VOLTAGE
ILJ
11
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8. CODE GROUPS
11
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9. OUTPUT OF RECEIVING SUBTRACTION CIRCUIT
10. RECEIVING STORAGE CONDENSER VOLTAGE
'J\
n
II. UNDELAYED CONTROL PULSE
n
Fig. 7 — Detailed wave forms for PCM Receiver (amplitude vs. time).
405
406 BELL SYSTEM TECHNICAL JOURNAL
Referring to Figs. 4 and 5, tlie "delayed control pulse" Curve 1 is the
principal timing pulse for the transmitting coder. It is used to sample the
audio wave and to start the step and timingpip generators. Two sets of
timingpips are produced; one, ("urvc 2, is used to generate the reference
step voltage while the other, Curve 7, is used for timing the digit pulses.
The reference step voltage, Curve 3, is used in the comparing circuit and in
the subtraction circuit. Curve 4 gives the output of the subtraction circuit,
while Curve 5 is the voltage on the storage condenser. The next plot gives
Curves 3 and 5 superimposed; the shaded area on this plot corresponds to
the time during which a pedestal voltage is generated. The pedestal voltage
is given by Curve 6, and the output of the pedestal modulator is given by
Curve 8. This last curve is a plot of the two code groups corresponding to
the two PAM pulses being coded .
In studying these wave forms it will be noted that the delayed control
pulse, the two sets of timingpips and the reference step voltage curves are
the same for each code group. On the other hand the storage condenser
voltage, the pedestal voltage, the group of code pulses, and the group of
pulses from the subtraction circuit are different for each code group.
It will be recalled that a pedestal voltage is produced during the time that
the condenser voltage exceeds the reference step voltage. The leading edge
of each pedestal pulse is generated by the falling part of the reference step
voltage. The trailing edge of each pedestal pulse is produced by the falling
part of the condenser voltage. This drop in condenser voltage is the result
of the operation of the subtraction circuit. The output of the subtraction
circuit depends upon the delayed digit pulse which has just been passed by
the pedestal pulse. Its magnitude depends upon the reference voltage step
that applies to the particular digit being transmitted. The function of the
delay in the feedback path is to allow the outgoing digit pulse to be com
pleted before the pedestal is terminated.
It is seen that the pedestal voltage contains the same information as the
transmitted code groups. Under ideal conditions the use of auxiliary
timing pulses would not be required. However, in a practical circuit the
leading edge of the pedestal varies, both as to relative timing and as to rate
of rise. Under these conditions the auxiliary timingpips permit accurate
timing of the outgoing PCM pulses, as well as constant pulse shape for the
input to the subtraction circuit.
Summarizing the foregoing it is seen that in the coder under discussion
a comparison is made for each digit between a reference voltage and the
voltage across a storage condenser. Initially the voltage across this con
denser represents the magnitude of the PAM pulse being coded. After
each digit I he voltage remaining on the condenser represents the magnitude
of the f)riginal PAM j)ulse remaining to l)e coded. A pedestal voltage is
TELEPHONY BY PULSE CODE MODULATION 407
obtained in the output of the comparing circuit whenever the storage con
denser voltage exceeds the reference step voltage.
This pedestal, if present, allows a timing pulse to be sent out as a digit of
the code group. This digit pulse is also delayed and fed back to a sub
traction circuit which reduces the charge on the condenser by a magnitude
corresponding to the digit pulse just transmitted. This process is repeated
step by step until the code is completed.
Synchronizing the two control pulse generators, one at the transmitter
and one at the receiver, is essential to the proper operation of the equipm.ent.
This may be accomplished in a variety of ways. The best method of syn
chronizing to use would depend upon the application. Although the control
could easily be obtained by transmitting a synchronizing pulse over the
line, the equipment would have been somewhat more complicated and for
these tests a separate channel was used to synchronize the control pulse
generators at the terminals.
Having thus established the timing of the receiving control pulse generator
shown in Fig. 6 relative to the received code groups, the receiver generates
a new set of waves as shown in Fig. 7. Except for delay in the transmission
medium, the first three curves are the same as those shown in Fig. 5 for the
transmitter. (1) is the delayed control pulse, (2) is the step timing wave,
and (3) is the reference step voltage. Curve 8 is the received code group
and (9) is the output current of the subtraction circuit. (10) gives the wave
form of the voltage across the receiving storage circuit, and (11) gives the
curve for the undelayed control pulse.
The receiver functions as follows: The storage condenser is charged to a
fixed voltage by each delayed control pulse. The charge on the condenser
is reduced by the output of the subtraction circuit. The amount of charge
that is subtracted depends upon which digit of the group produces the sub
traction pulse. This amount is measured by the reference step voltage.
At the end of the code group the voltage remaining on the condenser is
sampled by the undelayed control pulse.
It is seen that the storage subtraction circuits in the transmitter and
receiver function in similar ways. In the transmitter the original voltage
on the condenser depends upon the audio signal, and after the coding process
this voltage is substantially zero. The receiver starts with a fixed maximum
voltage and after the decoding process the sample that is delivered to the
output lowpass filter is given by the voltage reduction of the condenser
during the decoding process. Except that the conditions at beginning and
end of the coding and decoding periods are dififerent as discussed above,
the subtraction process is the same for both units.
The monitoring decoder in the transmitter operates in the same manner
described above, except that it employs the various waves already generated
for other uses in the transmitter (see Fig. 4).
408
BELL SYSTEM TECHNICAL JOURNAL
Experimental Results
An experimental system was set up as shown in Fig. 8. The pulse code
modulator, radio transmitter, and antenna comprised the transmitting
terminal; while an antenna, radio receiver and pulse code demodulator were
used for the receiving terminal. A short airpath separated the terminals.
The transmitter used a pulsed magnetron oscillator and the receiver em
ployed a broadband superheterodyne circuit. The results obtained with
this system were similar to those obtained by connecting the pulse code
RADIO
TRANSMITTER
RADIO
RECEIVER
PULSE CODE
MODULATOR
PULSE CODE
DEMODULATOR
AUDIO
INPUT
A
O O—
AUDIO
OUTPUT
Fig. 8— Block diagram of PCM system.
modulator and demodulator together without the radio equipment. In
fact, unless a large amount of attenuation was inserted in the path the
presence of the radio circuit could not be detected.
It was possible to adjust the PCM transmitter so that different numbers
of digits could be produced. A brief study was made of the number of
digits required. It was found that, with regulated volume, a minimum
of three or four digits was necessary for good intelligibility for speech though,
surprisingly enough, a degree of intelligibility was obtained with a single
one. With six digits both speech and music were of good quality when
regulated volume was used. Even with six digits, however, it was possible
to detect the difference between PCM and direct transmission in AB tests.
This could be done most easily by a comparison of the noise in the two
systems. If unregulated volume were used several more digits would proba
bly be desirable for high quality transmission.
In listening to the speech transmitted over the PCM system one obtained
the impression that the particular sound patterns of a syllable or a word
TELEPHONY BY PULSE CODE MODULATION 409
could be transmitted with three or four digits. If the volume range of the
talker varied it would be necessary to add more digits to allow for this
variation. Over and above these effects, however, the background noise
which is present to a greater or lesser extent in all communication circuits,
is quantized by the PCM system. If the size of the quanta or amplitude
step is too large the circuit will have a characteristic sound, which can easily
be identified. Since the size of the quanta is determined by the number of
digits, it is seen that the number of digits required depends not alone upon
the speech but also upon the background noise present in the input signal.
Summarizing, experimental results obtained indicate that at least 3
digits are desirable for a minimum grade of circuit and that as many as
6 or more will provide for a good quality circuit. If we wish to transmit a
nominal speech band of 4000 cycles, PCM requires the 8000 pulses per
second needed by any timedivision system, multiplied by the number of
digits transmitted. The extra bandwidth required for PCM however,
buys some real advantages including freedom from noise, crosstalk and
signal mutilation, and ability to extend the circuit through the use of the
regenerative principle.
The writer wishes to acknowledge the assistance of Mr. A. F. Dietrich
in the construction and testing of the PCM equipment discussed in this
paper.
Some Results on Cylindrical Cavity Resonators
By J. P. KINZER and I. G. WILSON
Certain hitherto unpublished theoretical results on cylindrical cavity reson
ators are derived. These are: an approximation formula for the total number
of resonances in a circular cylinder; conditions to yield the minimum volume cir
cular cylinder for an assigned (^; limitation of the frequency range of a tunable
circular cylinder as set by ambiguity; resonant frequencies of'the elliptic cylinder;
resonant frequencies and ^ of a coaxial resonator in its higher modes; and a brief
discussion of fins in a circular cylinder.
The essential results are condensed in a number of new tables and graphs.
Introduction
THE subject of wave guides and the closely allied cavity resonators was
of considerable interest even prior to 1942, as shown in the bibliography.
It is believed that this bibliography includes virtually everything published
up to the end of 1942. During the war, many applications of cavity reso
nators were made. Among these was the use of a tunable circular cylinder
cavity in the TE 01« mode as a radar test set; this has been treated in pre^
vious papers. ^'^ During this development, a num.ber of new theoretical
results were obtained; some of these have been published.^ Here we give
the derivation of these results together with a number of others not previ
ously disclosed.
In the interests of brevity, an effort has been made to eliminate all
material already published. For this reason, the topics are rather discon
nected, and it is also assumed that the reader has an adequate background
in the subject, such as may be obtained from a study of references 3 to 7
of the bibliography, or a text such as Sarbacher and Edson.**
A convenient reference and starting point is afforded by Fig. 1, taken from
the Wilson, Schramm, Kinzer paper. This figure also explains most
of the notation used herein.
Acknowledgement
In this work, as in any cooperative scientific development, assistance and
advice were received from many individuals and appropriate appreciation
therefor is herewith extended. In some cases, explicit credit for special
contributions has been given.
Contents
1. Approximation formula for number of resonances in a circular cylin
drical cavity resonator.
2. Conditions for minimum volume for an assigned ().
410
NORMAL WAVELENGTHS
"WWWf
SAME AS TM MODES
C Vui = VELOCITY OF
ELECTROMAGNETIC WAVES
IN DIELECTRIC
f = FREQUENCY
SAME FORM AS FOR
CYLINDER
Tom HAS DIFFERENT
VALUES
Ic shape factors for recta
? 411
)\n mode
xial reso
N A
y are ob
(1)
The dis
t, can be
quency /o
e the true
es being a
iped curve
1 approxi
mces of a
Bolt^ and
? to apply
e from the
ionant fre
represent
to find the
R = '^
C
ators
of re
culai
circu
resor
discu
Th
THI
of
It is beli
up to th
nators w
cavity ir
vious pa
results v^
the dem
ously dis
In the
material
nected, a
in the su
of the bi
A conv
the Wils
of the nc
In this
advice w(
therefor i
contributi
1. Appi
drica
2. Cone
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 411
3. Limitation of frequency range of a tunable cavity in the TE Oln mode
as set by ambiguity.
4. Resonant frequencies of an elliptic cylinder.
5. Resonant frequencies and Q of higher order modes of a coaxial reso
nator.
6. Fins in a circular cylinder.
Approximation Formula for Number of Resonances in a
Circular Cylinder
From Fig. 1, the resonant frequencies of the cylindrical cavity are ob
tained from the equation:
In which r is written in place of f/m , to simplify the equations. The dis
tribution of the resonant frequencies, starting with the lowest, can be
approximated by a continuous function
where N represents the total nunter of resonances up to a frequency /o
or a wavelength Xo . This is bcur.d lo be en approxirraticn, since the true
function F is discontinuous (or stepped) by virtue of the resonances being a
series of discrete values. For practical purposes, if /*' fits the stepped curve
so that the steps fluctuate above and below F, it will be a useful approxi
mation.
Derivation of such a formula as applied to the acoustic resonances of a
rectangular box has recently been a subject of investigation by Bolt^ and
Maa.'" Only slight modifications of their method need be made to apply
to the {^resent situation.
MuUiply (1) thru by (
TTflA" 2 , /wan
.7) •■ +[2L
Hence, if a point ( r, — — J is plotted on the A'l' plane the distance from the
origin to this point will be —  and hence a measure of the resonant fre
c
quency. If all such points are plotted, they will form a lattice represent
ing all the possible modes of resonance. The problem, then, is to find the
number of lattice ]X)ints in a quadrant of a circle with radius, R = — — .
412 BELL SYSTEM TECHNICAL JOURNAL
The values of the Bessel zero, r, are not evenly spaced along the X axis;
indeed the density, or number per unit distance, increases as r increases.
Let the density be p{x). Then the problem becomes one of finding the
weight of a quadrant of material whose density varies as p{x).
Suppose the expression for M, the number of zeros r, less than some value
X, is of the form
M = Ax"^] Bx
whence, by dififerentiation,
p{x) = 2Ax^B.
The weight, IF, of the quadrant of a circle of radius R is then, by integra
tion,
W =\aR^ + ^ BR^.
3 4
2L . . 2LW
Since there are — lattice points per unit distance along the Y axis,
ira iro.
is apparently the total number of points in the quadrant. However, there
are two small corrections to consider. First is that in this procedure a
lattice point is represented by an area and for the points along the X axis
Tra . . . .
half the area, i.e., a strip — wide lying in the adjacent quadrant, has been
omitted. Second is that the restriction w > for TE modes eliminates
half the points along the X axis. As it happens, these corrections just
cancel each other. Thus we have
^  3 xr 2 X^
in which
7 = ^^ S = ^aL Xo = ^
4 /o
From a tabulations^ of the first 180 values of r, the empirical values A =
0.262, B = Q were obtained. This gives
V
N = 4.39 z .
Ao
Subsequently, from an analysis of over a thousand modes in a "square
cylinder" (a = L), Dr. Alfredo Baiios, formerly of M.I.T. Radiation Lab
oratory, has calculated the empirical formula
N = 4.38 3 + 0.089 ;2 (2)
Aq Aq
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 413
from which A — 0.262,, B = 0.057. These values give better agreement
with the 180 tabulated values of r.
There is a twofold degeneracy in a circular cylinder for modes with
■^ > 0, which is removed, for example, when the cylinder is made elliptical.
The total number of modes, then, counting degeneracies twice, is about 2N,
which brings (2) in line with the general result that, in any cavity resonator,
Stt V
the total number of modes is of the order — r^ .
3 Ao
Minimum Volume of Circular Cylinder for Assigned Q
In practical applications of resonant cavities, the conditions of operation
may require high values of Q which can be attained only by the use of high
order modes. The total number of modes, most of which are undesired,
can then be reduced only by making the cavity volume as small as possible,
consistent with meeting the requirement on Q.
It will be shown that, for a cylinder, operation in the TE 01m mode very
probably gives the smallest volume for an assigned Q.
Statement of Problem
When the relative proportions (the shape) of a cavity and the mode of
oscillation are fixed, both the Q and the volume, V, of the cavity are func
tions of the operating wavelength, X. Since we are primarily interested
in the relationship between Q and V, with X fixed, some simplification can
be made by eliminating X as a parameter. This may be done by a change of
8 V
variables to ()  and — , respectively; to simplify the typography, these
A A
quantities will be denoted by single symbols:
We are, consequently, interested in the following specific problem:
In a circular cylindrical resonator, which is the optimum mode
family and what is the corresponding shape to obtain the smallest
value of W for a preassigned value of P?
A rigorous solution cannot be obtained by the methods of elementary
calculus, since P is not a continuous function of the mode of oscillation.
However, a possible procedure is to assume continuity, and examine the
relation between P and W under this assumption. If sufficiently positive
results are obtained, the conclusions may then be carried over to the dis
continuous (i.e., the physical) case with reasonable assurance that, except
414 BELL SYSTEM TECHXICAL JOURNAL
perhaps for special \'alues, the correct answer is obtained. W'e proceed on
this basis.
Solntion
To permit a more coherent presentation of the arguments, only their
general outline follows. More mathematical details are given later.
We start with the formulas for (^  (= i^) as given in Fig. 1.
A
The lirst operation is to show that, under comparable conditions, i.e.,
X, r, n tixed, the TE Oniii modes give the highest values of P. That this is
j)lausib!e can be seen in a general manner from the equations as they stand.
For the TE modes, if ( — 0, the numerator of the fraction is largest. Also,
P simplities, and the denominator roughly reduces the e.xpression in square
brackets to the 1 2 power. Now compare this expression with those for
the TM modes. That for the TM modes (// > 0) is smaller because of the
factor (1 + R) in the denominator. Finally, that for the TAf modes (;/ =
0) is still smaller, because 1 < (1 + pRY'.
This leaves only the TE Omii modes to be considered, and the next step
is to show that ;;/ = 1 is the most favorable value. Since the relation be
tween P and ir is com{)licated, a j)arameter cp is introduced, with (p dehned
by
tan (^ = pR. (3)
The resulting parametric equations are:
r 1
P = ^ ^^— (4)
^TT .•? ,1.3
COS v? +  sm (f
p
pr^ 1
47r cos ip sm ip
For each of the discrete values of r and n (;/ is related to p) then, plots
of P vs W can be prepared as shown in Fig. 2 for the TE 01 » modes.
Inspection of Fig. 2 shows that the best value of Q does not correspond
to a minimum of W or a maximum of P for a given value of ;/, but rather to
a point on the "envelope" of the curves. To get the envelope, we assume
p to be continuous and proceed in the standard manner. It turns out that,
by solving (4) f(^r p in terms of 7^ / and v?, substituting the resulting e.x
(9 IF
pression in TF, and setting  = an equation is obtained which, when
^^p
Sf)lvcd for <p, gi\'es the \'alucs of ^p which lie on the en\clo]u\
SOME RESULTS OX CYLINDRICAL CAVITY RESONATORS
415
/
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1
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(
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416 BELL SYSTEM TECHNICAL JOURNAL
We next substitute this expression for <p in W and calculate — assuming
dr
now that r is continuous, and find that W has no minimum. Practically,
this means that the smallest value of r should be used, i.e., the TEOln mode.
Finally, since from Fig. 2 it is seen that the envelope is reasonably smooth
8
for values of ^  > 1, the expression for <p derived on the assumption of
continuous p is used to obtain a simple relation of great utility in practical
cavity design.
Details of solution
In (3), since R must be finite for a physical cylinder, < tan (p < oo ,
< sin v? < 1, and < cos v? < 1. Hence we may always divide by
sin (p or cos <p. Note that (p ranges between 0° and 90°,
From Fig. 1,
2d2\1/2
whence
^ ^ 2r(l + p'R')
, . 2prR
k sin (p = — —
a
(6)
^ cos ^ = — . (7)
We define W by:
a
3 ,3
X3 4R 87r3 ^^^
Substituting (6) and (7) in (8),
pr^ 1
W = ^2 —2 r— . (5')
47r cos cp sin <p ^
Substitution of (3) into the expression for Q (= P) for the TE modes as
A
given in Fig. 1 yields, after some manipulation
2x 3
COS
(p \  sin^ ^ + ( COS ^ —  sin ^ ) (^/r)^sin^ <p
P \ P /
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 417
To show that any value of ^ > reduces P below its value when ^ = 0,
let
a = cos^ (p {•  sin^ <p
P
b = { cos (f —  sin ^ 1 sin^ ip
c = {l/r)\
It suffices to show that
a a \ he
where the question is in doubt because h may take on negative values. If
the inequality is to be valid, it is necessary only that (i + a) > 0, that is,
cos «^ > 0. Hence, for the TE modes, only I — ^ needs be considered. For
this case, the expression for P simplifies to
r 1
P =
For the TM modes, there is similarly obtained
27r 3 , 1 . 3 ' (4')
cos ^ +  sm (^
P =
P =
r
1
2K , 1 . w > (9)
cos V? +  sm ip
r cos (p
2K , 1 . « = 0. 10)
cos v? + ;r sm <p
Ip
It is easy to show, since cos ^ < 1 and sin ^ < 1, that both (9) and (10)
are less than (4').
Hence we have shown that, under comparable conditions, i.e., r and p
constant, the TE Omn modes have higher values of P than any others.
There is one flaw in the argument, viz., r takes on discrete values and cannot
be made the same for all modes. It is conceivable, therefore, that for some
specific values of P, a mode other than the TE Omn can be found which
gives a smaller W than either of the two "adjacent" TE Omn modes, one
having a value of r higher, the other lower, than the supposed highP
mode. This situation requires further refinement, and hence complication,
in the analysis; we pass over this point.
Having so far indicated that the TE Omn modes are the best, our next
objective is find the best value of m, if possible.
418 BELL SYSTEM TECHNICAL JOURNAL
By use of the parametric equatiuns (4) and (5), Fig. 2 has been ])lotted
for r = ^.S^ (TE 01» modes) and values of n from 1 to 9. This drawing
shows that, for each discrete value of r, minimum IT P is given by points
on the "envelope" of the family of curves.
The standard method of obtaining the envelope is to express If as a
function of /' with )i as parameter (r is assumed fixed, for the moment),
■J 7,'
i.e., ir = F(P, //),an(l then set — = 0. However, in this case it is easier
dn
to express IT = G(P, <p) and (p = H{ti), whence
dF ^dG d^
dn dtp dn
fir" fi /»
and the envelope is obtained by setting r = provided t 5^ 0. We
d<p on
proceed, therefore, as follows.
Assume p is continuous, and solve (4) for p, obtaining;
sin^ tfi
2^  cos ^
Now substitute (11) in (5). This gives TT' as a function of P and (p'.
3
47r
sm (p
cos ^ \ j~p  cos^ ip
(12)
rJll
To solve — = 0, we dilTerentiate and simplifv. This yields
dip
5 cos (^ — 3 cos"* tp = —  . (13)
irP
Substituting (13) back into (11) yields
2 sin <p
P = ^
3 cos^ ip
(14)
The situation so far is that, with P and r assigned, W lies on the en
velope and is a minimum when v? satisfies (13); p is then given by (14).
Obviously, for (13) to hold, it is necessary that
2^<>
'•'() obtain the best value of ;, the ])rocedure is to differentiate ir„n„ with
respect to r, assuming now that r is continuous, and examine for a mini
SOME RESULTS 0\ CYLINDRICAL CAVITY RESONATORS 419
mum. W'c can, however, first differentiate (12) by setting
dW _ dW dW dip
dr dr d(p dr
dW
and then substitute from (13). However, when (13) is satisfied, — = 0.
o<p
This process yields
dW ^ r (2  3 cos^ <p)
dr IT 9 sin (p cos^ (p
This shows r to be positive, when cosV < I • Hence — = corresponds
dr dr
to a maximum, rather than a minimum.* If cos(p < f, that is,^ > 35°16',
then r should be as small as possible. The smallest r is 3.83, for the TE
01;/ modes. For r = 3.83, and (p > 35°, from (13) there is obtained P >
0.75.
s
The analysis thus indicates that, for values of P = () greater than 0.75,
A
the TE 01;/ mode yields the smallest ratio W/P or V/Q.
An interesting and simple relation between /a and R for minimum W/P
can easily be derived from the foregoing equations. Substitute (14) back
into (6), thereby obtaining
■ *^^ (15)
3 a cos^ p
Now use (7) with (15) to eliminate cos p, replace k by 27r/X, and r by 3.83,
its numerical value for the TE 01;; modes. This gives
^] R = 2.23
or by substituting X =  , c = 3 X 10 ,
(fa) R  20.1 X 100.
This useful relation was first discovered by W. A. Edson.
Some further discussion is of interest. It is realized that a number of
points have not been taken care of in a manner entirely satisfactory mathe
matically, but nevertheless important practical results have been obtained.
As an example, since p and r can assume only discrete values, there are
* It is for this reason that the determination of the stationary values of ]V{r, [>, f),
subject to the constraint P(r, p, ^) = constant, by La Grange multipliers fails to yield
the desired least value of W/P.
420 BELL SYSTEM TECHNICAL JOURNAL
specific situations where some mode other than the TE Oln gives a smaller
W/P. For example, it may be shown that for P between 0.97 and 1.14
the TE 021 mode yields a smaller W than the TE 013 or TE 014 modes.
However, the margin is small, and for larger P, the TE 02n modes become
progressively poorer.
Limitation on Frequency Range of Tunable Cavity as Set
BY Ambiguity
In the design of a tunable cylindrical resonant cavity intended for use
in the TE 0\n mode, the requirements on Q may dictate a diameter large
enough to sustain TE 02n' or TE 03n' modes. Also, the range of variation
of cavity length may be such that the TE 01 (w + 1) mode is supported. As
the cavity is required to tune over a certain range of frequency, the maximum
frequency range possible in the TE 01« mode without interference from the
TE 01 (w + l)t or any TE 02 or TE 03 modes is of interest. The interference
from the TE 0\(n\ 1) limits the useful range of the TE 01« by the presence
of extraneous responses at more than one dial setting for a given frequency
or more than one frequency for a given dial setting. In applications so far
made, it has been possible to eliminate extraneous responses from the TE 02
and TE 03 modes, but crossings of these modes with the main TE Oln mode
have not been permitted. No designs have had diameters sufficiently large
to support TE 04 modes.
The desired relations are easily obtained by simple algebraic manipula
tion of equation (1). For simplicity in presentation of the results, we in
troduce some symbols applicable to this section only:
A = r^T B = r^T = 2.247 X 10=^"
Ao = value of A for TE 01« modes = 13.371 X 10
/ = A/Ao
:Vo = (a/Ly at low frequency end of useful range of TE 01m mode
maximum/
frequency range ratio =
minimum /"
The values of A and / depend upon the interfering mode under considera
tion. For the TE Oln modes, A = 44.822 X lO'", / = 3.3522.
The two typical cases of interest are shown on Fig. 3. For case I, am
t It is easy to show that the extra,neous respo^nse from the TE 01 (m — 1) mode is not
limiting. The proof depends on the inequality n* > (« f 1) (w — 1).
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS
421
/
/
/
//
^
f\i
/ /
(0
/
• ' ' '.*. ■ • ' \>'^ ' • . .*."•.'■ •^^. ■.*•■*•■
C*
/
•■'■'v^;^>i^\'\''''''iy^'':':?r//.'
II
3)
m = 4/
/
W'M^:0B0iM
n
3/
W00MiiM&
2

'mUKti.
—
TE 02
n=4/
/
3 /
^
/
u
I
i^'^^'^
TE 01
£
^
 —
<_
— ""^
Xo
Xo
Fig. 3 — Mode chart illustrating types of interference with TE 01« mode,
biguity from TE 01 (w + 1) mode, it is found that
Curves of F for this case are shown on Fig. 4.
The maximum value of F is obtained when Xo = oo and is
i^ max —
n + 1
422
BELL SYSTEM TECHNICAL JOURNAL
..''
/
/
TEOII^
/
/
/
/
/
•
012^
4
/
/ ^
^
■ 01^3 —
^
/,
^
— ^ ■
/f
f /
/
/
/ /
/
/
A
/
^■
y
0.2 0.4 0.6 0.8 I.O t.2 1.4 1.6 1.8 2.0 2.2 2.4
^ (minimum)
Fig. 1 — Curves showing maximum value of frequency ratio without interference from
TE 01 (?z + 1) mode (case I of Fig. 3).
Table I. — Cast II: Maximum Frequency Range Ratio, t\ for TE Uln Mode wlien Limited
by Mode Crossings with TE 02m and TE 02{m+I) Modes.
n = 3
M =4
n =
12
F
(''/■f)min
F
("/^'min
F
(«/i'min
1
1.198
1.323
1.086
0.966
1.008
0.313
2
1.242
1.080
1.013
0.316
3
1.019
0.322
4
1.027
0.331
5
1.037
0.343
6
1.051
0.360
7
1.071
0.384
8
1.104
0.418
9
1.168
0.471
10
1.345
0.564
SOME RESULTS OX CYLIXDRICAL CAVITY RESONATORS 423
For case II, range limited by mode crossings, it is found that
A  .4o
•To =
F' =
Bin'  w'2)
or  ■»/)[»/  {n' + 1)']
Some values for this case are given in Table I.
The formulas above are general and may be used for any pair of mode
types by using the appropriate values for A and /.
The Elliptic Cylinder
In the design of high Q circular cylinder cavity resonators operating in
the TE 01;/ mode, it is desirable to know how much ellipticity is tolerable,
so that suitable manufacturing limits may be set. The elliptical wave
guide has already been studied, notably by Brillouin^ and Chu,^^ but the
results are not in suitable form or of adequate precision for the present
purposes. More recently tables" have become available which permit the
calculation of some of the properties of the elliptical cylindrical resonator.
The elliptical cavity involves Mathieu functions, which are considerably
more complicated than l^essel functions. ^^ The tables give the numerical
coefficients of series expansions, in terms of sines, cosines, and Bessel func
tions, of the Mathieu functions up to the fourth order. These tables have
been used for the calculation of some quantities of interest in connection
with elliptical deformations of a circular cylinder in the TE 01« mode.
The Ellipse
All mathematical treatments of the ellipse (including the tables men
tioned above) use the eccentricity, e, as the quantity describing the amount
of departure from the circular form. The eccentricity is the ratio
distance between foci
e = . . .
major axis
This is not a quantity subject to direct measurement, hence we here in
troduce and use throughout the ellipticity, E, defined as
_ difference between major and minor diameters
major diameter
It is clear that the ellipticity is easily obtained directly.
Again, many results are given in terms of the major diameter. Since we
are interested in deform.ations from circular, and in such deformations the
424
BELL SYSTEM TECHNICAL JOURNAL
perimeter remains constant, while the major diameter changes, we have
expressed our results in terms of an average diameter, defined as
_ perimeter
Figure 5 shows the elHpse and various relations of interest.
Y
P=PERIMETER
e=ECCENTRICITY =
_ Co
a
E=ELLIPTICITY=^^^' .
A= AREA = TTab
D = "average" DIAMETER = £:
TT
b=aYie2 = a(iE)
A=Tra2'Yie2=Tra2 oe)
Fig. 5 — The ellipse
Elliptic Coordinates and Functions
The elliptic coordinate system is shown on Fig. 6. Following Stratton,'^
we have used ^ in place of the table's z, since we wish to use z as the coor
dinate along the longitudinal axis. Stratton also uses tj = cos if as the angu
lar coordinate; this is frequently convenient.
Analogous to cos (6 and sin (d in the circular case, there are even and
odd* angular functions, denoted by
^Sf{c, cos <f>) and °Sf{c, cos ^)
which reduce to cos Id and sin Id respectively when c
are even and odd* radial functions, denoted by
'Jf^c, k) and "Jfic,
* For ^ = 0, only even functions exist.
