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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
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THE BELL SYSTEM TECHNICAL JOURNAL
Published quarterly by the
American Telephone and Telegraph Company
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EDITORS
R. W. King J. O. Perrine
EDITORIAL BOARD
W. H. Harrison
O. B Blackwell
H. S. Osborne
J. J. PiUiod
O. E. Buckley
M. J. KeUy
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•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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SILICON CRYSTAL RECTIFIERS
17
» * CRYSTAL
PARTS I I RECTIFIER
(ENLARGED)
RETAINING PLUG
Fig. 7 — Converter for wave guide circuits as installed in the radio frequency unit of
the AN/APQ13 radar system. This was standard equipment in 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
^^
'"'
/
^
'r'
^^
•
l
■^
^
^
-^
/
/
^^■7
^
■^^"^
/
/
'^A
0-9
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K
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1
cu
rrent c?
1
an
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1
JT 1
ics
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N AMPE
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RE
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silicon (
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:ryE
0-3
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J
however, until certain improved materials, processes, and techniques were
developed.
In the engineering development of improved cr>'stal rectifier materials
and jjrocesses, basic data have been acquired which make it possible to
alter the properties of the rectifier in a predictable manner so that tlie units
may now be engineered to the specific electrical requirements desired by
the circuit designer in much the same manner as are modern electron tubes.
This has led not only to improvements in performance but also to a diver-
sification in types and applications.
The simplified equivalent circuit for the point contact rectifier, shown
in Fig. 11, provides a basis for consideration of the various process features.
In Fig. 11, Cb represents the electrical capacitance at the boundary between
the point contact and the 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
■^
^
'■^
/
■^'
\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 .■
:■ t ■:
:• z:-
■■- ^.■■
.'z'.-.
; o ••
.'z ■■■
. O . ■'
■ ■ to-
II
,■.■■ 01 :
>t
ctz
'OZ>
h- ■
10
■ UJ • .
. H ■
:• u ;•
• UJ •:
'.-!■.
•; Q •
• o •;
irz.
. ■ Q •.
■. _J ■.
■• cr. ■■
■. > •
•. UJ ■
;■ Q ;.
■ : >- ■
•• O .
■ a.-
■: Q- .'•
. u ■•
• 1- ■•.
. o ■■■
m
ao
: < ■■
. _j ■
: <-!■
• Q--/
■ Q ■•■.;
orr
'.■ ^ .
•.■ cc '.
•: t^ '■'.
' a. ■
• CEO
■•' o ."
:o v
(CK-
:o(r
.cr • ;
<z
.. UJ ■
, o ■
•o ■■
.cDi-:
y
-
-lO'
• o .
■■o •:
<^
■X-
• o ..
vo ::•
.-. o-
■•fO .',
■_]0.
8
• •- • 1 ■ ■
'.'.■.■•-.•
". • ■ .
,
OCT
1942
DEC
1942
MAR APR
1945
SEPT
JAN JULY
SEPT
NOV
APR
1941
1942
1943
1944
1945
DATE
* note: — 6 DECIBELS IS THE MINIMUM 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
//. = 0
k = ^ ^ kl-^ kl
A
^1
2r , _ nv
D ' ~ L
r = ;;;"' root of J f{x) = 0 for TM Modes.
= m"' root of /;.(.v) = 0 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 0 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
0
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 = 0 is first discussed. For ^ = 1 , /i(0) = 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 = 0 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( = 0 (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) = 0 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 - 0 - ^) .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
0
.1
.2
.3
.4
.5
1291
.6
.7
.8
.9
0
0
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
0
+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)
0
.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
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
0
0
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
0
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
0
.1
.2
.3
0152
.4
.5
.6
0604
.7
.K
M
0
0
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
0
.1
9983
.2
.3
.4
9734
..s
9589
.6
.7
.8
.9
0
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
0
+ 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
+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
0
.2
.3
0006
.4
.5
.6
0044
.7
0069
.8
0102
.9
0
+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
0
.4
.5
.6
.7
0006
.8
.9
0
+0
0
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
+0
0
0
0
0
0
0
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
0
.2
.3
.4
.5
.6
.7
.8
.9
0
0
0
0
0
0
0
0
0
0
1
0
0
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
0
.8
0
.9
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
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
0
+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
+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
+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
+0
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
+0
0
0
0
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
0
.7
.8
.9
0
+0
0
0
0
0
0
0
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 0 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 — 0 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 = -— = 0 ;
dx2
dxi „
dX3
dX2 .
9X2 n
„2 = = 0
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— = 0
oxi 0X2 dXz
^3 = ^^ = .866; m2=^=-.5; «2 = ^^ = 0 (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 = 0
cn ; ^12 ; ^13 ; cu ; fis = ~<^25 ; ^le = 0
c\2 ; C21. — c\\ ; C23 = c\i ; C24 = — '"14 ; C25 ; ^26 = 0
Cn ; C20 = C\3 ; f33 ; ("34 = 0; czh — ^\ C36 = 0
(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 = 0
612 = 0; 622 = 6U ; 623 = 0 (131)
613 = 0; €23 = 0; €33
flu ; fin = — //ii ; //13 = 0; //i4 ; //I5 = 0; //i6 = 0
//21 = 0; //22 = 0; //23 = 0; //24 = 0; h, = -hu ; //26 = -hn (132)
//3l = 0; //32 = 0; //33 = 0; //34 = 0; hy, - 0; //36 - 0
PIEZOELECTRIC CRYSTALS IN TENSOR FORM
115
(133)
Cn ; Ci2 ; C\3 ; Cu ; cis = 0; cie = 0
Cn 5 ^22 = ^11 ; ^23 = C\3 ; C24 = Ci4 ; C25 =0; C26 = 0
Ci3 ; ^23 = Ciz ; (^33 ; C34 = 0 ; C35 = 0 ; Cae = 0
fi4 ; C24 = — Ci4 ; r34 = 0; <:44 ; C45 = 0; r46 = 0
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 , 0 , €13
0 , €22 , 0
ei3 , 0 , €33
€11,0 ,0
0 , 622 , 0 (134)
0 ,0 , €33
€11,0 ,0
0 , €„ , 0
0 ,0 , €33
€11,0 ,0
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 = 0 (Class 2)
0 ,0 ,0 , //14 , 0 , /?16
hii , lin , fhz ,0 , //26 , 0
0 ,0 ,0 , //34 , 0 , /;,6
hn , Ih2 , hn ,0 , /7i6 , 0
0 ,0 ,0 , /724 , 0 ,ht
hi , /'32 , /'33 , 0 , hsB ,0
Monoclinic prismatic (center of symmetr>0 h = 0 (Class 5)
0 ,0 ,0 , /7i4 , 0 ,0
0 ,0 ,0 ,0 ,//26,0
0 ,0 ,0 ,0 ,0 ,//36
0 ,0 ,0 ,0 ,/;i6,0
0 ,0 ,0 , //24 , 0 ,0
/?31 , //32 , //33 , 0 ,0 ,0
Orthorhombic bipyramidal (center of s}mmetr>-) // = 0 (Class 8)
0 , 0,0, liu , liib , 0
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 , 0,0, -//15, /7l4,0
//31 , -/'31 , 0 , 0,0, //36
0 ,0 ,0 , Ihi , //15 , 0
0 ,0 ,0 ,//l5, -//i4,0
//31 , //31 , //33 , 0 , 0 ,0
Tetragonal scalenohedral (Class 11) / I 0 ,0 ,0 , liu ,0 ,0
quaternar\'. A' and I' binary ,^ ,^ ,, ,, , n
^ ' 0 , 0 , 0 , 0 , //i4 , 0
0 ,0 ,0 ,0 ,0 , //36
Tetragonal trapezohedral (Class 12) jO ,0 ,0 , Im , 0 ,0
Z quaternar^^ A' and F binar^^ 0.0,0,0, -/;. , 0
I 0 , 0 , 0 , 0 , 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 ,0 , /;,5 , 0
0 .0 .0 ,//i5,0 ,0
/-■■U , /?.l , //33 , 0 ,0 ,0
Ditetragonal bipyramidal (center of symmetry) // = 0 (Class 15)
Trigonal pyramidal (Class //u , — //u , 0 , hu , /?i5 , —fi'n
16) Z trigonal axis / a / / /
— //22, /^22 , 0 , /;i5 —hu,—fin
hn , //;u , //3.3 , 0 , 0 , 0
Trigonal rhombohedral (Class 17) center of symmetry, // = 0
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 , 0
/?14 ,
0 ,
()
0 ,
0 ,0
0 ,
-Ihi ,
-hn
0 ,
f) , 0
0 ,
0
0
//ll.
-//11,()
0 ,
0
, — A22
-//22,
//22 , 0
0 ,
0
, -hn
0 ,
0 , 0
0 ,
0
, 0
0 ,
(» , 0
0 ,
//15
-//22
-//22,
//?2 , 0
//15,
0
0
hu ,
Ihl , //33
0 ,
0
. 0
1) center of symmetry.
// =
0
hn,
-//u ,0
, 0 ,
0
, 0
0 ,
0 ,0
,0 ,
0
, -hn
0 ,
0 , 0
,0 ,
0
, 0
0 ,
0 , 0
, Ihi ,
//15
, 0
0
0 , 0
, flu ,
— hu
, 0
hi ,
//31 , //33
, 0 ,
0
, 0
0 ,
0 , 0
, //14 ,
0
, (•
0 ,
0 ,0
,0 ,
-//14
, 0
0 ,
0 ,0
,0 ,
0
, 0
1 18 BELL SYSTEM TECH NIC A L JOURNA L
Hexagonal bipyramidal (Class 25) center of symmetry, /? = 0
Dihexagonal pyramidal (Class 26) .Y
hexagonal Y plane of symmetry
0 ,0 ,0 ,0 ,/7i5,0
0 ,0 ,0 ,/7i5,0 ,0
h\ , //31 , /'33 , 0 ,0 ,0
Dihexagonal bipyramidal (Class 27) center of symmetry, h = 0
Cubic tetrahedral-pentagonal-dedo-
cahedral (Class 28) A', V, Z binary
0 ,0 ,0 ,hu,0 ,0
0 ,0 ,0 ,0 , //i4 , 0
0 ,0 ,0 ,0 ,0 ,/;,4
Cubic pentagonal-icositetetrahedral (Class 29) ^ = 0
Cubic, dyakisdodecahedral (Class 30) center of symmetry, // = 0
Cubic, hexakisletrahedral (Class 31)
X, I', / quaternary alternating
0 ,0 ,0 , /;i4 , 0 ,0
0 ,0 ,0 ,0 , //i4 , 0
0 ,0 ,0 ,0 ,0 ,/7i4
Cubic, hexakis-octahedral (Class 32) center of symmetry, // = 0
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 0 du dn —Id^i
— dvt d^i 0 </i5 —du —2dn
dn dsi d33 0 0 0
^u -dn 0 du 0 0
0 0 0 0 -du -2dn
0 0 0 0 0 0
dn -dn 0 0 0 -2^22
-da (/22 0 0 0 -2dn
0 0 0 0 0 0
^11 -dn 0 0 0 0
0 0 0 0 0 -2dn
0 0 0 0 0 0
(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
0
fl5
0
The s tensor is
(Classes 3, 4 and 5) 12
moduli
Cl2
Co.i
C03
0
C2b
0
entirely analo-
gous
C\3
Co.3
C33
0
Csb
0
0
0
0
Cii
0
C4f,
Cl5
f"25
<"36
0
(^55
0
0
0
0
C46
0
^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
0
0
0
The s tensor is
en
C22
C23
0
{)
0
entirely analo-
gous
Cn
C23
C33
0
0
0
0
0
0
C44
0
0
0
0
0
0
(-"55
0
0
0
0
0
0
CbG
Cn
C\2
Cn
0
{)
Cu
The s tensor is
C\2
Cn
Cn
0
0
— Cu
entirely analo-
gous
Cn
Cn
C33
0
0
0
0
0
0
C44
0
0
0
0
0
0
Cu
0
^16
-C16
0
0
0
Cf,e,
Cn
C\2
Cn
0
0
0
The i- tensor is
Cu
Cn
Cn
0
0
0
entirely analo-
gous
("13
Cn
C33
0
()
0
0
0
0
Cii
0
0
0
0
0
0
(44
0
0
0
0
0
0
(-"6fi
C\\
Cl2
Cn
("14 -
-t-25
0
The 5 tensor is
cu
Cn
Cn
— ("14
("26
(^
analogous ex-
cept that 546 =
Cn
Cn
C33
0
n
0
2^25 , ■^56 = 2^14 ,
Cu
-(14
0
-(■44
0
'25
^66 = 2 (511 — ^12)
— f26
(-25
0
0
("44
("14
0
0
0
("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
0 0
Cn
C\1
C\3
0
0
Cxi
Cn
0
0
0
Cn
Cn
Cn
Cn
Cn
C\z
0
0
Cn
Cn
Cn
0
0
0
Cn
Cn
Cn
Cn Cu
Cn — Cu
C33 0
Cn
Cn
C3Z
0
0
0 0
Cn
Cn
Cn
0
0
0
Cn
Cn
Cn
0
C44
0
0 0 0 0
0
0
0
C44
0
0
0
0
0
0
0
0
0
C44
Cu
Cu
Cn-
Cn
2
0
0
0
0
Cii
0 0 0 0 0
0
0
0
Cn
0
0
0
0
0
Cn Cn
0
0
0
0
C44
0
0
0
0
0
0
0
0
0
^11 ~ Cn
2
0
0
0
0
0
Cn
0
0
0
0 0 0 0
Cn ~ Cn
0 0
0 0
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 = 0 (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 0 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 0 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 0 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
0
-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 0 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 ,
0 ,
0
•^12 ,
522,
523,
524 ,
0 ,
0
•^13 ,
■^23 ,
533 ,
534 ,
0 ,
0
•^14 ,
■^24 ,
Sm ,
544 ,
0 ,
0
0 ,
0 ,
0 ,
0 ,
555 ,
5&6
0 ,
0 ,
0 ,
0 ,
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 0 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
0
0
0
0
0
0
0
0
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)
0 0
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 , 0
0, e,,, 623 (182)
0 , €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 = 0 (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 = 0 (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 0 (/).
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) \
0 ,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,„ = 0
2» Amn + kl (n -j- m — 3) Am-l.n+l
+ ^1 (w — w + 3) A „,-!,„+! + 2kon Am-\.n = 0
2m ki Amn -\- {m — n — 3) Am-l,n^l
+ (m -f n + 3)A„.+i,„+i + 2^ow ^m,„+i = 0
2 m h Amn + {m -]r n — 3) Am-i.n-\
-\- {m — n -\- 3)Am+l,n-l + 2^oW A^.n-l = 0
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
0 Q2 0.4 0.6 08 1.0 1.2 1.4 1.6 1.8 2 0
^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.
< 0
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 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
0 = 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
0 ■>!■
3 0
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 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
0 — 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
0 = 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
z
,<
y,n-2
KT'
\
v\
\
\
\
\i
>
^
\
1
1-
c^
-v^
^
Cr
^ ^
■^,
PL
.ATir
guM
-
-100
0 100 200
TEMPERATURE °C
300
400
Fig. 2. — Logarithm of specific resistance versus temperature for three thermistor ma-
terials as compared with platinum.
whose electrical conductivity at or near room temperature is much less than
that of typical metals but much greater than that of typical insulators.
While no sharp boundaries exist between these classes of conductors, one
might say that semiconductors have specific resistances at room tempera-
ture from 0.1 to 10* ohm centimeters. Semiconductors usually have high h
negative temperature coefKicients of resistance. As the temperature is
increased from O^C. to 300°C., the resistance may decrease by a factor of a
thousand. Over this same temperature range the resistance of a typical
metal such as platinum will increase by a factor of two. Figure 2 shows
how the logarithm of the specific resistance, p, varies with temperature, T,
in degrees centigrade for three typical semiconductors and for platinum.
PROPERTIES AND USES OF THERMISTORS
175
Curves 1 and 2 are for Materials No. 1 and No. 2 which have been ex'ten-
sively used to date. Material No. 1 is composed of manganese and nickel
oxides. Material No. 2 is composed of oxides of manganese, nickel and
cobalt. The dashed part of Curve 2 covers a region in which the resistance-
temperature relation is not known as accurately as it is at lower tempera-
tures. Curve 3 is an experimental curve for a mixture of iron and zinc
2
U 10-
2
5
y 10-
/
r
y
/
/
—
/
/
)
/
/
-^
--
-y—
— ^
/^ —
f
v^
/
/
/
/
/
/
/
J. —
~A~
- --
t 7.
'szr
/
/
" /
/
.r
/
/
/
/ J
/
/
/
/
/
/
/ /'
' /
/
/
3.0
xiO''
temperature: °k
Fig. 3. — Logarithm of the si)ecific resistance of two thermistor materials as a function
of inverse absolute temperature. See equation (1).
oxides in the proportions to form zinc ferrite. From Fig. 2 it is obvious
that neither the resistance R nor log R varies linearly with T.
Figure 3 shows plots of log p versus l/T, for Materials No. 1 and No. 2.
These do form approximate straight lines. Hence
BlT
Pooe or p = poe
(,bIt)-{bitq)
(1)
where T = temperature in degrees Kelvin; p„ — p when T = oo or \/T = 0;
P{i = p when T = To ; e = Naperian base = 2.718 and 5 is a constant equal
to 2.303 times the slope of the straight lines in Fig. 3. The dimensions of B
176 BELL SYSTEM TECHNICAL JOURNAL
are Kelvin degrees or centigrade degrees; it plays the same role in equation
(1) as does the work function in Richardson's equation for thermionic
emission. For Material No. \, B — 392()C°. This corresponds to an elec-
tron energy equivalent to 3920 11600 or 0.34 volt.
While the curves in Fig. 3 are approximately straight, a more careful
investigation shows that the slope increases linearly as the temperature
increases. From this it follows that a more precise expression for p is:
, T — c PIT
p = A 1 6 or
log p = log .1 - r log T + D/2.303r (2)
The constant c is a small positive or negative number or zero. For Ma-
terial No. 1, log A = 5.563, < = 2.73 and D = 3100. For a particular
form of Material No. 2 log .1 = 11.514, c = 4.83 and D = 2064.
If we define temperature coefilicient of resistance, a, by the equation
a = {\/R) {(IR/dT) (3)
it follows from equation (1) that
a = -B/r. (4)
For Material No. 1 and T - 300°K, a - -3920/90,000 = -0.044. For
platinum, a — +0.0037 or roughly ten times smaller than for semiconduc-
tors and of the opposite sign. From equation (2) it follows that
«= -{D/D- (c/T). (5)
From equation (3) it follows that
a = (1 2.303) {(flogR'dT). (6)
For a discussion of the nature of the conductivit}^ in semiconductors,
it is simpler and more convenient to consider the conductivity, a, rather
than the resistivity, p.
a = \/p and logo- = —log p. (7)
The characteristics of semiconductors are brought out more clearly if the
conductivity or its logarithm are plotted as a function of \/T over a wide
temj:;erature range. Figure 4 is such a j)lot for a number of silicon sam-
ples containing increasing amounts of impurity. At high temperatures
all the samples have nearly the same conductivity. This is called the
intrinsic conductivity since it seems to be an intrinsic properly of silicon.
At low temperatures the conductivity of different sami:)les varies by large
factors. Tn this region silicon is said to be an impurity semiconductor.
For extremely i)ure silicon only intrinsic conductivity is present and the
PROPERTIES AND USES OF THERMISTORS
177
resistivity obeys equation (1). As the concentration of a particular im-
purity increases, the conductivity increases and the impurity conductivity
predominates to higher temperatures. Some impurities are much more
effective in increasing the conductivity than others. One hundred parts
per million of some impurities may increase the conductivity of pure silicon
at room temperature by a factor of 10^ Other impurities may be present
7 '0
O
310
I
o
bio-
o
§'0-
o
—I
\
\
2
\
1
V
/^^"^
\
1
^^.
..^
0
\
"-~-^
"^
^
s
*""***
^^
•
\,
^
-1
\. ..
S
=s—
^
X
\ —
-^
\
s.
S,
A
N
^
s.
■?
\
\
■\
>^
\ \
\ \
\ ^
X
-3
\
\
""^^
— \—
-Vi—
\
\,
\
V
V
4
\
\
s.
^
^^^-^
-5
\_
>-^
-\ —
"^*^«<
\
g
i-i-
\
o
s
-6
7
V
1
l'
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
— \i
s^- —
X,
N
v^
^k. -
■
N
X.
\^
^^
V^
V.
^
*
t^
^
^,
K
\\\.
^
>v
\
1:
X-
V
=»^
\v
^
^
— *%
\ ^
\l
v
^S
\
X
s.
\
\^
k
\
\,
\
N
^i^^
\
-^^^
\
\
^
t
V —
\
\
\
— ^
\
>
\
-^
\
0
0
0 —
OJ
0
\
1^
^
0
1
0
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"
-
-
-
~
-
—
-
-
-
r-
-
—
-
-
—
-
10^
,
10*
J
•
-
10*
o
?n
f\J
ijlO*
5
.
|l03
z
"~
W
•
w
'
y id'
o
s
10°
,
I0-'
irrZ
6
xlO'^
'"0 I 2 3 4
B IN X AT 25t
Fig. 7.— Logarithm of the resistivity of various semiconducting materials as a func-
tion of B in equation (I). The quantity, B, is proportional to the temperature coefiicient
of resistance as given in equation (4).
Physical Properties of Thermistors
One of the most interesting and useful properties of a thermistor is the
way in which the voltage, F, across it changes as the current, /, through
it increases. Figure 8 shows this relationship for a 0.061 centimeter diam-
eter bead of Material No. 1 suspended in air. Each time the current is
PROPERTIES AND USES OF THERMISTORS
183
changed, sufficient time is allowed for the voltage to attain a new steady
value. Hence this curve is called the steady state curve. For sufficiently
small currents, the power dissipated is too small to heat the thermistor
appreciably, and Ohm's law is followed. However, as the current assumes
larger values, the power dissipated increases, the temperature rises above
ambient temperature, the resistance decreases, and hence the voltage is less
than it would have been had the resistance remained constant. At some
current, !„ , the voltage attains a maximum or peak value, Vm • Beyond
/^^
Xso
h
\
\
6(fS.
^^s^
I
100
^^^--
'
..
""""^55
2
0.5
0 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 = 0 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 / = 0 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
'^ ■
— — ''
0
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.
0
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
0
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 0 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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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
/THERMISTOR
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
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THE BELL SYSTEM TECHNICAL JOURNAL
Published quarterly by the
American Telephone and Telegraph Company
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EDITORIAL BOARD
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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 0 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 0 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 = 0
of a coordinate system and let the uniphase wave front coincide with the
line X — f. Let us assume that one point 0 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 0 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 0 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 0 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.
0
■- 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) = 0 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 0 < 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; = 0 (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] — 0 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 = 0 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 0 to co when R ranges from 0 to co and also when b ranges
from 0 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 = 0 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 = 0 to 2.8).
from 0 to 2.8 and the parameter b ranging by steps from 0 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 — 0 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 (/^ = 0 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 = 0 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 = 0 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^ 0 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 = 0 can have no area under it.
P(0;1) denoting (by convention) the value, or values, of P{R;b) when
R — 0 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 /? = 0 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) = 0 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) — 0 slI R — 0 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 0 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 = 0 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 = 0 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? = 0 (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 0
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 0 + 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 0 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 0 to 1.4 and the parameter b ranging by steps from 0 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
0 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 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 = 0 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 = 0 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
0 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 — 0 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 0
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 — 0 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 = 0 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 = 0 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 = 0 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
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 = 0 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 = 0 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? = 0 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 = 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 = 0 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;\) = 0 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 = 0 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 0
(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? = 0 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
''' Li-6cos2dr
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
\
\
1
k
\
\
Uvi
\v
\\J\
\\
vv
\
\\
N
\
^
\\
^ \
:^i
\
1
\
V
\\
\^
^v\
\
\
\
N]
V \ •*
\ ■f\<
^'
t$;
V
\
V
:S^
^
x\
\
<3\\
V
V^
^^
^1
^
v.
4
o'
^
\
\
^\
^\
^
i
\,
q
ij
\
\
\
\\
^
s^
^
\
\,
^
sV
;:$
NNV
^
X
,^
:^
^
^
^ —
^
^
CUMULATIVE DISTRIBUTION FUNCTION, Q(e;b)
Fig. 7.1 — Cumulative distribution function for the angle.
Thus, the fact that the curve for ^ = 0 is a straight Hne, whose equation is
(3(0 ;0) = e/2-w = 07360°, {b = 0),
corresponds to the fact that for 6 = 0 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,
GdUdV = 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 ~ 0
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 0 to 1,
Tc ranges from 0 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 0 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 found 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 0 to 1, b/Tc ranges from 2/3 to 0; Tc/b from 3/2 to cc ; and Tc from
0 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 0 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
0
TIME,
FREQUENCY SPECTRUM, g (f)
-6C -4C -2C 0 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 = 0
WHERE f = Val
AVERAGE =2EL
1
E
1
-
L 0 +L TIME,t
WHERE f = I/2L
«->
/
\ AVERAGE = 0
<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
"^ 0
_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 0 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 0 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
~^"^---
0 21 4T 6T 8T lOT 12T
TlME.t
0.6
UJ 0.4
""^-^ FREQUENCY
^^^^ SPECTRUM
O
1-
^^^
O0.2
Hi
OC
cc
UJ 0
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
0 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 0
^o.sr
Q
3
5^0,4
<
liJ 0.3
>
o
h-
Q.
<
^
E
1
\
-L 0 L
TIME,t — *
---^
"V
y
^
UlJ
Q
Q-
<
\
\
\
\
\
\
\
-5L -3L -L 0 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) = 0 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 0 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
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Copyright, 1947
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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, 0 to 9 inclusive. In a
binary system, the digits can have only two values, either 0 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 0 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 0 I 0 ; 1 I 0 I
J — L
16 ; 0 I 0 1 2 ; 0
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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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
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Fig. 5 — Detailed wave forms for PCM Transmitter (amplitude vs. time).
403
404
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
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? 411
)\n mode
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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 > 0 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 --- = 0 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, 0 < tan (p < oo ,
0 < sin v? < 1, and 0 < 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 ^ > 0 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 > 0 (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- = 0 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 -— = 0 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 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, 0
* 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) ~ 0 for TE
modes and that '"J/ic. a) = 0 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, 0 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, 0 sin k^z cos ut
Q
■\/ 1 ~2
■Et, = —^3 S'((c, r))J({c, 0 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,^) — 0 for ^-^ modes
J'^ifyO = 0 for TE modes.
428
BELL SYSTEM TECIIMCAL JOiRXAL
TAULK 111 Rout Valiks ok Kauial Elliptic Cylinder Functions
Mode
c
r
e
E
i
TEOl
0
3.8317
0
0
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
0
3.8317
0
0
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
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
0.05214
0.10397
0.15516
0.20542
0.25446
0.30203
0.34788
0
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
0
6.706
0
0
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
0
6.706
0
0
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
0
8.015
0
0
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
0
8.015
0
0
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\
0
2.4048
0
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
0
1.8412
0
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 foci
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.
0 - 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 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 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 — > 0 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 — > 0 as x ^ 0. Furthermore Zt{r) remains
finite as t? -^ 0. Hence H -^ 0^^ and — '^ x^~^. Therefore, for / > 0,
n
— — > 0 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) — > 0 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' -^
0 as ?7 -^ 0. Hence, () - — > 0 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 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
0
OJg,
^
"^
^
^
^
^ '
0.14
— '
' '
^^
-^
x-'^
of
= 0.16
-^
0 18
^
^
- —
, 1
0.20
^
~
0.22
^
/
^
J3.24
■
/
/
^
^"
0.2
76
■
■— ■
0.26
^
0 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.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 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 ^ > 0 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 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.
0 13?
Fig. 11 — Vector voltage diagram for maximum vector sum.
0 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 0
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
0
■^
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 0
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 = 0
(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
/
^
0 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 0.1 0.2 0.3 0.4 0.5 0 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 = 0 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) = 0 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 0 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 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 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^ = 0
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 =
0 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 < 0 . (8.2)
For stable oscillation at amplitude V we require
Ge[V] + Gi = 0 (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 0 = 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^ = 0 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
0 5
,0.2
'0.1
0
;,.o
I 0 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 0 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.'\.'?
0 <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
0
50
40
30
20
10
0
-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'-
0
oo
u,a
-I
-20
o in
/iiJ
< UJ
-4 0
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 0
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 = 0
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 = 0
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 = 0 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^ = 0
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 0 power contour lies not on the Gi = 1 — G2 contour but on the locus of
Ao)
values Gi = cos — wot — d .
In order to determine the power contours when the transit time rather
than the transit angle is held constant we make use of (9.3) with addition of
resonator loss. In normalized coordinates ((9.6) and (9.12)) and for a phase
angle of electronic admittance 86 we have
Gi + G2 = '^^^^ cos 89 . (9.18)
From (9.5) and (9.13) we have for the power output
Gi 2XJi{X) ,_ . .
Along any constant frequency contour 86 is constant and has the value
given by (9.15) in terms of wo and coqt. Hence, it will be convenient to plot
(Gi + G2) vs X for various values of 86 as a parameter. This has been
done in Fig. 35. The angle 86 has been specified in terms of a parameter A
which appears in the Rieke diagrams as a measure of frequency deviation.
^=^^ (9.20)
ye Wo
In terms of the parameter A
86 = (y,/2A/)(coor)/l . (9.21)
Once we have the curves of Fig. 35 we can find the power for any point
on the impedance performance chart. We may, for instance, choose to
find the power along the constant frequency contours, for each of which
A (or 86) has certain constant values. We assume some constant resonator
loss G2 . Choosing a point along the contour is merely taking a particular
value of Gi . Having 86, G2 and Gi we can obtain A^ from Fig. 35. Then,
knowing A^, we can calculate the power from (9.19).
REFLEX OSCILLATORS
519
In constructing an impedance performance chart we want constant power
contours. In obtaining these it is convenient to assume a given value of
G2 . We will use G2 = -3 as an example. Then we can use Fig. 35 and
a95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
I
0.50
0.45
0.40
0.35
0.30
0.25
11^
xN
\,
^^ N
. N^=o
\\
\
K\
^
^^
N
>N\\
V
N
^"
\^
\
.WW
^"
\
\^
N,
\
\
i
3^67,
v
^
\
\^
\^
^^
4.36
.\
vV
'^
^
^
rv
1
G2 = 0.3
i
"^
^\
^^
^
sV
^^
X
^
\x
m
^
^
0 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 = 0 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 0 power, the
sink margin, is about 2.3. This sink margin is nearly independent of the
resonator loss for oscillators loaded to give maximum power at unity stand-
ing wave ratio, as has been discussed and illustrated in Fig. 10. If the sink
margin must be increased or the pulling figure must be decreased^" the coup-
'" The pulling figure is arbitrarily defined as the maximum frequency excursion pro-
duced when a voltage standing wave ratio of v 2 is presented to the oscillator and the
phase is varied through 180°.
REFLEX OSCILLATORS
521
0.46
^=0
0.42
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0.40
0.38
0.36
0.34
f-
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0.12
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0.10
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0.08
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w
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3.87
\
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0.06
0.04
III 1
/
\
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w
If
f
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i
0.02
IL
^.36
i
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^
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—
\
\i
0
r
\
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\
1
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\\\i
0.05 0.10 015
0.20 0.25 0.30 0.35 0.40 0.45 0.50
NORMALIZED LOAD CONDUCTANCE, Gi
0.55 0.60 0.65 0.70
Fig. 36. — Normalized power vs normalized load conductance for various values of the
parameter A which gives the deviation in drift time from the optimum drift time. These
curves are computed for the case G2 = .3. Optimum drift angle equal to 15.5 n radians
and a ratio of characteristics resonator admittance to small signal electronic admittance
of 90 is assumed.
522
BELL SYSTEM TECHNICAL JOURNAL
ling can be reduced so that for unity standing wave ratio the load conduct-
ance appearing at the gap is less than that for optimum power.
Finally, in making measurements the load impedance is usually evaluated
at a point several wavelengths away from the resonator. If performance is
plotted in terms of impedances so specified, the points on the contours of
LOAD POWER G2 = 0.3
MAX.LOAD POWER G2=0.3
LOAD POWER G2=0.3
MAX. LOAD POWER 6^=0
I
^<m^)
ye
POWER
Fig. 37. — A Rieke diagram for a reflex oscillator having a lossy resonator, taking into
account the variation of drift angle with frequency pulling. This results in closed power
contours.
Fig. 38 appear rotated about the center. As the line length in wavelengths
will be different for different frequencies, ]:)oints on different frequency
contours will be rotated by different amounts. This can cause the contoirs
to overlap in the region corresponding to the zero admittance region of Fig,
38. With very long lines, the contours may overlap over a considerable
region. The multiple modes of oscillation which then occur are discussed
in somewhat different terms in the following section.
REFLEX OSCILLATORS
523
Figure 39 shows the performance chart of Fig. 38 as it would appear with
the impedances evaluated at a point 5 wavelengths away from the resonator.
Figure 71 of Section XIII shows an impedance performance chart for 2K25
reflex oscillator.
— -i^K^i
Fig. 38. — The Rieke diagram of Fig. 37 transformed to apply to the oscillator loaded
for optimum power at unity standing wave.
B. Frequency — Sensitive Loads — Long Line Efect
When the load presented to a reflex oscillator consists of a long line mis-
matched at the far end, or contains a resonant element, the operation of a
reflex oscillator, and especially its electronic tuning, may be very seriously
affected.
For instance, consider the simple circuit shown in Fig. 40. Here Mr
is the characteristic admittance of the reflex oscillator resonator as seen
from the output line or wave guide and Ml is the characteristic impedance
of a line load I long, so terminated as to give a standing wave ratio, <r.
524
BELL SYSTEM TECHNICAL JOURNAL
In the simple circuit assumed there are essentially three variables; (1)
the ratio of the characteristic admittance of the resonant circuit, Af« to
LOAD POWER
Fig. 39. — The Rieke diagram of Fig. 38 transformed to include the effect of a hne five
wave lengths long between the load and the oscillator.
Mr M|_
Fig. 40. — Equivalent circuit of a lossless resonator, a line and a mismatched load.
that of the line, Mi, . This ratio will be called the external Q and signified
hyQ,
Q, = Mn/M^ . (9.23)
REFLEX OSCILLA TORS
525
For a lossless resonator and unity standing wave ratio, the loaded Q is equal
to Qe ■ For a resonator of unloaded Q, Q» , and for unity standing wave
ratio, the loaded Q, obeys the relation
\/Q = \/Qe + 1/(3..
(9.24)
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 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 0 = 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 0 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 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
0 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
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
0 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
0
(b)
0 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 0
\ \ \
\ 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-
554
BELL SYSTEM TECHNCLAL JOURNAL
'^h
Fig. 58. — External view of the W.E. 707-A reflex oscillator tube. This tube is intended
for use with an external cavity and was the first of a series of low voltage oscillators.
trode and an accelerating electrode Ch which is a mesh grid formed on a
radius. 'J'he gun design is based on the principle of maintaining boundary
conditions such that a rectilinear electron beam will flow through the
resonator gap. The resonator grids Gi and G3 are mounted on copper discs.
REFLEX OSCILLATORS
555
One of these has a re-entrant shape to minimize stray capacitance in the
resonant circuit. These discs are sealed to glass tubing which provides a
ELECTRODE
Fig. 59. — X-ray view of the W.E. 707-A shows the method of applying an external
cavity tuned with a piston.
vacuum envelope. The discs extend beyond the glass to permit attach-
ment to the external resonant chamber. The shape of the repeller is chosen
556
BELL SYSTEM TECHNICAL JOURNAL
to })r()vide as nearly as possible a uniform field in the region into which the
beam penetrates.
A wide variety of cavity resonators has been designed for use with this
oscillator. An oscillator of this construction is fundamentally capable of
oscillating over a much wider frequency range than tubes tunable by means
of capacitance variation. The advantage arises from the fact that the inter-
action gap where the electron stream is modulated by the radio frequency
field is fixed. As discussed in more detail in Section X, this results in a
slower variation of the modulation coeflficient with frequency and also a
slower variation of cavity losses and gap impedance than in an oscillator in
w'hich tuning is accomi)lished by changing the gap spacing. A cavity
designed for wide range frequency coverage using the 707A tube is shown in
Fig. 60. Using such a cavity it is possible to cover a frequency range from
1150 to 3750 megacycles. The inductance of the circuit is varied by moving
the shorting piston in the coaxial line. For narrow frequency ranges,
Fig. 60. — Sketch showing a piston tuned circuit for the VV.E. 707-A which will permit
operation from 1150 to 3750 mc.
cavities of the type shown in Fig. 61 are more suitable. In such cavities
tuning is effected by means of plugs which screw into the cavity to change
its effective inductance. Power may be extracted from the cavity by means
of an adjustable coupling loop as shown in Fig. 61.
The 707A was the first reflex oscillator designed to operate at a low voltage
i.e. 300 volts. This low operating voltage proved to be a considerable
advantage in radar receivers because power supplies in this voltage range
provided for the i.f. amplifiers could be used for the beating oscillator as
well. Operation at this voltage was achieved by using an interaction gap
with fine grids, which limits the penetration of high frequency fields. This
results in a shorter effective transit angle across the gap for a given gap
spacing and a given gap voltage than for a gap with coarse or no grids.
Hence, for a given gap spacing a gcod modulation coeflficient can be ob-
tained at a lower voltage. Moreover, since drift action results in more etii-
cient bunching at low voltages, a larger electronic admittance is obtained
than with an open gap. This gain in admittance more than outweighs the
I
REFLEX OSCILLATORS
557
greater capacitance of a gap with fine grids, so that a larger electronic tuning
range is obtained than with an open gap. The successful low voltage
operation of the 707A established a precedent which was followed in all the
succeeding reflex oscillators designed for radar purposes at the Bell Tele-
:707
Fig. 61. — A narrow tuning range cavity for the W.E. 707-A of the t}'pe used in radar
systems. The inductance of the cavity can be adjusted by moving screws into it. This
view also shows the adjustable coupling loop.
phone Laboratories. The 707A is required to provide a minimum power
output of 25 milliwatts and a half power electronic tuning of 20 megacycles
near 3700 megacycles. The power output and the electronic tuning are in
excess of this value over the range from 2500 megacycles to 3700 megacycles
in a repeller mode having 3f cycles of drift.
558 BELL SYSTEM TECHNICAL JOURNAL
C. A Reflex Oscillator With An Integral Cavity— The 723
The need for higher definition in rachir systems constantly urges eperation
at sliorter wavelengths. Thus, while radar development proceeded at 3CC0
megacycles, a program of development in the neighborhccd cf lO.COO
megacycles was undertaken. Although waveguide circuit techniques were
employed to some extent at 3000 megacycles, the cumbersome size cf the
guide made its use impractical in the receiver and hence coa.xial techniques
were employed. The \" by h" guide used at 10,000 megacycles is con-
venient in receiver design and also desirable because the loss in coaxial con-
ductors becomes excessive at this frequency. Hence, one of the first
requirements on an oscillator for frequencies in this range was the adaptabil-
ity of the output circuit to waveguide coupling.
In considering possible designs for a 10,000 megacycle oscillator the simple
scaling of the 707A was studied. This appeared impractical for a number
of reasons. The most important limitation was the constructional diffi-
culty of maintaining the spacing in the gap with sufficient accuracy with
the glass seaUng technique available. Also, variations in the capacitance
caused by variations in the thickness of the seals caused serious difficul-
ties in predetermining an external resonator. Contributing difficulties
arose from the power losses in the glass within the resonant circuit and the
problem of making the copper to glass seals close to the internal elements.
Consideration of these factors led to a new approach to the problem, in
which the whole of the resonant circuit was enclosed within the vacuum
envelope. This required a different mechanism for tuning the resonator,
since variation of the inductance of a cavity requires relatively large dis-
placements w^hich are difficult to achieve through vacuum seals. The
alternative is to vary the capacitance of the gap. Since the gap is small a
relatively large change in capacitance can be achieved with a small dis-
placement. This sort of tuning permits the use of metal tube construc-
tional techniques, and these were applied.
As a matter of historical interest an attempt at this technique made at the
Bell Telephone Laboratories is shown in Fig. 62. This device was held to-
gether by a sealing wax and string technique and was net tunable in the first
version. It oscillated successfully on the pumps, however, and a second
version was constructed which was tuned by means of an adjustable coaxial
line shunting the cavity resonator. Adjustment of this auxiliary line gave
a tuning range of 7.5%. Such a tuning method is fraught with the com-
plications outlined in Section IX.
An early reflex oscillator tube of the integral cavity type designed at the
Bell Telephone Laboratories was the Western Electric 723A/B.
REFLEX OSCILLATORS
559
This design was superseded later by the W.E. 2K25 which has a greater
frequency range and a number of design refinements. From a construc-
tional point of view the two types are closely similar, however, and to avoid
duplication the later tube will be described to typify a construction which
served as a basis for a whole series of oscillators in the range from 2500 to
10,000 Mc/s.
Fig. 62. — An early continuous!}' pumped metal reflex oscillator tuned with an
external line.
Before proceeding to a description of the 2K25 tube it seems desirable to
recapitulate in more detail the design objectives from a mechanical point
of view. These were:
1. To provide a design which would lend itself to large scale production
and one sufficiently rugged as to be capable of withstanding the rough
use inherent in military service.
2. To provide output means which permit coupling to a wave guide in
such a manner that installation or replacement could be accomplished
in the simplest possible manner.
3. To provide a tuning mechanism for the resonant circuit which, while
simple, would give sufficiently fine tuning to permit setting and holding
560 BELL SYSTEM TECHNICAL JOURNAL
a frequency within one or two parts in 10,000. In addition, in order
to avoid field installation it was desired to have the tuning mechanism
cheap enough to be factory installed and discarded with each tube.
4. The oscillator was required to be compact and light in weight to
facilitate its use in airborne and pack systems.
Figure 63 shows a cross-section view of the final design of the 2K25 reflex
oscillator. The resonant cavity is formed in part by the volume included
between the frames which support the cavity grids and also by the
volume between the flexible vacuum diaphragm and one of the frames.
This diaphragm also supports a vacuum housing containing the repeller.
The electron optical system consists of a disc cathode, a beam electrode and
an accelerating grid. These are so designed as to produce a slightly con-
vergent outgoing electron stream. The purpose of this initial convergence
is to offset the divergence of the stream caused by space charge after the
stream passes the accelerating grid and to minimize the fraction of the elec-
tron stream captured on the grid frame on the round trip. The repeller
is designed to provide as nearly as possible a uniform retarding field through
the stream cross-section.
Power is extracted from the resonant circuit by the coupling loop and is
carried by the coaxial line to the external circuit. The center conductor of
the coaxial line external to the vacuum is supported by a polystyrene in-
sulator and extends beyond the outer conductor to form a probe. Coupling
to a wave guide is accomplished by projecting this probe through a hole in
that wall of the wave guide which is perpendicular to the E lines so that
the full length of the probe extends into the guide. The outer conductor
is connected to the wave guide either metallically or by means of an r.f.
bypass or choke circuit. A more detailed section on such coupling methods
will be given later.
The tube employed a standard octal base modified to pass the coaxial line.
Thus if a standard octal socket is similarly modified and mounted on the
wave guide it is possible to couple the oscillator to the wave guide and
power supply circuits simply by plugging it into the socket, just as with
any conventional vacuum tube.
The tuning means for this type of oscillator tube presented a serious
problem. This will be appreciated when it is realized that the mechanism
must permit setting frequencies correctly to within one megacycle in a device
in which the frequency changes at the rate of approximately 200 megacycles
per thousandth of an inch displacement of the grids. In other words, the
tuner was required to make possible the adjustment of the grid spacing to
an accuracy of five miljionths of an inch. The design of the mechanism
adopted was originated by Mr. R. L. Vance of these Laboratories. The
operation of the tuner can be seen from an examination of the cross-section
RESONATOR
FLEXIBLE
DIAPHRAGM
TUNER SCREW
TUNER BOW
REPELLER
CAVITY GRIDS
ACCELERATING
GRID
BEAM-FORMING
ELECTRODE
_ TUNER
BACK STRUT
COAXIAL
OUTPUT LEAD
Fig. 63. — A 3 centimeter reflex oscillator with an internal resonator. This tube is a
further development of the earliest internal resonator reflex oscillator designed at the Bell
Telephone Laboratories.
561
562 BELL SYSTEM TECHNICAL JOURNAL
and external views of Figs. 63 and 67. On one side of the tube a strut extend-
ing from the base is attached to the repeller housing. This strut acts as a
rigid vertical support but provides a hinge for lateral motion. On the
oppos'te side the support is provided by a pair of steel strips. These are
clamped together where they are attached to the vacuum housing support
and also where they are attached to a short fixed strut near the base. A
nut is attached rigidly to the center of each strip. One nut has a right and
the other a left handed thread. A screw threaded right handed on one half
and left handed on the other half of its length turns in these nuts and drives
them apart. The mechanism is thus a toggle which, through the linkage
provided by the repeller housing, serves to move the grids relative to one
another and thus to provide tuning action.
The 723A/B was originally designed for a relatively narrow band in the
vicinity of 9375 megacycles. It operates at a resonator voltage of 300 volts
and the beam current of a typical tube would be approximately 24 milliam-
peres. The design was based on the use of repeller voltage mode which
with the manufacturing tolerances lay between 130 and 185 volts at 9375
megacycles. It is difficult to establish with certainty the number cf cycles
of drift for this mode. Experimental data can be fitted by values of either
6| or 7f cycles and various uncertainties make the value calculated from
dimensions and observed voltages equally unreliable. This value is. how-
ever, of interest principally to the designer and of no particular mxoment in
application. The performance was specified for the output line cf the os-
cillator coupled to a f" x \\" wave guide so that the probe projected full
length into the guide through the wider wall and on the axis of the guide.
With a matched load coupled in one direction and a shorting piston ad-
justed for an optimum in the other the oscillator was required to deliver a
minimum of 20 milliwatts power output at a frequency of 9375 megacycles.
Under the same conditions the electronic tuning was required to be at
least 28 megacycles betw^een half power points.
For reasons of continuity a more detailed description of the properties of
the 3 centimeter oscillator will be given in a later section. The 723A/B
oscillator served as the beating oscillator for all radar systems operating
in the 3 centimeter range until late in the war when the 2K25 supplanted
it. At the time that the 723A/B was developed the best techniques and
equipment available were employed. In retrospect these were somewhat
primitive and of course this resulted in a number of limitations of per-
formance. Since the tubes designed as beating oscillators commonly
served as signal generators in the development of ultra-high frequency
techniques and equipment the wartime designer of such oscillators usually
found himself in the position of lifting himself by his own bootstraps. In
spite of these limitations the later modifications of the 72v3A/B which led to
REFLEX OSCILLATORS 563
the 2K25 did not fundamentally change the design but were rather in the
direction of extending its performance to meet the expanding requirements
of the radar art. The incorporation of the resonant cavity within the
vacuum envelope resulted in a major revision of the sccpe of the designer's
problems. He assumed a part of the burden of the circuit engineer in that
it became necessary for him to design an appropriate cavity and predeter-
mine the correct coupling of the oscillator to the load. The latter trans-
ferred to the laboratory a problem which in the case of separate cavity oscil-
lators had been left as a field adjustment.
D. A Reflex Oscillator Designed to Eliminate Hysteresis — The 2K29
As service experience with the external cavity type of reflex oscillator
was gained a number of limitations of such a design became apparent.
The ditficulties arose primarily from the conditions of military application.
A typical difficulty was the corrosion of cavities and copper flanges under
the severe tropical conditions met in some service applications. The diffi-
culty of maintaining a moistureproof seal in a cavity tuned by variation
of the inductance made it very difficult to alleviate this condition. The
success of the all metal technique in the three centimeter range suggested
the application of the same principles to the design of a 10 centimeter
oscillator and this was undertaken.
Mechanically, the problem was straightforward, but an extrapolation of
the electrical design of the 723A/B to 10 centimeters suffered frcm a fatal
defect. The difficulty, previously described in Section VHI, was the dis-
continuous and multiple valued character of the output as a function of the
repeller voltage. Reference to Fig. 19 will indicate the operational prob-
lems which would arise in an oscillator in which the hysteresis existed in
marked degree. The a.f.c. systems were such that in starting the repeller
voltage would start from a value more negative than required for cscilla-
tion and decrease. As the repeller voltage decreased through the range
where oscillation would occur the frequency would of course cover a range
of values. When the repeller voltage reached a value such that the fre-
quency of the oscillator had a value differing from the transmitter by the
intermediate frequency the steady shift of the repeller voltage would be
stopped and would then hunt over a limited range about the value required
to maintain the difference frequency. When adjusted for operation this
condition would pertain with the repeller voltage at a value such that the
oscillator would be delivering maximum power. If under operating con-
ditions the frequency required of the oscillator by the system drifted to that
corresponding to the amplitude jump at B, any further drift of frequency
could not be corrected. Thus, one effect of the hysteresis is to limit the
electronic tuning range. As a second possibility, let us assume that the
564 BELL SYSTEM TECHNICAL JOURNAL
frequency has drifted so that the oscillator is operating in a range between
A and H of Fig. 19. If now the operation of the system is momentarily
interrupted the a.f.c. system will start hunting. This is done by returning
to the non-oscillating repeller voltage just as when operation is initiated.
When the hunting repeller voltage passes through the value between A
and B from the non-oscillating state no oscillation occurs and hence the
a.f.c. cannot lock in and the system becomes inoperative. Thus it is im-
perative that hysteresis be kept, as a minimum requirement, outside the
useful electronic tuning range.
As indicated in Section \TII it was found that the electronic hysteresis
occurred when the electron stream made more than two transits across the
gap. Thus an added objective of the design of the 10 centimeter metal
oscillator became the achievement of an electron optical system which
would limit the number of transits to two while insuring that the maximum
number of electrons leaving the cathode would make the two transits with
a minimum spread in zero signal transit time.
Figure 64 shows a sectional view of the final design adopted. The elec-
tron optical structure differs from that of the 723A/B in a number of
respects. The first grid of the 723x'\/B has been eliminated and one of the
cavity grids now plays a dual role in simultaneously serving as an accelerat-
ing grid. The grids are curved towards the cathode, which has a central
spike. This arrangement is intended to produce a hollow cylindrical elec-
tron stream. It will be observed that the second grid is larger in diameter
than the first and that the repeller has a central spike. The design is such
that the cylindrical beam entering the repeller region is caused to diverge
radially, so that in re-traversing the gap after its reversal in direction it
impinges on and is captured by the frame supporting the first grid.
The repeller design was determined by using an electrolytic trough to
determine the potential plots for a number of trial configurations. Then
by making point by point calculations of the electron paths the best con-
figuration was chosen. A typical example of such path tracing is shown in
Fig. 65. This figure shows the equipotential lines and the trajectories
computed for electrons on the inner and outer boundaries of the outgoing
stream. The method of calculating the trajectories has been described by
Zworykin and Rajchman.^" It assumes that space charge may be neg-
lected. Fig. 65 shows that the repeller design is such that the cylindrical
outgoing stream is focussed upon its return onto the frame supporting the
first grid. The cathode spike prevents emission from the central portion
of the cathode, since it would be difficult to prevent electrons from this
portion from returning into the cathode space. A second requirement on
12 V. K. Zworykin and J. A. Rajchman, Proc. of I.R.E., Sept. 1939, Vol. 17, No. 9, pp.
558-566.
TUNER BOW-
FLEXIBLE
DIAPHRAGM
TUNER SCREW
RESONATOR
COUPLING LOOP
-REPELLER
GRIDS
CATHODE SPIKE
CATHODE
_CATHODE
i HEATER
BEAM -FORMING
ELECTRODE
_ TUNER
BACK STRUT
Fig. 64. — Section view of the W.E. 2K29 reflex oscillator shows the electron optical
system used in eliminating hysteresis. (Fig. 65) This tube has an internal cavity and is
designed for the frequency range from 3400 to 3960 mc/s.
565
566
nELL SYSTEM TECHNICAL JOURNAL
the design was that the spread in the transit angle for zero signal should be
small. This requirement is not as stringent as might be expected. The
contribution to the electronic conductance by a current element whose
APPROXIMATE
ZERO
EQUIPOTENTIAL
Fig. 65. — Repeller design for eliminating hysteresis in a reflex oscillator. The electrons
make only two transits through the gap. The repeller does not return them to the cathode
region hut to the edge sup])orting one of the grids of the gap. Equipotcntials determined
from an electrolytic trough investigation are shown and the electron trajectories com-
puted from these equipotcntials.
transit angle deviates from the optimum by a small angle Ad varies as cos A^,
so that even for spreads in angle as great as ± 30° the effect is not serious.
The design illustrated in Fig. 64 was strikingly effective in reducing
hysteresis. Fig. 66 shows repeller characteristics for the original design
which was extrapolated from the 723A/I5 and for the design of Fig. 64,
REFLEX OSCILLATORS
567
In addition to improving the electronic tuning characteristics the design
was found to be more stable against variations in load, as would be ex-
pected from the discussion of Section VIII.
The arrangement adopted provided a prototype electron optical system
which was used in a whole series cf reflex oscillators designed for radar and
communication systems. These tubes were the 726A, 726B and 726C,
2K29, 2K22, 2K23 and 2K56.
The output line of these tubes is intended to couple through an adapter
to either a coaxial line or a wave guide. In the first case the adapter serves
to couple the output line of the tube to the cable, in some instances through
^
^^''
^
>^>.
^.-
//
/
"^
\n
1
^
/
\
/
\
1
'J
[
i
'
\
\
1
f
- 2K29
-
—
- 1349-XQ
«
\
1 /
\
\
/ /
\
Jl
f
1
i
120 130 140
NEGATIVE REPELLER VOLTAGE
Fig. 66. — Use of the electron optical system shown in Fig. 65 eliminated the bad fea-
tures of the repeller characteristics of the earlier 1349XQ in which the electrons were re-
turned to the repeller region and gave the repeller characteristic of the final 2K29
(dashed line).
an impedance transformer and in some instances directly. As practice
developed it became standard to design for optimum oscillator output
characteristics with output line coupled to a 50 ohm resistive impedance.
In the second case the adapter serves to couple the tube output line to the
guide through a transducer.
A t>'pical example of a reflex oscillator incorporating this construction
is the 2K29. This tube is intended to cover the frequency range from 3400
to 3960 Mc/s. An external view of this tube is shown in Fig. 67. It will
be observed that the center conductor of the output line extends beyond the
polystyrene supporting bushing. Fig. 68 shows an adapting fitting which
568
BELL SYSTEM TECHNICAL JOURNAL
permits the oscillator to be coupled to a fifty ohm coaxial cable. The
center conductor of the tube output line projects into the split chuck,
while contact is made to the outer conductor of the tube output line by the
spring contact fingers F. The coupling unit can be mounted in a standard
octal socket so that the tube can be coupled simultaneously to the power
supplies and the high frequency circuit by a simple plug in operation. It
is frequently desirable to insulate the outer conductor of the oscillator from
Fig. 67. — Metal reflex oscillator with enclosed resonator designed for operation from
3400 to 3960 mc/s.
^TO FIT TYPE"N" POLYSTYRENE
/( FITTINGS INSULATOR
CHUCK FOR INNER
COAXIAL CONDUCTOR
/ , CONTACT TO OUTER
■^^^' ■ '( ' ' ' '|\ / COAXIAL CONDUCTOR
Fig. 68. — A transducer for connecting the output lead of the 2K29 to a 50 ohm cable.
the line for direct voltages while maintaining the high frequency contact.
This can be accomplished with either an insulating sleeve of low capacitance
or a modification of the design which incorporates a high frequency trap
in the outer conductor. In some instances it is necessary to insulate the
center conductor of the tube from the line for direct voltages. This caji be
done with a concentric condenser. The characteristic impedance of the
section of the coupler may be made the same as that of the line by main-
taining the proper ratio of the diameters of the condenser and the outer
REFLEX OSCILLATORS
569
conductor or may be arranged so that the condenser serves simultaneously
as an impedance transformer to transform the impedance of the line to that
required for best performance of the oscillator. A general discussion of
the problems involved in such coupling designs will be given in a later sec-
tion.
One of the primary considerations in the design of a reflex oscillator is
the choice of resonator characteristics. The various controlling factors
have been outlined in previous sections. In Section X it was shown that
the power output and the electronic tuning optimize at different gap transit
160
10
^150
tx.
D
U
140
Q
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O
f£!i30
(0
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5
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3960
-REQUIRED RANGE >)
1
ELECTRONIC
TUNING
._ TUNED LOAD
— j- —
^
_ 50-C
)HM CABLE
""■"
"■~»^_
^
t"
z
:^
—^
^ — ^
'"-->
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J
/
■~^-~
-~^^
*''*■*>•.
/
POWER OUTPUT
~- TUNED LOAD
- 50-OHM CABLE
s*^^
i .^
A .
w
//
O"
LU LU
10>
40 Lut
y UJ
zo
DO.
z<
3500 3600 3700 3800
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 69. — Power output and electronic tuning vs frequency for the 2K29 reflex oscillator
The solid lines give the performance into a load adjusted for optimum power output at
each frequency. The dashed lines show the performance obtained when the tube is
coupled to a 50 ohm load by means of the fitting of Fig. 68.
angles. It is therefore necessary to compromise with the ultimate use in
mind. In a beating oscillator for a radar receiver, uniformity of the elec-
tronic tunmg is of greater importance than uniformity of power output,
since an adjustment of the coupling to the crystal permits some variation
of the latter quantity. Hence, the resonator characteristics are chosen to
provide as nearly uniform electronic tuning as possible. Fig. 69 shows the
electronic tuning and power output characteristics for the Western Electric
2K29 tube. These are shown for two conditions. The solid lines show the
power output and electronic tuning measured into an adjustable load which
570 BELL SYSTEM TECHNICAL JOURNAL
made it possible to present to the oscillator at each frequency the admittance
into which the oscillator delivered maximum power output. The solid
lines show the power output and electronic tuning over the band with the
oscillator connected to a load presenting a resistive 50 ohm impedance to
the coupling unit of Fig. 68. The problems involved in obtaining such
performance will be outlined in the next section.
E. Broad- Band Reflex Oscillators— The 2K25
As experience with the design of radar systems and components developed
and as a better understanding of the operation and limitations of the in-
dividual components was achieved, a great deal of effort was directed
toward simplifying and making more reliable the number of adjustments
required to optimize the performance of a system. As an illustration of
this problem as related to the beating oscillator, the development problem
of the 2K25 will be described. As the number of radar systems in the three
centimeter range increased it became apparent that to avoid self-jamming
it would be desirable to assign frequencies to various sets operating, for
example, in a fleet unit. Secondarily, the over-all band of the three centi-
meter range was widened to cover from 8500 to 9660 Mc/s. Prior to this
the 723A/B had been essentially a spot frequency oscillator and had been
primarily tested as such. As was so frequently the case with tubes for
military requirements, it was desired that the ultimate tube be interchange-
able with an existing tube, in thjs case the 723A/B, and hence the im-
provements had to be effected within its framework.
Changes in the electronic design from that of the 723A/B produced an
improved performance in the 2K25. These were a mcdification of the
electron gun which increased the effectiveness of the electron stream and
the elimination of a resonance of the region containing the electron gun
which coupled with the resonant cavity and in some cases impaired the
performance over the wide band. Beyond this the problem concerned the
determination of an output coupling system which would provide the de-
sired properties. This will be described in detail.
One of the most serious difficulties which occurred in early radar receivers
arose from the general failure to appreciate the effect of the load impedance
on the performance of an oscillator. This j)roblem has been discussed in
Section IX. In early radar receivers the method of coupling the beating
oscillator to the crystal was dictated mainly by mechanical convenience
rather than electrical considerations, and as a result most of the discontinu-
ities of performance due to bad load conditions which are discussed in
Section IX occurred at one time or another in most of the systems. For
instance, users were much surprised to hnd that a beating oscillator which
could be tuned to frequencies both above and below that required for
I
REFLEX OSCILLATORS 571
reception of the signal might yet fail to oscillate at the desired frequency.
The convenience of simple replaceability of the local oscillator became in
this instance a cross for its designer, since an exchange of tube would fre-
quently eliminate the effect. This led to the obvious conclusion on the part
of the user that the oscillator was defective, in spite of the fact that the pre-
sumably defective tube had passed the test specifications and would operate
satisfactorily in some other receiver. The plain fact, in the light of later
knowledge, was that the tubes were being improperly used, so that the
usual range of manufacturing variations was not tolerable.
The appreciation of this fact led to a new approach to the problem of
coupling the beating oscillator to a waveguide. In early designs the oscil-
lator was decoupled from the load by withdrawing the probe from the wave-
guide. This presented the oscillator with an uncontrolled admittance with
disastrous results in many cases. The new approach, proposed by the
group working with Dr. H. T. Friis at the Bell Telephone Laboratories, was
that of designing the receiver so that the beating oscillator could be oper-
ated into an essentially fixed impedance. The crystal was in this case
loosely coupled to form a part of this load, so that variations in its impedance
and the impedance looking toward the TR tube were largely masked. A
great many further refinements in the design of the receiver have since
been proposed, but this basic principle of defining the load into which the
oscillator is required to operate is fundamental to all. In the interests of
simplicity of use it appeared to be desirable to endeavor to pre-plumb the
coupHng of the oscillator to the wave guide. The tube designer in this
instance found himself perforce, as so frequently occurs in dealing with
micro-waves, a circuit designer — an instructive and illuminating experience
which might happily be reversed.
The wave guide coupling was made separate from the tube, both to
preserve the plug in feature of the tube and to maintain its interchange-
ability with the 723A/B. As a further simplification it was desired that the
coupling should require no adjustments. A convenient fixed load admit-
tance to present to this coupler is the characteristic admittance of the wave
guide, since this can readily be maintained fLxed over a wide band. The
problem, then, is the apparently straightforward one of transforming the
guide admittance to the admittance which the oscillator requires for opti-
mum performance. Actually, the problem is complicated by the fact that
the optimum admittance will vary throughout the band. The electronic
admittance varies with frequency even for a fixed drift angle, because the
modulation coefficient of the gap varies as the oscillator is tuned. The
losses of the resonator also vary with frequency, both because of skin effect,
the depth of penetration of the high frequency currents changing with fre-
quency, and because the circulating currents in the resonator are a function
572 BELL SYSTEM TECHNICAL JOURNAL
of the effective capacitance of the gap. As the capacitance increases in
tuning to lower frequencies the I R losses therefore increase.
The objective of the coupling design may not be to obtain maximum
power output at all points in the band but rather to obtain uniformity of
electronic tuning and power output. An additional requirement in some
cases is that a minimum sink margin as defined in section Mil should be
maintained. This is equivalent to the problem arising in magnetron design
of controlling the pulling figure.
In designing an appropriate wave guide coupling a number of variables
are at one's disposal. In the case of the W.E. 2K25 the variables available
are, the length of the tube probe exposed within the guide, the oflfset of the
probe from the axis of the guide, and the distance from the probe to the
shorting piston in the guide. In addition, the characteristics of the output
line of the tube are adjustable, and, finally, the coupling of the loop to the
resonator can be adjusted. As one might expect, there is a large number of
solutions with so many variables available. The most desirable solution
is one which provides a low standing wave ratio in all parts of the coupling
system. The method employed in the present case was to design a wave
guide to coaxial transducer which would provide a smooth broad band
transition from the wave guide characteristic admittance to the admittance
of the coaxial line. In the ideal case, the characteristic admittance of the
coaxial line to the loop should be maintained as uniform as possible. Struc-
tural considerations in the present case led to discontinuities which had to
be appropriately balanced in the final transducer. Dr. W. E. Kock of the
Bell Telephone Laboratories has given an expression which, for thin probes,
relates the probe length, the offset and the distance of the backing piston
when given the characteristic admittance of the coaxial line and the dimen-
sions of the guide between which a match is required. Once such a trans-
ducer has been obtained, the admittances which must be presented to it in
order to obtain maximum power from the oscillator are measured over the
band. From such measurements it is then possible to determine the cor-
rections in the loop size and in the transducer to obtain a given sink margin
throughout the band. This last step actually involves a certain amount of
cut and try in an effort to obtain satisfactory performance in all respects.
Figure 70 illustrates the transducer developed for the W.E. 2K25 oscillator
for use with \" by \" wave guide. All tests made on this tube are specified
in terms of operation in this coupler and with a load having the characteris-
tic impedance of the \" x \" guide presented to the coupler.
Figure 71 shows a performance diagram for a typical W.E. 2K25 oscillator
operating in the coupler of P^ig. 70. The reference plane for the diagram is
"J. B. Fisk, H. D, Hagstrum and P. L. Hartman, "The Magnetron as a Generator of
Centimeter Waves", B. S. T. J. Vol. XXV No. 2, pp. 167-348 (April, 1946).
REFLEX OSCILLATORS
573
'not the plane of the grids of the oscillator but is instead a more accessible
reference plane external to the tube, in this instance the plane through the
wave guide perpendicular to its axis which includes the tube probe. It will
be observed that the sink margin in the case illustrated was equal to 5.5.
At the frequency at which this diagram was obtained, the minimum sink
margin permitted by the test specification is 2.5. The variation in this
margin results from a variety of causes. As shown in Section III the sink
margin is determined by the ratio of the total load conductance to the small
signal electronic conductance. The total load conductance consists of the
U 0.734" >|
. 1 M
SECTION ON A-A
Fig. 70. — A broad band coupling designed to connect the 2K25 to a 1" x §" wave guide
conductance representing the resonator losses and the conductance arising
from the wave guide load transformed through the coupling system. Hence,
the coupled load will be subject to variations in the loop dimensions, the
characteristics of the couphng line and the transducer. The resonator loss
will differ from tube to tube because of the variation in the heating of the
grids and resonator by the electron stream, and there will be variations aris-
ing from other causes. The electronic conductance varies from tube to
tube primarily because of the spread in beam current and secondarily as a
result of such factors as variations in the modulation coefficient of the gaps,
non-uniformities in the drift space causing a spread in the transit time and
574
BELL SYSTEM TECHNICAL JOURNAL
the like. The sum total of these variations necessitated the maintenance of
an average sink margin of 6.7 times in order to insure a minimum of 2.5.
Figure 72 illustrates further characteristics of the 2K25 oscillator and
coupling. These data are the results of standing wave measurements look-
ing towards the coupler with the tube inoperative. From such "cold test"
A?!,
■ —
0
+ 2
\
-0.1
\
0.2
\
1
1
1
-0.3
1
1
1
/
- /
/
0.5
/
\ 1
\ 1
/
w
/
- /
/
A /
J<v
/ \i ^
I.0\
''i
W
2
P
5/
^7^\
y
FREQUENCY
POWER
^
~i5/
'A^C'
MODE A
F = 9350 MC
Fig. 71. — A Rieke diagram for the 2K25 connected to the load by the coupling of Fig.
70. This diagram was obtained for a repeller mode having a drift angle of 15.5 -k radians
at a nominal frequency of 9360 mc/s.
measurements one may determine the intrinsic (Jo of the resonator and the
external Qk . The former is a measure of the resonator losses while the
latter is a measure of the tightness of coupling of the oscillator to the load.
The values of Qo measured from a cold test have little significance in an
oscillator in which heating of the resonator and especially of the grids is
appreciable. This is particularly true of oscillators tuned by variation of
II
REFLEX OSCILLATORS
575
the capacitance of the interaction gap. It is possible to make hot tests
in which the thermal conditions cf operation are estabhshed without the
interaction effects of the beam, but these measurements are not available
for the 2K25. The external Qb is not affected, at least to a iirst order, by
thermal effects in the resonator. The third curve of Fig. 72 shows the ratio
of the power delivered when a matched load is coupled to the coupler to the
power delivered to a load which presents optimum impedance to the oscil-
100
^ 90
8500
9660
1
1
-*\
POWER FRACTION
1
1
1
1
■
>
DC
3 80
Q
d 70
a.
uj 60
s
o
a
5 50
3
5
X 40
<
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t-
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a.
LU
Q. 10
0
1
1
1
->
"^
1
1
1
/
<- — —
■"~r-
"^^
1
1
V ]
Qo^
*^
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/
r
'''4..^
^^-'
.---'
"
y
y^
1
— -i
^^.-
■-"Qext
7-=
1
1
1
1
1
1
1
1
1
-►^ 600 D
8800 9000 9200 9400
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 72. — Variation of the percentage of maximum power output delivered, unloaded
Q, Qo , and external (J, Qe . as functions of frequency for the 2K25 when coupled to the
characteristic admittance of the 1" x \" guide with the coupling of Fig. 70. The power
variation is for a mode having 15.5 -k radians drift.
lator. These data are given for the normal operating repeller mode as dis-
cussed below.
It was pointed out early in this work that the available power output and
electronic tuning have a contrary variation with respect to the number of
cycles of drift in the repeller space. Consequently, this is one of the most
important and exasperating parameters of the tube. Fig. 73 is a diagram
illustrating the characteristics of a typical W.E. 2K25 oscillator. The
abscissa is the repeller-cathode voltage which, for a fixed resonator voltage,
determines the drift angle. Thus, as this voltage is made increasingly
negative, successive modes of oscillation appear, corresponding to consecu-
576
BELL SYSTEM TECHNICAL JOURNAL
tive decreasing values of n. Our best determination for the value of n
for each mode is given. The base lines are displaced vertically on a uniform
wavelength scale, so that the variation of repeller voltage with wavelength
is indicated. The power output increases with decreasing values of n
but the half power electronic tuning for each repeller mode has a contrary-
variation. The repeller mode chosen as providing the best compromise be-
tween power output and half power electronic tuning is the 7f cycle mode.
The design of the coupling unit and all the primary characteristics of the
tube are based on the use of this mode. It will be observed that repeller
§4
O
UJ 2
>
I-
LU
cr 4
MOD
\ J
E A
/
F IN MEGACYCLES
PER second:
/
-^
r
A
/
\
T
8500
/
,-^
8770
/'
^
\
/
^^
\
/
\
CYCLES
DRIFT
H
«l
^1
';
.^
\
9050
d
^^
\
/■\
/
\
ELECTRONIC |
TUNING IN '
MC/SEC= 90
61
48
1
30.,^-
N,
9390
/"
'\
/'^
r
^
/'
\
9660
/^
/"
\
/
A
/
0 20 40 60 eo 100 120 140 160 180 200 220 240 260 280 300 320
NEGATIVE REPELLER VOLTAGE
Fig. 73. — Operation of the 2K25 in various repeller modes and at various frequencies
when connected to characteristic admittance of a 1" x ^" guide by the coupling of Fig. 70.
modes having n values less than 6 do not appear in Fig. 73. For values of
w = to 0, 1, 2 and possibly 3, this is because the conductance representing
the resonator losses is in excess of the electronic conductance. For the
values of w = 4 and 5 the coupled load conductance plus the resonator loss
conductance in the specified transducer are in excess of the electronic con-
ductance. Conversely for modes having n values in excess of 7 the coupling
is weaker than desired.
Fig. 74 illustrates the broad band characteristics for a typical W.E. 2K25
tube operating in the coupling of Fig. 70 into a matched load. In Fig. 74
REFLEX OSCILLATORS
577
are shown the power output, half power electronic tuning, and sink margin
as functions of frequency for the 7f cycle mode.
F. Thermally Tuned Reflex Oscillators — -The ZK45
The trend toward the simplification of radar systems to the fewest possible
adjustments, coupled with the ever present possibility of enemy jamming,
led to the attempt to produce a system which was described as a single knob
system. The ultimate objective of such a system was ability to shift the
frequency of the transmitter at will with a single control. This puts the
chief burden on the receiver, which must automatically track with the
REQUIRED RANGE-
8800 9000 9200 9400 9600
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 74. — Variation of electronic tuning, power output and sink margin with frequency
for the 2K25 in a repeller mode having 15.5 ir radians drift. Characteristic admittance
load and coupling of Fig. 70.
transmitter. The problem is much simplified by designing as many of the
components as possible so that retuning is not required when the frequency
is shifted within the requisite band. In the case of the beating oscillator
it was necessary to devise a mechanism which would permit rapid automatic
control over a frequency range of 1160 mc. This range was many times in
excess of any immediately realizable electronic tuning range. It is of course
apparent that such a method of tuning will also lend itself readily to use in
many apphcations in which, although the transmitter frequency is nominally
fixed, the system is required to operate under such extreme conditions that
the sum total of the possible frequency deviations is in excess of the available
578 BELL SYSTEM TECHNICAL JOURNAL
electronic tuning range. Examples of such systems are installations in high
altitude air craft, in which wide variations in temperature and pressure
may be expected.
It was highly desirable to have the frequency control electrical. One
means of obtaining such a control is through motion of the resonator grids
produced by the thermal expansion of an element heated electrically. A
step in this direction was taken in the Sperry Gyroscope Company 2K21
oscillator in which the resonator was tuned by the thermal expansion of a
strut heated by passing a considerable current through it.
At the Bell Telephone Laboratories the design for a thermally tuned
beating oscillator was based on a method which permitted the control of a
small current at a high voltage. In general, controlled high voltages are
easily available both from power supplies and from control circuits. Fur-
ther, it seemed desirable that the control of the heating should require no
power. These considerations suggested that the heating of the thermal
tuning element be accomplished by electron bombardment. Through the
use of a negative grid to regulate the bombardment, the tuning control
became a pure voltage adjustment. The bombardment method made it
possible to utilize configurations in the tuner which would have been less
practical if resistance heating had been employed.
An early reflex oscillator incorporating these ideas was the Western
Electric 2K45 vacuum tube. Fig. 75 shows an external view of the tube
which, except for the output coaxial line, looks like a forshortened 6L6
vacuum tube. The plug-in feature of the earlier mechanically tuned os-
cillators was maintained in this oscillator, which was designed to couple to
the waveguide circuit through the same transducer developed for the 2K25.
Figures 76 and 77 are cross-sectional views of the 2K45 made at right
angles to each other. The thermal tuning mechanism is contained in the
upper part of the structure. It is a bimetallic combination consisting of a
U shaped channel and a multi-leaf bow. The channel is formed of a material
with a large coefficient of expansion and a high resistance to slow permanent
deformation or creep at elevated temperatures. At the ends of this channel,
tabs bent down at right angles to the channel axis, provide rigid vertical
support for the channel without interfering with axial expansion. These
tabs are connected to the resonator, which in turn is supported by the
vacuum envelope as closely as possible in order to minimize the thermal
impedance of the path. This connection also serves to cool the channel
ends. The multi-leaf bow is welded to the channel at its end. The leaves
are made of a material having a low coefficient of expansion and, as they are
fastened to the channel at its ends, they remain cool and do not expand
appreciably as the channel is heated. The purpose of the multi-leaf con-
REFLEX OSCILLATORS
579
struction of the bow is to reduce the internal stresses produced by bending
during the tuning action.
A cathode, control grid and a pair of focussing wires are supported by
micas in a position facing the open side of the U channel. The channel
serves as an anode for the cathode current, which is controlled by the grid.
The focussing wires beam the cathode current into the anode. The grid is
proportioned so that under all operating conditions it remains negative,
and the control system need supply no power to it.
The heating of the channel by the electron bombardment causes it to
expand with a large differential with respect to the bow. As a result the
bow flattens out and its center moves toward the channel. The purpose
of this construction is to provide a magnification of the expansion of the
Fig. 75. — An external view of the W.E. 2K45 — an early thermally tuned reflex oscillator
designed by the Bell Telephone Laboratories.
channel which by itself would provide insufficient motion. The cross
member welded to the center of the bow and the vertical struts transmit the
motion of the bow to the diaphragm which supports one of the cavity grids.
The action is illustrated in Fig. 78 which shows a series of X-ray views of an
operating tube. Thus, the first view shows the conditions for no power
applied to the tube, the second, for the tube operating but with the tuner
grid biased to cut-ofT. The following views show the behavior with progres-
sive increases in the power into the tuner channel.
The ramifications in the design of a thermal tuner are many and the possi-
ble configurations of the mechanism will depend greatly on the individual
requirements of the application. It is possible, however, to lay down some
basic principles. These are concerned with positiveness of action and speed
of tuning. With regard to the first, it may seem anomalous to speak of
580
BELL SYSTEM TECHNICAL JOURNAL
TUNER YOKE
CONNECTOR WIRE
FOCUSING WIRES
TUNER GRID
TUNER CATHODE
CAVITY GRIDS
COUPLING LOOP
CATHODE HEATER
■COAXIAL OUTPUT
LEAD
Fig. 76. — Internal features of the W.E. 2K45: section through the output lead and
normal to the tuner cathode and strut.
