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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 Oxide-Catliode 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 Semi-Inhnite 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 Semi-Infinite 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., High-Vacuum Oxide-Cathode 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, High-Vacuum Oxide-Cathode 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 Semi-Infinite 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, High-Vacuum Oxide-Cathode, C. E. Fay, page 818. 

R 

Radar: High -Vacuum Oxide-Cathode 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 Semi-Infinite Unit Slope of Attenuation, 

page 870. 
Tyrrell, W. A . and G. E. Mueller, Polyrod Antennas, page 837. 

V 

Vacuum, High-, Oxide-Cathode 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 

NEW YORK 



50^ per copy $1.50 per Year 



THE BELL SYSTEM TECHNICAL JOURNAL 

Published quarterly by the 

American Telephone and Telegraph Company 

195 Broadway J New York, N. Y- 






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 
A. B. Clark 
S. Bracken 



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•i«i«>«i^i«a«i«>«>«i« 



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 2-14 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- 
high-frequency and microwave communications techniques then under 
way at the Holmdel Radio Laboratories of Bell Telephone Laboratories. 

' A crystal rectifier is an assymmetrical, non-linear 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 semi-conductor. 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 
ultra-high 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 resistor-like 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. 1934-1943. 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 semi-conductors 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 semi-conductor 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 n-t)"pe or negative 



SILICON CRYSTAL RECTIFIERS 5 

insert tended to give better performance as microwave converters while 
the p-type, 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 non-uniform. 
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 semi-conductor, 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 platinum-iridium 
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 d-c 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 high-frequency by-pass 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. 95-102, 1939; Carl R. Englund, "Dielectric Con- 
stants and Power Factors at Centimeter Wave Lengths," Bell Sys. Tech. Jour., v. 23, pp. 
114-129, 1944; lirainerd, Koehler, Reich, and WoodrulT, "Ultra High Frequency Tech- 
niques," D. Van Nostrand Co., Inc., 250-4th 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 



lO-i 



10-2 



10-3 









































^ 


.' 




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 















10-8 



10-6 



10-5 lO"'^ 

CURRENT IN AMPERES 



10-3 



10-2 



10-1 



Fig. 4 — Direct-current 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 1940-1941 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 4-70°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 cone-shaped 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 high-frequency 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 air-borne 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 silica-filled bakelite, and 
has spot welded to it a 0.002-inch 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 65-ohm 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 proof-tests. 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 non-linear 
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 signal-to-noise properties of the unit 
at the operating frequency, the power handling ability, and the uniformity 
of impedance are important factors. Tlie signal-to-noise 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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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 B-29 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 air-borne \ 
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 signal-to-noise 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. 419-422; 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 proof-tests 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 proof-test 
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 transmit-receive 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 
transmit-receive 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. 48-101. Jan. 1946. 



SILICON CRYSTAL RECTIFIERS 21 

Specification proof-test levels are, of course, not design criteria. Since 
the units are generally used in combination with protective devices, such 
as the transmit-receive 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 d-c spike proof- 
; test or peak powers of a magnitude comparable with that used in the multiple 
I flat-top d-c pulse proof-test. 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 signal-to-noise 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 low-power 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 high-frequency 
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 low-level 
performance is a function of the resistance at low voltages and the direct- 
current output for a given low-power radio frequency input. These may 
be combined to derive a figure of merit which is a measure of receiver 
performance.^ 

Typical direct-current 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. 61-15, March 16, 1943. 



22 



BELL SYSTEM TECHNICAL JOURNAL 



that by changes in processing routines the direct-current 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 signal-to-noise 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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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 semi-conductor, Rn the non-linear resistance at 
this boundary, and /^s is the spreading resistance of the semi-conductor 



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 by-pass 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 by-pass 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 (NON-LINEAR 
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 semi-conductor. 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 semi-conductor 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 low-power 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 low-power 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 
high-level powers, b\- direct-current tests, and by simple 60-cycle 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 by-pass 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 lower-fre- 



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-- 
nal-to-noise performance in each frequency band . 

Use of the improved materials and processes produced rather large im- 
provements in the d-c rectification ratio, conversion loss, noise, power 
handling ability, and uniformity. Typical direct-current 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 



10-2 

































REVERSE 
CURRENT 








FORWARD 
CURRENT 










s/ 




^"^ 


' 






,J 


"^ 


y 








' y 


■x"* 





■^ 




^ 


'■^ 










/ 




■^' 











































\0-^ lO"'* 10" 

CURRENT IN AMPERES 



10-2 



Fig. 14 — Improvement in the direct-current rectification characteristics of 
sihcon crystal rectifiers in a four-year 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 .■ 




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:• z:- 
■■- ^.■■ 
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; 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- 












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vo ::• 




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■•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 RECEIVER-NOISE 
FIGURE ATTAINABLE WITH A DOUBLE DETECTION RE- 
CEIVER EMPLOYING A CRYSTAL CONVERTER AND A 
5-DB INTERMEDIATE-FREQUENCY 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 3000-megacycle 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 high-frequency 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 POWER-HANDLING 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 INTERMEDIATE-FREOUENCy 
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 1942-1945. 

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 non-linear 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 non-military character, at 
microwave as well as lower frequencies, where its sensitivity, low capacitance, 
freedom from aging effects, and its small size and low-power 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 second-order 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 p-direction. 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 f-J({x) 
-logsm^^ = j ^j'^^dx. (9) 

The right-hand 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) 

T-2 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 two-way 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 

ATG-30» 


^ 




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 e-22'/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 0-90" 












\ ^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 term-by-term, 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)]). .VS-44, .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 r-z = 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 , r-j , 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 
r-i ; the value p2 corresponding to r^ for values of .v from bi to a point bi be- 
tween r-i 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 



J-iix) 



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 


18-16 


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 electro-optic 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. 1-72, 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 piezo-optical 
effect and the electro-optical 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] d-q 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 



25|3 — 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^ + X56-5 + 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 oi-Js + ^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//l65i5-6) 

-(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 + dirnT-i. + 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 IC-i^ 


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) 



Si-i — 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 + <"1132-S'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 • 



t-1131 



(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 


= -5212-2 


; 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 ~ "^13-1 ~ "^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 



A-K'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 (9-i) 
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 c-jkC 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^e-rnkf (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 tetrahedral-pentagonal-dedo- 
cahedral (Class 28) A', V, Z binary 



,0 ,0 ,hu,0 ,0 
,0 ,0 ,0 , //i4 , 
,0 ,0 ,0 ,0 ,/;,4 



Cubic pentagonal-icositetetrahedral (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, hexakis-octahedral (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 - 