0. Similarly, there
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS
425
which both reduce to Jf(kip) when c ^ 0. In the above, c is a parameter
related to the elHpticity.* The tables do not give values of the functions,
but rather give numerical coefficients
Di and Fi
of expansions in series of cosine, sine and Bessel functions, which permit one
to calculate the elliptic cylinder functions. The coefficients, of course,
Fig. 6 — Elliptic coordinate system
depend on the parameter c; the largest value of c in the tables is 4.5, which
corresponds to an ellipticity of 39% in a cylinder operating in the TE 01//
mode.** For this case, Bessel functions up to Jn(x) and Juix) are needed
for calculating the radial function. It is clear that calculations on elliptic
cylinders have not been put on a simple basis.
* Not to be confused with c = velocity of electromagnetic waves; the symbol c is
here carried over from the published tables.
** An ellipticity of 39% means that the difference between maximum and minimum
diameters is 39% of the maximum diameter. For a given c, the ellipticity depends oii
the mod^.
426 BELL SYSTEM TECHXICAL JOURNAL
Field lujiialions
The equations for the fields arc easily obtained from section 6.12 of
Stratton's book, and are given in Table II, which is selfexplanatory, except
for the quantity c, which we now proceed to discuss.
Resonaiil Frequencies
The ellij)tic c\linder has the major diameter, 2a, and the focal distance;
2c[) . The equation of its surface is then cx{)ressed bv ^ = ^ — a. On
this surface, £, must vanish. This requires that '"J f{c, a) ~ for TE
modes and that '"J/ic. a) = for TM modes. The series expansions are
in terms of c^ as variable. Let ca ~ rf,n or r^,,, be the roots of the above
^ r .
equations. Then — =  (dropi)ing the subscripts f, m). Xow, in working
out the solution of the differential equations, it turned out that c — Coki.
, f
Here ^i is one component of the wave number, kj. Hence ^i =  . Further
a
more, the eccentricitv is e = — =  . The indicated procedure is: 1) choose
a r
a value of c; 2) laid the various values of r for which the radial function or
its derivative is zero; 3) then calculate the corresponding eccentricity and
resonant frequency. Notice that for a given value of c, the values of r
will depend on the mode, and hence so will the eccentricity.
We now wish to express our results in terms of the ellipticity and the
average diameter. To convert eccentricity to ellipticity, we use
£ = 1  Vf ^^•
The perimeter of the ellipse is given by P = ■iaE(e) where E(e) is the com
plete elliptic integral of the second kind.tt
In terms of the average diameter we find
*l
2r£(e) "[
2s
or calling the C[uantity in brackets s, A'l = — . This is now in the same form
as ki for a circular cylinder of diameter D. The quantity 5 is the recipro
cal of Chu's ■^.
t It is recalled that
2ir / , r tiTT
^ = _ =, V)fe2 + k^ ; ki =  ; k, =— ,
X 1 ' a L
tt This is tabulated as E(a) in Jahnke & Emde, p. 85, with a = sin^e.
SOME RESULTS OX CVLIXDKICAL CAVITY RESOXATORS 427
We liave calculated and give in Table III values of r, e, E and s for several
values of c and for a few modes of special interest. For three cases, "TE 01,
"TM 11 and "TM 11, we have determined an empirical formula to fit the
calculated values of ^. These are also given in Table III.
TE Modes
TABLE II. Elliptic Cylinder Fields
Et = —k i/ ^ S((c, r])J((c, sin k.iZ cos cot
r •Y/t2 _ \
Er, = k A/ S(,{c, ri)j'({c, t) sin k:i z cos ut
y e 1
\/>^  1
^j = ^3 >5'^(c, t])] \{c, f) cos k>, z sin wt
H.q = kz S({c, ri)J((c, ^) COS kiZ sin wt
q
11 z = klSfic, Ti)J(,{c, t) sin hz sin ut
TM Modes
\/^2 — 1
E^ = —kz Siic, ri)J({c, sin k^z cos ut
Q
■\/ 1 ~2
■Et, = —^3 S'((c, r))J({c, sin ^3 3 cos wt
1
Ez = k'l S((c, 7))J ({c, l) cos hz cos (Jit
H^ = —k 4 /  S'((c, ri)Jp{c, i:) cos ^3 z sin coi
/y/t2 _ J
 "S^Cc, j/jZ/Cc, $) cos h z sin wi
Notes:
Derivatives are with respect to ^ and 77.
Sf and // carry prefixed superscripts, e or 0, since they may be either even or odd.
q = Co Vl^ — rf' c = coki
Kl = «3 = 7" « = ^1 + «j
a L
2co is distance between foci of ellipse.
a is the semi major diameter of the ellipse,
r^ „, is the value of c$ that makes
J l{c,^) — for ^^ modes
J'^ifyO = for TE modes.
428
BELL SYSTEM TECIIMCAL JOiRXAL
TAULK 111 Rout Valiks ok Kauial Elliptic Cylinder Functions
Mode
c
r
e
E
i
TEOl
3.8317
3.8317
0.2
3.8343
0.05216
0.001361
3.8317
0.4
3.8423
0.10410
0.005434
3.8318
0.6
3.8558
0.15561
0.012181
3.8324
0.8
3.8753
0.20643
0.021539
3.8337
1.0
3.9015
0.25631
0.033406
3.8366
1.2
3.9349
0.30496
0.047636
3.8417
1.4
3.9763
0.35209
0.064033
3.8500
1.6
4.0264
0.39738
0.082346
3.8624
2.0
4.154
0.4814
0.12351
3.902
3.0
4.634
0.6474
0.2378
4.101
4.0
5.29
0.756
0.346
4.42
4.5
5.66
0.795
0.393
4.62
5 =
3.8317 + 4.33 E^ + \.9E^
^TM 11
3.8317
3.8317
0.2
3.8330
0.05218
0.001362
3.8304
0.4
3.8370
0.10425
0.005449
3.8265
0.6
3.8436
0.15610
0.012259
3.8201
0.8
3.8532
0.20762
0.021791
3.8113
1.0
3.8658
0.25868
0.034036
3.8003
1.2
3.8818
0.30913
0.048981
3.7874
1.4
3.9015
0.35884
0.066599
3.7727
1.6
3.9253
0.40761
0.086844 .
3.7568
4.5
5.13
0.878
0.520
3.91
3.8317  0.96£ + 1.1 /^^
^TM 11
0.2
0.4
0.6
0.8
1.0
1.2
1.4
3.8317
3.8356
3.8474
3.8670
3.8944
3.9298
3.9731
4.0243
0.05214
0.10397
0.15516
0.20542
0.25446
0.30203
0.34788
0.001361
0.005419
0.012111
0.021326
0.032918
0.046701
0.062462
3.8317 + 0.95E + 2.2E^
3.8317
3.8330
3.8370
3.8436
3.8530
3.8654
3.8809
3.8997
'TE 22
6.706
6.706
0.4
6.712
0.0596
0.00178
6.706
0.8
6.729
0.1189
0.00709
6.705
1.2
6.756
0.1776
0.01590
6.702
1.6
6.788
0.2357
0.02817
6.693
2.0
6.826
0.2930
0.04389
6.677
"TE 22
6.706
6.706
0.4
6.712
0.0596
0.00178
6.706
0.8
6.730
0.1189
0.00709
6.706
1.2
6.762
0.1775
0.01587
6.708
1.6
6.810
0.2350
0.02799
6.715
2.0
6.877
0.2908
0.04323
6.729
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS
429
Mode
c
r
e
E
s
•r£32
8.015
8.015
0.4
8.020
0.0499
0.00124
8.015
0.8
8.035
0.0996
0.00497
8.015
1.2
8.059
0.1489
0.01115
8.014
1.6
8.093
0.1977
0.01974
8.013
2.0
8.135
0.2459
0.03070
8.010
"TEH
8.015
8.015
0.4
8.020
0.0499
0.00124
8.015
0.8
8.035
0.0996
0.00497
8.015
1.2
8.060
0.1489
0.01115
8.015
1.6
8.097
0.1976
0.01972
8.018
2.0
8.146
0.2455
0.03061
8.022
'TMQ\
2.4048
0.2
2.4090
0.08302
0.4
2.4216
0.16518
0.6
2.4431
0.24559
0.8
2.4739
0.32337
1.0
2.5149
0.39762
'TEn
1.8412
0.2
1.8416
0.10860
0.4
1.8430
0.21704
0.6
1.8452
0.32516
0.8
1.8484
0.43280
1.0
1.8527
0.53975
Notes:
Superscripts e and o on mode designation signify even and odd.
c is parameter used in the Tables (Stratton, Morse, Chu, Hutner, "Elliptic Cylinder
and Spheroidal Wave Functions")
r is the value of the argument which, for TM modes, makes the radial function zero
and, for TE modes, makes its derivative zero.
e is the eccentricity of the ellipse;
_ distance between foc i
major diameter
E is the ellipticity of the ellipse;
difference between major and minor diam.
major diameter
5 is the root value, referred to the "average diameter"; it is related to r by:
_ r perimeter
IT major diameter
The quantity 5 is also related to the cutoff wavelength in an elliptical wave guide
according to:
_ perimeter of guide
cutoff wavelength
Resonator Q
Although the calculation of the root values is straightforward and not
overly laborious, the same cannot be said for the integrations involved in
the determination of resonator Q. The procedure is obvious: The field
430 BELL SYSTEM TECHNICAL JOURNAL
equations are given; it is only necessary to integrate H^dr over the volume
and IPda over the surface and get Q from
2 / ^'^'
Q = I (16)
j H^da
with 5 = skin depth, a known constant. Unfortunately the integrations
cannot at present be expressed in closed form. A numerical solution can
be obtained by a combination of integration in series and of numerical
integration.
The calculations have been made for the ^TE 01 mode with c — 2.0, for
which r = 4.154. This value of c corresponds in this case to an ellipticity
of about 12%; in a 4" cylinder this would amount to 1/2" difference between
largest and smallest diameters. Evaluation* of the integrals yields:
Hdr = 12.307 k^L + 12.294 klL
v
H"d<7 = 49.228 k^ + 0.1619 kiHL + 6.6847 kiL
s
Substituting k] — and kg = — ^ o^^^ obtains, finally
Q8 = 0.471 D
1 + 0.1622 nR"
,1 + 0.0039 «2i?2 ^ 0.1529 n'R^
For a circular cyhnder,
'1 + 0.1681 nR"
Qc8 = 0.5 D
1 + 0.1681 n'R
Comparison of these two formulas for Qd shows that the losses in the end
plates {nR term) are less with respect to the side wall losses in the ellip
tical cylinder. The net loss in Q8, as described by the reduction in the mul
tiplier from 0.5 to 0.471, is thus presumably ascribable to an increase in side
wall losses (stored energy assunied held constant). The additional term
in n^R in the denominator is responsible for the difference in the attenuation
frequency behavior of elliptical vs circular wave guide as shown by Chu,
Fig. 4. Incidentally, these results agree numerically with those of Chu.
* Numerical integration was by Weddle's rule; intervals of 5° in ^ and 0.1 in x were used.
The calculations were made bj^ Miss F. C. Larkej'.
SOME RESULTS Oi\ CYLINDRICAL CAVITY RESONATORS 431
Corresponding expressions for the resonant wavelength are
ttD 0.805 D
X = 
a/\ + hnD\ ^1 + 01622 «2i22
\2sL/
0.820 D
Vl +0.1681 w2/?2
As an example, take n = 1, R = 1, then
(Circular) Qc5 = 0.500 D X^ = 0.759 D
(Elliptical) Q8 = 0.473 D X = 0.747 D
Ratio = 0.946 Ratio  0.984.
Conclusions
The mathematics of the elliptic cylinder have not yet been developed to the
point where the design of cavities of large ellipticity could be undertaken.
On the other hand, sufficient results have been obtained to indicate that the
ellipticity in a cavity intended to be circular, resulting from any reasonable
manufacturing deviations, would not have a noticeable effect on the reso
nant frequencies or Q values, at least away from mode crossings.
Full Cylindrical Coaxial Resonator
The full coaxial resonator has been of some interest because of various
suggestions for the use of a central rod for moving the tuning piston in a
TE OUi cavity.
The cylindrical coaxial resonator, with the central conductor extending
the full length of the resonator, has modes similar to the cylinder. In
fact, the cylinder may be considered as a special case of the coaxial. The
indices /, m, n have much the same meaning and the resonant frequencies
are determined by the same equation (1). However, now the value of r
depends in addition (see Fig. 1) upon 77, where
_ diameter inner conductor _ ^
diameter outer conductor a '
The problem now arises of how best to represent the relations between
/, a, b and L. The r's depend on tj; so one possibility is to determine their
values for a given 77 and then construct a series of mode charts, one for each
value of 77.
A more flexible arrangement is to plot the values of r vs 77 and allow
the user to construct graphs suitable for the particular purpose in hand.
An equivalent scheme has been used by Borgnis.^^
It turns out that as 77 — ^ 1, r(l — 77) —> ftiir, for the TM modes and the
432 BELL SYSTEM TECHNICAL JOURNAL
TE Omn modes, and r{\ — rj) ^ {m — l)x for all other TE modes. For
the former modes, r becomes very large as r; — > 1, that is, as the inner con
ductor fills the cavity more and more, the frequency gets higher and higher.
For the TE (In modes, however, as the inner conductor grows, the f re
queue}' falls to a limiting value. This is discussed in more detail by
Borgnis.^^
Figure 7 shows r(l — 77) vs 77, for a few of the lower modes; the scale for 77
between 0.5 and 1.0 is collapsed since this region does not appear to be of
great engineering interest. A different procedure is used for the roots of
the TE (hi modes. Figure 8 is a direct plot of r vs 77 for a few of the lower
modes. In this case, r ^ f as 77 h^ 1.
Distribuiion of Normal Modes
The calculation of the distribution of the resonant modes for the coaxial
case follows along the lines of that for the cyhnder, as given previously.
The difference lies in the distribution of the roots r, which now depend upon
the parameter r,. The determination of this latter distribution offers
difficulties. There is some evidence, however, that the normal modes will
follow, at least to a first approximation, the same law as the cylinder, viz.:
V
N = 4.4 ^
Ao
with some doubt regarding the value of the coefficient.
 in Coaxial Resonator
X
The integrations needed to obtain this factor are relatively straightfor
ward, but a little complicated. The final results are given in Fig. 1.
The defining equation is (16); the components of H are given in Fig. 1.
The integrations can be done with the aid of integrals given by McLachlan^^
and the following indefinite integral :
which can be verified by differentiation, remembering that y = Zi{x) is a
solution of y" +  y' f ( 1   ) y = 0.
X \ xJ
7.0
\
1
^.
\
^.
—
:
TM 12
_
6.2
TM02_
6.0
^
(
5.4
5.2
^;
\
\
\\
A
4.4
V
\
\
4.2
\
v\l
\
\,
4.0
3.8
\
\
\
\
V
\
TE12^
\,
\
TM 21
3.6
^
s
^
^^
^
3.4
^
TE 01
..^
TM11
^
;:C
K
3.2
—
"■ 
.
J^^
^
.
—
 —
^^^^
3.0
2.8
2.6
TM 01
/
(
2.4
\ — 1 — 1
— 1 — 1 —
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 1.0
INNER CONDUCTOR DIAMETER
I ~ OUTER CONDUCTOR DIAMETER
Fig. 7 — Full coaxial resonator root values r^^ (1 — »?)
433
TE 41
— l~l
I — 1
5.2
5.0
4 6
"
^^
\
\
4.6
4.4
4.2
4.0
3.8
3.6
e
£^'3.4
3 3.2
\
\
TE3I
\
"~~
■\
^v^
\
V
\
\
\
\
TE 21
\
O 3.0
O
Ct
2.8
2.6
2.4
2.2
2.0
1.8
1.6
^
"*^^
^^
^^
V
\
\
V
\
■
■~~
^
TEll
■
..^
"~~~
\
1.0
1
— 1 — 1
ii.
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
INNER CONDUCTOR DIAMETER
n~ OUTER CONDUCTOR DIAMETER
Fig. 8 — Full coaxial resonator root values r.
434
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 435
An investigation needs to be made of the behavior of the formulas as
77 — > before any conclusion may be drawn regarding their blending
into those for the cylinder. For TE modes with ^ = 0, the term involving
jj
— disappears, hence no question arises. Consider then / > 0, and let
X = Tjr for the discussion following. From expansions given in McLachlan,
it is easy to show that, for small x
J({x) =
<^)^';""©'
T \X/ X
X^
Since, from Fig. 1,
A =
J'({r) _ Jiiv) _ Ji(x)
2i{(  1) !
y'lir) y'tinr) Y({x)
it is found, upon substitution of the approximations given above:
That is, Zt{x) '~ x^ and hence — > as x ^ 0. Furthermore Zt{r) remains
finite as t? ^ 0. Hence H ^ 0^^ and — '^ x^~^. Therefore, for / > 0,
n
— — > as 77 — > 0.
Hence, the expression for Q  for the coaxial structure reduces to that for
the cylinder, for any value of (, in the TE modes.
For the TM modes, and for ^ > 0, an entirely similar argument shows
that H' remains hnite as 7? — > 0. Hence, the expression for Q  for these
A
modes also reduces to that for the cylinder.
For the TM modes, and with / = 0, we have
Zo(x) = 7i(.r) + 7o(t)
F,(x)
For X — )■ 0, /i(.v) — > and Jq{x) ^ 1, hence for small x,
yo{x)
436
BELL SYSTEM TECHNICAL JOURNAL
Now substitute the approximate values of the I' for small x. The result is
Since Zo(r) is tinite, it follows that
•qH' ' — '
1
a; log
('3^
and it is easily shown that r)II' — > <» as r; ^ 0. On the other hand, rfH' ^
as ?7 ^ 0. Hence, ()  — > as 77 — ^ 0. On the other hand, for tj = 0, a
A
0.50
0.45
q: a.
0.40
0.30
Q 0.25
z
o
'' 0.20
0.15
^ 0.10
0.05
/
r
^
A
^
y
Q 4 =0.30
/
X
y
X
y
y'
y
aaj.
y
^^
y^
0.40
^
s
^'^
^
\
\,
^
c
3.45
.
..^
\
■
'
0.50
^
\
, ■

^
^
^
\
\
[max. 0.656
2
^DU^
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
_a_
L
R=r
Fig. 9 — Coaxial resonator. TE Oil mode Contour lines of ()
A
perfect cylinder exists whose (J  is not zero. It is concluded that the ex
X
pression for Q  does not apply for small 7/ for the TM modes with /" = 0.
A
s
Thus it is seen that the expressions for the factor (() ) reduce to those
A
given for the cylinder, when t; = 0, except for TM modes with /* = 0.
For these latter cases, the factor approaches zero as 7/ approaches zero,
because 77//' increases without limit. This means that an assumption
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS
437
liJ
0.40
1
1
LiJ
III
<
5
<
0.35
Q
Q
O
a
n
0.30
1
1
U
o
a
n
0.25
z
7
o
o
o
o
0.20
or
QC
Ul
Ol
7
1
0.15
o
d
0.10
0.05
OJg,
^
"^
^
^
^
^ '
0.14
— '
' '
^^
^
x'^
of
= 0.16
^
18
^
^
 —
, 1
0.20
^
~
0.22
^
/
^
J3.24
■
/
/
^
^"
0.2
76
■
■— ■
0.26
^
0.2 0.4 6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.S
^ L
Fig. 10 — Coaxial resonator. TE 111 mode Contour lines oiQ_
3 0.25
Q
Z
o
O 0.20
cr 0.10
f'
— n
A.
O
/
/
1
/
/
y
^
/d
/
/
/
/
/
1
/
7
7
%
/
/
/
/
/
/
/
/
Y
4
//
/
^
J"'
/
/,
/
/
/
^
■^
^
//
/
/
y
/
^^
>^
r/
y
/
^
^
^^
2i^
 0.16
________
1
1
^
"^^^^i
::::^
^

•:^ —
0.2 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Fig. 11 — Coaxial resonator. TM Oil mode Contour lines of Q,
438
BELL SYSTEM TECHNICAL JOURNAL
which was made in the derivation of the Q values is not valid for small tj;
that is, the fields for the dissipative case are not the same as those derived
on the basis of perfectly conducting walls.
The expressions for the factor are rather complicated, as it depends on
several parameters. When a given mode is chosen, the number of param
eters reduces to two, 77 and R. Contour diagrams of ()  vs 77 and R are
A
given on Figs. 9, 10, 11 and 12 for the TE Oil, TE 111, TMOll and TM 111
Fig. 12— Coaxial resonator. TM HI mode Contour lines of Qj
modes. As mentioned above, the true behavior of ()  for the TM Oil
mode for small rj is not given by the above formula, so this contour diagram
has been left incomplete.
Fins in a Cavity Resonator
The suppression of extraneous modes is always an important problem
in cavity design. Among the many ideas advanced along these lines is the
use of structures internal to the cavity.
It is well known that if a thin metallic fin or septum is introduced into a
cavity resonator in a manner such that it is everywhere perpendicular to
the £lmes of one of the normal modes, then the field configuration and
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS
439
frequency of that particular mode are undisturbed. For example, Fig. 13
shows the £lines in a TE llw mode in a circular cylinder. If the upper
half of the cylinder wall is replaced by a new surface, shown dotted, the
field and frequency in the resulting flattened cylinder will be the same as
NEW SURFACE PERPENDICULAR
TO E LINES LEAVES REST OF
FIELD AND FREQUENCY UNALTERED
Fig. 13 — E Lines in TE lire mode
^ORIGINAL CYLINDER
t'
Fig. 14— "TE 01m" mode in halfcylinder
before. Indeed, they will also be the same in the crescentshaped resonator
indicated in the figure.
Except for isolated cases, all the other modes of the original cylinder will
be perturbed in frequency since the old fields fail to satisfy the boundary
conditions over the new surface. Furthermore, if the original cylinder was
440
BELL SYSTEM TECHNICAL JOURNAL
circular, its inherent double degeneracy will be lost and each of the original
modes (with minor exceptions) will split into two.
Although the frequency and fields of the undisturbed mode are the
same, the Q is not necessarily so. For example, Fig. 14 shows a ""TE 01«
mode" in a half cylinder.*
It is easy to calculate Q  for this case. The result is
(1 + p'R'f"
in which
Ki = 1.290 A'2 = 0.653
(17)
Here A'l and K2 are constants which account for the resistance losses in
the flat side. For the full cavity, shown dotted in Fig. 14, eq. (17) holds
with A'l = A'2 = 0. If the circular cavity has a partition extending from
the center to the rim along the full length, (17) holds with the values
of A'l and A'2 halved. If a tin projects from the rim partway into the in
terior, still other values of A'l and A'2 are required. It is a simple matter
to compute these for various immersions; Fig. 15 shows curves of A'l and
K2 . The following table gives an idea of the magnitudes involved:
mode: r£ 0,1,12 R = 0.4
8
Fin, % a
^4
Ratio
0%
2.573
1.0
10
2.536
.985
20
2.479
.965
50
2.04
.79
100
1.47
.57
The question now is asked, "Suppose a longitudinal fin were used, small
enough to cause only a tolerable reduction in the Q. Would such a fin
ameliorate the design difficulties due to extraneous modes?"
Some of the effects seem predictable. All modes with ^ > will be split
to some extent, into two modes of different frequencies. Consider the
TE I2n mode, for example. There will be one mode, of the same frequency
as the original whose orientation must be such that its £lines are perpendicu
lar to the fin. The Q of this mode would be essentially unchanged. There
will be a second mode, oriented generally 90° from the first, whose £lines
will be badly distorted (and the frequency thereby lowered) in the vicinity
* Solutions for a cylinder of this crosssection are known and all the resonant fre
quencies and Q values could be computed, if they had any application.
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS
441
of the fin. It would be reasonable to expect the Q of this mode to be appre
ciably lowered because of the concentrated field there. If two fins at 90°
were present, there would be no orientation of the original TE \2n mode
which would satisfy the boundary conditions. In this case both new modes
0,0.40
(M
0.35
UJ
/
/
/
/
/
/
.^
/
/"
/
7
'
K2/
/
/
/
f
/
/
/
/
/
f
/
/
/
/
/
/
/
/
/
/
^
^,.
'
fin
O 0.20
$ 0.15
0.05
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
WIDTH OF FIN
RADIUS OF CYLINDER
Fig. 15 — Constants for calculation of Q of TE Oln mode in cylinder with longitudinal
would be perturbed in frequency from the original value. If both fins were
identical, the perturbations would be equal and a double degeneracy ensue.
Similar effects would happen to the other types of modes.
The major advantage derivable from such effects would appear to be in
extraneous transmissions. The fin serves to orient positively the fields in
4^2 BELL SYSTEM TECHNICAL JOURNAL
the cavity, and the input and output couphng locations can then be appro
priately chosen. On the basis that internal couplings are responsible for
mode crossing difficulties, one might hazard a guess that a real fin would
increase such couplings.
Another application of fins might be in a wave guide feed in which it is
desired to establish only a TE Oni wave. In this case, Q is not so important
and larger fins can be used. If these extended virtually to the center and x
of them were present (with uniform angular spacing) all types of wave trans
mission having / less than x/2, x even or / less than x, x odd, would be sup
pressed. This use of fins is an extension of the wires that have been
proposed in the past.
Conclusion
It is hoped that the foregoing, which covers some of the theoretical work
done by the author during the war, will be of value to other workers in
cavity resonators. There is much that needs to be done and hardly time
for duplication of effort.
Bibliography
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Wiley and Sons, (1943).
9. R. H. Bolt, "Frequency Distribution of Eigentones in a ThreeDimensional Con
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10. DahYou Maa, "Distribution of Eigentones in a Rectangular Chamber at LowFre
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a^jproximation formula.
11. I. G. Wilson, C. W. Schramm, J. P. Kinzer, "High Q Resonant Cavities for Micro
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Phys., 9, pp. 583591 (1938).
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SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 443
15. J. A. Stratton, "Electromagnetic Theory," McGrawHill, (1941).
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Additional Bibliography
19. J. J. Thomson, "Notes on Recent Researches in Electricity and Magnetism," Oxford,
Clarendon Press, 1893, — §300 gives the resonant frequencies of the TE modes in a
cylinder with a/L = 0; §315316 consider two concentric spheres; §317318 treat
of the Q of the spherical cavity.
20. Lord Rayleigh, "On the passage of electric waves through tubes or the vibrations of
dielectric cylinders" Phil. Mag.; 43, pp. 125132 (1897) .^Considers rectangular
and circular crosssections.
21. A. Becker, " Interf erenzrohren fiir elektrische Wellen," Ann. d. Phys., 8, pp. 2262
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22. R. H. Weber, " Elektromagnetische Schwingungen in Metallrohren," Ann. d. Phys.,
8, pp. 721751 (1902)— Abstract in Set. Abs., 6A, No. 96 (1903).
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Ann. d. Phys., 25, pp. 337445 (1908) — A part of this article deals with the solution
of the equations for the sphere; also shown are the E and H lines for the lowest
eight resonant modes.
25. H. W. Droste, " UltrahochfrequenzUbertragung langs zylindrischen Leitern und
Nichtleitern," TFT, 27, pp. 199205, 273279, 310316, 337341 (1931)— Abstract
in Wireless Engr., 15, p. 617, No. 4209 (1938).
26. W. L. Barrow, "Transmission of Electromagnetic Waves in Hollow Tubes of Metal,"
Proc. I.R.E., 24, pp. 12981328 (1936)— A development of the equations of propa
gation together with a discussion of terminal connections.
27. S. A. Schelkunoff, "Transmission Theory of Plane Electromagnetic waves," Proc.
I.R.E., 25, pp. 14571492 (1937)— Treats waves in free space and in cylindrical
tubes of arbitrary crosssection; special cases; rectangle, circle, sector of circle and
ring.
28. L. J. Chu, "Electromagnetic Waves in Elliptic Hollow Pipes of Metal," Jour. App.
Phys., 9, pp. 583591 (1938) — A study of field configurations, ci;itical frequencies,
and attenuations.
29. G. Reber, "Electric Resonance Chambers," Communications, Vol. 18, No. 12, pp.
58 (1938).
30. F. Borgnis, " Electromagnetische Eigenschwingungen dielektrischer Raume," Ann.
d. Phys., 35, pp. 359384 (1939). Solution of Maxwells equations for rectangular
prism, circular cylinder, sphere; also derivations of stored energy and Q values.
31. W. W. Hansen, "On the Resonant Frequency of Closed Concentric Lines," Jour. App.
Phys., 10, pp. 3845 (1939). — Series approximation method for TM OOp mode.
32. R. D. Richtmyer, "Dielectric Resonators," Jour. App. Phys., 10, pp. 391398 (1939).
33. H. R. L. Lamont, "Theory of Resonance in Microwave Transmission Lines with
Discontinuous Dielectric," Phil. Mag., 29, pp. 521540 (1940).— With bibliography
covering wave guides, 19371939.
34. E. H. Smith, "On the Resonant Frequency of a Type of Klystron Resonator," Phys.
Rev., 57, p. 1080 (1940).— Abstract.
35. W. C. Hahn, "A New Method for the Calculation of Cavity Resonators," Jour. App.
Phys., 12, pp. 6268 (1941). — Series approximation method for certain circularly
symmetric resonators.
36. E. U. Condon, "Forced Oscillations in Cavity Resonators," Jour. App. Phys., 12
pp. 129132 (1941). — Formulas for coupUng loop and capacity coupling.
37. W. L. Barrow and H. Schaevitz, "Hollow Pipes of Relatively Small Dimensions,"
A.I.E.E. Trans., 60, pp. 119122 (1941). — Septate coaxial wave guide and cavity
resonator, based on bending a fiat rectangular guide into a cylinder.
444 BELL SYSTEM TECBNICAL JOURNAL
38. H. Konig, "The Laws of Similitude of the Electromagnetic Field, and Their Appli
cation to Cavity Resonators," Wireless Engr., 19, p. 216217, No. 1304 (1942).
"The law of similitude has strict validity only if a reduction in dimensions hy the
factor \/m is accompanied by an increase in the conductivity of the walls bv the
factor w." Original article "in Ilochf; tech u. Elek:akus, 58,' pp. 174180 (1941).
39. S. Ramo, "Electrical Conce[)ts at Extremely High Frequencies," Electronics, Vol. 9,
Sept. 1942, pp. 3441, 7482. A nonmathematical description of the physical
phenomena involved in vacuum tubes, cavity resonators, transmission lines and
radiators.
40. J. Kemp, "Wave Guides in Electrical Communication," Jour. I.E.E., V. 90, Pt. Ill,
pp. 90114 (1943). — Contains an extensive hsting of U. S. and British patents.