REFLEX OSCILLATORS
581
TUNER BOW
TUNER CHANNEL-
REPELLER
DIAPHRAGM CONNECTOR
FLEXIBLE DIAPHRAGM
RESONATOR
HEAT BLEEDER-
FOCUSING ANODE
BEAM -FORMING
ELECTRODE
Fig. 77. — A section of the W.E. 2K45 cut parallel to the tuner cathode and strut.
positive action in a device which has thermal inertia. What is actually
meant, however, may be best indicated by the following: Let us suppose that
some power, Pa must be dissipated in the tuner in order to hold the oscillator
at a frequency, /„ . Now, suppose that the tube is operating with power
less than Pa and that we are required to switch to frequency /„ . In order
to switch rapidly, we apply power in excess of Pa to the tuner until the fre-
quency has reached fa , at which time we switch to the power Pa required
582
BELL SYSTEM TECHNICAL JOURNAL
25° C
Pt = 0 WATTS
Ta=280*' C
3. Pt= 1.2 WATTS Ta= 382 ° C 4. Pj = 2.1 WATTS Ta=438°C
-L
i^S
1 "iS
^^B
1 T "'
". I* i Hi
^r \Mm
Pt = 4.0 WATTS
Ta=525'
Pt = 6.0 WATTS
556°C
Fig. 78. — A series of x-ray photographs of the thermal tuning mechanism of an operating
2K45. These show the motion of the bows for successively increasing power inputs to
the tuner channel.
to sustain /„ . \\'e say that the control is positive if no overshoot occurs, i.e.,
if the frequency does not continue to change for a time and then return to
REFLEX OSCILLATORS 583
fa ■ ^^ e might equally well have illustrated this by an example in which
the tuner was cooling. In order to avoid overshoot it is necessary that no
thermal impedance exist between the heat source and the expanding element.
Thus, as an example of a tuner in which overshoot will occur, one may cite
an expanding strut in the form of a tube heated by a resistance heater in-
ternal to the tube. In order to transfer heat from the resistance heater to
the tubing the former must of necessity operate at the higher temperature.
Hence, in the example given above, when the power is switched to the sus-
taining value heat will continue to be transferred to the tubing until the
heater and the tubing arrive at the same temperature. To minimize the
thermal impedance the heat should be generated within the body of, or on
the surface of, the expanding element. The resistance heating by current
passing through the expanding element illustrates the first case and heating
by electron bombardment the second.
The second principle is quite obvious when once stated. If a rapid shift
infrequency is to be obtained at any point within the required tuning range,
then the potential tuning range must be considerably in excess of that re-
quired. Thus, if the tube is operating near one of the required frequency
limits and one demands that it go to the limit, the shift will progress very
slowly in the absence of excess range. If, however, it is possible to overdrive
the limit, the time required will be materially shortened. On the basis of
actual tube design this requires that the safe maximum or full on power into
the tuner must be considerably in excess of the power required to hold the
tuner at the frequency band limit nearest the full on condition. The tuning
mechanism must be capable of continuous operation under the full on condi-
tion in case this accidentally persists. Further, the power required to hold
the tuner at the other end of its band must be considerably in excess of zero
in order that rapid cooling may occur near this limit. It is not necessary
that the tuner produce motion for power inputs outside the band limits;
the essential condition is that the rates of heating and cooHng near these
limits should have values considerably greater than zero.
It is always possible to purchase heating speed by the expenditure of
power in available over-drive. The cooling speed, on the other hand, is con-
trolled by the temperature difference between the source and sink, the heat
capacity of the tuner and the mechanism of cooling. Two methods of cool-
ing are available, conduction and radiation. For small amounts of motion
and in circumstances where large forces are not required, conduction cooling
can provide a satisfactory answer. In cases in which a large amount of
motion is required, as in the 2K45, conduction coohng imposes a number
of serious restrictions. The expanding element must be made from a ma-
terial having a large coefhcient of expansion and necessarily must be long.
Unfortunately, alloys having large expansion coefficients are very poor con-
584 BELL SYSTEMJTECHNICAL JOURNAL
ductors of heat. In the case of conduction the cooHng rate will depend on
the ratio of the length times the heat capacity divided by the cross-section.
For radiation cooling the rate depends on the heat capacity divided by the
radiating area and is independent of the length except as the heat capacity
depends upon this factor. Radiation cooling has the advantage that it per-
mits more freedom of structural shape and the material may be chosen on
the basis of strength. In the operating range of the 2K45 the heat radiated
is approximately four times that conducted away.
The automatic frequency control circuits which have been used with
thermally tuned beating oscillators in radar receivers have been of a full
on or off type so that they do not continuously hold the frequency at a fixed
reference difference from the transmitter frequency. The reason for this
in part is that a pulse transmitter gives the required reference information
only during the pulse. Thus with an on-off control circuit if at a given
pulse a correction of frequency is demanded the power into the tuner is
turned full on or off, depending on the direction of the correction, and left in
this condition until the occurrence of the next pulse which again determines
the direction of the control. The result is that the frequency of the beating
oscillator continually hunts about the transmitter frequency. The control
system must be designed to hold the hunting deviation within tolerable
limits. It is of course possible to work within narrow-er limits than full
on or off tuner power. One advantage of full on or off control is that it
results in the minimum tuning time between required frequencies since the
tuning rate is at all times held to the maximum possible value.
Under certain conditions the performance of a thermal tuner may be
completely described by a time constant. In general, however, this is not
the case and the information required in designing a control system is con-
cerned first with initiation of operation and second with the factors deter-
mining the magnitude of the hunting deviation. For initiation of operation,
one is concerned with the time required for the oscillator to reach the operat-
ing frequency. The quantities known as the cycling times give upper limits
for this quantity. The cycling times are to a certain extent arbitrarily
defined as indicated in the following. There are two band limit frequencies,
fc , the limit requiring the lesser power input, Pc , and corresponding to a
tuner temperature, Tc , and fh , the limit for which the power input will be
Ph , and the temperature, Th . The cycling time for heating, n, , is defined
and measured in the following manner. The power input to the tuner is
set to Pc and held at this value until equilibrium is established. The power
input is then switched to the maximum allowed value, Pm and the time
interval, n , between the instant of switching and the instant at which the
frequency of operation has reached fh is measured. Correspondingly, the
cycling time Tc for cooling is measured by setting to Pn until equilibrium is
REFLEX OSCILLATORS 585
established and then switching the tuner power off and measuring the inter-
val, Tc , until the operating frequency reaches /c . These quantities are of
importance in determining the "Out of Operation" time in case the frequency
reference of the control system is momentarily lost, so that the control starts
cycling in order to re-establish the reference.
While the cycling times can be taken to give an indication of the average
speed of tuning, more detailed information is required to determine the
hunting deviation. This demands a knowledge of the instantaneous tuning
rates which will result at any point in the band when the power is switched
full on or off. These rates vary through the band since the overdrive avail-
able, for example, on heating will decrease as the operating frequency ap-
proaches the limit nearest to the maximum drive.
In the following, an outline will be given of the factors which must be
considered in designing a thermally tuned reflex oscillator. The 2K45 will
be used as an illustration. Our first consideration concerns the time re-
quired for the tuner to heat and cool between given temperatures. In
Appendix XI expressions are derived for the cycling times th and Tc . The
expressions applicable to the 2K45 are:
CT
T, = ^^ [F,(Trn) - F,{Trc)\ (13.1)
CT
To = ^4 [F.iT^c) - F^{Tsk)\ (13.2)
2KT\
where the symbols are defined in the appendix. The functions Fi and F2
are plotted in terms of the reduced temperatures, Tr and Ts in Figs. 79 and
80. In the analysis conduction cooHng is neglected and it is assumed that
the whole of the expanding element operates at the same temperature.
Because of these limitations the theory is largely qualitative. It will be
observed that the cycling time, th , is proportional to the ratio of the heat
energy stored in the tuner at the maximum equilibrium temperature to the
rate of loss of energy at this temperature. It is therefore apparent that this
equilibrium temperature should have the maximum possible value, and also
that the heat capacity of the tuner should be kept to a minimum. Assuming
for simplicity that the frequency of oscillation is proportional to the tem-
perature, so that a given temperature difference is proportional to the fre-
quency, one sees by examining the function Fi that it is desirable to keep
the reduced temperatures Trh and Trc small compared to 1. Under these
circumstances, the cycling time n will have its minimum value and will be
more or less independent of the reduced temperatures. If we examine the
expression for the cycling time for cooling, Tc , we observe that this is
proportional to the ratio of the heat stored in the tuner at the equilibrium
586
BELL SYSTEM TECHNICAL JOURNAL
0.7
1
0.6
/
0-5
y
/
0.4
y
0.3
y^
y^
0.2
0.1
^y^
^y^
0
^^
0.5
Tr
Fig. 79. — A plot of a function used in determining the time required for the tempera-
ture of a thermal tuner to rise between two given values when the tuner is cooled by radia-
tion alone. The abscissae are given in terms of a reduced temperature.
25
\
]
y
2.2
^ 2.1
H
\
\
\|
\
1.9
1.8
\
s
\
x^
.^^
1.7
1.6
' —
1 1
1.0 1.1 1.2 1.3 1.4 1.5 1.6 '.7 1 e 1.9 2.0 2.1 2.? 2.3
Fig. 80.— A plot of a function used in determining the time required for the lcmj)era-
ture of a thermal tuner to fall between two given values when cooled by radiation alone.
The abscissae are given in terms of a reduced temperature.
REFLEX OSCILLATORS
587
temperature Tq corresponding to zero bombardment power divided by the
rate of heat loss at this temperature. This indicates a rather paradoxical
result, that the temperature for zero bombardment power should be as high
as possible. This arises from the dependence on radiation cooling. We are
limited in setting the value of To by the form of the function F^ , which
requires that the reduced temperatures T^c and Tsh should be very large
compared to 1. Since the true temperatures corresponding to these reduced
values must simultaneously be small compared to Tm , it is apparent that
we are not completely free to make Tq large, and a compromise must be
worked out.
8500
1
9660
1
16
1 >
»
15
/
X
14
y
/
in
-J 13
5
/
?,2
y"
/^
z
O 11
^
y
9 10
a.
o
y
y
9
8
^^
J^
^-^
''^\
7
8800 9000 9200 9400
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 81. — Variation of the frequency of resonance vs gap displacement for the W.E'
2K45 resonator. The vertical lines show the required tuning range.
In determining a practical design for a thermal tuner, the first charac-
teristic which must be known is the variation of the resonant frequency of
the oscillator cavity as a function of gap displacement. It is apparent that,
for the highest speed of tuning, the rate of change of frequency with gap
displacement should have the maximum possible value. However, this
tuning characteristic is dictated by the performance requirements of the
tube as an oscillator and hence is not available as a variable in the design.
Fig. 81 shows the variation of frequency with gap displacement for the
2K45 resonator. The required range is indicated.
When the required motion is known a choice may be made of a mechanism
588
BELL SYSTEM TECHNICAL JOURNAL
for a tuner. There is a certain amount of arbitrariness in choosing the
limiting dimensions of the tuner. If the expanding element is to be short
it is necessary either to operate over a very wide range of temperatures or
else use some mechanical means of amplifying the motion obtained over a
more limited range. Since the more limited is the required temperature
range the greater is the tuning speed, it is obviously advantageous to use
mechanical amplification. As previously pointed out the electron bombard-
ment method of heating and radiation cooling are especially suitable to such
a mechanism because of the freedom of design they permit. Previous dis-
TEMPERATURE
Fig. 82. — The ideal type of deflection vs temperature characteristic desired for a ther-
mally tuned oscillator. The motion 6 is that required to shift the resonant frequenc\-
of the cavity through its required band.
cussion has shown that the temperatures corresponding to zero and ma.ximum
power input must be separated by wide margins from the temperatures
corresponding to the limits of the tuning range. Any motion which occurs
in these margins is unnecessary and in general undesirable. Ideally, the
tuning mechanism should have a characteristic as shown in Fig. 82. The
type of tuning mechanism chosen for the 2K45 is a first appro.ximation to
such a characteristic, as is shown in Fig. 83, which gives a family of charac-
teristics corresponding to various initial offsets of the bow for a given length
of bow. The bows are formed to a sinusoidal shape. This structure gives
REFLEX OSCILLATORS
589
a large mechanical amplitication of the expansion of the tuner. The tuner
is made as long as will fit conveniently into the tube envelope. Further
arbitrary decisions are required with regard to the power which can be ex-
5
O
o
a. 22
O 18
(0
14
_)
5
z
12
*
o
CD
10
II
o
7
6
O
\-
u
6
BOW OFFSET IN INCHES : y
^
0.040— ^x^'^
0.036
^'/^
\ i
^
'^->-*
"f
/
0.033
/ 1
/ 1
' 1
/
/ 1
'
/
u
4
1
/
7 /
1
1
j\
1
1
5
0.028
1
/ 1
/
/
'■■/
1
1
1
1
f
i
i
/
1
1
1
4
/
/
f 1
1
1 /
t
/]
0.024
//
r.
//
/
/
%
/ y
r
0
/.
''v
/
0.019/'
// .
/
/ /
A
'//.
^A
/
\
1
1
/ 1
I'//
^^/
• ^
1
1
1
(^
1
1
Tol
!tc
Th!
.Tm
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700
TEMP2RATURE IN DEGREES CENTIGRADE
Fig. 83. — A family of deflection vs temperature characteristics for the type of tuning
mechanism used in the W.E. 2K45. The parameter is the initial oflset of the bows from
the channel at room temperature.
pended in operating the tuner. \\ ith such decisions made, the tuner design
can then be completed as a compromise of a great many variables. Limita-
tions on the strength of the tuner materials at elevated tetnperatures deter-
590
BELL SYSTEM TECHNICAL JOURNAL
mine a maximum safe operating temperature. The anode material is
chosen to have a large coefficient of expansion and resistance to "creep"
at elevated temperatures. In choosing the form of the tuner it is necessary
to keep constantly in mind the necessity of maintaining the minimum ratio
of heat capacity to radiating surface. A minimum operating temperature
of the tuner is determined by heat flow to it from extraneous sources, l-'igure
84 shows the temperature as a function of bombardment power input for the
2K45 tuner. It will be observed that the heat from sources other than
bombardment produces a considerable rise in temperature. One principal
source of uncontrolled heating is radiation from the tuner cathode. This
y\
2 5 3 0 3.5 4 0 4.5
ANODE POWER IN WATTS
Fig. 84. — The tuner anode temperature as a function of the bombardment power.
The temperature rise at zero boml^ardment power is caused by radiation from the thermal
cathode and extraneous sources.
is minimized by keeping the cathode as remote from the anode as is con-
sistent with the required electronic characteristics.
When the maximum and minimum operating temperatures and the length
of the channel are determined the remaining problem is to determine and
offset for the tuning bow which will provide the optimum tuning characteris-
tics. We wish to obtain characteristics such that the heating time n and
the cooling time Tc are approximately equal and of a minimum value. The
choice of the bow offset also involves a choice of an initial gap spacing for the
resonator. On Fig. 83 the boundary values for the limiting temperatures Tq
and Tm are indicated by vertical lines. \\ ith any given bow offset which
corresponds to a particular tuning characteristic a limit is set on the initial
gap spacing by the requirement that the total motion of the bow between
REFLEX OSCILLATORS
591
room temperature and r,„ shall not exceed the initial gap spacing. From
our previous analysis we know that to provide maximum tuning speed we
desire to make the temperature intervals Tc — To and T,,, — Tk as large as
possible. For any given tuning characteristic these intervals may be ad-
justed by a variation of the initial gap spacing subject to the limitation just
10
-
\
THEORETICAL
\
11
1
ij
\
ll
1
1
11
coc
DLING
\
HEATING
ll
1
\
\
\
\
\
\
1
1
1
\,
^ \
//
V
y /
y /
y
y
-•
~^
j
^^-'
^^'
1
1
1
6 ■ -
420 440
DEGREES CENTIGRADE
Fig. 85. — The cycling time of the W.E. 2K45 as a function of the temperature of the
channel at the band frequency limit requiring the smaller tuner power. The solid lines
are the experimental curves, the dashed lines are theoretical results. One point on the
heating time is fitted in order to determine the heat capacity of the tuner.
imposed. Since Tc and Th are interrelated for a given bow characteristic
by the fact that they correspond to a specific increment of motion 8 deter-
mined by the cavity design, we may study the effect of shifting the interval
8 along the tuner characteristic by varying the initial gap offset. We may
specify this in terms of value of Tc . The result of such a study is shown in
Fig. 85. The optimum offset corresponds to the temperature at which the
592 BELL SYSTEM TECHNICAL JOCRNAL
curves for t/, and r,. cross o\cr. Ihe optimum bow offset is then the value
which provides the minimum value at the crossover. From the theory
given earlier it is possible to compute these curves. If we had analytical
expressions for the motion of the tuner with temperature and for the varia-
tion of frequency with gap spacing it should be possible to obtain completely
theoretical curves.
As a test of the theory of heating and cooling it is sufficient to use the
experimental curves for the motion of the tuner with temperature and the
variation of frequency with gap spacing in conjunction with the heating
and cooling curves of Figs. 79 and 80 calculated from equations (F^.l) and
(13.2). The value of A' which must be determined in order to obtain
numerical values from the curves of Figs. 79 and 80 may be determined
from Fig. 84 by using the relationship
,, Pa - Pb
K =
n - n
where Pa and Pb are the bombardment power inputs corresponding to anode
temperatures Ta and Tb ■ There is no ready means for directly determining
the heat capacity C. However, if one point on either the heating or cooling
curves is fitted to the experimental data the value of C may be determiined
and the remainder of the points computed. The results of such a computa-
tion are show^n by the dashed lines in Fig. 85. In view of the restrictive as-
sumptions of a uniform anode temperature and the neglect of all conduction
cooling the agreement in general form is reasonably good.
The W.E. 2K45 includes a number of advances in reflex oscillator tech-
nique over the 2K25. It will be observed in Figs. 76 and 77 that the electron
optical system employed in the gun differs from that used in the 2K25. In
the 2K25 a gun producing a rectilinear beam was employed. In the 2K45
the gun consists of a concave cathode surrounded by a cylindrical electrode
and a focussing anode. The design of this type of gun was originated by
Messrs. A. L. Samuel and A. E. Anderson at these laboratories. The design
is such as to produce a radial focus beam which converges into the cylindrical
section of the focussing anode, /fter the beam enters the focussing anode
its convergence is decreased by its own space charge, and the beam passes
through the grids at api)ro.\imately the condition of minimum diameter.
J^etwecn the second grid and the repeller the beam continues to diverge
radially on the outbound and return trij)s. \ he intention of the design is
that the beam shall ha\ e diverged sufficiently so that the maximum possible
fraction will recross the gap within a ring having an inner diameter equal
to the first grid and outer diameter equal to the second jzrid. Fnder these
conditions only a small fraction of the beam will return into the cathode
region, the remainder being captured on the su])p()rt of the first grid after tlic
REFLEX OSCILLATORS 593
second transit of the gap. This tends to ehminate electronic tuning hystere-
sis and the repeller characteristic of the 2K45 is essentially free of this
phenomenon. This gun design has the further advantage that it avoids the
necessity for the first grid used in the 2K25. This eliminates the current
interception on this grid with a resulting increase in the effective current
crossing the gap. This type of gun also permits the design of a more efficient
resonator by reducing the grid losses.
A second variation in design from the 2K25 is that in the 2K45 the second
grid moves with reference to the repeller. This has the advantage of reduc-
ing the variation of the repeller voltage for optimum power with resonator
tuning. The drift angle in a uniform repeller field is given by
e-^Jf (13.3)
where /is the spacing between the repeller and the second grid of the resona-
tor. If the same drift angle 6 is maintained at all frequencies in the band,
then the repeller voltage must vary. If ( is fixed as/ increases V r must also
increase in order to maintain a fixed fraction. If /varies and increases as /
decreases then a smaller variation in V r is required and in the particular
case that Ovaries inversely with/ the repeller variation may be made zero.
Usually other requirements determine the variation of /and it is not always
possible to make the variation zero. In the case of the 2K45 the variation
over the band is approximately half the amount which would occur if /were
fixed.
The output coupling and fine of the 2K45 were designed so that the
oscillator would provide the desired characteristics in the same waveguide
adapter as designed for the 2K25. The power output as a function of
frequency for a typical tube is shown in Fig. 86a. Curves A, B and C of
Fig. 86 show the variation of power output with cavity tuning when the
repeller voltage is set for an optimum at the indicated frequency and held
fixed as the cavity tuning is varied. The frequency shift between half power
points in this case is very much wider than with repeller tuning. This is a
consequence of the fact that whereas in repeller tuning both the frequency
and the drift time change in a direction to shift the transit angle away from
the value for maximum power, with cavity tuning only the frequency
changes. Moreover, the fact that the repeller to second grid spacing in this
design varies with frequency tends to reduce the variation of the drift angle
with frequency. The envelope of the curves .4, B and C gives the power
output as a function of frequency when the repeller voltage is adjusted to
an optimum at each frequency.
Fig. 86b gives the half power electronic tuning as a function of frequency
measured statically and also dynamically with a 60 cycle repeller sweep.
The difiference arises from thermal effects.
;q4
BELL SYSTEM TECHNICAL JOURNAL
50
3(0
o5
uj=! 20
Q.Z
- 10
(a)
/•
•*'
>. i---
'-»«,
«-^ >.'•'"
^***te^
.^
A
/
/
/
/
B
/-
"n
^
X
'^"^^■.
'
-.
/
/
/
N
>{
\
\
■^x.
■
:a: O
U i/l lil
O 80
; o 70
-
^
(b)
%
.^
^
STATIC
-^
^
^
DYNAMIC {60-CYCLE SWEEP)
■^
^^
.
8800 9000 9200 9400 9600
FREQUENCY IN MEGACYCLES PER SECOND
9800
Fig. 86. — Characteristics of the W.E. 2K45 reflex oscillator. Fig. 86-a shows the varia-
tion of the power output as a function of frequency in two cases. The curves A, B and C
illustrate the power variation with frequency when the repeller voltage is set for the opti-
mum at the indicated frequency and held tixed as the cavity tuning is changed. The
envelope of these curves shows the power variation with frequency when the repeller
voltage is maintained at its optimum value for each frequency.
In Fig. 86b the electronic tuning is shown, in one case where the repeller voltage is
shifted between half power points so slowly that thermal equilibrium exists at all times
and in the dynamic case in which the repeller voltage is shifted at a 60 cycle rate.
NEGATIVE REPELLER VOLTAG
FOR OPTIMUM POWER OUTPU
o o o o o c
-^
^
^
^
^
8800 9000 9200 9400
FREQUENCY IN MEGACYCLES PER SECOND
9800
Fig. 87. — Repeller voltage for optimum power as a function of frc(|ucncy for the W.E.
2K4.S oscillator.
Fig. 87 shows ihc variation of repeller voltage for optimum power with
frequency. This variation is so nearly linear that it has been proposed that
REFLEX OSCILLATORS
595
a potentiometer properly ganged with the transmitter in radar systems
would provide optimum output throughout the band. T his is an advantage
over electronic tuning, since the signal to noise performance of the receiver
depends in part on the beating oscillator power into the crystal.
In a thermally tuned tube it is necessary to provide safeguards against
excessive power input to the tuning strut since this might produce a per-
manent deformation and impair tuner operation. A simple method for
obtaining such protection is to use low-frequency cathode feedback produced
by a cathode resistor. With a cathode biasing resistor of 725 ohms the
grid may be held 15 volts positive with respect to the more positive end of
the cathode biasing resistor indefinitely without damage to the tuner when
the normal plate voltage of 300 volts is applied.
The grid control characteristics for a typical 2K45 shown in Fig. 88 were
obtained while using the cathode biasing resistor. These characteristics
may be given in two ways. In one case the repeller voltage is held fixed
and the characteristics are given over a range between half power points.
It will be observed in this case that the characteristics are discontinuous
because of the electronic tuning resulting from the repeller voltage shifts
between ranges. For the other case the repeller voltage is maintained at
its optimum value at each frequency.
In either case, one striking feature is the essential linearity of the variation
of frequency with grid voltage. This is of considerable importance in many
frequency stabihzing systems and represents an advantage of thermal tuning
over electronic tuning. In the case of electronic tuning, as shown in Section
VII, the rate of change of frequency with repeller voltage varies rapidly
as the repeller voltage shifts away from the optimum value. Since frequency
stabilization is essentially a feedback amplifier problem in which the rate of
change of the frequency with the control voltage enters as one of the factors
determining the feedback, it is apparent that the frequency stabilization
will vary as the repeller voltage is shifted. In contrast, for the case of
thermal tuning, because of the linearity of frequency with grid voltage, the
stabilization will be independent of the frequency. It should not be for-
2K45 Operating Conditions
Heater Voltage
Resonator Voltage ....
Klystron Current ....
Repeller Voltage Range
Tuner Current
Tuner Power
Th (9660-8500 Mc/s) . .
Tc (8500-9660 Mc/s) . .
Normal
Maximum
6.3
6.8 Volts
300
350 Volts
22
30 mA
-60 to -175
-350 Volts
Oto25
mA
7.0 Watts
6.0
9.0 Sec
6.0
9.0 Sec
596
BELL SYSTEM TECHNICAL JOURNAL
gotten, however, that thermal tuning is inherently slower in action than
electronic tuning, since the latter is capable of frequency correction rates
limited, for practical purposes, only by the control circuits, whereas in
9500
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o
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tr
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\
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V
\
V
\
\
\
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\
1,
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-6 -4 -2 0
TUNER GRID VOLTAGE
Fig. 88. — Frequency as a function of the grid voltage for the W.E. 2K45 oscillator"
The dashed lines show the variation when the repeller voltage is held fixed for the mid-
range value and the solid lines shows the variation with the repeller voltage maintained
at its optimum value for each freciuency. These characteristics apply when a 725 ohm
biasing resistor is connected in series with the tuner cathode.
thermal tuning the thermal inertia of the tuning strut limits the tuning speed
to rates of the order of 100 mc/sec^ for the 2K45.
The oj)erating conditions for the 2K45 are given in the table at the
bottom of i)age 595.
REFLEX OSCILLATORS 597
G. An Oscillator With Waveguide Out put— The 2K50
Late in the war it became apparent that there was an urgent need for
radar systems which would permit a very high degree of resolution. Such
resolution requires the use of the shortest wavelengths possible, and as a
practical step development work was undertaken in the neighborhood of
1 cm.
Work at these Laboratories led to a tube which produced over 20 milli-
watts and was thermally tunable over the desired frequency range by means
roughly similar to those employed in the 2K45. This tube had a wave guide
output. It had no grids; a sharply focused beam passed through a .015"
aperture in the resonator. The tube operated at a cavity voltage of 750
volts.
Work by Dr. H. V. Neher at the M.LT. Radiation Laboratory resulted
in a design for an oscillator using grids which operated at a resonator voltage
of 300 volts. At the request of the Radiation Laboratory, the Bell Tele-
phone Laboratories undertook such development and modification as was
necessary to make the design conform to standard manufacturing techniques.
This work was carried out with the close cooperation of Dr. Neher.
Figure 89 shows an external view of the tube and Fig. 90 a cross sectional
view of the final structure. There are two striking departures in this tube
from the designs previously described. One of these is that the axis of
symmetry is no longer parallel to the axis of the envelope but instead is
perpendicular to it. This construction makes possible in part the other
striking feature of the tube, which is the wave guide output. A number
of factors combine to make this type of output desirable and prac-
tical. The resonant cavity for a wavelength near L25 cm. becomes ex-
tremely small. Were loop coupling used this would necessitate a very small
coupling loop and also a very small diameter output line. The small dimen-
sions with loop coupling would require tolerances extremely difficult to
maintain with conventional vacuum tube techniques. On the other hand,
the wave guide used at L25 cm. is of dimensions (.170" x .420") such that
a wave guide output with a choke coupling can readily be incorporated in a
standard vacuum tube envelope.
The wave guide coupling is accomplished by means of a tapered wave
guide which couples to the cavity through a non-resonant iris. The guide
tapers in the narrow dimension only, from the iris to a circular output
window. The tapered guide couples to the window by means of a circular
half wave choke. The VSWR introduced by the window is 1.1 or less.
External to the tube, there is an insulating fitting which permits the tube
to be coupled directly to the guide by means of a second choke coupling.
This makes it possible to operate the shell of the tube at a different potential
598
BELL SYSTEM -TECH MCA L JOIRNAL
Fig. 89.— The 2K50 — a reflex oscillatjr with thermal tuning and a wave guide output
or operation in the 1 centimeter range
GUN CATHODE
ENLARGED DETAILS
Fig. 90. — Internal features of the 2K50.
REFLEX OSCILLATORS
599
than the guide. This is desirable in a radar receiver for circuit reasons,
which require that the cathode of the oscillator be at ground potential.
The iris size is a compromise chosen to provide sufficient sink margin
throughout the band. An iris coupling inherently varies with frequency
and provides a weaker coupling at lower frequencies. Hence, since a suffi-
\^
^S)^
LO
^
lo
.<?/
f= 24,464 MEGACYCLES
PER SECOND
Fig. 91. — A performance diagram for the 2K50 at the high frequency band limit. This
diagram shows loci of constant power as a function of the admittances presented at the
plane of the tube window. Admittances are normalized in terms of the characteristic
admittance of the wave guide.
cient sink margin must be provided at the wavelength where the coupling
is a maximum, this means that an e.xcess of sink margin exists at the low
frequency end of the band. This is illustrated by the impedance perform-
ance diagrams of Figs. 91 to 93.
The 2K50 presented a difficult mechanical problem which will be appreci-
ated when the minute dimensions of the resonant cavity are observed in
600
BELL SYSTEM TECHNICAL JOURNAL
Fig. 90. The electron optical system consists of a concave cathode, a
cylindrical beam electrode, and a grid concave towards the cathode. This
produces an electron beam which converges into a conical nose and through
the cavity grids to the repeller space. The repeller, which returns the beam
across the gap, is rigidly held in a mica supported in a cylindrical housing
connected to a diaphragm which serves as one wall of the resonator. The
0 "~
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vk
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^
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^c
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"7 \
!kh
io\
<5
^^
JiS
f = 23,984 MEGACYCLES
PER SECOND
Fig. 92. — Performance diagram for the 2K,S() at the mid-hand frequencj-.
cylindrical housing is connected to a thermal tuning mechanism consisting
of a simple framed structure in the shape of a right triangle. The base of
the triangle is a heavy piece of metal brazed to the cavity block and a bleeder
shoe which in turn is brazed to the bulb. One leg of the triangle can be
heated by electron bombardment controlled, as in the 2K45, by a negative
grid. The other leg is heated only by conduction. Since both legs are
made from the same material, general heating of the structure will produce
REFLEX OSCILLATORS
601
only a second order effect. Because of the small motion required to tune
the 2K50 through its range, the tuner dimensions permit reliance on conduc-
tion cooHng.
M
4^
f=23504 MEGACYCLES
PER SECOND
Fig. 93. — Performance diagram for the 2K50 at the low frequency band limit.
The characteristics of the thermal tuner of the 2K50 differ considerably
from the 2K45. In Appendi.x XI expressions are given for the heating and
cooHng times as
kTc
c , Th
(13.4)
(13.5)
It is desirable that the cycling times should be equal. Equating (13.4)
and (13.5) one obtains
P„, = k{Tn+ r.). (13.6)
602 BELL SYSTEM TECHNICAL JOURNAL
This states that the maximum j)ermitted temperature, r„, = Pm/k, shall
exceed the temperature Tk by the same temperature difference as that by
which Tr exceeds the sink temperature. From (13.4) and (13.5) it can be
seen that to minimize the heating and cooling times the heat capacity
should be small and the heat conductivity of the strut should be large. It
is also evident that the ratio TjJTc should be as nearly unity as possible.
This requires that the tuner should produce the required displacement in
the smallest temperature interval possible. If a given temperature diflfer-
ence is required to produce the necessary motion, then from a speed stand-
point it is desirable to make both Th and Tc large in order that their ratio
shall be nearly unity. The allowable temperature is usually limited by
constructional considerations.
Over the normal tuning range of a reflex oscillator we have previously
shown that the tuning characteristic may be represented by
X = a \/Cf + C(x) (13.7)
where a is a constant
Cf is a lumped fixed capacitance
C{x) = ^
X
jS isa constant.
Hence one can show that for small changes A.v from .Vo
AX = -i^" (13.8)
XoCo
Co = C/+ C(.Vo). (13.9)
One may also show that for the type of tuner employed and the small
motions involved in the 2K50 the displacement of the grids as a function
of the temperature diflference T — To will be
.V = .Vo - H(T - To) (13.10)
whence
AX = "M^^ZlzJi.) . (13.11)
-Tq Co
If at time / = 0, x = .to , T = To , \ — Xn , we have for heating
X' _ Ao = i^f^ - ToVl - e-'"'-'). (13.12)
.To Co \k I
If we give Pi its maximum value l\n then at / = oo the temperature of
Pi
the strut will be -^ = T„, ,X = X,„
k
REFLEX OSCILLATORS 603
and AX = X„ - Xo = W^ iT„ - To) (13.13)
or X - Xo = (X„ - Xo)(l - e-^'""^). (13.14)
Thus the behavior of this type of tuner may be described by a time constant
which is given by r = - . This has been verified experimentally for the
k
2K50, in which this constant has been found to have a typical value of 1.3
seconds.
The instantaneous tuning rates at a given wavelength based on full on-
full oflf operation can be shown to be
f = --^(/-/J heating (13.15)
at T Jm
f = -7(/o-/) cooling (13.16)
at Tjo
where /o is the frequency at zero tuner power
/„ is the frequency at maximum permitted drive.
Figure 94 shows the instantaneous tuning rate as a function of frequency
on heating and cooling.
Typical power output versus frequency characteristics for the 2K50 are
shown in Fig. 95. Curve A shows the power output with the repeller voltage
optimized at each frequency while Curve B gives the variation when the
repeller voltage is set for an optimum at the center of the band and_^held
fixed as the frequency changes. For constructional reasons the spacing
between the repeller and second cavity grid is fixed in the 2K50 so that on
a proportional frequency basis the range between half power points with
fixed repeller voltage is smaller for the 2K50 than for the 2K45.
Figure 96 shows the frequency vs. grid voltage characteristics for the
2K50. For normal operation with full on-full ofif operation the grid voltage
is switched between zero and cutoff.
H. A Millimeter Range Oscillator
During the latter stages of work on the 2K50 development, work was
started on an oscillator for a wavelength range around .625 cm. The design
of this developmental tube known as the 1464XQ was undertaken.
There are several difficulties in going from 1.25 cm. to .625 cm. Greater
accuracy of construction is required and the cathode must be operated at a
higher current density. The greatest difficulty arises from the fact that
the grids cannot be directly scaled in size from those used in tubes for longer
wavelengths.
604
BELL SYSTEM TECHNICAL JOURNAL
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23.6 23.8 24.0 24.2 24.4 24.6
FREQUENCY IN MLOMEGACYCLE S PER SECOND
Fig. 94.— Computed instantaneous tuning rates on heating and cooling for the 2K50
oscillator. These results are based on a time constant of 1.3 seconds and on the assump-
tion of "full on or off" operation.
to
1-
—
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GACYC
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. o — ■ r-.- -"--^".v.— ^o>,.^o ^. ..li,, ^iv,M.« usLiuatui. (curves .1, D ana c
Illustrate the power variation with frequency when the repcllcr voltage is set for the opti-
mum at the indicated frequency and held fixed as a cavity tuning is changed. The
envelope of these curves shows the power variation with frequency when the repeiler
voltage is maintained at its optimum value for each frequency.
REFLEX OSCILLATORS
605
Let us consider the factors involved in scaling from a tube operating at a
given frequency to a smaller tube operating at a higher frequency. If the
cathode is operated space-charge-limited and the anode voltage is the same
as for the larger tube, the total electron current will be the same and each
grid wire will intercept as much current and hence receive the same power
to dissipate as in the larger tube. Suppose the length of a grid wire in the
larger tube is /o and in the smaller tube the length of the corresponding wire
is h . Suppose that all the other dimensions of the smaller tube, including
24 8
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8 24.6
LU
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£ 24 4
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-I 24.2
U
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u
5 24 0
LU
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O
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LU
a 23.4
LU
tr
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23.2
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s.
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-12 -10 -8 -6
TUNER GRID VOLTAGE
Fig. 96. — Frequency vs tuner grid voltage for the 2K50.
the diameter of the grid wire, are reduced in the ratio h/U . We do not
know a priori whether or not the temperature distribution along the grid
wires of the smaller tube will be the same as that for the larger tube; suppose,
however, that it is. Then if To is the temperature at some typical point,
say, the hottest, on the grid wire of the larger tube, and Ti is the temperature
at the similar point on the smaller tube, the power the wire loses by radiation
in the large tube, P^o and in the small tube, Pt\ are given by
Pr^ = Al\T\.
(13.17)
(13.18)
606 BELL SYSTEM TECHNICAL JOURNAL
Here A is nearly constant for given tube geometry and materials. In
radiation the power lost varies as the area, which varies as / , and as the
temperature to the fourth power.
The power lost by end cooling, for the large and the small tube, P«o and P.i
will be given by
P«o = BloT^ (13.19)
Pei = BhTi. (13.20)
Here B is another constant. These relations express the fact that the
power lost by end cooling (thermal conduction) varies as cross sectional
area divided by length and hence as / and as temperature difiference, taken
as proportional to T.
Now, in scaling the tube the power to be dissipated has been kept constant.
Further, in making the tube small, the hottest point of the grid cannot be
run hotter than the melting point of the wire; in fact, it cannot be run nearly
this hot without unreasonable evaporation of metal. Suppose we let the
grid in the smaller tube attain the maximum allowable temperature Tm
and let the power the wire must dissipate be P. Then for the large tube
P = Pro-\- Peo = {AloTl + B)loTo (13.21)
and for the smaller tube
P --= Pri+ Pel = (AhTl + B)hT,„. (13.22)
Hence, the smallest value h can have without running the grid too hot is
given by the equation
To (AIqTI + B)
Tm{AhTl+B)
We see that if /o is very small.
^1-^0^" ',::z z- (13.23)
AloT:n«B
AhT^n « B
(13.24)
Numerical examples show that this is so for a tube such as the 2K50. This
means that nearly all of the power dissipated by the grid is lost through end
cooling, not radiation.^* Further, in the 2K50 the grid is already operating
near the maximum allowable temperature. Hence, nearly. To = Tm and
the smallest ratio in which the tube can be scaled down without overheating
the grids is approximately unity. This means that in making a tube for
.625 cm. the grid wire cannot be made half the diameter of the wire used in
"The fact that one kind of dissipation predominates in both cases justifies the as-
sumption of the same temperature distriliution in both cases.
REFLEX OSCILLATORS 607
the 2K50. In fact, the diameter of the grid wire can be made but little
smaller. Thus, in the 1464XQ the grid is relatively coarse compared with
that in the 2K50. This results in a reduced modulation coefficient and
hence in less efficiency.
In going to .625 cm., resonator losses are of course greater. The surface
resistivity of the resonator material varies as the square root of the fre-
quency. Surface roughness becomes increasingly important in increasing
resistance at higher frequencies. Further, in order to provide means for
moving the diaphragm in tuning, it was necessary in the 1464XQ to use a
second mode resonator (described later) and this also increases losses over
those encountered at lower frequencies.
Development of the 1464XQ was stopped short of completion with the
cessation of hostilities. However, oscillation in the range .625-. 660 cm.
had been obtained. The power output varied from 2-5 milliwatts between
the short wave and long wave extremes of the tuning range. The cathode
current was around 20 ma, the resonator voltage 400 volts. The tube
operated in several repeller modes in a repeller voltage range 0 to — 180 volts.
Figures 97 and 98 illustrate features of the 1464XQ oscillator. Figure 98
is a scale drawing of the resonator and repeller structure. The electron
beam is shot through two apertures covered with grids of .6 mil tungsten
wire. These grids are 80% open and are lined up. The aperture in the
grid nearest the gun is 23 mils in diameter and the second aperture is 34 mils
in diameter. The repeller is scaled almost e.xactly from the 723A 3 cm. reflex
oscillator. A second mode resonator is used. The inner part, a, of Fig. 98
is about the size of a first mode resonator. This is connected to an outer
portion, c, by a quarter wave section of small height, b, which acts as a
decoupling choke. The resonator is tuned by moving the upper disk with
respect to the lower part, thus changing the separation of the grids. The
repeller is held fixed. Power is derived from the outer part, c, of the reso-
nator by means of an iris and a wave guide, which may be seen in the section
photograph Fig. 97. There is an internal choke attached to the end of the
part of the wave guide leading from the resonator. This is opposed to a
short section of wave guide connected to the envelope, and in the outer end
of this wave guide there is a steatite and glass window of a thickness to
give least reflection of power.
I. Oscillators for Pulsed Applications — The 2K23 and 2K54
All the reflex oscillators described in the preceding sections have been low
power oscillators intended for beating oscillator or signal oscillator applica-
tions. Some limitation on the power capability of these oscillators in the
form previously described is set by the power handling capacity of the grids.
If the tubes are pulsed with pulse durations which are short compared with
■THERMAL TUNER
HEATER
THERMAL TUNING
STRUT
TUNING YOKE
STRUT
REPELLER
QUARTER -WAVE
CHOKE
CHOKE
STEATITE AND GLASS
WINDOW
Fig. 97. — An experimental thermally tuned retlex nscillatdr. the 1464, designed for
operation between the wave lengths of .0 and .7 cms.
008
REFLEX OSCILLATORS
609
the thermal time constant of the grids, then the peak power input to the
oscillators may be increased over the continuous limit by the duty factor,
provided that the voltages applied are consistent with the insulation limits
of the tubes and that the peak currents drawn from the cathode are not in
excess of its capacity.
An application for a pulsed oscillator arose in the AN/TRC-6 radio
system/'' This was an ultra-high frequency military communication system
using pulse position modulation to convey intelligence. With the high gain
which may be achieved with antennas in the centimeter range, the power
necessary in the transmitter for transmission over paths limited by Hne of
sight is of the order of a few watts peak. In beating oscillator applications
the power output is of secondary importance to electronic tuning, so that
the reflex oscillators previously described were designed to operate with a
drift time in the repeller space such as to provide the desired tuning. In a
REPELLER --■»■
\ INNER"RESONATOR"
\ -- -^ DECOUPLING SECTION^ J
'OUTER NON-CRITICAL PORTION OF RESONATOR^
Fig. 98. — The resonator and repeller structures of the oscillator shown in Fig. 97.
pulsed transmitter electronic tuning is unnecessary and indeed undesirable,
since it leads to frequency modulation on the rise and fall of the pulse. In
section III it is shown that for maximum efficiency with a given resonator
loss there will be an optimum value for the drift time. If there were no
resonator loss this time would be f cycles, which is the minimum possible.
By utilizing the optimum drift angle and taking advantage of the higher
peak power inputs permitted by pulse oscillator it was possible to obtain
peak power outputs of the order of several watts using the same structure
as employed for the beating oscillators previously described without exceed-
ing the power dissipating capability of such a structure. From the stand-
point of military convenience this was a very desirable situation for reasons
of simplicity of tuning and ease of installation.
^^ A Multi-channel Micro-wave Radio Relay System, H. S. Black, W. Beyer, T. J.
Grieser, and F. \. Polkinghorn, Electrical Engineering Vol. 65, No. 12, pp. 798-806, Dec.
1946.
610
BELL SYSTEM TECHNICAL JOURNAL
The first tube designed for this service was the 2K23. It was based on
the design employed in the 2K29 and operated with a drift time of If cycles
in the repeller space. A severe limitation on the performance was set by
by the requirement that a single oscillator should cover the frequency range
from 4275 to 4875 Mc/s. It is shown in Section X that in an oscillator tuned
by changing the capacitance of the gap the efficiency will vary considerably
{2 500
1 450
-I
1
2 400
a 350
t-
■D
O
0:300
Ul
9 250
r--
REQUIRED RANGE
^
1
^^^
^
'^'
^y
^
J^
>^
4300
4400 4500 4600 4700
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 99. — Variation of the peak power output vs frequency for the 2K23 reflex oscillator.
This tube was designed for pulse operation withja duty factor of 10 in a repeller mode
having 3.5 t radians drift.
I'ig. 100. — Modulator circuit for use in connection with reflex oscillators showing
means for applying pulse voltage to the repeller to reduce frequency modulation during
pulsing.
over such a tuning range. In the case of the 2K23 the variation of the peak
pow-er output with frequency is shown in Fig. 99. The duty factor at which
the tube was used (the ratio of the time between pulses to the pulse length)
was 10. In the AX TRC-6 application the tube was operated on a fixed
current basis; i.e., the pulse amplitude was adjusted to a value such that
the average current drawn was 15 ma. The schematic of the circuit em-
ployed in pulsing oscillators in this way is shown in Fig. 100. The resonator
REPLEX OSCILLATORS ■ 611
of the oscillator is operated at ground and the cathode of the oscillator
is pulsed negative with respect to this ground. The repeller voltage is
referenced from the cathode and this reference is maintained during the
j)ulse. Some frequency modulation occurs during the rise and fall of the
pulse because of the changing electron velocity. This can be reduced in
part by applying a part of the pulse in the repeller circuit proportioned
in such a way as to tend to maintain the drift time independent of the
cathode to resonator voltage. Satisfactory performance is achieved in
this way.
As mentioned previously, the intelligence is conveyed by pulse position
modulation. The AN/TRC-6 system uses time division multiplex to provide
eight communication channels. The multiplexing is achieved by trans-
mitting a four micro-second marker pulse which provides a time reference
followed by eight one micro-second pulses. The time of each of the latter
pulses is independently varied in position with reference to the marker
pulse.
The time interval from the marker to each pulse could be measured to
either the leading or trailing edge of the pulse. In Section XII it is shown
that the leading edge of the r.f. pulse will be subject to what is commonly
called "jitter" because of the random time of rise which will result if oscilla-
tion starts from shot or Johnson noise. Conceivably oscillation might be
started by shock excitation of the resonant circuit by the pulsed beam cur-
rent. However, in Appendix X it is shown that the initial excitation pro-
duced by shot noise in the beam exceeds that induced by the current tran-
sient by a factor of approximately 100. The trailing edge of the pulse will
not be subject to this form of jitter provided two conditions are met. First,
the pulse duration must be long enough so that oscillation builds up to full
amplitude during the pulse. Second, the receiver must have a sufficient
bandwidth so that the transient which occurs on reception of the leading
edge has fallen to a small value by the time the trailing edge is received.
Since these conditions were met in the AN/TR('-6 system, the trailing edge
of the pulse was used.
In the latter stages of the development of the AN/TRC-6 system it was
decided to remove the restriction that the required tuning range should be
covered with a single transmitter tube. This made possible the achievement
of a design which would i)rovide an improved performance throughout the
band. In order to improve the circuit efficiency of the resonator, the new
designs were based on the oscillator structure which was employed in the
2K4.S. It has been pointed out previously that this makes possible a con-
siderably higher resonant impedance of the cavity, partly because of the
reduced gap capacitance and also because the smaller first and second grids
reduce the resonator losses. These effects were reflected in the higher
efficiency obtained in the 2K54 and 2K55.
612
BELL SYSTEM TECHNICAL JOURNAL
The three tubes, tlie 2K2v^, 2K54 and 2K55 were inteiuled to couple to
wave guide circuits. Although direct coupling to the guide through a wave
guide output would have been desirable in some ways, it would have re-
sulted in a very large and awkward structure. It seemed best, therefore, to
retain the coaxial output feature of tubes designed for beating oscillator
service. From a standpoint of convenience, it would have been desirable
0.485"
Fig. 101. — A transiluccr ck-si^iicd to adiipt tlic 2K2,\ 2K.^4 and 2K55 oscillators to a
terminated wave guide load. If the hack piston of this coujiling is set at a distance of
1.08" from the center of the prohe the impedance presented to the oscillator line is 50
ohms with a 1 dl) variation over the re(|uired fre(|uency range.
to have had the wave guide probe an integral part of the tube, as is the case
in the 2K2vS. However, the length of a quarter wave probe at the operating
wave lengths of the AN/TRC-C system made such a design hazardous.
Because of this a transducer was developed into which the tube could be
plugged. This is shown in Fig. 101. The central chuck e.xtends the center
conductor of the output coa.xial ot" tiu' tube into the guide. It had been
REFLEX OSCILLATORS
613
originally intended to design this transducer so that all elements would be
fixed. It was found, however, that an adjustable back piston setting per-
mitted compensation for manufacturing tolerances in the tubes as well as
permitting the presentation of a better impedance to the oscillator over the
band than was attainable with a fixed transducer. This is illustrated in
Fig. 102 which shows the power output versus frequency for a typical 2K54
and a typical 2K55 as a function of frequency for three cases. In one case,
as shown by curve B, the power output is delivered at all frequencies into
the optimum impedance, i.e. the impedance into which the oscillator will
deliver maximum power. Curve A shows the power output delivered into
■REQUIRED RANGE
- REQUIRED RANGE
4200 4300 4400 4500 4600 4700 4600 4900
FREQUENCY IN MEGACYCLES PER SECOND
fig. 102. — Peak power output vs frequenc\- for the 2K54 and 2K55 oscillators for sev-
eral load conditions. Curves A give the performance obtained when the tubes operate
into a characteristic admittance load threugh the coupling of Fig. 101 with the end plate
fixed so that the admittance presented to the tubes is constant to within 1 db over the
frequenc}- range. Curves B give the performance obtained when the optimum imjjedance
is presented to the oscillators throughout the band. Curves C give the performance ob-
tained when the tubes are coupled to a characteristic admittance load with the coupling
unit of Fig. 101 and with a back piston adjusted to the best value at each frequency.
a transducer which has the back piston fixed at a distance from the probe,
so chosen that the impedance seen by the oscillator is fiat over the band to
within 1 db. Curve C shows the power output when the back piston of the
transducer is adjusted so that the tube delivers maximum power. It can
be seen that the performance obtained under the latter circumstances is very
nearly as good as that obtained when the optimum impedance is presented
to the oscillator.
Froon Sections III and IX we would expect that, since the coupling system
is such as to give maximum power throughout the band, the sink margin
should be slightly greater than 2. Figures 103 to 106 give impedance per-
formance diagrams for 2K.S4 and 2K55 at the four transmitting frequencies
614
BELL SYSTEM TECHNICAL JOURNAL
of the AN/TRC-6 system. From this it can be seen that the sink margin is
approximately 2 at all frequencies. In a transmitter tube another factor,
which is of small importance in a beating oscillator, becomes of interest.
This factor is the {lulling figure, which is defined as the ma.ximum frequency
FREQUENCY
POWER
Fig. 103. — Rieke diagram for the 2K54 at a nominal frequency of 4500 megacycles.
The point at the unity vsivr condition is obtained by adjusting the repeller voltage and the
back piston of the coupling of Fig. 101 to the values which gave maximum power. These
conditions were then held fixed for the remainder of the chart.
excursion which will be produced when a VSWR of 3 db is presented to the
transmitter and the phase is varied over 180°. Fig. 107 gives the pulling
figure as a function of frequency for the 2K54 and 2K55. The requirements
on the pulling figure for the 2K54 and 2K55 were not severe, since in the
AN/TRC-6 system the tube is coupled to the antenna by a very short wave
guide run and, furthermore, the antennas are fixed.
REFLEX OSCILLATORS
615
Investigation of the pulling figure of early models of the 2K55 led, how-
ever, to the disclosure of one unforeseen pitfall arising from the existence of
electronic hysteresis. It had at first been considered that electronic hystere-
sis would not be of importance in the transmitter tube, where the feature of
electronic tuning was of no importance. This might be true in a CW oscilla-
FREQUENCY
POWER
Fig. 104. — Rieke diagram for the 2K54 oscillator at a nominal frequency of 4350
megacycles. The unity iswr point was obtained as described in Fig. 103.
tor, but in a pulsed oscillator the existence of hysteresis resulted in an un-
foreseen reduction of the sink margin. Since the oscillator is being pulsed,
for each pulse the oscillating conditions are being re-established. Although
the cathode-repeller voltage need not vary during the pulsing, the fact that
the cathode-resonator voltage is being changed means that for each pulse
the drift angle in the repeller space varies on the rise and fall of the pulse.
The effect during the rise of the pulse is the same as though the repeller
616
BELL SYSTEM TECHNICAL JOIRNAL
voltage were made less negative. In other words, on the rise of each pulse
the situation is equivalent to that in a C'lT oscillator when, for a fixed
resonator voltage, one starts with a repeller voltage too negative to permit
oscillation and then reduces the repeller voltage until oscillation occurs.
Let us now suppose that the hysteresis is such that under these circumstances
1.0
/
V
/ /
\
/ /
V
\ \
1
V
1
7~~~~7
v \ /
\
\\^
U—
-5
0
G
Ml
-
50
0.2
C
>
50
JooA
7\
F
i^SvL
^
\
\
^
\
\
\
3 \
^
£
FREQUENCY
POWER
^
Fig. 105. — Rieke diagram for the 2K55 oscillator at a nominal frequency of 4800 mega-
cycles. The unity vsicr point was obtained as described in Fig. 103.
the amplitude of oscillation would suddenly jump to a large value. We
would then obtain a variation of peak jjower output as a function of the
repeller voltage shown in Fig. 108. Ordinarily, the repeller voltage would
be adjusted so as to obtain maxmum power output as, for example, with
the repeller voltage V m ■ Next, let us sui)|K)se that a variable impedance
is presented to the oscillator with the repeller \'oltage held fixed at value
V R\ , as would be done, for example, in obtaining an impedance performance
REFLEX OSCILLA TORS
15"
.65^'
\\^
G
M|_
0.2
^N
\
0.5
^^
y\ "^"^
,^
^\^f'
?^
y^}
?^
""/Tx ~
-~
///T
i:o~
i
••5.
J- ""''^^
2^
^^
3_j
617
FREQUENCY
POWER
^^
Fig. 106. — Rieke diagram for the 2K55 oscillator at a nominal frequency of 4650 mega-
cycles. The unity v$xvr point was obtained as described in Fig. 103.
Q 16
o
u
il 14
15 LU
50 '^
Q-li) 10
Z 8
./
/
/
2K55
'**
2K
54
/
/
/
"""N
\
/
r
^
-
4200
4400 4500 4600 4700
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 107. — Pulling figures for the 2K54 and 2K55 reflex oscillators as functions of fre-
quency. With a unity I'^wr load the repeller voltage and back piston of the coupling of
Fig. 101 were adjusted for a maximum power and held fixed for the pulling figure meas-
urements.
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
/ /
1 1
1 1
1
1 1
1 j
I 1 Vri
^
NEGATIVE REPELLER VOLTAGE
Fig. 108. — The variation of power output with repeller voltage for a pulsed re 'ex
oscillator exhibiting hysteresis.
MAXIMUM POWER
-^N
\
\
0^ ;o
1 0.2
"^ 1
[-0,3
j-0.4
\ 1
Fig. 109. — The elYect of hysteresis on the Ricke diagram of a pulsed retlex oscillator.
The hysteresis shown in Fig. 108 can result in failure of a jjulsed oscillator to operate in
the lightly shaded jwrtion of the Riekc diagram as well as the heavily shaded portion
corresponding to the normal sink.
618
1
REFLEX OSCILLATORS
619
diagram. The effect of varying the impedance on the repeller characteristic
in Fig. 108 is to shift the whole characteristic to the right or left, depending
upon the phase of the impedance, as well as to change its general form as
shown by the dotted curve. It can be seen from this that if the hysteresis
is sutliciently bad and if the pulling figure exceeds a particular value, one
(Joe
UJuj
200
180
160
140
120
100
-REQUIRED RANGE-
■1
ZZ
oo
7m
40
z"'
DOC
t-Ui
n
30
o
5W)
?0
iro
t-v
oo
UJ<
-Jo
10
UJiu
5
0
180
160
-,</)
n •-
i-t-
■^5
140
oS
i?n
<-fc
^z
100
(a)
>^
^
""
-^
^^
-T
■^
1 1
^
• —
■ —
(b)
'
^^
^ .
|_
1 1
(0
1
^
^^
-^^""^
^
^
^
^
4400 4500 4600 4700 4800
FREQUENCY IN MEGACYCLES PER SECOND
Fig. 110. — Performance characteristics of the 2K22 operated into a 50 ohm load.
effect of the hysteresis will be to reduce the sink margin, and this is found
to be the case. Fig. 109 shows an impedance performance diagram obtained
with pulsed operation for an early model of the 2K55 in which the hysteresis
was excessive. From this it can be seen that the area of the sink on the
diagram is very greatly increased and also that the sink margin is reduced
from the theoretical value of somewhat in excess of 2 to a value of less than
C20 BELL SYSTEM TECHNICAL JOURNAL
1.1 although maximum power occurs at unity \'S\\'R. A modification in
the design reduced the hysteresis and eliminated this efifect as shown by
Figs. 103 to 106.
In addition to the transmitter tube for the AN/TRC-6 system, it was
necessary to design a beating oscillator. This tube is known as the Western
Electric 2K22. Its design was scaled from the 2K29, previously described.
The tube was designed to operate into a 50 ohm impedance and can be
coupled by a coaxial adapter to a 50 ohm line or by means of the transducer
of Fig. 101 to a wave guide. When the back piston of the transducer is set
at a distance of 1.080" from the probe center, the impedance presented to
the oscillator is 50 ohms with approximately a. 1 db variation over the
frequency range. Fig. 110 gives the performance characteristics of the
2K22 operating into a 50 ohm load. The AN/TRC-6 system using these
tubes provided a military communication system during the war. A
description of this service has been given. Also, models of the AN/TRC-6
system have been put into service to provide telephone communication be-
tween Cape Cod and Nantucket and also between San Francisco and
Catalina Island.
J. Scope of Oscillator Dccelopment at the Bell Telephone Laboratories
The reflex oscillators discussed in the foregoing sections were developed
primarily for beating oscillator service, and in one instance for a transmitter.
Refex oscillators also received wide application in test equipment. The
best-known application of this type was in the spectrum analyzer, in which
the electronic tuning characteristic of the oscillators made possible the dis-
play of output spectra, and especially of the spectra of magnetron oscillators,
on an oscilloscope. This greatly facilitated the development of the mag-
netron. The reflex oscillator also was widely used as a signal generator, and
the ease of frequency adjustment particularly suited it to this application.
In some signal generators it was desired to pulse the reflex oscillator at low
power levels. As an alternative to the method previously described, in
which the voltage between the cathode and resonator was pulsed, it is
possible to leave this voltage fixed and to pulse the voltage between the
repeller and cathode. In this case the repeller-cathode voltage is set at a
base value at which the tube will not oscillate and the pulse varies this
voltage to the oscillating value.
The oscillators which have been described have been chosen to indicate
various features of their development. In addition to these, a number of
other oscillators were developed to meet various service needs. Figure 111
shows a chart giving the frequency ranges of these tubes. Oi the eleven
beating oscillators of the reflex oscillator type on the Army-Navy preferred
1
1464 ]
-
1
2K50
-
-
2K25
AND 2K45
1449
---
, 2K26 , _
■2K27
1^ 2K55 "1
2K23
2K22;
-
( 2K54 1
■:.:;2K56
2K29
- —
726A
-
- ■
. -
707A .
7078
■ IN .
SUITABLE
CAVITY
•
726B
L^?,^^,. J
-4.5^
OSCILLATORS BY CODE NUMBERS
Fig. 111.— Reflex oscillators developed at the Bell Telephone Laboratories. The
loxes around the tube numbers show the frequency range covered.
621
622 BELL SYSTEM TECIINLCAL. JOURNAL
list of electron tubes for l'M5, nine were developed at the Bell Telephone
Laboratories.
APPENDIX I
Resonators
In thinkin<f about resonators it is imjM)rtant in order to avoid confusion
to keep a few fundamental ideas in mind. One of the most important is
that we must not use the notion of scalar potential in connection with fluc-
tuating magnetic fields. Electric fields produced by fluctuating magnetic
fields cannot be derived from a scalar potential, and in the presence of such
fields to speak about the potential at a point is hopelessly confusing.
The idea of voltage as the line integral of electric field along a given path
between two points is useful, but it must be remembered that the voltage
depends on the path chosen. Consider, for instance, an ordinary 60-cycle
transformer with the secondary wound of copper tubing. For a path from
one secondary terminal to the other through the center of the tubing the
voltage (integral of field times distance) is zero. For a path between ter-
minals outside of core and coil, the voltage between terminals is d\l//dl,
where i/' is the magnetic flux linkage of the path and the coil, counting each
line of force as many times as the path encircles it.
If resistance drop is neglected the work done in moving a charge through
a conductor is zero. The line integral of an electric field around a closed
path is d\l//dt. If part of the path is through a conductor, or through a
space where there is no electric field, the voltage along the rest of the path
(as between portions of the conductor) isdip/dt. For paths linking different
amounts of flux, the voltage will be different. In the case of low frequency
transformers, all paths linking the terminals and lying outside of the core
and coil link practically the same amount of flux, and there is little am-
biguity about the voltage. In reflex oscillators the electrons travel from
one field free region to another along a certain path and this determines
the path along which the voltage should be evaluated.
To review: the voltage between two points is the integral of the field along
the path times distance, and refers to a certain path. If the path begins
and ends in a lield free region, the voltage is d\p/dt, where \p is the magnetic
flux linking tlie chosen path and a return path through the field free region.
To this should be added that high frequency currents and fields penetrate
the surface of metals only a fraction of a thousandth of an inch in the
centimeter range, so that the interior of a conductor is field free, and fields
inside of a metal enclosed space cannot produce fields outside of that space
"* The electric field can, of course, be derived from a scalar and a vector i)otential.
'^ .Vs an exani])le, for copper the field is reduced to (1/2.72) of its value at the surface
KEFLEX OSCILLA TORS
623
In considering resonators we should further note that the magnetic flux
must be produced by a flow of current, either convection current or dis-
placement current, around the lines of force. In a transformer this can be
identified as the current flowing in the coils. In the resonators used in
reflex oscillators it is the current flowing in the walls and, as displacement
current, across the gap and from one face of the resonator to the other.
Two axially symmetrical resonators suitable for use in reflex oscillators
are shown schematically in Figs. 112 and 113. The resonator in Fig. 112
has grids and might be used with a broad unfocused electron beam at a low
d-c voltage; that shown in 113 has open apertures and might be used with a
COUPLING
LOOP
H
COAXIAL LINE
Fig. 112. — An oscillator cavity with grids and loop coupling to a coaxial line.
IRIS WAVE-GUIDE
Fig. 113. — An oscillator cavity without grids and with iris cou[)ling to a wave guide
focused high voltage electron beam. In Fig. 112, the resonator is coupled
to a coaxial line by means of a coupling loop or coil; in Fig. 113 the resonator
is coupled to a wave guide by means of a small aperture or "iris."
Let us consider the resonator of Fig. 112 in the light of what we have just
said. A magnetic field flows around the axis inside of the resonator. There
is an electric field between the top and bottom inside surfaces of the resona-
at a depth.
6 = 3.82 X W-'^^^X
(al)
Here X is wavelength in centimeters. It may also be convenient to note that the surface
resistivity of a centimeter square of resonator surface is, for copper,
R = .045/Vx
(a2)
This means that if a current of I amperes flows on a surface over a width \V and a length
/, the power dissipation is
F = r-iv./w
(a3j
For other non-magnetic metals, both 6 and R are ])r()p()rti()nal to the square root of the
resistivity with respect to that of copper.
624 BELL SYSTEM TECHNICAL JOVRNAL
tor. There is no electric field outside of the resonator (except a very little
that leaks out near the grids). To take a charge from one grid to another
outside of the resonator require? no work. Thus the voltage across the gap
is #/(//, where \p is the magnetic flux around the axis. Actually, there
is a little magnetic flux between the grids, and hence the voltage near the
edges of the grids is a little less than that at the center.
Current flows as a displacement current between the grids, and as con-
vection current radially out around, and back along the inside of the resona-
tor to the other grid. This current flow produces the magnetic field that
links the axis.
Part of the magnetic flux links the coupling loop. If this part is \pi and
if the coupling loop is open-circuited, the voltage across the coupHng loop
will be d-^tjdt.
J '~W^ 0
^rn3
4
L2i
c-L
^3g
Fig. 114. — Equivalent circuit for a resonator having 3 modes of resonance.
In the resonator of Fig. 113, power leaks out through the iris into the wave
guide. Part of the wall current in the resonator flows out through the hole
into the guide; part of the magnetic flux in the resonator leaks out into the
guide.
In dealing with resonators as resonant circuits of reflex oscillators, to
which the electron stream and the load are coupled, we are interested in
the gap and output impedances. For a clear and exact treatment, the
reader is referred to a paper by Schelkunoff. No exhaustive treatment
of the problem will be given here, but a few important general results will
be given.
If the resonator is lossless, the impedance looking into the loop ma>- be
represented exactly by an equivalent circuit indicated in Fig. 114. As
coils used at low frequencies arc really not simply ideal "inductances,"
(an idealized concept), but have many resonances (ascribed to distributed
capacitance), so the resonator has many, in fact, an intinity of resonances.
In the equivalent circuit shown in Fig. 114, only ^ of these are represented,
'" S. A. Schelkunoff, Representation of InipccUuicc I'luutions in Tfrnis nf Resonant
Frequencies, Proc. I.R.E., 32, 2, pp. 83-90, Fcl)., 19-44.
REFLEX OSCILLATORS 625
by the inductances U , Li , U and the capacitances Ci , C2 , C3 . These
resonant circuits are coupled to the terminals by mutual inductances Wi ,
W2 , W3 . In series with these appears the inductance measured at very low
frequencies Lq , the self inductance of the couphng loop. The circuit in
Fig. 1 14 may be regarded as a symbolic representation to be used in evaluat-
ing Z, just as a mathematical expression may be a symbolic representation
of the value of an impedance.
In practical cases, the resonances are usually considerably separated in
frequency, and near a desired resonance the efifect of others may be neglected.
In addition, if the Q is high we may add a conductance Gr across the capaci-
tance to represent resonator losses (Fig. 115). It would be equally legiti-
mate to add a resistance in series with L. In Fig. 115 a load impedance
Zt has been added. Fig. 115 is a very accurate representation of a slightly
lossy resonator, a low loss coupling loop, and a load impedance. The
Fig. 115. — Etjuivalent circuit showing connection between the oscillator gap regarded
as one pair of terminals and the oscillator load for an oscillator resonator having only one
resonant frequency near the frequency of operation.
meaning of L and C will be made clearer a little later. We will now clarify
the meaning of m. Suppose no current flows in the coupling loop (Z = co ).
Let the peak gap voltage be V . The peak voltage across m will be
F„, = mV/L (a4)
In a resonator, if a peak voltage V across the gap produces a peak flux \p,n
linking the coupling loop when no current flows in the coupling loop, then
F„, - d^p^Jdt = tnV/L (a5)
This defines m in terms of magnetic field, and L.
Figure 115 is also a quite accurate representation of Fig. 113. In this
case the "terminals" are taken as located at the end of the wave guide.
Le is the inductance of the iris, which will vary with frequency.
When we are interested in the impedance at the gap as a function of fre-
quency, we may equally well use the equivalent circuit of Fig. 116. Here
G i^ represents conductance due to load; Gr represents conductance due to
resonator loss. The total conductance, called Gc, is
Gc = Gu-\r Gl (a6)
626
BELL SYSTEM TECHNICAL JOlRyAL
1 he circuit of Fij:;. 116 does not tell us how (> i, varies with variation in load
impedance Z i. . I'urther, L and C in this circuit are changed in changing
Zl , and include a contribution from the inductance of the couj)ling loop
(which should not be very large).
Fig. 116. — A simplified ecjuivalent circuit of an oscillator resonator.
Several resonator parameters are vitally important in discussing reflex
oscillators. These will be discussed referring to Fig. 116. G i. and G«
have already been defined. The resonant radian frequency a-o is of course
a-„ = (LCr'" (a7)
A very important quantity will be called the characteristic admittance M
M = (C/Lf-
(a8)
This quantity is important because at a frequency Aw off resonance the
admittance of the circuit is verv nearlv
I' = G -\- jIMAw/wu
AcO = CO — (x\)
The "loaded" Q of the circuit \\ ill be referred to merely as Q and is
{) = M/Gc.
The unloaded () is
()„ = M/G,i
A quantity which may be called the "external ()"is
(),. = M/G,
We see
l/C>o+ \/Qk= 1/()
(a9)
(alO)
(all)
(a 12)
(al3)
The energy stored in the magnetic lield at zero voltage across the resonator,
and the energy stored in the electric field at the Nollage maximum are both
IF., = (1/2) F'C
= (l/2)F'M/con
(al4)
(al5)
REFLEX OSCILLATORS 627
Here V is the peak gap voltage. Expressions (al4) and (al5) are valuable
in making resonator calculations from exact or approximate field distribu-
tions. They define C, L and M in terms of electric and magnetic field.
The energy dissipated per cycle is
Wj = 7rF'C/con (al6)
Hence, we might have written Q as
Q = 2wWJ]W (al7)
This is one popular definition of Q.
In (a8)-(al7) we usually assume that there is no appreciable energy
stored in the load or the field of the coupling loop, so that M is considered
as unafifected by load. The effect of "high ()" loads with considerable
energy storage is considered in a somewhat different manner in Sec. IXB.
It must be emphasized that the expressions given above are valid for
high Q circuits only (a Q of 50 is high in this sense). Expression (al7) is
often used as a general definition of Q, but it is not complete without an
additional definition of the meaning of resonance in a low Q circuit with
many modes. Schelkunoff uses another definition of Q. Unforced oscilla-
tions in a damped circuit can be represented as a combination of several
terms
F:e^" + V,^'' + • • • • (al8)
* pi = ai+jcoi (a 19)
Schelkunoff takes the Q of the ni\\ mode as
Qn = w„/«n (a20)
This is at least a consistent and complete definition. The reader can easily
see that it accords with the definitions given for high ()'s in connection
with the circuit of Fig. 116.
Sometimes there may be a complicated circuit between the gap and a
coixial line or wave guide. In this case, the circuits intervening between
thi gap and the line can be regarded as a 4 terminal transducer (Fig. 117).
The constants of this transducer will vary with frequency. No further
consideration of this generalized treatment will be given, as it is well covered
in books on network theory. A particular representation of the transducer
will be pointed out, however. If the impedance in the line is referred to a
special point, one parameter can be eliminated, giving the equivalent circuit
shown in Fig. 118. If the gap is short-circuited, the impedance is zero
and the impedance at the special point to be chosen on the line is R\ the
special point may be chosen as the potential minimum with the gap shorted.
628 BELL SYSTEM TECHNICAL JOURNAL
N, the voltage ratio of a perfect transformer, is usually complex. The
impedance ratio A^A^* is real. If we are not interested in the relation of
output phase to gap phase, we may disregard the phase angle of N, and
deal only with the impedance ratio, the absolute value of A^ squared, which
we will call N . Thus, using this equivalent circuit and disregarding the
phase of A^ we can for our purposes reduce the number of independent
parameters from the usual 6 for a passive 4 terminal network to 4,
Y(= G + jB), 7V^ and R. This reduction can greatly simplify the algebra
and arithmetic of microwave problems. The circuit of Fig. 118 has an
additional advantage; if we choose our impedance reference point on the
output line or guide to be the suitable point nearest to the actual output
TRANSDUCER
LINE OR
WAVE-GUIDE
Fig. 117.— A 4-terniinal transducer is the most general connection between the oscilla
tor gap and a line or wave guide.
I
SPECIAL POINT
ALONG LINE
O
Fig. 118. — One circuit which will represent all the properties of a general 4-terminal
transducer. This circuit consists of an admittance Y shunting the gap, a perfect trans-
former of complex ratio N and a series resistance R.
loop or iris, the circuit represents fairly accurately the frequency dependence
of the output impedance if we merely take Y as
Y = C;« + ico(C - 1/coL) (a21)
Near resonance we may use the simpler form.
I' = Gh + jlMAco/coo (a22)
Something has already been said in a general way about the evaluation of
L and C" in terms of the electric and magnetic field distribution in a resonator.
It is completely outside of the scope of this paper to consider this subject
at any length; the reader is referred to various books. ' ' The discon-
tinuity calculations of Whinnery and Jamieson are also of great value in
" Electromagnetic Waves, S. A. Schelkunoff, Van Nostrand, 1943.
*" Fields & Waves in Modern Radio, Ramo & Whinnery, Wiley, 1944.
" Microwave Transmission Data, Sperry (lyroscope Company, 1944.
" J. R. Whinnery and H. W. Jamieson, Ecjuivalcnt Circuits for Discontinuities in
Transmission Lines, Proc. I.R.E., i2, 2, pp. 99-114.
REFLEX OSCILLATORS 629
making resonator calculations. These can be profitably combined with
disk transmission line formulae. ' "
The writers would like to point out that in the present state of the art
the testing of resonator calculations by models is important. Models
need not be made of the size finally desired. If all dimensions are made A^
times as large, the wavelength will be A^ times as great. The characteristic
admittance M will be unchanged. If the material is the same, and the
surface is smooth and homogeneous, Q and \/Gr , the shunt resonant
resistance, will be \/7V^ times as great.
It is perhaps a needless caution to say that the accuracy of a metliod of
resonator calculation cannot be judged by its mathematical complexity or
the difl&culty of using it. Methods of calculation which are simple and
may seem to make unduly broad approximations are sometimes better
founded than appears on the surface, and complicated methods, exact if
carried far enough, may be so unsuited to the problem as to give very bad
answers if used in obtaining approximate values.
APPENDIX II
Modulation Coefficient
In this appendix the effects of space charge are neglected.
The modulation coeflficient /3 is defined as the peak energy in electron volts
an electron can gain in passing through the field of a gap divided by the
peak r-f voltage across the gap. If an electron were transported across the
gap very quickly when the r-f voltage was at a maximum the energy in
electron volts gained by the electron would be equal to the peak r-f voltage.
Thus, the modulation coeflficient can also be defined as the ratio of the peak
energy actually gained to the energy which would be gained in a very quick
transit at the time of maximum voltage.
In this appendix modulation coefficient will be considered only for r-f
voltages small compared with the d-c accelerating voltage.
If an electron gains an energy /3 times the r-J voltage V across the gap,
the work done on it is ^eV electron volts. By the conservation of energy,
an induced current must flow between the electrodes of the gap, transferring
a charge-|8e against the voltage V and hence taking an amount of energy
^eV from the circuit. Pursuing this argument we see that the modulation
coefficient /3 times the electron convection current in the beam, q, gives the
current induced in the gap by electron flow. In a circuit sense, there is fed
into the gap, as from an infinite impedance source, an induced current
We will assume that the gap involves a region in which the field along the
electron path rises from zero and falls to zero again. This region is assumed
630 BELL SYSTEM TECHNICAL JOURNAL
to be small compared with a wa\elength and to have little a-c magnetic
field in it, so thai we can pretty accurately represent the field in this re-
stricted region as the gradient of a potential. Along the path the potential
is taken as the real part of
V(x)e^''' (61)
For small gap voltages, to first order, the time that an electron reaches a
given position may be taken as unaffected by the signal, so that
/ = .r/w„ + /„ (62)
Then the gradient along the path is
-,- = Real 7'(.v)e>(-'«o+'o«o)_ .^^
dx
The change in momentum in passing through the field may be obtained by
integrating the force on an electron times the time through the field. Let
points a and b be in the field-free region to the left and right of the gap.
Then we have
r''
A(x) = Real ^ |/'(.v)^/-/"o+-'o^ ^^. ^^^-^
A{x) = Real''— [ V'(x)e'''' dx (b5)
y = co/tiQ . (b6)
The integral will be a complex quantity. The exponential factor involving
the starting time In will rotate this. A(.v) will have a maximum value when
the rotation causes the vector to lie along the real axis, and thi^ maximum
value is thus the absolute value of the integral. Hence
A(x)„,ax =^ -
I V'(x)e''
''a
dx
(b7)
For a-c voltages small compared with the voltage specifying the speed
Uo , the energy change is proportional to the momentum change. For an
electron transported instantly from one side of the gap to the other, the
momentum change can be obtained by setting y = 0.
dx
(b8)
REFLEX OSCILLATORS
631
Here I' is the voltage between a and b. Hence, the modulation coefficient
jS is given for small signals by
/3 = (1/F)
•'a
dx
V = V{b) - V{a)
7 = w/z'o
Wo = V27/F0
(b9)
(MO)
(Ml)
(M2)
Thus Uii is the electron velocity.