-t-25 





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 

+ 2c23ktS-a + 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^ '"" dx-dX-dx{ ' 

. , , (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 so-called "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 so-called "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 , p-i , 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 piezo-optical 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 electro-optical 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 -8-4 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(r-25) + 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 lO-i 
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""'(r-25) + 7.4 X 10""(T-25)' - 3.13 X 10 "'(T-25) 

{X direction) 

= 43.7 X 10~*(T-25) + 8.16 X 10''(T-25)' - 3.60 X 10~'''(T-25)' 

(I' direction) 

= 49.4 X 10~'(r-25) + 1.555 X 10"'(r-25)' - 2.34 X nr'\T-25) 

{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° ,Y-cut 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*^ X-cut 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 



-8-4 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° F-cut 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. 744-750, 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 non-conducting 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 non-conduction 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 
negative-feedback amplifier. The abrupt transitions from non-conducting 
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 non-conducting 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 
chopped-up 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 frequency-selective circuit, which delivers a smoothly 
varying function of time. Knowing the spectrum of the chopped input 
to the selective network and the steady-state 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 non-linear 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 cross-modulation 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 single-frequency 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 two-frequency modulation 
products. It is found that the integer-power-law 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. 127-206, April, 1941. 

. ^ W. R. Hcnnctt, Response of a Linear Rectifier to Signal and Noise, Jour. Acous. Soc. 
Amer., Vol. 15, pp. 164-172, 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 single-frequencj' 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. 228-243, 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 y-axes. 
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 y-axis. 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 single-frequency 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 two-frequency 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 



E-InR 



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 d-c. 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 d-c 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 so-called 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 r-f or 
i-f 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 low-frequency group /;/ containing all the frequencies comparable 
with those in the spectrum of .1 (/). The components of this group flow 
through R. 

2. A high-frequency 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(0-1-1 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 high-frequency 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, ^ (1-21) 



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 low-frequency components of Er by Ei/, we have: 

£,/ = ^ (1.23) 

IT 

or 

A (/) = wE,f (1.24) 

* See, for example, the top curve of Fig. 9-25, p. 311, H. J. Reich, Theory and AppHca- 
tions of Electron Tubes, McGraw-Hill, 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 half-wave 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 low-frequency 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 two-frequency 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 direct-current 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. Single-Frequency 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. 619-621, 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. X-a-c; 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 amplitude-modulated wave 
or to the detection of a frequency-modulated wave after it has passed through 
a slope circuit. 

A case of considerable practical interest is that of an amplitude-modulated 
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^'' (, ...l-A .2-. 



where X = V/P. Note that K is a constant equal to the direct-voltage 
output if P is constant. If P varies slowly with time compared with the 
high-frequency 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 amplitude-modulated (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 — power-law rectifier as an envelope detector with low-imped- 
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 — power-law 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 — power-law rectifier as an envelope detector with impedance 
of signal generator low except in signal band. 




Fig. 10. — Performance of 3/2 — power-law 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 — power-law 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 — power-law rectifier as envelope detector with impedance 
of signal generator low except in signal band. 

3. Two-Frequency Inputs 

The general formula for the coefficients in the two-frequency 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. 
745-780, 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 power-law 
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 
zero-power, linear, and square-law detectors, in which /(z) is proportional 
to z", z , and z" respectively. The zero-power-law 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 low-order 





OS pt+Qcos qt 

RESPONSE OF LIMITER 


_A_ 


I 


m\mm 


■kw////////A 




m 




"^ TIME ». ^ 




f 



Fig. 13. — Response of biased total limiter to two-frequency 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 d-c 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 zero-power-law 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 zero-power-law 
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 - 



Zero-Power Rectifier or Total-Limiter 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. 32-48, 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 Hnear-rectifier 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,n-l 

-{- ki (m -\- n -{- 3)Am+i,n~i + 2kon .4„,+i,„ = 

2» Amn + kl (n -j- m — 3) Am-l.n+l 

+ ^1 (w — w + 3) A „,-!,„+! + 2kon Am-\.n = 

2m ki Amn -\- {m — n — 3) Am-l,n^l 

+ (m -f n + 3)A„.+i,„+i + 2^ow ^m,„+i = 
2 m h Amn + {m -]r n — 3) Am-i.n-\ 

-\- {m — n -\- 3)Am+l,n-l + 2^oW A^.n-l = 

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 third-order product A21 is of considerable importance in the design 
of carrier ampHfiers and radio transmitters, since the (2/> — 9)-product is 
the cross-product 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 Unear-rectifier coefficients give the Fourier 
series expansion of the admittance of a biased square-law 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 d-c for the zero-power law 
and for the d-c 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)hkiZra-3 + (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. 
1203-1209, 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 

• a|Q- 








^ 


\. 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(24-l)(23-2i) [^^ ^' + '^'(^^ - ^^^" 



(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 

H-0.4 

< 



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 full-wave biased zero-power-law 
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 full-wave biased zero-power-law 
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, 1825-28, 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. — D-c. 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_^ 


J-i— 


■^ 



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 third-order sum and difference 
products for the biased zero-power-law 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 H-function. 

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 power-series expan- 
sion in ^oi we find : 
Zero-Power- 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 full-wave unbiased zero- 
power-law 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 Two-Frequency 
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 audio-frequency 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 ultra-high-frequency 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 
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-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- 





















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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 
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bio- 



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xlO"' 



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 





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OJ 





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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 evi-i- 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" 


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'"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 



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^^s^ 














I 










100 




















^^^-- 










' 
















.. 






















""""^55 


2 




























































0.5 

















































































5 10 

MILLIAMPERES 

Fig. 8. — Static voltage-current 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 voltage-current 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. 



Ro-SQPOO OHMS 



T = 300 K 



1000 




o-* 



10- 



10- 



10- 



10 



10-2 

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 three-quarter 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 one-half 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 
noise-free 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 feed-back 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 volt-ampere 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: resistance-temperature, volt-ampere, and current- 
time or d^mamic relations. 

Resistance-Temperature 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 resistance-temperature 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 long-time 
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. 


— Temperature-Resistance 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 rod-shaped 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 one-quarter 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 d-c 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 one-tenth 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 -t-75 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 d-c 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. 