41. H. A. Wheeler, "Formulas for the Skin EiTect," Proc. I.R.E., 30, pp. 412424 (1942)—
Includes: a chart giving the skin depth and surface resistivit} of several metals
over a wide range of frequency; simple formulas for H.F. resistance of wires, trans
mission lines, coils and for shielding effect of sheet metal.
42. R. C. Colwell and J. K. Stewart, "The Mathematical Theory of Vibrating Mem
branes and Plates," J.A.S.A., 3, pp. 591595 (1932) — Chladni figures for a square
plate.
43. R. C. Colwell, "Nodal Lines in A Circular Membrane" J.A.S.A., 6, p. 194 (1935)—
Abstract.
44. R. C. Colwell, "The Vacuum Tube Oscillator for Membranes and Plates," J.A.S.A.,
7, pp. 228230 (1936) — Photographs of Chladni figures on circular plates.
45. R. C. Colwell, A. W. Friend, J. K. Stewart, "The Vibrations of Symmetrical Plates
and Membranes," J.A.S.A., 10, pp. 6873 (1938).
46. J. K. Stewart and R. C. Colwell, "The Calculation of Chladni Patterns," J.A.S.A.,
11, pp. 147151 (1939).
47. R. C. Colwell, J. K. Stewart, H. D. Arnett, "Symmetrical Sand Figures on Circular
Plates," J.A.S.A., 12, pp. 260265 (1940).
48. V. O. Knudsen, "Resonance in Small Rooms," J.A.S.A., 4, pp. 2037 (1932)— Ex
perimental check on the values of the eigentones.
49. H. Cremer & L. Cremer, "The Theoretical Derivations of the Laws of Reverberation,"
J.A.S.A., 9, pp. 356357 (1938)— Abstract of Akustische Zeits., 2, pp. 225241,
296302 (1937) — Eigentones in a rectangular chamber.
50. H. E. Hartig and C. E. Swanson, "Transverse Acoustic Waves in Rigid Tubes,"
Pliys. Rev., 54, pp. 618626 (1938) — Experimental verification of the presence of
acoustic waves in a circular duct, corresponding to the TE and TM electromag
netic waves; shows an agreement between calculated and experimental values of
the resonant frequencies, with errors of the order of ± 1%.
51. D. Riabouchinsky, Comptes Rendus, 207, pp. 695698 (1938) and 269, pp. 664666
(1939). Also in Science Abstracts A42, j^364 (1939) and A43, 7^1236 (1940).—
Treats of supersonic analogy of the electromagnetic field.
52. F. V. Hunt, "Investigation of Room Acoustics by Steady State Transmission Meas
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,53. R. Bolt, "Standing Waves in Small Models," J.A.S.A., 10, p. 258 (1939).
54. L. Brillouin, "Acoustical Wave Propagation in Pipes," J.A.S.A., 11, p. 10 (1939) —
Analogy with TE waves.
55. P. E. Sabine, "Architectural Acoustics: Its Past and Its Possibilities," J.A.S.A., 11
pp, 2128, (1939). — Pages 2628 give an illuminating review of the theoretical work
in acoustics.
56. R. H. Bolt, "Normal Modes of Vibration in Room Acoustics: Angular Distribution
Theory," J.A.S.A., 11, pp. 7479 (1939). — Eigentones in rectangular chamber.
57. R. H. Bolt, "Normal Modes of Vibration in Room Acoustics: Experimental Investiga
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58. L. Brillouin, "Le Tuyau Acoustique comme Filtre PasseHaut/' Rev. D'Acoiis., 8,
pp. 111 (1939). — A comparison with TM waves; some historical notes, tracing the
inception of the theory back to 1849.
59. E. Skudrzyk, "The Normal Modes of Viijration of Rooms with NonPlanar Walls,"
J.A.S.A., 11, pp. 364365 (1940).— Abstract of Akustische Zeits., 4, p. 172 (1939).—
Considers the equivalent of the TAl 00/) mode.
60. G. M. Roe, "Fre((uency Distribution of Normal Modes," J.A.S.A., 13, pp. 17
(1941). — A verification of Maa's result for a rectangular room, and an extension
to the cylinder, sphere and several derived shapes, which leads to the result that the
number of normal modes (acoustic) below a given frequency is the same for all
shapes.
SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 445
61. R. S. Bolt, H. Feshbach, A. M. Clogston, "Perturbation of Sound Waves in Irregular
Rooms," J.A.S.A., 14, pp. 6573 (1942) — Experimental check of eigentones in a
trapezoid vs calculated values.
Abstracts of Foreign Language Articles in Wireless Engineer
62. H. Gemperlein," Measurements on Acoustic Resonators," 16, p. 200, No. 1504 (1939),
63. M. Jouguet," Natural Electromagnetic Oscillations of a Cavity," 16, p. 511, No. 3873,
(1939).
64. M. S. Neiman, " Convex Endovibrators," 17, p. 65, No. 455, (1940).
65. F. Borgnis, "The Fundamental Electric Oscillations of Cylindrical Cavities," 17,
p. 112, No. 905, (1940). See also Sci. Abs., B43, No. 343 (1940).
66. H. Buchholz, "UltraShort Waves in Concentric Cables, and the "HollowSpace"
Resonators in the Form of a Cylinder with PerforatedDisc Ends," 17, p. 166, No.
1301 (1940).
67. J. Aliiller, "Investigation of Electromagnetic Hollow Spaces," 17, p. 172, No. 1379
(1940).— Sci. Abs., B43, No. 857 (1940).
68. V. I. Bunimovich, "An Oscillating System with Small Losses," 17, p. 173, No. 1380
(1940).
69. M. S. Neiman, "Convex Endovibrators," 17, p. 218, No. 1743 (1940).
70. M. S. Neiman, "Toroidal Endovibrators," 17, p. 218, No. 1744 (1940).
71. H. Buchholz, "The Movement of Electromagnetic Waves in a ConeShaped Horn,"
17, p. 370, No. 3009 (1940). — Cavity formed by cone closed by spherical cap.
72. O. Schriever, "Physics and Technique of the HollowSpace Conductor," 18, p. 18
No. 2, (1941).— Review of history.
73. F. Borgnis, "Electromagnetic HollowSpace Resonators in Short Wave Technique,"
18, p. 25, No. 61, (1941).
74. T. G. Owe Berg, "Elementary Theory of the Spherical Cavity Resonator," 18, p. 287,
No. 1843 (1941).
75. F. Borgnis, "A New Method for measuring the Electric Constants and Loss Factors of
Insulating Materials in the Centimetric Wave Band," 18, p. 514, No. 3435 (1941). —
An application of the cylindrical cavity resonator.
76. V. I. Bunimovich, "The Use of Rectangular Resonators in UltraHighFrequency
Technique," 19, p. 28, No. 65 (1942). Use in 17 cm oscillator.
77. V. I. Bunimovich, "A Rectangular Resonator used as a Wavemeter for Decimetric
and Centimetric Waves," 19, p. 37, No. 176 (1942).
78. M. Watanabe, "On the Eigenschwingungen of the Electromagnetic Hohlraum," 19,
p. 166, No. 927 (1942).
79. F. Borgnis, "The Electrical Fundamental Oscillation of the Cylindrical TwoLayer
Cavity," 19, p. 370, No. 2306 (1942). Considers cylindrical resonator with two
concentric internal cylinders of different dielectric constant.
80. W. Ludenia, "The Excitation of Cavity Resonators by SawTooth Oscillations," 19,
p. 422423, No. 2641 (1942).
81. Ya. L. Al'pert, "On the Propagation of Electromagnetic Waves in Tubes," 19, p.
520, No. 3181 (1942). — Calculation of losses in a cylindrical wave guide.
82. V. I. Bunimovich, "The Propagation of Electromagnetic Waves along Parallel Con
ducting Planes," 19, p. 520, No. 3182 (1942). — Equations for Zo and attenuation of
rectangular wave guide, and resonant frequency and Q of rectangular cavity.
83. C. G. A. von Lindern & G. de Vries, "Resonators for UltraHigh Frequencies," 19,
p. 524, No. 3206 (1942). — Discusses transition from solenoid to toroidal coil to
"single turn" toroid, i.e., toroidal cavity resonator.
Abstracts in Science Abstracts
84. L. Bouthillon, "Coordination of the Different Types of Oscillations," A39, No. 1773
(1936) .^General theory of mechanical, acoustic, optical and electric oscillations.
85. Biirck, Kotowski, and Lichte, "Resonance Effects in Rooms, their Measurement and
Stimulation," A39, No. 5226 (1936).
86. G. Jager, "Resonances of Closed and Open Rooms, Streets and Squares," 40A, No
306 (1937).
87. K. W. Wagner, "Propagation of Sound in Buildings," A40, No. 2199(1937).— Trans
mission through a small hole in a wall.
88. M. Jouguet, "Natural Electromagnetic Oscillations of a Spherical Cavity," 42A, No.
3822 (1939).
89. H. R. L. Lament, " Use of the Wave Guide for Measurement of Microwave Dielectric
Constants," 43 A, No. 2684 (1940).
Precision Measurement of Impedance Mismatches
in Waveguide
By ALLEN F. POMEROY
A method is described for determining accurately the magnitude of the reflection
coeflicient caused by an inipeiance mismatch in waveguide by measuring the
ratio between incident and reflected voltages. Reflection coeflicients of any
value less than 0.05 (0.86 db standing wave ratio) can be measured to an accuracy
of ± 2.5%.
TONG waveguide runs installed in microwave systems are usually
*— ' composed of a number of short sections coupled together. Although
the reflection at each coupling may be small, the effect of a large number in
tandem may be serious. Therefore, it is desirable to measure accurately
the very small reflection coefficients due to the individual couplings.
A commonly adopted method for determining reflection coefficients in
phase and magnitude in transmission lines has been to measure the standing
wave ratio by means of a traveling detector. Such a system when carefully
engineered, calibrated and used is capable of good results, especially for
standing waves greater than about 0.3 db.
Traveling detectors were in use in the Bell Telephone Laboratories in
1934 to show the reactive nature of an impedance discontinuity in a wave
guide. A traveling detector was pictured in a paper^ in the April 1936
Bell System Technical Journal. Demonstrations and measurements using
a traveling detector were included as part of a lecture on waveguides by
G. C. Southworth given before the Institute of Radio Engineers in New
York on February 1, 1939 and before the American Institute of Electrical
Engineers in Philadelphia on March 2, 1939.
Methods for determining the magnitude only of a reflection coefficient
by measuring incident and reflected power have been developed by the Bell
Telephone Laboratories. A method used during World War II incorporated
a directional coupler^. The method described in this paper is a refinement
of this directional coupler method and is capable of greatly increased accu
racy. It uses a hybrid junction^ to separate the voltage reflected by the
mismatch being measured from the voltage incident to the mismatch.
Each is measured separately and their ratio is the reflection coefficient.
The problem to be considered is the measurement of the impedance
mismatch introduced by a coupling between two pieces of waveguide due to
differences in internal dimensions of the two waveguides and to imperfec
tions in the flanges. The basic setup might be considered to be as shown
in Fig. 1. The setup comprises a signal oscillator, a hybrid junction, a
446
MEASUREMENT OF IMPEDANCE MISMATCHES
447
calibrated detector and indicator, a termination Z', a piece of waveguide
EF (the flange E of which is to be part of the couphng BE to be measured)
and a termination Z inserted into the waveguide piece EF so that the
reflection coefl&cient of the couphng BE alone will be measured. In addi
tion a fixed shorting plate should be available for attachment to flange B.
Four cases are considered :
I. Termination Z and Z' perfect, only one coupling on hybrid junction.
II. Termination Z imperfect, termination Z' perfect, only one coupling
on hybrid junction.
III. Termination Z perfect, four couplings on hybrid junction.
IV. Termination Z imperfect, four couplings on hybrid junction.
SIGNAL
OSCILLATOR
TERMINATION Z'
A
1
HYBRID
JUNCTION
^N\
'■ D
C
CALIBRATED
DETECTOR &
INDICATOR
TERMINATION Z
\AA/
rr
Fig. 1 — Block schematic for cases I and II.
It is assumed in all cases that:
1. The hybrid junction has the properties as defined in the discussion of
case I.
2. The signal oscillator absorbs all the power reflected through arm A of
the hybrid junction.
3. The calibrated detector and indicator absorb all the power transmitted
through arm C of the hybrid jimction.
4. The oscillator output and frequency are not changed when the hybrid
junction arm B is shortcircuited.
5. The attenuation of waveguide may be neglected.
I. Termination Z and Z' Perfect, Only One Coupling on
Hybrid Junction
In this case the hybrid junction, termination Z' and termination Z,
as shown in Fig. 1, are all considered to be perfect. This means for the
hybrid junction that its electrical properties are such that the energy from
448 BELL SYSTEM TECHNICAL JOURNAL
the oscillator splits equally in paths AD and AB. The half in AD is com
pletely absorbed in the perfect termination Z' . The half in AB is partly
reflected from the impedance mismatch due to the waveguide coupling BE
and the remainder is absorbed in the perfect termination Z. Again due to
the properties of the perfect hybrid junction, the impedance presented by
the arm B when arms A and C are perfectly terminated is also perfect,
and the reflected energy from waveguide coupling BE splits equally in
paths BA and BC. The part in BA is absorbed by the oscillator. The
part in BC representing the voltage reflected from the coupling BE is meas
ured by the calibrated detector and indicator. The magnitude of the inci
dent voltage may be measured when the waveguide piece EF is replaced
by the fixed shorting plate.
It is convenient to measure voltages applied to the calibrated detector
and indicator in terms of attenuator settmgs in db for a reference output
indicator reading. Then the ratio expressed in db between incident and
reflected voltages (hereafter called W) is
W2 (due to the coupling BE) = Ai  A2 (1)
where Ai is attenuator setting for incident voltage and A2 is attenuator
setting for reflected voltage.
Both reflection coefficient and standing wave ratio may be expressed in
terms of 11'. For if
X = voltage due to incident power (2)
and Y = voltage due to reflected power, (3)
Y
then reflection coefficient = — (4)
and voltage standing wave ratio = .—^. p—  (5)
Since Widb) = 20 logio .^l (6)
W
1 + antilog —
then in db, standing wave ratio = 20 logio (7)
W
1 + antilog —
Standing wave ratio plotted versus W is shown in Fig. 2. Reflection coeffi
cient versus W can be found in any "voltage ratios to db" table.
II. Termination Z Imperfect, Termination Z' Perfect, Only One
Coupling on Hybrid Junction
In Fig. 1, if the termination Z is not perfect, there will be two reflected
voltages from branch B. The vector diagram of the voltage at C might be
MEASUREMENT OF IMPEDANCE MISMATCHES
449
represented as in Fig. 3, where vector 01 represents the voltage reflected
from couphng BE and vector 12 represents the voltage reflected from the
termination Z. To make measurements, termination Z should be movable
40
30 40
W IN DB
Fig. 2 — Standing wave ratio (SWR) versus W.
and the magnitude of its reflection coefiicient be the same at a given position
of rest for either direction of approach, and be the same for positions of rest
over an interval of a half a w^avelength in waveguide.
The reflected voltage is measured twice, once for minimum output as the
position of the termination Z is adjusted and again for maximum output.
Then
Fn^in = Fb  F. and V^^ = Vb+V, (8)
450 BELL SYSTEM TECHNICAL JOURNAL
where Vb is voltage reflected from coupling BE and V ^ is voltage reflected
from termination Z.
Equations (8) can be solved for Vh and V ^ for
V 4 V ■ V — V
F' max 1^ ' mm i t' ' max ' mm /rvN
5 = and V^ = (.9)
The incident voltage is measured as before. Therefore, using equation (6)
W = 20 log 1^1 and W" = 20 log L^' (10)
where W is due to coupling BE, W" is due to termination Z and Va is
incident voltage.
^
*•""
"^"^^
y
^v
•
/ N
/
/ \
/
/ \
/
/ \
/
1
1
/ 1
\
\
'l
1
/
/
\
/
\
/
\
/
\
y
Fig. 3 — Vector diagram of voltages reflected from coupling BE and termination Z.
A more practical solution involving only addition, subtraction and the
use of the characteristics in Fig. 4 is now presented. The settings of
the detector attenuator for incident voltage, minimum output and maxi
mum output might be yli , Az and Ai^ .
Then Wz = ^i  ^3 and 1^4 = A^ A,
(11)
But Wz = 20 log j^° and W, = 20 log , „  ^ , ° „ .
(12)
T
and WzW, = 20 log ' 7 ^ '/ = 20 log %
 1 + antilog 2Q
(13)
where 20 log 'y^' = T = W"  W (14)
MEASUREMENT OF IMPEDANCE MISMATCHES
451
40
20
1
0.8
O 0.6
o
0.4
0.2
0.1
o.oe
0.0 6
0.04
0.02
0.01
i
V
i
\
\
\
\.
>
L >.
V s.
\,
\ ^
s.
\
\
\
T
\,\
I
\
1+ ANTILOG ^
\
\
S^a ■
1 + ANTILOG ^
1
^
\^2 20 LOG
1
1
ANTILOG
>x
^ 20
N
s.
SS.
\,
NS.
\,
^
\
V
N
^
\
\
\
^
K
\
r, = 20L0G
1
\
\
' T
ANTILOG ^
X
^
\
V
\,
\
v
\
\
V
V
\,
\
\
N
\
\
\
\
\
10 20 30 40 50
TIN DB
Fig. 4— F, , Fi and Wz  W^
There is an Fi{T) = 20 log /l + \ \
antilog
20>
and an FiiT) = 20 log
1 
1
antilog
20
such that W = Wi + Fi= W^  F^
W" = T+W, + F,= T+ W^  F2
and Fi\ F2= W3  Wi
60
70
(15)
(16)
452
BELL SYSTEM TECHNICAL JOURNAL
Figure 4 shows Fi , F2 and their sum TT'3 — TI'4 plotted versus T. It may
be noted that Wz — Tr4 versus T has the same values as SWR versus W
in Fig. 2.
Using equations (16) and Fig. 4, TI'' and W" may be evaluated for the
particular values of Ws and Wi in equation (11). In the evaluation, if
there is uncertainty as to which reflection coefficient belongs to the wave
guide coupling BE and which belongs to the termination /., a termination
with a different magnitude of reflection coefficient should be used and the
technique repeated. The reflection coefficient which is the same in the
two cases is of course that due to the waveguide coupling BE.
SIGNAL
OSCILLATOR
A~ p
TERMINATION
Z
MOVABLE
SHORTING
PISTON
1 1
— 1 1—
VARIABLE
ATTENUATOR
1 1
HYBRID
JUNCTION
^11 ''l 1 "1
vw
' 'e ' 'g '
_ _C
CALIBRATED
DETECTOR &
INDICATOR
Fig. 5 — Block schematic for cases III and IV.
It is assumed in the above solution that multiple reflections between
the two impedance mismatches are inconsequential. Appendix A outlines
a procedure for evaluating the maximum probable error due to multiple
reflections.
III. Termination Z Perfect, Four Couplings on Hybrid Junction
In this case the setup might be as shown in Fig. 5. This setup differs
from that shown in Fig. 1 in that the hybrid junction has four couplings
shown, termination Z' has been replaced by a variable attenuator and a
movable shorting piston, and the waveguide coupling FG is to be measured
instead of coupling BE. The hybrid junction and the termination Z are
assumed to be perfect as defined for case I.
Since it is the object of the measuring method to measure impedance
mismatches in branch B, it is desirable to make the voltage at C depend only
on power reflected from branch B. This is accomplished by adjusting
MEASUREMENT OF IMPEDANCE MISMATCHES 453
branch D so that the voltages due to the flanges of the hybrid junction are
cancelled.
The vector diagram of the voltage at C might be represented as in Fig. 6.
Vector 01 represents the voltage at C when input is applied to A , due to
the impedance mismatch at the coupling BE. Vector 12 represents that
due to the mismatch at coupling D. Vector 23 represents that due to
the mismatch at the variable attenuator, (which will usually change in
magnitude and probably in phase for different settings). Vector 30 repre
sents the voltage at C due to the cancelling voltage from the branch D.
Its phase can be varied by changing the position of the movable shorting
piston. Its magnitude can be varied by changing the setting of the variable
attenuator. When the adjustment is accomplished effectively no power
reaches the detector. It is necessary that the reflection coefficients of
Fig. 6 — Vector diagram of voltages at terminal C.
couplings A, B, and C be small so that multiple reflections caused by them
will not affect the accuracy of measurement.
The reflected power from coupling FG may be measured when wave
guide GH is connected to waveguide EF as shown in Fig. 5 and termination
Z is located within waveguide GH. The detector attenuator setting might
he A5 . The incident power may be measured as before when termination
Z is withdrawn from the waveguide EF and the piece of waveguide GH is
replaced by a fixed shorting plate.
Wi, (due to reflection coefficient of the coupling FG) = Ai — A^ (17)
IV. Termination Z Imperfect, Four Couplings on Hybrid Junction
In Fig. 5 if the movable termination Z is not perfect, there will be two
reflected voltages in branch B when the adjustment is being made. The
vector diagram of the voltage at C might be as in Fig. 7. This is the same
as Fig. 6 except that a new vector 05 represents the voltage due to the
mismatch of the movable termination Z. The adjustment is accomplished
the same as in the last section except that the criterion is to have no change
in detector output as the movable termination Z is moved axially over a
454 BeIl system TECHNICAL JOURNAL
range of a half a wavelength in waveguide. As for the last case it is neces
sary that the reflection coefficients of the couplings A, B and C be small if
good accuracy is desired.
When measuring the coupling FG the procedure and evaluation are the
same as for case II.
Part of a laboratory setup as used at about 4 kilomegacycles is shown in
Fig. 8. It includes a hybrid junction, a variable attenuator, a movable
shorting piston, a straight section of waveguide and a movable termination
which consists of a cylinder of phenol resin and carbon with a tapered section
at one end. It is mounted in a phenolic block so that it may be moved
axially in the wave guide.
Fig. 7 — Part of a laboratory setup as used at 4 kilomegacycles.
In cases III and IV if the hybrid junction has "poor balance" so that
voltage appears at C when input is applied to arm A even though B and D
are perfectly terminated, the adjusting procedure will cancel this voltage
as well. Measuring accuracy will not be impaired provided the other
assumptions are fulfilled.
Measuring TI'— A Fitting Which Does Not Admit of Measuring
Each End Separately
A piece with a configuration unsuited to the preceding technique may be
measured by connecting it between two straight pieces of waveguide such
as between flanges F and G in Fig. 5. The IT due to the vector sum of
the reflection coefficients of the coupling at one end, any irregularities and
the coupling at the other end, is measured. Due to the distance between
the mismatches, the vector sum will vary over the frequency band of
interest.
MEASUREMENT OF IMPEDANCE MISMATCHES
455
m.
r:
IP
o
456 BELL SYSTEM TECHNICAL JOURNAL
Accuracy
There are three important sources of error. The first is lack of proper
adjustment. The second is that due to the detector attenuator calibration.
The third is that due to multiple reflections.
Experience and care can almost eliminate the first source. The second
source may have a magnitude of twice the detector attenuator calibration
error. In equations (1) and (17) this is readily apparent. The evaluation
of W using equations (16) introduces negligibly more error provided IFs — Wi
is made large by proper choice of the magnitude of the reflection coefficient
of the termination Z. The possible errors due to multiple reflections be
tween the waveguide impedance discontinuity being measured and an
imperfect termination are discussed in Appendix A. If the impedance
presented by the arm B of the hybrid junction is not perfect, energ>^ re
flected from the hybrid junction will be partly absorbed in the termination
and cause an error in the measurement. If the magnitude of this reflection
coefficient is known, the maximum error may be computed.
If a detector attenuator calibration error of ±0.1 db is assumed to be the
only contributing error, it is possible to measure the W due to an impedance
mismatch to an accuracy of ±0.2 db provided the W is greater than 26 db.
These numbers correspond to measuring a standing wave ratio of any value
less than 0.86 db to an accuracy of ±0.02 db or reflection coefficients of any
value less than 0.05 to an accuracy of ±2.5%.
APPENDIX A
Maximum Probable Error Due to Magnitude of Reflection
Coefficient Being Measured When Measuring a
Waveguide Coupling
The purpose of this appendix is to derive equations so that the maximum
probable error due to multiple reflections may be calculated. The assump
tions may not be rigorous but the mathematical treatment appears to
represent a reasonable approximation. It is assumed that there is no dissi
pation in waveguide EF, waveguide GH and in coupling FG.
The electrical relations of the coupling FG and the movable termination
Z might be represented as in Fig. 9, where Ka = characteristic impedance
of waveguide EF and Kh = characteristic impedance of waveguide GIL
The first few multiple reflections from the two discontinuities, coupling
FG and termination Z, can be illustrated as in Fig. 10.
Evaluation of the magnitudes of the reflections can be accomplished as
outlined in paragraph 7.13, page 210 in the book "Electromagnetic Waves"*
by S. A. Schelkunoff.
* Published by D. Van Nostrand, Inc., New York City, 1943.
MEASUREMENT OF IMPEDANCE MISMATCHES
457
FG
Z
< >
^^b
wv
Fig. 9 — Relation between coupling FG and termination Z.
Vq
^c
Vb
^f
^h
^g
Vci
Vn
Vp
^k
h
where r —
Fig. 10 — Multiple reflections from two planes of discontinuity.
Va = Incident voltage
Vb = rVa
Kb — Ka
Kb + Ka
Vo= Va\ Vb^ Vail + r)
V,= e~'^''Vc = e'^' Vail + r)
Ve = zVd = ze'^" Vail + r)
where z =
Z  Zb
Z + Zb
i2^L
Ve = ze^'" Vail + r)
Vn = —rVf = ze
i2pL
F„(l + r)ir)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
458 BELL SYSTEM TECHNICAL JOURNAL
^^''' ' = KTTKb (27)
Vh = Vf+V„ = 2^'=^^^ Va(l + r)(l  r) (28)
V, = e<P^V, = ze''^'V.{l + r){r) (29)
F„.  zV, = z'^e''^'^ Va(l + /•)(;) (30)
F„ == e'"^^7„. = sV'^^ F.(l + r){r) (31)
Kp = rVn = 2^r''^^F„(l + r)(r)'^ (32)
V, = V„+ Vp= 2V'*^^Fa(l  0(r) (33)
For purposes of analysis it is now assumed that further multiple reflections
are negligible.
13?
Fig. 11 — Vector voltage diagram for maximum vector sum.
3 2 1
* « ^
Fig. 12 — Vector voltage diagram for minimum vector sum.
Equations (19), (28) and (33) are the reflected voltages that combine
vectorially to be measured. If ^L = 0, 7r, 2x , ■ • • nw then the vector
voltage diagram might appear as in Fig. 11. If BL =, — , — , • • •
— then the vector voltage diagram might appear as in Fig. 12.
The followmg example illustrates the calculations involved in computing
the errors due to the magnitude of the reflection coefiicient being measured.
The assumptions are such that an appreciable error is computed. If one
assumes r = 0.316 and z = 0.282, then from equation (6) TIV = 10 db
and T^. = 11 db. In Figs. 11 and 12,
vector 01 = r, vector 12 = z(l — r'), vector 23 = rs(l — r) (34)
then
TFo_i = 10 db, IFi2 = 11.00 + 0.92 = 11.92 db,
and IFos = 10.00 +22.00 + 0.92 = 32.92 db (35)
In order to evaluate vector 02 in Fig. 11 (the vector sum of vectors 01
and 12), one calculates their difference T.
T = 11.92  10.00 = 1.92 db (36)
For T = 1.92 db, /'i = 5.10 db (37)
therefore W02 = 10.00  5.10 = 4.90 db (38)
MEASUREMENT OF IMPEDANCE MISMATCHES 459
In order to evaluate vector 03 in Fig. 11 (the vector difference of vectors
02 and 23) one calculates their difference T.
T = 32.92  4.90 = 28.02 db (39)
For T = 28.02 db, F^ = 0.36 db (40)
therefore TFo3 = 4.90 dz 0.36 = 5.26 db = TF4 (41)
In order to evaluate vector 02 in Fig. 12 (the vector difference between
vectors 01 and 12), one uses T from equation (36).
For T = 1.92 db, Fo = 14.10 db (42)
therefore I['o2 = 10.00 + 14.10 = 24.10 db (43)
In order to evaluate vector 03 in Fig. 12 (the vector difference between
vectors 02 and 23), one calculates their difference T.
T = 32.92  24.10 = 8.82 db (44)
For T = 8.82 db, F. = 3.93 db (45)
therefore TF03 = 24.10 + 3.93 = 2S.03db = IF3 (46)
Using equation (16)
WsWi= 22.77 db, r = 1.24 db, Fi = 5.40 and therefore W = 9.66 db.
Since we started by assuming Wr = 10 db, the error amounts to 0.34 db.
References
1. Page 120, "Transmission Networks and Wave Filters," T. E. Shea. Published by D.
Van Nostrand, Inc., New York City, 1929.
2. "Hyperfrequency Waveguides — General Considerations and Experimental Results,"
G. C. Southworth, Bell System Technical Journal, April, 1936.
3. "Directional Couplers." W. W. Mumford, Proceedings of the Institute of Radio Engineers,
Februar>' 1947.
4. "Hybrid Circuits for Microwaves," W. A. Tyrrell. A paper accepted for publication
in the Proceedings of the Institute of Radio Engineers.
5. "Note on a Reflection Coefficient RIeter," Nathaniel I. Korman, Proceedings of the
InstiliUe of Radio Engineers and Waves and Electrons, September 1946.
6. "Probe Error in StandingWave Detectors," William Altar, F. B. Marshall and L. P.
Hunter, Proceedings of the Institute of Radio Engineers and Waves afid Electrons,
January 1946.
7. Pages 20 to 24, "Practical Analysis of UHF Transmission Lines — Resonant
Sections — Resonant Cavities — Waveguides," J. R. Meagher and H. J. Markley
Pamphlet published by R. C. A. Service Company, Inc., in 1943.
8. "Microwave Measurements and Test Equipments," F. J. Gaffney, Proceedings of the
Institute of Radio Engineers and Waves and Electrons, October 1946.