It is sometimes convenient to integrate by parts, giving the mathematical
expression for /3 a different form
^ = (1/F)
V'{x)e^
h
as F'(.v) is zero at a and h
13 = (1/F)
- ~ Vixy"" d.y
31 -la
- [ F"(.r)e^'^" rfx
T ''a
(bl3)
An interesting and important case is that of a uniform held between
grids. Let the tirst grid be at .v = 0 and the second at .v = d. There is an
abrupt transition to a gradient V/d at .v = 0, and another abrupt transition
to zero gradient at x = d. Thus, the integral (bl3) is reduced to these two
contributions, and we obtain
/3 =: (1/F)
F
yd
1 - e
jyd
(bl4)
This is easily seen to be
/3 = sin (7 d/2)/(y d/2) ' (bl4)
This function, the modulation coefficient for fine parallel grids, is plotted in
Fig. 119.
Sometimes apertures, as, circular apertures, or long narrow slits are used
without grids. There are important relations between the modulation
coefficient for a path on the axis and one parallel to the axis for such systems."^
In a two-dimensional gap system with axial symmetry, if the modulation
coefficient for a path along the axis is /?o , the modulation coefficient for a
path y away from the axis is
^y = ^0 cosh yy
(bl5)
-■' These relations first came to the attention of the writers through unpublished work of
D. P. R. Petrie. C, Strachey and P. J. Wallis of .Standard Telephones and Cables.
632
BELL SYSTEM TECHNICAL JOURNAL
In an axially symmetrical electrode system, if the modulation coefficient
on the axis is /3o , the modulation coefficient at a radius r is
/3. = ^0 h (yr)
(bl6)
Here In is a modified Bessel function.
It is easy to see why (bl5) and (bl6) must be so. The field in the gap can
be resolved by means of a Fourier integral into components which vary
0.9
^
\
\
0.8
\,
\
\,
\
z
s.
y 0.6
IL
U.
8 0.5
s.
s
\
X
2
o
•^ 0.4
_l
O0.3
2
>
\,
\
\
\
0.2
0.1
0
V
\
V
\
^
X
0 0.5 1.0 1.5 0.2 2.5 0.3 3.5 0/J 45 0.5 5.5 0.6 6.5 0.7
TRANSIT ANGLE, yd, IN RADIANS
Fig. 1 19. — Modulation coefficient for fine parallel grids vs transit angle across the gap in
radians.
^ = I sin {yd/2) /{yd/2) \, y = w/uo = 3nO/xV\^ .
sinusoidally along the axis and as the hyperbolic cosine (in the two-dimen-
sional case) of the same argument normal to the axis or as the modified
Bessel function (in the axially symmetrical case) of the same argument
radially. When the integration of (b9) is carried out, only that portion of
the Fourier integral representation for which the argument is 7.V con-
tributes to the result, and as that part contains as a factor cosh 7.V or
Io{yx) (bl5) and (bl6) are established.
The simple theory of velocity modulation presented in Appendix III
makes no provision for variation of modulation coefficient across the beam.
If we confine ourselves to very small signals, we find that the factor which
appears is /3^. We may distinguish two cases: If the distance from the axis
REFLEX OSCILLATORS 633
of symmetry were the same for both transits of the electron, we would do
well to average /3 . If the electrons got thoroughly mixed up in position
between their two transits, we would do well to average jS and then square
the average. We will present both average and r.m.s. values of /3. The
averaged value of ^ will be denoted as ^a , the r.m.s. value as /3s ; the value
on the axis will be called j3o .
From (bl5) and (bl6) we obtain by simple averaging for the two dimen
sional case,
yy
0s = /3o
n^i^^ + i
(bl8)
and for an axially symmetrical case
/3„ - 0,2I,(yr)/iyr) (bl9)
0. - ^oUliyr) - Iliyr)]'" (b20)
It is convenient to rewrite these in a slightly different form, using (bl5)
and (bl6).
0a = 0y (tanh yy/yy) (b21)
sinh 2yy -\- 2yy
0s = 0y
]
(b22)
_27y(cosh 27y + 1)
0a = 0r2h{yr)/{yr)h[yr) (b23)
0.. = 0\\ - l\{yr)/ll{yr)]"' (b24)
Now consider two similar cases: two pairs of parallel semi-inhnite plates
with a very narrow gap between them, and two semi-infinite tubes of the
same diameter, on the same axis and with a very narrow gap between them
(.see Fig. 120). For electrons traveling very near the conducting surface,
V is zero save over a very short range at the gap, and the modulation coeffi-
cient is unity. Thus, by putting /3y = jS^ = 1, we can use expressions
(bl5)-(b24) directly to evaluate 0a , 0s and 0^ for the configurations de-
scribed. These quantities are shown in Figs. 121 and 122.
Suppose, now, that the gap between the plates or tubes is not very small.
In this case, we need to know the variation of potential with distance
across the space d long which separates the edges of the gap in order to get
the modulation coefficient at the very edge of the gap, 0y or 0r .
If the tubing or plates surrounding the gap are thick, we might reasonably
634
BELL SYSTEM TECHNICAL JOURNAL
I I (very narrow gap)
y OR r
Fig. 120. — A gap consisting of pairs of semi-infinite i)lanes or semi-infinite tubes.
0.9
0.8
<Q
yo.6
O0.3
>
^
y^
X ^
V
y 1 /f^ ^
2^
I
\
^
/^
V/^///W//////^/Ai///WM//M//Ar
\
\^
>•>
\
\
\,
N
V
^
^
\
p^
^
\
\
'
P>0^
\
■
"^
1.0 1,5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
HALF DISTANCE BETWEEN PLANES, Ty, IN RADIANS
7.0
Fig. 121. — Modulation coefiicient for two semi-infinite pairs of parallel planes with the
edges very very close together, plotted vs the half distance between planes in radians.
/3n is the modulation coefficient for electrons travelling along the axis, i^a is the average
modulation coefficient and /J^ is the r.m.s. modulation coefiicient. The separation of the
l)lanes is 2v, % = 1/ cosh yy,
13,. = tanh yy/yy.
a.
-r^
sinh 27y + 2yy
j 27y(cosh 2yy -\- 1)
]'
assume a linear variation of potential with distance in the space between
them. In this case, (b9) gives
^y or I3r = I'\(yd) = sin (yd/2)/iyd/2)
(h25)
This is the same function shown in I'"ig. 119.
If the tube wall or plates are very thin, one may, following Petrie, Strachey
and W'allis"^' assume a |)ot('ntial N'arialion between the edges of the gap of
REFLEX OSCILLATORS
635
0.9
,0.7
V 0.6
O0.3
VZZZZZZZZ^TZZ.
\ ~
2.0 2.5 3.0 3.5 4.0 4.5 5.0
RADIUS OF TUBE, Tr, IN RADIANS
Fig. 122. — Modulation coefficient for two semi-in finite tubes separated by a very small
distance, plotted vs the radius of the tube in radians. j3n is the modulation coefficient on
the axis, (3o is the average modulation coefficient and /3s is the root mean square modulation
coefficient, r is the radius of the cylinders.
So = \IU{yr)', /3„ = 2/i(7r)/Tr/c(7'-),
ft = [1 - 7?(7'-)//o(7'-)]^
the form
In this case, (b9) gives
,r 1 . -1 2x
V — - sin — -
IT a
0y or Br = F,(yd) = .h(yd/2)
(b26)
(b27)
Both Fi{yd) and Fiiyd) are plotted vs. yd in Fig. 123.
Figures 121, 122 and 123 cover fairly completely the case of slits and
holes. The same methods may be used to advantage in making an ap-
proximate calculation taking into account the effect of grid pitch and wire
size on modulation coefficient.
Assume we have a pair of lined up grids, as shown in Fig. 124. Approxi-
mately, the potential near the left one is given as
V ^ V[x/2 + (aV[/4Tr)
•(/Sh
2tx
— COS
2 Try
(b28)
636
BELL SYSTEM TECHNICAL JOURNAL
-a
0.5
»55»^^
^
'^'^
r/ ■'/// ■///'/■//] y//////////Av/A
\
\>
^
Kd-H
Y///Ay//////////\ y//////////////A
\
\,
\
\
\
F2(rd)\
\
\F|(rcl)
' ■■ "'
\
\
\
\
\
\
\
\
/
y
\
\
\
V
/
\
-^
^
0 0.5 1.0 15 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7 0
SEPARATION, yd, IN RADIANS
Fig. 123. — If the planes or tubes considered in Figs. 121 and 122 arc separated bv ap-
preciable distance, the modulation coefficients given in those figures must be multiplied
by a factor F{y d). In this figure, the multiplying factor is evaluated and plotted vs the
separation in radians for two assumptions — that the gap has very blunt edges (Fiiyd))
and that the gap has very sharp edges (Ftiyd)).
This is zero far to the left and Vix far to the right. This expression is
useful only when the wire radius ;- is quite small compared with the separa-
tion a. Midway between wires,
r = V[x/2 + (a l'(/2T)ln
V" = (V'lT/la) sech
2 cosh
J TT.V
a
vx~\
a J
(b29)
(b30)
If we use (b30) for each grid, the values of V" for the two grids will overlap
somewhat. However, let us neglect this overlap, apply (b30) at each grid,
and using (bl3), integrate for each grid from — j^ to -{- x , giving
sech" -^ e^"^' dx
(b31)
/3o = (l/DO'iTr/Ta) |1 -e^''\\ f
\ J—o
-/{sin (yd/2)/iyd/2)\Go{ya)
f = V[/iV/d) (b32)
Goiya) = i I [ "e^^^"^-^'" sech'w du = (7a/2)/sinh {ya/2) (b33)
J— 00 I
where
REFLEX OSCILLATORS
637
^"
[-»- VORd J
(GAP)
Fig. 124.^A gap consisting of lined up grids.
:=:::;
n = i
^G (ra); APPROX. FOR
MESH GRID
■-^
^"a9~
■
■
o.s"
""-
■-^
^""^"^
0 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
ra
Fig. 125.— A factor used in obtaining the modulation coefficient of lined up grids vs
tlie wire spacing in radians, n is the fraction open. The curve for n = 1 also applies
approximately for a mesh grid.
Suppose we average over the open space of the grid. If the grid is a frac-
tion n open, from (bl7) we see that the average over the open space will be
obtained by substituting for Go a quantity G{ya, n) given by
G(ya,n) = {sinh (w7a/2)/(;i7c/2)}Go(7a)
(b34)
In Fig. 125, Giya,n) is plotted vs. ya for n = 1, .9, .8. This about covers
the useful range of values. It should be positively noted that the average
is over the open area of the grid and applies to current getting through.
It remains to evaluate the factor/. Suppose x = 0. At the surface of
the grid wire, y = r, the radius of the wire,
8V = (F(a/4x) In [2 (l - cos ^) = iV[a/2n) In [2 sin ^'1
63d BELL SYSTEM TECHNICAL JOL'RNAL
As we have already assumed r is small, we may as well write
8V = {V[a/2w)2.3 logm (j^^ (b35)
This emphasizes the sign of 6 \' .
According to (b28), the grid [)lane appears from a distance to be at zero
potential. Thus,
Vui - 28V = V (b36)
and from (h.^2)
f = il + (a/w (I) 2 J logu, (a/27rr)}"' (b37)
If we go back over our results, we have for lined-up singly-wound grids,
from (b31), (b34) and (b37), the average modulation coefficient
^„ = /Isin {ya/2)/iya/2)]G(ya,n) (b38)
The quantity sin (ya/2) can be obtained from P'ig. 119, G(ya,n) is plotted
in Fig. 125, and/ can be calculated from (b37) above.
It must be emphasized again that these expressions are good only for
very line wires (;- « a), and get worse the closer the spacing compared
with the w'ire separation. It is also important to note that G{ya,n) indi-
cates little reduction of (3„ even for quite wide wire separation. Now yd
will be less than 27r, as /3„ = 0 at 7^ = 27r. As a approaches d in magnitude,
the assumptions underlying the analysis, in which the integration around
each grid was carried from — ac to 00, become invalid and the analysis is
not to be trusted.
It is very important to bear one point in mind. If we design a resonator
assuming parallel conducting planes a distance L apart at the gap, and then
desire to replace these planes with grids without altering the resonant fre-
quency, we should space the grids not L apart but
d - t'L (hm
apart to get the same capacitance and hence the same resonant frequency.
Mesh grids are sometimes used. To get a rough idea of what is expected,
we may assume the potential about a grid to be
V = V[x/2 + (aT'(/87r) In \2 (^cosh — - cos — 'M
+ (aT'(/87r) In [2 (cosh ~''--' - cos """M
(1)40)
REFLEX OSCILLATORS 639
Here the grid is assumed to lie in the y, z, plane. This gives a mesh of wires
about squares a on a side, the wires bulging at the intersections. We take
r to be the wire radius midway between intersections.
We see that /3o will be the same in this case as in the case of a parallel wire
grid. Thus the added wires, which intercept electrons, haven't helped us
as far as this part of the expression goes.
As a further appro.ximation, an averaging will be carried out as if the
apertures had axial symmetry. Averaging will be carried out to a radius
giving a circle of area a\ The steps will not be indicated.
Further a factor analogous to / will be worked out. Again, the steps
will not be indicated. The results are
^a = gisin {yd/2)/{'yd/2)\Gy{ya) (b41)
Ci-ya) = 2h{ya/V^)/(ya/\/^)Go{ya) (b42)
g = 1 + (.365 a/d) (log.o (a/Tr) - .69) (b43)
The quantity 6'i(7,a) is plotted in Fig. 125 for comparison with the parallel
wire case. It should be emphasized that these expressions assume r « a,
and that Gi(ya) is really only an estimate based on a doubtful approximation.
The indications are, however, that the only beneficial affect of going from a
parallel wire grid to a mesh with the same wire spacing lies in a small de-
crease in 5F (a small increase in the mu of the grid), while by doubling the
number of wires in the parallel wire grid, a can be halved, both raising mu
and increasing G(ya,n).
APPENDIX III
Approximate Treatment of Bunching
We assume that the conditions are as shown in Fig. 126 where the elec-
tron energy on first entering the gap is specified by the potential Vo . Across
the gap there exists a radio frequency voltage, V sin co/. The ratio
of the energy gained by the electron in crossing the gap to the energy
which it would gain if the transit time across the gap were zero is
called the modulation coefficient and is denoted by a factor, /3. We assume
that the modulation coefficient is the same for all electrons. We also neg-
lect the effects of space charge throughout. After leaving the gap the
F/e + Vo
electrons enter an electrostatic retarding field of strength Eq
I
2' This analysis follows the method given In- Webster. J. Ann. Phys. 10, Julv 1939, nn
501-508. ' - 'M
j540
BELL SYSTEM TECHNICAL JOURNAL
such that the stream flow is reversed and caused to retraverse the gap.
The round trip transit time, Tq , in the retarding field when a signal exists
across the gap is then given by
2 \/277(Fo + &V sinoih)
(cl)
C "7 It
where 77 = - X 10 = 1.77 X 10 in practical units and h is the time of
m
first entry of the gap. The time of return to the gap will be
h= h^- Ta (c2)
More accurately /i and h are measured from the central plane of the gap in
which case a second term should be added to (cl) corresponding to motion at
RESONATOR
Fig. 126.— Diagram of a. reflex oscillator showing quantities used in the treatment ol
bunching.
constant velocity. This term is, however, very small and will be neglected
here. If ii is the current returning to the gap and 7o the uniform current
entering the gap on its first transit, then from conservation of charge one may
write
h dh = /.. dh (c3)
In what follows it will be first assumed that /? and h are related by a single
valued function. At the end of this appendix it will be shown that the
analysis is also valid where the relating function is multiple valued.
\Vc now make a Fourier series analvsis of ;'•) in order to determine the
REFLEX OSCILLATORS 641
(c4)
harmonic distribution. Thus
ii =00+^1 cos (w/o -\- ip) -\- a2 cos 2(w/2 -\- <p) -\- • • •
+ ^1 sin (a'/i + ^) + ^2 sin 2iwti + ^) + • • •
where
1 r .
On = - I h cos 11 (uk -\- (f) dwU
TT J-ir
1 r"
bn = - I k sin n (cj/j + ^) do^h
■W J-v
Using (cl) to (c3) we change our variable to /i obtaining
fln = - / /o cos «w I /i + ^ + ip\ dooti
JM , fl 2co VvTo 0^F
Let oj/i = Wi coTo = — - — =6 X = -^-—
2Fn
(c5)
(c6)
- / h cos n Idi + if + d\ I + ~ sin di
X
^ — sin' ^1 +
^de,
(c7)
(c7) cannot be evaluated in closed form without further restriction. The
first order theory may be obtained by assuming that ^ <<C -. It is
6 2
not suflficient to assume that — « A'. The latter assumes that the third
6
and higher terms of the expansion are small compared to the second. Let
the integrand be denoted as io cos nx. The quantity to be evaluated is the
argument of a trigonometric function where the total angle is of less impor-
tance than the difference nx — 2tmr where m is the largest integer for which
the difference is positive. The condition first expressed requires that the
contribution of the third and higher terms to the difference phase shall be
small. The restriction requires that
n (^J wro « T (c8)
This is a more stringent requirement than
(c9) requires only a small modulation depth while (c8) imposes a restriction
on both the modulation depth and the drift time.
642 BELL SYSTEM TECHNICAL JOURNAL
With the restriction (c8) imposed we obtain
a„ = ~ j h cos n Idi + (f i- 8 1 + ~ sin di j ddi (clO)
If we let V? = —6 all coefficients b„ will be zero and
a„ = 2(-\)"IJ„(Xn), (/,, = /,, (ell)
Thus the first order expansion for the current returning throujj;h the gap is
/, = /„ [(1 - 2Jr(X) cos a>(/2 - Tu)
(cl2)
+ IJ.ilX) cos 2co(/2 - To) • • • ]
Our principal interest is in the fundamental component, which in complex
notation is given by
{hlf = -2I,JiiX)e'''^''~''^ (cl3)
It is shown in Appendix II that the circuit current induced in the gap will
be given, if account is taken of the phase reversal of k resulting from the
reversal of direction of the beam, by
The gap voltage at the time of return will be v — V sin o^t^ or in complex
notation
Hence the electronic admittance to the fundamental will be
In the foregoing it was assumed that h was a single valued function of
/i . We may generalize by writing (c3) as
Y^I.dl^ = hdto (cl6)
For sufficiently large signals there may be several intervals <//i which con-
tribute charge to a given interval dt2 and hence we write a summation for the
left hand side of (cl6). \\ hen the Fourier analysis is made and the change
in variable from I2 to /1 is made the single integral breaks up into a sum of
integrals. In Fig. 127 we plot time /, on a vertical scale with the sine wave
indicating the instantaneous gaj) voltage. Displaced to the right on a ver-
tical scale we plot time Aj . The solid lines connect corresponding times in
the absence of signal for increments of time dl^ and df-: . \\ hen sufficiently
REFLEX OSCILLA TORS
643
arge signals are applied some of the electrons in the original interval dti
will gain or lose sufficient energy to be thrown outside the original cor-
responding interval dt^ as for example as indicated by AB. If we consider
a whole cycle of the gap voltage in time /o it is apparent that, under steady
state conditions, for every electron which is thrown outside the correspond-
ing cycle in (2 another from a different cycle in /i is thrown in whose phase
differs by a multiple of Itt as for example CD. In summing the effects of
these charge increments the difference of 2ir in starting phase produces no
physical efifect. This is of course also true mathematically in the Fourier
analysis of a periodic function since in integrating over an interval 2-k it is
immaterial whether we integrate over a single interval or break it up into a
Fig. 127. — Diagram showing the relation between /i , the time an electron crosses the
gap for the first time, and t-i , the time the electron returns across the gap.
sum of integrals over intervals — tt to a, 2x;zi + a to lirih + b, Itth-' + 6 to
27r;/o -f- (-, etc. where the subintervals sum up to 2ir. Hence we conclude
that the preceding analysis is also valid up to (c7) for signals sufficiently
large so that k and /i are related by a multiple valued function and is valid
beyond that point provided that we do not violate (c8).
APPENDIX I\'
Drift Angle .as .a. Function of Frequency and \'oltage
Let r be the transit time in the drift space. Then the drift angle is
^ = COT (dl)
For changes in voltage (resonator or repeller), both r and co will change.
644 BELL SYSTEM TECHNICAL JOURNAL
Thus
Ae/e = Aco/w + At/t
(d2)
= Aw/W + ((dT/dV)/T)AV
As shown in Appendix VT, the derivative of t with respect to repeller
voltage, dr/dVK , is always negative, while the derivative of t with respect
to resonator voltage, dr/dVo , may be either negative or positive. For a
linear variation of potential in the drift region, dr/dVo is zero when V^ =
Vo and negative for smaller values oi Vr .
APPENDIX V
Electronic Admittance — Non-Simple Theory
A closer treatment of the drift action in the repeller space follows, in
which are considered the changes which occur as the voltage on the cavity
becomes large.
The additional terms to be considered come from an evaluation in series
of the higher-order terms of (c7), which were neglected in Appendix III.
Only the fundamental component of current will be considered, although
other terms could be included if desired. The integrals of interest may be
rewritten from (c7), using the relation ip = —0, as follows:
ai = - [ cos (d, -I- X sin 01 - i ^ sin' di
IT J-TT \ ^ W
+ :; — T sm 6i -f
2 6-
'^=^/-/^"t
I X' . 2
1 + X sm 01 — - — sm di
2
I ^ X • 3 „ I
-f - — - sm 6i -\-
Jddi
jddi
(el)
(e2)
We shall hereinafter neglect terms of higher order in ^ than those explicitly
u
shown here. With this neglect, we can expand the trigonometric functions,
obtaining
di
-' j cos (dr + X sin e,) jl - ^ -^ sin' 0i -f • • • 1 i0i
- - [ sin (0, -f X sin di) (e3)
. _ ^ ^. sin' di -\- l~ sin' 0i + • • • \ ddi
/>, = -/" sin (^1 + X sin 0,)
REFLEX OSCILLA TORS
IX'
645
sin^ 01 + • • • U^i
+ ^° f cos (01 + X sin di)
TT J-ir
r 1 ^' • 2 . , 1 ^' • :i . .
(e4)
(/^i
Now of these terms, not all give contributions; some integrate to zero since
the integrand is an odd function of di. Rewriting with those terms omitted,
(e5)
ax^^^ \ cos (01 + X sin dM\ - ^^ sin' 0i + • • • J rf0,
- ^^ I sin (01 + X sin 0i) [5^ sin' 01 + • • • J rf0i
bx = -^f cos (01 + X sin 0,) I -^ y sin' 0i + ■ • • 1 rf0i . (e6)
Evaluation of these terms is formally simplified by the following relation-
ships, each obtained by differentiation of the previous one:
-2Ji{X) = 1 f cos (01 + A' sin 0i) dOi
IT J-w
-2/i(X) - -- f sin (01 + A' sin 0i) sin 0i ddi
IT J-TT
-2Ji(X) = -- [ COS (01 + A sin 0i) sin' 0i ddi .
(e7)
(e8)
(e9)
Continuation of this process gives all the terms of interest in (e5) and (e6).
Hence
(elO)
(ell)
Therefore the expression for the fundamental component of the beam
current may be written as follows, passing to complex notation:
(^'2)/ ■^ Oi cos (w/o — 0) + ^1 sin (aj/2
r'^e-*/o('-2/i +iy // + ^, {x'j[" + IX' jn^ +
(el2)
646
BELL SYSTEM TECHNICAL JOURNAL
Following usual conventions, the real part of the complex expression repre-
sents the physical situation, and the exponential time function will usually
be omitted in what follows.
Similarly to Appendix III, the induced current in the gap is
/2 - - ^(^2)/
The gap voltage is still the same, viz.
T- • , T- .;(w<2-(t/2)) 1 2'oX y(a)<.,-
1 sni wto ^^V e = H — e -
^ e
(t/2))
(el4)
Accordingly, the electronic admittance to the fundamental will be
V _ -^2 _ /o ,2 d j((irl2)-e)
V Vo X
(^' -f ■'"- Kf-'"' + ?■"')) ^^^-^^
The argument of /i and its derivatives is understood to be .Y. The
derivatives can be evaluated in terms of Jo and /i by repeated use of the
Bessel function recurrence relations (see Jahnke-Emde, Funktionentafeln,
p. 144). The result is
Vo A
(■'.-i[(-f)^--^»]
-^[ux' + x')A-ix'jo'^+ •••)
(el6)
The real part of this will be of interest; it is
^ I ^0 ^2 0 . ^/, , cot^r/. X
I lux' + x')j, - ixVo])
(el7)
The power generated by the electron stream is given by
P = -hGeV
= —lol'o ~r I sin
K-
A - ^, ((X' + X\r, - 2A'\Ao)]
+
cos 6
[0-T)"-f4
(el8)
Jo +
REFLEX OSCILLATORS 647
Of this power, only a part is usefully delivered to the load admittance
Gl , the rest being dissipated because of the circuit loss conductance Gr .
The power lost in the circuit is
P« = iG«F^ = ^4^«^' (el9)
Accordingly, the useful power delivered to the load is
P, = p - p^ (e20)
A quantity of interest is the maximum useful power which can be ob-
tained from the reflex oscillator; this is given by
^^ = 0 - — ^ (e21)
dX dd
These two conditions are expressed by the next two equations.
2X/o + 1 cot 0 (-XVo - XVi) + I (ax' - iX')/o
a t7" \
-(-'^-*&)-'^-
^ sin 0
(e22)
XJo + (-4 + XVi + 0 cot e-2J, - ""^ a-^X' - 2X)Jo
a
+ (4 - ix' + iX')J,) + 1 (-fXVo + HX' + X')JO (e23)
Gu
jS2/o "0sin&
One may note that even if one sets G« = 0 and neglects terms in — the
o-
second of these equations is a little difi"erent from the corresponding one
of Appendix III, so that a slightly different phase angle is predicted for
maximum generated power. The result of Appendix III was
e cot e = 1
predicting a phase angle 6 slightly less than d„ = (;/ + f)27r. However, the
zero order result here is
^ cot ^ = 2 - ^- = - .892 (e24)
predicting a phase angle a trifle larger than 6,, , in the approximation of
Appendix III.
648
BELL SVSTE.Vf TECHNICAL JOVRNAL
The equations (e22) and (e23) may look as if drastic measures would now
be needed, but a parametric solution is sur{)risingly easy; one need only
solve (e22) for G h , and substitute back into the power expression (e20).
The results are
Gh =
(:
^ §' e sin e (iJo + "V i-x-'Jo - XA\
(e25)
Pl = -/oT'o ?sin 6 llJ, - AVo + ~
+
Gl = ^Ge — Gr =
X'
/l
+i
y7l + (-fX' +
\X')J^
(e26)
^0 ,p2 ^ • air 1 V 7 I cot 0
— ^ - sm d\Ji - ^X/o +
[(•-?)^-(-f-|>»]
+ -[1XV, + (-3%X^ +
AX^)/o )
One further convolution is necessary, because the equations (e25) and
(e26) are still subject to the optimum phase angle condition (e23). Since
we are here carrying only terms as far as — , approximations are in order.
I'Vom (e24) we get the hint that 6 cot 6 is of the order of unity, so that
terms in are of the same order as those in — . Accordingly an ap-
proximate solution is obtainable by adding (e22) and (e2v^) and neglecting
these small terms. The result is
'\ herefore the optimum phase angle is given by
. . FiX)
sin 0 = - 1 +
e,, = (« + f)27r
\t\X)\'
2dl
(e28)
(e29)
fe30)
KEFLEX OSCILLATORS 649
cot 0 1 „.„v , ^^,
From computation it turns out that in the range of interest, the quantity
F{X) does not differ from (—1) by more than 20%.
The desired approximate solution comes now from substituting the ex-
phcit phase optimum (e29) to (e32) back into (e25)-(e27). The results are:
X2 /2/i 1 \
Pl = /oFo— ( Y - -^0 + ^J^(X)] (e34)
iO 2^n /2-/l
G,. = :jT(S'y ^^Y - ■'» + ^•^>W ) (<=35)
^5.(X))
The S-functions are given by
Si(X) = (-F - ^^ - ^ + LX' - ^^M-^i
6-2(A') = [-F' - F— + 4F +
8 8 /
+ ("--'f- - - -x*)-L'
\ 2 4 4 /Z
The equations (e33)-(e35) have the following meaning: they presup-
pose that the load Gl has been adjusted for maximum useful power in the
presence of circuit loss Gk , and that the drift angle is also optimum. Then
the useful power is given parametrically in terms of the circuit conductance
by equations (e3>3) and (e34), while (e3i5) gives the required optimum load
conductance, also in terms of the parameter A'.
The results may be expressed as a chart of useful power, plotted against
the value of resonator loss conductance. This is done in Fig. 128.
One may also be interested in the maximum power which could be gen-
(e36)
(e37)
(e38)
650
BELL 51.bT£.U TECHMLAL JUiK.\AL
1
60
55
- 50
UJ
u
a.
S! 45
Z
a a
II
f^ 35
>
>
i 30
u.
u.
Ul
5 25
Q.
1-
3
o
20
15
10
5
0
\
\
\
\
\
\ \i
\ 1
\ \
\ I
—
\
\
I
\
V^
\
1"^^
^^^
■^^^
-2 0 1 23456789 10
GrVq
lo
Fig. 128. — Plot of clTiciency vs a parameter proportional to resonator loss for several
repeller modes.
n
0
1
2
3
n
0.48
0.22
0.14
0.105
Fig. 129. — Maximum etVicienc\- for several repeller modes.
erated if tlic resonator were perfect. This comes from setting Gh = 0,
calculating the resulting value of useful power. The results are compared
with the simpler theory in Fig. 12*^.
REFLEX OSCILLA TORS
651
2 0.60
0.55
~^
\
\
V
\
^
\
^
\
^
\
\
\
\
\
V
\
w
^i
\^
Y
\\ THEORY {'^-^)
\\
\
^'
i
\
n=o/
%
/
^
■X^
\ ""y
\
y
1.2 1.6 2.0 2.4 2.8
BUNCHING PARAMETER, X
Fig. 130. — Relative electronic admittance vs l)unching parameter for several repeller
modes.
652
BELL SYSTEM TECHNICAL JOURNAL
240
200
160
140
uj 80
a.
\3
Z 60
O 40
a 20
-20
-60
■100
/
/
y
/
/
/
/
i
/
/
n=o
/
^^
y
ss;
-m-tr:
*^^
:nr
1
2_
\
\
"SIMPLE THEORY (n-OO) 1
I
1
I
1
1
1
1
1
11
1
\
\
\
04 0 8 I
2 1.6 2.0 2.4 2.8
BUNCHING PARAMETER, X
Fig. 131. — Phase of electronic admittance vs bunching parameter for .several repeller
modes.
REFLEX OSCILLATORS
653
ye 0.3
' ■■
^\n=o
SIMPLE "IT^^^v
THEORY *''^-°°J
:\
^^55:5
^
^
^
\^
0.2 0.3 0.4 0.5 0.6 0.7 0.8
Gr
ye
Fig. 132. — Optimum load conductance divided by small signal electronic admittance vs
resonator loss conductance divided by small signal electronic admittance for several re-
peller modes.
I.U
0.9
0.8
^
^
^
^
^^
Gc
ye 0.7
'y
^
0.6
^
^^
f
^^^
^^^
n=o^
<^^
o.e>
r
^
^, SIMPLE /n_ool
THEORY \^-^)
0.4
0.2 0.3 0.4 0.5
Gr
ye
0.6 0,7
0.8
I.O
Fig. 133. — Optimum total circuit conductance divided by small signal electronic ad-
mittance vs resonator loss conductance divided by small signal electronic admittance for
several repeller modes.
Further data which may be determined include the variation of the mag-
! nitude of the electronic admittance with gap voltage, in Fig. 130; also
its phase, in Fig. 131. The optimum load conductance is plotted as a
654
BELL SYSTFM TECHNICAL JOURNAL
30
26
i-
z
111
U 24
a.
a
? 22
'^|Ql 20
I
\
I
\
\
\
\
)
y
\
\
^
\
\
^
^,
\
^=0
V
\
\
\2
^
--
5
■ —
7
5^
^
^
IS-""^
--
-^
'"_
15
--
0 0.5 t.O 1.5 2.0 2.5 3.0 3.5 4.0
EFFECTIVE DRIFT ANGLE, F N, IN CYCLES PER SECOND
Fig. 134.— Efliciency vs elTective drift angle for several degrees of resonator loss.
REFLEX OSCILLATORS
655
function of resonator loss in Fig. 132, and the total conductance, load plus
loss, in Fig. 133. The next Fig. 134, plots efficiency versus mode number
20
0.8
0.6
0.5
0.4
0.3
V
\
\
\
\\
V.
V
-V^
-
V-
\^
\
Vvj,
V
\
\
.\
V
1
\
\\
K
1
\
\
1
\
\,
n=o
A
\
3\
\
\
\
-
\
\
\
-
\
\
\
\
\
\
\
\
\
\
\
\
\
'
\
\
\
\
\
\
\
\
\
I
5 6
GrVo
Fig. 135. — Efficiency vs a parameter proportional to resonator loss for several repeller
modes.
with resonator loss as a parameter, while the next Fig. 135, plots efficiency
versus resonator loss with mode number as a parameter.
Most of these graphs include for comparison the results of the simpler
theory, and it can be seen that the deviations indicated by the second
656 BELL SYSTEM TECHNICAL JOURNAL
order theory are ordinarily rather small even for the first two modes of
operation, and are quite negligible for higher modes.
APPENDIX VI
General Potential Variation in the Drift Space
Suppose that the potential of the drift space is given by V{x), where
^ = 0 at zero potential and ar = / at the gap. Then the transit time from
the gap to zero potential and back again is
Imagine now that the entire drift space is raised by a very small amount
AF. The zero potential point will now occur at
X = -AF/F'(0) (f2)
where
F'(.t) = dV/dx (f3)
Hence the new transit time will be
TO + Ar = (2/^2^) f PTV ^^l at/v (f"^)
J-^vlv'm [V(x) + AF>
Now let
z= x-^ AF/F'(0) (f5)
Then, including first order terms only, if V{x) can be expanded in a Taylor's
series about 0,
To + At
0 [F(2) - [F'(z)/F'(0)]AF + AF]^
(2/V27?) I
Jo
-n/-./i-\(t^ ^' -±- Av [^ KF-(.)/F-(0)) - l]dz
AF \
+ F'(0)[F(/)]V
Whence
, At ,,, ,-. I 1 ^ rq(r(.)/F-(0)) - \\dz
In computing F it should be noted that by definition the gap voltage pre-
REFLEX OSCILLATORS 657
\iously referred to as Fo is
Vo=V{() (f8)
In the notation as it has been modified, the transit time is
-=(^/^2~,)jf^. (f9)
For a constant retarding field in the drift space of magnitude Eq , we can
write
vEo(ti/2) = V = a/2VF (flO)
Here F is the total energy with which electrons are shot into the drift
space. From (flO)
Tl
_ 2V'27?F
r\E,
/
= dr/dV =
2V2r7F
vEo
In the notation used earlier this
is
/
To
fe)=
27
(fll)
(fl2)
Now we will compare r' from (f7) with ri, the rate of change of transit time
for a linear field, taking ri for the same resonator voltage F(/) and the same
transit time, given by (fl), as the nonlinear field. The factor F relating
t' and Tl will then be
F-r/
Tl
^^^^'^l\v'mV{C)\' (fl4)
rHv'{z)/v'(o) - i]dz\ [^ dz y
"^io 2[V(z)]i jJo [F(2)]0 •
If an electron is shot into the drift space with more than average energy,
its greater penetration causes it to take longer to return, but it covers any
element of distance in less time. Consider a case in which the gradient of
the potential is small near the zero potential (much change in penetration
for a given change in energy) and larger near the gap. The first term in
the brackets of (fl4) will be large, and the second is in this case positive.
This means that in this type of field the increased penetration per unit energy
and the effect of covering a given distance in less time with increased energy
work together to give more drift action than in a constant field. However,
658 BELL SYSTEM TECHNICAL JOURNAL
we might have the gradient near the gap less than that at the zero potential.
In that case the second term in the brackets would be negative. This means
a diminution in drift action because the penetration changes little with
energy while the electron travels faster over the distance it has to cover.
To show how large this effect of weakening the field near the zero potential
point may be, we will consider a specific potential variation, one which
approximates the field in a long hollow tubular repeller. The field con-
sidered will be that in which
V{z) = {e' - \)/e^ (fl5)
We obtain
{e^ - D'
tan-' {e^ - l)'
F=(e^-l) + . Z^j":,^- m)
Now
F'(0) = e-f (fl7)
V'(z) = 1 (fl8)
Hence
/ 1 \ \V'(0) '
This shows clearly how the effective drift angle is increased as the field at
the zero potential point is weakened. For instance, if V'(0) = ^, so that
the field at the zero potential point is ^ that at the gap, the drift effective-
ness for a given number of cycles drift is more than doubled (F = 2.27).
There is another approach which is important in that it relates the varia-
tions of drift time obtained by varying various voltages. Suppose the gap
voltage with respect to the cathode is Fo and the repeller voltage with re-
spect to the cathode is — Fh . Now suppose Fo and Vr are increased by a
factor a, so that the resonator and repeller voltages become aFoand —aVn.
The zero voltage point at which the electrons are turned back will be at the
same position and so the electrons will travel the same distance, but at each
point the electrons will go a times as fast. If, instead of introducing the
factor a, we merely consider the voltages F« and Fn to be the variables, we
see that the transit time can be written in the form
T = Fr/^(F«/Fo) (f20)
REFLEX OSCILLATORS 659
The function F{V r/Vo) expresses the effect on r of different penetrations of
the electron into the drift field and the factor V^" tell us that if V h and
Fo are changed in the same ratio, the drift time changes as one over the
square root of either voltage.
We can differentiate, obtaining
dr/dW = VrF'iVn/V,) (f21)
dr/dV, - ~Vr{{Vn/V,)F'{Vr/V,) + {\/2)F{Vr/V,)) (f22)
If the electron gains an energy ^V fh crossing the gap, the effect on r
is the same as if Fo were increased by /3F and V r were changed by an
amount — jSF, because in an acceleration of an electron in crossing the gap
the electron gains energy with respect to both the resonator (where the
energy is specified by Fo for an unaccelerated electron) and with respect to
the repeller {—Vr for an unaccelerated electron). We may thus write
dr/di^V) = dr/dVn - dr/dVR
= -Fr[(l + Vr/Vo)F'{Vr/Vo) + hF(VR/Vo)] (f23)
This expression (f23) is for the same quantity as (f7). Fo of (f23) is
F(0 of (f7). Expressions (f21), (f22) and (f23) compare the effects on
drift time of changing the repeller voltage alone, as in electronic tuning, the
resonator voltage alone, and of accelerating the electrons in crossing the
gap. As making the repeller more negative always decreases the drift
time, we see that the two terms of (f22) subtract, and usually | dr/dVo \
will be less than | dr/dYR \ . In fact, for a linear variation potential in
the drift space and for Fo = Vr , dr/dVo = 0. Weak fields at the zero
potential point make the absolute value of F'(Vr/Vo) larger and hence
tend to make both ] dr/dVR | and | dr/di^V) \ larger. However, these
quantities are not changed in quite the same way.
The reader should be warned that (fl4) and (f20)-(f23) apply only for
fields not affected by the space charge of the electron beam. For instance,
suppose we had a gap with a fiat grid and a parallel plane repeller a long
way off at zero potential. If edge effects and thermal velocities were
neglected, we would have a Child's law discharge. The potential would be
zero beyond a certain distance from the repeller, and we would have
V(z) = Az^
According to (fl4), F should be infinite. There is no reason to expect
infinite drift action, however, for the drift field, which is affected by the
fluctuating electron density in the beam, is a function of time, and (fl4)
does not apply.
660 BELL SYSTEM TECHNICAL JOURNAL
APPENDIX \'II
Ideal Drift Field
The behavior of reflex oscillators has been analyzed on the basis of a uni-
form field in the drift space. It can be shown that this is not the drift
field which gives maximum efficiency. The field which does give maximum
efficiency under certain assumptions is described in this appendix.
Consider a reflex oscillator in which a voltage V appears across the gap.
This voltage causes an energy change of /3F cos di for the electron crossing
the gap. Here di is the phase at which a given electron crosses the gap for
the first time. The effect of the drift space is to cause the electron to re-
turn after an interval Ta where Ta is a function of this energy.
Ta = fm cos e,) (gl)
Thus, each value of Ta will occur twice every cycle (lir variation of ^i). We
will have
01 = co/i (g2)
da = w(/i 4- Ta)
= 01 + <p{d,)
(p(di) = OiTa (g4)
(g3)
Here h is the time at which an electron first crosses the gap and (/i -f Tq)
is the time of return to the gap. ^i and 6a are the phase angles of the voltage
at first crossing and return.
The net work done by an electron in the two crossings is
W = |SF4-cos e, + cos {d, + <p{e,))] (g5)
If the beam current has a steady value /o , the power produced will be
P = (^17o/27r) I [-cos e, + cos (01 + ^{d,))\ dd, . (g6)
The integral of cos d\ is of course zero. Further, from (g4) we see that
<^(0i) = -^(-^i)
Hence
p = {fiVh/2-K) I Icos (-01 -f ip{e,) -f cos (0, + .p(0i))1 de^
Jo
(g7)
= (/3K/o/27r) f cos <^(0i) cos 0i (/0i .
^0
REFLEX OSCILLATORS
661
As cos ^(^i) cannot be greater than unity, it is obvious that this will have its
greatest value if the following holds
0 < ^1 < 7r/2, .^(^i) = Inir, cos ^{6^) = + 1 (g8)
7r/2 < ^1 < TT, ip{ei) = (2m + l)ir, cos (^(^i) = -1 (g9)
These conditions are such that for a positive value of cos <p the gap voltage
is accelerating giving a longer drift time than obtains for a negative value
of cos ^(^i) for which a retarding gap voltage is required. Thus, physically
we must have
2n > 2w + 1
(glO)
The simplest case is that for m = 1 and m = 0, so that in terms of the gap
voltage
V <0, <p{di) = TT
V > 0, ifidi) = Itt
This sort of drift action is illustrated by the curve shown in Fig. 136
(gll)
2 TT
GAP VOLTAGE, V
Fig. 136. — Ideal variation of drift time in the repeller region with resonator gap voltage.
The problem of finding the variation with distance which would give this
result was referred to Dr. L. A. MacCoU who gave the following solution:
Suppose Vo is the voltage of the gap with respect to the cathode and $ is
the potential in the drift space. Let
Xo = \/2r]Vo/u (gl2)
Here co is the operating radian frequency. Let x be a measure of distance
in the drift field.
4> = Foil - (x/.vo)'], 0 < X < .To
$= Foil - [(x/.Tor+ l]74(.v/.vo)'|,
This potential distribution is plotted in Fig. 137.
X > Xo
(gl3^
662
BELL SYSTEM TECHNICAL JOURNAL
1.0
■\
s.
0.6
OA
0.2
j£_ 0
Vo
0.2
a4
0.6
• 0.8
1.0
\
V
\
>
\
\
X^
S
\
\
\
s.
\
0 0.2 0.4 0.6 0.8 1.0 !.2 1.4 1.6 1.8 2.0 2.2 24
X
^0
Fig. 137. — Variation of potential in the repel ler region vs distance to give the charac-
teristics shown in Fig. 7.1.
Fig. 138.— Electrodes to achieve approximately the potential variation in the drift
region shown in Fig. 7.2.
KEFLEX OSCILLATORS 663
The shapes for electrodes to realize this field may be obtained analytically
by known means or experimentally by measurements in a water tank. The
general appearance of such electrodes and their embodiment in a reflex
oscillator are shown in Fig. 138. Here C is the thermionic cathode forming
part of an electron gun which shoots an electron beam through the apertures
or gap in a resonator R. The beam is then reflected in the drift field formed
by the resonator wall, zero potential electrode I and negative electrode II,
which give substantially the axial potential distribution shown in Fig. 137.
Small apertures in the resonator wall and in electrode I allow passage of the
electron beam without seriously distorting the drift field. Voltage sources
Vi and V-i maintain the electrodes at proper potentials. Either suitable
convergence of the electron beam passing through the resonator from the
gun or axial magnetic focusing will assure return of reflected electrons
through the resonator aperture. In addition, the aperture in electrode I
forms a converging lens which tends to offset the diverging action of the
fields existing between the resonator wall and I, and between I and II.
R-f power is derived from resonator R by a. coupling loop and line L.
APPENDIX VIII
Electronic Gap Loading
If a measurement is made of gap admittance in the presence and in the
absence of the electron beam passing across it once, it will be found that the
electron stream gives rise to an admittance component Y. The susceptance
is unimportant, but the conductance G can have a noticeable effect on the
efficiency of an oscillator.
Petrie, Strachey and Wallis have provided an important expression for
this gap conductance due to longitudinal fields when the r-f voltage is small
compared with the beam voltage Vo-f In this analysis it is presumed that
the fields in the beam are due to the voltages on the electrodes only and not
to the space change in the beam.' This analysis is of such importance that
it is of interest to reproduce it in a slightly modified form. We will first
consider the general cases of interaction with longitudinal fields and will
then consider transverse fields also.
A. Longitudinal Field
Assume a stream of electrons flowing in the positive x direction, constitut-
ing a current — /o , bunched to have an a-c convection current component
2' These expressions were communicated to the writers through unpubUshed but widely
circulated material by D. P. R. Petrie, C. Strachey and P. J. Wallis of Standard Tele-
phones and Cables Valve Laboratory.
28 The expressions are valid in the presence of space charge, but as the field is not known,
they cannot be evaluated.
664 BELL SYSTEM TECHNICAL JOURNAL
i\ . Now if /i is the time a particle passes Xi and h the time the particle
passes .V2 , the convection current at .vj will be
h = (-/o +h)~ + /o. (hi)
a/2
This merely states that the charge which passes .\\ in the time interval dti
will pass .r2 in the time interval dt^ . Suppose the electrons are accelerated
at .Ti by a voltage
If Vo is the voltage specifying the average speed of the electrons, the veloc-
ity will be
V = (2,,Fo)'(l + {V/Vo)e"'''f (h2)
We then have
/2 = /i + t(1 + (F/Fo)e''"'0"*
(h3)
r = x{2-nV,r
dh 2(1 + (F/Foy-'O" ^ ^
Now assume (F/Fo) <3C 1. If we neglect higher powers than the first
we can replace /i by
h = ti — T
and obtain
and from (hi), neglecting products of two a-c quantities
i, = i,-J^Ve^-'^^-\ (h6)
Suppose we consider an electron stream travelling through a longitudinal
field of potential fluctuating as <?"'" and of magnitude F(.v), where V{x)
may be complex. Then the current at .T2 due to the action of the field at
Xi is, omitting for convenience the factor e^" ,
dn = •' — — - — 7(.T2 - Xi)e dxi (h7)
7 = co/wo (h8)
uo = (2r,Fo)*. (h9)
REFLEX OSCILLATORS 665
Hence, the convection current at x^ is
^' = ^ r y'(-^i)y(^^ - xi)^~'''''~"' dx, . (hio)
The power flow from the electron stream to the circuit is
P = 1 f V'{Xi)it dx2 (hll)
P = {^ [ r V'{x,)V'*{x,)y{x^ - x,)^-'^'''-''^ dx, dx2 . (hl2)
4 K 0 "Loo J—oo
We have in (hl2) an expression, based on the physics of the picture, for
power flow from the electron stream to the circuit. This expression is a
product of the electron convection current, due to bunching, and the electric
field, and the product is integrated from x — — oo tox= +oo, which is
merely a way of including all the a-c fields present. We could as well have
integrated between two points a and b between which all a-c fields lie.
We will now go through some strictly mathematical manipulations of
(hi 2), designed to transform it to a more handy form. In the following
steps we may regard Xi and x^ as merely two different variables of integra-
tion, disregarding completely their physical significance.
Suppose in the .Vi .Tj plane, Xi is measured + to the right and .V2 , + up-
wards. Then the plane is divided into two portions by the line x^ = X] ,
and we are integrating over the upper left portion. If we reverse the order
of integration we obtain
p = ill [ I V'{x^)V'*{x,)y{x2 - x,)e^''^"-''^ dx^dx,. (hl3)
Let us (a) interchange the variables of integration x-i and Xi and (b) take
the conjugate of P. We obtain
P* = T^l \v'{x,)V'*{x,)y{x,-x,)e'''''-'''Ux,dx,. (hl4)
The real part of P is ^ (P -t- P*) ; hence
Real = ^J^ r r V'{x^)V'*{x,h{x, - x,)e'''''-''' dx, dx, . (hl5)
O VQ J— «o J— 00
Let us consider the quantity
A = [ V'ix)e''''dx (hl6)
666 BELL SYSTEM TECHNICAL JOURNAL
= f°° r V'{x,)V'*(xOe'''''--''' dx.dx,
J— 00 •f— 00
d\A
-00 •'—00
i2 • ,» 00 ^00
= i ( ( V'(x,)V'*{x^)y(x, - x,)e^''^'-^^' dx.dx, . (hl8)
P J— 00 •'— 00
^7
Hence, we see
B. Transverse Field
Suppose we consider the additional power transfer because of deflections.
There will be two sources of energy transfer. First, imagine a fluctuating
y component of velocity, y. Let i(o be the x component of velocity and — 7o
the convection current to the right. In a distance <fjc this will flow against
the potential gradient in the y direction a distance
dy = (y/uo)dx (h20)
and the power flowing to the held from the beam will be
dP = -^-^^(r/uo)dx. (h21)
2 dy
This is not the total power transfer, however. The beam will also suffer
a displacement y in the y direction. Now the x component of field varies
with displacement; hence the beam will encounter a varying field. We
can write the instantaneous power transferred from the beam to the field.
Let {V)i be the instantaneous value of V and (y)i be the instantaneous
value of y. The instantaneous power will be
dp= -
e-^' +§&■»■)-
Let us compare this with the instantaneous power transferred from the
beam to the field by a fluctuating convection current (i)i
dp = ^' {i)i . \ (h23)
We see that according to our convention that ■
F = VI* (h24)
we may meS^oiwir\[^lQ^'^'iiai lo labio arii snignBfi'J . ix = ic in oias
The y gradient of the potential at .Ti produces a velocity at .T2
y, = ^ r (^JL) e--^--^'' dx, . (h26)
Wo J-00 \oy/i '
It produces a displacement at Xo
y2 = ^ r (^) (X2 - xOe-^-''^'-'^' dx, . (h27)
Wo J_oo \^y/i
Writing the total power as
p = p^^ P^_ (h28)
We have the two contributions from (h22) and (h23) using (h9)
and
4Fo ^-00 ^-«
Again, we will turn to mathematical manipulation disregarding the
physical significance of the variables. If we change the order of integra-
tion, (h30) becomes
4Fo
Integrating with respect to Vo by parts we obtain
-£/:(as)>'-'--)-
The first term is zero because I — — I is zero at .Vi = — 00 and (.V2 — Xi) is
\dy/
t668 BELL S]fjm3M\nmM!fICXi\Ek^0URNAL
zero at Xa = .ri . Changing the order of integ?^rb)i^ro(iio|«biiw \Bm aw
(h33)
From (h29) and (h30) we see that the total power is
^ = > 47, £ £ (^), (^)>(- - -'^""■"'■' "- '- ■ *^*'
By the same means resorted to in connection with (hi 2) we find
Real^-^T^J/J-' (h35)
B = f'—e''^ dx. (h36)
J- 00 dy
We can go a step further. We have
A
= f — e^'^^'dx. (h37)
J- 00 ox
Now
Integrating by parts
dy dy
^= r P^e^'"" dx. (h38)
dy J- 00 oxdy
f "^ ^ e^>- dx. (h39)
J- 00 OV
ar
dy
The first term is zero, and we see that
I iJ I' = (I dA/ay IVt') (1,40)
Reai^J.^'l^^/fl'/^r (h4,)
C. Electronic Gap Loading
In (hl9) and (h41) we have expressed the power flow from the electron
stream to the circuit in rather general terms. What we want immediately
is the quantity (conductance) giving the power flow from the circuit to the
stream for a single gap. Assume we have a single gap with unit peak r-f
voltage across it. The power absorbed by the electron stream can be
attributed to a shunt conductance such that niiijdo av/ airii moi^ fanA
Also, in tHis<:^^^ l4^|4s slnicly^-^, jthe m<>dui4lii)i(i"<?x)etfficient. Hence, the
conductariredWtb action of tlielongitiid'in^i'fitlds is, from (hl8), simply
2 ?i -•->- '
(h43)
4Fo ^7
And, due to the action of transverse fields there is another conductance,
from (h41)
'470^7 \7'
G = Gi + Go
y^ W /
(h44)
(MS)
These are surprisingly simple and very useful relations.
It is interesting to take an example which will indicate both effects.
We have from (b24) for tubes of radius ro with a narrow gap between them
f^: = ^;[l - /I(7ro)//o(7'-o)]. (h46)
Accordingly, the part of the conductance due to the longitudinal field is
G, = Fz.(7^o)/o/4Fo (h47)
Fdyro) = -yd^l/dy
(h48)
IjMV _ (h(yro)\ _ (hiyro)y-
Myro)/ ^ " V/o(7ro)/ Voiyro)/ _ "
= -2
Similarly, the part of the conductance due to the transverse field is
Gt = FTiyro)Io/'iVo
Friyro) = -yT~~i
dy To
1 p/1 d '
■iL[\yJr^:
2rdr.
From (bl6) we obtain
^r = Ioiyr)/Io(yro)
■si. [;ar^')
hiyrp) _ ll{yro)~\
hiyro) ll(yro)j
" 'T' = hiyr)/ hiyro) _
7 dr To
2r dr
(h49)
(h50)
(hSl)
(h52)
««) BELL S^mm^^^mSl^mXl^fOURNAL
And from this we obtain Ir.riJ ri3ua SDnBiDubnoo Anuria e ot bsJudiiJiu
■»■" ^8 Id)
[
moil ,;>i ebim^Mn^nMMiyc/ noh-ir, M'^'Um
The total conductance is s,
Gi + G2 = (Fz,(7ro) + /v(7^o))/o/4Fo.
(h53)
ni ,02! A
rrdiDubnoo
(h54)
In Fig. '139, (G1F0//0), (G2F0//0) and (GF0//0) are plotted vs 7^0 .
It may be seen that while the conductance due to transverse fields may be
negative, the total conductance is always positive.
r''"
^x
'
/
/
/
\
'\
/
/
^
"N
.^
4^)
/
/
/
\
■ !
/
>
/
\.
'v
/
/
/
S
/G|V
v.
^-.
>^
/
/
/
rj-
"-^^
/
1
>
""^^
"•-^
1 /
/ /
'"■"•
i
1 /
[%
_^'
^'
1
■
■
1
If
y^
1/
/
/'
\
/
/
\
/
\
/
004
>
k^
/
r
15 2 0 2,5 3.0 3 5 A.0 4.5 5.0 5;
RADIUS OF TUBES, 7 Tq , IN RADIANS
Fig. 139. — Gaj) loading factor vs radius of tubes forming gap measured in radians.
The tul)es are supposed to he filled Avith uniform electron flow. The curve involving
G\ is that for longitudinal effects, that involving Gi is for transverse effects, and that
involving G is for both combined
This example tends to exaggerate the effects of transverse fields because
the beam is assumed to fill the whole tube. In an actual case the beam
REFLEX OSCILLATORS
671
would probably not fill the whole tube, and the effect of transverse fields
would be less. ■ i •
Perhaps a more useful expression is one involving longitudinal fields
only; that for infinitely fine parallel grids. In this case, if the separation
is ^
^' = sin^' iy(/2)/{yt/2y
(h55)
and, from (h43)
2Fo (t^/2) L (t^/2) ' ' J
In Fig. 140, (GVo/Io) is plotted vs (yC/2). The negative conductance
region beyond yt - 2x, familiar through Llewellyn's work with diode
oscillators, is of less interest in connection with reflex oscillators.
0 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
GRID SEPARATION, 7l, IN RADIANS
Fig. 140. — Gap loading factor for fine parallel grids vs grid separation in radians.
. It is of some interest to compare the electronic gap loading with the small
signal electronic conductance due to drift action. Assume, for instance,
we have fine parallel plane grids for which yf=ir. From (h2) we get
(GVo/h) = .202.
As the current crosses the gap twice we should count the current involved
as twice the d-c beam current /&
From (2.1)
G = .404 h/Vo
^ = .633
/3^ = .400.
6>2 BELL SYSTEM TECHNICAL JOURNAL
If we assume 3.75 cycles of drift, then from (2.4) the magnitude of the small
signal electronic admittance is
y, = ^'hO/lV, (h57)
- 4.71 h/Vo .
Thus, in this example, the gap loading is about 1/10 of the small signal elec-
tronic admittance.
D . Bunching in the Gap
An unbunched stream will become bunched due to a single transit across
an excited gap. Expression (hlO) gives us a means for calculating the ex-
tent of this bunching. As an example, we will consider the case of fine
parallel grids separated by a distance (. Then the gradient is given by
V'(xr) = V/f. (h58)
from Xi = 0 to Xj = X2 = A and by zero elsewhere. Thus
2Vaf Jo
klV = (/o/Fo)(l/2)[l - UMi^ - e^'^W^. (h59)
It should be noted that for large values of 7 ^
i<,IV = (l/2)(/o/Fo)r^'"^. (h60)
For our previous example, if 7 ^ = tt,
i^lV = .592 (/o/Fo)r''''". (h61)
If the current is referred to the center of the gap instead of the second
grid, we obtain a current i such that
ijY = .592 {h/Vo)e-''-'\ (h62)
Now, the electrons constituting current / will drift 3.75 cycles and will
return across the gap in the opposite direction. To get the induced circuit
current / we take this into account and multiply by ^
I/V = -^{i/V)e-'''^'-''^
(1^63)
= -.375(/o/Fo)r^'''.
This is very nearly of the same size as the electronic gap loading.
The general conclusion seems to be that for typical conditions encoun-
tered in reflex oscillators, gap loading and bunching in the gap are small
and probably less important than various errors in the theory.
REFLEX OSCILLATORS 673
APPENDIX IX
Losses in Grids
Although the general problem of resonator loss calculation is not treated
in this paper, grids seem peculiar to vacuum tubes and losses in grids will
be discussed briefly.
Assume that we have a pair of grids of some mesh or network material,
parallel, circular, and of a diameter compared with the wavelength small
enough so that variation of voltage over the grids may be neglected. The
capacitance inside of a radius r is
C = iirr Id
(il)
e = 8.85 X 10 " farads/cm.
Here d is the separation between the grids. For unit r.m.s. voltage across
the grids, the power dissipated in the part of both grids lying in the range
dr at r is
dP = l{i^C)\R dr/lirr)
(12)
= CO 6 irRr dr I d .
Here R is the surface resistivity. The total power is equal to V G, where G
is the conductance measured at the edge of the grids, and is
2 2 T, r-D/2
G = P = "^^^^ [ r" dr
CO e ttR
'd^ Jo ■ " (i3)
o:'e\RD'/64d\
It is interesting to express co in terms of X, the wavelength, c the velocity of
light, and then to put in numerical values
G = (l/16)AVi?Z)Vx'<^' (i4)
G = 1.39 X m~'RDy\^d\ (15)
Now for the copper, the surface resistivity is
Re = .045/ VX (i6)
where X is measured in cm. Suppose the grid material is non-magnetic and
has N times the low frequency resistivity of copper. Then for it, the sur-
face resistivity will be A'^^ times that given by (16). Suppose that the diam-
eter of the grid wires is 2r and the distance center to center is o. If current
flowed on one half of the wire surface only, the surface resistivity parallel
to the wires would be
R = N^Rcia/irr). (i7)
674 BELL SYSTEM TECHNICAL JOURNAL
If we use this as the surface resistivity in (i5) we obtain
G = 1.99 X 10"' (fj \/N/X/D*\'d\ (i8)
It is a somewhat remarkable fact that if we work out the loss for a pair
of grids with surface resistivity R in one direction and infinite resistivity
in the other direction (parallel wire grids) we get just twice the conductance
given by (i5). Thus, it appears to be roughly true that if we have a given
parallel wire grid, adding wires between the original wires and adding wires
perpendicular to the original wires should have about the same effect in
reducing circuit loss.
APPENDIX X
Starting of Pulsed Reflex Oscillator
If a reflex oscillator is turned on gradually, the voltage from which the
oscillations build up is certainly that due to shot noise in the electron stream.
However, if the current is turned on as a pulse of short time of rise, it might
be that the microwave voltage produced by the high frequency components
of the pulse would be larger than the voltage produced by shot noise, and
hence that oscillations would build up from the transient produced by the
pulse and not from shot noise. This would be important because presum-
ably the voltages produced by the pulse are always the same and related in
the same manner to the time of application of the pulse; thus, in buildup
from the transient of the pulse there would be no jitter.
In an effort to decide from which voltage the oscillations build up, we
will consider Johnson noise and shot noise voltages.
Associated with a mode of oscillation of the resonator there is a mean
stored energy kT. If L and C are the effective inductance and capacitance
of the mode and P and V^ are the mean square current and voltage
kT = Yl/I + 'v^C/2. ' (jl)
On the average, half of the energy is in the capacitance and half in the in-
ductance; thus
7^= kT/C
(J2).
= kTuo/M.
Here M is the characteristic admittance of the mode and con is the resonant
radian frequency.
As an equivalent circuit for the mode we may use conductance (/ in shunt
with an inductance L and a capacitance C. The impressed Johnson noise
•3af-Ve?it-'lbrla£feao«tM»il!hn#iibn£ ,\A arniJ r> ir. ,,1 suIbv £ oi sraii riliw yhfianil
bsJoaini adi lo sviJcgan srli .qs^di KsmnjL Jnanuo mulsi b ad Iliw giadl^
JnanuD b3:tD9ini srii oJ Joaqasi nii// t arrin Tihb arii yd bgyfibb bnB Jnsnua
-Sappioye tH^ (hsfa-ealt 4roB8att^ fehb'ig^p isf0/§)(aiC(Di©rpot /)jitvjt4iittaii(DDha:aiiBEts).
flirftlieadihpeiajt MdJfwlb9h®lt2iasrEeti'(udssp£((l:e:^cHaogeiT^dliotioii aif (Kbibe)ntJae
dRipm9se)[^ishs6fnoisEicti»5TitjpffirfJuattitlbaand'i^djjth feiiM He'^^"
avis ,T^v«-"r>'' ' ■ ^» = 2e{2Io)B
= 4eIoB.
Thus, we see that the shot noise voltage will be the Johnson noise voltage
times the factor
G5)
J2 eh
kTG
11,600 h
T G'
The
conductance G
is given by
G = M/Q.
it us
take reasonable values,
T
= 293° K (20° C)
h
= .2 amperes
M
= .06 mhos
Q
- 400 (loaded 0.
The values of M and Q are about right for the 2K23 pulsed reflex oscillator.
We obtain
{? = 5.3 X 10'. (j6)
I7
Thus, shot noise is very much more important than Johnson noise.
From (j2) and (j5) we see that the shot noise voltage squared may be
expressed as
Vl = '"^'^'^ (j7)
In pulsing a reflex oscillator, the voltage and current will be raised simul-
taneously; however for the sake of simplicity in calculation, we will assume
that the voltage is constant and the injected current is zero at / = 0, rises
?,e86 BELL SV}STmrK^I\MBm£A'^SAM)URNAL
linearly with time to a value I a at a time A/, ani^ rt4nrt5iia&xlcw6^ajjti{>^er(f^i§5.
, There will be a return current across the-^ap, the negative of the injected
current and delayed by the drift time r wim respect to the injected current.
.(afJnjcaflaiilattiatgvth^ tosponnse-.tc^Vtliis qpgplidd tfMiprenb, Aa>atiexi Q^lwillobiqaS-
^slilnlediitia) be (inf]tnilbv[actuaUy^)tihqr;shaJib3ieffis1tariee[Iwlll'ibfe ipiasmtiifvea^lifeh
the current is smaH ifitd rtega;thT€i;U'bbDijthfi|Cjn"5eiitil^onadS)fIar|[esesinpgh
so that the electronic conductance is larger in magnitude than the circuit
conductance. The assumption of zero conductance should, however, give
us an idea of the transient which would be effective in starting the oscillator.
If a charge dq is put onto a capacitance C = M/coo forming part of a
resonant circuit of frequency coo , the subsequent voltage across the circuit
will be
dV - ^ e'"'' dq. g8)
We see that for times late enough so that the injected and returned current
are both constant, the voltage due to our assumed current will be
•^' / r'
dto
M \Jq At Jm
- C ^'-^-^ e^"^'-'»' dt, - f e^"^'-"^' dt}\
Ci9)
Integrating, we find
'■ = {liik) (' - ^"")(e-'"" - 1). am)
If we have n + 3/4 cycle drift,
e-^'""=~j. (jll)
The extreme value of («""""" — l)is— 2. For this value we would obtain
I T |2 2/o rin\
M^At^uiQ
From this and (j7) we obtain
Taking the values
VI ^ eQAt\l
Tp 2/o
e = 1.59 X l(r^^ coulombs
/o = .2 amperes
Q = 400 (loaded Q)
At = .2 X 10"'' seconds
Wo = 25 X 10^ radians/second.
Gi^^)
REFLEX OSCILLATORS 677
We obtain
p^, = 99.2. (jl4)
The indications are that oscillations will built up from shot noise rather
than from the high frequency transients induced by the pulse.
The preceding analysis assumes that the full shot noise voltage will
appear across the resonator soon enough to override the pulse. The shot
noise voltage will reach full amplitude in a time after application of the
current of the order of Q/f^ = lirQ/wo . For the values given above
27r(2/coo = .05 X 10^
as this is a considerably smaller time than the .2 microseconds allotted for
the buildup of the pulse, the assumption of full shot noise voltage is pre-
sumably fairly accurate.
APPENDIX XI
Thermal Tuning
Two extreme conditions may exist
(1) Cooling is by radiation alone
(2) CooHng is by conduction alone.
(1) Radiation Cooling
The rate of change of temperature on heating will be given by
^ = 1 [p, - /cri. (ki)
Where
T is the absolute temperature of the expanding element.
C is the heat capacity of the element.
K is the radiation loss in watts/ (degree Kelvin) .
Pi is the power input to the tuner in watts.
It is assumed that the temperature of the surroundings is constant and the
power radiated to the expanding element is included in Pi . Let
Pi = KT\ and Tr = ^ . (k2)
Tr is then a reduced temperature since Tg is the temperature which the
expanding element would reach at equilibrium for a given power input, i.e.
(pY'
r« = I — * 1 . r, is then always less than 1 .
678 BELL SYSTEM TECHNICAL JOURNAL
Upon substitution of {kl) in {ki)
^Il=^ Tl[l - Tt\ (k3)
which may be integrated to
C
2^y,-3 [tan ' Tr + tanh ' 2%] = / + /o (k4)
where /o is a constant of integration. Let
h\{Tr) = [tan~'r, + tanh~'r,]. (k5^
This function is plotted in Fig. 79. In order to determine the cycHng time
for heating n assume
At ^ = 0, Tr = TrC i.e. Trc = Tc/T^
t = Th, Tr — Trh Trh = Th/Tm
where
Tm is the value of the temperature Tg corresponding to the maximum
power input Pm ■
Th is the temperature corresponding to one band limit.
Tc is the temperature corresponding to the other band limit.
Tu > T,.
The cycUng time for heating n, is then
C _
Th = riT^rr^s [tau"' Trh " tau"' Trc + tanh"' Trh — tanh~^ Trc\ (k6)
= 2"^ ^PiiTrh) - F,(Trc)] (k7)
which gives the time required for the expanding element to rise in tempera-
ture from Trc to Trh , i-e. from Tc to Th .
If we reduce the power input and wish to determine the cooling time the
analysis is similar. If the i)()wer input from electron bombardment is
reduced to zero there will still be i)ower input to the tuner. The residual
power which is kept to the minimum possible level comes from such sources
as heat radiated from the cathode and general heating of the envelope by
the oscillator section.
Let Pa be the value of the reduced power inj)ut.
REFLEX OSCILLATORS • 679
Then -"^Jl = l(KT' - Po) ~ fk8)
at C
or ^' = -^ TtiTt - 1). (k9)
Here T, = ^ and Po = KTo (klO)
To
where Po is the power from other sources than direct bombardment. In
this case T, is always greater than 1.
Integration yields
C
IKTi
[tan"' Ts + ctnh"' T,] ^ t -^ k . (kll)
Let FoXTs) = [tan"'r. + ctnh~Y,]. (kl2)
This function is plotted in Fig. 80. To determine the cycling time for
cooling assume
At time
/ =0,
Ts — Tsh
i.e.
Tsh — Th/To
t = Tc,
T = T
Tsc = TjTo
Then
c
Tc = o;^7-3 [FiiTsc)
- F,(Tsk)]
{
(kl3)
giving the time for the contracting element to cool from temperature Tsk
to Tsc', i.e. from T;, to Tr .
(2) Conduction Cooling
The rate of change of temperature on heating will be given by
where
T is the temperature difference between the tuning strut and the
heat sink.
C is the heat capacity of the strut.
k is the conduction loss in watts/°C.
Pi is the power into the tuner.
The solution of (kl4) is then,
P
k
(^-)
P<-T = (^- To]e-^"''' (kl5)
680 BELL SYSTEM TECHNICAL JOURNAL
where Tq is the temperature diflference at / = 0.
If
the temperature difference from the sink is Tq = Tc at the cooler
limit of the band.
Th is the temperature difference from the sink at the hotter limit of the
band.
Pm is the maximum permitted input, then the cycling time for heating
On cooling
from which
f=-e' *>')
T = Toe-^'""^' (kl8)
Tc = ^ log. ^\ (kl9)
Acknowledgments
A large number of people have contributed to the work described in this
paper. The program was carried on under the general direction of J. R.
Wilson whose support and advice are gratefully acknowledged. J. O.
McNally gave more specific direction to these efforts and is responsible for
many of the design features. We are greatly indebted to him for his close
support and encouragement.
Members of the Bell Laboratories technical staff concerned with various
portions of the developments were E. M. Boone, K. Cadmus, R. Hanson,
C. A. Hedberg, P. Kusch, C. G. Matland, H. E. Mendenhall, R. M. Ryder
and R. C. Winans. R. S. Gormley of the Western Electric Company made
considerable contributions to the work on the 2K25.
The mechanical design, construction and specifications for manufacture
were carried out by a group of engineers under the direction of V. L. Ronci.
This group included D. P. Barry, F. H. Best, S. 0. Ekstrand, G. B. Gucker,
C. Maggs, W. D. Stratton, R. L. Vance and E. J. Walsh. L. A. Wooten
and his associates cooperated in numerous chemical problems throughout
this work.
The authors wish to make particular acknowledgement of the work of
two technical assistants, Miss Z. Marblestone and F. P. Dreschler, whose
REFLEX OSCILLATORS 681
diligent and painstaking efforts were a direct contribution to the successful
completion of most of the designs described herein.
The NDRC Radiation Laboratories at the Massachusetts Institute of
Technology extended very close cooperation both on problems of design and
applications of the tubes. A. G. Hill and J. B. H. Kuper were particularly
helpful with technical advice and support. H. V. Neher was responsible
for the initial design of the 2K50 and assisted on its final development at
the Bell Laboratories. G. Hobart assisted on the 2K50 at M.I.T. and also
on the final development as a resident at Bell Laboratories.
Professor R. D. Mindlin on leave from Columbia University served as a
consultant on vibration and stress analysis problems particularly on the
design of diaphragms and mechanical and thermal tuning mechanisms.
In the preparation of the manuscript the authors are particularly in-
debted to R. M. Ryder for his extension of the simple bunching theory
which is included as a part of this paper. L. A. MacColl provided the
solution for several mathematical problems.
Abstracts of Technical Articles by Bell System Authors
Investigation of Oxidation of Copper by Use of Radioactive Cu Tracer} J.
Bardeen, W. H. Brattain, and W. Shockley. A very thin layer of radio-
active copper was electrolylically deposited on a copper blank. The surface
was then oxidized in air at 1000°C for 18 minutes, giving an oxide layer with
a thickness of 1.25 X 10"^ cm. After quenchirg, successive layers of the
oxide were removed chemically, and the copper activity in each layer was
measured. The observed self-diffusion of radioactive copper in the oxide
agrees quantitatively with a theory based on the following assumptions: (a)
The oxide grows by diffusion of vacant Cu+ sites from the outer surface of
the oxide inward to the metal, (b) The concentration of vacant sites as the
oxygen-oxide interface is independent of the oxide thickness, and drops
linearly from this constant value to zero at the metal boundary, (c) Ac-
companying the inward flow of vacant sites, there is a flow of positive elec-
tron holes such as to maintain electrical neutrality, (d) Self-diffusion of
copper ions takes place only by motion into vacant sites. The results give
a fairly direct confirmation of the theory of oxidation first suggested by
Wagner.
A New Magnetic Material of High Permeability} (). L. Boothby and
R. M. BozoRTH. This paper describes the preparation, heat treatment, and
properties of supernialloy, a magnetic alloy of iron, nickel, and molybdenum.
In the form of 0.014 in. sheet it has an initial permeability of 50,000 to
150,000, a maximum permeability of 600,000 to 1,200,000, coercive force of
0.002 to 0.005 oersted, and a hysteresis loss of less than 5 ergs/cm^ /cycle at
B = 5000. Transformer cores made of insulated 0.001 in. tape, spirally
wound, have about the same initial permeability and a maximum permeabil-
ity of 200,000 to 400,000. The alloy has a Curie point of 400°C and appears
to have an order-disorder transformation temperature somewhat above
500°C.
Magnetoresistance and Domain Theory of Iron-Xickel Alloys? R. M.
BozoRTH. Measurements of change of electrical resistivity with magneti-
zation and with tension are reported for iron-nickel alloys contai'ii-g 40 to
100 i)er cent nickel. When the magnetostriction is negative (81 to 100 per
' Jour, oj Chemical Ptiysics, December 1946.
'^ Jour. Applied Physics, February 1947.
' Phys. Rev., Dec. 1 and 15, 1946.
682
ABSTRACTS OF TECHNICAL ARTICLES 683
cent nickel), tension (a) decreases resistivity, and magnetic field (H) in-"
creases it. Domain theory predicts the ratio a/H at which the resistivity
is equal to that of the unmagnetized specimen, and the theory is accurately
confirmed. Measurements are made in transverse as well as longitudinal
magnetic fields, and the difference between the resistances so measured is
shown to be independent of the distribution of domains in the unmagnetized
state; the erratic results reported in the literature are thus explained and
avoided. When magnetostriction is positive, the limiting changes of resis-
tivity with field and tension are sometimes found to be different; this is
shown to be caused by the variation of magnetostriction with crystallo-
graphic direction.
A Wide-Titning-Range Microwave Oscillaior Tube.* John W. Clark and
Arthur L. Samuel. This paper describes a reflex-type velocity- variation
oscillator tube with a wide tuning range in the microwave band. The tube
will oscillate from 20C0 to 13,000 megacycles, but practical tuning considera-
tions limit the band in any one circuit to a two-to-one frequency range.
The problems involved in the design and a description of the various elements
are given.
Accelerated Ozone Weathering Test for Rubber} James Crabtree and
A. R. Kemp. Light-energized oxidation and cracking by atmospheric ozone
are the agencies chiefly responsible for the deterioration of rubber outdoors.
Since these processes are separate and distinct, it is proposed to distinguish
between them in the evaluation of rubber for resistance to weathering. An
accelerated test for susceptibility to atmospheric ozone cracking is discussed.
Apparatus for conducting the test and for measurement of ozone in minute
concentration is described in detail.
Measurements in Communications.^ N. B. Fowler. For convenient
reference, some of the more common measurement units and scales used in
communication engineering are presented in tabular form together with sup-
plementary explanatory text. Included in the table, which also indicates
the limitations involved, are quantities used in measuring power, volume,
circuit noise, sound, light, radio fields, crosstalk coupling, and certain other
transmission concepts.
An Improved 200-Mil Push-Pull Density Modulator.'' J. G. Frayne,
T. B. Cunningham and V. Pagliarulo. A completely new variable-
'' Proc. I.R.E. and Waves and Electrons, January 1947.
^ Indus. & Engg. Cliemistry. .inalytical Edition, December 1946.
^ Electrical Engineering, February 1947.
^ Jour. S.M.P.E., December 1946.
684 BELL SYSTEM TECHNICAL JOURNAL
density modulator utilizing a three ribbon push-pull valve is described.
The entire valve is sealed by the force of the Alnico V permanent magnet on
the Permendur pole pieces. Signal is applied to the center ribbon and noise-
reduction currents are applied to the outer ribbons. True class A push-pull
operation is obtained from the two component single ribbon valves by the
use of an inverter prism which aligns the modulating and noise-reduction
edges of each aperture.
An anamorphote condenser lens is used to eliminate lamp filament stria-
tions at the valve ribbon plane. An anamorphote objective lens gives a 4: 1
reduction of the valve aperture in the vertical plane at the film and a 2:1
reduction along the length of the sound track. A meter is supplied to meas-
ure exposure as well as setting up "bias." A photocell monitor is supplied
and a "blooping" light for indicating synchronous start marks.