Volt-Ampere 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 one-third 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 
resistance-power 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 d-c as well as 
a-c 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 direct-current or 
low-frequency 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 wave-guide 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 d-c 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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I0-* 2 4 6 810"^ 2 4 6 810'^ 2 4 6 8I0'' 2 4 6 8 | 

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 broad-band 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 volt-ampere 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 volt-ampere 
characteristic, it is possible to provide snap and lock-in 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 volt-ampere 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~> ^ 




10-2 



4 6 8I0-' 



2 4-68! 2 46 8|0' 

CURRENT IN MILLIAMPERES 



4 6 810^ 



Fig. 24. — Characteristics of a typical thermistor manometer tube, showing the effect 
of gas pressure on the volt-ampere and resistance-power 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 6|0 



2 4 6 810-2 2 4 6 B|0- 

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 resistance-value 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 re-use 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 audio-frequercy range. If a thermistor is biased at a point on the 
negative slope portion of the steady-state volt-ampere 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 steady-state curve and dV/dl will be 
negative. As the frequency of the applied a-c 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 d-c 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 453-67. 

2. Semi-conductors and Metals (book), A. H. Wilson. The University Press, Cam- 

bridge, England, 1939. 

3. The Modern Theory of Solids (book), Frederick Seitz. McGraw-Hill 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 887-9. 

6. Uber die Elektrizitatsleitung Anorganischer Stofle mit Elektronenleitfahigkeit, Wil- 

fried Meyer. Zeitschrift Fur Physik, Volume 85, 1933, pages 278-93. 

7. Thermal Agitation of Electricity in Conductors, J. B. Johnson. Physical Review, 

Volume 32, July 1928, pages 97-113. 

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 197-223. 

9. Automatic Temperature Control for Aircraft, R. A. Gund. AIEE Transactions, 

Volume 64, 1945, October section, pages 730-34. 

10. The Bridge Stabilized Oscillator, L. A. Meacham. Proc. IRE, Volume 26, October 

1938, pages 1278-94. 

11. Frequency Stabilized Oscillator, R. L. Shepherd and R. O. Wise. Proc. IRE, Vol- 

ume 31, June 1943, pages 256-68. 

12. A Pilot-Channel Regulator for the K-1 Carrier System, J. H. Bollman. Bell Labora- 

tories Record, Volume 20, No. 10, June 1942, pages 258-62. 

13. Thermistors, J. E. Tweeddale. Western Electric Oscillator, December 1945, pages 

3-5, 34-7. 

14. Thermistor Technics, J. C. Johnson. Electronic Industries, Volume 4, August 1945, 

pages 74-7. 

15. Volume Limiter for Leased-Line Service, J. A. Weiler. Bell Laboratories Record, 

Volume 23, No. 3, March 1945, pages 72-5. 



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 intensity-frequency-time 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 
cathode-ray 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 Cathode-Ray 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 sub-bands by filters. 
The average speech energy in the sub-bands 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 d-c. 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 intensity-frequency-time 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 two-part report is designed to 
present a quick over-all 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 
Driving-Impedance 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 step-voltage 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, 
1922-24; Asst. Prof, of Physics, Stanford University, 1924. Engineering 
Dept., Western Electric Company, 1924-1925; Bell Telephone Laboratores, 
1925-. Mr. Becker has worked in the fields of X-Rays, 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 Electro-Chemical Engineering, Pennsylvania State 
College, 1918; U. S. Army, 1918 (2nd Lieutenant, Signal Corps); Vacuum 
tube development, Westinghouse Lamp Company, 1919-21; Instructor in 

217 



218 BELL SYSTEM TECHNICAL JOURNAL 

Physics, University of Colorado, 1921-1922. Department of Development 
and Research, American Telephone and Telegraph Company, 1922-27; 
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, 1921-25. Bell Telephone Labora- 
tories, 1925-. Mr. Wilson has been engaged in the development of am- 
plifiers for broad-band 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 



50i per copy $1.50 per Year 



THE BELL SYSTEM TECHNICAL JOURNAL 

Published quarterly by the 

American Telephone and Telegraph Company 

195 Broadway^ New York, N. Y. 



EDITORS 

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EDITORIAL BOARD 

W. H. Harrison O. E. Buckley 

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BOOKS AND PERIODICALS 
FOR WORLD RECOVERY 

The desperate and continued need for American publica- 
tions to serve as tools of physical and intellectual recon- 
struction abroad has been made vividly apparent by 
appeals from scholars in many lands. The American Book 
Center for War Devastated Libraries has been urged to 
continue meeting this need at least through 1947. The 
Book Center is therefore making a renewed appeal for 
American books and periodicals — for technical and scholarly 
books and periodicals in cdl fields and particularly for 
puhlicalions of the past ten years. We shall particularly 
welcome complete or incomplete files of the Bell System 
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The generous support ^^■hich has been given to the Book 
Center has made it possible to ship more than 700,000 
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this amount before the Book Center closes. The books 
and periodicals which individuals as well as institutional 
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and peace. 

Ship your contributions to the American Book Center, 
% The Library of Congress, Washington 25, D. C, freight 
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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 .\nti-Aircraft 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 SCR-545 Radar " Search" and "Track" Antennas 307 

15.2 The AN/TPS-IA Portable Search Antenna 309 

16. Airborne Radar Antennas 312 

16. 1 The AN/APS-4 Antenna 312 

16.2 The SCR-520, SCR-717 and SCR-720 .Antennas 313 

16.3 The AN/APQ-7 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 two-fold. 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 nineteen-thirties 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 
non-directive and consecjuently produces a relatively weak, field at 
a distance. When the wave-length 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 two-fold, 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 wave-guide. 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:, McGraw-Hill 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 

KJT-Aji = 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 = -^— = 1-5 (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) w-e 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) 

J-al2 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 4-n 



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 

k-2 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 mid-band. 

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 
so-called 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 non-uniform 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 'spill-over' 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 spill-over. 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 by-product 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.^ 



\ V-FEED 

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 non-directive 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) 




[{l|||i|||M||||l ""i>i|{|||||l||||l 




(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 over-all 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 anti-aircraft 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 over-all 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 s3-stem 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 non-stabiUzed 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 bushy-brain 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 cross-over < 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 

SJ-1 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 cross-over < 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 SJ-1 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 SJ-1 /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 


680-720 MC 


680-720 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 SCR-296 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 half-wave 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 short-wave 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 Anti-Aircraft 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 Anti-aircraft 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 anti-aircraft 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 pointer-trainer 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 anti-aircraft 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 anli-aircraft 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 115-volt, 60 cycle, single phase motor to which was coupled 
a two-phase 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 24-inch 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 transmitter-receiver (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 Anti-Aircraft 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 anti-aircraft radar 
is shown in Fig. 57. 