Reflex Oscillators
By J. R. PIERCE and W. G. SHEPHERD
Table of Contents
I. Introduction 463
II. Electronic Admittance — Simple Theory 467
III. Power Production for Drift Angle of (m + f) Cycles 470
IV. Effect of Aiijiroximations 479
V. Special Drift Fields 480
VI. Electronic Gap Loading 482
VII. Electronic Tuning — Arbitrary Drift Angle 484
VIII. Hysteresis 493
IX. Effect of Load 512
A. Fixed Loads 513
B. Frequency Sensitive Loads — Long Line Effect 523
C. Effect of Short Mismatched Lines on Electronic Tuning 531
X. Variation of Power and Electronic Tuning with Frequency 537
XI. Noise Sidebands 542
XII. Buildup of Oscillation 545
XIII. Reflex Oscillator Development at the Bell Telephone Laboratories 550
A. Discussion of the Beating Oscillator Problem 550
B. A Reflex Oscillator with an External Resonator — The 707. . 553
C. A Reflex Oscillator with an Integral Cavity — The 723 558
D. A Reflex Oscillator Designed to Eliminate Hvsteresis — The 2K2^ 563
E. Broad Band Oscillators— The 2K25 '. 570
F. Thermally Tuned Reflex Oscillators— The 2K45 577
G. An Oscillator with WaveGuide Output— The 2K50 597
H. A Millimeter— Range Oscillator— The 1464 603
I. Oscillators for Pulsed Applications— The 2K23 and 2K54 607
J. Scope of Development at the Bell Telephone Laboratories 620
Appendices
I. Resonators 622
II. Modulation Coeflficient 629
HI. Approximate Treatment of Bunching 639
IV. Drift Angle as a Function of Frequency and Voltage 643
V. Electronic Admittance — Nonsimple Theory 644
VI. General Potential Variation in the Drift Space 656
VII. Ideal Drift Field 660
VIII. Electronic Gap Loading 663
IX. Losses in Grids 673
X. Starting of Pulsed Reflex Oscillators 674
XI. Thermal Tuning 677
Symbols
A A measure of frequency deviation (9.20).
B Bandwidth (Appendix 10 only, ij3)).
B Susceptance
Bi Reduced susceptance (9.7).
Be Electronic susceptance.
C Capacitance
C Heat capacity (A'l).
D Reduced gap spacing (10.3).
460
REFLEX OSCILLA TORS 461
Ea Retarding field in drift space.
F Drift effectiveness factor (5.4).
G Conductance
Gi, G2 Reduced conductances (9.6), (9.12).
Ge Gap conductance of loaded resonator.
G< Electronic conductance.
Gl Conductance at gap due to load.
Gr Conductance at gap due to resonator loss.
H Efficiency parameter (3.7).
Hm Maximum value of // for a given resonator loss.
/ Radiofrequency current.
h Current induced in circuit by convection current returning across gap.
h DC beam current.
A' Resonator loss parameter (3.9).
A" Radiation loss in watts/(degree Kelvin)'' {k2).
L Inductance.
M Characteristic admittance (a8).
Ml Characteristic admittance of line.
Ml/ Short line admittance parameter (9.38).
N Drift time in cycles.
N Length of line in wavelengths (Section IX only).
N Transformer voltage ratio.
P Power.
Q Equation (a 10).
Qe External (? (a11).
Qo Unloaded Q (a12).
R Surface resistance (o2).
5 Scaling factor (9.17).
T Temperature.
V Radiofrequency voltage.
V Potential in drift space (Appendix VI only).
I'o DC beam voltage at gap.
Vr The repeller is at a potential (— I'r) with respect to the cathode.
W Work, energy (Appendix I).
W Reduced radian frequency (10.5).
A' Bunching parameter (2.9).
V Admittance.
Yc Circuit admittance.
I\ Electronic admittance.
Y L Load admittance.
Y R Resonator admittance.
Z Impedance.
Zl Load impedance.
a Distance between grid wire centers.
d Separation between grid planes or tubes forming gap.
e Electronic charge (1.59 X 10''^ Coulombs).
/ Frequency.
/ Factor relating to effective grid voltage (b37).
i Radiofrequency convection current.
72 Radiofrequency convection current returning across gap (c4).
{12) f Fundamental component of /•> .
j V1
k Boltzman's constant (1.37 X 10^^ joules/degree A).
k Conduction loss watts/degree C (yfe14).
/ Length.
m Mutual inductance.
;« Electronic mass (9.03 x 10"' gram sevens).
n Repeller mode number. The number of cycles drift is n { } for "optimum
drift".
p Reduced power (9.5).
r Radius of grid wire, radius of tubes forming gap.
t Time, seconds.
Uf, DC velocity of electrons.
462 BELL SYSTEM TECHNICAL JOURNAL
v Total velocity (A]:)pen(li.\ VIII only).
V Instantaneous gap voltage
'ii' Real part of frequency (12.1).
X Coordinate along heani.
y :\ rectangular coordinate normal to .v.
_v Half separation of planes forming s\mmelrical gap.
3'c Magnitude of small signal electronic admittance.
z A rectangular coordinate normal to x (Appendix II).
c A variable of integration (Appendix VI).
a Negative coeflicient of the imaginary part of frequency (12.1).
/3 Modulation coeflicient.
/3o Average value of modulation coefScient.
/3o Modulation coeflicient on axis.
/3r Modulation coefficient at radius r from axis.
0s Root mean squated value of modulation coefficient.
/3y Modulation coeflicient at distance y from axis.
7 7 = oi/iio.
e Dielectric constant of space (8.85 x 10~" farads/cm).
6 Drift angle in radians.
6g Gap transit angle in radians.
X Wavelength in centimeters.
<!> A phase angle.
i Reduced potential (g13).
a Voltage standing wave ratio.
T Transit time.
T Time constant of thermal tuner.
TH Cycling time on heating.
Tc Cycling time on cooling.
\}/ Magnetic flux linkage.
w Radian frecjuency.
THE reflex oscillator is a form of longtransittime tube which has
distinct advantages as a low power source at high frequencies. It
may be light in weight, need have no magnetic focusing lield, and can be
made to operate at comparatively low voltages. A single closed resonator
is used, so that tuning is very simple. Because the whole resonator is at
the same dc voltage, high frequency bypass difhculties are obviated.
The frequency of oscillation can be changed by several tens of megacycles
by varying the repeller voltage ("electronic tuning") This is very ad
vantageous when the reflex oscillator is used as a beating oscillator. The
electronic tuning can be used as a vernier frequency adjustment to the
manual tuning adjustment or can be used to give an allelectrical autcmatic
frequencycontrol. Electronic tuning also makes reflex oscillators serve
well as frequency mcdulated sources in low power transmitters.
The single resonator tuning property makes it possible to construct (iscil
lators whose mechanical tuning is wholly electronically controlled. Such
control is achieved by making the mechanical motion which tunes the cavity
subject to the thermal e.xjiansion of an element heated by electron bom
bardment.
The efficiency of the reflex oscillator is generally low. The wide use of
the 707li, the 723A, the 726A and subsequent Western Electric tubes
shows that this defect is outweighed by the advantages already mentioned.
REFLEX OSCILLATORS 463
The first part of this paper attempts to give a broad exposition of the
theory of the reflex oscillator. This theoretical material provides a back
ground for understanding particular problems arising in reflex oscillator
design and operation. The second part of the paper describes a number
of typical tubes designed at the Bell Telephone Laboratories and endeavors
to show the relation between theory and practice.
The theoretical work is presented first because reflex oscillators vary so
widely in construction that theoretical results serve better than experi
mental results as a basis for generalization about their properties. While
the reflex oscillator is simple in the sense that some sort of theory about it
can be worked out, in practice there are many phenomena which are not
included in such a theory. This leaves one in some doubt as to how well
any simplified theory should apply. Multiple transits of electrons, different
drift times for different electron paths and space charge in the repeller
region are some factors not ordinarily taken into account which, neverthe
less, can be quite important. There are other effects which are difficult to
evaluate, such as distribution of current density in the beam, loss of elec
trons on grids or on the edges of apertures and dynamic focusing. If we
could provide a theory including all such known effects, we would have a
tremendous number of more or less adjustable constants, and it would not
be hard to fit a large body of data to such a theory, correct or incorrect.
At present it appears that the theory of reflex oscillators is important in
that it gives a semiquantitative insight into the behavior of reflex oscilla
tors and a guide to their design. The extent to which the present theory
or an extended theory will fit actual data in all respects remains to be seen.
The writers thus regard the theory presented here as a guide in evaluating
the capabilities of reflex oscillators, in designing such oscillators and in
understanding the properties of such tubes as are described in the second
part of this paper, rather than as an accurate quantitative tool. Therefore,
the exposition consists of a description of the theory of the reflex oscillator
and some simple calculations concerning it, with the more complicated
mathematical work relegated to a series of chapters called appendices.
It is hoped that this so organizes the mathematical work as to make it
assimilable and useful, and at the same time enables the casual reader to
obtain a clear idea of the scope of the theory.
I. Introduction
An idealized reflex oscillator is shown schematically in Fig. 1. It has,
of course, a resonant circuit or "resonator."^ This may consist of a pair of
grids forming the "capacitance" of the circuit and a single turn toroidal
1 For a discussion of resonators, see Appendix I. It is suggested that the reader consult
this before continuing with the main work in order to obtain an understanding of the circuit
philosophy used and a knowledge of the symliols employed.
464
BELL SYSTEM TECHNICAL JOURNAL
coil forming the "inductance" of the circuit. Such a resonator behaves
just as do other resonant circuits. Power may be derived from it by means
of a couphng looj) hnking the magnetic field of the single turn coil. An
electron stream of uniform current density leaves the cathode and is shot
across the "gajV' between the two grids, traversing the radiofrequency held
in this gaj) in a fraction of a cycle. In crossing the gaj) the electron stream
is velocity modulated; that is, electrons crossing at different times gain
ZERO —
EQUIPOTENTIAL
SURFACE
OUTPUT LINE
Fig. 1. — An idealized reflex oscillator with grids, shown in crosssection.
different amounts of kinetic energy from the radiofrequency voltage across
the gap." The velocity modulated electron stream is shot toward a negative
repeller electrode which sends it back across the gap. In the "drift space"
between the gap and the repeller the electron stream becomes "bunched"
and the bunches of electrons passing through the radio frequency lield in
the gap on the return transit can give up power to the circuit if they are
returned in the proper phase.
^ The most energy any electron gains is jiV electron volts, where V is the peak radio
frequency voltage across the gap and /3 is the "modulation coelTicicnt" or "gap factor",
and is always less than unity, /i dci)ends on gap configuration and transit angle across
the gap, and is discussed in Appendix II.
REFLEX OSCILLATORS
465
The vital features of the reflex oscillator are the bunching which takes
place in the velocity modulated electron stream in the retarding field be
tween the gap and the repeller and the control of the returning phase of the
bunches provided by the adjustment of the repeller voltage. The analogy
of Fig. 2 explains the cause of the bunching. The retarding drift field may
Fig. 2. — The motion of electrons in the repeller space of a reflex oscillator may be lik
ened to that of balls thrown upward at different times. In this figure, height is plotted
vs time. If a ball is thrown upward with a large velocity of I'l at a time Ti, another with
a smaller velocity at a later time To and a third with a still smaller velocity at a still later
time Ti the three balls can be made to fall back to the initial level at the same time.
be likened to the gravitational field of the earth . The drift time is analogous
to the time a ball thrown upwards takes to return. If the ball is thrown
upward with some medium speed Vo , it will return in some time /o . If it is
thrown upward with a low speed Vy smaller than ro , the ball will return in
some time /i smaller than /o . If the ball is thrown up with a speed ^2
greater than Vq , it returns in some time /o greater than /o . Now imagine
three balls thrown upward in succession, evenly spaced but with large,
466
BELL SYSTEM TECHNICAL JOURNAL
medium, and small velocities, respectively.^ As the ball first thrown up
takes a longer time to return than the second, and the third takes a shorter
time to return than the second, when the balls return the time intervals
between arrivals will be less than between their dei)artures. Thus time
position "bunching" occurs when the projection velocity with which a
uniform stream of particles enters a retarding iield is progressively decreased.
Figure 3 demonstrates such bunching as it actually takes place in the
retarding field of a reflex oscillator. :\n electron crossing the gaj) at phase A
RF VOLTAGE
ACCELERATING
FOR ELECTRONS
FROM CATHODE
RF VOLTAGE RETARDING
FOR ELECTRONS
FROM CATHODE
FOR ELECTRONS
RETURNING TOWARD
/ CATHODE \
T T
Fig. 3. — The drift time for transfer of energy from the bunched electron stream to the
resonator can be deduced from a plot of gap voltage vs time.
is equivalent to the first ball since its velocity suffers a maximum increase,
an electron crossing at phase B corresponds to the ball of velocity ^o where
for the electron Vq corresponds to the d.c. injection velocity, and finally an
electron crossing at j^hase C corresponds to the third ball since it has suffered
a maximum decrease in its velocity. The electrons tend to bunch about the
electron crossing at phase B. To a tirst order in this process no energy is
taken from the cavity since as many electrons give up energy as absorb it.
The next step in the process is to bring back the grou])ed electrons in
such a phase that they give the maximum energy to the r.f. field. Now,
f of a cycle after the gap voltage in a reflex oscillator such as that shown in
Fig. 1 is changing most rai)i(lly from accelerating to retarding for electrons
^ The reader is not advised to try this experimentally unless he has juggling experience.
REFLEX OSCILLATORS 467
going from the cathode, it has a maximum retarding value for electrons
leturning through the gap. This is true also for If cycles, 2f cycles, n + f
cycles. Hence as Fig. 3 shows if the time electrons spend in the drift space
is 11 + f cycles, the electron bunches will return at such time as to give up
energy to the resonator most effectively.
II. Electronic Admittance — Simple Theory
In Appendix III an approximate calculation is made of the fundamental
component of the current in the electron stream returning through the gap
of a reflex oscillator when the current is caused by velocity modulation
and drift action in a uniform retarding field. The restrictive assumptions
are as follows:
(1) The radiofrequency voltage across the gap is a small fraction of the
dc accelerating voltage.
(2) Space charge is neglected. Amongst other things this assumes no
interaction between incoming and outgoing streams and is probably the
most serious departure from the actual state of affairs.
(3) Variations of modulation coefficient for various electron paths are
neglected.
(4) All sidewise deflections are neglected.
(5) Thermal velocities are neglected.
(6) The electron flow is treated as a uniform distribution of charge.
(7) Only two gap transits are allowed.
An expression for the current induced in the circuit (/3 times the electron
convection current) is
(0Vd\ j{ut6)
i = 2h^J,[^^Je^'^''\ (2.1)
Here the current is taken as positive if the beam in its second transit across
the gap absorbs energy from the resonator. The voltage across the gap
at the time the stream returns referred to the same phase reference as the
current is v — Ve~^ "'"" ' . Hence the admittance appearing in shunt
with the gaps will be
_ 21 (,13 (^Vd\ ,((W2)9) (r. r.\
For small values of V approaching zero this becomes
_ h^'O j((,r/2)e) _ J((ir/2)9) /^ ,,
i es — r>[T ^ Jef^ \^Jj
ZV
468
BELL SYSTEM TECHNICAL JOURNAL
Here we define 1%, as the small signal value of the admittance, and }v
as the magnitude of this quantity. If we plot the function Yes in a complev
admittance plane it takes the form of a geometric spiral as shown in Fig. 4.
CONDUCTANCE, G
Fig. 4. — The negative of the circuit admittance (the heavy vertical line) and the small
signal electronic admittance (the spiral) are shown in a plot of susceptance vs conductance.
Each position along the circuit admittance line corresponds to a certain frequency. Each
position along the spiral corresponds to a certain drift angle.
Such a presentation is very helpful in acquiring a qualitative understanding
of the operation of a reflex oscillator.
In Appendi.x I it is shown that the admittance of the resonant circuit in
the neighborhood of resonance is very nearly
Vh = Gr + i2MAco/w
(2.5)
where Gr is a constant. The negative of such an admittance has been
plotted in Fig. 4 as the vertical line A'B'. Vertical positjon on this line is
REFLEX OSCILLATORS
469
proportional to the frequency at which the resonator is driven. The condi
tion for stable oscillation is
W + F, = 0.
(2.6)
I.U
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
■^
X
>
\
\
\
\
>
\
\
\
\
V
\
\
1.0 1.5 2.0 2.5 3.0
BUNCHING PARAMETER, X
Fig. 5. — Relative amplitude of electronic admittance vs the bunching parameter X
The bunching parameter increases linearly with radio frequency gap voltage so that this
curve shows the reduction in magnitude of electronic admittance with increasing voltage.
We may rewrite (2.2) for any given value of 6 as
where
F{V) =
2/i
(13 Vd)
(2Fo) _ 2/i(X)
2Fo
X
The quantity
X =
2Fo
(2.7)
(2.8)
(2.9)
is called the bunching parameter. A plot of the function F{V) vs X is
shown in Fig. 5. For any given value of 6 and for fixed operating conditions
470 BELL SYSTEM TECHNICAL JOURNAL
it is a function of V only and its action is clearly to reduce the small signal
value of the admittance until condition (2.6) is satistied. It will be observed
that this function affects the magnitude only and not the phase of the
admittance.
Thus, as indicated in Fig. 4, when oscillation starts the admittance is
given by the radius vector of magnitude jc , terminating on the spiral,
and as the oscillation builds up this vector shrinks until in accordance with
(12.6) it terminates on the circuitadmittance line A'B', which is the locus
of vectors (— Vr). The electronic admittance vector may be rotated by a
change in the repeller voltage which changes the value of 6. This changes
the vertical intercept on line A'B', and since the imaginary component of
the circuit admittance, that is the height along A'B', is proportional to
frequency, this means that the frequency of oscillation changes. It is this
property which is known as electronic tuning.
Oscillation will cease when the admittance vector has rotated to an angle
such that it terminates on the intersection of the spiral and the circuit
admittance line A'B'. It will be observed that the greater is the number of
cycles of drift the greater is the electronic tuning to extinction. \Miile it is
not as apparent from this diagram, it is also true that the greater the number
of cycles of drift the greater the electronic tuning to intermediate power
points. Vertical lines farther to the left correspond to heavier leads, and
from this it is apparent that the electronic tuning to extinction decreases
with the loading. By sufficient loading it is possible to prevent some repeller
modes (i.e. oscillations of some n values) from occurring. Since losses in
the resonant cavity of the oscillator represent some loading, some modes
of low n value will not occur even in the absence of external loading.
III. Power Production for Drift Angle of (« + ) Cycles
Now, from equation (2.2) it may be seen that Ye will be real and negative
for d = On = (n + 4)27r. Because 6 also appears in the argument of the
Bessel function this value of 6 is not exactly the value to make the real
component of Ye a maximum. However, for the reasonably large values
of n encountered in practical oscillators this is a justifiable approximation.
Suppose, then, we consider the case of n + f cycles drift, calling this an opti
mum drift time. Using the value of n as a parameter we plot the magni
tude of the radiofrequency electron current in the electron stream returning
across the gap given by equation (2.1) as a function of the radiofrequency
voltage across the gap. This variation is shown in Fig. 6. As might be
expected, the greater the number of cycles the electrons drift in the drift
space, the lower is the radiofrequency ga]) voltage required to ])r(){luce a
given amount of bunching and hence a given radio frequenc) electron
current. It may be seen from Fig. 6 that as the radiofrequency ga}) voltage
REFLEX OSCILLATORS
471
is increased, the radiofrequency electron current gradually increases until a
maximum value is reached, representing as complete bunching as is possible,
after which the current decreases with increasing gap voltage. The maxi
mum value of the current is approximately the same for various drift times,
but occurs at smaller gap voltages for longer drift times.
The radiofrequency power produced is the voltage times the current.
As the given maximum current occurs at higher voltages for shorter drift
POWER DISSIPATED X
CIRCUIT AND LOAD/
/
/
/
/
Q^S
RADIOFREQUENCY GAP VOLTAGE, V'
Fig. 6. — Radio frequency electron convection current / and the radio frecjuency power
given U]3 by the electron stream can be plotted vs the radio frequency gap voltage V for
various drift times measured in cycles. Maximum current occurs at higher voltage for
shorter drift times. For a given number of cycles drift, maximum power occurs at a
higher gap voltage than that for maximum current. If the power produced for a given
drift time is higher at low voltages than the power dissipated in the circuit and load
(dashed curve), the tube will oscillate and the amplitude will adjust itself to the point at
which the power dissipation and the power production curves cross.
times, the maximum power produced will be greater for shorter drift times.
This is clearly brought out in the plots of power vs. voltage shown in Fig. 6.
The power dissipated in the circuits and load will vary as the square cf
the radiofrequency voltage. Part of this power will go into the load coupled
with the circuit and part into unavoidable circuit losses. A typical curve
of power into the circuit and load vs. radiofrequency voltage is shown in
Fig. 6. Steady oscillation will take place at the voltage for which the power
production curve crosses the power dissipation curve. For instance, in
Fig. 6 the power dissipation curve crosses the power production curve for
472 BELL SYSTEM TECHNICAL JOURNAL
If cycles drift at the maximum or hump of the curve. This means that
the circuit impedance for the dissipation cur\'e shown is such as to result in
maximum production of power for If cycles drift. For 2f cycles drift and
for longer drifts, the power dissipation curve crosses the power production
curves to the right of the maximum and hence the particular circuit loading
shown does not result in maximum power production for these longer drift
times. This is an example of operation with lighter than optimum load.
The power dissipation curve might cross the power production curve to
the left of the maximum, representing a condition of too heavy loading for
production of maximum power output. The power dissipation curve in
Fig. 6 lies always above the power production curve for a drift of f cycles.
This means that the oscillator for which the curves are drawn, if loaded to
give the power dissipation curve shown, would not oscillate with the short
drift time of f cycles, corresponding to a very negative repeller voltage.
In general, the conclusions reached by examining Fig. 6 are borne out in
practice. The longer the drift time, that is, the less negative the repeller,
the lower is the power output. For very negative repeller voltages, how
ever, corresponding to very short drift times, the power either falls off.
which means that most of the available power is dissipated in circuit losses,
or the oscillator fails to operate at all because, for all gap voltages, the power
dissipated in circuit losses is greater than the power produced by the elec
tron stream.
Having examined the situation qualitatively, we want to make a some
what more quantitative investigation, and to take some account of circuit
losses. In the course of this we will find two parameters are very important.
One is the parameter X previously defined by equation (2.9), which ex
presses the amount of bunching the beam has undergone. In considering a
given tube with a given drift time, the important thing to remember about
X is that it is proportional to the rJ gap voltage V . For 6 = 6,, expression
(2.2) is a pure conductance and we can express the power produced by the
electron stream as one half the square of the peak rf voltage times the cir
cuit conductance which for stable oscillation is equal to the negative of the
electronic conductance given by (2.2). This may be written with the aid
of (2.9) as
. P = 2(hVo/en)XJ,(X). (3.1)
Suppose we take into account the resonator losses but not the power lost
in the output circuit, which in a well designed oscillator should be small.
If the resonator has a shunt resonant conductance (including electronic
loading) of Gr , the power dissipated in the resonator is
P, = V'Gr/2. (3.2)
REFLEX OSCILLATORS 473
Then the power output for dn is
P = 2(/oFo/0„)X/i(X)  V^Ga/2. {3,.i)
The efficiency, 77, is given by
P 2
■n =
From (2.4) and (2.9)
Hence we may write
V =
P
DC
^^^'W"^] (^^^
'i^ = ~ X'. (3.5)
2/0 Fn ye
^)lH«ff]
(w + 3/4)
(3.6)
TT
Let us write r] = ~ where N = (» + f). We may now make a generalized
examination of the effect of losses on the efficiency by examining the function
H = (l/7r)[AVx(X)  iG,/ye)Xy2]. (3.7)
Thus, the efficiency for 6 =0„ is inversely proportional to the number of
cycles drift and is propotional to a factor H which is a function of X and
of the ratio Gnlye , that is, the ratio of resonator loss conductance to small
signal electronic conductance.'* For w + f cycles drift, the small signal
electronic conductance is equal to the small signal electronic admittance.
For a given value of Gn/ye there is an optimum value of X for which H
has a maximum Hm ■ ^^'e can obtain this by differentiating (3.7) with
respect to A^ and setting the derivative equal to zero, giving
XJo(X)  (Gn/ye)X =
(3.8)
Jo{X) = (G,/ye).
If we put values from this into (3.7) we can obtain Hm as a function of
Gnhe ■ This is plotted in Fig. 7. The considerable loss of efficiency for
values of Gn/ye as low as .1 or .2 is noteworthy. It is also interesting to
note that for Gnlye equal to \, the fractional change in power is equal to
the fractional change in resonator resistance, and for Gs/ye greater than \,
the fractional power change is greater than the fractional change in resonator
resistance. This helps to explain the fall in power after turnon in some
tubes, for an increase in temperature can increase resonator resistance
considerably.
^ An electronic damping term discussed in Appendix VIII should be included in resona
tor losses. The electrical loss in grids is discussed briefly in Appendix IX.
474 BELL SYSTEM TECHNICAL JOURNAL
In the expression for the admittance, the drift angle, 6, appears as a fac
tor. This factor plays a double role in that it determines the phase of the
admittance but also in a completely independent manner it determines,
in part, the magnitude of the admittance. 6 as it has appeared in the
foregoing analysis, which was developed on the basis of a linear retarding
field, is the actual drift angle in radians. As will be shown in a later section,
certain special repeller fields may give effective drift action for a given angle
greater than the same angle in a linear field. Such values of effective drift
angle may have fractional optimum values although the phase must still be
such as to give within the approximations we have been using a pure con
ductance at optimum. In order to generalize the following work we will
speak of an effective drift time in cycles, N e = FN, where N is the actual
drift time in cycles, n \ f , and F is the number of times this drift is more
effective than the drift in a linear field.
Suppose we have a tube of given /3^, 7o , Fo and resonator loss Gr and wish
to find the optimum effective drift time, FN, and determine the effect on the
efl&ciency of varying FN. It will be recalled that for very low losses we may
expect more power output the fewer the number of cycles drift. How
ever the resonator losses may cut heavily into the generated power, for
short drift angles. With short drift angles the optimum load conductance
becomes small compared to the loss conductance so that although the
generated power is high only a small fraction goes to the useful load. There
is, therefore, an optimum value which can be obtained using the data of
Fig. 7. We define a parameter
K = ^«G. (3.9)
which compares the resonator loss conductance with the small signal elec
C K
tronic admittance per radian of driftan gle. Then in terms of A', — = — .
Je B
Hence, for a fixed value of K, various values oi 6 = lirFN define values of
/^
— . When one uses these values in connection with Fig. 7 he determines
Je
the corresponding values of //„, and hence the efficiency, r] = — ^ . These
values of r] arc plotted against FN as in Fig. 8 with values of A' as a param
eter. In this })lot A' is a measure of the lossiness of the tube. The opil
mum drift angle for any degree of lossiness is evident as the maximum
of one of these curves.
The maximum power outputs in various repeller modes, « = 0, 1 etc.
and the repeller voltages for these various power outputs correspond to
discrete values of n and FN lying along a curve for a particular value of A'.
REFLEX OSCILLATORS
475
Thus, the curves illustrate the variation of power from mode to mode as the
repeller voltage is changed over a wide range.
Changes in resonator loss or differences in loss between individual tubes
of the same type correspond to passing from a curve for one value of K to a
curve for another value of K.
0.50
0.40
E
I
qT 0.10
o
t3 0.08
<
u.
>0.06
g 0.05
o
,7 0.04
LU
0.03
v
Hrr>
—

■^
\45°
'^"^ FN
^""«».„^^ \
\
^
\
^
S

\
V


\
\
\"
)
\
\
\
1
1
\
0.05
0.1
ye
Fig. 7. — Efficiency factor Hm vs the ratio of resonator loss conductance to the small
signal electronic admittance. Efficiency changes rapidly with load as the loss conductance
approaches in magnitude the small signal electronic admittance. The efficiency is in
versely proportional to the number of cjxles drift.
It will be observed from this that although, from an efficiency standpoint,
it is desirable to work at low values of drift time such low drift times lead
to an output strongly sensitive to changes in resonator losses.
Perhaps the most important question which the user of the oscillator may
ask with regard to power production for optimum drift is; what effect does
the external load have upon the performance? If we couple lightly to the
oscillator the rf voltage generated will be high but the power will not be
extracted. If we couple too heavily the voltage will be low, the beam will
not be efficiently modulated and the power output will be low. There is
476
BELL SYSTEAf TECBNICAL JOURNAL
apparently an optimum loading. Best output is not obtained when the
external load matches the generator impedai.ce as in the case of anamplifier.
6.0
5.5
Z 5.0
O
a.
ff4.5
Z
^4.0
o
2
UJ
G 3.5
IL
U.
UJ
2.0
1.5
1.0
0.5
\
\
\\
>lo''''" ye
\
\
\
s.
\
\
L KO
\
Os
\
\
\
\
•v
\
S,5
\^
\,
/
X
\
\
\^
\
^
^
/
.m^
V
J
/
^
/
15
~~
/
''
20
/
f
y
/
/
'^
/
r
/
^
123456789 10
EFFECTIVE DRIFT, FN, IN CYCLES PER SECOND
Fig. 8. — Efficiency in per cent vs the effective cycles drift for various values of a para
meter A" which is proportional to resonator loss. These curves indicate how the power
output differs for various repeller modes for a given loss. Optimum power operating
points will be represented l)y points along one of these curves. For a very low loss resona
tor, the power is highest for short drift times and decreases rapidly for higher repeller
modes. Where there is more loss, the power varies less rapidly from mode to mode.
We return to equation (3.7) for // and assume that we are given various
n
values of — . With these values as parameters we ask what variation in
efficiency may be expected as we vary the ratio of the load conductance,
C C
Gl, to the small signal admittance, y«. When 1 = 1 oscillation
ye ye
REFLEX OSCILLATORS
477
will just start and no power output will be obtained. We can state the
general condition for stable operation as 1% + Fc = 0, where Y c is the
vector sum of the load and circuit admittances. For the optimum drift
time this becomes
Gc
ye
2/i(X)
X
(3.10J
ye /
~^
V
Y
N
I
Y
\
\
/
0.^
—
^
s
\
\
\
/
V
\
\
f)
/
04
V
\
\
\
\
1/
/
N
\,
\
\
\
¥
\
\
\
\
0.1 0.2 0.3 0.4 0.5 6 0.7 0.8 0,9 10
ye
Fig. 9. — EiEciency parameter U vs the ratio of load conductance to the magnitude of
the small signal electronic admittance. Curves are for various ratios of resonator loss
conductance to small signal electronic admittance. The curves are of similar shapes and
indicate that the tube will cease oscillating {U = 0) when loaded by a conductance about
h' ice as large as that for optimum power.
where
ye
Gl + Gr
(3.11)
G c
Hence for a given value of — we may assume values for — between zero
ye ye
Q
and 1 — — and these in (3.11) will define values of X. These values of X
ye ^
substituted in (3.7) will define values of H which we then plot against the
assumed values for — , as in Fig. 9. Thus we have the desired function of
ye
the variation of efhciency factor against load.