Mathematical analysis of the exposure produced by the modulating ribbon
is appended as well as a similar analysis of the four ribbon push-pull valve
which the new valve supersedes.
Factors Governing the Intelligibility of Speech So^mds^ N. R. French
and J. C. Steinberg. The characteristics of speech, hearing, and noise are
discussed in relation to the recognition of speech sounds by the ear. It is
shown that the intelligibility of these sounds is related to a quantity called
articulation index which can be computed from the intensities of speech and
unwanted sounds received by the ear, both as a function of frequency.
Relationships developed for this purpose are presented. Results calculated
from these relations are compared with the results of tests of the subjective
effects on intelligibility of varymg the intensity of the received speech, alter-
ing its normal intensity-frequency relations and adding noise.
Short Duration Auditory Fatigue as a Method of Classifying Hearing Im-
pairment.'^ Mark B. Gardner. Earlier studies have classified deafness
cases into two general groups, those having functional disorders of the middle
ear and those having impairments resulting from atrophy of the nerve fibers
terminating along the basilar membrane (conductive and nerve deafness
types, respectively). Such classifications have been made using bone con-
duction threshold measurements and unilateral loudness balance results as
the basis for differentiation. Bone conduction results, however, are often
subject to considerable error while the unilateral loufbiess balance technique
can only be applied to individuals ha\'ing one normal and one impaired ear.
These limitations introduce a need for a completely independent monaural
method of classifying deafness types. This is particularly true for the selec-
^Jour. Acoiis. Soc. Amer., January 1947.
'Jour. Aeons. Soc. Amer., January 1947.
ABSTILACTS OF TECHNICAL ARTICLES 685
tion of candidates suitable for the fenestration operation for the restoration
of hearing in otosclerosis (immobilized stapes). The present paper is con-
cerned with an investigation of short time auditory fatigue as a method of
obtaining an impairment analysis. In this study, it was found that the
fatigue of the conductively deafened observer was similar to the normal
observer except the onset of fatigue was shifted by the amount of the thresh-
old loss. For the nerve deafened observer, on the other hand, the onset of
fatigue was found to occur at normal intensity levels. The occurrence of
excessive fatigue in one of the nerve type impairment cases investigated
appears to offer additional information on the nature of the lesion.
A Sampling Procedure for Design Tests of Electron Tubes {Sponsored by
Joint Electron Tube Engineering Council)}^ S. W. HoRROCKS, P. M.
DiCKERSON, H. F. Dodge,* E. R. Ott, H. G. Romig,* W. B. Rupp, J. R.
Steen, R. E. Wareham, and A. K. Wright. The Committee on Sampling
Procedure was established on July 21, 1943 as part of the Electron Tube
Section of the Radio Manufacturers Association (RMA). The purpose of
the Committee is to develop sampling methods and to act in an advisory
capacity towards standardization of Sampling Procedures throughout the
Electron Tube industry. This Committee was later embodied as a main
Committee of the Joint Electron Tube Engineering Council of the RMA
and NEMA. This Council was established in 1945 to handle all engineering
matters for the Electron Tube industry for both trade associations. Radio
Manufacturers Association and National Electrical Manufacturers Asso-
ciation.
One of the earliest projects handled by the Committee was the develop-
ment of a statistically sound sampling inspection procedure for so-called
"design tests" of electron tubes. In general, design tests relate to character-
istics that are normally quite stable and are relatively less important to the
consumer. The nature of these tests is such that only relatively small
samples are practicable. The Joint Army-Navy Specification JAN-IA
incorporated a sampling plan for design tests allowing (1) not more than 10%
of the sample tubes to contain design test defects of any one kind and (2)
not more than 20% of the sample tubes to contain design test defects of any
kind. Because of the extremely wide range in lot sizes for different classes
of electron tubes, such a simplified sampling plan was m effect too strict for
small lot sizes and too liberal for large lot sizes. Moreover, no distinction
was made in the relative seriousness of different kinds of design test defects.
The Committee accordingly set about to prepare a sampling inspection plan
that would be relatively free of these faults. The new procedure covers all
'" Industrial Quality Control, November 1946.
* Of Bell Tel. Labs.
686 BELL SYSTEM TECHNICAL JOURNAL
aspects of the acceptance problem and provides an operationally definite
criterion for reducirg inspection for a product whose quality is regularly well
controlled within the intent of the specification.
The procedure developed by the Committee was approved by J.E.T.E.C.
(Joint Electron Tube Engineering Council), was approved by the JAN
Committee in September 1945, and is reproduced in full in this article. It
will be noted that the procedure provides for two Acceptable Quality Levels
(AQL), namely 6% defective and 3% defective for individual design test
characteristics. Each design test of a particular type of electron tube is
classified as either a Standard Design Test with an AQL = 6% or a Special
Design Test with an AQL = 3%. For any design test, if product sub-
mitted for inspection has quality equal to the AQL, the chances of acceptance
are of the order of 94 to 98 out of 100. If quality runs consistently better
than the AQL, reduced inspection is permitted, thus serving as an incentive
for the manufacturer to strive for better quality. The operatirg characteris-
tics of the sampling plans involved are appended to this article and show the
degree to which the plans will discriminate for various levels for submitted
quality.
Resonant Circuit Modulator for Broad Band Acoustic Measurements}^
Gordon Ferrie Hull, Jr.* A modulation method is described whereby
a broad band frequency response is obtained for recording of sound. In
particular low frequency sound approaching zero c.p.s. can be recorded.
The theory of the resonant circuit modulating principle is first discussed
followed by a description of the apparatus which was constructed for this
purpose.
Quality Reporting — Putting Inspection Results to Work}- Harold R.
Kellogg. Quality reportirg is an integral part of the general inspection
problem. It cannot be divorced from the logic and aims of an overall inspec-
tion program. A discussion of quality reporting should therefore include
consideration of (1) inspection procedures, including the collection of data;
(2) appraisal of the data; (3) reporting and publicizing results. This out-
lines the program as it is discussed in this paper.
Properties of Monoclinic Crystals}'^ W. P. Mason. Two crystals of the
monoclinic s])henoidal class have been found which have modes of vibration
with zero temperature coefiicients of frequency, and this jiroperty together
^^ Jour. Applied Physics, December I'Mft.
* This research was carried out while the author was a member of the Technical Stall
of the lieii Telephone Laboratories, Inc., Murray Hill, New Jersey.
'^ Induslrial (Juatity Control, November 1946.
'^ Phys. Rev., Nov. 1 and 15, 1946.
ABSTRACTS OF TECHNICAL ARTICLES 687
with the high electromechanical coupling and the high Q's make it appear
probable that these crystals may have considerable use as a substitute for
quartz which is difficult to obtain in large sizes. These crystals are ethylene
diamine tartrate (CeHijNiOe) and dipotassium tartrate (K-jC4H.i06, |HcO).
Complete measurem.ents of the elastic, piezoelectric, and dielectric constants
of the dipotassium tartrate (DKT) crystal are given in this paper. The
crystal has 4 dielectric constants, 8 piezoelectric constants, and 13 elastic
constants. A discussion is given in the appendix of the method of measuring
these constants by the use of 18 properly oriented crystals.
An Acoustic Constant of Enclosed Spaces Correlatable ivith Their Apparent
Liveness}^ J. P. Maxfip:ld and W. J. Albersheim. An acoustic constant
called liveness is derived, which constant is correlatable with the acoustic
properties of the enclosed space and with the distance between the sound
source and the listener. This constant represents the ratio of a time integral
of the energy density of the reverberant sound to the unintegrated energy
density of the direct sound. The validity of this constant is substantiated
by empirical data. Certain subjective effects of monaurally reproduced
sounds as a function of the liveness of its pick-up conditions are briefly
discussed.
Directional Couplers}^ W. W. Mumford. The directional coupler is a
device which samples separately the direct and the reflected waves in a
transmission line. A simple theory of its operation is derived. Design data
and operating characteristics for a typical unit are presented. Several appli-
cations which utilize the directional coupler are discussed.
Theory of the Beam-Type Traveling-Wave Tnhe}^ J. R. Pierce. The
small-signal theory of the beam traveling- wave tube has been worked out.
The equations predict three forward waves, one increasing and two attenu-
ated, and one backward wave which is little affected by the electron stream.
The waves are partly electromagnetic and partly disturbance in the electron
stream. The dependence of the wave propagation coefficients on voltage,
current, circuit loss, and the other properties of the transmission mode which
propagates energy and the^^cut-off transmission modes is given. Expres-
sions for gain and noise figure and an estimate of power output are given.
Appendix A gives an expression for the field in a uniform transmission system
due to impressed current (as, of an electron stream) in terms of the para-
meters of the transmission modes. Appendix B calculates the propagation
^^ Jour. Aeons. Soc. Amer., January 1947.
i^Froc. I.R.E., February 1947.
18 Pw. I.R.E., February 1947.
688 BELL SYSTEM TECHNICAL JOURNAL
constant and the field for unit power flow for the gravest mode of hehcal
transmission system.
Traveling-Wave Tubes}'' J. R. Pierce and L. M. Field. Very-broad-
band ampUfication can be achieved by use of a traveling-wave type of circuit
rather than the resonant circuit commonly employed in amplifiers. An
amplifier has been built in which an electron beam traveling with about 1/13
the speed of light is shot through a helical transmission Ime with about the
same velocity of propagation. Amplification was obtained over a band-
width 800 megacycles between 3-decibel points. The gain was 23 decibels
at a center-band frequency of 3600 megacycles.
Attenuation of Forced Drainage Effects on Long Uniform Structures}^
Robert Pope. When forced drainage is applied to an underground metallic
structure to provide cathodic protection, the greatest effects on the structure
and earth potentials occur in the vicinity of the drainage point and anode.
These effects taper off as the distance from the drainage point increases and
even in the relatively simple case of a long, uniform structure, the manner in
which these effects taper off or attenuate is quite complex. However, by
making a few justifiable assumptions, relatively simple equations are devel-
oped which provide sufficiently accurate results in most practical cases.
Furthermore, the simple equations bring out more clearly the relative im-
portance of the various factors involved than do the more rigorous equations.
The approximate equations have been used with fair success in predicting
the effects of drainage on underground telephone cables in conduit and on
buried coated cables. They should apply quite accurately to coated pipes,
and there are examples of reasonably good application on some bare pipes.
The soil and structure characteristics which enter into the equations are
discussed, and the units used established.
Alkaline Earth Porcelains Possessing Low Dielectric Loss}^ M. D. Rig-
TERINK and R. O. Grisdale. Alkaline earth porcelains have been prepared
from mixtures of clay, flint, and synthetic fluxes consisting of clay calcined
with at least three alkaline earth oxides. These porcelains possess excellent
dielectric properties, have low coefficients of thermal expansion, are white,
and are especially valuable as bases for deposited carbon resistors for which
they were developed. Their characteristics make it probable that other uses
will be found for materials of this type.
An illustrative composition is 50.0% Florida kaolin, 15.0% flint {?>15
'Tr0C. LR.E., February 1947.
" Corrosion, December 1946.
" Jour. Amer. Ceramic Soc, March 1, 1947.
ABSTRACTS OF TECHNICAL ARTICLES 689
mesh), 35.0% calcine (200 mesh). The composition of the calcine is 40.0%
Florida kaolin, 15.0% MgCOs, 15.0% CaCOs, 15.0% SrCO., 15.0% BaCOs,
calcined at 1200°C. The electrical properties of this body at 1 mc. are Q at
25°C, 2160; Q at 250°C, 280; Q at 350°C, 90; specific resistance at 150°C,
1013-5 ohm-cm. and at 300°C, 10^°'' ohm-cm.
A Coaxial-Type Water Load and Associated Power-Measuring Apparatus.'^''
R. C. Shaw and R. J. Kircher. This paper presents a description of a
coaxial- type water load and associated equipment suitable for measuring
peak pulse powers of the order of a megawatt. Water-cell loads have been
designed to operate at wavelengths of from 10 to 40 centimeters, where the
average dissipation is of the order of 3C0 watts. Ordinary tap water is used
in the load to dissipate the radio-frequency power.
The Ammonia Spectrum and Line Shapes Near L25-cm Wave-Length}^
Charles Hard Townes. The ammonia "inversion" lines near 1.25-cm
wave-length are resolved, their widths being decreased at low pressures to
2C0 kilocycles. Line shapes, intensities, and frequencies are measured and
correlated with theory. Calculated intensities and Lorentz-type broadening
theory fit experimental results if frequency of collision is fifteen times greater
than that measured by viscosity methods. Splitting due to rotation is in
fair agreement with a recalculation of theoretical values. A saturation efi'ect
is observed with increase of power absorbed per molecule and an interpreta-
tion made.
Non-Uniform Transmission Lines and Reflection Coefficients}" L. R.
Walker and N. Wax. A first-order differential equation for the voltage
reflection coefi&cient 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.
Temperature Coefficient of Ultrasonic Velocity in Solutions^ G. W.
WiLLARD. Extensive measurements have been made, at ten megacycles, of
the temperature dependence of ultrasonic velocity in liquids and liquid mix-
tures. All smgle liquids tested, except water, were found to have large
negative temperature-coefficients in the temperature range of zero to 80°C.
Water has a large positive coefiicient at room temperature, decreasing to
zero at 74°C and then becoming negative (with a peak velocity of 1557
20 Proc. I.R.E. and Waves and Electrons, Januar>' 1947.
" Phys. Rev., Nov. 1 and 15, 1946.
'''^ Jour. Applied Physics, December 1946.
'^ Jour. Acous. Soc. Amer., January 1947.
690 BEI.L SYSTEM TECHNICAL JOURNAL
meters/sec). Solutions in water of various other liquids (and of some solids)
give parabolic velocity vs. temperature curves like that for water but with the
peak velocity and peak temperature values shifting with the concentration
of the solution. In general increasing the concentration raises the peak
velocity sHghtly and lowers the peak temperature markedly from the values
for water alone. It has also been found possible by compounding three-
component solutions to adjust the values of the peak velocity and peak
temperature independently within a narrow range of velocities and a wide
range of temperatures.
Measurements of Ultrasonic Absorption and Velocity in Liquid Mixtures}^
F. H. Willis. The absorption {a) and velocity (V) of sound in liquid
mixtures were measured at four frequencies {v) in the range 3.8 to 19.2 mc,
using the Debye-Sears-Lucas-Biquard optical technique improved by the
addition of a differential photoelectric cell indicator. This improvement
permitted the use of lower sound intensities together with a wider sound
beam than in the visual extinction method, thus improvmg conditions with
respect to cavitation and beam distortion. In the mi.xtures investigated,
a/v^ was found to be independent of frequency within the accuracy of the
method, and there was no measurable dispersion of acoustic velocity. An
absorption peak at intermediate concentrations not shifting with frequency
was found in mixtures of acetone and water, and of ethyl alcohol and water,
but was not in evidence in mixtures of acetone and ethyl alcohol, and of
glycerol and water. The absorption peaks await theoretical explanation.
Measuring Inter-Electrode Capacitances?^ C. H. Young. New bridge,
developed for measurement of extremely small values in high frequency
tubes, useful to two-billionths of a microfarad.
^^Jour. Acous. Soc. Amer., January 1947.
25 Tele-Tech, February 1947.
Contributors to this Issue
William M. Goodall, B.S., California Institute of Technology, 1928;
Bell Telephone Laboratories, 1928 — . Mr. Goodall has worked on re-
search problems in connection with the ionosphere, radio transmission
and early radio relay studies, radar modulators, and more recently micro-
wave radio relay systems.
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 tele-
phone repeaters; during the war his attention was directed to investigation
of the mathematical problems involved in cavity resonators.
J. R. Pierce, B.S. in Electrical Engineering, California Institute of
Technology, 1933; Ph.D., 1936. Bell Telephone Laboratories, 1936 — .
Engaged in study of vacuum tubes.
Allen F. Pomeroy, B.S. in E.E., Brown University, 1929. Public
Service Electric and Gas Company, Electrical Testing Laboratory, 1923-
1925, 1926; Weston Electrical Instrument Corporation, 1927; Bell Tele-
phone Laboratories, 1929 — . Since 1936 Mr. Pomeroy has been principally
occupied in developing equipments to measure attenuations, phase shifts,
envelope delays, and reflection coefhcients for systems suitable for television
transmission, and during the war in the development of radar testing equip-
ment.
W. G. Shepherd, B. S. in Electrical Engineering, University of Min-
nesota, 1933; Ph.D. in Physics, University of Minnesota, 1937. Bell
Telephone Laboratories, Inc., 1937 — . From 1937 to 1939 Dr. Shepherd
was engaged in non-linear circuit research. Since 1939 he has been engaged
in the design of electron tubes.
I. G. Wilson, B.S. and M.E., University of Kentucky, 1921. Western
Electric Co., Engineering Department, 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.
691
VOLUME XXVI OCTOBER, 1947 no. 4
THE BELL SYSTEM
TECHNICAL JOURNAL
DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS
OF ELECTRICAL COMMUNICATION
rubAc imfff
The Radar Receiver I. W. Morrison, Jr. 693
High-Vacuum Oxide-Cathode 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-Infinite Unit
Slope of Attenuation D.E. Thomas 870
Abstracts of Technical Articles by Bell System Authors . . . 900
Contributors to this Issue 904
AMERICAN TELEPHONE AND TELEGRAPH COMPANY
NEW YORK
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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. PilUod F. J. Feely
«■■■■««■■>■
SUBSCRIPTIONS
Subscriptions are accepted at $1.50 per year. Single copies are 50 cents each.
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Copyright, 1947
American Telephone and Telegraph Company
PRINTED IN U. S. A.
The Bell System Technical Journal
Vol. XXVI Oclober, 1947 No. ./
The Radar Receiver
By L. W. MORRISON, JR-
Table of Coxtf.nts
Introduction 694
1. Radar Receiver Design Considerations 695
1.1 The Military Radar System 695
1 .2 Tlie I'linctioii of the Radar Receiver 696
1.21 Characteristics of the Radar Receiver Input Signal 698
1.22 Character of the Output of a Radar Receiver 701
1 .3 Composition of the Radar Receiver 703
2. Radar Receiver Component Design 706
2.1 The Radar Receiver Input Circuit 706
2.1 1 Input Signal Characteristics 707
2.12 Input Circuit Noise Considerations 707
2.13 1000-Mc Radio-Fref[uenc_\' Amplifier Design 709
2.14 The Radar Converter 712
2.15 The Radar Receiver Beat Oscillator 721
2.16 Typical Radar Input Circuit Designs 725
2.2 The Radar Intermediate Fref|uency .\mpiifier 731
2.21 IF Amplifier Ref|uirements 731
Hand Width 731
Gain Characteristics 733
Intermediate Midband Fre(|uenc\' 734
The Second Detector 735
2.22 IF Amplifier Input Circuit Design 735
2.23 Interstage Circuit Design 739
2.24 Second Detector Design 743
2.25 Typical Component Designs 743
2.3 The Radar Video .Vmplifier 749
2.31 fiain-Fre(|uency Considerations 749
2.i2 CJain-Amplitudc Considerations 751
2.^?< D-c Restoration Methods 753
2.34 T\])ical Radar Receiver Video Amplifier Circuits 755
2.4 The Radar Indicator 757
2.41 Classification of Radar Dis])la\' T\pes 757
2.42 The Cathode-Ray Tube '. . .' 762
Klectrostatic Deflection Txpe 763
Magnetic Deflection Type 764
Characteristics of the Fluorescent Screen 765
2.43 ry])ical Radar Indicator Component Designs 767
2.5' The Radar Sweep Circuit 773
2.51 Function 773
2.52 The Timing Wave Form (ienerator 776
The Multivibrator 776
Ty]3ical Timing Wave Circuits 779
2.53 The Sweej) Wave Form Generator 781
2.54 The Sweej) .\m|)lifier 784
2.6 (Circuits fc^r Radar Range and Hearing Measurement 789
2.61 Flectronic Hearing Marker Circuits 790
2.62 Range Marker Circuits 792
Fi.xed Range Markers 792
693
694 BELL SYSTEM TECHNICAL JOURNAL
Variable Range Marker Circuits 793
2.7 Automatic Frequency Control and Automatic Gain Control 800
2.71 Automatic Frequency Control 800
Function and Requirements 802
AFC Circuit Design Considerations 804
Typical AFC Circuit Designs 805
2.72 Automatic Gain Control 809
2.8 Radar Receiver Power Supplies 810
2.81 Primary Power Sources 810
2.82 Low Voltage Power Supplies 811
2.83 High Voltage Power Supplies 814
Conclusion 816
Introduction
THE spectacular development of radar during World War II remains
an outstanding achievement in the history of communications and the
allied electronic sciences. With military necessity furnishing the required
driving force and through the full interchange of technical knowledge among
all interested workers in this field, it has been possible to extend our visual
senses far beyond the horizons considered quite inelastic only a few years
ago. The potentialities of radar in the peacetime world and the future
application of radar design principles and techniques to the communications
and allied fields justify a review of some further details of this wartime de-
velopment.
The performance and design aspects of radar receivers will be considered
in this paper. For this purpose, the radar receiver will be defined as that
assemblage of components within the radar system which is required to
detect, amplify, and present the desired information as gathered at the radar
location. The input signals to the radar receiver consist of radio-frequency
pulses containing information regarding the area under observation by the
radar system, together with coordinate data defining further characteristics
of this observed area. The output of the radar receiver is most commonly
an optical presentation of this composite information, but in certain appli-
cations the output is further converted into electrical or mechanical signals
for specific use. In general, the output of a radar receiver is presented in a
form capable of immediate analysis and use.
Though the functional boundaries of the radar receiver are by the above
definition quite distinct, the e.xact detailed composition of the receiver and
the specific component designs are influenced by a considerable number of
factors. The successful performance of a radar receiver is dependent to a
large degree on the nicety with which these individual components are
assembled into the system as a whole.
A study of the principal factors which influence the design of the radar
receiver is j^resented in the following sections, followed by a more detailed
exposition of the principal design aspects of the various components associ-
ated within the receiver. Illustrative equipment descriptions arc included
THE RADAR RECEIVER 695
where military security permits. All of the specific equipment examples
presented here have been chosen from radar systems that have been de-
veloped within the Bell Telephone Laboratories and manufactured for the
services by the Western Electric Company. This latter limitation excludes
many interesting experimental developments w^hich have not been produced
in quantity and which have not, therefore, been substantially employed by
the services during the war period.
It should be observed that the rapid development and successful employ-
ment of radar systems during World War II have come about through the
cooperation and coordination of various governmental and military agencies,
many research and development organizations, and countless individual
workers. It is, therefore, an impossible task to assign individual credit for
the details of the development which is here described. Radar has reached
its present state of development through the efforts of many, not only those
employed specifically on radar projects during the war years, but also those
technical workers in the communications and allied fields in the years prior
to the war who so adequately supplied the firm basic foundation upon which
to build.
1. Radar Receiver Design Considerations
1.1 The Military Radar System
The specific use and area of operation of the radar system are two basic
factors which exercise a profound influence on the receiver design. It is,
therefore, pertinent to consider some common classifications of radar sys-
tems as employed during the war years.
A convenient functional classification of military radar systems may be
made as follows:
A. Search or Navigation
This classification may include warning of the presence of enemy surface
vessels or aircraft, navigation by location of landmarks, and reconnaissance.
B. Missile Control
This function includes radar systems to control gunfire and release bombs
or missiles.
C. Aircraft Interception
This classification may be considered as an airborne combination of search
and missile control, but is separated here because of the special radar design
problems encountered.
As the detailed electrical performance requirements are primarily in-
696 fiELI. SYSTEM TECUM CM. JOVRXAL
iiuenccd l)y the above functional classilication of radar systems, the me-
chanical equij)ment design or arrangement is likewise considerably in-
fluenced by the area of use of the radar system. As an illustration of this
factor, radar systems may be alternatively classified according to the area
of operation as follows:
A. Ground Equipmoit
The ecjuipment design of a ground radar system must include provisions
foi portability, protection against damage in movement over rough terrain,
and for operation under extreme weather conditions.
H. Wival Surface ]'essel Equipnwiil
Radar equipment employed on surface vessels is subject to extreme at-
mospheric corrosive conditions, severe shock from gunfire and, in some
cases, partial immersion in sea water. Large naval vessel installations
often involve extreme distances between the location of the various com-
])onents of the radar system.
(". Airbonie Eijuipmoit
Mechanical equipment features for aircraft use must include provision
for operation over rapidly fluctuating conditions of temperature and atmos-
pheric pressure. \'ibration conditions coupled with strict weight require-
ments result in many additional ])roblems from the equipment designer's
standpoint.
]*'igures 1, 2, and 3 indicate tyi)ical radar equipments employed on the
ground, sea, and air, some component designs of which will be reviewed in
later sections of this paper.
1.2 Tl:c !•' unction of the Radar Receiver
The basic function of a radar receiver is to translate and present the in-
formation received at the radar location in a desirable form. This requires
that the receiver contain provisions to enable:
A. Conversion of the received signals, originally of a microwave fre-
quency character, to signals in a frequency region more convenient io
utilize.
li. .Amplification of the extremely low energy signals as received to
ami)litu(les useful to the observer.
(". ("orrclation of all other a\ailal)lc pertinent data with the received
microwave signals to allow determination of the complete coor-
dinates and other desired characteristics of the target under observa-
tion.
THE RADAR RECEIVER
697
Fig. 1.— Radar Ec|ui|>ment.-"Mark 20. This mobile iiiuipuKiU oiHiatiiig at lOOO nu
is employed for searchlight control purposes.
698
BELL SYSTEM TECHNICAL JOURNAL
D. Presentation of all desired information to the observer in a form cap-
able of immediate analysis and use.
With these functional requirements in mind, it is in order to examine the
character of the information available at the input and that required at the
output terminals of a radar receiver.
1.21 Cliaracterislics of the Radar Receiver Input Signal
In general, the signal radiated by the radar transmitting antenna and
received by the receiving antenna consists of intermittent pulses of energy
SJ ANTENNA BEARING INDICATOR
PLAN POSITION
INDICATOR
RANGE UNIT
RANGE INDICATOR
ANTENNA STEERING WHEEL
Fig. 2. — SJ Suljinarine Radar. Operating position of radar equipment in conning
tower of U. S. Nav\' submarine.
at microwave frequencies. The methods employed in the generation and
propagation of these radar microwave signals have been described else-
where.^' ^ For our purpose, it sulBces to state that microwave frequencies
extending from 700 me to 10,000 mc are commonly employed. Pulse widths
of 0.5 microsecond to 5 microseconds at rey^etition rates extending from 100
j)ps to 2000 i)ps are encountered in modern military radar systems, the
'"The Magnetron as a Generator of Centimeter Waves," J. B. Fisk, H. D. Hagstrum
and P. L. Hartman, Bell System Technical Journal, Vol. XXV, April 1946.
^" Radar Antennas," H. T. Friis and W. D. Lewis, Bell System Techuical Journal,
Vol. XXVI, April 1947.
THE RADAR RECEIVER
699
INDICATORS
iNDiCAFOR AMPLIFIERS
J : i JUNCTION
\~^ |r« BOX
SELF-RELEASE
PLUG
TRANSMITTER
Fig. 3— Components of Lightweight AN/APS-4. Airborne search and interception
radar equipment.
700 BELI. SYSTEM TECH. \ UAL JOIR.XA/.
exact choice of these parameters being dictated })y the specific a])plication
of each type of equipment.
It has been customary to employ a common antenna system for both the
transmitting and receiving functions, the necessary protection of the sensi-
tive receiver input circuits from the high power transmitted pulse being
furnished by a TR switch or tube.^ The gas discharge TR switch assembly
attenuates the energy fed to the input terminals of the receiver for the
duration of ttie microwave outgoing pulse. At the time of decay of this
transmitting pulse, the TR switch is arranged to offer low attenuation be-
tween the antenna and the receiver to any received signal.
The received microwave signal will be found to fluctuate in amplitude
between extremely wide limits. This amplitude characteristic of a re-
ceived radar signal is affected by the size and composition of the target,
the power of the radiated outgoing pulse, the distance or range to the target,
and miscellaneous propagation effects. Military requirements necessitate
designing the radar receiver to perform successfully with the minimum re-
ceived signal, while not unduly compromising this performance when sig-
nals of relatively high energy content are encountered.
Other characteristics of the transmitted microwave signal, such as pulse
shape and repetition rate, are chosen to enable maximum {)erformance to
be attained for the specific application to be covered. The proper treat-
ment of these miscellaneous characteristics of the radar signal is of basic
concern to the receiver designer.
The primary basic information which can be derived from the charac-
teristics of the received radar signal itself consists of data concerning the
range to the target under observation. This range data is made available
by a measurement of the elapsed time between the departure of the out-
going pulse and its return after reflection from the target and consideration
of the velocity of electromagnetic wave propagation.
To determine the complete coordinates of a radar target, correlation of
the range information, as determined above, and the direction of radiation
from the antenna is necessary. Signals containing information as to the
instantaneous attitude of the antenna with respect to chosen reference axes
are, therefore, to be considered as essential inputs to the radar receiver.
Though the natural coordinate system of radar is of a polar form, many
specific applications of radar systems require conversion of this information
into other forms more convenient of use. I-'or example, while in gun-])oint-
ing radar applications, it is desirable to present the final information in a
j)olar coordinate form, corresponding to the aiming axes of the guns in many
airborne radar bombing systems, it is necessary that the presentation be
'"'I'lic (ias- Discharge Transmit- Receive Switcli," A. L. .Saimiel, J. W . (lark and \\ .
W. .Vluniford, Bell System Tcciinicol Journal, Vol. X.W, Jaiuiar> 1*M6.
THE RADAR RECEIVER 701
made in terms of rectangular or other convenient coordinate systems re-
ferred to the ground itself. In certain applications, the characteristics of
the display or presentation device requires that coordinate conversion func-
tions be included within the radar receiver. The conversion and proper
presentation of all radar system coordinate information will, therefore, be
considered a necessary function of the radar receiver.
Additional forms of radar receiver input signals encountered are those
primarily associated with the specific application of the radar system.
Reference coordinate axes data obtained from compasses or gyroscopes are
among the most common of these. Fn gun-fire control and bombing appli-
cations a considerable quantity of computed data must be accepted by the
receiver. These data may include predicted quantities which must be
jiresented in addition to the usual received present-time radar information.
In the case of airborne radar systems, provision must be included to properly
display navigational beacon and identification signals. The beacon is
operated by the radar transmitter in the aircraft and returns a coded signal
at a frequency slightly removed from the normal radar band. All aircraft
radar receivers are required to adequately detect and projierly display this
information. It has also become common practice to require provision
within the radar receiver for display of interrogator-response signals as
employed for military identification purposes. The identification equi])-
ment (IFF) proper is not a radar system component and, therefore, it will
not be considered here.
1.22 Character of the Oulpul of a Radar Receiver
The output of a radar receiver is required to be availaljlc to the observer
in a form which will permit immediate analysis and use of a maximum
of the received information. The consideration of some additional charac-
teristics of the radar information available and the military applications
will furnish a basis for choice of presentation means in the receiver.
Because of the inherent ])ace of the constantly changing tactical mili-
tary scene, the basic requirements imposed upon the radar presentation
device are severe. For example, the use of radar in an aircraft, or on the
ground or sea directed against aircraft involves a process of obtaining in-
formation on targets having relative velocities upward of 500 feet per
second. If the radar system under consideration is being emj)loyed to fur-
nish data for the release of bombs or to direct gunfire, a fraction of a second
represents dimensions comparable to the target size. Such considerations
adequately emphasize the extreme imj)ortance of retaining the "immediacy"
characteristic of the information through the presentation device.
Another factor influencing the design of the presentation components in a
radar receiver is found in consideration of the extreme complexity of the
702 BELL SYSTEM TECHNICAL JOURNAL
received radar information. Tliis conii)lexity is created out of the compara-
tively limited resolution available and from the military need for presenta-
tion of detailed information concerning large areas during small time
intervals.
The effect of limited resolution on the choice of presentation means of a
radar receiver can be appreciated by considering the following. The micro-
wave pulse employed in modern radar systems has a duration in time cor-
responding in linear range dimensions of hundreds of feet, while the beam
width of commonly employed radar antenna systems likewise includes
hundreds of feet of target at useful ranges. Thus, the inherent radio-
frequency "resolution" is limiting to an extent that, while it usually enables
one to determine the coordinates of the centroid of the target, it will not
furnish adequate information as regards the exact size or shape characteris-
tics of the target. The radar response of an area of mihtary interest is a
function of electrical conductivity and other related characteristics, rather
than of the military importance of the target. These factors indicate
strongly that the human observer must be required to supply a certain
function of interpretation, and that the chosen radar presentation means
should be such that this is possible. An effective illustration of this situa-
tion is found in the descriptive term "navigation by constellation" which
was common among the radar operators in the long-range bombing forces.
Here the navigation to and the orientation with respect to the military
objective was often possible only through the interpretation of strong
radar responses from known landmarks. Offset bombing, where the
bombing radar operator carried out his observations on a satisfactory
radar target in the vicinity of the final objective and introduced the known
offset coordinates in the computed release point so as to strike the military
objective, was found to be a successful method of partially overcoming this
basic limitation of World War II radar equipments.
Modern warfare is concerned with rapid movement and extremely large
area operations. The display of continuous information regarding these
large areas is a basic military requirement of the modern radar system.
For specific military applications the radar viewpoint may often be re-
stricted to selected limited areas with increased demands on detail and on
reproduction of changing information during small time intervals.
The above considerations have led to a choice of presentation means for
the military radar system which is of an optical nature and has the essential
characteristics of motion i)ictures. Such a display of complex information,
in general, allows the observer to concentrate his attention at any time on
any desired region of interest, to orient himself with respect to the broad
features of the complete area, and to be cognizant of changes in the scene as
they occur.
THE RADAR RECEIVER 703
The cathode-ray tube has been most universally employed as the display
device in the modern radar system. The incoming electrical information
from the various receiver inputs is here electronically converted into a
visual form capable of being modified over small intervals of time as the
change in the radar scene occurs. Multiple presentations having various
map-scale factors are found in many modern radar systems to enable de-
tailed examination of a small magnified target area, while retaining the
ability to observe the broad area features at will.
The use of radar for the purpose of control of gunfire, release of bombs, or
steering of a vessel or aircraft requires that essentially continuous coordinate
information be transmitted from the radar observer to the device under his
control. This is usually accomplished through the registration of mechan-
ical or projected electronic markers upon the visual radar display, this
process of successive alignment furnishing the required information to the
controlled device. In the case where automatic means of maintaining
coincidence between marker and target are employed it should be observed
that the original selection of the target and the initial coincidence adjust-
ment still remains a matter for interpretation on the part of the human oper-
ator, and here again the visual radar display form is desirable.
The presentation means of a radar receiver has, therefore, been chosen
to allow complete display of the data as quickly as it is received and in a
form most convenient to the understanding of the human operator. In
a broad sense, the radar system output terminal conditions and require-
ments are similar to those encountered for any communication system
i.e. to supply the human observer with all the information available to the
system in a form which will permit maximum usefulness with a minimum of
delay.
1.3 Composition of the Radar Receiver
For convenience in the discussion to follow, the generalized radar receiver
will be partitioned as shown in Figure 4. This particular choice of com-
ponent division is somewhat arbitrary, but is chosen because of its func-
tional simplicity. Mechanical, and in some cases electrical, conditions
encountered in any particular radar system design often indicate physical
arrangements which will dififer considerably from the arrangement illus-
trated.
The converter component of the radar receiver has, as its primary func-
tion, the conversion of the received microwave signal to an intermediate
frequency region where further amplification and discrimination is possible.
The converter consists of a beating oscillator operating in the microwave
frequency region and a nonlinear element, which at the higher radar fre-
quencies consists of a point-contact crystal element. At the lower radar
nn
lUiLL SYSTEM TECIIMCM. J01R.\AL
frequencies it is customary to employ vacuum tube radio-frequency ampli-
t'lers preceding the converter element, and in these cases, similar vacuum
tubes are employed as the nonlinear element. The output frequency of the
converter commonly ranges from 30 mc to 100 mc. Because of the com-
parative difficulties experienced in the transmission of microwave energy
over transmission lines or waveguides as contrasted with the problem at the
lower intermediate frequency region, it is standard practice to locate the
converter in close proximity to the antenna and transmitter portions of the
radar system.
The intermediate frequency (IF) amplifier and associated second detector
unit following the converter in Fig. 4 is required to obtain the necessary
Fig. 4. — Schematic diagram of principal C()mi)niu"iits ol" a military radar recei\-ei
amplilication to the wanted signal, to sui)ply discrimination against un-
wanted signals, and to finally convert the desired signal to a video form for
presentation purposes. The usual gain required of a modern radar inter-
mediate amplifier is of the order of 100 db. The band width of the ll-'
ami)lirier is usually chosen between 1 mc and 10 mc depending on the specific
radar system requirements. The techniques of construction developed for
high gain radar IF amplifiers have resulted in compact component designs
which are complete units in themselves, caj)al)le of being integrated into
various radar systems.
The video amplifier characteristics are dependent to a large degree on
llic parti( iilar type of display system associated with it. Its primary func-
THE RADAR RECEIVER 705
tion is to amplify the video signal output of the second detector, together
with some other signals to be impressed on the cathode-ray tube. In this
lield the principles of design follow closely those developed for television
{)rior to the war. Band widths of the order of 1 mc to 5 mc are commonly
employed and output voltages ranging from 10 volts to 250 volts are re-
quired by the cathode-ray tube. Controllable nonlinear amplitude-gain
characteristics are occasionally included in these video circuits to enhance
the contrast or improve the apparent signal-to-noise performance.
The display device uni\'ersally employed in modern military radar systems
is the cathode-ray tube. The electro-optical response characteristics of this
device may be chosen over quite wide limits to suit the specitic needs of a
particular system. In slow scanning systems, use of long time-decay optical
characteristics of the sensitive screen is found to be advantageous, while in
other cases of high-speed scanning, shorter persistence-type screens are
employed. Screen diameters commonly employed range from 2" to 12"
with a wide variety of color characteristics available to tit the detailed re-
quirement of the radar system. A variety of radar presentation forms are
employed to display most conveniently the received information for the
specitic application at hand. These display types differ i)rimarily in the
manner in which the radar tield coordinates are presented. The deflection
methods employed may be of an electrostatic or magnetic nature with com-
binations of each occasionally encountered.
The sweep circuit components of the radar receiver generate the electrical
time wave forms necessary to display the received data properly. Here
again the television art has supplied a technical background for these
specialized electronic circuits. The great number of display types em-
ployed, requiring varied wave forms, has resulted in the development of a
myriad of specialized sweep circuits whose apparent complexity is the result
of varied combinations of elemental electronic circuits.
The range and time-marker circuits are required to interpret the coor-
dinate data available to the receiver and to prepare this data for display
ill the desired form. Here again, television techniques have been employed
and enlarged upon as the radar systems became more comjilex. A large
number of specialized electronic circuit forms has resulted.
The automatic frequency control (AFC) and automatic gain control
(A(1C) com])onents of a radar receiver have a function not unlike those ele-
ments found in most radio-communication systems. The automatic ad-
justments of the tuning and gain of the radar receiver has contributed greatly
to the successful employment of radar under practical military conditions
and have now become practically indispensable in the art. The circuit
designs follow, in general, the techniques i)reviously employed in radio
communications ihoutrh it will be <)l)ser\-ed that the character of the signal
706 BELL SYSTEM TECHNICAL JOURNAL
and the close association of the receiver and the transmitter in the radar
system introduces some additional new considerations.
The power supply components are shown divided into two types "low
voltage" and "high voltage." This is a convenience which is desirable
because of the different design problems encountered and the quite different
equipment and circuits employed. The low d-c voltages required for the
operation of the electronic components of a radar system vary from 100
volts to 500 volts with both polarities with respect to ground often required.
The cathode- ray tube and "keep alive" circuits of the TR tube require
voltages from 1000 to 7000 volts usually at low current. The voltage
regulation of these power supply components to permit stable radar system
operation under the extreme and variable military operating conditions en-
countered presents a problem of considerable magnitude to the radar re-
ceiver designer.
2. Radar Receiver Component Design
2.1 The Radar Receiver Input Circuit
The conversion of the received microwave radar signal to a lower fre-
quency region where more efficient amplification is possible represents an
extremely important function of the radar receiver. The basic military
requirement of radar, that of providing the greatest possible useful sensi-
tivity, depends fundamentally on the efficient handling of the low-energy
microwave received signal in the input of the radar receiver. The micro-
wave character of the received signal and the extremely low amplitudes en-
countered contribute to the difficulties of radar input circuit design.
The techniques of microwave transmission available to the communica-
tions engineer have but recently been developed and are at this time still
extremely limited in comparison to those methods commonly employed at
the normal radio frequencies. Even short physical connections required
between elements in the microwave region become "electrically long," a
matter of several wavelengths in the usual case, and here the design prob-
lem becomes acute. Network element of inductance, capcitance, and re-
sistance are not available in the form so efficiently employed at the lower
frequencies. Waveguides and cavities at radar frequencies replace them,
and the design of selective frequency networks in the microwave region
becomes a matter of precise arrangement and control of complex mechanical
forms.
Suitable means for vacuum tube ami)lilication at frequencies above 1000
mc have not been available to date; this represents another extremely re-
stricting situation for the radar receiver designer.
THE RADAR RECEIVER 707
2.11 Input Signal Characteristics
The low amplitude of the signal at the input terminals of the radar
receiver requires that this signal be efficiently utilized. The power of the
received signal at this point under somewhat idealized free space assump-
tions is given by the following:
Pr =
16Tr'~D^
Where G = Power gain of common transmitting and receiving antenna
P = Transmitted power of radar system
Ae = Equivalent flat plate area of target (This represents an equiva-
lent flat plate normal to the incident beam which reradiates
all impinging energy)
D = Range to the target.
The two following sample computations are illustrative of the military
radar system conditions:
1. Naval Vessel Search Radar System
Frequency = 3000 mc
Target range = 25 nautical miles
Target-Destroyer (Effective flat plate area = 0.03 sq meter at 3000 mc)
(This value has been determined from a study of target response with
military radar systems)
Power gain of Antenna = 30 db
Transmitter Peak Power = 100 kw
Received Peak Power = 13 X 10"" watts
2. Airborne Search Radar System
Frequency = 10,000 mc
Target Range = 70 nautical miles
Target-Destroyer (Effective flat plate area = 0.2 sq meters
at 10,000 mc)
Power Gain of Antenna = 30 db
Transmitter Peak Power = 100 kw
Received Peak Power = 1 1 X 10^" watts
A reduction of the available received signal power, as computed above,
is to be expected in practice due to multiple path effects and absorption and
refraction effects over the propagation path.
2.12 Input Circuit Noise Considerations
While it is possible to conceive of providing sufficient gain within a radar
receiver to meet any desired sensitivity requirement, this sensitivity caimot
usefully be employed beyond certain limits as determined by the amplitude
70S HELL SYSTKM TF.CIIMCAL JOIRXAL
of the noise disturhaiues al the in])ut teriniiials of the receiver. Xoise
disturbances ma}- be defined as the resultant unwanted interfering electrical
energy at the y)oint under consideration and includes contributions due to
atmospheric disturbances, unwanted radiation from adjacent electrical
efjuipment, microi)honic disturbances, and noise due to vacuum tubes and
thermal agitation. At microwave frequencies we are usuallv concerned
only with the thermal agitation and tube noise rontribulion. .Atmospheric
disturbances at radar frequencies are negligible and microphonic and elec-
trical interferences from adjacent electrical equipment can, bv proper and
sufficient engineering, be reduced to any desired level.
It has been shown- ■'' that the thermal noise (Johnson noise) voltage which
appears at the input terminals of a radio or radar recei^'er is determined bv
the value of the resistance component of the generator imj^edance at this
point. For ma.ximum transfer of signal power the load termination is re-
quired to be equal to the internal impedance of the generator and for this
condition the total thermal noise power delivered to the load is given by
Ps = KTB (watts)
where:
A' = Boltzman's constant = 1.38 X !()"-•' Joule/degree abs.
T = Absolute temperature in degrees
B = Bandwidth under consideration in cycles i)er second; and the signal-
to-noise ratio is given by
d" "" IF^rii ^'^ numeric)
where Ps = ma.ximum available signal jjower.
If the signal generator referred to is followed by any 4-terminal network
representative of a converter element, an amplifier, or a passive network,
the effective signal-to-noise ratio at the output terminals of the network will
be modified. To obtain a measure of this ef^"ect we may assign a figure of
merit, /•', to the network called the "noise figure" of the network and deline
this as the ratio of the available signal-to-noise ratio at the signal generator
terminals to the a\ailable signal-to-noise ratio al the output terminal of
the network.
'his may be written as:
'"The .Misolutf Sinsitivilx of Radio Rcici\HTs," 1), (). Ndiih. A'. ( '. .1. Rcviri,'. \'ol.
\1. January 1^42.
^"Xoisc Fif^urt-s of Radio RirriviTS." II. T. I'liis. I'rm-. I. K. E.. X'ol. M. Xo. 7. Jiii\
1944.
THE RADAR RECEIVER 709
where:
p ,
G = ~ , by definition the "gain" of the network.
or we may write:
Fj,u = FKTBG watts.
It is common practice to express F in decibels given by the relation 10 \ogu^F.
For the case of two generalized networks in tandem following the signal
generator and having the same effective bandwidth we may similarly write:
P„„,., = FaG,Gt.KTB + {Fo - DG^KTB
= U\, + tt^jG^G,, KTB watts.
where the subscripts refer to networks a and b.
The effective noise hgure of such a system is given by:
/.' - (f + ^''" ~ ^
' system — I ' « T , ,
This expression indicates the im{)ortance of the gain of the lirst network in
the over-all system noise performance.
As an illustration, a noise figure of 11 db and a loss of 6 db may be con-
sidered as acceptable performance for a typical silicon crystal converter
operating at M)i)() mc. If the following input circuit of the IF amplifier
has an effective noise performance represented by a noise figure of vSdb , the
over-all system noise figure will be found to be 13 db. The reduction in sys-
tem performance due to the noise contribution of the input stage of the
IF am])lifier here is ai)proximately 2 db. If the system performance must
be improved by increasing the power of radiation, the importance of this
secondary noise contribution is apparent.
A comparison of the noise figures of a point-contact crystal converter
element and the ()L-2C4() Lighthouse vacuum tube used as an amplifier
and as a converter element is given in Fig. 5. At frequencies below lOOO mc
there is a definite advantage in employing a \acuum tube as a radio-
frequency (Rl*) amplifier preceding the nonlinear clement.
2.13 1000 MC Radio-Frequency Amplifier Design
In the design of military radar systems for the lOOO mc operating range
the GL-2C4() vacuum tube has been employed rather universally in con-
verter circuits.'' The essential design features of this special purpose triode
""The Lighthouse Tul)e," IC. D. McArthur and IC. F. l^cterson, Rroc. of Natiotuil
Electronics Conference, Vol. 1, 1044.
710
BELL SYSTEM TECHNICAL JOURNAL
are illustrated in Fig. 6. The basic advantage of the GL-2C40 tube for use
at high frequencies is found in its construction, whereby the tube elements
are arranged to form an integral portion of the external circuit with a mini-
mum of mechanical disturbance. At these frequencies external coupling
circuits of the transmission line type are usually employed. This tube
when operated at an anode potential of 250 volts has a mutual conductance
of approximately 6000 micromhos and an amplification factor of vS5. It has
been customary to employ one or two stages of RF ampliiication associated
50
FREQUENCY IN MEGACYCLES PER SECOND
100 200 300 500 1000 2000 3000 5000
LIGHTHOUSE
TUBE AMPLIFIER
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CRYSTAL CONVERTER
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\
LIGHTHOUSE^
TUBE CONVERTE
R ^^
V
\
"V.
'x
\
\
1
—I —
\
1
500 300 200 150 100 80 60 40 30 20 15
WAVELENGTH IN CENTIMETERS
Fig. 5. — Comparison of noise figures for point-contact silicon crystal rectifier and
GL2C40 vacuum tube.
with the nonlinear element and a beating oscillator using this same tube.
The reduced performance of the GL-2C40 tube as a converter element as
indicated in Fig. 5 is not of imj)ortance, if sufficient gain is provided prior to
the actual conversion process, and the ability of this vacuum tube to operate
at higher levels is a positive advantage.
The electrical circuit design techniques employed in this frequency region
are based on the use of transmission line elements in place of the more or
less orthodox luni])ed clement configurations at the lower frequencies. The
difficulties experienced in i)racti(al designs of radar converters of this type
revolve about successfully terminating and satisfactorily isolating each stage
so that confusing and inefficient interaction effects are minimized.
THE RADAR RECEIVER
711
The simplified schematic of a typical radio-frequency amplifier which
employs a GL-2C40 vacuum tube operating at approximately 1000 mc is
given in Fig. 7, together with an outline of the mechanical arrangement of
the tuning elements. This amplifier is of the tuned-grid, tuned-plate type
ANODE CONNECTION
GRID
CONNECTION
ACTUAL SIZE
ENLARGED SECTION
Fig. 6. — Constructional Features of the GL-2C40 (Lighthouse) vacuum tube.
of circuit where the input and output circuits consist of coaxial transmission
line elements tuned by sliding plunger or ring elements, and is so constructed
to include the CiL-2C4() tube as an integral part of the tuning structure.
The input coupling to the amplifier is achieved by means of a probe extend-
"12
HEI.I. SYSTEM I FA II MCA I. .101 R.\.\L
ing into the input or grid cavity and is i)r()i)ortioned for operating out of a
5()-ohm im])edance cable connection to tiie radar antenna system. The
output coupHng is obtained from a coupling loop located in the region be-
tween the grid and i)late concentric sleeves. The gain of this RF amplifier
design is 12 db when operating at a frequency of approximately lOOO mc
and the noise tigure of this component is 14 db.
2.14 The Radar duivcrter
The basic converter is illustrated in l^'ig. 8 and may be defined as a device
which has two input pairs and one output pair of terminals and is charac-
OUTPUT
^'i^^ 7.- l()0()-mc Radio Frc(|iR'nc\ Aniplilicr. Siniplilicd schematic (iia,i,nam.
tcrizc'l further by the property of (leli\ering an outjiul signal, which, in
terms of amplitude of one of the in])ut signals, is essentially linear but
which has an output frequency related to the sum or difference of the two
input frequencies. This frequency conversion is obtained by the use of an
element which has a nonlinear Noltage-current relationship and upon which
is impressed the two in[)ut signals. The desired sum or difference frequency
signal is then selected and utilized as the wanted signal output.
The basic configuration of the coinerter shown in l'"ig. 8 employs a non-
linear element with network coupling means to ])rovide efBcient transfer ot
signal power into and out of the component and a beating oscillator with its
associated coupling network to suppK' the rc(|uire(l additional input tre-
THE RADAR RECEIVER
713
quency. One or more of the networks involved may, of course, be arranged
alternatively in a parallel contiguration if desired. The nonlinear element
may be a vacuum tube with properly chosen operating conditions or a point-
contact silicon crystal rectifier. The beating oscillator in the modern
military radar converter employs special types of vacuum tubes such as the
GL-2C49 type previously mentioned or the single-cavity velocity modulated
types'^ at the higher radar frequencies.
\ typical voltage-current characteristic of a point-contact silicon rectifier
is shown in Fig. 9. This nonlinear characteristic may be expressed mathe-
matically as a power series with coefficients whose amplitude decreases
quite rapidly with the order of the associated term. If two sinusoidal
voltages represented by frequency /i and f-2 are simultaneously impressed
NON- LINEAR
IMPEDANCE
SIGNAL
INPUT
INPUT
NETWORK
BEATING
OSCILLATOR
NETWORK
OUTPUT
NETWORK
I-F
OUTPUT
BEATING
OSCILLATOR
Fig. 8. — Basic configuration of a radar converter.
upon such a nonlinear element, the resultant current flowing in the output
impedance will produce output signals of the form nfi ± mf-i where // and m
are integers including zero. The effective amplitude of each of these modu-
lation products is related to the magnitude of the power series coefficients.
In the radar converter under consideration j\ may represent the received
signal frequency and fi represents the beat oscillator frequency. The
difference terms of the above e.xpression are of the greatest importance here
because the desired output signal frequency has been chosen at a low value
as compared with the input received radar signal. The selection of the
wanted tirst-order difference term is accomplished by the frequency selec-
tivity characteristics of the converter output coupling network and the
following IF amplitier.
A major design problem encountered in the practical development of
microwave radar converters is one concerning the design of the coupling
networks. The previously referred to limitations of microwave network
^"Reflex Oscillators," J. R. Pierce and \V. G. Shepherd, Bell Svstem Technical Journal,
Vol. XXVT, July 1947.
714
BELL SYSTEM TECHNICAL JOURNAL
techniques and the difficulty in maintaining precise control of each element
over the required band of frequencies are the fundamental problems of the
radar converter designer. The network problem here is concerned with
realizing the desired frequency conversion with a minimum of dissipation of
the useful signal. This implies that coupling of the nonlinear element to
the input and output terminals must be achieved in such a fashion that
matched impedance conditions result for the signals in their respective fre-
quency regions. Since the output signal has been shown to appear as a
9
8
7
6
5
4
3
2
/
/
1
/
/
/
I
1
/
/
0
--H
y
0.1 0.2 0.3
POTENTIAL l^
0.4 0.5
VOLTS
Fig. 9. — Typical voltage-current characteristic of a point-contact silicon crystal
rectifier.
number of energy concentrations extending over a wide frequency range, it
would appear that an efficient design of output coupling network, would
involve optimizing the impedance relationshi})S over this wide-frequency
band. Several factors tend to simplify this problem by restricting the out-
put frequency band which must be considered in a practical radar converter
design. First, the greatest amount of energy is found to be present in the
lower order modulation products because of the rapid reduction in amj)litu(le
of the higher order terms in the equivalent expression for the nonlinear
element. This results in a concentration of energy at the input, output,
and beat oscillator frequencies and their respective second harmonic regions.
Second, the ratio of /i to /a in a microwave radar converter is of necessity
THE RADAR RECEIVER
715
essentially equal to one, this factor also contributing to a narrowing of the
frequency regions of interest to those around the input, the beat oscillator,
and the output values. The third factor, which is of assistance to the con-
verter designer, is the effective separation of the input and output circuits
by the loss of the nonlinear element. Where a vacuum tube is employed as
the nonlinear element, the interaction of these circuits may be made quite
negligible and, in the case of the crystal rectifier the inherent loss of this
element, which may be of the order of 6 db and undesirable from a radar sys-
tem performance standpoint, does simplify the converter network design.
It has been found in practice that it is sufficient to consider the impedance
conditions at the internal terminals of the converter networks in the fre-
quency regions which include /i,/i — fo^fi +/2 and 2/0 — /i.
I-F
OUTPUT
Fig. 10. — 1000-mc Vacuum Tube Radar Converter. .Simplified schematic diagram.
The matter of etficient transfer of power from the local beating oscillator
to the nonlinear element is of secondary importance generally because of the
relatively large amount of power available. This condition is advantageous
allowing mismatch loss in this branch to effectively minimize the unwanted
interaction effects with input and output circuits.
Figure 10 illustrates the schematic and certain constructional features of
a vacuum-tube converter which has an operating frequency of approxi-
mately 1000 mc. The similarity of circuit and mechanical arrangement to
that of the radio-frequency amplifier unit, shown in Fig. 7, is apparent.
In the case of the GL-2C40 converter the two input frequencies are similarly
probe coupled to the grid-cathode circuit, maintaining optimum impedance
conditions for the signal input and locating the beat oscillator probe so as
716
BEI.L SYSTEM TKCHXICAL JOlh'.XAL
to effect an impedance mismatch and thus reduce the interaction between
this circuit and the input circuit. The output coupling network, in this
case operating at 60 mc, consists, of an inductive impedance tuned with a
variable condenser. The outj)ut of this converter is fed into the following
IF amplifier by means of a 75-ohm coaxial transmission line. The loss of
this converter unit, defined as the ratio of the power of the wanted 60-mc
output signal to the signal power impressed ui)on the input terminal, is
t 368A
(800 MC)
IN 21 r
(3000 Mcn
GL446
I (1000 MC)
IN23
(10.000 MC)J
IN21A
(3000 MC)J
IN25 r
(1000 MOi
IN23A
(10.000 MC) I
IN28
(3000 MC)
IN21B
(3000 MC)1
IN26
^24.000)
IN23B
10,000^
A MC y
RECEIVER NOISE FIGURE CALCU-
LATED FOR POOREST ACCEPTABLE
CRYSTAL UNIT UNDER EACH SPECI-
FICATION. THE FOLLOWING I-F
AMPLIFIER IS ASSUMED TO HAVE
A NOISE FIGURE OF 5 DECIBELS.
Fig. 11. — Chronological development of point-contact silicon crystal rectifiers and
associated receiver noise jjerforniance.
approximately 6 db. 'Ilie noise ligurcof a tyi)ical unit of this tyi)e isa])pr().\i-
mately 20 db, this value being of secondary importance since sufticient gain
is provided in the associated radio-frequency amplifier.
At the higher radar frequencies the noise performance characteristics of
silicon crystal-point contact rectifiers are considerably better than that of
vacuum tubes available during the ])ast war years. Figure 11 outlines the
chronological developments of these units with respect to the receiver noise
THE RADAR RECEIVER
717
performance. The performance of two types of vacuum tubes are included
for comjmrison purposes. The development details of the silicon crystal
unit have been discussed elsewhere.*^ This development, together with the
corresponding magnetron and reflex oscillator studies, assumes a status of
major importance in the progress of the development of military radar equip-
ment during World War II, allowing the designers to extend the frequency
of operation upward with increased system performance resulting.
It is sufficient to limit our attention to the frequency region 3000 mc and
above in the consideration of the silicon crystal converter. It is apparent
that the absence of suitable vacuum tube radio-frequency amplifiers in this
region imposes a strict requirement on the efficiency of conversion of this
component.
Fig. 12.— 3000-mc Crystal Converter of an early design.
Historically, one of the fiist of the microwave converters of the silicon
crystal type, operating at 3000 mc and which was employed in military radar
equipment, is shown in Fig. 12 together with an illustration of the mechanical
arrangement^ in Fig. 13. In this early model the silicon crystal element was
mounted in a cartridge type unit, a method which proved quite satisfactory
and was followed for the remainder of the war years. The input tuning
circuit of the converter here illustrated consists of a coaxial transmission
line element having a length of approximately three-quarters of a wave-
length and adjustable to enable fine control of tuning. The nonlinear ele-
s'' Development of Silicon Crystal Rectifiers for Microwave Radar Receivers,"
J. H. bcatf and R. S. Ohl, Bell System Technical Journal, Vol. XXVI, January 1947
'"'Microwave Converters," C. F. Edwards, To be published in a forthcoming issue of
rroceedtngs I.R.E.
718
BELL SYSTEM TECHNICAL JOURNAL
ment is located at the high-impedance end of this transmission line, while
the other end is essentially short-circuited at the input frequency by means
ot a small by-pass condenser element built into the IF output transmission
line. The input line is couf)led into this tuned circuit by means of a variable
coupling probe and the local beat oscillator in turn is similarly coupled to
the input line. The point of coupling of the beat frequency oscillator input
is so arranged as to introduce an effective mismatch and thus provide ade-
quate isolation of this and the input circuit. The output circuit of this con-
verter includes the by-pass condenser previously referred to, together with
the input transformer of the tirst IF amplitier stage. The average loss of a
BY- PASS
CONDENSER
BEATING-
5) OSCILLATOR
INPUT
SIGNAL
INPUT
Fig. 13. — 3000-mc Crystal Converter. Schematic diagram.
3000-mc crystal converter of this type was 6 db and a noise figure of 11 db
was realized.
Another design of crystal converter which was developed during the early
part of the war, and whose basic form was employed in many military radar
equii)ments operating in the region of 1(),(K)() mc, is shown in Fig. 14. Here
the silicon crystal cartridge is positioned within the waveguide with its axis
parallel to the E vector plane and at a point approximately one-cjuarter of a
wavelength from the short circuiting j)iston which terminates this assembly.
The II'' output is obtained from the coaxial line mounting structure shown
which offers a low impedance to the injjut frequency by virtue of its equiva-
lence to a one-half wavelength element with a short circuit at the far end.
THE RADAR RECEIVER
719
The dielectric supporting rings shown form an RF by-pass element minimiz-
ing the loss of input signal power in the IV output network. In this design
the beating oscillator energy is introduced into the waveguide by mounting
the reflex oscillator tubes adjacent to the waveguide in such a fashion that
the output probe is inserted into the waveguide cavity at a point removed by
an odd one-quarter of a wavelength from the face of the TR output iris.
This assures reflection of the local oscillator energy which travels toward
the TR tube directing it toward the crystal element. The degree of coupling
of the reflex oscillator circuit is varied by adjustment of the distance that
the probe is inserted within the waveguide. For airborne applications an
additional oscillator tube is included for beacon reception. This basic form
LOCAL OSCILLATORS
&
p^^^^^^r^^z^^
»
©^^-?
rl
"W'^^v/ vl^^^''^''^^''^'^'''' u ^''^'-'-'''^'-^'-'-^'^'-'-'^
ADJUSTABLE
CRYSTAL PISTON
\ I
cssmsssssussssis
ESSSJ
\\\\\\v\\\\':Tvr
t
mX
4
x^xmsssmsisssssxsiisi^ssssssxsiixs
nX
4
COAXIAL ^^
INPUT FILTER H^
I-F
OUTPUT
Fig. 14. — 10,000-mc Crystal Converter. Schematic diagram.
of radar converter was employed in large numbers in the military airborne
radar field during World War II.
A third type of crystal converter design which was developed in the latter
period of the war is illustrated in Fig. 15. A basic difiference in this struc-
ture is found in the use of a waveguide hybrid junction often referred to as a
"magic tee." This junction has an electrical performance characteristic
at microwaves similar to that of the hybrid coil common to low frequency
communication circuits, i.e., a 4-pair terminal network with an internal
configuration such that power applied to any one pair of terminals will appear
equally at two other pairs of terminals, but will not be available at the re-
maining pair of terminals. Referring to Fig. 15, it should be noted that
power applied to the input waveguide will appear equally in the output
branches but is balanced out of the beat oscillator branch. In a similar
fashion, the beat oscillator power will appear equally in the two output
720
BELL SYSTEM TECIIXICAL JOIRXAL
branches but will not appear in the input waveguide. Certain impedante
matching adjustments are obtained through the use of the matching rods
positioned as shown.
The method of insertion of the crystal cartridge into the waveguide in this
TRANSFORMER
TUNING
CAPACITOR
I-F OUTPUT
BALANCE TO
UNBALANCE
I-F TRANSFORMER
BEATING- OSCILLATOR
INPUT
SIGNAL INPUT
BEATING -OSCILLATOR
MATCHING ROD
BEATING - OSCILLATOR
INPUT
Fig. l.S. — Balanced crystal converter em])l()\iiig wave-guide hybrid junction.
design follows the scheme eniployc.l in the converter just described. How-
ever, here two crystal elements are em])loyed in a balanced form which
necessitates a balanced-to-unbalanced imj)e(lance transformation of the TF
output signal for transmission over a coa.xial line to the input stage of the
II"' ami)litier. The degree of balance obtainable here is necessarily a fiuic-
THE RADAR RECEIVER 721
tion of the similarity of the crystal elements as well as the other elements
shown. In practice, no particular difficulty was experienced in maintenance
of sufficient balance with the improved production control of crystals during
the latter part of the war program.
The advantages of the balanced radar converter here described are two-
fold. First, the signal power dissipation in the beating oscillator branch is
reduced to a minimum and conversely the beating oscillator power fed back
into the antenna and reradiated is reduced. Second, the noise sidebands,
which are associated with the outi)ut frequency of the reflex oscillator, are
reduced effectively in the IF output branch by the degree of balance avail-
able. This local oscillator noise sideband contribution is normally respon-
sible for a definite degradation of the over-all radar receiver noise perform-
ance and hence, the use of a balanced converter will contribute to improved
l)erformance. An additional advantage of the balanced converter is the
minimizing of signal branch impedance variation effects on the beating oscil-
lator load impedance and, therefore, its frequency. The variation of the
antenna impedance during the scanning cycle has in this design little effect
on the tuning of the receiver.
2.15 The Radar Receiver Beat Oscillator
Vov microwave radar receiver purposes the selection of the local beat
oscillator within the converter assembly has been essentially limited to two
types of tubes, both of which were developed during the past war period.
I'\)r radar systems operating at frequencies of 1000 mc and lower, the
(iL-2C4() lighthouse triode has served quite satisfactorily, while at fre-
quencies above 1000 mc, the single-cavity reflex oscillator tube has been
extensively employed. Both of these tube types have adequate power
output and frequency stability characteristics to meet th© normal radar
system requirements.
Some of the desirable characteristics of a beat oscillator tube for use in
military radar receivers can be listed as follows.
A. At least 20 milliwatts of useful output power is desirable. In the case
of the silicon crystal rectifier element, the applied power is limited to
approximately 1 mw; however, the availability of additional oscillator
power enables the converter designer to effectively isolate the beat
oscillator and signal branches by simple inpedance mismatch means.
H. The frequency stability of the ideal beat oscillator tube must be in-
herently good, or convenient means to automatically contrt)l this
frequency must be ])rovided. The maximum allowable radar receiver
frequency variation due to all causes is of the order of 1 mc. In terms
of the beating oscillator frequencies employed in the radar systems of
the past war, this represents an allowable oscillator frequency variation
722
BELL SYSTEM TECHNICAL JOURNAL
from all causes of from 0.1% to .01%. The possible influencing fac-
tors include temperature, atmospheric i)ressure, supply voltage, and
load impedance variations with time, and mechanical shock and
vibration.
C. It is extremely desirable that frequency control of the beat oscillator
be available by remote electrical means. The use of automatic tuning
control of a military radar receiver has proved a necessity during the
past war and, as will be discussed in a later section, the rates of change
of frequency encountered are found to be quite great. This requires
essentially that an electrical control method of continuously adjusting
the beat oscillator frequency be employed to obtain satisfactory re-
ceiver performance.
OUTPUT
Fig. 16. — 1000-nic Radar local beat oscillator employing GI^-2C40 vacuum tul)c —
chematic diagram.
D. It is desirable that the output of the beating oscillator tube be free
from noise. In the usual radar system, if the output frequency of the
beat oscillator is modulated with noise, a reduction in receiver per-
formance will result. As previously discussed, the development of
the balanced converter has provided the converter designer with some
relief from this noise source and in this case this requirement assumes
less importance.
A beat oscillator arrangement utilizing the GL-2C40 lighthouse tube, as
develoj)ed for a military radar system operating in the 1000 mc region, is
indicated in Figure 16. This assembly is quite similar mechanically to the
RF and converter components employed in this frequency region and
described previously. The positive feedback necessary to sustain oscilla-
tion is provided by means of a feedback coupling probe as shown. The
oscillator out])ut is available by means of a pick-up loop inserted into the
THE RADAR RECEIVER 723
plate-grid cavity and thence to the output coaxial Hne. The frequency
stabiUty of this particular design has been quite satisfactory due to the con-
siderable development effort expended on the mechanical design features of
this assembly.
Another radar receiver beating oscillator which has been extensively
employed in military equipment designs operating at 3000 mc and higher
during the past war period is the reflex velocity-modulated oscillator. The
2K25 type, which is a typical single-cavity reflex oscillator for operation in
the 10,000-mc region, is illustrated in Fig. 17. The general principle of
operation is quite straightforward, but the complete theory of operation is
exceedingly complex and has been described in detail elsewhere.^" A beam
of electrons of relatively uniform velocity and density is projected from a
cathode surface toward a cavity space defined in part by two grids and then
toward a repeller electrode beyond. The presence of the oscillatory poten-
tial between the two cavity grids acts to impart initial velocities to the
electron stream in accordance with the cavity radio-frequency potentials
existing at the time of crossing of this gap. Under certain operating condi-
tions between these grids and the repeller electrode, which is maintained
at a negative potential, "bunching" of electrons occur and upon the return
of the electrons to the cavity region under the retarding influence of the
repeller electrode, a certain amount of power may be extracted and utilized
externally. As might be expected from this cycle of events, the optimum
operating conditions necessary for reinforcement of oscillation within the
cavity are related to the time required to return the electrons to the cavity
with reference to the instantaneous oscillatory radio-frequency potential of
the cavity. Thus, numerous modes of oscillation are found in this type of
reflex velocity-modulated oscillator which are related to each other by
integral numbers of periods of the osciUatory frequency and the transit time
of the electrons. In the practical application of the reflex oscillator the
number of useful modes are limited to perhaps two or three, the external
power output available at the additional theoretical modes being reduced
by dissipative conditions within the oscillatory system. The relation of
repeller potential to the appearance of these modes is illustrated in Fig. 18
for the case of a 2K25-type oscillator. It should be observed here that the
frequency at the maximum power output condition for each mode is the
same, i.e. the frequency associated with the cavity dimensions for that series
of modes. The cavity dimensions of the reflex oscillator tube are varied
by the application of external mechanical pressure regulated by an adjustable
tuning strut as shown in Fig. 17. The power for external use is obtained
from a coupling loop located withiji the cavity region and transmitted
through the base by means of a coaxial lead.
" Pierce and Shepherd, Loc. Cit.
724
BEIJ. SYSTEM TECHSKM. JOIRXM.
TUNER BOW
RESONATOR
FLEXIBLE DIAPHRAGM
COUPLING LOOP
ACCELERATING GRID
ACTUAL SIZE
ENLARGED SECTION
I'if,'. 17. Conslructifinal dcl.iils of the I\|h' 2K25 sin>,'li' cavil \ ii-llcx oscillalor for us
al lO.OOO iiK .
THE RADAR RECEIVER
725
The single-cavity velocity-modulated oscillator is admirably suited to
electrical remote control of its oscillation frequency by means of the potential
applied to the repeller element, thus lending itself to automatic frequency
control in a simple manner. Figure 19 indicates the frequency and power
output versus repeller voltage characteristic of a typical 2K25 10,000-mc
reflex oscillator when operating at a normal mode previously shown in
Fig. 18. It is customary to define the electronic tuning range of a reflex
oscillator as that range of frequencies over which the power output exceeds
MOD
E A
/
F IN MEGACYCLES
PER second:
/
-^
r
A
/
\
6 500
/
8770
0
/■
^
r
\
/
'^
\
/
\
54
o
/
\.
9050
Y^\
'
r^
\
\
r
\
ELECTRONIC
TUNING IN '
MC/SEC= 90
A l/K
z^
n\
0 20 40 60 60 100 120 140 160 180 200 220 240 260 280 300 320
NEGATIVE REPELLER VOLTAGE
Fig. 18. — Typical output power modes vs. repeller voltage for a 2K25 type reflex
oscillator.
50% of the maximum power output. The tube whose characteristic is
illustrated in Fig. 19 would accordingly have an electronic tuning range
of 53 mc.
2.16 Typical Radar Input Circuit Designs
An example of a radar receiver input circuit operating in the 1000-mc
frequency range is shown in its final mechanical form in Fig. 20. This
particular design was employed in several military ground radar systems
including the Mark-20 Searchlight Control equipment of Fig. 1, and in modi-
fied form in several naval fire control systems. It consists of two stages
of radio-frequency amplification, a converter stage, and a local beating
726
BELL SYSTEM TECHNICAL JOURNAL
oscillator all of which employ the GL-2C4() lighthouse tube. This particular
equipment example has been developed in the form of a sliding drawer to
assure ease of maintenance but with no sacrifice in rigidity of mechanical
assembly so necessary in this type of equipment. P'urther mechanical
details of the cavity and tube structures of each component are shown in
Fig. 21 and are generally similar to the examples previously described.
These cavity structures are heavily silver-plated to assure good electrical
and thermal conductivity. The problem of adequate conduction of heat
from the GL-2C40 vacuum tube under the severe ambient temperatures
x'"'
,"''
"•^^
J
/
i
f
N
\
\
\
\/
/
/ RELATIVE
POWER OUTPUT
/
/
y
\
/
/
/
/
^
\
\
\
1
1
1
^
\
\
\
1
1
1
1
X
,^'^EQUE^JCY
DEVIATION
\
\
\
t
1
I
/
/
\
\
\
1
1
/
\
\
1/
/
s
\
\
-142
-ie6
-170
-146 -150 -154 -156 -162
D-C REPELLER POTENTIAL IN VOLTi
Fig. 19. — Power output and frequency deviation vs. repeller voltage for 2K25-type re-
flex oscillator.
encountered in military service is extremely important. Positive locking
devices are provided for all adjustments in this mechanical design.
Figure 22 indicates the schematic diagram of this design. It may be
observed that separation filters arc employed extensively to assure negligible
interaction effects between the various stages through the common power
supply connections.
The over-all performance of this {)articular design of a radar receiver
input circuit is as follows:
Input rrcfjuency 1000
Output Intermediate Frequency.
Over-all \Va\m\ Width
Over-all Ciain
Over-all Noise Figure
Injjut Impedance
Output IF Impedance
60 mc
5 mc
18(11)
14 dl)
-SO ohms
75 ohms
THE RADAR RECEIVER
727
The above performance is representative of the Kmits which all manu-
factured units are required to meet and also indicates the field performance
which must be maintained for satisfactory military service. Under labo-
ratory conditions and with a certain compromise of stability and ease of
Fig. 20. — 1000-mc Radar receiver input circuit design, including two stages of radio
frequency amplification, converter, and beat oscillator.
adjustment, improved performance over the figures given here can be
expected.
The design for manufacture of a converter assembly typical of airborne
equipment methods is illustrated in Fig. 23. Here the equipment design
reflects the fundamental requirement of radar equipment for aircraft — that
of providing compact units of sufficient rigidity with a minimum of weight.
This particular converter unit was employed in the AN/APS-4 Airborne
728
BELL SYSTEM TECHNICAL JOURNAL
Search and Interception radar equipment shown in Fig. 3 and operates
within the 10.0('0-mc frequency band. The automatic frequency control
tm^
mmmmmmmwmmmmmmmmmmmimmmiiimmiV''
rig. 21. Mccluiniciil dclail.s ui cavity sUucUircs cmi)loyc(J in 1000 nic radar receiver
input circuit.
circuit and two IF preamplifier stages are also included here as an integral
part of the converter assembly.
The schematic diagram of the converter portion of this assembly is given
in Fig. 24. It should be observed that the basic arrangement of the crystal
TEE RADAR RECEIVER
729
converter is similar to a form described in a previous section. The signal in
this example is introduced into the converter section of the waveguide
through an iris following the TR tube. Two 2K25 reflex oscillator tubes
are mounted upon this waveguide with their output probes extending into
the guide proper. The crystal cartridge is located near the end of the wave-
guide with an adjustable piston terminating the guide. A waveguide to
_iH(-'
R-F
AMPLI-
FIER
jHh'
jH(-
T
_jHH
-ORJir-
Fig. 22. — 1000-mc Radar Receiver Input Circuit. Schematic diagram.
coaxial transforming section employs the crystal as an extension of its axial
element with an adjustable capacitance element for tuning purposes.
It will be observed here that a shutter is included in this design, whereby
the crystal element can be isolated from the signal input end of the wave-
guide. This is a most necessary device on all radar converters employing
crystal converter elements to prevent accidental overload of the crystal.
When the transmitter is not in operation and accordingly the TR tube "keep
alive" is not energized, there is a possibility of subjecting the crystal element
to signals of sufficient amplitude to destroy its characteristic. These over-
730
BELL SYSTEM TECHNICAL JOURNAL
load si<];iials may occur as the result of direct pick-up of an adjacent radar
system or in certain cases atmospheric discharges. To prevent this a
mechanical shutter is inserted into the converter which offers effective
attenuation to any signal input. This shutter is withdrawn only if the
}>roper TR "keep alive" voltage is available.
LOCAL oscillators:
BEACON-1 r-RADAR
waveguide -
(to magnetron)
SCREEN CAP ON
CONVERTER CRYSTAL
JACK
R-T Box/l-»^^i^H»r . ««iH|L. X
.^WKml ,.' ..,. vtmmm "i r-t box
i.r
_..^^.»^_ - -T-RBOX
T-R BOX ^BBfl^^^H^
-> -.,—-■
WAVEGUIDE "~f
(to antenna)
Fig. 23. — 10,000-mc Converter and IF j)reanii)lilk'r assemhl}' lor AN, Al'.S-4 airl)orne
radar ecjuipment.