15. Land Based Radar Antennas 
15.1 The SCR-545 Radar ''Search'' and "Track" Antennas''' 

The SCR-545 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 anti-aircraft 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 wave-length 
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 coaxial-waveguide 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 SCR-545 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 SCR-545 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 SCR-545 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/TPS-IA 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/TPS-IA 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 APS-4 Anten)ia^ 

AN/APS-4 was designed to provide the Navy's carrier-based 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 bomb-shaped 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/.\PS-4 Antenna. 



16.2 The SCR-520, SCR-717 and SCR-720 Antennas-' 

The antenna shown in Fig. ol is typical of the type used with the SCR-520 
and SCR-720 aircraft interception (night fighter) airborne radar equip- 
ment, as well as the SCR-717 sea search and anti-submarine 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— SCR-520 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 take-offs 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 SCR-717 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/APQ-7 Radar Bombsight Antenna^^ 

Early experience in the use of bombing-through-overcast 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/APQ-7 radar 
equipment, operating at the X-band of frequencies. This equipment 
provided facilities for radar navigation and bombing. 

The AN/APQ-7 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/APQ-7 AntennaMounted on B24 ;Bomber. 



CHOKE JOINT 
COUPLING 




SLIDING 
SURFACES 



Fig. 63— AN/APQ-7 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/APQ-7 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/APQ-7 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 
co-workers 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 2-dimensional 
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 r-variate, 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,(7-plane, 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 chance-variable') 
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,o--plane 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,o--plane, 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 non-infinite 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, 0-2 = 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 0-34) 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 , 0-34). (1.5) 

These equations serve to define the above-mentioned 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 \ 0-34) is a 'distribution function' for p individually, with the 
understanding that a' is restricted to the range a^-to-ai ; 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 r-variate 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 0-34) 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 single-integral 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 , 0-12) = 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), (1-18) 

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), (1-24) 



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,r-axes; 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, ii-plane 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, r-plane; and P{u, v) is the ordinate 
from any point u,v in the u,v-p\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'<vi-dV). (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 0-to-5(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 0-to-7r/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 so-called '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 (53-A) 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' (3-5) ^ = ^^ = 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 above-mentioned argument bR~/(l — b'^) = T not exceeding 
the largest values of X in the above-cited 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. 698-713), 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 . 0-5) (l -5)(3-7) 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 1-62 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. 45-72, 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 
'non-effective' 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 well-known 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 1-dimensional 
'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 right-hand maximum value a.t R = 0+, whence this 
m-aximum 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/il-b'). 



(3.22) 



uj 0.8 

I 

UJ 

O 
5 0.4 

gO.2 

»- 
o 

z 

2 













MAX P(R;b) 






■;^ 




























"^ 


Rr 










" 







Vi-b2 














"~~~~ 


■^ 


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.S3-155. (Cases of more than one variate are treated on pp. 155-174 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(l-b^')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{r-b) = P{r;0) = ^^expf-ij. (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 r-range 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.e-j< 

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 6-range (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 single-integral 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 


r-i/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, 0-to-l. 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 
(53-A), (56-A), (52-A), (55-A) and (22-A), 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 (22-A) 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. 255-332. (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 one-third 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 one-third 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 


" 


.0016 


.0015 


.0017 


.0016 


.0012 


.ODIO 


.0011 


.0039 


-.0002 


-.00019 


K 


.19 


.18 


.20 


.19 


.15 


.12 


.14 


.53 


-.03 


-.03 


0.6 


.6977 


.6880 


.6799 


.6686 


.6531 


.6324 


.6055 


.5721 


.5578 


.54851 


<( 


.6992 


.6892 


.6814 


.6698 


.6540 


.6334 


.6065 


.5764 


.5572 


.54831 


(( 


.0015 


.0012 


.0015 


.0012 


.0009 


.0010 


.0010 


.0043 


-.0006 


-.00020 


K 


.22 


.17 


.22 


.18 


.14 


.16 


.17 


.75 


-.11 


-.04 


0.8 


.5273 


.5167 


.5081 


.4969 . 


.4826 


.4656 


.4477 


.4316 


.4261 


.42371 


" 


.5290 


.5183 


.5099 


.4982 


.4840 


.4672 


.4488 


.4357 


.4266 


.42355 


" 


.0017 


.0016 


.0018 


.0013 


.0014 


.0016 


.0011 


.0041 


.0005 


-.00016 


II 


.32 


.31 


.35 


.26 


.29 


.34 


.25 


.95 


.12 


-.04 


1.6 


.07730 


.07986 


.08207 


.08522 


.0891 


.0938 


.0990 


.1042 


.1070 


. 10960 


" 


.07727 


.07988 


.08210 


.08536 


.0892 


.0938 


.0989 


.1042 


.1069 


. 10958 


'< 


-.00003 


.00002 


.00003 


.00014 


.0001 


.0000 


-.0001 


.0000 


-.0001 


-.00002 


" 


-.04 


.03 


.04 


.16 


.11 


.00 


-.10 


.00 


-.09 


-.02 


2.0 


.01832 


.02153 


.02394 


.02681 


.0301 


.0337 


.0375 


.0414 


.0435 


.04550 


" 


.01823 


.02145 


.02383 


.02685 


.0302 


.0338 


.0376 


.0415 


.0436 


.04552 


<( 


-.00009 


- .00008 


-.00011 


.00004 


.0001 


.0001 


.0001 


.0001 


.0001 


.00002 


(( 


-.49 


-.37 


-.46 


.15 


.a 


.30 


.27 


.24 


.23 


.04 



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 0-4 = 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- ^ j-qj] • (6-5) 




MmP{d;b)/MsixP(6;b) ^ (l-6)/(l + 6), (6.6) 

P{e;b)/MiixP{e;b) = P{d;b)/PiO°;b) = {l-b)/{l-b 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)/(l-6cos2^). (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 




/ 


/ 


r 




















^ 






/ 


















^ 


-^ 


f/^ 














II 

0) 
0) 

d 




Ol 




-^ 


^ 


/ / 


/ J 


^^^ 














o ^ 

a 
o 


y 


^ — 


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/(1-6 cos 20). 