478
BELL SYSTmr TkCSNlCAL JOURNAL
From the curves of Fig. 9 it can be seen that the maximum efficiency is
obtained when the external conductance is made equal to approximately
half the available small signal conductance; i.e. ^{je — Gr). This can
be seen more clearly in the i)l()t of Fig. 10. Equation (3.8) gives the condi
tion for maximum efficiency as
Gn
 /o(X).
Gl
ye
— ^y^
0.4
0.5 0.6
ye
Fig. 10. — The abscissa measures the fractional excess of electronic negative conductance
over resonator loss conductance. The ordinate is the load conductance as a fraction of
electronic negative conductance. The tube will go out of oscillation for a load conductance
such that the ordinate is equal to the abscissa. The load conductance for optimum power
output is given by the solid line. The dashed line represents a load conductance half as
great as that required to stop oscillation.
If we assume various values for — these define values of Xo which when
substituted in
^
Je
Gc
ye
Gr
ye
2/i(Zo)
Xn
 /o(Xo)
(3.12)
give the value of the external load for ojitimum power. We plot these data
against the available conductance
1 
Gr
= 1
MX)
(3.13;
as shown in Fig. 10.
In Fig. 10 there is also shown a line through the origin of slope 1/2.
It can thus be seen that the optimum load conductance is slightly less than
half the available small signal or starting conductance. This relation is
independent of the repeller mode, i.e. of the value FN. This does not mean
REFLEX OSCILLATORS
479
that the load conductance is independent of the mode, since we have ex
pressed all our conductances in terms of je , the small signal conductance,
and this of course depends on the mode. What it does say is that, regard
less of the mode, if the generator is coupled to the load conductance for
maximum output, then, if that conductance is slightly more than doubled
oscillation will stop. It is this fact which should be borne in mind by the
circuit designer. If greater margin of safety against "pull out" is desired
it can be obtained only at the sacrifice of eflficiency.
ye
I.U
0.9
0.8
^
^
^^
y^
0.7
0.6
^
y
y
y
0.5
y^
04
O.b
Gr
ye
Fig. 11. — The ratio of total circuit conductance for optimum power to small signal
electronic admittance, vs the ratio of resonator loss conductance to small signal electronic
admittance.
An equivalent plot for the data of Fig. 10, which will be of later use, is
shown in Fig. 11. This gives the value of — for best output for various
values of — .
ye
IV. Effect of Approximations
The analysis presented in Section II is misleading in some respects. For
instance, for a lossless resonator and N = \ cycles, the predicted efficiency
is 53%. However, our simple theory tells us that to get this efficiency, the
radiofrequency gap voltage V multiplied by the modulation coefficient /3
(that is, the energy change an electron suffers in passing the gap) is I.OI8V0 •
This means that (a) some electrons would be stopped and would not pass the
gap (b) many other electrons would not be able to pass the gap against a
retarding field after returning from drift region (c) some electrons would
480 BELL SYSTEM TECHNICAL JOURNAL
cross the gap so slowly that for them /3 would be very small and their effect
on the circuit would also be small (d) there might be considerable loading of
the resonator due to transit time effects in the gap. Of course, it is not
justifiable to apply the small signal theory in any event, since it was derived
on the assumption that /ST' is small compared with Fo .
In Appendix IV there is presented a treatment by R. M. Ryder of these
Laboratories in which it is not assumed that /3r«Fo . This work does
not, however, take into account variation of /3 with electron speed or the
possibility of electrons being turned back at the gap.
For drift angles of If cycles and greater, the results of Ryder's analysis
are almost indistinguishable from those given by the simple theory, as may
be seen by examining Figs. 128135 of the Appendix. His curves approach
the curves given by the simple theory for large values of n.
For small values of n, and particularly for f cycles drift, Ryder's work
shows that optimum power is obtained with a drift angle somewhat different
from n + f cycles. Also, Fig. 131 shows that the phase of the electronic
admittance actually varies somewhat with amplitude, and Fig. 130 shows
that its magnitude does not actually pass through zero as the amplitude is
increased.
The reader is also referred to a paper by A. E. Harrison.
The reader may feel at this point somewhat uneasy about application of
the theory to practice. In most practical reflex oscillators, however, the
value of w is 2 or greater, so that the theory should apply fairly well. There
are, however, so many accidental variables in practical tubes that it is well
to reiterate that the theory serves primarily as a guide, and one should not
expect quantitative agreement between experiment and theory. This will
be apparent in later sections, where in a few instances the writers have made
quantitative calculations.
V. Special Drift Fields
In the foregoing sections a theory for a reflex oscillator has been developed
on the assumption that the repeller field is a uniform retarding electrostatic
field. Such a situation rarely occurs in practice, partly because of the diffi
culty of achieving such a field and partly because such a field may not return
the electron stream in the manner desired. In an effort to get some in
formation concerning actual drift fields, we may extend the simple theory
already presented to include such fields by redefining X as
X = ^VFe/2Vo. (5.1)
Here the factor F is included. As defined in Section /// this is the factor
which relates the effectiveness of a given drift field in bunching a velocity
^ A. E. Harrison, "Graphical Methods for Analysis of Vrlocitv Modulation Bunching."
Proc. I.R.E., 33.1, pp. 2032, June 1945,
REFLEX OSCILLATORS 481
modulated electron stream with the bunching effectiveness of a field with
the same drift angle 6 but with a linear variation of potential with distance.
Suppose, for instance, that the variation of transit time, r, with energy
gained in crossing the gap V is for a given field
dr/dV (5.2)
and for a linear potential variation and the same drift angle
(dr/dV),. (5.3)
Then the factor F is defined as
F = (dT/dV)/(dT/dV),. (5.4)
In appendix V, F is evaluated in terms of the variation of potential with
distance.
The efficiency is dependent on the effectiveness of the drift action rather
than on the total number of cycles drift except of course for the phase re
quirements. Thus, for a nonlinear potential variation in the drift space
we should have instead of (3.7)
■n = H/FN. (5.5)
In the investigation of drift action, one procedure is to assume a given
drift field and try to evaluate the drift action. Another is to try to find a
field which will produce some desirable kind of drift action. As a matter
of fact, it IS easy to find the best possible drift field (from the point of view
of efficiency) under certain assumptions.
The derivation of the optimum drift field, which is given in appendix VH,
hinges on the fact that the time an electron takes to return depends only on
the speed with which it is injected into the drift field. Further, the varia
tion in modulation coefficient for electrons returning with different speeds
is neglected. With these provisos, the optimum drift field is found to be
one in which electrons passing the gap when the gap voltage is decelerating
take IT radians to return, and electrons which pass the gap when the voltage
is accelerating take lrr radians to return, as illustrated graphically in Fig.
136, Appendix VH. A graph of potential vs. distance from gap to achieve
such an ideal drift action is shown in Fig. 137 and the general appearance of
electrodes which would achieve such a potential distribution approximately
is shown in Fig. 138.
With such an ideal drift field, the efficiency of an oscillator with a lossless
resonator is
Vi = (2/7r)(/3F/Fo). (5.6)
482 BELL SYSTEM TECHNICAL JOURNAL
For a linear potential variation in the drift space, at the optimum rf gap
voltage, according to the approximate theory presented in Section III the
efficiency for a lossless resonator is
r? = (.520)(/3F/Fo). (5.7)
Comparing, we find an improvement in efficiency for the ideal drift tield in
the ratio
■m/r) = 1.23, (5.8)
or only about 20%. Thus, the linear drift field is quite effective. The
ideal drift field does have one advantage; the bunching is optimum for all
gap voltages or, for a given gap voltage, for all modulation coefficients since
ideallv an infinitesimal af voltage will change the transit time from tt to 27r
and completely bunch the beam. This should tend to make the efficiency
high despite variations in /3 over various parts of the electron flow. The
hmitation imposed by the fact that electrons cannot return across the gap
against a high voltage if they have been slowed up in their tirst transit across
the gap remains. '
This last mentioned limitation is subject to amelioration. In one type of
reflex oscillator which has been brought to our attention the electrons cross
the gap the first time in a region in which the modulation coefficient is small.
If the gap has mesh grids, a hole may be punched in the grids and a beam of
smaller diameter than the hole focussed through it. Then the beam may be
allowed to expand and recross a narrow portion of the gap, where the modula
tion coefficient is large. Thus, in the first crossing no electrons lose much
energy (because /3 is small) and in the second crossing all can cross the gap
where /3F is large and hence can give up a large portion of their energy^
\T. Electronic Gap Loading;
So far, attention has been concentrated largely on electronic phenomena
in the drift or repeller region. To the long transit time across the gap
there has been ascribed merely a reduction in the effect of the voltage on the
electron stream by the modulation coefficient /3. Actually, the long transit
across the gap can give rise to other effects.
One of the most obvious of these other effects is the production of an elec
tronic conductance across the gap. If it is positive, such a conductance
acts just as does the resonator loss conductance in reducing the power out
put. Petrie, Strachey and Wallis of Standard Telephones and Cables have
treated this matter in an interesting and rather general way. Their work,
in a slightly modified form, ap])cars in Ajipcndix \'III, to wliicli the reader
is referred for details.
REFLEX OSCILLATORS 483
The work tells us that, considering longitudinal iields only, the electron
flow produces a small signal conductance component across the gap
7 =  (6.2)
Here ^ is the modulation coefficient and Uo is the electron speed. 7o and
Vq are beam current and beam voltage. If the gap has a length d, the
transit angle across it is 6g = yd and (6.1) may be rewritten
It is interesting to compare this conductance with the magnitude of the
smallsignal electronic admittance, ye ■ In doing so, we should note that
the current crosses the gap twice, and on each crossing produces an elec
tronic conductance. Thus, the appropriate comparison between loss con
ductance and electronic admittance is IGehlje ■ Using (6.3) we obtain
Usually, the drift angle Q is much larger than the gap transit angle Qg .
Further, if we examine the curves for mcdulation coefficient /? which are
given in Appendix II, we find that {dl3^/ddg)/l3''^ will not be very large. Thus,
we conclude that in general the total loss conductance for longitudinal fields
will be small compared with the electronic admittance. An example in
Appendix VIII gives {IGehlj^ as about 1/10. It seems that this effect
will probably be less important than various errors in the theory of the reflex
oscillator.
Even though this electronic gap leading is not very large, it may be in
teresting to consider it further. We note, for instance, that the conductance
GeL is positive when jQ" decreases as gap transit angle increases. For paral
lel fine grids this is so from Qg = to ^^ = 27r (see Fig. 119 of Appendix II).
At Qg = Itt, where /3 = 0, dfS'^/ddg = 0, and the gap loading is zero. In a
region beyond dg — 2x, d^'^/ddg becomes positive and the gap conductance
is negative. Thus, for some transit angles a single gap can act to produce
oscillations. For still larger values of dg , Gcl alternates between positive
and negative. Gap transit angles of greater than lir are of course of little
interest in connection with reflex oscillators, as for such transit angles /3 is
very small.
For narrow gaps with large apertures rather than fine grids, d^^/ddg
484 BELL SYSTEM TECHNICAL JOURNAL
never becomes very negative and may remain positive and the gap loading
conductance due to longitudinal fields be always positive. In such gaps,
however, transverse fields can have important effects, and (6.3) no longer
gives the total gaj) conductance. Transverse fields act to throw electrons
approaching the gap outward or inward, into stronger or weaker longitudinal
tields, and in this manner the transverse felds can either cause the electrons
to give up part of their forward velocity, transferring energy to the reso
nator, or to pick up forward velocity, taking energy from the resonator.
An analysis of the effect of transverse fields is given in Appendix VIII, and
this is applied in calculating the total conductance, due both to longitudinal
and to transverse fields, of a short gap between cylinders with a uniform cur
rent density over the aperture. It is found that the transverse fields con
tribute a minor part of the total conductance, and that this contribution
may be either positive or negative, but that the total gap conductance is
always positive (see Appendix \TII, Fig. 140).
The electron flow across the gap produces a susceptive component of
admittance. This susceptive component is in general more difficult to cal
culate than the conductive component. It is not very important; it serves
to affect the frequency of oscillation sHghtly but not nearly so much as a
small change in repeller voltage.
Besides such direct gap loading, the velocity modulation and drift action
within a gap of fine grids actually produce a small bunching of the electron
stream. In other words, the electron stream leaving such a gap is not only
velocity modulated but it has a small density modulation as well. This
convection current will persist (if spacecharge debunching is not serious)
and, as the electrons return across the gap, it will constitute a source of elec
tronic admittance. We find however, that in typical cases (see Appendix
VIII, (h59)(h63)), this effect is small and is almost entirely absent in gaps
with coarse grids or large apertures.
Secondary electrons produced when beam electrons strike grid wires and
grid frames or gap edges constitute another source of gap loading. It has
been alleged that if the frames supporting the grids or the tubes forming a
gap have opposed parallel surfaces of width comparable to or larger than the
gap spacing, large electron currents can be produced through secondary
emission, the r/ field driving electrons back and forth between the opposed
surfaces. It would seem that this phenomenon could take place only at
quite high rf levels, for an electron would probably require of the order of
100 volts energy to produce more than one secondary in striking materials
of which gaps are usually constructed.
VH. Electronic Tuning — Arbitrary Drift Angle
So far, the "on tune" oscillation of reflex oscillators has been considered
except for a brief discussion in Section II, and we have had to deal only with
REFLEX OSCILLATORS 485
real admittances (conductar.ces). In this section the steady state operation
in the case of complex circuit and electronic admittances will be discussed.
The general condition for cscillaticn states that, breaking the circuit at any
point the sum of the admittances looking in the two directions is zero. Par
ticularly, the electronic admittance Ye looking from the circuit to the
electron stream, must be minus the circuit admittance Yc , looking from the
electron stream to the circuit. Here electronic admittance is used in the
sense of an admittance averaged over a cycle of oscillation and fulfilling the
above condition.
It is particularly useful to consider the junction of the electron stream
and the circuit because the electronic admittance Ye and the circuit admit
tance Yc have very different properties, and if conditions are considered
elsewhere these properties are somewhat mixed and full advantage cannot
be taken of their difference.
The average electronic admittance with which we are concerned is a
function chiefly of the amplitude of oscillation. Usually its magnitude
decreases with increasing ampUtude of oscillation, and its phase may vary
as well, although this is a large signal effect not shown by the simple theory.
In reflex oscillators the phase may be controlled by changing the repeller
voltage. The phase and magnitude of the electronic admittance also vary
with frequency. Usually, however, the rate of change with frequency is
slow compared with that of the circuit admittance in the vicinity of any one
resonant mode. By neglecting this change of electronic admittance with
frequency in the following work, and concentrating our attention on the
variation with amplitude and repeller voltage, we will emphasize the im
portant aspects without serious error. However, the variation of electronic
admittance with frequency should be kept in mind in considering behavior
over frequency ranges of several per cent.^
The circuit admittance is, of course, independent of amplitude and is a
rapidly varying function of frequency. It is partly dependent on what is
commonly thought of as the resonator or resonant circuit of the oscillator,
but is also profoundly affected by the load, which of course forms a part of
the circuit seen from the electron stream. The behavior of the oscillator is
determined, then, by the electronic admittance, the resonant circuit and
the load. The behavior due to circuit and load effects applies generally
to all oscillators, and the simplicity of behavior of the electronic admittance
is such that similarities of behavior are far more striking than differences.
We have seen from Appendix I that at a frequency Aw away from the
resonant frequency wo where Aw<<Ca;o , the admittance at the gap may be
expressed as:
Yc = Gc + i2MAa;/a;o. (7.1)
* Appendix IV discusses the variation of phase with frequency and repeller voltage.
The variation of phase of electronic admittance with frequency is included in Section IX A.
486
BELL SYSTEM TECHNICAL JOURNAL
Here the quantity M is the characteristic admittance of the resonator,
which dej^ends on resonator shape and is unaffected by scaling from one
frequency to another. Gc is the shunt conductance due to circuit and to
load. Ye as given by (7.1) represents to the degree of aj^proximation re
quired the admittance of an)^ resonant circuit and load with only one
resonance near the frequency of oscillation.
It is ])rofitable to consider again in more detail a complex admittance
plot similar to Fig. 4. In Fig. 12 the straight vertical line is a plot of (7.1).
Ye = ye(2J,(X)/x)eJ^Q
Ae
UJo = (LC)'/2
Y = G+j2MAuj/u)o
CONDUCTANCE, G *■
Fig. 12. — The resonator and its load can be represented as a shunt resonant circuit
with a shunt conductance G. For frequencies near resonance, the conductance is nearly
constant and the susceptance B is proportional to frequenc\', so that when susceptance is
plotted vs conductance, the admittance Y is a vertical straight line. The circles mark off
equal increments of frequency. The electronic admittance is little affected by frequency
but much affected by amplitude. Tne negative of an electronic admittance Y ^ having a
constant phase angle \6 is shown in the figure. The dots mark off equal amplitude steps.
Oscillation will occur at a fref[uency and amplitude specified by the intersection of the
curves Y and — Ye ■
The circles mark equal frequency increments. Now if we neglect the varia
tion of the electronic admittance with phase, then the negative of the small
signal electronic admittance on this same plot will be a vector, the Iccus of
whose termination will be a circle. The vector is shown in lig. 12. The
dots mark off admittance values corresponding to equal amplitude incre
ments as determined by the data of Fig. 5.
Steady oscillation will take place at the frequency and amplitude repre
sented by the intersection of the two curves. If the phase angle 16 of the
— Ye curve is varied by varying the repeller voltage, the point of intersection
will shift on both the I'c curve and the — !'« curve. 'I'hc shift along the
REFLEX OSCILLATORS 487
I'(. curve represents a change in frequency of oscillation; the shift along the
— Yc curve represents a change in the amplitude of oscillation. If we know
the variation of amplitude with position along the — 1% curve, and the varia
tion of frequency with position along the Y ,■ curve, we can obtain both the
amplitude and frequency of oscillation as a function of the phase of — 1% ,
which is in turn a function of repeller voltage.
From (2.3) and (2.7) we can write — Ye in terms of the deviation of drift
angle M from n + f cycles.
 Fe = yXlJ^)/Xy^\ (7.2)
The equation relating frequency and Ad can be written immediately from
inspection of Fig. 12.
2MAco/coo = Gc tan Ad
Aco/wo = {Gc/2M) tan A0 (7.3)
Aco/wo =  (1/2(3) tan M.
Here Q is the loaded Q of the circuit.
The maximum value of Ad for which oscillation can occur (at zero ampli
tude) is an important quantity. From Fig. 12 this value, called A^o , is
obviously given by
cosA^o = Gc/ye = {Gc/M)(M/ye) (7.4)
= (M/ye)/Q.
From this we obtain
tan A^o = ± {Q'(ye/My  1)\ (7.5)
By using (7.3) we obtain
(Aa,/coo)o = ± (h) iye/M) (1  {M/yeQYf (7.6)
or
(Aco/a;o)o  ±(§) (y./M) (1  {Gc/yeYf. {1.1)
These equations give the electronic tuning from maximum amplitude of
oscillation to zero amplitude of oscillation (extinction).
The equation relating amplitudes may be as easily derived from Fig. 12
Gl + (2MAa,/co)2 = y; {2J,{X)/xy (7.8)
at
Ao) = let X = Xo . Then
Aco/a'o = {ye/2M) {{2J,{X)/XY  {2J ,{X ,) / X ,Y)\ (7.9)
188 BELL SYSTEM TECHNICAL JOIRNAL
It is of interest to ha\'e the value of Aw wo at half the i)o\ver for Aw = 0.
At half power, X = A'o/\/2, so
(Ac., o;o)i = (:ye/2M)((2/i(Xo/V2)/(Xo/\/2))  {IJ ,{X ,) / X ,y)\ (7.10)
For given values of modulation coefficient and Fn , X is a function of the
rf gap voltage V and also of drift angle and hence of A0, or repeller voltage
(see Appendix IV). For the fairly large values of d typical of most reflex
oscillators, we can neglect the change in A^ due directly to changes in M,
and consider X as a direct measure of the rJ gap voltage V, Likewise
Ve is a function of drift time whose variation with A0 can and will be dis
regarded. Hence from (7.9) we can plot (X/A'o) vs. Aw/coo and regard this
as a representation of normalized power vs. frequency.
Let us consider now what (7.3) and (7.9) mean in connection with a given
reflex oscillator. Suppose we change the load. This will change Q in
(7.3) and A'o in (7.9). From the relationship previously obtained for the
condition for maximum power output, Gn/ye = /o(Xo), we can find the
value of A^o that is, A' at Ao; = 0, for various ratios of GrIj^ . For Gr — ^
(zero resonator loss) the optimum power value of A^o is 2.4. When there is
some resonator loss, the optimum total conductance for best power output
is greater and hence the optimum value of A^o is lower.
In Fig. 13 use is made of (7.3) Aw/wo in plotted vs. A0 (which decreases
as the repeller is made more negative) for several values of (), and in Fig. 14,
(7.9), is used to plot (A7.A0)" vs. (2M/ye)Aw/ajo , which is a generalized
electronic tuning variable, for several values of Xo . These curv^es illustrate
typical behavior of frequenc} vs. drift angle or repeller voltage and power
vs. frequency for a given reflex oscillator for various loads. In practice,
the S shape of the frequency vs. repeller voltage curves for light loads
(high Q) is particularly noticeable. The sharpening of the amplitude vs.
frequency curves for light loads is also noticeable, though of course the cusp
like appearance for zero load and resonator loss cannot be reproduced ex
perimentally. It is important to notice that while the plot of output vs.
frequency for zero load is sharp topped, the plot of output vs. repeller volt
age for zero load is not.
Having considered the general shape of frequency vs. repeller voltage
curves and power vs. frequency curves, it is interesting to consider curves of
electronic tuning to extinction ((Aa'/ao)o) and electronic tuning to half power
((Aw/coo)i) vs. the loading parameter, {MjyeQ) = Gdye . Such curves are
shown in Fig. 15. These curves can be obtained using (7.7) and (7.10).
In using (7.10) X can be related to Gdye by the relation previously derived
from 2J\iX)/X = Gc/ye and given in Fig. 5 as a function of A'. It is to
be noted that the tuning to the half power point, (Aoo/a'o)> , and the tuning
to the extinction point, (Aa)/coo)o , vary quite differently with loading.
REFLEX OSCILLATORS
489
10^ X4
\:
s
\
\ Ql = ioo
200 N^
V150
\
X
r>^
\
^^
^
^
;^.
•^^
\
:?^>
\
\
\
\
\
\
60 50 40 30 20 10 10 20 30 40 50 60
ANGLE, AG, IN DEGREES
Fig. 13. — A parameter proportional to electronic tuning plotted vs deviation from
optimum drift angle M for various values of loaded Q. For lower values of Q, the fre
quency varies rapidly and almost linearly with M. For high values of Q, the frequency
curve is S shaped and frequency varies slowly with A^ for small values of A5.
1.0
^
/
\
\
Xo = 2.40, (B = 0.43)
MAX. POWER WITH ZERO/'
RESONATOR LOSS/
.'/
;^
V
^.
x
y/.
Xo=
(t)
\^
Xo = 1.6
{^A
^\
I
\
1.0 0.8 0.6
0.4 0.2
/2M'\ Au
0.4 0.6
Fig. 14. — The relative power output vs a parameter proportional to the frequency
deviation caused by electronic tuning, for various values of load. For zero loss and zero
load, the curve is peaked. For zero loss and ojjtimum load, the curve has its greatest
width between half power points. For zero loss and greater than optimum load, the curve
is narrow.
490
BELL SYSTEM TECHNICAL JOURNAL
The quantity
(Aoj/coo) I
has a maximum value at Gc/yc = .433(X = 2.40), which is the condition
for maximum power output when tlie resonator loss is zero.
In Fig. 11 we have a plot of Gc/jc vs. GR/je for optimum loading (that
is loading to give maximum power for A0 = 0). This, combined with the
^
\
"v
\
....
\
/
^^
"^
"~^
^X^
K
\
.
\
\
\
N
\
\ \
\ \
\ \
\ \
\ \
\ \
\
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M _ Go_
yeQL ye
Fig. 15. — A parameter proportional to electronic tuning range vs the ratio of total
circuit conductance and small signal electronic admittance. The electronic tuning
to extinction (Aco/a)o)o is more affected by loading than the electronic tuning to half power
points (Aaj/wo)f .
curves of Fig. 15, enables us to draw curves in the case of optimum leading
for electronic tuning as a function of the resonator loss. Such curves are
shown in Fig. 16.
From Fig. 16 we see thai with optimum loading it takes very large reso
nator losses to affect the electronic tuning range to half power very much,
and that the electronic tuning range to extinction is considerably more
affected by resonator losses. Turning back to Fig. 7, we see that power is
affected even more profoundly by resonator losses. It is interesting to
REFLEX OSCILLATORS
491
compare the effect of going from zero less to a case in which the less con
ductance is \ of the small signal electronic conductance (Gr = ydT). The
table below shows the fraction to which the power cr efficiency, the elec
"^
^
(AO)]
^
V
\
\
~".
N
\
~'^^.
\
^
N \
N \
\ \
\ \
\ \
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M _ ^
yeQo ye
Fig. 16. — The effect of resonator loss on electronic tuning in an oscillator adjusted for
optimum power output at the center of the electronic tuning range. A parameter pro
portional to electronic tuning is plotted vs the ratio of the resonator loss to small signal
electronic admittance. The electronic tuning to extinction is more affected than the
electronic tuning to half power as the loss is changed.
tronic tuning range to extinction, and the electronic tuning range to half
power are reduced by this change.
Power, Efficiency (77)
.24
Electronic Tuning to Extinction
(Aw/a,o)o
.76
Electronic Tuning To Half Power
(Am/ojo) 1
From this table it is obvious that efforts to control the electronic tuning by
varying the ratio — are of dubious merit.
492
BELL SYSTEM TECHNICAL JOURNAL
One other quantity may be of some interest; that is the phase angle of
electronic tuning at half power and at extinction. We already have an
expression involving A^o (the value at extinction) in (7.4). By taking ad
vantage of (3.10) and (3.8) (F'igs. 5 and 11), we can obtain Ado vs. Gr/ye
70
50
30
LU 20
^
^^
^
^^
*^^^^
"\
^0
*^^
\
\,
2
"""
\
"'"H^
'X
x\
\
0.4 0.5 0.6
M _ Gr
Qoy ye
0.8
Fig. 17. — The phase of the drift angle for extinction and half power vs the ratio o
resonator loss to small signal electronic admittance.
{ = M/Qye) for optimum loading. By referring to Fig. 12 we can obtain the
relation for A6i (the value at half power)
Gc = ye [2Ji{Xo/V2)/{Xo/\/2)] cos A^j.
However, we have at A^ =
Gc = ye [2/,(A'o)/Xo].
Hence
cos AOi =
JiiXo)
V2MXo/V~2)
(7.11)
(7.12)
(7.13)
Again, from (3.10) and (3.8) we can express A'o for optimum j^ower at Ad =
in terms of Gc/ye • In Fig. 17, A^o and A6{ are plotted vs. Gc/ye for
optimum loading.
REFLEX OSCILLATORS
VIII. Hysteresis
493
All the analysis presented thus far would indicate that if a reflex oscillator
is properly coupled to a resistive load the power output and frequency will
be singlevalued functions of the drift time or of the repeller voltage, as
illustrated in Fig. 18. During the course of the development in these labora
tories of a reflex oscillator known as the 1349XQ, it was found that even if
NEGATIVE REPELLER VOLTAGE >■
Fig. 18. — Ideal variation of power and frequenc\ with repeller voltage, arbitrary units.
the oscillator were correctly terminated the characteristics departed vio
lently from the ideal, as illustrated in Fig. 19. Further investigation dis
closed that this departure was, to a greater or less degree, a general charac
teristic of all reflex oscillators in which no special steps had been taken to
prevent it.
The nature of this departure from expected behavior is that the output is
not a single valued function of the repeller voltage, but rather that at a given
repeller voltage the output depends upon the direction from which the repel
494
BELL SYSTEM TECHNICAL JOURNAL
ler voltage is made to approach the given voltage. Consider the case illus
trated in Fig. 19. The arrows indicate the direction of repeller voltage vari
ation. If we start from the middle of the characteristic and move toward
more negative values of repeller voltage, the amplitude of oscillation varies
continuously until a critical value is reached, at which a sudden decrease in
NEGATIVE REPELLER VOLTAGE »
Fig. 19.— A possible variation of power and frequency with repeller voltage when there
is electronic hysteresis. The arrows indicate the direction of variation of repeller voltage.
amplitude is observed. This drop may be to zero amplitude as shown or to a
finite amplitude. In the latter case the amplitude may again decrease con
tinuously as the repeller voltage is continuously varied to a new critical
value, where a second drop occurs, etc. until finally the output falls to zero.
In every observed case, even for more than one drop, the oscillation always
dropped to zero discontinuously. Upon retracing the repeller voltage varia
tion, oscillation does not restart at the repeller voltage at which it stopped
but remains zero until a less negative value is reached, at which point the
REFLEX OSCILLA TORS
495
oscillation jumps to a large amplitude on the normal curve and then varies
uniformly. The discontinuities occur sometimes at one end of the charac
teristic and sometimes at the other, and infrequently at both. It was first
thought that this behavior was caused by an improper load/ but further
investigation proved that the dependence on the load was secondary and
the conclusion was drawn and later verified that the effect had its origin in
the electron stream. For this reason the discontinuous behavior was called
electronic hysteresis.
In any selfexcited oscillator having a simple reasonant circuit, the os
cillating circuit may be represented schematically as shown in Fig. 20.
Here L and C represent the inductance and capacitance of the oscillator.
Gr is a shunt conductance, representing the losses of the circuit, and Gi is
the conductance of the load. Henceforth for the sake of convenience we
•Gr
Fig. 20. — Equivalent circuit of reflex oscillator consisting of the capacitance C, induct
ance L, the resonator loss conductance Gr, the load conductance G^ and the electronic
admittance W ■
will lump these and call the total Gl ■ Ye represents the admittance of the
electron stream. Such a circuit has a characteristic transient of the form
V = Voe"
(8.i:
where
Ge+Gi
2C
and
Vlc'
Oscillations will build up spontaneously if
Geo + Gi < . (8.2)
For stable oscillation at amplitude V we require
Ge[V] + Gi = (8.3)
(8.2) and (8.3) state that the amplitude of oscillation will build up until
nonlinearities in the electronic characteristics reduce the electronic con
ductance to a value equal and opposite to the total load plus circuit con
ductance. Thus, in general
Ye = G,o/'i(F) + jBeoF^iV) (8.4)
' See Section IX.