The performance characteristics of this converter and IF j)rcamp]iher
unit is as follows:
Radar Frequency 10,000 mc
IF Frequency 60 mc
Conversion Loss . .' 6. 5 db
Noise Figure of IF Preamplifier 4. 5 db
Noise Figure of Converter 8. 6 db
Noise Figure of Converter and Preamplifier 1 1 .9 db
IF Hand Width 4 mc
These values are rcpre.sentative of the performance under conditions of
average crystals and average IF iiijiut lubes.
THE RADAR RECEIVER
731
2.2 The Radar Inlermediate Frequency Amplifier
The intermediate frequency (IF) amplifier component of the radar re-
ceiver has as its function the selection and amplification of the received
signal following its conversion to the intermediate frequency. The further
conversion of the IF signal to a video form suitable for use in the display
device is usually included as an integral part of the IF amplifier and will,
therefore, be considered here as an additional function of this component.
2.21 IF Amplifier Requirements — Band Width
The frequency-selectivity characteristic of the radar receiver is deter-
mined effectively by the IF amplifier, since the pieceding converter and
other microwave components offer little or no selectivity. The receiver
FROM
fv<AGNETRON
TUNING
PLUG
TO l-F
AMPLIFIER
Fig. 24. — Schematic diagram of 10,000-mc AN/APS-4 converter unit.
band width required to adequately transmit the wanted signal, while restrict-
ing the noise contributions, is an extremely important factor in the radar
system design and represents, therefore, a basic consideration in the design
of the IF amplifier.
The receiver band width required to adequately transmit the basic re-
ceived pulse information can be determined by a consideration of the fre-
quency-energy characteristics of a radar pulse. The microwave pulse is
created by the sudden application of a high-energy pulse to the magnetron
microwave generator. It has been found sufficient to treat this output
envelope as a simple rectangular pulse in band width computations. Fig-
ure 25 indicates the amplitude, time, and frequency relationships which
exist for an idealized rectangular radar pulse envelope. It may be observed
732
BELL SYSTEM TECHNICAL JOURNAL
that approximately 75% of the energy contained in the idealized original
pulse will be available after transmission through a band-pass structure of
band width dimension of — cycles per second. A further doubling of the
T
2
band width to — cycles per second will increase the available energy by only
T
about 15%. In the case of the practical radar-pulse envelope which usually
is characterized by a trapezoidal form with finite rise and decay intervals,
the energy contained at frequencies outside of a — band is reduced some-
r
what over the idealized case illustrated in Fig. 25.
- — r— *
(a)
FREQUENCY
Fig. 25. — Amplitude-time and energy-frequency relationships for a rectangular
radar pulse envelope.
The signal-to-noise ratio of the radar receiver is dependent on the over-all
receiver band width as indicated in the previous section of this paper. It
is extremely important then to restrict the IF band width as much as pos-
sible, consistent with adequate transmission of the signal itself. The final
band width choice is that value where a further increase would result in a
noise increase greater than the corresponding signal improvement and where
a band width reduction would diminish the signal by a greater increment
than the nofke. The exact determination of the optimum radar receiver
banfl width must be carried out using the linal display device in making
the signal-to-noi.se comparisons. If the raflar system is to be employed for
search pur])oses where echo presence is the primary measure of the perform-
ance of the equipment, the optimum over-all receiver band width has been
found to be of tfie order of cvcles per second resulting in an IF ampli-
T
2
fier band width of — cycles per second to adequately tnmsniit tlie double-
T
THE RADAR RECEIVER 733
sideband signal at this point. In the case of fire-control radar equipments,
where precision range measurement is required, it is desirable to determine
the range by reference to the leading or lagging edge of the radar received
pulse. Here the optimum band width may be considerably greater than
the value indicated above for a simple search system to assure a
minimum rise time of the displayed pulse and an accordingly more precise
determination of the position of the pulse. Additional factors which influ-
ence the radar receiver band width value in a particular system design
involve the frequency stability of the microwave generator and the local
beating oscillator, the sensitivity characteristics of the automatic frequency
control system, the frequency stability characteristics of the IF amplifier
itself, and finally the desire to permit ease of tuning by the radar operator.
The phase distortion introduced by the IF amplifier is of secondary in-
terest in the case of radar systems. The faithful reproduction of all char-
acteristics of the received radar pulse is usually not of extreme importance,
suice with few exceptions the criterion of presence is all important. The
detailed form of the transmission characteristic is likewise not extremely
critical, the usual "rounded" IF transmission characteristic, however, con-
tributing somewhat to ease in tuning the radar receiver.
Gain Characteristics
A consideration of the converted input signal levels encountered and the
video output level desired indicates the IF amplifier gain requirement. The
input signal to the IF amplifier is determined by the type of converter em-
ployed and the presence or not of radio-frequency amplification preceding
the IF section and the absolute noise power resulting. The video level at
the second detector must be maintained at a sufiiciently high level so that
microphonic disturbances within the remaining video components are neg-
ligible and low enough to assure satisfactory detection without serious over-
load effects. Undesired feedback at the IF frequency also tends to limit
the practical gain which can be introduced into the IF amplifier. In the
military radar systems of the past war period the usual maximum gain
associated with the IF amplifier was of the order of 110 db with a maximum
detector output level of approximately 1 volt rms of input circuit noise.
The extreme variation of the level of the desired radar signal makes neces-
sary that provision for a large gain control variation be included in the IF
amplifier design. This gain control often involves automatic, as well as
manual, adjustment and commonly a gain variation range of the order of
80 db is required.
Another consideration which enters into the IF amplifier design and is
associated with gain features, is the behavior of the amplifier in the presence
of extremely large radar signals or enemy "jamming" signals. Optimum
734 BELL SYSTEM TECHNICAL JOURNAL
protection against complete "blocking" of the IF amplifier under these
conditions involves the use of extremely sniall-valued filter or by-pass ele-
ments which are associated with the grid and cathode circuits to present
very short time constants and, accordingly, assure recovery of the amplifier
in fractions of a microsecond after overloading by a large pulse signal. The
gain control method is also chosen to minimize possible overloading by reduc-
ing the gain of the amplifier at a point as far forward in the radar receiver
chain as possible.
Intermediate Midband Frequency
The choice of the intermediate frequency for a radar receiver is essentially
a compromise between the need for reduction of unwanted external inter-
ference and noise, and the desire to realize maximum performance in terms
of gain and noise figure.
The tendency to employ a high IF midband value arises from a considera-
tion of the character of the certain local beating oscillator noise sidebands.
As mentioned previously, the noise sideband output decreases as the fre-
quency interval from the oscillator frequency is increased and it is apparent
that a high value for the IF will be advantageous. In the case of balanced
converters where the oscillator noise sidebands are reduced by the circuit bal-
ance, this factor assumes less importance. A moderately high IF is also
advantageous in the elimination of the IF signal from the final detected video
output which must be accomplished by the use of low-pass filters. The
automatic frequency-control problem is somewhat simplified by the use of
a high IF. The wider separation of the desired tuning point and the image
response and the reduction of interfering TR pulse energy is a positive help
in the performance of the automatic frequency-control circuits and will be
discussed in detail in a later section.
There are also a number of factors which indicate that a low value of
midband IF may be desirable. The noise figure of the input stage of the
IF amplifier is generally better at a low frequency, though this improvement
tends to be quite small for the band widths employed in the military radar
system. A more important advantage of a low IF value is found in the
improvement of the absolute frequency stability of the IF transmission char-
acteristic under the influence of variations of tube and circuit capacitance.
Intermediate midband frequencies of 30 mc and 60 mc have been employed
in the majority of military radar systems developed in the United States
during World War II. In general, 30 mc has been employed quite exten-
sively in naval and ground radar equipments for fire control where radar
pulse widths of the order of one microsecond are used. For airborne radar
ecjuipments with the emphasis on compactness, weight, and the trend toward
higher microwave transmission frequencies, 60 mc has proven to be an
extremely popular intermediate frequency value.
THE RADAR RECEIVER 735
The Second Detector
The final conversion of the IF signal to a video form is accomplished by
simple detection. This process is usually associated with the IF amplifier
because of the relative ease by which the video signal can be transmitted to
the following display circuits, usually physically removed from the IF ampli-
fier, as compared with the transmission problem which exists at the inter-
mediate frequency.
Two types of second detector circuits which are commonly employed in
radar receiver design are the linear diode rectifier and the plate circuit detec-
tor, both similar in form to those employed in prewar television practice.
Several factors must be considered in the choice of the second detector
operating characteristic. Linear detection of the signal is desirable from
the standpoint of realizing the greatest possible visibility of weak radar
signals in the presence of noise of comparable amplitude. In the case of
lobing radar signals where bearing determinations are made by comparison
of the return signal amplitude for two bearing conditions of the antenna
radiation, the characteristic of the second detector enters to affect the sensi-
tivity of azimuth response in two ways. The lobing sensitivity is increased
by the use of a square law detector, however, the presence of a "jamming"
signal of the CW type which may be located off axis can introduce a false
bearing indication, which is not present when a purely linear detector is
employed. In general, the linear detector has been employed in most radar
systems developed during the past war period.
The detailed circuit design of the IF amplifier can be conveniently sepa-
rated into three quite distinct parts. These include the input circuit design,
the interstage arrangement, and the design of the second detector circuit.
2.22 IF Amplifier Input Circuit Design
The primary consideration in the IF amplifier input circuit design is the
effect of this stage on the over-all receiver noise figure. As indicated pre-
viously, this over-all receiver noise figure is dependent on the performance
characteristics of the first or input IF amplifier stage to a degree dependent
on the loss of the converter stage preceding it. In the military radar field,
employing microwave frequencies of 3000 mc and greater, the crystal con-
verter is universally employed and the over-all noise performance is deter-
mined largely by the IF input stage. The use of high-gain pentodes in the
IF amplifier assures that noise contributions from the following stages
are negligible.
Figure 26 represents an equivalent input circuit of the IF amplifier con-
venient for discussion of the noise performance of this circuit. Here the
noise contribution, exclusive of the signal source, is observed to be composed
j of two sources, one due to shot noise of the first IF stage referred to the grid
736
BELL SYSTEM TECHNICAL JOURNAL
circuit and a second noise source related to active grid loading effects. ^^
The optimum IF amplifier input circuit design involves primarily the
selection of impedance transformation means which results in the maximum
over-all signal-to-noise performance for the radar receiver. It can be shown
that this optimum value of impedance transformation is not that value which
\
K
1
<Rs
1 \/t
^^tx
\\
9
\
Mvs
|f^g
LL -:-:-: -^--
"^
1 6AK5 ^
pn
V
M
It
\
^
T
—J
1
/
/
7
^
;^
\
Rg =
/
7^
6
5
\
S>
10,000 (100 MC)y
/
\
^^
—
y
4
3
^
30,00C
)(60 W
c)
r
-
^
/
S
"S^oo (low freq.)
2
\
1
0
1
1
—
1
1
600 600 1000
2000
IN OHMS
10,000
Fig. 26. — Simplified equivalent input circuit of an intermediate frequency amplifier
and noise-figure vs signal source impedance.
maximizes the signal but rather is a condition where a definite signal mis-
match obtains. This may be understood by an inspection of the character-
istics of the two fictitious noise sources illustrated. The shot noise con-
tribution of the vacuum tube employed in the input IF amplifier stage is
here represented by the series noise generator Vt . Under conditions of
shorted grid this is the only effective noise contribution, and this procedure
offers a simple method for determination of the magnitude of this ctTcct.
''"Considerations in the Design of a Radar IF Ami)lifier," Andrew L. HopiJcr and
Stewart E. Miller, Proc. /. R. £., November 1947.
THE RADAR RECEIVER 737
The additional noise contribution under the condition where an impedance
is placed in the grid circuit is shown by the noise generator V g ■ This noise
source is related to the active grid loading. The resistance Rq represents
this effective loading which is due to transit time and tube lead inductance
effects and, therefore, has a value which is associated with the frequency of
operation and the particular design of tube employed. At very low^ fre-
quencies, i?G — ^ 00, the shot noise effects are entirely controlling, and the
optimum IF ampUfier input grid circuit condition for minimum noise figure
is that condition where the impedance of the signal source approaches an
infinite impedance. As the frequency of operation is increased, Rg assumes
a finite decreasing value and the optimum signal source impedance is given
by the relationship illustrated in Fig. 26. It should be noted that the
frequency values associated with the above characteristic are indicative of
the performance of the 6AK5 pentode, one of the most satisfactory of avail-
able tubes for this purpose. It should be observed that the input IF ampli-
fier stage noise figure is independent of the impedance of the signal source
providing that a perfect or lossless transformer can be employed to achieve
the optinum impedance transformation indicated.
The realization of proper impedance transformation characteristic in the
radar IF amplifier input circuit is basically a network problem. The input
grid impedance which can be mamtained over the desired band of frequencies
is limited by the total capacitance present in the grid circuit. For narrow
IF band widths, particularly at the lowei frequencies, the realizable imped-
ance at the grid may be in excess of the active grid loading value and here
the noise performance indicated in Fig. 26 may be realized. However, for
wide IF band widths at normal radar IF midband values the maximum grid
circuit impedance which can be achieved under the limitation of the grid
circuit capacitance will be less than the value associated with the active grid
loading and the noise figure obtained will be somewhat higher than the opti-
mum shown. This restrictive design condition is referred to as "band width
limited." In modern military radar receivers this condition normally
obtains.
To achieve the maximum input coupling network efficiency it is extremely
desirable to minimize all parasitic capacitances under control of the designer
and to employ the most effective network arrangement available. The
double-tuned transformer and autotransformer networks are commonly
employed for this purpose. Any loss in the impedance transforming net-
work will result in further degradation of the noise performance, since this
loss is effective in reducing the signal energy while having no effect on the
tube noise level. The magnitude of the impedance transformation ratio
required in a typical radar design case for optimum noise figure is approxi-
mately seven times, where a crystal converter of the type shown in Fig. 14
738
BELL SYSTEM TECHNICAL JOURNAL
is employed, and the optimum grid impedance at 60 mc as shown in Fig. 26
is approximately 3000 ohms.
The vacuum tube employed in an input stage of an IF amplifier should
have the following characteristics to achieve the optimum performance.
The tube should be capable of providing sufficient gain to assure that the
noise contribution of the following stages is negligible. The input con-
ductance of the selected tube at the intermediate frequency should be as low
as possible indicating generally that physical size should be small to assure
small transit time effects and low lead inductance values. The noise output
of the tube itself should be a minimum, this characteristic being somewhat
controllable by proper emission characteristics. The characteristics of the
most desirable of the vacuum tubes available to the radar receiver designer
during the past war period for IF amplifier purposes is given in Table I.
Table I. — Principal Characteristics of IF Amplifier Tubes
Heater
Power
Watts
Plate
Current
ma
Total
Power
Consump-
tion
Watts
Nominal
Transcon-
ductance
Micromhos
Interelectrode Capaci-
tances Micromicrofarads
Band
Merit
Bo
mc
Type
Control Grid
to Heater,
Cathode,
Screen
Suppressor
Grid
Plate to
Heater,
Cathode,
Screen,
Suppressor
Grid
6AC7
6AG7
6AG5
717A
6AK5
2.84
4.10
1.89
1.10
1.10
10
30
7.2
7.5
7.5
4.7
9.6
3.0
2.3
2.3
9000
11000
5100
4000
5000
11
13
6.5
4.9
3.9
5
7.5
1.8
3.8
2.8
89
85
95
73
117
The development of the 6AK5 pentode was the direct result of tlie necessity
for improved radar IF amplifier performance, and details of this develop-
ment and the performance of this tube have been described elsewhere. ^-
The noise present in a pentode is greater than in a triode, primarily due to
the presence of additional grid structures. Because of this fact, a number
of attempts have been made to employ triodes in the input stage of the IF
amplifier. To prevent oscillation due to large positive feed-back present
through the plate-to-grid interelectrode capacitance, neutralization methods
have been employed. For moderate IF band widths at 60 mc such experi-
mental designs have shown an improvement in noise figure of slightly more
than 1 (lb over the pentode design; however, the criticalness of the neiilrali/a-
tion scheme and the difiiculties in extending the ])erformance to wider II'"
band widths has not allowed this design to be adojiled extensively to
military radar e([ui])ment during the ])ast war period.
'2 "Characteristics of Vacuum TuIjcs for Radar Intermediate Frequencv .Xmijlifiers,"
G. T. Ford, Bell System Teclmical Journal, Vol. XXV, July 1946.
THE RADAR RECEIVER 739
2.23 Interstage Circuit Design
The design of the IF amplifier interstage circuit is basically concerned with
achieving the greatest possible gain per stage while not compromising the
frequency characteristic and stability, or complicating the structure to an ex-
tent where it will be difficult to realize the designed measure of performance
under military operating conditions. The problem can be resolved into the
choice of the vacuum tube and the selection of an interstage coupling
network.
The gain of a pentode operated into a 2-terminal parallel resonant network
which exhibits a maximum impedance Ro at resonance is given by G^Ro
under the restriction Ro < < Rp . If a band width AF is defined as that
band of frequencies where the magnitude of the impedance Z of the network
Ro .
is equal to or greater than ^^ it can be shown that
The gain-band width product of an amplifier stage is a measure of perform-
ance of the stage and serves as a criterion for tube performance if a standard
load impedance as above is adopted. Then the band merit Bg for this con-
dition will be given by
B, = AFG„, Ro = r-^ (On. Ro) = ^.
The band merit Bo of a tube has the dimensions of frequency and may be
interpreted as the frequency at which the voltage gain of the vacuum tube is
at unity with a plate load impedance restricted solely by the sum of the plate
and grid tube capacitances. The ideal IF amplifier vacuum tube will
exhibit a high band merit figure, stable operating characteristics over the
life of the tube, uniform characteristics during the production period, small
size for compact amplifier use and for resulting mechanical rigidity, and
finally low-power consumption, desirable from both the power supply stand-
point and heat dissipation considerations. The actual types of vacuum
tubes generally employed for radar IF amplifiers during the past military
program in order of their availability were the 6AC7, 717A, and finally the
6AK5. The improvement in performance achieved through this succession
of developments can be observed by reference to Table I and the band merit
and power consum])tion figures for these tube types.
Figure 27 illustrates three types of interstage coupling networks which
have been commonly employed in radar IF amplifiers for military purposes.
The synchronous single-tuned network design has an advantage of simplicity
I of construction and permits relatively simple realignment procedures under
740
BELL SYSTEM TECHNICAL JOURNAL
the limited resources of military field conditions. This type of interstage
has been employed quite widely in radar equipments designed during World
War II. To achieve the maximum gain per stage it is necessary to restrict
the total shunt capacitance of the interstage circuit to the unavoidable ele-
ments due to tube and circuit arrangement. Additional capacitance con-
tributions are avoided by the use of a variable inductance element adjustable
through the use of movable magnetic cores to resonate the network to the
desired midband IF value. The shunt resistance element is chosen to
achieve the desired band width.
SYNCHRONOUS
SINGLE TUNED
STAGGERED
SINGLE TUNED
DOUBLE
TUNED
Fig. 27. — Typical IF Amplifier Interstage Circuits. Simplified schematics.
The band width required of each individual interstage circuit of a multi-
stage amplifier of this type to meet an over-all band width requirement of
B cycles is given by
B = AF\/2^"^ - 1
where AF represents the band width of the indivichial interstage network,
defined as the band over which the response is within 3 db of the midband
IF value, and N is the number of interstage circuits employed. As the
individual interstage band width is increased to achieve the desired over-all
value, the gain per stage is reduced and a greater number of stages is required
THE RADAR RECEIVER
741
to meet the over-all gain requirement. The over-all gain of a multistage
amplifier employing synchronous single- tuned interstage networks is
given by
\ IttCtB )
where Ct represents the total interstage shunt capacitance and B is the over-
all band width requirement. Table II presents the individual interstage
band widths and the maximum over-all gain obtainable for multistage IF
amplifiers having a 5 mc over-all band width requirement. Here the use of
the 6AK5 pentode is assumed and the total interstage shunt capacitance is
assumed to be 12 micromicrofarads. It should be observed that unavoidable
misalignment of circuits, aging of tubes, and other such effects all tend to
reduce the idealized computed performance under the practical military
radar conditions and must be considered thoroughly in the design.
Table II. — Interstage Band Width and Over-all Gain of Multistage IF Amplifiers
No. of Amplifier
Stages
Synchronous Single-Tuned Interstage
Double-Tuned Interstage
Interstage
Band Width
mc
Over-all Gain
db
Interstage
Band Width
mc
Over-all Gain
db
1
2
4
6
8
10
5
7.8
11.5
14.3
16.6
18.7
24
37
61
80
96
110
5.0
6.2
7.6
8.6
9.3
9.8
27
47
87
125
162
198
The double-tuned interstage network configuration shown in Fig. 27 is
a somewhat more efficient circuit form than the single-tuned variety just
discussed, and because of this improved performance has been employed to
about the same extent as the synchronous single-tuned type during the past
war. Its performance advantage lies in the basic fact that the transmission
response curve for this structure has a flat-top characteristic resulting in a
slower rate of over-all band width reduction as these circuits are cascaded.
The ability to separate the output plate and the input grid circuit capacitances
and the elimination of the plate-to-grid coupling capacitor with its additional
parasitic capacitance to ground results in a greater gain-per-stage perform-
ance. In this structure the resonant frequency of both primary and second-
ary circuits corresponds to the midband IF and the conditions of equal Q
of primary and secondary circuits and critical coupling are assumed. These
conditions result in a smooth flat-topped response characteristic having
optimum gain performance. The relationship between the individual inter-
742 BELL SYSTFAT TECIIXICAL JOURNAL
stage band width and the o\-cr-all band width of a multistage ami)lifier
employing double-tuned interstage circuits is given by
B = A/^V4(2'/^ - 1)
which is illustrated in Table II togetlier with the corresponding over-all gain
of multistage IF amplifiers of this design.
The third type of interstage network arrangement illustrated in Fig. 27
represents a method employed to realize improved performance of the single-
tuned interstage network type by resonating alternate interstages at fre-
quencies above and below the desired midband IF value. This stagger-tuned
interstage design permits greater gain per stage together with an increased
over-all band width for each pair of amplilier stages over that obtained with
the synchronous single-tuned design. In the case of IF amplifiers having
six or more stages a variation of this stagger-tuned method can be employed
where three successive interstages are considered as a design unit and the
individual interstage resonances are adjusted below, above, and centered at
the midband IF respectively. To afiford a measure of the improved perform-
ance of a stagger-tuned IF amplifier we may consider the relative performance
of a 6-stage IF amplifier of 5 mc over-all band width employing the 6AK5
vacuum tube. An individual interstage band w'idth of 7.2 mc and an over-
all gain of 116 db will result from the use of stagger-tuned interstage circuits
while reference to Table II indicates the synchronous single-tuned design
would have an over-all gain of only 80 db while the double-tuned design
would result in an over-all gain of 125 db. The use of a triple stagger-tuned
design would produce a 6-stage amplifier having approximately the same
gain performance as the double-tuned example above.
The choice of the interstage network configuration to be employed in a
radar IF amplifier must be made considering the circuit efiiciency, the gain-
frequency stability behavior, and with due regard for the ever-present prob-
lem of maintenance of performance under the field conditions of modern
warfare. From a standpoint of circuit ellhciency alone, it has been shown
that the synchronous single-tuned interstage network is decidedly inferior
to the more complex forms, but the obvious simplicity of construction of this
type and the possibility of adjustment and realignment with simple methods
available in the field is a strong recommendation for its adoi)tion in military
radar IF ampliliers. The double-tuned circuit has a considerable advantage
in circuit performance over the case above, but some portion of this increased
elTiciency must be sacrificed since it is impractical to construct this network
with adjustable elements. Here the normal variations in interstage capaci-
tance with replacement of vacuum tubes or aging effects must not be allowed
to reduce the over-all amplifier performance below the design limit. The
solution to this problem must be achieved by design of each interstage circuit
THE RADAR RECEIVER 743
to obtain a somewhat wider band under average tube conditions so that
under subnormal but acceptable tube conditions the over-all performance of
the amplifier is still within requirements. The use of stagger-tuned inter-
stage designs will also result in increased performance over the basic syn-
chronous single-tuned variety, but the maintenance of the over-all perform-
ance of a radar IF amplifier of this type involves relatively complex measure-
ments not always possible under military conditions.
2.24 Second Detector Design
The final conversion of the IF signal to the video form, as required by the
following radar display device, is accomplished by simple detection or recti-
fication. For this purpose either a diode rectifier or a triode operating as
plate circuit detector is usually employed. The second detector design
follows the practices generally developed for television receivers prior to the
war. The diode second detector method has the advantage of simplicity
with no plate supply voltage being required, but the performance of such a
detector is somewhat limited for the frequencies employed in radar systems.
The linearity of rectification of a diode depends on maintaining a high load
impedance relative to the internal impedance of the tube. The external
load impedance is limited, by the presence of tube and parasitic circuit
capacitance and the video band width required, to somewhat less than 1000
ohms for the typical radar case. The internal impedance of the usual avail-
able diode is of the order of several hundred ohms so that the linearity of
detection suffers. The low value of the diode load resistance is also reflected
in the termination of the last IF amplifier stage and affects the gain of
this stage.
The plate circuit detector often employed consists of a triode operated
near plate current cutoff. Here the detector load impedance is effectively
isolated from the plate circuit of the last IF amplifier stage. The linearity
that can be obtained from this type of second detector is essentially the same
as with the usual diode detector.
The polarity of the detected video output signal may be chosen of either
sign by proper circuit arrangement for convenience in the video amplifying
and limiting circuits which follow. It is desirable to reduce the ampHtude
of the IF signal which appears at the output of the second detector to prevent
overload and interference in the video amplifier following. This is accom-
plished commonly by the inclusion of a low-pass filter of simple form in the
output circuit of the second detector.
2.25 Typical Component Designs
In the military radar system design it has been observed that a maximum
video output signal of the order of one volt of noise is desirable as a design
744 BELL SYSTEM TECHNICAL JOURNAL
objective resulting in an over-all gain requirement of the order of 110 db.
If the radar system employs RF amplification, the entire IF amplification
may be provided in one unit. However, in radar systems operating above
1000 mc it has proved advantageous to provide the total IF gain required in
two separate amplifier sections. The IF preamplifier assembly is commonly
designed to be mounted adjacent to the crystal converter located in the
transmitter portion of the radar system and usually consists of two stages
of IF amplification. The main IF amplifier is usually located at some
distance from the preamplifier, commonly associated with the indicator
components of the radar receiver. The main IF amplifier assembly includes
the second detector circuit and occasionally one stage of video amplification
is included.
The IF preamplifier location as described above is quire desirable, elimi-
nating the need for a long transmission line connecting the IF output circuit
of the crystal converter to the input stage of the IF amplifier. As has been
discussed previously, the impedance transformation employed in the IF
input stage is chosen to realize optimum signal-to-noise performance. The
output impedance of the crystal converter is normally of the order of
400 ohms. To assure negligible impedance reflection losses in this circuit,
any connecting cable employed would have to be designed to present a char-
acteristic impedance of this order of magnitude which is inconvenient. The
practical solution as employed in past military radar systems is obtained by
locating the IF preamplifier in close proximity to the converter. The
absence of long leads at this IF input stage is also advantageous in reducing
the interference pickup into this low signal level point. After moderate
amplification the output of the IF preamplifier is usually fed over a 75-ohm
coaxial transmission line to the main IF amplifier.
Figure 28 illustrates the converter and IF preamplifier assembly as em-
ployed on the AN/APQ-13 and AN/APQ-7 airborne radar bombing equip-
ments operating at 10,000 mc. The local oscillator and silicon crystal con-
verter are arranged in a manner similar to a basic type previously described
in this paper. The IF out})ut of the crystal converter is introduced directly
into the preamplifier assembly without exposure. This preamplifier is
arranged to offer two stages of amplification employing the 717A pentode
and using a double-tuned input, interstage, and output network. Figure 2^
indicates the circuit arrangement. The gain of this IF preamplifier is 30 db
and an IF band width of 6 mc is provided. The outi)ut transformer network
is arranged to operate into a 75-ohm coaxial transmission line. It should be
observed that provision is here included to disable the preamplifier by a|)pli-
cation of a positive pulse to the cathode circuit of the second amplifier tube.
This feature reduces the gain of the IF ])reamplilier during the short interval
coincident with the outgoing radar pulse, which assures that the TR tube
THE RADAR RECEIVER
745
"spike" which precedes conduction will be attenuated and, therefore, less
interfering with AFC operation. Further details of this effect will be dis-
cussed in a later section of this paper.
CRYSTAL
CONTAINFR
WITH
CRYSTAL
Fig. 28. — Converter and IF preamplifier assembly for AN/APQ-13 and AN/APQ-7
airborne radar bombing equijiments.
CRYSTAL
CURRENT TEST,
POINT
+300 VOLTS
Fig. 29. — Simplified schematic diagram of IF preamplifier component of Figure 28.
Another example of equipment design of an airborne radar converter and
preamplifier assembly has been illustrated previously in Fig. 23. In this
design the 6AK5 tube is employed and single-tuned interstages and auto-
transformer input and output networks are employed; however, the general
arrangement is quite similar to the design previously described.
The remainder of the IF amplifier gain required is of the order of 80 to
\
746
BELL SYSTEM TECHNICAL JOURNAL
100 db which is usually provided in a main IF amplifier that can be located
conveniently within the receiver indicator portion of the radar system. The
main IF amplifier is commonly designed as a complete shielded unit, required
by the high-gain concentration and desirable from the standpoint of allowing
the same unit to be used in several radar systems. Three IF amplifiers are
shown in Fig. 30 which well illustrates the technological development in this
field during World War II. The first amplifier employing 6AC7 tubes was
developed at the beginning of the war, has an over-all gain of 95 db with an
appro.ximate band width of 2 mc, and employs synchronous single-tuned
Fig. 30. — T\-pical IF amplifier c(|ui[Miient designs for military radar ap])lications.
interstage networks. This design was employed extensively in early mili-
tary radar equipments for land, sea, and air use. It has a total power con-
sumption of 31 watts and weighs 2 pounds 4 ounces. The second amplifier
illustrated was de\-el()ped early in the war primarily for airborne search and
interception radar systems and employs 71 7A pentodes with a double-tuned
and single-tuned interstage combination of networks to produce a gain of
85 (11) with an over-all band width of 4 mc. The total power consumption is
11 watts and the weight here has been reduced to 1 jiound 14 ounces. This
design of li' aniplilier with minor modil'ications was employed on the major-
TEE RADAR RECEIVER
747
ity of airborne bombing radar equipments produced by the Western Electric
Company during World War II. The third IF amplifier design illustrated
was developed somewhat later in the radar program for specific application
to the AN/APS-4 light-weight airborne search and interception radar equip-
ment. This amplifier employs 6AK5 tubes with synchronous single-tuned
interstage coupling networks realizing an over-all gain of 100 db for a band
width of 2 mc. The power consumption here is 14.5 watts and the weight
has been reduced to 9 ounces. This amplifier design has been employed in
a number of airborne radar equipments during the later period of the
past war.
Fig. 31. — IF amplifier design as employed in AN/APS-4 airborne radar equipment.
Figure 31 illustrates further mechanical constructional details of the 6AK5
amplifier described above, and the schematic circuit arrangement is given
in Fig. 32. Five stages of amplification and a second detector of a modified
plate circuit type is included. A positive polarity video output is obtained
from the cathode of the second detector and a single-section video low-pass
filter attenuates the IF signal which appears at the detector output. A
variable gain control voltage is applied to the plate circuits of the first three
stages of the amplifier.
The mechanical arrangement of the components of this amplifier has been
devised with a view to achieving optimum frequency and gain stability.
748
BELL SYSTEM TECHNICAL JOURNAL
^
THE RADAR RECEIVER 749
This equipment design features short and rigid connections and the use of
silvered-mica button-type by-pass elements which are mechanically an-
chored in slots cut into the chassis and soldered in place. The entire unit is
arranged to plug into a multipin socket which supplies all power and receives
the video signal output. The IF signal input is arranged for plug-in connec-
tion at the opposite end of the chassis.
The adjustable inductance elements shown are wound on forms having an
approximate diameter of \" and the small variation in inductance required
to compensate for circuit variations is achieved by the use of tuning screws
as illustrated. These coils are adjusted in manufacture by a comparison
technique employing factory standards of the same form. The completed
amplifier is aligned with mean capacitance tubes and all tuning screws locked
and sealed. Sufficient design margin of gain has been included in this design
to enable meeting the radar system gain requirements with a complete set
of ''low-limit" tubes.
2.3 The Radar Video Amplifier
The video amplifier of the radar receiver, which follows the IF amplifier
and second detector, has as its function the final preparation of the received
and detected signal for display. This process involves amplification of the
signal in its now video form, introduction of additional coordinate signals
and wave forms required for proper display, and often includes modification
of the original amplitude characteristics of the signal itself to enhance the
presentation. The radar video signal is quite similar in many respects to the
television video signal and the circuit technology, therefore, parallels the
television art in many respects. Two characteristics of the radar video
signal result, however, in somewhat less stringent demands on the radar video
amplifier design. The lowest frequency of concern in radar video practice
is related to the repetition rate which rarely is found to be less than 250 pps.,
while it is customary to design television systems to adequately transmit
signals of the order of 1 cps. The requirement of faithful reproduction of the
radar pulse shape is usually of secondary importance; the quality of presence
alone usually sufficing to meet the radar system design objective. In certain
fire-control radar systems, however, the radar system band width must be
adequate to reproduce the received pulse to an exactness of the order main-
tained in standard television practice. In general these somewhat reduced
transmission requirements for radar purposes result in a desirable economy
of circuit elements and power consumption.
2.31 Gain-Frequency Consideralions
The limiting performance of a video amplifier can conveniently be evalu-
ated by a consideration of the transmission problem at the extremities of the
750
BELL SYSTEM TECHNICAL JOURNAL
video band of frequencies. At high frequencies the gain-frequency char-
acteristic of a \'ideo ampliiier stage employing pentodes as illustrated in
Fig. ii is gi\en by
Vi + p/ll
where fu represents the frequency at which the relative gain has been reduced
vS db over the value achieved at the video midband region. This cut-off
frequency relationship is similar in form to that encountered in the radar II''
(a)
10
10^
IC^ 10-^ 10^ 10-^
FREQUENCY IN CYCLES PER SECOND
Fig. 33. — Radar video amplifier gain vs. frequency relationships.
amj)lilier design previously reviewed. In the video amplifier design the
vacuum tube band merit Z?,, again determines the limiting performance of the
amplifier, but since the associated video interstage circuit elements con-
tribute considerably to the total circuit parasitic capacitance by reason of
their large i)hysical size, the effective band merit of a vacuum tube for video
purposes must be considered in terms of the total tube and circuit capaci-
tances. The additional consideration in vacuum tube choice for radar Nideo
amplifiers is one of load capacity, since the output signal voltage required
for indicator use may range upward to several hundred volts.
In a somewhat analogous manner to the relationships discussed in the
THE RADAR RECEIVER 751
design of IF amplifier interstage networks, the video amplifier performance at
high frequencies can be improved by the use of more complex 2- and 4-termi-
nal interstage networks. In military radar systems, how^ever, the added
performance realized is more than offset by the undesirability of the addi-
tional circuit elements required and the more complex maintenance problems
that arise, so that usually only the simple resistance-coupled interstage
design is encountered in radar systems.
At the low-frequency extreme of the video band the gain performance of
the simple video amplifier is related to the product of the series interstage
plate-grid coupling capacitance and the input resistance of the following
grid circuit. The low-frequency cut-off of a video amplifier is again defined
as the frequency where the gain has fallen 3 db over the value at midband
1
frecjuencies and is given by/^ = . The highest value of Rg that can
be employed is related to the grid current characteristics of the vacuum tube
chosen. The use of a large value of Cc is undesirable for two reasons.
First, the interstage shunt parasitic capacitance increases as a physically
larger condenser is employed, which results in poorer high video frequency
performance. Second, the use of large coupling capacitances is undesirable
from the standpoint of increase in susceptibility to blocking or paralysis in
the presence of large signals or enemy jamming. The low-frequency gain
response can be improved by certain proportioning of the plate, screen, and
cathode by-pass elements also resulting in somewhat less possibility of un-
desirable feedback through the common power supply impedance.
In certain military radar system designs, multistage negative feedback
\ideo amplifiers have been employed. Here considerably greater trans-
mission band width may be realized with the simple interstage network
design and an order of improvement in stability results. The feedback
amplifier design in these cases usually involves common cathode feedback
impedance between the first and third stages.
2.32 Gain- Amplitude Considerations
The use of nonlinear gain versus amplitude characteristics in a video
amplifier is a condition peculiar to the radar system and represents a con-
siderable departure from established television practice. The factors that
indicate the desirability of this treatment of the signal involve the behavior
of the amplifier under the extreme range of received radar and jamming
signals encountered and the electro-optical characteristics of certain radar
indicator cathode-ray tubes.
Definite amplitude limiting of the video radar signal is commonly included
in military radar systems. By introducing amplitude limiting at an early
part of the video amplifier, complete amplifier paralysis is avoided when
752 BELL SYSTEM TECHNICAL JOURNAL
extremely large overloading signals are encountered. These signals may
represent strong radar return echoes from objects in the vicinity of the radar
antenna or may be due to enemy jamming signals. The ability of the radar
receiver to recover in a short time following such serious overloading is an
extremely important design consideration. In the case of jamming signals
of a continuous-wave or long-pulse form, their effectiveness can be minimized
by the inclusion of a high-pass network at the input of the video amplifier.
In the case of radar systems which employ intensity modulated displays
of the B, C, and PPI forms the maximum useful brightness that can be
attained is limited by "blooming," a phenomenon in which the cathode-ray
tube spot on the fluorescent screen undergoes a sudden defocussing when the
brightness is increased beyond a critical value. In addition, halation effects
are quite pronounced in these long-persistence cascade layer screens and
contribute to an undesired masking effect when large areas of extreme bright-
ness are encountered. In these cases it is extremely important to limit the
maximum ampUtude of the signal that can be impressed upon the indicator.
The usual radar video amplifier includes an amplitude limiting stage located
early in the amplifier chain whose operating conditions are such as to be
driven to plate current cut-off by signals which exceed a preselected
amplitude.
The range of useful brightness of the cascade screen radar indicator is
severely limited in comparison with the extreme amplitude range of the
received radar signals. As has been discussed, the maximum useful bright-
ness has been seen to be limited by halation and defocussing effects while the
minimum brightness threshold is controlled by halation and ambient viewing
conditions. These limitations of the viewing tube result in a criticalness
of adjustment required of the radar operator to achieve the optimum per-
formance of the radar system. In an effort to improve the general repro-
duction efficiency of the military radar system, several circuit forms have
been devised whereby the amplitude of the indicator signal is related to the
received radar signal in a nonlinear fashion. In certain instances two paral-
lel amplifier paths have been provided where one path operates in a normal
fashion until overload is reached when the second transmission path, de-
signed to properly transmit the higher amplitude signals, becomes effective.
In this manner two relatively linear amplification regions are provided with
a step or amplitude limiting region interposed. Such a nonlinear circuit
arrangement has been referred to as "duo-tone", indicative of the two
reproduction regions employed. Another video nonlinear characteristic
which was employed in a certain airborne radar bombing equipment de-
veloped toward the end of the war was of a logarithmic form realized by a
two-path amplifier design. In general this nonlinear treatment of the
radar signal amplitude has proven capable of reducing the critical adjust-
THE RADAR RECEIVER 753
ment which has in the past been required of the radar indicator and in this
respect has contributed some measure of improved performance under
miUtary operating conditions.
2.33 D-c Restoration Methods
It is pertinent to examine the exact form of the video signal encountered
at the output of the video amplifier as it exists available for indicator use.
The presence of series coupling condensers in the video amplifier has re-
moved the d-c component from the signal as detected and, therefore, the
average value of the signal is zero. In this form the amplitude of the posi-
tive and negative signal excursions are dependent on the form of the signal
itself. If such a signal is impressed upon an indicator of the intensity
modulated type the average brightness of the scene will remain constant,
and the presence of several large amplitude signals will tend to drive any
accompanying weak signals below the useful reproduction threshold and
effectively fail to reproduce them. In the case of an A-type display where
the video signal deflection modulates the beam, the no-signal base line will
assume a position on the screen dependent on the video signal form. For
these applications it is required that the d-c component be restored to the
signal before display. In many other parts of the radar system d-c restora-
tion is required to enable utilizing to the fullest extent the load capabilities
of vacuum tubes under the conditions of varying duty cycles of the im-
pressed wave forms. In sweep circuit design a considerable economy of
power is achieved by operating the amplifier tubes at or near plate current
cut-off for no-signal conditions. Through the medium of d-c restoration,
the signal excursions are confined to positive regions only and then regard-
less of the duty cycle the signal range of amplitude impressed upon the tube
is maintained within desired limits.
Figure 34 illustrates three circuit forms which are employed to "re-
insert" the d-c component of an a-c video signal. The diode restorer com-
monly employed in radar systems is shown in Fig. 34a. The impressed
input wave, assumed to have an average value of zero as shown, will cause
the diode to conduct whenever the signal polarity is negative. During this
diode conducting period the condenser C will be charged rapidly, the full
effective negative peak signal voltage appearing across its terminals. Dur-
ing the following positive excursion of the signal this voltage difference
will be applied effectively in series with the signal. The time constant
RC is chosen large with respect to the period of the signal repetition rate
and thus maintains this additive bias for the remainder of the signal cycle.
Since the effective time constant during the diode conducting period is
extremely small valued, limited only by the conductive internal resistance
of the diode itself, an extremely small negative excursion time will suffice to
754
BELL SYSTEM TECUNTCAL JOURNAL
1 A n n A' r
0
0 -| 1 1 p
B B'
, 1
^H T7-; 1
restore the grid circuit to reference zero potential. This particular d-c
restorer circuit form is referred to as a positive restorer, indicative of the
final polarity of the restored signal. A simj)le reversal of the diode elements
will reverse the polarity of the restored signal.
Figure 34b illustrates the usual radar circuit form of a negative d-c
restorer, where the diode is eliminated, the normal vacuum tube grid circuit
Fig. 34. — Typical D-c Restorer Circuit Forms. Simplified schematic diagrams.
serving here to fulfill this function. Here again an impressed signal form
having an average value of zero is assumed, and a negative polarity re-
stored signal is desired at the grid of the amplitier tube. During periods
of positive excursions of the input signal the vacuum tube grid will conduct
since it is normally operated at zero bias. The series condenser C is ac-
cordingly charged relative to the positive signal peak amplitude and this
value of potential will be additively combined with the signal during nega-
tive excursions in a similar manner to the diode restorer action just de-
scribed.
.A. third form of d-c restoration is known as a clamper or synchronized
d-c restorer and is illustrated in Fig. 34c. Here a diode bridge circuit
THE RADAR RECEIVER 755
is arranged to be normally nonconductive except during the application of
a clamping pulse bias introduced as shown. During this clamping interval,
the grid circuit point of the condenser is re-established to reference poten-
tial by the low impedance of the conducting diode circuit. At the time of
decay of the clamping pulse wave forms the operation of this circuit follows
the principles of the d-c restorer types just described. This circuit has
been employed less extensively than the preceding simple d-c restorer
methods because of the relatively more complex arrangement, but has an
advantage in that the impressed signal may be clamped to a convenient
reference potential at any particular repetitive point in the cycle.
2.34 Typical Radar Receiver Video Amplifier Circuits
The radar receiver video amplifier signal output is required to modulate
the indicator by either position or intensity change. In the A type of
display the video signal is usually impressed upon a pair of vertical de-
flection plates of an electrostatic type of cathode-ray tube to present the
amplitude characteristics of the signal while the range to the target is
displayed as the horizontal coordinate. The maximum video signal am-
plitude required here to deflect the beam satisfactorily is usually of the order
of several hundreds of volts. In the case of B, C, and PPI forms of display
the radar video signal is required to intensity modulate the cathode-ray
tube. Here a maximum video signal amplitude of 50 volts is commonly re-
quired by the radar indicator.
In certain military radar system applications it is desirable to locate the
indicator component at some distance from the main radar receiver and
video amplifier assembUes. This requirement is commonly encountered
in large naval vessel installations where the main radar components may be
located below deck and the indicator mounted as a part of the gun pointing
mechanism. In such cases video amplifier designs employing video trans-
former coupling between the output amplifier stage and a coaxial trans-
mission line and between the line and the indicator circuit proper, have
proven to be entirely successful.
The development of video pulse transformers for radar purposes repre-
sents a considerable advance in the art of communication transformer de-
sign. The greatly improved wide frequency band performance of these
components is the result of the employment of improved magnetic core
materials such as supermalloy having relative permeabilities upward of four
times that available in the permalloy materials, improved techniques of
coil winding distribution, and the use of additional network elements in the
final configuration. Figure 35 illustrates the constructional features of
such a video pulse output transformer which has a band width extending
756
BELL SYSTEM TECHNICAL JOURNAL
from 100 cycles to 7 megacycles as employed in a naval fire-control radar
equipment.
Fig. 35.— Typical designs of radar receiver video frequency transformers.
! +300 VOLTS -
^WV
Fig. 36. — Simplified schematic diagram of video amplifiers as employed in AN/APQ-7
airborne radar equipment.
Figure 36 illustrates a video amplifier circuit arrangement as develojKxl
for the AN/APQ-7 radar bombing equipment during the latter period of
the past war. This system employed two GPI type indicators, one of
which was located at a remote station of the aircraft and included also an
THE RADAR RECEIVER 757
A-type indicator which was employed for certain conveniences in operation
and maintenance of the equipment. This circuit design includes a main
video amplifier for the ground plan indication and a separate amplifier for
the A-type display, both of which are of the negative feedback type. The
limiting amplifier is included as the second stage with negative d-c restora-
tion included in this grid circuit and diode d-c restoration at the grid of the
last stage. To provide sutlficient output signal level with the wide video
band width required it was necessary to employ two 6A(j7 tubes in parallel
in the final output stage of the main video amplifier. The local indicator is
fed from the j)late circuit while the remote navigator's display is fed by
means of a low impedance coaxial transmission line. The video gain control
is essentially an adjustment of the video amplitude limiting level, the actual
signal amplitude being previously adjusted by the IF amplifier gain control.
The over-all gain of the main video amplifier is approximately 32 db with a
band width of approximately 5 mc. The over-all gain of the A-type display
amplifier circuit is approximately 43 db with a useful band width of ap-
proximately 6 mc.
2.4 The Radar Indicator
The radar indicator assumes a position of extreme importance in the
components of the radar receiver. Here with a few specific exceptions all
of the electrical information which has been obtained regarding the area
under observation is finally correlated and converted into an optical display
for use by the radar observer. In the discussion thus far, only the received
radar microwave signal properly selected, amplified, and finally converted
to the video form has been discussed in detail. The preparation of the addi-
tional coordinate and reference data necessary to properly present the com-
plete scene is reviewed in the following sections. In this section the charac-
teristics of the presentation will be reviewed from the standpoint of the
requirements imposed by the various radar applications. The electro-
optical characteristics of the display device are also discussed.
2.41 Classification of Radar Display Types
The number and types of display methods which have been developed for
military radar systems during World War II are the result of the varied
specific applications to which radar has been subjected. These types of
displays are in general related directly to the functional classification of
military radar systems previously discussed. It is of interest to consider
at this time the various types of indicators which have become common in
the radar field. Figure 37 illustrates the basic characteristics of the most
important types of radar presentations.
Basically the three coordinates which determine the position of the target
758
BELL SYSTEM TECHNICAL JOURNAL
AMPLITUDE
THE RADAR RECEIVER 759
in space and which are determinable from a single radar observation loca-
tion are the range to the target, azimuth angle with respect to a chosen
direction axis, and elevation angle as measured from a convenient reference
plane. The classification of radar displays shown in Fig. 37 results from the
fact that the only available and convenient display device has the property
of resolving only two such coordinates simultaneously. The radar display
problem is then one of selecting the most important two coordinates for the
specific radar application and choosing the presentation means accordingly.
For example, if the radar system under consideration is to be employed on a
surface naval vessel against similar naval vessel targets, it follows that
elevation angle radar information is redundant and, therefore, type-A or B
display patterns are quite satisfactory and are in fact the typical presenta-
tions which have been universally employed for surface target fire-control
applications. The basic A-type indicator presents range-only data, but
for fire-control purposes a modified form is often employed with lobe switch-
ing by which accurate training of the radar antenna is possible and bearing
information is thus secondarily obtained.
For airborne radar search and bombing applications the presentation is
concerned with targets, one coordinate of which is known by other than radar
means. Since all targets of interest are in this case located on the ground
plane, the relative location of which is determinable by reference to the al-
timeter and a gyroscopic artificial horizon within the aircraft, it is sufficient
here to present all information as a 2-dimensional map. The presence of
targets and to some extent their composition is observable as an intensity
modulation of the field of view. For this type of application the PPI or its
more exact successor the GPl form of presentation is extensively employed.
For military radar applications where fire-control information is desired
pertaining to targets which are not confined to a definite plane all three de-
terminable coordinates must be known and, therefore, presented to the radar
observer. In certain instances this requirement has been fulfilled by the
use of multiple displays each presenting the information regarding one or
two coordinates and in cases where gun training is accomplished through
separate operators for each coordinate axis, range, bearing, and elevation,
this method has proven entirely satisfactory. During World War II the
fast moving and highly maneuverable aircraft target has required a more
direct and, therefore, faster system of gun pointing. In these cases, the
operator has been provided with a display which electronically duplicates a
sighting telescope and which merely requires the operator to position the
gun (and associated radar antenna) until the target image is centered. To
introduce a measure of range to the target the size or form of the target
"spot" is often varied in accordance with the range data. For defense
against low-level aircraft attacks this admittedly crude range information
760 BELL SYSTEM TECHNICAL JOURNAL
has proven quite satisfactory. Similar presentation methods developed
for airborne aircraft interception rachir equijjments have employed, in
addition, separate instruments to notify the i)ilot or gunner at the time when
the range to the target was proper for firing of the guns. Another variation
in method of obtaining accurate range information simultaneous with the
elevation and azimuth data is through the employment of automatic range
tracking. In this case after identihcation and selection of the target has
been made and the initial coincidence accomplished, the operator is then free
to track in elevation and azimuth with the automatic tracking device con-
tinuing to furnish the changing range data to the gun.
Figure 37 indicates the fact that included in these display forms are varia-
tions which are a function of the deflection coordinates peculiar to the display
device itself. The factors which determine this choice are related to the
required form of the presentation from the standpoint of military use, the
characteristics of the particular display device available and the mechanical
form of the antenna scanning system.
It should be observed that a number of minor variations in the exact
presentation is available to the radar system designer within the general
classification indicated in Fig. 37. As mentioned previously, the A-type
display may be modified to indicate azimuthal pointing errors. In this case,
sometimes referred to as a K-type display, the radar system employs an
antenna capable of producing two beams of radiation, available one at a
time, with azimuthal bearings diiTering by the order of the beam width.
Two signals, each of which is associated with one position of the radiated
beam, are displayed in the basic A-type form with one slightly displaced in
the range coordinate with respect to the other. By "steering" the antenna
until the amplitude response of the desired target appears equal for each
image, the target bearing is determined as the direction line bisecting the
two antenna lobes.
It is often desirable to limit the display to a small area or to a small se-
lected range interval to enable magnification of this particular portion of the
scene. The accurate measurement of range for fire-control purposes can
be accomplished on a conveniently small indicator screen by expanding only
a selected small range inter\al of interest. The loss of information at other
ranges under these conditions is unimportant. In certain airborne apjili-
cations it is desirable to present large area information for navigational
purposes, but at the time of starting the radar bombing attack the area of
interest is limited to a narrow sector extending outward from the plane in
the direction of the attack. Here a selected sector may be expanded with a
])r<)bable increase in accuracy of indixidual largi't identification and final
bombing accuracy.
In the "range only" classification of I'ig. 37 the J-typc of display has an
THE RADAR RECEIVER 761
advantage of expansion of the range information by a factor of approxi-
mately three times for the same size screen. The A and J types are em-
ployed extensively in fire-control radar equipments.
In the range versus azimuth class of radar presentations the B scan his-
torically preceded the other forms shown. Its application was found
originally in airborne radar systems for interception purposes. It was com-
monly employed in conjunction with an auxiliary C-type indicator for
target elevation determination. The B type of display suffers from a dis-
tortion due to the reproduction of polar coordinate information directly
on a rectangular coordinate field. This particular form of distortion is not
of major importance where only a few isolated aircraft targets are to be dis-
played, and in the case of guiding an aircraft to intercept the target the
relative expansion of the azimuth scale at short ranges may be a slight
advantage. When the B type of presentation is employed for navigation
and observation of ground features this inherent field distortion becomes
very objectionable when map comparisons of the radar image are required.
The B type of display has also been employed extensively on narrow
sector rapid scanning naval fire-control radar systems.
The plan position indicator (PPI) type of display was developed to over-
come the objectionable distortion of the B-type display and to afford a
method of presenting a 360° azimuthal pattern when rotating antenna struc-
tures were employed. This form of display essentially replaced the B-type
display for aircraft search radar systems and has been since universally
employed for ground and naval vessel search systems. Here the linear range
trace on the screen is directed outward from the center of the tube, its
radial position being synchronized with the instantaneous bearing of the
scanning antenna. The map presentation is exact for ground and naval
vessel radar locations and for low-flying aircraft radar systems the dis-
tortion is negligible, since the slant radar range to the target at low altitudes
is essentially comparable to the range as measured on the ground plane. As
the altitude of a radar equipped aircraft is increased, the map distortion of
the simple form of PPI display also becomes quite objectionable and several
modified forms of this display can be employed to improve the presentation.
One of these involves delaying the time of start of the linear range sweep by
a time interval corresponding to the propagation time of the radar pulse to
the ground and return. In this manner a simple but desirable improve-
ment in display is realized. As the military requirements during the later
period of the past war became more exacting with the emphasis on high-
altitude radar bombing, the remaining distortion of the delayed PPI presen-
tation was found undesirable, and the necessity for accurate map display
directly beneath the aircraft resulted in the development of the ground plan
indicator (GPI). In this type of display the range trace is deflected as a
762 BELL SYSTEM TECHNICAL JOURNAL
nonlinear function of time; its exact time function being dependent on the
altitude of the aircraft. The altitude information is obtained from the air-
craft altimeter and may be manually or automatically introduced into the
radar receiver to produce the proper form of sweep function. These modi-
lied forms of PPI presentation were employed extensively in the large bomb-
ing through overcast radar program which attained a status of major
importance toward the later portion of World War II.
The elevation versus azimuth classification of display forms is essentially
restricted to fire-control and aircraft interception radar applications. As
previously noted, the C-type display was developed early in the military
radar program and has somewhat the same characteristics as the B scan in
terms of the distortion which results in the display of polar coordinate data
in a rectangular coordinate field without proper mathematical conversion.
In the case of aircraft interception radar applications, this type of display
is quite satisfactory and has been employed quite extensively for this
purpose.
The moving spot (MS) form of radar display is usually associated with
a radar system in which conical scanning or lobing is employed. Here the
source of radiation of the antenna is arranged and rotated so as to provide
a beam whose path describes a cone. If the target is located on the axis of
this cone of radiation the signal response will be essentially constant for all
instantaneous positions of the beam. If the target is positioned to one side
of the cone's axis the received radar signal will be modulated at the fre-
quency of the conical scanning process and the degree of modulation will be
related to the angle between the conical axis and the bearing toward the
target. This modulation information is utilized within the radar receiver
to position an optical image on the face of the indicator screen in accordance
with the direction of the target. In radar systems employing this form of
indication the observer positions his radar antenna, and accordingly the
associated weapon, to center the target image on the indicator. Mechanical
or electronic cross hairs are employed as the reference axis. A measure of
range to target information is often introduced into this form of display
by assigning an arbitrary but distinctive size or shape to the target spot
which can be varied in accordance with the range to the target being ob-
served.
2.42 The Cathode-Ray Tube
The cathode-ray tube is without serious competition as the ideal radar
indicator, i)rimarily because of its unique high-frequency electro-optical
response characteristic. Since the radar presentation requirements are not
unlike those encountered in television practice, it is natural that the cathode-
ray tube development for radar purposes should have progressed along simi.
TEE RADAR RECEIVER
763
lar lines originally established by prewar television. Two general types of
cathode-ray tubes have been commonly employed in the military radar
program, electrostatic and magnetic, these classifications being indicative of
the deflection and focussing method employed.
Electrostatic Deflection Type
A typical form of electrostatic-type of cathode-ray tube suitable for radar
indicator purposes is shown in Fig. 38. In this tube type the electrons
emitted from an indirectly heated cathode surface are initially formed into
a beam by passage through an aperture which serves as a beam density or
ultimate brightness control element. Following this, the electrons proceed
through another aperture which is maintained at a positive potential with
respect to the cathode. This first anode together with the following second
anode forms an electron lens system which focuses the beam on the fluores-
ELECTRON
GUN V,
I
CATHODE _[
HEATER
CONTROL
GRID
SECOND
ANODE
DEFLECTION PLATES:
VERTICAL HORIZONTAL
FLUORESCENT
SCREEN
ji y
^ FIRST
FOCUSING
ANODE
HIGH -VOLTAGE ,^
AUXILIARY ANODE
II j i ANOUb ' I
Fig. 38. — Schematic diagram of an electrostatic type cathode-ray tube.
cent screen surface. The relative potentials of these elements serve to
enable focussing of the beam by electrical means. The deflection of the
beam for scanning purposes is here accomplished by the introduction of an
electric field formed by the application of potential across the deflection
plates shown. Two pairs of plates enable separate horizontal and vertical
deflection to be employed. The plates are formed as shown to enable ob-
taining the maximum deflection per unit of electrical potential applied with-
out interference with the beam under large deflection conditions. The
high-voltage auxiliary anode is provided in certain tube types to further
accelerate the beam after deflection without an appreciable reduction of
deflection sensitivity and results in an image of increased brightness.
To realize optimum performance of an electrostatic-type cathode-ray
tube several precautions must be observed. Serious defocussing of the beam
as it is deflected will result if the average potential of the pair of deflection,
plates is allowed to vary substantially from the value present at the second
anode. To minimize this effect, balanced sweep deflection amplifiers are
764
BELL SYSTEM TECHNICAL JOURNAL
commonly employed in radar receivers employing electrostatic deflection,
and the second anode is maintained at the average potential of these de-
flection plates. Astigmatism, the selective focussing in one direction on the
screen at the expense of focus in the other, results from the mechanical
limitations whereby the electric fields of the two pairs of deflection plates
cannot be made effective at the same point within the tube. Some im-
provement can be realized in this respect by operating the pairs of deflecting
plates at slightly different average potentials.
The limitation in deflection response at high frequencies is a function
of the total deflection circuit capacitance. To eliminate the blocking con-
densers with their considerable parasitic capacitance to ground it is common
radar indicator practice to operate the deflection plates directly from the
COILS
FLUORESCENT
/ SCREEN
CATHODE-.
HEATER
CONTROL GRID
r^s
CENTERING FOCUS DEFLECTION
I
1 r i_
I I
I I
HlGH-VOLTAGE
ANODE
Fig. 39. — Schematic diagram of a magnetic type cathode-ray tube.
plates of the sweep amplifier tubes and then accordingly operate the cathode
of the cathode- ray tube at a high negative potential.
Magnetic Deflection Type
A typical form of a magnetic-type cathode-ray tube is illustrated in Fig.
39. In this tube the electron gun structure comprises a heater, cathode,
control grid, and second or screen grid which, together with the second
anode, usually formed by an aquadag coating within the glass envelope and
which is maintained at a high positive potential, roughly delineates the
beam. The second grid in this case serves to shield the control grid from
the high potential second anode and results in an improved control grid
characteristic. The beam of electrons is focussed through the use of a mag-
netic field located as shown in Fig. 39, and produced by direct current flow-
through a coil or by a |)crmanent magnet structure. The centering of the
beam u{)on the screen under no deflection conditions is accomjilishcd by the
use of a distorting field commonly introduced as a part of the deflection coil
and fo( ussing assembly. The deflection of the beam for a rectangular coor-
THE RADAR RECEIVER 765
dinate display system is here accomplished by magnetic deflection fields
perpendicularly disposed and produced by a pair of deflection coils located
at the junction of the neck and bulb as shown. In the polar form of dis-
play (PPI) usually only one deflection coil is employed for the ])roduction
of the radial sweep, the angular deflection being produced by rotation of this
coil about the neck of the cathode-ray tube in synchronism with the rotation
of the antenna.
It is of interest to compare the relative characteristics of the electrostatic
and the magnetic-type of cathode-ray tubes from the standpoint of their
application to radar indicators. The electrostatic-type of cathode-ray
tube has a distinct advantage of lighter total weight for the tube itself and
the associated deflection circuits required, which in the case of airborne
radar equipment design is an important factor in its favor. In general, it
is desirable to present a large radar display field. For the larger diameter
cathode-ray tubes the magnetic-type tube has an advantage of shorter
over-all length which has proven an important equipment design factor for
airborne radar equipment where the available operating space is severely
limited. The magnetic-type cathode-ray tube requires weighty focussing
and deflecting assemblies and large power supplies to furnish the heavy
deflection current required, but because of the higher anode voltages which
may be employed here, the screen brightness achieved is considerably greater
than that available in the usual electrostatic type of cathode-ray tube. If
the anode potential of an electrostatic type of cathode-ray tube is increased
and hence the screen brightness, the deflection sensitivity is seriously im-
paired and a difticult deflection amplifier design results. From a deflection
point of view, it is possible to achieve somewhat better performance in re-
producing extremely high-speed sweeps by employing an electrostatic-type
of indicator which has somewhat less serious parasitic elements which act
to limit the high-frequency response. The final choice of type of cathode-
ray tube for the radar indicator is dependent on the specific detailed con-
siderations of the system in hand. Xo general and fast rules governing this
decision are evident.
Characteristics of the Fluorescent Screen
The fluorescent screen of the cathode-ray tul>e upon which the final radar
information is converted into the desired visual form consists of a deposit
of certain materials which exhibits fluorescence when bombarded with a
high-velocity electron stream. Phosphorescence, the continued emission
of visible light after bombardment has ceased, is also a property of all of
these screen materials. These screen materials, referred to as "phosphors",
have characteristics dependent on their physical form as well as their chem-
ical composition.
766 BELL SYSTEM TECHNICAL JOURNAL
Two general types of phosphors have been commonly employed in military
radar systems classified according to their phosphorescence characteristics.
The medium persistence class of phosphors exhibit decay times of the order
of milliseconds and are composed of a single layer of Willemite (zinc ortho-
silicate). This type of screen exhibits a green luminous response and is
employed extensively in reproducing high-speed wave forms such as encoun-
tered in high-speed scanning systems, and for general A-type presentation
purposes. The visible hght output decays to the order of 1% of its initial
value in approximately 50 milliseconds after electron excitation ceases.
For photographic purposes other phosphor compositions of the same per-
sistence class whose useful light output has a higher actinic value are com-
monly employed.
The long-persistence phosphors are composed of two layers of screen mate-
rial which combination exhibits sustained phosphorescence, the visible light
output decaying very slowly after cessation of bombardment. The double
layer or cascade screen consists of an innermost layer subject to the direct
influence of the electron stream which is composed of a silver activated zinc
sulphide and a second layer adjacent to the glass envelope which consists of
copper activated zinc cadmium sulphide. The first-named material
fluoresces with an extremely brilliant blue light under bombardment and
exhibits a rather rapid decay characteristic. The second layer is in turn
excited by the blue radiation from the first layer and responds with a yellow
visible emission which persists for a matter of several seconds after excita-
tion ceases. The initial blue flash is appreciably absorbed by the second
phosphor layer, but usually further optical attenuation is required to prevent
eyestrain and degradation of night vision of the observer. This is commonly
provided by the use of an amber optical filter placed over the screen face.
The long-persistence characteristics available in this type of tube have
proven invaluable in military radar systems which feature slow antenna
scanning. In many of these systems the time between successive scans of
the target may be of the order of a second or more and only through the use
of the cascade-type long-persistence tube can the image be retained for this
period of nonexcitation. Another property of the long persistence class of
cathode-ray tube screens which is of advantage for radar purposes is an
accumulative increase in brightness with successive scans of the target.
Since the target image is usually repetitive as regards position on the screen,
the image brightness will increase with successive scans while because of the
random character of noise no such increase in noise image will result, and a
small but evident signal-to-noise improvement obtains. The long-persist-
ence type of screen characteristic is employed in the majority of military
radar indicators of the B, C, and PPI-types.
Another general characteristic of the cathode-ray tube screen which influ-
THE RADAR RECEIVER
767
ences the over-all system performance is the range of useful brightness avail-
able. The extreme variation in the radar response of targets in an area
under observation has been discussed previously. The inability of the
cathode-ray tube screen to convert this extreme range of electrical signals to
a correspondingly large optical brightness range has been a restriction on
the performance of military radar systems. Isolated measurements of the
useful brightness range available in a cascade screen cathode-ray tube indi-
Fig. 40. — Typical magnetic focus and deflection coil designs as developed for military
radar purposes.
cates that it is of the order of 10 to 1. The development of the logarithmic
video amplifier and "duo-tone" is an attempt to improve this situation.
In general this limited useful brightness range of the radar indicator results
in a critical adjustment of the operating region of the indicator tube. In a
practical military radar system operating under wartime conditions the
inclusion of such a critical adjustment always effectively results in a reduc-
tion of performance from the optimum achieved in the laboratory.
2.43 Typical Radar Indicator Component Designs
Figure 40 illustrates a few typical component designs of magnetic deflec-
tion and focus coil structures developed for specific military radar applica-
768 BELL SYSTEM TECHNICAL JOURNAL
tions during the past war. The deflection coils illustrated include both air
and permalloy core structures as used in rectangular and polar types of
displays. Where the radar system involves extremely short time interval
sweep wave forms, the maximum inductance which can be employed in the
deflection coil is limited by the power supply voltages available, and in these
indicator designs the air-core type of deflecting coil is usually employed.
Where the sweep wave form is relatively slower the permalloy core types
have been extensively employed with an effective improvement in deflection
sensitivity. For the PPI form of display a toroidal coil structure has been
devised which contains two distributed windings connected in an opposing
sense. The internal leakage flux of such a structure is essentially uniform
and, therefore, satisfactory for magnetic cathode-ray deflection purposes.
The usual PPI type coil structure is arranged to mount within a large ring
ball bearing to enable rotation around the neck of the tube with provisions
being included on the coil housing for slip rings to afford connection to the
deflection coil proper.
With the increased emphasis on extremely accurate radar presentations,
which developed during the later war years especially in connection with the
radar bombing program, the design and manufacturing tolerances allowable
in connection with the large scale production of these magnetic deflection
coils were severely reduced. Figure 41 illustrates the constructional details
of a deflection coil as employed in the AN/APQ-7 radar bombing equipment.
The presentation in this instance is of the GPI type employing rectangular
coordinate deflection and extremely fast sweep wave forms. This deflection
coil structure employs accurately formed open air-core windings which are
initially adjusted and cemented to concentric phenol plastic cylinder forms.
This design features a vernier rotation adjustment of the horizontal and
vertical pairs of coil assemblies to meet a manufacturing scanning require-
ment of 90° ± 0.5°.
Two examples of focussing and centering structures for magnetic-type
cathode-ray tube radar indicators are included in Figure 40. The focus coil
consists of a simple winding located axially about the neck of the tube with a
shielding magnetic structure containing an annular air-gap which restricts
the external field to a region including the cathode-ray tube electron beam.
This structure is designed to produce a uniform magnetic field distribution
in the complete area of the beam to avoid defocussing effects. In certain
early-design airborne radar cquii)ment applications where the equipment
was subjected to extreme variations in ambient temperature over short
periods of operation some difiiculty in maintaining optimum focus was
experienced. This defocussing, due to the change in coil resistance with
ambient temi)erature and in part to dissipation in the winding proper, is
minimized in the designs shown by the introduction of a varistor element
TEE RADAR RECEIVER
769
mounted in close proximity to the winding whose resistance temperature
characteristic was chosen to compensate for similar characteristics of the
winding proper. The beam centering structure included in the designs
shown employs two pairs of coils arranged upon a closed magnetic core
structure adjacent to the focus winding. The perpendicularly disposed
magnetic fields are produced by direct current in the same fashion as dis-
cussed in connection with the deflection coil assembly.
Figure 42 illustrates the operation of a permanent magnet type of focus-
sing and centering structure developed during the war and which was em-
Fig. 41. — Constructional details — deflection coil design for AN/APQ-7 radar bombing
equipment.
ployed extensively in airborne radar applications. Here a permanent ring
magnet is equipped with a variable internal mechanical magnetic shunt
whereby the focussing magnet field can be varied as shown. The centering
of the beam is accomplished by means of a centering ring located as shown
capable of being mechanically controlled along two perpendicular paths.
Mechanical linkage arrangements provide location of the focus and center-
ing adjustment controls on the indicator panel convenient to the operator.
The permanent magnet type structure has the advantage of maintenance of
proper focus and centering under conditions of extreme variations in ambient
temperature. Similar permanent magnetic type structures have been em-
770
BELL SYSTEM TECHNICAL JOURNAL
ployed to permanently deflect the beam for off-center displays such as the
B-type and certain modified sector scanning PPI forms. Here the ampli-
CATHODE-RAY
TUBE
'^/mac
CENTERING- ^'^ u^^- MAGNETIC
RING * ' SHUNT
FRONT AND BACK
POLE PIECES
(a)
Fig. 42.— Operational diagram of permanent magnet type of focussing and centering
structure for radar indicator.
tude of deflection required is approximately equal to the radius of the screen
and this precludes the use of the usual beam centering structure located near
the gun structure portion of the neck of the tube. To prevent physical
THE RADAR RECEIVER lU
interference by the neck of the tube with the deflected beam such off-center
display deflecting structures are mounted close to the junction of the neck
and bulb of the cathode-ray tube.
As indicated previously the blue-flash characteristic of the excitation of
the first layer of the cascade-type screen is usually reduced in intensity
through the use of an optical filter placed between the screen of the cathode-
ray tube and the observer. These optical screens are usually constructed of
an amber-colored transparent plastic whose particular optical transmission
characteristics are chosen in accordance with the particular phosphor and
speed of scanning employed. It is common practice to engrave general
range and direction reference lines on such screens, and in many cases
variable edge lighting of these engraved screens is employed to enhance the
display.
In certain applications of radar, namely, airborne reconnaissance and
bombing operations, it is desirable to obtain a photographic record of the
display to be used for training, briefing, or damage assessment purposes. It
is customary in these cases to employ a stop-frame moving picture camera
attached to the indicator and exposed periodically as desired. Figure 43
indicates the design of a radar indicator viewing attachment which was
employed in connection with an airborne bombing radar equipment toward
the end of the past war. In this design a partially silvered mirror is located
at a 45° angle with respect to the axis of the cathode-ray tube. An illumi-
nated image of an adjustable course marker located below the mirror is
observed superimposed upon the radar presentation with negligible parallax
distortion. A portion of the light of the radar image is also reflected from
the surface of the partially silvered mirror, and together with the direct
image of the course marker is available for photographic recording by the
camera mounted as shown. Automatic exposure at any preselected time
interval is provided, the exact exposure time being controlled by impulses
derived from the azimuthal scanning mechanism of the radar antenna. The
photographic recording of radar displays has become a matter of prime im-
portance in the modern military operations and has added another consid-
eration for the military radar indicator designer.
Figure 44 indicates an equipment arrangement of a PPI indicator as
employed in the AN/APQ-13 radar bombing system. In this system the
indicator is designed for convenient overhead mounting in the restricted
radar operating space available in modern bombing aircraft. The deflection
coil in this equipment is rotated about the neck of the 5" diameter long-
persistence cathode-ray tube by a geared selsyn motor energized from a
similar selsyn unit mechanically linked to the rotating radar antenna. Per-
manent magnet focussing and centering is employed in this particular indi-
cator. Figure 45 illustrates a somewhat similar packaging arrangement for
772
BELL SYSTEM TECHNICAL JOURNAL
-ILLUMINATED COURSER
AND SCALE
Fig. 43. — ATechanical arrangement of a radar indicator viewing and photographic
attachment for AN/APQ-7 radar bomliing equipment.
SCALE -ILLUMINATION
CONTROL r-CATHODE-RAY TUBE
indicator housing j
(cover removed)
RUBBER
["gasket
VISOR CLAMP
VISOR
junction box
(cover removed)
Fig. 44. — Indicator design for .VN/Al'Q-13 radar homhing eciuipment.
THE RADAR RECEIVER
773
an airborne radar low-altitude bombing equipment as employed extensively
during the past war. Here the display employed is of the B variety utilizing
a 5" long-persistence cathode-ray tube. The focussing and centering
magnetic fields here are produced by coils located as shown. The azimuth
sweep voltage is obtained by means of a potentiometer mechanically geared
to the sector scanning antenna. The permanent off axis deflection of the
zero range line of the B display is here obtained by the use of a permanent
magnet yoke mounted at the junction of the neck and bulb of the cathode-
ray tube.
Figure 46 illustrates a typical form of an A-type indicator as developed
for the SH Naval and Mark 16 mobile fire-control radar equipments. This
unit includes provision for receiver tuning and video limiting adjustment
convenient to the observer. The SH and Mark 16 radar systems employ
Fig. 45. — Mechanical design of AN/APQ-5 type B radar indicator.
both lobing of the antenna for precise azimuth bearing determination and
also continuous rotation of the antenna for general search purposes. The
indicator shown in Fig. 46 is employed when the lobing system is in opera-
tion and features a precision range sweep as well as the full range display.
The ecjuipment design here reflects the severe mechanical requirements
which must be satisfied for electronic equipment which is to be employed on
naval vessels or for trailer mobile ground service.
2.5 The Radar Siveep Circuit
2.51 Function
The sweep circuit components of a radar receiver are required to generate
the specific voltage or current wave forms necessary to properly display the
radar-received information in the desired form. These wave forms must
also be actuated by or related to the various coordinate or computed types
of data furnished to the radar receiver. The actual detailed configuration
of the circuit cmi)loyed for this purpose is dependent on the form of the input
774.
BELL SYSTEM TECHNICAL JOURNAL
Fig. 46. — A-type indicator design for SH Naval radar equipment.
THE RADAR RECEIVER 775
information available and upon the indicator deflection methods to be
employed.
The sweep deflection problems may be separated into two quite distinct
categories dependent on the speeds of operation involved. The slow-speed
deflection systems originate with the required deflection of the beam of the
indicator cathode-ray tube in accordance with the instantaneous position
of the axis of propagation of the antenna structure. Since the antenna
structures with which we are concerned at radar frequencies are of compara-
tively large dimensions, their motional velocities are extremely limited and
accordingly the electrical information describing these slow mechanical
changes contains only low-frequency components. The deflection problem
associated with this information, in general, offers little difficulty to the radar
receiver designer. Commonly employed methods of slow-speed sweep
deflection include the use of potentiometers, selsyn generators or variable
capacitors mechanically linked to the deflection axes of the antenna struc-
ture and associated circuits of relatively simple form whereby the electrical
changes in the characteristics of these devices are more or less directly
impressed upon the proper deflection axis of the indicator. In the case of
the PPI form of display, it is quite common to synchronize the rotation of
the deflection coil about the axis of the cathode- ray tube with the azimuth
bearing of the antenna through the use of selsyn motors or servo mechanisms.
In general, slow-speed sweep deflection problems are associated with bearing
coordinate data only.
The determination of range to the target, on the other hand, requires high-
speed scanning whereby the time interval encompassing the time of propaga-
tion and return of the radar pulse over the selected range interval must be
completely displayed upon the indicator screen. The total time interval
available to deflect the beam for range measurement purposes may extend
from 2500 microseconds which represents a range measurement of approxi-
mately 240 miles down to perhaps 6 microseconds representing an expanded
interval of approximately 1000 yards useful in certain fire-control applica-
tions. Here, with extremely small times available for deflection purposes,
the radar receiver designer is faced with difficult circuit design problems
where the usual negligible parasitic circuit elements now severely restrict
the circuit performance. In the following discussion, therefore, emphasis
will be placed upon the design factors involved in the development of high-
speed radar sweep wave forms.
The radar sweep circuit can be considered as providing the following
functions :
1. Generation of time wave forms.
2. Generation of display sweep wave forms.
3. Amplification of sweep wave forms.
776 BELL SYSTEM TECHNICAL JOURNAL
2.52 The Timing Wave Form Generator
The generation of the timing wave forms consists of the preparation of
specitic voltage wave forms for use in the following sweep generator in ac-
cordance with timing information available at the radar receiver input.
These timing data may consist of a pulse coincident or related to the out-
going radar microwave pulse serving as a reference for range display or, in
the case of direction sweeps, may consist of signals related to the instanta-
neous position of the antenna.