(6.11) 



PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 



349 







■ 


























































































1 
















\ 










' 






o 














/ 


1 












\ 


/ 




1 


/ 


/ , 
















C\j/ 

d/ 


/ 


1 






/ 














/ 


m / 
61 


J 




/ 


/ 


/ 










/ 


1 


0.5 




/ 


/ 


/ 


\ 




/ 




/ 




/ 


<0 / 

6/ 


/ 


/ 


\ 










/ 




/ 




V 


/ 


/ 










' / 


/ 


/ 


} 


/ 


o 


/ 






/ 




// 


'/ 


/ 


/ 


/ 


/ 


'01 

d 






/ 


/ / 


// 


// 


/ 


/ 


y 


/ 




\ 






/// 


O 


/ 




y 


/ 






/en 
/ o 

/ 'I 




l/// 


// 


/ 


^ 


^ 


X 






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 0-integral 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 well-known 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 i-6cos2d 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) 



I-e 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 





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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/2-w = 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 one-fourth the area of the entire scatter diagram. Hence 
Q(360°;b) = 4Q{90°;b) = 1, which is evidently correct. 

Acknowledgment 

The computations and curve-plotting 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/a-6"). 

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 6-range, 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 end-value of the 
Tc-range, 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 Tc-range corresponding to the 6-range, 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/{l-b), (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. 91-98, Dec. 1944; H. S. Black "AN-TRC-6 A Microwave Relay Sys- 
tem", Bell Labs. Record, V. 33, pp. 445-463, 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 Non-linear Coils", Bell Labs. Record, 
Vol. 23, pp. 445-463, 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 magnitude-time 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 straight-forward, for it is apparent that the 
various components in the frequency spectrum must add up to the desired 
magnitude-time 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- 
tude-time 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 
magnitude-time 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 magnitude-time function of a complex wave is the 
sum of all the components of the frequency spectrum, we have only to mul- 
ti])ly this magnitude-time 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 magnitude-time 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 magnitude-time 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 non-periodic 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 non-periodic 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 magnitude-time function. The mag- 
nitude-time 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 magnitude-time function and a 
unit sinusoid at each frequency. However, the actual transformations 
in the case of the non-periodic 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- 
nitude-time 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 magnitude-time 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. 

Non-Linear 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 non-linear components, in cases where the transmission through the 
non-linear elements is independent of frequency. Under these conditions, 
the magnitude-time representation of the wave can be used in computing 

'Whittaker and Robinson, Calculus of Observations. 



364 



BELL SYSTEM TECHNICAL JOURNAL 



llie transmission over each non-linear 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 non-linear elements as the modulators and limiters generally 
encountered are substantially independent of frequency. 

Frequency Spectrum of the Single Pulse 

The single pulse is a non-periodic 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 magnitude-time 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 



MAGNITUDE-TIME 
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 magnitude-time 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 1-kJT 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 U-AT2 


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 



TITTTITfTTITrTTrrn-rTT-n-T-r.-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 n-l 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 C-V 



(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 non-periodic. 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 magnitude-time 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 two-position 
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 d-c 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( 1-k — 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 




2C-V 












^- 






































• 








































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 "on-off" 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 Magnitude-time function e{t) is given by the d-c 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 non-periodic 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 J-oo / 

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 . n-K (. . 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 d-c 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 J-c 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: 1922-1926} 
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 8-channel microwave 
relay system is described. Known to the Army and Navy as AN/TRC-6, 
the system uses radio frequencies approaching 5,000 megacycles. At 
these frequencies, there is a complete absence of static and most man-made 
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 four-millionth 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/TRC-6 system provides are high-grade telephone 
circuits and can be used for signaling, dialing, facsimile, picture transmission, 
or multichannel voice frequency telegraph. Two-way voice transmission 
over radio links totaling 1,600 miles, and one-way 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 angle-of-arrival measurements 
and by observations of multiple -path transmission. 

The Ejffect of Non-Uniform 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 non-uniformly 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 non-uniform 
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 current-density 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 non-homogeneities 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 well-defined fine-weather 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 Cathode-Ray Tube for Vieiving Continuous Patterns? J. B. Johnsox. 
A cathode-ray 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. 

Metal-Lens 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 snapping-shrimp, 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 Photo-Cell Employing a PJwio-Conductive Effect in Silicon. 
Some Properties of High Purity SiliconP G. K. Teal, J. R. Fisher, and 
A. W. Treptow. a pure photo-conductive effect was found in pyrolytically 
deposited and vaporized silicon films. An apparatus is described for 
making bridge type photo-cells by reaction of silicon tetrachloride and 
hydrogen gases at ceramic or quartz surfaces at high temperatures. The 
maximum photo-sensitivity occurs at 8400-8600A 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 
non-conducting 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. 

Non-Uniform Transmission Lines and Reflection Coefficients}^ L. R. 
Walker and N. Wax. A first-order differential equation for the voltage 
reflection coefficient of a non-uniform line is obtained and it is shown how 
this equation may be used to calculate the resonant wave-lengths 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, 1917-18; Fellow of the American 
Scandinavian Foundation, 1919; Columbia University, 1919. Western 
Electric Company, 1920-25; 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, 
1906-07. Western Electric Company, Engineering Department, 1907-11. 
American Telephone and Telegraph Company, Engineering Department, 
1911-19; Department of Development and Research, 1919 34. Bell 
Telephone Laboratories, 1934-. Mr. Hoyt has made contributions to the 
theory of loaded and non-loaded 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, 1934-35. R.C.A. \'ictor Manufacturing 
Company, 1935-36; 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 

NEW YORK 



50^ per copy $1.30 per Year 



THE BELL SYSTEM TECHNICAL JOURNAL 

Published quarterly by the 

American Telephone and Telegraph Company 

195 Broadway^ New York, N. Y. 



EDITORS 
R. W. King J. O. Perrine 

EDITORIAL BOARD 

W. H. Harrison O. E. Buckley 

O. B. Blackwell M. J. KeUy 

H. S. Osborne A. B. Clark 

J. J. PiUiod F. J. Feely 

SUBSCRIPTIONS 

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Copyright, 1947 
American Telephone and Telegraph Company 



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 OX-OFF pulses. 2" 
amplitude values can be represented by an n digit binary number code. For a 
nominal 4 kc. speech band these n OX-OFF pulses are transmitted 8000 times a 
second. Experimental ef|uipment 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 three-unit code appears to be necessary for a 
minimum grade of circuit, while a six- or seven-unit 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 
8-channel 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 time-division 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 time-division 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 time-division 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 ON-OFF or two-position pulses. 2" discrete levels can be 
represented by a binary number of n digits.* Thus, PCM represents each 
quantized amplitude of a time-division sampling process by a group of 
ON-OFF 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 time-division 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 low-pass 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 d-c 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 low-pass 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 
ON-OFF 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 ON-OFF 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 signal-to-noise 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 five-digit code. For a time-division 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 low-pass 



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 ON-OFF 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 lOO-link system must huNc a signal-to-noise ratio 20 db 




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399 



400 BELL SYSTEM TECHNICAL JOURNAL 

better than the complete system. For PCM, however, with regenerative 
repeaters the required signal-to-noise 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 5-digit 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 single-link 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 signal-to-noise 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 d-c 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 ON-OFF 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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402 BELL SYSTEM TECIIMCAL JOLRXAL 

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 16-unit 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 
16-unit 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 16-unit 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 8-unit 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 



Jl 



il 



i 



i 



11 



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8. CODE GROUPS 



Fig. 5 — Detailed wave forms for PCM Transmitter (amplitude vs. time). 