496 BELL SYSTFAf TECHNICAL JOURNAL
wliere
Ve = G.0 + jBrO (8.5)
is the admittance for vanishing; amplitude, wliicli is taken as a reference
value. The foregoing facts are familiar to an}' one who has worked with
oscillators.
Now, condition (8.,\) ma} be satisfied although (8.2) is not. Then an
oscillator will not be selfstarting, although once started at a sulTiciently
large amplitude its operation will become stable. An example in common
experience is a triode Class C oscillator with fixed grid bias. In such a case
■ F{Vi) > /<(()) (8.6)
holds for some Fi .
As an example of normal behavior, let us assume that F(V) is a continu
ous monotonically decreasing function of increasing V, with the reference
value of V taken as zero. Then the conductance, G> = G(oF{V) will vary
with V as shown in Fig. 21. Stable oscillation will occur when the ampli
tude Vi has built up to a value such that the electronic conductance curve
intersects the horizontal line representing the load conductance, Gi . G,o
is a function of one or more of the operating parameters such as the elec
tron current in the vacuum tube. If we vary any one of these parameters
indicated as X„ the principal effect will be to shrink the vertical ordinates
as shown in Fig. 21 and the amplitude of oscillation will assume a series
of stable values corresponding to the intercepts of the electronic conductance
curves with the load conductance. If, as we have assumed, F{V) is a
monotonically decreasing function of F, the amplitude will decrease con
tinuously to zero as we uniformly vary the parameter in such a direction as
to decrease Geo . Zero amplitude will, of course, occur when the curve has
shrunk to the case where Gco = Gl . Under these conditions the power
output, ^GlV, will be a single value function of the parameter as shown in
Fig. 22 and no hysteresis will occur.
Suppose, however, that F{V) is not a monotonically decreasing function of
V but instead has a maximum so that G,qF{V) appears as shown in Fig. 23.
In this case, if we start with the condition indicated by the solid line and
vary our parameter A' in such a direction as to shrink the curve, the ampli
tude will decrease smoothly until the parameter arrives at a value of A'5
corresponding to amplitude Fsat which the load line is tangent to the maxi
mum of the conductance curve. Further variation of A' in the same direc
tion will cause the amplitude to jump to zero. Upon reversing the direction
of the variation of the parameter, oscillation cannot restart until X arrives
at a value A'4 such that the zero amplitude conductance is equal to the load
conductance. When this occurs the amplitude will suddenly jump to the
REFLEX OSCILLATORS
497
, ,
^"\X
Ge = Geo (x) F M
QJ
<J)
^
111
U
z
^ N.
\
<
H
o
\
Z)
\
a
\^ \
z
\^ \
o
u
^ ^^\
\ \
o
^^^^^ >
V ^V \
z
^^^^
>v \ \ NEGATIVE OF LOAD
o
^\,
N. N^ \^ CONDUCTANCE, Gl
K
"^N.
^\. ^V \.
u
UJ
^^^
"^ N. \ \
_J
^s^ ^v ^v^ ^^
UJ
"^^^
AMPLITUDE OF OSCILLATION, V *
Fig. 21. — A possible variation of electronic conductance with amplitude of oscillation
for the general case of an oscillator. Arbitrary units are employed. Different curves
correspond to several values of a parameter A' which determines the small signal values of
the conductance. The load conductance is indicated by the horizontal line. Stable
oscillation for any given value of the parameter A' occurs at the intersection of the elec
tronic conductance curve with the load line Gl
Fig. 22.
21 apply.
BUNCHING PARAMETER, X *•
Variation of power output with the parameter X when the conditions of Fig.
498
BELL SYSTEM TECHNICAL JOURNAL
value Vi . Under these conditions the power output will appear as shown in
Fig. 24, in which the hysteresis is apparent.
Let us now consider the conditions obtaining in a reflex oscillator. Fig.
1 shows a schematic diagram of a reflex oscillator. This shows an electron
gun which projects a rectilinear electron stream across the gap of a resonator.
y^ N. Ge ^ Geo (X) F (V)
^^x^ ^\ \ \\ NEGATIVE OF LOAD
y^^ X^ \ \ \ CONDUCTANCE, Gl
1%^
AMPLITUDE OF OSCILLATION, V — — *■
Fig. 23. — Variation of electronic conductance with amplitude of oscillation of a form
which will result in hysteresis. The parameter A' determines the small signal value of the
conductance. The horizontal line indicates the load conductance.
After the beam passes through this gap it is retarded and returned by a uni
form electrostatic field. If we carry out an analysis to determine the elec
tronic admittance which will appear across the gap if the electrons make one
round trip, we arrive at expression 2.2 which may be written
Fe =
lo^'eMX)
[sin 6 \ j cos d]
(8.7)
where
X =
REFLEX OSCILLATORS
499
This admittance will be a pure conductance if = 0o = (« + f ) 27r. As
we have seen, in an oscillator designed specifically for electronic tuning, n
usually has a value of 3 or greater and the variations M from 6 arising from
ll
p4GlV2
BUNCHING PARAMETER, X »•
Fig. 24. — A curve of power output vs parameter X resulting from the conductance
curves shown in Fig. 23 and illustrating hysteresis.
repeller voltage variation are sufficiently small so that the efifect of M in
varying .Y may be neglected. In this case we may write
Ge = Je L„ COS ^^
cv
ye =
c =
2Fo
M
(8.8)
The parameter which we vary in obtaining the repeller characteristic of
the tube is Ad. The variation of this parameter is produced by shifting the
repeller voltage Vr from the value Fro corresponding to the transit angle
do . Since as is shown, Fig. 25a, ^ decreases monotonically as V
increases, no explanation of hysteresis is to be found in this expression.
Fig. 25b shows the smooth symmetrical variation of output with repeller
voltage about the value for which A^ = which is to be expected.
500
BELL SYSTEM TECHNICAL JOURNAL
Now suppose a second source of conductance Gei exists whose amplitude
function is of the form illustrated in Fig. 26a. Let us suppose that for the
1.0
0.9
LU
u
Z 0.8
<
I
o
3 0.7
Q
Z
o
O0.6
O
gO.5
cr
''04
_I
LU
,,,0.3
====:
^
REPELLER VOLTAGES:
K
(a)
^
H
.^
^
^v.
vro
"^
^v
^
^
^4
NEGATIVE OF
LOAD CONDUCTANCE ,Gl
"
■^
u^
[S
\,N
k
^I'oi
''^
>>
N
1
sN
^
1
s
<:
^
s
^^
^
Vsj
voj
^
^
•s^
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
1.0 1.5 2.0 2.5 3.0
AMPLITUDE OF OSCILLATION IN ARBITRARY UNITS
^
>s
(b)
/
/
/
/
x AV^]>)
V
/
1 ! 1
kAVr2>
\
/
1 1
L 1.
Vr3
.i\
1 1
^\
/
i i ! 1 ^
I* 1 AVr4]
1 1 1 !
\^
/
1
\
/
Vroi
\
/
\
NEGATIVE REPELLER VOLTAGE »
Fig. 25. — a. Variation of electronic conductance with amplitude of oscillation for an
ideal oscillator. The parameter controlling the small signal electronic conductance is the
re])eller voltage wl'ich determines the transit angle in the repcilcr region. The horizontal
line indicates the load conductance.
b. The variation of power output witli tlie repeller voltage which results from the
characteristics of Fig. 25a.
value of ^0 assumed tlie phase of this coiukiclaiue is such as to oppose
Gel , Gel may or may not be a function of Id. For the sake of simplicity let
us assume that G^o varies with \d in the same way as Gei . The total conduc
REFLEX OSCILLATORS
501
0.8
V
0.6
5
,0.2
'0.1
;,.o
I 8
0.6
RESELLER VOLTAGES:^
(a)
/^^^^
:
./
^^v:"/^
^°^
N
S,
y
^.>^
■iWf
N
C^
y
^ f^5"\.^^V
NEGATIVE OF
, ! 1 'V^ \ \ V
^ "
LOAD CONDUCTANCE ,Gl
^^
?u>^
'4
P^
V^
^<i<:i^
C3^
V4 V5
:vo 1
^^
Gei "Ge2
(b)
s^l
^
<,
^^^"X^ 1 j ! , ! 1 i
'0 1.5 2 2.5 3.0 3.5
AMPLITUDE OF OSCILLATION IN ARBITRARY UNITS
1
UZ
v§
(c)
LU>
1a. 4
y
y
[""""'■^
N
V
it
acD
<< 3
/
/
\
N
/
Vroj
N
"'.^
•
U<
—
 AV
lA*"^
4
0^ 1
vli i
1
1 \a.'\.'?
<J
1 L_
r4
 f »'
NEGATIVE REPELLER VOLTAGE 
Fig. 26. — a. Curve Ga shows the variation of electronic conductance with amplitude
of oscillation for an ideal reflex oscillator. Curve Ge2 represents the variation of a second
source of electronic conductance with amplitude. The difference of these two curves
indicated GeiGti shows the variation of the sum of these two conductance terms with
amplitude.
h. Electronic conductance vs amplitude of oscillation when two conductance terms
exist whose variation with repeller voltage is the same.
c. Power output vs repeller voltage for a reflex oscillator in which two sources of con
ductance occur varying with amplitude as shown in Fig. 26b.
502 BELL SYSTEM TECHNICAL JOURNAL
tance d = Ge\ — Gti will appear as shown. As the repeller voltage is varied
from the optimum value the conductance curve will shrink in proportion to
cos A0, and the amplitude of oscillation for each value of M will adjust itself
to the value corresponding to the intersection of the load line and the con
ductance plot as shown in Fig. 26b. When the load line becomes tangent,
as for amplitude F4 , further variation of the repeller voltage in the same
direction will cause oscillation to jump from F4 to zero amplitude. Cor
respondingly, on starting oscillation will restart with a jump to Vz . Hence,
two sources of conductance varying in this way will produce conditions pre
viously described, which would cause hysteresis as shown in Fig. 26c.
The above assumptions lead to hysteresis symmetrically disposed about the
optimum repeller voltage. Actually, this is rarely the case, but the ex
planation for this will be deferred.
Fig. 27 shows repeller characteristics for an early model of a reflex oscil
lator designed at the Bell Telephone Laboratories. The construction of this
oscillator was essentially that of the ideaUzed oscillator of Fig. 1 upon which
the simple theory is based. However, the repeller characteristics of this
oscillator depart drastically from the ideal. It will be observed that a
double jump occurs in the amplitude of oscillation. The arrows indicate
the direction of variation of the repeller voltage. The variation in the fre
quency of oscillation is shown, and it will be observed that this also is dis
continuous and presents a striking feature in that the rate of change of fre
quency with voltage actually reverses its sign for a portion of the range. A
third curve is shown which gives the calculated phase A0 of the admittance
arising from drift in the repeller field. This lends very strong support to
the hypothesis of the existence of a second source of conductance, since this
phase varies by more than 180°, so that for some part of the rangelhe repel
ler conductance must actually oppose oscillation. The zero value phase is
arbitrary, since there is no way of determining when the total angle is
{n + f)27r.
Having recognized the circumstances which can lead to hysteresis in the
reflex oscillator, the problem resolves itself into locating the second source
of conductance and eliminating it.
A number of possible sources of a second conductance term were in
vestigated in the particular case of the 1349 oscillator, and most were found
to be of negligible importance. It was found that at least one important
second source of conductance arose from multiple transits of the gap made
by electrons returning to the cathcde region. In the case of the 1349 a de
sign of the electron optical system which insured that the electron stream
made only one outgoing and one return transit of the gap eliminated the
hysteresis in accordance with the hypothesis.
REFLEX OSCILLATORS
503
Inasmuch as multiple transits appear to be the most common cause for
hysteresis in reflex oscillator design, it seems worthwhile to obtain a more
detailed understanding of the mechanism in this case. Other possible
z9
100
90
80
70
60
50
40
30
20
to
50
40
30
20
10
10
20
30
40
^
^^^
.^
/
■\
N
1
f
/
/
/
/
:
1
(a)
\
\
/
>
1
r
^^.
,^''
V,
X
'
^
y;^
^
Af^
x'
^'
/
^
"^
.^'
^
(b)
/
^>
■le
/
f
/
,
■'"
<D
<
7
80
1
UJ
60
a.
TUJ
n>
40
n"
H>
oo
20
't'
oo
u,a
I
20
o in
/iiJ
< UJ
4
UJ u
(/I UJ
<n
60
QZ
UJ
>
80
^7
1 20
NEGATIVE
130 140
REPELLER VOLTAGE
Fig. 27. — Amplitude, frequency and transit phase variation with the repeiler voltage
obtained experimentally for a reflex oscillator exhibiting electronic hysteresis. The
arrows indicate the direction of variation of the repeiler voltage.
mechanisms such as velocity sorting on the repeiler will give rise to similar
effects and can be understood from what follows.
In the first order theory, the electrons which have retraversed the gap
are conveniently assumed to vanish. Actually, of course, the returning
stream is remodulated and enters the cathode space. Unfortunately, the
504 BELL SYSTEM TECHNICAL JOURNAL
conditions in the cathode region are very complex, and an exact analysis
would entail an unwarranted amount of effort. However, from an approxi
mate analysis one can obtain a very simple and adequate understanding of
the processes involved.
Let us examine the conditions existing after the electrons have returned
through the gap of the idealized reflex oscillator. In the absence of oscilla
tion, with an ideal rectilinear stream and ideally fine grids all the electrons
which leave the cathode will return to it. When oscillation exists all elec
trons which experience a net gain of energy on the two transits will be cap
tured by the cathode, while those experiencing a net loss will not reach it,
but instead will return through the gap for a third transit, etc. In a prac
tical oscillator even in the absence of oscillation only a fraction of the elec
trons which leave the cathode will be able to return to the cathode, because
of losses in axial velocity produced by deflections by the grid wires and vari
ous other causes. As a result, it will not be until an appreciable amplitude
of oscillation has been reached that a major proportion of the electrons
which have gained energy will be captured by the cathode. On the other
hand, there will be an amplitude of oscillation above which no appreciable
change in the number captured will occur.
The sorting action which occurs on the cathode will produce a source of
electronic admittance. Another contribution may arise from space charge
interaction of the returning bunched beam with the outgoing stream. A
third component arises from the continued hunching , ^suiting from the iirst
transit of the gap. From the standpoint of this third component the reflex
oscillator with multiple transits suggests the action of a cascade amplifier.
The situation is greatly complicated by the nature of the drift field in the
cathode space. All three mechanisms suggested above may combine to
give a resultant second source. Here we will consider only the third com
ponent. Consider qualitatively what happens in the bunching action of a
reflex oscillator. Over one cycle of the r.f. field, the electrons tend to bunch
about the electron which on its first transit crosses the gap when the field
is changing from an accelerating to a decelerating value. The group re
crosses the gap in such a phase that the field extracts at least as much energy
from every electron as it gave up to any electron in the group. When we
consider in addition various radial deflections, we see that very few of the
electrons constituting this bunch can be lost on the cathode.
Although it is an oversimplification, let us assume that we have a linear
retarding field in the cathode region and also that none of the electrons are
intercepted on the cathode. To this order of ai)pr()ximation a modified
cascade bunching theory would hardly be warranted and we will consider
only that the initial bunching action is continued. Under these conditions,
REFLEX OSCILLATORS 505
we can show that the admittance arising on the third transit of the gap will
have the form
F: = +7o ^' Al^ [sin e, + j cos d,] (8.9)
where /o is the effective d.c. contributing to the third transit, dt = 6 \ Be
is the total transit angle made up of the drift angle in the repeller space, 6,
and the drift angle in the cathode space dc . As before, assume that the
small changes in dt caused by the changing repeller voltage over the elec
tronic tuning range exercise an appreciable effect only in changing the sine
and cosine terms. Then we may write
Y'e=G'e+ jB'e = y'e ^^^^ [siu Ot + j COS 9t] (8.10)
where
If Ad = di  dto
Ci'e = y'e ^'^^jf^P [sin 0,0 cos ABt + cos 0,o sin A0,]. (8.11)
C2 V
Now
AFr
Ad = waT + Aw To
Vr + V,
Ada = AuTc (8.12)
AVr
Adt = CjOoT + ACOTO + AcOTc •
Fr+ Fo
We observe that the phase angle of the admittance arising on the third
transit varies more rapidly with repeller voltage (i.e., frequency) than the
phase angle of the second transit admittance. This is of considerable im
portance in understanding some of the features of hysteresis.
Let us consider (8.11) for some particular values of ^ccr di . We remem
ber that 6 1 is greater than 6 and hence Co > Ci . Since this is so, the limit
. . . lAiaV) .... ^ , , ,,^,, 2/i(CiF)
mg 1 unction — will become zero at a lower value ot l than — — .
C2 F CiV
We will consider two cases 6 1 — (« + 4)27r and dt — (// + f)2x. These
506
BELL SYSTEM TECHNICAL JOURNAL
correspond respectively to a conductance aiding and bucking the conduct
ance arising on the first return. In case 1 we have
^, /2/i(C2F)
(8.13)
2Jl(CiV)
Ge  ye Ky
5
y ^e  ye C2V
AMPLITUDE OF OSCILLATION. V
Fig. 28. — Theoretically derived variation of electronic conductance with amplitude o^
oscillation. Curve Ge represents conductance arising from drift action in the repeller
space. Curve Gi represents the conductance arising from continuing drift in the cathode
region. G" represents the conductance variation with amplitude which will result if
Ge and Ge are in phase opposition.
and case 2
^, , /2/i(C2F)
C2 V
(8.14)
Figure 28 illustrates case (2) and Fig. 29 case (1). If cos M , and cos 16
varied in the same way with repeller voltage, the resultant limiting function
would shrink without change in form as the repeller voltage was varied,
and it is apparent that Fig. 28 would then yield the conditions for hysteresis
and Fig. 29 would result in conditions for a continuous characteristic.
If Fig. 28 applied we should e.xpcct hysteresis symmetrical about the opti
mum repeller voltage. We recall, however, that in Fig. 27 hysteresis
REFLEX OSCILLATORS
507
occurred only on one end of the repeller characteristic and was absent on
the other. The key to this situation lies in the fact that M t and A6 do not
vary in the same way when the repeller voltage is changed and the fre
quency shifts as shown in (8.12). As a result, the resulting limiting function
does not shrink uniformly with repeller voltage, since the contribution
Ge changes more rapidly than G^ . Hence we should need a continuous
series of pictures of the limiting function in order to understand the situa
tion completely.
\^Gg = Ge+ Gg
V
■ ^ r.^,,< 2J, (C2V)
.V^
\
"^
^^^^^^"'^
AMPLITUDE OF OSCILLATION, V >
Fig. 29. — Theoretically derived curves of electronic conductance vs amplitude of oscil
lation. Curve G" shows the variation of the resultant electronic conductance when
the repeller space contribution and the cathode space contribution are in phase addition.
Suppose we consider Fig. 29 and again assume in the interests of simplicity
that Mt and A0 vary at the same rate. In this case we observe that in the
region aa' the conductance varies very rapidly with amplitude. This would
imply that in this region the output would tend to be independent of the
repeller voltage. If we refer again to Fig. 27 we observe that the output is
indeed nearly independent of the repeller voltage over a range.
We see that these facts all fit into a picture in which, because of the more
rapid phase variation of 6 1 than 6 with repeller voltage, the limiting function
at one end of the repeller voltage characteristic has the form of Fig. 28.
accounting for the hysteresis, and at the other end has the form of Fig. 29,
508 BELL SYSTEM TECHNICAL JOURNAL
accounting for the relative independence of the output on the repe'.Ier
voltage.
In what has been given so far we have arrived qualitatively at an explana
tion for the variation of the amplitude. There remains the explanal i' ,,
for the behavior of the frequency. In this case we plot susceptance as a
function of am])litude and, as in the case of the conductance, there will be
several contributions. The primary electronic susceptance will be given by
Be = ye ^^^ sin e. (8.15)
Hence, as we vary the parameter M by changing the repeller voltage the
susceptance curve swells as the conductance curve shrinks. The circuit
condition for stable oscillation is that
Be + 2iAcoC = 0. (8.16)
A second source of susceptance will arise from the continuing drift in the
cathode space. Referring to equation (8.10) we see that this will have the
form
Be = ye—p^j^— c^&Qt (8.1/)
C2 V
and corresponding to equation (8.11) we write
B'e = y'e ' ^ ' ^ [cos 0,0 cos ^^ i  sin dt^ sin ^^t\. (8.18)
C2 V
Consider the functions given by (8.18) for values oi 6 1 — (n + l)2r and
(« + f)27r as functions of V. These are the extreme values which we
considered in the case of the conductance. The ordinates of these curves
give the frequency shift as a function of the amplitude.
In case 1 we have
Be = —ye ' T/ ^^" ^^' (.^l^)
C2 y
and case 2
„/ /2/i(C2F) . /o lr»\
Be = ye ' „ sm Adt . (8.20)
C2 V
The total susceptance will be the sum of the susceptance appearing across
the gap as a result of the drift in the repeller space and the susceptance
which appears across the gap as a result of the cascaded drift action in the
repeller region and the cathode region. If sin Adt and sin Ad varied in
the same way with the repeller voltage, the total susceptance would expand
REFLEX OSCILLATORS
509
or contract without change in form as the repeller voltage was varied. In
Figs. 30 and 31a family of susceptance curves are shown corresponding
respectively to cases 1 and 2 above for various values of A0( , assuming
that Ml and A0 vary in the same way with the repeller voltage. As the
(J
1
(a)
Ae.L=4),= o^;::;:=: ^r:;::::;^^^
V
57^ ^^
___ — ■ — ' ^ ^\r
"^^^
V5 V4 V3V2V,
AMPLITUDE OF OSCILLATION, V
Fig. 30. — a. Theoretical variation of electronic conductance vs amplitude of oscillation
in the case in which two components are in phase opposition. The parameter is the re
peller transit phase. It is assumed that the two contributions have the same variation
with this phase.
h. Susceptance component of electronic admittance as a function of amplitude for the
case of phase opposition given in Fig. 30a. The parameter is the repeller phase. The
dashed line shows the variation of amplitude with the susceptance shift.
repeller voltage is varied the amplitude of oscillation will be determined
by the conductance Umiting function. In the case of the susceptance we
cannot determine the frequency from the intersection of the curve with a
load line. The frequency of oscillation will be determined by the drift
angle and the amplitude of oscillation. The amplitude variation with
510
BELL SYSTEM TECHNICAL JOURNAL
angle may be obtained from Fig. 30a, which gives the conductance family.
This gives the frequency variation with angle indicated by tlie curve con
AMPLITUDE OF OSCILLATION, V *■
Fig. 31. — Theoretical variation of the susceptaiice components of electronic admittance
vs amplitude of oscillation for the case in which two components of electronic susceptance
are in phase addition.
necting the dots of Fig. 301). On the assumption that A0, and A0 vary at
the same rate with repeller voltage a symmetrical variation about A0 =
will occur as shown in Fig. 30b. However, from the arguments used con
REFLEX OSCILLATORS 511
cerning the conductance the actual case would involve a transition from
the situation of Fig. 30b to that of Fig. 31. If a discontinuity in amplitude
occurs in which the amplitude does not go to zero, it will be accompanied
by a discontinuity in frequency, since the discontinuity in amplitude in
general wall cause a discontinuity in the susceptance. If this discontinuity
in susceptance occurs between values of the amplitude such as Va and Vh
of Fig. 30, we observe that the direction of the frequency jump may be
opposite to the previous variation. We also observe that if the rate of
change of susceptance with amplitude is greater than the rate of change of
susceptance with Ad, then in regions such as that lying between zero ampli
tude of Vb the rate of change of frequency with A0 may reverse its direction.
One can see that because of the longer drift time contributing to the third
transit the conductance arising on the third transit may be of the same
order as that arising on the second transit. In oscillators in which several
repeller modes, i.e., various numbers of drift angles, may be displayed, one
finds that the hysteresis is most serious for the mcdes with the fewest cycles
of drift in the repeller space. One might expect this, since for these mcdes
the contribution from the cathode space is relatively more important.
Some final general remarks will be made concerning hysteresis. One
thing is obvious from what has been said. With the admittance conditions
as depicted, if all the electronic operating conditions are fixed and the load
is varied hysteresis with load can exist. This was found to be true experi
mentally, and in the case of oscillators working into misterminated long lines
it can produce disastrous effects. Where hysteresis is severe enough, it
will be found that what we have chosen to call the sink margin will be much
less than the theoretically expected value. An illustration of this is given
in Fig. 109.
The explanation which we have given for the hysteresis in the reflex
oscillator depends upon the existence of two sources of conductance. This
was apparently a correct assumption in the case studied, since the elimina
tion of the second source also eliminated the hysteresis. It is possible,
however, to obtain hysteresis in a reflex oscillator with only a single source.
This can occur if the phase of the electronic admittance is not independent
of the amplitude. Normally, in adjusting the repeller voltage the value
is chosen for the condition of maximum output. This means that the drift
angle is set to a value to give maximum conductance for large amplitude.
If the drift angle is then a function of the amplitude, this will mean that for
small amplitude it will no longer be optimum. Thus, although the limiting
function ^ tends to increase the electronic conductance as the ampli
tude declines, the phase factor will oppose this increase. If the phase factor
depended sufficiently strongly on the amplitude, the decrease in Gr caused by
512 BELL SYSTEM TECHNICAL JOURNAL
the phase might outweigh the increase due to the function ^ '^ . Asa
CiV
result the conductance niiglit have a maximum value for an amplitude
greater than zero, leading to the conditions shown in Fig. 23, under which
hysteresis can exist.
The first order theory for the reflex oscillator does not predict such an
effect, since the phase is independent of amplitude. The second order
theory gives the admittance as
_ ^ihO 2Ji(X) y(e_(^/2)) /., _ 1
.. (8.21)
■ I i\(A + 1)  X' ^^  ^^ (2  A)  X ^1^
The quantity appearing outside the brackets is the admittance given by the
first order theory. The second order correction contains real and imaginary
parts which are functions of A" and hence of the amplitude of oscillation.
Thus, for fixed dc conditions the admittance phase depends upon the am
plitude of oscillation and hence hysteresis might occur. It should be ob
served that the correction terms are important only for small values of the
transit angle 9. In particular, this explanation would not suffice for the
case described earlier since the design employed which eliminated the hys
teresis left the variables of equation (8.21) unchanged.
IX. Effect of Load
So far we have considered the reflex oscillator chiefly from the point of
view of optimum performance; that is, we have attempted chiefly to evaluate
its performance when it is used most advantageously. There has been some
discussion of nonoptimum loading, but this has been incidental to the
general purpose of the work. Oscillators frequently are worked into other
than optimum loads, sometimes as a result of incorrect adjustment, some
times through mistakes in design of equipment and quite frequently by
intention in order to take advantage of particular properties of the reflex
oscillator when worked into specific nonoptimum loads.
In this section we will consider the effects of other than o])timum loads
on the performance of the reflex oscillator. We may divide this discussion
into two major subdivisions classified according to the type of load. The
first type we call fixed element loads, and the second variable element loads.
The first type is constructed of arbitrary passive elements whose constants
are independent of frequency. The second category includes loads con
structed of the same tyi)e of elements but connected to the oscillator by
lines of suflicient length so that the frequency variation of the load admit
tance is appreciably modified by the line.
REFLEX OSCILLATORS 513
A. Fixed Element Loads
In this discussion it will be assumed initially that M, the phase angle of
— Ye , is not affected by frequency. The results will be extended later to
account for the variation of A0 with frequency. A further simplification is
the use of the equivalent circuit of Fig. 118, Appendix I. Initially, the
output circuit loss, R, will be taken as zero, so the admittance at the gap
will be
Yc = Gr^ 2jM^oi/oi + Yl/N\ (9.1)8
Here, Gr is the resonator loss conductance, M is the resonator characteristic
admittance, and Fj, is the load admittance.
We will now simplify this further by letting Gk =
F. = 2iMAco/co + Yl/N\ (9.2)
From Fig. 12 we see
GJN^ = yA2Ji(X)/X] cos Ad (9.3)
B, ^ 2MAC. ^ _y^i2MX)/X] sin Ad . (9.4)
Now it is convenient to define quantities expressing power, conductance and
susceptance in dimensionless form.
p = X^G^/2.Smye (9.5)
Gi = GjWye (9.6)
^1 = Bz./7V2y«. (9.7)
The power P produced by the electron stream and dissipated in G^, is related
to p
e^>
P = (^^7 P (9.8)
In terms of p and Gi , (9.3) can be written
p = (l/1.25)(2.5/>/Gi)~Vi[(2.5/^/Gi)1 cos A0. (9.9)
By dividing (9.4) by (9.3), we obtain
Aco/coo = (Gi/2A'W) tan A^  BlI2X'^M (9.10)
 (2M/ye)Aco/a'o = Gi tan AQ ^B,. (9.11)
* To avoid confusion on the reader's part, it is perhaps well to note that we are, for the
sake of generality, changing nomenclature. Hitherto we have used F/, to denote the
load at the oscillator. Actually our load as the appendix shows is usually coupled by
some transformer whose ecjuivalent transformation ratio is 1/A'^, so that the admittance
at the gap will be YiJN^.
514
BELL SYSTEM TECHNICAL JOURNAL
Equations (9.9) and (9.11) give the behavior of a reflex oscillator with
zero output circuit loss as the load is changed. It is interesting to plot
this behavior on a Smith chart. Such a plot is known as a Rieke diagram
or an impedance performance chart. Suppose we iirst make a plot for
A^ = 0. This is shown in Fig. 32. Constant p contours are solid and, as
Fig. 32. — Theoretical Rieke diagram for a reflex oscillator operating with optimum
drift angle. The resonator is assumed lossless. Admittances are normalized in terms of
the small signal electronic admittance of the oscillator so that oscillation will stop for unity
standing wave.
can be seen from the above, they will coincide with the constant conductance
lines of the chart. Constant frequency curves are dashed and, for M = 0,
they coincide with the locii of constant susceptance. The numbers on the
frequency contours give values of (2M/ye)(Aco/wo). The choice of units is
such that Gi = 1 means that the load conductance is just equal to small
signal electronic conductance which, it will be recalled, is the starting condi
tion for oscillation. Hence, the d = 1 contour is a zero power contour.