The basic wave forms employed in this connection consist of rectangular
pulses where the duration of the pulse may be controlled to serve as a
measure of time, extremely short-duration pulses useful as time markers, and
various combinations of these. In general, these wave forms are character-
ized by their nonsinusoidal form. The generation of these nonsinusoidal
wave forms is accomplished by a number of specialized electronic circuits,
which though apparently quite complex can be resolved generally into a
combination of relatively simple basic circuit forms.
The Multivibrator
The "trigger" or multivibrator circuit was developed nearly thirty years
ago and provides the fundamental circuits for the sweep circuit designer.
Figure 47a illustrates a simple historical form of a trigger circuit which is
of the Eccles- Jordan type. The essential current-voltage relationship which
characterizes this circuit and all circuits employed for this purpose is a nega-
tive resistance characteristic which exists over a limited portion of the
operating range of the device. In the case of the electronic circuit shown in
Fig. 47a this negative-resistance characteristic is bounded by two stable
limiting conditions. Referring to the trigger circuit of Fig. 47a, the chrono-
logical order of operation can be described as follows: Assume Vi is con-
ducting a somewhat larger current than To so that the ])otential at the plate
of Vi is lower than the corresponding point at I'o due to the voltage drop
across the plate resistor Ri. This condition further implies that the grid
potential of l^ as determined by the connection from the plate of I'l through
the coupling resistor R^ is lower than that at the grid of I'l. Similarly the
grid potential of V\ is at a higher positive potential, due to its connection
with the ])late of ]'•_.. The action is cumulative and results in stablizing the
circuit under the condition where the plate current of 1% is entirely cut off
and the voltage drop across I'l is less than the grid bias voltage Ec.
\{ now a voltage is imi)ressed across the input terminals of eitlier a positive
or negative form, the circuit will be driven away from this stable equilibrium
condition as follows. Assume now that a large ])ositive jnilse be applied
to the circuit shown. The tube I'l which is operating in a conducting con-
THE RADAR RECEIVER
777
dition will not be afifected but the grid of V2 will be raised in potential by the
amplitude of the enabling pulse. The plate current flow in V2 under this
influence will reduce the plate potential of V2 and accordingly will tend to
decrease the positive bias of Vi . The accompanying plate current reduction
of V\ will increase its plate potential and this will result in increasing the
grid potential of Vi through the coupling resistance Rz . Again the cumula-
tive effect will be to abruptly cut off the plate current of Vi and operate V-i .
Thus an abrupt switching of this electronic circuit results when a single
enabling pulse is impressed upon it. The wave form across one of the plate
V , —
r
1^ • 1^
1^
^r^^r^
INPUT
(a)
Fig. 47. — Basic Multivibrator Circuit Forms.
resistances is of a rectangular form whose duration is determined by the time
interval which exists between the two applied excitation pulses.
A basic modification of the fundamental trigger circuit which has been
found most useful in radar sweep circuit is given in Fig. 47b. Here the
original circuit form has been modified so as to permit a complete cycle of
operation upon excitation by a single actuating pulse. The duration of the
cycle is here internally controlled by the arrangement and value of the
circuit elements. This form of multivibrator is characterized by having one
stable equilibrium condition and is known as a "one shot" type.
The chronological order of operation of this circuit type may be considered
as follows: Vi is normally conducting heavily because of the large positive
grid potential impressed u\^on it by the plate sujiply battery and the connec-
778 BELL SYSTEM TECHNICAL JOURNAL
tion through R% . The plate current of Vi flowing through the common
cathode resistor R^ results in a large effective bias applied to Vo which con-
tinues to maintain V2 in a cut-off condition. If a negative pulse of relatively
short duration is impressed upon the grid of Vi this tube will be driven to-
ward cut-off with an attendant increase in the plate potential of Vi . This
positive increase in voltage will be impressed upon the grid of V2 causing V2
to conduct plate current. The resulting decrease in the potential at the
plate of V2 further decreases the grid potential of Vi through the coupling
condenser C2 . This action progresses until Vi is driven beyond plate cur-
rent cut-off and V2 is conducting. This condition remains as long as the
discharge of the condenser C2 through Rs will maintain the grid of Vi at a
net negative potential. When the condenser C2 has discharged sufficiently
to allow the grid of Vi to increase above the cut-off value, Vi will again
conduct and the resultant action will reduce and eventually cut off the plate
current of V2 . The duration of the cycle of operation is here shown to be
dependent on the time constant of the circuit R3 d and may accordingly
be controlled as desired by proper selection of these elements. The return
of the grid of Vi to a very high positive voltage point in the circuit has a
definite advantage which may be considered as follows: A variation of the
grid voltage of V\ required to cut off the plate current will influence the time
duration of the cycle of operation. Here the time rate of change of the
grid voltage has been made extremely large by the choice of the return to the
high-voltage supply. Thus, an order of magnitude increase in the duration
stability of the circuit is achieved.
A further modification of the trigger circuit furnishes the third general
type of multivibrator employed in the radar receiver field. This circuit
form, called the "free-running" type, has the property of presenting two
unstable limiting conditions and accordingly will produce sustained oscilla-
tions of a nonsinusoidal form. Figure 47c illustrates this circuit arrange-
ment. The essential circuit change over that given in Fig. 47b, is seen to be
the elimination of the stable equilibrium condition of Fi by the absence of a
positive potential on the grid of Vi .
In the free-running type of multivibrator shown, the duration of operation
of a particular tube is related to the time of discharge of the coupling con-
denser and the grid resistance associated with the tube. If a different time
constant is chosen for each tube circuit, an unsymmetrical wave form, i.e. —
a pulse- to-no-pulse interval ratio other than one, can be produced. In gen-
eral, the free-running multivibrator is seldom used in this basic form because
of the limited repetition-rate stability of this circuit. It is customary, how-
ever, to trigger this free-running type of multivibrator with short-duration
pulses having a slightly higher repetition-rate than that determined by the
multivibrator circuit constants. In this manner the repetition-rate may be
THE RADAR RECEIVER
779
externally controlled as desired. It is also possible to synchronize this par-
ticular form of circuit at a submultiple of the externally available trigger
repetition frequency.
Pentodes and other now available multi-element vacuum tubes, where the
multivibrator interstage coupling involves additional control elements, are
commonly employed in the modern radar receiver. Wave forms other than
the basic rectangular pulse forms appearing at the plate terminals of the
multivibrator circuit are available at various other points in the circuit and
are often employed in specific applications.
One other basic form of pulse producing electronic circuits is known as
the ''blocking-oscillator" type: two typical examples of which are illustrated
in Fig. 48. Here the positive feedback of energy required to produce the
multivibrator characteristic is realized through the use of a single vacuum
+ B
\smj —
Fig. 48. — Typical Blocking-Oscillator Circuit Forms.
tube and a transformer feedback circuit. This form may be described as an
oscillatory vacuum tube circuit where the grid circuit is so arranged to be
driven negative after one or more cycles of operation. This results in an
intermittent oscillation and the production of nonsinusoidal wave forms
similar to those produced by the general multivibrator circuits previously
described. The basic advantage of the blocking oscillator circuit form is one
of economy of vacuum tubes and attendant power supply reduction.
Typical Timing Wave Circuits
The practical military equipment requirements of World War II with the
emphasis on compactness and low-power consumption has resulted in the
development of a myriad of specialized circuits which reflect the ingenuity
of the electronic circuit designer and the basic flexibility of the modern
vacuum tube. In general, however, these circuit developments are quite
similar operationally to the basic forms here described.
Figure 49 illustrates a typical circuit arrangement of the sweep timing
I portion of a PPI indicator as employed in a naval search radar equipment.
780
BELL SYSTEM TECHNICAL JOURNAL
In this particular radar system the transmitting magnetron is pulsed from a
free-running modulator and, therefore, the controlling timing reference pulse
for sweep purposes must be obtained from the modulator circuit. In many-
other military radar systems it has proven desirable to time both the trans-
mitter modulator and the receiver sweep circuits from a common controllable
repetition frequency source. As shown in Fig. 49, a positive synchronizing
pulse as obtained from the transmitting modulator is delivered to the radar
receiver for range timing reference and here applied to the "clipper" portion
of the timing circuit. It was considered here desirable to clip or limit the
timing pulse to gain freedom from timing instability, due to possible ampli-
tude variations and to eliminate any possible negative excursions of the
timing pulse which might cause faulty operation of the following multi-
vibrator circuit. The multivibrator shown is a modified form of the "one-
STOP PULSE
l+B — !+B
Fig. 49. — Radar Sweep Timing Circuit. Simplified schematic diagram.
shot" type described previously. The grid of W is normally maintained at
a positive potential through its connection to the positive plate supply source
and accordingly Vi is normally cut off.
Upon application of the positive synchronization pulse to Fi and the
resultant lowering of the plate potential of Vi the grid of F3 is driven below
cut-off decreasing the voltage drop across the common cathode resistance
and causing ¥> to conduct. This condition will be maintained until the
coupling condenser has discharged sufficiently to permit V^ to again conduct.
In the circuit here described, however, this controlling time constant has
been selected to be somewhat larger than tlie total period of the sweep rate
and the termination of the sweep timing [julse is accomplished by an external
stoj) pulse applied as shown. This stop pulse is developed in the following
sweep amplifier circuits not liere shown and is controlled directly by the
deflection current. Details of this stop-pulse timing and generation is given
in a later section.
The out|)Ut of this timing circuit shown here then is observed to consist
of a rectangular i)ulse whose leading edge is related to the time of the out-
THE RADAR RECEIVER 781
going radar pulse and whose duration has been controlled by the limits of
deflection desired on the radar indicator tube. This form of sweep circuit
is known as a "start-stop" type and has proven extremely satisfactory as
employed in a number of military radar ecjuipments designed during the
past war period.
2.53 The Sweep Wave Form Generator
The sweep wa\'e form generator is required to generate the specitic voltage
or current time functions required to properly deflect the electron beam of
the cathode-ray display device. The timing of the interval of this sweep
wave form is provided by the timing or synchronizing circuits just described.
In general, it has been required that the range sweep wave form amplitude
be essentially a linear function of time over the range interval under observa-
tion. During the latter portion of the war, certain airborne applications of
radar did require that a specific nonlinear wave form be employed, but the
commonly employed displays (A, B, C, and PPI) are usually operated with
linear range deflection sweep circuits.
The basic method of obtaining a sweep voltage wave form which increases
with time is illustrated in Fig. 50a. In this circuit Fi is normally operat-
ing at little or no bias and, therefore, due to the large voltage drop across the
plate resistor R, the plate potential of Vi is considerably lower than the plate
supply voltage B. If a negative rectangular pulse is applied to the grid of Vi
the tube will be abruptly driven to cut-off and, due to the current flow-
through the condenser C, the plate potential will rise exponentially as indi-
cated to eventually assume the value of the supply voltage B. At the time
of end of the negative driving pulse, I'l will again conduct and the potential
at the plate of Fi will be abruptly reduced as shown.
There are several methods employed in radar sweep circuits to improve
the linearity versus time of the fundamental exponential sweep wave form.
The first of these takes advantage of the fact that the initial rise of the expo-
nential wave form in the limit is a linear function of time. By using only
a small portion of the wave form shown and supplying later ampiilication to
produce the desired deflection, a simple improvement results. This form
of linear sweep generation represents the original and by far the most com-
mon of the types employed in military radar systems during the past war.
Figure 50b illustrates a method of improving the linearity of the sweep
wave form whereby the exponential wave form generated by the basic
condenser charging operation is modified through the use of feedback. As
shown here the asymptotic value of the exponential charging voltage has
been increased by a factor of (ju + 1) and the effective time constant of the
charging circuit has likewise been increased by the same factor. The use.
of an amplifier in the feedback circuit having an effective gain of 50 would
782
BELL SYSTEM TECHNICAL JOURNAL
result in an improvement in linearity comparable with the circuit of Fig. 50a
where the plate supply voltage was increased by the same factor.
I+B
n F'
^lT
(a)
;c E,
I+B
>R
"LT
(b)
LOW-
IMPEDANCE
OUTPUT
VOLTAGE
\
T P ^
^C--R-*^ RC(1+JLL)
\
!
\Ec
1 — ^~
-^v^
" \
/
■ TIME,t— ^
Fig. 50. — Radar sweep generation circuits — ^basic form and modified by negative
feedback to improve linearity.
Fig. 51. — Circuit employed to improve linearity of sweep wave form b}- integration
method.
A furtlier method of improving the linearity of the generated sweep wave
form is illustrated in Fig. 51 where an additional correction voltage is super-
imposed on the exponential sweep wave form. This correction voltage is
THE RADAR RECEIVER
783
derived by integration of the sweep wave form as impressed upon the ele-
ments i?2, C2 and the voltage appearing across C2 is effectively superimposed
upon the output wave form. As employed on an airborne bombing radar
equipment, a circuit similar to that shown in Fig. 51, was employed where a
residual nonlinearity of less than 0.5% was achieved and maintained under
severe military operating conditions.
In certain instances it is desirable to generate a sweep wave form which
has a specific nonlinear time characteristic. An illustration of one such
case as applied to airborne radar is given in Fig. 52. Here the airborne
radar display was required to present a nondistorted ground plan which in
^y-^'
GROUND RANGE, GR=VS^-H^
Fig. 52.— Development of hyperbolic sweep wave form for true ground plan radar
presentation.
turn required that the range sweep wave form be of a hyperbolic form. The
start of the display sweep must be delayed in time corresponding to the time
of propagation and return of the radar pulse between the aircraft and the
ground. This delay may be produced by the use of a multivibrator of a
convenient form, actuated by a pulse coincident with the outgoing radar
pulse, and where the duration of the multivibrator pulse is controlled either
manually or automatically by reference to the aircraft's altimeter.
The hyperbolic sweep wave form illustrated may be approximated mathe-
matically as the sum of a linear and a series of exponential terms. In this
particular application, it was found sufficient to consider a linear and two
additional exponential terms only to satisfy these specific requirements.
Figure 53 indicates the method employed to generate this specific wave
form. As indicated, the desired theoretical hyperbolic sweep function has
784
BELL SYSTEM TECHNICAL JOURNAL
an infinite starting slope which cannot be provided with the practical limi-
tations of frequency band width and power available so that here the actual
delay used was chosen as 0.9 H, resulting in evident but acceptable distortion
in the display in the area directly beneath the aircraft. Figure 53b indi-
cates the fundamental circuit method of generating this wave form. The
linear term is generated across the capacitance Co and the series current
flowing through the additional elements Ri , Ci and Ri , Co supplies the two
SWEEP , ,
TUNING J |_
PULSE
l"'ig. 53. — H}'perl)olic sweep wave form generation — simplified schematic diagram.
additional exponential voltage wave forms. The resistances Ki and R^
arc required to be variable, their value being determined by the altitude of
the aircraft. The i)ractical form of the circuit employed is outlined in
Fig. 53c, which includes the additional resistor required to enable mo(lif}'ing
the rate of rise of the sweep wave form in accordance with the selected
interval of ranges to be displayed.
2.54 The Sivcep Amplijier
The remaining portion of the radar sweep circuit is concerned with the
amplification of the j)roperly timed and generated sweep wave forms to
THE RADAR RECEIVER
785
assure adequate deflection voltage or current for display purposes. The
sweep amplilier for range deflection purposes is essentially a specialized form
of video amplifier which must be capable of wide band transmission to ade-
quately reproduce the short time sweep wave forms and whose output char-
acteristics are such as to properly supply the high voltage or current signals
as required by the radar display system. Two general sweep amplifier de-
sign problems are presented for the two basic radar indicator types. The
!+B
Fig. 54. — -Range sweep anii)lilier circuit schematics for electrostatic-type radar displays.
electrostatic type cathode-ray tube generally requires a balanced to ground
deflection signal of moderately high amplitude while the magnetic type cath-
ode-ray tube requires a large deflection current for its operation.
Figure 51a illustrates a simplified schematic of a range sweep deflection
amplifier to be employed with an electrostatic type-A radar display. Here
the previously generated sweep wave form is impressed upon the grid of
Fi and after amplification a portion of the signal of opposite polarity and of
amplitude comparable with the input signal at the grid of Vi is applied to
the grid of Fo . The plate circuit of each tube is connected directly to the
deflection plate of the electrostatic cathode-ray tube. In this instance,
786
BELL SYSTEM TECHNICAL JOURNAL
the average potential of the horizontal plates of the indicator is maintained
at a value determined by the d-c plate potentials and, as indicated pre-
viously, this same potential should be applied to the second anode of the
cathode-ray tube to avoid defocussing effects. Another variation of a
phase inverter amplifier which is commonly employed in radar sweep cir-
cuits is illustrated schematically in Fig. 54b. In this instance, a common
cathode impedance is employed to accomplish similar excitation of the
balanced output tubes. If one grid is excited the plate current flow of this
e,(t)
It
^^
62 (t)
K
63 U)
TlME,t »•
Fig. 55. — -Voltage-current-time relationships for magnetic deflection structures.
tube through the cathode resistance serves to excite the second tube and a
balanced output sweep signal voltage will result. The values of the plate
resistors in this form of circuit are of unequal values and must be adjusted
to produce the desired balanced output. The additional control illustrated
in the grid circuit of V^ may be employed to serve as a d-c positioning con-
trol.
The sweep amplifier design considerations involved for the magnetic
deflection type of cathode-ray tube radar indicator are somewhat more
involved, due primarily to the character of the amplifier load impedance.
In the case of magnetic deflection the fuial flux density, and accordingly the
sweep current through the deflection coil, is required to be proportional to
THE ILiDAR RECEIVER 787
the deflection time function desired. If a linear deflection function is
assumed, as shown in Fig. 55, producing a linear sweep, the necessary form
of the appHed voltage wave will vary depending on the inductance, the
resistance and the parasitic capacitance of the coil circuit. These condi-
tions are illustrated in Fig. 55. It is entirely possible to generate sweep
voltage functions of the forms indicated here for application to a linear
amplifier and deflection coil circuits and in fact such an approach was em-
ployed in early military radar designs of World War II. It has, however,
proven more convenient to employ negative feedback amplifiers whereby
the deflection coil current output is maintained proportional to the applied
voltage at the input of the sweep amplifier. In this manner, a sweep
generator voltage wave form can be employed which has the characteristics
desired of the final deflection.
A simplified schematic of a feedback sweep amplifier to be employed in
connection with a magnetic deflection radar indicator is shown in Fig. 56.
In this example the impressed sweep wave form voltage having the essential
characteristics of the desired deflection time function is impressed upon
the grid of T'l . This sweep form is amplified and the deflection coil current
of the output stage which flows through the 80-ohm cathode resistance
common to the first and third stages produces a voltage drop proportional
to this current which is effectively applied between the cathode and grid
of the first stage thus completing the negative feedback loop. If sufficient
forward gain and adequate feedback is provided, the deflection coil current
can be made to assume the essential characteristics of the original im-
pressed voltage sweep wave form. It should be observed that the grid of
the third stage is biased negatively beyond plate current cut-off to insure
that the deflection coil current has an initial value of zero before the start
of the sweep. If this condition is not observed, the zero range point on the
indicator will be a function of the d-c current of the output stage and in the
case of a PPI form of display, the zero range region will assume the charac-
teristics of an open circle and map distortion at the short ranges will result.
In this amplifier circuit, application of the sweep signal to the grid of F3
will not result in deflection current flow until the tube is driven above cut-
off. During this time the feedback is not effective and the over-all gain of
the amplifier is at its maximum value. Due to the inductive characteristics
of the amplifier load impedance, the initial rise in current will be delayed
slightly with respect to the applied voltage and accordingly a further delay
of the feedback voltage is introduced by the use of a time constant in the
common feedback network. The result is a delaying of the applied feedback
voltage with a corresponding period of maximum gain of the amplifier which
tends to produce a sharp increase in deflection coil current at the time of the
788
BELL SYSTEM TECHNICAL JOURNAL
start of the sweep. After this short interval of time, the feedback becomes
effective and the output current and input voltage corresi)ondence obtains.
It is essential in this ty{)e of circuit tliat the feedback be determined en-
tirely by the deflection coil current if optimum oj)eration is to be obtained.
It should be observed that this condition requires that the screen current
which also normally flows through the feedback impedance does not con-
+ 300 VOLTS I
' + 600 VOLTS
DEFLECTION
COIL
= LPR4+L2R4-L, R3
L2R4 = i-iRs
\^K-G = '-PR4
WHICH IS DEPENDENT ON PLATE
CURRENT ONLY
Fig. 56.— Radar range sweei) amplifier cmiiloying negative feedback and screen grid
bridge circuit — simplified schematic.
tribute to the net feedback. Figure 56 illustrates the bridge circuit which
has been devised to accomi)lish this. .\ xoltage from the screen of the third
stage is directly a{)])lied to the cathode of the iirst stage, this voltage being
equal in magnitude and of opposite jiolarity to that which appears across
the feedljack impedance due to the third stage screen current (low. This
bridge circuit operation is independent of tlie absolute screen \'ollage value.
To insure identical starting potential conditions regardless of the duration
of the range sweeps in use, d-c restoration is emplo^-ed in the grid circuit ot
the last stage. The action here is similar to the o|)eration described pre-
viously in connection with \ideo am])liner design. The delay inherent in
THE RADAR RECEIVER
789
the magnetic deflection circuits must be carefully considered in the over-
all radar receiver design, if the display is required to reproduce short-range
information. In such cases, it is customary to insert delay networks in the
video channel introducing a delay to the received signal equal to that present
in the indicator deflection system, or to "pre-pulse" the deflection circuits
prior to the time of the outgoing radar pulse.
It is desirable from a power consumption and display appearance stand-
point to limit the range deflection of the radar indicator only to that ampli-
tude required to adequately fulfull the display requirements. A method
commonly employed is indicated in Fig. 57. Here a measure of the current
flow through the deflection coil, and accordingly the amplitude of the de-
flection of the cathode-ray tube beam upon the screen, is obtained from the
feedback voltage of the sweep amplifier. This voltage is impressed upon a
INPUT (FROM SWEEP-
AMPLIFIER FEEDBACK
RESISTANCE)
TT
STOP- PULSE
OUTPUT
(TO SWEEP
MULTIVIBRATOR)
--SWEEP- AMPLITUDE
CONTROL
Fig. 57. — Range sweep stop pulser circuit for limiting sweep deflection.
"sweep-stop" pulser which upon rising to a preselected value corresponding
to the desired sweep amplitude is caused to trigger this circuit. The output
pulse of this circuit is then employed to operate the sweep limiter portion
of the sweep timing multivibrator previously described, thus terminating the
sweep timing pulse proper.
2.6 Circuits for Radar Range and Bearing Measurement
In this review of radar receiver design principles only the presentation of
the received radar signal in a form convenient to the observer has been
considered. To fully utilize the complete radar information available,
determination of the complete coordinates is necessary with an exactness
which is determined by the specific use of the data and by the capabilities
of the radar system itself. This section will be devoted to a review of the
methods employed to generate electronic markers necessary for the deter-
mination of range and azimuth and elevation angles. These markers in-
clude both the fixed variety, whereby the approximate coordinates of a
radar target can be determined by inspection, and steerable markers by
790 BELL SYSTEM TECHNICAL JOURNAL
which means precise coordinate data necessary for most military applica-
tions are determinable. As indicated previously, the optical tilterscommonly
employed over the screen of the radar indicator often serve as a medium for
display of range, azimuth or elevation coordinate markings: however,
these methods are seldom completely satisfactory in military fire-control
radar systems because of errors introduced by the ever-present size or posi-
tion variations in the electronic display field. Their use has been strictly
limited to search or reconnaisance radar systems.
2.61 Electronic Bearing Marker Circuits
The bearing marker methods reviewed here are applicable generally to
both azimuth and elevation angle determination. A method of azimuth or
elevation bearing determination which can be associated with a lobing
antenna system and an A-type indicator has been mentioned previously.
This method remains an extremely precise system which has the desirable
advantage of simplicity. During the latter part of the war, automatic
tracking was applied to this method where the actual comparison of the
lobes of the selected target signals was carried out electronically and the
resultant antenna steering information utilized as the final bearing data. In
a strict sense, however, only an indication of error in antenna training is
observable to the operator on the radar equipment proper. The exact
bearing data must be obtained from a measurement of the position of the
antenna itself.
In the case of the continuous scanning systems employing B, C, or PPI
presentations, it is common practice to provide a steerable electronic marker
which can be superimposed upon the display field and by which means rela-
tively exact azimuth and elevation angles can be determined by target and
marker coincidence. This electronic marker method has the advantage that
it is subject to the same size and position display field distortion influences
as the received pulse signal, thus eliminating this source of error.
A circuit arrangement which has been employed in connection with a
naval vessel radar search system to brighten a selected and variable range
trace of the PPI indicator to serve as an electronic azimuth marker is given
in Fig. 58. In this example the rotating antenna structure is equipped
with a small permanent magnet j)ole piece whose cyclic excursions past a
sealed magnetic reed relay cause a jxiir of contacts to close indicating coin-
cidence. The relay structure is likewise mounted on a ring which can be
rotated with respect to the scanning axis of the antenna. The relative bear-
ing of a target is thus determinable by a knowledge of the angular position
of the relay ca])sule with respect to the lubber line of the vessel. The cir-
cuit of Fig. 58 i^roduces one brightened range trace for each revolution of
the antenna upon closure of the bearing marker relay contacts and is ar-
THE RADAR RECEIVER
791
ranged to be unaffected by any subsequent chatter or false switch closures.
The pedestal generator which includes vacuum tubes Fi and V2 which are
normally operated below plate current cutoff produces upon closure of the
bearing marker switch contact a rectangular negative pulse having a dura-
tion of 10,000 microseconds. This pulse operation is independent of addi-
tional chatter effects following the initial contact closure. The following
single-cycle bearing mark multivibrator is normally held inoperative by the
bias voltage developed on the cathode of F3 . The grid of Vz is continu-
ously excited with the range sweep start pulses of an amplitude insufficient
to actuate this multivibrator circuit. The presence of the 10,00()-micro-
second pedestal is sufficient, however, to allow the following range sweep
OUTPUT,
BEARING-MARK
SIGNAL (TO GRID
OF INDICATOR)
O 1
BEARING-'^/
MARKER /
SWITCH A
+300 VOLTS
Fig. 58. — Electronic azimuth bearing marker circuit — simplified schematic.
Start pulse lo operate the multivibrator. The output of this circuit is
then a 55()-microsecond pulse which represents a time shghtly longer than
the maximum range to be displayed (60,C0) yards) but shorter than the
period of the sweep repetition rate. This positive 550-microsecond pulse
is applied to the modulating grid of the PPI indicator tube through an
adjustable trace brightness control element.
Another convenient azimuth display method which has been extensively
employed in naval and airborne radar systems involves the use of "true
North" presentations. Here the PPI azimuth indication is presented in
terms of a compass reference, the actual instantaneous position of the range
trace on the screen representing the compass direction of the antenna beam.
In the indicator previously illustrated in Fig. 44 the compass information
is introduced by means of a dilTerential synchro inserted in the antenna-
indicator synchronizing connections whose angular displacement is pro-
792
BELL SYSTEM TECHNICAL JOURNAL
portional to the instantaneous heading of the aircraft. In the SL radar
indicator shown in Fig. 61 the compass information is introduced by means
of a mechanical differential rotation of the indicator deflection coil propor-
tional to the angular position of the compass repeater mechanism.
2.62 Range Marker Circuits — Fixed Range Markers
A simplified schematic diagram of a convenient fixed range mark gener-
ator circuit which has been extensively employed on airborne radar search
systems is given in Fig. 59. Here the radar system requires 1- and 5-
statute mile fixed range markers. The sweep start multivibrator pulse is
applied to the grid of Fi as shown. In the absence of a signal, this tube
-1-300 VOLTS
INPUT,
SWEEP-RANGE
START PULSE
RANGE
MARKER
OUTPUT
O
(-375 VOLTS
RANGE-MARKER
SWITCH
L, (80.9KC) c,
Fig. 59. — Fixed range marker circuit — simplified schematic.
operates at effectively zero bias, and because of the large plate current flow
reduces the effective plate potential of F2 and the grid potential of F3 to a
low value. Since the cathode of F3 is subject to a large positive potential,
this tube is cut off and the oscillatory circuit shown is inoperative. Upon
application of the negative start pulse, Fi is cut off for the duration of this
pulse and the attendant rise in the F3 grid potential permits the oscillator
circuit to function. The series resonant elements Li Ci and L2 Ci determine
the frequency of oscillation by providing a high value of positive feedback at
the series resonant frequency. The output of F3 which consists of approxi-
mate sinusoidal pulses is differentiated by means of the air-core transformer
shown. The differentiated pulses are then applied to a cathode follower
amplifier stage biased to cut off where the output is limited to the desired
positive fixed range mark pulses for indicator display. By careful choice of
circuit elements and equipment arrangement, this simple circuit form has
THE RADAR RECEIVER
793
produced an entirely satisfactory range marker signal for radar search
systems.
Variable Range Marker Circuits
Variable range marker circuits are employed where more precise range
information is required for missile control applications of radar. Here the
observer may position the electronic range mark to obtain coincidence with
the selected target, and from an associated calibration of this positioning
control, determine the range coordinate. For search or reconnaissance
purposes it is often desirable to determine range with somewhat more ac-
curacy than is afforded by the display of fixed markers, and for this purpose,
Fig. 60. — Variable range marker circuit of moderate precision — simplified schematic.
several designs of moderate precision variable range marker circuits have
been developed and employed during the past war period.
Figure 60 illustrates the circuit operation of a variable range mark gen-
erator of moderate precision. This circuit operation depends on a pulse
obtained from the transmitting modulator to serve as the zero time or
range reference. This is applied to a single-cycle multivibrator which pro-
duces a rectangular pulse whose leading edge is coincident with the time of
the synchronizing pulse and whose duration is somewhat greater than the
maximum range measurement required. A saw-toothed voltage wave form
is generated in the following RC wave generator by means similar to the
sweep wave form generation described in a previous section of this paper.
The coincidence circuit which follows consists of a vacuum tube biased be-
low cut-off whose exact cut-off bias is determined by the range mark poten-
tiometer setting. This coincidence circuit is thus inoperative until the
saw-toothed input signal has reached the value of the cut-off voltage, at
which time this circuit functions and produces a sharp decrease in its plate
potential. The effective time delay which is here produced with respect
794 BELL SYSTEM TECHNICAL JOURNAL
to the time of the synchronizing pulse is observed to be a function of the
rate of change of the saw-toothed wave form and the setting of the range con-
trol potentiometer which may be calibrated directly in units of range to the
target. The following range mark generator differentiates this coincidence
circuit output wave form and furnishes the desired amplification. Zero
range calibration is here provided by employing a sample of the zero time
refererce pulse and introducing this voltage into the range control circuit.
^^MBte. RANGE-YARDS-
^^i^HHik RANGE DIAL-i
Fig. 61. — -Transmitter-receiver-indicator assembl\- as designed for SL-Naval Search
Radar equipment.
Figure 61 illustrates the transmitter-indicator assembly of the SL naval
vessel search radar system. This system employs a PPI display with avail-
able range sweeps of 5, 25 and 60 nautical miles. The assembly shown to the
right of the main unit contains a variable range marker circuit of the type
just described. This range mark is positioned by means of the control
located toward the top of this unit and its calibration is observable through
a window located on the top panel. Here a ma.ximum measuring range in-
terval of 40,000 yards is available. Tn this application, the RC elements of
the wave generator are enclosed within an oven and maintained at a constant
temperature by thermostatic means. The accuracy achieved in this ex-
ample, without recourse to calibration means involving targets at known
THE RADAR RECEIVER
795
range, is zb 200 yards at the maximum range with an accuracy of ± 100
yards for targets within 5 nautical miles.
For more precise determination of range than is afforded in the circuit
just described, two methods have been extensively employed. The iirst
method involves the production of a known time delay by actual measure-
ment of the time of propagation of an acoustical wave through a liquid me-
dium. Here the physical length of path is varied to produce the variable
delay. The second method involves the phase shifting of a known precise
sinusoidal frequency standard which bears a fixed phase relationship to the
time of the outgoing radar pulse.
The "liquid delay tank" variable range unit over-all operation may be
observed by reference to Fig. 62. The zero time range reference is obtained
in the form of a pulse coincident with the outgoing radar pulse. This pulse
actuates the one-cycle multivibrator shown to produce a sharp high-ampli-
DELAY TANK
CONTROLn5:g3gztL [jZ\/\/\A/V--~-r[:^
SYNC
PULSE
INPUT
MULTI-
VIBRATOR
CIRCUIT
AUTOMATIC
GAIN
CONTROL
TRIMMER
CIRCUIT
MULTI-
VIBRATOR
CIRCUIT
RANGE-
PULSE
OUTPUT
to tp
tntp
Fig. 62. — Liquid delay tank type of precision variable range unit — block diagram of
operation.
tude output pulse, here relatively independent of amplitude and form char-
acteristics of the synchronizing pulse and which is applied directly to the
delay tank. This delay tank consists of a suitable container filled with a
mixture of iron-free ethylene glycol and water so composed as to produce a
zero temperature-velocity coefficient at 135°F, at which temperature the
liquid is maintained by thermostatically controlled electrical heaters. In
this temperature region the temperature-velocity characteristic is such as
to produce a decrease of velocity of 0.1% for a temperature variation of
14°F. Located at one end of this tank is a quartz crystal, approximately |"
square and 0.040" in thickness, mounted securely on a brass plate which
serves as one electrode and which is immersed in the liquid. A similar
crystal element is attached to a lead-screw carriage and located so that the
face of this crystal is parallel to the fixed crystal. The distance between the
crystal faces can be varied by rotation of the lead screw. The sharp voltage
wave ai)plied to the transmitting crystal causes it to oscillate in a damped
vibration at its natural frequency for longitudinal waves which in this case
796
BELL SYSTEM TECHNICAL JOURNAL
is of the order of 1.4 megacycles. The mounting plate and surrounding
liquid serves to highly dampen this oscillation. A short vibrational wave
train is projected through the liquid toward the receiving crystal. The
amplitude of this disturbance is only slightly attenuated by viscous dissipa-
tion for the maximum path length here employed. The large area of the
crystal relative to the wave length results in a highly directive radiation and
is reflected in a parallelism requirement for the crystal faces of the order of
.01". The voltage developed across the receiving crystal upon application
of this delayed acoustical wave consists of a main response followed by minor
disturbances due to re-reflections between the crystals.
Fig. 63. — Liquid delay tank t}-pe of precision varialjle range unit.
The following amplilier shown in Fig. 62 is required to increase the .005-
volt received signal to appro.ximately 20 volts. This gain supplied is con-
trollable by means of an automatic gain control circuit so as to provide a
relatively constant amplitude of the first response signal. The following
trimmer circuit consists of a pentode operating below cutoff such that a
signal of at least 20 volts is required for plate current flow. Since the AGC
circuit operates to adjust the gain of the amplifier to this condition, only the
first and highest response peak is transmitted to the final range j^ulse multi-
vibrator circuit where a sharp narrow rectangular pulse is produced to be
employed in the following indicator circuit.
Figure 63 is a photograph of the liquid-tank type of variable range unit as
developed and manufactured early in the past war and employed extensively
in naval fire-control radar systems. This unit includes provision for a
THE RADAR RECEIVER 797
maximum range measurement of 40,000 yards with an accuracy of ± 40
yards at this range under normal field operating conditions.
The use of the liquid range unit is practically restricted to ground and
naval vessel application because of its weight and the problems of handhng
these critical liquids. Another variable range unit development was ini-
tiated to meet the same accuracy requirements as above, but to be more suit-
able for aircraft and other extreme ambient applications. The phase-
shifting type of variable range unit whose operation is illustrated in Fig. 64
was the result of this effort.
The input start-stop single-cycle multivibrator circuit produces a rectang-
ular pulse output wave form whose leading edge is coincident with the time
of the outgoing radar pulse and whose duration encompasses the maximum
range time to be measured, in this example 270 microseconds.
The timing wave generator and associated phase shifting circuit is shown
schematically in Fig. 65. The resonant frequency of the oscillatory circuit
is 81.955 kc which period represents an equivalent radar range interval of
2000 yards. An initial d-c current of approximately 10 ma is present in the
Li Ci circuit in the absence of input start signals. Upon application of the
negatively poled rectangular start-stop pulse V\ and V2 are abruptly driven
to cutofif and the energy associated with the magnetic field of Li produces
local current flow and oscillation at a frequency determined by Li C\ .
The initial circuit conditions here are the same as the zero voltage condition
for each cycle of a sustained oscillation and the behavior of the oscillatory
system is the same as for the case of sustained oscillation. The absolute
average potential of the oscillation is maintained constant regardless of
the magnitude of the duty cycle. Positive feedback of the timing
wave is included in the F3 cathode connection in such a manner that uni-
form amplitude of the timing period throughout the active period results.
The purpose of the remaining circuits shown in Fig. 65 consisting of W ,
Vi and F5 is to produce four output timing wave voltages whose relative
phases differ by 90°. These voltages are to be later combined in such a
manner that continuous phase shift of the output timing wave results.
Two quadrature voltages are here produced by the use of LR and CR net-
works so proportioned that C0L2 = — ^ = R2 at 81.955 kc. The desired
C0C2
four timing wave outputs are produced by the use of the phase inverter
stages V4 and F5 .
The method here employed to combine four quadrature voltages to enable
continuous relative phase shift of the resultant output is illustrated in Fig.
66. This phase shifter capacitor consists of four quadrant shaped stator
sectors which are equal in area and shape and which are mounted parallel
to a ring stator as shown. A carefully shaped eccentric dielectric vane rotor
798
BELL SYSTEM TECHNICAL JOURNAL
<?-
Str
H-
THE RADAR RECEIVER
799
is provided whose rotation between the stator elements affects each quad-
rant stator capacitance in a like manner. As illustrated the resultant out-
put voltage which appears from the ring stator to ground has a phase shift
relative to any applied wave which varies linearly with angular displace-
ment of the condenser shaft.
The function of the following amplifier shown in Fig. 64 is to provide a
high-impedance termination for the phase-shifting condenser, and to pro-
vide increased amplitude of the timing wave. The pulse generator which
follows limits or clips the applied timing wave, and differentiates the re-
!B+
FROM
START- STOP
CIRCUIT
TIMING- WAVE
GENERATOR PHASE- SHIFTERS
Fig. 65. — Timing wave generator circuit of phase shifter t}-pe range unit.
sultant wave form. The output here consists of trains of alternate positive
and negative timing pulses.
The pulse selector component shown in Fig. 64 enables obtaining delay
intervals greater than 12.2 microseconds the value associated with 360°
phase shift of the timing wave. An increasing exponential saw-toothed
wave form is generated starting at zero time reference by an RC circuit
having a time constant of the order of 800 microseconds. The timing pulses
are applied additively with this exponential to the grid of a vacuum tube
operating below cutoff, its exact value of bias being determined by the
setting of a potentiometer control. At the time that the grid signal ampli-
tude exceeds this critical cut-off bias value, this tube conducts abruptly as
800
BELL SYSTEM TECHNICAL JOURNAL
shown and an output pulse is produced whose position on a time scale is
determined by the additive phase shift of the timing wave and the setting of
the pulse selector potentiometer. By mechanically gearing the potentiom-
eter and phase-shifting condenser, the final rotation of the control shaft will
result in an output pulse whose delay will increase uniformly and correspond
to 2000 yards per revolution of the control. F'urther amplification is fur-
nished in the output amplifier shown.
Figure 67 illustrates the final equipment features of a phase-shifting type
of variable range unit as developed for naval vessel radar system application.
It will be observed that this unit is mechanically interchangeable with the
liquid range unit shown in Fig. 63.
RELATIVE PHASE
OF INPUT VOLTAGE
ECCENTRIC
ROTOR
RING
STATOR
EQUIVALENT
CIRCUIT
Fig. 66. — Schematic outline of operation of phase shifter condenser.
In this model, the parallel resonant timing wave oscillatory circuit is
maintained at a constant temperature by employing an electrically heated
oven. Measurements made on this design indicate that the range shift
error to be expected for a "warm-up" period of 6 minutes was .003% or 15
yards at 45,000 yards range. After 6 minutes time, thermal equilibrium is
reached and the total range error will be less than 20 yards at 45,000 yards
range. The unit here illustrated has been universally employed in the
majority of naval vessel fire-control radar systems of the past war and these
basic circuit principles have served for range measurement in other apj^li-
cations including precision radar bombing.
2.7 Automatic Frequency Control and Atitomalir Gain Control
2.1 \ Automatic Frequency Control
The automatic frequency control (AFC) of the local beating oscillator to
THE RADAR RECEIVER
801
assure correct tuning of the radar receiver has become an extremely im-
portant feature of the military radar system and the successful solution of
Fig. 67.— Precision variable range unit of the continuous phase shifter type.
this problem has contributed greatly to the practical success of radar during
World War II, by assuring consistent optimum system performance imder
802 BELL SYSTEM TECHNICAL JOURNAL
severe military field conditions. In the case of the usual radio-communi-
cation system, some knowledge of character and extent of the information
which is being transmitted is available to the receiving location which may
serve to evaluate the receiver operating performance, but in the case of the
military radar system such reference data is not always available. The
usual military operating conditions for radar systems are extremely severe
which, in general, tends to degrade the performance. Mistuning of the
radar receiver and the attendant reduction of the performance of the system
must be immediately evident to the operator even under conditions where no
radar signal returns are present.
During the first years of the past war, this tuning problem was recognized
and the initial attempts at solution involved the inclusion of receiver tuning
indicators to serve as an indication of adjustment. As the radar systems
became more complex and with the trend toward the use of higher trans-
mission frequencies the necessity for completely automatic continuous
tuning adjustment of the receiver became increasingly evident and the
present types of automatic frequency control devices were developed. It
has been determined that in the specific case of airborne radar bombing
equipment operating at 10,000 mc that the automatic frequency control of
the receiver tuning is an absolute necessity, since the radar operator cannot
under the normal military operating conditions maintain the system per-
formance in this regard to a small fraction of the optimum.
Functions and Requirements
The basic reference for a radar automatic frequency control system must
be the transmitter frequency since it is required that the receiver be properly
tuned under the condition where no radar return signals are available.
Either the frequency of the transmitter magnetron or the local beating
oscillator frequency may be adjusted from an electrical error signal whose
characteristics are related to the tuning point. Magnetrons whose fre-
quency was conveniently controllable by remote electrical means were not
then available so that the later method has been universally apj)lied in mili-
tary radar systems developed during the past war period.
It is ])ertinent to review the nature and extent of the frequency instability
of a radar .system to derive the requirements to be imposed upon an AFC de-
vice. The sources of frequency instability are associated with the trans-
mitter as well as the receiver elements of a radar system. The magnetron
frequency is determined in part by the physical dimensions of its oscillatory
cavity structure, and as would be expected, ambient temperature and
pressure conditions exert a decided influence. For example, a typical
thermal coefficient of frequency for a magnetron may be as high as 200 kilo-
cycles per degree Centigrade which ()\-er the range of ambient tcnii)eralures
THE RADAR RECEIVER 803
to be considered important for military equipment may result in a frequency
shift of 20 mc from the time the equipment is turned on until thermal equi-
librium is established. The magnetron frequency is extremely sensitive
to its terminating load impedance. This termination in a radar equipment
is composed of the antenna, the interconnecting RF transmission line, and
the duplexing devices. The typical radar antenna system employs rotating
joints or connecting devices to enable transfer of RF power to the antenna
proper while it is mechanically operated over its scanning cycle. These
connections cannot be made to present an entirely uniform impedance over
their entire mechanical operating range and thus introduce variable im-
pedance irregularities to the magnetron generator. The frequency of these
impedance variations may range from a fraction of a cycle per second to
perhaps 60 cps. The input impedance of the antenna proper is dependent
on the extent and character of nearby obstructions in the radiation path.
The characteristics and form of the radome employed to protect the antenna
contribute to the variable impedance characteristics of the antenna and
thus influence the magnetron frequency. An additional instability in mag-
netron operation which is introduced through power supply variations within
the modulator and transmitter portion of the system must also be considered
in the detailed design of the AFC system.
The receiver itself is responsible for a major contribution to the frequency
instability characteristics of the radar system. The local beat oscillator
frequency is critically dependent on the physical dimensions of its oscillatory
structure and on the supply voltages. The effects of temperature and at-
mospheric pressure on the frequency of a reflex oscillator of the types pre-
viously described is considerable. For example, a thermal coefficient of
0.25 mc per degree Centigrade, typical of the 10,000-mc tubes, will produce a
total excursion of perhaps 25 mc over the range of ambients experienced in
military equipment. In the case of supply voltage variations, a 5-mc fre-
quency shift will result for a 1% change in anode and repeller potential for a
typical 10,000-mc reflex oscillator. Another source of receiver frequency
instability is associated with the shift of the IF amplifier frequency selectiv-
ity characteristic with tube aging and operating conditions.
If the operating requirements for an AFC system are now reviewed from
a consideration of these factors, it will be observed that for a radar system
operating at the higher frequencies a total effective frequency change of
perhaps as much as 50 mc may be encountered whose rate of change, in
general, will be relatively slow and may be classified generally as effects
due to "warm up". In addition fast variations of frequency will be present
whose rates of frequency change may extend from 1 mc per second per
second to 1000 mc per second per second. At the lower radar frequencies
804
BELL SYSTEM TECHNICAL JOURNAL
these frequency variations will be somewhat lower with an attendant reduc-
tion of the range of operation required of the AFC circuit.
AFC Circuit Design Considerations
To obtain a measure of the basic frequency reference for AFC purposes
the direct approach is evident. A sufiiaciently attenuated sample of the
outgoing radar pulse may be obtained from the transmitter and after sepa-
RADIO
FREQUENCY
INTERMEDIATE r
FREQUENCY
»■
■
LOCAL
OSCIL
-ATOR
(a)
TO
BEAT-OSCILLATOR
REPELLER
A^^— 1
TUNING
CONTROL
I +150 VOLTS (b) I -300 VOLTS
Fig. 68.— Radar AFC system — block diagram and circuit arrangement.
rate conversion but, under the influence of the regular receiver beat oscil-
lator, may be employed as a true sample of the outgoing signal as it exists
with the normal receiver IF channel. The separate AFC converter method
has been employed in some military radar equipment but has the disad-
vantage that additional conversion components are required. A second
method is more economical of equipment and has been extensively employed
during the past war period, but this later type of AFC circuit has limitations
wiiich are imposed on it by the character of the IF signal as normally avail-
able in a radar receiver. Figure 68a illustrated the essential elements of
such an AFC system for a radar receiver. It will be here observed that a
THE RADAR RECEIVER S05
sample of the IF signal after normal conversion and some amplification is
applied to a frequency sensitive discriminator circuit and the resulting error
signal is employed to readjust the beat oscillator frequency. The outgoing
radar pulse is normally attenuated effectively by the TR circuit, and thus
the remaining signal available for AFC purposes is due to inherent leakage
of the TR elements. As previously indicated, the frequency spectrum of
the outi)ut ''spike" of the TR device extends over a wide frequency range,
due primarily to the small finite delay in the breakdown of the TR tube.
The energy frequency distribution characteristic of this spike is to a large
degree independent of the magnetron frequency and, therefore, must be
considered as an undesirable masking signal and accordingly reduced to a
noninterfering level. As previously indicated this is usually accomplished
by disabling one or more of the IF amplifier input stages for a short time
interval coincident with the outgoing radar pulse.
With a signal available which is related to the frequency of the outgoing
pulse, the remainder of the AFC design is concerned with the utilization of
this information to accomplish the automatic tuning of the radar receiver.
To determine the frequency gain characteristic of the discriminator circuit
it is pertinent to examine the frequency repeller potential relationship of the
local beat oscillator. This relationship for a 2K25-type reflex oscillator
operating at 10,000 mc shown in Fig. 19 is approximately 2 mc/volt and is
representative of the tubes of this type. This quantity provides an indi-
cation of the "loop gain" required for a satisfactory AFC circuit.
Typical AFC Circuit Designs
Figure 68b illustrates the essential elem.ents of a radar AFC discriminator
and amplifier circuit. This consists of an input circuit which is required to
furnish the means for frequency measurement, rectifier elements to convert
this frequency deviation information to a proportional voltage error signal,
followed by an amplifier to increase the amplitude of this signal to the re-
quired level to adequately control the frequency of the local beat oscillator.
The operation of the discriminator input circuit may be observed by refer-
ence to the vector diagram of Fig. 69a. The input circuit, essentially a
double-tuned transformer having a low value of mutual inductance, serves
to couple the AFC rectifier to the preceding IF amplifier tube. The
resonance frequency of both primary and secondary circuits is main-
tained at the desired midband IF tuning point, in this example, 60 mc. The
output of the balanced secondary winding of this input network is applied to
a balanced rectifier shown in the vector diagram as E^ and Ei . In addition
a portion of the IF signal voltage which appears across the primary winding
is also applied to each element of the balanced rectifier. At resonance, the
primary and secondary voltages assume a quadrature relationship as indi-
806
BELL SYSTEM TECHNICAL JOURNAL
cated. The vector relationships for frequencies above and below this reso-
nance, as shown, result in an amplitude change across the rectifier circuit.
Figure 69b illustrates a typical rectified voltage versus frequency character-
istic of such an array. The location of the actual crossover zero voltage point
is determined only by the resonance of the secondary circuit over the limited
range under consideration here. The primary resonance contributes essen-
tially only to the symmetry of the voltage output versus frequency charac-
teristic. The introduction of the time constant elements in the detector
output circuit integrates the pulse output and are chosen with due regard
Fig. 69. — Operation of the AFC circuit — vector relationships in input circuit and out-
put voltage vs. frequency characteristic.
to the maximum frequency rate of change which this circuit must control.
The d-c amplifier shown is normally biased to operate at ma.ximum gain
consistent with stability, to produce the maximum sensitivity to frequency
change and to accordingly achieve the least threshold deviation from the
ideal tuning condition. Provision for disabling the AFC circuit is included
to enable initial manual adjustment of the beat oscillator repeller potential.
With the circuit shown here, failure of the AFC circuit proper will result in
the return of the rc[)cller potential to that value originally selected by initial
tuning and further manual control may be used.
It is often convenient to describe the effectiveness of an AFC circuit in
terms of its "pull in" range and its "hold in" characteristics. "Pull in"
THE RADAR RECEIVER 807
will be defined as the ability of the AFC circuit to restore proper tuning con-
ditions with sudden application of the signal. The "hold in" characteristic
of the AFC system will be defined as the ability of the circuit to maintain
proper tuning conditions as slow changes occur in the frequency of the con-
trol signal. In the AN/APS-4 airborne radar equipment previously re-
ferred to, which employs an AFC circuit similar to the form here described,
the "pull in" range is approximately zt 5 mc from the 60-mc midband value
and the "hold in" range includes the entire tuning range of the reflex oscil-
lator employed which is of the order of ± 40 mc. This example will main-
tain the tuning within 0.5 mc of the desired tuning point over the range of
conditions encountered in wartime aircraft apphcations.
In some applications use has been made of a frequency scanning process
whereby the AFC output voltage, in the absence of a suitable controlling
signal within the IF band, is caused to vary periodically in a saw-tooth
fashion thus causing the local beat oscillator frequency to vary, sweeping
across the complete tuning range of the receiver. When the desired signal
frequency is produced the AFC then functions in the normal manner.
This form of circuit was employed in certain radar equipments developed
during the early part of the war and a somewhat similar oscillatory AFC
circuit has been employed in connection with later developed thermally
tuned reflex oscillators and reported elsewhere. ^^
An automatic frequency control unit designed in connection with the
AN/APQ-7 radar bombing equipment which operates at 10,000 mc is illus-
trated in Fig. 70 and Fig. 71. The basic operation of this equipment ex-
ample is similar to the d-c amphfier type previously described but includes
certain modifications important for this particular application. In this cir-
cuit the plate potential of the first IF stage of the AFC unit is obtained as a
positive pulse from the transmitting modulator thus enabling the AFC cir-
cuit only during the short interval of time encompassing the outgoing radar
pulse. This arrangement assures that no detuning of the receiver will result
from spurious or nearby signals after the radar pulse has been transmitted.
The rectifier elements here consist of triodes operating near plate current
cut-off which results in improved hnearity of detection. The d-c amplifier
portion of this AFC circuit is arranged somewhat differently from theexample
previously discussed, including in this case balancing controls to account for
tube and circuit variations. At the condition of resonance, in this case 60
mc, the voltages applied to each grid of the amplifier are equal and the net
repeller potential is determined entirely by the manual control value.
The overall output range of voltage for this circuit is ± 20 volts, which in
this application represents a ± 40-mc frequency change for the associated
" "Considerations in the Design of Centimeter-Wave Radar Receivers", Stewart E.
Miller, Proc. I. R. R., Vol. 35, No. 4, April, 1947.
808
BELL SVSTJUf TECHNICAL JOURNAL
5ZOG
OU- +- '.
Ph
THE RADAR RECEIVER 809
reflex oscillator. Variations in the amplitude of the controlling signal are
of less importance by virtue of the biasing action of this d-c amplifier circuit.
The performance of this design includes maintenance of the tuning point of
this receiver to within ± 0.25 mc of the center of the IF band which in this
case is 60 mc.
2.72 Automatic Gain Control
Automatic gain control (AGC) of a selected radar signal is often required
in military radar systems employed for fire control or aircraft interception
purposes. In the case of the common search radar system, AGC is seldom
required. The radar receiver AGC function is quite similar to that required
of this circuit in the usual radio communication system, i.e., automatic am-
Fig. 71. — AFC component design as employed in AN/APQ-7 airborne radar system.
plitude stabilization of the desired signal. For the radar receiver case,
however, the desired signal must be selected on a time interval basis.
In the usual type of automatic tracking radar system, the target is selected
by manual alignment of a range and/or bearing "gate". This gating process
's essentially a modulation process by which the complete received radar
pulse signals are modulated with a rectangular pulse synchronized with
the outgoing radar pulse. The modulating pulse or gate has a finite ampli-
tude only over the time interval under observation. In this manner all
received information, except that occurring during the selected time interval,
is effectively rejected and the automatic gain control circuit operation is
entirely defined by the data present during this time interval.
The remainder of the automatic gain control circuit is concerned with
the measurement of the amplitude of the selected signal, usually by a peak
voltage measurement, the averaging of this measurement over a convenient
time interval, and the production of a suitable gain control voltage to be im-
pressed upon the radar receiver IF or video amplifier circuit. The detailed
810 BELL SYSTEM TECHNICAL JOURNAL
AGC circuit design is dependent to a major extent upon the dynamic
characteristics of the associated automatic tracking device. The subject
of automatic tracking design principles cannot be reviewed here and accord-
ingly more detailed AGC design consideration must also await inclusion in
such a future report.
2.8 Radar Receiver Power Supplies
The remaining components of the radar receiver to be here reviewed con-
sist of the power supplies necessary to produce the various d-c potentials as
required for the operation of the electronic components of the receiver. The
principal design problems associated with these components arise from the
relatively poor stability characteristics of the prime sources of power avail-
able at the military scene and the rather severe output voltage requirements
to adequately serve the precision nature of radar reception, display, and
measurement. The supply voltages necessary for a military radar receiver
range from low bias potentials upward to 5000 volts for cathode-ray tubes
and TR application with both polarities often required.
2.81 Primary Power Sources
The characteristics of the primary source of power available for the mili-
tary radar system are dependent on the area of use of the equipment. In
the case of mobile ground radar installations, the gasoline engine driven
generator represents the typical primary power source. In the case of large
mobile radar systems it is customary to employ ILS-volt — 60-cycle primary
power, while in certain more portable designs 115-volt — 400-cycle primary
power has proven satisfactory. The frequency and output voltage of a
gasoline engine driven alternator cannot be maintained within the narrow
limits desired by the radar receiver and here the major burden of precise
voltage regulation must be carried by the electronic regulated power supply
within the radar receiver.
For naval vessel radar installations 115-230-volt — 60-cycle primary power
is commonly available on the larger vessels. For PT and similar smaller
craft, certain radar installations have been employed operating from 24-48
volts direct current with motor generator sets supplying 60-cycle or 400-cycle
power for the radar system. In the case of undersea craft, the storage
battery is em])loyed as a primary source of ])ower and motor generator sets
are employed to obtain 115 volts, 60 cycles in most instances.
The primary source of ])ower for aircraft radar purposes is either a low
voltage (27 volts) d-c generator driven by the aircraft engine or an alter-
nator similarly driven. If d-c power is available, an additional motor gen-
erator set may be emj)loyed to furnish the 115-volt — 400 to 800-cycle power
for the radar equipment use. Voltage regulators of the carbon pile compres-
THE RADAR RECEIVER 811
sion or electronic types are usually employed here, resulting in a nominal ±
3% voltage stabilization. The extreme variable electrical loads imposed
upon the aircraft power system by electrically operated gun turrets and
other combat equipment result in increased emphasis being placed on ade-
quate regulation capabilities of the radar receiver power supplies. In
addition, the ever-present requirement of minimum weight for aircraft
equipment results in motor generator designs employing a minimum of
magnetic material and usually results in a variation in output voltage wave
shape with load. This factor must also be considered in the detailed
design of the aircraft radar receiver power supplies.
During the initial airborne radar development program, the power
frequencies in common use were 400 cps and 800 cps. The British use of
direct coupled aircraft engine driven alternators produced variable fre-
quency output voltages ranging from 12C0-2400 cps dependent on the en-
gine speed. In connection with the electronic warfare standardization pro-
gram for equipment to be used jointly by our allies, the aircraft radar system
was required to operate over the entire range of power frequencies extending
from 400 cps to 2400 cps. All of the airborne radar equipment developed
during the later half of World War II was designed to meet these variable
power frequency requirements.
2.82 Low Voltage Power Supplies
The electronic regulated power supply has been universally employed to
furnish the stable low voltages as required by the radar receiver. Here the
output voltages required extend from 50 volts to 600 volts with maximum
direct current required extending upward to 500 ma.
The basic electrical circuit arrangement of such a power supply is shown
in Fig. 72. In this circuit, the regulating element consists of a variable
series impedance, furnished in the form of a vacuum tube and resistance
combination, whose magnitude may be controlled electrically from an error
signal associated with the output voltage of the power supply and a reference
voltage. The control circuit consists of a bridge network which includes a
constant voltage gas-discharge tube as one element. A d-c amplifier is
connected across the output terminals of this bridge circuit and serves to
amplify the error signal for use in the regulating element. If the output
voltage of the power supply varies from the desired value for any cause, the
error signal appears at the output terminals of the bridge circuit, due to the
effective unbalance of this circuit at all voltage levels except the reference
value. The error signal after sufficient amplification is impressed upon the
grid of the series regulating tube with a polarity such that a corrective
impedance variation results. The degree of regulation obtainable is a func-
tion of the loop gain provided and the absolute stability of the output voltage
812
BELL SYSTEM TECH NICA LpOURNAL
is determined primarily by the constancy of the reference voltage derived
by the use of the gas-discharge tube. By incorporating wide frequency
band characteristics to the looj) gain elements, the maximum rate of change
of regulation can be extended, and this circuit becomes very effective in re-
ducing the fundamental and harmonics of the primary supply voltage. The
effective impedance of a radar receiver power supply of this type is of the
order of one ohm, a factor of extreme importance in reducing the unwanted
interaction between the various receiver components by coupling due to
this common impedance. Other variations in circuit arrangement for
regulated power supplies are occasionally employed, the most common of
which involves the use of a vacuum tube as a shunt regulating element as
contrasted with the series arrangement shown here. In certain low-current
SERIES
REGULATION TUBE
Fig. 72. — Simplified circuit schematic of low-voltage power supply — Series regulation
type.
applications the use of gas-discharge tubes of constant voltage characteris-
tics are occasionaUy employed in a shunt circuit configuration.
Figure 73 illustrates the equipment arrangement of a radar receiver power
supply as employed for an airborne radar bombing system. In this example
is included one non-regulated and three regulated rectifier power supplies
with output voltages of +600, +300, +120 and -300 volts available for the
radar receiver. The forced ventilation feature shown here is required to
prevent extreme temperature rise of the components under high altitude
conditions in the presence of considerable heat dissipation by the rectifier
and series regulating circuits. Each power supply is designed as a separable
subchassis within the over-all enclosure to provide for ease in manufacture
and testing of the unit. The weight of this complete unit as installed in a
military aircraft is approximately 50 pounds.
THE RADAR RECEIVER
813
Fig. 73. — Low-voltage power supply for AN/APQ-7 airborne radar system— Mechani-
cal features.
Fig. 74.— Low-voltage power supply for AN/APQ-5 low altitude radar bombing equip-
ment.
Figure 74 indicates the mechanical design features of a typical airborne
radar receiver power supply of the series regulated type as employed in the
AN/APQ-5 radar bombing equipment. This example illustrates the em-
phasis which is placed on compactness and lightweight construction of air-
borne radar components.
814 BELL SYSTEM TECHINAL JOURNAL
To obtain the maximum dependable performance from the various power
transformers and coils employed in the radar receiver power supplies under
the severe military held conditions, a considerable development program
was carried on throughout the past war. At the beginning of this program
the only available method of insuring adequate transformer winding insula-
tion under extreme liumidity conditions involved the sealing of the structure
in a metal container. P^or aircraft service, where weight is of prime con-
cern, this added weight could not be tolerated so that here an open type of
structure was extensively employed, with the protection to the winding
being furnished in the form of several coats of varnish followed by an enamel
overcoat. With the increased emphasis on high-altitude oj)eration of mili-
tary aircraft and the rapid temperature and pressure changes involved, a
development [)rogram was instituted to improve the open-type transformer
sealing process. The result of this program has produced the Flexseal
process whereby the service life of this type of power transformer has been
increased as much as ten times that obtained with the varnish process for-
merly employed. This process involves a multiple-dip varnish coating
method where the varnish is thickened by the addition of talc, a very
tine low-gravity wettable inert filler. This process results in the formation
of a relatively thick plastic shell which completely surrounds the trans-
former structure. One of the features of this process is found in its sim-
plicity, whereby no special equipment was required to carry out the pro-
cedure, an important factor during a w^artime production program. Figure
75 illustrates a number of typical open-type power transformers which
were employed on various military radar projects, all of which incorporate
the Flexseal treatment for improved service life.
2.83 High Voltage Power Supplies
The high voltage, as required for radar receiver cathode-ray tube indicator
purposes, varies from 2000 volts to 5000 volts, and for the TR tube keep-
alive potentials of the order of 1000 volts must be provided. In these cases,
however, the d-c current requirements are quite small and generally no
regulation means are required for stabilization of the voltage, the stability
of the primary source of power usually being sufficient.
The design problems encountered in this type of power supply are con-
cerned {)rimarily with the requirements of reliability of operation under
severe military ()])erating conditions, and furlhcr require that only circuit
elements having well-defined factors of safety be employed in such appli-
cations.
Figure 76 illustrates a number of typical high-voltage transformer designs
which have been employed in mihtary radar systems during the past war
period. Both air insulated and oil immersed types of structures are shown.
THE RADAR RECEIVER
815
Fig. 75. — Flexseal treated open-core type power transformers as developed for air-
borne radar application.
Fig. 76. — Power transformer designs for high voltage radar power supi)ly applica-
tion— Air-insulated and oil-filled types are included.
The use of air insulation in a high-voltage transformer results in a relatively
low coupling coefiticient between primary and secondary windings and
816
BELL SYSTEM TECUNICAL JOURNAL
accordingly restricts the range of power frequencies over which satisfactory-
regulation and operation can be expected. With the emphasis on power
supply frequencies extending from 400 cps to 2400 cps for aircraft purposes,
this air insulated type of structure was abandoned in favor of oil immersed
types. The primary disadvantage of the oil immersed type of transformer
is the increased weight of the unit.
A high-voltage power supply design as produced for airborne radar system
application is shown in Fig. 77. In this example +4900 volts is supplied
for cathode-ray tube indicator purposes and — 1000 volts is available for the
TR tube keep-alive circuit. This unit employs an air insulated type of
Fig. 77. — High-voltage power supply for airborne radar receiver application— pressur-
ized type.
high-voltage transformer and by the use of a hermetically sealed enclosure
operated at sea-level atmospheric pressure, satisfactory performance at high
altitudes is realized.
Conclusion
The complete technical story of radar is of a magnitude comparable to a
detailed report of the military campaigns of this past global war. During
this period, the Bell Telephone Laboratories developed for manufacture more
than 70 specialized radar systems for the Armed Services. It has been
possible here only to review the major technical considerations which in-
THE RADAR RECEIVER 817
fluence design of a military radar receiver and to present a few radar circuit
and equipment illustrations of the specialized technology that has resulted.
It is a tribute to the ingenuity and industry of the workers in the radar iield
that this technology, developed under extremely accelerated and difficult
conditions, will have a permanent value in future communication systems
design .
The material used in this paper represents the concrete contributions of
countless individual workers within and without the Bell Telephone Labora-
tories. The equipment products illustrated are the product of concentrated
effort on the part of the staff of the Western Electric Company who pro-
duced these specialized and complex radar systems in quantities and within
schedules necessary for the successful prosecution of the war. It is regret-
table that it is an impossible task to assign individual credit for these spe-
cific mil'tary radar contributions.
High-Vacuum Oxide-Cathode Pulse Modulator Tubes
By C. E. FAY
Introduction
TN practically all pulsed oscillators such as those used in radar,
-*■ some means must be provided to apply the pulse voltage to the oscil-
lator circuit. In many early radars, a high-vacuum modulator was used
for this purpose. The pulse was generated at low power level and then
amplified by means of one or more stages employmg high vacuum tubes.
The final stage was required to block, or cut off the d-c supply voltage with
no pulse applied, and to permit as much as possible of the d-c voltage to
appear on the oscillator during the pulse. Since most radar oscillators
operate at pulse voltages of from 5 to 20 kv and require currents of several
amperes during the pulse, the requirements of the modulator tubes are quite
severe. Standard transmitting^ tubes were used at first, the higher power
tubes having the necessary voltage rating and having in general a fair
amount of cathode emission. Tubes were operated in parallel to provide
the required amount of current. Practically all of these tubes were of the
thoriated tungsten filament type. For example an early army radar, the
SCR268,^ employed 8 tubes in parallel having a total filament power of
1040 watts to provide a pulse current of about 10 amperes. The use of
such equipment in portable or airborne service would be obviously imprac-
tical because of the large power consumption, bulk, and weight. In an
attempt to provide tubes more suited to this type of service, those described
in this paper were developed.
Tube Requirements
The function of the high-vacuum modulator tube essentially is to act as a
switch U) turn the pulse on and off at the transmitter in response to a con-
trol signal. The best device for this purpose will be the one which requires
the least signal power for control and which allows the transfer of power w ith
the least loss, from the transmitter power source to the oscillator.
If the oscillator must be supplied with a pulse of voltage E„ and current
/;„* or power Epfp, then the voltage w-hich must be supplied by the trans-
mitter power su[)ply will be E^ = Ep-\- c,,, I'ig. 1, if e'p represents the voltage
• It is assumed here that the pulse is rectangular in shape. This is usually the desired
shape and is fairly well ajjproximated in most cases.
818
PULSE MODULATOR TUBES
819
drop in the modulator tubes necessary to allow current Ip to pass. The
E
plate efficiency of the modulator is then simply —■ and the power dissipated
in the modulator tube plate is IpCp during a pulse. The average power dis-
sipated in the plate is then IpCp multiplied by pulse length and by pulse
frequency. The heat storage capability of the plate is ordinarily great
enough that the average power is all that needs consideration.
The conditions imposed on the modulator tube are somewhat analogous
to those of a class C amplifier at low frequency. The main difference is
that the angle of operation is very small, and there is usually no appreciable
backswing of plate voltage since the load is essentially a resistance. Typical
1
m >
t
y
1
i
en
k
ip
I
1
*
Y
Ip
I
Fig. 1 — Current and voltage relations in a pulse modulator tube.
modulator circuits are shown in Fig. 2. It is sometimes found desirable to
employ a shunt inductance across the oscillator in the interest of a sharp
cutoff of the pulse on the oscillator, particularly where capacitances to
ground of various circuit components are appreciable. This results in an
additional current demand on the modulator tube since the current through
the inductance must also be supplied. The oscillator is often coupled to
the modulator circuit by means of a transformer in order that desirable im-
pedances are realized in each circuit.
Design Considerations
It was apparent on first consideration of the high-vacuum modulator
problem that use of oxide coated cathodes would be of enormous advantage
820
BELL SYSTEM TECHNICAL JOURNAL
in keeping power requirements down. Heretofore the use of oxide cathodes
in high voltage power tubes had been found very difficult, particularly
where filamentary cathodes were employed. Any spark or momentary
discharge in operation usually resulted in the burning out of the relatively
fragile filaments. This result was caused mainly by the fact that a con-
siderable amount of energy was of necessity available from the power supply
equipment. However, in pulse service it is possible to limit the amount of
energy available so that a momentary tube breakdown will not result in
damage to a reasonably rugged equipotential cathode. Also, in the interest
OSCILLATOR
MODULATOR
(a) SERIES MODULATOR CIRCUIT
CURRENT-LIMITING
IMPEDANCE STORAGE
^±jO ^^b MODULATOR CAPACITOR 03^,,,^,^,
(b) SHUNT- MODULATOR CIRCUIT
Fig. 2 — Typical pulse modulator circuits.
of conserving control power it is desirable to build high perveance tubes
which require very close control-grid to cathode spacings. This is much
more easily accomplished with rigid cathode structures rather than fila-
mentary cathodes, especially for service conditions under which extreme
shock and vibration may be encountered.
Conservation of drive power requires that the modulator tube have high
power-gain. This is most easily provided in the tetrode which provides a
high over-all amplification factor with reasonable drive characteristics.
Lining up the control-grid and screen-grid wires is of course advantageous
in the interest of minimizing screen dissipation and getting the largest
possible jiortion of the cathode current to the plate. The control-grid to
PULSE MODULATOR TUBES 821
plate capacitance is of little importance as long as it does not store much
energy, there being little chance for oscillation in such a circuit. While it is
desirable to operate with as low a minimum plate voltage as possible, it is of
little additional advantage to bring the plate voltage below the screen volt-
age if the screen voltage is about 1000 volts and the supply voltage 15 kv.
It was therefore thought permissible to increase plate to screen spacing
beyond the optimum for best characteristics in the interest of high voltage
and screen dissipation ratings.
The insulation in the tube between plate and other electrodes must be
capable of withstanding the full supply voltage plus a comfortable margin.
This dictates that if internal insulators are used they must have long path
and that the bulb must have sufficient length to prevent flash-over ex-
ternally.
The 701A Vacuum Tube
At the time of this development a tube was very urgently needed for a
Navy radar application.^ Since speed was of prime importance it was de-
cided to take parts of a standard oxide-cathode beam-power tetrode,
Western Electric 350A, and mount them in a stiucture capable of with-
standing the required voltage; 12 kv in this case. Accordingly a cruciform
structure was designed in which four sets of 350A electrodes were mounted
on ceramic members attached to a molded glass dish-stem as shown in
Figure 3. The four cathodes have a total coated area of approximately 14
square centimeters. A molybdenum plate of cruciform section mounted
from a lead-in at the top of the bulb was used. This construction elimi-
nated internal insulators between plate and grids other than the bulb.
The control-grid of the 350A is normally gold plated to inhibit primary
emission. This feature was retained in the 701A and the screen-grid also
gold plated. The plate to screen-grid spacing was -increased over that
normally used in the 350A in order to allow somewhat better cooling of the
grids and to allow greater clearance for high voltage reasons. This made the
characteristics depart from good "beam tube" performance but at the high
voltage condition of operation this was of little consequence. Character-
istic curves of the 701A indicating performance under both high voltage
and low voltage conditions are shown in Fig. 4. Since no experience was
available at the time of this development to indicate what currents could
safely be drawn from the cathodes under pulse conditions, the matter of
rating these tubes was mainly guesswork since time was not available to
await the outcome of life tests under various conditions. The ratings put
on the 701 A are as shown in Table I.
For the immediate application in hand, which required 12 ampere pulses
at about 10 kv, it was decided to specify two 701 A tubes operating in paral-
822
BELL SYSTFAr TRCIJNJCAL JOI'RNAL
rig. 3 — The 701 A vacuum Uihc.
PULSE MODULATOR TUBES
823
Eg =
50 VOLTS
EsgCVOLT
1200 1
s) =
>
\000,— -
— '
/
^
800,- —
— -
/
/
^
t
400
V
!^^-^
J200
800 ~
'-- = -
Q- 1.2
y 0.4
Esg =
250 VOLTS
^'^
^
/
Eq (volts:) =
-10
/
^
/
-20
"^^
-30
-40
^
fe
r^
-0
0 0.5 1.0 1.5
plate potential
2.0 2.5 3.0
J KILOVOLTS
PLATE CURRENT
SCREEN CURRENT
AVERAGE CUT-OFF BIAS
Ip =0.2 MA
^^
\
^.
V Esg (volts) =
^V
«»
"^t^
>^
^v
\.
^.
^oo
''v
■N
^v
^
^^
0
^v
V
^^
"V
*\,
"-V
^^o
^\
^^
r^
^\
^v_
^
*s
"^
N
•s,
>
^
1.5 2.0 2.5 3.0 0 4 6
PLATE POTENTIAL IN KILOVOLTS
Fig. 4 — Characteristics of the 7()1A vacuum tube.
824
BELL SYSTEM TECHNICAL JOURNAL
lei. The pulse in this case was trapezoidal having a base width of about
4n seconds and a top width of about 1.75 n seconds; repetition rate was 1600
per second. When the tubes were operated under these conditions there
was customarily some sparking within the tube in the first few minutes and
then it apparently "aged in," and operated satisfactorily. At the rated
heater voltage, the cathodes operated at about 800°C (brightness). This
temperature would normally provide a cathode life of more than 1000 hours.
Life tests in the laboratory indicated satisfactory performance for about
2000 hours. Reports from the Navy were difficult to obtain but those
which were obtained indicated similar results. End of life was caused by
both loss of cathode emission and by primary grid emission. Mechanically
the tube proved to be reasonably rugged for normal service. However,
shocks sustained in shipment of tubes caused mechanical misalignment in
Table I
Table of Ratings of Oxide -Cathode
Pulse Modulator Tubes
Tube
Heater
Voltage
Heater
Current
Peak
Plate
Voltage
Peak
Screen
Voltage
Peak
Plate
Current
Plate
Dissi-
pation
Screen
Dissi-
pation
Max.
Duty
Cycle for
Peak
Plate
Current
Capacitances
Cin
Cout
Cgp
701A
715A
715B
5D21
426XQ
Volts
8
27
26
26
8
Amperes
7.5
2.15
2.10
2.10
7.5
KV
12.5
14
15
20
25
KV
1.2
1.2
1.25
1.25
1.5
Amperes
10
10
15
15
20
Walls
100
60
60
60
150
Walls
15
8
8
8
15
0.005
0.002
0.001
0.001
0.001
56
35
35
35
46
mm}
11.5
7
7
7
7.5
3.2
1.2
1.2
1.2
0.6
some cases, indicating a need for a more rugged structure for use in the
armed services.
The 715A Tube
The advent of airborne radar made the development of high power light-
weight transmitters an urgent requirement. In this case long life was some-
what subordinate to lightweight and small dimensions. Ruggedness was
also a requirement. The electronic properties of the 701A tube were well
suited for airborne radar but the large bulk was an extreme disadvantage.
Work was begun on a tube using the same cathodes as the 701A but having
a simpler and more rugged mechanical structure. Out of this evolved the
715A tube. In this tube the cathodes were placed side by side and en-
veloped by a single control-grid, screen-grid and plate. In order to provide
the necessary ruggedness and to keep the grids cool, heavier grid wires
were used and they were wound on very heavy supports of high heat-
PULSE MODULATOR TUBES
825
conductivity material. Both grids were gold plated as in the 701A. All
electrodes were mounted between two specially shaped ceramic insulators
which provided a relatively long path between plate and grids. This
structure is shown in Fig. 5. Heat radiating fins were attached to the ends
^
Fig. 5 — The 7 15 A vacuum tube.
of the control-grid and screen-grid supports. The plate is molybdenum
with zirconium coating on its outside surface. This coating was employed
to increase the thermal emissivity of the plate in the interest of a low operat-
ing temperature. It also serves to absorb some gas. The cathode, heater
and grid terminals of the tube were brought out in the moulded-glass 4-Pin
826
BELL SYSTEM TECHNICAL JOURNAL
16
5
< 10
50 VOLTS
Esg (VOLTS) =
1000
0
/
80
0
0
_600_
\
>
^^^ 1200
> —
Eg-
150 VOLTS
Esg CV0LTS)= \202- '
/
^
^^000
'/
^
&oo___
p —
/
f ^
_600___
/
^—
,^
y
r
\1200
4 00.-.^>^
^^
0.5 1.0 1.5 2.0 2.5 3.0
PLATE CURRENT
SCREEN CURRENT
2 00 VOLTS
Esg CVOLTS) = \200,-—
/
\000.