403 



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BELL SYSTEM TECHNICAL JOURNAL 



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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 timing-pip generators. Two sets of 
timing-pips 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 timing-pips 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 timing-pips 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 low-pass 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 air-path separated the terminals. 
The transmitter used a pulsed magnetron oscillator and the receiver em- 
ployed a broad-band 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 A-B 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 time-division 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 two-fold 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 + p-R-Y'-. 

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 







































































/ 




1 


1 


1 














1 






1 
1 
1 


1 




















1 
1 


( 




















1 


/ 


















1 


1 










1 


r^ 


10 




1 

1 

no 


1 










c N 




\ 




1 

( 
1 
1 
1 
1 
1 
1 


J 

/ 

/ 


1^1 












\ 


^ 


L 


1 

1 
1 
1 


/ 

/ 

/ 






,'(\j 












^^^^ 


1*"--^ 


/ 








_ 










^ 


— \ ^< 




\y 


^-'1 




ij 

c 












V 


"\-. 




/ / 
/ / 


















\- 


■"r"" 


/ 1 't 


















V, 


/ 


1 1 






















/ 1 

/ » 
/ " 




















1 


/ '' 




1 
















\_ 


^ '' 1 




I 
/ 


















/'/ 




/ 






















/ 


















1 

1 

1 




/ 
/ 




















J 




















1 


/ 




















V.- 


/ 





























O 



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 



2-K , 1 . w > (9) 

cos V? + - sm ip 



r cos (p 



2-K , 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 high-P 
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 10-0. 

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^-\'\'''-'-''i-y^'':-':?r//.'- 




II 

3) 


m = 4/ 


/ 






W'M^:-0B0--i-M 


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=aYi-e2 = a(i-E) 
A=Tra2'Yi-e2=Tra2 o-e) 



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 self-explanatory, 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: 



H-dr = 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 {n-R 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 + 0-1622 «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 \ x-J 



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'ti-nr) 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 half-cylinder 



before. Indeed, they will also be the same in the crescent-shaped 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 cross-section 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 

1. E. I. Green, H. J. Fisher, J. G. Ferguson, "Techniques and Facilities for Radar Test- 

ing." B.S.T.J., 25, pp. 435-482 (1946). 

2. I. G. Wilson, C. W. Schramm, J. P. Kinzer/'High Q Resonant Cavities for Micro- 

wave Testing" B.S.T.J., 25, pp. 408-434 (1946). 

3. J. R. Carson, S. P. Mead, S. A. Schelkunoff, "Hyper-Frequency Wave Guides — 

Mathematical Theory," B.S.T.J., 15, pp. 310-333 (1936). 

4. G. C. Southworth, " Hyperf requency Wave Guides — General Considerations and 

Experimental Results," B.S.T.J., 15, pp. 284-309 (1936). 

5. W. W. Hansen "A Type of Electrical Resonator," Jour. A pp. Phys., 9, pp. 654-663 

(1938). — A good general treatment of cavity resonators. Also deals briefly with 
coupling loops. 

6. W. W. Hansen and R. D. Richtmyer, "On Resonators Suitable for Klystron Oscil- 

lators," Jour. A pp. Phys., 10, pp. 189-199 (1939). — Develops mathematical methods 
for the treatment of certain shapes with axial symmetry, notably the "dimpled 
sphere," or hour glass. 

7. W. L. Barrow and W. W. Mieher, "Natural Oscillations of Electrical Cavity Reso- 

nators," Proc. I.R.E., 28, pp. 184-191 (1940). An experimental investigation of 
the resonant frequencies of cyhndrical, coaxial and partial coaxial (hybrid) cavities. 

8. R. Sarbacher and W. Edson, "Hyper and Ultrahigh Frequency Engineering," John 

Wiley and Sons, (1943). 

9. R. H. Bolt, "Frequency Distribution of Eigentones in a Three-Dimensional Con- 

tinuum," J.A.S.A., 10, pp. 228-234 (1939) — Derivation of better approximation 
formula than the asymptotic one; comparison with calculated exact values. 

10. Dah-You Maa, "Distribution of Eigentones in a Rectangular Chamber at Low-Fre- 

quency Range," J.A.S.A., 10, pp. 235-238 (1939)— Another method of deriving an 
a^jproximation formula. 

11. I. G. Wilson, C. W. Schramm, J. P. Kinzer, "High Q Resonant Cavities for Micro- 

wave Testing," B.S.T.J., 25, page 418, Table IK (1946). 

12. L. Brillouin, "Theoretical Study of Dielectric Cables," Elec. Comm., 16, pp. 350- 

372 (1938)— Solution for elliptical wave guides. 

13. L. J. Chu, "Electromagnetic Waves in EUiptic Hollow Pipes of Metal," Jour. App. 

Phys., 9, pp. 583-591 (1938). 

14. Stratton, Morse, Chu, Hutner, "Elliptic Cvlinder and Spheroidal Wave Functions," 

M.I.T. (1941). 



SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 443 

15. J. A. Stratton, "Electromagnetic Theory," McGraw-Hill, (1941). 

16. F. Jahnke and E. Emde, "Tables of Functions," pp. 288-293, Dover Publications 

(1943). 

17. N. VV. McLachlan, " Bessel Function for Engineers," Clarendon Press, Oxford (1934). 

18. F. Borgnis, "Die konzentrische Leitung als Resonator," Hochf: tech u. Elek:akus., 

56, pp. 47-54, (1940). — Resonant modes and Q of the full coaxial resonator. For 
long abstract, see Wireless Engineer, 18, pp. 23-25, (1941). 

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; §315-316 consider two concentric spheres; §317-318 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. 125-132 (1897) .^Considers rectangular 
and circular cross-sections. 

21. A. Becker, " Interf erenzrohren fiir elektrische Wellen," Ann. d. Phys., 8, pp. 22-62 

(1902)— Abstract in Set. Abs., 5, No. 1876 (1902)— Experimental work at 5 cm. 
and 10 cm. 