Any larger values of Gi will not permit oscillation to start, so the Gx contour
'P. H. Smith, "Transmission Line Calculator," Electronics, Jan. 1939, pp. 2931
REFLEX OSCILLATORS 515
bounds a region of zero power commonly called the "sink," since all the
frequency contours converge into it. The other zero power boundary is
the outer boundary of the chart, Gi = 0, which, of course, is an open circuit
load. The power contours on this chart occur in pairs, except the maximum
power contour which is single. These correspond to coupling greater than
and less than the optimum.
The value of Gi for any given power contour for A0 = may be deter
mined by referring to Fig. 9. We are assuming no resonator loss so we use
the curve for which Gulje = 0. From (9.5), ii p = 1 we have Gt/N^ye =
2.5/X which, substituted in (9.3), gives XJi{X) = 1.25. This is just the
condition for maximum power output with no resonator loss. From this
it can be seen that we have chosen a set of normalized coordinates. Hence,
in using Fig. 9, we have p = H/Hm, where Hm = .394 is the maximum gen
eralized efficiency. Thus, for any given value of p we let H in Fig. 9 have
the value .394/> and determine the two values of Gi corresponding to that
contour.
From Fig. 32 we can construct several other charts describing the per
formance of reflex oscillators under other conditions. For instance, sup
pose we make M other than zero. Such a condition commonly occurs in
use either through erroneous adjustment of. the repeller or through inten
tional use of the electronic tuning of the oscillator. We can construct a
new chart for this condition using Fig. 32. Consider first the constant
power contours. Suppose we consider the old contour of value pn lying
along a conductance line Gin . To get a new contour, we can change the
label from pn to pm = pn cos A0, and we move the contour to a conductance
line Gn = Gm cos A0. That this is correct can be seen by substituting these
values in (9.9). Consider a given frequency contour lying along Bi .
We shift each point of this contour along a constant conductance line Gi„
an amount B^ = Gin tan M. It will be observed that this satisfied (9.11).
In Fig. 33 this has been done for tan M — \, cos A0 = ■s/ll'l.
Now let us consider the effect of resonator loss. Suppose we have a
shunt resonator conductance Gr . Let
G. = Gnhe. (9.12)
Then, if the total conductance is G„ , the fraction of the power produced
which goes to the load is
/ = {Gn  G,)/Gn = Gi/(Gi + G,) (9.13)
accordingly, we multiply each power contour label by the fraction/. Then
we move all contour points along constant susceptance lines to new values
G„. = Gn G2 (9.14)
In Fig. 34, this has been done to the contours of Fig. 32, for Gz = .3.
516
BELL SYSTEM TECHNICAL JOURNAL
The diagrams so far o])tained have been based on the assumption that A0
has been held constant. To obtain such a diagram experimentally would be
extremely difficult. It would require that, as the frequency changed through
load puUing, and hence the total transit angle d = IttJt changed, an adjust
ment of the repeller voltage be made to correct the change. In actual
practice, Rieke diagrams for a reflex oscillator are usually made holding the
LOAD POWER AG
LOAD POWER Ae=
Fig. 33. — A transformation of the Rieke diagram of Fig. 32 showing the effect of shifting
the drift angle away from the optimum l)v 45°.
transit time r constant or in other words, with fixed operating voltages.
What this does to the basic diagram of Fig. 32 is not difficult to discover,
I)rovided that bd is sufficiently small so that we may ignore the variations
of the Bessels functions with bd. We will tirst investigate the effect of fixed
repeller voltage on the constant frequency contours. To do this we will
rewrite (*X11), rei)lacing A0 by A^ + bd and expand.
Aco
ACOT = COoT
Wo
(9.15)
REFLEX OSCILLATORS
517
POWER INTO LOAD FOR 62= 03
MAX. POWER INTO LOAD FOR 62= 0.3
LOAD POWER G2 = 0.3
___ A = (2M1 (AOJ^
Fig. 34. — A transformation of the Rieke diagram of Fig. 32 to show the effect of the
resonator loss if the phase angle is assumed to be optimum.
In rewriting (9.11) we will also replace Gi by Gi + G^ , to take resonator loss
into account. We obtain for very small values of hd
(2M/3;,)(Aco/a'o) = ((Gi + G2) tan A^ + B,)S (9.16)
S = 1/(1 + (Gi + G2)wor/(2M/>;,) cos^ A^)
S = 1/(1 + wor/2() cos A^). (9.17)
Q is the loaded Q of the oscillator.
To obtain the new constant frequency contours in the case of A^ =
we shift each point of the old contour from its original position at a sus
ceptance B,, along a constant conductance line G^,, to a new susceptance line
B,n = B„/S. This neglects a second order correction. It will be observed
that for small values of the conductance Gi near the outer boundary, the
frequency shifts will be practically unchanged, but near the sink where the
518 BELL SYSTEM TECHNICAL JOURNAL
conductance Gi is large the effect is to shift the constant frequency contours
along the sink boundary away from the zero susceptance line to larger sus
ceptance values. Hence, the constant frequency contours no longer coincide
with the constant susceptance contours, not even for A0 = 0.
The change in the power contours is considerably more marked. As the
frequency of the oscillator changes the transit angle is shifted from the
optimum value by an amount bd = (Aco/coo)c<;or. Thus the electronic
conductance is reduced in magnitude by a factor cos — coot. In particular.
Wo
for the sink contour where the load conductance is just equal to the elec
tronic conductance we see that when the repeller voltage is held constant
the power contour lies not on the Gi = 1 — G2 contour but on the locus of
Ao)
values Gi = cos — wot — d .
In order to determine the power contours when the transit time rather
than the transit angle is held constant we make use of (9.3) with addition of
resonator loss. In normalized coordinates ((9.6) and (9.12)) and for a phase
angle of electronic admittance 86 we have
Gi + G2 = '^^^^ cos 89 . (9.18)
From (9.5) and (9.13) we have for the power output
Gi 2XJi{X) ,_ . .
Along any constant frequency contour 86 is constant and has the value
given by (9.15) in terms of wo and coqt. Hence, it will be convenient to plot
(Gi + G2) vs X for various values of 86 as a parameter. This has been
done in Fig. 35. The angle 86 has been specified in terms of a parameter A
which appears in the Rieke diagrams as a measure of frequency deviation.
^=^^ (9.20)
ye Wo
In terms of the parameter A
86 = (y,/2A/)(coor)/l . (9.21)
Once we have the curves of Fig. 35 we can find the power for any point
on the impedance performance chart. We may, for instance, choose to
find the power along the constant frequency contours, for each of which
A (or 86) has certain constant values. We assume some constant resonator
loss G2 . Choosing a point along the contour is merely taking a particular
value of Gi . Having 86, G2 and Gi we can obtain A^ from Fig. 35. Then,
knowing A^, we can calculate the power from (9.19).
REFLEX OSCILLATORS
519
In constructing an impedance performance chart we want constant power
contours. In obtaining these it is convenient to assume a given value of
G2 . We will use G2 = 3 as an example. Then we can use Fig. 35 and
a95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
I
0.50
0.45
0.40
0.35
0.30
0.25
11^
xN
\,
^^ N
. N^=o
\\
\
K\
^
^^
N
>N\\
V
N
^"
\^
\
.WW
^"
\
\^
N,
\
\
i
3^67,
v
^
\
\^
\^
^^
4.36
.\
vV
'^
^
^
rv
1
G2 = 0.3
i
"^
^\
^^
^
sV
^^
X
^
\x
m
^
^
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
BUNCHING PARAMETER, X
Fig. 35. — Curve of load plus loss conductance vs bunching parameter X for various
values of a parameter A which gives the deviation in the drift time from the optimum
time. The load and loss conductance are normalized in terms of the small signal elec
tronic admittance. The horizontal line represents a loss conductance of G2 = .3.
(9.19) to construct a family of curves giving p vs Gi with A (or 86) as a
parameter. In a particular case it was assumed that
M/y, = 90
COOT = 27r(7 + f).
520 BELL SYSTEM TECHNICAL JOURNAL
These values are roughly those for the 2K25 reflex oscillator. Figure 36
shows p vs Gi for the particular parameters assumed above. The curves
were obtained by assuming values of Gi for an approj)riate .1 and so obtain
ing values of .V from Fig. 35. Then the power was calculated using (9.19)
and so a curve of j)ower vs d for a })articular value of .1 was constructed.
Figure 37 shows an impedance performance chart obtained from (9.16)
and Fig. 36. In using Fig. 36 to obtain constant power contours, we need
merely note the values of Gi at which a horizontal line on Fig. 36 intersects
the curves for various values of A. Each curve either intersects such a
horizontal (constant power) line at two points, or it is tangent or it does not
intersect. The point of tangency represents the largest value of A at which
the power can be obtained, and corresponds to the points of the crescent
shaped power contours of the impedance performance chart. The maximum
power contour contracts to a point.
Along the boundary of the sink, for which p — 0, X = and we have from
(9.18)
Gi = cos bd  Gi. (9.22)
The results which we have obtained can be extended to include the case
in which Id 9^ 0. Further, as we know from Appendix I, we can take into
account losses in the output circuit by assuming a resistance in series with
the load. In a welldesigned reflex oscillator the output circuit has little
loss. The chief effect of this small loss is to round off the points of the
constant power contours.
In actually measuring the performance of an oscillator, output and fre
quency are plotted vs load impedance as referred to the characteristic
impedance of the output line. Also, frequently the coupling is adjusted so
that for a match (the center of the Smith chart) optimum power is obtained.
We can transform our impedance performance chart to correspond to such a
plot by shifting each point G, B on a contour to a new point
Gi = G/Gxaax
Bi = B/Gmas
where Gmax is the conductance for which maximum power is obtained.
Such a transformation of Fig. 37 is shown in Fig. 38.
It will be noted in Fig. 38 that the standing wave ratio for power, the
sink margin, is about 2.3. This sink margin is nearly independent of the
resonator loss for oscillators loaded to give maximum power at unity stand
ing wave ratio, as has been discussed and illustrated in Fig. 10. If the sink
margin must be increased or the pulling figure must be decreased^" the coup
'" The pulling figure is arbitrarily defined as the maximum frequency excursion pro
duced when a voltage standing wave ratio of v 2 is presented to the oscillator and the
phase is varied through 180°.
REFLEX OSCILLATORS
521
0.46
^=0
0.42
i
^
1
fe
k
0.40
0.38
0.36
0.34
f
KS
A
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cr O20
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ill
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ill
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0.12
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0.10
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0.08
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3.87
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0.06
0.04
III 1
/
\
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If
f
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0.02
IL
^.36
i
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—
\
\i
r
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1
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\\\i
0.05 0.10 015
0.20 0.25 0.30 0.35 0.40 0.45 0.50
NORMALIZED LOAD CONDUCTANCE, Gi
0.55 0.60 0.65 0.70
Fig. 36. — Normalized power vs normalized load conductance for various values of the
parameter A which gives the deviation in drift time from the optimum drift time. These
curves are computed for the case G2 = .3. Optimum drift angle equal to 15.5 n radians
and a ratio of characteristics resonator admittance to small signal electronic admittance
of 90 is assumed.
522
BELL SYSTEM TECHNICAL JOURNAL
ling can be reduced so that for unity standing wave ratio the load conduct
ance appearing at the gap is less than that for optimum power.
Finally, in making measurements the load impedance is usually evaluated
at a point several wavelengths away from the resonator. If performance is
plotted in terms of impedances so specified, the points on the contours of
LOAD POWER G2 = 0.3
MAX.LOAD POWER G2=0.3
LOAD POWER G2=0.3
MAX. LOAD POWER 6^=0
I
^<m^)
ye
POWER
Fig. 37. — A Rieke diagram for a reflex oscillator having a lossy resonator, taking into
account the variation of drift angle with frequency pulling. This results in closed power
contours.
Fig. 38 appear rotated about the center. As the line length in wavelengths
will be different for different frequencies, ]:)oints on different frequency
contours will be rotated by different amounts. This can cause the contoirs
to overlap in the region corresponding to the zero admittance region of Fig,
38. With very long lines, the contours may overlap over a considerable
region. The multiple modes of oscillation which then occur are discussed
in somewhat different terms in the following section.
REFLEX OSCILLATORS
523
Figure 39 shows the performance chart of Fig. 38 as it would appear with
the impedances evaluated at a point 5 wavelengths away from the resonator.
Figure 71 of Section XIII shows an impedance performance chart for 2K25
reflex oscillator.
— i^K^i
Fig. 38. — The Rieke diagram of Fig. 37 transformed to apply to the oscillator loaded
for optimum power at unity standing wave.
B. Frequency — Sensitive Loads — Long Line Efect
When the load presented to a reflex oscillator consists of a long line mis
matched at the far end, or contains a resonant element, the operation of a
reflex oscillator, and especially its electronic tuning, may be very seriously
affected.
For instance, consider the simple circuit shown in Fig. 40. Here Mr
is the characteristic admittance of the reflex oscillator resonator as seen
from the output line or wave guide and Ml is the characteristic impedance
of a line load I long, so terminated as to give a standing wave ratio, <r.
524
BELL SYSTEM TECHNICAL JOURNAL
In the simple circuit assumed there are essentially three variables; (1)
the ratio of the characteristic admittance of the resonant circuit, Af« to
LOAD POWER
Fig. 39. — The Rieke diagram of Fig. 38 transformed to include the effect of a hne five
wave lengths long between the load and the oscillator.
Mr M_
Fig. 40. — Equivalent circuit of a lossless resonator, a line and a mismatched load.
that of the line, Mi, . This ratio will be called the external Q and signified
hyQ,
Q, = Mn/M^ . (9.23)
REFLEX OSCILLA TORS
525
For a lossless resonator and unity standing wave ratio, the loaded Q is equal
to Qe ■ For a resonator of unloaded Q, Q» , and for unity standing wave
ratio, the loaded Q, obeys the relation
\/Q = \/Qe + 1/(3..
(9.24)
50.5
1.5 2.0 2.5 3.0
CONDUCTANCE, G
Fig. 41. — Susceptance vs conductance for a resonator coupled to a 50 wave length line
terminated by a load having a standing wave ratio of 2. Characteristic admittance of
the resonator is assumed to be equal to 100 in terms of a line characteristic admittance of
unity. The circles mark off relative frequency increments
Aco
coo
103,
where coo is the frequency' of resonance.
(2) the length of the line called 6 when measured in radians or n when
measured in wavelengths, (3) the standing wave ratio a.
Figures 41 and 42 show admittance plots for two resonant circuits loaded
by mismatched lines of different lengths. The feature to be observed is the
loops, which are such that at certain points the same admittance is achieved
at two different frequencies. It is obvious that a line representing —Ye
526
BELL SYSTEM TECHNICAL JOURNAL
may cut such a curve at more than one pohit : thus, oscillation at more than
one frequency is possible. Actually, there may be three intersections per
loop. The two of these for which the susceptance B is increasing with fre
quency represent stable oscillation; the intersection at which B is decreasing
with frequency represents an unstable condition.
The loops are of course due to reactance changes associated with varia
tion of the electrical length of the line with frequency. Slight changes in
tuning of the circuit or slight changes in the length of the line shift the loops
up or down, parallel to the susceptance axis. Thus, whether the electronic
admittance line actually cuts a loop, giving two possible oscillating fre
quencies, may depend on the e.xact length of the line as well as on the ex
DO.l
BETWEEN POINTS
k
k^
J/
0.5 0.6 0.7
CONDUCTANCE, G
Fig. 42. — Susceptance vs conductance for line 500 wave lengths long terminated by a
load having a standing wave ratio of 1.11. Circles mark off relative frequency increments
of 10"''. Characteristic admittance to the resonator equals 100.
istence of loops. The frequency difference between loops is such as to
change the electrical length of the line by onehalf wavelength.
The existence or absence of loops and their size depend on all three pa
rameters. Things which promote loops are:
Low ratio of Mr/M ^ or Qe
Large n or 6
High 0
As any parameter is changed so as to promote the existence of loops, the Y
curve first has merely a slight periodic variation from the straight line for a
resistiveiy loaded circuit. Further change leads to a critical condition in
which the curve has cusps at which the rate of change of admittance with
frequency is zero. If the electronic admittance line passes through a cusp,
REFLEX OSCILLATORS 527
the frequency of oscillation changes infinitely rapidly with load. Still
further change results in the formation of loops. Further change results in
expansion of loops so that they overlap, giving more than three intersections
with the electronic admittance line.
Loops may exist for very low standing wave ratios if the line is sufficiently
long. Admittance plots for low standing wave ratio are very nearly cy
cloidal in shape; those for higher standing wave ratios are similar to cycloids
in appearance but actually depart considerably from cycloids in exact form.
By combining the expression for the near resonance admittance of a tuned
circuit with the transmission line equation for admittances, the expression
for these admittance curves is obtained. Assuming the termination to be
an admittance I'V which at frequency wo is do radians from the resonator,
1 \j{Yt/Ml) tan 0o(l + Aco/wo)
The critical relation of parameters for which a cusp is formed is important,
for it divides conditions for which oscillation is possible at one frequency
only and those for which oscillation is possible at two frequencies. This
cusp corresponds to a condition in which the rate of change with frequency
of admittance of the mismatched line is equal and opposite to that of the
circuit. This may be obtained by letting Yt be real.
Yt/Ml > 1, do = nir where n is an integer.
The standing wave ratio is then
a = Yt/Ml . (9.26)
The second term on the right of (9.25) is then
\1 +_;o tan ^oAco/coo/
For very small values of Aco we see that very nearly
72 = MlW  i(cr2  l)0oAco/a'o] • (9.28)
Thus for the rate of change of total admittance to be zero
2Mh = Ml{c'  1)60
% = 2{Mj,/ML)(a'  1)
= 2Q^/{a'  1) . (9.30)
Thus, the condition for no loops, and hence, for a single oscillating frequency,
may be expressed
00 < IQeHo"  1) (9.31)
528 BELL SYSTEM TECHNICAL JOIRNAL
We will remember that ^o is the length of line in radians, a is the standing
wave ratio, measured as greater than unity, and Qe is the external Q of the
resonator for unity standing wave ratio.
Replacing a given length of line by the same length of wave guide, we fnd
that the phase angle of the reflection changes more rapidly with frequency,
and instead of (9.31) we have the condition for no loops as
e < 2(3^(1  (X/Xo)2)/(a^  1) (9.32)
'^ < Vl +2Qe(1  (X/Xo)2)/0o
Here X is the free space wavelength and Xn is the cutoff wavelength cf the
guide.
Equations (9.32) are for a particular phase of standing wave, tl at is, for
relations of Yt and 6o which, produce a loop symmetrical abcve the C axis.
Loops above the G axis are slightly more locped than Iccps belcw the G
axis because of the increase of do with frequency. For reasonably Icng lines,
(9.32) applies quite accurately for formation of loops in any position; for
short lines locps are cf no consequence unless they are near the G axis.
An imporant case is that in which the resonant lead is ccupled to the
resonator by means of a line so short that it may be considered to have a
constant electrical length for all frequencies of interest. The resonant
load will be assumed to be shunted with a conductance equal to the charac
teristic admittance of the line. As the multiple resonance of a long mis
matched line resulted in formation of many locps, so in this case we would
rightly suspect the possibility of a single loop.
If the resonant load is , f, etc. wavelengths from the resonator, and
both resonate at the same frequency, a loop is formed symmetrical about the
G axis. Figure 43 is an admittance curve for resonator and lead placed 5
wavelength apart. Tuning either resonator or load moves this loop up
or down.
If the distance from resonator to resonant load is varied above or below a
quarter wave distance, the loop moves up or down and expands. This is
illustrated by an eighth wavelength diagram for the same resonator and load
as of Fig. 43 shown in Fig. 44.
When the distance from the resonator lo the resonant load, including
the effective length of the coupling loop, is 5, 1, 1^, etc. wavelengths, for
frequencies near resonance the resonant load is essentially in shunt with
the resonator, and its effect is to increase the loaded Q of the resonator. An
admittance curve for the case is shown in Fig. 45. In this rase the loo])s
REFLEX OSL'ILLA TORS
529
have moved considerably away in frequency, and expanded tremendously.
There are still recrossings of the axis near the origin, however, as indicated
in this case by the dashed line which represents 2 crossings, in this case
about 4% in frequency above and below the middle crossing if the length of
the line t is X/2.
Mp=I 00
CD 0.25
\
4^=0.5X103
BETWEEN POINTS
\
\
\
\
\
/^
^
?
"^^
_^
.
i
!
/
I
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
CONDUCTANCE, G
Fig. 43. — Susceptance vs conductance for two resonators coupled by a quarter wave line.
The resonator at which the admittance is measured has a characteristic admittance of 100
in terms of a line characteristic admittance of unit}. The other resonator has a character
istic admittance of 200 and a shunt conductance of unity. The circles mark off relative
frequency increments of 5 X 10"' in terms of the resonant frequency.
As a sort of horrible example, an admittance curve for a high () lead 50
wavelengths from the resonator was c( mputcd and is shown in Fig. 46.
Only a few of the loops are shown.
Admittance curves for more complicated circuits such as impedance trans
formers can be computed or obtained experimentally.
530
BELL SYSTEM TECHNICAL JOURNAL
As has been stated, one of the most serious effects of such mismatched
long line or resonant loads is that on the electronic tuning. For instance,
consider the circuit admittance curve to be that shown in Fig. 47, and the
minus electronic admittance curve to be a straight line extending from the
origin. As the repeller voltage is varied and this is swung down from the
\B axis its extreme will at some point touch the circuit admittance line
r
 i=S^^
■^
rT
>
M = 1 00
M5=200
\
^=0.5X103
BETWEEN POINTS
\
^^
V
A
^
N
\
\
/
A
/
V
J
/
\
V
/
,:
J
^
0.25 0.50
0.75 1.00 1.25
CONDUCTANCE, G
Fig. 44. — Susceptance vs conductance for the same resonators as of Fig. 43 coupled
by a oneeighth wave line.
and oscillation will commence. As the line is swung further down, the
frequency will decrease. Oscillation will increase in amplitude until the
— Ye line is perpendicular to the I' line. From that point on oscillation
will decrease in amplitude until the — Ye line is parallel to the Y curve
on the down side of the loop. Beyond this point the intersection cannot
move out on the loop, and the frequency and amplitude will jump abruptly
to correspond with the other intersection. As the — 1% line rotates further,
REFLEX OSCILLATORS
531
amplitude will decrease and finally go to zero when the end of the — Ye
line touches the V curve. If the — Ye line is rotated back, a similar phe
nomenon is observed. This behavior and the resulting electronic tuning
characteristic are illustrated in Figs. 47 and 48. Such electronic tuning
Mr=I 00
Ml=1
Ms=200
m 0.25
'
[ f^^ 0.5X103
' 1^0
BETWEEN POINTS
<
1
j OTHER CROSSINGS
,,AT 2 ±47o
1* IN FREQUENCY (
1
(
(
'
O.a.^ 0.50 0.75 I.OO 1.25 1.50 1.75 2.00
CONDUCTANCE, G
Fig. 45. — Susceptance vs conductance for the same resonators as of Fig. 43 coupled
by a onehalf wave line. The dash line indicates two other crossings of the susceptance
axis, at frequencies ±4% from the resonant frequency of the resonators.
characteristics are frequently observed when a reflex oscillator is coupled
tightly to a resonant load.
C. Effect of Short Mismatched Lines on Electronic Tuning
In the foregoing, the effect of long mismatched lines in producing addi
tional multiplewesonant frequencies and possible modiness in operation has
532
!2 0.5
2.5
BELL SYSTEM TECHNICAL JOURNAL
h« l=50X ^
G = :.
Mr=IOO
Ml=i
Ms=200
/
'■■''
/
\
b.
/
\
j
/
^
^
\,
\
/
y
\
. \
/
\ \
\V
/
\
y\
/\
7^
b
\
\
h
N
L J
1
A
1
\
/
\
\
/
\
\
y
^ /
\
\
V
•^
^
y
/
\
/
\
V
y
/
V
^
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
CONDUCTANCE , G
Fig. 46. — Susceptance vs conductance for the resonators of Fig. 43 coupled by a line
50 wave lengths long.
been explained. The effect of such multiple resonance on electronic tuning
has been illustrated in Fig. 48.
Tf a short mismatched Hne is used as the load for a reflex oscillator, there
REFLEX OSCILLA TORS
53^
may be no additional modes, or such modes may be so far removed in fre
quency from the fundamental frequency of the resonator as to be of little
CONDUCTANCE , G •
Fig. 47. — Behavior of the intersection between a circuit admittance line with a loop
and the negative of the electronic admittance line of a reflex oscillator as the drift angle is
varied (circuit hysteresis).
REPELLER VOLTAGE ►
Fig. 48. — Output vs repeller voltage for the conditions obtaining in Fig. 47.
importance. Nonetheless, the short line will add a frequencysensitive
reactance in shunt with the resonant circuit, and hence will change the char
acteristic admittance of the resonator.
Sii BELL SYSTEM TECHNICAL JOURNAL
Imagine, for instance, that we represent the resonator and the mismatched
line as in shunt with a section of Hne N wavelengths or 6 radians long mis
terminated in a frequency insensitive manner so as to give a standing wave
ratio <r. If Ml is the characteristic admittance of the line, the admittance
it produces at the resonator is
Y,=M,f±4^^. (9.33)
1 + ja tan 6
Now, if the frequency is increased, 6 is made greater and Y is changed.
{1 j j(T tan&)2
We are interested in the susceptive component of change. If
Vz. = Gl+JBj^ (9.35)
we find
»Bjm = M, " ~ "'Y'r ^ytf """ ' • (936)
(1 + 0 nan^ 6)
Now, if frequency is changed by an amount df, 9 will increase by an a mount
6(df/f) and Bl will change by an amount
dB:^ = {dBJdd){2T,N){df/f). (9.37)
We now define a parameter Mm expressing the effect of the mismatch as
follows
TidB^/dd) = Mm. (9.38)
Then
dBj^ = INMuidf/f). (9.39)
If the characteristic admittance of the resonator is Mr , then the characteris
tic admittance of the resonator plus the line is
M = Mji\ NMm. (9.40)
If, instead of a coaxial line, a wave guide is used, and Xo and X are the cutoff
and operating wavelengths, we have
dB^ = 2NMM{df/f)(l  (X/Xo)2)^ (9.41)
and
ikr = M« + NMm(1  (X/Xo)2)^ (9.42)
In Fig. 49 contour lines for Mm constant are plotted on a Smith Chart
(reflection coefficient plane). Over most of the plane Mm has a moderate
REFLEX OSCILLATORS
535
positive value tending to increase characteristic admittance and hence
decrease electronic tuning. Over a very restricted range in the high admit
tance region Mm has large negative values and over a restricted range
outside of this region Mm has large positive values.
Fig. 49. — Lines of constant value of a parameter. Mm shown on a chart giving the con
ductance and susceptance of the terminating admittance of a short line. The parameter
plotted multiplied by the number of wave lengths in the line gives the additional charac
teristic admittance due to the resonant effects of the line. The parameter Mm is of course
for terminated lines (center of chart).
This is an appropriate point at which to settle the issue: what do we mean
by a "short line" as opposed to a "long line." For our present purposes,
a short line is one short enough so that Mm does not change substantially
over the frequency range involved. Thus whether a line is short or not
depends on the phase of the standing wave at the resonator (the position
536
BELL SYSTEM TECHNICAL JOURNAL
on the Smith Chart) as well as on the length of the line. Mm changes most
rapidly with frequency in the very high admittance region.
As a simple example of the effect of a short mismatched line on electronic
tuning between half power points, consider the case of a reflex oscillator
with a lossless resonator so coupled to the line that the external Q is 100
and the electronic conductance is 3 in terms of the line admittance. Sup
pose we couple to this a coaxial line 5 wavelengths long with a standing wave
ratio cr = 2, vary the phase, and compute the electronic tuning for various
100
50
0.04 0.06 008 010 0.12 QW ai6 0.18 0.20 022 Q24 0.26
VOLTAGE STANDING WAVE RATIO PHASE IN CYCLES PER SECOND
Fig. 50. — The normalized load conductance, the characteristic admittance of the resona
tor and the normalized electronic tuning range to half power plotted vs standing wave
ratio phase for a particular case involving a short misterminated line. The electronic
tuning for a matched line is shown as a heav\' horizontal line in the ilot of (Aw/coo)! .
phases. We can do this by obtaining the conductance and Ml from Fig.
49 and using Fig. 15 to btain (Aw/wo)j . In Fig. 50, the parameters
GlIJc (the total characteristic admittance including the effect of the line),
A'', and, finally, (Aaj/wo)j have been plotted vs standing wave phase in
cycles. (Ac<j/ajo)j for a matched load is also shown. This example is of
course not tyi:)ical for all reflex oscillators: in some cases the electronic tuning
might be reduced or oscillation might stop entirely for the standing wave
phases which produce high conductance.
1
REFLEX OSCILLATORS 537
X. Variation of Power and Electronic Tuning with Frequency
When a reflex oscillator is tuned through its tuning range, the load
and repeller voltage being adjusted for optimum efficiency for a given drift
angle, it is found that the power and efiiciency and the electronic tuning
vary, having optima at certain frequencies.
When we come to work out the variation of power and electronic tuning
with frequency we at once notice two distinct cases: that of a fixed gap
spacing and variable resonator (707A), and that of an essentially fixed
resonator and a variable gap spacing (723A etc.); see Section XIII.
Here we will treat as an example the latter case only.
The simplest approximation of the tuning mechanism which can be ex
pected to accord reasonably with facts is that in which the resonator is
represented as a fixed inductance, a constant shunt "stray" capacitance
and a variable capacitance proportional to 1/rf, where d is the gap spacing.
The validity of such a representation over the normal operating range has
been verified experimentally for a variety of oscillator resonators. Let
Co be the fixed capacitance and Ci be the variable capacitance at some
reference spacing di . Then, letting the inductance be L, we have for the
frequency
CO = (L(Co + Ci d,/d))K (10.1)
Suppose we chocse di such that
Co = Ci. (10.2)
Then, letting
d/di = D (10.3>)
a'l = (ILCor = 27r/i (10.4)
w/a;i = IF. (10.5)
IF = 2'(1 + \/D)~K (10.6)
We find
This relation is shown in Fig. 51, where D is plotted vs TF. It is perfectly
general (within the validity of the assumptions) for a proper choice of refer
ence spacing di . We have, then, in Fig. 51a curve of spacing D vs re
duced frequency IF.