—
^
/
8C
0,,.— -
- —
/
60ii
A
aoO^
- —
400-
\
\I2C
0
1
^ — —
—
0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0
PLATE POTENTIAL IN KILOVCLTS
Fig. 6" ("lianictcristics of tlic 715.\ \-;Ki.iuni tulu'.
base, and the |)latt' terminal out tin- top of the bulb. This provided a
very rigid structure which could stand extreme shock and vibration condi-
i
PULSE MODULATOR TUBES
827
tions. Although this structure sacrificed something in electronic perform-
ance over the 701 A it still was quite satisfactory as a pulse modulator.
0.2 0.4 0.6 0.8 1.0 1.2
0.5 1.0 1.5 2.0 2.5 3.0 O
PLATE POTENTIAL IN KILOVOLTS >.
a. Lu
05
Q< 2
Esq (VOLTS) =
^ /o
Ep=
2000 VOLTS
y
/
.^
^
/^
^
/^
tt>^
-
O-500
"J -600
MAXIMUM CUT-OFF BIAS
-^
-\
"~~~-
Esg (VOLT
s) =
~\
^
^"""*-««&o
^^
>
■^
-..^
^^
0
"^.-,,___^^
•
"~-
^
^^Oo
**
^^
50 100 150 200 250 300
GRID POTENTIAL IN VOLTS
0 4 8 12 16 20 24
PLATE POTENTIAL IN KILOVOLTS
Fig. 6 (Continued)
The characteristics of the 715A tube applicable to both high voltage and
low voltage operation are shown in Fig. 6. It was found that in spite of
828
BELL SYSTEM TECHNICAL JOURNAL
the internal ceramic insulators, satisfactory operation could be obtained at
voltages as high as 15 kv. Since the 715A was designed primarily for air-
borne applications it was found desirable to design the heater to operate
directly from the aircraft's storage battery which was a 24 volt battery.
Fig. 7 — The 715H vacuum tube.
It was also desirable that the equipment be operable when the charging
generator was not ruiming, at which point the voltage might be as low as 22
volts, and also when the generator was chargmg and the voltage as high as
28.5 volts. This required a compromise in the design of the heater which
PULSE MODULATOR TUBES
829
resulted in operation of the cathode at somewhat higher than normal tem-
perature under rated conditions. The ratings of the 715A tube are given in
Table I.
0 U
Fig. 8 — The 5D21 vacuum tube.
The 715B Tube
Some applications developed which required a peak pulse current slightly-
greater and of longer duration than that for which the 715A was rated.
Meanwhile more experience with the 715A and improvements in processing
techniques indicated that a higher peak current rating was justifiable pro-
viding the grid temperatures were not increased.
830
BELL SYSTEM TECHNICAL JOURNAL
The 715B tube is essentially the same structure as the 715A except that
larger radiating fins are attached to the ends of the grid support wires.
Figure 7 shows the structure of the 715B tube. The characteristics were
identical but the ratings were changed, as indicated in Table I. The life
obtained in laboratory life tests under rated conditions averaged between
500 and 1000 hours. Failure was usually caused by grid emission or loss of
cathode emission.
(OSCILLATOR
LOAD)
SIGNAL
SCREEN
PLATE
II
1]
1
1 1
1
L.
GRID
it
Fig. 9 — Non-linear coil modulator circuit with illustration of the current and voltage
relations in the 5D21 vacuum tul)e.
TiiE 5D21 Tube
In response to the demand for further improvement of this structure in its
ability to withstand higher voltage, the 5D21 tube was developed, Fig. 8.
It is of the same family as the 715A and 715B. It was found that higher
voltages could be used if the grid cooling radiators were removed from the
top end of the tube. The cooling of the grid was maintained by providing
copper wire connections from the bottom ends of the grid support wires to
the base seals. This and the use of a specially designed plate terminal cap
enabled the voltage rating to be raised to 20 KV.
PULSE MODULATOR TUBES
831
Fig. 10 — The experimental 426X() vacuum tube.
The 5D21 tube also found application as the control tube in non-linear
coil type modulators/ Here the function of the tube was to permit passage
832
BELL SYSTEM TECHNICAL JOURNAL
Eg=
0 VOLTS
Esg (volts;) =
^
—
1 1200
~~ 1000
_22_00
12
< 10
=? 6
O 2
50 VOLTS
Esg (VOL1
1200
rs) =
/
1000
800 _
^
^
600
V
1200
" — — -
Eg=
100 VOLTS
Esg (VOLTS) =
,^
120C
1000
,
t
/^
8C
0
/
^
600
4 00
-^
400"
'\1200
Eg =
200 VOLTS
Esg (V0LTS>
^
1
/
/^
\ooo.
— -
/
^
800---
—
/
60C
--
/
y^
aoo
•—-
r
\1200
\
400'
-^*.
r
0 0.5 1.0 1.5 2.0 2.5 3.0
PLATE CURRENT
SCREEN CURRENT
Eg.
250 VOLTS
Esg (volts) =
j2^
—
/
3000,
-— •
r
8
00,— ■
-—•
/
1
600- —
-—
/
^
aoo,— -— '
-—
1
f
Vi
\
\
DO \
r* —
1200
■"'"■
0.5 I.O 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0
PLATE POTENTIAL IN KILOVOLTS
Fig. 11 — Characteristics of the 426XQ vacuum tube.
PULSE MODULATOR TUBES
833
of a moderately high current through an inductance and then suddenly to cut
off the current and withstand the resulting voltage which built up across the
circuit. A schematic of this type of circuit is shown in Fig. 9. In this cir-
cuit the tube is required to pass about two amperes peak plate current
which builds up over a period of about 150 microseconds. The d-c. volt-
age under these conditions is about 1000 to 4000 volts and the screen volt-
age may be obtained from the same source through series resistance. The
grid is driven only slightly positive. Screen-grid dissipation is one of the
limiting factors in this type operation. Primary emission from the screen
50 100 150
GRID POTENTIAL
200 250
N VOLTS
S 300
200 400 600 600 1000
PLATE POTENTIAL IN VOLTS
Fig. 11 (Continued)
grid, being present when the plate rises to its very high potential, tends to
discharge the circuit prematurely, the energy wasted appearing as heat at
the plate.
The 426XQ Tube
Since there was considerable demand for tubes capable of operating at
voltages as high as 25 KV, a lube was developed to operate at this voltage.
The limit of the 5D21-715B type structure seemed to have been reached at
about 20 KV. It was also desirable to increase the current rating of the
834 BELL SYSTEM TECHNICAL JOUkNAL
tube since in a laboratory test equipment a pulse power of 1.5 to 2 megawatts
was needed. The laboratory number 426XQ tube sho\vn in Fig. 10 was the
result. Four of these tubes in parallel were capable of providing pulses of
about 1.75 megawatts. In the 426XQ tube, the bulb used on the 701 A
was employed and the plate supported entirely from its terminal in this
bulb. The same four cathodes were used, but were spaced farther apart
than in the 715-type tube. Two separate control-grids and two screen-
grids were used, each pair encompassing two cathodes. This allowed a
reduction of dissipation per grid compared to the 715-type, otherwise simi-
lar techniques were employed. The characteristics are shown in Fig. 11.
The tentative ratings applied to the 426XQ are given in Table I. The
allowable peak plate current was increased for this tube because the tech-
nique of processing had improved so that a higher level of cathode activity
was consistently realized. Also the greater spacing between cathodes and
use of two sets of grids resulted in better grid cooling. The tube was not
used in any radar equipment, because by the time it was available the trend
in radar equipment was toward small, compact apparatus in which spark
gap and transmission line modulators^ found considerable application.
The 426XQ proved very satisfactory in laboratory test equipment. One
set of these tubes operated for somewhat more than 2000 hours.
The Chief Problems
The difficulties experienced with this series of oxide-cathode pulse modu-
lator tubes can be divided into three general classifications, namely: spark-
ing, cathode emission, and grid emission.
The sparking in these tubes can roughly be divided into two types,
which may be called inter-electrode sparking and cathode sparking. Inter-
electrode sparking is a discharge between two electrodes of the tube caused
by the momentary breakdown of the insulation between them or by a gas
discharge. If the breakdown of insulation is caused by light deposited films,
the resultant discharge usually causes removal of the film and cures the
trouble automatically, provided no other damage is done to the tube, (ias
discharges from isolated pockets may be initiated by the high fields or by
bombardment by stray electrons. If these pockets are not numerous they
are usually dissipated after a few minutes of tube operation such that fur-
ther sparking is very intermittent and probably not of sulficient intensity
to interfere with operation. The gas so released is ordinarily taken up by
the getter in the tube so that operation is not subsequently impaired.
Cathode sparking may be caused by positive ion bombardment of the
cathode or by poor adherence of cathode material when subject to electro-
PULSE MODULATOR TUBES 835
static fields. This type of sparking usually does not clear up and when it
becomes serious the tube must be replaced.'' It can be aggravated by spark-
ing in the oscillator part of the radar system. There is some evidence to
indicate that very high rates of rise of the pulse current drawn from the
cathode may tend to produce cathode sparking. At rates of rise in excess
of about .SO amperes per microsecond per scjuare centimeter of cathode area
a tendency for increased sparking has been noticed.
Cathode emission, here as in any other tube, is governed by cathode
temperature and other considerations such as quantity and kind of gas in
the tube, the core material, coating material, and techniques of processing.
No attempt will be made to consider these factors in this paper as they are
sufficiently complex that no very clear cut dissertation can be given. Stand-
ard core materials and coatings were employed with good results. It was
found that the double carbonates (Ba, Sr) were less subject to sparking than
the triple carbonates (Ba, Sr, Ca). The cleanliness and previous treatment
of the other parts of the tube seemed to be the major factor in deteimining
the level of emission obtained.
Primary grid emission, or thermionic emission from the control-grid and
screen-grid, was one of the most difficult problems in the development and
production of these tubes. Many trials were made using different materials
and coatings on the grids, but from all considerations gold was found to be
the most satisfactory. The grids in all the tubes described here are gold
plated or gold clad molybdenum. It is not considered that the use of
molybdenum for the core material is necessary, it being used here mainly
because it seemed to be the most economical material that had sufficient
stiffness to maintain grid alignment. Materials that tend to alloy with
gold easily are not suitable as it was found that gold alloys were not as good
as pure gold on the grid surface. The limitation involved in the use of gold
is that the temperature of the grid must be kept low enough that evapora-
tion of gold is not serious. This temperature limit is probably about 700°C.
If gold is evaporated, the grid soon loses its coating and primary emission
builds up rapidly. Also, the cathode emission seems to be poisoned by the
gold vapor. ^
Acknowledgment
The author wishes particularly to acknowledge the contributions of his
immediate associates, Messrs. H. L. Downing, J. W. West, and J. E. Wolfe
in the development of this series of tubes. Many otliers also made im-
portant contributions. We are also indebted to the M.I.T. Radiation
Laboratory for data from life tests which they conducted on many of these
tubes.
836 BELL SYSTEM TECHNICAL JOURNAL
References
1. R. Colton, Radar in the U. S. Army, Proc. I.R.E., Vol. 33, pp. 740-750, November
1945.
2. The SCR-268 Radar, Editorial, Electronics, Vol. 18, pp. 100-109, Sept. 1945.
3. W. C. Tinus and W. H. C. Higgins, Early Fire-Control Radar for Naval Vessels,
Bell Sys. Tech. Jour., Vol. 25, pp. 1-47, Jan. 1946.
4. E. Peterson, Coil Pulsers for Radar, Bell Sys. Tech. Jour., Vol. 15, pp. 603-615, Oct.,
1946.
5. F. S. Goucher, J. R. Havnes, VV. A. Depp, and E. J. Ryder, Spark Gap Switches for
Radar, Bell Sys. Tech.' Jour., Vol. 15, pp. 563-602, October 1946.
6. E. A. Coomes, The Pulsed Properties of Oxide Cathodes, Jour. Applied Phys.. Vol.
17, pp. 647-654, .Vugust 1946.
7. J. Rothstein, The Poisoning of Oxide Cathodes by Gold, (Abstract) Phys. Rev. Vol.
69, lst/15th June 1946, p. 693.
Polyrod Antennas
By G. E. MUELLER and W. A. TYRRELL
The polyrod, a new form of microwave endfire antenna, is described. This
consists of a properly shaped dielectric rod protruding from a metal waveguide.
For applications requiring moderate gain, it possesses desirable electrical and
mechanical properties. It is useful as a unit antenna in broadside arrays on
account of its low crosstalk into adjacent polyrods. This paper describes work
done from 1941 to 1944 at the Bell Telephone Laboratories, Holmdel, N. J.
Important individual contributions are acknowledged in some of the footnotes.
A report of this development has been withheld from earlier publication for
reasons of military security.
1. Introduction
A UNIFORM rod (or "wire") of dielectric material without metallic
-^ ^ boundaries is a well-known type of single conductor transmission
line. In this kind of waveguide, a portion of the energy travels along in
the space outside the rod. At discontinuities, including those caused by
proximity to other objects, radiation takes place. For this reason, the di-
electric waveguide has not become generally useful as a transmission medium,
this need havmg been satisfied by the hollow metal pipe. The tendency
toward radiation inherent in the dielectric guide is turned to advantage,
however, in a new form of radio antenna. Here the objective is to encourage
radiation from all parts of the dielectric rod. Li progressing along the rod,
therefore, power is gradually transferred from within the dielectric to the
space outside. At a point where the transfer has been effectively com-
pleted, the rod can be terminated abruptly. By proper design, this radiat-
ing structure is an endfire antenna. Since it has been most often fabricated
from polystyrene, it has become known as the polyrod antenna. It is
especially useful for microwaves.
We must now review and examine certain features of dielectric rod trans-
mission and of endfire antenna theory, for their bearing on polyrod design
and performance.
2. Dielectric Wire Transmission^
A dielectric rod can be energized with an infinite variety of transmission
modes. These are in general hybrid waves- possessing transverse and longi-
[ tudinal components of both E and //. We shall here be concerned only
1 Hondros and Debye, Ann. der Fhvs., Vol. 32, pj). 465-476; J. R. Carson, S. P. Mead
and S. A. Schclkunoff, B.S.TJ., Vol. 15, pp. 310-333, 1936; G. C. Southworth, B.S.T.J.,
Vol. 15, pp. 284-309, 1936; S. A. Schelkunoff, "Electromagnetic Waves," pp. 425-428,
D. Van Nostrand, New York, 1943.
2 Except in the case of circular symmetry. Cf. Schelkunoff, loc. cit., pp. 154, 425.
837
838
hELI. SYSTR\r TFXnxrCAL JOURNAL
with tlic lowest mode,'' tluit which is the coiuUerpart of the dominant wave*
in a metal pipe. If a dielectric-lihed metal guide is excited at the dominant
mode, and if the metal shield is abruptly terminated, the wave energy will
continue on in the unsheathed dielectric rod and will be confined almost
exclusively to the lowest hybrid mode. This is, indeed, the most common
way of exciting the dielectric wire.
The extent to which the power is concentrated within the dielectric
is a function of the rod diameter and dielectric constant. This is shown^
in Fig. 1. If the curves for the two different dielectric constants are re-
DVe
plotted against tlie effective diameter,
, they become more nearly
14
1.2
1.0
0.6
0.4
0.2
r
/
/
r
/
f
/
/
/
e-2.5y
/
/
f
^
^
y
y
y
/
^
-
'
^^
10^
-
«'
-^
<
_J
0
0.004 0.01 002 004 0.1 0.2 0.4
1.0 2.0 4.0
10 20 40
Fig. 1 — Ratio of power inside Wi to jiowcr outside Wo for a cylindrical
dielectric wire.
coincident. A universal curve cannot be given, however, because the field-
retaining effect of a dielectric-air interface increases with increasing dielec-
tric constant.
The phase velocity within the rod is also a function of the diameter and
dielectric constant, as shown^ by Fig. 2. Wlien — is very small comi)ared
A
with unity, the rod exerts negligible guiding action, and the transmission
is close to that in free s[)ace. For rods of large diameter, the power is con-
' Unliki' all olJuT modes in a dielectric wire and all modes in a conduclinij; pil)e, the
lowest dielectric wire mode llieoreticall>- has its cutoff at zero fre(|uency. ("f. SchelkunofI,
loc. cil., p. 428.
■•That is, the TEn mode in circular i)ipc or the TKio mode in rectangular \n\ic.
^ Figs. 1 and 2 are based on calculations by Dr. Marion C. Gray.
POLY ROD ANTENNAS
839
fined almost entirely within the rod, and the phase velocity approaches that
in an unbounded medium of the same dielectric constant. By choosing
intermediate values of
D
can be varied between these limits.
3. Endpire Antennas
We consider a linear array of isotropic radiators, infinite in number but
so closely spaced as to occupy a finite length. We assume that the radia-
tors are uniformly excited from a feed line, a transmission line parallel to
the array phasing the various elements according to phase velocity on the
line. The radiation pattern is given by*^
sin 7r(p cos 6 — 0)
7r(p cos 0-/3)
(1)
\
\\
r
--
e = 2.5
Vj
4.0
I
10
?
0
32.5
0.2 0.4
0.8 1.0 1.2
Fig. 2 — Normalized phase velocity for a cjlindrical dielectric wire.
where r = relative field strength
6 = angle with respect to the array axis
p = length of array in free space wavelengths
27r/3 = phase shift in radians in the feed line from one end of array to
the other end.
The pattern is symmetrical about the array axis.
Plotted from (1), Fig. 3 shows the pattern of a six w^avelength radiator,
p = 6, for selected values of p. When /3 = p (= 6 in this case), phase
velocity along the feed line is equal to free space velocity, and the resulting
pattern is endfire. With /3 = p + 0.5 (^ 6.5) the pattern remains endfire
and the major lobe becomes sharper. For (8 < p and /3 > p + 0.5 (as shown
by /3 = 5.0, 5.5, 7.0) the pattern deteriorates into a forward conical beam.
6 R. M. Foster, B. S. T. J., Vol. 5, p. 307, 1926.
840
BELL SYSTEM TECHNICAL JOURNAL
The gain of a uniformly excited endfire antenna^ with ^ — p \s 4p. For
13 9^ p, the gain can be written
S = 4.4p. (2)
/
\
/
\
/3 - 7.0
/
\
/
\
/
\
, /
■\
/
\/
\
\
y
/
\/
\
/
V
X
y
V
\
/
^"^
N,
p = 6.5
/
\
y^
\ /
r
N
V /
-^
/
^
/
V
V
N
/^
\
5
<0.;
^
"
"^
p = 6.0
/
\
/
\
^— V
^
V
v
^
-—s.
y
k' '
\
^
\
p = 5.5
/
^
— "
\
/
\
/
^
/
\
A
\
/ ^
\
/
^\
p = 5.0
/
\
/
\
/
>
\
/
f
\
^
^
"^
^
V
"^
? 60
10 0 10
DEGREES OFF AXIS
60 +
Fig. 3— Directional i)attcrns of a six wavelength (p = 6) continuous arra>-.
The factor A is given graphically in Fig. 4 as a function of 1-k{§ - p), the
phase lag.* The highest gain occurs for a phase lag of approximately tt
radians relative to free space transmission, that is, for /3 = p + 0.5, in con-
formity with the patterns of Fig. 3. For a short radiator, ,1 is about 2;
with increasing antenna length, .1 approaches 1.8.
^ SchelkunolT, loc. cit. p. 347.
* Fig. 4 and ('(luation (5) were supplied by I>i
H. T. Friis.
POLYROD ANTENNAS
The width of the major lobe is given by
Beam Width =
B
841
(3)
The constant B depends on /3 — p and on the manner in which beam width
is defined. For width in degrees between half power points, and with
j3 — p = 0.5 for maximum gain, B is computed from (1) to be about 60.
/3-p
-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
< 1.2
0.6
/
^
>
^
P =
2- /
V
A
\
10,
7
\\
/
/
\
\
/
y
/
^
/
/-
^
^
-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2
xTT ADVANCED
PHASE LAG IN RADIANS
0.2 0.4 0.6 0.8 1.0 1.2 1.4
RETARDED >TT
Fig. 4 — Gain factor yl as a function of phase lag in endfire arrays.
If a sinusoidal variation in excitation voltage along the radiator is super-
posed on the constant amplitude assumed for (1), we get
I sin 7r(p cos Q — ^) , ,. . cos 7r(p cos 0 — (3)
r = \ a ; r- ::t + U ~ O)
7r(p COS 0-/3)
1 - 4(p COS 9 - ^y
(4)
where a is defined in Fig. 5. This figure gives patterns of a six wavelength
radiator according to (4) for various values of a. Here 0 is fixed at 6.5
for maximum gain. Tapering symmetrically away from the center de-
creases the minor lobes. The gain is also decreased, but to a lesser extent.
842
BELL SYSTEM TECHNICAL JOURNAL
Exponential tapering comes about from heat losses and radiation losses
in the feed line. With attenuation a per wavelength, (1) becomes**
2 cosh ap — 2 cos 27r(cos d — ^)
ay- + 47r2(p cos d - /3)2
(5)
20 25 30
DEGREES OFF AXIS
Fig. 5 — Effect of sinusoidal tapering of power upon directional characteristic of a six
wavelength continuous array.
Feed line attenuation increases slightly the minor lobe amplitudes and tills
in the nulls. Exponential tai)ering caused by radiation can be reduced or
eliminated if the coupling of the radiating elements to the feed line is gradu-
ally increased along the line.
4. The Polyrod Antenna
It has been found e.xperimentally that a suitably proportioned dielectric
rod can act as an efficient endtire radiator. A complete understanding of
* hoc. cit.
POLY ROD ANTENNAS
843
its operation involves the solution of Maxwell's equations subject to the
boundary conditions appropriate to the configuration. An analysis of this
sort is not available because of its mathematical complexity. However, a
satisfactory explanation of polyrod operation, especially for engineering
purposes, can be obtained by establishing analogies with array theory,
coupled with existing knowledge about transmission in uniform dielectric
wires. In this treatment by analogy, we remain essentially ignorant of the
local fields in the vicinity of the dielectric, the role played by the discon-
tinuities at both ends of the antenna, and other detailed features. We do
have, however, a working theory which predicts closely the features of the
radiation as observed at a distance. Under these circumstances, insistence
upon a rigorous field solution has not so far appeared necessary.
-10 0 10
DEGREES OFF AXIS
Fig. 6 — Data on polyrods of uniform rectangular cross-section gX by ^X.
Experimental data have been obtained at frequencies in the vicinity of
3000 megacycles except for Fig. 9, representing work at 9000 megacycles.
For the sake of generality, these results are presented in dimensions of X,
the free space wavelength. In all cases, polyrods have been energized from
a dielectric filled metal guide whose conducting sheath is abruptly termi-
nated, the dielectric continuing on as the radiator.
The earliest form of polyrod^ was a polystyrene rod of uniform rectangular
cross-section, about |X by §X. Figure 6 shows the gains and directional
patterns measured for such rods in three different lengths. In a plane
normal to the axis, the radiation is approximately isotropic. The observed
gains are proportional to length. They are greater than 4p by a factor of
»The earliest work on polyrods was done in 1<M1 hv Dr. G. C. Southworth. C:f. his
r. S. Patent 2,206,923 issued in 1940.
844 BELL SYSTEM TECHNICAL JOURNAL
about 1.4. Phase velocity in these rods was not measured and is not
available from theory. Referring to Fig. 4, however, we must assume at
least 0.47r radians of phase retardation to explain the increased gain. When
the pattern for the 6X rod is compared with the sharpest pattern (j8 = 6.5)
in Fig, 3, the observ^ed characteristic is sharper than expected even with a
phase retardation of tt. The amplitudes of minor lobes are in good agree-
ment. Attenuation, as revealed by the amplitudes at minima in the
patterns, is apparently appreciable but not serious.
The principal defect of the uniform polyrod is the strong minor lobes.
This is remedied by tapering the amplitude of radiation symmetrically
about the midpoint, as suggested in Fig. 5. To obtain such tapering let
us start at the waveguide end with a relatively thick rod. From Fig. 1,
this tends to retain a larger fraction of the power and should therefore not
radiate so strongly. Let us decrease the cross-section gradually in pro-
gressing along the rod, thus increasing the power radiated. Upon reaching
a point near the center, we find the power in the rod already considerably
diminished by the radiation which has already taken place. Beyond this
point, gradually decreasing radiation is automatically secured with a uni-
form cross-section as a result of previous radiation.
This line of reasoning, calling for a polyrod tapered down in cross-section
only in the first half of its length, is verified experimicntally. Since detailed
field analysis is not available for the polyrod, the most favorable proportions
have been found empirically. Three examples will be described.
Figure 7 shows a 6X rectangular polyrod linearly tapered for a little more
than half its length from a base ^X square to a rectangular section |X by ^X.
the remainder being uniform. The tapering is confined to the magnetic
plane. Measured phase velocity and directional pattern are included in
Fig. 7. By reference to Fig. 5, the observed minor lobe amplitudes cor-
respond to a value of a somewhat less than 0.5. The gain, considerably
improved over the uniform rod, nnplies from (2) a value of 1.86 for A in
remarkable agreement with Fig. 4.
Figure 8 shows data on a 6X cylmdrical polyrod linearly tapered for about
half its length from a diameter of 0.5X to 0.3X with the remainder uniform.
The pattern is very similar to that of the preceding example; the gain is
slightly reduced, and A = 1.66. From Fig. 1, e = 2.5, about half the
power is internal for — = 0.5, while less than one-tenth is internal for 0.3.
X
Agreement between Figs. 2 and 8 for phase velocity is fairly good.
Figure 9 gives information'" about an 8.65X radiator which resembles the
" Supplied by Mr. C. B. H. Feldman.
POLYROD ANTENNAS
845
conical-cylindrical design of Fig. 8, but which is longer and is tapered for
slightly less than half its length. The minor lobes (solid curve) are all lower
than 0.125, a marked improvement over Fig. 8. From the measured gain,
A is 1.82.
Regardless of whether the cross-section is^square, rectangular, or round,
radiation is nearly isotropic about the axis of^the polyrod. For the patterns
-;o 1.0
< _i
a. >
Ouj
2JJ0.9
I
Q.
0.8
0.6
^
->■
r " '
1
/
T
—^
— y
i
i -
c
^ c
o
■ — "
0.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5
DISTANCE FROM FEED, I , IN WAVELENGTHS
kl
^
GAIN = 16.5 DB
s
\
\
n.
\
.^
w
V
V
\
iy
V
^
-10 0 10
DEGREES OFF AXIS
Fig. 7 — Data on a 6X tapered rectangular polyrod.
in Fig. 6-9, beam widths in degrees between half -power points correspond
to values of B in (3) between about 50 and 60.
The characteristics of polyrods can thus be correlated with array theory
for isotropic radiators continuously distributed along an axis. There are,
to be sure, minor discrepancies which might become more serious in a dif-
ferent range of polyrod proportions. For the lengths and cross-sections
tested, however, equations (1) to (5) describe polyrod performance very
satisfactorily for engineering purposes.
846
BELL SYSTEM TECHNICAL JOURNAL
<3 1.0
q:>
OiiJ
< 0.9
1
}
J
h
^
1.5 2.0 2.5 3.0 3.5 4.0 4.5
DISTANCE FROM FEED, I, IN WAVELENGTHS
^\
^
GAIN = 16 DB
\
\
X
k
\
A
^
r
y
V
\/
V
A
-50 -40
-10 0 10
DEGREES OFF AXIS
Fig. 8 — Data on a 6X tapered cylindrical polyrod.
5C
SINGLE POLYROD
p= e.55\ G = 18 DB
7^-
"'' \
* SPACED 2.25A, ONLY
CENTER ANTENNA DRIVEN
THREE POLYRODS-*/
P= 8.65X G = 17DB /
\\
\\
V
/
\
/
\
''N
r^^
/-\;
'^
/
C>
-10 O 10
DEGREES OFF AXIS
Fig. 9 — Data on an 8.65X tajjered cylindrical polyrod, including elTect of adjacent similar
polyrods.
POLY ROD ANTENNAS
847
COMPLETE SOCKET
ASSEMBLY
GASKET
Fig. 10 — Polyrod and waveguide feed details.
848
BELL SYSTEM TECHNICAL JOURNAL
5. Construction and Operational Details
Figure 10 shows a production model of a polyrod for 3000 megacycles,
with means for matching it to a rectangular waveguide.^^ A two-iris
transformer is used with a resulting width of 4% between the 1 db standing
wave points. The clamping illustrated is designed to maintain a firm grip
on the rod despite tendencies of the polystyrene to cold flow.
Another type of coupling is indicated in Fig. 11. Here the polyrod is
still fed from a waveguide but this is in turn transformed to a coaxial line.
The composite can thus be regarded as a coaxial to polyrod coupling. The
coaxial line taps at point b onto the short-circuited antenna a-b-c at a point
POLYSTYRENE
ROD
COAXIAL
FEED
Fig. 11 — Coaxial feed for polyrod.
chosen to match the characteristic impedance of the coaxial line. The back
end of the waveguide is short circuited by a metal cap a quarter wavelength
behind the transverse wire antenna. A movable coaxial plunger provides
tuning. This arrangement has a bandwidth of 1% to the 1 db standing
wave points.
The frequency response of a polyrod is inherently broad. The directive
pattern varies slowly with both phase velocity and amplitude distribution
along the axis. As shown in Figs. 1 and 2, these quantities are slowly vary-
ing functions of X over a considerable range of polyrod proportions. At
" Developed by Mr. D. H. RiiiR.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
DISTANCE FROM FEED, I, IN WAVELENGTHS
1.0
^
H
MATERIAL
STYRAMIC
NOISE
GAIN
IN DB
16.5
/
P
1
\
HARD
RUBBER '^-^
/ c
1 I
5 \
ACETATE
BUTYRATE 90
4 i
i 1
t 1 '
/
*N
V \
o'
k?
^
^'^
^i
-50 -40 -30 -20 -10 0 10 20 30 40 50
DEGREES OFF AXIS
Fig. 12 — Effect of dielectric loss on polyrod performance.
^V
1
I N.
\ s
'x.
\^-^.
k rj a< H >.
-5
-10
r ^
r
"
1
\
S
\
s
\
\
\
\
\ REVERSE
-15
\
\
\
\
\
10
LU
ffl -20
O
\
- J^.
;- -;
;^^
\
\
ELECTRIC POLARIZATION IN
SAME PLANE (WORST CASE)
z
\
-I
<
iTl
S-30
O
\
\
\
\
I IN SAME
\ DIRECTION
\ ,
L
\
-40
\
Y
\
^-i
\
V
V
\J
>.
"''--
-50
M
X
^
0.5 1.0 1.5 2.0 2.5 3.0 3.5
SPACING, d, IN WAVELENGTHS
Fig. 13 — Crosstalk, between polyrods.
849
4.0 4.5 5.0
850
BELL SYSTEM TECHNICAL JOURNAL
POLY ROD ANTENNAS 851
present, the usable bandwidth is therefore limited primarily by the frequency
response of the coupling arrangements from polyrod to waveguide or coaxial
line.
We have been exclusively concerned so far with plane polarized radiation.
A circularly symmetrical polyrod such as in Fig. 8 can be used equally well
to radiate circularly polarized waves. To do this, the polyrod is fed from
a waveguide in which circulary polarized dominant waves are generated by
means of a 90° phase shift section.^-
The effect of dielectric loss upon polyrod performance is shown in Fig. 12,
to be compared with Fig. 7. The power factors are: Styramic, 0.0005;
hard rubber, 0.003; acetate butyrate, 0.020; pol3^styrene, 0.0002. Mate-
rials having power factors less than 0.001 are satisfactory for polyrod an-
tennas.
Figure 13 shows the crosstalk between adjacent polyrods, that is, the
power received in one radiator when the other is energized. For polyrods
pointing in the same direction, separations greater than a wavelength insure
low mutual coupling. This makes the polyrod attractive as the element
in broadside arrays. Proximity to other undriven polyrods affects the gain
and directional pattern to a greater extent, as shown in Fig. 9.
More generally, the performance of a polyrod is affected by proximity
to any metallic or dielectric objects. The gain and pattern must be deter-
mined empirically for each new configuration. It has been found that a
metal rod can be plated parallel to a polyrod without seriously affecting its
behavior so long as a separation of a wavelength or more is maintained.
Sheets of dielectric material can be brought even closer without adverse
effect so long as large surfaces are not in direct contact with the polyrod.
These and other experiences suggest that the polyrod is relatively unaffected
by nearby objects.
Tests have been made of the effect of fresh and salt water in the form of
a spray or solid stream playing on a polyrod. Provided that puddles do not
formi on the surface, as can happen with rectangular polyrods, the effect is
a decrease of 1 to 2 db in gain under the worst conditions.
In conclusion, for microwave applications involving moderate gains of
15 to 20 dh, the polyrod assumes a convenient physical form and displays
high electrical efficiency. It is less subject to disturbance by nearby ob-
jects than might be expected. It is especially useful as an element in broad-
side arrays. As an example of such arrays. Fig. 14 shows a 42 rod steerable
beam antenna used in an important type of Navy fire control radar.
'- For a discussion of this subject, cf. A. G. Fox, "An Adjustable Waveguide Phase
Changer," to be published in Proc. I. R. E.
Targets for Microwave Radar Navigation
By SLOAN D. ROBERTSON
The effective echoing areas of certain radar targets can be calculated by the
methods of geometrical optics. Other more complicated structures have been
investigated experimentally. This paper considers a number of targets of practi-
cal interest with particular emphasis on trihedral and biconical comer reflectors.
The possibility is indicated of using especially designed targets of high efficiency
as aids to radar navigation.
Introduction
IT NOW seems likely that radar, developed during the war, will find in-
creasing application as a navigational aid for aircraft and surface vessels.
In fact there are good reasons for expecting that peace-time radar can be
made even more efficient than its war-time prototype.
There are two ways of improving radar performance. One may concen-
trate on the radar set proper with the object of increasing either the power of
the transmitter or the sensitivit}^ of the receiver. Or, one may take steps to
improve the echoing efficiency of the targets. The latter was, of course, not
possible during the war since most of the targets of interest were controlled
by the enemy. It is a purpose of the present paper to consider the design
of various targets of high echoing efficiency and wide angular response which
may be placed at strategic points as aids to radar navigation. The ideal
reflector to serve as a "beacon" or "buoy" for guiding radar-equipped air-
craft or ships would present a highly effective area to incident radiation over
a full 360° in azimuth, and would also be operative over a fairly broad verti-
cal angle. The value of a particular target for navigational purposes may
therefore be considered in terms of two factors: effective area, and angular
response.
The echo received by a radar from a particular target can be calculated
by the formula:^
W. = Wr^^^ (1)
where Wr = echo power available at the terminals of the radar antenna.
Wt = power launched by radar.
Ar = effective area of radar antenna assuming that the same an-
tenna is used both for transmission and reception.
' This equation follows directly from Equation (1) of a paper !)>• H. T. Friis, "A Note
on a Sim])le Transmission Formula," J'yoc. I.K.E., Vol. 34, pp. 2.S4-256, May 1946. The
radar transmission formula is ol)tained by applying Friis' formula twice; first to the trans-
mission from the radar to the target, then to the transmission from the Uirget to the radar.
852
TARGETS FOR MICROWAVE RADAR NAVIGATION 853
A eff = efifective area of target.^
X = wavelength.
d = distance between radar and target.
The above formula applies to the case where free-space propagation prevails;
that is, where multiple path or anomalous transmission effects are absent.
It is apparent from the formula that, at a given wavelength and range, the
received echo power can be increased by increasing the transmitted power,
the size of the antenna, or the effective area of the target. The present paper
will consider only the latter.
In some cases the effective area of a target can be calculated from simple
geometrical optics. For the more complicated structures it is always pos-
sible to measure the effective area by comparing the signal reflected by the
object in question to the signal reflected from a simple target of known
effective area.
Flat Plates
The simplest target for which the effective area can be calculated is a flat
metal plate oriented so as to be perpendicular to the incident radiation. It
can be demonstrated that a flat plate with all linear dimensions large in pro-
portion to the wavelength of the incident radiation has an effective area
which is substantially equal to its geometrical area. Diffraction effects at
the edges of such a plate are small in comparison with the energy reflected
from the central portion of the plate.
Flat plates, however, have the serious disadvantage that, in order to create
strong echoes, they must be maintained accurately perpendicular to the
incident rays. At other angles of incidence the echoes fall off rapidly. For
this reason flat plates are of limited value as targets for use in navigation.
DrsEDRAL Corner Reflectors
A dihedral corner reflector consists of two perpendicular, plane conducting
surfaces which are usually arranged so that they intersect along a common
line. Figure 1 shows a typical dihedral reflector. The dihedral reflector
has the important property that a ray which enters the corner will experience
a reflection from each of the surfaces and will return in the direction from
which it came, provided of course that the entering ray lies in a plane which
is perpendicular to the line of intersection of the planes which form the
2 The term "effective area" as used in this paper refers to the equivalent flat plate area
of a target. The echoing effectiveness of a target may alternatively be expressed in terms
of the cross section of an equivalent isotropic reflector as described by Schneider, "Radar,"
Proc. I.R.E., Vol. 34, p. 529, August 1946. The alternative unit is called the "scattering
cross section" and is frequently denoted by the symbol <x, although Schneider uses S. The
two quantities are related by the equation a = iir A^eff/X-. Both units are useful. For
most of the targets considered in the present paper, Aeff does not vary with \ and is there-
fore preferable.
854
BELL SYSTEM TECHNICAL JOURNAL
corner. The latter restriction constitutes the principal objection to the
practical use of dihedrals. The path of a typical ray is shown in Fig. 1.
Fig. 1 — Dihedral comer reflector.
A A.
A
r
I
I
I
I
I
b
I
I
I
I
I
I
I
i
e=20'
I IK 2 Variation of effective area of a dihcclral with aspect angle.
Tlie effective area of a dihedral reflector depends upon both the size of the
reflector and the orientation of the reflector with respect to the incident rays.
iMgure 2 shows how the effective area varies as the dihedral is rotated about
the line of intersection of the two planes. The elTective areas for the differ-
ent orientations are shown by the shaded regions in the lower part of the
TARGETS FOR MICROWAVE RADAR NAVIGATION
855
figure. For a reflector having the dimensions shown in the figure the
effective area for different angles of incidence 6 can be calculated by the for-
mula.
A eff =2 ah sin (45° - 6)
where d is always considered positive and less than 45°.
Figure 3 shows the polarization of the reflected ray for differently polarized
incident rays. For our purpose, the incident rays may be assumed to enter
the left side of the reflector shown in the figure and the reflected rays may be
assumed to emerge from the right. It is apparent that if the incident ray
is polarized either parallel or perpendicular to the line of intersection of the
two surfaces the reflected ray will be polarized in the same plane as the inci-
Fig. 3 — Polarization effect in a dihedral reflector.
dent ray. If the incident ray is polarized at an angle of 45° to the line of
intersection, the reflected ray will be polarized perpendicularly to the
incident ray. In the latter case the signal received back at the radar will
not ordinarily be accepted by the same antenna which launched the incident
radiation.
Trihedral Cornek Reflector
Assume that three reflecting surfaces AOB, AOC, and BOC are placed so
as to form the right-angled corner illustrated in Fig. 4. In general, electro-
magnetic waves, upon striking an interior surface of the device, will undergo
a reflection from each of the three planes and return in a direction parallel to
856
BELL SYSTEM TECHNICAL JOURNAL
and with the same polarization as the incident ray. The path of a typical
ray is shown by line 1, 2, 3, the particular ray chosen having entered the
reflector along a line perpendicular to the plane of the paper. Points 1 and
3 represent the initial and linal points of reflection, respectively, whereas
point 2 represents the intermediate reflection point.
Two important conclusions can be drawn from a careful inspection of the
path 1,2,3 ; namely, the projections of points 1 and 3 are diametrically oppo-
site on a circle drawn about point O as a center, and points 1 and 3 appear
to be images of point 2 ; i.e., the ingoing ray at 1 appears to be directed toward
Fig. 4— Trihedral corner reflector showing the paths of typical rays.
the image of point 2 in plane AOB, and the outcoming ray at 3 appears to
come from the image of point 2 in plane AOC.
Not all rays falling upon a corner reflector of tinite dimensions will be re-
flected in the direction of the source. For example, a ray striking point 4 in
Fig. 4 may be reflected successively at points 4 and 5, but if the plane BOC
is not sufficiently extended it will not undergo the necessar>^ third reflection
required to return the ray in the incident direction.
'J'he portion of the ])rojccted cross-section of a corner reflector which is
able to return incident radiation to the source is called the ''effective area."
It is, of course, a function of the aspect, that is to say, the angle at which the
TARGETS FOR MICROWAVE RADAR NAVIGATION
857
reflector is being viewed, as well as the geometrical configuration of the
reflector. For some of the simpler configurations the effective area can be
readily determined by the following procedure.
Project the aperture of the reflector through the apex O to form the image
(A' B' C of Fig. 4); then project the aperture and its image upon a plane
perpendicular to the incident rays. The area common to the projections of
the aperture and its image is equal to the effective area. The effective area
of the triangular reflector of Fig. 4 is, therefore, represented by the hexagon
a b c d e f . Only those rays, perpendicular to the plane of the paper, which
-r — 1 ST IMAGE
2 ND IMAGE
-3 RD IMAGE
Fig. 5 — Determination of effective area of trihedral comer reflector.
fall inside the hexagon will be returned. Exactly the same procedure is used
in determining the effective area for other aspect angles.
The above rule must, however, be applied with caution. Situations arise
in which rays falling upon the area determined by this method do not return
to the source. Figure 5 shows a reflector in which this difficulty is encoun-
tered. This reflector differs from the previous reflector in that it has a notch
cut in one of the reflecting surfaces. The projection of the aperture upon the
plane of the paper is indicated by the solid line; that of its image by the
dotted line. According to the rule of the preceding paragraph, one would
expect the effective area to be defined by the total shaded area of the figure.
858 BELL SYSTEM TECHNICAL JOURNAL
Such, however, is not the case. It was stated earlier that the ingoing rays
appear to be directed toward the images of the intermediate reflecting
points. This requires that the images of the intermediate reflecting points
fall inside of the efi"ective area. In Fig. 5, the images of the notch fall inside
of what would otherwise be the effective area. Since the notch is incapable
of serving as an intermediate reflector, the more lightly shaded areas are
excluded from the effective area. In the absence of the notch, a ray entering
at 1 would be reflected at 2 and emerge at 3. In the presence of the notch,
however, it passes through plane AOC and escapes in the direction of 4.
Therefore, in order to determine the effective area of a corner reflector of
arbitrary shape and aspect, one must take account of three loci of points
defined by the aperture as follows:
1) The aperture itself
2) The locus of points determined by taking the direct mirror image of
each point of the aperture wath lespect to each of the two surfaces of the
trihedral not containing the point. For example, pomt D of Fig. 5 will have
the images D' and D" with reference to planes AOB and BOC, respectively.
The complete locus of points determined in this way is represented by the
dot-dash line of Fig. 5.
3) Locus of points on aperture after each has been assumed to have been
projected through the vertex. This image is pictured by the dotted lines of
Fig. 5.
These three images of the aperture can, for simplicity, be referred
to as the first, second, and third images, respectively. The effective area
is the area common to the projections of the first, second, and third images
of the aperture upon a plane passing through the apex of the reflector and
perpendicular to the incident rays. For a given aperture and aspect, a cor-
ner reflector can theoretically be replaced by a flat plate located at the apex.
The size and shape of the flat plate will vary with the aspect as well as with
the configuration of the aperture. The above procedure has been of consid-
erable aid in studying reflectors having apertures of arbitrary shape.
Although the graphical analysis just given is sufficient to enable one to
compute the effective area of a reflector for any aspect angle, it is frequently
more conveneint to determine the complete response pattern of a reflector
experimentally. Most of the experimental results reported in this paper
were obtained with a 1.25 centimeter radar arranged as shown in Fig. 6.
Echo levels were measured on the screen of a type-A indicator using a cali-
brated intermediate frequency attenuator to restore the signal to an arbi-
trary reference level. It is believed that the levels measured in this way are
accurate to within ± \ decibel. The coordinate system used in recording
and presenting the data is given in Fig. 7. The reflector was mounted On a
lurntablc which could be rotated al^out horizontal and vertical axes.
TARGETS FOR MICROWAVE RADAR NAVIGATION
859
Curves for the response patterns of a corner reflector of triangular aperture
are shown in Fig. 8. These curves were obtained with a reflector constructed
of silver-painted plywood whose aperture was in the form of a 24-inch
equilateral triangle. It had been previously determined that, with suitable
paints, reflectors of this construction behaved exactly as though they were
made of sheet metal.
Depending upon its angle of arrival, a ray may be reflected by a corner
reflector in one of four ways. If the angle is too oblicjue, the ray may not
be returned in the direction of the source at all. If the incoming ray is
^;V7T7777777777777777777777777Z777777777777777777777777777777777777777777777}
(<. ,000 FEET ■ ^>l
Fig. 6 — Arrangement of apparatus for measuring effective areas of targets.
SYMMETRIC AXIS
OF CORNER REFLECTOR
TO
RADAR
Fig. 7 — Coordinate system used in presenting data.
exactly perpendicular to one of the three reflecting planes, it will be returned
to its source after only one reflection. Should the ray arrive in a direction
exactly parallel to one of the three planes, it will again be returned in the
direction of its source but in this case it is reflected twice as in a dDiedral.
This particular mode of reflection is illustrated by the sharp peaks at the
extremities of the curv^es in Fig. 8. For all remaining angles of approach the
ray will be returned after three reflections in the manner already described.
The central regions of the curves represent this type of reflection which is of
principal interest in practice.
860
BELL SYSTEM TECHNICAL JOURNAL
The effective area of the triangular trihedral reflector along the symmetric
axis id = O, <l) = 0) can be computed from the geometry of Fig. 4.
A eff = 0.289 ^ (3)
where C is the length of one side of the aperture such as CB. The eflfective
area at other aspect angles can be computed by relating the echo level at the
aspect in question with that along the symmetric axis.
0 = -4O°
0 = 0°
/
^
^
\
A
/
■N.
/
/
\/i
/-
^
/
\
0
50
40
30
20
10
0
50
40
0 = -3O°
4^" Sa-
-^f- \-
0 = 10°
A
^
^
n
V
/
\
\
0 = -2O°
0 = 20°
\ .
—
-—
^^
,^
—
— ■
V,
V
\i
1
/\
/
^
\
A
1
I
I
\
'\
0=-1O°
/I
/
^
^
N,
/t
/
N
y
\
0=30°
r\
^^
^^
^^
^
^^
f\
/
V.
^
-40 -30 -20 -10
10 20 30 40 -40 -30 -20
ANGLE, e, IN DEGREES
10 20 30 40 50
Fig. 8— Echo-response patterns of a triangular trihedral reflector.
It should be pointed out that the eflfective area for trihedral reflections is
independent of wavelength where the reflector is large enough so that
geometrical optics prevail. If the wavelength is increased, however, the
sharp dihedral peaks at the edges of the pattern will be broader.
In the case of the triangular corner reflector the response levels for aspect
angles of 30°, as measured from the symmetric axis, are down by 10 decibels.
For many applications a flatter response pattern is desirable.
TARGETS FOR MICROWAVE RADAR NAVIGATION
861
The present investigation led to the discovery that the response pattern
of a corner reflector can be modified by a suitable alteration of the geometri-
cal configuration of the aperture. There is even the suggestion that the
response can, to a certain degree, be made to conform to a somewhat arbi-
trary pattern \\ ithin a region extending to approximately 30° from the princi-
pal axis. The procedure for accomplishing this has, so far, been one of trial
and error since the difficulties of a general mathematical solution appear to be
(a) (b)
Fig. 9 — Compensated comer reflector.
Fig. lO^Dimensions of face of compensated reflector.
insurmountable, at least in a practical sense. For practical purposes, how-
ever, it is comparatively easy to conduct a few graphical experiments in order
to design a reflector having the desired response pattern.
Figures 9a and 9b show two views of a modified corner reflector which w as
designed to have a relatively flat response characteristic out to angles of 30°
from the central axis. Each of the three sides of the reflector, instead of
being triangular as formerly, has the contour shown in Fig. 10. The
shaded regions of Fig. 10 represent the surface which has been added.
862 BELL SYSTEM TECHNICAL JOURNAL
In Fig. Qa. one is assumed to be looking into the reflector along the sym-
metric axis. The efi'ective area is represented by the shaded hexagon.
Evidently, the effective area of the modified reflector is identical to the
effective area of the original triangular reflector ABC. Therefore, for this
particular aspect, the effective area has not been changed by the addition
of the material at the corners.
Figure 9b is a view of the reflector at 0 = 30°, 0 = 0°. Again, the shaded
region represents the effective area, and the parallelogram abed is the effec-
tive area of the reflector before modification. The modification has evi-
dently introduced a substantial gain in effective area for this aspect. A
graphical com.parison of the effective areas of Figs. 9a and 9b shows them
to be of comparable magnitude.
With the dimensions defined as in Fig. 10, a corner reflector was con-
structed with a= b= 17". The response curves of this reflector are plotted
in Fig. 11, along with the curves of the ordinary triangular reflector. A
substantial improvement in response is exhibited by the com^pensated re-
flector. In the region extending out to 30° from the axis, the response level
varies by no more than a couple of decibels. The response appears to rise
slightly in the vicinity of 20°. This could, perhaps, be reduced by a more
appropriate shaping of the sides of the reflector.
The variation of the response curve with the ratio ^ has been studied
briefly. It appears that a value of ^ = 1 is about right, for the 30° contour
to equal the axial response. If - < 1, the reflector will only be partially
corrected; if ^ > 1, it will be overly corrected. In the uncorrected reflector
with triangular aperture, a = 0-
If b/a = CO , that is, if a = 0, one would expect to obtain a response curve
having a minimum value on the axis and rising to a maximum on either side.
A reflector having these properties is illustrated in Figs. 12a and 12b.
Again Fig. 12a is the axial aspect, whereas Fig. 12b is the 30° aspect. In
the former, the effective area should be zero; in the latter, it has the value
represented by the shaded portion. A reflector of this kind, in which b =
34" and a = 0, was constructed and tested. The experimental results are
shown in Fig. 13. The minimum is, perhaps, not as low in value as expected
because of residual reflections from the support upon which the reflector was
mounted. As expected, however, the curve passes from a minimum on the
axis to maxima on either side.
The above examples serve to illustrate some of the results which can be
realized with trihedral reflectors. We have seen that the response character-
istic can be controlled by appropriate modifications of the geometrical con-
figuration of the aperture.
Experiments were performed in order to determine the reduction in echo
caused by errors in the internal angles at the corner.
TARGETS FOR MICROWAVE RADAR NAVIGATION
— COMPENSATED REFLECTOR TRIANGULAR REFLECTOR
863
70
60
50
40
30
20
10
0
70
60
50
40
30
[IJ 20
a 10
a
z 0
~ 70
/■
"^
/'■
A
0 = - 40°
1
/^
—
^^"~
■*~^
"*N
\
1
/ y
/ /
/
\
\)
0 = 0°
_
__
/
•'
f
"~~-
'^v
\
/
f
>
\
0 = -3O°
y
<r
—
::>
V
j
-a-
n\^
1
1
1
0 = -2O°
„^'
""
~^-
-.^
^
\/
\j
0=-lO°
^/
y'
-"
1
^N.
\ .
~*
V'
^^
I
0=10°
i
/
^
— -
^
^
^
l\
,
^x^
N \
I
,'
/
s,
J '
0 = 20°
^
---
/
/
y
---'
IN
s.
/I
I
^A
,-•*'
^--
..\
/
0 = 30°
-40 -30 -20 -10
10 20 30 40 -40 -30 -20 -10
ANGLE. 0, IN DEGREES
10 20 30 40 50
Fig. 11 — Resi)onsc of compensated reflector compared witli that of triangular reflector.
In the first set of experiments only one of the mternal angles was altered
from its nominal value of 90°. Figure 14 shows the apparatus used in the
864
BELL SYSTEM TECHNICAL JOURNAL
w 20
- 0
-I 60
-I 50
O
Fig. 12 — Modified reflector having minimum response on axis.
60
If)
aj50
m
uj 40
Q
z
- 30
_i
tu
uj 20
O
F, 10
jl
^
^
y
/
\J
[y
\
0 = 0°
-40 -30 -20 -10
10 20 30 40
r
^
\
i
1
\
<P - -10°
/
— -—
-^
^
/
\
<p = -20°
--^
^
/
^
^
\
<t>- 10°
/
\
0 = 20°
-40 -30 -20
20 30 40 -40 -30 -20
ANGLE. G, IN DEGREES
10 20 30 40
Fig. 13 — Response patterns of reflector of Fig. 12.
TARGETS FOR MICROWAVE RADAR NAVIGATION
865
experiment. The 24-inch reflector was constructed of silver-painted ply-
wood and hinged along the intersection of the two upper surfaces so that the
angle a could be varied at will. A series of response patterns were taken for
various values of a. These are shown in the lower part of Fig. 14. It will
be observed that one effect of changmg a is to lower the echo level. This
appears, however, to be accompanied by a somewhat flatter response curve.
The radar'used in this experiment had a wavelength of 1.25 centimeters.
WAVELENGTH, \ =
1.25 CENTIMETERS
30
Z 0
~ 50
uj 40
I 30
a = 69°
0=0°
a =90°
0=0°
\
^
y^
—
V.
\/
^
X
V
IV
/^
N
Ji
'1/
\)
1 V
V \
20
a = 9i°
0=0°
_>*
I
/
'\
A
\/
\j
a = 92°
0=0°
\ ,
A
^
.A
\/
v'
-40 -30 -20 -10
10 20 30 40 -40 -30 -20 -10
ANGLE, e, IN DEGREES
10 20 30 40
Fig. 14— Effect of an error in one of the corner angles of a trihedral ujion its performance.
A second series of experiments was conducted in which all three internal
angles were varied simultaneously. The curves shown in the upper half of
Fig. 15 show the axial echo level for various sizes of reflector as a function
of the internal angles a. It will be noted that for the 24-mch reflector an
angular error of \ degree results in an echo-level reduction of two decibels,
while the same angular error in the 9f inch reflector produces only a negli-
gible reduction. The lower part of the figure shows some response patterns
866
BELL SYSTEM TECHNICAL JOURNAL
taken with a 24-inch triangular reflector for several values of a. A wave-
length of 1.25 centimeters was used in obtaining both sets of curves. Later,
similar measurements were made at a wavelength of 3.2 centimeters. It
was found that the loss of signal is a function of the Imear error of the aper-
ture in wavelengths rather than the angular error in degrees. Thus a given
angular error in a 9f-inch reflector at a wavelength of 1.25 centimeters will
produce the same loss in signal as the same angular error in a 24-inch reflector
operating at a wavelength of 3.2 centimeters.
I IN
NCHES
'-^>
^^
^'^~*>«,
e=o°
30
^
/
^
r ■
9i
-
20
10
0
—
89 90 91
ANGLE, a, IN DEGREES
I = 24 INCHES
a IN DEGREES =
/
\ ,
>^o
f
1
i
W
/'
.A
^
'^
N
9lN^
^1
v
y
y
\
^2
V
^
y
^
^
ANGLE, 4) =0 DEGREES
WAVELENGTH, A =
1.25 CENTIMETERS
-50 -40 -30 -20 -10 0 10 20 30 40 50
ANGLE, e, IN DEGREES
Fig. 15 — Effect of an error in all three comer angles upon the performance of a trihedral.
Spheres and Cylinders
Formulas for the effective areas of spheres and cylinders which have
dimensions large in comparison with a wavelength have been supplied by
J. F. Carlson and S. A. Goudsmit of the Radiation Laboratory.^ The
effective area of a sphere of radius R is given by
where X is the wavelength.
For a cylinder of radius R and height L, both large with respect to a wave
length, the effective area for rays perpendicular to the axis of the cylinder is
/^
2
'Unpul)lishcd Report
(4)
A = L
/'-
(5)
TARGETS FOR MICROWAVE RADAR NAVIGATION
867
It will be of interest to compare the sphere and the cylinder with corner
reflectors and flat plates. The response pattern of a sphere is ideal in that it
is uniform in all directions. Unfortunately its effective area is small in com-
parison with that of corner reflectors or flat plates having the same cross-
sectional area. The cylinder has a symmetrical response pattern in the plane
FLAT PLATE
WAVELENGTH, X
= 10 CM
Fig. 16— A comparison of several representative targets having equal effective areas.
perpendicular to the axis hut is very sharp ni the plane of the axis. The
ef^'ective area of a cylinder is intermediate between that of a corner reflector
and a sphere. Figure 16 is a scale view of a flat plate, a corner reflector, a
cylinder, and a sphere all having an effective area of one square foot at a
wavelength of 10 centimeters. For shorter wavelengths the flat plate and
868
BELL SYSTEM TECHNICAL JOURNAL
the corner reflector would remain the same size, whereas the cyUnder and
the sphere would have to be larger in order to maintain the same effective
areas. At a wavelength of 1 centimeter the sphere would have to have a
radius of about 60 feet.
S^ 10
i\
\-\.2bCM
J
^
^
.A
M
r
y
1
M
" \
40 -30 -20 -10 0 10 20 30
ANGLE, 0, IN DEGREES
Fig. 17 — Properties of the biconical comer reflector.
BicoNiCAL Corner Reflector
A reflector, combining the 360° horizontal response characteristic of the
cylinder with a vertical response like that of the corner reflector, was evolved
and is illustrated in Fig. 17. As shown here, the device consists of two coni-
cal surfaces placed in juxtaposition such that the generatrices of one cone
intersect those of the other at right angles. The operation of the reflector is
somewhat like that of a dihedral corner reflector in that a ray, upon striking
one of the cones, is reflected to the other and then returns in the direction of
TARGETS FOR MICROWAVE RADAR NAVIGATION 869
the source. The "Biconical" reflector may perhaps be likened to a cylinder
which automatically orients itself so that the impinging rays are always
perpendicular to the axis.
A biconical reflector was constructed of sheet metal having the dimensions
indicated in Fig. 17. The vertical response pattern was measured and is
plotted in the lower portion of the figure. Because of the circular symmetry
of the reflector, the vertical response curve shown will be equally valid for all
angles of azimuth. At </> = 0° the reflector exhibited an effective area of
0.16 square feet. The measurements were made at a wavelength of 1.25
centimeters.
In the above experiment the incident radiation was polarized in the plane
of the axis of the cones. In another test with the polarization perpendicular
to the axis the received echo was reduced by four decibels. This effect is not
as yet entirely explained. It probably results from a depolarizing effect
similar to that encountered in the dihedral corner reflector, complicated
however by the curvature of the cones.
Only a limited amount of data is available for predicting the effective area
of a biconical reflector over a wide range of sizes and wavelengths. The
available data indicate that to a rough approximation and for a given polari-
zation the effective area varies directly as the square root of the wavelength
and as the three-halves power of the diameter of the cones, assuming that
the height of the reflector is approximately equal to the diameter.
Tables of Phase Associated with a Semi-Infinite Unit
Slope of Attenuation
By D. E. THOMAS
This paper presents tables of the phase associated with a semi-infinite unit slope
of attenuation. The phase is given in degrees to .001 degree with an accuracy of
± .001 degree and in radians to .00001 radian with an accuracy of d= .000015
radian. The method of constructing the tables and a brief analysis of the errors
are given. An appendix, which gives a detailed explanation with specific exam-
ples of the use of the tables in determining the phase associated with a given
attenuation characteristic or the reactance associated with a given resistance
characteristic by means of the straight line approximation method given in Bode's
"Network Analysis and Feedback Amplifier Design," is included for the benefit of
those who are not already acquainted with this method. The Appendix also
presents an example of a non-minimum phase network^ in which the minimum
phase determined from the attenuation characteristic fails to predict the true
phase of the network.
THE method described by Bode^ for the determination of the phase
associated with a given attenuation characteristic or the reactance
associated with a given resistance characteristic has proved to be an ex-
tremely useful laboratory and design tool. In this method the attenuation
(or real) characteristic, plotted versus the log of frequency, is approximated
by a series of straight lines. The phase (or imaginary component) is then
determined by summing up the individual contributions of each elementary
straight line segment to the total phase (or imaginary component).
The most elementary straight line characteristic which can be used to
construct a given straight line approximation is that in which the attenua-
tion plotted against the log of frequency is constant on one side of a
prescribed frequency, /o, and has a constant slope thereafter. Such a
characteristic has been called by Bode a "semi-infinite constant slope"
characteristic.^ A semi-infinite unit slope of attenuation or one in which j
the attenuation changes 6 dh per octave, or 20 dh per decade is shown in
Fig. 1. The phase associated with this attenuation characteristic is plotted
in Fig. 2} The independent variable was chosen as///o for values of/ less
than /o and /o// for values of / greater than /o to keep it finite for all values
of/ and in order to show the phase plotted exactly as it is given in the tables
to follow. The phase associated with a semi-infinite constant slope of
' For a complete discussion of minimum phase see Hendrik W. Bode, "Network Analysis
and Feedback Amplifier Design," D. Van Nostrand Company, Inc., New York, N. Y.,
1945.
2 Ibid: Chap. XV, page 344.
■VIbid:Chap. XIV, page 316.
•Il)id: Chap. XIV page 317.
870
TABLES OF PHASE
871
/
/
.<5^
/
/
/
f/
'
o't
\r /
"^ /
A , CONSTANT TO 4r- -0
f 1
1
1
0.1 0.15 0.2 0.3 0.4 0.6 0.8 I 1.5 2 3
RATIO [j^]
Fig. 1 — Semi-infinite unit'slope of attenuation.
4 5 6 7 8 9 10
50
_) 40
<
lij 30
<
I
0. 20
10
0
NOTE: BOTH ORDINATE SCALES APPLY TO
EITHER ABSCISSA RATIO
y
y
/
y
/
/
-
/
/
J
/
-
/
/
y
/
y
/
y
/^
--i-n
^TT
u
0.2 0.4 0.6 0.8 1 0.8 0.6 0.4 0.2
I. J. fo
fo
-^
^_.l
RATIOS
Fig. 2 — Phase associated with semi-infinite unit slope of attenuation of Fig. 1.
lattenuation of the same character as the semi-infinite unit slope of attenua-
jtion of Fig. 1 but of slope k, is k times the phase given in Fig. 2.
872 BELL SYSTEM TECHNICAL JOURNAL
Bode points out,^ however, that the building up of the complete im-
aginary characteristic from a single primitive curve, namely a semi-infinite
real slope, suffers from the disadvantage that the phase contributions of the
individual slopes may be rather large positive and negative quantities,
even though the net phase shift is fairly small. In order to avoid this dis-
advantage, Bode recommends that the individual finite line segments which
constitute the straight line approximation to the real characteristic be
regarded as the elementary characteristics used in the summation of the
total phase. He then gives a series of charts, plotted as a function of ///"o,
of the phase associated with a finite line segment having a 1 db change in
attenuation and with a ratio of the geometric mean frequency (/o) of the
two termmal frequencies of the finite line segment to the lower terminal
frequency as a parameter (ratio designated a).
However, problems have arisen where, even with the finite line segment
phase charts, the phase contributions of the various elements were suth-
ciently large and nearly equal positive and negative quantities that diffi-
culties in interpolation between the curves for the various values of a, given
on the charts, resulted in a sufficient lack of precision that the quantity
being sought was lost.
Because of the usefuhiess of the method in question, and with its applica-
tion to a wider variety of problems, means of increasing its over-all precision
and simplification of computation have constantly been sought. It had
occurred to several engineers independently that a table of phase versus
frequency for a semi-infinite unit slope of attenuation would prove extremely
useful. The phase in radians at frequency /c, associated with a semi-
infinite unit slope of attenuation commencing at frequency /o, is given by
Bode as®
£W=?(.., + | + |+...) (1)
where:
7o Wo
The computation time required to determine the phase at a given frequency
by summation of the above series is such, that the work required to get the
phase at a sufficient number of points and to a sufficient number of sig-
nificant figures to prepare an adequate table proved to be sufficient to dis-
courage this procedure.
5 Ibid: Chap. XV page 338.
"Ibid: Chap. XV, page 343.
TABLES OF PHASE
873
The derivative of (1) above, however, proves to be quite simple and easy
to evaluate. It is given by Bode as:
dB 1 -
-— = — log
dXc TTXc
I — Xc
= l{'^hh-)^^^<'-
(2)
(2a)
It therefore seemed that since the phase had already been computed by the
Mathematical Research Group of the Bell Telephone Laboratories, Inc., at a
B(X), PHASE ASSOCIATED WITH SEMI-
1
INFINITE UNIT SLOPE OF ATTENUATION
B(x
■ A .^ !/
C+AX) — y -+-
STRAIGHT- LINE APPROXIMATION TO
6^6 /
ELEMENT OF B(x)
■\'--p\--
,)/ \
''*' y ' '
^^
^^*
./ 1 '
CD
^
^r 1 1
^— '
1 ^*
^r ' (
UJ
1 *'
>
1 '
to
<
I
Q.
^,''' iSjB^
y
d B / Ax K 1
^(xc+^^]ax I
B(xc)
i_ 1
__^*r^ 1
^^^"^^ ' ' 1
' > 1
' 1 1
1 1 1
xc
Xc+AX
■^0
Fig. 3 — Element of Fig. 2 for///o < 1 expanded qualitatively.
considerable number of points, using the infinite series expansion of (1)
above the function in the regions between known values of phase could be
constructed by assuming the intervening curve of phase as a function of
a; = - to be a series of straight lines having the slope given by (2) above
over intervals Ax of x made sufficiently small that the resultant straight
line approximation would approach the true phase curve to the desired
degree of accuracy for the table contemplated.
874
BELL SYSTEM TECHNICAL JOURNAL
In order to evaluate the errors involved in such a procedure let us refer
to Fig. 3 where a segment of the desired phase function to be constructed is
qualitatively represented on a large scale. It is assumed that the phase at
Xc, B{xc), is known and that it is desired to determine the error diB in phase
computed for Xc + Ax when it is assumed that the phase curve is a straight
line from B{xc) at Xc, to Xc + Ax having a slope, — ( ^"c + ^ ) , the slope
dx
of the true phase curve a,t x = Xc -r — ■
Then :
dE I Ax^
hB ^ B(xc + Ax) - B(xc) - ^ U'<= + y ) ^^
where;
9 1 -v* T
B(x,) = - X. + ^ + ?^ +
5
25
Bixc + Ax) = - [{xc + Ax) + i{xl + 3x1 Ax + SxoAx^ + Ax^)
TT
+ ^(^c + Sxt Ax + lOxl A.T- + lOxl Ax^ + 5xc Ax* + Ax^) +
B{xc + Ax) — B{Xc) = -
, x;A.r , XcAx'
Ax + -^ — + —
3 3
Ax , XcAx , 2XcAx , 2XcAx^ , avAx'
^95
+
+
Ax"
5 +25 +
A.v I]
,tri 2w
2n— 2 n=oo 2re— 1
+ Ax 2^
</x
,. + f,A.
9
/ 1
-Ax
il + :;
TT
V 3
1 n^l 2W + 1
' n(2n — l)x\
Ax\ /Ax
+ Ax^ XI
•Vc + 2x,
3 (In + 1)
2
+
+ ^ I -v: + 4x
2
TT
4 .
x. Ax
. , XoAx XcAx^ A.x^ ..^ —
Ax + + + — + +
3 3 12 5
2.V, Ax"
yjJi'/' ^Jkvv
A 4 .5
^^c^^ XcAx , Ax
10 10 80
]
n=oo 2n— 2
AxE ""
„^ 2w - 1 ±? 2w + 1
n=«; /r^ i \ 2n— 2
3 y n{2n - l)x,
+ ^-^ £l " 4(2;. + 1) +
(3)
]
l['-+-Kf)--:(f)" -•(¥)■+(¥)>-)
TABLES OF PHASE 875
Since A.r will be small compared to unity and since an error function is
being computed it is permissible to take only the 1st term of the difference
between the true phase and the computed phase, i.e. the Ax^ term, and drop
all higher order terms of Ax.
Then :
81 B
/r, ■i\ 2.n—l n^aa /n < \ ^n-
n{2n — l)Xc A .3 \^ "(2''^ ~ ^)-^<=
Ax' Z ^" ~ '^ : - A.r^ Z
3{2n + 1) ■ ,tt 4(2w + 1)
A.V -^ n\2n — \)Xc
(4)
E
6x ;;rt 2w + i
The equation (4) above for h^B gives only the error for a single increment
A.T of X = ///o. If the phase is known at x — Xa and x = .Xb and it is desired
to determine the phase at points between x = Xa and x — Xb then since 81B
always has the same sign the errors due to successive increments of x will be
cumulative and the total error at x = x & will be n times the average of the
diB errors of each increment of Ax between Xa and Xb where n is the total
number of equi-increments of x taken between Xa and Xb- However, since
the individual 81B errors decrease as the cube of Ax, the individual errors
will decrease as the cube of the number of increments taken between the
two frequencies at which the phase is known, whereas the cumulative 81B
error will increase only in proportion to the iirst power of n. Therefore,
the net result will be a vanishing of the cumulative error inversely as the
square of the number of frequency increments taken to approximate the
curve in the interval in question. It therefore follows that the accuracy
of the proposed method of building up the function, in so far as the phase
at the terminals of the straight line segments is concerned, is limited only
by the number of increments of frequency selected for the summation.
In order to determine the actual magnitude of errors to be expected 81B
was computed for Xc = A and Ax — .02 and found to be only .000015 degree.
Since the total number of .02 intervals needed to be used between previously
computed values of 5 is 5, the total cumulative error in this region for
increments of this magnitude will not be greater than .0001 degree, which
is entirely satisfactory, since the accuracy being sought is ± .0005 degree in
B. For Xc = .9 and Ax = .005 the 81B error proves to be only .00001 degree
and since in this region the value of B has already been determined at .01
intervals by the more accurate series expansion technique referred to above,
only two increments are necessary between known values of B and therefore
the 81B error is sufficiently small.
Having determined the order of magnitude of intervals necessary to keep
81B errors small, let us examine the errors due to the departure of the straight
line approximation from the true curve in the interval between Xc and Xc +
Ax. Since 81B will be very small it is anticipated that the maximum value
876 BELL SYSTEM TECHNICAL JOURNAL
Ax
of 52-B (see Fig. 3) will occur in the vicinity of Xc -\- — . 52^ at this point
may be determined as shown below,
where :
V^^ + 2 y' 2 xL h 2{2n - 1) ^ ^"^ „4( 2(2/^ + 1) + J'
dB
dx
Again retaining only the first term of the error function and dropping all
higher order terms of Ax
2n-l -1
n, r n=oo 2n— 1 n=«
TT L «=i 4(2w + 1) „=1
nxr.
„ri 2(2w + 1)
(6)
■ 2 n=» 2n-l ^ '^
_ Ax "^ nXc
~ ~ 1^ hi 2n + r
52jB proves to be negative and considerably larger than 8iB for the same
magnitude of interval. Therefore the computed B will always exceed the
true phase in the interval x,- to x^ + Ax except above a value of x very near
to Xc + Ax where the straight line approximation crosses the true phase
curve. When Xc = .35 and Ax = .02, 82B is found to be —.0005 degree
from (6) above, and for Xc = .91 and Ax — .005, 82B is also found to be
— .0005 degree. The 82B errors are therefore found to be much more im-
portant than the 81B errors. 82B errors are not accumulative, however,
and therefore increments of Ax of the above order of magnitude prove to be
sufficiently small to give the accuracy being sought, namely ± .0005
degree in B.
An evaluation of the 81B and 82B errors for values of Xc greater than .9
is difficult due to the slowness of convergence of the series giving these errors.
For values of x^ between .9 and unity, however, the frequency of known
values of /^determined from (1) above and available as check points is suffi-
cient to check the adequacy of intervals insofar as 81B errors are concerned.
Furthermore an analysis similar to that given above for the determination
of the 5i/> and 82B errors shows that an interpolation of the slopes computed
for construction of the tables in question, to give the intervening slopes
necessary to cut the increments of Ax in half will give check points at Xc +
Ax Ax
— frequencies, with a 81B error (Xc + -7^ is then the termination of a straight
TABLES OF PHASE 877
line segment since the Aa; interval has been halved) of comparable order of
magnitude to the 8iB error for the original interval selected and therefore
small in comparison to the 52^ error for the original Ax interval. This
technique was therefore used in checking the adequacy of the intervals in
so far as 52^ errors are concerned in the region Xc = .9 to Xc = 1.0.
Using the procedure outlined above the phase associated with the semi-
infinite unit slope of attenuation of Fig. 1 was computed for values of /less
than /o and is given as a function of ///o in Table I in degrees and in Table
III in radians. For values of/ greater than/o the phase was computed as a
function of /o// utilizing the odd symmetry behavior of the phase char-
acteristic of Fig. 2 on opposite sides of ///o = 1, and this phase is tabulated
in Table II in degrees and in Table IV in radians. For the other type of
semi-infinite unit slope of attenuation in which the attenuation slope is
constant and equal to unity at all frequencies below /o and the attenuation
is constant for all frequencies above /o (with the constant slope of attenua-
tion intersecting the /o axis at the same point as the constant attenuation
line) the same tables can be used by reading the values of phase for///o < 1
from the/o// tables and the values of phase for/o// < 1 from the ///o tables.
The intervals over which the straight line approximation to the true phase
was assumed are given below:
.02 from .00 to .40
.01 " .40 " .70
.005 " .70 " .92
.002 " .92 " .98
.001 " .98 " .996
.0005 " .996 " .998
.0002 " .998 " .999
.0001 " .999 " .9998
.00005 " .9998 " 1.0000
The points at which the cumulative sum of the straight line increments
of phase was corrected to the phase as determined from (1) above are listed
below :
Every
.1
from .00
to
.40
u
.05
.40
"
.80
"
.02
.80
"
.90
"
.01
.90
"
.99
and at .996,
.998,
.999, and 1.000
A study of the errors based on the error analysis discussed above indicates
that the computed values of B in degrees are accurate to ± .0005 degree and
since there is an additional possibility of ± .0005 degree error in dropping
all figures beyond the third decimal place, the over-all reliability of the degree
tables is d= .001 degree. Similarly the computed values of B in radians are
accurate to ± .00001 radian and since there is an additional possibility of
± .000005 radian error in dropping all figures beyond the fifth decimal
878 BELL SYSTEM TECHNICAL JOURNAL
place, the over-all reliability of the radian tables is ± .000015 radian.
Since the function tabulated was constructed by a series of straight line
approximations to the true phase, interpolation to get the phase for values
of ///o or/o// between those given in the tables in problems where this is
necessary, will result in the same accuracy as that given for the tabulated
values.
Murlan S. Corrington'^ of Radio Corporation of America has computed
the phase in radians for the semi-infinite unit slope of attenuation of Fig. 1
for approximately 100 values of ///o using equations 15-9 and 15-11 of
Bode's "Network Analysis and Feedback Amplifier Design" and has given a
table of these values to five decimal places. Where the values of Table III
difi"er from Corrington's values, his value is given as a superscript. Since
his approach is the more exact one, it is assumed that where a difference
exists, his value is correct. The differences have a maximum value of one
figure in the fifth decimal place which is consistent with the accuracy of
± .000015 radian given for Table III. However, linear interpolation of
Corrington's values to get the function to three figures in///o, which preci-
sion in //'/o is really needed to utilize five figure accuracy in B, will result
in errors considerably larger than those of Table III for the higher values of
///o.
Acknowledgment
The writer wishes to thank Miss J. D. Goeltz who carried out the calcula-
tions of the basic Tables and of the illustrative examples of this paper.
APPENDIX
Use of Tables I to IV' in Determining Phase from Attenu.a.tion or
Reactance from Resistance
The first step in determining the phase associated with a given attenua-
tion characteristic using the tables described in the basic paper is to plot
the attenuation as a function of log frequency to a suitable scale. Such an
attenuation characteristic is illustrated in Fig. 4a. The attenuation char-
acteristic is then approximated by a series of straight lines such as are shown
in dotted form. The number of straight lines used will depend upon the
accuracy desired in the resultant })hase. As a rule, an apjn'oximation to the
attenuation which does not depart by more than dz .5 db will give a resultant
phase which does not depart by more than ± 3° from the true phase.
If we now examine the straight line attenuation apj^roximation of Fig. 4a,
'Murlan S. Corrington, "Tabic of the JntcKral - / • dl" K.C.A. Review
IT Jo I
September, 1946, page 432.