22. R. H. Weber, " Elektromagnetische Schwingungen in Metallrohren," Ann. d. Phys., 

8, pp. 721-751 (1902)— Abstract in Set. Abs., 6A, No. 96 (1903). 

23. A Kalahne, "Elektrische Schwingungen in ringformigen Metallrohren," Ann. d. 

Phys., 18, pp. 92-127 (1905).— Abstract in Sci. Abs., 8A, No. 2247 (1905). 

24. G. Mie, "Beitrage zur Optik triiber Medien, spezieU kolloidaler Metallosungen," 

Ann. d. Phys., 25, pp. 337-445 (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, " Ultrahochfrequenz-Ubertragung langs zylindrischen Leitern und 

Nichtleitern," TFT, 27, pp. 199-205, 273-279, 310-316, 337-341 (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. 1298-1328 (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. 1457-1492 (1937)— Treats waves in free space and in cylindrical 
tubes of arbitrary cross-section; 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. 583-591 (1938) — A study of field configurations, ci;itical frequencies, 
and attenuations. 

29. G. Reber, "Electric Resonance Chambers," Communications, Vol. 18, No. 12, pp. 

5-8 (1938). 

30. F. Borgnis, " Electromagnetische Eigenschwingungen dielektrischer Raume," Ann. 

d. Phys., 35, pp. 359-384 (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. 38-45 (1939). — Series approximation method for TM OOp mode. 

32. R. D. Richtmyer, "Dielectric Resonators," Jour. App. Phys., 10, pp. 391-398 (1939). 

33. H. R. L. Lamont, "Theory of Resonance in Microwave Transmission Lines with 

Discontinuous Dielectric," Phil. Mag., 29, pp. 521-540 (1940).— With bibliography 
covering wave guides, 1937-1939. 

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. 62-68 (1941). — Series approximation method for certain circularly 
symmetric resonators. 

36. E. U. Condon, "Forced Oscillations in Cavity Resonators," Jour. App. Phys., 12 

pp. 129-132 (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. 119-122 (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. 216-217, 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. 174-180 (1941). 

39. S. Ramo, "Electrical Conce[)ts at Extremely High Frequencies," Electronics, Vol. 9, 

Sept. 1942, pp. 34-41, 74-82. A non-mathematical 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. 90-114 (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. 412-424 (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. 591-595 (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. 228-230 (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. 68-73 (1938). 

46. J. K. Stewart and R. C. Colwell, "The Calculation of Chladni Patterns," J.A.S.A., 

11, pp. 147-151 (1939). 

47. R. C. Colwell, J. K. Stewart, H. D. Arnett, "Symmetrical Sand Figures on Circular 

Plates," J.A.S.A., 12, pp. 260-265 (1940). 

48. V. O. Knudsen, "Resonance in Small Rooms," J.A.S.A., 4, pp. 20-37 (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. 356-357 (1938)— Abstract of Akustische Zeits., 2, pp. 225-241, 
296-302 (1937) — Eigentones in a rectangular chamber. 

50. H. E. Hartig and C. E. Swanson, "Transverse Acoustic Waves in Rigid Tubes," 

Pliys. Rev., 54, pp. 618-626 (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. 695-698 (1938) and 269, pp. 664-666 

(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- 

urements," J.A.S.A., 10, pp. 216-227 (1939). 
,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, 21-28, (1939). — Pages 26-28 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. 74-79 (1939). — Eigentones in rectangular chamber. 

57. R. H. Bolt, "Normal Modes of Vibration in Room Acoustics: Experimental Investiga- 

tions in Non-rectangular Enclosures," J.A.S.A., 11, pp. 184-197 (1939). 

58. L. Brillouin, "Le Tuyau Acoustique comme Filtre Passe-Haut/' Rev. D'Acoiis., 8, 

pp. 1-11 (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 Non-Planar Walls," 

J.A.S.A., 11, pp. 364-365 (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. 1-7 

(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. 65-73 (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, "Ultra-Short Waves in Concentric Cables, and the "Hollow-Space" 

Resonators in the Form of a Cylinder with Perforated-Disc 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 Cone-Shaped Horn," 

17, p. 370, No. 3009 (1940). — Cavity formed by cone closed by spherical cap. 

72. O. Schriever, "Physics and Technique of the Hollow-Space Conductor," 18, p. 18 

No. 2, (1941).— Review of history. 

73. F. Borgnis, "Electromagnetic Hollow-Space 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 Ultra-High-Frequency 

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 Two-Layer 

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 Saw-Tooth Oscillations," 19, 

p. 422-423, 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 Ultra-High 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 Micro-wave 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 inipe-iance 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 short-circuited. 

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 0-1 represents the voltage reflected 
from couphng BE and vector 1-2 represents the voltage reflected from the 
termination Z. To make measurements, termination Z should be movable 



4-0 




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 Wz-W, = 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 0-1 represents the voltage at C when input is applied to A , due to 
the impedance mismatch at the coupling BE. Vector 1-2 represents that 
due to the mismatch at coupling D. Vector 2-3 represents that due to 
the mismatch at the variable attenuator, (which will usually change in 
magnitude and probably in phase for different settings). Vector 3-0 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 0-5 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)i-r) 



(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 0-1 = r, vector 1-2 = z(l — r'-), vector 2-3 = rs-(l — r-) (34) 

then 

TFo_i = 10 db, IFi-2 = 11.00 + 0.92 = 11.92 db, 

and IFo-s = 10.00 +22.00 + 0.92 = 32.92 db (35) 

In order to evaluate vector 0-2 in Fig. 11 (the vector sum of vectors 0-1 
and 1-2), 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 W0-2 = 10.00 - 5.10 = 4.90 db (38) 



MEASUREMENT OF IMPEDANCE MISMATCHES 459 

In order to evaluate vector 0-3 in Fig. 11 (the vector difference of vectors 
0-2 and 2-3) 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 TFo-3 = 4.90 dz 0.36 = 5.26 db = TF4 (41) 

In order to evaluate vector 0-2 in Fig. 12 (the vector difference between 
vectors 0-1 and 1-2), one uses T from equation (36). 

For T = 1.92 db, Fo = 14.10 db (42) 

therefore I['o-2 = 10.00 + 14.10 = 24.10 db (43) 

In order to evaluate vector 0-3 in Fig. 12 (the vector difference between 
vectors 0-2 and 2-3), 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 TF0-3 = 24.10 + 3.93 = 2S.03db = IF3 (46) 

Using equation (16) 

Ws-Wi= 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. "Hyper-frequency 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 Standing-Wave 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. Build-up 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 Wave-Guide 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 — Non-simple 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, ij-3)). 