The parameter which governs the power and eflicency is Gn/ye . We
have
Cs/jc = (G«/i8')(2Fo//o0). (10.7)
As Fo , /o and 6 will not vary in tuning the oscillator, we must look for varia
ton in Gu and (3'^.
538
BELL SYSTEM TECHNICAL JOURNAL
For parallel plane grids, we have
l/)82 = (V2)Vsin2 {ej2) (10.8)
where 6g is the transit angle between grids. We see that in terms of W
and D we can write
dg = diWD .
(10.9)
lU

\
//


V
\
/
^7

\,
/y
/
S
^
.02
J
''/
\
/;
/
\
\,
/
^
/■
^^
^
wi
\
\
'\
^
/
;^'

,^
^


y
^^
\
\,
—
—
D^'
V
\
y'
'* y
\
^
y
/
WD
\
^,
y
X
/
\
0.1
/
/
/
\
%
V
i
0.6 0.7 0.8 0.9 1.0 I.I 1.2 1.3 1.4
RELATIVE FREQUENCY, W
Fig. 51. — Various functions of relative frequency W and relative spacing D plotted vs
relative frequency.
Here B\ is the gap transit angle at a spacing d\ and a frequency TFi . So
that we may see the effect of tuning on 1//3, WD has been plotted vs IF
in Fig. 51 and l//3^ has been plotted vs Qg in Fig. 52.
We now have to consider losses. From (9.7) of Appendix IX we see that
the grid loss conductance can be expressed in the form
Gg = GgyW^D^ (10.10)
Here Ggi is the grid loss conductance a.t d = di and co = wi .
Finally, let us consider the resonator loss. If the resonator could be
represented by an inductance L with a series resistance R, at high frequencies
the conductance would be very nearly
REFLEX OSCILLATORS
If R varies as co', we see that we could then write
G^ = GnWK
Here Gli is the conductance at a frequency wi .
S39
(10.11)
(10.12)
1000
800
600
100
80
60
05 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
TRANSIT ANGLE, Gg , IN RADIANS
Fig. 52. — The reciprocal of the square of the modulation coefficient is a function of the
gap transit angle in radians for the case of fine parallel grids.
As an opposite extreme let us consider the behaviour of the input conduct
ance of a coaxial line. It can be shown that, allowing the resistance of
such a line to vary as oj , the input conductance is
Gt = ^C0*CSC2(C0//C).
(10.13)
Here t is the length of the line and C is the velocity of propagation. If
Gl given by (10.12) and Gi of 10.13 give the same value of conductance at
some angular frequency wi then it will be found that for values of t typical
of reflex oscillator resonators the variation of G( with w will be significantly
I less than that of Gl • Although typical cavities are not uniform lines
I (10.13) indicates that a slower variation than (10.12) can be expected.
It will be found moreover that the shape of the power output vs frequency
i curves are not very sensitive to the variation assumed. Hence as a rea
sonable compromise it will be assumed that the resonator wall loss varies as
540
BELL SYSTEM TECHNICAL JOURNAL
Suppose that at D — 1, i.e.
Gs = GsxW~\ (10.14)
In Fig. 51 ir~ has been plotted vs W.
Now let us take an actual example.
{d = d\, (j> = oji)
6 = 2
G,a = .inyye
Gs, = .()95/ye
The information above has been used in connection with Figs. 51 and 52
and ratio of resonator loss to small signal electronic admittance, Gr/jc,
has been plotted vs IF in Fig. 53. A 2K25 oscillator operated at a beam
Gr
ye
1.0
\
'
/
\,
1 t / 1
0.9
s.
s
.
1
08
s.
/
\,
J
0.7
0.6
^
V
/
X
«s^
j
/
'
y 1
0,5
; ^*****
^^^ 1
0.76 0.80 0.84
0.88 092 0.96 1.00 1.04
RELATIVE FREQUENCY, W '
Fig. 53. — Computed variation of ratio of resonator loss to small signal electronic ad
mittance vs relative frequency W for certain resonator parameters assumed to fit the
characteristics of the 2K25.
voltage, Fo , of 300 volts had a total cathcde current /d of 26 ma. This
current passed three grids on the first transit and back through the third
grid on the return transit. On a geometrical basis, h^^^ of the cathode
current should make this second transit across the gap. Th,us the useful
beam power was about
Po = (.53) (300) (.026) = 4.1.
If we assume a drift efifectiveness factor F of unity, then for tb.e 7 cycle
mode, the efficiency should be given by Um divided by 7f . //„, is plotted
as a function of Gn/y, in Fig. 7. Thus, we can obtain rj, the efficiency, and
hence the power output. This has been done and the calculated power
output is plotted vsIFin Fig. 54, where IF = 1 has been taken to correspond
to 9,000 mc. It is seen that the theoretical variation of output with fre
quency is much the same as the measured variation.
REFLEX OSQLLATORS
541
Actually, of course, the parameters of the curve were chosen so that it
corresponds fairly well to the experimental points. The upper value of W
at which the tube goes out of oscillation is most strongly influenced by the
value of di chosen. We see from Fig. 51 that as TI' is made greater than
unity WD increases rapidly and hence, from Fig. 52, /3^ decreases rapidly,
increasing Gnlye . On the other hand, as IF is made smaller than unity,
jS approaches unity but the grid loss term W'/D"^ increases rapidly, and
this term is most effective in adjusting the lower value of IF at which oscilla
tion will cease. Finally, the resonator loss term, varying as IF~\ does not
change rapidly and can be used to adjust the total loss and hence the opti
mum value of Gu/ye and the optimum efficiency.
It is clear that the power goes down at low frequencies chiefly because in
moving the grids very close together to tune to low frequencies with a fixed
nductance the resonator losses and especially the grid losses are increased.
50
45
40
[135
<
I 30
2 25
?20
UJ
% '5
o
'^ 10
y . \i
/ • \
%.A_ X
0.76 0.80 0.84
0.88 Q92 0.96 1.00 1.04
RELATIVE FREQUENCY, W
Fig. 54. — Computed curve of variation of power in milliwatts with relative frequency W
for the parameters used in Fig. 53. The circles are experimental points. The curve has
been fitted to the points by the choice of parameters.
In going to high frequencies the power decreases chiefly because moving the
grids far apart to tune to high frequencies decreases /3. Both of these
effects are avoided if a fixed grid spacing is used and the tuning is accom
plished by changing the inductance as in the case of the 707A. In such
tubes there will be an upper frequency limit either because even with a
fixed grid spacing ^ decreases as frequency increases, or else there will be a
limit at the resonant frequency of the smallest allowable external resonator,
and there will be a lower frequency limit at which the repeller voltage for a
given mode approaches zero; however, the total tuning range may be 3 to 1
instead of around 30% between extinction points, as for the 2K25.
542 BELL SYSTEM. TECHNICAL JOURNAL
. The total electronic tuning between halfpower points at optimum load
ing, 2(A/)i , can be expressed
2(A/)j = (fye/M)(2AWo,o)/(ye/M). (10.15)
We can obtain (2Aw/coo)/iye/M) from Fig. 16.
If we assume a circuit consisting of a constant inductance L and a capaci
tance, the characteristic admittance of the resonator is
M = 1/coL = Itt/iPF (10.16)
and
2(A/)i = 27rWJ,'LyX2AW^o)/(ye/M) (10.17)
and we have
ye = /327o(2xAO/2Fo . (10.18)
Here A^ is the total drift in cycles.
A rough calculation estimates the resonator inductance of the 2K25 as
.30 X 10~ henries. Using the values previously assumed, /o = (.53)(.026),
Fo = 300, N = 7f , and the values of Gulyc^"^ and j\ previously assumed,
we can obtain electronic tuning.
A curve for half power electronic tuning vs TF has been computed and is
shown in Fig. 55, together with experimental data for a 2K25. The experi
mental data fall mostly above the computed curve. This could mean that
the inductance has been incorrectly computed or that the drift effectiveness
is increased over that for a linear drift field, possibly by the effects of space
charge. By choosing a value of the drift effectiveness factor other than
unity we could no doubt achieve a better fit of the electronic tuning data
and still, by readjusting Gg\ and Gs\ , fit the power data. This whole pro
cedure is open to serious question. Further, it is very hard to measure such
factors as Ggx for a tube under operating conditions, with the grids heated by
bombardment. Indirect measurements involve many parameters at once,
and are suspect. Thus, Figs. 54 and 55 are presented merely to show a
qualitative correspondence between theory and experiment.
XI. Noise Sidebands in Reflex Oscillations
In considering power production, the electron flow in reflex oscillators
can be likened to a perfectly smooth flow of charge. However, the discrete
nature of the electrons, the cause of the familiar "shot noise" in electron
flow engenders the production of a small amount of rf power in the neigh
borhood of the oscillating frequency — "noise sidebands". Thus the energy
spectrum of a reflex oscillator consists of a very tall central spike, the power
output of the oscillator, and, superposed, a distribution of noise energy
having its highest value near the central spike.
REFLEX OSCILLA TORS
S43
Such noise or noise "sidebands" can be produced by any mechanism which
causes the parameters of the oscillator to fluctuate with time. As the mean
speed, the mean direction, and the convection current of the electron flow
all fluctuate with time, possible mechanisms of noise production are numer
ous. Some of these mechanisms are:
(1) Fluctuation in mean speed causes fluctuation in the drift angle and
hence can give rise to noise sidebands in the output through frequency
modulation of the oscillator.
90
u uj 50
LU
o
UJ UJ
2z

(> •
•
/ \ •
•\
20
0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 '
RELATIVE FREQUENCY, W
Fig. 55. — Computed variation of electronic tuning range in megacycles vs relative
frequency W. The curve is calculated from the same data as that in Fig. 54 with no
additional adjustment of parameters. Points represent experimental data.
(2) If the drift field acts differently on electrons differently directed,
fluctuations in mean direction of the electron flow may cause noise sidebands
through either amplitude or frequency modulation of the output.
(3) Low frequency fluctuations in the electron convection current may
amplitude modulate the output, causing noise sidebands, and may frequency
modulate the output when the oscillator is electronically tuned away from
the optimum power point.
(4) High frequency fluctuations in the electron stream may induce high
frequency noise currents in the resonator directly.
Mechanism (4) above, the direct induction of noise currents in the reso
nator by noise fluctuations in the electron stream, is probably most impor
544 BELL SYSTEM TECHNICAL JOURNAL
tant, although (3) may be appreciable. An analysis of the induction of
noise in the resonator is surprisingly com])licated, for the electron stream
acts as a nonlinear load impedance to the noise power giving rise to a com
plicated variation of noise with frequency and with amplitude of oscillation.
On the basis of analysis and experience it is pcssible, however, to draw
several general conclusions concerning reflex oscillator noise.
first, it is wise to decide just what shall be the measure of noise. The
noise is important only when the oscillator is used as a beating r scillator,
usually in connection with a crystal mixer. A power P is supplied to the
mixer at the beating oscillator frequency. Also, the oscillator supplies at
signal frequency, separated from the beating oscillator frequency by the
intermediate frequency, a noise power P„ proportional, over a small fre
quency range, to the bandwidth B. An adequate measurement of the
noisiness of the oscillator is the ratio of P„ to the Johnson ncise po\^er, kTB.
The general facts which can be stated about this ratio and seme explanaticn
of them follow:
(1) Electrons which cross the gap only once contribute to noise but not
to power. Likewise, if there is a large spread in drift angle amcng various
electron paths, some electrons may contribute to noise but not to power.
(2) The greater the separation between signal frequency and beating
oscillator frequency (i.e., the greater the intermediate frequency) the less
the noise.
(3) The greater the electronic tuning range, the greater the ncise for a
given separation between signal frequency and beating oscillator frequency.
This is natural; the electronic tuning range is a measure of the relative mag
nitudes of the electronic admittance and the characteristic admittance of
the circuit.
(4) The degree of loading affects the noise through affecting the bunching
parameter X. The noise seems to be least for light loading.
(5) Aside from controlling the degree of loading, resonator losses do not
affect the noise; it does not matter whether the unused power is dissipated
inside or outside of the tube.
(6) When the tube is tuned electronically, the noi?e usually increases at
frequencies both above and below the optimum power frequency, but the ■
tube is noisier when electronically tuned to lower frequencies. At the opti ^
mum frequency, the phase of the pulse induced in the circuit when an elec
tron returns across the gap lags the pulse induced on the first crossing by
270°. When the drift time is shortened so as to tune to a higher frequency,
the angle of lag is decreased and the two pulses tend to cancel; in tuning
electronically to lower frequencies the pulses become more nearly in phase.
An approximate theoretical treatment leads to the conclusion that aside
from avoiding loss of electrons in reflection, or very wide spreads in transit
REFLEX OSCILLATORS 545
time for various electrons, (see (1) above) and aside from narrowing the
electronic tuning range, which may be inadmissable, the only way to reduce
the noise is to decrease the cathode current. This is usually inadmissable.
Thus, it appears that nothing much can be done about the noise in reflex
oscillators without sacrificing electronic tuning range.
The seriousness of beating oscillator noise frcm a given tube depends, of
course, on the noise figure of the receiver without beating oscillator noise
and on the intermediate frequency. Usually, beating oscillator ncise is
worse at higher frequencies, partly because higher frequency oscillators have
greater electronic tuning (see (3) above). At a wavelength of around
1.25 cm, with a 60 mc I.F. amplifier, the beating oscillator ncise may be
sufficient so that were there no other noise at all the noise figure cf the
receiver would be around 12 db.
Beating oscillator noise may be eliminated by use of a sharply tuned filter
between the beating oscillator and the crystal. This precludes use of elec
tronic tuning. Beating oscillator noise may also be eliminated by use of a
balanced mixer in which, for example, the signal is fed to two crystals in the
same phase and the beating oscillator in opposite phases. If the LF. output
is derived so that the signal components from the two crystals add, the
output due to beating oscillator noise at signal frequencies will cancel out.
There is an increasing tendency for a number of reasons to use balanced
mixers and thus beating oscillator noise has become of less concern.
XII. Buildup of Oscillation
In certain applications, reflex oscillators are pulsed. In many of these
; it is required that the rf output appear quickly after the application cf
' dc power, and that the time of buildup be as nearly the same as possible
: for successive applications of power. In this connection it is important to
study the mechanism of the buildup of oscillations.
In connection with buildup of oscillations, it is convenient to use complex
frequencies. Impedances and admittances at complex frequencies are
given by the same functions of frequency as those at real frequencies.
Suppose, for instance, the radian frequency is
oj = ic — ja (12.1)
This means the oscillations are increasing in amplitude. The admittance
!of a conductance G at this frequency is
y = G
The admittance of a capacitance C and the impedance of an inductance L are
V = jo:C = juC + aC (12.2)
Z = jcoL = jivL + aL (12.3)
546 BELL SYSTEM TECHNICAL JOURNAL
In other words, to an increasing oscillation reactive elements have a "loss"
component of admittance or impedance. This "loss" component corre
sponds not to dissipation but to the increasing storage of electric or magnetic
energy in the reactive elements as the oscillation increases in amplitude.
The admittance curves plotted in Figs. 4146 may be regarded as contours
in the admittance plane for a = 0. If such a contour is known either by
calculation or experiment, and it is divided into equal frequency increments,
a simple construction will give a neighboring curve for w = w — jAa where
Aa is a small constant. Suppose that the change in F for a frequency
Acoi is AFi . Then for a change —jAa
AY = j — Aa. (12 .4)
•^ Awi ^
Thus, to construct from a constant amplitude admittance curve an admit
tance curve for an increasing oscillation, one takes a constant fraction of
each admittance increment between constant frequency increment points
(a constant fraction of each space between circles in Figs. 4146), rotates it
90 degrees clockwise, and thus establishes a point on the new curve.
This construction holds equally well for any conformal representation of
the admittance plane (for instance, for the reflection coefficient plane repre
sented on the Smith chart).
The general appearance of these curves for increasing oscillations in terms
of the curve for real frequency can be appreciated at once. The increasing
amplitude curve will lie to the right of the real frequency curve where the
latter is rising and to the left where the latter is falling. Thus the loops
will be diminished or eliminated altogether for increasing amplitude oscilla
tions, and the low conductance portions w^ill move to the right, to regions
of higher conductance. This is consistent with the idea that for an increas
ing oscillation a "loss" component is added to each reactance, thus degrading
the "Q", increasing the conductance, and smoothing out the admittance
curve.
The oscillation starts from a very small amplitude, presumably that due
to shot noise of the electron stream. For an appreciable fraction of the
buildup period the oscillation will remain so small that nonlinearities are
unimportant. The exponential buildup during this period is determined
by the electronic admittance for very small signals.
As an example, consider a case in which the electronic admittance for
small signals is a pure conductance with a value of — ye . Here the fact that
that the quantity is negative is recognized by prefixing a minus sign.
Assume also that the circuit admittance including the load may b'^ ex
pressed as in (a22) of Appendix I, which holds very nearly in case there
is only one resonance in resonator and load. Then for a complex frequency
Wo — jao the circuit admittance will be
REFLEX OSCILLATORS 547
Yc = Gc+2Mao/wo (12.5)
Thus in this special case we have for oscillation
yco = Gc+ IMaJwo (12.6)
and
ao = ^{Y,oGc) (12.7)
The amplitude, then, builds up initially according to the law
V = Voe""'. (12.8)
If the amplitude does not change too rapidly, the buildup characteristic
of an oscillator can be obtained stepbystep from a number of contours
for constant a and from a — Ye curve marked with amplitude points. The
Ye curve might, for instance, be obtained from a Rieke diagram and an
admittance curve.
Consider the example shown in Fig. 56. Fig. 56a shows curves con
structed for complex frequencies from the admittance curve for the resonant
circuit for real frequency. In addition the negative of the electronic ad
mittance is shown. Oscillation will start from some very small amplitude,
V = Vo , and buildup at an average rate given by a = 2.5 X 10~ until
F = 1. Let Vo = .1. Then the interval to buildup from F = .1 to
F= lis
In
Ah =
©
2.5 X 10«
= .92 X 10"^ seconds.
From amplitude 1 to amplitude 2 the average value of a will be 1.5 X 10'
and the time interval will be
At. =
1
Similarly, from 2 to 3
Ah =
1.5 X 10«
.46 X 10"^ seconds.
M
.5 X 106
.80 X 10"^ seconds.
The buildup curve is shown in Fig. 56b.
Similarly, from a family of admittance contours constructed from a cold
impedance curve, and from a knowledge of frequency and amplitude vs time,
548
BELL SYSTFAf TECHNICAL JOURNAL
Ye can be obtained as a function of time. It may be that in many cases the
real part of the frequency is nearly enough constant during buildup so that
only the amplitude vs time need be known . As the input will commonly be a
function of time for such experimental data, I\. vs time will yield I'«at vari
GIVEN gapI
VOLTAGE, Vl^
3
RATE OF
BUILDUP,
OL =
1 XIO^
2 X 10^
(a)
CONDUCTANCE, G
2
1
(b)
0.5 KG 1.5 2.0 2.5 3.0
TIME, t, IN MICROSECONDS
Fig. 56. — a. A plot of the circuit admittance (solid lines) for various rates of buildup
specified by the parameters a. The voltage builds up as e"' . The circuit conductance is
greater for large values of a. The negative of the electronic admittance is shown by the
dashed lines. The circles mark off the admittance at which various amplitudes or voltages
of oscillation occur. The intersections give the rates of buildup of oscillation at various
voltages. By assuming exponential build up at a rate s])ecified by a between the voltages
at these intersections, an api)ro.\imate liuildu]) can be constructed.
h. A build up curve constructed from the data in Fig. 56a.
ous amplitudes and inputs. Curves for various rates of applying input will
yield tables of Ye as a function of both input and amplitude.
It will be noted that to obtain very fast buildup with a given electronic
admittance, the conductance should vary slowly with a. This is the same
as saying that the susceptance should vary slowly with co, or with real fre
quency. For singly resonant circuits, this means that av/M should be large.
Suppose the admittance curve for real frequency, i.e. a = 0, has a single
REFLEX OSCILLA TORS
549
loop and is symmetrical about the G axis as shown in Fig. 57. Suppose the
— Ye curve lies directly on the G axis. The admittance contours for increas
ing values of a will look somewhat as shown. Suppose buildup starts on
Curve 2. When Curve 1 with the cusp is reached, the buildup can con
tinue along either half as the loop is formed and expands, resulting either of
the two possible frequencies of Curve 0. l^resumably in this symmetrical
1
\ \ \
\ 1 \ RATE OF
\ \ \ BUILDUP,
\ \ \ a
"> \
i \ \
\ \ \
\ \ \
\ X'V
\/ \ \N
^/ \ \ '
/ V / /'
/ ^y. JJ
/ / "/
/ /
/ / /
/ / /
/ / /
' 1 /
' ' /
/ / /
CONDUCTANCE, G *
Fig. 57. — Circuit admittance vs circuit conductance in arbitrar} units for different
rates of buildup at turnon. When the buildup is rapid {a = 2) the admittance curve
has no loop. As the rate of buildup decreases the curve sharpens until it has a cusp a = 1.
As the rate of buildup further decreases the curve develops a loop {a = 0). There may
be uncertainty as to which of the final intersections with the a = Q line will represent
oscillation.
case, nonsynchronous fluctuations would result in buildup to each frequency
for half of the turnons. If one frequency were favored by a slight dis
symmetry, the favored frequency would appear on the greater fraction of
turnons. For a great dissymetry, buildup may always be in one mode,
although from the impedance diagram steady oscillation in another mode
appears to be j)ossible.
550 BELL SYSTEM TECHNICAL JOURNAL
In the absence of hum or other disturbances the buildup of oscillations
starts from a randomly fluctuating voltage caused by shot noise. Thus,
from turnon to turnon some sort of statistical distribution may be expected
in the time t taken to reach a given fraction of the final amplitude. In un
published work Dr. C R. Shannon of these laboratories has shown that in
terms of «<> , the initial rate of buildup, the standard deviation br and the
root mean square deviation (5t')^ are given by
5t = .38/«o (12.9)
(572)1/2 ^ ^^^^^ ^2.10)
Thus the "jitter" in the successive positions of the rf pulses associated with
evenly spaced turnons is least when the initial rate of buildup, given by Oo ,
is greatest.
Such conditions do not obtain on turnoff, and there is little jitter in the
trailing edge of a series of rf pulses. This is of considerable practical
importance.
XIII. Reflex Oscillator Development at the Bell Telephone
Laboratories
For many years research and development directed towards the genera
tion of power at higher and higher frequencies have been conducted at the
Bell Telephone Laboratories. An effort has been made to extend the fre
quency range of the conventional grid controlled vacuum tube as w^ell as
to explore new principles, such as those embodied in velocity variation
oscillators. The need for centimeter range oscillators for radar applications
provided an added impetus to this program and even before the United
States entry into the war, as well as throughout its duration, these labora
tories, cooperating with government agencies, engaged in a major effort to
provide such power sources. The part of this program which dealt with
high power sources for transmitter uses has been described elsewhere. This
paper deals with low power sources, which are used as beating oscillators in
radar receivers. In the following sections some of the requirements on a
beating oscillator for a radar receiver will be outlined in order to show^ how
the reflex oscillator is particularly well suited for such an application.
A. The Beating Oscillator Problem
The need for a beating oscillator in a radar system arises from the neces
sity of amplifying the very weak signals reflected from the targets. Imme
diate rectification of these signals would entail a very large degradation in
signal to noise ratio, although providing great simplicity of operation. It
would also lead to a lack of selectivity. Amplification of the signals at the
" See Appendix 10.
REFLEX OSCILLATORS 551
signal frequency would require centimeter range amplifiers haying good
signal to noise properties. No such amplifiers existed for the centimeter
range, and it was necessary to beat the signal frequency to an intermediate
frequency for amplification before rectification. For a number of reasons,
such intermediate frequency amplifiers operate in the range of a few tens
of megacycles, so that the beating oscillator must generate very nearly the
same frequency as the transmitter oscillator.
In radar receivers operating at frequencies up to several hundred mega
cycles, conversion is frequently achieved with vacuum tubes. For higher
frequencies crystal converters have usually been employed. With few ex
ceptions, the oscillators to be described were used with these crystal con
verters which require a small oscillator drive of the order of one miUiwatt.
In general it is desirable to introduce attenuation between the oscillator and
the crystal to minimize effects due to variation of the load. Approximately
13 db is allowed for such padding so that a beating oscillator need supply
about 20 milliwatts. Power in excess of this is useful in many applications
but not absolutely necessary. Since the power output requirements are
low, efficiency is not of prime importance and is usually, and frequently
necessarily, sacrificed in the interest of more important characteristics.
The beating oscillator of a radar receiver operating in the centimeter
range must fulfill a number of requirements which arise from the particular
nature of the radar components and their manner of operation. The inter
mediate frequency amplifier must have a minimum pass band sufficient to
amplify enough of the transmitter sideband frequencies so that the modu
lating pulse is reproduced satisfactorily. It is not desirable to provide much
margin in band width above this minimum since the total noise increases
with increasing band width. It is therefore necessary for best opera
tion that the frequency of the beating oscillator should closely follow fre
quency variations of the transmitter, so that a constant difference frequency
equal to the intermediate frequency is maintained.
This becomes more difficult at higher frequencies, inasmuch as all fre
quency instabilities, such as thermal drifts, frequency pulling, etc. occur as
percentage variations. Some of the frequency variations occur at rapid
rates. An example of this is the frequency variation which is caused by
changes in the standing wave presented to the transmitter. Such varia
tions may arise, for instance, from imperfections in rotating joints in the
output line between the transmitter to the scanning antenna.
For correction of slow frequency drifts a manual adjustment of the fre
quency is frequently possible, but instances arise, notably in aircraft installa
tions, in which it is not possible for an operator to monitor the frequency
constantly. Rapid frequency changes, moreover, occur at rates in excess
of the reaction speed of a normal man. Hence for obvious tactical reasons
552 BEI.I. SYSTEM TECHMCM. JOCRXAL
it is imperative that the difference frequency between the transmitter and
tlie beating oscillator should be maintained by automatic means. As an
illustration of the problem one may expect to have to correct frequency
shifts from all causes, in a 10,000 megacycle system, of the order of 20 mega
cycles. Such correction may be demanded at rates of the order of 100 mega
cycles per second per second.
Although the frequency range of triode oscillators has since been some
what extended, at the time that beating oscillators in the 10 centimeter
range were lirst required the triode oscillators available did not adequately
fullill all the requirements. In general the tuning and feedback adjust
ments were complicated and hence did not adapt themselves to autcmatic
frequency control systems. \'elccity variation tubes of the multiple gap
type which gave more satisfactory performance than the tricdes existed in
this range. These, however, generally required operating voltages of the
order of a thousand volts and frequently required magnetic tields for focus
sing the electron stream. The tuning range obtainable by electrical means
was considerably less than needed and, just as in the case of the tricde oscil
lator, the mechanical tuning mechanism did not adapt itself to automatic
control. These dilTiculties fccussed attention on the refiex oscillator, whcse
properties are ideally suited to automatic frequency control. The feature
of a single resonant circuit is of considerable importance in a military applica
tion, in which simple adjustments are of primary concern. The repeller
control of the phase of the negative electronic admittance which causes
oscillation provides a highly desirable vernier adjustment of the frequency,
and, since this control dissipates no power, it is particularly suited to auto
matic frequency control. Furthermore, since the upper limit on the rate of
change of frequency is set by the time of transit of the electrons in the repeller
field and the time constant of the resonant circuit, both of which are gen
erally very small fractions of a microsecond, very rapid frequency correction
is possible.
As the frequency is varied with the repeller voltage, the amplitude of
oscillation also varies in a manner ])reviously described. The signal to noise
j)erformance cf a crystal mixer depends in part on the beating oscillator
level and has an c jitimum value with respect to this parameter. In conse
quence, there are limitations on how much the beating oscillator power
may depart from this ( ptimum value. This has a bearing on the oscillator
design in that the amount of amplitude variation permitted for a given
frequency shift is limited. The usual criterion of perfomance adopted has
been the electronic tuning, i.e. the frequency difference, between points for
a given re])eller m( dc at which the i^ower has been reduced to half the maxi
mum value.
Reception of the wrong sideband by the receiver causes trouble in con
I
REFLEX OSCILLATORS 553
nection with automatic frequency control circuits in a manner too compli
cated for treatment here. In some cases this necessitates a restriction on
the total frequency shift between extinction points for a given repeller mode.
The relationship between half power and extinction electronic tuning has
been discussed in Section \TI.
In addition to the electrical requirements which have been outlined,
military applications dictate two further major objectives. The first is the
attainment of simple installation and replacement, which will determine, in
part, the outward form of the oscillator. The second is low voltage opera
tion, which fundamentally affects the internal design of the tubes. In some
instances military requirements conflict with optimum electronic and circuit
design, and best performance had to be sacrificed for simplicity of construc
tion and operation. In particular, in some cases it was necessary to design
for maximum flexibility of use and compromise to a certain extent the
specific requirements of a particular need.
In the following section we will describe a number of reflex oscillators
which were designed at the Bell Telephone Laboratories primarily to meet
military requirements. These oscillators are described in approximate
chronological crder of development in order to indicate advances in design
and the factors which led to these advances.
The reflex oscillators which w'ill be described fall into two general classi
fications determined by the method employed in tuning the resonator. In
one category are oscillators tuned by varying primarily the inductance of the
resonator and in the other are those tuned by varying primarily the capaci
tance of the resonator. The second category includes two types in which
the capacitance is varied in one case by external mechanical means and in
the second case by an internal means using a thermal control.
B. A Rejiex Oscillator With An External Resonator — The 707
The Western Electric 707A tube, which was the first reflex oscillator
extensively used in radar applications, is characteristic of reflex oscillators
using inductance tuning. It was intended specifically for service in radar
systems operating at frequencies in a range around 3000 megacycles. Fig. 58
shows a photograph of the tube and Fig. 59 an xray view showing the inter
nal construction. A removable external cavity is employed with the 707A
as indicated by the sketch superimposed on the xray of Fig. 59. Such
cavities are tuned by variation of the size of the resonant chamber. Such
tuning can be considered to result from variation of the inductance of the
circuit.
The form of this oscillator is essentially that of the idealized oscillator
shown in Fig. 58. The electron gun is designed to produce a rectilinear
cylindrical beam. The gun consists of a disc cathode, a beam forming elec
554
BELL SYSTEM TECHNCLAL JOURNAL
'^h
Fig. 58. — External view of