TABLES OF PHASE
879
we see that it can be constructed by adding a number of semi-infinite con-
stant slopes of attenuation as shown in Fig. 4b. The first of these will be a
semi-infinite slope of magnitude ki commencing at the first critical frequency
f
0
fl
f2
f3
f
4
(a)
/
V
ATTENUATION (A)
STRAIGHT-LINE APPROXIMATION TO A
5
0
/
\
\
//s
LOPE
= K,
V-
K2
-5
\\
\
\
^
*^3.,
-10
^.
;^K.
-15
20
\
V
,1
1
\
^
. 1
(b)
/
/
/
c
?1
f
7
(
/
/
/
/
I
\
\
K3
\
s.
\
\
^
\
\
1
\
\
\
1 ,
0.6 0.8 1 1.5
FREQUENCY (f )
5 6 7 8 9 10
Fig. 4 — (a) Straight line approximation to attenuation characteristic, (b) Individual
semi-infinite constant slopes of attenuation which add to produce the straight line approxi-
mation of Fig. 4(a).
/o. The second will be a semi-infinite slope of magnitude —^1 commencing
at the critical frequency /i which must be added to correct for the fact that
the first straight line of slope +^1 does not extend to infinity, but terminates
at the critical frequency /i, where the straight line approximation assumes a
880 BELL SYSTEM TECHNICAL JOURNAL
new slope. In order to achieve this new slope a semi-infinite slope of mag-
nitude ko, commencing at frequency /i, must be added. This process is
continued up the frequency scale until the entire straight line approxima-
tion is constructed.
The total phase d{f) at a particular frequency/ is then given by the sum :
of the phase at frequency/ associated with each of the semi-infinite constant
slopes of attenuation which together make up the straight line ap-
proxmation.
Thus:
Oil) - ^1^0 - kA + kidi - kiSi + hG2 - hds + kids - kSi
or for the general straight line approximation having slopes
ki , ki J • • • k„
d{f) - h (do - ^l) + h (dl - ^2) + • • • ^n (^„-l - On)
where :
dn is the phase at frequency/ associated with the semi-infinite unit slope
of attenuation commencing at frequency /„ and extending to / = 00
and is read from Tables I or III for / < /« and Tables II or IV for
/ >/",
and
kn is the slope of the straight line approximation between /„_i and /„ \
given by:
7 ■'^n .^n— 1
20 log f-
Jn-l
where:
An is, the attenuation at frequency /„ on the straight line approximation.
Note that in Fig. 4a the attenuation is constant from zero frequency to
the first critical frequency /o- In many problems, there is a constant slope
below frequency /i to frequency zero. In that event, the initial critical
frequency, /o, will be zero, and 60 will be 90°. (/o// = 0 at all finite fre-
quencies.) When this occurs, ^1 must be determined by choosing a finite
frequency /o and taking the ratio of attenuation change between /o and /i
to 20 log of the ratio of /i to/o. Similarly, the attenuation is constant in
the illustration from the top critical frequency fi to infinity, whereas in
many problems the attenuation will have a constant slope extending from
the top critical frequency to infinity. In these cases, the top critical fre-
quency will be infinity and the final angle 0„ will, of course, be zero. Here
again the final slope k„ must be determined over a finite portion of this
infinite slope.
TABLES OF PHASE 881
It will also be noted that in the illustration given the characteristic is
approximated, commencing at zero frequency, by a series of semi-infinite
slopes, each of which is a constant times the characteristic of Fig. 1 of the
basic paper, for which Tables I to IV were computed. The characteristic
could have been approximated just as well with a series of semi-infinite
constant slopes, commencing at / = oo and going down in frequency, each
having a flat attenuation above a critical frequency /„ and constant slope
at frequencies below. In summing the phase for such an approximation
Tables I to IV may be used by reading the angles for ///„ from the /o//
tables and vice versa as indicated in the basic paper.
As an illustration of the above procedure, consider the determination of
the phase associated with the characteristic given by 20 log ] Z [ shown in
Fig. 5. The characteristic is first approximated by a series of straight lines
as shown in dotted form. The critical frequencies and values of ^ = 20
log I Z I at these critical frequencies are then read from the straight line
approximation^ and the slopes of the various straight line segments deter-
mined as illustrated in Table V.
Having determined the slopes of the various segments of the straight line
approximation, the phase at any desired frequency is summed as illustrated
in Table VI where the phase for/ = 1.5 is summed.
The mesh computed value of d for the network in question is plotted in
Fig. 6 and it will be noted that the phase summation of Table VI checks the
true value to within the accuracy to which the phase can be read from the
curve. The identical procedure is followed in determining the phase at
any other frequency. As an illustration of the accuracy of the method, the
phase was determined at a considerable number of frequencies and the results
shown as individual points in Fig. 6. The straight line approximation to
20 log I Z I of Fig. 5 was of the order of ± .25 db and, in accordance with the
estimated accuracy of the method given above, the maximum departure of
the phase summation from the true phase is approximately ± 1.5°.
A much simpler approximation than that of Fig. 5 may be used without a
great loss in accuracy. For instance, a five-line approximation determined
by the critical frequencies of Table VII will match 20 log | Z | to within
approximately ± .5 rfZ* and therefore should give a phase summation
within ± 3° of the true phase. The phase was actually summed at 12 fre-
quencies chosen at random for this five-line approximation and the maxi-
mum departure of the summed phase from the true phase was 3.2°. With
experience in use of the method, simpler approximations can be used and
; the phase determined more accurately than the limits of accuracy of the
summation at individual frequencies by plotting the individual summations
* The original plot was expanded and had much greater scale detail than can be shown
, with clarity on a single page plate.
Table I — Degrees Phase (±.001°) for Semi-Intinite AxTENtrATiON Slope k == If </o
///o
0
1
2
3
.109
4
.146
5
6
7
8
9
.00
.000
.036
.073
.182
.219
.255
.292
.328
.01
.365
.401
.438
.474
.511
.547
.584
.620
.657
.693
.02
.730
.766
.803
.839
.875
.912
.948
.985
1.021
1.058
.03
1.094
1.131
1.167
1.204
1 . 240
1.277
1.313
1.350
1.386
1.423
.04
1.459
1.496
1.532
1 . 569
1.605
1.642
1.678
1.715
1.751
1.788
.05
1.824
1.861
1.897
1.934
1.970
2.007
2.043
2.080
2.116
2.153
.06
2.189
2.226
2.262
2.299
2.335
2.372
2.409
2.445
2.482
2.518
.07
2.555
2.591
2.628
2.664
2.701
2.737
2.774
2.810
2.847
2.884
.08
2.920
2.957
2.993
3.030
3.066
3.103
3.140
3.176
3.213
3.249
.09
3.286
3.322
3.359
3.396
3.432
3.469
3.505
3.542
3.578
3.615
.10
3.652
3.688
3.725
3.762
3.798
3.835
3.871
3.908
3.945
3.981
.11
4.018
4.054
4.091
4.128
4.164
4.201
4.238
4.274
4.311
4.347
.12
4.384
4.421
4.457
4.494
4.531
4.568
4.604
4.641
4.678
4.714
.13
4.751
4.788
4.824
4.861
4.898
4.934
4.971
5.008
5.044
5.081
.14
5.118
5.155
5.191
5.228
5.265
5.302
5.338
5.375
5.412
5.449
.15
5.485
5.522
5.559
5.596
5.632
5.669
5.706
5.743
5.779
5.816
.16
5.853
5.890
5.927
5.963
6.000
6.037
6.074
6.111
6.148
6.184
.17
6.221
6.258
6.295
6.332
6.369
6.405
6.442
6.479
6.516
6.553
.18
6.590
6.626
6.663
6.700
6.737
6.774
6.811
6.848
6.885
6.922
.19
6.959
6.996
7.033
7.070
7.106
7.143
7.180
7.217
7.254
7.291
.20
7.328
7.365
7.402
7.439
7.476
7.513
7.550
7.587
7.624
7.661
.21
7.698
7.735
7.772
7.809
7.846
7.883
7.920
7.957
7.994
8.032
.22
8.069
8.106
8.143
8.180
8.217
8.254
8.291
8.329
8.366
8.403
.23
8.440
8.477
8.514
8.551
8.589
8.626
8.663
8.700
8.737
8.774
.24
8.811
8.849
8.886
8.923
8.960
8.998
9.035
9.072
9.109
9.147
.25
9.184
9.221
9.259
9.296
9.333
9.370
9.408
9.445
9.482
9.519
.26
9.557
9.594
9.631
9.669
9.706
9.744
9.781
9.818
9.856
9.893
.27
9.931
9.968
10.006
10.043
10.080
10.118
10.155
10.193
10.230
10.267
.28
10.305
10.342
10.380
10.417
10.455
10.492
10.530
10.568
10.605
10.643
.29
10.680
10.718
10.755
10.793
10.830
10.868
10.906
10.943
10.981
11.018
.30
11.056
11.094
11.131
11.169
11.207
11.244
11.282
11.320
11.358
11.395
.31
11.433
11.471
11.508
11.546
11.584
11.622
11.659
11.697
11.735
11.772
.32
11.810
11.848
11.886
11.924
11.962
12.000
12.037
12.075
12.113
12.151
.33
12.189
12.227
12.265
12.303
12.341
12.379
12.416
12.454
12.492
12.530
.34
12.568
12.606
12.644
12.682
12.720
12.758
12.797
12.835
12.873
12.911
.35
12.949
12.987
13.025
13.063
13.101
13.139
13.177
13.215
13.254
13.292
.36
13.330
13.368
13.406
13.445
13.483
13.521
13.559
13.598
13.636
13.674
.37
13.713
13.751
13.789
13.827
13.866
13.904
13.942
13.981
14.019
14.057
.38
14.096
14.134
14.173
14.211
14.250
14.288
14.327
14.365
14.404
14.442
.39
14.481
14.519
14.558
14.596
14.635
14.673
14.712
14.750
14.789
14.827
.40
14.866
14.905
14.943
14.982
15.021
15.059
15.098
15.137
15.175
15.214
.41
15.253
15.292
15.330
15.369
15.408
15.447
15.486
15.525
15.563
15.602
.42
15.641
15.680
15.719
15.758
15.797
15.836
15.875
15.914
15.953
15.991
.43
16.030
16.070
16.109
16.148
16.187
16.226
16.265
16.304
16.343
16.382
.44
16.421
16.460
16.500
16.539
16.578
16.617
16.657
16.696
16.735
16.774
.45
16.813
16.853
16.892
16.931
16.971
17.010
17.050
17.089
17.128
17.168
.46
17.207
17.247
1 7 . 286
17.326
17.365
17.405
17.444
17.484
17.523
17.563
.47
17.602
17.642
17.681
17.721
17.761
17.800
17.840
17.880
17.919
17.959
.48
17.999
18.039
18.078
18.118
18.158
18.198
18.238
18.277
18.317
18.357
.49
18.397
18.437
18.477
18.517
18.557
18.597
18.637
18.677
18.717
18.757
.50
18.797
18.837
18.877
18.917
18.958
18.998
19.038
19.078
19.118
19.158
.51
19.198
19.239
19.279
19.320
19.360
19.400
19.441
19.481
19.521
19.562
.52
19.602
19.642
19.683
19.723
19.764
19.804
19.845
19.885
19.926
19.967
.53
20.007
20.048
20.088
20.129
20.170
20.211
20.251
20.292
20.333
20.373
.54
20.414
20.455
20.496
20.537
20.578
20.619
20.660
20.701
20.741
20.782
882
Table I — Continued
0
20.823
21.234
21.648
22.063
22.481
22.901
23.324
23 . 749
24.177
24.607
25.041
25.478
25.917
26.361
26.807
27.257
27.711
28.169
28.631
29.097
29.568
30.043
30.524
31.010
31.502
32.000
32 . 504
33.015
33.533
34.059
34.594
35.138
35.691
36.256
36.832
37.422
38.026
38.647
39.287
39.949
40.638
41.358
42.120
42.938
43.846
1
20.864
21.276
21.689
22 . 105
22.523
22.943
23.366
23.792
24.220
24.651
25.085
25 . 522
25.962
26.405
26.852
27.302
27.757
28.215
28.677
29.144
29.615
30.091
30.572
31.059
31.551
32.050
32.555
33.066
33.586
34.113
34.648
35.193
35.747
36.313
36.891
37.482
38.088
38.710
39.352
40.017
40.708
41.432
42.199
43.024
43.945
20.906
21.317
21.731
22.147
22.565
22.986
23.409
23.834
24.263
24.694
25.128
25.566
26.006
26.450
26.897
27.348
27.802
28.261
28.724
29.191
29.663
30.139
30.621
31.108
31.601
32 . 100
32 . 606
33.118
33.638
34.166
34.702
35.248
35.804
36.370
36.949
37.542
38.149
38.773
39.418
40.085
40.779
41.507
42.278
43.111
44.045
20.947
21.358
21.772
22.189
22.607
23.028
23.451
23.877
24.306
24.738
25.172
25.610
26.050
26.494
26.942
27.393
27.848
28.307
28.770
29.238
29.710
30.187
30.669
31.157
31.651
32.150
32.657
33.170
33.690
34.219
34.756
35.303
35.860
36.428
37.008
37.602
38.211
38.837
39.483
40.153
40.850
41.582
42.359
43.199
44.148
20.988
21.400
21.814
22.230
22.649
23.070
23.494
23.920
24.349
24.781
25.216
25.654
26.095
26.539
26.987
27.438
27.894
28.353
28.817
29.285
29.757
30.235
30.718
31.206
31.700
32.201
32.707
33.221
33.743
34.272
34.810
35.358
35.916
36.485
37.067
37.662
38.273
38.901
39.549
40.221
40.921
41.657
42.439
43.288
44.253
21.029
21.441
21.855
22.272
22.691
23.112
23 . 536
23.963
24.392
24.824
25 . 259
25.698
26.139
26.584
27.032
27.484
27.939
28.399
28.863
29.332
29.805
30.283
30.766
31.255
31.750
32.251
32.758
33.273
33.795
34.325
34.865
35.413
35.972
36.542
37.125
37.722
38.3.34
38.965
39.615
40.290
40.993
41.733
42.521
43.378
44.361
21.070
21.482
21.897
22.314
22.733
23.155
23.579
24.006
24.435
24.868
25.303
25.742
26.183
26.628
27.077
27.529
27.985
28.445
28.910
29.379
29.853
30.331
30.815
31.305
31.800
32.301
32.810
33.325
33.848
34.379
34.919
35.469
36.029
36.600
37.184
37.783
38.397
39.029
39.681
40.359
41.066
41.809
42 . 603
43.469
44.473
21.111
21.524
21.939
22.356
22.775
23.197
23.621
24.048
24.478
24.911
25.347
25.786
26.228
26.673
27.122
27.574
28.031
28.492
28.957
29.426
29.900
30.379
30.864
31.354
31.850
32.352
32.861
33.377
33.901
34.433
34.974
35 . 524
36.086
36.658
37.244
37.844
38.459
39.093
39.748
40.428
41.138
41.887
42.686
43.561
21.152
21.565
21.980
22.397
22.817
23.239
23.664
24.091
24.521
24.954
25.390
25.830
26.272
26.718
27.167
27.620
28.077
28.538
29.003
29.473
29.948
30.428
30.913
31.403
31.900
32.403
32.912
33.429
33.954
34.486
35.028
35.580
36.142
36.716
37.303
37.904
38.522
39.157
39.815
40.497
41.211
41.964
42 . 769
43.655
21.193
21.606
22.022
22.439
22.859
23.281
23 . 706
24.134
24.564
24.998
25.434
25.873
26.316
26.762
27.212
27.665
28.123
28.584
29.050
29.521
29.996
30.476
30.961
31.453
31.950
32.453
32.963
33.481
34.006
34.540
35.083
35 . 636
36.199
36.774
37.362
37.965
38.584
39.222
39.882
40.567
41.285
42.042
42.854
43.750
(refer to table below)
.9960
44.473
9984
44.763
.9992
44.871
.9965
44.530
9985
44.776
.9993
44.886
.9970
44.589
9986
44.789
.9994
44.900
.9975
44.649
9987
44.802
.9995
44.915
.9980
44.711
998S
44.816
.9996
44.931
.9981
44.724
9989
44.829
.9997
44.946
.9982
44.737
9990
44.843
.9998
44.963
.9983
44.750
9991
44.857
.9999
1.0000
44.980
45.000
883
Table II— Degrees Phase (±.00r
FOR Semi-Infinite Attentjation Slope k = 1
/>/o
/»//
0
1
2
3
4
5
6
7
8
9
.00
90.000
89.964
89.927
89.891
89.854
89.818
89.781
89.745
89.708
89.672
.01
89.635
89.599
89.562
89.526
89.489
89.453
89.416
89.380
89.343
89.307
.02
89.270
89.234
89.197
89.161
89.125
89.088
89.052
89.015
88.979
88.942
.03
88.906
88.869
88.833
88.796
88.760
88.723
88.687
88.650
88.614
88.577
.04
88.541
88.504
88.468
88.431
88.395
88.358
88.322
88.285
88.249
88.212
.05
88.176
88.139
88.103
88.066
88.030
87.993
87.957
87.920
87.884
87.847
.06
87.811
87.774
87.738
87.701
87.665
87.628
87.591
87.555
87.518
87.482
.07
87.445
87.409
87.372
87.336
87 . 299
87.263
87.226
87.190
87.153
87.116
.08
87.080
87.043
87.007
86.970
86.934
86.897
86.860
86.824
86.787
86.751
.09
86.714
86.678
86.641
86.604
86.568
86.531
86.495
86.458
86.422
86.385
.10
86.348
86.312
86.275
86.238
86.202
86.165
86.129
86.092
86.055
86.019
.11
85.982
85.946
85.909
85.872
85.836
85.799
85.762
85.726
85.689
85.653
.12
85.616
85.579
85.543
85.506
85.469
85.432
85.396
85.359
85.322
85.286
.13
85.249
85.212
85.176
85.139
85.102
85.066
85.029
84.992
84.956
84.919
.14
84.882
84.845
84.809
84.772
84.735
84.698
84.662
84.625
84.588
84.551
.15
84.515
84.478
84.441
84.404
84.368
84.331
84.294
84.257
84.221
84.184
.16
84.147
84.110
84.073
84.037
84.000
83.963
83.926
83.889
83.852
83.816
.17
83.779
83.742
83 . 705
83.668
83.631
83.595
83.558
83.521
83.484
83.447
.18
83.410
83.374
83.337
83.300
83 . 263
83.226
83.189
83.152
83.115
83.078
.19
83.041
83.004
82.967
82.930
82.894
82.857
82.820
82.783
82.746
82.709
.20
82.672
82.635
82.598
82.561
82.524
82.487
82.450
82.413
82.376
82.339
.21
82.302
82.265
82.228
82.191
82.154
82.117
82.080
82.043
82.006
81.968
.22
81.931
81.894
81.857
81.820
81.783
81.746
81.709
81.671
81.634
81.597
.23
81.560
81.523
81.486
81.449
81.411
81.374
81.337
81.300
81.263
81.226
.24
81.189
81.151
81.114
81.077
81.040
81.002
80.965
80.928
80.891
80.853
.25
80.816
80.779
80.741
80.704
80.667
80.630
80.592
80.555
80.518
80.481
.26
80.443
80.406
80.369
80.331
80.294
80.256
80.219
80.182
80.144
80.107
.27
80.069
80.032
79.994
79.957
79.920
79.882
79.845
79.807
79.770
79.7J3
.28
79.695
79.658
79.620
79..S83
79.545
79.508
79.470
79.432
79.395
79.357
.29
79.320
79.282
79.245
79.207
79.170
79.132
79.094
79.057
79.019
78.982
.30
78.944
78.906
78.869
78.831
78.793
78.756
78.718
78.680
78.642
78.605
.31
78.567
78.529
78.492
78.454
78.416
78.378
78.341
78.303
78.265
78.228
.32
78.190
78.152
78.114
78.076
78.038
78.000
77.963
77.925
77.887
77.849
.a
77.811
77.773
77.735
77.697
77.659
77.621
77.584
77.546
77.508
77.470
.34
77.432
77.394
77.356
77.318
77.280
77.242
77.203
77.165
77.127
77.089
.35
77.051
77.013
76.975
76.937
76.899
76.861
76.823
76.785
76.746
76.708
.36
76.670
76.632
76.594
76.555
76.517
76.479
76.441
76.402
76.364
76.326
.37
76.287
76.249
76.211
76.173
76.134
76.096
76.058
76.019
75.981
75.943
.38
75.904
75.866
75.827
75.789
75.750
75.712
75.673
75.635
75.596
75.558
.39
75.519
75.481
75.442
75.404
75.365
75.327
75.288
75.250
75.211
75.173
.40
75.134
75.095
75.057
75.018
74.979
74.941
74.902
74.863
74.825
74.786
.41
74.747
74.708
74.670
74.631
74.592
74.553
74.514
74.475
74.437
74.398
.42
74.3,59
74.320
74.281
74.242
74.203
74.164
74.125
74.086
74.047
74.009
.43
73.970
73.930
73.891
73.8.52
73.813
73.774
73.735
73.696
73.657
73.618
.44
73.579
73.540
73.500
73.461
73.422
73.383
73.343
73.304
73.265
73.226
.45
73.187
73.147
73 . 108
73.069
73.029
72.990
72.950
72.911
72.872
72.832
.46
72.793
72.753
72.714
72.674
72.635
72.. 595
72.5.S6
72.516
72.477
72.437
.47
72.398
72.3.S8
72.319
72.279
72.239
72.200
72.160
72.120
72.081
72.041
.48
72.001
71.961
71.922
71.882
71.842
71.802
71.762
71.723
71.683
71.643
.49
71.603
71.563
71.. 523
71.483
71.443
71.403
71.363
71.323
71.283
71.243
.50
71.203
71.163
71.123
71.083
71.042
71.002
70.962
70.922
70.882
70.842
.51
70.802
70.761
70.721
70.680
70.640
70.600
70..S59
70.519
70.479
70.438
..S2
70.398
70.. 358
70.317
70.277
70.236
70.196
70.155
70.115
70.074
70.033
.53
69.993
69.952
69.912
69.871
69.830
69.789
69.749
69.708
69.667
69.627
..54
69.. 586
69.545
69.504
69.463
69.422
69.. 381
69.340
69.299
69.259
69.218
884
Table II — Continued
fo/f
0
1
2
3
4
5
6
7
8
9
.55
69.177
69.136
69.094
69.053
69.012
68.971
68.930
68.889
68.848
68.807
.56
68.766
68.724
68.683
68.642
68.600
68.559
68.518
68.476
68.435
68.394
.57
68.352
68.311
68.269
68.228
68.186
68.145
68.103
68.061
68.020
67.978
.58
67.937
67.895
67.853
67.811
67.770
67.728
67.686
67.644
67.603
67.561
.59
67.519
67.477
67.435
67.393
67.351
67.309
67.267
67.225
67.183
67.141
.60
67.099
67.057
67.014
66.972
66.930
66.888
66.845
66.803
66.761
66.719
.61
66.676
66.634
66.591
66.549
66.506
66.464
66.421
66.379
66.336
66.294
.62
66.251
66.208
66.166
66.123
66.080
66.037
65.994
65.952
65.909
65.866
.63
65.823
65 . 780
65.737
65.694
65.651
65.608
65 . 565
65.522
65.479
65.436
.64
65.393
65.349
65.306
65.262
65.219
65.176
65.132
65.089
65.046
65.002
.65
64.959
64.915
64.872
64.828
64.784
64.741
64.697
64.653
64.610
64.566
.66
64.522
64.478
64.434
64.390
64.346
64.302
64.258
64.214
64.170
64.127
.67
64.083
64.038
63.994
63.950
63.905
63.861
63.817
63.772
63.728
63.684
.68
63.639
63.595
63.550
63.506
63.461
63.416
63.372
63.327
63.282
63.238
.69
63.193
63.148
63.103
63.058
63.013
62.968
62.923
62.878
62.833
62.788
.70
62.743
62.698
62.652
62.607
62.562
62.516
62.471
62.426
62.380
62.335
.71
62.289
62.243
62.198
62.152
62.106
62.061
62.015
61.969
61.923
61.877
.72
61.831
61.785
61.739
61.693
61.647
61.601
61.555
61.508
61.462
61.416
.73
61.369
61.323
61.276
61.230
61.183
61.137
61.090
61.043
60.997
60.950
.74
60.903
60.856
60.809
60.762
60.715
60.668
60.621
60.574
60.527
60.479
.75
60.432
60.385
60.337
60.290
60.243
60.195
60.147
60.100
60.052
60.004
.76
59.957
59.909
59.861
59.813
59.765
59.717
59.669
59.621
59.572
59.524
.77
59.476
59.428
59.379
59.331
59.282
59.234
59.185
59.136
59.087
59.039
.78
58.990
58.941
58.892
58.843
58.794
58.745
58.695
58.646
58.597
58.547
.79
58.498
58.449
58.399
58.349
58.300
58.250
58.200
58.150
58.100
58.050
.80
58.000
57.950
57.900
57.850
57.799
57.749
57.699
57.648
57.597
57.547
.81
57.496
57.445
57.394
57.343
57.293
57.242
57.190
57.139
57.088
57.037
.82
56.985
56.934
56.882
56.830
56.779
56.727
56.675
56.623
56.571
56.519
.83
56.467
56.414
56.362
56.310
56.257
56.205
56.152
56.099
56.046
55.994
.84
55.941
55.887
55.834
55.781
55.728
55.675
55.621
55.567
55.514
55.460
.85
55.406
55.352
55.298
55.244
55.190
55.135
55.081
55.026
54.972
54.917
.86
54.862
54.807
54.752
54.697
54.642
54.587
54.531
54.476
54.420
54.364
.87
54.309
54.253
54.196
54.140
54.084
54.028
53.971
53.914
53.858
53.801
.88
53.744
53.687
53.630
53.572
53.515
53.458
53.400
53.342
53.284
53.226
.89
53.168
53 . 109
53.051
52.992
52.933
52.875
52.816
52.756
52.697
52.638
.90
52.578
52.518
52.458
52.398
52.338
52.278
52.217
52.156
52.096
52.035
.91
51.974
51.912
51.851
51.789
51.727
51.666
51.603
51.541
51.478
51.416
.92
51.353
51.290
51.227
51.163
51.099
51.035
50.971
50.907
50.843
50.778
.93
50.713
50.648
50.582
50.517
50.451
50.385
50.319
50.252
50.185
50.118
.94
50.051
49.983
49.915
49.847
49.779
49.710
49.641
49.572
49.503
49.433
.95
49.362
49.292
49.221
49.150
49.079
49.007
48.934
48.862
48.789
48.715
.96
48.642
48.568
48.493
48.418
48.343
48.267
48.191
48.113
48.036
47.958
.97
47.880
47.801
47.722
47.641
47.561
47.479
47.397
47.314
47.231
47 . 146
.98
47.062
46.976
46.889
46.801
46.712
46.622
46.531
46.439
46.345
46.250
.99
46.154
46.055
45.955
45.852
45.747
45.639
45.527
(refer
to table below)
.9960
45.527
.9984
45.237
.9992
45.129
.9965
45.470
.9985
45.224
.9993
45.114
.9970
45.411
.9986
45.211
.9994
45.100
.9975
45.351
.9987
45.198
.9995
45.085
.9980
45.289
.9988
45.184
.9996
45.069
.9981
45.276
.9989
45.171
.9997
45.054
.9982
45.263
.9990
45.157
.9998
45.037
.9983
45.250
.9991
45.143
.9999
1.0000
45.020
45.000
885
Table III — Radians Phase (±.000015) for Semi-Infinite Attenuation Slope k = If </
///o
0
.00
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
.30
.31
.32
.34
.35
.36
.37
.38
.39
.40
.41
.42
.43
.44
.45
.46
.47
.48
.49
.50
.51
.52
..S3
.54
0.00000
0.00637
0.01273
0.01910
0.02547
0.03184
0.03821
0.04459
0.05097
0.05735
0.06373
0.070132
0.07652
0.08292
0.08932
0.09574^
0.10215
0.10858
0.11501
0.12145
0.12790
0.13436
0.14082
0.14730
0.15379
0.16029
0.16680
0.17332
0.17985
0.18641"
0.19296
0.19954
0.20613
0.21274'^
0.21935
0.226W0'
0.23265
0.239332
0.24601
0.25274'
0.25946
0.26621
0.27299
0.27978
0.28660'
0.29345
0.30032
0.30721-
0.31414
0.32109
0.32807
0..«508
0.34212
0.,U919
0.35629
0.00064 0.00127
0.00700 0.00764
0.01337
0.01974
0.02611
0.03248
0.03885
0.04523
0.05160
0.05799
0.06437
0.07076
0.07716
0.08356
0.08996
0.09638
0.10279
0.10922
0.11565
0.12210
0.12854
0.13501
0.14147
0.14795
0.15444
0.16094
0.16745
0.17398
0.18051
0.18706
0.19362
0.20020
0.20679
0.21340
0.22002
0.22666
0.23332
0.24000
0.24669
0.25341
0.26013
0.26689
0.27367
0.28047
0.28729
0.29414
0..^0101
0..^0791
0.31483
0.32179
0.32877
O..S3578
0.34282
0.34990
0.35701
0.01401
0.02037
0.02674
0.03311
0.03949
0.04586
0.05224
0.05862
0.06501
0.07140
0.07780
0.08420
0.09061
0.09702
0.10344
0.10987
0.11630
0.12274
0.12919
0.13565
0.14212
0.14860
0.15509
0.16159
0.16810
0.17463
0.18116
0.18772
0.19428
0.20086
0.20745
0.21406
0.22068
0.22733
0.23398
0.24067
0.24736
0.25408
0.26081
0.26757
0.27435
0.28115
0.28797
0.29482
().,^()170
0.30860
0.3 15, S3
0.32248
0.32947
0.33648
0.343.S3
0.3.S()61
0.35772
0.00191 0.00255 0.00318
0.00828 0.00891 '0.00955
0.01464 0.0 1.S28
0.02101
0.02738
0.03375
0.04012
0.04650
0.05288
0.05926
0.06565
0.07204
0.07844
0.08484
0.09125
0.09766
0.10408
0.11051
0.11694
0.12339
0.12984
0.13630
0.14277
0.14925
0.15574
0.16224
0.16875
0.17528
0.18182
0.18837
0.19493
0.20152
0.20811
0.21472
0.22135
0.22799
0.23465
0.24134
0.24803
0.25475
0.26148
0.26824
0.27503
0.28183
0.28866
0.29551
0.302,^9
0.30929
0.31622
0.32318
0.33017
0.33719
0.34424
0.35132
0.3.S844
0.02165
0.02802
0.03439
0.04076
0.04714
0.05352
0.05990
0.06629
0.07268
0.07908
0.08548
0.09189
0.09830
0.10472
0.11115
0.11759
0.12403
0.13048
0.13695
0.14342
0.14990
0.15639
0.16289
0.16941
0.17593
0.18247
0.18903
0.19559
0.20218
0.20877
0.21538
0.22201
0.22866
0.23532
0.24200
0.24870
0.25542
0.26216
0.26892
0.27571
0.28251
0.28934
0.01592
0.02228
0.02865
0.03503
0.04140
0.04778
0.05416
0.06054
0.06693
0.07332
0.07972
0.08612
0.09253
0.09894
0.10537
0.11179
0.11823
0.12468
0.13113
0.13759
0.14406
0.15055
0.15704
0.16354
0.17006
0.17659
0.18313
0.18968
0.19625
0.20283
0.20943
0.21605
0.22268
0.22932
0.23599
0.24267
0.24937
0.25610
0.26284
0 . 26960
0.27639
0.28319
0.29003
0.29620 0.29688
0.3(M08 0.. •50377
0.30998
0.31692
0.32388
0.33087
0.33789
()..U495
0.35203
0.35915
0.31068
0.31761
0.32458
0.33157
0.3,^860
0.34.S65
0.35274
0.3.S986
0.00382 0.00446 0.00509
0.01019 0.01082 0.01146
0.01719
0.02356
0.02993
0.03630 0.03694
0.01655
0.02292
0.02929
0.03566
0.04204
0.04841
0.05479
0.06118
0.06757
0.07396
0.08036
0.08676
0.09317
0.09959
0.10601
0.11244
0.11888
0.12532
0.13178
0.13824
0.14471
0.15119
0.15769
0.16419
0.17071
0.17724
0.18378
0.19034
0.19691
0.20349
0.21009
0.21671
0.22334
0.22999
0.23666
0.24334
0.25005
0.25677 0.25744
0.01783
0.02419
0.03057
0.04267
0.04905
0.05543
0.06182
0.06821
0.07460
0.08100
0.08740
0.09381
0.10023
0.10665
0.11308
0.11952
0.12596
0.13242
0.13888
0.14536
0.15184
0.15834
0.16484
0.17137
0.17789
0.18444
0.19099
0.19757
0.20415
0.21076
0.21737
0.22401
0.23065
0.23732
0.24401
0.25072
0.26351
0.27028
0.27706
0.28388
0.29071
0.29757
0.30446
0.31137
0.31831
0.32527
0.33227
0.33930
0.34636
0.35345
0.3605810.36129
0.04331
0.04969
0.05607
0.06245
0.06885
0.07524
0.08164
0.08804
0.09445
0.10087
0.10729
0.11372
0.12016
0.12661
0.13307
0.13953
0.14601
0.15249
0.15899
0.16549
0.17202
0.17855
0.18509
0.19165
0.19823
0.20481
0.21142
0.21803
0.22467
0.23132
0.23799
0.24468
0.25139
26419
27095
27774
28456
29140
29826
30515
31206
31900
32597
0.33297
0.34000
0.34707
0.35416
0.0057c
0.0121C
0.0184«
0.0248::
0.0312C
0.03755
0.0439!
0.0503v'
0.05671
0.06305
0.0694?
0.0758J
0.0822?
0.08865
0.0950«
0.10151
0.1143;
0.12083
0.1272^
0.1337:
0.1401{
0.1466(
0.1531^
0.1596^
0.1661
0.17261
0.1792(
0.1857,
0.1923(
0.19881
0.2054'
0.2120}
0.2186<
0.2253'
0.2319;
0.2386(
0,2453-
0.2520(1
0.25811 0.2587<|
26486
27163
27842
28524
29208
29S94
305S4
31275
31970
32667
0.33368
0.34071
0.34777
0.354cS7
0.36201
2655-
2723
2791(
2859:
2927(
2996:
3065:
3134^.
3203^
3273:
0.3343J
0.3414:
0.3484}
0.3555}
0.3627:
Superscripts — Corrington's values.
886
i
Table III — Continued
0
0.36343
0.37061
0.37782
0.38507
0.39237
10.39970
0.40708
0.41450
0.42197
0.43705
0.44467
0.45234
0.46008
0.46787
10.47573
0.48365
0.49164
0.49970
0.50784
0.51605
.0.52436
0.532745
0.54123
0.54981
t 0.55850
0.56730
0.57622
0.58526
0.59445
0.60378
iO. 61327
0.62293
0.63278
0.64284
0.65313
0.66368
0.67452
0.68569
0.69724
10.70926
lO. 72183
0.73513
; 0.74942
'0.76527
1
0.36415
0.37133
0.37855
0.38580
0.39310
0.40044 0.40117
0.36487
0.37205
0.37927
0.38653
0.39383
0.40782
0.41524
0.42272
0.43024
0.43781
0.44544
0.45312
0.46086
0.46866
0.47652
0.48444
0.49244
0.50051
0.50866
0.51688
0.52519
0.53359
0.54208
0.55068
0.55938
0.56819
0.57712
0.58618
0.59538
0.60472
0.61423
0.62391
0.63378
0.64387
0.65418
0.66476
0.67562
0.68683
0.69843
0.71049
0.72313
0.73651
0.75092
0.76698
0.40856
0.41599
0.42347
0.43100
0.43857
0.44620
0.45389
0.46164
0.46944
0.47731
0.48524
0.49325
0.50132
0.50948
0.51771
0.52603
0.53444
0.54294
0.55154
0.56025
0.56908
0.57802
0.58709
0.59631
0.60567
0.61519
0.62489
0.63478
0.64489
0.65523
0.66583
0.67672
0.68797
0.69961
0.71172
0.72443
0.73790
0.75243
0.768743
0.36559
0.37277
0.38000
0.38726
0.39457
0.40191
0.40930
0.41674
0.42422
0.43175
0.43934
0.44697
0.45466
0.46242
0.47023
0.47810
0.48604
0.49405
0.50213
0.51030
0.51854
0.52686
0.53528
0.54380
0.55241
0.56113
0.56996
0.57892
0.58801
0.59723
0.60661
0.61615
0.62587
0.63578
0.64591
0.65628
0.66691
0.67783
0.68911
0.70080
0.71297
0.72574
0.73930
0.75397
0.77053
0.36631
0.37350
0.38072
0.38799
0.39530
0.40265
0.41004
0.41748
0.4249
0.432510.43327
0.44010
0.44774
0.45544
0.46319
0.47101
0.47889
0.48684
0.49485
0.50295
0.51112
0.51937
0.52770
0.53613
0.54465
0.55327
0.56201
0.57085
0.57982
0.58892
0.59816
0.60756
0.61711
0.62685
0.63678
0.64693
0.65733
0.66798
0.67894
0.69026
0.70199
0.71421
0.72706
0.74070
0.75552
0.77236
0.36702
0.37422
0.38145
0.38872
0.39603
0.40339
0.41079
0.41823
0.42572
0.44086
0.44851
0.45621
0.46397
0.47180
0.47968
0.48763
0.49566
0.50376
0.51194
0.52019
0.52854
0.53697
0.54551
0.55414
0.56288
0.57174
0.58072
0.58984
0.59909
0.60850
0.61808
0.62783
0.63779
0.64796
0.65837
0.66906
0.68006
0.69141
0.70319
0.71547
0.72838
0.74213
0.75709S
0.77425
0.36774
0.37494
0.38217
0.38945
0.39677
0.40413
0.41153
0.41898
0.42647
0.43402
0.44162
0.44927
0.45698
0.46475
0.47258
0.48047
0.48843
0.49647
0.50457
0.51276
0.52103
0.52938
0.53783
0.54637
0.55501
0.56377
0.57264
0.58163
0.59076
0.60003
0.60945
0.61905
0.62882
0.63880
0.64899
0.65943
0.67015
0.68118
0.69257
0.70439
0.71673
0.72971
0.74356
0.75867
0.77620
0.36846
0.37566
0.38290
0.39018
0.39750
0.40486
0.41227
0.41972
0.42723
0.43478
0.44238
0.45004
0.45776
0.46553
0.47337
0.48127
0.48923
0.49727
0.50539
0.51358
0.52186
0.53022
0.53868
0.54723
0.55588
0.56465
0.57353
0.58254
0.59168
0.60097
0.61041
0.62002
0.62981
0.63981
0.65003
0.36918
0.37638
0.38362
0.39091
0.39823
0.40560
0.41301
0.42047
0.42798
0.43554
0.44315
0.45081
0.45853
0.46631
0.47415
0.48206
0.49004
0.49808
0.50621
0.51441
0.52269
0.53106
0.53953
0.54809
0.55676
0.56553
0.57443
0.58345
0.59260
0.60190
0.61136
0.62099
0.63080
0.64082
0.65106
0.66050 0.66156
0.67124 0.67233
0.68230 0.68342
0.69373
0.70560
0.71800
0.73106
0.74501
0.76028
0.69490
0.70681
0.71927
0.73240
0.74646
0.76192
0.36989
0.37710
0.38435
0.39164
0.39897
0.40634
0.41375
0.42122
0.42873
0.43629
0.44391
0.45158
0.45930
0.46709
0.47494
0.48285
0.49084
0.49889
0.50702
0.51523
0.52352
0.53190
0.54038
0.54895
0.55763
0.56642
0.57532
0.58436
0.59353
0.60284
0.61231
0.62196
0.63179
0.64183
0.65210
0.66262
0.67343
0.68456
0.69607
0.70804
0.72055
0.73377
0.74794
0.76358
(refer to table below)
9960
0.77620
.9984
0.78125
.9992
0.78315
9965
0.77720
.9985
0.78148
.9993
0.78340
9970
0.77822
.9986
0.78171
.9994
0.78366
9975
0.77928
.9987
0.78195
.9995
0.78392
9980
0.78036
.9988
0.78218
.9996
0.78419
9981
0.78058
. 9989
0.78242
.9997
0.78446
9982
0.78080
.9990
0.78266
.9998
0.78475
9983
0.78103
.9991
0.78290
.9999
1.0000
0.78505
0.78540
887
Table IV — Radians Phase (±.000dl5) for Semi-Infinite ATTENtJATiON Slope k = 1
f>fo
fo/f
00 1.57080 1.57016
.01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.11
.12
.13
.14
.15
.16
.17
.18
.19
.20
.21
.22
.23
.24
.25
.26
.27
.28
.29
.30
.31
.32
.33
1 . 56443
1.55806
55170
1.54533
1.53896
1.53258
52621
51983
51345
1.50706
1.50067
1.49428
1.48788
1.48147
1.47506
1.46864
1.46222
1.45579
1.44934
1.44290
1.43644
1.42997
1.42349
1.41701
1.41050
1.40400
1.39747
1.39094
1.38439
1.37784
1.37125
1.36467
1.35806
56379
55743
55106
54469
53832
1.53195
1.52557
1.51919
1.51281
1.50642
1 . 50003
1.49364
1.48724
1.48083
1.47442
1.46800
1.46157
1.45514
1.44870
1.56952 1.56889 1.56825
1.56316
1.55679
1.55042
1.54405
1 . 53768
1.53131
1 . 52493
1.51855
1.51217
1.50578
1.49939
1.49300
1.48660
1.48019
1.47378
1.46736
1.46093
1.45450
1.44805
1.44225 1.44161
34 1.35144
1.34480
1.33815
1.33147
1.32478
1.31806
1.31134
1.30458
1.29781
1.29101
1.28419
1.27735
1.27048
1 . 26358
1 . 25666
1.24971
1.24273
1.23572
1.22868
1.22161
1.21450
1.43579
1.42933
1.42284
1.41636
1.40985
1.40335
1.39682
1.39029
1.38374
1.37718
1.37060
1.36401
1.35740
1.35078
1.34413
1.33748
1.33080
1.32411
1.31739
1.31066
1.30391
1.29713
1 . 29033
1.28351
1.27666
1.26979
1 . 26289
1.25596
1.24901
1 . 24203
1 . 23502
1.22797
1.22090
1.21379
1.43514
42868
1.42219
1.41571
1.40920
1.40270
1.39617
1.38963
1.38308
1.37652
1.36994
1.36335
1.35673
1.56252
1.55615
1.54979
1.54342
1.53704
1.53067
1 . 52430
1.51792
1.51153
1.50515
1.49875
1.49236
1.48596
1.47955
1.47314
1.46672
1.46029
1.45385
1.44741
1.44096
1.43450
1.42803
1.42155
1.41506
1.40855
1.40204
1.39551
1.38J
1.38242
1.37586
1.36928
1.36269
1.35607
1.35011 1.34945 1.34878
1.34347
1.33681
1.33013
1.32344
1.31672
1.30999
1.30323
1.29645
1 . 28965
1.28282
1.27597
1.26910
1.26220
1.25527
1.24831
1.24133
1.23431
1.22726
1.22019
1.21307
56188
1.55552
1.54915
1.54278
53641
53003
52366
1.51728
1.51089
1.50451
1.49811
1.49172
1.48532
1.47891
1.47249
1.46607
1.45964
1.45321
1.44677
1.44031
1.43385
1.42738
1.42090
1.41441
1.40790
1.40139
1.39486
1.38832
1.38177
1.37520
1.36862
1.36203
1.35541
56761
56125
55488
54851
1.54214
1.53577
1.52940
1 . 52302
1.51664
1.51026
1.34280
1.33614
1.32946
1.32277
1.31604
1.30931
1.30255
1.29577
1.28897
1.28214
1.27529
1.26841
1.26150
1.25457
1.24762
1 . 24063
1 . 23361
1.22656
1.21948
1.21236
1.34214
1.33548
1.32879
1.32209
1.31537
1.30864
1.30187
1 . 29509
1.28828
1.28145
1.27460
1.26772
1.26081
1 . 25388
1.24692
1 . 23992
1 . 23290
1.22.585
1.21876
1.21165
1.50387
1.49748
1.49108
1.48468
1.47827
1.47185
1.46543
1.45900
1.45257
1.44612
1.43967
1.43320
1.42673
1.42025
1.41376
1.40725
1.40074
1.39421
1.38767
1.38111
1.37455
1.36796
1.36136
1.35475
1.34812
1.34147
1.33481
1.32812
1.32142
1.31470
1.30796
1.. SOI 20
1 . 29441
1.28760
1.28077
1.27391
1 . 26703
1.26012
1.25318
1 . 24622
1.23922
1.23220
1.22514
1.21805
1.21093
1 . 56698
1.56061
1.55424
1.54787
1.54150
1.53513
1 . 52876
1.52238
1.51600
1 . 50962
1.50323
1.49684
1.49044
1.48404
1.47763
1.47121
1.46479
1.45836
1.45192
1.44548
1.43902
1.43256
1.42608
1.41960
1.56634
55997
1.55361
54724
54087
1.53450
1.52812
1.52174
1.51536
1.50898
1.50259
1.49620
1.48980
1.48339
1.47698
1.47057
1.46414
1.45772
1.45128
1.44483
1.43837
1.43191
1.42544
1.41895
1.41311 1.41246
1.40660
1.40008
1.39356
1.38701
1.38046
1.37389
1.36730
1.36070
1.35409
1.34745
1.34081
1.33414
1.32746
1.32075
1.31403
1.30729
1.30052
1.29373
1 . 28692
1 . 28008
1.27323
1 . 26634
1 . 25943
1.25249
1.24552
1.23852
1.231. SO
1 . 22444
1.21734
1.21022
1.40595
1.39943
1.39290
1.38636
1.37980
1.37323
1.36665
1.36004
1.35343
1.34679
1.34014
1.33347
1.32679
1.32008
1.31336
1.30661
1.29984
1 . 29305
1 . 28624
1.27940
1.27254
1 . 26565
1.25874
1.25179
1.24482
1.23782
1 . 23079
1.22373
1.21663
1.20950
1.56570
1.55934
1.55297
1 . 54660
1 . 54023
1.53386
1.52748
1.52111
1.51472
1 . 50834
1.50195
1.49556
1.48916
1.48275
1.47634
1.46993
1.46350
1.45707
1.45063
1.44419
1.43773
1.43127
1.42479
1.41831
1.41181
1.40530
1.39878
1.39225
1.38570
1.37915
1.37257
1.36599
1.35938
1.35277
,56507
,55870
,55233
,54596
.53959
53322
1 . 52685
1.52047
1.51409
1.50770
1.50131
1.49492
1.48852
1.48211
1.47570
1.46929
1.46286
1.45643
1.44999
1.44354
1.43708
1.43062
1.42414
1.41766
1.41116
1.40465
1.39813
1.39160
1.38505
1.37849
1.37191
1.36533
1.35872
1.35210
1.34613 1.34546
1.33948
1.33280
1.32612
1.31941
1.31268
1.30594
1.29916
1.29237
1.28556
1.27872
1.27185
1.26496
1.25804
1.25110
1.24413
1.23712
1 . 23009
1.22302
1.21592
1.20879
1.33881
1.33213
1.32545
1.31873
1.31201
1.30526
1.29849
1.29169
1.28487
1.27803
1.27116
1.26427
1.25735
1.25040
1.24343
1.23642
1 . 22938
1.22231
1.21521
1.20808
888
Table IV — Continued
Uli
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
95
.96
.97
.98
.99
0
1.20736
1.20019
1.19297
1.18572
1.17843
1.17110
1.16372
1.15630
1.14883
1.14131
1.13375
1.12613
1.11845
1.11072
1 . 10293
1.09507
1.08715
1.07916
1.07110
1.06296
1.05474
1.04644
1.03805
1.02957
1.02099
1.01230
1.00350
.99458
.98553
.97635
.96702
.95753
.94787
.93801
.92795
.91766
.90712
.89628
.88511
.87355
.86154
.84896
.83567
.82138
.80553
1
1 . 20664
1 . 19946
1.19225
1.18499
1.17770
1.17036
1 . 16298
1.15555
1 . 14808
1 . 14056
1 . 13298
1.12536
1.11768
1 . 10994
1.10214
1.09428
1.08635
1.07836
1.07029
1.06214
1.05391
1.04560
1.03721
1.02871
1.02012
1.01142
1.00261
.99368
.98462
.97542
.96507
.95657
.94689
.93701
.92693
.91662
.90604
.89518
.88397
.87237
.86031
.84766
.83428
.81988
.80381
1 . 20593
1 . 19874
1.19152
1 . 18426
1.17696
1 . 16962
1.16224
1.15481
1.14733
1.13980
1.13222
1.12459
1.11690
1.10916
1.10135
1.09349
1.08556
1.07755
1.06947
1.06132
1.05309
1.04477
1.03636
1.02786
1.01925
1.01054
1.00172
.99278
.98370
.97449
.96513
.95561
.94591
.93601
.92591
.91557
.90496
.89407
.88283
.87119
.85907
.84637
.83290
.81836
.80206
1.20521
1 . 19802
1 . 19080
1.18353
1.17623
1.16888
1.16149
1.15406
1 . 14658
1 . 13904
1.13146
1.12382
1.11613
1 . 10838
1 . 10057
1.09270
1.08476
1.07675
1.06866
1.06050
1.05226
1.04393
1.03551
1.02700
1.01839
1.00967
1.00083
.99188
.98279
.97356
.96418
.95464
.94493
.93501
.92488
.91452
.90389
.89296
.88168
.87000
.85783
.84505
.83150
.81683
.80027
1 . 20449
1.19730
1 . 19007
1.18280
1.17550
1
16815
1 . 16075
1.15331
1 . 14582
1.13829
1 . 13070
1.12306
1.11536
1 . 10760
1.09978
1.09191
1.08396
1.07594
1.06785
1.05968
1.05143
1.04309
1.03467
1.02615
1.01752
1.00879
.99994
.99097
.98187
.97263
.96324
.95368
.94395
.93401
.92386
.91347
.90281
.89185
.88054
.85658
.84374
.83009
.81528
.79844
1.20377
1 . 19658
1 . 18935
1 . 18208
1.17476
1.16741
1 . 16001
1.15257
1 . 14507
1.13753
1 . 12994
1.12229
1.11459
1 . 10682
1.09900
1.09112
1.08316
1.07514
1.06704
1.05886
1.05060
1.04226
1.03382
1.02529
1.01666
1.00791
.99905
.99007
.98096
.97170
.96229
.95272
.94297
.93301
.92284
.91242
.90174
.89074
.87938
.86761
.85533
.84241
.82867
.81371
. 79655
1 . 20306
1 . 19586
1 . 18862
1.18135
1.17403
1 . 16667
1.15927
1.15182
1 . 14432
1.13677
1.12917
1.12152
1.11381
1 . 10604
1.09821
1.09032
1.08236
1.07433
1.06622
1.05804
1.04977
1.04142
1.03297
1.02443
1.01578
1.00703
.99816
.98916
.98004
.97077
.96134
.95175
.94198
.93200
.92180
.91136
.90064
.88962
.87823
.86641
.85407
.84108
.82724
.81212
. 79460
1.20234
1.19514
1.18790
1.18062
1.17330
1.16593
1.15853
1.15107
1.14357
1 . 13602
1 . 12841
1.12075
1.11304
1.10526
1.09743
1.08953
1.08156
1.07352
1.06541
1.05721
1.04894
1.04058
1.03212
1.02357
1.01491
1.00615
.99726
.98826
.97912
.96983
.96039
.95078
.94099
.93099
.92077
.91030
.89955
.88850
.87706
.86519
.85280
.83974
.82579
.81051
20162
19442
18717
17989
17256
1.16519
1.15778
1.15032
1 . 14282
1.13526
1 . 12765
1.11999
1.11227
1 . 10448
1.09664
1.08874
1.08076
1.07271
1.06459
1.05639
1.04811
1.03973
1.03127
1.02271
1.01404
1.00526
.99637
.98735
.97819
.96889
.95944
.94981
.93999
.92998
.91973
.90924
.89846
.88737
.87590
.86398
1.20090
1 . 19369
1 . 18645
1.17916
1.17183
1 . 16446
1.15704
1 . 14958
1 . 14207
1 . 13450
1 . 12689
1.11922
1.11149
1.10371
1.09586
1.08794
1.07996
1.07191
1.06378
1.05557
1.04727
1.03889
1.03042
1.02185
1.01317
1.00438
.99547
.98644
.97727
.96796
.95848
.94884
.93900
.92896
.91870
.90818
.89737
.88624
.87473
.86276
.85153 .85025
.83839 .83703
.82434 .82286
.80888 .80722
(refer to table below)
.9960
0.79460
.9984
0.78954
.9992
0.78765
.9965
0.79360
.9985
0.78931
.9993
0.78739
.9970
0.79257
.9986
0.78908
.9994
0.78714
.9975
0.79152
.9987
0.78885
.9995
0.78688
.9980
0.79044
.9988
0.78862
.9996
0.78661
.9981
0.79022
.9989
0.78838
.9997
0.78633
.9982
0.78999
.9990
0.78814
.9998
0.78605
.9983
0.78977
.9991
0.78789
.9999
1.0000
0.78575
0.78540
889
890
BELL SYSTEM TECHNICALpOURNAL
of phase versus frequency and drawing a smooth curve weighting the points
in accordance with the errors known by experience to occur for various types
(fo.Ao)
f,0= OoWi'o.A'k)^
0.2 0.3 0.4 0.6 0.8 1 1.5 2 3 4 5
FREQUENCY (f)
6 8 10 15 20 30 40 50
Fig. 5-20 log I Z|.
of departures of the straight line approximation from the exact char-
acteristic.
Although the degree and db relationship is applicable to attenuation and
phase computations, nepers and radians are proper theoretical units which
can be used in other problems^ For instance, Tables III and I\' give the
* Bode, "Network Analysis and Fcedljack Amplifier Design," C'hapter XV, page 340.
%
TABLES OF PHASE
891
reactance in ohms associated with a semi-infinite unit slope of resistance
where a unit slope of resistance is one in which a one-ohm change in resistance
Table V
Tabulation of Critical Points and Determination of Slopes of Straight Lines
x\pPROXIMATING CHARACTERISTIC OF FiG. 5
n
fn
An
An — A„-l
20 log ^
kn
0
.13
0
1
.433
-1.40
-1.40
10.45
-.134
2
1.19
-5.55
-4.15
8.78
-.473
3
1.38
-5.55
.00
1.287
0
4
1.62
-4.30
+ 1.25
1.393
+ .897
5
1.96
-.45
+3.85
1.655
+2.326
6
2.20
-.45
.00
1.003
0
7
3.00
-6.70
-6.25
2.694
-2.320
8
5.00
-13.40
-6.70
4.437
-1.510
9
20.0
-26.00
-12.60
12.04
-1.046
10'
40.0
-32.0
-6.0
6.02
10
00
-1.0
Note that/'io = 40.0 is chosen to get ^lo over a finite section of the semi-infinite slope
extending to / = =o .
Table VI
Summation of Phase Associated with 20 log | Z | of Fig. 5 at / = 1.5
fn from
fn
/
0n
07.-1 - e„
kn from
kn(fin-\ — e„)
Table V
f
fn
Degrees
Degrees
Table V
Degrees
0
.13
.087
86.824
1
.433
.289
79.357
7.467
-.134
-1.00
2
1.19
.793
58.349
21.008
-.473
-9.94
3
1.38
.920
51.353
6.996
0
4
1.62
.926
39.029
12.324
+ .897
+ 11.05
5
1.96
.765
30.283
8.746
+2.326
+20.34
6
2.20
.682
26.450
3.833
0
7
3.00
.5
18.797
7.653
-2.320
-17.75
8
5.00
.3
11.056
7.741
-1.510
-11.69
9
20.0
.075
2.737
8.319
-1.046
-8.70
10
00
0
.000
2.737
-1.00
-2.74
2 kn
[On-l - On)
= -20.43
? (/ = 1.5)
= -20.4°
Note that for/o to/s the ratio of/ to/„ must be taken /„// to be less than unity and On
is therefore read from Table II, whereas for/t to/io the ratio must be taken ///„ and 9n is
therefore read from Table I.
occurs between frequencies which are in the ratio e = 2.7183. The same
technique described above for the determination of the phase associated
892
BELL SYSTEM TECHNICAL JOURNAL
Table VII
Critical Points for Fivk Line Approximation to Characteristic of Fig. 5
n
/,.
An
0
.25
0
1
1.40
-5.8
2
2.10
0
3
3.00
-7.0
4
10.0
-20.0
5'
40.0
-32.0
5
oo
0
-5
<
-10
-15
-20
-25
UJ
lU
y)-35
UJ
O
2-40
W.45
UI
_l
Z-50
<
UJ
5-55
I
-60
-65
-70
-75
-
PHASE ANGLE (6)
^-^
APPROXIMATION TO 20 LOG IZI
N
k.,
^
V
A
N
/
\\
\
\
/
\
^
■>^
1
>
^
\
1
+J
W^^ j
z^lzieJ® ;
' U
_L .1 .V2 -L <,
— -Jf -JT-r s'
-85
-90
^
0
1 0
15 0
2 0
3 0
4 0
5 0
6
0
8
1
5 2
A
5
FREQUENCY (f)
Fig. 6 — Phase associated with 20 log | / | of Fig. 5.
TABLES OF PHASE
893
with a given attenuation characteristic may therefore be used to determine
the reactance associated with a given resistance characteristic. The only
Table VIII
Tabulation of Critical Points and Determination of Slopes of Straight Lines
Approximating Resistance Characteristic of Fig. 7
n
fn
Rn
Rn - Rn-l
2.303 log -^
fn-l
kn
0
.078
1.000
0
1
.185
.912
-.088
.864
-.102
2
.290
.805
-.107
.450
-.238
3
.900
.400
-.405
1.133
-.357
4
1.20
.400
0
—
5
1.50
.547
+ .147
.2231
+ .659
6
1.67
.840
+ .293
.1074
+2.728
7
1.84
1.280
+ .440
.0969
+4.54
8
1.92
1.280
0
9
2.20
.335
-.945
.1361
-6.94
10
2.45
.094
-.241
.1076
-2.24
11
2.85
.015
-.079
.1512
-.52
12
5.00
.000
-.015
.562
-.027
Table IX
Summation of Reactance Associated with Resistance of Fig. 7 at/ = 1.0
n
/„ (From
Table
VIII)
fn
f
/
Xn
Ohms
Xn-i — Xn
Ohms
kn (From
Table VIII)
kn (Xn-1 - Xn)
Ohms
0
.078
.078
1.52111
1
.185
.185
1.45257
.06854
-.102
-.0070
2
.290
.290
1.38439
.06818
-.238
-.0162
3
.90
.900
.91766
.46673
-.357
- . 1666
4
1.20
.833
.58801
.32965
5
1.50
.667
.45004
.13797
+ .659
+ .0909
6
1.67
.599
.39897
.05107
+2.728
+ . 1393
7
1.84
.543
.35844
.04053
+4.54
+ . 1840
8
1.92
.521
.34282
.01562
9
2.20
.455
.29688
.04594
-6.94
-.3188
10
2.45
.408
.26486
.03202
-2.24
-.0717
11
2.85
.351
.22666
.03820
-.52
-.0199
12
5.00
.200
.12790
.09876
-.027
- .0027
^kn
(X„_i - X„)
= -.1887
X(f
= 1.0) = -.
189 Ohm
difference is that the slopes of the straight lines approximating the resistance
plotted on a log frequency scale are determined by the expression below :
Rn — Rn-l Rn ~ Rn-l
k =
log*
where:
fn-.
2.303 log
A
fn-l
Rn is the resistance at/„ on the straight line approximation to R.
894
BELL SYSTEM TECHNICAL JOURNAL
Ofv|U^ool^ooou^■<tu^
cO-OOO-*-*00C0(»)Cn^
(/) (^oojoq^tTtmoqcvjajfoooo
z
o
°- _?S§8oor-'l-c\JO"i'no
^ o66d-^ — -- — — f^f^ifju^
P
a:
cr
4f
'o
cr
cr
5^
„—
6
A
^
-
=—
<o
cr
a5
£
-.^
^*^s.
cr
— i^
t^
//
/
oo
ii
1 Q-
X Q.
O <
'^^
Q. O
ol -1 O
Z t -1
< ^ <
J
/
/
<!
1 — VvV— 1
\{
3
5^
//
11
I -J-
If
/l
fl N
II
1 N i
c
c
1 o
11
i ^
;]
o
a.
(I
^f
RESISTANCE (R) IN OHMS
TABLES OF PHASE
895
\
\
\
\
\
V
\
V
x.
^^
-
i=>
c
^^
N
a.
o
^^
a. z
o a.
Si
si
si
Q^
^i
i£
tr 1-
UJ LU
t- Q
LU
Q X
O
\
\
A
V
\
— VA — 1
)
\
11
K
•p
\{
^
o
11
X
+ A
tr
II
ISJ I
/
/
/
/
REACTANCE (X) IN OHMS
896
BELL SYSTEM TECENICALJOVRNAL
As an example of the determination of the reactance associated with a
given resistance characteristic, consider the resistance characteristic of Fig.
7 and the straight line approximation shown in dotted form. The slopes
of the straight lines are determined as illustrated in Table VTII.
Having determined the slopes of the various straight lines of the approxi-
mation, the reactance can be summed at any desired frequency. As an
illustration the reactance is summed at/ = 1.0, in Table ^X.
The mesh computed reactance of the network of Fig. 7 is plotted in Fig.
8 and the reactance summed for/ = 1.0 is seen to be within .01 ohm
of the true reactance. The reactance was summed at a considerable number
of frequencies and the results plotted as individual points in Fig. 8. The
degree of approximation to the true reactance should be similar to the
1 "A^ 1
\\\ 1
. I.
l( r
'/
1 \\
P
p
i+d <
2 <
<
>
> —
(p^jw)
_ i + d
" 2P
k
3
T r
Figr. 9 — Parallel T network.
degree of approximation to the original resistance and this is borne out by
the example where the straight line approximation to the resistance char-
acteristic is within ± .03 ohm and the maximum departure of the reactance
determined from the straight line approximation is ± .025 ohm.
As was pointed out in the attenuation example a much simpler straight
line approximation to the resistance characteristic would have resulted in a
reactance determination without too much greater error than the deter-
mination of the illustration.
A word of caution is necessary in connection with the use of the straight
line approximation method discussed above. The true phase or reactance
is reliably obtained only in those cases where the problem in question is a
minimum phase one. In order to illustrate the failure of the method in
those problems in which non-minimum phase conditions exist consider the
parallel T network of Fig. 9. The transfer impedance Z012 defined by the
TABLES OF PHASE
897
(wo.A'o)(wo=o)
(0 10
o 8
M°N4
N
V
Zoi2
Z012 1 „
ie
i + dp + p^
\
Zoo Zoo r p(i + p)
20L0G^^ FOR Cl = -I-^, OR d = -4-
Zoo
STRAIGHT-LINE APPROXIMATION TO 20 LOG -^^
Zoo
O CRITICAL POINTS ON APPROXIMATION
\
(Wl,A,
)
>
\
CRITICAL POINTS
n ojn An
0 0
0' 0.05 26.0
1 0.10 20.0
2 0.29 10.2
3 0.50 2.7
4 0.69 -3.7
5 0.98 -15.0
6 1.02 -15.0
7 1.51 -5.7
8 2.75 - 1.5
9 10.00 0
\
\
(a)2,A2)
\
\
\
N
J (a
3.A3
)
\
(U)9,^/
'J
v\
(wa
Aa)^
^
r^
^
■^^
6(c
04,/
'^4)
/
/
c
A
7.A7)
1
i
1
li
\
\
I
. .1..
.1..
' 1 \
(W5,A5)i
1 1 1 1
/(cJg.Ae)
1
...1—
_L.
0.05 0.1 0.15 0.2 a3 0.4 0.6 0.8 1 1.5 2 3 4 5 6 7 8 10
I 7 I
Fig. 10 — 20 log -^ for network of Fig. 9.
ratio of the open circuit voltage £2 to the open circuit driving current I\ is
given by:
_ 1 1 ^ d \ -^dp^ p"
^''' - 22+d p{l+p) •
898
BELL SYSTEM TECHNICAL JOURNAL
-320
/
d
>-
—
__^
/
/ Zo,2 Zoi2 ^je_ i+dp + p2
' Zoo Zoo Pd+p)
FOR d = + -^
FOR a = -J-
O e DETERMINED FROM STRAIGHT-
LINE APPROXIMATION TO
20 LOG ^^ , FOR
Zoo
d = + j, OR d=-4:
/
/
/
»^ (
\
"^
^
-^
^
^N,
^.,
\
\
t
t
t
1
1
1
1
t
I
\
\
\
\
\
\
\
\
"-x^^
1^
1
1
■■"
'-•-.
1
1
0.1 0.15 0.2 0.3 0.4 0.6 0.8 I 1.5 2 3 4 5 6 7 8 9 10
Fig. 11 — Phase angle of -^ — for network of Fig. 9.
If we take the ratio of Zmo to its value for w = oc then:
Zoi2
zT
Zoi2
zT
^,e ^\+dp-^ p'
p{\ + p)
TABLES OF PHASE 899
20 log
log
is plotted in Fig. 10 for d = +1/4 and it is apparent than 20
for d = —1/4 is identical. This identity does not hold for 9,
Zoi2
Zoi2
z^
however. This is shown in Fig. 11 where 6 for d = +1/4 and 6 for d =
— 1/4 are plotted.
The real characteristic of Fig. 10 was then approximated by a series of
straight lines determined by the critical points listed and the phase asso-
ciated with this straight line approximation summed. The phase so deter-
mined is plotted as individual points in Fig. 11. It is seen that this summa-
tion determined the phase of the function in question for d = +1/4 but
completely failed to do so for d = — 1/4. The function for d = — 1/4 is an
example of a non-minimum phase function for which the above technique
fails to determine the phase of the function from its attenuation
characteristic.'"
There are certain instances where the above technique can be usefully
applied in connection with non-minimum phase systems in spite of the
failure of the method to predict the total phase. ^^ However, the necessity
of checking for non-minimum phase conditions and, if such exist, deter-
mining whether the above method of computing phase is at all applicable, is
illustrated by the non-minimum phase example above.
1" This is the anticipated result since the function is identified as a non-minimum phase
function by the fact that it has two zeros falling in the right half p plane.
" Bode, "Network Analysis and Feedback Amplifier Design," Chap. XIV, page 309.
Abstracts of Technical Articles by Bell System Authors
Television Network Facilities} L. G. Abraham and H. I. Romnes. Tele-
vision networks, like sound broadcasting networks, must be available to
make distribution of high quahty programs economical. For television cir-
cuits interconnecting studios in different cities, coaxial cable and radio relay
are the most suitable methods. For short distance transmission balanced
wire pairs also may be used. Local conditions will control the type circuit
selected.
Protective Coatings on Bell System Cables} V. J. Albano and Robert
Pope. The practice of placing some Bell System cables directly in the
ground without the use of conduit was introduced in about 1929. Since
bare cable thus installed would be subject to the corrosive action of soils,
or damage from lightning or gophers, suitable protective coatings to guard
against these hazards had to be developed. Seven types of such coverings
are described, and their particular field of application is indicated.
Surface States and Rectification at a Metal Semi-Conductor Contact.^ John
Bardeen. Localized states (Tamm levels), having energies distributed in
the "forbidden" range between the filled band and the conduction band, may
exist at the surface of a semi-conductor. A condition of no net charge on
the surface atoms may correspond to a partial filling of these states. If the
density of surface levels is sufficiently high, there will be an appreciable
double layer at the free surface of a semi-conductor formed from a net
charge from electrons in surface states and a space charge of opposite sign,
similar to that at a rectifying junction, extending into the semi-conductor.
This double layer tends to make the work function independent of the height
of the Fermi level in the interior (which in turn depends on impurity con-
tent). If contact is made with a metal, the difference in work function be-
tween metal and semi-conductor is compensated by surface states charge,
rather than by a space charge as is ordinarily assumed, so that the space
charge layer is independent of the metal. Rectification characteristics are
then independent of the metal. These ideas are used to explain results of
Meyerhof and others on the relation between contact potential differences
and rectification.
' Electrical Engineering, May 1947.
2 Corrosion, May 1947.
^Phys. i?ei»., May 15, 1947.
900
ABSTRACTS OF TEC H NIC A L ARTICLES 901
Plating on Aluminum} R. A. Ehrhardt* and J. M. Guthrie. This
article describes tests made to develop a satisfactory process for producing
adherent electrodeposits on aluminum alloys using a zincate immersion pre-
treatment.
Since the major interest was the fabrication of aluminum structures by the
use of lead-tin solders the adherence of the deposit was determined by meas-
uring the strength of soldered joints.
Excellent results were obtained with commercially pure aluminum and
copper bearing alloys and satisfactory results with magnesium and silicon
bearing alloys.
Corrective Networks.^ F. L. Hopper. A type of fully compensated con-
stant resistance network is described which provides a larger family of
equalization characteristics particularly suited to corrective use in rerecord-
ing as determined by aural monitoring.
Speclrochemical Analysis of Ceramics and Other Non-Metallic Materials.^
Edwin K. Jaycox. The procedure described is applicable to the quanti-
tative spectrochemical analysis of ceramics, ashes, ores, paints, and other
non-metallic materials for the determination of most of the common metals
and their oxides. These include: aluminum, boron, barium, beryllium, cal-
cium, copper, chromium, iron, lithium, magnesium, manganese, sodium,
lead, silicon, titanium, zinc, and zirconium, in the general range of 0.30-70.0
per cent. Samples in the form of a fine powder are mixed one part of sample
to 10-100 parts of a suitable metal oxide which serves as a bufiFer, diluent,
and internal control. Carbon dust is added to this mixture for its additional
buffering effect. Spectra are obtained of the samples and of an appropriate
series of standards. Determinations of the amount of element sought are
made, in most cases by the well known internal standard technique, in others
by the simple comparison standard procedure.
The Spectrochemical Analysis of Nickel Alloys."^ Edwin K. Jaycox. A
procedure is described for the analysis of nickel alloys for copper, iron, lead,
magnesium, manganese, silicon, titanium, and zinc in the range 0.005-0.30
per cent and for boron in the range 0.0003-0.03 per cent. Samples are taken
into solution with dilute nitric acid, evaporated to dryness, and baked at
400°C. The resulting dry nitrate-oxide powder is mixed with pure carbon
dust which acts as a buffer and diluent. Aliquots of each sample and of a
^ The Monthly Review, American Electroplaters Society, April 1947.
* Of Bell Tel. Labs.
^Jour. Soc. Motion Pic. Engrs., March 1947.
^ J our. Optical Soc. Amer., March 1947.
''Jour. Optical Soc. Amer., March 1947.
902 BELL SYSTEM TECHNICAL JOURNAL
series of standards are excited in the direct current arc, and their spectra
recorded on the same plate. Determinations of the amounts of constituent
elements present in the sample are made by measuring the logarithm of the
ratio of the relative intensities of a line of the element sought to that of a
nickel control line by the general internal control technique.
Measurement of the Viscosity and Shear Elasticity of Liquids by Means of a
Torsionally Vibrating Crystal.^ W. P. Mason. This paper describes a
method of measuring viscosities of liquids at high frequencies by means of
oscillating cylinders, in which a torsionally vibrating crystal generates a
viscous wave in the medium to be measured. Both a reactance and a resist-
ance loading occur in the crystal which lowers its frequency and raises the
measured resistance at resonance. The viscosity may then be determined
by measuring the changes in the properties of the crystal. By varying the
voltage on the crystal, the shearing displacement can be varied and hence
the viscosity can be measured as a function of shearing stress. Measure-
ments on light oils over a viscosity range from 0.01 poise to 10 poises check
within a few per cent when made with rough temperature-control conditions.
Considerations in the Design of Centimeter-Wave Radar Receivers.^ Stew-
art E. Miller. A review of the radar duplexer and receiver, as developed
during the war, is presented. Attention is devoted to the principles of oper-
ation and typical circuit arrangements employed in the duplexer, the crystal
converter, the local-oscillator injection circuits, the intermediate-frequency
amplifier, and the automatic-tuning unit. Emphasis is placed on methods
found advantageous in the 1 -centimeter and 3-centimeter wavelength re-
gions. The interrelation between the various receiver components in deter-
mining the over-all receiver noise figure is shown analytically, and typical
performance numbers are given.
Experimental Rural Radiotelephony}^ J. Harold Moore, Paul K.
Seyler and S. B. Wright. The first rural party-line telephone service
utilizing radio installations operating on the subscribers' premises was under-
taken experimentally in the vicinity of Cheyenne Wells, near the eastern
border of Colorado. Radio links have been used to supply regular telephone
service to eight ranches since August 20, 1946. The development of a
standard rural radiotelephone system will be aided materially by the expe-
rience gained from these experiments.
8 Transactions A.S.M.E., May 1947.
^Froc.LK.E., April 1947.
'° Electrical Engineering, April 1947.
ABSTRACTS OF TECHNICAL ARTICLES 903
Alkaline Earth Porcelains Possessing Low Dielectric Loss}^ M. D. Rigter-
INK and R. O. Grisdale. Alkaline earth porcelains have been prepared
from mixtures of clay, flint, and synthetic fluxes consisting of clay calcined
with at least three alkaline earth oxides. These porcelains possess excellent
dielectric properties, have low coefhcients of thermal expansion, are white,
and are especially valuable as bases for deposited carbon resistors for which
they were developed. Their characteristics make it probable that other uses
will be found for materials of this type.
An illustrative composition is 50.0% Florida kaoHn, 15.0% flint (325
mesh), 35.0% calcine (200 mesh). The composition of the calcine is 40.0%
Florida kaolin, 15.0%, MgCOs, 15.0% CaCOa, 15.0%^ SrCOs, 15.0%o BaCOs,
calcined at 1200°C. The electrical properties of this body at 1 mc. are Q at
25°C., 2160; Q at 250°C., 280; Q at 350°C., 90; specific resistance at 150°C.,
1013-5 ohm-cm. and at 300°C., lO^^-^ ohm-cm.
Attenuation of Drainage Effects on a Long Uniform Structure with Distributed
Drainage}^ J. M. Standring, Jr. This paper discusses the general be-
havior of forced drainage currents on long uniform underground commu-
nication cables with particular regard to the case where drainage is applied at
regular intervals. Expressions are developed for the structure-to-earth po-
tential which is caused by uniformly spaced drainers when the power supply
is from variable e.m.f. sources, such as rectifiers, and also for the case where
fixed e.m.f.'s, such as galvanic anodes, are employed.
"/owr. Amer. Ceramic Society, March 1, 1947.
12 Corrosion, June 1947.
Contributors to this Issue
Clifford E. Fay, B.S. in Electrical Engineering, Washington University,
1925; M.S., 1927. Bell Telephone Laboratories, 1927-. Mr. Fay has been
engaged principally in the development of power vacuum tubes for radio
purposes.
Laurence W. Morrison, B.S. in Electrical Engineering, University of
Wisconsin, 1930. Graduate work, 1930-31. Bell Telephone Laboratories,
1931-. Mr. Morrison was engaged in the development of telephone and
television terminal equipment for the coaxial system to 1941. Dunag the
past war he was concerned with the development of various radar sys^ms as
project engineer. Since 1945 he has been in charge of a group concerned
with the development of television transmission over wire facilities.
G. E. Mueller, B.S., Missouri School of Mines and Metallurgy-, 1939;
M.S., Purdue University, 1940. Bell Telephone Laboratories, 1940-46.
Mr. Mueller was engaged in television and radio research. During the war
he worked on radar antenna development. Mr. Mueller is now Assistant
Professor of Electrical Engineering at the Ohio State University.
Sloan D. Robertson, B.E.E., University of Dayton, 1936; M.Sc, Ohio
State L^niversity, 1938; Ph.D., 1941, Instructor of Electrical Engineering,
University of Dayton, 1940. Bell Telephone Laboratories, 1940-. Dr.
Robertson was engaged in microwave radar work in the Radio Research
Department during the war. He is now engaged in fundamental microwave
radio research.
D. E. Thomas, B.S. in Electrical Engineering, Pennsylvania State College,
1929; M.A., Columbia University, 1932. Bell Telephone Laboratories,
1929-. On Military leave from 1942-46 with U. S. Army Signal Corps and
U. S. Army Air Forces. Mr. Thomas has been engaged in investigations of
submarine telephone cable systems.
W. A. Tyrrell, B.S., Yale University, 1935; Ph.D., 1939. Bell Tele-
phone Laboratories, 1939-. Dr. Tyrrell has been engaged in waveguide
research, principally in the field of microwave loss measurements. During
the war he developed a number of waveguide components for Navy radar.
904
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