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, G-2 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. 

/ Radio-frequency current. 

h Current induced in circuit by convection current returning across gap. 

h D-C beam current. 

A' Resonator loss parameter (3.9). 

A" Radiation loss in watts/(degree Kelvin)'' {k-2). 

L Inductance. 

M Characteristic admittance (a-8). 

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 (? (a-11). 

Qo Unloaded Q (a-12). 

R Surface resistance (o-2). 

5 Scaling factor (9.17). 

T Temperature. 

V Radio-frequency voltage. 

V Potential in drift space (Appendix VI only). 
I'o D-C 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 (b-37). 

i Radio-frequency convection current. 

72 Radio-frequency convection current returning across gap (c-4). 

{12) f Fundamental component of /•> . 

j V-1 

k Boltzman's constant (1.37 X 10^^ joules/degree A). 

k Conduction loss watts/degree C (yfe-14). 

/ 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, D-C 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 (g-13). 

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 long-transit-time 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 by-pass 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 all-electrical autcmatic 
frequency-control. 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 semi-quantitative 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 radio-frequency 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 cross-section. 

different amounts of kinetic energy from the radio-frequency 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 



R-F VOLTAGE 
ACCELERATING 

FOR ELECTRONS 
FROM CATHODE 



R-F 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 radio-frequency voltage across the gap is a small fraction of the 
d-c 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{ut-6) 



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 comple-v 
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 circuit-admittance 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 radio-frequency electron current in the electron stream returning 
across the gap given by equation (2.1) as a function of the radio-frequency 
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 radio-frequency 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 radio-frequency ga}) voltage 



REFLEX OSCILLATORS 



471 



is increased, the radio-frequency 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 radio-frequency 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 



RADIO-FREQUENCY 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 radio-frequency 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. radio-frequency 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 r-J 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 r-f 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 turn-on 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 r-f 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 K-O 
















\ 




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 
radio-frequency 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. 128-135 of the Appendix. His curves approach 
the curv-es 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. 20-32, 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 l-rr 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 r-f 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 a-f 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 
small-signal 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 space-charge 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 r-f 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)e-J^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 l-ig. 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 
r-f 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 r-J 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'/a-o)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 single-valued 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 self-excited 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 
non-linearities 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 self-starting, 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 w-e vary any one of these parameters 
indicated as X„ the principal effect will be to shrink the vertical ordinates 
as show-n 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 





l-l 


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 
k---AVr2----->| 


\ 












/ 








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- 

U-Z 
















v§ 












(c) 


LU>- 

1-a. 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 Gei-Gti 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. 




T-UJ 




n> 


40 


n-"- 




H-> 




oo 


20 










't'- 





oo 




u,a 




-I 


-20 


o in 




/iiJ 




< UJ 






-4 


UJ u 
(/I UJ 




<n 






-60 


Q-Z 




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 ex-peri- 
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 d-c 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 non-optimum 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 non-optimum 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. 29-31 



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 G-z = .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, 














\ 


\ 




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3^67, 




v 


^ 














\ 




\^ 


\^ 












^^ 


4.36 




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vV 


'^ 













^ 


^ 


rv 




1 




G2 = 0.3 




i 














"^ 


^\ 


^^ 


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^^ 


















X 


^ 




















\x 


m 




















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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 well-designed 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 



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\\ 


\\ 


0.06 
0.04 


III 1 


/ 








\ 










\ 


\ 


\ \ 


w 


If 


f 




V 




\ 







^ 




\ 


\ 


\ 


i 


0.02 


IL 


^.36 




i 






\ 




^ 




\ 


— 


\ 


\i 





r 


\ 










\ 




\ 






1 


\ 


\\\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) 



5-0.5 




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 



10-3, 



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- 



D-O.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 one-half 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^=0.5X10-3 
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^-^ 



■^ 



r-T 



> 



M = 1 00 



M5=200 



\ 




-^=0.5X10-3 
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 one-eighth 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^^ -0.5X10-3 
' 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 one-half 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 frequency-sensitive 
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 """ ' • (9-36) 

(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 


''/ 
















\ 
















/; 


/ 


















\ 


\, 










/ 


^ 


/■ 










^^ 


^ 








w-i 


\ 


\ 


'\ 


^ 


/ 


;^' 














- 
















,^ 


^ 














- 




- 












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^ = GnW-K 
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 half-power 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 r-f 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 non-linear 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 band-width 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. Build-up of Oscillation 

In certain applications, reflex oscillators are pulsed. In many of these 

; it is required that the r-f output appear quickly after the application cf 

' d-c power, and that the time of build-up be as nearly the same as possible 

: for successive applications of power. In this connection it is important to 

study the mechanism of the build-up of oscillations. 

In connection with build-up 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. 41-46 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. 41-46), 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 
build-up period the oscillation will remain so small that nonlinearities are 
unimportant. The exponential build-up 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 (a-22) 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,o-Gc)- (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 build-up characteristic 
of an oscillator can be obtained step-by-step 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 build-up at an average rate given by a = 2.5 X 10~ until 
F = 1. Let Vo = .1. Then the interval to build-up 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 10-6 



.80 X 10"^ seconds. 



The build-up 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 build-up 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 

BUILD-UP, 

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 build-up 
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 build-up 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 liuild-u]) 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 build-up 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 build-up starts on 
Curve 2. When Curve 1 with the cusp is reached, the build-up 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 
\ \ \ BUILD-UP, 

\ \ \ 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 build-up at turn-on. When the build-up is rapid {a = 2) the admittance curve 
has no loop. As the rate of build-up decreases the curve sharpens until it has a cusp a = 1. 
As the rate of build-up 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 build-up to each frequency 
for half of the turn-ons. If one frequency were favored by a slight dis- 
symmetry, the favored frequency would appear on the greater fraction of 
turn-ons. For a great dissymetry, build-up 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 build-up of oscillations 
starts from a randomly fluctuating voltage caused by shot noise. Thus, 
from turn-on to turn-on 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 build-up, 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 r-f pulses associated with 
evenly spaced turn-ons is least when the initial rate of build-up, given by Oo , 
is greatest. 

Such conditions do not obtain on turn-off, and there is little jitter in the 
trailing edge of a series of r-f 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 ex-plore 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 micro-second, 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 x-ray view showing the inter- 
nal construction. A removable external cavity is employed with the 707A 
as indicated by the sketch superimposed on the x-ray 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- 



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BELL SYSTEM TECHNCLAL JOURNAL 




'^h 




Fig. 58. — External view of